Handbook of Nitride Semiconductors and Devices, GaN-based Optical and Electronic Devices (Handbook of Nitride Semiconductors and Devices (VCH)) (Volume 3)

Hadis Morkoc¸ Handbook of Nitride Semiconductors and Devices

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Hadis Morkoç

Handbook of Nitride Semiconductors and Devices Vol. 3: GaN-based Optical and Electronic Devices

The Author Prof. Dr. Hadis Morkoç Virginia Commonwealth University Dept. of Electrical and Computer Engineering Richmond, VA USA

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at . # 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting Thomson Digital, Noida, India Printing Strauss GmbH, Mörlenbach Binding Litges & Dopf GmbH, Heppenheim Cover SPIESZDESIGN GbR, Neu-Ulm Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-40839-9

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Contents Preface

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Color Tables XIX 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.7.1 1.7.2 1.8 1.9 1.9.1 1.9.1.1 1.9.1.2 1.9.1.3 1.9.1.4 1.9.1.5 1.9.2 1.9.3 1.9.4 1.10 1.10.1 1.10.2 1.11

Light-Emitting Diodes and Lighting 1 Introduction 1 Current-Conduction Mechanism in LED-Like Structures 4 Optical Output Power 6 Losses and Efficiency 7 Current Crowding 11 Packaging 15 Perception of Visible Light and Color 22 Visible-Light Terminology 27 Luminous Efficacy 29 Chromaticity Coordinates and Color Temperature 30 Inroads by LEDs 33 Nitride LED Performance 37 LEDs on Sapphire Substrates 38 Blue and Green LEDs 40 Amber LEDs 47 UV LEDs 48 Resonant Cavity-Enhanced LED 55 Effect of Threading Dislocation on LEDs 57 LEDs on SiC Substrates 60 LEDs on Si Substrates 61 LEDs Utilizing Rare Earth Transitions 62 On the Nature of Light Emission in Nitride-Based LEDs 64 Pressure Dependence of Spectra 65 Current and Temperature Dependence of Spectra 67 LED Degradation 70

Handbook of Nitride Semiconductors and Devices. Vol. 3. Hadis Morkoç Copyright # 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40839-9

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1.12 1.13 1.14 1.14.1 1.14.2 1.15 1.15.1 1.15.2 1.15.3 1.15.4 1.16 1.17 1.17.1 1.17.2 1.17.3 1.17.4 1.17.4.1 1.17.4.2 1.17.4.3 1.17.4.4 1.17.4.5 1.17.4.6 1.17.5 1.17.5.1 1.17.5.2 1.17.6

2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.4 2.4.1 2.4.2 2.4.2.1 2.4.2.2 2.4.2.3

LED Efficiency 76 Monochrome Applications of LEDs 82 Luminescence Conversion and White-Light Generation with Nitride LEDs 85 Color as Related to White-Light LEDs 87 Color Rendering Index 88 Approaches to White-Light Generation 91 White Light from Three-Chip LEDs 91 White Light from Four-Chip LEDs 97 Combining LEDs and Phosphor(s) 100 Other Photon Conversion Schemes 111 Toward the White-Light Applications 114 Organic/Polymeric LEDs (OLED, PLED) 122 OLED Structures 124 Charge and Energy Transport Fundamentals 132 Properties of Organic Crystals 135 Light Emission Dynamics 139 Nonradiative Recombination 143 Internal Conversion 143 Intersystem Crossing 143 Singlet Fission 143 Aggregation and Davydov Splitting 144 Charge-Transfer Excitons 145 OLED Devices 146 White OLEDs 147 Displays 151 Lighting with OLEDs 155 References 157 Semiconductor Lasers 169 Introduction 169 A Primer to the Principles of Lasers 172 Waveguiding 175 Refractive Index of GaN and AlGaN 176 Refractive Index of InGaN 180 Analytical Solution to the Waveguide Problem 180 Numerical Solution of the Waveguide Problem 184 Far-Field Pattern 191 Loss, Threshold, and Cavity Modes 194 Optical Gain 196 A Glossary for Semiconductor Lasers 208 Optical Gain in Bulk Layers: A Semiconductor Approach 215 Relating Absorption Rate to Absorption Coefficient 216 Relating Stimulated Emission Rate to Absorption Coefficient 217 Relating Spontaneous Emission Rate to Absorption Coefficient 217

Contents

2.4.2.4 2.4.3 2.5 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.6.6 2.6.6.1 2.6.7 2.6.7.1 2.6.7.2 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19

3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.4.1 3.1.5 3.1.6 3.2 3.3

Fermi’s Golden Rule, Stimulated and Spontaneous Emission Rates, and Absorption Coefficient Within the k-Selection Rule 217 Gain in Quantum Wells 231 Coulombic Effects 235 Numerical Gain Calculations for GaN 239 Optical Gain in Bulk GaN 239 Gain in GaN Quantum Wells 241 Gain Calculations in Wz GaN Q Wells Without Strain 241 Gain Calculations in Wz Q Wells with Strain 241 Gain in ZB Q Wells Without Strain 244 Gain in ZB Q Wells with Strain 245 Pathways Through Excitons and Localized States 246 Measurement of Gain in Nitride Lasers 257 Gain Measurement via Optical Pumping 258 Gain Measurement via Electrical Injection (Pump) and Optical Probe Method 261 Threshold Current 262 Analysis of Injection Lasers with Simplifying Assumptions 263 Recombination Lifetime 264 Quantum Efficiency 267 GaN-Based LD Design and Performance 267 Gain Spectra of InGaN Injection Lasers 280 Near-UV Lasers 290 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs) 293 Polariton Lasers 308 GaInNAs Quaternary Infrared Lasers 312 Laser Degradation 315 Applications of GaN-Based Lasers to DVDs 327 A Succinct Review of the Laser Evolution in Nitrides 330 References 334 Field Effect Transistors and Heterojunction Bipolar Transistors 349 Introduction 349 Heterojunction Field Effect Transistors 352 Electron Transport Properties in GaN and GaN/AlGaN Heterostructures 353 Heterointerface Charge 356 Electromechanical Coupling 366 Analytical Description of HFETs 369 Examples for GaN and InGaN Channel HFETs 375 Numerical Modeling of Sheet Charge and Current 389 Numerically Calculated I–V Characteristics 398 The s-Parameters and Gain 401 Equivalent Circuit Models, Deembedding, and Cutoff Frequency 417

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3.3.1 3.3.2 3.3.2.1 3.3.2.2 3.3.3 3.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.5.5.1 3.5.5.2 3.5.5.3 3.5.5.4 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.6.5 3.6.5.1 3.6.5.2 3.6.6 3.6.7 3.6.8 3.6.9 3.7 3.8 3.9 3.9.1 3.9.1.1 3.9.1.2 3.10 3.11 3.11.1 3.11.2 3.11.2.1 3.11.2.2 3.11.2.3 3.11.2.4 3.12 3.12.1

Small-Signal Equivalent Circuit Modeling 420 Large-Signal Equivalent Circuit Modeling 434 Nonlinearities 436 Dispersion and Temperature Effects 447 Cutoff Frequency 454 HFET Amplifier Classification and Efficiency 457 AlGaN/GaN HFETs 463 Experimental Performance of GaN FETs 465 Power Amplifiers and Low-Noise Amplifiers (LNAs) 478 Drain-Voltage and Drain-Breakdown Mechanisms 483 Field Plate For Spreading Electric Field to Increase Breakdown Voltage 494 Anomalies in GaN MESFETs and AlGaN/GaN HFETs 497 Effect of the Traps in the Buffer Layer 500 Effect of Barrier States 509 Field-Assisted Emission from the Barrier Traps 516 Defect Mapping by Kelvin Probe and Effect of Surface States 520 Electronic Noise 531 Shot Noise 532 Generation–Recombination Noise 533 Thermal Noise 534 Avalanche Noise 536 Low-Frequency Noise (1/f Noise) 537 Nonfundamental 1/f Noise 537 Fundamental 1/f Noise 538 High-Frequency Noise 541 Treatment of Noise with FET Equivalent Circuit 544 1/f Noise in Conjunction with GaN FETs 550 High-Frequency Noise in Conjunction with GaN FETs 558 Dielectrics for Passivation Purposes or Gate Leakage Reduction 563 Heat Dissipation and Junction Temperature 570 Hot Phonon Effects 577 Phonon Decay Channels and Decay Time 581 LO Phonon Decay Channels in Wurtzitic GaN 584 Implications for FETs 593 InGaN Channel and/or InAlN Barrier HFETs 609 FET Degradation 612 Reliability Measurements 617 GaN HFET Reliability 619 Gate Current 624 Metallurgical Issues 628 Hot Electron and Hot Phonon Issues 629 Other Reliability Issues 633 Heterojunction Bipolar Transistors 635 HBT Fundamentals 638

Contents

3.12.1.1 3.12.1.2 3.12.1.3 3.12.1.4 3.12.1.5 3.12.1.6 3.12.1.7 3.12.2 3.13 3.14

4 4.1 4.1.1 4.1.1.1 4.1.1.2 4.1.2 4.1.2.1 4.1.2.2 4.1.2.3 4.1.2.4 4.1.3 4.1.3.1 4.1.3.2 4.1.4 4.1.5 4.1.5.1 4.1.5.2 4.1.5.3 4.1.5.4 4.1.5.5 4.1.5.6 4.1.6 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6 4.2.7

Current Transport Mechanism Across the Heterojunction 639 Current Transport Mechanism Across Base 648 Electron Velocity Overshoot in the Collector Space Charge Region 657 Current Gain and Recombination Current of HBTs 659 Current Gain at High Currents 661 Emitter Current Crowding Effect 662 Noise in Bipolar Transistors 662 Nitride-Based HBTs 665 Concluding Comments 671 Appendix: Sheet Charge Calculation in AlGaN/GaN Structures with AlN Interface Layer (AlGaN/AlN/GaN) 672 References 675 Ultraviolet Detectors 709 Introduction 709 Principles of Photodetectors 713 Current and Voltage Response to Incident Radiation 718 Photoconductive Detectors 719 p–n-Junction Photovoltaic Detectors 720 Noise in Detectors 729 Thermal Noise 729 Shot Noise 730 Generation–Recombination Noise 731 1/f Noise 731 Quantum Efficiency 733 Quantum Efficiency in Photoconductors 733 Quantum Efficiency in a p–n-Junction Detector 734 Responsivity 738 Signal-to-Noise Ratio, Noise Equivalent Power, and Detectivity 738 Thermal Limited 740 Shot Current Limited 740 Generation–Recombination Limited 740 Background Radiation Limited 741 Noise in a p–n-Junction Detector 741 Detectivity for a p–n-Junction Detector 742 Surface and Bulk Recombination in Detectors 743 Particulars of Deep UV Radiation and Detection 744 Solar UV Radiation 746 Stratospheric Ozone and UV Absorption 747 Computational Method Called Plexus 747 UV Transmission of the Atmosphere 748 Number of Solar UV Photons Reaching Lower Altitudes 749 Atmospheric Detection Range 751 Inevitable and Unavoidable Losses of Photons 754

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4.2.8 4.2.9 4.2.10 4.3 4.3.1 4.3.2 4.4 4.4.1 4.4.2 4.4.2.1 4.4.2.2 4.4.2.3 4.4.2.4 4.4.2.5 4.4.2.6 4.4.2.7 4.4.2.8 4.5 4.6

Practical UV Sensor Detection Ranges 756 Available UV Sensors 757 Design Requirements for UV Solar-Blind Imaging Detectors Si and SiC-Based UV Photodetectors 762 Silicon-Based UV Photodetectors 763 SiC-Based UV Photodetectors 764 Nitride-Based Detectors 767 Photoconductive Detectors 768 Photovoltaic Detectors and Junction Detectors 775 GaN and AlGaN-Based Schottky Barrier Photodiodes 776 Metal–Semiconductor–Metal Detectors 782 p–n- and p-i-n-Junction Detectors 789 AlGaN/GaN Heterojunction Detectors 796 AlGaN Detectors Including the Solar-Blind Variety 801 AlGaN/GaN MQW Photodetectors 806 Heterojunction Phototransistors 808 GaN Avalanche Photodetectors 810 UV Imagers 814 Concluding Comments 819 References 820 Index

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Appendix 847

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Preface This three-volume handbook represents the only comprehensive treatise on semiconductor and device fundamentals and technology under the overall umbrella of wide bandgap nitride semiconductors with comparison to GaAs when applicable. As it stands, the book is a reference book, a handbook, and a graduate text book all in one and would be beneficial to second-year graduate students majoring in semiconductor materials and devices, graduate research assistants conducting research in wide bandgap semiconductors, researchers and technologists, faculty members, program monitors, and managers. The philosophy of this endeavor is to present the material as much clearly and comprehensively as possible, so that there is very little need to refer to other sources to get a grasp of the subjects covered. Extreme effort has been expended to ensure that concepts and problems are treated starting with their fundamental basis so that the reader is not left hanging in thin air. Even though the treatise deals with GaN and related materials, the concepts and methods discussed are applicable to any semiconductor. The philosophy behind Nitride Semiconductors and Devices was to provide an adequate treatment of nitride semiconductors and devices as of 1997 to be quickly followed by a more complete treatment. As such, Nitride Semiconductors and Devices did not provide much of the background material for the reader and left many issues unanswered in part because they were not yet clear to the research community at that time. Since then, tremendous progress both in the science and engineering of nitrides and devices based on them has been made. While LEDs and lasers were progressing well even during the period when Nitride Semiconductors and Devices was written, tremendous progress has been made in FETs and detectors in addition to LEDs and lasers since then. LEDs went from display devices to illuminants for lighting of all kinds. Lasers are being implemented in the third generation of DVDs. The power amplifiers are producing several hundred watts of RF power per chip and the detectors and detector arrays operative in the solar-blind region of the spectrum have shown detectivities rivaling photomultiplier tubes. The bandgap of InN has been clarified which now stands near 0.7 eV. Nanostructures, which did not exist

Handbook of Nitride Semiconductors and Devices. Vol. 3. Hadis Morkoç Copyright # 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40839-9

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during the period covered by Nitride Semiconductors and Devices, have since become available. The technological breakthroughs such as epitaxial lateral overgrowth, laser liftoff, and freestanding GaN were either not fully developed or did not exist, neither did the highly improved quantum structures and devices based on them. In the interim period since then, the surfaces of nitrides and substrate materials, point defects and doping, magnetic ion doping, processing, current conduction mechanisms, and optical processes in bulk and quantum structures have been more clearly understood and many misconceptions (particularly, those dealing with polarization) identified, removed and/or elucidated. The handbook takes advantage of the fundamental and technological developments for a thorough treatment of all aspects of nitride semiconductors. In addition, the fundamentals of materials physics and device physics that are provided are applicable to other semiconductors, particularly, wurtzitic direct bandgap semiconductors. The handbook presents a thorough treatment of the science, fundamentals, and technology of nitride semiconductors and devices in such a width and depth that the reader would seldom need to engage in time-consuming exploration of the literature to fill in gaps. Last but not the least, the handbook contains seamless treatments of fundamentals needed or relied on throughout the entire book. The following is a succinct odyssey through the content of the three-volume handbook. Volume 1, Chapter 1 discusses the properties of nitride-based semiconductors with plenty of tables for reference. Volume 1, Chapter 2 treats the band structure of III–V nitrides, theories applied to determining the band structure, features of each theory with a succinct discussion of each, band structure of dilute III–V semiconductors doped with N, strain and stress, deformation potentials, and in-depth discussion of piezo and spontaneous polarization with illustrative and instructive artwork. Volume 1, Chapter 3 encompasses substrates that have been and are used for growth of nitride semiconductors, mainly, structural and mechanical (thermal) properties of those substrates, surface structure of planes used for growth, and substrate preparation for growth. Orientation and properties of GaN grown on those substrates are discussed along with commonly used surface orientations of GaN. The discussion is laced with highly illustrative and illuminating images showing orientations of GaN resulting through growth on c-plane, a-plane, m-plane, and r-plane substrates whichever applicable and the properties of resulting layers provided. The treatment segues into the discussion of various growth methods used for nitrides taking into account the fundamentals of growth including the applicable surface-oriented processes, kinetics, and so on, involved. A good deal of growth details for both OMVPE and MBE, particularly, the latter including the fundamentals of in situ process monitoring instrumentation such as RHEED, and dynamics of growth processes occurring at the surface of the growing layer are given. Of paramount interest is the epitaxial lateral overgrowth (ELO) for defect reduction. In addition to standard single multistep ELO, highly attractive nanonetwork meshes used for ELO are also discussed. Specifics in terms of growth of binary, ternary, and quaternaries of nitride semiconductors are discussed. Finally, the methods used to grow nanoscale structures are treated in sufficient detail.

Preface

Volume 1, Chapter 4 focuses on defects, both extended and point, doping for conductivity modulation and also for rendering the semiconductor potentially ferromagnetic segueing into electrical, optical, and magnetic properties resulting in films, with sufficient background physics provided to grasp the material. A clear discussion of extended defects, including line defects, are discussed with a plethora of illustrative schematics and TEM images for an easy comprehension by anyone with solid-state physics background. An in-depth and comprehensive treatment of the electrical nature of extended defects is provided for a full understanding of the scope and effect of extended defects in nitride semiconductors, the basics of which can be applicable to other hexagonal materials. The point defects such as vacancies, antisites, and complexes are then discussed along with a discussion of the effect of H. This gives way to the methods used to analyze point defects such as deep level transient spectroscopy, carrier lifetime as pertained to defects, positron annihilation, Fourier transform IR, electron paramagnetic resonance, and optical detection of magnetic resonance and their application to nitride semiconductors. This is followed by an extensive discussion of n-type and p-type doping in GaN and related materials and developments chronicled when applicable. An in-depth treatment of triumphs and challenges along with codoping and other methods employed for achieving enhanced doping and the applicable theory has been provided. In addition, localization effects caused by heavy p-type doping are discussed. This gives way to doping of, mainly, GaN with transition elements with a good deal of optical properties encompassing internal transition energies related to ion and perturbations caused by crystal field in wurtzitic symmetry. To get the reader conditioned for ferromagnetism, a sufficient discussion of magnetism, ferromagnetism, and measurement techniques (magnetic, magneto transport, magneto optics with underlying theory) applied to discern such properties are given. This is followed by an in-depth and often critical discussion of magnetic ion and rare earth-doped GaN, as well as of spintronics, often accompanied by examples for materials properties and devices from well-established ferromagnetic semiconductors such as Mn-doped GaN and Cr-doped ZnTe. Volume 2, Chapter 1 treats metal semiconductor structures and fabrication methods used for nitride-based devices. Following a comprehensive discussion of current conduction mechanisms in metal semiconductor contacts, which are applicable to any metal semiconductor system, specific applications to metal-GaN contacts are treated along with the theoretical analysis. This gives way to a discussion of ohmic contacts, their technology, and their characterization. In particular, an ample discussion of the determination of ohmic contact resistivity is provided. Then etching methods, both dry (plasma) and wet, photochemical, process damage, and implant isolation are discussed. Volume 2, Chapter 2 deals with determination of impurity and carrier concentrations and mobility mainly by temperature-dependent electrical measurements, such as Hall measurements. Charge balance equations, capacitance voltage measurements, and their intricacies are treated and used for nitride semiconductors, as well as a good deal of discussion of often brushed off degeneracy factors.

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Volume 2, Chapter 3 is perhaps one of the most comprehensive discussions of carrier transport in semiconductors with applications to GaN. After a discussion of scattering processes in physical and associated mathematical terms, the methods discussed are applied to GaN and other related binaries and ternaries with useful ranges of doping levels, compositions, and lattice temperatures. Comparisons with other semiconductors are also provided when applicable. This treatment segues into the discussion of carrier transport at high electric fields applicable to field-effect transistors, avalanche and pin (biased) photodiodes. This is followed by the measurement of mobility and associated details, which are often neglected in text and reference books. The discussion then flows into magnetotransport beyond that present in standard Hall measurements. Low, medium, and high magnetic field cases, albeit only normal to the surface of the epitaxial layers, are treated. The treatise also includes cases where the relaxation time, if applicable, is energy-dependent and somewhat energy-independent. The discussion of the magnetotransport paves the way for a fundamental and reasonably extensive discussion of the Hall factor for each of the scattering mechanisms that often is not treated properly or only in a cursory manner in many texts leading to confusion. After providing the necessary fundamentals, the transport properties of GaN are discussed. This gives way to the discussion of various scattering mechanisms in two-dimensional systems that are relied on in high-performance FETs. For determining the mobility of each layer (in the case of multiple layer conduction), quantitative mobility spectrum analysis including both the fundamentals and experimental data obtained in nitride semiconductors is discussed. The quantum Hall effect and fractional quantum Hall effect in general and as germane to GaN are discussed along with parameters such as the effective mass determined from such measurements. Volume 2, Chapter 4 is devoted to p–n junctions, beginning with the discussion of band lineups, particularly, in the binary pairs from the point of view of theoretically and experimentally measured values. Current conduction mechanisms, such as diffusion, generation-recombination, surface recombination, Poole–Frenkel, and hopping conductivity are discussed with sufficient detail. Avalanche multiplication, pertinent to the high-field region of FETs, and avalanche photodiodes, are discussedfollowed by discussions of the various homojunction and heterojunction diodes based on nitrides. Volume 2, Chapter 5 is perhaps the most comprehensive discussion of optical processes that can occur in a direct bandgap semiconductor and, in particular, in nitride-based semiconductors and heterostructures inclusive of 3, 2-, and 0-dimensional structures as well as optical nonlinearities. Following a treatment of photoluminescence basics, the discussion is opened up to the treatment of excitons, exciton polaritons, selection rules, and magneto-optical measurements followed by extrinsic transitions because of dopants/impurities and/or defects with energies ranging from the yellow and to the blue wavelength of the visible spectrum. Optical transitions in rare earth-doped GaN, optical properties of alloys, and quantum wells are then discussed with a good deal of depth, including localization effects and their possible sources particularly media containing InN. The discussion then leads to the

Preface

treatment of optical properties of quantum dots, intersubband transitions in GaNbased heterostructures, and, finally, the nonlinear optical properties in terms of second and third harmonic generation with illuminating graphics. Volume 3, Chapter 1 is devoted, in part, to the fundamentals of light emitting diodes, the perception of vision and color by human eye, methodologies used in conjunction with the chromaticity diagram and associated international standards in terms of color temperatures and color rendering index. Specific performances of various types of LEDs including UV varieties, current spreading or the lack of related specifics, analysis of the origin of transitions, and any effect of localization are discussed. A good deal of white light and lighting-related standards along with approaches employed by LED manufacturers to achieve white light for lighting applications is provided. The pertinent photon conversion schemes with sufficient specificity are also provided. Finally, the organic LEDs, as potential competitors for some applications of GaN-based LEDs are discussed in terms of fundamental processes that are in play and various approaches that are being explored for increased efficiency and operational lifetime. Volume 3, Chapter 2 focuses on lasers along with sufficient theory behind laser operation given. Following the primer to lasers along with an ample treatment based on Einstein’s A and B coefficients and lasing condition, an analytical treatment of waveguiding followed by specifics for the GaN system and numerical simulations for determining the field distribution, loss, and gain cavity modes pertaining to semiconductor lasers are given. An ample fundamental treatment of spontaneous emission, stimulated emission, and absorptions and their interrelations in terms of Einstein’s coefficients and occupation probabilities are given. This treatment segues into the extension of the gain discussion to a more realistic semiconductor with a complex valence band such as that of GaN. The results from numerical simulations of gain in GaN quantum wells are discussed, as well as various pathways for lasing such as electron-hole plasma and exciton-based pathways. Localization, which is very pervasive in semiconductors that are yet to be fully perfected, is discussed in the light of laser operation. Turning to experimental measurements, the method for gain measurement, use of various laser properties such as the delay on the onset of lasing with respect to the electrical pulse, dependence of laser threshold on cavity length to extract important parameters such as efficiency are discussed. The aforementioned discussions culminate in the treatment of performance of GaN-based lasers in the violet down to the UV region of the optical spectrum and applications of GaN-based lasers to DVDs along with a discussion of pertinent issues related to the density of storage. Volume 3, Chapter 3 treats field effect transistor fundamentals that are applicable to any semiconductor materials with points specific to GaN. The discussion primarily focuses on 2DEG channels formed at heterointerfaces and their use for FETs, including polarization effects. A succinct analytical model is provided for calculating the carrier densities at the interfaces for various scenarios and current voltage characteristics of FETs with several examples. The 2-port network analysis, s-parameters, various gain expressions, circuit parameter extraction of equivalent circuit parameters, for both low and high rf power cases, temperature and dispersion

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effects are discussed in detail. Experimental performance of GaN-based FETs and amplifiers is then discussed followed by an in-depth analysis of anomalies in the current voltage characteristics owing to bulk and barrier states, including experimental methods and probes used for cataloging these anomalies. This is followed by the employment of field spreading gate plates and associated performance improvements. This segues into the discussion of noise both at the low-frequency end and high-frequency end with sufficient physics and practical approaches employed. The combined treatment of various low-frequency noise contributions as well as those at high frequencies along with their physical origin makes this treatment unique and provides an opportunity for those who are not specialists in noise to actually grasp the fundamentals and implications of low- and high-frequency noise. Discussion of high-power FETs would not be complete without a good discussion of heat dissipation and its physical pathways, which is made available. Unique to GaN is the awareness of the shortfall in the measured electron velocity as compared to the Monte Carlo simulation. Hot phonon effects responsible for this shortfall are uniquely discussed with sufficient theory and experimental data. Power dissipation pathways from hot electrons to hot LO phonons followed by decay to LA phonons and in turn to heat transfer to the bath are treated with sufficient physics. In particular, the dependence of the hot phonon lifetime on the carrier concentration and its implications to carrier velocity is treated. The effect of lattice matched AlInN Barrier layers vis a vis AlGaN barrier layers on the hot LO phonon lifetime and carrier velocity is treated. A section devoted to reliability with specifics to GaN based high power HFETs is also provided. Such effects as surfaces, carrier injection by the gate to the surface states and the resultant virtual extended gate, surface passivation, interplay of temperature, strain, and electric field and their combined effect on reliability are treated in detail. Finally, although GaN-based bipolar transistors are not all that attractive at this time, for completeness and the benefit of graduate students and others who are interested in such devices, the theory, mainly analytical, of the operation of heterojunction bipolar transistors is discussed along with available GaN based HBT data. Volume 3, Chapter 4 discusses optical detectors with special orientation toward UV and solar-blind detectors. Following a discussion of the fundamentals of photoconductive and photovoltaic detectors in terms of their photo response properties, a detailed discussion of the current voltage characteristic of the same, including all the possible current conduction mechanisms, is provided. Because noise and detectors are synonymous with each other, sources of the noise are discussed, followed by a discussion of quantum efficiency in photoconductors and p–n junction detectors. This is then followed by the discussion of vital characteristics such as responsivity and detectivity with an all too important treatment of the cases where the detectivity is limited by thermal noise, shot current noise, generation-recombination current noise, and background radiation limited noise (this is practically nonexistent in the solar-blind region except the man-made noise sources). A unique treatment of particulars associated with the detection in the UV and solar-blind region and requirements that must be satisfied by UV and solar-blind detectors, particularly, for the latter, is then provided. This leads the discussion to various UV detectors

Preface

based on the GaN system, including the Si- and SiC-based ones for comparison. Among the nitride-based photodetectors, photoconductive variety as well as the metal-semiconductor, Schottky barrier, and homo- and heterojunction photodetectors are discussed along with their noise performance. Nearly solar-blind and truly solar-blind detectors including their design and performance are then discussed, which paves the way for the discussion of avalanche photodiodes based on GaN. Finally, the UV imagers using photodetectors arrays are treated. It is fair to state that I owe so much to so many, including my family members, friends, coworkers, colleagues, and those who contributed to the field of semiconductors in general and nitride semiconductors in particular, in my efforts to bring this manuscript to the service of readers. To this end, I thank my wife, Amy, and son, Erol, for at least their understanding why I was not really there for them fully during the preparation of this manuscript, which took longer than most could ever realize. Also, without the support of VCU, with our Dean R. J. Mattauch, Assistant Dean Susan Younce, Department Chair A. Iyer, and my coworkers and students, it would not have been possible to pursue this endeavor. Special recognitions also go to Dr N. Izyumskaya for reading the entire manuscript for consistency in terms of figures, references, and so on, which had to have taken perseverance beyond that many could muster; Dr Ü. Özgür for being the bouncing board and proofing many parts of the book, particularly chapters dealing with optical processes, lasers and magnetism; my colleague P. Jena for reading and contributing to the band structure section; my coworker Professor M. Reshchikov for his contributions to the point defects and doping sections; Professor A. Baski for her expert assistance in obtaining microprobe images; Dr D. Huang for his many contributions to the quantum dots section; Dr Y-T Moon for his assistance in current crowding; C. Liu for her assistance with ferromagnetism; Prof. A. Teke for reading the chapter on detectors; Dr. R. Shimada for her contributions to the surface emitting laser section; Dr. J.-S. Lee for his help in updating the LED chapter; Dr Q. Wang for her help in generating the accurate ball and stick diagrams in Volume 1, Chapter 1; Dr V. Litvinov for calculating the energy levels in quantum wells; students Y. Fu, Fan Qian, X. Ni, and S. Chevtchenko for their contributions to various sections of the book with proofing equations, redoing calculations, and so on; and to J. Leach who took it upon himself to be the local expert in the latest in semiconductor and organic LEDs and helped with the chapter on LEDs and read the chapter on transport as well as proofread some of the other chapters; Ms G. Esposito for reading a large portion of the text for English. Undergraduate students K. Ngandu, D. Lewis, B. D. Edmonds, and M. Mikkelson helped in reading various parts of the manuscript as well as helping with the artwork. Unbeknown to them, many graduate students who took classes from me helped in many immeasurable ways. In terms of the non-VCU colleagues, special thanks go to Professors R. M. Feenstra, A. Matulionis, A. Blumenau, P. Ruterana, G. P. Dimitrakopulos, P. Handel, K. T. Tsen, T. Yao, P. I. Cohen, S. Porowski, B. Monemar, B. Gil, P. Le Febvre, S. Chichibu, F. Tuomisto, C. Van de Walle, M. Schubert, F. Schubert, H. Temkin, S. Nikishin, L. Chernyak, J. Edgar, T. Myers, K. S. A. Butcher, O. Ambacher, A. di Carlo, F. Bernardini, V. Fiorentini, M. Stutzmann, F. Pollak, C. Nguyen, S. Bedair, N. El-Masry, S. Fritsch, M. Grundman, J. Neugebauer, M. S. Shur,

XVII

XVIII

Preface

J. Bowers, J. C. Campbell, M. Razhegi, A. Nurmikko, M. A. Khan, J. Speck, S. Denbaars, R. J. Trew, A. Christou, G. Bilbro, H. Ohno, A. Hoffmann, B. Meyer, B. Wessels, N. Grandjean, and D. L. Rode; and Drs Z. Liliental-Weber, P. Klein, S. Binari, D. Koleske, J. Freitas, D, Johnstone, D. C. Look, Z.-Q. Fang, M. MacCartney, I. Grzegory, M. Reine, C. W. Litton, P. J. Schreiber, W. Walukiewicz, M. Manfra, O. Mitrofanov, J. Jasinski, V. Litvinov, Jan-Martin Wagner, K. Ando, H. Saito, C. Bundesmann, D. Florescu, H. O. Everitt, H. M. Ng, I. Vurgaftman, J. R. Meyer, J. D. Albrecht, C. A. Tran, S.-H. Wei, G. Dalpian, N. Onojima, A. Wickenden, B. Daudin, R. Korotkov, P. Parikh, D. Green, A. Hansen, P. Gibart, F. Omnes, M. G. Graford, M. Krames, R. Butte, and M. G. Ganchenkova for either reading sections of the book, providing unpublished data, or providing suggestions. Many more deserve a great deal of gratitude for willingly spending considerable time and effort to provide me with digital copies of figures and high-quality images, but the available space does not allow for individual recognition. They are acknowledged in conjunction with the figures. In a broader sense, on a personal level, it gives me great pleasure to recognize that I benefited greatly from the counsel and support of Professor T. A. Tombrello of Caltech. I also would like to use this opportunity to recognize a few of the unsung heroes, namely, Dr Paul Maruska and Professor Marc Ilegems who truly started the epitaxy of nitrides with the hydride VPE technique independently, and Dr S. Yoshida and Professor T. Matsuoka for their pioneering work in AlGaN and InGaN, respectively. Richmond, VA January 2008

Hadis Morkoç

XIX

Color Tables

Figure 1.1 InGaN LEDs spanning the spectral range from violet to orange. Courtesy of S. Nakamura, then with Nichia Chemical Co. Ltd. (This figure also appears on page 3.)

Figure 1.2 The LED materials and range of wavelength of the emission associated with them. The color band indicates the visible region of the spectrum. (This figure also appears on page 4.)

Handbook of Nitride Semiconductors and Devices. Vol. 3. Hadis Morkoç Copyright
2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40839-9

XX

Color Tables

Figure 1.11 (a) Cross-sectional schematic flipchip mounted high-luminance LED package with Ag back-reflector. Electrostatic discharge protection is integrated into the Si submount (circa 1997). The package is able to handle power dissipation associated with 350 mA current injection with resulting LED lumens of 20–40 lm. (b) Artistic rendition of the package inclusive of plastic lens. In this flip-chip model, the substrate

can be removed and the exposed N-polarity GaN can be made dark by chemical etching or a polymeric photonic crystal can be placed on what is now the top surface for better photon collection. Polymeric photonic crystal can be produced by laser lithography (holography) to reduce the cost of fabrication. Courtesy of Lumileds/Phillips. (This figure also appears on page 18.)

Color Tables

Figure 1.15 Diagram of a human eye showing its various structures along with the optical path of vision. (This figure also appears on page 22.)

XXI

XXII

Color Tables

Figure 1.16 A schematic showing rods that are sensitive to low-level gray light and cones that are sensitive to color. In this rendering, light enters the eye from the bottom. The photons travel through the vitreous fluid of the eyeball and penetrate the entire retina, which is about half a millimeter thick, before reaching the photoreceptors: the cones and rods that respond

to light (the colored and black cells attached to the epithelium on top). Signals then pass from the photoreceptors through a series of neural connections toward the surface of the retina, where the ganglion cell nerve fiber layer relays the processed information to the optic nerve and then to the brain [27]. (This figure also appears on page 24.)

Color Tables

Color matching functions

Figure 1.17 Luminous efficacy of monochromatic radiation, K(l), for the human eye under light (photopic vision) and dim (scotopic vision) conditions. The band indicates the color at visible wavelengths. The maximum luminous efficacy, Km, for photopic vision is 683 lm W
1 and occurs at 555 nm. (This figure also appears on page 27.)

Z(λ)

1.5

Y(λ)

1.0

X(λ)

0.5

0.0

400

500 600 Wavelength (nm)

700

Figure 1.19 CIE 1931 XYZ color matching functions (standard colorimetric observer sensitivity plots for red (x), green (y), and blue (z) color perception cones of human eye. Courtesy of Lumileds/ Philips. (This figure also appears on page 31.)

XXIII

XXIV

Color Tables

Figure 1.20 (a) The CIE 1931 xy chromaticity space (horizontal and vertical axes denote the x and y coordinates, respectively) and (b) the CIE 1976 UCS u0 v0 chromaticity space. Courtesy of Lumileds/Philips. (This figure also appears on page 32.)

Color Tables

Figure 1.20 (Continued )

XXV

Color Tables 100 Fluorescent light

AlIn GaP/ GaP Red Orange

–

Halogen

Luminous efficacy (lm W-1)

XXVI

AlIn GaP/ GaAs Red Orange AlGaA s/AlG aAs Red

–

Unfiltered incandescent

10

1.0

AlGaAs /GaAs Red Thomas Edison's GaAs P:N first bulb GaP:N Green Red – yellow AlQ 3 GaP:Zn ,O Molecular solids Red

0.1

1970

1975

1980

1985

White InGaNGreen

InGaN Blue

Polymers

SiC Blue

GaAsP Re d

Yellow

1990

PPV

1995

2000

Year Figure 1.23 Evolution of all LED performance with some benchmarks against commonly used lamps. Both red and white LEDs were projected to produce luminous efficiencies of about 150 lm W
1 by about the years 2015–2020 as compared to the value of about 100 lm W
1

achievable by fluorescent bulbs and 10 lm W
1 by incandescent lamps. However, the figure for white LEDs was attained in 2007, well ahead of projections. Courtesy of Lumileds/Philips. (This figure also appears on page 36.)

Figure 1.46 Photographs of red (a), green (b), and blue (c) emissions in LED fabricated from MBE-grown GaN doped with Eu (a), Er (b), and Tm (c) [79–81]. (This figure also appears on page 62.)

Color Tables

Figure 1.47 Laterally integrated GaN:RE thin-film ELD containing the three primary colors fabricated with the SOG liftoff technique: (a) optical microscopy photograph of the GaN ELD showing the three-color integration; (b) blue, green, and red emission under DC bias from ELD GaN devices doped with Tm, Er, and Eu, respectively [83]. (This figure also appears on page 63.)

XXVII

XXVIII

Color Tables

Figure 1.58 High-power LED packages for lighting applications. (a) LUXEON circa 1998. (b) LUXEON K2 circa 2006. Courtesy of Lumileds/Philips. (This figure also appears on page 74.)

Figure 1.62 Evolution of LED-based traffic signals and a comparison of LED to incandescent traffic lights. Incandescent bulbs consume 135 W and must be replaced every 6 months. The LED alternatives, on the other hand, consume 15 W and would have to be replaced only every 120 months (2002 figures). Using red traffic lights as

an example, because of their priority, the number of LEDs for each traffic light went down from 700 per traffic light in 1993 to 12–18 in 2003. The latest LED count is similar to the improvement experienced for green-LED-based traffic lights over the years. Courtesy of Lumileds/Philips. (This figure also appears on page 82.)

Color Tables

Figure 1.63 Comparison of a thin-film flip-chip (TFFC) white LED to conventional halogen and HID lamps used for low-beam automotive forward lighting (headlight) applications. The top row is the lit visual image each of the three technologies compared. The lower top row represents the color-scaled luminance image.

Munsell samples

Reflectance factor

0.8

0.6

8 1 7

0.4

2 6

0.2 0.0

The scales for the halogen filament and HID arc are the same. The scale for the LED is different and is indicated at the left. The table lists average luminance, source flux, input power, and useful flux (utilization percentage) in the application. Courtesy of Lumileds/Philips. (This figure also appears on page 84.)

4 5

400

500

600

3

700

Wavelength (nm) Figure 1.65 Munsell samples spectra for determining the color rendering index. Courtesy of M. E. Coltrin of Sandia National Laboratory. (This figure also appears on page 89.)

XXIX

Color Tables 0.9 520 530

0.8

InGaN Green LED 510

0.7

550 GaP Green LED

Y el low

Green

en

y-Color coordinate

(Y 1-x,Gd x)3 (Al 1-y,Ga y) 5O 12:Ce

-gre

0.6

AlInGaP Green LED

570

500

0.5 Yellow 590 3000K

0.4

2500K 5000K

Blue-green

k pin

White

0.3

490

es ad Sh

6000K

o

ur fp

Orange

epl

620 700

Red

AlGaAs Red LED

Red-Purple

0.2 Blue Purple

0.1 470 InGaN Blue LED 450

0 0

0.1

440

0.2

0.3

0.4

0.5

0.6

0.7

x-Color coordinate Figure 1.66 The chromaticity diagram along with available commercial LED performance data. Clearly, blue and green InGaN LEDs constitute the two important legs of the triad, the three primary colors that are needed for full-color displays. Moreover, the output of an optically pumped YAG medium doped for yellow

emission is shown with data points indicative of various Gd concentrations. The broken line that connects the blue LED to one particular composition of the YAG medium indicates the range of warm white colors that can be obtained. (This figure also appears on page 90.)

1.4

605nm

1.2

Intensity (a.u.)

XXX

540nm

1.0 0.8

455nm Photopic curve

0.6 0.4 0.2 0.0

400 450 500 550 600 650 700

Wavelength (nm) Figure 1.67 White-light output emission spectrum from a threecolor multichip LED. Courtesy of M. E. Coltrin of Sandia National Laboratory. (This figure also appears on page 92.)

Relative power

Color Tables 1.2

x

0.3812

1.0

y

0.3795

0.8

CCT(K)

4011

0.6

∆ uv Ra

0.001

0.4 0.2

80

Efficacy(lm W-1) 399

0.0 400

600

500

(a)

(b)

700

Wavelength (nm)

0.9 Spectrum locus Planckian locus Illuminant A Illuminant D65 Mixture Blue Green Red

520 0.8

530

Green 510

540

0.7

550

0.6

560 Yellow 570

500 0.5

y

580 590

Blue-green

0.4

K

0K 00 00K 0 20

480 0.1

P ur

Blue

0K

00 10

490

0.2

20 00K

30 0 50

0.3

600 610 640

1000K Red

p le

470 0 (c)

0.0

0.1

0.2

0.3

0.4

x

0.5

0.6

Figure 1.69 The effect of optimizing the wavelengths of a three-chip white-light LED. (a) The spectral response of the three LEDs used for mixture. (b) Correlated color temperature, color rendering index (Ra), efficacy, and (c) the chromaticity coordinates (x, y) for the white light

0.7

0.8

are shown as the solid square near 4000 K. Because of the CRI, Ra of about 80, this is an example of white light with good color rendering [129]. (This figure also appears on page 94.)

XXXI

XXXII

Color Tables

Figure 1.70 Contour plots for LE of radiation and CRI of a trichromatic Gaussian white-light source with linewidth of 5 kT at 300 K for different wavelength combinations. Courtesy of E. F. Schubert [139]. (This figure also appears on page 95.)

Color Tables

Figure 1.71 Contour plots for LE of radiation and CRI of a trichromatic Gaussian white-light source with linewidth of 8 kT at 300 K for different wavelength combinations. Courtesy of E. F. Schubert [139]. (This figure also appears on page 96.)

XXXIII

XXXIV

Color Tables

Figure 1.77 The relationship between the linewidth (top), wavelength (center), and luminous efficacy (bottom) in the BGGRR approach wherein a blue LED source is used to pump green and red phosphors to achieve white light. Courtesy of M. E. Coltrin, Sandia National Laboratories. (This figure also appears on page 102.)

Color Tables

A

C

Intensity (a.u.)

B

400

450

500

550

600

650

700

Wavelength(nm)

Power conversion efficency (%)

Figure 1.84 Emission spectra of phosphors of the triphosphor blend: A is Eu3 þ :(Sr, Ba, Ca)5(PO4)3Cl (blue); B is (Ce3 þ , Tb3 þ ): LaPO4 (green); and C is Eu3 þ :Y2O3 (red) [129]. (This figure also appears on page 109.)

80

InGaNAlInGaPAlGaAsInGaAs Requirement for 150 lm W-1 RGB white

LED Vertical cavity Laser diode Edge -Emitting Laser diode

60 40 Fluorescent

20

Incandescent

0 400

500

600 700 800 Peak wavelength (nm)

900

Figure 1.86 Best power conversion efficiencies reported for laser LEDs across the visible spectrum. For comparison and benchmarking, the efficiencies of unfiltered incandescent light bulbs and fluorescent light bulbs are also indicated. The vertical arrows indicate vertical emitters while the horizontal ones indicating edge emitters. Courtesy of Lumileds/Philips. (This figure also appears on page 114.)

XXXV

XXXVI

Color Tables

Feedback sensors Controller

Target color point

RG B Backlight B

R

G

B

R

G

Figure 1.87 White point control in a backlight with R, G, and B LEDs. The LEDs inject the light into the backlight, where the light is mixed to create white. The color point of the white is measured using three sensors with approximate

B

Power supply B R G

the X, Y, and Z color matching functions. A controller is used to compare the signals from the color sensors with those from the target white point. Courtesy of Lumileds/Philips. (This figure also appears on page 117.)

Figure 1.90 LED backlighted 18 in flat panel computer screen and a schematic representation of illumination. Courtesy of Lumileds/ Philips. (This figure also appears on page 120.)

Color Tables

Figure 1.92 A flexible, full-color organic electroluminescent display (OLED) built using organic thin-film transistor (TFT) technology with a plastic substrate. The 2.5 in prototype display, which is 0.3 mm thick and weighs 1.5 g without the driver, supports 16.8 million colors at

a 120  160 pixel resolution (80 ppi, 0.318 mm pixel pitch). Press release by Sony Corp., http:// www.sony.co.jp/SonyInfo/News/Press/200705/ 07-053/index.html. (This figure also appears on page 123.)

Figure 2.1 Photograph of a Sony InGaN laser array with an output power of 6.1 W at 407 nm in action. Courtesy of Drs S. Goto and Shigetaka Tomiya of Sony Corporation. (This figure also appears on page 170.)

XXXVII

XXXVIII

Color Tables

Figure 2.2 A schematic representation of how a DVD operates. The diagram at the upper left of the figure shows the pits and their shapes with larger pits causing reduced reflection. The disk itself is of the two-layer type, which doubles the capacity. The holographic lens makes it possible to focus on the upper or the lower layer. (This figure also appears on page 171.)

Color Tables

Figure 2.69 (a) Schematic cross section of the micro-LED structures, (b) optical microscope image of a 20 mm mesa with a 3 mm nonoxidized aperture (dark area), and (c) electroluminescence from a micro LED under forward bias (250 mA, 5.8 V) [206]. (This figure also appears on page 308.)

Figure 2.70 Schematic showing the interaction of cavity mode dispersion and exciton polariton dispersion leading to Rabi splitting. (This figure also appears on page 310.)

XXXIX

Color Tables L3

10

L2

L1

9

T=300K

PL (a.u.)

XL

10

7

10

5

10

3

10

1

10

-1

3.40

3.46

Energy (eV) Figure 2.73 Emission spectra at pump powers from 20 mW to 2 mW at 0 C. Spectra integrated over 10 ms show multiple emission line [214]. (This figure also appears on page 313.)

Color Tables

Figure 2.79 FESEM and FEAES images based on carbon and oxygen mapping for two LD samples. Courtesy of C. C. Kim, L.G. Electronics. (This figure also appears on page 323.)

XLI

Color Tables

0.14 0.12 V =0 V G

Coupled Uncoupled

0.10 Drain current (A)

XLII

V = -1.0 V

0.08

G

0.06

V = -2.0 V

0.04

V = -3.0 V

G

G

0.02

V = -4.0 V G

0.00 0

1

2

3

4

Drain voltage (V)

Figure 3.10 Calculated output characteristic for the Al0.2Ga0.8N/ GaN HFET with parameters described in the text with low field mobility of 1000 cm2 V
1 s
1 and a saturation velocity of 107 cm s
1, (Voff ¼
4.9 V without the electromechanical coupling and
4.83 V with mechanical coupling). (This figure also appears on page 380.)

5

Color Tables

35 30

48

25 20

46

15

Gain (dB)

Output power (dB m)

40 Peak power=108 W CW Power density=4.5 W mm-1

50

10 44 32

(b)

5

f=2 GHz, VDS=52 V

34

36

38

40

42

44

0

Input power (dB m)

Figure 3.64 (a) A photograph of a package device exhibiting 108 W of CW output power at 2 GHz. (b) Output power and gain versus input power drive of the same device measured at 2 GHz at a drain bias of 52 V. Courtesy of Primit Parikh, Cree Santa Barbara Technology Center. (This figure also appears on page 474.)

XLIII

Color Tables

40

50

51 W

Gain DE PAE Pout

45

30

40

25

35

20

30

15

25

10

20

5

15 f = 6 GHz pulsed

0

10 18

(b)

22

26

30

34

38

42

Input power (dB m)

Figure 3.69 (a) Photograph of an 8 mm gate periphery amplifier exhibiting a total pulsed power of 51 W at 6 GHz. (b) The output power, drain efficiency, gain, and power-added efficiency versus the input power of the same device. Courtesy of Primit Parikh, Cree Santa Barbara Technology Center. (This figure also appears on page 480.)

Output power (dB m)

35

Gain (dB), DE (%), PAE (%)

XLIV

Color Tables

44

24 P1dB (dBm) SS Gain (dB)

22

40

20

38

18

36

16

34

14

32

12

30

10 1.5

(b)

1.6

1.7

1.8

1.9

2.0 2.1

2.2

2.3

2.4

Frequency (GHz)

Figure 3.70 (a) Photograph of a broadband amplifier. (b) Power and small-signal gain versus frequency in the range of 1.8–2.2 GHz, showing an 11 W over the entire bandwidth with a 17 dB gain with 0.15 dB ripple for wireless communications. Courtesy of John Palmour, Cree, Inc. (This figure also appears on page 481.)

Small signal gain (dB)

P1dB (dBm)

42

XLV

Color Tables

15

V DS = 9 V, IDS=75 mA mm-1

Gain

2.5

10

2.0 NF (LNA)

7.5

1.5 1.0

5 Fmin (device)

2.5 0 8 (b)

3.0

9

10 Frequency (GHz)

11

Figure 3.71 (a) A photograph of the gain and (b) the NF of the amplifier (NF (LNA)) and minimum noise figure at the device level (Fmin). Courtesy of Primit Parikh, Cree Santa Barbara Technology Center. (This figure also appears on page 482.)

0.5 0.0 12

NF (dB)

12.5 Gain (dB)

XLVI

Color Tables

Figure 3.72 A set of photographs of low-noise amplifiers along with their scattering parameters (s21 representing forward gain, s11 representing input, and s22 representing output) versus frequency. Courtesy of Primit Parikh, Cree Santa Barbara Technology Center. (This figure also appears on page 483.)

XLVII

XLVIII

Color Tables

Figure 3.83 Schematic representation of RF current lag superimposed on top of DC drain (solid lines) and RF (dashed lines) I–V characteristics with a DC load line. (This figure also appears on page 500.)

0.5

T =10oC T =20oC T =30oC T =40oC T =50oC T =60oC T =70oC T =80oC T =90oC

0.1

EA = 0.22 ± 0.01 eV

0.4

-1

-2

A = 410 ± 30 s K

0.3

0.2

eT

-2

-1

-2

(s K )

∆ID(t) / ID

SS

Color Tables

0.01 0

(a)

30

60

90

t (µs)

120

0.1 0.09 2.6

150

2.8

(b)

3.0

3.2

3.4

-1

1000/T (K )

Figure 3.91 Temperature dependence of the (b) Experimentally determined values of eT
2 trap emission rate. (a) The difference between versus the inverse temperature. The emission rate is extracted by fitting an exponential decay the steady state and the actual drain current after switching the gate voltage VG from
11 function to the data using Equation 3.254. to 0 V in temperature window ranging from 10 Courtesy of O. Mitrofanov [286]. (This figure also to 90  C and the source–drain bias of 12 V. appears on page 512.)

Figure 3.139 The electric field distribution between the gate and the drain of an HFET having a dielectric surface layer, including between the gate metal and the semiconductor, for cases with (a) and without the field plate (b). The gate–source voltage and the drain–gate voltage are
4 and 50 V, respectively. The field plate extension is 0.4 mm. Courtesy of R. Trew. (This figure also appears on page 613.)

XLIX

L

Color Tables

Figure 3.140 Variation of cross-sectional electric field distribution in an AlGaN/GaN HFET without an SiNx passivation layer under the gate (a) and with an SiNx passivation layer (b) for gate bias voltage of
5 V and drain bias voltage of 40 V. The unit of color bar is MV cm
1. Courtesy of Y. Nanishi. (This figure also appears on page 614.)

Figure 3.141 Calculated electric field distribution with undoped AlGaN surface layer with SiN passivation (a) and with doped GaN surface layer with SiN passivation. Courtesy of T. Kikkawa, Fujitsu Laboratories [451]. (This figure also appears on page 615.)

Color Tables

0.20 Measured VGS=0V Modeled

IDS(A mm-1)

0.16

0.12

0.08 VGS=-2.5V 0.04 VGS=-5V 0.00 0

1

2

3

4

(a)

5 6 VDS(V)

7

8

9

10

0 ×100 VGS =-1 V

-2×10-5

VGS =-3 V

IGS(A)

-4 ×10 -5 -6×10 -5

VGS =-5 V -8 ×10 -5 Measured -1 ×10 -4

Modeled

-1.2 ×10 -4

0×100

(b)

2×10 -5

VGS =-7 V 4×10 -5

6×10 -5

8×10 -5

VDS (V)

Figure 3.151 Simulated and measured (a) IDS–VDS characteristics and (b) IGS versus VDS characteristics for an AlGaN/GaN HFET (red lines, measured data; blue, modeled data) Courtesy of R. Trew. (This figure also appears on page 625.)

10×10 -5

LI

Color Tables

Drift velocity (107cm s-1)

LII

1.5

No degeneracy No hot phonon No self heating

AlGaN/GaN 300 K Hot phonons

1.0

Degeneracy

No degeneracy No self heating No self heating

0.5

0.0

Self heating

0

+Hot phonons +Degeneracy +Self heating

5 10 15 Electric field (kV cm-1)

Figure 3.155 The effect of hot phonons and selfheating on the carrier velocity versus electric field as determined by Monte Carlo simulations. The black line depicts the velocity–field relationship that can be expected when hot phonon, selfheating, and degeneracy effects are neglected altogether. The blue line is for the case where only

the hot phonon induced velocity degradation is considered. Green line considers degeneracy in addition to hot phonons and the red line considers hot phonons, degeneracy, and selfheating. Courtesy of A. Matulionis. (This figure also appears on page 631.)

Color Tables

ROIC

p-contact p-GaN or p-AlGaN n-contact

n-absorbing AlGaN

n-transparent AlGaN Sapphire substrate (a)

UV illumination

Detector mesas

Sapphire substrate AlGaN/GaN heterojuction

Bump bond Electronics boards

Si readout chip (CMOS multiplexer with N×N units)

(b) Figure 4.5 (a) Schematic diagram of a back-illuminated nitridebased detector butted to an Si readout circuit by In bumping. (b) A more detailed view of (a). Courtesy of M. Reine of BAE systems. (This figure also appears on page 717.)

Unit cell

LIII

Color Tables

Figure 4.12 Cartoon showing how the O3 ozone layer prevents the solar radiation to reach the sensor. Courtesy of R. J. Manning of BAE systems. (This figure also appears on page 748.)

1015

Solar irradiance (photons/s-1 cm-2 nm-1)

LIV

1014 1013 1012 1011 1010 109 108 107 106

0.24 atm cm

105 104

0.30 atm cm

103

0.36 atm cm

102

280

285

290

295

300

305

Wavelength (nm) Figure 4.13 The spectrum of the Sun’s radiation in terms of photon flux per unity bandwidth for three different atmospheric masses (atm cm) ranges. (This figure also appears on page 749.)

310

315

Color Tables

Figure 4.57 A top view of GaN avalanche photodetector with 200  200 mm2 in size under a probe station and under bias, left. The magnified image of the active area of the device under reverse bias shows regions of microplasma breakdown, assumed to be associated with highly dislocated regions, right. The reverse bias equal to 80 V. Courtesy of Dr. A. Osinsky. (This figure also appears on page 811.)

Figure 4.60 UV reflection image of a US dollar coin taken with 256  256 AlGaN solar-blind FPA( f ) in Table 4.6. Courtesy of M. Reine BAE systems [17]. (This figure also appears on page 817.)

LV

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1 Light-Emitting Diodes and Lighting Introduction

Owing to nitride semiconductors primarily, which made possible emission in the green and blue wavelengths of the visible spectrum, light-emitting diodes (LEDs) transmogrified from simple indicators to high-tech marvels with applications far and wide in every aspect of modern life. LEDs are simply p–n-junction devices constructed in direct-bandgap semiconductors and convert electrical power to generally visible optical power when biased in the forward direction. They produce light through spontaneous emission of radiation whose wavelength is determined by the bandgap of the semiconductor across which the carrier recombination takes place. Unlike semiconductor lasers, generally, the junction is not biased to and beyond transparency, although in superluminescent varieties transparency is reached. In the absence of transparency, self-absorption occurs in the medium, which is why the thickness of this region where the photons are generated is kept to a minimum, and the photons are emitted in random directions. A modern LED is generally of a double-heterojunction type with the active layer being the only absorbing layer in the entire structure inclusive of the substrate. Such LEDs have undergone a breathtaking revolution that is still continuing, since the advent of nitride-based white-light generation for solid-state lighting (SSL) applications. Essentially, LEDs have metamorphosed from being simply indicator lamps replacing nixie signs to highly efficient light sources featuring modern technology for getting as many photons as possible out of the package. In the process, packaging has changed radically in an effort to collect every photon generated within the structure. Instead of just employing what used to be the standard 5 mm plastic dome to focus the light, the device package is now a high-tech marvel with even holographically generated (employing laser lithography, which is maskless and convenient for periodic patterns) polymeric photonic crystals placed on top or flip-chip mounts (after peeling the GaN structure from the sapphire substrate) with the blackened N-polarity surface for maximum light collection. Furthermore, the area of the device as well as the shape of the chip is designed for maximum
etendue, a measure of the optical size of the device. Furthermore, device packaging also had to adopt strategies not only to remove the heat generated by Handbook of Nitride Semiconductors and Devices. Vol. 3. Hadis Morkoç Copyright
2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40839-9

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the process but also to deal with the thermal mismatch between the chip and the heat sink owing to the Joule heating effect resulting from the current levels in the vicinity of 350 mA. It should be pointed out that nitride LEDs are fabricated on the polar Ga-face of GaN. Therefore, the quantum wells (QWs) used are subjected to quantum-confined Stark shift (red) due the electric field induced by spontaneous and piezoelectric polarization. The latter is severe for increased InN mole fraction in the lattice, in particular, for green LEDs. This results in reduced emission efficiency because of reduced matrix element (lowered overlap integral between the electron and hole wave functions that are pushed to the opposing sides of the quantum well). In fact, the carrier lifetime increases from some 10 ns in bulk InGaN to as high as about 85 ns in a quantum well corresponding to green wavelength. While the same situation is present in lasers, much lower InN compositions and much higher injection levels mitigate the situation to some extent. A quick fix that helps to some extent is to use vicinal substrates even with tilt angles as small as 1
to reduce the polarizationinduced field. To really combat this issue, nonpolar surfaces such as the a-plane GaN is explored. However, the quality of the films is much inferior to those on the c-plane GaN, owing in part to the severe structural mismatch between the r-plane sapphire and a-plane GaN and small formation energy of stacking faults. An additional, aggravating issue is that not much In can be incorporated on this plane, preventing the achievement of blue and green wavelength emission. Research on other orientations such as growth of m-plane GaN has begun. For further information, growth of a- and m-plane GaN is discussed in Volume 1, Chapter 3, and the issue of polarization is discussed in Volume 1, Chapter 2. Elaborating further, as LEDs became brighter and white light generating varieties became available, the role of LEDs shifted from being simply indicator lights to illuminators. The advent of nitride LEDs made white light possible with perfect timing, just when handheld electronic devices such as cell phones and digital cameras became popular, and energy cost increased. In these gadgets, LEDs are used not only for background illumination but also as flashlights, particularly in cell phones. Additionally, LEDs penetrated the automotive industry (aircraft industry is going to follow) in a major way with every indicator and/or background light source, with the exception of headlights, being of LEDs. In the year 2002, with nearly $2 billion in sales worldwide, about 40% accounted for mobile electronics, 23% for signs, and 18% for automotive. The mobile electronics market is mainly of the whiteLED type, which is made possible solely by nitride LEDs. The market continues to experience rapid growth. Retail lighting, shelf lighting, flashlights, night lighting, traffic signaling, highway moving signs, outdoor displays, landscape lighting, and mood lighting have all gone the way of LEDs. The power savings made possible by LEDs in the year 2002 amounted to nearly 10 TW year1 with potential savings approaching 35 TW year1, which will ease the tax on the environment by reducing the greenhouse gas emission. The next frontier for LEDs is to conquer the general illumination, which is underdeveloped, with fierce competition that will bring the best out of those who are going to make this possible. Nitride-based LEDs with InGaN

Introduction

Figure 1.1 InGaN LEDs spanning the spectral range from violet to orange. Courtesy of S. Nakamura, then with Nichia Chemical Co. Ltd. (Please find a color version of this figure on the color tables.)

active regions span the visible spectrum from yellow to violet, as illustrated in Figure 1.1. The three types of LEDs are surface emitters, which are divided into those with plastic domes and those with varieties of flat surface-mount, lacking the dome, edge emitters, generally intended for fiber-optic communications, and superradiant or superluminescent devices, which are biased not quite to the point of lasing but biased enough to provide some gain and narrowing of the spectrum. Antireflection coatings or some other measures are taken to ensure that the device does not lase. Among the applications of LEDs are displays, indicator lights, signs, traffic lights, printers, telecommunications, and (potentially) lighting, which requires emission in the visible part of the spectrum. While saturated-color red LEDs can be produced using semiconductors such as GaP, AlGaAs, and AlGaInP, the green and blue commercial LEDs having brightness sufficient for outdoor applications have so far been manufactured with nitride semiconductors. Figure 1.2 exhibits the various ternary and quaternary materials used for LEDs with the wavelength ranges indicated. The color bar corresponds to the visible portion of the spectrum. We should also mention that another wide-gap semiconductor, ZnO, with its related alloys is being pursued for light emission, as it is a very efficient light emitter. However, lack of convincingly high p-type doping in high concentration has kept this approach from reaching its potential so far [1]. Even though there is still some discussion of the fundamentals of radiative recombination in InGaN LEDs, the basics of LEDs will be treated first, assuming that the semiconductors of interest are well behaved. This will be followed by the performance of available nitride LEDs and their characteristics. The discussion is completed with succinct treatments of the reliability of nitride-based LEDs, and of organic LEDs (OLEDs), which have progressed to the point that indoor applications are being considered.

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Figure 1.2 The LED materials and range of wavelength of the emission associated with them. The color band indicates the visible region of the spectrum. (Please find a color version of this figure on the color tables.)

1.1 Current-Conduction Mechanism in LED-Like Structures

Consider an AlGaN(p)/GaN(p)/AlGaN(n) double-heterojunction device that is forward biased. The carrier and light distribution in the active layer are depicted schematically in Figure 1.3. For simplicity, let us assume that a double-heterojunction device is one in which all the carriers recombine in the smaller bandgap active region. In reality, recombination takes place in the active layer, some fraction of the recombination is nonradiative, and at the two heterointerfaces on both sides of the active layer that is nonradiative. Here, the larger bandgap AlGaN n- and p-layers are doped rather heavily so that no field exists in these regions. The treatment here will be developed in a manner similar to that of Lee et al. [2] and Wang [3]. Because the active layer is p-type, we will be dealing with minority electron carriers. The continuity equation for electrons can be written as D

q2 n nn0 qn þg ¼ ;  t qx 2 qt

ð1:1Þ

where n and n0 represent the minority-carrier concentration and the equilibrium minority-carrier concentration, respectively. The terms D, g, and t represent the electron diffusion length, the generation rate, and the carrier lifetime, respectively, and x and t have their usual meaning. If the active layer were n-type, the same equations would apply with the minority electron parameters replaced with the minority hole parameters. Under steady-state conditions and high injection levels, such as the case for LEDs, the time dependence vanishes, the generation rate and the equilibrium minority carrier concentration can be neglected, and the continuity expression reduces to D

d2 n n  ¼ 0: dx 2 t

ð1:2Þ

This second-order differential equation can be solved with appropriate boundary conditions that can be arrived at by considering the rate of change in the carrier

1.1 Current-Conduction Mechanism in LED-Like Structures

n(x) = n 0 + Α n exp(-x/L)

w

n-Contact

n-AlGaN n(x), Electron concentration

p-AlGaN

p-(In,Ga)N

p-Contact

P = P 0 exp(-αx)

w

Power

x

x

Figure 1.3 The spatial carrier and light distribution in a double-heterostructure LED structure.

concentration at each side of the active p-layer. The general solution of the continuity equation is given by n(x) ¼ A exp


x
x  þ B exp L L

ð1:3Þ

or in the p-region n(x) ¼ A sinh


wx 
wx  þ B cosh : L L

ð1:4Þ

Here, L is the diffusion length L ¼ (Dt)1/2, and the constants A and B can be found subject to the boundary conditions as described below.

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The rate of change in carrier concentration at x ¼ 0 is the difference between the injection rate and the interface recombination rate. The rate of change in carrier concentration at x ¼ w is the difference between the injection rate at x ¼ w and the interface recombination rate at x ¼ w: 

 J (0) vs n(0) dn  ¼ diff  qD dx x¼0 D



 J (w) vs n(w) dn  ¼ diff  qD dx x¼W D

at x ¼ 0

ð1:5Þ

and at x ¼ W;

ð1:6Þ

where q is the electronic charge, Jdiff is the diffusion current density, and vs (cm s1) is the interface recombination velocity. It is assumed that Jdiff(w) is negligible in the case when the p-layer is thicker than the diffusion length. One must keep in mind that the rate of change in the minority carrier is always negative. The solution to the continuity equation subject to the above boundary conditions is ) pffiffiffiffiffiffiffiffiffi rffiffiffiffi( cosh[(wx)=L]þvs t=Dsinh[(wx)=L] J diff (x ¼ 0) t p ffiffiffiffiffiffiffiffi ffi : n(x) ¼   q D v2s (t=D)þ1 sinh(w=L)þ[2vs t=Dcosh(w=L)] ð1:7Þ Here, Jdiff (x ¼ 0) can be assumed to be the terminal current as the hole injection is negligible, given the very small intrinsic carrier concentration. The average electron concentration in the active region can then be calculated from the integral: nave

ðw 1 Jteff : ¼ n(x)dx ¼ qw w

ð1:8Þ

0

Substitution of the electron concentration (Equation 1.7) into Equation 1.8 leads to an effective carrier lifetime, which reduces to 1 t1 þ2 eff ¼ t

vs vs 1 ¼ t1 rad þ tnrad þ 2 w w

ð1:9Þ

if w/L < 1 and v2s (t=D)  1. In addition, in the absence of interface recombination, the effective lifetime would reduce to t, which is related to radiative and nonradiative 1 recombination times through t1 ¼ t1 rad þ tnrad . 1.2 Optical Output Power

As seen by the electron-density expression, the electron density and thus the photon density are reduced in the area away from the junction. Consequently, increasing the

1.3 Losses and Efficiency

active layer thickness does not lead to a continually increasing optical power. In addition, the light generated in the active layer itself is self-absorbed in the active layer. Here, it is assumed that the rest of the structure is a larger bandgap semiconductor, which would not be absorbing. The photon flux density can be approximated by a Gaussian function of the form   4(ll0 )2 S(l) ¼ S0 exp ; (Dl)2

ð1:10Þ

where S is the number of photons per unit time per unit volume with S0 representing the same at the center of the spectrum. At a given point x, in the active layer, S0 ¼ Dn(x)=trad  n(x)=trad , with trad being the radiative lifetime. Recognizing that the photon energy equals hn ¼ hc=l, the power is given by ð¥ S(l) dl; ð1:11Þ P ¼ Ahc l 0

where A is the cross-sectional area. With further manipulation and substitutions Ð w connecting the photon density to the carrier density in the form of S(l) ¼ t1 rad 0 n(x)exp[a(l)x]dx, we obtain ð¥

ðw d l n(x)exp[a(l)x] P ¼ Ahc d x; l trad 0

ð1:12Þ

0

with a(l) being the absorption coefficient, which is of course a function of wavelength, l.

1.3 Losses and Efficiency

One must grapple with the fact that the photons generated in the active layer are emitted in all directions with only a fraction of them escaping the device to reach the human eye. To combat this loss of photons, which relates to the collection efficiency, various packaging designs have been developed. For example, in 5 mm (the epoxy dome diameter) LEDs, the mounting scheme also involves a back-reflector to reflect the light back into the epitaxial composite and then out of the surface for collection. In this regard, the transparent nature of sapphire substrates is very advantageous in that the wavelength of interest is not absorbed as it traverses the structure as shown in Figure 1.4. The backside of the sapphire substrate must be thinned by polishing to facilitate breaking of the wafer into the LED chips and, in the context of this discussion, to eliminate absorption at the otherwise rough surface. Thus, absorption of photons emitted down into the semiconductor structure can be reduced by utilizing layers and substrates transparent to the radiant wavelength. This is coupled with a highly reflective back at the bottom face (substrate in the case of normal mount

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Transparent metal p-Electrode n-Electrode Ga(In,Al)N

Sapphire

Reflector Figure 1.4 Schematic diagram of an LED intended for as much light extraction as possible with a back-reflector and a transparent substrate as is employed in some InGaN-based devices.

and top epitaxial layer in the case of flip-chip mount). The absorption by the metal ohmic contact in the way of the light ray can be reduced by using transparent contacts such as indium tin oxide (ITO). Absorption within the LED (hA in terms of efficiency), critical-angle loss (hc), and reflections (hF) (Fresnel loss) represent the main sources of loss. These loss factors are not yet considered in the derivation of Equation 1.12. It should be noted that in hot intensity LEDs designed for illumination, the substrate is completely removed and the flipped chip is mounted on metal alloy that plays a dual role as reflector and heat sink, as discussed in Section 1.5. In GaAs diodes with a GaAs substrate, about 85% of the photons generated are absorbed. If transparent substrates are used, such as GaP, only about 25% of the photons are lost. In the case of InGaN diodes, the entire structure with the exception, of course, of the active layer itself, is transparent, and therefore, absorptive losses are almost eliminated. As for the reflection at the semiconductor–air interface, when light passes from a medium with refractive index n2, which is the active layer here, to a medium with refractive index n1, being air in this case, a portion of the radiation is reflected at the interface. This loss, which is called the Fresnel loss, is given in the case of normal incidence by

n2  n1 2 : ð1:13Þ R¼ n2 þ n1 The Fresnel loss efficiency [4] can be defined as hF ¼ 1  R. The critical angle for total reflection qc – total reflection taking place above this angle – is determined by Snell’s law

n1 : ð1:14Þ qc ¼ sin1 n2 For GaAs and GaP, these angles are 16
and 17
, respectively. For GaN–air interface, the critical angle is about 21
, 24
, and 25
at the wavelengths of 365, 450, and 520 nm, respectively. The critical loss efficiency can be expressed as hc ¼ sin2 qc or 1  cos qc.

1.3 Losses and Efficiency

If the efficiency term associated with internal losses including interface recombination and self-absorption is denoted by hA, then hopt ¼ hFhchA would represent the efficiency of the total power extraction. The optical power at the central wavelength l0 can be obtained as ðw Ahc Ahc n(x)exp(a0 x)d x ¼ Jteff : P0 ¼ l0 trad ql0 trad

ð1:15Þ

0

Recognizing that hc/l0 represents the photon energy and if the photon energy is xph given in electron volts, one can define the internal quantum efficiency as hint ¼

P0 : Ixph

Utilizing Equation 1.15 for the power, we obtain



hc I 1 teff hint ¼ : ¼ t ql0 Ix trad t1 rad ph eff

ð1:16Þ

ð1:17Þ

Multiplying the internal quantum efficiency by the combined loss and efficiency factors, the external quantum efficiency becomes teff hext ¼ hopt hint ¼ hopt ; ð1:18Þ trad which is about 10% for UV and blue GaN-based diodes. In the case where there are ohmic losses, the term xph must be replaced by the energy corresponding to the applied voltage qVappl. Then, the external quantum efficiency will assume the form

xph teff hext ¼ hopt : ð1:19Þ trad qV app The optical power extracted from the LED is given by

hc I : P0 ¼ hopt ql0 trad t1 eff

ð1:20Þ

For a double-heterojunction LED, where the active layer is the only absorbing layer in the entire structure on a transparent substrate, the internal absorption term, hA, including interface recombination, has been determined to be [2] hA ¼

 



 1 w w 2 V 2s þ1 sinh þ2V s cosh L L   

  1þV s 1þaL w 1V s 1exp w  exp  1þaL 1aL L L 

 1aL w ;  1exp w exp L L

with Vs ¼ vsL/D.

ð1:21Þ

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1

Efficiency and power coefficient

s=100

ηΑ

s = 1000

0.8 ηΑ Coeff.V

0.6

0.4

Coeff.V

s = 100 s = 1000

0.2

0 0

0.1

0.2

0.3

0.4

0.5

Active layer thickness (µm) Figure 1.5 The efficiency reduction term caused by interface recombination and bulk absorption in an otherwise ideal GaNbased LED for surface-recombination velocities of 100 and 1000 cm s1. The coefficient term relates the output power to the injection current.

Figure 1.5 exhibits hA as a function of the active layer thickness for two surfacerecombination velocities (100 and 1000 cm s1). The other parameters used are for GaN, even though all the LEDs are made of InGaN (center wavelength: 450 nm; electron mobility: 600 cm2 V1 s1). The effective carrier lifetime is as indicated (radiative lifetime ¼ 2  109 s, absorption coefficient a ¼ 105 cm1, and refractive index ¼ 2.6). Moreover, the coefficient in front of the injection current in Equation 1.15 relating the power to the injection is also plotted. In the absence of available data, what would be plausible was chosen based on the assertion that the GaN surface is reasonably inert. Further consideration was given to the observation that the Schottky barrier height seems to become higher with an increased work function of the metal. Figure 1.6 displays the same parameters as a function of the surface-recombination velocity in the range of 1–10 000 cm s1 for several thicknesses of the active layer ranging from 3.5 to 20 nm. Having done the analysis, we must recognize that the underlying assumption made is that the carrier motion in the active layer is driven by conventional diffusion. However, the InGaN active layer utilized in an LED is highly clustered and textured, and far from ideal for the diffusion-limited current to be applicable. In fact, these clusters may be responsible for the carrier localization and therefore the enhanced radiative recombination, which may explain the efficient light emission even in the presence of extremely high concentrations of defects. The expressions above are meant to provide the reader with a guide to which parameters are important and what role they play in the device operation. Appropriate carrier lifetimes, when available,

1.4 Current Crowding τe1=0.5 ns, τe2=1 ns w 1=3.5 nm, w 2=20 nm 100

τe2 & w2 τe1 & w1 τe2 & w1

10-2

τe1 & w2 -3

10

τe2 & w2 τe1 & w1

10-4

Power coefficient

Efficiency and power coefficient

10

Efficiency

τe1 & w2 -1

τe2 & w1

10-5 0

2×10

3

4 ×10 3

6 ×10 3

8 ×10 3

1 ×10

4

Surface recombination velocity (cm s-1) Figure 1.6 The parameters of Figure 1.5 as a function of the surface-recombination velocity for active layer thicknesses ranging from 3.5 to 20 nm. Here, te and w represent the lifetime and active layer thickness, respectively.

can be used in conjunction with the expressions provided here to arrive at characteristics representative of devices available today.

1.4 Current Crowding

Current spreading is prevalent in many semiconductor devices including bipolar transistors, which provided impetus for investigations leading to the understanding of the phenomenon and the means to combat it, and of particular interest LEDs and laser diodes fabricated on insulating substrates. Current crowding robs the device of its optimum performance, as well as causing premature breakdown, and as such it is an important and complex process [5, 6]. Most GaN-based lightemitting diodes are grown on insulating sapphire substrates and thus employ mesa structures with lateral contact geometries of the anode and cathode electrodes. The uniform current spreading in such mesa structures can be expected when the p- and n-cladding layers are thick and highly conductive [5]. However, the growth of thick and highly conductive p-type AlInGaN films is more difficult, compared to that of the n-type films. Thus, it is intuitively expected that the current in the mesa structure will crowd near the edge of the p-type metal electrode. Nakamura and Fasol [7] used a Ni/Au transparent electrode on the p-type cladding layer to help uniformly spread the current in the mesa structure of GaN-based

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LEDs. To date, the p-transparent electrode is generally used for GaN-based LEDs. It can be, in turn, expected that the highly conductive p-transparent electrode may cause a current crowding at the edge of p-type mesa near the n-type electrode when the sheet resistance of n-type GaN cladding layer is larger than that of the p-transparent electrode. In 1999, Eliashevich et al. [8] demonstrated experimentally that the insufficient n-GaN conductivity in InGaN LEDs caused current crowding near the edge of the mesa adjacent to the n-contact pad, resulting in a degradation of the LED performance. This kind of current crowding, which can be caused by employing the highly conductive p-transparent electrode, can be minimized by (i) adjusting the device parameters related to the current-spreading length, Ls [9], (ii) optimizing the configuration of p- and n-metal electrodes [10], or (iii) a combination of the methods. Calculations of current crowding in lateral p-side-up mesa structures utilizing GaN/InGaN grown on insulating substrates have been carried out by Guo and Schubert [11] using an equivalent circuit model shown in Figure 1.7. The model includes the p-type contact resistance and the resistances of the n-type and p-type cladding layers with the assumption that the p-type metal contact has the same electrostatic potential at every point. The p–n-junction region was approximated by an ideal diode. The developed model revealed an exponential decrease of the current density with distance from the mesa edge. That is, the model could quantitatively explain the current crowding effect near the mesa edge when the sheet resistance of the p-type transparent electrode is zero. Later, an advanced model to explain the current crowding phenomenon was presented by Kim et al. [9]. The model included the lateral resistance component of the p-type transparent electrode in the equivalent circuit of the LED, which is different from the earlier model developed by Guo and Schubert [11]. The expanded model revealed that the current distribution of the LED was critically dependent on the sheet resistance of p-type transparent electrode, that is, the film thickness of the p-transparent electrode, and that theoretically uniform current spreading would be possible when the sheet resistances of the p-transparent electrode and

p-pad tt

t rc rp

p

n-pad n

rt

tp

Vj rn dx

x=0

tn

dx x=l

Figure 1.7 Equivalent LED circuit, with a p-pad as the physical ground that can be used to model current crowding [9].

1.4 Current Crowding

n-type layer are identical. The expanded model developed by Kim et al. [9] is introduced as follows. Figure 1.7 shows the schematic LED structure with lateral injection geometry. In this structure, important distributed components of the total series resistance can be categorized into the lateral resistance component of the n-layer, rn, and the transparent electrode, rt, and the vertical component of the p-layer, rp, and the p-contact, rc. It is noted that the nontransparent p-pad and the transparent p-electrode layer are discriminate. Applying the assumption that the p-pad is grounded, the current continuity equation applied to the circuit in Figure 1.7 leads to the following two basic equations: d2 V n rn ¼ J; dx 2 tn

d2 V t rt ¼ J: dx 2 tn

The relation between Vn and Vt can also be expressed as follows:

eV j þV t : V n ¼ V j þRv I 0 exp kT

ð1:22Þ

ð1:23Þ

The parameters Vn, Vt and rn, rt are the lateral voltage drops and the electrical resistivities of the n-layer and p-transparent electrode, respectively, and J is the current density across the p–n-junction region. Vj is the junction voltage drop, I0 is the reverse saturation current, and Rv is the vertical resistance of the area element w dx. Using the above three equations, Kim et al. [9] derived the following diode equation: 2 0 13   ffi

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  rn rt 1 J(x) ¼ J(0)exp4x @ (rc þrp tp )   A5; ; p-pad; þ; n-pad; tn tt ð1:24Þ where J(0) is the reverse saturation current density at the mesa edge, and the () sign holds for the p-pad and the (þ) sign for the n-pad as physical grounds, respectively. As a result, the current-spreading length, Ls, can be expressed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi  rn rt 1 ð1:25Þ Ls ¼ (rc þrp tp )   : tn tt Figure 1.8 shows the calculated current distribution of the LED, indicating that it is possible to achieve a perfectly uniform current distribution at the critical condition of rt/tt ¼ rn/tn. The current-spreading length equation given above illustrates that the current distribution in the LED structure can be controllable by adjusting the device parameters included in the expression, which is fruitful in designing efficient high-power III-nitride LEDs. Because the conductivity of the p-layer is relatively low, which is the crux of the current-spreading problem, coupled with all on-top contact configuration, attempts have been made to reduce the current crowding effect. However, it must be mentioned that vertical device structure thin-film LEDs, including the top and

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x = 0 (n-pad)

(p-pad) x = l

ρt / tt=ρn / tn=50 Ω (perfectly uniform current spreading)

1.0

Ls = infinite

Guo et al. Calculated by Kim et al.

J(x) (a.u.)

0.8

0.6

1266 µm

1266 µm 45 Ω

55 Ω 0.4

895 µm

895 µm

40 Ω 633 µm 30 Ω 70 Ω 517 µm 517 µm 80 Ω 20 Ω 448 µ m 400 µm 10 Ω 100 Ω Ls = 400 µm Ls = 231 µm ρt / tt = 200 Ω p-transparent electrode equipotential: ρt / tt = 0 Ω

60 Ω

0.2

0.0 0

633 µm

200

400

600

800

1000

x (µm) Figure 1.8 Calculated current distributions versus the lateral length x in a LED. The parameters used in the calculation rn ¼ 0.01 W cm, tn ¼ 2 mm [9].

bottom contact configurations, have been developed for high-efficiency and highly luminous devices that obviate the current-spreading problem to a large extent. Nevertheless, for completeness and instructional value, a few approaches that have been used to help alleviate current spreading in surface oriented devices are discussed here. For example, Jeon et al. [12] have shown that pþ/nþ GaN tunnel junctions (TJs) inserted into the upper cladding layers of conventional devices allow the use of an n-type GaN in place of a p-type GaN as the top contact layer. The reverse-biased tunnel junction helps with lateral current spreading without semitransparent electrode and spatially uniform luminescence exhibiting an improved radiative efficiency. With the same goal in mind, an nþ short-period superlattice [13] and a p-InGaN tunneling contact layer on low conductive p-GaN [14] have been employed for uniform current distribution. Again, in the same vein an undoped GaN spacer layer [15] and a modulation-doped AlGaN/GaN heterostructure [16] have also been inserted. The current crowding effect, which can be reduced by employing a p-transparent electrode, can also be minimized by improving the metal contact configuration. This is especially important in high-power AlInGaN LEDs with a large chip size. Krames et al. [10] reported an interdigitated contact geometry, wherein each set of p-metal contacts is surrounded by two n-metal contact fingers. The interdigitated configuration reduces the spreading distance required by the current in the n-GaN layers, resulting in a more uniform current distribution on a large-area LED chip [17]. Figure 1.9 shows the schematic (a) planar and (b) cross-sectional views of a large-area LED structure employing the interdigitated contact geometry, which shows symmetric current paths below the p-metal pads.

1.5 Packaging

p-pad

(a)

n-pad p-transparent metal p-GaN

p-pad n-pad

n-GaN

Sapphire (b) Figure 1.9 Schematic (a) plane and (b) cross-sectional views of a 1  1 mm2 power AlInGaN LED employing interdigitated contact geometry [10].

1.5 Packaging

Packaging plays multiple roles in that it mechanically interfaces the LED chip to its operating interface, inclusive of electrical connections; in many cases, it focuses and increases the light extracted from the LED, dissipates the heat generated, and provides protection, both in physical terms and against stray fields. Packaging is configured for a given device design. It should be pointed out, however, that LEDs are nowadays competing in areas that are not traditional for them and as such the standard packaging design may not be appropriate for all devices. The standard LED package is the 5 mm diameter epoxy dome with which everyone is familiar. Device design and packaging go hand in hand. To increase light extraction from a p–n-junction, the active layer is placed close to the surface; the entire layer structure outside of the active layer is made of a transparent (larger bandgap) semiconductor that transmits the emission wavelength. Moreover, a dome of lower refractive index (lens) is placed on top of the device, which increases the collection cone and causes the photons entering it to strike the domed surface at or near a normal angle with an escape certainty of unity (Figure 1.10). For extracting as much of the light generated

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Gold wire

LEDchip

Epoxy dome lens

Reflector cup

Cathode lead

Anode lead

(a)

Figure 1.10 (a) Schematic representation of a standard 5 mm epoxy domed LED for enhanced light collection as well as focusing of the emitted light (circa 1970). This cross-sectional schematic also identifies the various important components in the package. (b) A photograph of

three 5 mm domed packaged LEDs. The longer lead represents the anode and is connected to the positive terminal of the battery for light emission. This package could be used at current levels of 30 mA with the LED producing 2–3 lm. Courtesy of Lumileds/Phillips.

1.5 Packaging

as possible, a transparent top ohmic contact coupled with, when applicable, a highreflectivity back contact is also employed. The dome increases the efficiency by about twice the square of the refractive index of the semiconductor. The dome also serves to focus the light, concentrating the radiation within the field of view. Various dome approaches are available, among which are hemispheres, truncated spheres, and paraboloidal types with narrowing radiation patterns. The cone defined by the total reflection angle in larger bandgap semiconductors, such as GaN, is larger and increases the light collection. In 5 mm LEDs, only about 60% of the light flux is captured within a 120
cone in air. For higher power LEDs, as compared to the 5 mm devices, the chip design must be improved [18]. For high-luminance LEDs, flip-chip mounting combined with Ag-based back-reflector techniques has been utilized [19, 20], an example of which, developed by Lumileds, is shown in Figure 1.11. These high-power devices are much larger than the conventional chip, with dimensions of 1  1 mm2 (i.e., 10 larger active area) or larger. The large area of the chip puts stringent requirements on the current spreading in the GaN:Si layers beneath the active region. It is impossible to spread current uniformly over the full distance of the chip through a GaN:Si layer that is only a few microns in thickness, but an interdigitated contact design is employed to separate the large area of the chip into segmented areas, or cells, wherein uniform current spreading and low series resistance is achieved. The flip-chip design described below addresses the issues pertinent to this high-power device. There are many advantages to the flip-chip LED (FCLED) design among which is the fact that light exits through the polished transparent sapphire substrate instead of an absorbing Ni/Au contact layer as for the case of top-emitting power and conventional AlGaInN LEDs [19]. Also, downward-propagating light is reflected up, increasing the light extraction, as shown in Figure 1.11. A second benefit is that the heat generated in the LED flows directly from the p–n-junction out through the Si submount, which forms the base of the package and has a thermal conductivity that is at least three times better than that of sapphire. This is important because under standard operating conditions, the power dissipation in the high-power chip is approximately 1 W (350 mA at 3.5 V), which is more than 10 times that for conventional LEDs in 5 mm LEDs [18]. The Si submount also contains two backto-back semiconductor diodes for electrostatic discharge (ESD) protection to which LEDs on sapphire are prone. Third, current spreading on the p-side of the device is handled by a thick p-contact, instead of the standard thin Ni/Au contact in conventional mounting schemes, reducing spreading resistance. In conventional mounting schemes, the Ni/Au contact must be made thin enough for light propagation. Thinner metal allows greater light transmission at the expense of increased current crowding, which lowers the injection efficiency of the devices by preferentially favoring the edges of the device and forcing the area in the middle to be pumped at lower injection currents. Finally, the outer housing, which provides support for a plastic lens, contains a concave recess that forms a cavity surrounding the chip. This cavity is backfilled with a soft silicone encapsulant (n  1.5), which provides increased light extraction while minimizing thermal expansion/contraction stress on the chip and wire bonds during operation.

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Figure 1.11 (a) Cross-sectional schematic flipchip mounted high-luminance LED package with Ag back-reflector. Electrostatic discharge protection is integrated into the Si submount (circa 1997). The package is able to handle power dissipation associated with 350 mA current injection with resulting LED lumens of 20–40 lm. (b) Artistic rendition of the package inclusive of plastic lens. In this flip-chip model, the substrate

can be removed and the exposed N-polarity GaN can be made dark by chemical etching or a polymeric photonic crystal can be placed on what is now the top surface for better photon collection. Polymeric photonic crystal can be produced by laser lithography (holography) to reduce the cost of fabrication. Courtesy of Lumileds/Phillips. (Please find a color version of this figure on the color tables.)

The improved extraction efficiency of the flip-chip LED provides 1.6 times more light when compared to top-emitting power LEDs and 10 times more light than conventional small-area (0.07 mm2) LEDs. The design also features low spreading resistance, mitigating current crowding. Additional attributes are low thermal

1.5 Packaging

resistance, stable and soft gel inner encapsulant, and controlled radiation patterns. As has been reported, this improved packaging led to output powers greater than 250 mW (1  1 mm2 device) and 1 W (2  2 mm2 device) at standard operating current densities (50 A cm2), corresponding to “wall-plug” efficiencies of 22–23% in the blue wavelength regime. Employing phosphors for the generation of white light, these same devices achieve luminous efficiencies higher than 30 lm W1. This package is well suited for the high-power Luxeon LEDs developed by Lumileds with electrical input power levels up to 5 W. For enhancing the light extraction efficiency, a thin-film LED structure, in which the substrate is removed, has been reported [21]. This particular technique employs a combination of laser liftoff (LLO) and photoelectrochemical (PEC) etching to produce a roughened top surface with cone-like features, from which the light emerges. The output power of an optimally roughened surface LED showed a twofold to threefold increase as compared to that of an LED without surface roughening. The combination of the thin-film LED concept with flip-chip technology can provide surface brightness and flux output advantages over the conventional flip-chip and vertical-injection thin-film LEDs [22]. An encapsulated thin-film flip-chip blue LED (TFFC-LED) (441 nm) exhibited an external quantum efficiency of 38% at a forward current of 350 mA. A white-LED lamp based on a YAG:Ce phosphor-coated device exhibited a luminous efficacy of 60 lm W1 at 350 mA with a peak efficiency of 96 lm W1 at 20 mA and a luminance of 38 Mcd m2 for 1 A drive current. Green (517 nm) LED devices exhibited a luminance of 37 Mcd m2 at 1 A. In the world of LEDs, it is clear that there are three guiding principles: convert all the electron–hole pairs into photons to the extent possible; collect all the photons, again to the extent possible, from the emitting surface; and minimize the Joule heating (heat loss). The former relies on device designs, such as the use of heterojunction, and material quality and is commonly measured by internal quantum efficiency. The second is addressed by the use of transparent layers, contacts, and substrates with the exception of the emitting region of the semiconductor that can be made thin by using double heterojunctions. In addition, the plastic lens, which not only focuses the light somewhat but also increases the collection angle from within the semiconductor, helps in this respect. Essentially, the optical size of a component, which is termed the
etendue, must be increased. As the need to extract more light from LEDs became more critical with the advent of LEDs being used not just for displays but illumination also, the shape of the semiconductor chip has also taken a central role. The third item can be addressed by reducing the series resistances arising from both the semiconductors and ohmic contacts. The evolution of chip designs within the framework of light extraction is shown in Figure 1.12. In the Lambartian design of Figure 1.12d, the
etendue that is given (in terms of mm2 sr) by. E ¼ pAn2 sin(q=2)

ð1:26Þ

is maximized. Here, A is the emission area of the optical component (in mm2), n is the refractive index of the medium surrounding the optical component, and q is the angle of emission. In the Lambartian design, the angle q ¼ p, maximizing E ¼ pAn2.

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Figure 1.12 Evolution of LED chip and chip shapes in the pursuit of extracting more of the photons generated within the chip. The circle represents the point of photon generation. Part (a) depicts the early designs with absorbing substrates, (b) indicates transition to the transparent substrates with back-reflectors, (c) depicts increased area for better collection of photon (larger
etendue), and (d) indicates

reshaping of the LED package for even more enhanced collection of photons. In the Lambartian design (d), the use of a thin quantum well emitting layer as opposed to a thick emitting layer has also been incorporated. From (a) to (d), an enhancement by a factor of 30 in the light flux has been realized. Courtesy of Lumileds/ Phillips.

Once the chip is removed from the substrates, the chip can be mounted on a metal alloy substrate, which has a very high thermal conductivity of 400 W m1 K1 and therefore allows high-current operation (see Figure 1.13 for a schematic) [23]. The chip has a patterned surface with “photon-injecting nozzle” microstructures to

Figure 1.13 A MVP-LED structure from Semi-LEDs. Courtesy of C. A. Tran [23].

1.5 Packaging

enhance light extraction in the forward direction. Eight LED chips packed together could provide 460 lm white light with a luminous efficacy of 58 lm W1 at 470 mA. The metal vertical photon LED (MVP-LED) has a p-down epitaxial structure mounted on a reflector layer that is attached to a metal alloy substrate. Even with the thin flip-chip package, one must still deal with the top-emitting surface, namely, the semiconductor–air interface reflection and the associated collection cone angle. Unless this is countered, only a fraction of photons headed toward the top surface would actually escape the semiconductor. To eliminate the above-mentioned internal loss issue of GaN LEDs and to extract more light from the device surface, GaN-based photonic crystal (PC)-LEDs fabricated using a laser holography (LH) method have been reported [24]. These structures are suitable for high-throughput and large-area processing. There are two kinds of PC-LEDs reported: top-loaded and bottom-loaded. For the top-loaded PC-LEDs [25], a conventional LED structure was grown first. Then using laser holography method, the photonic crystal pattern was generated, as shown in Figure 1.14. The resultant LEDs exhibited significant improvements in light extraction, up to 1.5 times that of planar LEDs without PC integration. Similar results were obtained for the bottom-loaded

Figure 1.14 (a) Schematic view of a top-loaded PC-LED, illustrating the vertical layer structure of the device. (b) Scanning electron microscope image for a top-loaded PC-LED device surface. The square-lattice air–hole array pattern was generated by the holographic double-exposure method. The lattice period of this specific example is 700 nm. Courtesy of H. Jeon [24].

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PC-LEDs, in which the photonic crystal was formed between the substrate and the epilayer. Another approach to reduce the internal optical loss is to employ a so-called “sidewall deflector” to reflect out the photons that are trapped between the air– GaN–sapphire waveguide. Typically, angles of 20–40
between the mesa sidewalls and the substrate were achieved by a photoresist reflow method [26]. Experimental results, including photoluminescence and near- and far-field patterns, show a strong additional emission along the sidewall edge, and the proposed LED structure enhanced the overall surface emission intensity by a factor of 2 for a sidewall angle of 30
. When a combination of photonic crystal and angled sidewall was employed, three times higher emission power was achieved.

1.6 Perception of Visible Light and Color

Human vision is a complex process, the understanding of which is still an evolving matter. It involves the simultaneous interaction of the eyes and the brain through a network of neurons, receptors, and other specialized cells. The initial steps in vision are the stimulation of light receptors in the retina, which lines the back of the eyeball, and transmission of electrical signals containing the vision information to the brain through the optic nerves. This information is processed in several stages, ending at the visual cortex of the brain. The human eye is equipped with a variety of optical elements including the cornea, iris, pupil, aqueous and vitreous fluids, variable-focus lens, and the retina, as illustrated in the schematic shown in Figure 1.15. Together, these elements work

Figure 1.15 Diagram of a human eye showing its various structures along with the optical path of vision. (Please find a color version of this figure on the color tables.)

1.6 Perception of Visible Light and Color

to form images of the objects in our field of view. The iris, which opens wide at low light levels and closes to protect the pupil and retina at very high levels, controls the amount of light entering the eye. As the light enters, it is first focused through the cornea and lens onto the retina, a filmy, multilayered membrane that contains millions of light-sensitive cells that detect the image and translate it into a series of electrical signals. These image-capturing receptors, which are termed rods and cones, are connected with the fibers of the optic nerve bundle through a series of specialized cells and tissues that coordinate the transmission of signals to the brain. It is interesting to note here that these tissues develop from a pouch of the embryonic forebrain, and therefore the retina is considered part of the brain. The basic structure of the retina can be compared to a three-layer cake, with the bodies of nerve cells arrayed in three rows separated by two layers packed with synaptic connections. The back of the retina contains the photoreceptive sensory cells, and being there, light rays must pass through the entire retina before reaching photoreceptive molecules to excite. The retina in mammals contains at least two types of photoreceptors, namely, rods and cones, but rods dominate. Rods are utilized for low-light vision and cones for daylight. At dusk, dawn, and in dimly lit places, rods provide gray vision without color. The cones are responsible for bright-colored vision. Most mammals have two types of cones, green-sensitive and blue-sensitive, but primates have three types of cones: red-sensitive, green-sensitive, and blue-sensitive. With our cone vision, we can see from gray dawn to the dazzling conditions of high noon. Most fish, frog, turtle, and bird retinas have three to five types of cones and consequently very good color vision. In cats and dogs, images focus to a central specialized area, appropriately called the area centralis, where the cones predominate. The retinas of mammals such as rabbits and squirrels, as well as those of nonmammals such as turtles, have a long, horizontal strip of specialized cells called a visual streak, which can detect the fast movement of predators. Primates as well as some birds have front-projecting eyes allowing binocular vision and thus depth perception; their eyes are specialized for good daylight vision and are able to discriminate color and fine details. Primates and raptors, such as eagles and hawks, have a fovea, a tremendously cone-rich spot devoid of rods where images focus. The fovea contains most of the cones, packed together as tightly as physically possible, and allows good daylight vision. More peripheral parts of the retina can detect the slightest glimmer of photons at night. Initially, the cone photoreceptors themselves can adapt to the surrounding brightness, and circuitry through the retina can further modulate the eye’s response. Similarly, the rod photoreceptors and the neural circuitry to which they connect can adapt to lower intensities of light. An artistic rendering of retina with photoreceptors is shown in Figure 1.16, where the photoreceptors are at the top of the schematic. Both rods and cones respond to light with a slow hyperpolarizing response, but report quite different image properties. Rods, detecting dim light, usually respond to relatively slow changes. Cones, dealing with bright signals, can detect rapid light fluctuations. As photons strike these photoreceptors, retinal molecules become fixed in the photoreceptors’ scotopsin (rods) or photopsin (cones) proteins, and these

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Figure 1.16 A schematic showing rods that are sensitive to low-level gray light and cones that are sensitive to color. In this rendering, light enters the eye from the bottom. The photons travel through the vitreous fluid of the eyeball and penetrate the entire retina, which is about half a millimeter thick, before reaching the photoreceptors: the cones and rods that respond

to light (the colored and black cells attached to the epithelium on top). Signals then pass from the photoreceptors through a series of neural connections toward the surface of the retina, where the ganglion cell nerve fiber layer relays the processed information to the optic nerve and then to the brain [27]. (Please find a color version of this figure on the color tables.)

molecules change their conformation in response to the photons. Once retinal molecules are exposed to light and undergo their conformational change, they are recycled into the dark row of cells at the back of the retina called the pigment epithelium [27, 28]. This tissue behind the retina is usually very dark because its cells are full of melanin granules. The pigment granules absorb stray photons, preventing their reflection back into the photoreceptors, which would cause images to blur. They also protect the cells from too much exposure to light radiation. The image continues to be broken into component elements at the first synapses of the visual pathway, those between photoreceptors and what are called the bipolar cells. Different bipolar cells have different types of receptors for the neurotransmitter glutamate, allowing the cells to respond to photoreceptor input differently. Some bipolar cells are tuned to faster changes and some to slower ones in the visual signal; some glutamate receptors resensitize rapidly and others more gradually.

1.6 Perception of Visible Light and Color

Consequently, the cells fire either quickly in succession or relatively slowly in response to the same amount of stimulation. Some receptors respond to glutamate by activating what is known as an OFF pathway in the visual process, detecting dark images against a lighter background. Other bipolar cells have inhibitory glutamate receptors, which prevent the bipolar cell from firing when the cell is exposed to the neurotransmitter. These receptors activate the ON pathway, detecting light images against a darker background [19]. Intricately wired neurons in the retina allow a good deal of image assembly to take place in the eye itself. The parallel sets of visual channels for the ON (detecting light areas on dark backgrounds) and OFF (detecting dark areas on light backgrounds) qualities of an image are fundamental to our seeing. Parallel bipolar channels transmit inputs to ganglion cells. In early stages of development, the architecture of the inner plexiform layer, which is full of synapses between bipolar and ganglion cells, shows that synaptic connections become segregated in distinct and parallel pathways. Connections occur between ON bipolar cells and ON ganglion cells and also between OFF bipolar cells and OFF ganglion cells in delineated portions of the inner plexiform layer. If the retina were simply to transmit opposite-contrast images directly from the photoreceptors to the brain, the resulting vision would probably be coarse, grainy, and blurred. Further processing in the retina defines precise edges to images and provides the means to focus on fine details. The fine-tuning of image perception starts at the first synaptic level in the retina, where horizontal cells receive inputs from many cones. Horizontal cells’ receptive fields become even broader because their plasma membranes fuse with those of neighboring horizontal cells at gap junctions. The membrane potentials of a whole sheet of cells become the same; consequently, horizontal cells respond to light over a very large area. Meanwhile, a single bipolar cell receives inputs from a handful of cones and thus has a mediumsized receptive field. Whereas a single bipolar cell with its OFF or ON light response would carry a fairly blurred response to its ganglion cell, horizontal cells add an opponent signal that is spatially constrictive, giving the bipolar cell what is known as center surround organization [27]. At the anatomical level, imaging techniques ranging from silver staining to electron microscopy and modern-day antibody staining have been applied to reveal the shapes and sizes of the retina’s cell types and how the different cells connect to form synapses. The optic nerve fibers could be stimulated to give traditional depolarizing action potentials, like those observed in other neurons. These “S potentials” are now known to originate with the photoreceptors and to be transmitted to horizontal cells and bipolar cells. The membrane hyperpolarization starts on exposure to light, follows the time course of a light flash, and then returns to the baseline value when the light is off. This reflects the counterintuitive fact that both rods and cones release neurotransmitters during the dark, when the membrane is depolarized and sodium ions flow freely across the photoreceptors’ cell membranes. When exposed to light, ion channels in the cell membranes close. The cells go into a hyperpolarized state for as long as the light continues to shine on them and do not release a neurotransmitter [28]. In the brain, the optic nerves from both eyes join at the optic chiasma where information from both retinas is

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correlated. From here, the visual information travels to the lateral geniculate nucleus where the signals are distributed to the visual cortex located on the lower rear section of each half of the cerebrum. The fovea centralis is located in an area near the center of the retina located on the optical axis of the eye. This area contains exclusively high-density tightly packed cone cells and is the area of sharpest vision. The density of cone cells decreases outside of the fovea centralis, and the ratio of rod cells to cone cells gradually increases. At the periphery of the retina, the total number of both types of light receptors decreases substantially, causing a dramatic loss of visual sensitivity at the retinal borders. This is offset by the fact that we constantly scan objects in the field of view, giving a perceived image that is uniformly sharp. Rod cells have peak sensitivity to green light (about 550–555 nm), although they display a broad range of responses throughout the visible spectrum. They are the most populous visual receptor cells, with each eye containing about 130 million cells. The light sensitivity of rod cells is about a thousand times that of cone cells. However, as mentioned above, the images generated by rod stimulus alone are relatively blurred and confined to shades of gray, similar to those found in a black-and-white soft-focus photographic image. Rod vision is commonly referred to as scotopic or twilight vision because in low light levels, it allows us to distinguish the shapes and relative brightness of objects, but not their colors. When all three types of cone cells are stimulated equally, we perceive the light as being achromatic or white. As an example, noon sunlight appears to us as white light because it contains approximately equal amounts of red, green, and blue light. An excellent demonstration of the color spectrum of sunlight is interception of the light by a glass prism, which refracts (or bends) different wavelengths to varying degrees, spreading out the light into its component colors. Our color perception is dependent upon the interaction of all receptor cells with light and this combination results in nearly trichromic stimulation. There are shifts in color sensitivity with variations in light levels, so blue colors look relatively brighter in dim light and red colors look brighter in strong light. This effect can be observed by pointing a flashlight onto a color print under dim ambient light, which will result in the red suddenly appearing much brighter and more saturated. In daylight, the human eye is most sensitive to the wavelength of 555 nm, with a maximum sensitivity of 683 lm W1. This is called photopic vision. In low-light and night situations, when scotopic vision is used the peak sensitivity changes, blueshifting to 507 nm. The maximum sensitivity for scotopic vision is 1754 lm W1. At the red and blue extremes, the sensitivity of the human eye drops dramatically [29–32]. Figure 1.17 shows the luminous efficacy, K(l), which represents the effectiveness of the radiant power of a monochromatic light source in stimulating the visual response for daylight vision (photopic vision) and night vision (scotopic vision). As seen in Figure 1.17, the efficacy curve falls drastically at both ends of the visual spectrum. This increases the requirements for the output power and the external quantum efficiency for emitters in the blue and red regions to achieve the same brightness or luminous performance offered by green-light sources and for practical visual displays.

1.7 Visible-Light Terminology

104 Scotopic vision Km=683 1

Photopic vision 102 101 0.5

100

Value, V ( λ )

Luminous Efficacy(lm W-1)

103

10-1

10-2 350

400

450

500

550

600

650

700

750 800

Wavelength (nm) Figure 1.17 Luminous efficacy of monochromatic radiation, K(l), for the human eye under light (photopic vision) and dim (scotopic vision) conditions. The band indicates the color at visible wavelengths. The maximum luminous efficacy, Km, for photopic vision is 683 lm W1 and occurs at 555 nm. (Please find a color version of this figure on the color tables.)

Another interesting feature of human eye is that it is extremely good at delineating edges. This manifests itself, for example, in two patterns having the same spectral power distribution appearing different depending on the background. In other words, the photopigment response in a small area does not determine the color perceived by human eye. The color appearance then depends on the shape of the image as a whole that is being viewed. This brings about an added complexity to the meaning of color and thus the illumination needed to bring it to being.

1.7 Visible-Light Terminology

The optical power generated by a light-emitting diode must excite the human eye. This brings into the discussion the color perception of the human eye, which has been standardized by the Commission Internationale de l’Éclairage or International Commission on Illumination (CIE) [33]. This commission produces charts used by the display society to define colors. Detection and measurement of radiant electromagnetic energy is called radiometry, which when applied to the visible portion of the spectrum involving the human eye is termed photometry. The nomenclature for the latter delineates itself from the former by adding the adjective luminous to the terms

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used for the former. For example, energy in the former is called the luminous energy in the latter. The former can be converted to the latter, and vice versa, if the perception of color by the human eye is understood. The terms employed to describe LED performance in terms of photometric terms are as follows: Brightness: A subjective term used to describe the perception of the human eye, ranging from very dim, on the one hand, to blinding, on the other. The relationship between brightness and luminance is very nonlinear. Luminance: The luminous intensity per unit area projected in a certain direction in SI units (cd m2). Luminous efficiency: The power in photometric terms, measured in lumens per watt, divided by the electric power that generates it. In short, it is lumens output divided by electric power input. To avoid confusion, luminous efficacy is used in the display field. In the LED literature, one finds the reference luminous performance for this term (lm W1). Wall-plug efficiency (power efficiency): Optical power output divided by the electrical power provided to the device, irrespective of the spectrum of that output. The electrical efficacy of a device is the product of the wall-plug efficiency and the spectral or optical efficacy. This is the most appropriate term when it comes to figuring out the energy usage. Electrical efficiency (hv): Represents the conversion of electrical energy to photon energy and is defined by photon energy divided by forward voltage multiplied by electronic charge, qVappl (%|el). The forward voltage applied is determined by the diode characteristics and should be as low as possible for the photon-emitting medium, which ideally is the bandgap of the medium if radiative recombination is through conduction band electrons with holes in the valence band. Resistive losses and electrode injection barriers add to the forward voltage. Luminous flux: Power of visible light in photometric terms. Luminous intensity: The luminous flux emitted from a point per solid angle. The unit is lumens per steradian, or candela (cd). This term is dependent on the package and the angle of measurement, and as such is not reliable. Other definitions having to do with the power efficiency, which are becoming increasingly relevant and important, are the following: Internal quantum efficiency (IQE): Ratio of the photons emitted from the active region of the semiconductor to the number of electrons injected into the p–njunction LED. Extraction efficiency (c): Ratio of photons emitted from the encapsulated chip into air to the photons generated in the chip. This includes the effect of power reflected back into the chip due to index of refraction difference between air and the device surface but excludes losses related to phosphor conversion.

1.7 Visible-Light Terminology

External quantum efficiency (EQE): Ratio of extracted photons to the injected electrons; as such, it is the product of the internal quantum efficiency, IQE, and the extraction efficiency, c. Color-mixing efficiency (hcolor): Losses incurring during mixing the discrete colors for white-light generation (not the spectral efficacy but just optical losses only). Color mixing could also be achieved in the fixture and optics. Scattering efficiency: Ratio of the photons emitted from the LED to the number of photons emitted from the chip, which accounts for the scattering losses in the encapsulant of the lamp. The specified total luminous flux F of a LED is determined for photopic vision and can be calculated through the relation F ¼ S(l)K(l)dl;

ð1:27Þ

where S(l) is the spectral power output of the LED and K(l) is the luminous efficacy of monochromatic radiation at wavelength l. A more relevant criterion to describe the performance of an LED is the luminous performance, which is the amount of electrical power converted to luminous power. This is to be contrasted to the conversion efficiency, which is the amount of electrical power converted to radiant power. 1.7.1 Luminous Efficacy

The luminous efficiency of light sources involves the efficiency of energy conversion from electrical power (W) to optical power (radiant flux in watts), followed by conversion by the eye sensitivity over the spectral distribution of light. The conversion by eye also accompanies a conversion of units, from radiant flux (W) to luminous flux (lumen ¼ lm), and is called luminous efficacy of radiation, having units of lm W1. The luminous efficacy of monochromatic radiation K(l) at wavelength l is shown in Figure 1.17 and is defined by K(l) ¼ KmV(l), where Km ¼ 683 lm W1 and V(l) is the spectral luminous efficiency of photopic vision defined by the CIE [34] and is the basis of photometric units. The value for Km is a constant given in the definition of the candela and is referred to as the maximum luminous efficacy of radiation. No light source can exceed this efficacy value, as shown in Figure 1.17. It should be noted that K(l) peaks at 555 nm (depicted as Km, which is equal to 683 lm W1) and falls off at both ends of the visible region. The values of K(l) are the theoretical limits of light source efficacy at each wavelength. For example, monochromatic light at 450 nm has a luminous efficacy of only 26 lm W1 (theoretical limit). For real light sources, including LEDs, the luminous efficacy of radiation, K, is calculated from its spectral power distribution S(l) by Ð¥ K m 0 Sl (l)V(l)dl Ð¥ ; ð1:28Þ K¼ 0 Sl (l)dl

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Standard incandescent Tungsten halogen Halogen infrared reflecting Mercury vapor Compact fluorescence (5–26W) Compact fluorescence (27–40W) Fluorescent (full size and U-tube) Metal halide Compact metal halide High pressure sodium White sodium

0

50

100

150

Efficacy, lm W-1 (lamp plus ballast) Figure 1.18 Efficacy of traditional light sources [129].

where Km ¼ 683 lm W1, which occurs at 555 nm for photopic vision, and V(l) is the spectral luminous efficiency of photopic vision being normalized to 1 at 555 nm. The total efficacy (lumens per electrical power, including ballast losses) of traditional light sources is summarized in Figure 1.18 [35]. Within a lamp type, the sources having higher wattage are generally more efficient than the ones with lower wattage. High-pressure sodium (HPS), metal halide, and fluorescent lamps are the most efficient white-light sources. Obviously, the spectral power distribution of white light producing LEDs should be designed to have high luminous efficacy. 1.7.2 Chromaticity Coordinates and Color Temperature

All color spectra detectable by human eye can be represented in a number of different three-dimensional spaces. To convert the light spectrum to color space, the color matching functions of CIE colorimetric observer functions are used, which are shown in Figure 1.19. These plots of functions X, Y, and Z are representative of the sensitivity of the human eye versus wavelength over the entire visible spectrum. The common conversion splits the color space into two variables for color and one for luminance, paving the way for color representation by two dimensions, often neglecting the luminance term. This reduces the color space into two-dimensional color spaces such as CIE 1931 xy chromaticity diagram (or space), shown in Figure 1.20a, and CIE 1976 UCS (uniform color space) chromaticity diagram (or space), shown in Figure 1.20b. The color of light is expressed by the CIE colorimetric system, which is a dynamic system in that improvements are made periodically to enhance its utility and render it

Color matching functions

1.7 Visible-Light Terminology

Z(λ)

1.5

Y(λ)

1.0

X(λ)

0.5

0.0

400

500 600 Wavelength (nm)

700

Figure 1.19 CIE 1931 XYZ color matching functions (standard colorimetric observer sensitivity plots for red (x), green (y), and blue (z) color perception cones of human eye. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

more representative [36]. The spectrum of a given light is weighted by the XYZ color matching functions [37], as shown in Figure 1.19. From the resulting three weighted integral values (called tristimulus values, X , Y, Z) the chromaticity color coordinates x, y are then calculated by x ¼ X =( X þ Y þ Z ), y ¼ Y =( X þ Y þ Z ). The basic two-dimensional representation of visible colors is the CIE 1931 xy chromaticity space, as shown in Figure 1.20a, which is by far the most commonly used two-dimensional color representation. However, it has one major drawback that the distance between two points in this space (Dxy) does not correspond to the perceived color difference. In the revised CIE 1976 UCS chromaticity space, this drawback has been resolved to a large extent, as shown in Figure 1.20b. The newer chromaticity diagram, although not perfect, is the preferred two-dimensional color representation. In this revised color space, the distance between two points, usually called Du0 v0, is a reasonable indicator of the perception of color difference. A typical value for the visibility limit by human eye is Du0 v0 ¼ 0.005. Any color of light can then be expressed by the chromaticity coordinate (x, y) on the CIE (x, y) chromaticity diagram, as shown in Figure 1.21 (still using the 1931 CIE standard), with commonly used colors indicated by their abbreviations. The boundaries of this horseshoeshaped diagram represent the plots of monochromatic light, called the spectrum locus, such as that expected from narrow bandwidth light source, which represents saturation color for that wavelength. Also plotted near the center of the diagram is the so-called Planckian locus, which is the trace of the chromaticity coordinate of a blackbody with its temperature ranging from 1000 to 20 000 K. The colors on the Planckian locus can be specified by the blackbody temperature, which is called the color temperature, expressed in kelvin. The colors around the Planckian locus from about 2500 to 20 000 K are regarded as white. The 2500 K color temperature corresponds to reddish (soft white) white and 20 000 K to bluish white. The point labeled “Illuminant A” is the color of a typical incandescent lamp,

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Figure 1.20 (a) The CIE 1931 xy chromaticity space (horizontal and vertical axes denote the x and y coordinates, respectively) and (b) the CIE 1976 UCS u0 v0 chromaticity space. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

and “Illuminant D65” represents the typical daylight, as standardized by the CIE [38]. The colors of most traditional lamps for general lighting fall in the region between the points from 2850 to 6500 K. The color shift along the Planckian locus is generously accepted or purposely varied for general lighting for desired mood. But the color shift away from the Planckian locus (greenish meaning toward green or purplish meaning toward purple) is hardly acceptable. For illustrative purposes, the chromaticity coordinates of a few common fluorescent lamps are shown in Figure 1.22 [39]. The color temperature is not used for color coordinates (x, y) off the Planckian locus. In this case, the term correlated color temperature (CCT) is used. CCT is the temperature of the blackbody whose perceived color most resembles that of the light source in question. Due to the nonuniformity of the (x, y) diagram, the ISO-CCT lines are not perpendicular to the Planckian locus on the (x, y) diagram shown in Figure 1.22. To calculate CCT, an improved chromaticity diagram, namely, the CIE

1.8 Inroads by LEDs

Figure 1.20 (Continued)

1960 (eV) diagram, later replaced by 1976 (u0 , v0 ) diagram [36], is used where the ISOCCT lines are perpendicular to the Planckian locus by definition. An important characteristic of the chromaticity diagram is that light stimuli on the diagram are additive. A mixture of two colors will produce a chromaticity coordinate falling on the line between their respective chromaticity coordinates. For example, the mixture of two sources of light with wavelengths at 485 nm (blue) and 583 nm (yellow–orange) each with a half-bandwidth of 20 nm results in an (x, y) value of (0.38, 0.38). This value is very close to the spectrum locus, producing a soft white color with a temperature of about 4000 K. This idea is the basis of the phosphor conversion LED (pcLED) in which a yellow-emitting yttrium aluminum garnet (YAG) phosphor is pumped with a blue LED, generating white light, as discussed in Section 1.15.3.

1.8 Inroads by LEDs

LEDs are coming! Though we have lived in the domain of fluorescent and incandescent bulbs for a long time, the brightness and efficiency of LEDs have come so far that they have begun to replace incandescent bulbs in special applications and to cause

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0.9 520 0.8

y-Color coordinate

Illuminants

540

510 InGaN colors

Spectrum locus Planckian locus

530

0.7 0.6

570

yG

500

AlInGaP colors gY 580 AlGaAs Y yO colors 2856 590 3500 O 4800 A 600 OPK 610 6500 rO B 620 E 10 000 D65 PK 650 C R 770 pPK W pR YG

0.5 bG

0.4

BG 0.3 490

gB

0.2

B

480

RP

pB bP

0.1 470 450 0 0.0

GaP colors

560

G

0.1

380 0.2

rP

P

0.3

0.4

0.5

0.6

x-Color coordinate Figure 1.21 CIE (x, y) chromaticity (color coordinate) diagram, commonly known color domain, Planckian blackbody radiation locus with its temperatures, several available white-light illuminants, and the wavelength range achievable with various LED semiconductor materials. Courtesy of Lumileds/Philips.

00 30

K 4000

6000 K 5000 K 000

0.35

K

0.40

10

y-Color coordinate

0.45

Planckian locus Illuminant A Illuminant D65 Fluorescent lamps

K

0.50

0.30

0.25

0.20 0.20

0.25

0.30

0.35

0.40

0.45

0.50

x-Color coordinate Figure 1.22 Chromaticity coordinates of common fluorescent lamps along with illuminants A and D65 [129].

0.7

0.8

1.8 Inroads by LEDs

many to project that LEDs could have applications not just in low-level lighting, but in general lighting. If so, the savings in fuel and pollutants would be tremendous, as LEDs are very efficient and environmentally green, unlike fluorescent bulbs that contain Hg and cannot be disposed off safely. For a detailed description of potential energy savings and related issues, the reader is referred to a report by Arthur D. Little, Inc., conducted for the US Department of Energy [40]. These devices have come a long way and are no longer just for low signal applications. They have had a “colorful” history, continually pushed by technological advances and pulled by key applications. The General Electric Corporation first demonstrated an LED in 1962, and LED products were first introduced in 1968 as indicator lights by Monsanto and then as electronic displays by Hewlett-Packard. Within a few years, LEDs replaced incandescent bulbs for indicator lamps, and LED displays made the Nixie tube obsolete. The initial performance of these LEDs was poor by today’s standards, around 1 mlm at 20 mA, and the only color available was deep red.1) Steady progress in efficiency made LEDs viewable in bright ambient light, even in sunlight, and the color range was extended to orange, yellow, and yellow/green and with the advent of nitrides was followed by blue and green. The evolution of LEDs over the years in terms of their efficiency is shown in Figure 1.23. Not only the efficiencies but brightness as well has come a long way since the 1960s. In fact the flux per LED (lumens/package) has been doubling every 2 years since 1968, with this figure being higher in the later years. LED flux has progressed from millilumens in the late 1960s to over 300 lm in the first decade of the twenty-first century. Until the timescale of about 1985, LEDs were limited to small-signal applications requiring less than 100 mlm of flux per indicator function or display pixel. Around 1985, LEDs started to enter the medium-flux power signaling applications, in the range of 1–100 lm (Figure 1.24). The first large-scale application of red LEDs was the newly required center high-mount stop lights in automobiles, but it took many more years for LEDs to conquer this application fully. The early versions had some 75 LEDs in a row or in a two-dimensional array. With increased brightness, the number of LEDs was reduced. By 1990, efficiencies reached 10 lm W1 in the GaAlAs material system, exceeding for the first time an equivalent red-filtered incandescent lamp. Nevertheless, even higher efficiencies were desired to further decrease the number of LEDs required per vehicle. Moreover, the GaAlAs system was limited in color to deep red with wavelengths longer than 640 nm. Currently, conversion efficiencies of commercial LEDs emitting in the red (650–660 nm) stands at 21% at 400 nm and red LED are said to have a conversion efficiency of 50% (Section 1.16), which compares with 75% of higher for edge-emitting lasers. These developments set the stage for the exploration of the GaAlInP quaternary material system with still higher efficiencies and a wider color range, encompassing the red to yellow/green region of the visible spectrum. The efficiency exceeded 1) For comparison, a 60 W incandescent lamp emits six orders of magnitude higher light flux (about 900 lm).

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100 Fluorescent light

AlIn GaP/ GaP Red Orange

–

Luminous efficacy (lm W-1)

Halogen

AlIn GaP/ GaAs Red Orange AlGaA s/AlG aAs Red

–

Unfiltered incandescent

10

1.0

AlGaAs /GaAs Red Thomas Edison's GaAs P:N first bulb GaP:N Green Red – yellow AlQ 3 GaP:Zn ,O Molecular solids Red

0.1

1970

1975

1980

1985

White InGaNGreen

InGaN Blue

Polymers

SiC Blue

GaAsP Re d

Yellow

PPV

1990

1995

2000

Year Figure 1.23 Evolution of all LED performance with some benchmarks against commonly used lamps. Both red and white LEDs were projected to produce luminous efficiencies of about 150 lm W1 by about the years 2015–2020 as compared to the value of about 100 lm W1

achievable by fluorescent bulbs and 10 lm W1 by incandescent lamps. However, the figure for white LEDs was attained in 2007, well ahead of projections. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

20 lm W1 in the 620 nm red/orange part of the spectrum. If the maximum possible efficiency of 150 lm W1 is nearly realized, LEDs could challenge even the very efficient yellow low-pressure sodium lamps. However, the lack of blue and true green prevented LED-based full-color displays from reality until the advent of GaN and related heterostructures, which paved the way for full-color displays, accent lighting, Small signal (monochrome)

1lamp/function

Power signal (monochrome)

1–100 lamps/function

Lighting (white)

≥lamps/function

0.001

0.01

0.1

1

10

100

Flux (lm) Figure 1.24 Flux and numbers of lamps required for various classes of LED applications: low–medium-flux “signaling” applications in which lamps are viewed directly and medium–high-flux “lighting” applications in which lamps are used to illuminate objects. Current LED lamps emit 0.01–10 lm of light. Courtesy of Lumileds/Philips.

1000

10000

1.9 Nitride LED Performance

moving signs, traffic lights, and so on as well as demonstrations of 100 lm W1 whitelight performance, which is considered illumination class.

1.9 Nitride LED Performance

Light-emitting diodes have undergone a tremendous advancement in performance and are now used in nearly every aspect of life. The future of many technologies including printing, communications, displays, and sensors depends profoundly on the development of compact, reliable, and inexpensive light sources. A primary goal of GaN research is to efficiently harness its direct energy bandgap for optical emission. Though the band-edge emission in GaN occurs at about 362 nm, which is in the UV, by appropriately alloying GaN with its cousins AIN and InN, the energy bandgap of the resulting Al(In)GaN can be altered for emission in the range of ultraviolet (UV) to yellow or even red. The first GaN LEDs were reported more than nearly three decades ago by Pankove et al. [41]. Due to difficulties in doping GaN p-type at that time, these LEDs were MIS LEDs, rather than p–n-junction LEDs. The electroluminescence (EL) of these LEDs could be varied from blue to yellow, depending on the doping of the insulator layer. Unfortunately, the measured efficiencies of these preliminary MIS LEDs were not sufficient to compete with the commercially available LEDs of that time. With the achievement of p-type doping, see Volume 1, Chapter 4, followed by the improvement in the quality of InGaN with its compositionally dependent tunable bandgap, blue (430 nm), green (530 nm), and later white LEDs became available. Now, LEDs cover practically the entire visible spectrum, enabling their entry into additional power signaling applications, such as traffic lights [42]. Nitride LED performance on sapphire and SiC substrates associated with large area and standard die sizes for UV-, blue-, green-, and white-light emission is tabulated in Table 1.1. Efficiencies for green and white LEDs stood at 60 and 80 lm W1 with white LEDs are beginning to exceed 150 lm W1 for high injection levels and still improving. These high-efficiency devices sport features for extracting the photons efficiently from the device employing flip-chip mounts, photonic crystals, and so on. It should be pointed out that due to the large In mole fraction in green LEDs, the strain-induced polarization is relatively large and thus the quantum-confined Stark shift is large. Consequently, the emission efficiency due to the reduced overlap of the electron and hole wave functions is lower than it would otherwise be. Naturally, this effect is reduced as the injection level is increased. Furthermore, the light intensity in green LEDs saturates as the injection current is increased for maximum power. This is attributed to the loss of localization, allowing the nonradiative recombination manifold to be available, and efficiency droop due to relatively large hole mass and the resultant electron leakage. When higher quality layers with reduced clustering/ localization are produced coupled with better device designs, this effect is expected to be reduced in which case the saturation would be due to thermal effects. The increased sensitivity of human eye to green cannot compensate for the inferior

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Table 1.1 Nitride LED performance for both commercial and developmental devices as of 2007.

Drive Die HighOutput Flux/ Luminous Drive Lifetime power Wavelength power LED efficiency current voltage size (V) (mm2) (h) Company LEDs (nm) (mW) (lm) (lm W1) (mA) UV

365 385 Blue 470 460 Green 530 530a White Blue þ phosphora

250 310 35

9.4

55 100 170

52.3 26.9 69.4

500 500 1000 700 300 1000 700

84

75

350

385

3.8 3.7 3.72 4.5 3.5 3.72 3.5

1·1 1·1 100 000 1·1 50 000 0.9 · 0.9 50 000

3.2

0.9 · 0.9 50 000

1·1 1·1

50 000 50 000

Nichia Nichia Lumileds Cree Nichia Lumileds Lumileds Cree

a

Commercial.

absolute quality of green LEDs versus the blue and in particular violet LEDs. In spite of this, efficiencies of about 60 lm W1 were achieved in the year 2005. For white LEDs that feature a blue LED and a yellow phosphor, efficiencies of 80 lm W1 are obtained in the flip-chip mount configuration with darkened surface for optimum light extraction out of the top surface. With multiple chips (4), the efficiency figure has been increased to 100 lm W1. Because the power level and the number of chips used could be variables in this figure of merit, it is more instructive to use and compare the total output power for a fixed chip size of, say, 1  1 mm2. 1.9.1 LEDs on Sapphire Substrates

The first GaN p–n-junction LED was demonstrated by Amano et al. in 1989 [43]. The fabricated device consisted of a Mg-doped GaN layer grown on top of an undoped n-type (n ¼ 2  1017 cm3) GaN film with the chemical Mg concentration estimated to be 2  1020 cm3. The electroluminescence of the devices was dominated by nearband-edge emission at 375 nm, which was attributed to transitions involving injected electrons and Mg-associated centers in the p-GaN region. Additionally, a small shoulder extending 20 nm, due to defect levels, was also observed. One of the timely advancements in the nitride effort has been the exploitation of double heterostructures (DHs) for light emission devices [44–46]. The advantage of DH LEDs over homojunction LEDs is that the entire structure outside of the active region where the light is generated is transparent, reducing the internal absorption losses. Furthermore, this cladding region serves as an interface for scattering light, thus minimizing the probability of total internal reflection within the device. Together, these two factors enhance the probability of escape for the light out of the device. In order to achieve other desired colors, InGaN alloys for emission media are required. While an increased InN mole fraction in GaN redshifts the spectrum, this

1.9 Nitride LED Performance

would be at the expense of introducing additional structural defects unless InGaN is made sufficiently thin, because it is not lattice matched to GaN. This is the picture in classical semiconductors whose properties are not dominated by inhomogeneities. In homogeneous semiconductors, lattice-mismatched films can be grown up to a certain thickness called the critical thickness for a given composition. Larger compositions correspond to smaller critical thicknesses. In view of this, there should be substantial effort devoted to optimization. In this vein, near-band-edge emission was also obtained for LEDs employing Si-doped InGaN quantum wells as the active region in a GaN/InGaN DH LED [47]. The In mole fraction content of the active layer was varied and resulted in a shift of the peak wavelengths of the device’s electroluminescence spectra from 411 to 420 nm. Impressively, researchers at Nichia Chemical [48] were later successful in reducing the thickness of InGaN emission layers to
about 30 A. With this achievement, InGaN quantum wells with InN mole fractions up to a maximum of 70% have been obtained, and light-emitting diodes with commercial capabilities are possible even in amber color, in addition to the blue and green. The amber LED is desirable owing to its performance being less temperature sensitive than that of the AlInGaP varieties. It should be mentioned, however, thinner quantum wells particularly with relatively thick barriers are prove to efficiency droop. A schematic representation of a typical early variety of nitride-based Nichia LED is illustrated in Figure 1.25. The light generated in the active InGaN layer traverses the chip without any appreciable absorption, as the entire LED chip is transparent to the radiation wavelength except for the self-absorption in the thin active layer. Due to

p-Electrode

ITO p-GaN p-AlGaN InGaN

n-AlGaN n-GaN

n-Electrode

n-GaN

Sapphire substrate Figure 1.25 Artist’s view of an LED with a transparent and conductive large-area contact made of ITO to the top p-type GaN for better light transmittance. In the wavelength range of interest, the transmittance through ITO is about 80%. In the early versions, semitransparent AuNi was used in place of ITO. Demand for better performance caused the abandonment of the semitransparent metal approach.

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the problematic nature of p-doping and the low hole mobility, the spreading resistance in the top layer is large. To combat this problem, only a small portion of the p-layer, where the wire bond is made, is covered with a thick metallization, with the rest being covered with a semitransparent metal contact, thin in this case. However, when the competition for getting the maximum number of photons became heated, a transparent top contact such as indium tin oxide (ITO) became popular. ITO can be e-beam evaporated on the partially processed chip layer. Followed by a qualityenhancing annealing step, transmission coefficients near 80% in the spectrum of interest can be obtained. The method naturally keeps the contact resistance to the p-layer low, by virtue of increased area of contact, and reduces current crowding and associated ill effects. It should be mentioned that in high-intensity applications, the flip-chip mount method is used in which the p-side is mounted face down on a metallic mirror and the light is collected from the n-side after the removal of the substrate, and therefore, there is no need for conducting oxides. A discussion of conducting oxides can be found in Volume 2, Chapter 1. It should be pointed out that internal field induced by polarization on c-plane GaN reduces internal quantum efficiency due to carrier separation and ensuing increased carrier lifetime. To circumvent this obstacle, nonpolar surfaces of GaN have been explored for light emission. Among them are the 1 1 2 0 a-plane and (1 1 0 0) m-plane GaN growth. Some pertinent details can be found in Volume 1, Chapter 3. 1.9.1.1 Blue and Green LEDs The blue and blue–green LEDs developed by Nichia Chemical initially relied on the transitions to Zn centers in InGaN (Figure 1.26). Although it was suggested [49] that Zn levels are deep, no direct evidence was provided as to whether the centers were direct Zn centers and/or deep levels produced due to the presence of Zn in the growth environment. Although the presence of Zn causes the film to be of high electrical resistivity, Zn centers situated about 500 meVabove the valence-band edge of GaN are Conduction band –

hν

Zn centers

Valence band Figure 1.26 Schematic depicting transitions from states near the conduction band to Zn centers in the earlier versions of commercial nitride LEDs. The Zn centers were also used in the original GaN LEDs fabricated in the 1970s. A schematic representation of optical transitions in Zn-doped and unintentionally doped InGaN LEDs.

1.9 Nitride LED Performance

Blue

EL intensity (a.u.)

100

Green

Zn Center

50

0 350

400

450

500

550

600

Wavelength (nm) Figure 1.27 Electroluminescence spectra of the Nichia blue and green LEDs. By way of comparison to the one with Zn centers, the contrast is drawn to the improved spectrum in devices relying on near band-to-band transitions.

very efficient centers for recombination. The addition of Zn was originally necessitated by the need to increase the wavelength to the desired values, and the limitation deemed present in the amount of In that could be added while maintaining a good crystalline quality. These LEDs had the undesirable characteristics of wide spectral widths and a saturation of the light output with injection current accompanied by a blueshift. Figure 1.27 plots the electroluminescence spectrum of one such blue LED, whose radiative transitions are from near the conduction band to the Zn centers, along with the electroluminescence of so-called quantum well devices not containing Zn centers. The large spectral width of the Zn devices spoiled the color saturation with the undesirable outcome that not all the colors could be obtained through color mixing. It should be noted that, in the quantum well approach, the term quantum well has been used very loosely, and in many cases, the InGaN layers are not thin enough for carrier confinement and the transitions rely on band-tail states near the conduction and/or valence bands. With the so-called quantum well approach, the In mole fraction can be extended to about 70%, which paves the way for excellent violet, blue, green, and yellow/amber InGaN LEDs. The commercial LEDs exhibit power levels of 5 and 3 mW at 20 mA injection currents for the wavelengths of 450 and 525 nm, respectively. Elimination of the Zn centers immediately led to the very important consequence that the FWHMs of the emission spectra were reduced to 20, 45, and 90 nm for blue, green, and yellow LEDs, respectively (Figure 1.28). For yellow, the InN molar fraction approaches 40% (70% has been reported in the past, on the assumption that the bandgap of InN is 1.9 eV). It should be stressed that an accurate determination of the

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Blue: 450, 20 nm

100

Green: 525, 45 nm

EL efficiency (a.u.)

Yellow: 590, 90 nm

50

0 -100

-50

0

50

100

∆λ0 (nm) Figure 1.28 Spectral linewidth of blue, green, and yellow InGaNbased LEDs, the so-called quantum well types. The increasing line broadening, due to strain and compositional inhomogeneities, is noticeable according to an increase in the InN mole fraction, as one goes successively from blue to yellow.

InN molar fraction is difficult due to inhomogeneities in composition and strain. This is even more so when the InN bandgap was not accurately known. The In mole fractions deduced from using the large-bandgap value of InN are 15–20, 40–45, and about 60% for the 450, 525, and 590 nm emission, respectively. However, with downward correction of the bandgap of InN, these figures are actually, in the same order, under 15, 25, and 40% (Volume 1, Figure 1.44). In Zn-free blue LEDs, the linewidth has a temperature-dependent as well as a temperature-independent term. The latter is dominant and is attributable to inhomogeneities in the semiconductor such as compositional and strain variations. The behavior in green LEDs is similar, but to a different extent. With advances in technology, the linewidths have been narrowed, which bodes well for accessing more of the colors defined in the CIE diagram. Despite the advances, the not so narrow linewidths of InGaN-based LEDs, particularly that of green, make it nearly impossible to achieve the full range of colors or true white with high color rendering index (CRI) by using the three-LED solution. In part, it is for this reason, a 3-phosphor and UV LED pump solution is being pursued with development effort focusing on producing phosphors that can be efficiently pumped with the near-violet UV LEDs afforded by InGaN, and perhaps GaN, emitting active layers. Figure 1.29a and b shows the dependence of EL spectra on injection current and ambient temperature of green InGaN LEDs, respectively. The blueshift observed with increasing current is attributed to band filling of the localized energy states caused by compositional fluctuations in the InGaN well layer.

1.9 Nitride LED Performance

120

(a)

T = 25°C

Relative intensity (a.u.)

100

I = 5 mA I = 20 mA

80

I = 80 mA

60

40

20

0 450

475

500

550

525

575

600

Wavelength (nm) 120 I=20mA 100

Relative intensity (a.u.)

o

T=-30 C o T=25 C o T=80 C

80 60 40 20

(b)

0 450

475

500

525

550

575

600

Wavelength(nm)

Figure 1.29 (a) The spectral dependence on operating current in an InGaN green LED at room temperature; note the blueshift caused by filling of the extended band-edge states or localized energy states. (b) Spectral dependences of InGaN green SQW structure on temperature at an injection current of 20 mA indicative of good temperature stability [42].

Contrary to injection current dependence, there is no discernable change of the EL with ambient temperature; additionally, these observations are consistent with those for blue LEDs. Comparing these observations to the InGaAlP red multiquantum well (MQW) LEDs (Toshiba TLRH 157P) could shed some light on the matter. The InGaAlP red LEDs show no dependence of the emission wavelength on injection

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44

70

GaInN blue-green LEDs λ=515 nm at 300 K and DC

60

Flux (lm)

50 40 30 20 Power FCLED Conventional LED Top-emitting power LED

10 0 0

200

400

600

800

1000

Current (mA) Figure 1.30 Flux (lm) versus current (mA) for a conventional LED, top-emitting power LED, and a power FCLED operated with DC drive currents. The power LED easily operates above 200 mA while the conventional LED fails. The FCLED operates at a higher efficiency than the top-emitting power LED exhibiting 48 lm at 1.0 A. Courtesy of Lumileds/Philips.

current due to the lack of localized states induced by compositional fluctuations. The same LEDs do show a redshift with increasing temperature, indicative of bandgap narrowing. The lack of a redshift in InGaN LEDs would imply that the temperature dependence of the transition energy or the bandgap of InGaN is not as pronounced as that for InGaAlP and/or carrier confinement is stronger. Much work has been done of late to increase the efficiency and the amount of light extracted from blue and green LEDs. Utilization of the flip-chip method discussed in Section 1.5 has ushered in considerable advantages over traditional LEDs. Figure 1.30 shows a plot of flux versus the drive current for a power FCLED with an area of 0.7 mm2 and a conventional LED with an area of 0.07 mm2. The flux for both LEDs is for CW operation at a peak wavelength of 515 nm. The output flux of a conventional LED peaks at 150 mA, where the epoxy-based 5 mm lamp package degrades dramatically in a few hours, eventually leading to catastrophic failure [50]. The performance of the conventional 5 mm package LED is limited at high currents caused by the smaller chip size and the high thermal resistance of the package, which is typically 145
C W1. In contrast, the flip-chip LED can be operated up to 1.0 A without significant power degradation or failure and sports a thermal resistance of approximately 14
C W1. Moreover, the flip-chip LED has higher efficiency, producing 16 lm with about 27 lm W1 efficiency at an injection current of 200 mA, which corresponds to a current density of about 30 A cm2. When the drive current of the flip-chip LED is increased to 1 A, a flux value of 48 lm at 445 nm is produced [19]. The forward voltage of the power flip-chip LED is 2.95 V at 200 mA, as compared to 3.15 V for the top-emitting power LED. Figure 1.31 is a plot of normalized external quantum efficiency versus peak wavelength (nm) for a top-emitting power LED and a flip-chip LED fabricated from the same wafer and subjected to a 350 mA pulse (10% duty factor), circa 2001. Pulsed operation was employed to eliminate effects associated with different thermal resistances between the top-emitting power LEDs and power flip-chip LEDs [19].

1.9 Nitride LED Performance

1.25

Normalized efficiency

Flip-chip power LED Top-emitting power LED

1.00 0.75

1.6×

0.50 0.25 0.00 450

470

490

510

530

Peak wavelength (nm) Figure 1.31 Comparison of flip-chip mount conventional topemitting mount LEDs (operated at 350 mA pulsed (10% duty factor) and from the same wafer) in terms of efficiency over the wavelength range covering blue and green. The flip-chip is 1.6 times brighter. Courtesy of Lumileds/Philips.

The flip-chip LED external quantum efficiency was shown to be 1.6 times greater than that of the top-emitting power LED, from the blue to the green region of the visible spectrum. The improvement in efficiency is also approximately constant from a drive current range of 25–1000 mA. Luminous efficiency for two different wavelengths, 502 and 527 nm, versus the drive current in 2  2 mm2 Lumiled LEDs is given in Figure 1.32, where efficiencies

Luminous efficiency (lm W-1)

140 527 nm 502 nm

120 100 80.0 60.0 40.0 20.0 0.00 0

200

400

600

800

Current (mA) Figure 1.32 Luminous efficiency for two different 2  2 mm2 Lumiled flip-chip LEDs operative at 502 and 527 nm as a function of the drive current. Courtesy of Lumileds/Philips.

1000

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Output power (W)

100 10-1

InGaN LEDs λ ~ 450nm

Operating pt. Power LED

10-2 Operating pt.

10-3 10-4 -3 10

5mm LED

10-2

10-1 100 Input power (W)

101

Figure 1.33 Optical output power versus input power for conventional and flip-chip mount LEDs operative at 450 nm. Courtesy of Lumileds/Philips.

20 16

Power LED

12 8 4

120 InGaN LED λ d ~ 530 nm 100

InGaN LED λ d ~ 530 nm Luminous flux (lm)

External quantum efficiency (%)

are about 120 lm W1 for 527 nm and 80 lm W1 for 502 nm devices, at low injection levels; at higher injection levels, the efficiency drops to between 20 and 40 lm W1. Luxeon LEDs deliver average lumen maintenance of 70% through 50 000 h under typical conditions. Output power for a conventional top-emitting diode and a flip-chip mount LED versus input power [51] operating at 450 nm is shown in Figure 1.33. External quantum efficiency versus drive current and luminous flux versus drive for a 5 mm conventional LED and a flip-chip LED emitting at 530 nm are shown in Figure 1.34. To summarize, typical conventional LED indicator lamps are 0.25 mm2 in chip size and are mounted in packages that can handle about 0.1 W electrical input power. High-performance LEDs have an output of 1–2 lm per device. Larger chips, up to about 4 mm2 in size, when packaged appropriately such as with flip-chip mounting, are capable of handling several watts of electrical input power and are available with optical outputs of tens of lumens and hundreds of milliwatts, as tabulated in Table 1.2.

80 60

Power LED

40 20

“5 mm” LED

“5 mm” LED 0 0

50 100 150 200

Current density(A cm-2)

0

50 100 150 200

Current density(A cm-2)

Figure 1.34 External quantum efficiency versus drive current and luminous flux versus drive for a 5 mm conventional LED and a flipchip LED emitting at 530 nm. Courtesy of Lumileds/Philips.

1.9 Nitride LED Performance Table 1.2 High-power InGaN LED performance for blue, cyan, and green.

Color and wavelength

Chip area (mm2) flip-chip mount

Drive J (A cm2)

Efficiency (lm W1)

Flux output (lm)

Blue, 473 nm Cyan, 506 nm Green, 519 nm

2·2 2·2 2·2

50 50 50

9.4 41.4 36

46 175 170

a

Keep in mind that this is a moving target with results improving daily. Therefore, the results represent a snapshot at a given time. Courtesy of Lumileds/Philips.

When the LEDs are driven at high currents, nearly 100 lm performance is obtained. Specifically, 108 lm intensity was obtained for a drive current of 700 mA for green LEDs. Similar results, even better of late, are available for white LEDs also. Such packages, which also feature photonic crystals in p-face-up mounts, are increasingly used for red, amber, and green traffic signaling lights. Even larger chips, emitting hundreds or even thousands of lumens at higher current densities per package, may be required for general illumination. Typically, high-performance devices are obtained in the laboratory followed by reproduction in development and production facilities. Values similar to the aforementioned one, or slightly lower, are made mention of in this book, representing variation among the laboratories. One thing is certain that improvements in epitaxial layer quality as well as packaging are vigorously pursued in the industry. The light output for green LEDs (which features the largest InN mole fraction) saturates at very high injection levels possibly due in part to the loss of localization, which renders the nonradiative recombination paths to be active. Heating effect of the junction may also increase the nonradiative recombination. When higher quality layers with reduced clustering/localization are produced, this effect is expected to be reduced. Also complicating the issue, albeit more so at the lower end of injection levels, is the polarization charge and associated Stark shift, which reduces the emission intensity by reducing the overlap of electron and hole wave functions, manifested as the increase of carrier lifetime by as much as a factor of 10, gets to be rather severe as the InN mole fraction is increased for green emission with its associated piezoelectric polarization induced by strain. Nonpolar surface orientations such as the a-plane, see Volume 1, Chapter 3 for a discussion of growth with this polarity, and also m-plane, again see Volume 1, Chapter 3 for growth details, are beginning to be explored. The a-plane variety suffers from low quality as it is produced on r-plane sapphire with highly mismatched interfaces, and In incorporation has proved to be difficult. High-quality a-plane growth can be obtained on a-plane bulk GaN and to some extent on SiC and ZnO. The former is discussed in terms of it optical performance in Volume 2, Chapter 5. 1.9.1.2 Amber LEDs Amber-colored LEDs are the domain of the InGaAlP material system. However, because the bandgap of the emitting layer is not very different from the larger bandgap carrier-confining layers, carrier leakage occurs. The small barrier

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is exacerbated by the temperature dependence of the output light of the amber LEDs. Because plenty of band discontinuity is available in the InGaN/GaN, the carrier leakage is not the major problem in this context. The main issue becomes one of whether high-quality InGaN layers with large In concentration could in fact be grown. If so, temperature stability of these amber LEDs would be superior to those based on the InGaAlP system. In this section, the properties of InGaN-based amber LEDs [42, 52] are described. Similar to blue and green LEDs, the amber InGaN
LED device structure developed by Nichia consists of a 300 A thick GaN nucleation buffer layer grown at low temperature (550
C), a layer of undoped GaN 0.7 mm thick, a layer of n-type GaN:Si 3.3 mm thick, a layer of undoped GaN (the current-spreading

layer) 400 A thick, an active layer of undoped InGaN 25 A thick, a layer of p-type
Al0.2Ga0.8N:Mg 300 A thick, and a layer of p-type GaN:Mg 0.2 mm thick. It was not possible to determine the exact value of the mole fraction of indium in the InGaN active layer due to the weak signal intensity in X-ray diffraction and photoluminescence measurements. However, using a bandgap of 0.8 eV for InN bandgap and with the appropriate bowing parameter (1.43 eV), the predicted composition becomes nearly 30% for amber color. The n-type GaN:Si layer in blue and green LEDs was replaced with a combination of an undoped GaN and a n-GaN:Si. The purpose of the undoped GaN layer having a relatively high resistivity between the InGaN active and n-GaN layers was to uniformly spread the current in the InGaN active layer. The characteristics of the LEDs were measured under direct current-biased conditions at RT, except for the measurement of the temperature dependence of the output power. The typical forward voltage was 3.3 V at a forward current of 20 mA. The peak wavelength and the FWHM of the emission spectra of the amber InGaN LEDs were 594 and 50 nm, respectively. Figure 1.35 shows the output power of amber InGaN and AlInGaP LEDs (type: HLMP-DL32, what was then Hewlett-Packard – now Lumileds) as a function of the ambient temperature from 30 to þ80
C. At 25
C, the output powers of amber InGaN and AlInGaP LEDs shown are 1.4 and 0.66 mW, respectively. When the ambient temperature was increased from RT to 80
C, the output power of amber AlInGaP LEDs decreased dramatically due to the carrier leakage or overflow caused by a small band offset between the active layer and cladding layers, which is dictated by the need to maintain lattice match between the layers of InGaAlP and the GaAs substrate on which they are grown [53]. Large bandgap discontinuity in the nitride system leads to very weak temperature dependence, as shown. When the ambient temperature is increased from RT to 80
C, the output power of amber InGaN LEDs decreases only to 90% of the room-temperature value, possibly due to additional nonradiative paths becoming available. 1.9.1.3 UV LEDs For many biological staining/imaging applications, biodetection, and even shortrange communications, LEDs operating in the UV region of the spectrum are of interest. Because the wavelength of interest is comparable to, and in many cases, smaller than the one that can be attained with GaN, thin confined layers of GaN and AlGaN active emitting layers are used to reduce emission wavelength. However, any LED-emitting radiation above the energy of violet light is technically considered a UV

1.9 Nitride LED Performance

2.0

Relative output power (a.u.)

1.8 Amber InGaAlP LED

1.6 1.4 1.2 1.0 0.8

Amber InGaN LED

0.6 0.4 -40

-20

0

20

40

60

80

Ambient temperature (°C) Figure 1.35 The normalized output power of amber InGaN and AlInGaP LEDs as functions of the ambient temperature from 30
C to þ80
C. The output power of each LED was normalized to 1.0 at 25
C, the crossing point in the figure [42].

LED, the fringe of which is accessible by low InN molar fraction InGaN. As mentioned throughout this book, some amount of In seems to be a requisite for localization-enhanced radiative emission. Because InGaN lends itself to composition fluctuations or clustering and thus reduced adverse effects of nonradiative recombination, the emission intensity goes down as the InN mole fraction is reduced near the GaN end of the binary, as shown in Figure 1.36 in the form of external quantum efficiency for Nichia LEDs as well as those that have been published by various groups and compiled by Kneissl et al. [54] with original data published in Refs [55–62]. Nevertheless, LEDs based on InGaN QW active layers just below the violet wavelengths have been developed and go under the nomenclature UV LEDs. Shorter wavelength LEDs deeper in the UV region rely on the quaternary, quantum wells with GaN, or AlGaN depending on the wavelength desired. Lowering the sample temperature can provide a glimpse of improvements in the efficiency of LEDs that can be had by reducing the defect concentration. At lower temperatures, the carrier localization is more effective and manifold to nonradiative recombination centers as phonon interaction is made unlikely. In this vein, Figure 1.37 shows the temperature dependence of the output power of InGaN LEDs having a room-temperature emission peak wavelength of 400 nm. The efficiency is reduced by a factor of 2 at room temperature as compared to low temperatures. This implies that reducing the nonradiative recombination centers could lead to a twofold improvement in LED performance. The LED structures utilizing low mole fractions of InGaN, GaN, or AlGaN emission layers are similar in fabrication geometry to those of the longer wavelength

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10 2 Others

External quantum efficiency (%)

Nichia 10 1

10 0

10 -1

10 -2 250

300

350

400

450

500

550

600

650

Emission wavelength (nm) Figure 1.36 The wavelength dependence of the external quantum efficiency in the GaN-based LEDs grown on sapphire substrates. One should keep in mind that the data improve in time. The point is clear that there is a precipitous drop in efficiency approximately below 375 nm corresponding to about 10% InN mole fraction. Compiled from data presented in Refs [42, 54].

InGaN varieties, of course with the exception of the composition of the emitting medium. The varieties possessing low mole fractions of InGaN [63], GaN, GaN/ AlGaN quantum well, AlGaInN quantum well [64], AlGaInN [65], and AlGaN [66] emission layers have been reported. By reducing the InN mole fraction in the lattice, the wavelength of operation can be shifted to the UV region of the spectrum. Because the InN mole fraction is lower, as indicated above, the emission intensity degrades. However, the low InN concentration also means that compositional fluctuation-induced localization is not as severe. Consequently, the injection current-induced blueshift should not be a serious problem. Figure 1.38 shows the EL spectral dependence of UV InGaN LEDs (having an emission peak wavelength of 380 nm at a current of 20 mA) on current and temperature. The reduced blueshift of the emission peak wavelength with increasing current (compared to InGaN green LED) confirms reduced localization energy of the carriers. Figure 1.38 also shows only slight shift of the emission peak wavelength with temperature, as compared to the stable case of the green and blue LEDs. As one reduces the InN content slightly to reduce the wavelength merely by another 5 nm, the picture changes substantially. Figure 1.39 shows the EL of a UV LED with an

1.9 Nitride LED Performance

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Integrated EL intensity (a.u.)

1.0 0.8 0.6 0.4 0.2 0.0 0

50

100

150

200

250

300

Temperature (K) Figure 1.37 The temperature dependence of the output power of InGaN LEDs with an emission peak wavelength of 400 nm [42].

emission peak wavelength of 375 nm. Contrary to the case of 380 nm LED, the emission peak wavelength does not show any change with increasing current. However, a more noticeable redshift of the emission peak wavelength, as compared to the 380 nm LED, is observed with increasing temperature. This spectral shift with temperature is comparable to the conventional AlInGaP LEDs, which could be explained by reduced barrier discontinuity. The UV LEDs with an emission peak wavelength of 375 nm have small, localized energy states resulting from small fluctuations in In composition. This means that the emission mechanism is dominated by a conventional band-to-band emission, as in AlInGaP LEDs. It should also be mentioned that the possibility of the bandgap narrowing with increased temperature should be considered as well. A discussion of the dependence of the

Relative intensity (a.u.)

100 80

(a)

120

T = 25 oC

(b)

100 I = 5mA I =20mA I = 80mA

60 40 20 0 350 360 370 380 390 400 410 Wavelength (nm)

Relative intensity (a.u.)

120

80

I=20mA T = -30 oC T = 25 oC T = 80 oC

60 40 20 0 350 360 370 380 390 400 410 Wavelength (nm)

Figure 1.38 The operating current (a) and ambient temperature (b) dependences of the EL of InGaN UV SQW structure LEDs with an emission peak wavelength of 380 nm [42].

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120

Relative intensity (a.u.)

Relative intensity (a.u.)

100

I = 5mA I =20mA I =80mA

80 60 40 20 0

120

T = 25 oC

(a)

I =20 mA

(b)

100

T =-30ºC T = 25ºC T = 80ºC

80 60 40 20 0

350 360

370

380

390

400

410

350

360

Wavelength (nm)

370

380

390

400

410

Wavelength (nm)

Figure 1.39 The operating current (a) and ambient temperature (b) dependences of the EL of InGaN UV SQW structure LEDs with an emission peak wavelength of 375 nm [42].

bandgap of nitride semiconductors on temperature can be found in Volume 2, Chapter 5. Large area, 1  1 mm2 GaN- and InGaN-based UV LEDs with optical power levels of 250 and 310 mW at 365 and 385 nm, respectively, with operating lifetimes of 100 000 have been achieved at Nichia Chemical. The forward current to attain this power level is 500 mA and the forward voltage is around 3.8 and 3.7 V, respectively. As mentioned above, deleterious effects of threading dislocations, through the nonradiative recombination centers that they introduce, increase with reduced InN mole fraction in an attempt to obtain UV emission. In this vein, investigations with reduced threading dislocation counts have been undertaken [57]. UV LEDs with InGaN multiquantum wells were fabricated on a patterned sapphire substrate (PSS) using a single growth process of metalorganic vapor-phase epitaxy. In this investigation, the grooves were made along the h1 1 2 0i direction of sapphire. The GaN layer grown by ELO on a patterned substrate (light-emitting polymers (LEPs)) has a dislocation density of 1.5  108 cm2. The LEDs fabricated using ELO on patterned sapphire substrates were flip-chip mounted on Si templates and exhibited power levels of about 15 mW at 382 nm for an injection current of 20 mA at room temperature. When the injection current increased to 50 mA, a power level of 38 mW resulted. To reduce the wavelength further, GaN, AlGaN, the quaternary, or the quantum wells of these emission layers must be used. Because the effect of dislocations is expected to be more severe, approaches such as incorporation of GaN templates for substrates have been explored. In one such investigation [66], an output power exceeding 3 mW at the peak wavelength of 352 nm for injection current of 100 mA under a bare-chip geometry was reported. The internal quantum efficiency was estimated at more than 80%. The maximum power exceeded 10 mW for a large current injection of 400 mA, with an operation voltage below 6 V. As the AlGaN mole fraction is increased for shorter wavelength operation, the energy barrier height at the junction of the emission and cladding layers must be

1.9 Nitride LED Performance

maintained to the fullest extent possible. Otherwise, carrier spillover and leakage reduces the quantum efficiency as is the case in conventional compound semiconductor systems based on As and P. Theoretically, the AlN mole fraction in barriers can be increased, but quality degrades as well as Mg doping becoming more difficult for p-type layer. One approach is to introduce a carrier-blocking layer between the p-type AlGaN layer and the active layer, which would allow one to get by without so high an Al mole fraction in the p-type AlGaN barrier layer. By doing so, Nishida et al. [67] improved the output power of an AlGaN-based UV LEDs by one order of magnitude to 1 mW at the emission peak wavelength of 341–343 nm. The quaternary alloy that allows lattice-matching conditions while providing bandgap variability is attractive for UV LEDs for its higher quantum efficiency. This may have the genesis in In-causing compositional fluctuations or impeding dislocation propagation. The UV emission is considerably enhanced by the In-segregation effect upon introducing 2–5% of In into AlGaN. Room-temperature intense UV emission in the wavelength range of 315–370 nm from quaternary InxAlyGa1xyN alloys grown by metalorganic vapor-phase epitaxy has been obtained [68]. The In incorporation in quaternary InxAlyGa1xyN is enhanced with the increase of Al content when using a relatively high growth temperature in the range of 830–850
C. Maximally efficient emission was obtained at around 330–360 nm from the fabricated quaternary InxAlyGa1xyN (x ¼ 2.0–4.8%, y ¼ 12–34%). The intensity of the 330 nm emission from quaternary In0.034Al0.12Ga0.85N was as strong [68]. By using In0.05Al0.34Ga0.61N/In0.02Al0.60Ga0.38N three-layer MQWs with approximately 1.4 nm well thickness, the EL emission was lowered to 320 nm [68]. The authors observed emission fluctuations of submicron size in cathodo luminescence images of Inx1Aly1Ga1x1y1N/Inx2Aly2Ga1x2y2N single QWs, which might be due to In segregation effect. The temperature dependence of photoluminescence emission for InAlGaN-based QWs was greatly improved in comparison with that of GaN- or AlGaN-based QWs. These quaternary structures have been used in UV LEDs in the form of multiple quantum wells, as discussed below. Almost all nitride-based LEDs have quantum wells or at least one in the active emission medium. In this section, devices utilizing multiple quantum wells for confinement-induced blueshift or growth regimes facilitated/mitigated by quantum wells are discussed. Quantum wells provide an efficient medium for carrier recombination; therefore, efficiency in a device can be improved. Moreover, the blueshift obtained by narrow quantum wells allows increasing the emission energy beyond the bulk bandgap of the emitting medium and tuning of the emission wavelength. The emission energy versus quantum well thickness in Al0.4Ga0.6N/GaN quantum wells is shown in Figure 1.40, which indicates that no matter how small the quantum well thickness is made, the maximum energy obtainable is about 3.84 eV, which corresponds to a wavelength of about 320 nm. This means that for wavelengths below this value, the active layer composition must be changed to either AlGaN or AlGaInN. As seen in blue and green LEDs, the active medium of emission is made of InGaN quantum wells. The same holds true for shorter wavelength emission in that remarkable progress has been made in the field of MQW-based UV LEDs, primarily grown on sapphire and SiC substrates [57]. Output powers greater than 1 mW at

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CL peak energy (eV)

3.9 3.8 3.7 3.6 3.5

0.6

0.8 1.0 1.2 1.4 1.6 Quantum well width (nm)

Figure 1.40 The emission energy versus quantum well thickness in Al0.4Ga0.6N/GaN quantum wells as determined by cathodoluminescence. Courtesy of M. Holtz and H. Temkin.

emission wavelengths of 340–350 nm have been achieved [66]. LEDs with shorter emission wavelengths in the range of 280–330 nm are of much interest [64, 65, 69] and require bulk AlGaN, AlInGaN, or quantum wells of the two in some combination, since GaN quantum wells, no matter how thin they are made, barely reach the longer end of this range. Higher output powers and shorter wavelengths are limited by difficulties associated with attaining high-quality Al-rich AlGaN, reduction in Mg doping activation in p-type large-bandgap alloys, and design, which ironically is probably the easiest part of the entire problem. The high deposition temperatures associated with conventional MOVPE make it difficult to control the composition and thickness of AlGaInN quantum wells. However, the compositional fluctuations could be mitigated by use of quantum wells and/or pulsed epitaxy, which when done led to LEDs emitting at 340 nm [64]. The expanded description of pulsed epitaxy in the context of the quaternary alloy can be found in Volume 1, Chapter 3. In this particular structure, the epilayers were deposited on c-plane sapphire or nþ-SiC substrates using a pulsed atomic layer epitaxy process for a better control of the composition and thickness. The overall structure consists of a 0.8 mm thick nþ-Al0.26Ga0.74N layer followed by a 30-period nþ-Al0.2Ga0.8N/Al0.16Ga0.84N multiple short-period struc
tures with periodicity of 30 A. The device active layers consisted of a quaternary Al0.15In0.02Ga0.83N/Al0.1In0.01Ga0.89N MQW, the barrier and well layer thickness of
which were each kept at 15 A and the number of wells was varied up to 10. A power level of 0.11 mW was achieved at a large bias current of 500 mA. In another approach, AlN/AlGaInN superlattices (SLs) were employed to produce LEDs emitting at 280 nm, using reactive MBE with ammonia [70]. The device is composed of n- (doped with Si) and p-type (doped with Mg) superlattices of AlN (1.2 nm thick)/AlGaInN (0.5 nm thick) to mimic large bandgap bulk. With these superlattices, and despite the high average Al content, hole concentrations of 0.7–1.1  1018 cm3 with corresponding mobilities of 3–4 cm2 V1 s1 and electron concentrations of 3  1019 cm3 with the mobilities of 10–20 cm2 V1 s1 were obtained at room temperature. The barriers are 1.2 nm thick and the wells are 0.5 nm thick, as calculated from growth rates. The Al content in the well is 0.1. The

1.9 Nitride LED Performance

level of In incorporation, 0.05%, is estimated from the redshift of cathodoluminescence (CL) spectra of AlGaInN compared to a reference of AlGaN. The n-type dopant is introduced intermittently, during the growth of AlGaInN wells only. The n-type structure contains 150 barrier/well pairs for a total thickness of less than 300 nm. The p-type SL is intended to be structurally identical to the n-type structure. It is uniformly doped with Mg evaporated from an effusion cell. The growth is terminated with a 5 nm thick quaternary p-type contact layer. The barrier and well dimensions are below their critical thickness and no additional dislocations appear to be generated in the SL itself. Electroluminescence results are strongly influenced by the design of the active region and the device fabrication procedure with Ni contacts of 70 and 500 mm diameter, the large diameter contact serving as the cathode. Light is collected with a UV-transparent fiber from the edge of the small dot, the p-type contact, and analyzed with a spectrometer. Although the spectrum has multiple peaks with the dominant emission occurring at 330 nm, a second peak at 280 nm is also present. When driven with pulsed current, up to 350 mA, the intensity of the 280 nm begins to dominate. When mesas were etched, the 330 nm emission decreased to a shoulder with the predominant emission at 280 nm. 1.9.1.4 Resonant Cavity-Enhanced LED The resonant cavity-enhanced (RCE) LEDs are interesting in that by use of Bragg reflectors, nearly monochromatic light (when single-mode operation is supported) can be obtained [71]. In addition, this device represents the first step toward the fabrication of vertical cavity lasers discussed in Section 2.14. Blue resonant cavity light-emitting diodes (RCLEDs) based on InGaN/GaN quantum well heterostructures have been reported [72]. Vertical microcavity devices with either one or both mirrors forming the cavity are patterned and high-reflectivity dielectrics, Bragg reflectors, have been fabricated. The active region in Ref. [72] was grown on 2.0 mm
GaN buffer layer and was composed of 10 InxGa1xN quantum wells (Lw ¼ 30 A) with
GaN barriers (LB ¼ 50 A), surrounded by Al0.07Ga0.93N upper and lower outer
cladding layers (approximately 3600 A and 0.5 mm, respectively). The In concentration was varied to span the wavelength range from about 430 to 480 nm. The
structures were capped by a 1000 A thick p-GaN contact layer. As could be expected, a good morphology is imperative for achieving a high-Q vertical cavity. Both one metallic and one dielectric distributed Bragg reflector (DBR) and two dielectric DBRs were employed to form the cavities. The schematic diagram of the latter is shown in
Figure 1.41. The process begins with the deposition of a patterned 1000 A thick layer of ITO, having a conductivity of 4  104 W cm and an absorption loss of approximately 2% in the 400–500 nm wavelength range, on the p-GaN cap layer of the asgrown nitride heterostructure for a transparent conducting layer [73]. Electron beam evaporation of a Cr/Al bilayer formed both the electrical contact to ITO and one optical reflector (R  0.8). Patterned SiO2 was used to define a current injecting aperture of 15–35 mm diameter. A 20-pair SiO2/HfO2 multilayer DBR was deposited atop the ITO film to create a high-reflectivity mirror (R > 0.999). The electrical injection to p-type nitride

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15–35µm (-)

HfO2/SiO2 DBR stack

n-Contact InGaN/GaN MQW

SiO2 ITO

p-Metal

(+) Copper stand

Conductive template

Figure 1.41 Schematic representation of a RCLED with two dielectric DBR mirrors. The device also features a low loss intracavity ITO p-current-spreading layer. Courtesy of A. Nurmikko [72].

was provided by the lateral current spreading via the ITO. A patterned multilayer metallization was applied to contact the ITO film outside the optically active area. Next, the entire structure was flip-chip mounted on a permanent host substrate (e.g., silicon or other electronically integrateable material). Pulsed excimer laser radiation at l ¼ 308 nm was directed through the backside of the sapphire substrate so as to induce its complete separation from the nitride heterostructure in a single 10 ns laser shot. Finally, a HfO2/SiO2 multilayer dielectric stack (DBR) was deposited directly onto the exposed n-AlGaN or n-GaN layer surface and patterned for completion of the optical resonator (consistent with the 15–35 mm effective aperture). Patterned Ti/Al contacts were used on the n-side of the junction. Spectral response of the two-mirror RCLED described above is shown in Figure 1.42, for which the light emission was collected along the optical axis of the device, within an approximately 5
forward solid angle. The cavity for the device is relatively thick, approximately 16l. The spectrum of the device encompassing two dielectric DBRs is centered at around l ¼ 430 nm and underscores the impact of a high-quality resonator on the definition of the cavity modes, with the inset (Figure 1.42) showing a particular mode. The modal linewidth is approximately 0.6 nm, implying a cavity Q-factor of approximately 750. This device, where the bottom reflector is made of a metallic mirror with an improved reflectivity of R > 0.9, is easier to fabricate and could be a good candidate for short-cavity blue RCLED. If higher power could ensue from a structure of this kind owing to better collection of photons, the structure with appropriate bandgap could be used to excite dyes for white-light generation discussed in Section 1.15.3. A three-terminal tunnel junction employed in a monolithic, electrically segmented dual wavelength blue–green LED consisting of two electrically isolated InGaN QWs

Emission intensity (a.u.)

1.9 Nitride LED Performance

428

360

380

400

420

440

430

460

432

480

500

Wavelength (nm) Figure 1.42 Emission spectrum of the RCLED device incorporating two dielectric DBR mirrors [72].

of different indium composition within a single vertical heterostructure has been demonstrated [74]. The device incorporated a pþþ/nþþ InGaN/GaN tunnel junction so as to operate a time-multiplexed two-color blue–green LED source operative at 470 and 535 nm (note that these are not necessarily consistent with the true definition of the color terminology, and the pþþ is used figuratively to indicate doping as high as possible). The nitride heterostructure used, in the form of an artistic view of the fabricated device, is shown in Figure 1.43. The tunnel junction segment is inserted between the two active InxGa1xN QW emitter sections, having a quantum well
thickness of Lw ¼ 30 A. Here, the tunnel junction (TJ), serves the purpose of electrically sectioning the nitride heterostructure into two independent LEDs and lateral current spreading for the bottom device. The diameters of the top and the bottom LEDs are 60 and 80 mm, respectively. In the LED injection regime for drive current levels of approximately 100 A cm2 or less, the TJ in the bottom device typically added about 1 V to the forward turn-on characteristics. Shown in Figure 1.44 are the spectral characteristics where the dashed curves depict the case for each LED being switched on in a time sequential manner and the solid line is for the LED operated as a simple two-terminal device with a constant voltage applied across the top p-GaN and lowest n-GaN. The electrical independence of the blue and green segments allows one to program the 470 and 535 nm LEDs for any time sequence up to speeds of 100 MHz. 1.9.1.5 Effect of Threading Dislocation on LEDs As discussed in Volume 1, Chapter 3, the ELO process can be used to reduce the threading dislocation density in the GaN buffer layer. This would then allow one to determine the effect of threading dislocations on LED intensity, and other relevant features as they have been reported [42]. Figure 1.45 shows the relative output power of UV (380 nm) InGaN and GaN LEDs produced using sapphire and ELO substrates as functions of forward current. The ELO and GaN on sapphire were reported to have average dislocation densities of 7  106 and 1  1010 cm2, respectively. Here, the

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Figure 1.43 Schematic view of the two-wavelength blue–green LED, indicating the active regions, the tunnel junction, and the bias arrangement. A plan view photograph of a device is shown at the top. Courtesy of A. Nurmikko [74].

Intensity (a.u.)

1.0 0.8 0.6 0.4 0.2 0.0

440

480

520

560

600

Wavelength (nm) Figure 1.44 Superposition of the emission spectra from the blue and green LEDs when they are activated sequentially (dashed lines), and their simultaneous activation as a two-terminal LED (solid line). Courtesy of A. Nurmikko [74].

1.9 Nitride LED Performance

Relative intensity (a.u.)

30

20

G In

aN

In

10

D LE

N Ga

on

D LE

EL

o

O

ap ns

ph

ire

GaN LED on ELO GaN LED on sapphire 0 0

10

20

30

40

50

60

Current (mA) Figure 1.45 Relative output power of UV InGaN and GaN LEDs as functions of forward current for LED chip size as large as 350  350 mm2, which covers many ELO stripes covering both higher and lower dislocation density regions [42].

average dislocation density of the ELO on sapphire was obtained by dividing the dislocation density of 2  107 cm2 on the window region by the ratio (stripe periodicity of 12 mm)/(window width of 4 mm) because the dislocation density on the SiO2 stripe region was very small. The 350  350 mm2 LED chip size caused many windows and SiO2 stripe regions to fall under each device, necessitating the use of a geometric average dislocation density for the ELO on sapphire. As the figure indicates, the UV GaN LED on ELO has a much higher (about twofold) output power than that on sapphire. This is because the magnitude of dislocation densities of a GaN LED on ELO is much smaller than that on sapphire. However, the 380 nm UV InGaN LED on ELO showed a smaller improvement in output power (25%) in comparison with that on sapphire at 20 mA. This is attributed to alloy composition fluctuation in InGaN and resultant carrier localization, which apparently does not occur to the same extent for GaN. This is consistent with the overall poorer performance of GaN emission layers as compared to InGaN discussed in Section 1.9.1.3. Therefore, any reduction in threading dislocation density is more effective for GaN emitting layer than the InGaN variety. Furthermore, radiative recombination is limited by hole injection and improvement of layer quality unless accompanied by enhanced hole injection in relation to electron injection, carrier leakage prevent attainment of improved light output. This localization picture, which we also touched upon in Section 1.9.1.1, is also consistent with poorer performance as the InN mole fraction in the alloy is reduced. This reduced InN fraction causes a reduction in the depth of the localized energy states, which degrades the effectiveness of carrier localization and therefore opens manifolds to nonradiative recombination centers. However, when the InN mole fraction is increased past the point corresponding to a wavelength of about 470 nm, the efficiency degrades again presumably due to the increased lattice mismatch

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between the GaN buffer layer and InGaN emitting layer. In addition, a combination of relatively high InN mole fraction and high temperatures used during growth, extensive compositional inhomogeneities and resultant localization may occur. This manifests itself as saturation in light output with increasing injection current as nonradiative recombination channels become available for recombination. The electron leakage also must be kept in mind. With reduced defects, increased pressure during MOCVD growth may be employed to reduce some of the aggravating conditions mentioned above. We should point out that InN is of special interest in light emission in that InN containing ternary has long radiative recombination lifetime compared to GaN and the emission intensity does not drop as much as that in GaN with increased temperature from low temperatures. Because of reduced or lack on InN in the lattice, the effect of InN in the emission process is of paramount importance in UV LEDs with very low InN mole fraction in the emission layer, GaN, or AlGaN emission layers depending on the emission wavelength. Somewhat relevant to the case is the fact that InN nanostructures have been prepared where the emission intensity shows little or no temperature dependence on temperature from low temperature to room temperature. This means that when and if InGaN active layers without defects and localization are available, one can expect much improved performance. We should mention that GaN layers are not as efficient light emitters at room temperature as InN containing active layers (providing that the InN mole fraction is not high). Naturally, the UV varieties do not feature InN in their active media. It is known that the PL efficiency for InN nanostructures do not degrade at room temperature in temperature-dependent PL measurement experiments. If the same would hold true due to confinement, one can then assume that if nanostructured GaN is used for the active medium for emission, much better performance of UV LEDs can be obtained [75]. 1.9.2 LEDs on SiC Substrates

SiC substrates have some advantages. Among them are the conductive natures of the substrates, which simplify LED fabrication and in particular packaging; generally, SiC has better quality epitaxial layers and has better thermal conductivity compared to sapphire. Moreover, the LEDs on SiC are less sensitive to electrostatic fields whereas special packaging and handling methods must be employed in LEDs on sapphire. Some high-brightness LEDs produce optical power levels of 15, 14, 11, and 9 mW at 460 (deep blue), 470 (blue), 505 (for green traffic lights), and 527 (for display green) at a forward current of 20 mA, respectively, in production levels. The forward voltage at 20 mA drive is between 3.7 and 3.8 V, and the die size is 300  300 mm2. The 460 nm LEDs can produce 30 mW of power in selected production runs at a forward current of 30 mA, which is remarkable. When the die size is increased to 900  900 mm2, an optical power output level of 150 mW is available at 470 nm for a drive current of 350 mA. The same size die designed for UV operation at 405 nm produces 250 mW of optical power at a drive current of 350 mA. The forward voltage for both large die devices is 3.7 V. The electrostatic discharge threshold is about 1000 V.

1.9 Nitride LED Performance

1.9.3 LEDs on Si Substrates

Bringing down the price of semiconductor material by improving the quality and rate at which epitaxial reactors produce semiconductor wafers is crucial to reducing the overall cost of LEDs. Other keys to cost reduction include designing low-cost revolutionary packages with high reliability and low thermal impedance, and increasing the area of substrates while reducing their cost [76]. Silicon shows promise as a substrate for nitride-based LED devices that emit across the spectrum from green into the ultraviolet. The possibility of LEDs built on Si substrates is exciting, but high-performance devices have not yet been demonstrated due to problems such as differences in the thermal coefficient of expansion between the deposited semiconductor and substrate, and lattice mismatching wherein the lattice sizes of the deposited semiconductor and the substrate are different to the extent that lattice defects cause significant amounts of energy to be thermalized. Unlike lasers, which sport waveguides, the photons in LEDs are emitted in random directions. Having a substrate and a buffer layer structure that do not absorb these photons allows them to be emitted back through the surface by back-reflectors, and so on. However, Si absorbs all the visible and UV photons, which causes efficiency degradation even for identical quality layers, as compared to a transparent substrates. With ELO on Si, discussed in Volume 1, Chapter 3, the layer quality has been improved substantially. But the absorbance issue is a fundamental one. Unless special precautions are undertaken, GaN and AlN films crack on Si substrates. Therefore, thick crack-free films become an important issue. With ELO and similar techniques, these problems have been somewhat assuaged. By using a low-temperature AlN:Si seed layer and two low-temperature AlN:Si interlayers for stress reduction in tandem, Dadgar et al. [77] reported crack-free GaN films of about 2.8 mm in thickness. Further, low turn-on voltages along with a series resistance of 55 W were observed for a vertically contacted diode. By in-situ insertion of a SixNy mask, the luminescence intensity was significantly enhanced to the point of producing 152 mW at a current of 20 mA and a wavelength of 455 nm. As in the case of sapphire and SiC substrates, the quaternary InGaAlN system, in the short-period multiple quantum well fashion, was applied to Si(1 1 1) substrates as well, for below 300 nm light emission [78]. The layers are composed of superlattices of AlGaN/GaN and AlN/AlGaInN and nearly identical to those reported on sapphire [70]. The LEDs show light emission between 290 and 334 nm. The roomtemperature EL spectrum of the carrier injection structure, a homojunction-like LED based on n- and p-type superlattices of AlN/AlGaInN, obtained with a DC current of 100 mA and Vf  25 V shows a well-defined peak at 290 nm, followed by a broader structure at 340 nm. This is in part due to the recombination region not being well defined and the entire structure being heavily doped. The translation is that more work needs to be done in the area of below 300 nm LEDs, in general, let alone whether Si is a viable substrate. However, even the discussion of this issue in itself is a mark of how remarkably the field has progressed. Exciting advances can, however, be made using orientations of Si which when etched allow the growing plane to be for example m-plane because growth takes place on the (1 1 1) surface of Si.

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1.9.4 LEDs Utilizing Rare Earth Transitions

The basics behind tunable wavelength emission in rare earth (RE) doped GaN are discussed in Volume 2, Chapter 5. Here, only the device aspects of rare earth doped GaN light-emitting diodes are discussed. Recent success in fabrication of electroluminescent devices (ELDs) utilizing red, green, and blue (RGB) emission from REs in GaN, as shown in Figure 1.46, has made possible the realization of a new generation of flat panel displays (FPDs) [79–81]. The lateral integration of ELDs

Figure 1.46 Photographs of red (a), green (b), and blue (c) emissions in LED fabricated from MBE-grown GaN doped with Eu (a), Er (b), and Tm (c) [79–81]. (Please find a color version of this figure on the color tables.)

1.9 Nitride LED Performance

doped with different REs has been recently demonstrated [82]. GaN:RE films were grown by radio-frequency plasma-assisted MBE on 2 in p-Si (1 1 1) substrates. The main fabrication steps for three-color integrated devices are as follows [83]: (1) spin SOG on Si wafer twice, then coat with photoresist (PR) and expose the PR pattern; (2) etch SOG with 0.1% diluted HF to form the SOG window for GaN:RE growth; (3) grow GaN:Tm in MBE system after the SOG was sufficiently outgassed; (4) SOG liftoff with HF revealing the GaN:Tm patterns; (5) repeat twice the procedures from (1) to (4), using GaN:Eu and GaN:Er; (6) produce PR patterns for ITO electrodes on GaN:RE pixels; (7) sputter-deposit a thin film of ITO; (8) liftoff PR and anneal samples in N2 ambient to form good contacts. Figure 1.47 shows a photograph of the laterally integrated three-color thin-film ELD from GaN:RE as well as the ELDs in operation under DC bias. The EL brightness

Figure 1.47 Laterally integrated GaN:RE thin-film ELD containing the three primary colors fabricated with the SOG liftoff technique: (a) optical microscopy photograph of the GaN ELD showing the three-color integration; (b) blue, green, and red emission under DC bias from ELD GaN devices doped with Tm, Er, and Eu, respectively [83]. (Please find a color version of this figure on the color tables.)

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0.9 520 0.8 510

GaN:Er (green)

540

y-Color coordinate

0.7 560

0.6 500 0.5

GaN:Eu (red)

EBU

0.4

600 610 640

Integrated pixels

0.3 0.2 0.1 0.0 0.0

480

0.1

GaN:Tm (blue)

460 0.2 0.3 0.4 0.5 0.6 x-Color coordinate

0.7

0.8

Figure 1.48 CIE x–y chromaticity diagram showing the locations of the blue, green, and red emission from the individually biased pixels in an integrated GaN:RE-based ELD and from simultaneously biasing all three pixels in the device. Also shown are the coordinates of the EBU-recommended phosphors [83].

values of the red, green, and blue pixels from the three-color integrated device are approximately 14, 45, and 3 cd m2, respectively. Figure 1.48 illustrates the full-color capability of RE-doped GaN integrated ELDs using the Commission International d’Eclairage (CIE) chromaticity diagram. The solid triangle in the diagram defines the full-color capability of emission from GaN doped with Eu (red), Er (green), and Tm (blue). The CIE triangle recommended by the European Broadcasting Union (EBU) is shown by a dashed line for comparison.

1.10 On the Nature of Light Emission in Nitride-Based LEDs

Although nitride LEDs were introduced as a commercial product in late 1993, there are still many unanswered questions regarding the optical emission processes responsible for their outstanding operation. One of the questions sending everyone to search for answers is the unusually high efficiency of light emission in the presence of large concentrations of defects in the material as well as a current-conduction mechanism that is inconsistent with band-to-band recombination. It is known that the radiative recombination in wide-bandgap semiconductors such as ZnS deposited on glass is very efficient. The mechanism responsible for this has been attributed to carrier localization and, of course, it is very enticing to make a parallel here.

1.10 On the Nature of Light Emission in Nitride-Based LEDs

1.10.1 Pressure Dependence of Spectra

In order to take a glimpse at the origin of the strong electroluminescence in nitridebased LEDs, and more importantly to understand the origin of light emission, Perlin et al. [84] examined the photoluminescence and electroluminescence emission in commercially available blue and green LEDs under hydrostatic pressure after they had been decapsulated. To complement the pressure experiments, Perlin et al. [85] also embarked on an extensive investigation of EL from blue and green LEDs over a broad current and temperature range in an effort to gain some insight into the genesis of radiative transitions. The blue and green LEDs exhibited similar behavior with the green ones accenting the anomalies mentioned. Consequently, the following discussion will be limited to the case of the green devices.
The green Nichia LED investigated has an undoped active layer consisting of a 30 A thick In0.45Ga0.55N layer sandwiched between n-GaN on the bottom and p-Al0.2Ga0.8N on the top layers. The lattice mismatch between the active layer and the barrier materials is about 6%, bringing into question whether the structure has a pseudomorphic character or the strain is relaxed by a large concentration of dislocations, on the order of 1010 cm2 [86]. Another perplexing aspect of these diodes is the 2.28–2.33 eV photon energy that is somewhat smaller than the 2.50–2.68 eV expected from the bandgap of In0.45Ga0.55N, although the clustered nature of InGaN makes the reported compositional dependence of its bandgap a suspect [87]. Nakamura et al. [88] suggested that this discrepancy could be caused by tensile strain in the quantum well, which had been induced by differences in the thermal expansion coefficients of the quantum well and barrier materials. Other suggestions involve unidentified and elusive localized states [89] or localized excitons [90]. In any case, no definitive answer has been given yet as to the nature of the recombination in the so-called Nichia single quantum well LEDs. Figure 1.49 exhibits the photoluminescence spectra of a green Nichia diode as a function of hydrostatic pressure. The observed Fabry–Perot interference fringes are

PL intensity (a.u.)

1.0

2 GPa

Green LED T = 300 K

0.8

0.2 GPa 4.2 GPa

0.6 0.4 0.2 0.0 2.1

2.2

2.3

2.4

2.5

2.6

Photon energy (eV) Figure 1.49 Photoluminescence spectra of a Nichia green LED at three indicated uniaxial pressures at 300 K obtained after decapsulation [84].

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Energy shift (eV)

GaN gap

40

InN gap

20

0

0.0

0.4

0.8

1.2

Pressure (GPa) Figure 1.50 Shift of the EL peak position as a function of the hydrostatic pressure for a Nichia green LED. The data are shown along with error bars and the expected shifts of the band edge for GaN (experimental) and InN (theory) are the labeled solid lines. The solid line through the data points is a guide to the eye [84].

indicative of the fact that the structure is mostly transparent to the particular radiation. The distance between fringe maxima is close to 50 meV (500 cm1) and gives a total cavity thickness, sapphire on the one side and air on the other, of approximately 5 mm, which agrees well with the total thickness of these GaN/InGaN/ AlGaN structures (4.6 mm). The presence of interference fringes can cause misidentification of the peak position, which is made worse as the fringe periodicity can change with pressure due to the pressure dependence of the refractive index (about 6 meV GPa1). For accuracy, the Gaussian peak profile and an oscillating function representing the interference fringes were fitted to the data, which paved the way to determine the peak shift in luminescence and electroluminescence with hydrostatic pressure, the latter of which is displayed in Figure 1.50 for a green LED. The observed linear shift with pressure was 12 meV GPa1 for the green LED, compared with 40 meV GPa1 (experimental) and 33 meV GPa1 (theoretical) for GaN and InN, respectively, and this indicates that the transition energy is not of traditional bandedge origin. The contribution of pressure to any confinement due to an increasing effective mass has been determined to be about 2 meV GPa1 [84]; it is very small. In short, one can conclude that the LED spectra do not follow the band edge. A pressure coefficient lagging behind the band edge can be expected from deep states. For example, transitions between uncorrelated electrons and holes, meaning those with wave functions that do not overlap, trapped in band-tail states caused by indium clusters/compositional fluctuations lead to pressure coefficients smaller than the band-edge value. Band-tail states can extend deep into the forbidden gap, and deep states have a pressure coefficient that is an average across the entire Brillouin zone. This average coefficient is much lower than that at the r-point direct bandgap. Localized excitons have also been postulated as being responsible for the transitions

1.10 On the Nature of Light Emission in Nitride-Based LEDs

Figure 1.51 Out-of-plane and in-plane compositional variation of InGaN in a GaN/InGaN well structure similar to that used in the Nichia LEDs. Courtesy of C. Ksielowski, Lawrence Berkeley National Laboratory.

in InGaN LEDs [90]. However, the observed behavior retains the same trend at high current injection levels where excitons would certainly dissociate and would therefore not be responsible. Localization effects such as the postulated quantum dots (QDs) are so far limited to casual observations of compositional variations [91]. Shown in Figure 1.51 is the compositional variation of InN in an InGaN quantum well, which is similar to that used in LEDs, manufactured by Nichia Chemical Co. Clearly, the molar fraction varies in the growth direction and in the plane of growth leading to clusters. Moreover, dots of the kind postulated would have pressure coefficients similar to the band edge. In short, the available pressure dependence indicates that the transitions are due to uncorrelated electron–hole pairs localized deep in band tails, which are most likely caused by inhomogeneous InN mole fraction and strain. Ironically, we would not be too illogical if we were to argue that it is precisely the presence of these band tails that is responsible for the extraordinary performance . 1.10.2 Current and Temperature Dependence of Spectra

Figure 1.52 exhibits the peak position of the EL emission at several temperatures from the green LED discussed above, with a clear shift to higher energies as the injection

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2.40

EL peak position (eV)

2.38

T = 300 K T = 220 K T = 150 K T = 15 K

2.36 2.34 2.32 2.30 2.28 2.26 10-7

10 -6

10 -5

10 -4

10 -3

10-2

Current (A) Figure 1.52 The peak position of the EL emission at several temperatures from a green Nichia LED indicating a clear shift to higher energies as the injection current exceeds 0.1 mA [85].

current approaches 0.1 mA. As the temperature is increased, the low-current plateau also shifts toward higher energies. This blueshift is about two orders of magnitude larger than what we would expect from the filling of the conduction band states. Consequently, the transition responsible must be due to states with very low density of states compared to the conduction band. This observation is also consistent with the premise that deeper states are the origin of the observed transitions. Deeper states here include band tails as well as other pseudocontinuous states. Interestingly enough, the energy of the emitted photons at the largest applied currents is quite close to the estimated separation between the confined states in a 3 nm thick In0.45Ga0.55N quantum well. Figure 1.53 depicts the same shift at 300 K up to a pulsed-current level of 2 A and accentuates the blueshift very well with injection current. Figure 1.54 plots the temperature dependence of the peak energy of the EL emission. The solid line depicts the expected trend of the band edge of GaN but shifted rigidly to represent the bandgap of InGaN. Figure 1.54 indicates that for a current level of 0.1 mA (preceding the band-tail filling), the EL emission undergoes a blueshift with increasing temperature. Between 15 and 300 K, this shift can be as large as about 70 meV, for the lowest applied current. It should also be stated that the emission bandwidth (130 meV at 1 mA, not shown) remains practically unchanged over the entire temperature range. The data presented above suggest that the radiative recombination does not directly involve the conduction and valence bands and, in the case of quantization, quantum well subbands. Though radiative recombination has been suggested to relate to excitons localized in regions containing large InN molar fraction that are caused by fluctuations of the indium contents in the active layer material, the data of Perlin support the band-tailing effect. On the contrary, the same data on high carrier injection

1.10 On the Nature of Light Emission in Nitride-Based LEDs T = 300 K

2.52

EL peak position (eV)

2.48

2.44

2.40

2.36

2.32 10-6

10-5

10-4

10-3

10-2

10-1

10-0

Current (A) Figure 1.53 The peak position of the EL emission at room temperature from the same green LED as in Figure 1.52, showing the blueshift at 300 K up to a pulsed-current level of 2 A accenting very well the blueshift with injection current [85]. ~ ∆ Band edge (a.u.)

Peak position (eV)

2.32

2.31

2.30

2.29

2.28 0

50

100

150

200

250

300

Temperature (K) Figure 1.54 The PL peak position of the same green LED as in Figure 1.52 measured at different temperatures. The PL peak shifts to higher energies with increasing temperature. This is to be contrasted to the band-edge energy dependence on temperature, which is depicted by the solid line. This line is intended to show the trend only with no attention paid to the absolute values [85].

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levels do not support the exciton premise. Particularly, the rapid blueshift of the electroluminescence with injection is indicative of a continuous density of states favoring band-tailing effects that are caused by strain and compositional fluctuations.

1.11 LED Degradation

Prior to the debut of GaN, the experience with all LED materials had been that acceptable quantum efficiencies can be obtained only when the defect concentration in the semiconductor is well below about 104 cm2. GaN, with defect concentrations about six orders of magnitude higher than this at the time of introduction of GaNbased LEDs, was not consistent with the trend of the time, and thus GaN was discounted by many LED manufactures. Even with large defect concentrations in the vicinity of 109 cm2, GaN LEDs, with longevity well over the minimum 10 000 h, required by the display society (CIE), were marketed in early 1994, which took many by surprise. In order to reduce the amount of InN employed in InGaN active emission layers, early devices utilized deep Zn centers in the 20 nm thick active layer to shift the wavelength to about 450 nm, which is defined as blue by CIE. These devices were shown to have more gradual and graceful aging than the AlGaAs diodes, particularly in steam tests. Nevertheless, degradation caused by p-metallization was notable. In the second wave of devices, the Zn centers were eliminated and the blue color was obtained by an increased InN mole fraction in the lattice, with an accompanying decrease in the active layer thickness to about 3–4 nm. These devices exhibit longevity well over that required by CIE. In addition to manufacturers’ own life testing, early independent life testing of Nichia LEDs was undertaken by Osinski et al. [92], Barton et al. [93], and Barton and Osinski [94]. This work was later extended to include improved single quantum well varieties with sufficient InN in the lattice so as to not rely on Zn centers [95]. In the initial degradation experiments of Osinski et al. [92], there were three types of devices, two containing Zn deep centers and wider InGaN layers, and a third set of devices that did not utilize Zn, but took advantage of the increased InN mole fraction to obtain the 450 nm blue radiation. The general trend amongst the 18 LEDs measured was for the output intensity to decrease at a faster rate within the first 50–100 h, and then at a slower rate over the remainder of the test. The output intensity of the two earlier model LEDs containing Zn increased within the first 50 h and then decreased during the remainder of the test. After the first 1000 h, the drive currents of the LEDs were increased to accelerate the test. The relative intensity of one of the older generation LEDs dropped to about onehalf of its initial value after approximately 1200 h of testing. In this case, the high current (70 mA) had indeed caused a rapid failure. The cause of this premature degradation was a crack in the LED, which isolated a part of the junction area from the p-contact. The remaining devices driven at the same current level, however, have performed much better. The degradation rate slowed after a relatively fast drop in the output (10–15% over the first 750 h).

1.11 LED Degradation

Figure 1.55 Secondary electron EBIC image of a shorted Nichia LED after the p-contact metal has been removed. Courtesy of M. Osinski, University of New Mexico.

To speed up the life test, electrical stress under high pulsed-current conditions was also applied, which resulted in a degradation of the I–V characteristics with some devices exhibiting a low-resistance ohmic short (40–800 W). Electron beam induced current (EBIC) imaging pointed to a conductive path extending from the surface (the p-contact metal) to the n-type side of the junction. The high forward current applied to this device caused metal from the p-contact to migrate across the junction, as is evident in the secondary electron EBIC image (Figure 1.55). The use of low-resistance and thermally stable ohmic contacts to the p-GaN layer can mitigate indiffusion of the ohmic contact elements along dislocations in nitridebased epilayers, leading to an electrical short of the p–n-junction. The use of a diffusion barrier in Ni/Au-based contacts showed superior long-term stability of turn-on voltage, leakage current, and output power [96]. Double-heterojunction green LEDs without the deep Zn centers were also stressed with pulsed currents of approximately 5 A with a 1 kHz repetition rate and a 104% duty cycle yielding an average power dissipation of 25 mW to eliminate heating. Three devices were stressed to failure with a sudden and complete loss of light output. The I–V characteristics were all linear with resistive shorts in the range 18–140 W. In both blue and green failed LEDs, optical micrographs taken after decapsulation showed severe damage to the plastic encapsulation, which could not be completely removed even in hot acids due to damage. The opaque encapsulant is to a great extent responsible for the faster light output degradation, leaving the cause of the slow degradation to nonradiative recombination center generation. It has also been noted that as the material’s quality improved, the high current-induced stress causes encapsulant damage earlier than the device [97]. The data presented above on the early vintage double-heterojunction LEDs indicate a possible connection between the large number of crystalline defects and a tendency for metal to migrate from the p-contact across the junction and short out the device. The newer generation, quantum well LEDs showed a significant improvement in resistance to this type of stress and revealed that the limitation may be in the plastic packaging material and not in the diode itself. The slow degradation in light output appears to be due to degradation in the transparency of the plastic package material

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caused by high current pulses, not likely due to the creation of crystalline defects. Constant current life tests have shown that the plastic encapsulant can change in transparency due to Joule heating. The conditions in the final stage of the CW life test showed that the package would be adversely affected by prolonged exposure to temperatures around 140
C. The analysis of LEDs subjected to pulsed-current stress showed that some regions on the LED may have exceeded 140
C during a single pulse, thereby damaging the plastic in that region. The observed rapid degradation occurs when the plastic degrades to the point where its conductivity increases and facilitates the burnout of the LED. At this point of low resistance, ohmic shorts are observed across the LED. Degradation testing of high-brightness green LEDs under high current electrical stress with current pulse amplitudes between 1 and 7 A, voltages between 10 and 70 V, a pulse length of 100 ns, and a 1 kHz repetition rate has been carried out [95]. When the current amplitude increased above 6 A, a fast degradation (on the order of 1 s), accompanied by a visible discharge between the p- and n-type electrodes occurred. Follow-up analysis revealed severe damage to the metal contacts, causing electrical shorts on the surface plane of the diode. Below 6 A, a slow degradation was observed in the form of a decrease in optical power and increase in the reverse current leakage. However, a rapid degradation, similar to that for higher current, occurred between 24 and 100 h. Overall analyses indicate that the degradation process begins with carbonization of the plastic encapsulation material on the diode surface, which leads to the formation of a conductive path across the LED and subsequently to the destruction of the diode itself. Additional DC aging experiments, following a schedule of tests under 20, 50, and 100 mA current stress, were carried out on more improved, in terms of semiconductor properties, blue LEDs packaged with encapsulants, as well as LED chips mounted without encapsulants, in order to delineate the role of encapsulant degradation [98]. In the process, three distinct failure modes were identified: (i) High current levels and, consequently, high junction temperatures, induce degradation of the epoxy encapsulant where it is in contact with the hot device surface, leading to opacity of the encapsulant. This degradation mechanism is faster, but does not apply to high intensity LEDs which do not employ epoxy encapsulant. (ii) High current and thus temperature degrades the semitransparent p-ohmic contact and the top surface of the p-GaN layer, which leads to increased series resistance. This leads to current crowding, reducing the optical power, again this does not apply to flip chip mounts and their transparent contacts which could have their own reliability issues in terms of transparency retention. (iii) Redistribution of electrically active deep levels, which already exist in control devices, and also generation of shallow traps, as probed by deep-level transient spectroscopy (DLTS). The decrease of optical power seems to correlate with reverse leakage current, which in turn is possibly due to an increase in the density of nonradiative recombination centers.

1.11 LED Degradation

Figure 1.56 Emission micrograph of a failed LED after decapsulation with severe damage to the plastic encapsulation, which could not be removed completely, even in hot H2SO4 [98].

Figure 1.56 shows an emission image of a blue LED aged for 250 h at 100 mA after removal of the epoxy encapsulant in hot H2SO4. High temperature and shortwavelength irradiation have damaged the encapsulant and left an opaque layer, which could not be removed in hot sulfuric acid. Comparative electrical measurements in control and stressed LEDs are shown in Figure 1.57. As mentioned earlier, the failure modes are as follows: (i) increase of the 10-2

(IV)

Unstressed blue LED Aged at 100 mA for 220 h

10-4

Current (A)

10-6 (III)

10-8 10-10 10-12

(II) (I)

10-14 -6

-4

-2 0 Voltage (V)

2

4

Figure 1.57 I–V characteristics of control and stressed (100 mA and 220 h CW) LEDs [98].

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reverse leakage current accompanied with optical power degradation; (ii) increase of the generation–recombination current for low forward-bias voltages; (iii) degradation of the device ideality factor; and (iv) increase of the parasitic series resistance, most likely due to degradation of the semitransparent p-type ohmic contact when used. The increased resistance causes current crowding with increasing current exacerbating device performance [98]. Extending the issue of longevity to white-light LEDs, Narendran et al. [99] tested the lifetimes of two groups of white LEDs. The LEDs in one group had similar junction temperatures but different amplitudes for the short-wavelength emission, and the LEDs in the second group had similar amplitudes for the short-wavelength emission but different junction temperatures. In the experiment, both the junction temperature and the amplitude of short-wavelength emission were affected by the degradation rate of the white LEDs. However, the effect of the temperature on the degradation of the LEDs was much more extensive than the short-wavelength amplitude. Furthermore, they also showed that some portion of the light circulates between the phosphor layer and the reflector cup, potentially increasing the epoxy-yellowing issue. In an effort to assure the reliability of the LEDs especially under the high temperature (>100
C) and extreme photon fluence (>50 W cm2), several improvements in base material and chip attachment technologies have been undertaken. Among them is a package that can allow 150
C (white) or higher (185
C, nonwhite) junction temperature operation, as provided by the LUXEON K2 package, an artistic view of which is shown in Figure 1.58 for junction tempera-

Figure 1.58 High-power LED packages for lighting applications. (a) LUXEON circa 1998. (b) LUXEON K2 circa 2006. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

1.11 LED Degradation

tures T < 120
C and T  150
C [100]. While on the topic, we should mention that the epoxy used in some packages suffers from yellowing, a degradation mechanisms that can be mitigated by substituting it with silicon resins or siliconepoxy resins with thermal stability and UV resistance for use in encapsulating LED [101]. The generation of additional nonradiative defect centers after aging was also confirmed in another investigation where electroluminescence, electron beam induced current, and cathodoluminescence investigations were brought to support this conclusion, albeit in early varieties of devices [102]. The aged InGaN/ AlGaN DH LED exhibited formation and propagation of dark spots as well as a crescent-shaped dark patch, which were acting as nonradiative recombination centers, determined by cathodoluminescence images. The degradation rates of the relative optical power under an injected current density of 0.1 kA cm2 were determined to be 1.1  103, 1.9  103, and 3.9  103 h1 at ambient temperatures of 30, 50, and 80
C, respectively. The activation energy of degradation was also determined to be 0.23 eV. Having said this, the point of changing the deeplevel picture is of some controversy, as earlier reports based on investigations utilizing deep-level transient spectroscopy, thermally stimulated capacitance, and admittance spectroscopy measurements performed on stressed (earlier version) devices showed no evidence of any deep-level defects that may have developed as a result of high current pulses [92]. This may be due to variability in stress testing or higher extended and point-defect concentration existing in devices before testing and thus earlier degradation than shown by more improved devices of later varieties. High junction temperature limits the performance of GaN LED, with the main degradation mechanisms arising due to the emission crowding and series resistance. Pending confirmation, the degradation process has been attributed to the presence of hydrogen, which can diffuse in the p-layer and generate Mg–H bonds with the acceptor atoms, thus compensating the overall active, already low, hole concentration and reducing device performance. The likely source of hydrogen could be from the passivation layer [103] typically deposited by PECVD on the LEDs for chip encapsulation and surface leakage current reduction. In phosphor conversion LEDs, in addition to degradation of LEDs themselves, the reliability of YAG and any encapsulant used enters into the picture. In the short history of pcLEDs, there have been speculation and some reports about lifetime threats. Especially for YAG:Ce, this appears strange, as this material has been proven to be extremely stable under most adverse conditions [104]. However, as the best argument is a reliability test, a meaningful ensemble of YAG-pcLEDs was put into a 60% relative humidity/85
C test chamber and driven at 50 A cm2. After 3000 h, no significant degradation of the lumen output was observed. These studies were conducted on pcLED in a special package, which did not contain an epoxy encapsulant. Previous tests on 5 mm LEDs with epoxy dome lenses, which are still common in the industry, failed often because of “browning” (oxidation) of the epoxy, reducing the transmission. It is not unlikely that the previous reports relate to results that actually tested the package materials [105].

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1.12 LED Efficiency

A rather important aspect in GaN LEDs in general is the reduction of output light with increasing injection current. The thermal component of the problem is reasonably well addressed with very advanced packaging technologies that not only effectively extract the photons generated but also dissipate the heat. However, all GaN LEDs suffer efficiency reduction when pumped at high injection currents, the absolute value of which depends on the layer quality and packaging, among others. Specifically, a good part of the reduction in efficiency in LEDs on c-plane sapphire with increasing injection current is due to reduction in the internal quantum efficiency, which in general involves opening up channels for nonradiative recombination. Carrier leakage when the band discontinuities in the quantum well regions are small can also contribute to such an effect. Consequently, the maximum internal quantum efficiencies are obtained at relatively low injection levels, sometimes as low as 10 A cm2 (see Figure 5 of Ref. [108] and Figure 8 of Ref. [107]). This is in contrast to the observations made in LEDs based on relatively well-developed and high-quality conventional III–V semiconductor material systems such as (In, Al, Ga)As and (Al, Ga)InP. As the current density is increased beyond that point at which the maximum injection efficiency is attained, a monotonic drop in quantum efficiency is noted in blue and green InGaN/GaN QW LEDs, even under short-pulse, low-duty-factor, and constant-temperature injection [106, 107]. However, just about all the applications of these LEDs require them to retain high quantum efficiencies at larger current densities (e.g., >50 A cm2). As mentioned throughout this chapter, these applications include, but are not limited to, projection displays [108], automotive headlights [100], and general lighting. The decay in quantum efficiency with increasing current for blue and particularly green InGaN/GaN LEDs has been attributed to many different mechanisms, among which are poor injection efficiency [109, 110], carrier delocalization from quantum dots [111], exciton dissociation [112], high plasma carrier temperatures (hot carriers) [113], and polarization effects together with electron-blocking layers [114]. Furthermore, the Auger nonradiative recombination [115] as well as methods to mitigate it [116] has been reported to cause the droop in efficiency at high injection levels. It should be noted the measurements performed at various laboratories, including that of the author, indicate that the intensity droop is not observed in PL experiments, but rather in EL experiments, indicating that reduction in the efficiency is not related to the MQW radiative recombination efficiency but to the loss of carriers by means such as recombination outside the quantum well region. This is generally termed as carrier leakage. The effect of heating due to increased current at high injection levels has been ruled out. The dependence of the efficiency droop on quantum well layer thickness has been investigated. Studies indicate that the droop is not as evident in wider quantum wells that have the downside of reduced overall intensity pointing to some unresolved issues [117]. In the case of the notion of Auger recombination, the Auger recombination coefficient that is estimated by fitting a third-order polynomial is on the order of

InP GaAs

InGaAs

InAs InGaSb

-23

InAsSb InSb

1.12 LED Efficiency

InGaAsP

1×10

-24

6

Auger coefficient, C (cm s-1)

1×10

-25

1×10

-26

1×10

-27

1×10

-28

1×10

-29

1×10

-30

1×10

-31

1×10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Energy band gap (eV) Figure 1.59 Auger coefficient C for direct-bandgap semiconductors in which it has been determined at room temperature. Data courtesy of J. Piprek.

1.4  1030 to 2.0  1030 cm6 s1, which is comparable to the one that has been measured for InP that has half the bandgap of InGaN emitting in the blue [115]. This is approximately four to five orders of magnitude larger than what can be expected of the bandgap of the blue of InGaN when extrapolated from the data shown in Figure 1.59 (also see Equation 1.29), in more conventional semiconductors ranging from InAsSb on the small-bandgap side to GaAs on the relatively large-bandgap side [118]. The four to five orders of magnitude disparity could in fact increase, considering that the dispersion between hole and electron masses in conventional III–V semiconductors is much smaller as compared to InGaN. We should mention that Auger recombination has been studied in the smallest bandgap triad of nitride semiconductors, InN [119]. Regardless, it is clear that the efficiency droop with increasing injection level points to carrier leakage or loss of some sort, which will take some time sort out. All of the above-mentioned processes with the exception of Auger recombination [120–123] are discussed in Volume 2, Chapter 5. It should be recognized that Auger recombination is an intrinsic property of a given semiconductor and is determined by the conduction and valence bands as well as the bandgap and its type, meaning direct or indirect. Further, because Auger recombination involves carrier recombination across the band and also carrier excitation to higher energies, the process involves many carriers and as such it becomes more important at high carrier densities. To a first degree, if the Auger recombination, which is more dominant at high injection levels and as in its simplest treatment goes with the third power of the carrier concentration, were to take place in GaN-based LEDs at current densities dwarfed by those in effect in lasers, to be consistent one can surmise that lasers in GaN could not be obtained if the layer qualities were in the same ball park.

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Because, this is not the case, it is more likely that the efficiency reduction with injection current is related to the material quality/particulars (e.g., localization, which at high injection levels would not be as effective) and heterojunction design (e.g., thin multiple quantum wells versus one relatively thick active layer considering the large hole effective mass) if the processing- and packaging-related issues are addressed effectively. It is likely that the Auger process involving defect levels could enter into the equation. One can then surmise that the current density at which the efficiency peaks would increase in time. Already preliminary results in author’s laboratory indicate that large hole effective mass and relatively low p-doping form the genesis and that measures can be taken to increase the current at which the efficiency peaks. Nevertheless, below we give a brief discussion of Auger recombination and how it is quantified for instruction purposes. In brief, Auger recombination depicts the process in which an electron in the conduction band recombining with a hole in the valence bands does not lose its energy radiatively. Instead, it gives off its energy to another electron, which is then excited to a higher energy level in the conduction band, or a hole, which is excited to a higher energy level in the valence band. As shown in Figure 1.60, the process naturally requires that both energy and momentum be conserved, necessitating indirect transitions to occur. Again, as shown in Figure 1.60, the process can also involve phonon participation. For example, if the process following the recombination is electron excitation to a higher energy, the process goes by the depiction of CCCH. The processes involving donor and acceptor states are not shown, for simplicity, but can be found elsewhere [124]. The dominant temperature dependence of the Auger recombination specific carrier lifetime (the inverse of Auger recombination rate) to a first extent is given by a modified expression of Beattie and Landsberg [120] by     E g (T) 3=2 1 þ 2M E g (T) t/ exp 1 þ M kT kT

ð1:29Þ

for a nondegenerate and intrinsic semiconductor. The terms Eg(T) and M represent the temperature-dependent bandgap and electron to hole mass ratio, respectively. For a semiconductor with a conduction band effective mass smaller than the valence band effective mass, the lifetime is determined by electron–electron collisions, namely, electron recombination with a hole followed by another electron excitation to a higher energy or electron excitation from the valence band to the conduction band coupled with hot electron relaxation to near the bottom of the conduction band with the entire process conserving energy and momentum. If the hole mass is smaller than the electron mass, which is not the case in GaN and related materials, the lifetime would be determined by hole–hole collisions, namely, electron recombination with a hole coupled with another hole excitation to a higher energy or electron excitation from the valence band to the conduction band, coupled with hot hole relaxation to near the bottom of the valence band, again with the whole process conserving both energy and momentum [120]. Typically, Auger recombination rate is higher, or the lifetime is smaller, for relatively small-bandgap materials, meaning the value of the ratio of energy gap/kT, in which

1.12 LED Efficiency

CHSH or IVBA

CCCH

CHLH

E

E

E

c

1′

2

1

1

1 (a)

2′

2

1′

1′

k

2′

k

hh lh

k 2

2′

so CCCHP

CHLHP

CHSHP E

E

E

1′

2

1

1

1 I

2′ q

1′ k

I

I

2′

1′ k

q 2

Figure 1.60 Auger processes involving the intraconduction band, interband, and intravalence band processes. Parameters c, hh, lh, and so represent the conduction band, heavy-hole valence band, light-hole valence band, and spin–orbit split-off band. Processes CCCH, CHSH, and CHLH indicate the conduction band–conduction band and conduction band–heavy-hole valence band, conduction band–heavy-hole valence band and split-off band–heavy-hole valence band, and conduction band–heavy-hole valence band and light-hole band–heavy-hole valence band transitions, respectively. These transitions conserve both momentum and energy (a).

(b) 2′ k

q 2

Processes similar to CHSH are also considered IVBA processes. Processes CCCHP, CHSHP, and CHLHP represent the conduction band–conduction band and conduction band–heavy-hole valence band with phonon interaction, conduction band–heavy-hole valence band and split-off band–heavy-hole valence band with phonon interaction, and conduction band–heavy-hole valence band and light-hole band–heavy-hole valence band with phonon interaction transitions, respectively. These latter transitions (b) conserve energy and momentum through the assistance of phonons. Patterned after Refs [125, 126].

case, electrons in the conduction band and holes in the valence band participate in the process. By the same token, Auger recombination is more important at high temperatures. To a first order, the traps do not participate in the process, although their taking part could be important in some cases. In large-bandgap semiconductors, the Auger process would depend on the doping level and become important in degenerate cases. At high doping concentrations, the electron wave function of adjacent impurities would overlap and delocalize the electrons (or holes), which

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increase the likelihood of Auger recombination. The energy of the excited carrier in the process would be dissipated by a cascade of optical phonon emission while conserving energy and momentum. The recombination process, in general, has been treated in Volume 2, Chapter 4. Here, we will extend the discussion to include the contribution to this process of Auger processes. Recall Volume 2, Equation 4.54 for a p-type semiconductor, repeated below for convenience: U¼

svth N t (pnn2i ) 1 (pn  n2i ) ¼ ; n þ p þ 2ni cosh[(xt xi )=kT] tr n þ p þ 2ni cosh[(xt xi )=kT]

ð1:30Þ

with t1 r ¼ svth N t , the average lifetime of the minority carriers. As mentioned above, Auger recombination involves three particles, namely, an electron and a hole that recombine in a band-to-band transition and give off the resulting energy to another electron or hole. Naturally, the expression for the net recombination rate in Auger recombination is similar to that of band-to-band recombination but with the exception that it must include the density of the electrons or holes, which receive the released energy from the electron–hole annihilation. In the case of electron excitation to a higher level in the conduction band, the Auger recombination rate can be expressed as   U Auger ¼ Cn n(pnn2i ) : ð1:31Þ In the case of a hole excited to a higher energy in the valence band, the Auger recombination is given by   U Auger ¼ Cp p(pnn2i ) ; ð1:32Þ where Cn and Cp represent the Auger recombination coefficients for electrons and holes, respectively. If both of the above processes take place, then the sum of Equations 1.31 and 1.32 must be used. In the case of high-level injection, the electron and hole concentrations, n and p, are nearly equal to each other and both are much larger than ni. Consequently, the Auger recombination rate would be proportional to n3. For comparison, the band-to-band radiative recombination rate (U bb ¼ B(pnn2i )), which at high injection levels would be proportional to Bn2, and the recombination by traps, Shockley, Hall, and Reed recombination (U SHR  t1 n (np np0 )), would be proportional to n. In simple terms, the total rate of change of electron concentration at high injection levels due to deep levels, band-to-band radiative recombination, and nonradiative Auger recombination can be expressed as U¼

dn ¼ Bdl nþBbb n2 þ BA n3 ; dt

ð1:33Þ

where Bdl, Bbb, and BA represent the deep-level recombination, band-to-band recombination, and Auger recombination coefficients, respectively, in the units of s1, cm3 s1, and cm6 s1, respectively. If there is generation, for example, by photoexcitation, then a generation term G must be added to Equation 1.33. Timeresolved luminescence experiments have been conducted to determine the decay

1.12 LED Efficiency

rates. The term dn=dt can be written as (dn=dIPL )(dI PL =dt) wherein the first term can be obtained from time decay of the PL experiments in which the photogenerated carrier concentration before decay is known. Specifically [115], a fit of the dependence of the PL intensity on the absorbed photon density, I (in terms of W cm2), for the total recombination rate can be used to determine the internal quantum efficiency hint through the relation hint ¼ Bn2/G, albeit without a unique combination of Bdl, Bbb, and BA coefficients (popularly referred to as the A, B, and C coefficients). Luckily, the B coefficient can be uniquely determined by fitting a decay curve at a known I and hint (I) in an iterative procedure. The carrier density n is determined from two points in a single light output intensity decay (L1 and L2 at time intervals t1 and t2) from n¼

2G(t2 t1 ) : 1(L2 =L1 )

ð1:34Þ

Knowledge, therefore, of n, hint , and G would pave the way to calculate the B coefficient from B ¼ hintG/n2, which in turn leads to the determination of the remaining coefficients, A and C. Because of the cubic dependence, the Auger recombination would become relatively more effective at very high injection levels, such as those experienced in lasers, particularly in small-bandgap semiconductors. Having larger bandgaps reduces the Auger recombination rate in nitride semiconductors, which should inherently be the case (Equation 1.29). Still, the carrier leakage/loss issue, which causes the efficiency droop at high injection levels, particularly for green LEDs, would have to be addressed satisfactorily. Another issue with the external quantum efficiency is that the external quantum efficiency drops sharply for wavelengths shorter than 360 nm [127], as shown in Figure 1.61. Typically, two reasons are thought to be responsible for the efficiency rolloff at longer peak wavelengths. First, a reduced crystal quality is expected for structures with higher In content as it is necessary to lower the bandgap of GaxIn1–xN. Second, a higher piezoelectric field for GaxIn1xN layers that are more highly strained with higher In content leads to a stronger separation of the electron and hole wave functions. On the shorter wavelength portion of the spectrum, the roll-off 10 2

EQE

10 0

Nitride -LED, η

(%)

10 1 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 200

300

400

500

600

Wavelength (nm) Figure 1.61 External quantum efficiency of GaN-based LEDs. Courtesy of H. Amano.

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is associated with reduced localization due to relatively low In mole fraction. If and when the defect concentration in GaN is substantially reduced, for example, when inexpensive bulk GaN substrates are available, one can discern whether InGaN is inherently more efficient in radiative recombination. How to improve the external quantum efficiency without significantly increasing the processing costs is still a challenging problem.

1.13 Monochrome Applications of LEDs

LEDs have already penetrated many monochrome applications. Among the most notable is the traffic light [128], a schematic of which is shown in Figure 1.62. Estimates are that a 12 in traffic light in the United States usually employs a 135 W long-life light bulb in combination with a red, yellow, or green filter. The most advanced LED based red varieties use 12–18 LEDs per traffic light and consume a total of about 14 W, which will slide downward with time, including power supply losses. A single-LED traffic light sells for about $110 compared with a $30 cost of an incandescent variety. The operating cost for electricity is approximately $10 per year for the LED variety compared with $90 for the incandescent model, and LED model pays for installation costs in less than 1 year, not to mention the long operating lifetime of the LED further reduces maintenance and emergency repair costs. By some estimates, there are some 10 M red/yellow/green traffic lights in the United States consuming approximately 400 MW. On an average, red lights are lit 65% of the time, 90% in the case of red arrows. Just converting all red lights to LEDs would

Figure 1.62 Evolution of LED-based traffic signals and a comparison of LED to incandescent traffic lights. Incandescent bulbs consume 135 W and must be replaced every 6 months. The LED alternatives, on the other hand, consume 15 W and would have to be replaced only every 120 months (2002 figures). Using red traffic lights as an example, because of their priority, the number

of LEDs for each traffic light went down from 700 per traffic light in 1993 to 12–18 in 2003. The latest LED count is similar to the improvement experienced for green-LED-based traffic lights over the years. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

1.13 Monochrome Applications of LEDs

reduce the US electricity consumption by approximately 250 MW. This would nearly double with green LED insertion. All large buildings with public access must have lighted emergency signs assisting in evacuation during emergencies such as power failure. These “Exit” signs are designed with two incandescent or compact fluorescent lamps consuming 15–30 W. Using approximately 100 inexpensive LEDs (the number of LEDs needed declines with improved brightness) is comparable in cost to the conventional variety but uses only 5 W, saving somewhere between $10 and $25 in annual electricity cost per sign, while reducing the size and cost of the stand-by battery [128]. As mentioned in Section 1.8, LEDs started to be used on the taillight of cars shortly after the center high mount stop lights (CHMSL) was made a mandatory feature in the United States in 1982. As of 1999, LEDs have reached a penetration of 30–40% of those cars equipped with a CHMSL. In the model year 2000, the first rear combination lights (taillight, brake light, and turn indicator light) emerged as LEDs on highend models in the United States and Europe. Other functions such as side markers and front turn indicators followed shortly thereafter. Shallow design that does not protrude into the trunk, styling freedom, reduced warranty cost, and reduced power consumption are among the reasons for preferring LED insertion. It is only a matter of time before the red taillights will fully convert from incandescent lighting to LEDs, with the yellow front beam indicators following in their footsteps. This will eventually give way to conversion of the white backup, license plate lights, background dashboard illumination. Estimates are that the average car will contain 1000 lm of LED flux, or even more: 300 red, 300 yellow, and 400 white. Operating these LED chips at 100 A cm2 will require about 20 mm2 of LED material per car. The conversion of the passenger car market from incandescent lighting to LEDs is quite sensitive to the cost differential. However, the truck and bus market is less cost sensitive and failed taillights require an immediate repair. As a result, the US truck market made a quick and nearly complete conversion of incandescent to LEDs several years ago. The very high brightness white LEDs are already beginning to replace conventional automotive headlamps in luxury model, which will definably segue into the other models in time. Halogen (55 W) and high-intensity discharge (HID) (35 W) lamps, which are the conventional automotive lamps, can be replaced by several thinfilm flip-chip LEDs (12–44 W). Moreover, the use of LEDs not only allows a more compact auto headlamp design but also a precisely defined emitting area owing to sharper far-field radiation patterns, which improve visibility while reducing glare for the oncoming traffic [100] (Figure 1.63). Decorative lighting is an area where not only savings occur but also the design flexibility is gained, with the use of LEDs. For example, multicolor landscape lighting with programmable and decorative color might someday soon be very popular. There are other applications as well, for example, the Australian branch of the McDonald’s restaurant chain started to outline the rooflines of its buildings with a chain of red LEDs. LEDs are significantly more energy efficient than the competing neon technology. Red LEDs are already at cost parity with neon lights and expectations are that similar cost parity will emerge for yellow, green, and blue LEDs in the not so distant future. There are three major groups of commercial enterprises that are

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Figure 1.63 Comparison of a thin-film flip-chip (TFFC) white LED to conventional halogen and HID lamps used for low-beam automotive forward lighting (headlight) applications. The top row is the lit visual image each of the three technologies compared. The lower top row represents the color-scaled luminance image.

The scales for the halogen filament and HID arc are the same. The scale for the LED is different and is indicated at the left. The table lists average luminance, source flux, input power, and useful flux (utilization percentage) in the application. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

interested in decorative lighting: fast food chains, gas stations, and hotels. It is a form of advertising and presentation, and all these groups wish to get the attention of potential customers driving by. Other target applications for LEDs are the large outdoor video screens and changeable displays for advertising. For instance, a 600 m2 video screen uses 3 million 5 mm LEDs. The LEDs are arranged in end-stackable tiles. The LED density is one lamp per 2 cm2 of board space. The 5 mm lamp itself has a cross section of 0.2 cm2, thus leaving 90% of the space empty. The LED is the technology of choice for large video screens because it is the technology with the lowest cost for the empty space between the pixels, basically, the cost of a two-sided printed circuit board, which is far less than any glass-based display technology. Additionally, LEDs are directly viewed and unfiltered, which makes LEDs more competitive than other display technologies, an advantage that is more appealing when it comes to monochrome applications. In these cases, there are no color-mixing losses for the LEDs, but there are additional filtering losses for incandescent lamps.2) 2) Variability in efficiency of the color filters used to produce various colors is of special interest. For example, the filter used in a red traffic light absorbs 90% of the white light. Roughly, the same is true for blue filters. However, the red filter of an automobile taillight has a wider

transmission band and results in an orange–red color. Yellow and green filters are fairly efficient and transmit a large fraction of the white spectrum. LEDs are inherently monochrome and do not suffer from filtering losses, as they do not need them.

1.14 Luminescence Conversion and White-Light Generation with Nitride LEDs

Luminous efficiency (lm W-1)

80

InGaN

2005 production

60

2000 production

40

Best reported result

2005 production 2000 production

20 0

GaAlInP

450

500

550 600 Peak wavelength (nm)

650

Figure 1.64 LED efficiency at 85
C junction temperature as a function of wavelength for the two dominant material systems (GaAlInP and AlGaInN). The best estimate for year 2005 production is also shown, which has been met since then, if not exceeded.

LEDs of reasonable efficiencies span almost the entire visible spectrum, with the exception of a narrow window in the yellow–green, paving the way for white-light sources. The improvements in materials quality and device design and packaging have led to LED efficiencies (in photometric terms (lm W1)) at 85
C junction temperature, which is shown in Figure 1.64. The efficiency of the GaAlInP material system (from red to amber) follows the luminous efficacy response. The same is true, albeit with lower efficiencies for the InGaN system. The long-wavelength end for the InGaN system (amber/yellow) and the short-wavelength end for the AlGaInP system fall short of the luminous efficacy curve. As shown in Table 1.3, LED efficiencies exceed those of filtered incandescent lamps by a large margin over the entire visible wavelength range except for yellow, where the two technologies are close to parity .

1.14 Luminescence Conversion and White-Light Generation with Nitride LEDs

Availability of violet and blue compact LED emitters has paved the way for alternative approaches to generate blue, green, and red primary colors. A blue or a violet LED can be used to pump a medium containing the desired color centers, dyes in organic and phosphors in inorganic materials, to generate the color(s) desired including white. For white LEDs to be accepted for indoor illumination, the CRI, see Section 1.14.2, must conform to the CIE standards. While the color temperature requirements can be met with phosphor-pumped LEDs, the strict CRI as well as the very high luminous efficacy is nearly impossible to meet. The former is due to very close adherence to produce the Planckian spectrum and the latter is due to the Stokes shift (loss in this

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Table 1.3 LED efficiencies in broad color ranges as compared

to those of filtered long-life incandescent lamps.

Colora

Filtered long-life incandescent efficiency (lm W1)

Year 2000 LED production (lm W1)

Red (627 nm) Yellow (580 nm) Green (528 nm)

1–6 4–8 3–10

16 10 48

Blue (470 nm)

1–4

White

12

13 at low drive I 20

Year 2006 (lm W1) 58 (LumiLED’s K2 @350 mA) 40 (Cree’s Xlamp @350 mA) 110 (Osram’s Golden Dragon @100 mA 20 (Osram’s Golden Dragon @100 mA) 100 (developmental, Cree/Nichia @20 mA)b

For nitride LED, see the results for high-current operation [128]. The lifetime is defined as the time when a reduction to 70% of the original flux is reached. The data from LumiLEDs are associated with its K2 emitter; data from Cree are represented by the Xlamp3 7090 and from Nichia by the i-LED, which emit at red, yellow, and white, respectively. None of the three companies produce a violet (400–420 nm) LED commercially (Table 1.2). a Wavelengths above for the year 2006 LEDs indicated. b LED lamps with above 75 lm W1 are seen to be commercially available in 2006 and are produced by Cree, Lumileds, and Nichia. Values of 100 lm W1 are beginning to appear. These high flux lamps feature in some cases photonic crystals for top mount and darkened emitting surface for flip-chip mount varieties.

case) associated with phosphor downconversion. LED bandwidth and in particular the wavelength of emission in the multiple-color LED approach must be precisely controlled, which is very stringent, as will be discussed soon. With broad bandwidth associated with green LEDs (in addition to the much reduced power available in the so-called green gap) and to a lesser extent for blue LEDs, it may not be quite feasible for a while to use a three- or four-LED approach to achieve white light meeting the CIE indoor illumination standards. If so, an inexpensive and attainable method for white-light generation may be to pump tricolor phosphors, which are known to have sharp emission linewidths, with LEDs. Alternatively, a blue source could pump two-color or three-color phosphors for whitelight generation. However, if one were to limit the phosphors to what has traditionally been available, one needs deep UV LEDs with high power (which would not lead to high luminous efficacy because of losses associated with Stokes shift during downconversion) or develop efficient phosphors that can be excited efficiently with blue LEDs then the less desirable UV LED approach that is accessible by the GaN system. The phosphors that can be pumped by GaN-based LEDs are under development and seem to be progressing well. When LED-based white light with very good CRIs (in the 90 percentile) is achieved, some estimates are that LEDs will be direct replacement for point sources such as incandescent lamps, while OLEDs might eventually replace area sources such as fluorescent lamps for back illumination although LEDs have captured this market in at least the mobile devices where brightness is a key issue.

1.14 Luminescence Conversion and White-Light Generation with Nitride LEDs

Both LEDs and OLEDs have been under development for special segments of the illumination/display markets. Solid-state lighting based on inorganic LEDs has the potential to fundamentally change the nature of lighting that human kind has experienced over the last century [129]. Since the introduction of the incandescent lamp in 1879, there has been a drive for brighter, cheaper, smaller, and more reliable light sources. In the United States, about 30% of all generated electricity is used for lighting, with about 40% of this being incandescent lighting and 60% being fluorescent lighting. This is representative of the global trend; consequently, significant improvements in lighting efficiency would have a major impact on worldwide energy consumption. Unfortunately, none of the conventional light sources (incandescent, halogen, and fluorescent) have improved significantly in the past several decades in terms of efficiency. Because an average of about 70% of the energy consumed by these conventional light sources is wasted as heat, which in many cases ends up only increasing the cooling required and consuming additional energy in the process, there is clearly room for improvement. With increasing demand and declining reserves for natural means of generating energy, such as gas and fossil fuel, the future for efficient solid-state lighting is very bright indeed, despite commercial interests and policy makers’ infatuation with Hg containing compact fluorescent bulbs. The relatively recent developments in LEDs, in terms of range of wavelengths, efficiency, and lifetime, are proving to have significant impact in the low-flux lighting technology, with theprojection that the same will also be true for general lighting. In this fluidic environment, it may be that lighting applications may even encompass lasers, particularly vertical cavity surface emitting lasers (VCSELs) when developed in conjunction with phosphors. VCSELs emit light perpendicular to the p–n-junction plane, as compared to emission in the plane of the junction for edge emitters, and provide wallplug efficiencies as high as 50%, albeit at 850 nm [130]. As far as nitrides are concerned, current injection VCSELs are not yet available, but its forbearer, the optically pumped variety operating at the near-UV wavelength of 384 nm, has been reported [131].3) With SSL, anticipated improvements to the quality of white lighting for general illumination include steady output color at all levels of illumination, ability to continuously vary output color, simplified and flexible design for mounting and fixtures, ease of integration, advanced building controls including daylighting, and low-voltage and safe power distribution. Developments of efficient and reliable blue and green LEDs using nitride-based alloys are significant for the realization of efficient white-light LEDs with excellent quality (high color rendering index). 1.14.1 Color as Related to White-Light LEDs

In order for LEDs to be used for general lighting, it is imperative that they have appropriate white color with good color rendering characteristics when illuminating objects. Needless to say, color rendering and efficacy represent the two most 3) Low temperature injection VCSELs have been reported during the interim period.

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important criteria for light sources used for general lighting. The US Energy Policy Act (EPACT 1992 [132]) specifies the minimum color rendering indices, which are discussed in the next section, and the minimum efficacy for common lamps. Color rendering depends solely on the spectrum of the source. Thus, the whitelight spectrum generated from LEDs needs to meet requirements of efficacy and color rendering. Desirable color rendering is best achieved by broadband spectra distributed throughout the visible region, uniformly as perceived by eye. However, the efficacy is best achieved by a monochromatic radiation at 555 nm (green), the wavelength where the human eye response reaches its maximum for daylight vision as discussed in Section 1.6. To a first extent, it appears that there is a trade-off between the two important criteria for white light: high-quality color rendering and high efficacy. For example, a low-pressure sodium lamp used in some highways and parking lots, having a light orange color, has an efficacy of about 200 lm W1, which is the highest among the available discharge lamps, but colors of objects are not distinguishable. Objects such as a red car not having orange pigments would appear gray. However, a xenon arc lamp, having a very similar spectrum as daylight and exhibiting excellent color rendering, has an efficacy of only 30 lm W1. The evaluation method for color rendering of light sources is well established by CIE, and since 1965, the color rendering index [133] has been widely used in the lighting industry. A succinct description of the fundamentals of the CIE colorimetric system [36] including the color rendering index is given along with a discussion of applications to white-light generation. The definitions of the terms in photometry and colorimetry used in this section follow those found in Ref. [134]. For further details of colorimetry, an overview of the CIE system of colorimetry is available in an article by Ohno [135]. 1.14.2 Color Rendering Index

Color rendering of a light source is characterized by comparing the appearance of various object colors under illumination by the particular light source versus that under reference illumination, daylight for correlated color temperature (CCT), > 5000 K, and Planckian radiation for correlated color temperature (CCT), < 5000 K. The smaller the color differences of the object colors from those determined under reference source, the better the color rendering. The standardized method, the CRI, is very well outlined by the CIE [133]. In this method, 14 Munsell [136, 137] samples of various colors, spectra for eight are given in Figure 1.65, including several saturated colors, are carefully selected and the color differences, denoted as DEi, of these color samples under the test illumination and under the reference illumination are calculated on the 1964 W U V uniform color space [133]. The process incorporates corrections for chromatic adaptation. Then, the special color rendering index, Ri, for each color sample is calculated using Ri ¼ 100  4.6DEi. Here, a figure of 100 represents the best color rendering index. The Ri value is an indication of color rendering for each particular color. The general color rendering index, Ra, is given as the average of Ri for the first eight color samples that have medium color saturation. With the maximum value of 100, Ra gives a scale

1.14 Luminescence Conversion and White-Light Generation with Nitride LEDs Munsell samples

Reflectance factor

0.8

0.6

8 1 7

0.4

6

0.2 0.0

2 4 5

400

500

600

3

700

Wavelength (nm) Figure 1.65 Munsell samples spectra for determining the color rendering index. Courtesy of M. E. Coltrin of Sandia National Laboratory. (Please find a color version of this figure on the color tables.)

that matches well with the visual impression of color rendering of illuminated scenes. For example [135], lamps having Ra values greater than 80 may be considered to be high quality and suitable for interior lighting, and Ra values greater than 95 may be suitable for visual inspection purposes. Thus, the spectral distribution of whitelight-generating LEDs should be designed to achieve the Ra value required for the application in mind. For comparison with conventional light sources, the CRI values (Ra values) of several common types of fluorescent lamps and HID lamps are tabulated in Table 1.4. To gain an insight about the extent of color saturation, the LED output is generally indicated on the chromaticity diagram, as depicted in Figure 1.66 (see also Figure 1.20). The oval near the center indicates various grades of “white light.” The line Table 1.4 General CRI of common lamps.

Source type

CCT

General color rendering index (Ra)

Daylight Cool White White Warm white Cool white deluxe Warm white deluxe Metal halide Metal halide, coated Mercury, clear Mercury, coated High-pressure sodium Xenon

6430 4230 3450 2940 4080 2940 4220 3800 6410 3600 2100 5920

76 64 57 51 89 73 67 70 18 49 24 94

The first six entries represent various types of incandescent bulbs listed here for comparison [129].

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90

0.9 520 530

0.8

InGaN Green LED 510

0.7

550 Y el

GaP Green LED

low

Green

en

y-Color coordinate

-gre

0.6

(Y 1-x,Gd x)3 (Al 1-y,Ga y) 5O 12:Ce

AlInGaP Green LED

570

500

0.5 Yellow 590 3000K

0.4

2500K 5000K

Blue-green

k pin

White

0.3

490

6000K

es ad Sh

o

ur fp

Orange

epl

620

AlGaAs Red LED

Red-Purple

0.2

700

Red

Blue Purple

0.1 470 InGaN Blue LED 450

0 0

0.1

440

0.2

0.3

0.4

0.5

0.6

0.7

x-Color coordinate Figure 1.66 The chromaticity diagram along with available commercial LED performance data. Clearly, blue and green InGaN LEDs constitute the two important legs of the triad, the three primary colors that are needed for full-color displays. Moreover, the output of an optically pumped YAG medium doped for yellow

emission is shown with data points indicative of various Gd concentrations. The broken line that connects the blue LED to one particular composition of the YAG medium indicates the range of warm white colors that can be obtained. (Please find a color version of this figure on the color tables.)

through the white-light region indicates the color diagram for white light with the accompanying color temperature (Planckian locus). The narrower the output spectrum of an LED, the closer its color is to the outer periphery. As the spectrum gets wider, the corresponding color on the chromaticity diagram is pulled toward the center, reducing the range of colors that can be obtained by the color-mixing scheme. Moreover, the output of a commercially available white LED constructed from a blue InGaN LED overcoated with a yellow light under blue photoexcitation emitting cerium (Ce)-doped yttrium aluminum garnet (YAG:Ce) [Y3Al5O12:Ce3þ (4f 0 )] inorganic phosphor (YAG) is marked with data points indicative of various Gd concentrations (see Section 1.15.3). The blue and red LEDs available commercially are almost saturated while the same cannot yet be said for green ones. The spectral broadening observed for green LEDs is attributed to compositional inhomogeneities, which get larger with increasing InN mole fraction. When used in conjunction with the available red and blue LEDs, present InGaN green LEDs provide the means for achieving some 70–80% of all the color possible.

1.15 Approaches to White-Light Generation

1.15 Approaches to White-Light Generation

An advantage of LEDs is that they are available in most wavelengths in the visible region of the electromagnetic spectrum, and the output spectrum from LEDs may be more flexible than that for the traditional discharge light sources whose output spectra depend on available phosphors and emissions from gas. In the case of multiple-chip LEDs, white light can be achieved by a mixture of one or more LEDs, with the aid of phosphors when needed, with different emission wavelengths. Combining the spectra from three-LED, particularly, four-LED chips could provide the best color rendering index, making the multicolor approach quite acceptable for general lighting. But the downside of this approach, in addition to the cost, is that it requires absolute control over the wavelengths of all the LEDs, relative power levels, and also the spectral width of emission. For example, in the four-LED approach, the relative intensities provided by blue, green, yellow, and red LEDs would have to be 14, 23, 22, and 41%, respectively, for a color rendering index in the high 90 percentile range. The red color power needs to be the highest because of the much reduced eye sensitivity to red color. The width of the emission spectra and the control of the emission wavelength that are available from many of the LEDs may not be sufficiently narrow in many cases when the wavelength accuracy to a fraction of a nanometer and bandwidth to a nanometer might be needed. White light generated by LED/phosphor combination is the least expensive and can also have reasonable color rendering (in the 70 percentile range) because the phosphors generally produce broadband radiation. Let us delve into generation of white light using what is dubbed the SSL-LEDs suitable for highquality general illumination. To summarize, for a CCT of 3000 K CRI values of 85, 97, and 99 can be obtained with three-, four-, and five-LED approaches, respectively. It should be noted, however, that there is a trade-off between the color rendering index and luminous efficacy in that the higher the color rendering index desired the lower the efficacy. This means that while the approaches with more color sources provide better color rendering index, the efficacy is lower. In addition, to attain very high color rendering index, wavelength accuracy to a fraction of nanometers and linewidths of about 1 nm are required. The narrower linewidths give slightly better luminous efficacy without a penalty in CRI. 1.15.1 White Light from Three-Chip LEDs

The three primary colors, red, green, and blue with identical intensities perceived by the eye, can be mixed together to generate white light. With currently available LEDs, the generation of white light can have luminous efficacies of approximately 45 lm W1. An example of a white-light spectrum produced by combining the outputs from a three-color multiple LED is shown in Figure 1.67. Note that because the human eye’s reduced response to the red color, the intensity of the red color LED

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92

1.4

605nm

Intensity (a.u.)

1.2

540nm

1.0 0.8

455nm Photopic curve

0.6 0.4 0.2 0.0

400 450 500 550 600 650 700

Wavelength (nm) Figure 1.67 White-light output emission spectrum from a threecolor multichip LED. Courtesy of M. E. Coltrin of Sandia National Laboratory. (Please find a color version of this figure on the color tables.)

must be higher to generate white light with acceptable color rendering index. LEDbased white-light sources have been in commercial production and consist of collectively housed LED chips, or arrays of different colored LED lamps, that is, multiple-chip LEDs. This three-color approach is a potentially very efficient, highquality white lighting approach, but the cost is expected to be high as it involves InGaN and AlInGaP technologies with each LED requiring different drives. This particular approach has a few problems. The perceived color may change with viewing angle due to the discrete wavelengths of light used. Multiple-LED chip requirements make this approach relatively expensive. Obtaining a consistent color across an array of such white pixels could also be a source of problem because the light intensity of LEDs and driving voltages tend to vary from diode to diode, and colortuning individual diodes is likely to be difficult. Temperature dependence of wavelength shift for each of the three diodes used may be different, causing color variation with temperature. Another consideration is the variation in operating life and/or degradation rate of different color LEDs. Because the intensity variations also lead to overall color change, uneven degradation of the three LEDs would lead to color change over time. For example, the light output level of AlGaAs-based LEDs is found to decrease by about 50% after 15 000–40 000 h of operation. This effect represents a serious challenge for multiple-chip LEDs where the white-light color rendering is critically dependent on the relative intensities of the separate red, green, and blue colors. However, multichip SSL-LEDs could offer the greatest versatility and the largest efficacies of all the approaches that are discussed here. When and if perfected, these approaches could potentially produce any color and any color temperature with as high a color rendering index as desired. The advantages can be summarized as follows: long-term and the most efficient, dynamic tuning of color temperature, excellent color rendering, and very large range of colors are available. The challenges include the following: vintage 2002 LEDs require color feedback to account for LED degradation with temperature and time (three different color LEDs respond differently), color mixing is somewhat involved, and there may be a little gap in the

1.15 Approaches to White-Light Generation

0.9

CIE spectrum locus Planckian locus LEDs (115, 25 oC) YAG:Ce Illuminants A, and B Illuminant D65

520 530

0.8

540

510

0.7

Green

550

y-Color coordinate

560

0.6

570

500

Yellow

0.5

580 590

0.4 0.3

Blue-green

3000 K 5000 K 10 000 K

490

0.2 0.1 0.0 0.0

White

Blue 480

600

2000 K

610

Red

640

20 000 K

Purple

470 450

0.1

0.2

0.3 0.4 0.5 0.6 x-Color coordinate

0.7

0.8

Figure 1.68 Mixture of two LEDs at 485 and 580 nm each with a half-maximum width of 20 nm. The (x, y) chromaticity coordinates of the resulting white light is shown as a solid diamond [129].

yellow–green region. In time, with further advances in LED technology, this issue would be mitigated. The chromaticity coordinates (x, y), CCT, CRI, and the luminous efficacy of radiation can be calculated if the spectral power distribution of a light source is known [129]. The CRI (Ra) of the white light produced by mixing the outputs from two 20 nm half-bandwidth LEDs shown in Figure 1.68 is calculated to be only about 4, which is very small. Consequently, standard two-chip LEDs with any combination of wavelengths cannot produce white light with an Ra value that is acceptable for general lighting applications. As discussed later in this section, the three-LED chip approach and in particular the four-LED approach (discussed in Section 1.15.2) are capable of producing much better color rendering, but the selection of peak wavelengths is critical in order to produce a CRI acceptable for general lighting. Simulations of three LEDs having peak wavelengths of 450, 550, and 650 nm, with their relative power adjusted to create white light with a color temperature of 4000 K, result in a color rendering index value that is not acceptable. Each LED is modeled using a Gaussian line shape function [138], with a half-bandwidth of 20 nm. In this case, the CRI (Ra) is only 37 with luminous efficacy of 228 lm W1 (theoretical maximum). Again, a value of Ra ¼ 37 is not acceptable for use in general lighting, except for limited outdoor use [129]. In terms of the three-LED solution, when the wavelengths of each of the three LEDs in the three-LED chip set are optimized and controlled precisely, general color rendering index can be improved substantially [129]. Figure 1.69 shows the result of a simulation of a three-LED chip set with peak wavelengths of 459.7, 542.4, and 607.3 nm leading to Ra ¼ 80 and a luminous efficacy of 400 lm W1, which represents

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94

1.2

x

0.3812

1.0

y

0.3795

0.8

CCT(K)

4011

0.6

∆ uv Ra

0.001

0.4 0.2

80

Efficacy(lm W-1) 399

0.0 400

600

500

(a)

(b)

700

Wavelength (nm)

0.9 Spectrum locus Planckian locus Illuminant A Illuminant D65 Mixture Blue Green Red

520 0.8

530

Green 510

540

0.7

550

0.6

560 Yellow 570

500 0.5

y

580 590

Blue-green

0.4

00

480 0.1

K

0K 00 00K 0 20

0.2

P ur

Blue

0K

10

490

20 00K

30 0 50

0.3

600 610 640

1000K Red

p le

470 0 (c)

0.0

0.1

0.2

0.3

0.4

x

0.5

0.6

Figure 1.69 The effect of optimizing the wavelengths of a three-chip white-light LED. (a) The spectral response of the three LEDs used for mixture. (b) Correlated color temperature, color rendering index (Ra), efficacy, and (c) the chromaticity coordinates (x, y) for the white light

0.7

0.8

are shown as the solid square near 4000 K. Because of the CRI, Ra of about 80, this is an example of white light with good color rendering [129]. (Please find a color version of this figure on the color tables.)

the theoretical maximum. A 20% LED chip efficiency leads to a total efficacy of 80 lm W1, comparable to typical fluorescent lamps [129]. This combination is acceptable for general lighting including indoor applications. Continuing on with the three-LED solution, the LE and CRI values for a wide range of wavelength combinations from GaInN blue and green LEDs and AlGaInP red LED (the trichromatic approach) have also been calculated by Chhajed et al. [139] in order to find the best wavelengths for a trichromatic source. These values in the form of contour plots for a trichromatic Gaussian white-light source with linewidth of 5 and 8 kT at 300 K for different wavelength combinations are shown in Figures 1.70 and 1.71. The 5 and 8 kT linewidth examples are shown to make the point how sensitive this approach is to not only the control of the wavelength of sources but also their linewidths. If the linewidth (DE) of emission in each LED is assumed to be 5 kT at 300 K, a luminous efficacy of 318 lm W1 and a CRI of 86 with the wavelength

1.15 Approaches to White-Light Generation

Figure 1.70 Contour plots for LE of radiation and CRI of a trichromatic Gaussian white-light source with linewidth of 5 kT at 300 K for different wavelength combinations. Courtesy of E. F. Schubert [139]. (Please find a color version of this figure on the color tables.)

combination (455, 530, and 605 nm for blue, green, and red, respectively) can be realized. If, however, a light source linewidth of DE ¼ 8 kT is assumed, a luminous efficacy of 300 lm W1 and a CRI of 93 can be obtained using the wavelength combination (455, 530, and 610 nm). In addition, the calculation showed a rapid decrease of the CRI value for even small deviations from the optimum wavelengths. Therefore, the control of the high luminous efficacy together with high-CRI white light requires an extremely precise degree of control over the wavelengths and the linewidths of the three LEDs forming the backbone of the trichrome approach. In reality, however, the efficiency of an LED is different for different wavelengths, and the availability of various wavelength-emitting LEDs is restricted. Despite these restrictions, optimum designs of white-light LEDs with available color LEDs for any desired CCT can be made. A four-LED chip set should give even better color rendering. Setting difficulties aside, if progress in other approaches is not made, this method may be interesting for producing white light with acceptable color rendering index. First, use of more colors leads to better control of white light with a high color rendering index in addition to being able to obtain millions of colors. Second, photons from each of the LEDs contribute directly to the white-light intensity, eliminating the need for photon conversion media and associated conversion losses.

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Figure 1.71 Contour plots for LE of radiation and CRI of a trichromatic Gaussian white-light source with linewidth of 8 kT at 300 K for different wavelength combinations. Courtesy of E. F. Schubert [139]. (Please find a color version of this figure on the color tables.)

Third, by changing the relative intensity of the different color LEDs, it is relatively easy to change the color and hue of this kind of light source for different applications. To reiterate, for this method to work well, the separate colors from the individual components must be mixed appropriately to achieve uniform white light keeping human eye response in mind. Substantial effort is required for the multichip solution to achieve over 200 lm W1 white light and a power conversion efficiency of approximately 50% inclusive of color-mixing losses, which is formidable. Despite the difficulties, consideration of this approach continues to the system level as graphically depicted in Figure 1.72 under a study for the US Department of Energy by Navigant Consulting, Inc. for the year 2005 with projected targets. The diagram assumes a correlated color temperature target of 4100 K (the equivalent CCT of a cool white fluorescent lamp), and a CRI of at least 80. The year 2005 LEDs typically had color temperatures in the range of 5000–6000 K, and usually a lower CRI that assumed. Attaining the stated target values (goals) will require more efficient LEDs across the most efficacious part of the spectrum (particularly green emitters) and greater improvements elsewhere in the system than those indicated. To reiterate, further complications arise as three or more different color components required have different voltage requirements, degradation characteristics and temperature dependencies both of which would change the color, necessitating

1.15 Approaches to White-Light Generation

2005 Target

2005 Target 1

75%

2 17%

85%

66%

1 Driver

2

LED

2a

IQE

2b

EE, χ EQE

70%

Total luminaire

3

90%

9% 50%

2c

Electrical efficiency

2d

Color mixing loss

Fixture 3 & optics

Lumens out

Figure 1.72 The year 2005 and target system efficiencies using the trichrome LED colormixing approach in which the target assumes a CCT of 4100 K and CRI of 80. The year 2005 CCT and CRI used are 5000–6000 K and 75, respectively. IQE, internal quantum efficiency; c, extraction efficiency; EQE, external quantum efficiency, which is a product of the internal

60%B 2a 20%G 80%R 50%B 2b 50%G 50%R 30%B 10%G 40%R

90%B 90%G 90%R 90%B 90%G 90%R 81%B 81%G 81%R

2c 80%

90%

2d 80%

90%

EQE

LED

Total white LED

Electrical power

17% 66%

quantum efficiency and the extraction efficiency; B, G, and R are for blue, green, and red, respectively. Patterned after NGLIA LED Technical Committee report prepared for the US Department of Energy “Solid-State Lighting Research and Development Portfolio,” March 2006.

sophisticated control systems. Although on the surface this looks like an approach that could provide very good color rendition, in practice it would be very tricky. For demonstration purposes, LED solutions to replicate the spectrum of a lighting industry standard CIE D65 illuminant have been developed, in particular by Lumileds/ Philips Lighting with the nomenclature of “light box.” In all LED approaches, RGB, amber, and white LEDs were used together to reproduce a spectrum closely matching that of the D65 illuminant in the spectral range of 420–650 nm. A correlated color temperature of 6705 K and a color rendering index of Ra ¼ 96 have been obtained. The spectra of the D65 standard illuminant and LED set are shown in Figure 1.73. In short, this method requires narrow FWHMs from each of the three LEDs, which are not available with nitride LEDs, particularly green, yet. The voltage requirements for the three LEDs are widely different, making the control system complicated. Furthermore, the aging characteristics for each of the three LEDs may be different. Consequently, a single UV LED pumping a tricolor phosphor option is preferable providing that phosphors for two or three primary colors, which can be efficiently pumped by a near UV or blue GaN LED, can be developed. Good progress is being made along these lines both on the LED front and on the phosphor front. 1.15.2 White Light from Four-Chip LEDs

It is clear by now that the higher the number of LEDs employed, the better the color rendering index and/or the luminous efficacy as they are related. Let us now discuss

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Intensity (a.u.)

CIE D65 Illuminant LED light box

380

480

580

680

Wavelength (nm) Figure 1.73 Spectra of the CIE standard D65 illuminant and RGB, amber, and white LED set showing good representation in terms of the correlated color temperature (6705 K) and a color rendering index of Ra ¼ 96. Courtesy of Lumileds/Philips.

the four-LED approach for white-light generation as shown in Figure 1.74a. Also shown, Figure 1.74b, are the wavelengths of four LEDs required for a range of luminous efficacy and CRI combinations covered in Figure 1.74a. As Figure 1.74a indicates, the maximum luminous efficacy drops with increasing CRI as expected. Also, 408 lm W1 is the maximum luminous efficacy that can be expected for a four-color SSL with CRI ¼ 90 (assuming 100% wall-plug efficiency for

408 lm/W

400 350

300

Red

Yellow

450

(b) 450

Green

(a)

Blue

Luminous Efficacy of Radiation (lm/W)

Luminous Efficacy (lm/W)

500

500

400 350

300 0

20

40

60

80

100

400

Color Rendering Index Figure 1.74 The luminous efficacy and color rendering index compromise (a) and the wavelength requirements for each of the four LEDs (b). Courtesy of M. E. Coltrin, Sandia National Laboratories.

500

600

Wavelength (nm)

700

1.15 Approaches to White-Light Generation 1.5

Relative Power

614 nm

530 nm

573 nm

1.0

463 nm

0.5

0.0 400

450

500

550

600

650

Wavelength (nm) Figure 1.75 The wavelengths and relative power levels required from each of the LEDs for the four-LED white-light generation with high color rendering index and luminous efficacy. Courtesy of M. E. Coltrin, Sandia National Laboratories.

semiconductor sources). The wavelengths required for the blue, green, yellow, and red LED sources in this scheme are 463 nm (B), 530 nm (G), 573 nm (Y), and 614 nm (R), as shown in Figure 1.75, with the relative required power level from each LED. Small differences in the wavelength of LED source can result in large changes in power fractions needed from each of the four LEDs. For example, for a CRI of 60, the power fractions for green and yellow need to be 15% (G) and 33% (Y). Increasing the CRI to 90 changes the power fractions to 23% (G) and 22% (Y). Clearly the wavelengths of the four LEDs are widely spaced across the visible wavelengths, 463 nm (B), 530 nm (G), 573 nm (Y), and 614 nm (R). In addition, the relative watt fractions for the LEDs are as follows: B, 14%; G, 23%; Y, 22%; and R, 41%. Such a premium demand on the red LED is due to the lackluster eye response to that particular color. Furthermore, the linewidths required are very narrow compared to the broadly varying Munsell samples (basis of CRI). The 1 nm linewidths (or 20 nm linewidths characteristic of LEDs) are nearly “d-functions” as compared to the Munsell sample spectra shown in Figure 1.65. The red wavelength is very important in the white-light generation in that shorter wavelengths improve luminous efficacy, as they are closer to peak eye sensitivity. However, a CRI of 90 is impossible for red LED wavelength l < 615 nm, as shown in Figure 1.76. Conversely, the longer wavelength improves CRI, but a luminous efficacy >286 lm W1 is impossible for l > 654 nm. If 100% efficient primary semiconductor LED sources are assumed, the 614 nm red corresponds to an efficacy of 408 lm W1, and 654 nm corresponds to an efficacy of 286 lm W1. If a more realistic 70% efficient primary semiconductor source is assumed, the 615 nm red is needed to reach an efficacy of 286 lm W1, but the AlInGaP efficiency drops as l gets shorter. We should mention that attaining blue, green, and yellow at >70% efficiency is a challenge, particularly the green.

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Figure 1.76 The sharp dependence of luminous efficacy on the wavelength of the red LED used in the four-color LED approach to white-light generation for a CRI of 90 and CCT of 3000 K. Courtesy of M. E. Coltrin, Sandia National Laboratories.

1.15.3 Combining LEDs and Phosphor(s)

This method for white-light generation from multiple-chip LEDs involves the use of one or more phosphors, such as the combined use of a blue LED and a yellow phosphor, which represents the least expensive solution and has seen considerable activity. This method is favored in that the technology already exists, is low cost, and requires small space. Already cool white with a color temperature of 5500 K and CRI of 70, and warm white with a color temperature of 3200 K and CRI of 90 have been reported to be available. In the approach called the RGGB, the red and blue primary semiconductor sources are used in conjunction with a green phosphor pumped by blue. An efficiency of 95%

1.15 Approaches to White-Light Generation

is assumed for the green phosphor (less the 15.4% Stokes loss), which is a very challenging goal. Narrower ranges of red wavelengths can be used with efficiencies of 80% (615 nm red) or 90% (626 nm red) to reach 286 lm W1. Broad linewidths are needed for the green phosphor (50–75 nm) as the red wavelength increases; the CRI improves to makeup for the “missing” short-wavelength red. The broad green phosphor used in this approach replaces the green and yellow LEDs in the fourLED approach. In the RRGGB approach, the blue primary semiconductor source is used to pump the green and red phosphors. An efficiency of 95% is typically assumed for both phosphors with Stokes losses: 24.2% (red) and 15.4% (green). Owing to the properties of the red phosphor, a very narrow range of red wavelengths is allowed for >286 lm W1. However, one needs a blue efficiency of 90% (if 615 nm red is used) or 100% (if 625 nm red is used) to reach the 286 lm W1 luminous efficacy mark. The broad linewidths are needed for green phosphor (50–70 nm) as red wavelength increases. Narrow linewidths (1–20 nm) are needed for red phosphor pumped by blue. The irony is that no phosphor system meeting these specifications is available with developments reportedly in place. Figure 1.77 shows the interrelationship between the linewidth, wavelength, and luminous efficacy, underscoring again the importance of the red wavelength. We discuss the details, particularly in terms of the phosphors, later in this section following the discussion of the BYY approach in which a blue LED source is used to pump a yellow phosphor. A more simplified approach wherein a blue LED source is used to pump a yellow phosphor (as opposed to green and red phosphors shown in Figure 1.77), termed the BYY approach, has gained considerable interest despite its low color rendering index for outdoor lighting in particular. Commercially available white LEDs are constructed from a blue InGaN LED overcoated with a yellow light emitting (under blue photoexcitation) cerium (Ce)-doped yttrium aluminum garnet (YAG:Ce) [Y3Al5O12:Ce3þ (4f 0 )] inorganic phosphor. These are called the white pcLEDs. In this approach, the InGaN LED generates blue light at a peak wavelength of about 460–470 nm, which excites the trivalent cerium Ce3þ:YAG phosphor that emits pale-yellow light, centered at about 580 nm with a full-widthat-half-maximum linewidth of 160 nm. The combination of the blue light from the LED, which is transmitted through the phosphor, and the pale-yellow light from the Ce3þ:YAG results in soft white light with a color temperature of, for example, 4600 K, as shown in Figure 1.78 for several drive currents. The emission spectrum of the YAG phosphor can be modified (tuned) by substituting some or all the yttrium sites with other RE elements such as gadolinium (Gd) or terbium (Tb). The RE3þ:YAG emission and absorption spectrum can be further engineered by replacing some or all of the aluminum sites by gallium. Instead of illuminating inorganic phosphors such as RE3þ:YAG, the blue-light emission from the InGaN LED can also be used to generate luminescence from organic polymers that are coated on the domed epoxy encapsulate of an InGaN LED lamp, but this approach has not made it to the marketplace yet and may never do so, owing to the success of the approach using solid phosphors.

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Figure 1.77 The relationship between the linewidth (top), wavelength (center), and luminous efficacy (bottom) in the BGGRR approach wherein a blue LED source with a halfwidth of 1 nm is used to pump green and red phosphors to achieve white light. Courtesy of M. E. Coltrin, Sandia National Laboratories. (Please find a color version of this figure on the color tables.)

The efficacy of the properly packaged (for heat removal as well as light extraction) production of phosphor-white LEDs was around 60 lm W1 with half lifetimes of 100 000 h in the year 2005 (the figures improved considerably by a factor of nearly 3 for efficacy by the year 2008). When compared to multiple-chip LEDs for red, green,

1.15 Approaches to White-Light Generation

1.4 10 -3 1.2 10 -3 1 A, 4600 K

W nm-1

1.0 10 -3 8.0 10 -4 6.0 10 -4 4.0 10 -4 2.0 10 -4 0.0 10 0

100 mA

380

480

580

680

780

Wavelength (nm) Figure 1.78 Emission spectrum of a white single-phosphor conversion LED, pcLED, (blue þ YAG:Ce) showing the blue peak of light leaking through the phosphor and the broader yellow peak from the phosphor. In a well-designed pcLED, the spectrum changes negligibly with drive current. The color temperature (CCT) stays at 4600 K perfectly dimmable. Courtesy of Lumileds/ Philips.

and blue color outputs, an advantage of the phosphor-white or hybrid-white phosphor conversion LED is that it only requires one blue LED (or ultraviolet, in which case the blue light must be emitted by another phosphor which would lower the efficiency). Also, conversion efficiencies of about 90% are possible in inorganic YAG-based converters without the bounds imposed on the active layer composition. In addition, white-light LEDs based on phosphors have been shown to be relatively insensitive to temperature, which is very desirable. While simple, there are several technological problems with this approach, at least with early designs. Among them is a halo effect of blue/yellow color separation due to the different emission characteristics of the LED (directional) and the phosphor (isotropic). Moreover, as mentioned above, the color rendering index is low, only about 75–85 for the cool white, and broad color “bins” are necessary to ensure reasonable product yields. Finally, most lamps have color points that do not lie on the blackbody curve, which is undesirable and eventually a color shift from blue to yellow with aging and variation in drive current is noted. In addition to the aforementioned challenges that mainly deal with the excitation source, there are other challenges associated with the phosphor material. White pcLEDs with blue LED and yellow phosphor for solid-state illumination have seen considerable advances of late, so much so that they are beginning to be competitive in some areas of not so stringent lighting. We already displayed the emission spectra of the yellow phosphor and blue LED combination versus the drive current in Figure 1.78. Note that the wavelength for the LED up to a current drive of 1 A remains unchanged. Shown in Figure 1.79 is the lumen output of a single-chip pcLED together with its efficiency versus drive current under pulsed (duty cycle of 1%) and CW conditions [104]. Color parameters of 4200 K for CCTand 75 for

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Efficiency pulsed

Intensity pulsed

Intensity CW

100

60

White, 4200 K

50

80

40

60

30 40 20

Intensity (lm)

Efficiency (lm W-1)

Efficiency CW

20

10 0.0 0

200

600 400 Current (mA)

800

0.0 1000

Figure 1.79 Injection current dependence in a white phosphor conversion LED, pcLED, (blue þ YAG:Ce) of photometric flux (lm) and luminous efficiency (lm W1) under 1% duty cycle pulsed and CW (DC) drive have reached values, which surpass incandescent and halogen sources. Courtesy of Lumileds/Philips.

Ra have been obtained. At the nominal operation point of 350 mA (50 A cm2), more than 32 and 35 lm/device have been achieved. This compares favorably with figures of 6–15 lm W1 offered by incandescent lamps, and 25–30 lm W1 offered by small halogen lamps. In standard one LED/one phosphor solution (pcLED), the thickness of the yellow phosphor through which the blue LED light travels changes the tint by changing absorption and therefore the yellow emission. This means that the tint depends on the viewing angle for the construct shown in Figure 1.80a. Conformal coating of the LED die with the yellow die shown in Figure 1.80b is designed to mitigate the “tint” change problem. The viewing angle dependence of the normalized CCT, with respect to on axis, for the phosphor slurry method of Figure 1.80a and conformal phosphor coating method of Figure 1.80b is shown in Figure 1.81, where a change of as large as 700 K in the CCT for the slurry case as opposed to 80 K for the conformal coating case is noted. The flux that can be obtained in the pcLED combination is given as f ¼ hint hextr hv e0;ph hQD hph hpkg P;

ð1:35Þ

where hint is the internal quantum efficiency, hextr is the photon extraction efficiency (% photons extracted per photon generated), hv is the electrical efficiency (photon energy divided by the injection energy), e0;ph is the luminous efficacy of phosphor/ LED combination in terms of lm W1, hQD is the quantum deficit due to Stokes shift (in terms of %), hph is the phosphor quantum efficiency, hpkg is the package photon extraction efficiency, and P is the electrical power applied (W). Improvement of the internal quantum efficiency depends on the materials quality and quantum well design. The extraction efficiency depends on the use of generated photons and can be

1.15 Approaches to White-Light Generation

Figure 1.80 (a) Conformal coating of the yellow die (b) versus the standard coating to avoid color tint dependence on the viewing angle. Courtesy of Lumileds/Philips.

improved by chip and package design. The phosphor quantum efficiency is dependent on the phosphor material and can be improved by progress in the phosphor science. Finally, the level of electrical power that can be applied depends on the chip and package design. All of these components are on the table for improvement for

CCT (normalized to on axis)

200 100

Conformal coating method

80 K CCT range

0 -100 -200 -300 -400 -500 -600

phosphor slurry method

-700 700 K CCT range -800 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90

Viewing angle, θ (degrees) Figure 1.81 Normalized (with respect to on axis) viewing angle dependence of the CCT in LED þ phosphor slurry construct and LED þ phosphor conformal coating construct. A viewing angle dependence by as much as 700 K in CCT in the slurry design versus only 80 K in the conformal coating method is noted. Courtesy of Lumileds/Philips.

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2005 Target LED

2005 Target 85% 90+%

2 13%

56%

1 Driver

2

LED

2a

IQE

2b

EE, χ

2c

Total luminaire

3 70%

90%

8% 45%

Fixture 3 & optics

2b 60%

90%

30%

81%

EQE

2c 80%

90%

Electrical efficiency

2d 70%

85%

2e 80%

90%

2d

Phosphor

2e

Scattering loss

Lumens out

Figure 1.82 The year 2005 and target system efficiencies using the phosphor conversion LED color-mixing scheme in which the year 2005 CCT and CRI used are 5000–6000 K and 75, respectively, and the target CCT and CRI used are 4100 K and 80, respectively. IQE, internal quantum efficiency; c, extraction efficiency; EQE, external quantum efficiency, which is a product of

90%

EQE

1

2a 60%

Total white LED

Electrical power

13% 56%

the internal quantum efficiency and the extraction efficiency; B, G, and R are for blue, green, and red, respectively. Patterned after NGLIA LED Technical Committee report prepared for the US Department of Energy “Solid-State Lighting Research and Development Portfolio,” March 2006.

increasing the flux obtainable from white LEDs based on the phosphor/LED combination. Consideration of the phosphor LED approach continues to the system level as graphically depicted in Figure 1.82 under a study for the US Department of Energy by Navigant Consulting, Inc. for the year 2005 with projected targets. The diagram assumes a correlated color temperature target of 4100 K (the equivalent CCTof a cool white fluorescent lamp) and a CRI of 75 for the target data. In the scenario, the driver (1) is assumed to have an efficiency of 85% for the year 2005. Ultimately, this can potentially be improved to a value greater than 90%. The other components in the system have more room for improved efficiency. Among them, the 2005 extraction efficiency of the LED chip was about 50%. The ultimate goal is to raise the extraction efficiency of the mounted, encapsulated chip to 90% with the use of thin-film vertical and flip-chip mounted LEDs. The areas with the greatest potential for improvement are the internal quantum efficiency, IQE (2a), and extraction efficiency, c (2b), of the LED chip, and the fixture and optics (3). In the year 2005, the phosphor conversion LED luminaire/system was approximately 8% efficient at converting electrical power into visible white light. If all targets stipulated in the projections are attained, the LED device (lamp) would be expected to have an efficiency of 56%, with a system efficiency of 45%. Similar to the trichromatic color-mixing approach, the electrical luminous efficacy (in lm We1) of the phosphor conversion LED approach can be calculated by multiplying the wall-plug efficiency (Wo/We) by the optical luminous efficacy

Optical spectral density (W nm-1)

1.15 Approaches to White-Light Generation 3300 K Luxeon spectrum 3300 K Blackbody spectrum

4 10 -4

3 10 -4

2 10 -4

1 10 -4

0 10 0 400

450

500

550

600

650

700

750

Wavelength (nm) Figure 1.83 Spectra of 3300 K blackbody radiation as well as that from a 3300 K Lumileds Luxeon white pcLED indicating close replication. Courtesy Lumileds/Philips.

(useful light out (lm)/optical power in Wo) of a phosphor. Similar to the trichromatic color-mixing LED approach, a practical goal for a phosphor converting LED lamp would be about 160 lm We1. Improving the phosphor efficiency and temperature performance could improve the efficacy even more. Various pcLED solutions replicating the blackbody radiation have been demonstrated. To reiterate, nominally white pcLEDs with 3200–3500 K correlated color temperature and color rendering indices greater than 90% have been attained. Shown in Figure 1.83 are the spectra of 3300 K blackbody radiation and those provided by 3300 K Luxeon lamp. Clearly, the replication of the blackbody radiation for this warm white is very good. Packaging has a sizeable effect on the operative lifetime of pcLEDs. For example, the white LEDs constructed in a 5 mm package exhibit degradation of more than 60% in the output power after 10 000 h of operation, whereas higher power Luxeon LEDs packaged in a high-power package with good heat sinking display much more gradual degradation. Some 80% of the output power is retained after 20 000 h of operation. As the materials’ quality is improved for better quantum efficiency, rugged fabrication procedures developed and employed along with packages with more efficient heat removal capability are being implemented, one would expect much longer lifetimes for a given degradation. Simply, as the lumens/W efficiency of the LED is improved along with advances in packaging, better lifetimes would result. Let us now discuss illumination based on a near-UV emitting chip (380–410 nm peak wavelength), and a blend of downconversion phosphors where the optical pump does not really form the blue primary color. A plethora of prior work in relation to fluorescent lamps is already available. Consequently, the white light generated by UV excitation should closely follow that of available fluorescent lamps, which utilize a triphosphor (red, green, and blue) blend with one important exception. Because there is a dearth of phosphor materials that are efficiently excited by a 380–410 nm source,

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particularly red- and green-emitting phosphors, it will be necessary to develop these materials. The human eye’s response to visible light (discussed in Section 1.6) suggests a lamp design departing from the traditional methodology of commercial incandescent lamps. The incandescent lamps try to replicate daylight as a continuum. Strong visual effects, such as higher perceived brightness per watt and better color rendering, can be attained if white light more closely resembles the three pure spectral colors while leaving the rest of the visible spectrum nearly empty, as in fluorescent lamps. Fluorescent lamps utilize phosphors as converters of UV emission from rare gas/ mercury discharge plasma into visible white light. In the 1970s, a blend of three phosphors called triphosphor or tricolor blend, which emits in the blue, green, and red spectral regions, paved the way for a new generation of white-light fluorescent lamps that simultaneously provided markedly high color rendering and high efficacy. The role of the phosphors in the triphosphor blend of the fluorescent lamp is to generate photons at wavelengths near the “three-peaked” spectral response of the human eye. The three narrow emission bands centered near 450 nm (blue component), 550 nm (green component), and 610 nm (red component) are the ideal “prime colors” for this purpose. The resulting white light has high efficacy and excellent color rendering. The individual phosphors used in the tricolor blend of a typical fluorescent lamp are listed in Table 1.5, while their respective emission spectra are shown in Figure 1.84. Note that the color temperature can be varied by changing the ratio of the power in the three components while restricting any changes in the peak wavelength emission of the three components The question here is whether a corresponding triphosphor blend, excitable at 380–410 nm, can be used to develop a phosphor-assisted white-light LED with high efficacy and good color rendering, as in the case of fluorescent lamps. Furthermore, with such a blend, when available, it should be possible to design white-light LEDs with a variety of color temperatures and with very high color rendering index. As indicated earlier, pumping schemes employing a source so far from the visible spectrum lead to high Stoke shift-related losses, degrading the luminous efficacy considerably. For this reason, the method described earlier wherein a blue source is used to pump green and red (not yet available) phosphors would be desirable.

Table 1.5 Phosphors used in the triphosphor blend of typical fluorescent lamps.

Phosphor 2þ

Eu :(Sr, Ba, Ca)5(PO4)3Cl Eu2þ:BaMg2Al16O27 (Ce, Tb)3þ:LaPO4 Eu3þ:Y2O3

Color

Emission bandwidth

Emission peak (nm)

Blue Blue Green Red

Broad Broad Narrow Narrow

450 450 543 611

Here, “Broad” and “Narrow” are defined loosely and intended to give a flavor of comparison. Phosphors get excited by 254 nm wavelength radiation [129].

1.15 Approaches to White-Light Generation

A

C

Intensity (a.u.)

B

400

450

500

550

600

650

700

Wavelength(nm) Figure 1.84 Emission spectra of phosphors of the triphosphor blend: A is Eu3þ:(Sr, Ba, Ca)5(PO4)3Cl (blue); B is (Ce3þ, Tb3þ): LaPO4 (green); and C is Eu3þ:Y2O3 (red) [129]. (Please find a color version of this figure on the color tables.)

Unlike the triphosphor blend used in fluorescent lamps, one of the issues with existing phosphors, excited by blue LEDs, is that the absorption in the blue by the available rare earth ions such as Ce3þ is low. This situation is reminiscent of the early days of the solid-state laser research using Nd3þ:YAG, where Nd3þ:YAG had a similar absorption problem when pumped by flash lamps, that is, not much absorption in the blue. It was later discovered that adding Cr to YAG increased the blue-light absorption and thus, through excited-state energy-transfer processes, energy is transferred from the Cr3þ ion to the Nd3þ ion, thereby increasing the overall efficiency. Similar ideas for increasing the blue absorption process for the triphosphor blend and/or other phosphors should be explored. Phosphors must strongly absorb at the wavelength of the LED radiation with absorption exceeding 90%; the intrinsic phosphor efficiency defined by the ratio of the emitted photons to absorbed photons must be high to lead to a quantum efficiency of 85% or higher; the phosphors should be compatible with operation in the LED and be easily manufactured; and the phosphors should display excellent lumen retention, which is defined as the change in lumens/brightness with time. It is imperative that the red phosphor employed displays an emission spectrum that has a narrow linewidth centered at or near 614 nm, requirements that are well met by trivalent europium Eu3þ:Y2O3, as shown in Figure 1.84. The quantum efficiency of the Eu3þ:Y2O3 phosphor for excitation wavelengths below 254 nm is close to unity and is the highest of all known phosphors used in lighting. However, photons with wavelengths of 380–410 nm are very poorly absorbed by this phosphor, which preclude this phosphor being used in a UV LED-based white-light source unless advances are made. Any future discovery of red emitters based on the Eu3þ luminescence for UV LED applications will invariably require sensitization of the medium for the following reason. The absorption of UV photons by the Eu3þ ion is via a charge-transfer transition involving the Eu3þion and the surrounding anions.

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It is known that the quantum efficiency of Eu3þ activated phosphors is low when the charge-transfer transition is centered at wavelengths longer than about 300 nm. Hence, for UV LED applications, where the phosphor is expected to absorb 380–410 nm radiation, sensitization of the Eu3þ luminescence is imperative. The particulars for green phosphors are different from those of red-emitting phosphors. Because the eye response to green light is of relatively higher luminosity, the narrowness requirement of the spectral emission in the green is relaxed. This relaxation allows for the identification of several candidates for green-emitting phosphors. The broadband emission of divalent europium ions (Eu2þ), which is via the 4f 6 5d ! 4f 7 optical transitions, is extensively tunable with emission wavelengths extending from the UV to red wavelength spectral regions. Moreover, the absorption by 4f 7 ! 4f 6 5d optical transitions usually extends throughout the ultraviolet. Although not favored because of large Stokes shift related losses, let us discuss the particulars surrounding blue phosphors as well for completeness. The Eu2þ-based blue phosphors absorb the UV LED radiation and emit at the required wavelength of 450 nm. However, the absorption for 380–410 nm radiation needs to be further improved by increasing the europium content in the phosphor formulation (recall that the phosphor composition has been optimized for absorption of the 254 nm radiation). The increase in europium concentration is limited by the efficiency loss arising from concentration quenching and by the high cost of europium. Nevertheless, for the specific application in UV LEDs, there are numerous subtle trade-off possibilities that require reoptimization of the phosphor composition. The advantages of white-light generation in the scheme where a UV LED is used to pump RGB phosphor(s) are as follows: white light is determined by the phosphors only and not subject to variation in the LEDs; they are simple to manufacture, at least in theory; they have a decent color rendering index (Ra ¼ 75); and the temperature stability of various phosphors is excellent. However, damaging UV leakage is of concern, the fundamental limit of conversion efficiency is determined by phosphors, there is a Stokes shift, and self-absorption is an issue. There are also challenges in that color uniformity is dependent on the angle at which the source is viewed, and packaging must be robust to UV radiation. In short, this approach uses output from a UV LED to pump several phosphors to simultaneously generate multiple colors. High color rendering indices, which are comparable to standard fluorescent lamps, can be realized. This approach also has the advantage of limited “tint” variation besetting the blue LED/yellow phosphor approach (unless conformal coating is used that mitigates the problem to some extent, see Figure 1.81). In addition, the ballast driver requirement is simple. Because the UV light is not used directly, it requires that the UV emitter efficiency be higher to account for conversion losses. In order to achieve 200 lm W1 white light, a power conversion efficiency of over 70% might be required for the UV LED. Other disadvantages include lower efficacy, need for new phosphor development, and potential UV packaging requirements, as alluded to below. Packaging is the holy grail of LEDs in general, and white lighting applications are no exception. Emulating the phosphor-coated fluorescent lamp glass tube application

1.15 Approaches to White-Light Generation

of a phosphor layer directly to the LED can be an efficient manner to enhance conversion. The individual phosphors may be dry blended and then added to a liquid suspension medium or they can be added to a liquid suspension, such as a nitrocellulose/butylacetate binder and solvent solution used in commercial lacquers. Other liquids including water with a suitable dispersant and thickener or binder such as polyethylene oxide can also be used. The phosphor-containing suspension can then be painted, or coated, or otherwise applied to the LED and dried. Alternatively, the phosphors can be combined to suitable liquid polymer systems, such as polypropylene, polycarbonate, or polytetrafluoroethylene, or, more commonly, epoxy resin or silicone, which is then coated or applied to the LED and dried, solidified, hardened, or cured. For optimum brightness, it is imperative that high-quality coatings with minimum defects are developed and that the phosphor blend about the chip is arranged in such a manner as to convert as much of the chip’s radiation into visible light as possible. Moreover, the absorption and reflection of the binder materials and the overall conversion efficiency have to be taken into account. Naturally, methods to minimize intrinsic efficiency loss are important. Packaging would not be complete unless it addressed the issues related to LED drive electronics, LED addressing, and “on-board” controllers for multichip LED systems. Before any commercial production implementation, circuit and electronic designs must be stable and possible to produce at low costs. Thus, if a low-cost replacement for fluorescent or incandescent lights is to be realized, the packaging issues must be addressed satisfactorily. In short, the issue dealing with the viewing angle can be addressed as has been done for the single-LED yellow-phosphor case by controlling the thickness of the phosphor. With progress in the phosphor technology, the UV light leakage may not be of a serious concern. Furthermore, UV LEDs are improving in their efficiency very well. It could be very likely that one LED/trichrome phosphor approach will end up being used for white light, with color temperature and rendering index that are acceptable providing, however, that conversion efficiencies of over 70% is achieved in the pump LED. Otherwise, the blue pumped two phosphor approach would be desirable as it does not suffer from as much stokes loss. 1.15.4 Other Photon Conversion Schemes

There are other photon conversion schemes for white-light generation. Among them are photon energy conversion techniques based on quantum effects using aggregates of small-sized (nanometer scale) semiconductor materials [140], photon-recycling semiconductor LEDs (PRS-LEDs), where a blue InGaN LED is joined to an AlGaInP top layer generating two complementary colors and hence white light [141], and exploitation of the high-power-narrow-bandwidth light output produced by UV lasers [142]. Another approach, in which the phosphor converter is eliminated, is to construct an integrated one-chip white LED by incorporating two InGaN epilayers emitting two wavelengths (blue and yellow) [143, 144]. Although not seriously considered, we discuss these approaches as well for the sake of completeness.

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Quantum size effects can be used to tune the wavelength of visible light with high quantum yields by manipulating nanometer-size semiconductor structures whose bandgap energy depends on the size of the nanoparticle due to the effect of quantum confinement of the electron–hole pair. This is an advantage for inorganic semiconductors as compared to traditional organic phosphors for lighting applications in that they are less likely to suffer degradation during electron or hole injection in electroluminescent displays. Many semiconductors have been used in the form of QDs to shift the wavelength of emission and GaN is no exception. However, II–VI semiconductors lend themselves to synthesis processes that are not expensive. A variety of techniques are available for producing quantum dots. One method is by molecular beam epitaxy (MBE), which can form the dots as well as coat them with a larger bandgap material. Due to the control and precision afforded by MBE, the early developments of quantum dots more or less had to be fabricated by MBE, but now that a good deal of understanding exists regarding the optical properties of quantum dots, other and less expensive techniques are gaining momentum. The most common approach to the synthesis of colloidal QDs is the controlled nucleation and growth of particles in a solution of chemical precursors containing the metal and the anion sources (controlled arrested precipitation) [145]. The controlled nucleation methods are easy to employ, do not require expensive capital equipment, and offer the quality needed for imaging [146]. As can be garnered, nanocrystals can be prepared at relatively modest temperatures [147]. Assuaging the process is the fact that the melting temperature drops with size as 1/r [148, 149]. The decrease in melting temperature with small sizes allows the synthesis of highly crystalline and faceted nanoparticles at temperatures compatible with wet chemical processing. Consequently, extremely high-quality inorganic nanoparticles have been prepared as colloids. Recent successes in the preparation of II–VI [150, 151] and III–V [152–154] (which are more difficult than II–VI dots) quantum dots illustrate the strengths of the colloidal preparation techniques. A common method for II–VI colloidal QD formation is rapid injection of a solution of chemical reagents containing the group-II and group-VI species into hot and vigorously stirred solvent containing molecules that can coordinate with the surface of the precipitated QD particles [145, 155]. As a result, a large number of nucleation centers are initially formed, and the coordinating ligands in the hot solvent prevent or limit particle growth via Ostwald ripening. Further improvement of the resulting size distribution in the QD particles can be achieved through selective precipitation [155], whereby slow addition of a nonsolvent to the colloidal solution of particles causes precipitation of the larger sized particles. This process can be repeated several times to narrow the size distribution of II–VI colloidal QDs to a small percentage of the mean diameter [155]. The II–VI semiconductors such as CdS have direct bandgaps and are of particular interest. Because high quantum yields of visible light are possible. For example, light emission intensities have been demonstrated from 3.0 nm CdS nanoparticles similar in photoluminescence intensity and position to those obtained from laser dyes such as Coumarin 500. Furthermore, the peak of the light emission can be shifted from about 430 nm to 700 nm by variation of both the size and the interface characteristics. The effect of the latter is demonstrated in Figure 1.85, where a coplot of the

1.15 Approaches to White-Light Generation

Absorbance (cm -1 )

4 CdS/ CdS ZnS

PL×10 -1

5

3

2

4 CdS/ZnS 3.5 3 2.5 2 CdS 1.5 1 0.5 0 350 400 450 500 550 600 650 700 Wavelength (nm)

1

0 200

300

400

500

600

700

800

Wavelength (nm) Figure 1.85 Plot of the UV absorbance and green photoluminescence spectra (inset) of nanosized CdS with and without a ZnS coating [129].

absorbance and fluorescence (inset) from a solution of CdS nanoparticles coated with a layer of ZnS to reduce nonradiative surface recombination (ZnS by itself has emission at 420 nm and is transparent to CdS emission) was shown. A redshift and enhancement of over a factor of 3 in the light emitted occurs due to the coating. The redshift is due to reduced confinement caused by ZnS, which does not provide an infinitely large energy barrier to electrons and holes, and increases in intensity are due to reduction of nonradiative processes. It is also possible to alter the peak energy and quantum efficiency by varying the excess ions at the nanoparticle interface. For example, intense blue–green emission at 488 nm occurs with an excess of Cd at the interface, while weaker red emission at 590 nm is observed with excess S at the nanoparticle surface. Thus, by variation of both size and interface chemistry, it is possible to obtain a wide range of output colors even with only a single semiconductor material. A critical challenge with these kinds of nanoparticles for LEDs is to improve the quantum efficiency. To date, an absolute determination of the energy conversion efficiency has not been made, but studying this phenomenon is attractive and may have a long-term payoff. While these dots have already found a niche in biological fluorescence labeling, and so on, LED applications using these dots have not yet been realized and their utility has not been demonstrated in the marketplace. As for the photon-recycling semiconductor LED approach, the maximum theoretical white-light efficacies using a blue InGaN LED wafer bonded to a sapphire substrate and a photon-recycling wafer (AlGaInP) are estimated [156] to be about 300 lm W1, with laboratory models exhibiting about 10 lm W1 of white light. Light generation using semiconductor lasers for excitation source of phosphors rather than LEDs may be advantageous. In one approach, remotely located UV lasers may excite phosphors that are painted on a wall to produce unusual lighting effects without any power connections in the wall. Unlike an LED, all of the 380 nm pump beam can easily be directed to the phosphor for photon conversion due to

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80

InGaNAlInGaPAlGaAsInGaAs Requirement for 150 lm W-1 RGB white

LED Vertical cavity Laser diode Edge -Emitting Laser diode

60 40 Fluorescent

20

Incandescent

0 400

500

600 700 800 Peak wavelength (nm)

900

Figure 1.86 Best power conversion efficiencies reported for laser LEDs across the visible spectrum. For comparison and benchmarking, the efficiencies of unfiltered incandescent light bulbs and fluorescent light bulbs are also indicated. The vertical arrows indicate vertical emitters while the horizontal ones indicating edge emitters. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

directionality of the beam. The best power conversion efficiencies reported for laser LEDs across the visible spectrum are shown in Figure 1.86. For comparison and benchmarking, the efficiencies of unfiltered incandescent light bulbs and fluorescent light bulbs are also indicated. The vertical arrows indicate vertical emitters while the horizontal ones indicate edge emitters. Again, in the quest to generate white light without the use of phosphor conversion, Huang et al. [157] achieved white light InGaN/GaN quantum wells LED structures using the prestrained MOCVD technique. This technique enhanced indium incorporation in InGaN/GaN QWs by generating a tensile strain in the barrier above the QW. White light emitted by the LED comprised two wavelengths (460 and 576 nm) from blue QW and yellow QWs. The color temperature at 50 mA injection was reported to be 5600 K. In yet another phosphorless approach, Wang et al. [158] used a strained relaxed InGaN underlying layer (UL) for white-light emission. This ULenhanced indium phase separation resulted in the formation of large-radius In-rich QDs in the InGaN QWs. These QDs emitted yellow light (570 nm) while blue light (440 nm) came from the rest of the same QWs. We should caution, however, that approaches of this kind do not lend themselves to the production environment due to poor reproducibility.

1.16 Toward the White-Light Applications

While the LED manufacturers are gunning for conventional illumination and they are not too far away, many lighting applications, which can be served by low-power incandescent or halogen lamps, can be served by LEDs as available or without much further improvement [128]. For instance, a 15 W incandescent bulb generates 120 lm,

1.16 Toward the White-Light Applications Table 1.6 White-LED efficiencies (and lifetimes) including

those that are of “50þ” lm W1 expected for year 2010. Lamp type

Power (W)

Efficiency (lm W1)

Lifetime (h)

Standard incandescent Standard incandescent Long life incandescent Halogen Halogen Compact halogen Compact halogen Standard fluorescent White LED 2000 White LED 2002 White LED 2005 Warm white LED 2005 White LED 2010 Green LED 2005 Blue LED 2005 Red LED 2005 Amber LED 2005

15 100 135 20 300 50 11 30 Any Any Any

8 15 12 12 24 12 50 80 20 30 45 20 100þ 53 16 42 42

1000 1000 5000 3000 3000 2500 10 000 20 000 50–100 kh 50–100 kh 50–100 kh 50 000 kh 50–100 kh 50 000 kh 50 000 kh 50 000 kh 50 000 kh

Any

Efficiency figures over 100 lm W1 have been reported for low injection levels with about 100 lm W1 being available at high injection levels in the development circles. This calls for better and defining standards of efficiency in place, as it is clear that higher efficiencies are obtained in smaller devices at lower injection levels. Courtesy of Lumileds/Philips.

while a 50 W compact halogen lamp generates 600 lm. In this low-flux range from 100 to 600 lm, incandescent and halogen lamps are relatively inefficient and the energy saving from using LEDs can be significant, particularly in cases when they are on during a significant portion of each day. Among these applications are shelf lighting, theater and stair lighting, accent lighting, landscape and path lighting, flashlights, and some aspects of underwater lighting. Let us briefly review the history of technologies developed for white-light generation to gain an understanding of the relevance of LEDs to the field. Just observing what goes on around us would immediately indicate that the last century of lighting has, by and large, been dominated by incandescent, fluorescent, and HID light sources (the latest performance of which are tabulated in Table 1.6). The first electrically powered light was discovered by Thomas Edison and Joseph Swan independently in 1879. The Edison incandescent light bulb utilized a carbonized sewing thread. The first commercial product, though, used carbonized bamboo fibers and operated at about 60 W for about 100 h and an efficacy of approximately 1.4 lm W1. The improvements that followed over time have raised the efficacy of the current 120 V, 60 W incandescent lamp to about 15 lm W1 for products with an average lifetime of 1000 h. Peter Cooper Hewitt patented the first low-pressure mercury vapor (MV) discharge lamp in 1901, which marked the first prototype of today’s modern fluorescent lamp. General Electric improved the original design and created the first practical

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fluorescent lamp, introduced at the New York and San Francisco World’s Fairs in 1939. Ever since that time, the efficacy of fluorescent lighting has improved and reached a range of approximately 65–100 lm W1, depending on lamp type and wattage, but the technology is very mature. Sir Humphry Davy obtained a glow from platinum strips by passing an electric current through them in 1801, and in 1810, he demonstrated a discharge lamp to the Royal Institution Britain by creating a small arc between two charcoal rods connected to a battery. This paved the way to the development of HID lighting. However, the first high-pressure MV lamp was not sold until 1932. Gilbert Reiling patented the first metal halide (MH) lamp in 1961. This particular lamp achieved an increase of lamp efficacy and color properties over the MV variety, which made it more suitable for commercial and street lighting. As in the case of the fluorescent bulb, the MH lamp was introduced at the 1964 World’s Fair. The first HPS lamp soon followed in 1965. The efficacy of HID lighting since that time has reached a range of approximately 45–150 lm W1, being highly dependent on lamp type and wattage. Getting back to the discussion on low-level lighting applications of LEDs and focusing on shelf lighting, nominally hot and bulky incandescent and halogen lamps are traditionally used on the underside of shelves to illuminate merchandise in many retail outlets. Fluorescent lamps require protection against the inherent high operating voltage. An LED-based solution is nearly ideal for this application with its cool, compact, efficient, dimmable, long operating life, and low-voltage lighting capability. Similar arguments can be made about theater and stair lighting, where the LED can effectively illuminate flights of stairs and gangways. The LEDs can be mounted into the stair steps or on walls. Wall-mounted units require a very directional beam, which is consistent with the superior directionality of LED-based designs, as a large fraction of the light from an incandescent light bulb in these kinds of applications is wasted. Accent lights are used in retail shops to highlight merchandise, while in homes the main application is decorative ceiling lighting or highlighting artwork. LED-based solutions in place of incandescent and halogen lamps will lower energy and maintenance cost, as well as reduce fire hazard. Directionality of LEDs is again desirable here. Landscape lighting provides esthetics or orientation in public places such as parks, gardens, or office grounds. Low-voltage operation and efficient operation by LEDs should reduce installation, maintenance, and operation costs. Flashlights are a perfect example of LED replacement for incandescent lamps in flashlights, which have chronically poor shock resistance, and many flashlights are discarded due to incandescent filament breakage let alone insatiable battery consumption by incandescent bulbs. The 40–60 lm that are needed for flashlights can easily be provided by an LED source adding only $0.05 lm1, which is quickly compensated for by extended battery life. Many applications of white LEDs, in fact in rapidly growing numbers particularly in handheld portable electronics, are in back-displayed illumination, which requires different packaging. A typical backlighting system consists of a light guide plate, into which the light is allowed to go in from the bottom side. The light guide plate is equipped with optical extraction features and/or a taper and a reflector on the backside to reflect the light from the light plate toward, for example, the liquid crystal

1.16 Toward the White-Light Applications

display (LCD) panel. If, however, an RGB all-LED array is considered, the light from the LEDs is mixed in the light guide plate to generate the white light. If LEDs of each color are connected serially and each type of LED is controlled, the color of the backlight can be changed by changing the respective intensities of each color element. The system can be operated in either open-loop or closed-loop configuration with the aid of sensors, which allow measurement of color white point and color temperature [159]. In the open-loop configuration, the color points of all the LEDs must be known a priori, as well as inclusion in the system control of the predetermined LED color shift with temperature and drive current. This would include having to take into account flux degradation of the LEDs over time. The closed-loop system relies on an optical feedback sensor, which in its most simple form consists of three Si diodes, each having a separate color filter. The color filters ideally would have the transmission curves of the X, Y, and Z color matching functions. In this scenario, the signals from the three diodes can be directly related to the actual white point of the backlight. In the backlight configuration depicted in Figure 1.87, R, G, and B LEDs are situated directly at the side of the light guide plate, with the drawback that the backlight region close to the edge has poor color and luminance uniformity [159]. This deficiency can be mitigated by employing a separate mixing light guide [160], as illustrated in Figure 1.88. In this configuration, the mixing light guide is positioned behind the main light guide plate, and an elliptical U-turn mirror is employed to reflect the light from the mixing plate to the extraction plate. The color uniformity of the light after mixing depends on the length of the mixing plate and the size of the white cluster in the RGB sequence. In Ref. [160], an LED pitch of 9 mm was employed, with a mixing plate length of 75 mm, to attain good uniformity. The slim

Feedback sensors Controller

Target color point

RG B Backlight B

R

G

B

R

G

Figure 1.87 White point control in a backlight with R, G, and B LEDs. The LEDs inject the light into the backlight, where the light is mixed to create white. The color point of the white is measured using three sensors with approximate the X, Y, and Z color matching functions. A

B

Power supply B R G

controller is used to compare the signals from the color sensors with those from the target white point. Courtesy of Lumileds/Philips. (Please find a color version of this figure on the color tables.)

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Figure 1.88 RGB LED backlighting configuration used, for example, in computer monitors and small LCD displays such as those found in handheld portable devices (handsets). A separate mixing plate is used to mix the light emanating from LEDs with the aid of a mixing

plate placed behind the main light guiding plate the purpose of which is to spread the light over the LCD panel. Moreover, a 180
U-turn mirror is used to reflect the light from the mixing plate out into the main light guide plate. Courtesy of Lumileds/Philips.

LED technology for LCD backlighting to produce very thin handset displays for office automation devices, and consumer electronics. The side-emitting microside LEDs stand 0.6 mm high and high-intensity surface-mount LEDs have been developed for larger backlight applications [161]. Due to production variables (tolerances), LEDs vary in color, linewidth, efficiency, and forward voltage required for a given current necessitating auto testing and sorting (grouping) of all LEDs. The sequence of placement of sorted LEDs must to be optimized for attaining high luminance and color uniformity, as discussed in a patent on the topic [162]. In the U-turn configuration shown in Figure 1.88, a color uniformity with color variation u0 v0 < 0.01 (depicting the maximum distance between the color points in the 1976 CIE diagram) can be achieved, with a total efficiency of 50% (the ratio of light out from the backlight over to the light out from the LEDs). The efficiency is limited due to the light guide plates, inevitably with absorption losses, and the losses involved in coupling the light into the plate. If thickness is less of a concern, which is not the case in handsets, it is advantageous to avoid using light guide plates all together. An example of this, called a direct backlight, is shown in

1.16 Toward the White-Light Applications

Figure 1.89 An example of a direct LED backlighting configuration where the LEDs are placed in a highly reflective cavity that consists of highly reflective walls and a diffuser on the front side. For good color and illuminance uniformity, the LEDs must have a broad intensity profile. In the example shown, the light from the top surface of LED is blocked by a white reflective dot. Courtesy of Lumileds/Philips.

Figure 1.89. Two of these units are then used to illuminate the backplate from both sides. A series of diffuser sheets and other electronics are also at the end of the LCD panel. As implied, these types of backlights are used in large displays, for example, LCD TVs [163]. The LUXEON backlight module designed specifically for this purpose features a flux density of 60–75 lm in1 in a standard LED array and 125 lm in1 at its best, which will become the standard shortly. The pitch distance between LEDs in the one-dimensional array is 9 mm. The LEDs are binned and matched to produce a CCT of 9000 K. The lifetime of the backlight module is over 50 000 h and requires about 3 W in1 drive power. Since the light guide plates have absorption losses and light coupling losses into the plates, direct backlight is more efficient if the thickness of the panel is a less important factor. This direct backlight is used in large display, as for example, LCD TVs [159]. As an alternative to a side emitter (also a regular Lambertian emitter can be used), a secondary lens to spread the light is used [164]. In an effort to achieve a high color and brightness uniformity, side emitters have been employed.

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Figure 1.90 LED backlighted 18 in flat panel computer screen and a schematic representation of illumination. Courtesy of Lumileds/ Philips. (Please find a color version of this figure on the color tables.)

A side-emitting LED, for example, the LUXEON side emitter, has peak intensity at an approximate angle of 80
with the surface normal [165]. In the simplest sense, a white LED or a combination of white LEDs can be used in a glass waveguide. From the get go, flat panels of the size either 40  136 or 102  126 mm2 became available for background lighting. But color parameters such as color rendition and dimming without color shift are not sufficient. Consequently, more elaborate schemes had to be developed for the LED approach to be viable and compete with fluorescent-based and also organic LED-based backlighting schemes. Fortuitously, advanced designs incorporating three-color LEDs have been developed with excellent color stability through active feedback and infinite dimming without color shift. The advances are such that flat panel color displays of 18 in size have been produced with three-color LEDs. The advances are such that flat panel color displaysof18 insize have beenproducedwiththree-colorLEDs asshownin Figure1.90. The Luxeon backlight module built for this purpose features a flux density of 60–75 lm in1 in a standard LED array and 125 lm in1 at its best, which has the potential to become the standard, at least a derivative of this. The pitch distance between LEDs in the one-dimensional array is 9 mm. The LEDs are binned and matched to produce a CCT of 9000 K. The lifetime of the backlight module is over 50 000 h and requires about 3 W in1 drive power. Since the light guide plates have absorption losses and light coupling losses into the plates, direct backlight is more efficient if the thickness of the panel is a less important factor. This direct backlight is used in large displays, as, for example, in LCD TVs [159]. Last but not least, these modules are free of Hg used in cold cathode fluorescent light bulbs (CCFLs), each LED module replacing four CCFL bulbs. In addition, simulated images that can be produced by CCFLs for back illumination and RGB LEDs for the same indicate the RGB solution producing much more vivid and pleasant colors. The penetration of LEDs into the signaling market provided the impetus for the development of higher power LEDs and lower cost. This performance/price

1.16 Toward the White-Light Applications

Figure 1.91 Historical and projected evolution of the performance (lm package1) and cost ($ lm1) for commercially available red LEDs. Note: CAGR ¼ compound annual growth rate. Both lines on the same numerical scale, but with different units. These data were compiled by R. Haitz from HP historical records [128].

evolution for red- and white-light LEDs is illustrated in Figure 1.91, covering the period from first LED sales in 1965 to a projected year 2020. As the figure indicates, flux per unit increased 20-fold per decade for the past 40 years and crossed the 10 lm level in 1998. While during the same period, the cost per unit flux ($ lm1) decreased by 10-fold per decade and reached 6 cents lm1 in 2000. At this price, the LEDs in a typical 20–30 lm center high-mount stop lights in automobiles contributes only $1.50 to the cost of the complete unit.4) The success of LED-based white-light sources suitable for general illumination critically depends on improving LED efficacies across the visible spectrum from 30 to 50 lm W1 to the desired 200 lm W1. For phosphors and other energy conversion methods, further research and development is required in order to understand and improve photon conversion efficiencies and output wavelengths. Moreover, improved packaging strategies have to be developed for all approaches, and applications exploiting the unique properties of lasers have to be explored. Putting it in other words, to produce 150 lm W1 of white light, which is necessary to compete with fluorescent lamps, SSL devices producing red, green, and blue light must operate with a power conversion efficiency of about 50%. Already red LEDs are nearly operative at these efficiencies, but nitride-based LEDs producing green light need to be improved

4) Although this cost is higher than that of an incandescent light bulb, it is low enough that desired compactness, styling freedom, and absence of warranty cost easily make up the difference.

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by a factor of 5–10, and those producing blue light need to improve by a factor of 2–3. The challenge to increase nitride efficiencies to 50% is formidable, but it is doable. In pursuit of progress along these lines, Lumileds (formerly Agilent that in turn formerly Hewlett Packard) obtained a 592 nm (amber) LED producing 480 lm W1 outputs. Furthermore, green and blue InGaN-based LEDs having efficacies in the range of 70 lm W1 are being reported by Cree Research, Inc. Estimates for successfully demonstrating a 200 lm W1 white-light levels range from 3 to 6 years. A second goal, in many ways more applicable, is taking laboratory-produced LEDs (producing 200 lm W1 white light) and commercializing at competitive costs. Table 1.6 compares current and projected efficiencies of white LEDs with those of the most widely used conventional white-light sources. The most popular incandescent lamps with a power rating of 60–100 W have an efficiency of around 15 lm W1 and a rated life of 1000 h, the efficiency of which drops off at lower power ratings or for lamps with a longer, 3000–6000 h, rated lifetimes. A similar pattern is in effect for halogen sources covering the range of 12–24 lm W1. Fluorescent lamps at 80 lm W1 are the most efficient white-light sources and dominate commercial and industrial lighting applications. Nichia achieved an efficacy of 150 lm W1 in a white-light lamptype LED having the CCT of 4600 K in 2006, but this efficacy was obtained at an extremely low injection current of 20 mA in a laboratory setting (one can deduct that this high efficacy did not hold up at high current drives). Commercial products showing 75 lm W1 and CCR of 6000 K at 350 mA are available by Cree as of 2007. Semi-LEDs, a California-based company with manufacturing facilities in Taiwan, also achieved luminous intensities ranging from 92 to 96 lm from LEDs driven at 350 mA of injection current. These efficiencies significantly exceed the standard 60–100 W incandescent lamps performance. The leveling off is based on the understanding and expectations based on the time of predictions and one should not be surprised if the actual performance turns out to be different.

1.17 Organic/Polymeric LEDs (OLED, PLED)

Application areas of nitride-based emitters and organic emitters overlap when it comes to indoor uses, background lighting, large-area illumination, and display screens, and in the past decade, a good amount of research has been dedicated to the organic emitters’ understanding [166, 167]. This overlap in application areas provided the impetus to provide some basics related to OLEDs/PLEDs. As there is a multitude of organic compounds with varying electrical and optical properties, the organic materials used in various layers of an OLED device are classified according to their physical structure. OLEDs have been made with small organic molecules and organic polymers [168, 169]. Although the two types of OLEDs are produced differently, in both cases, the organic layers are highly disordered, which opens up the possibility that they can be produced inexpensively in large areas. Systems employing small organic molecular compounds are typically deposited by vacuum deposition [170], although developments in an organic vapor-phase deposition (OVPD) technique show promise

1.17 Organic/Polymeric LEDs (OLED, PLED)

for high-definition features to be deposited, which are critical in terms of production if the molecular system is to be the method of choice for thin OLED displays [171]. For polymer-based OLEDs, the organic medium can be sprayed onto a substrate along with a carrier that must be evaporated, and techniques such as those in spin coating, screen printing [172], and ink-jet printing [173] have been used to produce polymer films; the result is a low-cost method of production, suitable for large-scale manufacture. Dendrimers, due to their lower degree of disorder and ability to efficiently transfer energy from a singlet state on its surface to a housed phosphorescing metal within them, also show much promise in OLED applications. It should also be noted here that this use of dendritic molecular architectures has been successfully utilized to produce “molecular” systems in the same fashion as the polymeric systems (i.e., low-cost spin coating techniques) [174]. This compromise appears promising for electrophosphorescent (a method of increasing efficiencies in OLEDs that has had more success in molecular systems over polymeric systems, see Section 1.17.3) OLEDs that are additionally efficient and easily fabricated devices. Fabrication techniques originally required that the organic materials be deposited onto a glass substrate in a batch process; however, advances are bringing to life one of the advantages of utilizing an organic material over an inorganic in its ability to be deposited on mechanically flexible substrates [175, 176]. The prospects in terms of devices possible when flexible substrates are used include not only illumination and display screens but also novel applications such as furniture, wallpaper, clothing, books, and so on. To demonstrate the agility and flexibility of OLEDs, a flexible display produced by Sony Corp. is shown in Figure 1.92. A difficulty with using flexible substrates, as compared to glass, stems from the fact that films are unacceptably permeable to water and oxygen, which degrade the organic layers of the OLEDs and cause delamination of the electrodes [177]. Reasons

Figure 1.92 A flexible, full-color organic electroluminescent display (OLED) built using organic thin-film transistor (TFT) technology with a plastic substrate. The 2.5 in prototype display, which is 0.3 mm thick and weighs 1.5 g without the driver, supports 16.8 million colors at

a 120  160 pixel resolution (80 ppi, 0.318 mm pixel pitch). Press release by Sony Corp., http:// www.sony.co.jp/SonyInfo/News/Press/200705/ 07-053/index.html. (Please find a color version of this figure on the color tables.)

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for the degradation are that oxygen will often attack bonds in the film, forming nonradiative recombination sites, and that efficient devices employ low work function cathode metals, such as calcium, which are highly susceptible to oxidation [178]. OLEDs typically made on glass substrates are covered with a second piece of glass to keep water and oxygen out. But for the flexible substrates, solutions include building hermetic barrier layers onto the films, designing efficient devices that do not require reactive electrodes, and designing organic layers that are inherently resistant to photooxidation. OLED displays have now entered the market in portable imaging products such as cellular phones, digital cameras, and car audio systems: applications that are currently served by traditional LCDs and vacuum fluorescent displays (VFDs). Because OLEDs are self-luminous, backlights are not required as in LCDs. OLEDs have very low power requirements and are thin, bright, and efficient. Some other attractive characteristics of these displays are emission wavelengths tunable by incorporation of suitable dyes in the emissive layers, low forward voltage compared to some other fluorescent displays (under 10 V), high emission efficiencies, which result in high brightness coupled with low power consumption, wide viewing angles, fast response (10 ms), thin and lightweight display construction, potential for low-cost manufacturing, low-temperature processing technology, compatibility with flexible substrate displays, and environment-friendly features in the form of reduced power consumption and lack of need for mercury-containing backlight assemblies [166]. Disadvantages, which are formidable, are chemical instability and degradation by UV light. As such, requirements of lowered cost, storage lifetimes of at least 5 years, and operating lifetimes of >20 000 h are to be met. To meet these goals, significant activity is being undertaken to optimize the emitting structure, improve encapsulation, and develop materials that are more resistant to chemical degradation, such as oxidation. Similar to the case of inorganic LED inclusive of performance projections into the future, Figure 1.93 presents a diagram for an OLED system and compares the current typical efficiency values for the individual system elements to a set of suggested program targets. The projected target values in some categories are highly optimistic, some reaching the perfect value of 100%, which would be the first time ever. Furthermore, it is not clear if these projected values would hold up at high intensities and if so for how long. If the target values were to come true, OLED look much better than the nitride-based white LEDs for illumination. 1.17.1 OLED Structures

In what follows, the intent is to be introductory. As such, references to specific schemes, molecular or polymeric will be limited. Although Figures 1.94 and 1.95 show a number of widely used p-conjugated small molecules and polymers, respectively [179], the trials involved within each of the schemes lie outside the scope of this book. The reader interested in detailed information, specific to particular schemes, can refer to Ref. [179].

1.17 Organic/Polymeric LEDs (OLED, PLED)

2005 Target

1

85% 90%

2 6%

Total luminaire

3 70%

1 Driver

70%

2

90%

Fixture 3 & optics

4% 57%

OLED

Electrode 2a resistive loss

2b

IQE

2c

EE, χ

2d

EQE Electrical efficiency

Lumens out 2e

Figure 1.93 OLED system efficiencies and projections into the future. The target CCT and CRI used are 3000–6000 K and 80, respectively, with 1000 cd m2. IQE, internal quantum efficiency; c, extraction efficiency; EQE, external quantum efficiency, which is a product of the internal quantum efficiency and the extraction efficiency; B, G, and R are for blue, green, and red, respectively. Note that target efficiencies in some categories are perfect, which is very optimistic. In addition, whether these target values would hold up at light intensities and, if so, for how long

Substrate scattering

2a

99%

98%

2b 25%

100%

2c 30%

80%

EQE

2005 Target

OLED

90%

8%

2d 80%

90%

2e 100% 98% Total white OLED

Electrical power

6% 70%

remain to be seen. Patterned after NGLIA LED Technical Committee report prepared for the US Department of Energy “Solid-State Lighting Research and Development Portfolio,” March 2006. Note 1: Electrode electrical loss – negligible for small devices, possible issue for large-area devices. Note 2: Includes substrate and electrode optical loss – negligible for glass and very thin electrodes but may be important for plastic or thicker electrodes. NGLIA OLED Technical Committee.

Although tris-(8-hydroxyquinoline)aluminum (Alq3) OLEDs or LEPs performance is noteworthy, OLEDs represented not much more than a scientific curiosity prior to the early work of Tang and Van Slyke [168]; however, their work with Alq3 is more or less responsible for the creation of the field. In the wake of rapid scientific progress, particularly in the operation lifetime, a bright future is now seen for organic emitters, indoor displays, background panels, and night-lights built around relatively large organic molecules. The large area, the physical flexibility, and the low cost are the attractive features offered by the organic technology. There are also efforts to fabricate transistors based on polymers with the hope of constructing displays having built-in control circuitry in much the same way as liquid crystal displays. Originally, devices were constructed simply by sandwiching an organic emissive layer between two electrodes. The organic material inevitably had vastly different mobilities for electrons and holes, and thus, recombination often occurred near one of the electrodes, resulting in considerable nonradiative recombination due to conformational defects. By incorporating multiple organic layers as transport layers for holes or electrons, recombination was effectively moved away from the electrodes, resulting in higher quantum efficiencies (multiple layers were initially difficult to achieve in the polymeric systems). By choosing the hole transport layer (HTL) or

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N O

\

N

N Al

O

N

N

N Cu N

O

N

(a) Alq 3

(b) Rubrene

CH3

N

N

(c) CuPc

CH3 N

N

(d) TPD

N

N

(e) (α-) NPB, (α-)NPD

N

R N

N R

(f) TDATA

(g) DPVBi

Figure 1.94 Structures of widely used p-conjugated small molecules: (a) tris-(8-hydroxy quinoline Al) (Alq3); (b) rubene (5,6,11,12tetraphenyl tetracene or 5,6,11,12-tetraphenyl naphthacene); (c) copper phthalocyanine (CuPc); (d) N,N0 -diphenyl-N,N0 -bis(3-

methyphenyl)-1,10 -biphenyl-4,40 diamine (TPD); (e) N,N0 -diphenyl-N,N0 -bis(1-naphthylphenyl)1,10 -biphenyl-4,40 -diamine (NPB, a-NPB, NPD, or a-NPD); (f) 4,40 ,400 -(tris(diphenyl amino) triphenylamines (TDATAs); and (g) 4,40 -bis(2,20 diphenylvinyl)-1,10 -biphenyl (DPVBi) [179].

electron transport layer (ETL) appropriately (i.e., mismatching the energy bands so that the HTL’s highest occupied molecular orbital (HOMO) corresponds to the work function of the anode or the ETL’s lowest unoccupied molecular orbital (LUMO) corresponds to the work function of the cathode), the layers can be made to serve as hole- or electron-blocking layers, increasing the likelihood of radiative recombination within the emitting layers. Similarly, due to the unavailability of high work function metals, electrons are easier to inject, but the number of holes must be matched with the number of electrons that are injected, for a mismatch will contribute to current but not emission. The typical multilayer OLED device is shown in Figure 1.96a along with an energy level diagram for a bilayer OLED in Figure 1.96b, and schematic representation of the charge-carrier injection (1), exciton formation (2), and radiative

1.17 Organic/Polymeric LEDs (OLED, PLED) O n

n

H3C O

(c) MEH-PPV

(b) PPV

(a) PPV

H N

H N

H N

n

(d.1) LEB

H N

N N

n

(d.2) EB O

CH3

n

O

N n

N

S

n O C OCH3

n n SO3H

(d.3) PNB

(e) PEDOT-PSS

(f) PVK

(g) PMMA

R′

C6H13

R R n

C 6H13

S

n n

R

R

R=C nH2n+1 R′

(h) m-LPPP

(i) P3AT

Figure 1.95 Structures of widely used p-conjugated and other polymers: (a) poly(paraphenylene vinylene) (PPV); (b) s (solid line along backbone) and p (“clouds” above and below the s line) electron probability densities in PPV; (c) poly(2-methoxy-5-(2-ethyl)-hexoy-1,4phenylene vinylene) (MEH-PPV); (d) polyaniline (PANI): (d.1) leucoeneraldine base (LEB), (d.2) emeraldine base (EB), (d.3) pernigraniline base

n

(j) PFO

R

(k) PDPA

(PNB); (e) poly(N-vinyl carbazole) (PVK); (g) poly(methyl methacrylate) (PMMA); (h) methyl-bridged ladder type poly(pphenylene) (m-LPPP); (i) poly(3-alkyl thiophenes) (P3Ats); (j) polyflorenes (PFOs); (k) diphenyl-substituted trans-polyacetylenes (t-(CH)x) or poly(diphenyl acetylene) (PDPA) [179].

recombination (3) in organic LEDs in Figure 1.96c [180]. The choice of suitable energy band mismatch, as well as the ability to get light out of the device once it is emitted, is the key to realizing efficient OLEDs; the latter applies to LEDs of all types. The device consists of an anode, a high work function metal to inject holes (usually ITO), a hole transport layer, an emissive layer (or layers – this could also possibly be the same layer as the HTL or ETL and could possibly be doped with fluorescent or phosphorescent dyes) [181], an electron transport layer, and a low work function cathode, usually a reflective thin-film metal to inject electrons (typically aluminum, calcium, or magnesium or metal alloys such as magnesium–silver and

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Low work function cathode

Electron transport layer ITO anode

Light emission layer

Glass substrate

(a)

Hole transport layer

Light out

Vacuum level φ Cathode LUMO

Cathode

LUMO

φ Anode

HOMO HOMO

Anode

Hole transport layer

Electron transport layer

(b)

(1)

Anode

LUMO

Cathode

Anode

Cathode

Anode

HOMO

(c)

LUMO

Cathode

LUMO

hν

HOMO

HOMO

(2)

(3)

1.17 Organic/Polymeric LEDs (OLED, PLED)

lithium–aluminum). Note here that one of the electrodes must be transparent in order for the light to escape – this is typically the ITO anode; however, progress is being made in the vein of making completely transparent OLEDs [182, 183]. In this case, a thin layer (10 nm) of electron injecting material is typically placed next to the organic layers, and the cathode is capped with ITO. Transparent OLEDs (TOLEDs) are also suitable for additional applications over traditional OLEDs such as helmet or windshield-mounted displays, “smart” windows that double as nightlights, or other novel sources of light. In order to maximize device performance, each of the organic films must be specially optimized for each color desired. The overall thickness of the device is less than 200 nm, so it cannot support itself and as such is often fabricated on a glass substrate, though flexible substrates are being developed. The more flexible substrates are improved, the less susceptible they become to atmospheric permeation, a lingering problem for the earlier versions. It has been estimated that the permeability of an OLED package should be less than 5  106 g m2 day1 at room temperature, in order to achieve a lifetime of 10 000 h, or organic materials will have to be engineered to be unaffected by the presence of water or oxygen [184]. Langowski [185] discussed the barrier, against oxygen and water vapor, requirements for various product sectors (defined by dotted lines) and performance of polymer-based flexible structures (defined by shaded areas) as depicted in Figure 1.94. Developments have resulted in a barrier coating that can be put over a plastic substrate that significantly retards the diffusion of oxygen and water, called Barix. This is a multilayer coating consisting of alternating layers of inorganic oxide to impede the diffusion of the water and oxygen, and a polyacrylate film that planarizes the oxide and impedes the propagation of defects through this barrier. The surface
roughness of the coated sample is less than 10 A, and testing showed that this barrier allowed only 2  106 g m2 day1 of water or oxygen through it. The OLEDs that were built on Barix-coated polyethylene terephthalate (PET) were shown to have half lives of 3800 h from an initial luminance of 425 cd m2, when driven at 2.5 mA cm2 [186]. As there is a plethora of available organic compounds from which to choose suitable layers from, OLEDs being produced are of many varieties and are classified according to the physical structure’s organization: molecule, oligomer, or polymer. The most important differences in the types are the methods that are used to fabricate them, and again polymers are more attractive in that they can be made at room temperature using spin coating, screen printing, or ink-jet printing techniques. The radiative recombination methods of each type of organic device are essentially the same. The relaxation of singlet states, although excitations leading to them are 3 Figure 1.96 (a) A typical OLED multi-layer device structure. (b) Schematic energy level diagram of a bilayer OLED device, (c) Schematic of the charge carrier injection (1), exciton formation (2) and radiative recombination (3) in organic LEDs [191].

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Figure 1.97 Barrier requirements against oxygen and water vapor, for various product sectors (dotted lines) and performance of flexible polymer based structures (shaded areas). Patterned after [185].

limited by spin to statistics to occur 25% of the time at best, were the original method of harvesting photons from organic LEDs. More recently, by using dyes (molecules or on-chain ligands for polymers) that allow phosphorescence to occur from the relaxation of excited triplet states, efficiencies have been brought up. Conjugated molecular systems, polymers and oligomers, have attracted considerable attention for their applications to OLEDs [187, 188]. Conductive polymers have also been explored for electronics applications, primarily FETs [189], and a detailed treatment has been presented by Christos and Dimitrakopoulos [190]. A review of conjugated polymers, particularly of their optical properties can be found in Ref. [191]. While discrete emitters based on organic molecules and polymers have been attained, the impact of these devices is expected to be in the region of large-area displays. In this vein, flat panel displays promise to offer good value to electronic products that incorporate displays for viewing information, and manufacturers have already demonstrated active-matrix displays as large as 40 in [192]. Aside from problems with efficiency and lifetime, which would be ameliorated by employing hole transport and electron transport layers, the first organic polymers to be shown to have light-emitting properties were not feasible for employment in the production of white light due to their inability to produce short wavelengths (blues). The ability to shorten the conjugation length by addition of acceptor side chains on the polymer was soon discovered, and this was used to engineer the wavelength of the emitted light to the desired length; similarly, molecules have been found that produce the shorter blue wavelength necessary for white-light emission. For practical applications, it is important to operate the devices at low voltages, consuming little power, and polymers have been more successful in reducing

1.17 Organic/Polymeric LEDs (OLED, PLED)

their operating voltages than their molecular competitors. One widely used electron transport material in OLEDs is Alq3; however, the effective mobility of electrons in Alq3 is reported to be as low as 1.4  106 cm2 V1 s1 at an electric field of 4  105 V cm1 [193]. When both the cathode and anode are perfectly injecting, low electron mobility in Alq3 becomes a limiting factor on voltage reduction. One can reduce the thickness of the Alq3 layer for lowering the operating voltage. However, the thickness reduction unavoidably results in high leakage current and low quantum efficiency. Hamada et al. [194] reported that bis(5-hydroxy flavonato) beryllium had better electron transport property than Alq3, whereas a bias of 8 V was still required to generate a luminance of 2000–3000 cd m2. Hung and Mason [195] reported that the use of copper phthalocyanine (CuPc) as the ETL combined with an ultrathin LiF/Al bilayer between the ETL and the emitting Alq3 layer resulted in a reduction in the operating voltage by about 40%: from 8 V required to generate a current density of 100 mA cm2 when Alq3 was used as the ETL to 5–5.5 V. Another ETL composed of a Cs:phenyldipyrenylphosphine oxide (POPy2) layer with an atom:molar ratio of 1 : 2 has been employed by Oyamada et al. [196], which resulted in a current conduction of 100 mA cm2 at 3.9 V. The formation of a CsAl alloy layer of Cs:POPy2/Al cathode interface and the charge-transfer complex between the Cs and POPy2 contributed to attaining enhancement of the efficiency of electron injection and transport. An extremely low driving voltage of 2.9 V at a current density of 100 mA cm2 and very high luminance at a low driving voltage have been demonstrated in an OLED: 1000 cd m2 at 2.4 V, 10 000 cd m2 at 2.8 V, and 920 000 cd m2 at 4.5 V. In the OLED, the use of p-doped alpha-sexithiophene and n-doped phenyldipyrenylphosphine oxide carrier transport layers generated free charge carriers by charge transfer from matrix to dopant molecules, resulting in an increase in electrical conductivities and formation of Ohmic contacts at metal/organic interface [197]. As stated earlier, an imbalance of charge injection will contribute to current but not emission. Due to the unavailability of high work function metals, electrons are easier to inject. Thus, the number of holes injected is the limiting factor in terms of excitons formed. Along these lines, but in an attempt to decrease the number of excimer complexes formed between polymer chains, Sainova et al. [198] introduced several different hole-transporting materials into a single-layer OLED. The OLEDs were prepared from poly(2,7-(9,9-bis(2-ethylhexyl))-co-(9,9-bis((3S)-3,7-dimethyloctyl)) fluorine) (PF C26) blended with several low molecular weight hole-transporting molecules (HTMs) – a triphenylamine derivative (TPTE), starburst amine (ST 755)) and N,N8-diphenyl-N,N8-bis-a-naphthylbenzidide (ST 16/7) at weight ratio 1–0.03. Due to the differing oxidation potentials (electron affinity), the HTMs acted as hole traps, as was evidenced by thermoluminescence (TL). A shift to a higher temperature (by 50 K) of the peak of the TL clearly indicates the existence of deep trapping sites. The sites effectively reduced hole mobility as well as established a space charge field, which allowed both increased quantum efficiency and a reduction in longwavelength emission. Obviously, a similar approach could be employed in molecularbased systems.

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Because the internal quantum efficiencies of phosphorescent OLEDs can now reach 100% [199], the natural direction that much of the current research is going is toward increasing the external quantum efficiencies, hex, because this figure has been typically estimated to be as low as 0.17–0.5 [200, 201]. This substantial loss arises due to light incident to the substrate–air interface or the organic–substrate interface at angles greater than the critical angles, that is, total internal reflection occurs (Figure 1.98). Various methods have been used to increase the external efficiencies of OLED devices (Figure 1.98b–f) including surface roughening [203], introduction of an aerogel layer [204], an integrated distributed feedback gratings [205], scattering microspheres [206], microlenses [207], mesa structures on which to build [208], or by use of microcavities [209–211]. Each of these methods have their own advantages and disadvantages in terms of cost, production feasibility, polarization effects, dispersion effects, changes in spectral linewidth, and of course extraction efficiency, and as such a method for use in an illumination application may not be feasible for a display application. For details on these and other mechanisms refer to Ref. [202] and the corresponding literature. 1.17.2 Charge and Energy Transport Fundamentals

Unlike the inorganic LEDs, where high degrees of crystallinity and close-packed atomic structure favor higher electronic conductivity, strong exchange interactions often do not exist in the disordered organics. Intermolecular separations are large as compared to separations of atoms in the inorganic’s lattices, because the organics interact by weak van der Waals or London type forces. This implies that molecular orbitals weakly overlap and intermolecular electron exchange is small. Hence, electron orbitals are localized (except in the case of the fully conjugated polymers) making some argue that hopping or tunneling are the favored methods of charge migration in the organics [212]. In organic materials with localized energy states, energy can be transferred without the transport of mass or charge. Fortuitously, energy transfer within an organic material is less sensitive than charge transport to structural disorders (traps). This stems from the fact that unlike the inorganics, organics do not exhibit conventional Mott–Wannier excitons. The more localized, mobile Frenkel excitons are the energytransfer mechanisms, and due to their strong Coulombic interactions, they can be thought of as essentially neutral and hence not subject to the all of the trapping mechanisms inherent in the quasiamorphous system [212]. Unlike the inorganic semiconductors, where thermal excitation can lead to substantial dark conductance, the dark conductivity of organic systems that are not intentionally doped varies vastly from family to family of organic solids. This could arise from intrinsic excitation (DE < 2 eV) or defect excitations (DE > 2 eV). From this stems the long debate of whether the band model (often under Bloch’s one-electron potential approximation) or a nonband model (hopping or tunneling) is appropriate for the discussion of carrier transport. Either way, charge carriers often suffer from

1.17 Organic/Polymeric LEDs (OLED, PLED)

Figure 1.98 (a) Schematic of losses reducing extraction efficiency; (b–f) various methods utilized to extract light from OLED device [202].

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Figure 1.98 (Continued)

very low mobilities. The inherent disorder of the system causes interaction between free carriers and the surrounding “lattice” via polarization effects including chargeinduced dipole interactions, dipole-induced dipole interactions, charge–dipole interactions, charge–quadrupole interactions, superpolarization effects, charge-induced

1.17 Organic/Polymeric LEDs (OLED, PLED)

Triplet manifold

Singlet manifold

S* SF S1 (1Bu) 2Ag

T2

IC IC

ISC

IC ABS IC

T1

Luminescence IC

Weak phosphorescence

SO (1Ag) Figure 1.99 Schematic of the radiative and nonradiative recombination processes [191].

quadrupole interactions, and energy contributions from higher order multipoles. Chemical impurities, whether intentional or not, affect the electrical properties in that trapping occurs as differing electron affinities (ionization potentials) behave as traps or antitraps. This brings us to the issue of recombination. The goal here for the LED is for the recombination to be radiative. Once the charges are injected into a device, depending on the relative alignment of the spins, phonon interaction, orbital overlap, and longrange order, electrons and holes form negative and positive polarons, as they are coupled with the lattice. These polarons migrate across the electron transport or hole transport layer to the emissive layer and form singlet or triplet excitons as they meet. In turn, either electroluminescence occurs due to the fluorescent decay of singlet excitons from the first excited state (S1) to the ground state (S0) or eventual phosphorescent decay of triplet excitons takes place, as shown in Figure 1.99. 1.17.3 Properties of Organic Crystals

Not only the organic emissive layers must convert energy to useful amounts of light but also the device must be built in such a manner that the emitted light gets out of the OLED device. Investigations are moving from a focus on increasing the brightness and optimizing the efficiency of emission within the organic layer of the LEDs to optimizing the out-coupling of light from the device structure by incorporation of microcavities, shaped substrates, or index matching mediums [213]. Conjugated polymers are semiconducting owing to alternating single and double carbon–carbon bonds. Single bonds are called s-bonds and double bonds include a s-bond and a p-bond. While the s-bonds can be found in all conjugated polymers, the p-bonds are formed from the remaining out-of-plane pz orbitals on the carbon atoms overlapping with neighboring pz orbitals. The p-bonds are the source of the

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semiconducting properties of these polymers. First, the p-bonds are delocalized over the entire molecule, and then, the overlap of pz orbitals actually produces two orbitals, a bonding (p) orbital and an antibonding (p ) orbital. In terms of the electronic structure, the optical properties are related to the frontier orbitals: the HOMO produced by the lower energy p-orbitals and the LUMO produced by the higher energy p -orbitals. The energy difference between the two levels is equivalent to the bandgap that determines the optical properties of the material such as photon absorption and emission. The bandgaps for most conjugated polymers are between HOMO and LUMO in the range of 1.5–3 eV, and the absorption coefficient is 105 cm1. Ab-initio and semiempirical calculations have been undertaken and attribute the HOMO to the bonding p and LUMO to the antibonding p molecular orbitals determined by the overlapping of pz orbitals [214, 215]. The body of knowledge of molecular orbitals indicates that any variation in overlap, such as geometrical modifications induced by the presence of substituent groups in polymers, interactions with solvents, and thermal effects, leads to modifications in the electronic structure, and thus optical properties [191]. Bathochromic (hypsochromic) shifts are observed in both the absorption and emission bands as a result of positive (negative) solvatochromism in organic materials. Similar to the inorganic class of LEDs, intensity can be related to temperature, with reduced vibrations at low temperatures causing an increase. Also, torsion of the chain in a polymer-based system reduces the overlap of pz orbitals and increases the gap between p and p orbitals, which results in shorter conjugation lengths and that in turn causes a hypsochromic absorption shift [216]. Optical transitions in solid-state mode crystals are well known to be due to Frenkel excitons [217–219]. Properties resemble those of the isolated molecules because excitons are strongly confined in a molecule and only the weak interaction with the surrounding molecules takes place, which leads to biexcitation formation. The main effect of intermolecular interactions is to split each molecular state into different crystalline states, termed Davydov splitting [217]. These are characterized by different symmetries, as shown in Figure 1.100. If there is more than one molecule per unit cell, and the molecules are related by symmetry operations, the crystal wave functions are constructed out of the subsets of nonequivalent molecules. This exercise leads to representation of the crystal states, which are either symmetric or antisymmetric combinations of the subset wave functions [219]. This splitting of molecular states in crystalline states with well-defined symmetry strongly affects the relaxation processes of molecular excitations. The electronic structure and transition levels between occupied and unoccupied excited states of molecules can be determined from various optical techniques (ultraviolet photoemission spectroscopy (UPS), X-ray photoemission spectroscopy (XPS) [220], and so on). An interesting feature found in the oligothiophenes is the crossing of states, with different symmetries, as a function of the chain length. The states of interest within the C2h symmetry possess 1Bu and 2Ag symmetry. As the chain length and the number of double bonds are increased, the lowest optical transition changes from an allowed (Bu) to a forbidden transition (Ag) [191]. Another

1.17 Organic/Polymeric LEDs (OLED, PLED)

bg bu

Energy

1B u

ag au

1A g

Molecule

Crystal

ag

Figure 1.100 Energy diagram (calculated by semiempirical methods) of the electronic states for an isolated molecule (left) and in solid state (right). The presence of intermolecular interactions determines a splitting of the molecular states in a number of crystalline states depending on the number of equivalent molecules in the unit cell [191].

feature of interest in unsubstituted oligothiophenes is the linear dependence of the S0 ! S1 transition energy on the inverse number of rings forming the chain [221] (Figure 1.101). Intuitively, a substantial redshift of the lowest electronic transition occurs as the chain length is increased, which is a result of the progressive extension of the p-delocalized states. Instead, the change in size of the singlet–triplet S0 ! T1 transition energies with chain length is much weaker than that for the S0 ! S1 excitation [221]. The singlet–triplet excitation is only lowered by 0.2 eV when going from the dimer to the hexamer, whereas a bathochromatic shift of 1.4 eV is observed for the singlet–singlet transition S0 ! S1. Such behavior, in fact, reflects the stronger confinement of the triplet exciton with respect to those associated with singlets. This trend is consistent with optically detected magnetic resonance (ODMR) data for polythiophenes, which indicate that the T1 triplet state barely extends more

Transition energy (eV)

4.5 4.0 3.5 3.0

S0 S1

2.5 2.0 1.5 1.0 0.1

T1 Tn S 0 T1 0.2

0.3

0.4

0.5

1/N Figure 1.101 Chain size dependence of the S0 ! S1 transition energy between the ground state (S0) and the first excited singlet state (S1), of the singlet–triplet transition energy S0 ! T1, and of the triplet–triplet transition energy T1 ! Tn [191].

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than a single thiophene unit [222]. A larger delocalization of the higher lying triplet state, Tn, leads to a more noticeable redshift of the T1 ! Tn transition versus 1/N as shown in Figure 1.101. The exciton binding energy of the conjugated polymers has been the subject of debate in the literature over the past decades. Reporters have proposed the values between a few kbT (kbT in the order of 105 eV) and 1 eV for the binding energy [223–225]. It seems that people accept that most conjugated polymers have a binding energy of about 0.3 eV–0.4 eV [224, 226]. Also relevant is the diffusion length of singlet excitons, which is in the range of 5–15 nm, and the radiative decay associated with them in a timescale of 100–1000 ps [227]. At first, the efficiencies of OLEDs were limited due to the materials used; radiation only resulted from singlet relaxations, relaxations conserving spin symmetry. These extremely rapid (1 ns) transitions are typically from singlet excited states to ground states, and the resulting photon emission is called fluorescence. Based on a spin statistics, the singlet excitation efficiency hs was believed to have a maximum value of 25%. In other words, only 25% of excitons were supposed to be singlets, which was the only mechanism for producing photons. The remaining 75% of the excitons would therefore have resulted in triplet states, which do not lead to fluorescence. Consequently, this was thought to impose a 25% fundamental limit on the internal quantum efficiency of electroluminescence. However, further studies showed that the 25% figure was no longer valid in that singlet excitation could be as high as 50% in some p-conjugated polymers. This was attributed to a higher cross section for an electron–hole pair to form a singlet state versus a triplet, due to the delocalized nature of charged particles in p-conjugated polymers [228]. Nonetheless, in any organic system, it has become common to introduce phosphorescing materials, which can produce optical emission from triplets, into emissive layers in order to take advantage of the large majority of excitons that are triplets. By obtaining radiation from both singlet and triplet excitons, internal quantum efficiency can theoretically reach 100%. It should be noted that this technique has found more success in the molecular rather than the polymeric LEDs. The problem with polymeric systems that include phosphorescing chromophores is that aggregation quenching reduces the quantum efficiency. The polymer systems are too disordered, and including chromophores on all chains yields a chromophorerich system. Interchain interactions, especially the formation of excited states (exciplexes), lead to reduced radiative recombination as well as a bathromic shift in the spectrum. By utilizing phosphorescing materials (dyes) in the emitting layers, luminous phosphorescence arises from the forbidden transitions that do not conserve spin symmetry. A sample of some of the molecules that have been used to achieve phosphorescence from triplet states is shown in Figure 1.102. Baldo et al. [229] showed that a phosphorescent yield of approximately 90% transfer of energy from Alq3 to PtOEP, which means that both singlets and triplets must participate in energy transfer. They went on to suggest that the transfer was due to the

1.17 Organic/Polymeric LEDs (OLED, PLED)

Figure 1.102 Chemical structure of important degenerate and nondegenerate conjugated polymers, reported to have been used to get phosphorescence from triplet states. The PtOX and the Ir (ppy)3 are the most successful and produce red and green light, respectively.

Dexter process (short range, 100 ms, and thus saturation of emitting sites will occur at high drive currents when the doping levels are low. High drive currents, additionally, render the phosphors prone to nonradiative recombination mechanisms: triplet– triplet annihilation, triplet–charge carrier interactions, and electric field induced triplet dissociation phenomena, and so the phosphorescing materials employed in the emitting layers should have the shortest lifetimes available. In order to understand the dynamics of triplet relaxations, Baldo and Forrest [231] used transient electrophosphorescence to observe either the diffusion of triplets through a conductive host material or evidence of exciton trapping or direct formation of excitons on phosphorescing guest materials. Let us first assume that triplet states are created. Now there are four processes that determine the overall efficiency of energy transfer between host and guest molecules: relaxation rates kG and kH of excitons from guest and host, respectively, and forward and reverse transfer rates of triplets from host to guest, kF and kR (Figure 1.103). The rate equations are then. dG ¼ kG GkR GþkF H dt and dH ¼ kH HkF HþkR G; dt

ð1:36Þ

where G and H are the densities of triplets in the guest and host. The solutions to these are biexponential decays of the form ð1:37Þ

G; H ¼ A1 exp[k1 t]þA2 exp[k2 t];

kF

Host

∆G kR

kH

Guest kG

Figure 1.103 Triplet dynamics in a guest–host system: the rates of forward and back transfer kF and kR are determined by the Gibbs free energy change (DG) and molecular overlap; also of interest are the rates of decay from the host and guest (kH and kG), respectively [231].

1.17 Organic/Polymeric LEDs (OLED, PLED)

where A1 and A2 are determined from initial conditions and the characteristic decay rates: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kF þ kR þ kH þ kG 4(kG þ kR )(kH þ kF )4kF kR : ð1:38Þ k1 ; k2 ¼  1 2 (kF þ kR þ kH þ kG )2 Apparently, to maximize guest phosphorescence, we first seek kG  kH. This means that the host material has a very large triplet lifetime, allowing guest triplet relaxation even if the energy of the guest triplets is higher than that of the host triplets. Secondly, we seek kF  kR  kH, which maximizes the population of guest triplets, thus reducing nonradiative losses from triplets in the host. At low temperatures (high temperatures enhance the rate of nonradiative triplet decay), kH þ kG can be determined using the transient phosphorescent decay. The terms kF and kR can be understood after one considers the processes that are occurring. Dexter interactions permit exciton hops from molecule to molecule when there is no change in spin. In this manner, triplet hopping can be thought of as simultaneous transfer of an electron and a hole. With correction of the reorganization energy and comparison of the rates of triplet transfer with those of electron and hole transfer, the triplet transfer rate has been demonstrated [232] to be related to the electron and hole transfer rates predicted by Marcus theory. Marcus theory reflects the Franck–Condon principle: electron motion is so rapid during an electronic transition that the atomic configuration of the reactant and product states is unchanged. Marcus theory recognizes that the rate of transfer is not limited by the electron transfer itself, but rather the formation of the activated complex that precedes the transfer. These lead one to reason that the most probable activated complex is the one with minimal energy of formation (G, Figure 1.103), under the restriction that the total energy of the complex remains unchanged during the transfer. Transfer probability then takes the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   4p3 (DG þ l)2 2 k¼ jMDA j exp ; 4lkB T h2 lkB T

ð1:39Þ

where MDA is the matrix element mixing donor and acceptor states. Measuring the phosphorescent spectra for the relaxed triplet state energies for both the donor and acceptor molecules gives DG, and if DG ¼ 0. Then the energy barrier in the forward and reverse directions is given by l. For small changes in the free energy, Marcus transfer via an activated complex behaves similar to an Arrhenius barrier of l/4. As DG increases, the rate does as well until resonance, and with further increases in DG, the rate decreases yielding the so-called “Marcus inverted region.” Dexter transfer additionally requires of the molecules that combined spin be conserved during energy transfer. As an example, 3

D* þ1 A ! 1 D þ 3 A* :

ð1:40Þ

Note that the singlet states in the host may be passed to the guest via the Dexter transfer, but if the spectral overlap is sufficient, long-range dipole–dipole or Foerster

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energy transfer dominates. When the donor is phosphorescent, Foerster transfer from triplet of acceptor to singlet of donor molecules can be a very efficient energytransfer method. 3

D* þ1 A ! 1 D þ 1 A* :

ð1:41Þ

This is useful in transferring triplet states to singlets of the acceptor, or by employing a phosphorescing guest in a phosphorescing host. Unlike triplet–triplet transfer, donor and acceptor molecules are well coupled in Foerster transfer; hence, the rate depends on the overlap of donor emission and acceptor absorption [231]. Baldo and Forrest [231] used the following materials: (a) N,N0 -diphenyl-N,N0 -bis(3methylphenyl)-[1,10 -biphenyl]-4,40 -dimine (TPD) (hole transport), (b) 2,9-dimethyl4,7-diphenyl-1,10-phenanthroline (bathocuproine or BCP) (electron transport), (c) 4,40 -N,N0 -dicarbazole-biphenyl (CPB) (hole transport), and (d) Alq3 (electron transport) as host materials and (e) fac tris(2-phenylpyridine) iridium [Ir(ppy)3] and (f) 2,3,7,8,12,13,17,18-octaethyl-21H,23H-porphine platinum(II) (PtOEP) as guest materials. The phosphor guests emit at 510 nm with a phosphorescent lifetime of 0.4 ms [229] and at 650 nm with a phosphorescent lifetime of 100 ms [233] for Ir(ppy)3 and PtOEP, respectively. The phosphorescent spectra of each of these materials were examined first to
determine their triplet energy levels (Table 1.7). Films with a thickness of 2000 A were subjected to 500 ps pulse at 1 Hz excitations from a N2 laser emitting at a wavelength of 337 nm, and transient analyses using a streak camera separated delayed fluorescence from phosphorescence. Turning now to diffusion of triplets in a host, Baldo and Forrest [231] intentionally left areas undoped between the phosphorescing zones and the exciton formation zones, forcing excitons to travel lengths of several hundred angstroms. By applying a short pulse, singlet and triplet excitons are created at the HTL/ETL interface. Then after excitation, the delay between fluorescence and phosphorescence is measured. Knowing that the delay must be attributed to either charge diffusion or triplet diffusion, a reverse bias is applied, effectively turning the charge diffusion to zero. Table 1.7 Material triplet energies and room-temperature lifetimes.

Material

Triplet energy (0.1 eV)

Triplet lifetime

PtOEP Ir(ppy)3 CBP BCP TPD Alq3c

1.9 2.4 2.6 2.5 2.3 2.0

110  10 msa 0.8  0.1 msb >1 s 0

Loss (a.u)

Energy

T=0

EC–EV

Fn–Fp J

1

Figure 2.4 (a) Transitions involved in excitation (process 1), followed by relaxation to the bottom of the conduction band (process 2 on the order of 10
12 s) and recombination with resultant photon emission (with a carrier lifetime – time interval before recombination – of about 10
9 s) (process 3), and electron occupying a hole state

created by process 1 (process 4); (b) schematic representation of absorption and emission processes in a semiconductor beyond transparency; (c) gain/loss diagram for a semiconductor for T ¼ 0 and T > 0 K for two different pumping levels beyond transparency.

which schematically shows the upward and downward transitions across the bandgap. Initially the top of the valence band is full of electrons. Optical excitation commences by absorbing a photon, step 1 in Figure 2.4a. To conserve momentum, as photon momentum is negligible, the excited electron must originate deep in the valence band. Nearly simultaneously (determined by the rate of dominant scattering mechanism, which is polar optic phonon scattering in compound semiconductors taking place at picosecond timescale), the excited electron thermalizes down to the bottom of the conduction band, step 2 in Figure 2.4a. This is the route to population inversion in that an electron exits at the bottom of the conduction band and a hole at the top of the valence band. This situation is retained for a time period of the order of 1 ns, which is spontaneous emission (recombination) time. In a time period equal to

2.2 Waveguiding

the lifetime, on the average, an electron recombines with a hole at the top of the valence band, step 3 in Figure 2.4a. We should mention that as soon as a photon is absorbed (for excitation depicted with step 1 in Figure 2.4a), an electron leaves the top of the valence band to fill the hole created in the process, step 4 in Figure 2.4a. The notion of transparency can be understood while examining the schematic diagram of Figure 2.4b where the separation between energy levels E2 and E1 (defining absorption) is larger than the photon energy produced through recombination from the bottom of the conduction band to the top of the valence band. Another point of interest is that once the semiconductor is made transparent and pumped, in fact beyond transparency, the optical electromagnetic wave can be amplified as the medium converts from being an attenuating one to an amplifying one. Leaving the details for later discussions, before transparency, the loss in the semiconductor is a function of energy as shown by the lower dashed line in Figure 2.4c. As the transparency is approached, by optical or electrical pumping, the loss would be reduced giving way to zero at the onset of transparency and negative (gain) beyond the onset of transparency, upper solid lines. To a first extent the spectral dependence of the gain follow that of the joint density of states (JDOS) in the semiconductor. The gain would change to loss again for photon energies exceeding the separation of quasi-Fermi levels. The gain deviates to some extent from the functional dependence of the joint density of states due to statistical spread as shown. As the pumping level is increased so is the separation between the quasi-Fermi levels and also the extension of the gain to higher energies. The high photon densities needed for laser operation cannot be maintained except in a waveguide cavity. This was indeed the problem faced in the early stages of GaAs laser development that was limited to bulk p–n-junctions with no waveguiding other than the one that could be obtained by index guiding due to injection. In addition, the volume that had to be pumped was huge, making it impossible for any operation except at cryogenic temperatures. With the advent of heterostructures, it has become possible to confine the carriers and the optical field to a small portion of the semiconductor, maximizing the stimulated emission; it makes possible room-temperature and CW operations. If the pumped region contains many nonradiative recombination centers, not only is the photon generation impeded but also the lattice is heated, which feeds the nonradiative processes because they generally require phonon-coupled nonradiative recombination. Just as in the case of any semiconductor, GaN-based materials have suffered and still do from such defects. To everyone’s delight though, even in the light of large structural defects, the material seems to be more robust against the production of nonradiative recombination centers, even in the presence of a large flux of high-energy photons characteristic of wide-bandgap semiconductors.

2.2 Waveguiding

One of the unique attributes of DH lasers is the built-in optical waveguide without which CW semiconductor lasers cannot exist. Waves suffering the total internal

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reflections are supported by the waveguide and are called the waveguide modes. As mentioned earlier, the refractive index of the narrower bandgap active layer at the operating wavelength is larger than the refractive indices of the wide-bandgap emitters forming the waveguide (discussed in Section 2.4.1). The transverse field distribution of such a mode, vertical to the growth direction, can be described by a cosine or a sine function inside the layer. It decays exponentially in the outer, lowrefractive-index media. The refractive index is therefore very critical here and is discussed in the next section. 2.2.1 Refractive Index of GaN and AlGaN

Knowledge of the dispersion of the refractive indices of semiconductors materials is necessary for accurate modeling and design of optical devices. The wurtzite GaN and related nitrides lack cubic symmetry and therefore have anisotropic optical properties. The anisotropy results in uniaxial birefringence, that is, two different refractive indices for polarizations parallel (no-ordinary) and perpendicular (ne-extraordinary) to the c-axis. Refractive index dispersion has been measured for GaN and AlGaN in a variety of ways (interferometry, ellipsometry, and prism coupling) and using samples grown by a variety of methods (MBE, OMVPE, and HVPE) with notable dispersion stemming from the uncertainties and idiosyncrasies of the various techniques used [6–13]. There are three major measurement techniques reported in the literature, which have been used to quantify the refractive indices of nitride semiconductors. Transmission (reflection) interferometry is the simplest of all, where the transmission (reflection) is measured as a function of wavelength [8]. Oscillations are observed in the transparent wavelength region of the sample due to interference of multiple reflections at the surface/interfaces. When half-integer multiples of the wavelength equal the optical path length in the sample, maxima occur in the spectrum. Because the product of the thickness and the index is measured by this technique, thickness values are required from other measurements to determine the refractive index. Thus, the accuracy of this technique mainly depends on the accuracy of the thickness measurement. Additional error is introduced due to interface roughness, buffer layers, and thickness variations that tend to dampen the oscillations and lead to determination of the peak/valley wavelengths and thus the refractive index. The waveguide calculation presented below utilized the refractive index determined by this method [14]. Spectroscopic ellipsometry detects the polarization change in the light beam reflected from the sample [10, 12]. The measured ratio of the reflected light for the TE and TM polarizations allows for determination of the refractive index with a higher accuracy as compared to the interferometric technique. However, the thickness of the film should be on the order of the wavelength to avoid multiple solutions. Consequently, thick films are not suitable for the ellipsometry method, if the thickness is not known from other measurements and does not fall into the region causing multiple solutions.

2.2 Waveguiding

Figure 2.5 Using a micrometer, the semiconductor films are pressed against the base of a rutile-phase TiO2 prism mounted on a rotational stage. The waveguide only supports modes with certain values of the wave vector component in the ~ x direction, kx. With the integer m denoting the mode number, there are distinct angles of incidence qm at which the

values of kx in the prism and the waveguide are equal. At these angles and for air gaps of l/2, the evanescent wave from the prism can couple (“tunnel”) efficiently across the air gap and excite a mode in the waveguide. If qm is measured for two or more modes, the thickness, W, and the refractive index, n, of the film can be obtained simultaneously.

The prism coupling method is a very accurate means for measuring the refractive indices and thickness of thin films [9, 11, 13, 15, 16]. The measurement technique is based on phase matching the light in the prism to the modes allowed in the waveguide formed by the higher index film sandwiched between two lower index layers. It produces a higher accuracy compared to the other two techniques mentioned above; however, it requires the use of laser beams, limiting the measurement to the available laser wavelengths. The experimental setup for the prism coupling technique is shown in Figure 2.5. The propagation constant in the geometry dealing with the prism method is given by
1=2 N  sin a cos 2 þ n2p
sin2 a sin 2; ð2:1Þ where kp and k0 are the magnitudes of the wave vector in the prism and in free space, respectively. The term 2 equals 44.98  0.01o. The propagation constants for the modes in the wavelength Nm are determined by the equation [17]. 1=2
¼ mp þ F0 (n; N m ) þ F2 (n; N m ); k0 W n2
N 2m

ð2:2Þ

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where

"  !#1=2 2 2 n 2p N m
nj : Fj (n; N m )  tan nj n2
N 2m
1

ð2:3Þ

Equations 2.2 and 2.3 allow the redundant measurement of both n and W through the minimization of the variance in W as a function of n. When the spin–orbit splitting is ignored, the refractive index of many III–V materials below the direct bandgap is approximately given by [8, 10, 18, 19] 3 n2 (
hw; x) ¼ A(x)y
2 [2
(1
y)1=2
(1
y)1=2 ] þ B(x) 2
hw ; with y ¼ E g (x)

ð2:4Þ

where Eg(x) is the bandgap energy, x is the composition (Al composition in this case), A and B are x-dependent parameters, and
hw is the optical energy. Brunner et al. [8] and Tisch et al. [10] have used this functional form to fit the index data. Another widely used and simpler functional form, known as the first-order Sellmeier dispersion formula, is also used to fit the refractive index data obtained by prism coupling as has been done by Bergmann et al. [9], Özg€ ur et al. [11], and Sanford et al. [13]: n2 (l; x) ¼ 1 þ

A0 (x)l 2

2

2

l
l0 (x)

:

ð2:5Þ

Here, l is the wavelength, and A0(x) and l0(x) are the composition-dependent adjustable parameters. In addition to a few reports on selected AlGaN layers, there have been about seven somewhat comprehensive studies performed on the refractive indices of AlxGa1
xN [8–13, 20]. Within this set of data, there is considerable discrepancy among the dispersion curves as a function of x. Brunner et al. [8] used transmission spectroscopy to measure the ordinary refractive index of AlGaN epilayers grown on sapphire by plasma-induced MBE. Tisch et al. [10] investigated the temperature dependence of the ordinary refractive index of AlGaN films grown by MOVPE on sapphire substrates using spectroscopic ellipsometry. Laws et al. [20] reported improved functional forms for AlGaN refractive index data in the literature. Vincent et al. [12] measured the ordinary refractive index of AlGaN films grown by MBE on Si substrates. Both transmission spectroscopy and prism coupling techniques have been applied by Sanford et al. [13] to measure the AlGaN films grown on sapphire substrates by HVPE and organometalic vapor phase epitaxy (OMVPE). Using the highly accurate prism coupling technique, Bergmann et al. [9] and Özg€ ur et al. [11] measured the dispersion of the ordinary (no(l)) and extraordinary (ne(l)) indices of refraction, systematically for two different sets of MOCVD- and MBE-grown wurtzitic AlxGa1
xN epitaxial layers with 0  x  0.2 and 0  x  1.0, respectively, throughout the wavelength region 457 < l < 980 nm. The ordinary and extraordinary indices measured at various laser lines for the MBE-grown samples

2.2 Waveguiding

Ordinary index (n 0)

2.45

GaN Al0.15Ga0.85N Al0.25Ga0.75N

2.35

Al0.35Ga0.65N Al0.44Ga0.56N Al0.77Ga0.23N AlN

2.25

2.15

2.05

Extraordinary index (ne)

2.40

2.30

2.20

2.10 500

600 700 Wavelength (nm)

800

Figure 2.6 Measured ordinary and extraordinary index dispersion for MBE-grown AlGaN samples with different Al compositions. The lines are a result of the Sellmeier parameterization with bandgap energy [11].

are shown in Figure 2.6. The data for each sample were fit to the first-order Sellmeier dispersion formula. The resulting values of A0 and l0 were fit to polynomials in x. A0 (x) ¼ B0 þ B1 x;

ð2:6Þ

l0 (x) ¼ C0 þ C1 x þ C2 x 2 : The Bi and Ci (i ¼ 0,1 . . .) coefficients are listed in Table 2.1. Table 2.1 B and C coefficients for the adjustable parameters in the

Sellmeier dispersion formula for the MBE samples parameterized in x for the ordinary and extraordinary refractive indices. Coefficient

no

ne

B0 B1 C0 (nm) C1 (nm) C3 (nm)

4.1446  0.0146
1.0021  0.0273 190.72  2.48
82.999  12.363
27.521  11.619

4.2957  0.0165
0.9817  0.0310 191.71  2.23
76.363  11.142
23.427  10.471

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The agreement between all GaN and AlN refractive index values in the literature is good [6–9, 21–25], but the discrepancies among all reported measurements grow for AlxGa1-xN as the Al content approaches 50%. Other than the less accurate index measurements, discrepancies among different measurements of refractive index dispersion are shown to be a consequence of differing growth conditions with concomitant variations in strain, compositional-induced inhomogeneities, and the corresponding bowing parameters [11, 13]. Of course, the largest source of relative uncertainty is the uncertainty in the accuracy of x, which X-ray and Auger spectroscopy data constrain to 10%. These discrepancies are reconciled when the Sellmeier relation is parameterized not by x but by bandgap energy [11]. Using the bandgap values obtained from absorption measurements, the Sellmeier coefficients were fit to second-order polynomials in Eg. The fitted curves, which are shown in Figure 2.6, reproduce the data to within approximately 0.007. 2.2.2 Refractive Index of InGaN

InGaN is extensively used as the barrier and quantum well layers in the active regions of diode lasers and light-emitting diodes. Even though there are reports on the refractive index of InN [26, 27], there is almost a complete lack of InGaN refractive index data. The main reason is that InGaN suffers from more material inhomogeneities as compared to AlGaN, as well as the lack of sufficiently thick, uniform InGaN layers. A phase separation results in In-rich regions, and an inhomogeneous dielectric function poses a significant problem for accurate determination of the refractive index. Therefore, the widely used method to obtain the InGaN refractive index relies on the shift of the GaN refractive index data with the assumption that the refractive index is identical at the band edge for both materials. To assess the accuracy of this approximate method, but in the realm of AlGaN, Laws et al. [20] calculated the AlGaN refractive index values and compared them with the results of Brunner et al. [8]. For low Al compositions, they obtain an accuracy within 1% over the energy range 3.0–3.42 eV. They conclude that it is reasonable to suppose that this approximate method can be used to estimate the refractive indices for the InGaN alloys with low In compositions. It should be mentioned that this is a zeroth-order approximation where quantum confinement, piezoelectric field, and compositional fluctuation effects are neglected. 2.2.3 Analytical Solution to the Waveguide Problem

Let us consider a three-slab symmetrical waveguide with the core parameters designated with the subscript 1 and cladding-layer parameters with the subscript 2. Although the treatment of the problem of waveguiding has been discussed in many texts, we follow the treatment described in Refs [28, 29], which is also similar to that in Ref. [30]. Total internal reflection of an electromagnetic wave or the wave guidance requires the refractive index of the guide to be larger than that for the cladding layers.

2.2 Waveguiding

Specifically, an electromagnetic wave with the incidence angle with respect to the normal equal to or larger than the critical angle, qc ¼ sin
1

n2 ; n1

ð2:7Þ

will suffer total internal reflection provided n1 > n2. The effective waveguide size is actually larger than its physical dimension due to the Goos–Haechen shift [31]. However, this effect is neglected in the present first-order treatment of the waveguide problem. Before we launch into the waveguide expressions, let us define the media parameters. The complex dielectric function (e) can be related to the refractive index n (nr and n
are also used throughout this book for the same) and the extinction coefficient K according to e ¼ e0 þ je00 ¼ e0 (n þ jk)2 ¼ e0 (n2
k2 þ j2nk):

ð2:8Þ

Because the power of an electromagnetic field propagating along the z-direction is proportional to exp[
2(j(n þ jk)k0 x)];

ð2:9aÞ

where k0 ¼

2p l0

ð2:9bÞ

is the free space wave vector. The absorption coefficient is then given by a ¼ 2kk0 ;

ð2:10aÞ

e00 : 2e0 n

ð2:10bÞ

k¼

It is clear that the imaginary part of the dielectric constant is responsible for the loss term. A TE-mode plane wave propagating along the z-direction within the guide has even and odd solutions. The analytical treatment here will be limited to the even-mode solution in a lossless core, as the waveguide problem has been covered in countless textbooks. Suffice it to say that the electric field and magnetic field components are related to each other with the following expressions: q E y ¼
jwm0 Hx qz q q E x
E z ¼
jwm0 Hy qz qx q E y ¼
jwm0 Hz qx


~ ¼ jwe~ and from rx H E

ð2:11Þ

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q H y ¼ jweE x qz q q Hx
H z ¼ jweE y : qz qx q
H y ¼ jweE z qx


ð2:12Þ

The even-solution field components for TE mode, in phasor notation, are given by E y ¼ E
cos(kx)exp(
jbz);

ð2:13aÞ

Hx ¼


j q b
E cos(kx)exp(
jbz); Ey ¼
wm0 qz wm0

ð2:13bÞ

Hz ¼

j q jk
E sin(kx)exp(
jbz): Ey ¼
wm0 qx wm0

ð2:13cÞ

Here, over lining means that the electric field amplitude, E, is used in phasor notation and the angular frequency, w, is related to the frequency by w ¼ 2pn, b2 ¼ w2me. The notion of guidance within the guide explicitly implies that the wave should decay outside the waveguide. For this to happen, the exponent in the x-direction should be real. Then,      kd d E y2 ¼ E
cos exp
g x
exp(
jbz); ð2:14Þ 2 2 where g 2 ¼ b2
n2 k20 , k0 ¼ 2p/l0, and l0 ¼ c/n, with c being the speed of light in vacuum. For the wave not to propagate or attenuate outside the core of the waveguide, g should be positive, which leads to the condition b2 >n22 k20 , which brings us back to the total internal reflection condition of Equation 2.7. Similarly, the magnetic field components outside the core are     
j q b
kd d E cos Ey ¼
exp
g x
exp(
jbz); ð2:15aÞ Hx2 ¼ wm0 qz wm0 2 2

Hz2 ¼

     j q jg
kd d E cos Ey ¼
exp
g x
exp(
jbz): wm0 qx wm0 2 2

ð2:15bÞ

Applying the boundary conditions that the y-component of the ~ E field and the ~ field must be continuous at x ¼ d/2, we get z-component of the H " #1=2 kd g b2
n22 k20 tan ¼ ¼ 2 2 2 2 k n1 k0
b

or

kd kd gd tan ¼ : 2 2 2

ð2:16Þ

2.2 Waveguiding

γ(d/2) m=0

m=1

m=2

m=3

6 d=0.8 µm 5 d=0.6 µm 4 λ=365 nm d=0.4 3

d=0.4 µm

n=2.58, Al0.1Ga0.97N

2 d=0.2 µm

n=2.7, Al0.03Ga0.97N

1

0

1

2

3

4

5

6

κ(d/2)

Figure 2.7 Graphical approach to determining the waveguide modes. The quarter circles represent the solution for Equation 2.16. The solution for Equation 2.17, with m as the mode parameter, is also shown. The intersections of the two sets of plots are then the solutions for modes supported by the waveguide.

Here, b is the unknown and its solution can be found from the transcendental equation either numerically or graphically. Recognizing k 2 ¼ n21 k20
b2 and g 2 ¼ b2
n22 k20 and adding them leads to the elimination of b2 as follows:  2  2  2 k0 d kd gd ¼ þ : ð2:17Þ (n21
n22 ) 2 2 2 The joint solution of Equation 2.16 is that which satisfies the guidance condition. Equation 2.17 represents a circle with radius r ¼ (n21
n22 )1=2 (k0 d=2) in the kd/2, gd/2 plane. Intersections of the graphical representations of Equations 2.16 and 2.17 are then the desired solutions, as illustrated in Figure 2.7. The left side of Equation 2.16, kd/2 tan(kd/2), is plotted as gd/2 to give the quarter circles in Figure 2.7 for waveguide thicknesses ranging from 0.2 to 0.8 mm. It is clear that, for small values of d, only the fundamental-mode solution, m ¼ 0, exists. Figure 2.8 depicts the field distribution of the fundamental and second-order modes. It is evident that the fundamental mode is necessary for sufficient confinement. The Figure 2.6 displays the refractive indices for AlGaN in the entire range of compositions. The InGaN data are not as complete, but to a first approximation, the

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Intensity

TE0

1.0

TE1

0.8 0.6 0.4 0.2

–0.6

–0.4

–0.2

0

0.2

0.4

0.6 λ=365 nm d=0.4

n=2.58, Al0.1Ga0.97N

n=2.7, Al0.03Ga0.97N Figure 2.8 Optical field profile of fundamental and second-order modes for TE radiation obtained from Equation 2.13.

refractive index of this ternary at the wavelength corresponding to its band edge can be assumed to be the same as that for GaN at the wavelength corresponding to its band edge. While this deduction is reasonable for AlGaN, the InGaN figures so derived may have more errors. 2.2.4 Numerical Solution of the Waveguide Problem

Assume for the general case that the number of layers is l. Each layer has the (complex) refractive index nj and the thickness Dtj. Because the dimension of the waveguide is much larger in the y-direction than in the x-direction, and z is the propagation direction, the amplitude of the electromagnetic oscillation for a certain mode in the waveguide may be written as E y (x; z; t) ¼ E y (x)ej(wt
Gz) ;

ð2:18Þ

where G is the complex propagation constant for the mode. For the jth layer, the wave equation describing the optical propagation is given by [32]

2.2 Waveguiding


q2 E y;j (x)
G2
k20 n2j E y; j (x) ¼ 0; qx 2

ð2:19Þ

where k0 ¼ 2p/l is the wave number in vacuum and nj is the refractive index of the jth layer, which may be complex. The solution of this equation can generally be written as E y;j (x) ¼ Aj eaj (x
tj ) þ Bj e
aj (x
tj ) ;

ð2:20Þ


1=2 aj ¼ G2
k20 n2j :

ð2:21Þ

where

Our task now is to find the propagation constant, G. It can be accomplished by using the boundary conditions for each mode. Here, for example, the TE mode requires that Ey(x) and its first derivative be continuous at each interface, which leads to E y; j (tj ) ¼ E y; j þ 1 (tj ); ð2:22Þ

q q E y; j (tj ) ¼ E y; j þ 1 (tj ): qx qx

From Equations 2.21 and 2.22 it can be shown that the amplitude coefficients Aj and Bj for adjacent layers are related as Aj þ 1 Bj þ 1

! ¼

(1 þ g j þ 1 )edj (1
g j þ 1 )e
dj

!

(1
g j þ 1 )edj (1 þ g j þ 1 )e
dj

Aj Bj

! ¼ Tj

Aj Bj

! ;

ð2:23Þ

where g j þ 1 ¼ aj/aj þ 1 and d j ¼ ajDtj. In addition, because the oscillation will be confined within the waveguide, we can assume that E y;0 (
1) ¼ E y;k ( þ 1) ¼ 0:

ð2:24Þ

Hence, we can derive a relation between the solutions for the first and the last layer, and obtain 0 Bk

! ¼ T WG

A0 0

! ¼

t11 t12 t21 t22

!

A0 0

! ;

ð2:25Þ

where T WG ¼ T k
1  T k
2    T 1  T 0

ð2:26Þ

is called the transmission matrix of the waveguide. It is easy to see that in order to satisfy the relation, t11 must be zero. This fact is taken advantage of to solve the wave equations by numerically searching on the complex surface for the value of Gj that gives the least absolute value of t11. After Gj has been obtained, Equation 2.20 can be utilized to

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calculate the electromagnetic wave whose mode squared gives the intensity distribution. The confinement factor, which is commonly represented by the symbol, G, is obtained by determining the ratio of the total intensity within the active layers to that within all layers. Another important parameter in lasers is the confinement factor, G, which is defined as the ratio of the optical power overlapping with the active layer to the total optical power in that mode. Mathematically, Ð d=2 2
d=2 E(x) dx G ¼ Ð1 ; ð2:27Þ 2
1 E(x) dx where E(x) is the distribution of the optical field amplitude in the x-direction. The squared exponent is indicative of the need to use optical power. The method described above has been applied to waveguides employed in lasers that have been reported so far, namely, those utilizing InGaN active regions. Potential laser structures relying on GaN active layers and AlGaN waveguide and cladding layers, with different mole fractions of course, have been treated too. Figure 2.9 1 W=0.08 µm W=0.1 µm W=0.12 µm

Al0.15Ga0.85N

0.8

GaN

W

W

Relative intensity (a.u.)

InGaN 400 Å 0.6

0.4

0.2

0 –0.4

–0.2

0

Distance (µm) Figure 2.9 The optical field (E field) distribution for a four-well MQW InGaN active layer, GaN waveguide, and AlGaN cladding layer containing 15% Al, which represent a Nichia InGaN laser. The variable parameter is the thickness W of the GaN on each side of the active layer and is one-half the total thickness of the waveguide.

0.2

0.4

2.2 Waveguiding

1 x=0.1 x=0.2 x=0.3

Alx Ga1-x N

Relative intensity (a.u.)

0.8

GaN

W

W

W=0.1 µm InGaN 400 Å

0.6

0.4

0.2

0 –0.4

–0.2

0

0.2

0.4

Distance (µm) Figure 2.10 The field for the waveguide depicted in Figure 2.9 where the thickness of the GaN guide on one side is fixed at 0.1 mm while the AlN mole fractions in the cladding layers are allowed to have the values 0.1, 0.2, and 0.3.

exhibits the optical field (E field) distribution for a MQW InGaN active layer (GaN waveguide) and an AlGaN cladding layer containing 15% Al. The variable parameter is the thickness W of the GaN film on each side of the active layer. The active layer consists of four wells made of InGaN with 15% In and barriers made of InGaN with 5% In having a total MQW-region thickness of 400 Å; it replicates a Nichia laser structure. Here, W is allowed to have the values 0.08, 0.1, and 0.12 mm. Figure 2.10 exhibits the field for the waveguide depicted in Figure 2.9, where the thickness of the GaN guide on one side is fixed at 0.1 mm while the AlN mole fractions in the cladding layers are allowed to have values of 0.1, 0.2, and 0.3. Figure 2.11 shows the field and the refractive index distribution calculated for the same waveguide (Figure 2.9). The effect of the 200 Å Al0.2Ga0.8N employed to keep the InGaN MQW from dissociating during the growth of the top waveguide has been neglected. To interrogate the effect of the 200 Å Al0.2Ga0.8N layer designed to prevent the dissociation of the InGaN MQW region during the high-temperature growth, Figure 2.12 displays the field and refractive index distributions taking this 200 Å thick Al0.2Ga0.8N layer into consideration. Figure 2.13 depicts the variation of the confinement factor with the half-guide thickness. The parameter is the mole fraction in the cladding layers. The other

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1 Intensity Refractive index W=0.1 µm

Relative intensity (a.u.)

400Å W

InGaN W

GaN

0.5 Al0.15Ga0.85N

0 0

0.2

0.4 0.6 Distance (µm)

0.8

1

Figure 2.11 The field and refractive index distributions calculated for the waveguide as in Figure 2.9. Parameters used are 2.536 for the effective index, amplification region 6.46%.

parameters of the waveguide are as shown in the inset. Figure 2.14 exhibits the confinement factor for a guide, the characteristics of which are shown in the upper left corner, with varying quantum well thickness. The expected linear dependence is apparent. Let us now turn our attention to potential laser structures relying on GaN active layers, and AlGaN waveguide and cladding layers. Figure 2.15 presents the optical field (E field) distribution for a 50 Å GaN active layer (Al0.03Ga0.97N waveguide) and A10.2Ga0.8N cladding layers. The variable parameter is the thickness W of the Al0.03Ga0.97N film on either side of the GaN active layer, which ranges from 300 to 600 Å. Figure 2.16 shows a plot of the field for the waveguide depicted in Figure 2.15, where the thickness of the GaN guide on one side is fixed at 0.05 mm while the AlN mole fraction in the cladding layers is allowed to have the values 0.1, 0.2, and 0.3. Figure 2.17 shows the field and refractive index distributions calculated for the waveguide with three Al0.03Ga0.97N/GaN MQW wells, a 700 Å Al0.03Ga0.97N/GaN guide, and two Al0.1Ga0.9N/GaN cladding layers. Figure 2.18 exhibits the variation of the confinement factor with the half-guide thickness. The parameter is the mole fraction in the cladding layers. The other parameters of the waveguide are as shown in the inset. Figure 2.19 depicts the confinement factor for a guide, the characteristics of which are given in the inset, with a varying QW

2.2 Waveguiding

1

3 Intensity Refractive index

W=0.1 µm

Relative intensity (a.u.)

400Å InGaN GaN 0.5

W

W

2.5

Al0.15 Ga 0.85N Al0.2Ga0.8N

2

0 0

0.2

0.4

0.6

0.8

1

Distance (µm) Figure 2.12 The field and refractive index distributions taking the 200 Å thick A10.2Ga0.8N layer into consideration, which is designed into the Nichia laser to prevent the InGaN MQW region from dissociation during the waveguide growth. This takes place at about 100–200  C above the InGaN deposition temperature. Parameters used are 2.534 for the effective index, amplification region 8.93%.

thickness for the half-guide thicknesses 400, 500, and 600 Å. The expected linear dependence is apparent. The method described above treats each layer as being the same, which means that the algorithm requires the same amount of computational time per layer. When the number of QWs in the active layer is very large, this method would become very timeconsuming, and would not be worthwhile because each QW layer is usually too thin to affect the final results. One way to simplify the calculation involving a large number of QWs is to treat all of them as one thick layer with an effective alloy content x that can be determined by the weighted average of all well and barrier layers. Once a single refractive index (effective refractive index) n is assigned to this thick layer, the procedure described above can be carried out. Because there is mutual coupling between adjacent QW layers, the effective index depends strongly on the structural parameters of the single layer as well as the value of x. Various methods have been advanced to determine the best value of n based on these parameters [33, 34]. It should be pointed out that the lack of experimental data, particularly, the refractive indices of InxGal
xN and to a lesser extent AlxGal
xN, necessitated linear interpolation.

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0.025

x=0.3

0.02

Confinement factor, Γ

x=0.2 x=0.15

0.015

x=0.1

Alx Ga1-x N

0.01

GaN

W W InGaN L z (35 Å)

0.005 0

200

400 600 800 1000 Waveguide thickness, W (Å)

1200

1400

Figure 2.13 Variation of the confinement factor with the halfguide thickness. The details of the guide are shown in the lower right corner of the figure.

0.2 Al0.15Ga0.85N W

GaN

Confinement factor, Γ

0.15

W=0.1 µm W

InGaN L z (Å)

0.1

0.05

0 0

100

200

300

400

500

Well thickness (Å) Figure 2.14 Confinement factor for a guide, the characteristics of which are shown in the upper left comer, with varying quantum well thickness.

600

700

800

2.2 Waveguiding

1.2

λ0 = 364 nm Al0.2Ga0.8N

1

Al0.03Ga0.97N

W= 300 Å W= 400 Å W= 500 Å W= 600 Å

W

W

Relative intensity (a.u.)

GaN 50 Å

0.8

0.6

0.4

0.2

0 –0.4

– 0.2

0

0.2

0.4

Distance (µm) Figure 2.15 Optical field (E field) distribution for a 50 Å GaN active layer, A10.03Ga0.97N waveguide, and A10.2Ga0.8N cladding layers of varying total thickness from 60 to 120 nm (W ¼ 30–60 nm).

While the most up-to-date data have been sought, the accuracy of the calculation is certainly limited by the uncertainty in the refractive index, particularly in InGaN, as the mole fractions employed are not accurately known by the practitioners. 2.2.5 Far-Field Pattern

Knowledge of the radiation characteristics, that is, the far-field pattern, of a semiconductor laser is necessary for proper collection of the radiated power. It is also a measure of the waveguiding properties that gives a reality check on the calculations of the field distribution and the validity of the parameters used such as the refractive indices that are functions of not only the composition but also the doping and the temperature (in an expanded view, the refractive index would be a function of strain and also the presence of any externally applied electric field, as in some optical, mainly passive, devices). Diffraction expressions must be used for a given mode or sets of modes to calculate the far-field pattern, which is beyond the scope of this text. The form of the radiation pattern emanating from a semiconductor laser is exhibited in Figure 2.20, where q? and q//represent the

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1.2 E distribution in 500 Å guide Alx Ga1–x N 1

x=0.1 x=0.2 x=0.3

Al0.03Ga0.97N 500 Å 500 Å

Relative intensity (a.u.)

GaN 50 Å

0.8

0.6

λ0 = 364 nm n GaN 2.8 Al0.03Ga0.97N 2.7 Alx Ga1-x N x=0.1 2.58 x=0.2 2.5 x=0.3 2.44

0.4

0.2 x=0.1

0 –0.4

–0.2

0 Distance (µm)

0.2

0.4

Figure 2.16 Field distribution for the waveguide of Figure 2.15 where the thickness of the GaN guide on one side is fixed at 0.05 mm while the AlN mole fraction in the cladding layers were allowed to have the values 0.1, 0.2, and 0.3.

full angles at half power in the directions perpendicular and parallel to the plane of the p–n-junction, respectively. Because the dimension perpendicular to the junction plane is small, considerable divergence of the beam occurs in this direction. To a first approximation, a uniform aperture radiates a single lobe of which the full width at half power, q?, changes inversely with the thickness W of the emitting region [35], as given by q? 

1:2l : W

ð2:28Þ

Similarly, for a diode with a strong confinement in the lateral direction such as in index-guided lasers of width Z, the beam width in the plane of the junction is, again for a uniform aperture, q== 

1:2l : Z

ð2:29Þ

There exist higher order modes, as well, that give rise to complex radiation patterns, which cannot be represented by Equations 2.27 and 2.28. The semiconductor is not

2.2 Waveguiding

1

3 Field Refractive index

Al0.1Ga0.9N, n=2.58 Refractive index

Relative intensity (a.u.)

λ0 = 364 nm

GaN 0.5 Al0.03Ga0.97N

x=0.1

0 – 0.4

–0.2

0 Distance (µm)

0.2

0.4

2.5

Figure 2.17 Field and refractive index distributions calculated for the waveguide with three-well cladding layers: Al0.03Ga0.97N/GaN MQW, 700 Å A10.03Ga0.97N guide, and Al0.1Ga0.9N.

impartial to which mode propagates, and the mode that has the largest fraction of its intensity distribution overlapping the gain region will have the largest gain. Moreover, the facet reflectivity has an effect, but it is mainly manifested in the polarization. For example, a plane wave propagating in a waveguide whose E field lies in the facet experiences a lower facet loss than other polarizations, culminates in the TE mode that is supported. The far-field pattern can be obtained by considering the TE waves, as we have done in conjunction with waveguiding, and solving the wave equation in free space, the details of which can be found in many textbooks including in Refs [30, 35–38], gives the following expression: Ð þ 1 2 I(q? ) cos2 q? 
1 E y (x; 0)exp( j sin q? k0 x)d x  ; ¼ Ð þ 1 2 I(0)  E y (x; 0)dx 

ð2:30Þ


1

where k0 is the magnitude of the propagation vector in free space and Ey is the y-component of the electric field, it is the only component because the field is a TE mode.

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0.045 x=0.1 x=0.2 x=0.3

x=0.3

Confinement factor,Γ

0.04

0.035

x=0.2

0.03

x=0.1

0.025 Alx Ga 1–x N

0.02

Al 0.03 Ga 0.97 N 0.015

W GaN

W

50 Å 0.01 100

200

300

400

500

600

700

Half-waveguide width (Å) Figure 2.18 Variation of the confinement factor with the halfguide thickness. The guide details are stated on the bottom right side.

2.3 Loss, Threshold, and Cavity Modes

In addition to losses germane to the semiconductor and the waveguide properties, the end losses must also be compensated for by amplification inside the laser cavity. There are also other losses distributed inside the cavity. For instance, losses associated with transitions of electrons (holes) inside the conduction (valence) band via absorption are called free-carrier losses. Other losses that obey the usual absorption law I(z) ¼ I0e
az, with I0 being the incident light intensity, a the absorption coefficient, and z the distance, also come into play. Recognizing that the distributed losses in the active layer and in the cladding layers (aa and ac, respectively) are not identical, we can write for the internal loss as ai ¼ Gatot ¼ Gaa þ (1
G)ac ;

ð2:31Þ

where atot represents the total loss and G denotes the total confinement factor, as defined by Equation 2.27. Qualitative conditions for the onset of laser action or the threshold condition are obtained when the gain or negative absorption, at the population inversion condition, compensates for all losses sustained during a round trip in the laser cavity. This is given by I0 R1 R2 exp[2(g th
ai )L] ¼ I0 :

ð2:32Þ

2.3 Loss, Threshold, and Cavity Modes

0.14 Γ versus Lz, Wg=400, 500, 600 Å

0.12

Confinement factor

0.1

0.08

0.06

0.04 Wg=400 Å

0.02

Wg=500 Å Wg=600 Å

0 0

50

100 150 Well width (Å)

200

Figure 2.19 Confinement factor for a guide, the characteristics of which are shown in the inset with varying quantum well thickness for half-waveguide thicknesses of 400, 500, and 600 Å.

Laser

Str

ipe

x

θ y

z

Figure 2.20 Schematic representation of the far-field characteristics of an edge-emitter injection laser.

θ

250

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The amplitude portion of which leads to g th ¼ ai þ

1 1 ; ln 2L R1 R2

ð2:33Þ

where gth is the threshold gain, ai represents the distributed losses, R1 and R2 are the reflection coefficients of the laser facets (which are generally made unequal as the light extraction from one facet is favored over the other), and L is the cavity length (i.e., the length of the laser diode). The first term in Equation 2.33 is the internal loss term while the second term represents the end losses from the facets. The phase portion of Equation 2.32, that is, the condition that a wave making a full round trip in the cavity must be in phase for sustained oscillations, leads to   4pnr L lm ¼ 2mp or simply m ¼ L; ð2:34Þ 2nr lm where m is an integer taking the values 1, 2, 3, . . ., nr is the refractive index, and lm and lm/nr are the free space and inside-the-cavity wavelengths, respectively. The longitudinal-mode spacing near the fundamental mode l0 (corresponding to m ¼ 1) can be obtained from the derivative of the phase condition as dml0 þ m dl0 ¼ 2Ldnr :

ð2:35Þ

By setting dm ¼
1 and substituting m from Equation 2.34, one can find the adjacentmode spacing [30, 39]: 2

Dl0 ¼

l0 : 2L[nr
l0 (dnr =dl0 )]

ð2:36Þ

Polynomial fits to the data presented in Figure 2.6 and can ameliorate the calculation of the longitudinal modes determined by Equation 2.36. In a modern semiconductor laser, the propagating mode is not entirely with the active layer, in which case Equation 2.33 must be modified to take the confinement factor into account: Gg th ¼ ai þ

1 1 ; ln 2L R1 R2

ð2:37Þ

where Ggth ¼ gnth is the modal gain with gnth being the intrinsic or material gain at threshold (intrinsic gain). All the waveguide losses other than the end loss will be lumped into ai ¼ Gaa þ (1
G)ac, where ai is the internal loss.

2.4 Optical Gain

The gain is a figure of merit that indicates how well an electromagnetic field is amplified as it traverses through the semiconductor laser medium. It is needed to

2.4 Optical Gain

overcome internal losses as well as external losses such as radiation out of the facets, which are also referred to as end losses. Calculation of the gain is rather complicated and requires a very accurate knowledge of the genesis of the lasing mechanisms as well as the semiconductor band structure that includes effective masses. The gain can be calculated from the spontaneous emission spectrum or the absorption spectrum if we know the recombination mechanism, that is, the electron–hole plasma or the excitonic origin. The gain expression is directly related to the band structure, the transition probabilities, and the occupation probabilities. At high injected electron and hole densities, which is the case in semiconductor lasers, Coulomb interactions are screened completely in small-bandgap semiconductors with low exciton binding energies. This implies that the excitons are dissociated. In such a case, electrons and holes can be treated independently as plasma, except for the many-body effects such as bandgap renormalization. As a first approximation, additional assumptions are made in the framework of which the electrons are assumed to interact with the electromagnetic field but not with other electrons and phonons; this is the basis for the so-called single-particle model. Let us now discuss the simultaneous processes, that is, spontaneous emission, stimulated emission, and absorption as they are eventually related to gas, solid, and semiconductor lasers. Among the most important optical processes taking place in semiconductors, which directly influence device operation, are the absorption and emission of photons. Absorption and emission spectroscopies can be employed to extract a plethora of useful information about a semiconductor, particularly in a direct-bandgap semiconductor. As such, these techniques are commonly taken advantage of to shed light on the materials’ properties and gather data that could be used for devices. The fact that a great majority of GaN-based layers are grown on transparent sapphire substrates paves the way for performing absorption measurements without the need for an often cumbersome substrate removal. Simply stated, the semiconductor is transparent to below-the-gap radiation while it absorbs the above-the-gap radiation. Population inversion, which is a nonequilibrium process, changes this. Excitonic absorption is superimposed on the top of band-to-band absorption, which makes it convenient to investigate the role of excitons in device features. It is instructive to determine the gain coefficient and the lasing conditions by considering the three above-mentioned governing processes, that is, spontaneous emission, absorption, and stimulated emission in a medium be it gas, solid state, or semiconductor. Relying on the treatment of Planck, Einstein [40, 41] treated stimulated emission per unit electromagnetic energy with energies between hn and h(n þ Dn) due to the transition from level 2 to level 1 by the coefficient B21, which describes the transition probability from state 2 to 1. For transitions from level 1 to level 2, absorption were described by the transition probability B12. The spontaneous emission from level 2 to level 1 is depicted as A21. These A and B coefficients are called Einstein’s A and B coefficients, see for example Refs [18, 42, 43]. In a two-level system, the rate of upward (1 ! 2) and downward (2 ! 1) transitions for a system in thermal equilibrium at temperature T, were expressed by

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Einstein [40] as r 21 ¼ B21 re (n)

and

r 12 ¼ B12 re (n):

ð2:38Þ

The term re(n)dn is the volume density of the electromagnetic energy in the frequency range n and n þ dn or in the energy range of hn and hn þ dhn. The coefficients B21 and B21 are called Einstein’s B coefficient, as alluded to above. In a semiconductor, occupation probabilities of levels 1 and 2 must also be considered in which case Equation 2.38 would be modified as r 21 ¼ B21 f 2 (1
f 1 )r(n)

and

r 12 ¼ B12 f 1 (1
f 2 )r(n);

ð2:39Þ

where f1 and f2 represent the occupation probabilities of level 1 and level 2, respectively. Therefore (1
f1) and (1
f2) represent the probabilities of levels 1 and 2 being empty, respectively. Essentially, if a transition is to occur between level 1 and level 2, there must be electrons available in level 1, indicated by f1, and there must be empty states available in level 2, indicated by (1
f2), to receive that electron making the transition from level 1 to level 2. If N2 and N1 represent the population levels (or photon occupation numbers) of levels 2 and 1, respectively, the rate of change in the population of level 2 through spontaneous emission (represents decay in population, which is why the – sign) can be written as  dN 2  ¼
A21 N 2 : ð2:40Þ dt  spon

Note that spontaneous emission is not coupled to the optical field and therefore does not depend on the photon density in the system having the same energy. The above rate equation simply indicates that the photon population in level 2 would decrease at a rate of A21 by which we can argue that the spontaneous emission lifetime is tsp ¼ (A21)
1. Typically, this lifetime is on the order of 10
9 s. Turning our attention to stimulated emission, which again involves transitions from level 2 to level 1, we can write the rate equation as  dN 2  ¼
B21 N 2 re (n): ð2:41Þ dt st Note that this process is coupled to the photons having the same energy in the system, which is the genesis for the re(n) term. The product re(n)N2dn represents the photon energy density in the frequency range of n and n þ dn. A decrease in the N2 population due to transition from level 2 to level 1 would be accompanied by an equal increase in N1, which means that Equation 2.41 can also be written as  dN 1  ¼ B21 N 2 re (n): ð2:42Þ dt st Doing the same for stimulated absorption whose proportionality constant is B12 (as absorption involves excitation of an electron from level 1 to level 2)   dN 2  dN 1  ¼ R12 ¼ B12 N 1 re (n) or ¼
R12 ¼
B12 N 1 re (n): dt  dt  absorp

absorp

ð2:43Þ

2.4 Optical Gain

Note that the stimulated absorption, which is what is involved here, depends on the photon density, having the same energy, in the system. Again, the product re(n)N1dn represents the photon energy density in the frequency range of n and n þ dn. Let us now define the expression governing re(n). The sum of spontaneous emission and stimulated emission represents the total downward transitions from level 2 to level 1. The total rate of these two processes can then be expressed as R21 ¼ [A21 þ B21 re (n)]N 2 :

ð2:44Þ

Under thermodynamic equilibrium, each upward transition must be balanced by a downward transition, in which case we can write R21 ¼ R12 YA21 N 2 þ B21 N 2 re (n) ¼ B12 N 1 re (n):

ð2:45Þ

Manipulation of Equation 2.45 leads to N2 B12 re (n) : ¼ N 1 A21 þ B21 re (n)

ð2:46Þ

Determination of the photon density requires knowledge of the number of modes that can be accommodated in a cavity [44]. To find this, we begin with the wave equation q2 E q2 E q2 E 1 q2 E þ þ ¼ : qx 2 qy2 qz2 (c=nr )2 qt2

ð2:47Þ


0 e
jkx and so on, the dispersion relation reduces to For an E field of the form Ex ¼ E


 (2p)2 (2pn)2 (2pnnr )2 ¼ : k2x þ k2y þ k2z ¼ 2 ¼ c2 (c=nr )2 l

ð2:48Þ

Application of the boundary conditions (field strength is zero at the walls of the cavity) leads to discretization of the wave numbers in x-, y-, and z-direction as my p mx p for my ¼ 1; 2 . . . ; kx ¼ for mx ¼ 1; 2; . . . ; ky ¼ b a mz p for mz ¼ 1; 2 . . . ; and kz ¼ c ð2:49Þ where, a, b, and c represent the cavity dimensions. Assuming a cubic cavity with a side dimension of a for simplicity, with the help of Equation 2.48, Equation 2.49 can be rewritten as m p 2 m p 2 m p 2 2pnn 2 y x z r þ þ ¼ ; ð2:50Þ a c a a where mx, my, and mz are integers describing available modes. Solving Equation 2.50 for frequency gives rise to     1=2 
c
2 2nr an 2 mx þ m2y þ m2z or m2x þ m2y þ m2z ¼ ¼ R2 : n¼ 2nr a c ð2:51Þ

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Therefore, there is one mode in a volume (dmx ¼ 1)(dmy ¼ 1)(dmz ¼ 1). The number of modes between n ¼ 0 and v can be found by finding the volume of 1/8th (if only the positive values of the integers are taken, we limit ourselves to only the 1/8th of the sphere) of a sphere with the radius R ¼ (2nan=c) (see Equation 2.51) as V¼

  1 4pR3 : 3 8

ð2:52Þ

Recognizing that each point represents two modes, TE and TM, we can find the total number of modes, that is, resonances between the frequencies 0 and n as N ¼ 2x

  1 4pR3 8pn3r n3 3 ¼ a : 3 3c 3 8

ð2:53Þ

It is, however, more typical to express the mode density (instead of the electromagnetic energy density introduced in Equation 2.38) in a frequency interval between n and n þ dn (in a frequency interval dn) and for a unit volume, which can be formulated by r(v)dv ¼

1 dN 8pn3 dv ¼ 3 r v2 dv: c V dv

ð2:54Þ

Note that r(v) denotes the photon mode density in units of m
3 whereas re (n) depicts the energy density in units of J m
3. In a semiconductor, the density of states would be considered to get to the photon density. In that case, if the momentum vector is described as ~ k  x^kx þ y^ky þ z^kz ;

ð2:55Þ

the density of allowed values of k in a volume V is the number of cubes of face 2p/a that can be fit in that volume in k-space. The density of states is then the number of states between the momentum values of k and k þ dk. Recognize that the volume of a spherical shell of thickness dk is 4pk2dk and the unit volume in k-space is (2p/a)3. The density of modes is the product of the volume in k-space and the number of states divided by the volume in real space (V ¼ a3). Doing so leads to dN(k) ¼

[2(4pk2 dk)=(2p=a)3 ] : V

ð2:56Þ

The factor 2 is picked up due to the presence of two different polarizations for a given photon of given momentum, that is, TE and TM modes. Rearranging Equation 2.56 with help of k ¼ (2p)=l ¼ (2pvnr )=c leads to dN(k) ¼

 2 k dk: p2

ð2:57Þ

Noting that dk ¼ (2pnr/c)dn[1 þ (n/nr)(dnr/dn)] (the term in bracket is due to the dispersion of the refractive index, namely, due to its frequency dependence) and

2.4 Optical Gain

with k ¼ (2p)/l ¼ (2pnnr)/c, Equation 2.57 can be rewritten as dN(v) ¼

    8pn3r v2 v dnr dv: 1 þ 3 c dv nr

ð2:58Þ

Often times, a group refractive index is defined to account for the dispersion in the refractive index. Equation 2.58 can be rearranged as    8pn2r n2 dnr dN(n) ¼ dn: nr þ n c3 dn

ð2:59Þ

Defining n
g ¼ nr þ n(dnr =dn) allows us to rewrite Equation 2.59 in the form of dN(n) ¼

8pn2r ng n2 dn: c3

ð2:60Þ

If the refractive index dispersion is neglected, n
g  nr , the density of modes of Equations 2.58 and 2.60 take the more familiar form of dN(n) ¼

8pn3r n2 dn: c3

ð2:61Þ

The number of modes in a volume V in a frequency interval of around a central frequency n is obtained by multiplying Equation 2.61 with volume V: dN 0 (n) ¼

8pn3r n2 V dn: c3

ð2:62Þ

Instead of the number of photons in the frequency spectrum, the number of photon energy quanta around a certain photon energy is useful. To do so we replace n ! E/h and dn ! dE/h in Equation 2.62, which gives rise to dN 0 (E) ¼

8pE 2 n3r V dE: h3 c 3

ð2:63Þ

The number of modes (photon density) per unit volume and per unit frequency (spectral density) is then the derivative of N0 with respect to n (or w.r.t. E if we use Equation 2.63) divided by the volume, which represents the spectral density:

r(v) ¼

1 dN 0 (n) 8pn2 n3r 1 dN 0 (E) 8pE 2 n3r or in terms of energy : r(E) ¼ : ¼ ¼ 3 3 c V dn V dE h c3 ð2:64Þ

Boltzmann statistics imply that the probability of a given cavity mode that lies between hn and hn þ hdn is proportional to exp(
hn/kT)hdn. The average energy per

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mode using the aforementioned distribution function is given by hEi ¼

hn E ¼ : ehn=kT
1 eE=kT
1

ð2:65Þ

The average number of photons for each mode is given by Equation 2.65 divided by hn, which leads to hri ¼

1 ehn=kT
1

¼

1 eE=kT
1

:

ð2:66Þ

The photon mode density can be expressed as the product of Equations 2.64 and 2.66 r(v) ¼

8pn2 n3r 1 : 3 hn=kT c e
1

ð2:67Þ

Similar to the transition from Equation 2.62 to Equation 2.63, transitioning Equation 2.67 to energy leads to r(E) ¼

8pE 2 n3r 1 : h3 c 3 eE=kT
1

ð2:68Þ

Moreover, maintaining the dispersion of the refractive index causes Equation 2.67 to be written in the form r(v) ¼

8pn2 n3r 1þ(v=nr )(dnr =dv) : c3 ehn=kT
1

ð2:69Þ

Equation 2.69 can also be expressed in terms of energy, which is more convenient for semiconductors in the form r(E) ¼

8pn3r E 2 1þ(E=nr )(dnr =dE) : exp(hv=kT)
1 h3 c 3

ð2:70Þ

Equations 2.69 and 2.70 represent the Planck’s formula (Planck’s blackbody radiation distribution law). Now that we have treated the spectral density of photons, we can begin the long journey of finding the lasing conditions. In terms of Einstein coefficients, the term A21N2 (see Equation 2.45) represents the spontaneous emission process with which the electromagnetic wave does not participate, and B21re(n)N2 denotes the stimulated emission process with the electromagnetic wave participating. Because the spectral width of the radiation is finite, the unit of the beam intensity is W m
2 per unit frequency interval. Energy balance in equilibrium requires that R12 and R21 be equal (R12 ¼ R21, see Equation 2.45), which determines the spectral distribution. In other words, the upward transition rate must be equal to the total downward transition rate at thermal equilibrium. First, in thermal equilibrium the Boltzmann statistics requires that

2.4 Optical Gain

   E 2
E 1 N 2 ¼ N 1 exp
or kT

 
hn N 2 ¼ N 1 exp : kT

ð2:71Þ

With the help of Equations 2.45 and 2.71, the requirement that the rate of upward transitions must be balanced by the rate of downward transitions (stimulated emission plus the spontaneous emission) in equilibrium can be rewritten as    
hn
hn ð2:72Þ þ B21 re (n)exp ¼ B12 re (n): A21 exp kT kT Solving Equation 2.72 for re(n), we obtain re (n) ¼

A21 exp[
hn=kT] A21 1 ¼ : B12
B21 exp[
hn=kT] B21 (B12 =B21 )exp(hn=kT)
1

ð2:73Þ

Forcing B21/B12 ¼ 1 (which has some basis when only the temperature-dependent parts are considered) brings Equation 2.73 in line with Planck’s formula, given in Equation 2.67 (but multiplied with hv since in Equation 2.67 r(n) is defined as density of modes with energy hv). Therefore, equating Equations 2.67 and 2.73 leads to a relationship between the Einstein’s A21 (associated with spontaneous emission) and B21 (associated with stimulated emission) coefficients as A21 ¼

8phn3 n3r B21 c3

or

B21 ¼

A21 c 3 8phn3 n3r

ð2:74Þ

(Note that A21 has unit s
1 and B21 has unit m3 J
1 s
2.) The term relating the A21 and B21 coefficients in Equation 2.74 represents the density of the electromagnetic waves with the frequency between hn and hDn inside the medium times hn; in other words, the density of the electromagnetic wave energy with the frequency between hn and hDn inside the medium. Let us now take the necessary steps to calculating an expression for the gain. Using Equations 2.43 and 2.44 for stimulated to spontaneous emissions, we can write R21 (st) B21 N 2 re (v) : ¼ R21 (sp) A21 N 2

ð2:75Þ

We should keep in mind that during laser operation r(v) is not described by Equation 2.67 in that r(v) is much larger than its equilibrium value. With the help of Equation 2.74, Equation 2.75 can be rewritten as R21 (st) c 3 re (n) : ¼ R21 (sp) 8phn3 n3r

ð2:76Þ

Considering the absorption and assuming B12 ¼ B21, we can write R21 (st) N 2 ¼ : R12 (ab) N 1

ð2:77Þ

For stimulated emission to exceed photon absorption, population inversion must be achieved, which means, from Equation 2.77, N2 > N1. In addition, for stimulated

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emission to far exceed spontaneous emission described in Equation 2.76, we must have large photon density, which can only be obtained in an optical cavity. Population inversion is interesting in that it is not consistent with usual semiconductors statistics at equilibrium, which calls for density reduction as the energy increases. In the case of population inversion, the density N2 at a higher energy must be smaller than that the density N1 at a lower energy. This simply means lasers do not operate in thermal equilibrium rather they operate in nonthermal equilibrium. Let us then consider an optical cavity (appropriately pumped as in lasers) within which an electromagnetic radiation is propagating along z-direction. Let us further assume that the optical intensity at a point z along the cavity is given by l and that the optical intensity at point z þ dz is given by I þ dI. Unlike the lossy media, the wave is amplified as it propagates due to stimulated emission being larger than the sum of spontaneous emission, absorption, and applicable losses. In a lossy medium, the exponent of the wave peroration would be expressed by exp(
az) with a being the absorption coefficient. However, in lasers, gain is attained, in which case the exponent is expressed in terms of exp(gz) with g being the gain coefficient that is a positive number (cm
1). The optical power intensity along the cavity is proportional to the coherent photon density, Nph, and the associated photon energy hv. The coherent photons travel along the cavity with a velocity, n ¼ c=nr . The gain coefficient can then be defined as g¼

dN ph nr dN ph dI ¼ : ¼ Idz N ph dz cN ph dt

ð2:78Þ

The difference between emission R21 ¼ [A21 þ B21re(n)]N2 (in semiconductors R21 ¼ [A21 þ B21re(n)] f2(1
f1)) and absorption R12 ¼ B12re(n)N1 (in semiconductors R12 ¼ B12re(n) f1(1
f2)) (see Einstein’s relations of Equations 2.44 and 2.45) leads to the net rate of change in the density of coherent photons as (neglecting spontaneous emission) dN ph dN ph  ¼ [N 2
N 1 ][B21 re (n)]: dt dt

ð2:79Þ

The intensity of the electromagnetic wave is related to the photon density per unit frequency times the velocity (given by dw/dk ¼ (2pdE)/(hdk) ¼ c/nr, see the discussion immediately following Equation 2.57): I(z) ¼ re (v)jz

c chv ¼ N ph (z) ¼ I 0 e
az : nr Dnnr

ð2:80Þ

(Note that Nph(z) is the density.) Then, dI(z) chn dN ph (z) chn dN ph (z) dt ¼ ¼ dz Dnnr dz Dnnr dt dz chn dN ph (z) nr hn dN ph (z) ¼ ¼ : c Dnnr dt Dn dt

ð2:81Þ

2.4 Optical Gain

Solving for

dN ph (z) dt

leads to

dN ph (z) Dv dI(z)
aDv ¼ ¼ I(z) dt hv dz hv :
aDv c re (v) ¼ hv nr

ð2:82Þ

We can now obtain optical gain by equating Equations 2.79 and 2.82, we obtain [N 2
N 1 ][B21 re (n)] ¼


aDv c r (v): hv nr e

ð2:83Þ

(Here, N1 and N2 are the densities, giving the rate unit as s
1 m
3, not just s
1. Rates are defined per unit volume.) Here, we neglect the contribution to the electromagnetic wave of the spontaneous emission as we are interested in propagation along the cavity and the spontaneous emission is random in direction. The absorption and emission processes normally occur over a spread of energies (or frequencies and wavelengths, the intervals of which are described by Dn and Dl). This implies that the gain would also have distribution in energy, expressed as g(n) (or energy and wavelength) and called the gain curve. Solving Equation 2.83 for the absorption coefficient a or
a (for the gain) at a central photon energy of hn0 leads to [45, 46]
a(hn0 ) ¼ g(hn0 ) ¼ [N 2
N 1 ]

B21 nr hn0 : cDn

ð2:84Þ

At threshold, the gain is called the threshold gain, as described in Equation 2.33. At threshold, [N2
N1] ! [N2
N1]th. Therefore, Equation 2.84 at threshold can be rewritten as [N 2
N 1 ]th ¼ DN th ¼ g th

cDn : B21 nr hn0

ð2:85Þ

Clearly, if [N2
N1] > 0, then g > 0 meaning the system has gain. On the other hand, if [N2
N1] < 0, g < 0 or the absorption coefficient a > 0. Utilizing Equation 2.74, Equation 2.85 can be rewritten as DN th ¼ g th

8pn2 n2r Dntsp 8pn2 n2r Dn ¼ g th : 2 c2 A21 c

ð2:86Þ

The second part of Equation 2.86 is derived noting that A21 is the spontaneous emission rate, which is equal to the inverse of the mean free time for spontaneous recombination (therefore spontaneous emission) rate, tsp, which is on the order of 10
9 s for compound semiconductors. The threshold gain can be obtained from Equation 2.86 as g th ¼

c 2 DN th : 8pn2 n2r Dntsp

ð2:87Þ

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Noting that the threshold carrier density can be related to threshold current density through (with d being the thickness of the pumped region) J th ¼

qDN th d : tsp

ð2:88Þ

Equation 2.87 can be written as g th ¼

c 2 J th : 8pqdn2 n2r Dn

ð2:89Þ

Because the quantum efficiency is less then unity, Equation 2.89 must be rewritten as g th ¼

c 2 J th h 8pqdn2 n2r Dn

ð2:90Þ

with h being the quantum efficiency. The threshold current then can be written as J th ¼ g th

8pqdn2 n2r Dn : c2 h

ð2:91Þ

Equation 2.91 is quite telling in that the narrower the gain curve (small Dn), the thinner the region to be pumped (small d afforded for example by quantum wells), the smaller the threshold current becomes. The gain curve line shape in InGaN is of a special concern, as due to homogeneous and inhomogeneous broadening, the linewidth is wide. If lasing occurred due to excitonic transitions, the gain curve would be very narrow, which is the attraction of ZnO with its 60 meV exciton binding energy as opposed to slightly over 20 meV for GaN, which persists even above room temperature. Having derived expressions for the lasing condition, let us now consider in which systems it is possible to have population inversion, that is, two-, three-, and four-level systems. Equations 2.40 and 2.41 represent the rate of change of population in level 2 due to spontaneous emission and stimulated emission, respectively, and Equation 2.43 represents the rate of change in population of level 1. Let us now set the rate equations for both level 2 and level 1 with all three processes: stimulated emission, absorption, and spontaneous emission. dN 2 ¼ B12 re (n)(N 1
N 2 )
A21 N 2 (Utilizing B21 ¼ B12 ) dt   dN 1 dN 2 dN 1 ¼ B12 re (n)(N 2
N 1 ) þ A21 N 2 ¼
: Note that dt dt dt

ð2:92Þ

It then follows that d(N 2
N 1 ) dDN ¼
¼ 2B12 re (n)(N 1
N 2 )
2A21 N 2 dt dt

(with DN ¼ N 1
N 2 ): ð2:93Þ

If N ¼ N1 þ N2, then dDN ¼
2B12 re (n)DN þ A21 (N
DN): dt

ð2:94Þ

2.4 Optical Gain

2

N2

Emission

Absorption

N1

1 (a)

3

Fast decay

N2 2

Laser transition

Pump transition N1

1 (b)

Figure 2.21 A schematic representation of a two-level system (a) and a three-level system (b).

Under steady state of Equation 2.94, the left side is zero, giving rise to DN ¼

N >0 1 þ (2B12 re (n)=A21 )

Always!

ð2:95Þ

This implies that N1 > N2 and the population inversion in a two-level system, shown in Figure 2.21a, cannot be achieved in practice. Let us now consider a three-level system, shown in Figure 2.21b, where the optical pumping takes place between levels 1 and 3 and further assume that electrons relax (decay) rapidly from level 3 to level 2 so fast that the population in level 3 can be assumed to be zero. The rate equations can then be written (similar to Equation 2.92) dN 2 ¼ B12 N 1 re (n)
A21 N 2 (Utilizing B21 ¼ B12 ) dt dN 1 ¼
B12 N 1 re (n) þ A21 N 2 : dt

ð2:96Þ

Following a fashion similar to that in the derivations leading to Equation 2.94, we obtain dDN ¼
B12 re (n)(N þ DN) þ A21 (N
DN): dt

ð2:97Þ

Again under steady state, the time rate of change, the left side, goes to zero. We, therefore, have 0 ¼
B12 re (n)(N þ DN) þ A21 (N
DN)

ð2:98Þ

and DN ¼ N

(A21 =B21 )
re (n) : (A21 =B21 ) þ re (n)

ð2:99Þ

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Unlike the two-level system, if re(n) > (A21/B21), DN < 0 and population inversion can be practically attained. The A21/B21 ratio can be determined from Equation 2.74. The case is somewhat similar in a four-level system, where the upward pumping takes place from level 0 (ground state) to level 3, electrons decay to level 2 with a very large rate, make a downward lasing transition to level 1, then decay to level 0 at a very high rate. The net change in carrier concentration can be obtained by again writing the rate equations for levels 1 and 2 and finding the difference between the populations of level 1 and level 2 populations. Doing so leads to DN ¼ N 1
N 2 ¼
N

(A21 =B21 ) B12 f 1 (1
f 2 )r(E 21 )

ð2:112Þ

and because B21 ¼ B12, Equation 2.112 implies that f 2 (1
f 1 ) > f 1 (1
f 2 ):

ð2:113Þ

Substituting Equation 2.105 into Equation 2.113 leads to the population inversion condition being expressed in terms of the Fermi levels associated with levels 2 and 1 as F 2
F 1 > E 2
E 1

or in the semiconductor terms F n
F p > E C
E V ¼ E g : ð2:114Þ

In words, Equation 2.114 states that the quasi-Fermi-level separation must be larger than the bandgap for population inversion and, therefore, stimulated emission and gain to take place. For lasing, the gain must be large enough to overcome internal and external losses. Equation 2.114 was developed by Bernard and Duraffourg [47] in their original proposal for stimulated emission. It also represents the transparency condition, meaning that the semiconductor itself no longer absorbs (this is in addition to the waveguide and end losses). This condition simply expresses the fact that the separation

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E

E

Fn

Fn

∆ Eg

Eg

Fp

(a)

Fp

∆

(b)

Figure 2.23 The approximate positions of quasiFermi levels in a laser structure under injection of minority carriers and at the threshold of lasing: (a) depicts the hypothetical case where the effective masses of electrons and holes are about the same. Strained layer systems approach this

case reasonably closely; (b) illustrates the more realistic case of asymmetric electron and hole effective masses. This case where the effective mass of holes is larger than that for electrons, represents all the III–V compound semiconductors used for lasers.

of the quasi-Fermi levels must exceed the bandgap energy and that the electron quasiFermi level would lie in the conduction band because the density of states in the conduction band is much smaller than that in the valence band. Because carrier injection is through a diffusion process, the concentration of injected carriers in a homojunction laser cannot exceed the carrier concentrations in the n- or p-emitters. Consequently, to satisfy the condition (Equation 2.114), the equilibrium Fermi level in the emitter must also be shifted toward the corresponding band, as shown in Figure 2.23 (somewhat similar information can also be found in Figures 2.4 and 2.22). This figure depicts two cases where in (a) the conduction and valence band effective masses are equal and (b) the valence band effective mass is larger than that of the conduction band causing the quasi-Fermi level to enter the conduction band for the lasing condition to be satisfied. Typically, the donor and acceptor concentrations in the n- and p-emitter layers are in the range of 1  1018–5  1018 cm
3. With a heterojunction confinement of the carriers and light, this point is a mute one. The less the volume to be pumped in, the smaller the injection current that is required to reach the transparency condition. Before the advent of heterostructures, the thickness of the pumped region needed to be comparable to the wavelength of the radiation, so the light traveling along this region diffracted severely into absorptive passive regions. This occurs despite the waveguiding effect that is provided by the decreased refractive index of the emitter caused by the high carrier concentration. To maintain a population inversion in the laser diode with an excited region thickness of 2–3 mm, current densities of about 20–30 kA cm
2 are required. The power to be dissipated by a laser diode at those current densities is so high that the p–n-junction lasers were able to

2.4 Optical Gain

operate only under pulsed excitation at room temperature, with a very low duty cycle of about 10
4 or less, to avoid overheating and catastrophic failure. A similar situation would also occur if the nonradiative processes would be so dominant that the threshold current remains high despite the thin active region employed in the case of a presentday nitride-based laser. Heterojunctions allow the much needed flexibility in the design of laser structures in that it is possible to confine the injected carriers to a very small region while providing a waveguide due to the favorable spatial variation of the refractive index. Referring to Volume 1, Figure 1.38, the bandgap of the ternary AlGaN alloy increases monotonically with increasing Al content. Thus, a GaN/AlGaN or InGaN/ GaN/AlGaN DH laser is essentially a combination of a layer of narrow- bandgap material straddled by two layers of wider bandgap n- and p-type materials. Under forward bias, carriers are injected from wide-bandgap layers (emitters) into the narrow-bandgap (active) layer, where they are confined. The maximum concentration of injected carriers in the active layer does not strictly depend on the equilibrium carrier concentration in the emitters. Even when the concentration of injected carriers exceeds those in the emitters, diffusion back into an emitter is inhibited by heterojunction barriers; it removes the need for very high doping levels in the emitters. In DH lasers, the conductivity type of the active layer also becomes less important. An n-type active layer will be assumed, as in InGaN lasers, and electrons are injected from the p-GaN/n-InGaN heterojunction. These electrons, which are confined to the active layer, make it negatively charged and attract additional majority (but nonequilibrium) carrier holes from the p-AlGaN emitter. In general, injection of nonequilibrium minority carriers from the p–n-heterojunction rearranges the potential at the opposite n–n (or p–p) heterojunction to facilitate the flow of nonequilibrium majority carriers into the active layer; thereby, they maintain the overall charge neutrality. The flow and confinement of carriers into the active region is illustrated in Figure 2.24a. For simplicity, the waveguide is indicated to consist of some composition of InGaN, one binary end point of which is GaN, and the active region to be same. The desired wavelength and the state of the technology determine the composition of InGaN in the active region. Depending on the active layer, the cladding layer composition is chosen for efficient carrier and light confinement (waveguiding). In the example shown, the injected electrons are restricted by the energy barrier at the nInGaN/p-AlGaN heterojunction. The height of this barrier must be sufficiently high as compared to the electron energy in the active layer for carrier confinement. If we neglect the rearrangement of the potential at the p–n-heterojunction due to the injection of electrons into the active layer, being on the order of a few kT, one can deduce from Figure 2.24a that the barrier for electrons is approximately equal to the bandgap difference between the constituents forming the heterojunction. It is given b. pn

DE C  DE C þ DE V ¼ DE g :

ð2:115Þ

This requirement applies to most heterojunctions formed by III–V compounds and their alloys. Following the same argument, the barrier for holes at the

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n-AlGaN

Electron

InGaN

p-AlGaN

hν

(a)

Hole injection Refractive index, n d (b) I Distance Light intensity, I

L

n-AlGaN GaN p-AlGaN

(c)

Distance Figure 2.24 The energy position of the band edges and quasi-Fermi level(s) versus distance (energy band diagram) for the p–n-junction of the injection laser under forward bias are shown in (a). The regions occupied by electrons in the conduction band and valence band are shown by the shaded areas. Note the electron

void area near the top of the valence band in the quantum well. Subparts (b) and (c) indicate the refractive index and the optical field, respectively. The inset shows a schematic representation of DH bulk GaN (for descriptive purpose only) laser with AlGaN cladding layers.

n–n-heterojunction or homojunction under a high forward bias on the n-(Al,Ga) N/n-(In,Ga)N side is given by DE nn V  DE C þ DE V ¼ DE g :

ð2:116Þ

Figure 2.24b and c exhibit the refractive index and optical field profiles in a DH laser. The inset in Figure 2.24 depicts the current flow, which is perpendicular to the junction. In DH structures with a small active layer thickness, the concentration of nonequilibrium carriers in the active layer increases and allows a population inversion to be obtained at lower currents, as the active layer is made thinner and attention is paid to radiation (optical field) confinement. Otherwise, the radiation leaks out and causes a steep increase in the threshold current. Similar arguments hold for quantum well lasers of the separate confinement variety, with constant and/ or graded refractive index, as depicted in Figure 2.25. The subparts (a), (b), and (c) of Figure 2.25 depict the energy versus position diagram, the refractive index profile, and the optical field profile, respectively. As in the case of double heterojunctions, subpart (a) indicates a schematic representation of the carrier injection, the

2.4 Optical Gain

p-AlGaN n-AlGaN

Electron

(a)

hν

Hole injection Refractive index, n

(b)

Light intensity, i

Distance

(c)

Distance Figure 2.25 Energy position diagram along with carrier injection/ confinement in an SCH structure. Subpart (a) indicates the schematic representation of carrier injection, confinement, and photon emission. Subparts (b) and (c) depict the refractive index profile (solid lines for SCH and dashed lines for graded index SCH, the latter has not been implemented in the nitride system under the discussion) and the field profile, respectively.

confinement and photon emission. Subparts (b) and (c) illustrate the refractive index profile (solid lines for separate confinement heterostructure (SCH) and dashed lines for graded index SCH, the latter has not yet been implemented in the nitride system under the discussion) and the field profile, respectively. 2.4.2 Optical Gain in Bulk Layers: A Semiconductor Approach

Let us now begin the journey of developing expressions for the gain in a semiconductor laser as we have for gas and solid state lasers (for example, see Equation 2.84). Referring to Equation 2.109, we can write the net stimulated emission rate as R21 jst ¼ B21 f 2 (1
f 1 )r(E 21 )
B12 f 1 (1
f 2 )r(E 21 ) ¼ B12 ( f 2
f 1 )r(E 21 ):

ð2:117Þ

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Using Equations 2.110 and 2.111, Equation 2.117 can be rewritten as R21 jst ¼ A21 (f 2
f 1 )

h3 c 3 8pE 221 n3r 1 A21 (f 2
f 1 ) ¼ E =kT : e 21
1 8pE 221 n3r h3 c 3 eE 21 =kT
1

ð2:118Þ

For its similarity to Equation 2.108 depicting the spontaneous emission rate (R21|sp), R21(stim), which popularly has assumed the nomenclature of “stimulated emission rate,” defined as R21 (stim) ¼ R21 jst (eE 21 =kT
1) ¼ A21 (f 2
f 1 ):

ð2:119Þ

Again the stimulated emission rate of Equation 2.119 (R21(stim)) describes the downward transition rate for stimulation of a similar photon. When this is multiplied by (eE 21 =kT
1)
1, R21|st, which represents the net stimulated emission rate, is arrived at. Notwithstanding customary use, to avoid confusion in this text, R21|st is used for rate equations. Solving the rate balance equation of Equation 2.109 for the absorption rate and recalling that B21 ¼ B12, we obtain R12 jabs ¼ B12 (f 1
f 2 )r(E 21 ):

ð2:120Þ

2.4.2.1 Relating Absorption Rate to Absorption Coefficient Let us now embark on relating the net absorption rate R12|abs to the absorption coefficient (when negative, it is called the gain coefficient). As we have done in conjunction with Equation 2.78 through Equation 2.84, which dealt with calculating the loss or the absorption coefficient, the negative of which is called the gain coefficient, g, mainly for gas and solid state lasers, consider similarly a plane wave propagating along the z-direction (length of the waveguide) represented by I0 exp(
az). The loss parameter a can then be expressed as


a( ¼ g) ¼

dI=dz : I

ð2:121Þ

The numerator in Equation 2.121 represents the net power emitted per unit volume while the denominator represents the power per unit area. The term in the denominator is simply the photon density distribution per unit frequency (spectral photon density) multiplied by the group velocity, c/nr (see Equation 2.80). The term in the numerator can be represented by R12|abs given in Equation 2.120. Rewriting Equation 2.121 in the light of the aforementioned discussion, we obtain for a (now recognizing its energy dependence) a(E 21 ) ¼

B12 (f 1
f 2 )r(E 21 ) B12 (f 1
f 2 ) ¼ ¼
g(E 21 ): (c=nr )r(E 21 )E 21 (c=nr )E 21

ð2:122Þ

(Note that in Equation 2.80, the intensity is the photon energy density distribution (notice the hn term) per unit frequency. The intensity should have units W m
2. Similarly, the denominator here involves the photon energy E21.)

2.4 Optical Gain

2.4.2.2 Relating Stimulated Emission Rate to Absorption Coefficient Note that in Equation 2.122, the dispersion term associated with the wavelength dependence of the refractive index, [1 þ (v=nr )(dnr =dv)], is taken as unity, and also note that B12 ¼ B21. It is clear that if (f1
f2) < 0, which is the case in population inversion, a(E21) < 0 and g(E21) > 0. The denominator in Equation 2.122 can be expressed in terms of the stimulated emission rate by utilizing Equations 2.111 and 2.119 as

a(E 21 ) ¼
g(E 21 ) ¼

A21 (f 1
f 2 ) h3 c3 h3 c2 ¼
R21 (E 21 ; stim): 3 3 (c=nr )E 21 8pE 21 nr 8pE 321 n2r ð2:123Þ

2.4.2.3 Relating Spontaneous Emission Rate to Absorption Coefficient The absorption coefficient, a(E21), can also be related to the spontaneous emission rate. From Equation 2.122, we obtain

B12 ¼ B21 ¼ a(E 21 )

E 21 (c=nr ) (f 1
f 2 )

or

a(E 21 ) ¼ B12 (f 1
f 2 )(nr =c)(1=E 21 ): ð2:124Þ

By utilizing Equation 2.108 (R21|sp ¼ A21f2(1
f1)) and Equation 2.111 (A21 ¼ B21 (8pE 321 n3r )=(h3 c 3 )), the spontaneous emission rate can be rewritten as R21 (E 21 )jsp ¼ a(E 21 )

f 2 (1
f 1 ) 8pE 321 n2r : f 1
f 2 h3 c 2

ð2:125Þ

With the help of Equation 2.124, Equation 2.125 can be rewritten as R21 (E 21 )jsp ¼ a(E 21 )

f 2 (1
f 1 ) 8pE 321 n2r : f 1
f 2 h3 c 2

ð2:126Þ

Using the Fermi–Dirac statistics given in Equation 2.105 for fi(i ¼ 1,2) and taking EFi as F1 and F2 for levels 1 and 2 (later these will be replaced with Fp and Fn, the quasiFermi levels for holes and electrons, respectively), R21 (E 21 )jsp ¼ a(E 21 )

8pE 321 n2r 1 : h3 c 2 exp[(E 21
F 2 þ F 1 )=kT]
1

ð2:127Þ

2.4.2.4 Fermi’s Golden Rule, Stimulated and Spontaneous Emission Rates, and Absorption Coefficient Within the k-Selection Rule Equations 2.123 and 2.126 relate the stimulated emission rate and the spontaneous emission rate to the absorption coefficient, respectively. With the help of the same equations, the stimulated emission rate and the spontaneous emission rate can be related to each other. The implications of these equations are that both emission rates can be determined if the absorption coefficient along with its energy (or wavelength) dependence is known. Fortunately, the absorption coefficient is a measurable

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quantity. Therefore, once measured, the emission rates can be determined. The absorption rate can also be calculated with numerical techniques. Because the absorption coefficient, spontaneous and stimulated emission rates can all be determined with knowledge of Einstein’s B coefficient (recall that the A coefficient can be calculated from the B coefficient), calculation of the B coefficient is sufficient to determine the absorption coefficient (the last part of Equation 2.124), spontaneous emission rate (Equation 2.125), and the stimulated emission rate (Equation 2.118, as A and B coefficients are related). Below, a succinct description is given of the calculations leading to Einstein’s B coefficient and/or the two emission rates. The B coefficient represents the interaction of electrons in the solid with the electromagnetic wave that requires a quantum mechanical treatment. Typically, one assumes a small perturbation due to the electron radiation interaction and applies timedependent perturbation theory. In this perturbation method, the properties are determined in the absence of the electromagnetic wave first and then alterations to the solutions are made to account for the effect of the electromagnetic wave. An efficient method of accomplishing this task is through the use of Fermi’s Golden rule [48], which relates the B coefficient to the wave functions for the level 1 initial (Y1 (~ r ; t)) and level 2 final (Y2 (~ r ; t)) states with an interaction Hamiltonian HI of the form [30, 49]:   p  p   B12 ¼ r ; t)jHI Y1 (~ r ; t)ij2 ¼ jMI j2 ; ð2:128Þ hY2 (~ 2
h 2
h where |MI| is the matrix element and represents the strength of the level 1–level 2 interaction. (Note that the B12 coefficient above has J s
1 unit.) The interaction Hamiltonian, HI, can be considered as having time-harmonic and spatial-dependent components and can be expressed as the product of the two. HI (~ r ; t) ¼ H I (~ r )exp(jwt): The term in brackets can be written in the integral form as ð r ; t)jHI jY1 (~ r ; t)i ¼ Y 2 (~ r ; t)H I Y1 (~ r ; t)d3 (~ r ): hY2 (~

ð2:129Þ

ð2:130Þ

V

Now that we set the stage up partially, the goal of calculating the B parameter has been reduced to finding the matrix element describing transitions between levels 1 and 2. In this vein, let us first consider the Schr€ odinger’s equation, without the effect of electromagnetic wave, of the form   2     
h h qYi (~
r ; t)
r ) Yi (~ r ; t) ¼
r2 þ V(~ 2m0 j qt   
h qYi (~ r ; t) or H oop Yi (~ r ; t) ¼
: ð2:131Þ j qt where m0 is the particle mass. The bracketed term on the left is the Hamiltonian operator, Hoop , without the effect of the optical field. The wave function Yi (~ r ; t) can also be assumed to be the product of a spatially dependent component and a

2.4 Optical Gain

time-dependent harmonic component and can be described as Yi (~ r ; t) ¼ Yi (~ r ) exp(jwi t):

ð2:132Þ

The Hamiltonian must be modified to include the effect of the electromagnetic field. This is best done with the help of the vector field ~ A , which is related to the magnetic field through ~ B ¼ rx~ A . Detailed derivation of the Hamiltonian with the effect of the optical field included is beyond the scope of this text. While a short description is provided here, similar to that done in Ref. [30], detailed discussion can be found in Ref. [49]. The Hamiltonian with the effect of the field included (basically, the momentum term needs to be changed) can be expressed as   1 Hop ¼ r ): ð2:133Þ (~ p
q~ A)2 þ V(~ 2m0 where ~ A is defined by ~ B ¼ rx~ A and ~ p ! (
h=j)r is the momentum operator. Neglecting the higher order term (q2 (~ A )2 ) in the squared term in Equation 2.133, the Hamiltonian with field can be expressed as      1 q
h ~ Hop ¼ A  r þ V(~ r ): ð2:134Þ

h2 r2
2 2m0 j The additional component picked up due to field interaction is then HIop ¼

    q
h ~
A r ; jm0

ð2:135Þ

which with the aid of the momentum operator ~ p ! (
h=j)r can be rewritten as HIop ¼

    q ~
A ~ p : m0

ð2:136Þ

To treat the new Hamiltonian, let us now consider a plane wave, with ~ E having an x ~ field having a y-component only (for simplicity) and propagating component and H along the z-direction, which can be described by ~ E ¼ x^E 0 exp[j(wt
kz z)] and ~ ¼ y^H0 exp[j(wt
kz z)] ¼ y^E 0 H



 kz exp[j(wt
kz z)]: m0 w

ð2:137Þ

The interested reader can find manipulations performed here in any electromagnetics and fields textbooks such as given in Refs [28, 29, 50–53]. Knowing that ~ E ¼
q~ A =qt, the ~ A vector field can be expressed as   E0 ~ exp[j(wt
kz z)] A ¼ j x^ w

E2 and j~ Aj2 ¼ ~ A ~ A ¼ 02 w

ð2:138Þ

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For obtaining the magnitude of the electric field E0, we need to utilize the Poynting vector by relating the electromagnetic flux to its real part. The time-averaged Poynting vector is expressed as follows 1 ~ ): h~ Si ¼ Re(~ E H 2

ð2:139Þ

With the aid of Equation 2.137, the amplitude of the time-averaged Poynting vector can be determined as   1 E 20 kz jh~ Sij ¼ : 2 m0 w

ð2:140Þ

Noting kz ¼ k ¼ w/(c/nr), c
2 ¼ m0e0, and that the flux for a photon is given by the product of the photon energy and the group velocity, we have     1 E 20 kz 1 2
hw c ¼ (E 0 nr e0 c) ¼ ; 2 m0 w 2 V nr

ð2:141Þ

where V is the crystal volume. Note that the Poynting vector represents the energy flux, which has W m
2 unit. From Equation 2.141, we find that E 20 ¼ 2
hw=Vn2r e0 . Therefore, Equation 2.138 can be written as j~ Aj2 ¼

2
h : Vn2r e0 w

ð2:142Þ

For the more general three-dimensional (3D) polarization vector ~ P , Equation 2.138 is replaced with   E0 ~ A ¼ j~ P exp[j(wt
~ k ~ r )]: w

ð2:143Þ

However, Equation 2.142 remains applicable. Einstein’s B coefficient (the transition probability) of Equation 2.128 may now be rewritten with the aid of Equation 2.132 (for the wave functions), Equation 2.136 (for the interaction Hamiltonian), and Equation 2.143 (for the vector potential) as B12 ¼

  1=2    p  q 2
h ~ hY2 (~ ~ r ) exp(jw t)j
j P exp[j(wt
k ~ r )] ~ p 2 2
h  m0 Ve0 n2r w 2  jY1 (~ r ) exp(jw1 t)i : ð2:144Þ

For harmonic condition of w ¼ w2
w1, the exponential time dependencies in Equation 2.144 go to unity paving the way to express the above equation as

2.4 Optical Gain



   pq2
h pq2
h 2 r )j~ pjY1 (~ r )ij ¼ jhY2 (~ jMj2 : hw Vm20 e0 n2r
hw Vm20 e0 n2r


B12 ¼

ð2:145Þ

where M is the momentum matrix element and the missing exp(
j~ k ~ r ) in the above equation has been taken, as unity as the values of r where wave functions have small values, that is, ~ k ~ r 1. Note that B12, in Equation 2.145, has J s
1 unit. It should be pointed out that the momentum matrix element can be construed as the average of incident light polarizations [54].

i 1 h jMx j2 þ jM y j2 þ jM z j2 with 3    
h q h
~ r )jexp(
jk ~ r) r )i (using~ p! jY1 (~ r: M x ¼ hY2 (~ j qx j

jMj2 ¼

ð2:146Þ

If the wave functions of the upper and lower levels are Bloch functions characterized k1 , respectively, the matrix element will contain a by propagation vectors ~ k2 and ~ d(~ k2
~ k1  ~ k ) to represent the momentum conservation. Because the transitions are vertical in the k-space and photon wave vector is very small, ~ k is typically neglected and ~ k2  ~ k1 . If the upper and lower states are monotonic functions of kb ¼ j~ k2 j ¼ j~ k1 j, as in the parabolic bands the values of E2 and E1 are uniquely determined by the photon energy hv of the incident light. We will use these in the determination of spontaneous and stimulated emission rates soon. Recalling Equation 2.111 [A21 ¼ (8pE 221 n3r )=(h3 c 3 )B21 ], Einstein’s A coefficient can now be written in terms of the momentum matrix element, M, as

A21 ¼

  8pE 221 n3r pq2
h 4pq2 E 21 nr jMj2 : jMj2 ¼ 3 3 2 2 Vm0 e0 nr
hw h c Vm20 e0 h2 c 3

ð2:147Þ

Note that we utilized the relation
hw ¼ E 21 . Also note that A21 has sm
3 unit. The expressions we have developed so far for spontaneous and stimulated emission rates are predicated on the assumption that transitions take place between levels 2 and 1. However, in a semiconductor, the available states in the valence band (representing level 1) and conduction band (representing level 2) are distributed over energy. Therefore, an extension of the previous treatment must be undertaken to include all the eligible states in the conduction and valence bands. We touched upon the density of states in three-dimensional and reduced dimensional structures in Volume 1, Figure 2.19 and associated text. It is customary to use nomenclatures of gn and gp, D(E), and dN in addition to r(E) for the density of states. To determine the density of states for electron, we take an approach similar to the density of modes in a cavity we treated earlier in Section 2.4. Briefly, the density of states is then the number of states between the momentum values of k and k þ dk. The unit volume in k-space confined within the boundaries of the k-vector is

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222

V ¼ (2p)3/a3 and the volume of a spherical shell of thickness dk is 4pk2dk. The density of allowed values of k in a volume V is the number of cubes of face (2p/a) that can be fit in that volume in k-space. Therefore, number of states is the product of volume in kspace and the number of states divided by the volume in real space (V ¼ a3). Doing so leads to [39, 42, 43]  2 [2(4pk2 dk)=(2p=a)3 ] k r(k)dk ¼ r(E)dE ¼ ¼ dk: ð2:148Þ p2 V The factor of 2 is picked up due to the spin-up and spin-down polarization of electrons (spin degeneracy). If spin conservation is applied, the factor 2 would not be included, halving the final value. Recalling that for a parabolic band we have (
hk)2 ¼ 2m n (E
E C );

ð2:149Þ

where m n is the conduction band effective mass (density-of-states effective mass). Equation 2.148 can be written as rC (E
E C )j3
d ¼

  1 2m n 3=2 (E
E C )1=2 2p2
h2

(Good for E  E C ):

ð2:150Þ

See for a plot (Volume 1, Figure 2.19) of the energy dependence of the density of states. Let us take a short diversion and treat the density-of-states problem in a quantum well with one degree of confinement. Similar to the three-dimensional case, the density of states is then the number of states between the momentum values of k and k þ dk (or energy E and E þ dE). The unit area in k-space confined within the boundaries of the k-vector is A ¼ (2p)2/a2 and the area of a spherical shell of thickness dk is 2pkdk. The density of allowed values of k in an area A is the number of squares of face 2p/a that can fit in that area in k-space. Therefore, the number of states is the product of volume in k-space and the number of states divided by the area in real space (A ¼ a2). Doing so leads to [39, 42, 43]     (2pkdk)=(2p=a)2 k ¼ dk: r(k)dk ¼ r(E)dE ¼ 2 A p

ð2:151Þ

The factor of 2 is picked up due to the spin-up and spin-down polarization of the electrons. With the help of Equation 2.149, and noting from Equation 2.148 that
h2 kdk ¼ m n dE, we can write the expression of the density of states for one of the quantum levels as rC (E)dE ¼

kdk m n m ¼
2 dE and thus rC (E
E n )j2
d ¼
n2 (Good for E  E n ): p ph ph ð2:152Þ

The density of states of a given quantum state En is therefore independent of energy and well thickness for as long as there is confinement. Therefore, at the second

2.4 Optical Gain

excited state energy, an equal quantity would be added to the density of states, and so on (see Volume 1, Figure 2.19). In which case, we can write rC (E)j2
d ¼

m n X u(E
E i ); p
h2 i

ð2:153Þ

where u(E
Ei) is the step function, which is zero except when E ¼ Ei, and i represents the ith confined level. It would be informative to at least make mention of the density of states for onedimensional (quantum wires) and zero-dimensional (quantum dots) as they may pertain to certain types of lasers, particularly, the quantum dot variety. Quantum dot lasers have many advantages, such as increased confinement (high characteristic temperature that makes the threshold current less sensitive to temperature and other beneficial effect in communications systems) and increased lifetime in excited states implying that the radiative lifetime would be dominant lifetime. The dot lasers have been implemented in the InGaAs/GaAs system. Although quantum dots in the GaN system have been explored, as discussed in Volume 2, Chapter 5, no report of lasing action was available at the time of writing this book. The density of states in a onedimensional system per unit length of the wire at T ¼ 0 is given by (following a treatment similar to that for three- and two-dimensional systems provided above) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m n rC (E
E n )j1
d ¼ (Good for E  E n ): ð2:154Þ p
h 2(E
E n ) The factor of 2 accounts for the spin degeneracy. Clearly, Equation 2.154 represents a spiked dependence on energy E ¼ En that repeats itself at every quantum state (see Volume 1, Figure 2.19). As for a zero-dimensional system, the density of states can be represented by Nd (E
En) at each of the quantum states. The coefficient N contains the spin degeneracy factor, any accidental degeneracy of the bound state involved, and the number of quantum dots per unit volume (see Volume 1, Figure 2.19) [39]. Likewise, the density of states for a three-dimensional system in the valence band is given by rV (E V
E) ¼

 3=2 1 2mp (E V
E)1=2 2p2
h2

(Good for E  E V );

ð2:155Þ

where m p is the hole effective mass (density-of-states effective mass) in the valence band. We are assuming that only one of the bands in the valence band is participating. The density-of-states equation of Equation 2.152 and Equation 2.154 can similarly be modified for the valence band density of states for a reduced dimensional system as we have done in Equation 2.155. In optical devices relying on band-to-band transitions, both valence and conduction bands are involved. Therefore, it is often convenient to define “joint density of states,” which can be used to calculate the emission and/or absorption rates. Utilizing Equation 2.148, we can define rjoint(E) as

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224

rjoint (E) ¼

 2 k dk ; p2 dE

which is evaluated at E ¼ hv ¼ E 2
E 1 :

Considering a parabolic band (Equation 2.149), we can write ! 1 1 2 kdk: þ dE ¼ dE 2
dE 1 ¼
h m n m p

ð2:156Þ

ð2:157Þ

Solving for dk/dE and substituting it into Equation 2.156, we obtain rjoint (hv)  rCV (hv) ¼

k mn mp : p2
h2 m n þ m p

ð2:158Þ

Using the band parabolicity described by Equation 2.101 for both the conduction and valence bands, which relate the momentum to energy, together with Equation 2.103, we can rewrite " #1=2 2m n m p
hk ¼ (hv
E g ) ¼ [2m r (hv
E g )]1=2 ; ð2:159Þ m n þ m p where m r ¼ (m n m p )=(m n þ m p ) is the reduced effective mass. Then, rCV (hv) ¼

    1 2m r 3=2 1 2m r 3=2 1=2 (hv
E ) or r (E) ¼ (E
E g )1=2 : g CV 2p2
h2 2p2
h2 ð2:160Þ

Let us now derive the expression for the absorption coefficient, which we did in conjunction with Equation 2.122 for a two-level system, but with the view that the density of states be more applicable to semiconductor systems. The rate of excitation from the valence band, level 1, to the conduction band, level 2, depends on the availability of states with electrons in the valence band, which is rV(EV
E)f1, and the availability of empty states in the conduction band, level 2, which is rC(E
EC)(1
f2). Here, EC and EV are the conduction band minimum and valence band maximum. They also represent the origin for the density of states for conduction and valence bands, respectively. The absorption coefficient at an energy
hw must then be the integral of Equation 2.122 over all the possible energies corresponding to the energy difference
hw, the energy difference between level 1 and level 2. 1 ð

a(
hw) ¼
1

B12 (f 1
f 2 )V 2 r (E V
E 1 )rC (E 2
E C )d(E 2
E 1

hw)dE 1 dE 2 ; (c=nr )(E 2
E 1 ) V ð2:161Þ

where V is the crystal volume. Note that, according to Equation 2.145, B12 has J s
1 unit. The term r(E)dE has m
3 unit; therefore, the integral over volume is needed twice, once over E1 and also over E2, justifying the term involving the volume squared (V2).

2.4 Optical Gain

It should be mentioned that the integral implies that we are taking into account all the transitions between the conduction and valence bands. Later, we will introduce the k-selection rules and consider only the direct transitions. Equation 2.161 must be massaged to fit the semiconductor mold, as has been done by Lascher and Stern [54] in their seminal paper. First, we begin by mentioning that d(E

hw) is used in Equation 2.161, where E ¼ E2
E1, to explicitly show that hw. When the rC(E
EC) and rV(EV
E) are separated by a given photon energy,
delta Dirac function is not incorporated, its effect is implicit in the density of states for the conduction and valence bands. Following the convenient methodology of Lascher and Stern [54], we can define a new energy variable, E 0 , with a reference point of E 0 ¼ 0 at the conduction band minimum. Making the substitutions E2 ! E 0 , E1 ! E 0
E, f1 ! f1(E 0
E), and f2 ! f2(E 0 ), which after making use of Equation 2.145 and the joint (or reduced) density of states allow us to rewrite Equation 2.161 as a(E) ¼

pq2
h 2 m0 e0 nr cE 2

1 ð

rCV (E 0 )jM(E 0 ; E 0
E)j2 [f 1 (E 0
E)
f 2 (E 0 )]dE 0 :

ð2:162Þ


1

(The coefficient together with the momentum matrix element has m2 units. Here, as we defined E0 ¼ 0 at the CBM as the reference, there is only one energy to integrate over, E0 . The term rCV(E 0 )dE 0 from Equation 2.160 has m
3 unit, making the overall units for the right hand side as m
1. As shown by Lasher and Stern [54], the volume term V does not appears when joint DOS is used. But when separate DOS are used, we need to involve the crystal volume. From here onward, we define the rates per unit energy per unit volume, in units sm
3 J
1.) Integration from
1 ! EV corresponds to the valence band, and integration from EC ! 1 corresponds to the conduction band. Because there is no density of states within the bandgap, there is no need of integrating over that range of energies. To avoid any possible confusion and making additional nomenclature changes that are germane to semiconductors, such as quasi-Fermi levels and breaking the integral into two, one for the valence band and the other for the conduction band, Equation 2.162 can be represented by a double integral such as pq2
hV a(E) ¼ 2 m0 e0 nr cE 2

1 ð1 ð

rC (E n )rV (E p )[f p (F p
E p )
f n (E n
F n )]jM(E n ; E p )j2 dE n dE p ; 0 0

ð2:163Þ where Fn and Fp represent the quasi-Fermi levels for electrons and holes, respectively. We should point out that the Fermi functions in Equation 2.163 represent the electron occupation probabilities. In addition, the nomenclature (En
Fn) and (Fp
Ep) are used to indicate that the Fermi statistics for electrons and holes is to consider the electron quasi-Fermi level (Fn) and hole quasi-Fermi level (Fp), respectively. In some cases, a Dirac delta function is used to explicitly indicate that the equation under question has nonzero values only when the energy between the upper and lower levels (En and Ep),is near the energy of interest. In addition, instead of the electron occupation probabilities for the upper and lower levels, the statistics for the upper level for electron

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226

occupancy and statistics for the lower level for hole occupancy are sometimes used. In other words, f(E1) depicts the hole occupation factor and [1
f(E1)] represents the missing hole or electron occupation factor in level 1. This is interchangeably done in the literature, which causes some confusion but doing so allows the easily recognizable Fermi–Dirac equation for holes using the quasi-Fermi level for hole. The knowledge of the matrix element would then pave the way for calculating the absorption coefficient at a given energy. Similarly, the spontaneous emission rate can be calculated using the matrix element for a semiconductor. Considering Equation 2.147, which related the Einstein’s A coefficient that is specific to transitions between two discrete levels, level 2 and level 1, and making the necessary changes to account for a band of states around these two levels in a semiconductor (as we have for the absorption coefficient), we can write, for the spontaneous emission rate in the case of a single valence band (with the aid of Equation 2.108) [54]: R21 (E)jsp ¼

4pq2 Enr V m20 e0 h2 c 3

1 ð

rC (E 0 )rV (E 0
E)jM(E 0 ;E 0
E)j2 f 2 (E 0 )[1
f 1 (E 0
E)]dE 0


1

and with joint density of states, within the direct transition selection rule R21 (E)jsp ¼

4pq2 Enr rCV (E)jM(E)j2 f 2 (E)[1
f 1 (E)]: m20 e0 h2 c 3

ð2:164Þ

(Note that in Ref. [54], the lower case rates are defined per unit volume per unit energy. That is why Lascher and Stern [54] have an extra J
1 in units. The above coefficients together with the matrix element have m3 s
1 units. When using the joint states, the integral and the volume V should be removed.) If the matrix element is the same for all the initial and final states, it can be taken out of the integral in Equation 2.164 and we then would have (including indirect transitions as well) [54] 4pq2 Enr R21 (E)jsp ¼ 2 2 hjMj2 iav V m0 e0 h c 3

1 ð

rC (E 0 )rV (E 0
E)f 2 (E 0 )[1
f 1 (E 0
E)]dE 0 :


1

ð2:165Þ Enr =m20 e0 h2 c3 ) hjMj2 iav V,

With the definition of a recombination coefficient B  (4pq Equation 2.165 can be rewritten as (B has units m3 s
1, as also stated by Lascher and Stern [54]) 2

1 ð

R21 (E)jsp ¼ B

rC (E 0 )rV (E 0
E)f 2 (E 0 )[1
f 1 (E 0
E)]dE 0 :

ð2:166Þ


1

(Note that this rate also has units m
3 s
1 J
1. The integral over energy gives the overall rate per unit volume.) The integration could be made of a double integral, one over the valence band energies and the other over the conduction band. It should be noted that when the valence band mixing occurs, the bands cannot be represented by the parabolic band

2.4 Optical Gain

approximation and the simple momentum conservation treatment does not strictly hold. Consequently, more rigorous numerical simulations must be undertaken. In quantum wells, the density of states is of a staircase form corresponding to each of the subband energies, which also must be taken into consideration. These are discussed in Section 2.4.3 in some detail. As in the case of spontaneous emission rate, with the aid of Equation 2.119, we can write the stimulated emission rate as R21 (stim) ¼

4pq2 Enr V m20 e0 h2 c3

1 ð

rC (E 0 )rV (E 0
E)jM(E 0 ; E 0
E)j2 [f 2 (E 0 )
f 1 (E 0
E)]dE 0


1

4pq2 Enr R21 (stim) ¼ 2 2 rCV (E)jM(E)j2 [f 2 (E)
f 1 (E)] (with the joint density of states): m0 e0 h c 3 ð2:167Þ Note that the Fermi functions in Equation 2.167 represent the electron occupation probabilities for levels 1 and 2. Referring to Equation 2.119, the net stimulated emission rate R21|st can be found from R21 jst ¼ R21 (stim)(eE=kT
1)
1 and doing so leads to 4pq2 Enr V(eE=kT
1)
1 R21 (E)jst ¼ m20 e0 h2 c 3

1 ð

rC (E 0 )rV (E 0
E)jM(E 0 ; E 0
E)j2


1

½f 2 (E 0 )
f 1 (E 0
E)]dE 0 and with the joint density of states: R21 (E)jst ¼

4pq2 Enr (eE=kT
1)
1 rCV (E)jM(E)j2 [f 2 (E)
f 1 (E)]: m20 e0 h2 c 3

ð2:168Þ

As in the case of spontaneous emission, if the matrix element is the same for all the initial and final states, it can be taken out of the integral and we then would have [54] 1 ð

R21 (stim) ¼ B

rC (E 0 )rV (E 0
E)[f 2 (E 0 )
f 1 (E 0
E)]dE 0 :

ð2:169Þ


1

If there were no selection rules, one can integrate over the conduction and valence band independently, and doing so would lead to the rate at which photons are spontaneously emitted in unit volume: ð ð2:170Þ Rsp ¼ R21 (E)j dE ¼ Bnp; sp

where n and p are electron and hole concentrations, respectively. This total rate is in now in m
3 s
1 units. If we further assume that the electrons in the conduction band (band 2) and holes in the valence band (band 1) are in equilibrium and can be characterized by quasi-Fermi levels, Fn and Fp, respectively (as in the case of the literature, they are also defined as EFn and EFp), then the Fermi–Dirac distribution

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j 2 Semiconductor Lasers

228

function in the conduction band, fC, for electrons is given by fC ¼

1 1 þ exp[(E C
F n )=kT]

ð2:171aÞ

and that for holes (not electrons) in the valence band 1 : fV ¼ 1 þ exp[(E V
F p )=kT]

ð2:171bÞ

(As mentioned above, Fn and Fp are the quasi-Fermi levels for electrons and holes, respectively. We should also mention that a multitude of nomenclature is used for depicting the upper and lower levels as E2, EC, and E0 for the upper level and E 1 ; E V ; and E 0
E for the lower level. This is really brought about by making the transition from using the generic nomenclature of levels 1 and 2 to the specific forms used in semiconductors, such as EC and EV.) The stimulated and spontaneous emission rates, regardless of any assumption that may be made about the matrix elements, are related to each other by

  (E
F n þ F p ) : ð2:172Þ R21 (E)jstim ¼ R21 (E)jsp 1
exp kT Another implicit approximation made so far is that the scattering within the band was neglected. Such scattering causes a broadening in the absorption coefficient and, thus, the gain coefficient [55]. This is typically accounted for by using a broadening factor in the form of L(E eh
E) ¼

1
h=tin ; p (E eh
E)2 þ (
h=tin )2

ð2:173Þ

where Eeh is the optical transition energy. The intraband relaxation time of the final state, tin, is not exactly known for Wz GaN, but a value of 10
13 s has been used with good results. However, gain calculations so far have assumed that this time constant is similar to those of the well-characterized ZB semiconductors such as GaAs for which it is on the order of 0.1 ps. The linewidth broadening expression can be generalized to take into account any scattering mechanism that causes broadening, in which case we replace
h=tin with the general broadening factor G(Eeh
E). This factor is energy dependent and must take into account processes in both the conduction and valence bands [56]. Alternative depictions of the broadening parameter are (1=p)[g=(E 21
E)2 þ g 2 ], hw)2 þ (g)2 ], where g is (1=p)[(
h=tin )=((E CV

hw)2 þ (
h=tin )2 )], and (1=p)[g=(E CV

the half linewidth of the Lorentzian broadening line shape. Taking this broadening into consideration, the absorption or the gain coefficient of Equation 2.162 can be written for the broadened case as ð g B (E) ¼ g(E)L(E eh
E)dE eh pq2
h ¼ 2 m0 e0 nr cE

1 ð

rCV (E)
1

1 h=tin
jM(E)j2 [f 2 (E)
f 1 (E)]dE eh : p (E eh
E)2 þ (
h=tin )2 ð2:174Þ

2.4 Optical Gain

Again, the occupation probabilities defined in the above equation are for electrons in levels 1 and 2. The term Eeh represents the actual bandgap (the effect of any strain must be included) in the bulk layer and the bandgap plus the conduction and valence band confinement energies in quantum wells. We should also mention that the gain (absorption coefficient) defined in Equation 2.162, and therefore in Equation 2.174, is based on the k-selection rule and for thick active layers (the bulk case). In addition, all the lasers produced and researched are in the form of quantum wells for reduced threshold and less temperature sensitivity. We will thus modify Equation 2.162, and therefore in Equation 2.174, and make it more applicable to GaN quantum wells, as done in Section 2.4.3. In that case, the integral of in-plane and out-of-plane wave vector and summation over possible quantum states will be used. In general, however, the single-particle model with k conservation cannot be used. This is particularly true in semiconductors with significant band mixing, which is the case in the valence band of GaN and related materials. In such a case, the gain expression, Equation 2.162, must be modified as follows (the subscript is dropped from the broadened gain, i.e., we will use g(E) instead of gB(E) from here onwards) [57] ð
p
hq2 h=tin CV 2 1 (k )jM j r g(E) ¼ xyz CV 2 p nr e0 m0 cE (E eh
E)2 þ(
h=tin )2 [f C (kxyz )þf V (kxyz )
1]dkxyz ð 2
kxyz CV 2 1 p
hq2 h=tin ¼ jM j [f (kxyz )þf V (kxyz )
1]dkxyz 2 2 p (E eh
E)2 þ(
h=tin )2 C nr e0 m0 cE p p
hq2 ¼ nr e0 m20 cE

1 ð

1 ð

kz ¼
1 k? ¼0

k2xyz p2

jMCV j2

[f C (k? ;kz )þf V (k? ;kz )
1] p
hq2 ¼ nr e0 m20 cE

1 ð

1 ð

kz ¼
1 k? ¼0


1 h=tin p (E eh
E)2 þ(
h=tin )2

2pk? dk? dkz 4pk2xyz


k? h=tin CV 2 1 jM j 2p2 p (E eh
E)2 þ(
h=tin )2

[f C (k? ;kz )þf V (k? ;kz )
1]dk? dkz :

ð2:175Þ

where fC(k?,kz) depicts the electron occupancy in the conduction band, and f V (k? ;kz ) depicts the hole occupancy in the valence band ([1
f V (k? ;kz )] depicts the electron occupancy in the valence band). Also, the term Eeh in the broadening term is a function of k?,kz. The density of states in three-dimensional systems is given by rCV (k) ¼ k2 =p2, as in Equation 2.148. The key point is that the infinitesimal volume is 4pk2xyz dkxyz ¼ dkz dky dkz ¼ 2pk? dk? dkz . The integral extends over k? (normal to the c-axis) and kz (parallel to the c-axis) that are possible for a certain energy E, and the summation is included in the case of multiple valence and conduction bands. The integration over momentum can be replaced with summations if numerical techniques are

j229

j 2 Semiconductor Lasers

230

employed. Neglecting momentarily the factor in front of the integral, the optical gain is actually the product of three terms. The term rCV is the joint density-of-states function, which is determined by the band structure. For a bulk semiconductor and within parabolic approximation, the density of states in the conduction band for 3D, 2D, and 1D systems are given by Equations 2.150, Equation 2.152, and Equation 2.154, respectively. To obtain the JDOS, the effective mass in these equations must be replaced with the reduced effective
1
1 mass, m
1 r ¼ mn þ m p where mn and mp are, respectively, the effective density of states masses in the conduction and valence bands, as depicted by Equation 2.160 for the 3D case (m e and m h nomenclature is also used). The effective density of states mass in the conduction band is the same as the electron’s effective mass. But due to band mixing, the same does not hold for the valence band. For details, see Volume 1, Chapter 2. While the conduction band in nitrides can be assumed parabolic, the valence band is much more complex with heavy- and light-hole band mixing. In this case, a numerical integral over the k-space is warranted. Because the electron’s effective mass is much smaller than the hole mass, JDOS is dominated by the electron mass in both the Wz and ZB structures, whereas the conduction band is s-like and isotropic. A reduced JDOS implies lower gain. But the same also requires a lower transparency current and thus a lower threshold current; this is the premise of strained-layer quantum well lasers in the InGaAs/GaAs system [58]. In the ZB case, the VBM is degenerate, and this is lifted in QW structures. This, in turn, leads to a reduced density of states and therefore lower transparency current. On the other hand, VBM in the Wz structure is separated into three twofold degenerate bands, one G9 and two G7 bands. These bands are not affected much by quantum wells grown in the [0 0 0 1] direction, as discussed in detail in Volume 1, Chapter 2. The factor MCV which is the squared optical transition matrix (OTM) element, is given by [33] MCV ¼ hFC j~ pjFV i;

ð2:176Þ

where ~ p is the momentum operator, and FC and FV are the actual electron and hole wave functions, respectively. Because ~ p is a vector, we expect M CV also to have three components along the x-, y-, and z-directions, with z being the growth direction. Because the splitting of the p-like states at the VBM is determined by the crystal field, the matrix elements are dependent on the symmetry. For the TE mode, the E field is in the plane of the junction and so is the matrix element MCV . For the TM mode, the E field is normal to the plane of the junction so is the matrix element. The dipole momentum matrix element for a Wz material in the c-direction is [59] !
h2 m0 m0 (E g þ 2D2 )(E g þ D1 þ D2 )
2(D3 )2 CV 2 ; ð2:177Þ
jM== j ¼ 2m0 m=e= m=c= (E g þ 2D2 ) ==

==

where me and mc represent the c-direction electron effective mass and the influence of higher energy bands on the effective mass, both along the kz direction, respectively.

2.4 Optical Gain

The D1, D2, and D3 parameters represent the diagonal and off-diagonal terms of the 6  6 Hamiltonian that describe the three valence bands. The momentum matrix element in the direction perpendicular to the c-axis is about the same as that along the ? c-direction, but with effective masses m? e and m c in the kx,y plane. With the quasicubic approximation, that is, Dcr ¼ D1, Dso ¼ 3D2 ¼ 3D3, the matrix element reduces to ! E g (E g þ Dcr þ Dso ) þ 23 Dcr Dso
h2 m0 CV 2

1 ; ð2:178Þ jM== j  = = 2m0 me E g þ 23 Dso where Eg ¼ 0.25 Ry, Dcr ¼ 72.9 meV ¼ 5.36 mRy, and Dso ¼ 5.17 meV ¼ 0.38 mRy. The dimensionless Rydberg (Ry) denotes energy in terms of the lowest H energy level (13.6 eV), that is, 1 mRy ¼ 13.6 meV. In the direction perpendicular to the c-axis (in-plane), the momentum matrix element is approximately expressed as 2 jMCV ? j 

  E g (E g þ Dcr þ Dso ) þ 23 Dcr Dso
2 m0 h :
1 ? 2m0 me E g þ Dcr þ 23 Dso þ (Dcr Dso =3E g )

ð2:179Þ

The third term in the integrals of Equations 2.174 and 2.175 is the occupation factor, which describes the carrier density distribution. Again, the stated form represents the electron occupation, fC(E), in the conduction band and hole occupation, f V (E), in the valence band, as defined in Equation 2.171. As in the case of the matrix elements, the Fermi level, too, is dependent on the density of states in the valence band where the degeneracy is governed by the crystal symmetry as well [36]. The joint density of states calls for an increase in the gain with increasing energy. On the other hand, increasing energy reduces the occupation factor. These two competing processes result in a maximum in the gain, the maximum gain or the peak gain, with respect to energy. In addition, the gain increases with the injected carrier density and thus the injection current. This is schematically depicted in Figure 2.26. 2.4.3 Gain in Quantum Wells

Due to the quantization in the z-direction (the growth direction), the crystal momentum in this direction takes on discrete quantized values and shifts up the effective bandgap. In the plane of growth, however, the band remains as in the bulk (Volume 1, Figure 2.6a and b) unless the in-plane effective masses change, such as in the case of ZB strained wells [58]. Another pertinent change is a staircase DOS function, as indicated in Volume 1, Figure 2.19b. Consequently, in writing the gain expression for a quantum well, the integral over E in Equation 2.162, and therefore in Equation 2.174, is replaced by a summation over all transitions between the conduction energy levels and the valence energy levels [60]. Let us now go through the important steps involved in determining the gain coefficient in quantum well lasers as treated, for example, by Fan et al. [61].

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Dens ity of states

Energy

Conduction band n, J

Energy

gmax

Filled states

ntrans, Jtrans

Eg Empty states Gain, g

Light hole

maximum gain, gmax

Heavy hole

Figure 2.26 Schematic representation of conduction, heavy-hole and light-hole band densities of states, and gain versus energy for increasing injection, and the peak gain as a function of injection charge and/or current in Wz GaN. Note the transparency injection charge or current.

The electron concentration in all the available, in-aggregate, states in the quantum well for a parabolic band can be obtained by integrating the product of the Fermi–Dirac distribution function and 2D density of states over possible energies as ð1 N¼ f (E)rC (E)dE: ð2:180Þ EC

With the help of Equations 2.153 and 2.171, we can write for the electron concentration

  m kT X (E i
E FC ) N¼ n2 ln 1 þ exp
C : ð2:181Þ kT p
h i If we convert the electron concentration to an equivalent three-dimensional form by dividing the above expression by Lz (this would allow the use of the conventional gain equations in the case of quantum wells), we have [61]

  mn kT X (E i
E FC ) ln 1 þ exp
C N¼ 2 ; ð2:182Þ kT p
h Lz i where Lz is the quantum well width, EFC is the electron quasi-Fermi level, which we have also depicted by FC or Fn, and E iC represent the quantized conduction band energy levels in the quantum well. The summation is over all the quantized levels in the conduction band that are occupied.

2.4 Optical Gain

Because the valence band is not parabolic, the hole concentration must be found through a numerical integration that takes the dispersion into consideration Xð X ð ð kxy 1 1 j j P¼ f V [E V (kx ; ky )] dkx dky r2D (kxy )f V [E V (kx ; ky )]dkxy ¼ p Lz 2pkxy j j Xð ð 1 j ¼ f [E (k ; k )]dkx dky ; ð2:183Þ 2L V V x y 4p z j j

where E V is the hole energy (not for electrons) in the valence subbands. As in the case of the conduction band, the summation is over all the valence band quantized states that are occupied. Note that the hole concentration also has been normalized to a three-dimensional form with the Lz term in the denominator. Similar to Equation 2.171, the Fermi–Dirac distribution function in the conduction band of the quantum well structure, fC, for electrons is given by fC¼

1

ð2:184aÞ

1 þ exp[(E iC
F n )=kB T]

and that for holes (not for electrons) in the valence band fV¼

1 j 1 þ exp[(E V
F p )=kB T]

;

ð2:184bÞ

where Fn and Fp are the quasi-Fermi levels for electrons and holes, respectively. We must state that the occupation probability of electrons in the valence is what enters into the gain expression. Because of that we must use f 1 ¼ 1
f V , which makes the f2
f1 term to be (f C þ f V
1) for semiconductors as in Equation 2.175. As shown schematically in Figure 2.27, the dependence of the gain on energy in quantum well lasers differs from that of the bulk, Figure 2.26, due to the unique staircase-like density of states in which the gain maximum occurs at the confinement energy. This is because the density of states remains constant while the occupation terms decrease exponentially (Volume 1, Figure 2.19b). The gain coefficient [62], when modified [55, 61], is then given by g(E) ¼

pq2
h X 2 m0 e0 nr cE i; j

ðð

1 1 h=t
jMCV j2 4p2 Lz p (E eh
E)2 þ (
h=t)2

 [ f C (k) þ f V (k)
1]dkx dky :

ð2:185Þ

Recall that the density of states in a two-dimensional system is constant for a given quantum state. It cannot be overstated that the occupation probabilities in the above equations are for electrons in conduction band and holes in the valence band. The above equation must be modified to include carrier scattering, bandgap renormalization, and many-body effects [63, 64]. Assuming that the conduction and valence bands are parabolic, neglecting band mixing of the heavy and light holes in the valence band, and assuming the occupation of only the heavy-hole band, the joint DOS is

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Conduction band n, J

Energy

Energy

Staircase Density of states

Filled states

Eg+ ∆Econf

gmax

ntrans, Jtrans

Empty states Gain, g

maximum gain, gmax

Staircase Density of states

Light hole

Heavy hole

Figure 2.27 Schematic representation of conduction, heavy-hole and light-hole band densities of states, and gain versus energy for increasing injection, and the peak gain as a function of injection charge and/or current in a Wz GaN quantum well structure. Note the transparency injection charge or current.

i; j

rCV (E) ¼


1 j mr u E
E iC
E V ; 2 ph Lz

ð2:186Þ

where u denotes the unity function. The reduced mass here represents the reduced effective DOS mass. In general, the valence band mass includes the effect of band mixing of the heavy- and light-hole bands, which is also applicable to nitrides. However, if the disparity between the heavy-hole-band mass and light-hole-band mass is very large, and the momentum conserving transitions are the only processes taking place, the DOS hole mass can be equated to the heavy-hole mass. The jMCV j2 term in Equation 2.185 is the squared OTM element [55]. As in the bulk case, Equation 2.128, the matrix elements MCV are calculated as i   E D MQW   MQW with i ¼ x; y; z; ð2:187Þ ¼ YnV ;k p^i YnC ;k MCV i where p^i is the momentum operator. For TE mode MCV ¼ M x or MCV ¼ M y and for TM mode MCV ¼ Mz . In an ideal system with no perturbation and at the k ¼ 0(G) point, the optical transitions from conduction subband states to the valence subband states obey the selection rule Dn ¼ 0. The squared OTM for the TE mode represents mainly the contribution from electrons to heavy-hole transitions, while for the TM mode the contribution comes from electrons to light-hole and split-off hole-band transitions. This is because Mz does not include the heavy-hole wave function, while Mx (or My)

2.5 Coulombic Effects

includes it as well as the light-hole and split-off-hole wave functions. However, a semiconductor laser with a high injection of carriers, particularly a nitride-based one, does not represent this case, and symmetry breaking transitions would be allowed. This necessitates numerical approaches, especially for gain calculations, which include the entire band structure over the momentum and energy spaces. This is also applicable to the calculation of the OTM elements. The optical transition energy in the case of quantum wells is given by E eh ¼ E enC þ E hnV þ E sg :

ð2:188Þ

where E sg is the bandgap energy in strained GaN. The GaN quantum well would be placed between AlGaN waveguide layers and therefore the effect of strain on energy must be considered. Note that GaN is being used as a model quantum well material. In reality, the bulk of the laser work is based on InGaN. However, the concepts being discussed here would apply to InGaN wells as well. The energy of the strained GaN is defined as (see Volume 1, Chapter 2 for an extensive discussion)   c 12 E sg ¼ E g þ 2DCV 1
ð2:189Þ exx ; d c 11 where DCV d is the deformation potential, c11 and c12 are the elastic constants, and exx ¼ (a
a0 )=a0 is the in-plane strain in terms of the lattice constant of strained GaN, a, and relaxed GaN, a0. The spontaneous emission rate is given by ðð
nq2 E X jMCV j2 1 h=t f C (kx ;ky )f V (kx ;ky ) dkx dky Rsp (E) ¼
2 2 2 4p Lz p (E eh
E)2 þ(
h=t)2 p h e0 m0 c 3 nC ;nV or, since the energy Eeh in the broadening term is a function of k?,kz, ðð
nq2 E X jMCV j2 1 h=t f (E)f (E) dkx dky : Rsp (E) ¼
2 C V 2 4p Lz p (E eh
E)2 þ(
h=t)2 p h e0 m20 c3 nC ;nV ð2:190Þ The radiative current density can be calculated from the spontaneous emission rate as ð ð2:191Þ J rad ¼ qLz Rsp (E)dE:

2.5 Coulombic Effects

In the case of wide-bandgap semiconductors such as those employed for GaN-based lasers, the exciton binding energies are comparable to kT at room temperature, and Coulombic effects may not be automatically neglected without further consideration [63]. As discussed in Ref. [65], the semiconductor ZnO has an exciton binding energy of 60 meV that is several kTs at room temperature. In particular, the role of

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excitons in laser action is discussed [65] with stimulated emission. In the intermediate excitation density regime in ZnO, emissions due to biexcitonic, exciton–exciton and exciton–carrier interactions may be observed. The inelastic collision between the excitons results in an exciton excited into a higher state and photons whose energy is lowered by the same amount below the gap energy minus the exciton binding energy (Equation 2.29 in Ref. [65]). At high excitation densities, electron–hole pairs form, which in the case of ZnO is in the low 1019 cm
3 range, comparable to that in GaN. If such is the case, then the parameters used in the gain expression, Equation 2.185, must be modified to reflect this. Moreover, broadening effects such as carrier scattering must be considered. As we did in conjunction with Equation 2.174, the traditional approach is to convolute Equation 2.185 with a spectral line-broadening or line-shape function [55, 63, 64]. The resulting gain spectrum converges to the square root dependence of the absorption coefficient is described in Volume 2, Equation 5.18 at low carrier densities for the bulk and the parabolic band cases. With increasing injected carrier density, a positive gain peak develops. Keep in mind that the linewidth broadening in GaN-based structures is much larger than that in more mature semiconductor technologies. In addition, due to large concentrations of defects in relation to other more developed semiconductor laser materials, the linewidth parameter would be affected by inhomogeneous broadening with is not represented in Equation 2.173. The gain in wide-bandgap semiconductors is smaller and, in general, the peak gain is further reduced due to spectral broadening. Both are implicit in many of the gain expressions commonly used. It is thus instructive to consider another form of the gain expression [66], given in Equation 2.89, but with considering the quantum efficiency, namely, g(
hw) ¼

h2 c 2 Jhint ; 8pqn2r (
hw)2 Dnd

ð2:192Þ

which takes into account line broadening. The terms have their usual meanings:
hw, Dn, and d represent the emission energy, the spontaneous emission spectral halfwidth in terms of frequency, and the thickness of the active region, respectively. At first glance, the larger the spontaneous spectral width, the smaller is the gain. Likewise, the higher the frequency, as is the case for short-wavelength lasers, the smaller is the gain. Generally, any increase in the bandgap is accompanied by a reduced broadening due to the large phonon energies and other factors. However, broadening caused by inhomogeneities prevents narrow spectral widths in GaN and related ternaries. This is also true for excitonic transitions in case they are involved in the lasing action. It is therefore imperative that nitride-based semiconductors be prepared to produce sharp spontaneous linewidths. Reducing the doping level in the semiconductor’s active region also reduces the linewidth with the adverse consequence that the low doping levels cause the parasitic resistance to increase. Fortunately, in quantum well semiconductor lasers, the active region that determines the linewidth, is very thin; even when left undoped, it does not cause the resistance to be unacceptably large.

2.5 Coulombic Effects

Coulomb interaction between oppositely charged carriers helps to keep them closer to each other. Because the recombination efficiency increases with a reduced distance between the recombining electrons and holes, an increase in the spontaneous emission rate and consequently in gain would result. The question, of course, is whether the injected carrier concentration in available GaN-based lasers is too large to allow this phenomenon to show up. Specifically, present GaN lasers require injected carrier concentrations in the range of 1019–1020 cm
3 for transparency; and as discussed in Section 2.9, a good deal of broadening due to inhomogeneities occurs in InGaN, which dampens the effect of a Coulombic interaction except perhaps in dynamical situations or as the gain builds up. Consequently, it is highly unlikely for this process to be of much importance at this stage of development in the sense that it is in ZnCdSe lasers. However, with improved materials quality, the carrier concentration that must be injected for transparency and beyond should be substantially reduced in which case the Coulombic interaction may indeed play a role [67]. In CdZnSe single quantum well lasers and at an injected carrier concentration of 5.5  1011 cm
3, the Coulombic interaction term increases the gain by a factor of 2.35 and is accompanied by a blueshift of 6 meV in its spectrum. Carrier scattering alone produces a blueshift in the gain spectrum, as the injected carrier concentration is increased. A similar behavior occurs when the cavity length is enlarged. Shorter cavity lengths increase the end losses and cause the threshold gain to become larger and thus they require higher injected carrier concentrations. Bandgap renormalization alone causes a redshift with increasing injection, which implies a redshift with reduced cavity lengths. The Coulomb interaction causes a blueshift with higher injected carrier concentration. All of the three factors involved lead to a blueshift of the gain with an injected carrier concentration and a reduced cavity length. The emission energy in CdZnSe quantum well lasers reveals a blueshift with decreasing cavity length [68]. This has been suggested as evidence that Coulomb attraction indeed plays a role in these lasers although the experimental emission energies lagged the calculated ones by about 10 meV. This has been attributed to inaccuracies in the structural parameters. There are other terms that can affect the gain among which is a form many-body effect, specifically the bandgap renormalization. This is brought about by the large carrier concentrations that screen the repulsive processes between the valence and conduction electrons. It causes a change in the electron distribution, which, in turn, results in the reduction of the transition energy (effective bandgap). This decrease is called the Coulomb hole contribution to the bandgap renormalization. At very high injection levels, which may not be sustainable, both the ionic and covalent contributions to the bandgap may change. The Coulomb hole contribution to the bandgap renormalization is expressed as [67–69] " sffiffiffiffiffiffiffiffiffiffi # 8pnd DE CH ¼
2E R aB ls ln 1 þ ; ð2:193Þ 3 ls aB where d is the thickness of the active layer with the product specifying the areal density of electrons, and ER, aB, and ls represent the Rydberg energy, Bohr radius for

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238

electrons (excitons if excitons dominate the gain process), and the screening length, respectively. The Bohr radius is given by aR ¼

4p
h2 e ; q2 m

ð2:194Þ

where e is the dielectric constant and m is the effective mass of the electrons (reduced mass of excitons in case the gain is governed by excitons). The Rydberg energy can be written as ER ¼


2 h : 2ma2R

ð2:195Þ

The screening length for quantum wells can be defined by [70]   q2 X mCj f C (E Cj ) mVj f V (E Cj ) 2 ls ¼ 2 þ ; dCj dVj p
h e j

ð2:196Þ

where p
h p dnj ¼ j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ j knj? 2mn E nj

and

p
h p dVj ¼ j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ j ; kVj? 2mV E Vj

ð2:197Þ

where knj? and kV~j ? represent the equivalent out-of-plane wave vector for the j state that would assume values 1, 2, and so on, depending on the number of quantized states that participate. For injection levels in the range between low and high 1018 cm
3, the bandgap reduction is roughly between 40 and 60 meV. Additional contributions to the bandgap renormalization come from the screened carrier exchange. Both of these contributions have been treated elsewhere in detail [67]. Thus, the exchange contribution to the renormalization can be expressed as DE sx

2E R ¼ ls

1 ð

dk== k== 0

1 þ (ls aB k2== =8pnd) 1 þ (k== =ls ) þ (ls aB k2== =8pnd)

[f C (E Cjk== ) þ f V (E Vjk== )]: ð2:198Þ

For injection levels in the range between low and high values of 1018 cm
3, the bandgap, reduction is roughly between 20 and 40 meV. Assuming carrier injection levels in the low to high 1018 cm
3, the total bandgap, renormalization due to the Coulomb-hole contribution and the screened carrier exchange correction adds up roughly to values between 60 and 100 meV at room temperature. A phenomenological expression devised for GaAs relates the bandgap shrinkage to the sum of the cubic powers of the electron and hole concentrations [30]. Figure 2.28 depicts schematically the spectral gain in a quantum well laser for a given injection level without the many-body effects (a). Carrier scattering causes a broadening of the gain accompanied with a reduction of the peak gain (b). The Coulomb interaction increases the gain due to a reduction of the carrier-to-carrier

2.6 Numerical Gain Calculations for GaN

(a)

g(hν) (a.u)

(c)

(b)

Eeh

(a) No carrier scattering (b) With carrier scattering (c) With Coulomb interaction

Energy (a.u) Figure 2.28 Spectral gain in a quantum well laser for a given injection level: (a) without the many-body effects, (b) with carrier scattering, and (c) with Coulomb interaction.

distance because of the attraction between recombining carriers. This reduces the distance between each pair and thereby increases the recombination efficiency (c).

2.6 Numerical Gain Calculations for GaN

Despite a relatively short debut, GaN-based lasers in general and their optical gain in particular have attracted a great deal of theoretical attention due to the success of these lasers in compact disk players. In fact, following the successful demonstration of bright LEDs, the theoretical work aimed at determining the valence band structure and the optical gain in both the zincblende and wurtzite phases of GaN has progressed in parallel with efforts to demonstrate the first laser. While the bulk of the initial attempts involved the ZB phase, possibly because of the availability of computer codes developed for other zincblende systems, the attention quickly gave way to the wurtzite phase as this phase appears more and more to be the technologically important one. Consequently, the treatment here will be focused on the Wz phase with some mention of the ZB phase. Due to the nonparabolicity of the valence band, the p-like states, strain, and band mixing, gain calculations for GaN require the knowledge of the full bands and numerical methods applied. Last but not least, the lasing mechanism must be known, that is, free electron–hole based or exciton based. Calculation of the nitride band structure was covered in Volume 1, Chapter 2, and the treatment of gain that follows, builds on our knowledge of the band structure. 2.6.1 Optical Gain in Bulk GaN

Several researchers have attempted to calculate the gain in bulk GaN assuming an ideal semiconductor with no inhomogeneities and under the assumption that

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240

transitions are governed by electron–hole recombination. Contributions due to a Coulombic interaction, such as excitons, have also been considered. The Coulombic interaction serves to keep the carriers of opposite charge closer together, which enhances the recombination efficiency and, therefore, the gain. Assuming the electron–hole plasma to be the governing process, Suzuki and Uenoyama [60, 71, 72] and Meney and O’Reilly [73] calculated the gain in bulk Wz GaN utilizing the band structure determined by an adaptation of the full-potential linearized augmented plane wave (FPLAPW) [74]. The hole masses of the uppermost two valence bands along the growth direction are very heavy [75]. The mass of the CH band is light due to a strong coupling with CBM through the kzpz, perturbation. In the x-direction, in the plane of the layers, CBM is coupled to the mixed state of the uppermost two valence bands and, thus, the hole mass of the mixed band is light. The lowest valence band, CH, supports the optical gain of the TM mode. For this mode to become important, an extremely large carrier injection is required. This paves the way for the dominance of the optical gain by the TE mode under normal conditions. In the ZB case, of the three valence bands, the uppermost two bands are degenerate and the hole mass is heavy in all directions. This causes DOS to be larger than that for the Wz phase as long as the crystal field splitting in the latter is larger than the spin–orbit splitting. This results in the gain being lower in the ZB phase than in the Wz one. Because the TM mode is supported by the light-hole and spin–orbit split bands, the TE and TM modes are both available in the ZB phase. Figure 2.29 shows the optical gain in Wz (for the TE mode with its E field in the c-plane) and in ZB bulk GaN (for the TE mode, and TM mode with its E field in the c-direction). The topic of gain in bulk lasers did not get as much attention as in quantum wells, which will be discussed in the next section.

3000

Bulk GaN

Maximum gain (cm–1)

TE mode TM mode 2000 Wurtzite 1000

Zincblende 0 5

10

15

Injected carrier density (1018 cm–3) Figure 2.29 Maximum gain versus injected carrier density for the TE mode in bulk Wz GaN, and TE (solid) and TM (dashed) modes in bulk ZB GaN [71].

20

2.6 Numerical Gain Calculations for GaN

2.6.2 Gain in GaN Quantum Wells

Quantum wells at the very least serve to reduce the volume of the semiconductor that must be pumped, leading to very low threshold currents. More importantly, by virtue of thin layers employed in quantum wells, coherently strained systems can be obtained. Strain-induced reduction of the in-plane hole mass in ZB structures leads to a much reduced transparency density and thus a reduced threshold. Consequently, the degrees of freedom in laser design have been enhanced immensely with the advent of quantum wells.ThefavorableeffectofstraingermanetoZBsystemsisnotpredictedtooccurinWz structures. The heterostructures used in lasers based on wide-bandgap nitrides are not lattice matched; this will eventually require the use of thin layers. In addition, InGaN has not proven to be easily produced in bulk form and necessitates thin layers, which are provincially referred to as quantum wells. In what follows, the optical gain in quantum wells will be discussed, first without strain and later with strain. 2.6.3 Gain Calculations in Wz GaN Q Wells Without Strain

As reported by Suzuki and Uenoyama [60], in bulk Wz GaN, the hybridization of the CH band with HH and LH bands [3] is negligible at VBM due to small spin–orbit splitting energies. In Wz GaN QWs though, the splitting between the CH bands, and HH and LH bands1), is more pronounced, but the band mixing is similar to that in bulk GaN, which implies that DOS is not substantially reduced and that the TM-mode gain is not supported, leaving the TE mode to be dominant. In structures with pronounced quantization, that is, with well widths comparable to the Bohr radius, the separation of the HH and LH bands is more pronounced, and VBM DOS is somewhat reduced. The resultant effect manifests itself as a noticeable gain production at relatively low carrier densities. Results of the simulations by Suzuki and Uenoyama [60] for a series of GaN/Al0.2Ga0.8N quantum wells are shown in Figure 2.30. As Equation 2.194 clearly indicates, the intraband scattering broadens the spectral gain and reduces the maximum gain. In the absence of any experimental value for this parameter, Suzuki and Venoyama [60] performed gain calculations for a series of relaxation times in the range of 0.01 ps to infinity for a 60 Å active layer. The results are presented in Figure 2.31 in the form of gain versus the injected sheet carrier concentration . 2.6.4 Gain Calculations in Wz Q Wells with Strain

As discussed in Volume 1, Chapter 2, the band structure and thus the gain are affected by strain. Calculations performed by Suzuki and Uenoyama [60] indicate that 1) The HH, LH, and CH depict heavy hole, spin–orbit split-light hole, and crystal field split valence bands.

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Maximum gain (cm–1)

3000

TE mode L z =30Å

40Å 50Å

60Å 70Å

2000 80Å

1000

0 5

10

15

Injected carrier density (1012 cm–2) Figure 2.30 Maximum optical gain in Wz GaN/A10.2Ga0.8N single QWs with an active layer width varying from 30 to 80 Å with respect to the injected areal carrier concentration. The initial slope of the maximum gain in the QW structure is somewhat larger than that in the bulk. Due to gain saturation, the slope is reduced with increasing injection of carriers [72].

the biaxial strain lowers the injected carrier density for a given maximum optical gain. As a companion to the strain-free case illustrated in Figure 2.30, the maximum gain for a series of SQW is depicted in Figure 2.32 for a 0.5% compressive biaxial strain. On the other hand, the tensile strain has an adverse effect on the gain. Figure 2.33 exhibits the effect of biaxial and uniaxial (in the c-plane) compressive and tensile strains. The uniaxial compressive strain is along the y-direction that is perpendicular to the direction of propagation (x-direction). The growth direction is the z-direction or the c-direction. The tensile uniaxial strain is perpendicular to the polarization vector, 3000

Maximum gain (cm–1)

L z=60 Å TE mode

τi=

0.1 ps

0.05 ps

2000

1000 0.01 ps 0 0

5

10

Injected carrier density

15 (1012 cm–2)

Figure 2.31 Effect of the intraband relaxation time, tI, on the gain for TE mode in a GaAl0.2Ga0.8N structure with a 60 Å active layer [60].

20

2.6 Numerical Gain Calculations for GaN

0.5% Compressive biaxial strain 3000

TE mode

40Å 50Å

30Å

60Å

Maximum gain (cm–1)

70Å

2000 L z =80Å

1000

0 5

10

Injected carrier

15

density (1012

cm–2)

Figure 2.32 Maximum optical gain in Wz GaAl0.2Ga0.8N single QW with an active layer width varying from 30 to 80 Å with respect to the injected carrier concentration for 0.5% compressive strain [72].

meaning along the x-direction. Clearly, both biaxial and uniaxial tensile strains reduce the gain while the compressive strain increases it. The effect of a biaxial compressive strain is, however, not sufficiently large to make a notable difference in the threshold current. Uniaxial strain in the c-plane only, if it were possible, would 3000 Relaxed

Maximum gain (cm–1)

TE mode L z =60 Å 2000

Tensile 0.5 %

Compressive 0.5 %

Uniaxial –1%

1000 Uniaxial +1% 0 0

5

Injected carrier

10

density (1012

cm–2)

Figure 2.33 Gain in a 60 Å active layer GaN/GaAl0.2Ga0.8N structure for the cases of relaxed active layer (without strain), with 0.5% compressive biaxial strain, with 0.5% tensile biaxial strain, and with
1% uniaxial compressive strain in y-direction only and þ 1% uniaxial tensile strain along the x-direction only [60].

15

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244

6000 5000

Maximum gain (cm–1)

(a) 2.6×1012cm–2

TE mode Wz, Lz = 26 Å GaN/Al0.2Ga0.2N

(b) 5.2×1012cm–2 (c) 7.8×1012cm–2

4000 3000

c

2000 1000

b

0

a

-1000 3.5

3.55

3.6

3.65

3.7

Energy (eV) Figure 2.34 Spectral gain in a 26 Å strained Wz GaN/GaAl0.2Ga0.8N quantum well structure for injected carrier volume densities of 1  1019, 2  1019, and 3  1019 cm
3 (corresponding to areal densities of 2.6, 5.2, and 7.8  1012 cm
2)

for TE polarization, as the gain for the TM mode is rather weak. The strain caused by the lattice mismatch between the barrier and the active material has been taken into consideration [76]. Reprinted with permission from IEEE.

cause a notable increase in the gain due to reduced DOS at VBM through anisotropic splitting in the kx–ky plane. The literature is fairly rich in terms of gain calculations for both Wz and ZB quantum well structures. The gain spectra in a 26 Å strained Wz GaN/GaAl0.2Ga0.8N quantum well structure for injected carrier volume densities of 1  1019, 2  1019, and 2  1019, corresponding to 2.6, 5.2, and 7.8  1012 cm
2 are plotted in Figure 2.34 for the TE mode. The gain maximum in this particular example is higher than that calculated by Suzuki and Uenoyama [60] for comparable injection levels. The discrepancy may have originated from the different band structure parameters employed. In another report, the strain caused by the lattice mismatch between the barrier and the active material has been taken into consideration [76] . 2.6.5 Gain in ZB Q Wells Without Strain

GaN grown in the [1 1 1] direction promotes the Wz polytype. It is therefore natural to consider a ZB quantum well that is grown along the [0 0 1] direction with the resultant quantization along the same direction. In an attempt to calculate the gain, Suzuki and Uenoyama [60] assumed this orientation for ZB GaN. As indicated on numerous occasions, quantization removes the degeneracy at VBM of a ZB structure (G point) and separates the HH and LH bands. Band mixing is affected in such a way as to reduce the HH mass, even below that of the LH, causing the DOS at VBM to drop, which would by itself lead to a reduction in the transparency density or the current. However, the relatively small LH and SH

2.6 Numerical Gain Calculations for GaN

Maximum gain (cm-1)

3000

Zincblende L z = 60 Å

TM mode

2000

TE mode 1000

0

10 5 Injected areal carrier density (1012 cm-2)

15

Figure 2.35 Optical gain in an unstrained 60 Å active layer ZB GaAl0.2Ga0.8N single quantum wells, which is loosely referred to as quantum well. The solid and dashed lines correspond to TM and TE modes, respectively [60].

separation in GaN leads to increased coupling between the two bands with the adverse effect of an increased mass and a density of states causing the transparency current in ZB to be larger than that in a Wz structure. This enhanced coupling also manifests itself by the emergence of a noticeable TM mode. Figure 2.35 displays the maximum gain in a 60 Å active layer ZB GaN/GaAl0.2Ga0.8N structure where the dominance of the TM mode is clearly seen. Compared to Figure 2.30, which represents the Wz polytype, the gain is somewhat smaller and suggests the inevitable conclusion that these calculations do not support the rationale for developing a ZB polytype for laser applications. In fact, unlike a Wz polytype, there has been no report of stimulated emission in ZB GaN. 2.6.6 Gain in ZB Q Wells with Strain

It is a well-known phenomenon that strained quantum wells of the GaAs and InP systems exhibit much reduced DOS in the valence band, a property that has successfully been exploited in the form of strained QW lasers [58]. Following the same path, the effect of compressive and tensile biaxial strains on the maximum gain has been considered by many researchers [61, 71, 76–79]. Shown in Figure 2.36 are the maximum gain for TE and TM modes versus the injected areal sheet carrier density in a ZB GaN/Al0.2Ga0.8N single well structure having a 60 Å active layer, which is relaxed and under either compressive or tensile strain. The compressive strain strongly reduces the TM-mode gain while enhancing the TE-mode gain in relation to the relaxed case. On the contrary, a tensile biaxial strain causes the reverse to occur. In short, this is a consequence of compressive strain lifting of the HH and spin–orbit hole band (SH), the two uppermost states (referred to as X and Y states),

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Maximum gain (cm–1)

3000

Zincblende L z = 60 Å TE mode

Compressive (0.5%)

2000 Relaxed

1000 Tensile (0.2%)

(a)

0

5 10 15 Injected areal carrier density (1012 cm–2)

Maximum gain (cm–1)

3000 Zincblende Tensile (0.2%) L z = 60 Å TM mode

Relaxed

2000

1000 Compressive (0.5%)

0 (b)

5 10 15 Injected areal carrier density (1012 cm–2)

Figure 2.36 Optical gain in a strained 60 Å active layer ZB GaN/ GaAl0.2Ga0.8N single quantum wells with (a) and (b) corresponding to TE and TM modes, respectively. Strain free, 0.5% compressive strain and 0.2% tensile strain cases are shown [60].

well above the LH state (referred to as the Z state), which supports the TE mode. On the other hand, the tensile strain lifts the LH state (the Z state or the pz state) above the HH and SH states (X and Y states or px and py states), which proceeds and supports the TM mode at the detriment of the TE mode. 2.6.6.1

Pathways Through Excitons and Localized States

2.6.6.1.1 Excitons In terms of excitonic properties, wide-bandgap semiconductors such as GaN, ZnSe, and ZnO-based structures have large exciton binding energies, ranging from 28 meV in GaN to greater than 40 meV in ZnCdSe QWs and 60 meV in ZnO, all of which are larger than the thermal energy at room temperature. They may

2.6 Numerical Gain Calculations for GaN

not dissociate at the carrier concentration levels that cause the semiconductor to reach transparency. Noting that the exciton oscillator strength scales roughly as the square of the exciton binding energy, Ex, which imparts robustness on excitons in these material systems. The robustness of 2D excitons against screening and thermal dissociation could imply a notable contribution to the optical gain even at room temperature [67, 80, 81]. Some 30–50% improvement in the threshold current has been predicted. Whereas the compositional/structural roughness in InGaN QWs and piezofield Stark effects have made it difficult to delineate excitonic features in optical spectra to date (the spontaneous polarization contribution is not significant at the InGaN/GaN interfaces, Volume 1, Chapter 2, thus the QWs used in nitride-based lasers). In contrast, the relatively superior quality, in this narrow sense, of the heteroepitaxial of II–VI structures allowed this delineation. Specifically, both excitonic and biexcitonic features have been clearly identified in ZnSe-based QWs up into a high-density regime at cryogenic temperatures, where stimulated emission and laser action commences. Naturally, the question then arises as to whether or not excitons would be involved in the lasing process taking place in these semiconductors at room temperature. In a high-density electron–hole system in a 2D configuration such as that exists in ZnSe and GaN-based laser heterostructures, the direct e–h Coulomb interaction is accompanied by exchange and correlation effects that profoundly increase in importance with the particle density. In a 2D system, the exciton phase-space filling density was given by Ding et al. [82] as n¼

1 ; pa2B

ð2:199Þ

which for GaN with an exciton Bohr radius of about 28 Å, see Equation 2.194, leads to a concentration of 3  1012 cm
2. This is comparable to the transparency carrier concentrations that one predicts for lasers fabricated in ideal GaN. Because of this high dephasing density, one cannot automatically rule out excitonic processes in GaN-based lasers. Moreover, the exciton binding energy increases in quantum wells, making excitons all the more important in such structures. It should be mentioned that the lack of high-quality heterostructures prevents such lasers from being fabricated at this point of time. It is not too early to explore the exact pathway(s) leading to exciton participation in the lasing process and the extent of it. Calculations have already been performed to determine the gain enhancement due to Coulombic (excitonic in this case) processes that improve the matrix element for interband transitions. In other words, the recombination efficiency is enhanced. Chow et al. [77] estimated the Coulombic enhancement to be about 30% in ZB GaNbased laser structures. Uenoyama [83] presented a three-level picture involving excitons where the point has been made that the excitonic enhancement is present but the supply of electrons and holes for the excitonic transitions must involve a coupling to the two (conduction and valence) bands. Because experience with GaN-based lasers is very limited, the effort expended in the ZnSe realm, due to its similarities, will be utilized to provide a basis for the discussion of any exciton participation.

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GaN

GaN

Electron

(In,Ga)N

hν = Eg–EB Eg

Hole Figure 2.37 Schematic representation of a three-level hypothetical laser involving an excitonic state and conduction and valence bands. In practical terms, excitons and electron–hole plasma both contribute to lasing in semiconductors with large enough exciton binding energy, such as ZnCdSe, GaN, and particularly ZnO.

Although the detailed expressions that account for the exciton participation can be found in Refs [63, 77, 83], let us first phenomenologically consider the ideal case in which there exist conduction and valence bands and a well-defined excitonic level at some 20 meV below the conduction band (in the test semiconductor GaN). The excitons are states bound through Coulombic interaction between holes and electrons. For the sake of argument, let us further assume that the band edge of the semiconductor is abrupt and that the exciton linewidth broadening is negligible. This simplified picture represents an ideal three-band or three-level lasing model in which the carriers in the conduction and valence bands, placed by the injection of carriers and/or absorption of optical radiation (Figure 2.37), provide fuel for the excitons. If the coupling between the two bands and the exciton state is strong, the supply of electrons and holes for excitonic transitions will be sufficient. In this case, the quasiFermi-level separation needs to be larger than only the bandgap minus the exciton binding energy. This is a condition rendering the semiconductor transparent to the radiation emanating from the excitonic recombination. As mentioned on several occasions, the excitonic recombination is more efficient than that of electrons and holes in the conduction and valence bands, respectively, due to the Coulombic interaction and the resulting spatial proximity of the carriers with opposite charges. This satisfies the condition that the recombination rate at the excitonic level be larger than the band-to-band recombination, to achieve the required population inversion. In the absence of broadening, this picture illustrates an ideal three level laser. It is obvious that in this ideal picture, a semiconductor laser based on

2.6 Numerical Gain Calculations for GaN

excitonic recombination would offer a very low threshold current. It is also clear that the lasing that would occur at the excitonic level would be redshifted with respect to the band edge by an amount equal to the exciton binding energy. In reality, however, homogeneous and inhomogeneous broadening coupled with thermal broadening cause dispersion of the excitonic transitions. This is particularly so in an alloy such as InGaN. It simply means that the ideal picture described above, while shedding muchneeded light, would not alone be sufficient to describe the mechanisms involved. In addition, exciton lasing without a clearly defined third level is unrealistic and is a point that was advanced by Uenoyama [83] and Cingolani [84]. Realistically, weakly localized states are much more likely to be the mechanism for lasing in the GaN system, as will be discussed below. Linewidth broadening in the form G(T) ¼ G0 þ G1 (T) þ

GLO exp(
hwLO =kT)
1

ð2:200Þ

has been used to describe the excitonic dispersion in group II–VI semiconductorbased lasers by Ding et al. [82]. Additional details in the context of localization can be found in Volume 2, Chapter 5. An expression similar to this was employed to describe the energy band dispersion in GaN by Petalas et al. [85]. For bandgap broadening, the first term depicts the temperature-independent broadening, such as that caused by compositional and/or strain-induced inhomogeneities, impurity and surface scattering of carriers, electron–electron interaction, and Auger processes. The second term is the homogeneous-broadening term resulting from carrier scattering due to acoustic phonons. The third term depicts scattering due to LO phonons, a phenomenon that is dominant in polar semiconductors such as GaN at high temperatures. However, due to the large LO-phonon energy of 91 meV in GaN, the impact of the third term is diminished somewhat. As for exciton linewidth broadening, the above description applies, with the exception that the term exciton must be substituted for carriers and electrons. Depending on the quality, the temperature-independent broadening term in GaN for the bandgap parameter could be as high as 50 meV, with the figure expected to be much larger for InGaN and other alloys of nitrides. Ding et al. [82] through a fitting to the experimentally observed line broadening due to excitons in ZnCdSe-based quantum wells, reported the GLO term to be 36 meV. This value was also used to predict a larger broadening. Reduced LO phonon scattering or exciton dissociation rates by LO phonons lower the broadening. Broadening caused by the aforementioned processes coupled with occupation factors and DOS distributions in the conduction and valence bands may collectively cause the gain spectrum to have its maximum at a value of energy different from the excitonic state under discussion. In such a case, a blueshift may occur. As mentioned above, in a 2D high-density electron–hole system, the direct e–h Coulomb interaction together with exchange and correlation effects increase in importance with the particle density. In a GaAs bulk crystal or quantum well, these two extreme limits are identified in the form of a low-density bound state (exciton) regime and high-density electron–hole plasma [80]. For the latter, the exchange and correlation effects are mainly through the bandgap renormalization effect, which

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redshifts the emission wavelength of the laser by up to several tens of millielectronvolts. In terms of the line shape or magnitude of the actual gain spectrum, the manybody effects can be accounted for by small corrections to the calculations that are in the realm of the one-electron framework with some phenomenological damping rates. More in-depth theoretical approaches with converging to the semiconductor Bloch equations have been developed and employed to describe the gain spectra of some traditional III–V-based lasers with compressive success [86]. The effectiveness of these methods considering the electron–hole plasma limit with the added e–h Coulomb interactions in the widegap semiconductor lasers has been investigated [87]. In another approach, one can also adopt a viewpoint that takes the opposite limit, which can also be used as the starting point, that is, non- or weakly interacting excitons subject to many-body interactions, which is more in line with experiments when considering, for example, the ZnSe QW or “disorder-free” future GaN QW laser. In experiments, the temperature is gradually increased from liquid helium to room temperature where the role of bound e–h pair states at low temperatures is fairly established. Approaching the problem from this angle presents significant challenges to the semiconductor Bloch equation approach due to the difficulty of including the screening by the bound states into the formalism. Yet, evidence in the ZnCdSe QWs suggests that the Coulomb correlations remain very potent when compared with, for example, a GaAs-based laser where the exciton binding energy is about 5 meV that is very small compared to those in wide-bandgap semiconductors under discussion. Localized excitons have been reported to have a defining role for the optical gain in the ZnCdSe QW [88] with independent supporting data [89]. Comparing gain/ absorption spectra for a ZnCdSe [90] and GaAs SQW diode laser (not shown) in the vicinity of the n ¼ 1 QW HH exciton transition [91] clearly shows the impact of excitonic effects in ZnCdSe. The gain spectra obtained for both classes of semiconductors by the same experimental technique of correlating edge stimulated emission with top spontaneous emission, indicate that as the injection current is increased, the HH and LH exciton features are rapidly bleached in the GaAs QW whereas the HH exciton resonance is partially bleached and the LH resonance remained nearly intact, even at the laser threshold in ZnCdSe, for details refer to Ref. [90]. Additional support for the impact of the e–h Coulomb effect at room temperature comes from investigation of ZnCdSe QW lasers in a high magnetic field normal to the QW layer plane [92]. In this experiment, the quantum well parameters were chosen to approach the quasi-two-dimensional limit to enhance the exciton binding energy, which, for the Cd concentration used, was estimated at least 40 meV based on earlier studies on the ZnCdSe/ZnSSe QW [93]. A linear Landau shift with magnetic field on the order of DEL 0.5 meV T
1 would be expected in a free-carrier system, either due to the shift of the lowest conduction band Landau level or the jumps of the Fermi level between such occupied levels, which is about one order of magnitude larger than experiments in the II–VI laser case. This discrepancy suggests that free electron–hole behavior, including longitudinal mode hops as the gain spectrum shifts, is most likely not the dominant mechanism. This and the good agreement in

2.6 Numerical Gain Calculations for GaN

the diamagnetic shifts measured for the n ¼ 1 HH exciton transition in ZnCdSe QWs formed the basis for suggesting participation of excitons in the laser gain. Because of relatively large exciton binding energies, the role of the Coulomb interactions is relevant in GaN-based lasers. Even though the present state of InGaN with its severe inhomogeneities preclude any meaningful study of this effect at this time, but when such effects are reduced, the Coulombic interactions would have to be investigated. In addition, as the push toward shorter wavelengths continue, GaN and AlGaN would have to be considered for lasing medium. These semiconductors would be expected to be increasingly stable against dissociation and screening. If so, one should be reminded of the factor of 2–3 enhancement of the peak gain for roomtemperature ZnCdSe QW lasers. Note that this comes at cost of an accelerated spontaneous (radiative) decay rate. The oscillator strength in the gain spectrum near the n ¼ 1 HH exciton resonance specifically enhances the peak gain coefficient, whereas the spontaneous emission rate is proportional to the integral of the corresponding spontaneous emission spectrum [81]. Inspired by investigations in the II–VI systems, calculations predicting enhancement due to excitonic effects in GaN-based lasers have also been performed [83]. For a complete treatment, electron–hole plasma, excitonic, and polarization effects must be included in terms of what one would term as intrinsic processes. The extrinsic processes such as band-tail states, localization, and so on, are difficult to model without a clear picture. The treatment presented so far is intended to give the reader a reasonable picture of what goes on in the operation of GaN-based laser. More complete treatments encompassing the above-mentioned intrinsic processes have been undertaken. In addition, the effect of inhomogeneous broadening may be approximated by a statistical average of the homogeneous gain spectra. The reader interested in a deeper treatment is recommended to consult those reports that are summarized in few publications including Ref. [94]. As discussed in Volume 2, Chapter 5, the polarization-induced field causes electron and hole wave functions to be skewed toward opposing ends of the quantum wells. A consequence of this is that the recombination lifetime is increased and its efficiency is reduced. Moreover, the emission energy is affected by the polarization field, which is referred to as the Stark effect. What is more is that these parameters are strongly dependent on injection of electron–hole pairs and plasma screening. In lasers, particularly, as they are driven from spontaneous emission, at relatively lower injection regime, to stimulated emission, at relatively high injection regime, the whole gambit of injections is sampled. Consequently, the gain, particularly, its build up, will be affected by polarization. However, once the high-level injection levels are achieved, the effect of polarization is not expected to be substantial, as discussed in Volume 1, Chapter 2. 2.6.6.1.2 Localized States What can be gleaned from the above discussion is that in InGaN QWs, the poor quality of the layers would rule out the idealistic three-level exciton-based laser. Even the less-than-idealistic case, where there is exciton participation with electron–hole pair to the extent allowed by the applicable physics, is also ruled out due to the poor quality of InGaN QWs. As mentioned, further erosion of Coulombic processes is caused by the fact that the injected carrier concentration

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GaN

GaN

Electron

(In,Ga)N

Eg

GaN Electron

GaN (In,Ga)N

hν = E g– ∆ E

hν =E g – ∆ E

Eg

Hole

Hole Localization due to compositional inhomogeneities (a) Figure 2.38 Simplified schematic representation of band structure involving localized states caused by (a) compositional and/or strain inhomogeneities and (b) assumed localized states caused, for instance by defects. In general, though, localized states can be a deep donorand/or a deep acceptor-like states Subpart (a) exemplifies the case where the localized state is caused by strain and/or compositional inhomogeneities, reducing the local effective

Localization near the conduction band only (b) bandgap. Local tensile strain reduces the effective bandgap, and if and when accompanied by a juxtaposition compressive strain, those parts under compressive strain would have an increased bandgap. Compositional and strain inhomogeneities as well as localized defect states may all occur in the same structure. In practice, these states really end up broadening the gain spectrum with its adverse affects as they appear as a continuum below the intrinsic band edges.

for the transparency is more than one order of magnitude larger than concentrations at which the screening processes would rule out Coulombic interaction between carriers. It should be mentioned that the lifetimes used in calculating the transparency current might be overestimated. In short, it is very likely that some other pathway is involved in GaN-based lasers. It is also very probable, and may, in fact, be supported by experiments, that localized states may be involved. In the case of a localization, states below the bandgap, as shown in Figure 2.38a and b, for compositional and/or strain inhomogeneities and Fermi-level fluctuations, the latter may be coupled to the conduction band where the electrons are supplied to the localized states, would be formed. Compositional and strain inhomogeneities as well as localized defect states all may occur in the same structure. Again, for lasing to occur, the quasi-Fermi-level separation must exceed the energy separation between the localized states and the valence band maximum, and the recombination rate must be larger than the band-to-band recombination rate. In alloys, the line between the issues of localized states and compositional fluctuations and other inhomogeneities may be fuzzy. In this case, the effective band edge of the conduction band and possibly the valence band would extend into the gap. The case of a high-density localized state is depicted in Figure 2.39. If the inhomogeneities are wide spread, band-tail states would result and cause a soft band edge in the absorption spectrum and a redshift (Stokes shift) in lasers (Figure 2.39). If the recombination rate between the localized states is not much larger than that for band-to-band transitions and/or

2.6 Numerical Gain Calculations for GaN

GaN

GaN Electron

(In,Ga)N

GaN

(In,Ga)N

Electron

GaN

hν = E g– ∆ E

hν = E g – ∆ E Hole

Hole

(a) Localization due to compositional inhomogeneities

(b) Localization near the conduction band only

Figure 2.39 (a and b) Schematic rendition of a case where the localized state concentration is sufficiently high for the substantial overlapping of those states that cause band-tail states. Assuming similar properties in the barrier materials, the carrier transport would be due to defect assisted tunneling.

the coupling between the conduction and valence band states is not sufficiently large to not limit the carrier recombination between the localized states, lasing at the localized level(s) will be curtailed. The lasing energy will be shifted toward that of the bandgap if everything else is equal. In the case of substantial overlap between these localized states, band-tail states would appear (Figure 2.40). As discussed in Section 1.10, also see Volume 2, Chapter 5, there is a general agreement that the compositional fluctuations on the spatial scale of the electron and hole Bohr radii in the blue and green QW LEDs enhance the overall light emission efficiency. This is so because of the strong localization that results from such localization that blocks the pathways to nonradiative recombination. However, the very presence of the large average strain and local fluctuations in the strain, together with the large piezoelectric coefficients in GaN/AlGaN heterostructures [95], make it difficult to determine the specific contributions of these effects to the nature of near band-edge states in InGaN wells, specific to lasing process. The piezoelectric-induced fields on the order of MV cm
1 not only influence the energies of the conduction and valence band edges considerably ( 100 meV) but also reduce electron–hole overlap and interaction at low injection levels. The former is of relevance to the one-electron matrix element while the latter influences many-body electronic phenomena, specifically the excitonic effects discussed above. Kollmer et al. [96] have studied the optical transitions in undoped InGaN QWs by time-resolved photoluminescence (PL) to argue that the observed spectral shifts are consistent with the built-in piezoelectric fields, refer to Volume 2, Chapter 5 for details. For large injection levels, these fields are reasonably well screened [97], but detailed experimental studies to address this question in the actual devices have so far been lacking. Details of polarization and associated screening issues can be found in Volume 1, Chapter 2.

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E

g c (E)

Ec Electron

hν = Eg–∆ E

gv (E)

Hole Ev

Figure 2.40 Schematic representation of band-tail states in the context of density of states due to large concentrations of localized states causing continuum-like extension into the intrinsic bandgap. The tail states lead to broadening of the optical response with respect to injection level and cause an increase in the threshold current density.

A number of experimental reports employing microscope and microprobe techniques that have provided real images of the In-induced clustering in InGaN epitaxial thin films and quantum wells have emerged. At one end of the spatial spectrum, Kisielowski et al. [98] have employed atomic resolution transmission electron microscopy to show evidence for atomic scale clustering in conjunction with strain in studies in InGaN/GaN QWs. In fact, an image obtained as part of this investigation is shown in this volume (Figure 1.51). In the follow-up studies, however, Monemar et al. [99] appear to dispute this. Although not as well resolved, both cathodoluminescence (CL) [100] and near-field scanning optical microscopy (NSOM) [101–103] have been used to acquire luminescence-based images on a spatial scale-down to 100 nm. Such a strikingly wide distribution of clusters on In-rich regions, possibly affected by possible QW thickness fluctuations, is again indicative of the fact that the system under question is not a random alloy one where spatial compositional (and crystal potential energy) fluctuations are confined to the atomic scale. Chichibu et al. [100] estimated the mean In cluster size to be less then 60 nm, although the CL images revealed In-rich and In-deficient regions having diameters of up to half a micrometer. As for the NSOM, it is an attractive alternative to CL and allows subwavelength spatial resolution in PL measurements with measurable carrier injection into the QW, together with simultaneous topographic imaging. An example of NSOM imaging obtained in the collection mode by Vertikov et al. [104] is shown in Figure 2.41. The issues of carrier diffusion, which are prevalent in CL, can

2.6 Numerical Gain Calculations for GaN

Figure 2.41 NSOM PL images obtained in collection mode from an undoped InGaN/GaN MQW for three different wavelengths. Traces (a), (b), and (c) correspond to PL NSOM images at l ¼ 450 nm (2.75 eV), l ¼ 460 nm (2.70 eV),

and at l ¼ 470 nm (2.64 eV), respectively. Markers “A” and “B” highlight regions of complimentary optical contrast at different emission wavelengths. Courtesy of A.V. Nurmikko [104].

be minimized in NSOM when investigating the local spectral variations in the PL emission on a sub-100 nm scale under high e–h pair injection levels encountered in lasers. The figure depicts typical near-field PL images taken from an undoped 30 Å thick InxGa1
xN QWs separated by 90 Å thick GaN barriers for three different wavelengths of emission. Figure 2.41a shows the NSOM image obtained on the higher energy side of the far-field PL spectrum at l ¼ 450 nm (corresponding to an energy of
hw ¼ 2:75 eV). Figure 2.41b was recorded at the center of the spectrum at l ¼ 460 nm (
hw ¼ 2:70 eV), while Figure 2.41c depicts the NSOM image taken at l ¼ 470 nm (
hw ¼ 2:64 eV) on the lower energy side of the PL spectrum [105]. Even a cursory look at images reveals darker and brighter regions extending several hundred nanometers in size. These observations suggest that the light emission in roomtemperature InGaN MQW lasers occurs from the extended states, once localized states are filled. The excitation and position dependence of the NSOM spectra in Figure 2.41a and b is noteworthy in this state-filling process. A significant part of the

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PL spectra at each particular point of the sample is broadened homogeneously on the scale of the instrument resolution. Nurmikko [81] argued that all localized states and a significant portion of extended states that are possibly mixed with higher energy localized states contribute to the local radiative recombination under high injection levels. Moreover, the regions with larger local band energy gap, meaning In-deficient regions, can easily become interconnected through the carrier diffusion. Relevant to localization is the carrier diffusion in the presence of causes of localization and its dependence on injection level. It is known now that at low injection levels, recombination is through localized states. However, at high injection levels electron–hole plasma recombination is dominant as in the case of wellestablished semiconductor laser materials such as the GaAs-based variety. In this vein, a somewhat different application of the NSOM imaging technique has been applied to determine the electron–hole (ambipolar) diffusion in InGaN MQWs under optical injection [104]. The method involves setting up of an excitation interference grating with two blue laser beams, on a spatial scale of about 200 nm, and the direct imaging of the variations in the PL intensity with the NSOM fiber tip as the high spatial resolution light collector. In these types of experiments, what is studied is the grating contrast in the PL profiles as they diminish due to carrier diffusion. Numerical fitting of such PL profiles with the solutions of the diffusion equation of 2
 Dn(x; t) ¼ Dp(x; t) / e
t=t þ g 0 e
t=t(1 þ KLD ) cos(2px=L)

ð2:201Þ

give diffusion length LD, which in the case of InGaN/GaN quantum wells is found to be 130 nm. We should point out that this apparent diffusion length might be dominated by topological properties of the layers and may not represent the intrinsic diffusion length. Dn(x,t) and Dp(x,t) are the excess electron and hole concentrations. Continuing on t is the (carrier density independent) recombination lifetime, g 0 is the contrast parameter of the excitation grating, and L is the grating period. Vertikov et al. [104] studied a number of InGaN MQW samples obtained from different laboratories to find that the diffusion lengths at room temperature could vary very widely under low e–h injection, reflecting the topological microstructure of individual MQWs. On the other hand, in the high injection regime typical of a diode lasers (>1019 cm
3), the values of diffusivity converge to those roughly expected for a “free” electron–hole gas (plasma) where the localization effects would not be as apparent. Localized-state arguments have implications on the current conduction as well. If the localized-states density in the barrier layers is also high with band-tail states resulting, it follows that the carrier-transport paths may not be limited to conduction and valence bands. Additional paths through these band-tail states/gap states could take place, which lead to photon assisted tunneling. Preliminary experimental current–voltage characteristics suggest that the current conduction in p–n-junctions is through photon-assisted tunneling, as discussed in Section 1.10. Because research on GaN-based lasers may still be construed as being in an embryonic state, in terms of understanding the processes that take place, and seriously complicated by inhomogeneities in the InGaN active layers employed, the ZnSe lasers,

2.6 Numerical Gain Calculations for GaN

having been developed much further than their GaN counterparts, will be used as examples to debate the likely mechanisms involved in lasing and to discount others. Ding et al. [82] by means of optical pumping experiments in ZnSe-based heterostructures, argued in favor of excitons for the gain mechanism. They calculated the gain by employing a phenomenological exciton-driven gain mechanism, which is in agreement with experiments conducted at 77 K. On the other hand, Diessel et al. [106] made a case for the exciton recombination in processes that lead to lasing but not partaking in the actual lasing process due to exciton bleaching. The main controversy appears to have its origins in the observation of excitonic peaks in photoluminescenceexcitation experiments with pumping intensities well in excess of the threshold intensity. However, Diessel et al. [106] argued that this observation is no more than an evidence for carrier generation via excitonic absorption pathways that are accessible during the leading edge of the excitation pulse. According to Diessel et al. [106] pulse probe experiments, where a pulse beam is used to prime the semiconductor while a probe beam gauges the optical activity at an excitation level engendered by the pump beam. It interrogates the excitonic activity more independently of the carrier density dynamics during the leading edge of the pump beam. Such experiments lead to exciton bleaching, which justifies the argument against the gain mechanism being solely governed by excitons and in favor of electron–hole plasma. This conclusion is consistent with independent report by Cingolani et al. [107]. Cingolani et al. [108] performed pump and probe experiments in high magnetic fields, which support the exciton bleaching thesis. It has been observed that shallow quantum wells, those with low Cd mole fractions, have reduced the enhancement of the exciton binding energy. They exhibit the typical free-carrier fueled lasing that occurs after excitons have been bleached and the shift of the lasing line with magnetic field follows the Landau shift; this is endemic of free carriers. Wells with high Cd mole fractions, whose exciton binding energies are larger than the LO-phonon energy, lase at power densities where the exciton resonances are not bleached out. The lasing-line shift follows the diamagnetic shift, which is very small. The set of samples contained wells of widths ranging from 3 to 20 nm and Cd mole fractions in the range of 10–30%. The results can satisfactorily be described by a self-consistent many-body model based on the mass action law that includes exciton and free-carrier energy renormalization. The excitons and free carriers have been described by a phase diagram analogous to the dissociation of a diatomic gas. The relative density of the two phases depends on the stability of the excitons. In short, both exciton and free-carrier fueled lasing can be present depending on the conditions, while the primary factor is the strength of the exciton binding energy. However, what still clouds the picture is the existence of localized states that can present signatures similar to excitons. As the Cd mole fraction is increased, so are the likelihood of inhomogeneities and the impact of localized states. 2.6.7 Measurement of Gain in Nitride Lasers

Calculation of the gain is rather convoluted, in that it requires an accurate knowledge of key parameters as well as of the critical mechanisms involved. For nitrides, the

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respective mechanisms and many of the key parameters have yet not been determined experimentally. They must therefore be calculated. It is thus imperative that gain measurements be made to provide the basis for obtaining an insight into the operation of nitride-based injection lasers even before they become widely available and to provide the needed calibration for the calculations. At the onset, it should be stressed that high-quality laser material is required to measure the pertinent parameters with the needed degree of confidence. Ironically, as is always the case, good material is not available in the early phases of development when it is needed the most. Because carrier injection is a major impediment, particularly so early during the early stage of development, due to the lack of a good p–n-junction and the many leakage pathways, optical-excitation/pumping is generally employed to glean an insight into the processes in new semiconductors, such as nitrides, under high excitation densities. 2.6.7.1 Gain Measurement via Optical Pumping The optical pumping experiment provides an excellent environment to investigate the genesis of stimulated emission in semiconductors, particularly if pump and probe experiments are carried out. Lack of metallization, ease of accessibility of the active layer with optical pump and probe beams, the absence of absorbing contacts, and the ease with which the sample lattice temperature can be changed are among the reasons for the attractiveness of the method. Moreover, optical pumping experiments do not require the formation of good p–n-junctions and complicated fabrication procedures and as such are precursors to the current injection laser development. One of the methods used for studying the optical gain is the stripe excitation technique, which means that the light excitation source is focused on a stripe region. If the sample is excited sufficiently, spontaneously emitted light traveling along the excited stripe gets amplified by the stimulated emission. The intensity of light in such a case is then given according to Frankowsky et al. [109] as

dI ¼ Gg(hn)I þ b0 Gr sp (hn); dz

ð2:202Þ

where rsp(hn) and b0 are the spontaneous emission rate and spontaneous emission factor, respectively. The other terms have their usual meanings. When solved for a stripe length of l with the boundary condition I(0) ¼ 0, we obtain I(l) ¼

b0 r sp (hn) [exp(Ggl)
1]: g(hn)

ð2:203Þ

The term b0 r sp (hn) represents the spontaneous emission intensity I0 and I(l) is the intensity of the amplified spontaneous emission. A word of caution, namely, Equation 2.203 does not hold for long stripe lengths due to saturation effects. To account for this, a saturation term can be added, in which case Equation 2.203 is modified as [110]   gI(l) 0 g(hn)l ¼ a I þ ln þ1 ; ð2:204Þ I0 where a0 is the saturation parameter.

2.6 Numerical Gain Calculations for GaN

Experimentally, the spectrum of the amplified spontaneous emission is measured for various values of l and used in conjunction with Equation 2.203 or Equation 2.204 (depending on the lengths of the excited stripe) to determine the gain. This can be repeated for various wavelengths and pump intensities to obtain the gain spectra as well as the intensity dependence of the gain. Of course, the nature of the transitions, which causes the optical gain, must be known before an earnest effort can be made to calculate it. To this extent, experiments and theory go hand in hand to examine the groundwork that has to be laid. While we await improved GaN-based heterostructures to become available so that appropriate experiments leading to the genesis of lasing can be performed, a good deal of information can be gleaned from ZnSe-based lasers. Based on what little data are available for GaN-based lasers, it appears that there is a good deal of similarity between ZnSe- and GaN-based lasers. Despite the raging controversy as to the genesis of the lasing mechanism in ZnSe-based lasers, there is a lot to be learned from the results of various investigations. In this vein, the results of an optical pumping experiment are displayed in Figure 2.42. The absorption and emission spectra along with the pump-beam spectrum concerning a ZnCdSe active layer MQW laser structure show a redshift of about 60 meV [82]. Though the data are sketchy at this point of time, GaN-based lasers also redshift with respect to the absorption spectrum, as discussed in detail in Section 2.11. Optically pumped stimulated emission has been achieved in GaN almost from the inception of epitaxial GaN in the early 1970s [111]. Over the years since then, the lattice temperature at which optically pumped stimulated emission is observed rose. However, with the advent of injection lasers the allure of optically pumped stimulated emission naturally languished. The threshold optical power required for stimulated

T=77 K Absorption

Intensity (a.u.)

Lasing emission

Pump laser

2.570

2.584

2.598

2.612

2.626

Energy (eV) Figure 2.42 Optical absorption spectrum near the n ¼ 1 C to HH exciton resonance for a six-period and 30 Å ZnCdSe MQW laser structure at 77 K. The pump laser spectrum and the laser emission are also displayed with a redshift of 60 meV [82].

2.640

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emission has decreased to the point where 40 kW cm
2 is presently sufficient to achieve stimulated emission at room temperature [112]. Frankowsky et al. [109] employed optical pumping to investigate the gain in GaN and InGaN heterostructures. The GaInN/GaN double heterostructures and quantum wells employed consisted of a 500 nm GaN buffer layer, a 25 Å thick GaInN active layer, and a 150 nm top confinement layer. The GaN/AlGaN double heterostructures had a 500 nm Al0.1Ga0.9N buffer layer, a GaN-active layer, and a 150 nm AlGaN top layer. The modal gain shows strong polarization of the TE mode (electric field in the plane of the sample), being the dominant mode due to the Wz nature of the crystal. This results from symmetry-induced splitting of the valence band and the almost exclusive coupling of the heavy hole to the TE polarized light. Optical gain as a function of pump power exhibited a single maximum close to the bandgap of the ternary InGaN, which is accompanied by an apparent redshift of the gain maximum with increasing pump power. The shape of the gain spectra is characterized by a sharp reduction at the higher energy side due to population statistics. At the lower energy side, the gain shows a sublinear plateau that deserves further attention. This is indicative of the fact that the transparency carrier concentration is very high and unlike that displayed in Figure 2.42, where the plateau observed in InGaN at the low-energy side is not apparent. Using available and/or estimated effective electron (0.22) and hole (1.64) masses (estimates vary between 0.7 and 2 for the hole masses with the latter being more accepted) and an interband momentum matrix element (9.3 eV), the injected carrier concentrations were estimated, knowing the gain, while letting the confinement factor and the broadening factor be adjustable parameters. The resultant carrier concentrations were deduced to lie between 1.3 and 3  1019 cm
3 for excitation levels of 300, 1000, and 1600 kW cm
2. The optical gain spectra for a GaN/AlGaN double-heterostructure sample, the active layer of which is relatively better known, with Lz ¼ 100 nm was investigated at three different pump-power levels. In contrast to the GaInN/GaN structure, two distinct gain maxima at about 3.33 and 3.24 eV were observed. Within the range of pump-power levels accessible, only the amplitude of the lower energy peak was found to depend strongly on the pump power. A comparison of the gain and the absorption spectra indicates that the higher energy gain peak appears within the low-energy tail of the absorption spectrum. A suggestion has been introduced that the lower energy peak is due to exciton-LO-phonon scattering, which is about 90 meVand is close to the LO-phonon energy. The weak structure observed, about 90 meV above the localizedexciton peak, has been interpreted as an anti-Stokes exciton-LO-phonon gain peak. It supports the case of excitonic gain due to localization at least in GaN. The gain spectra have also been measured by the optical pumping method in the representative Nichia InGaN/GaN DH laser structures. The reason for the absence of gain spectral broadening with increasing optical pump intensity is not yet clear. One of the structures exhibits a narrower gain spectrum (sample 1) than the other (sample 2). Reportedly, sample 1 has more InGaN compositional fluctuations. To ascertain whether localization prevails, Cingolani et al. [112] investigated the stimulated emission peak in a separate GaN/AlGaN confinement MQW heterostructure as a function of magnetic field and temperature. In parallel, photolumi-

2.6 Numerical Gain Calculations for GaN

nescence and photoluminescence-excitation measurements were conducted with appropriate line-shape fitting to determine and identify the peaks in the spectra. The peak positions will be contrasted later to the lasing line in an effort to determine if there is a redshift and if so to what extent. In the experiments of Cingolani et al. [112], the broad spontaneous emission generated by pulsed photopumping evolves from the low-energy tail of the CW-PL spectrum (not shown, see the first edition for the figure), which suggests a dominant localized-exciton recombination. At threshold, a sharp spike appears at 3.34 eV (300 K) and grows superlinearly with an increasing photogeneration rate. On the average, the lasing line is redshifted by about 110 meV below the n ¼ 1 gap of the QW, which is too large to be accounted for by bandgap renormalization at threshold, and is suggestive of population inversion occurring at the low-energy tail of inhomogeneously broadened density of states, that is, lasing from localized state. To test this hypothesis further, the sample was subjected to a high magnetic field. The recombination mechanism could be identified by exploiting the difference in the diamagnetic shift of the free- carrier states (linear in a field with about 0.3 meV T
1 blueshift) and the excitonic states (very small and quadratic in field, with about 0.1 meV T
2 blueshift). The wavelength shift in the emission line was negligible in comparison to the calculated exciton diamagnetic shift and the free-carrier shift. The data support the hypothesis that lasing occurs through inhomogeneously broadened localized states in the sample under investigation. Their origin can be attributed to strain inhomogeneities and/or defect states. 2.6.7.2 Gain Measurement via Electrical Injection (Pump) and Optical Probe Method Optical pump-probe experiments used to deduce the gain spectra can be replicated in injection devices as well. In this case, the injected carriers render the semiconductor transparent, or push the active region of the semiconductor to even beyond transparency, while a wavelength-selective probe pulse gauges the gain/loss activity. If the semiconductor provides any gain at all, the probe pulse would be amplified with the degree being dependent on the magnitude of the gain. The complications are that one must be content with getting the probe pulse into the device without any unaccountable absorption external to the gain medium. The advantage is that an injection device that is much closer to an actual laser is used and renders the results much more applicable to injection lasers. Kuball et al. [113] applied this method to a light-emitting device with an InGaN active layer. As is commonly done, the absorption spectrum was obtained by using the device as a detector where the dependence of the photocurrent on the incident light with varying wavelength would represent the functional form of the absorption spectrum. For the particular InN mole fraction, the absorption peak in a single pass occurred at 3.3 eV and indicated an absorption coefficient of (5  0.5)  104 cm
1 at this wavelength. Though the structure contained five 25 Å quantum wells, no confinement was evident and the structure was characterized by a significant low-energy tail. With the help of absorption as calibration for an absolute measurement, the gain/loss spectrum of the device under forward injection at 233 K was deduced, along with the superimposed EL spectrum. The dependence of the maximum differential gain at the peak of the differential absorption on current density was also recorded. At low injection levels there is what

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appears to be a residual plateau as the device reaches transparency. At higher injection levels, the expected linear behavior is observed before saturation sets in, which at this point is most likely due to thermal effects as opposed to intrinsic saturation of the joint density of states. At a current density of J ¼ 3.5 kA cm
2, the single-pass total differential gain is about 6  10
3, leading to a gain coefficient of about 4800 cm
1 (the total active layer thickness of 125 Å multiplied by the gain per unit length yields the total single-pass gain). A point of concern is the large linewidths and the spectral position of the gain relative to the joint density-of-states (absorption) and the EL spectra. The gain bandwidth is estimated at 100 meV, as compared to about 30 meV in ZnCdSe. With increasing injection, the gain emanates in the low-energy tail of the rather graded absorption edge. This is similar to observations with injection lasers of Nakamura et al. [114], which are discussed in Section 2.12.

2.7 Threshold Current

The threshold gain can be related to the threshold carrier density through Equation 2.86, and the threshold current can be related to the threshold density through Equation 2.87 if the carrier lifetime is known. The gain (negative of the absorption coefficient) can be related to the spontaneous emission rate as in Equation 2.127. Once the spontaneous emission rate is calculated versus energy, the radiative current supporting the spontaneous emission rate of, for example, Equations 2.127 and 2.190 can be expressed by Equation 2.191, repeated here for convenience: 1 ð

J rad ¼ q

Rsp (E)dE

(A cm
3 ):

ð2:205Þ

E cv

In the case of a 2D system, this expression should be replaced by 1 ð

J rad ¼ qLz

Rsp (E)dE

(A cm
2 );

ð2:206Þ

i;j

E cv

where Lz is the total quantum well thickness. One approach for finding the gain and current in an injection laser is to start with a certain concentration of injected carriers and calculate the quasi-Fermi levels. Knowledge of the quasi-Fermi levels paves the way for determining the occupation probabilities. One can then calculate the gain as a function of energy using Equation 2.175 for bulk and Equation 2.185 for quantum wells, which relies on momentum selection rule but includes band mixing. The prerequisite, of course, is that one needs to have an accurate knowledge of the lasing mechanism (i.e., whether it is based on the singleparticle model in the framework of electron–hole plasma or not, i.e., excitonic, while the derivations so far in this chapter are based on the former), the band structure, and the relevant matrix elements associated with the active layer. From the gain expression,

2.8 Analysis of Injection Lasers with Simplifying Assumptions

the spontaneous emission rate and thus the radiative current can be calculated for that particular gain. Thecurrent should be consistent with the injectedcarrier concentration assumed at the beginning of this procedure. The radiative current density must be divided by the internal quantum efficiency to get the current density.

2.8 Analysis of Injection Lasers with Simplifying Assumptions

The calculations mentioned above, which were performed for GaAs lasers, together with experimental data culminated in the conclusion that the gain is linearly proportional to the injected carrier concentration for a range of carrier densities and thus current, which is called the linear region. This holds reasonably well for bulk lasers where the density of states, and thus the gain, does not saturate. In quantum well lasers, the absorption coefficient is a staircase function of energy. Nevertheless, the linear approximation has been shown to explain the observed results away from the extremes even in quantum wells. It is therefore very useful to the understanding and the diagnosis of semiconductor lasers. The gain of a semiconductor laser in the linear region can be expressed as follows [30, 57]: g ¼ A0 (n
ntransp ) ¼ b( J
J transp );

ð2:207Þ

where n is the injected carrier concentration and ntransp the injected carrier concentration to render the semiconductor transparent, J is the current density, and Jtransp is the current density that causes the semiconductor to be transparent. A0 and b are the differential gain and the gain coefficient, respectively. At the threshold, the net modal gain can be expressed as g th ¼ A0 (nth
ntransp ) ¼ b( J th
J transp ):

ð2:208Þ

The injected carrier concentration can be written as n¼

g þ ntransp ; A0

ð2:209Þ

g

and at threshold nth ¼ Ath0 þ ntransp : The injected carrier concentration needed to reach threshold can be calculated by    1 ai 1 1 n¼ þ ð2:210Þ þ ntransp : ln A0 G 2GL R1 R2 The current density is given by

 qd qd g þ ntransp ; J¼ n¼ ts ts A0

ð2:211Þ

where d represents the thickness of the pumped region and ts the carrier recombination lifetime, which is related to radiative and nonradiative recombination lifetimes through

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1
1 t
1 s ¼ trad þ tnonrad :

At the threshold, the current density reduces to

    qd qd 1 ai 1 1 þ ln þ ntransp : J th ¼ nth ¼ ts ts A0 G 2GL R1 R2

ð2:212Þ

ð2:213Þ

When the quantum efficiency is less than 1, that is, h ¼ ts =tr , the threshold current density can alternatively be written as

    qd qd 1 ai 1 1 J th ¼ þ nth ¼ þ ntransp ð2:214Þ ln htr htr A0 G 2GL R 1 R2 or J th ¼

   1 ai 1 1 þ þ J transp : ln b G 2GL R1 R2

ð2:215Þ

Here, the transparency current is calculated with the injected carrier concentration and the radiative lifetime. In reality, this figure would increase by the inverse of the efficiency or the factor of radiative lifetime over the carrier lifetime. From Equation 2.210, one can relate the gain to measurable quantities such as the current density and the differential gain by g¼

ts A0 þ A0 ntransp : qd

ð2:216Þ

The threshold gain can be obtained by replacing the current density with that at the threshold.

2.9 Recombination Lifetime

The carrier recombination lifetime is an important parameter, in that, if measured accurately, it gives a qualitative assessment of recombination processes vis- a-vis radiative and nonradiative varieties. Moreover, as indicated in the preceding section, the lifetime is needed in relating the current to important parameters such as the transparency carrier density. The lifetime ts can be measured from the delay td of the optical pulse emanating from the laser in response to a current pulse. This is determined from the rate equation that states that the time rate of change of carriers is equal to the rate of carrier injection minus the rate of recombination. Mathematically, we have dn kJ kJ qR(n) kJ n ; ¼
R(n) ¼
n¼
dt qd qd qn qd ts (J)

ð2:217Þ

where J, d, n, and ts represent the current density (assumed to be uniform across the active layer), the active layer thickness, the injected carrier concentration, and the carrier lifetime that is dependent on current. The term R(n) is the recombination

2.9 Recombination Lifetime

rate and k is the injection efficiency that ranges between 0 and 1 and accounts for the leakage current and the carrier overflow. The value of kJ relates to the part of the current that participates in the radiative recombination. Assuming a current-independent recombination time, the solution of Equation 2.217 with the boundary condition n(0) ¼ 0, meaning that the laser is not prebiased, can be written as    kJts
t n(t) ¼ 1
exp : ð2:218Þ qd ts At the onset of laser oscillations, the carrier concentration reaches the threshold value nth ¼

kJ th ts : qd

ð2:219Þ

Referring to the time needed to reach threshold and providing that k does not change with the injection current, one obtains td ¼ ts ln

J : J
J th

ð2:220Þ

InthecasewhenthelaserisprebiasedwiththecurrentdensityJ0, which isdonetoreduce td, the delay between the electrical pulse and the onset of lasing, the delay time becomes td ¼ ts ln

J
J 0 : J
J th

ð2:221Þ

A plot of the delay time against the logarithmic term gives a straight line, from which ts can be extracted. Once ts is determined and knowing the thickness of the pumped region, the injected carrier concentration can be calculated for a given current density above threshold from Equation 2.211. Here, the lifetime is assumed to be constant in the range of injection levels that are above threshold, and very high. In general, the recombination lifetime is not independent of current. The currentdependent lifetime can be determined from the 3 dB point of the modulation bandwidth through [115–118]   F0 ð2:222Þ F(w) ¼ 10 log j1 þ jwtj for a series of injection currents. Here, F(w) and F0 depict the frequency response and DC response of the laser power, respectively. The differential lifetime decreases with injection current as the excess carrier concentration increases, especially between a low-injection level and the level of transparency. Similarly, the injected carrier concentration can also be deduced from this method. When Auger recombination can be neglected, the lifetime can be expressed as 1 4bkJ ¼ a2 þ : t2 qd

ð2:223Þ

From a plot of t
2 versus J, one obtains a and bk/d. Again, from the dependence of the lifetime versus injection current, the excess electron concentration can be calculated from

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2.5

Delay time, τd (ns)

2.0

1.5

1.0

0.5

0 1.0

1.1

1.2

1.3

1.4

1.5

ln[J/(J–Jth)] Figure 2.43 The delay of the laser oscillation onset with respect to the current pulse as a function of the current drive level in an InGaN MQW Nichia laser. Courtesy of S. Nakamura, then Nichia Chemical Ltd.

ðJ k n( J) ¼ t( J 0 )dJ 0 : qd

ð2:224Þ

0

The recombination lifetime can be obtained using the delay between the optical output and the electrical input in response to an electrical pulse and Equation 2.220. Figure 2.43 depicts the delay time versus the natural log of the injection current density divided by the injection current density above the threshold current density, as described by Equation 2.220 [114]. For this particular device, the threshold current density was 7 kA cm
2 and the total active layer thickness was 150 Å. A fit to the data of Figure 2.43 results in a carrier lifetime of 5 ns. It should be mentioned that this figure is unreasonably long, and the underlying assumption that the recombination lifetime is current independent in deriving Equation 2.220 may not be applicable. From the knowledge of the threshold current, the total thickness of the entire pumped region, which is 150 Å, and the lifetime just determined, one can calculate the carrier injection level at threshold, which comes out to be about 1  1020 cm
3; this is extremely high. There are only three data points in Figure 2.43 that do not allow one to determine if, indeed, the delay at the onset of the lasing oscillation varies linearly with the parameter on the abscissa. Because the carrier lifetime decreases as the injection level is increased, a method not requiring a constant lifetime, such as in the modulation bandwidth measurements described by Equation 2.222, Equation 2.223, and Equation 2.224, is warranted.

2.11 GaN-Based LD Design and Performance

2.10 Quantum Efficiency

The external differential efficiency is a very useful parameter from which the internal quantum efficiency and, in turn, the internal loss can be determined. The differential quantum efficiency is simply the rate of change of the optical power with respect to the injection current. If the power is measured from one facet, the differential figure is doubled to account for two facets, provided the reflectivities of both mirrors are the same. The external efficiency of a laser is simply related to the internal efficiency through hext ¼ hint

end losses (1=2L)ln[1=R1 R2 ] ; ¼ hint total loss ai þ (1=2L)ln[1=R1 R2 ]

ð2:225Þ

which reduces to
1
1 h
1 ext ¼ hint þ hint

ai : (1=2L) ln [1=R1 R2 ]

ð2:226Þ

In other words, the external quantum efficiency beyond threshold is measured as the rate of change of the optical power converted to current versus the injection current. One method to obtain the internal quantum efficiency is to determine the external quantum efficiency of a series of lasers with varying cavity lengths. Plotting the inverse of the measured external quantum efficiency as a function of the cavity length leads to a straight line. In the limit of an infinitely long cavity, the end loss is zero, at which point the internal quantum efficiency is the same as the external efficiency. From the slope of the straight line, one can obtain the internal loss. The threshold gain is then equal to the sum of the internal loss just deduced and the end loss [(1/2L) In(1/RlR2)] that can be calculated knowing the facet reflectivities and the cavity length. Figure 2.44 depicts the results of such an exercise with a plot of the inverse of the external quantum efficiency versus the cavity length. The data fit to the straight-line expression 155.4x þ 3.1. The inverse of 3.1 is the internal quantum efficiency, which is about 33%. Using a reflectivity of 50% for the facets (coated), a confinement factor of 2.5%, and with the aid of Equation 2.37, one finds the gain at threshold to be 3200 cm
1.

2.11 GaN-Based LD Design and Performance

The first InGaN QW diode laser heterostructures were grown on thick GaN buffer layers on the (0 0 0 1) c-face of sapphire with large densities of extended defects and presumably point defects. The most commonly reported edge-emitting laser diode among nitrides is that made of SCH/MQW InGaN/AlGaN heterostructure in the form of index-guided mesa structure. Because nitrides are nearly impervious to wet chemical etches, dryetching techniques are employed to fabricate the structure, among which are the

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25

20 y=155.4x+3.1

η

-1 d

15

10

5

0

0.00

0.04

0.08

0.12

Cavity length, L (cm) Figure 2.44 Plot of the inverse of the external quantum efficiency against the cavity length for a series of lasers with all the other parameters remaining the same. Courtesy of S. Nakamura, then Nichia Chemical Ltd.

electron cyclotron resonance (ECR) plasma, chemically assisted ion beam etching (CAIBE), or standard reactive ion etching varieties. All use chlorine chemistry and can produce acceptable etch quality for the vertical walls of the few micrometer-wide lateral (ridge) waveguide and cavity facets. As already mentioned, in the case of the HVPE-grown thick buffer layers or SiC substrates, the optical cavity resonator can be formed by the cleaving of freestanding 100 mm films. Nakamura reported a CW InGaN MQW diode laser operating for several hundred hours with a threshold current density of 3.6 kA cm
2 in this type of high defectdensity environment. A cursory response to this tolerance against large concentration of defects in devices such as laser diodes where the injection currents and junction temperatures are both high has to do with the large covalent bonding energy of GaN and other mechanical properties of the material. However, even a semiconductor material such as GaN is no match to the demands imposed by long-term operation of the laser if it contains many defects. For truly long-lived blue and UV diode lasers (lifetime >104 h), the dislocation density had to be lowered through the defect reduction method of epitaxial lateral overgrowth (ELO), discussed in Volume 1, Chapter 3. This is in spite of the fact that the thermal load even in today’s very best InGaN diode lasers is still very high, due in part to the high threshold current density in the edge-emitting devices. Innovative approaches are now beginning to be implemented to address these issues. By incorporating the ELO process, Nakamura et al. [119] improved laser diode longevity well beyond the 100 h of the CW mode that, at the time, appeared to be the limit achievable with conventional layer structures. In standard ELO, a SiO2 mask

2.11 GaN-Based LD Design and Performance

layer is used that presumably adversely affects heat removal. The use of a variant of ELO wherein the SiO2 is removed after patterning the base layer allowed CW operation lifetimes in excess of 1000 h under high-power ( 50 mW) CW operation [120]. Researchers at Sony Laboratories have correlated the photoluminescence efficiency, surface profile of cleaved laser facets, and laser performance on ELO-based templates [121]. Naturally, the PL intensity inclusive of wing and direct growth regions showed increased values when the overall dislocation density was lowered. Likewise, the spatially resolved PL confirmed enhanced efficiency in the coalesced wing regions. As for the laser fabrication, the sapphire substrate was thinned down to about 100 mm to facilitate laser diode facet formation by cleaving. The AFM images showed the facet in ELO material to have a smoothness of about 1 nm as compared to 10 nm or so achieved in non-ELO material. The laser diodes so fabricated exhibited lifetimes beyond 1000 h at under 30 mW CW output. In addition to the ELO process alone, very thick HVPE-grown layers [122] can be incorporated into the process to lower the defect density and thus increase the longevity of laser diodes. As an example representing this philosophy, Nagahama et al. [123] explored a series of cases. One case involves just the ELO followed by laser structure. The other case involves the use of freestanding GaN in conjunction with ELO followed by laser structure. In the latter case, HVPE to synthesize a very thick (200 mm) GaN on top of the ELO template followed by removal of the sapphire substrates (using a technique such as pulsed UV excimer laser ablation from the sapphire side [124, 125]) and polishing from the originally substrate side down to about 150 mm thick freestanding GaN substrate. This is then followed by another ELO process and laser heterostructure of the active device. Because this is a GaN system in totality, cleaving techniques can be used, as opposed to dry etching, to form the facets along the {1 2 2 0} direction. Cathodoluminescence images show how the dislocations are reduced with the increased complexity of the ELO-based substrates/ templates, specifically, in terms of their density in GaN in the wing regions. The lifetimes at 30 mW average power increased from a reported 700 h in simple ELO to 15 000 h in two ELO, substrate removal and HVPE process, at a device case temperature of 60  C [81]. This experiment clearly illustrates the importance of reducing dislocations to increase the longevity of violet diode lasers. Heat removal is of major concern in laser diodes, typically in excess of 500 mW, and a substrate with poor heat conductivity such as sapphire presents problems. Just to reiterate, the thermal conductivity of sapphire at room temperature is about 0.5 W cm
1 K
1, which degrades rapidly with temperature, compared to about 2.3 W cm
1 K
1 for high-quality GaN and 4.9 W cm
1 K
1 for SiC. Although it is not an issue for SiC, the nonconducting sapphire prevents the present edge-emitting diode laser structures from being flip-chip mounted for heat sinking. Growth of laser structures on thick HVPE freestanding templates results not only reduces the defect concentration, but also helps in more efficient removal of heat. Because laser ablation has been used successfully to remove GaN layers from sapphire substrates, the same technique can be employed to not only remove the entire laser layered structure but also mount it on very efficient heat-removing substrates such as copper [126]. It

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follows that growth on SiC substrates allows for vertical current flow, eliminating any current crowding effect, similar to that discussed in conjunction with LEDs in Section 1.4, and for much more efficient heat removal, not to mention better quality material as layers on SiC. The active region of violet edge-emitting nitride-based diode lasers is based on the InGaN/GaN/AlGaN MQW SCH, grown either on sapphire, which is nonconducting, or SiC, which is conducting. Figure 2.45a sketches the compositional variation of a laser layered structure with InGaN MQW active layers fabricated by Nakamura et al. [114]. The structure combines the generic features of a separate confinement edge-emitting diode laser within the constraints imposed by the growth of the AlGaN, GaN, and InGaN cladding, waveguide, and QW active layers, respectively [81]. In addition, a pair of modulation-doped superlattices on both n- and p-side of the junction and the active MQW region sandwiched between AlGaN waveguide layers are typically employed for increased carrier concentration, particularly on the p-side. In the OMVPE process used to grow the structure, a wide range of temperatures is required for growing the three different materials (AlGaN, GaN, and InGaN), from about T ¼ 600  C (later developments indicate higher temperature leading to better quality films, see Volume 1, Chapter 3 for details) for the low-temperature buffer layer and 800  C for InGaN to nearly T ¼ 1100  C for AlGaN. If ELO is used, the details of which are discussed in Volume 1, Chapter 3, additional temperature schedules would have to be employed. The layer thicknesses and mole fractions are chosen for carrier- and lightconfinement while avoiding strain-induced cracks. When the AlGaN cladding layers exceed certain values for a given mole fraction, cracks occur. The exact figures are sample dependent because of varying degrees of strain relaxation based on the defect concentration of the starting AlGaN. On the other hand, thin and/or low AlN nitride mole fraction cladding layers are not so effective in guiding the radiation. One is then forced into making an undesirable compromise unless other means are found. As discussed in Ref. [127], the use of GaN/AlGaN short-period superlattices (SLSs) that emulate bulk AlGaN can ameliorate this compromise in the designer’s favor because thicker layers and/or higher mole fractions can be employed if the bulk cladding layers are replaced with SLS layers. The superlattice approach to modulation doping is particularly useful for the p-doping in terms of lateral and vertical transport, and it has been shown to have a role as a “dislocation” wall as well. This idea has been implemented by researchers at Nichia Chemical which follows the conceptual development and experimental demonstration in the GaAs system. Moreover, modulation doping, where the dopants are placed in the larger bandgap material, was also employed in cladding layers (particularly p-type) to reduce the effective binding energy of the dopant impurities and increase the carrier concentration over that obtainable with bulk layers, as shown for a Sony laser, Figure 2.45b, and see Volume 1, Chapter 4, for a discussion on p-type doping. In fact, uniformly doped SLS layers would automatically lead to modulation doping and benefit from carriers transferred from the larger bandgap material as well as those given off from the ionized impurities. The p-type layers benefit from this concept to a larger extent because of deep Mg acceptors. The carrier transport through the heterojunction barriers relies on tunneling. The structure in Figure 2.45a and b also takes advantage of

2.11 GaN-Based LD Design and Performance

20% 8% 8% AlGaN ~0.4 µm GaN

0.1 µm

3 µm 0.1 µm 5%

InGaN

0.5 µm

0.1 µm

2%

15% p-type

n-type

(25 Å/50 Å)×3 200 Å (a)

p-AlGaN SL cladding layer p-AlGaN electron blocking layer GaInN interlayer MQW GaN optical guiding layer

EC

n-AlGaN cladding layer

(b) Figure 2.45 (a) Schematic diagram of the conduction band edge structure of an InGaN MQW active layer Nichia laser structure. Courtesy of S. Nakamura, then Nichia Chemical Ltd. (b) Schematic of the conduction band edge of the Sony laser with an InGaN interlayer leading to a characteristic temperature of 146 K. The AlGaN interlayer variety led to higher characteristic temperature of 235 K [130, 131].

lateral growth, as shown in Figure 2.46a and b for Nichia and Sony lasers, respectively. The details of ELO growth are discussed in Volume 1, Chapter 3 for defect reduction. Unlike the case of LEDs, the range of InN molar fraction for InGaN diode lasers at practical current threshold has been rather limited (x 0.10–0.15), with typical

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p-Contact

p-GaN p-Al0.14Ga0.86N/GaN modulation doped SLS p-GaN guide p-Al0.2Ga0.8N protection layer n-In0.02Ga0.98N/ In0.15Ga0.85NMQW n-GaN guide

SiO2

n-Al0.14Ga0.86N/GaN modulation doped SLS n-In0.1Ga0.9N strain balance layer

n-Contact n-GaN buffer

SiO2 GaN LT buffer

(0 0 0 1) Sapphire substrate

(a) p-electrode (Pd/Pt/Au) Si: 45 nm SiO2: 40 nm p-AlGaN/GaN SLs

~ 7 µm

GaInN interlayer MQW n-GaN n-AlGaN

n-electrode (Ti/Pt/Au)

n-GaN (ELO-GaN layer) n-GaN (seed layer) ~ 5 µm

(0 0 0 1) Sapphire substrate

(b) Figure 2.46 (a) Cross-sectional view of a Nichia InGaN/GaN/ AlGaN DH laser structure, which takes advantage of short-period superlattice cladding layers and ELO. (b) Cross-sectional view of a Sony InGaN/GaN/AlGaN DH laser structure, which takes advantage of ELO, not shown, as well as growth on patterned sapphire. The details of the active area and cladding layers are shown in Figure 2.45b [131].

wavelength of emission close to the value l ¼ 410 nm. However, great strides have been made in reducing the wavelength down to the 300 nm range by paying attention to AlGaN growth parameters, namely, the requisite substrate temperature, which is much higher than that required for GaN growth and beyond the reach of conventional reactors.

2.11 GaN-Based LD Design and Performance

A different version of ELO wherein troughs are etched all the way down into the sapphire substrate after GaN growth has also been explored. In the subsequent growth, the standing GaN stripes are used as the seed layer to grow the ELO template on which the rest of the laser structure was grown, ELO through textured sapphire substrates [128]. An implementation of this concept along with ridge guide laser structure by the group at Sony is shown in Figure 2.46b. The width of the wing extends to 13.5 mm to facilitate the formation of the active region of the layer in the less defective wing region. The thickness of the ELO-GaN layer was kept below 5 mm to suppress wafer bowing and possible cracking of the layers [129, 130]. Grown on the ELO–GaN template is 2 mm thick GaInN/GaN/AlGaN multiple quantum well separate confinement heterostructure (MQW-SCH). An InGaN spacer layer, interlayer, was placed between the GaInN-MQW active layer and the Mg-doped AlGaN electronblocking layer as an optical guide [131]. A 1.5 mm wide ridge stripe was placed on the wing region and coated with a Si on SiO2 stack to obtain kink-free high-output power [132]. The ridge depth was adjusted to yield q|| of 9 on top of the aforementioned divergence perpendicular to the junction, q ? ¼ 22 . The laser cavities of length 600 mm were formed by cleaving along the {1 1 0 0} facets of GaN. The front and rear facets were treated with an antireflection film of 10% reflectivity and a highreflection film of 95% reflectivity. As for ohmic contacts, Pd/Pt/Au was used for the p-side electrode and Ti/Pt/Au was evaporated on the n-side contact layer. The (Al,Ga)N waveguide layers, more typically made of just GaN, may feature an additional thin AlGaN layer, as shown in Figure 2.45a, for preventing InGaN decomposition during high-temperature growth of the structure above the MQW section. In addition, the same layer is also useful in blocking (particularly when the barriers in effect are small) the electron/hole overshoot off of the active QW region, serendipitously positioned near the gain medium. Variants of this structure have also been explored [130] where an additional InGaN spacer layer, interlayer, is placed between the quantum well region and the p-AlGaN current blocking layer, as shown in Figure 2.45b. As an example of the device performance of the InGaN MQW violet laser, operating in the 405–415 nm wavelength range, Figure 2.47a plots the light and voltage versus current characteristics of such a Nichia InGaN laser at 20  C with a threshold current of 16 mA and a forward voltage of about 5 V. Figure 2.47b depicts the light and the voltage versus current characteristics of a Nichia ridge laser at 20  C with a threshold current of 50 mA and forward voltage of about 5 V relying on the deposition of an unpatterned c-plane onto a sapphire substrate. Figure 2.47c displays the light and voltage versus current characteristics of another Nichia ridge laser at 20, 60, 80, and 100  C. The laser structure of Figure 2.47c exhibits a room-temperature threshold current of 75 mA (forward voltage at threshold is 4–5 V, not shown). With further refinement of the ELO process, the extended defect concentration in the laser was reduced from about 108 cm
2 to about 106 cm
2, and other device processing procedures such as formation of facets and more stable lasing threshold with respect to ambient temperature have been reported [133, 134]. The characteristic temperature, T0, of the device shown in Figure 2.47d is estimated to be 132 K. The larger this figure the less the temperature dependence of the threshold current, as Jth exp(T/T0),

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6

2.5

5

2.0

4

1.5

3

1.0

2

0.5

1

Forward voltage (V)

Output power (mW)

274

0

0.0 10

0

30

20 Current (mA)

(a) 3

10 CW 20 °C

2 6

4

Voltage (V)

Output power (mW)

8

1 2

0

0 0 (b)

10

20

30

40

50

60

70

Injection current (mA)

Figure 2.47 (a) Light and voltage versus current characteristics of a Nichia ridge laser at 20  C with a threshold current of 16 mA and a forward voltage of about 5 V. Epitaxial lateral growth over SiO2 stripes (ELO) and modulation doping have been employed. (b) Light and voltage versus current characteristics of a Nichia ridge laser at 20  C with a threshold current of 50 mA and a forward voltage of about 5 V relying on direct unpatterned growth on the c-plane of a sapphire substrate. (c) Light versus current characteristics of a Nichia ridge laser at 20, 60, 80, and 100  C. The laser structure of (b) exhibits a room-

temperature threshold current of 75 mA and forward voltage of 4–5 V [112]. (d) Light versus current characteristics of a Sony ridge laser, built on ELO structures, between 20 and 80  C, featuring an InGaN spacer layer (with mole fraction less than that in the barriers of the three period quantum well region) on top of the MQW region [134]. (e) Light versus current characteristics of a Sony ridge laser, built on ELO structures, between 20 and 80  C, featuring an AlGaN spacer layer on top of the MQW region [130].

2.11 GaN-Based LD Design and Performance

10 9 Output power (mW)

8

100 °C

80 °C

7

60 °C

6 20 °C

5 4 3 2 1 0 0

150

100

50

(c)

250

200

Current (mA)

55 o

T 0 = 132 K

40 C

o

CW

60 C

Output power (mW)

50 o

o

80 C

20 C

40

30

20

10 0 0 (d)

50

100

150

Current (mA)

Figure 2.47 (Continued)

and a larger figure is generally associated with better carrier confinement. By replacing the InGaN interlayer with an AlGaN variety on top of the MQW region, the Sony ridge lasers built on ELO structures led to light versus current characteristics shown in Figure 2.47e taken between 20 and 80  C case temperatures. This led to a characteristic temperature, T0, of 235 K. Traditionally, with everything else being equal, quantum well devices faired better than others on this front. These devices operated more than 1000 h under CW operation at a power level of 30 mW and ambient temperature of 60  C. The temperature of operation and power levels are more or less what is required for CD writing applications. As mentioned, insertion of an InGaN and a AlGaN

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120

T 0=235K

o

40 C

o

CW

60 C

100 o

Output power (mW)

20 C

80 oC

80

60

40

20

0 0

50

(e)

100

150

Current (mA)

Figure 2.47 (Continued )

interlayer on top of the quantum well region led to characteristic temperatures of 146 and 235 K, respectively [130]. Some of these ridge waveguide structures have dimensions of 2  600 mm, and high-reflectivity facet coating is often employed to reduce the threshold current. The threshold current density in many of the laser diodes was, approximately, 4 kA cm
2 at a voltage of approximately 5.5 V, assuming uniform current flow. Differential slope efficiencies as high as 1.0 W A
1 have been obtained by the Nichia group and 1.21 W A
1 by the Sony group, the latter of which is shown in Figure 2.48, which compares standard and ELO process-based laser diodes. In quick laboratory tests, output powers of about 100 mW have been observed under high injection of about 200 mA in devices on freestanding GaN templates. Output power levels as high as 400 mW at an injection level of 400 mA under pulsed conditions have also been obtained in devices on ELO GaN. The level required for the writing process in most current optical disk media is about 50 mW, perhaps a little smaller, depending on developments. In more improved diodes, following the initial flurry of activity, the threshold current densities have been reduced to the range of 2–3 kA cm
2 in leading laboratories (Nichia, Sony, Xerox) and the slope efficiency increased up to about 1.5 W A
1 in the best cases. Likewise the threshold voltages have dropped somewhat to the 4–5 V range, still dictated in part by the quality of the p-contacts. Utilizing linear arrays of the violet InGaN MQW, a total output power exceeding 3 W has been reported by Goto et al. [135]. When an array of 11 laser chips was formed, each with 4 stripe emitters monolithically integrated (a total of 44 lasers), a maximum light

2.11 GaN-Based LD Design and Performance

40 20oC CW On ELO-GaN 1.21 W A–1

Output power (mW)

30

On sapphire 0.92 W A–1

20

10

0 0

50

100

150

Current (mA) Figure 2.48 L–I characteristics of lasers obtained at room temperature produced with and without the ELO process [134].

output power of 4.2 W under CW operation at 25  C was achieved (with a lifetime of 1000 h with 1 W CW output power at 25  C), demonstrating the highly uniform quality of the fabricated laser chips [136, 130]. Due to the small ridge width, a rather stable fundamental transverse-mode operation is possible; however, in order to avoid the kinks, characterized as the dependence of the far-field pattern on the injection level, the ridge waveguide design requires high accuracy in the index-of-refraction data as well as high precision in the ridge etch depth. To combat these constraints, Kijima et al. [137] explored an approach utilizing spin-on-glass (SOG) insulator in place of the more traditional SiO2 as the dielectric insulator at the mesa edges to stabilize the transverse mode. The lightabsorptive characteristics of this useful process material constrain and stabilize the lowest transverse mode. In another investigation, far-field full width at half-maximum (FWHM) angles of 8 and 27 , parallel and perpendicular to the junction plane, respectively, have been reported, with an aspect ratio of around 3.4 [134]. Yet in a later investigation, an InGaN spacer layer, interlayer, was placed between the GaInN-MQW active layer and the Mg-doped AlGaN electron-blocking layer as an optical guide to improve the laser characteristics. These two layers together affect the perpendicular beam divergence, q ?, and chosen thicknesses led to approximately 22 for this angle in the far-field pattern in the Sony device [130]. An additional attention paid to the depth of the ridge yielded q|| of 9 in the far-field pattern, giving an aspect ratio of 2.44, with the lowest value [138] being about 2.3. Figure 2.49 shows the far-field pattern showing FWHM angles parallel and

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25 oC CW 30 mW

θ// = 8.8 o

Intensity (a.u.)

θ⊥ = 20.8o

–40

–30

–20

–10

0

10

20

30

40

Angle (degree) Figure 2.49 Far-field emission pattern from the transverse-mode stabilized device at 30 mW output power and at 25  C [130].

perpendicular to the junction plane of 8.8 and 20.8 , respectively, amounting to an aspect ratio of about 2.36. Although there is a significant variation in the reported values of cavity losses in InGaN MQW laser diodes, the loss coefficient has been deduced from threshold current density variation as a function of resonator length and/or facet reflectivity. Such an exercise suggests that the typical value for the optical losses is in the range of 20–50 cm
1. A part of this loss is certainly attributable to low quality of facets formed in devices formed on sapphire substrates where cleaved facets are not available. The ternary of InGaN plays many roles in laser diodes. In addition to wavelength tuning, InGaN provides the medium in which the recombination efficiency is relatively high as compared to GaN. This is due to InGaN phase separation and/or compositional inhomogeneities, which form the basis for InN-rich regions of higher quality where the radiative recombination takes place in conjunction with LEDs, as discussed in Section 1.10. However, beyond a certain molar fraction, the lattice mismatch and technical difficulties of growing high-quality InGaN erase the gain garnered from compositional inhomogeneities. Thus, the threshold current density increases rapidly with the indium concentration. An increase of more than a factor of 5 over the range from about 410 to 450 nm has been reported [139]. The optimum number of quantum wells has been determined empirically to be on the order of 2–3, dictated by the modal gain required to offset the cavity losses. In the GaAs-based LDs, one quantum well is sufficient to offset losses and leads to very low threshold currents. Many of the other details in optimizing the SCH design and the active QW region have also been determined empirically, also with a limited amount

2.11 GaN-Based LD Design and Performance

of the kind of device modeling typical of other III–V-based lasers, in part, due to uncertainty in pertinent basic parameters. One such parameter is the problem of precise determination of the band offsets at the InGaN/GaN heterojunction due to uncertainties in the determination of the molar fraction, compositional inhomogeneities, fluctuations in quantum well thickness, and polarization charge, not to mention the not-so-sharp band edge of InGaN, some of which is touched upon in Volume 2, Chapter 4. This has a sizeable impact on the optical gain. If the common anion approach is applied, the electronic confinement in valence band is perhaps on the order of 50 meV or less, while the conduction band offset might be expected to be on the order of 180–270 meV for the typical average In composition (x 0.10–0.15) in the laser devices. The technology and empirical device considerations seem to converge on an optimum QW thickness on the order of 40 Å for the laser diode. Naturally, only a limited amount of systematic investigation exists concerning the designs of the SCH/ MQW lasers based on nitride semiconductors and the phenomenon of current leakage. Yet, the reports suggest that finite leakage current competes with the radiative recombination current in both the laser diodes, pointing also to a finite confinement factor being an issue, and to an extent, more than initially anticipated for LEDs that become more prominent at high-injection levels for high-intensity operation and longer wavelengths such as green. High In mole fraction containing InGaN suffers from not only the lattice mismatch and associated strain, but also connected to them is the compositional inhomogeneities and inferior material quality causing degradation of the active layer. It has, therefore, been difficult to achieve lasers at relatively longer wavelengths. However, Miyoshi et al. [140] extended the wavelength of CW operation to 488 nm by basically optimization growth conditions, building on the 482 nm laser reported earlier [141]. On the short wavelength side, 365 nm lasers have been reported [142]. The (Al,In,Ga)N heterostrucutures were grown on low-dislocation density c-plane freestanding GaN substrate initially grown on GaAs substrates by OMVPE. The structure of buried ridge geometry is nearly the same as that reported in a previous work. As alluded to above, the threshold current density of the lasers was observed to be dependent on the lasing wavelength. The wavelength of operation has been expanded to approximately 430 nm in 2001, 460 nm in 2003, 480 nm in 2005, and 488 nm in 2007. The threshold current density was 3.3 kA cm
2 and the operating current and the voltage at an output power of 5 mW were 71 mA and 5.3 V, respectively. The lifetime test of CW-operated 488 nm LDs was carried out at a case temperature of 25  C under automatic power control (APC) with an output power of 5 mW. The lifetime is defined as the estimated time required for the operating current of the LD to reach 1.3 times the initial operating current. After 1000 h of operation, the lifetime was estimated to be over 10 000 h. Sapphire has been the early choice for lasers, to a large extent influenced by the initial LED technology development on sapphire and lack of cost-effective alternatives. However, those who have access to high-quality SiC have pursued this material as a substrate. Among the advantages of SiC is its high thermal conductivity, availability of conductive substrates that allow electrically vertical

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devices to be fabricated, and lack of rotation of GaN with respect to SiC that allows cleaved mirrors. From the first day, InGaN diode laser spectral output on sapphire substrates exhibited a complex-longitudinal-mode pattern where the presence of subcavities is apparent with a length less than the physical length of the optical cavity. The inconsistency between the mode spacing and the cavity length has led to suggestions that certain crystalline defects, such as cracks that are frequently induced within the growth of the AlGaN waveguide layers formed unwanted subcavities that provide additional optical feedback. However, good progress has been made since the early stages of development to create stable, single-longitudinal-mode blue and violet lasers. This issue dealing with longitudinal modes is not of prime importance in the realm of optical storage applications, but the single transverse mode is required for such applications. By way of an example of lasers on SiC substrates, where substantial progress has also been made, an InGaN MQW diode laser grown on SiC substrate with a single-longitudinal-mode operation has been obtained during early developments [143]. Since then, the emphasis has been on reducing the threshold current and increasing the longevity. Lasers on SiC substrates are competitive with those on ELO GaN. Still, even with SiC there is a lattice mismatch of nearly 4% and thermal mismatch leading to tensile strain and treacherous cracking. As an onlooking observer could have predicted, the necessity of native substrates for laser development once again was proven, and this time in the nitride material system. The issue before us is what technique will be the method of choice for producing GaN substrates. Growth on ZnO, GaAs, Si, and sapphire substrates of freestanding GaN as well as ammonothermal growth and growth from liquid phase are being explored. Time will tell which one(s) of these approaches would provide the best quality at an affordable cost.

2.12 Gain Spectra of InGaN Injection Lasers

The gain spectra can be obtained employing, for example, the method developed by Hakki and Paoli [144, 145]. In this method, the device is pumped to just below threshold, and the gain is calculated from the interference fringes. This can be done as a function of injection current and wavelength, that is, R1 R2 exp(
2aL) ¼

power at peak ; power at valley

ð2:227Þ

where a is the negative loss, which is equal to the modal gain minus the internal loss. In other words,     1 1 power at peak ln : ð2:228Þ þ ln
a ¼ Gg
atot ¼ 2L R 1 R2 power at valley Having already found the internal loss, the gain can be calculated from Equation 2.228 as a function of wavelength for a given current. The peak power and the valley power

2.12 Gain Spectra of InGaN Injection Lasers

Ith=45 mA PA P B

Light intensity (a.u.)

30 mA

VA 400 nm

401

10 mA

395

400

405

410

Wavelength (nm) Figure 2.50 Light intensity of a ridge laser versus wavelength below the threshold current (45 mA), for current levels of 10 and 30 mA, indicating the modes used in calculating the modal gain through the method of Hakki and Paoli [144]. Courtesy of S. Nakamura, then Nichia Chemical Ltd.

can be determined at any wavelength. It is customary to average the intensity of the two adjacent peaks for the peak power as shown in Figure 2.50. The minimum (VA) straddled by the two adjacent maxima (PA and PB) is taken as the power at valley. The output power of the ridge laser (Figure 2.47b) versus wavelength above threshold for current levels of 50 and 53 mA is shown in Figure 2.51. The modal gain versus the injection current for an InGaN laser at the wavelength of 400.2 nm is depicted in Figure 2.52. For the InGaN laser employed in this experiment, the threshold current density is 8.8 kA cm
2, the internal loss is 46 cm
1, and the recombination lifetime is 3.5 ns. The methods used in the determination of these parameters are those described in Sections 2.9 and Sections 2.10. Superimposed on the figure are measurements of a ZnSe-based laser for comparison [146]. The modal gain of the InGaN laser can be represented by a straight line expressed by
180 þ 0.03 J (cm
1). It should be noted that the first term is constant whereas the second term is a function of the injection current. This is in agreement with the form of Equation 2.208, which applies to the linear operating regime. The constant term and the slope can be utilized to determine the differential gain coefficient and the transparency current, which for the device under test are 5.8  10
17 cm2 and 9.3  1019 cm
3, respectively. Again, the question arises how to find the carrier lifetime, as was alluded

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Ith=45mA

Light intensity (a.u.)

53 mA

50 mA

395

405

400

410

Wavelength (nm) Figure 2.51 Light intensity of a ridge laser versus wavelength above the threshold for current levels of 50 and 53 mA. Courtesy of S. Nakamura, then Nichia Chemical Ltd.

Net modal gain, Γg– αi (cm-1)

50

I th 0

I transp

I th

-50

-100

InGaN, λ=400.2 nm

-150

ZnCdSe, λ =508 nm -200 0

10

20

30

40

50

Current (mA) Figure 2.52 Net modal gain versus the injection current for an InGaN laser at a wavelength of 400.2 nm, which can be expressed as 0.03J – 180 (cm
1). InGaN laser data: courtesy of S. Nakamura, then Nichia Chemical Ltd.

60

2.12 Gain Spectra of InGaN Injection Lasers

to in Section 2.9. If the carrier lifetime were about 0.5 ns, the transparency density would reduce to about 1  1019 cm
3, which is much closer to the calculations of Suzuki and Uenoyama [60] for GaN. The single-carrier model indicates the transparency carrier density to be around 5  1018 cm
3. When InGaN is employed for the active layers, the carrier injection density at transparency should even be smaller. In defective materials, it is possible to have transparency currents that are extremely high because of nonradiative recombination and the carrier leakage, but a figure in the 1020 cm
3 range appears questionably high. As compared to the well-established laser material GaAs, this high e–h pair density can only be partly accounted for by the somewhat higher effective masses with regard to population inversion and only partially because the effective masses in ZnSe and GaN are of comparable magnitude and yet the threshold current density as well as the e–h pair density in a ZnCdSe QW laser are more than one order of magnitude lower. Therefore, a more thorough treatment of factors that influence the optical gain of the InGaN laser diodes is warranted. The answer may lie with the spectrally very broad band-edge luminescence emission in the InGaN QW system, even at cryogenic temperatures with linewidths on the order of 100 meV for In concentration x > 0.20. Up to the injection levels of at least 1018 cm
3, the luminescence spectrum, both CW and time resolved, shows evidence for localized electron–hole pair states that reflect the large energy range available for such localization, as discussed in detail in Volume 2, Chapter 5, and in Refs [147–149]. Here, the manner in which this localization affects the optical gain and transparency current is discussed for convenience. The optical properties of thin InGaN QWs show notable deviation from a usual random alloy. A proverbial and qualitative thesis that has been established through many microprobe studies is that the tendency toward In clustering may have its genesis in the finite degree of thermodynamic immiscibility of the InN and GaN constituents in a normal solid solution at the near-800  C growth temperature, a topic discussed in Volume 1, Chapter 3. These compositional anomalies would profoundly affect the band-edge electronic states that supply the carriers needed for optical emission in LEDs and lasers, specifically, optical gain in the latter [81]. Such a compositional disorder is suggested to pave the way for a competition between localized and extended electronic states for electronic excitations. Having made the point, there is also a possibility of disorder due to quantum well thickness fluctuations. In particular, the low-temperature photoluminescence and cathodoluminescence spectroscopy investigation has led Monemar et al. [99] to argue that quantum well thickness fluctuations and not the compositional disorder are responsible for the observed near-band-edge optical properties. It is plausible that both processes to varying degrees have an impact on the final process. Typically, the mean values of InN mole fraction in the lasers are in the range of 0.1–0.2, unless dictated otherwise due to the desired wavelength of operation. High-resolution electron microscopy studies and other methods used to determine the InN molar fraction in InGaN QWs point to variations that are not programmed in. Terms such as clustering and quantum dots have been liberally used to describe the situation germane to such clusters [148]. When this type of nanoscale heterogeneous semiconductor forms the active laser material, a question that arises concerns the

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competition between localization and many-electron correlations within the available electron–hole pair states, given the very high pair densities required even in the best of present devices to reach the lasing threshold (>1019 cm
3). Having discussed the particular complications involved in dealing with InGaN, let us turn our attention to the gain. There are a few methods for determining the modal gain. Nakamura and Fasol [150] employed one of these models, the Hakki and Paoli method, to deduce the gain as a function of wavelength for a series of injection currents up to below the threshold. Kuball et al. [113] used an electrical injection– optical probe method to study the formation of optical gain in InGaN QW p–n-junction heterostructures. Using the data, the value of net modal gain versus wavelength for injection currents of 10, 30, 50, and 53 mA has been obtained [114]. Clearly, 53 mA is too high as this is past the onset of lasing, in which case the gain spectra are skewed in favor of the lasing mode. The transparency current determined at zero net modal gain is about 50 mA. Since these early investigations, considerable insight into the optical gain spectra in MQW InGaN laser structures has been gained by taking advantage of the spontaneous emission spectra of the laser diode, in conjunction with its threshold characteristics. In this method, the fundamental relationships between spontaneous emission, stimulated emission, and absorption is utilized [151]. For a gain medium possessing such anomalies as the InGaN QW, the “Henry” approach is more appropriate to get at the fundamental gain characteristics and their relationship with radiative processes. Song et al. [143] carried out the observation of the wavelength dependence of gain in InGaN QW diode lasers (x ¼ 0.15), the result of which is that there is a pronounced extension of the gain spectra associated with the n ¼ 1 QW transition into the low-energy region. At threshold, a finite gain was observed below the n ¼ 1 QW transition by as much as 200 meV below, indicating a degree of broadening that is most uncharacteristic of other more established semiconductor laser material such as GaAs. A peak gain coefficient of approximately 3200 cm
1 was measured. Subsequently, Kneissl et al. [126] measured the gain spectra over a range of indium compositions in QW InGaN lasers, all near the violet region of the spectrum, as shown in Figure 2.53. Of particular interest is the radical change in the spectra of optical gain from the case where the InN mole fraction is 0.08 to the case where it is 0.12. Clearly, increasing InN mole fractions extends the modal gain to include lower energies. Because lasing occurs at the wavelength where the gain is at a maximum, providing that losses are overcome, wide gain spectra requires injection of large electron–hole pairs without necessarily utilizing them for lasing. As detailed by Song et al. [143], with increasing injection, the transparency condition is reached relatively easily. In fact, these levels of injection are not very different from that of the LED regime. With increasing current, gain emerges over the wide spectral range owing to the participation of corresponding range of electronic states [81]. The spectral position of the peak gain blueshifts somewhat for higher injection levels, but considerably less than anticipated from a one-electron state-filling picture, which may be due to manybody bandgap renormalization effects. In addition to extrinsic losses, there are some mechanisms intrinsic to a material that do not clearly allow a sharp band edge at high

2.12 Gain Spectra of InGaN Injection Lasers

(a) 100 6.3 kA cm -2 5.6 kA cm

Modal gain (cm-1)

50

-2

4.6 kA cm -2 3.5 kA cm

-2

2.3 kA cm -2

0

-50

-100 2.9

1.2 kA cm -2

In0.08Ga0.92N MQW (1 =400 nm)

3.0

3.1

3.2

3.3

Energy (eV)

(b) 100 11.5 kA cm -2 10.0 kA cm -2 8.3 kA cm -2

Modal gain (cm-1)

50

6.7 kA cm -2 3.4 kA cm -2 1.7 kA cm -2

0

-50

-100 2.7

In0.12Ga0.88N MQW (1=421nm)

2.9

2.9 Energy (eV)

3.0

3.1

Figure 2.53 Gain/absorption spectrum of the InGaN MQW diode laser of two different In compositions, at varying injection levels at room temperature [126].

injection levels. The consequence of this is that while the spectrally integrated gain is quite large, its peak value that determines the lasing is low [81]. These above discussions may be viewed as an extension of arguments reported consistently about carrier localization due to strain and compositional inhomogeneities [113, 147, 148, 152]. These anomalies paved the way for radiative recombination

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processes at the lowest interband transition in the InGaN QW to be considerably influenced by localized e–h pair states at room temperature, over an energy range that is as high as an order of magnitude larger than what one would estimate for a random alloy. These inhomogeneities are discussed in detail in Volume 1, Chapter 3. Suffice it to say that some of these are somewhat germane to the nature of InN in the lattice, and others have to do with the particular growth method and substrates employed. There is no question, thought, that all available optical data on InGaN QWs and thin films display to some extent the striking extension of the band-edge states. One consequence of this is that excitonic features in absorption at the n ¼ 1 QW states have not been unambiguously identified in InGaN. This is in contrast to GaN. However, these inhomogeneities, which would normally be unwanted, may also have provided just the added improvement in radiative recombination efficiency, albeit not yet on par with standards established by other laser semiconductors, to obtain lasing even with inferior characteristics that are under discussion. For comparison, gain spectra associated with wide-gap ZnCdSe QW lasers, as discussed in Section 2.6.6.1.1, show strong excitonic enhancement of the peak gain and an overall optical response at the n ¼ 1 HH exciton with pronounced Coulomb correlations in evidence. Such effects would clearly be masked by the inhomogeneous contributions in the InGaN QW with the end result of making it difficult to isolate predicted many-body interactions [87, 153, 154] in the dense e–h system within the active region of the blue laser diodes. As mentioned above, it is a necessity to fill the localized states prior to the buildup of a sufficient population inversion for threshold gain in the present devices. Because transparency is reached at a rather low injection level (n ¼ p 1018 cm
3), it may be possible to reduce the threshold current by designing a laser resonator with very low optical losses. The near clamping of the Fermi level at higher injection levels may be due to a significant increase in the effective density of states [80]. However, the question whether these states are still localized or extended is discussed in Section 2.6.6.1.2. Suffice it to say that InGaN QWs for mole fractions xIn < < 0.1 behave nearly as a random alloy, but the extent of In compositional anomalies increases very rapidly once the In concentration increases beyond 0.1. The works of Kneissl et al. [126] and others [81] shown how the gain spectra indeed get significantly narrower as the In concentration is reduced, which typically corresponds to lasers operating in the 395–405 nm range. However, for adequate electronic/optical confinement, the Al concentration in the cladding layers must be increased, adding another challenge. As the spectra displayed in Figure 2.50 indicate, many modes are supported by a gain versus energy (wavelength)–cavity combination. Even though only one mode appears when the injection current is at 53 mA, single-mode operation is not maintained, as the current is changed and/or the temperature is controlled. Figure 2.54 illustrates the mode hopping behavior with current. It appears that there is a slight blueshift at each mode as the injection current is increased, which appears to be indicative of the filling effects. On the other hand, Figure 2.55 exhibits mode hopping behavior as the device temperature is increased from 20 to 60  C. As expected, a redshift is clearly observed with increasing temperature.

2.12 Gain Spectra of InGaN Injection Lasers

Peak wavelength (nm)

401.0

400.5

400.0 150

160

180

170

190

Current (mA ) Figure 2.54 Mode hopping behavior of an InGaN laser with an injection current in the range of 150–190 mA. Courtesy of S. Nakamura, then Nichia Chemical Ltd.

During the evolution of GaN laser longevity, many types of devices exhibited varying lifetimes. For example, lasers grown without the ELO process, discussed in Volume 1, Chapter 3, had operating lifetimes in the tens of hours under CW testing at room temperature. With the incorporation of ELO in the structures, lifetimes 410

CW 1 mW

Peak wavelength (nm)

409

408

407

406

405 20

30

40

50

Temperature (°C ) Figure 2.55 Mode hopping as the case temperature is increased from 20 to 60  C in an InGaN laser, with the redshift being clearly observed with increasing temperature. Courtesy of S. Nakamura, then Nichia Chemical Ltd.

60

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150 N=13

Operating current (mA)

60 ºC 30 mW 100

MTTF = 15 000 h

50

0 0

500

1000

Time (h) Figure 2.56 Evolution of the current through 13 CW Sony InGaN lasers with initial threshold currents in the range of 60–70 mA. The injection current was adjusted to maintain the light output at 30 mW at 60  C. The MTTF is estimated at 15 000 h [155].

increased steadily and dramatically. For example, early versions of ELO-based lasers produced by Nichia operated at room temperature up to about 3000 h under CW testing. These life-testing improved devices, where the current drive is automatically adjusted to maintain a CW per facet power of 2 mW, led to lifetimes of up to about 1000 h without any marked degradation. The Sony devices [155], again utilizing the ELO process improved on this figure tremendously with extrapolated room-temperature lifetimes of about 15 000 h. Figure 2.56 displays the evolution of the current through 13 InGaN MQW lasers over 1000 h when the per facet output power is kept at 30 mW at 60  C. The power levels and temperature are acceptable for commercial write and definitely read applications. As is the case with any commercial laser, the lifetime and far-field pattern issues will much dominate future discussions as well as the understanding of the governing processes of lasing. In the context of the latter, the absorption, as determined by photocurrent experiments, for example, by using the LD device as a detector, PL excitation, and stimulated emission spectra of a laser are presented in Figure 2.57. Clearly, there is substantial (nearly 190 meV) redshift of the laser line with respect to the absorption spectrum, caused by localized states. The absence of localized states in the excitation spectrum may have to do with the small joint density of states. Resonant measurements may have been warranted to observe absorption by localized states. In weakly localized systems, the resultant proximity of electrons and holes would increase the injection efficiency. Strongly localized states, which may not occur here as this case requires strong localization potentials, and the spread in the reciprocal space would reduce the recombination efficiency and thus increase the carrier lifetime. In contrast to the GaN case, the Stokes shift in ZnCdSe lasers is about 30 meV or less. This redshift is most likely due to the lasing action that takes place at localized states. Redshift in laser spectra is also observed when the device is driven at

2.12 Gain Spectra of InGaN Injection Lasers

EL

Intensity (a.u.)

Photocurrent

PLE

300

350

400

450

Wavelength (nm) Figure 2.57 Absorption spectrum as determined by photocurrent (using the LD device as a detector) and PL excitation and the stimulated-emission spectrum of an InGaN laser. Courtesy of S. Nakamura, then Nichia Chemical Ltd.

higher injection currents for larger optical power output. In one of the Sony devices [155], the dominant mode wavelength increased from 409.2 to 410.2 nm as the output power increased from 5 to 70 mW at 25  C. In parallel developments, InGaN/GaN/AlGaN double heterostructures grown on SiC substrates have been reported to operate first pulsed with a duty cycle of 0.1% and later CW at room temperature [156]. Unlike the laser and LED structures on sapphire substrates, cursory investigations purportedly led to the conclusion that compositional inhomogeneities are not present. Moreover, the redshift that is prevalent in lasers on sapphire was presumably absent. The relative ease with which SiC substrates can be cleaved (still much harder than GaAs) manifested itself in sharp linewidths on the order of a few angstroms and attested to the quality of the cavity. However, as in the case of lasers on sapphire, the mode spacing in the spectral response of the lasers did not correspond to the physical cavity length. This anomaly, which is endemic to all nitride-based lasers, may have its roots in some internal reflections within the cavity. As discussed throughout Chapter 3, the barrier for dislocation motion in GaN is large, but an actual value has not yet been determined. Therefore, only at the upper end OMVPE and HVPE growth temperatures can the dislocations be moved. At lower temperatures, such as the MBE growth temperatures, the dislocation motion is only in the form of threading dislocations propagating along the c-direction for energy minimization. These are the perfect edge dislocations that are predominant in MBE with comparatively much fewer screw and mixed dislocations. High-quality layers can indeed be grown if GaN buffer layers are produced by OMVPE or HVPE freestanding GaN templates (prepared by

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either HVPE or high-pressure liquid method) are used. Using OMVPE-grown buffer layers, laser structures have been grown by reactive MBE with ammonia as the active nitrogen source. In a series of papers originating from Sharp Laboratories, room-temperature CW operation of lasers prepared by RMBE on freestanding GaN have been reported, first, in the form of pulsed lasers [157, 158], followed later by CW operation [159] Electron cyclotron resonance-based plasma etching was employed to form the mirrors. The laser epitaxial design consisted of a separate confinement heterostructure wherein the active region consisted of three In0.1Ga0.9N quantum wells with a nominal thickness of 2 nm. Characteristic of MBE, the p-type doping of GaN and AlGaN cladding regions was obtained without the use of postgrowth thermal annealing. Ridge waveguide lasers with 2.2 mm 1000 mm laser cavities having highly reflective facet coatings led to a continuouswave threshold current of 125 mA, corresponding to a threshold current density of 5.7 kA cm
2. At the threshold, the forward voltage was 8.6 V and lasers were reported to emit at a wavelength of 405 nm for 3 min at 20  C under CW conditions. Preparation of MBE-grown lasers on freestanding GaN templates allowed cleaved facet (consisting of {1 1 0 0} planes) to become attainable [160]. Atomic force microscopy was applied to determine that the facets were indeed smoother than the cleaved facets on other substrates. The OMVPE laser structures grown on bulk GaN prepared using the high-pressure melt growth technique took advantage of low dislocation densities, in the 105 cm
2 range as opposed to 107 cm
2 range for templates prepared by ELO using OMVPE and 106 range for cm
2 for freestanding GaN templates prepared by HVPE, and demonstrated pulse power levels of 1.3–1.9 W per facet (50% reflectivity) in response to 30 ns long pulses in 15 mm  500 mm-wide stripes [161]. From thermally accelerated electrical aging tests, the activation energy of the dominant degradation mechanisms was found to be 0.32 eV. Catastrophic optical damage was not observed up to power density of 40 MW cm
2. It was conjectured that the low defect density GaN substrates led to high-pulse power lasers [162].

2.13 Near-UV Lasers

The case was made in the prologue about short-wavelength laser being a crucial component in high-density optically recorded and read storage medium. In these diffraction-limited applications, the shorter the wavelength, the higher is the density that provides the impetus for exploring lasers with small InN content in the active emission medium and eventually just GaN and beyond. Other applications of UV lasers include many aspects of biology for activating labels, aspects of life sciences, compact biochemical-labs-on-chip, and so on. Exploration of such lasers is also of interest from the basic science point of view, including materials developments. The physics of optical gain and electrical characteristics of very wide-bandgap semiconductors, with bandgaps near and beyond 4 eV, are to a large extent unexplored, and so are the laser diodes operating from the violet into the near-ultraviolet spectral

2.13 Near-UV Lasers

range [81]. In this chapter, only the developments specific to lasers are discussed, leaving the developments in violet and UV LEDs to Section 1.9.1.3. Plenty of mention was made of the characteristic of InGaN that it is needed for increased radiative recombination energy, and yet its presence causes inhomogeneities that lead to the broadening of gain spectra. Also it is mentioned, both in this chapter and to a larger extent in Section 1.9.1.3, that increased InN mole fraction beyond a certain point and below a certain point brings one into a realm where there is diminishing if not insufficient gain for lasing. For InGaN QW ternary lasers, the threshold current density increases rapidly for wavelengths beyond about 420 nm. The same is true also for wavelengths below about 400 nm because of, to a large extent, the reduced recombination efficiency of the emitting medium and to some extent the weakened electronic confinement. As discussed in detail in Section 1.9.1.3, the efficiency of violet LEDs drops significantly as the In content is reduced. This has been attributed to a loss of electron–hole localization in the InGaN QWs and resulting paths to nonradiative recombination centers. The nitride semiconductor system encompassing GaN, InN, and AlN binaries is very similar to the GaAs, InAs, and AlAs system, in that heterojunctions for light and carrier confinement are possible in a wide range of wavelengths. For simplicity of the arguments, the AlGaInN nomenclature is used to describe the domain of the binaries, ternaries, and quaternaries formed by the nitride semiconductor system. In going from the violet to the ultraviolet lasers, a binary GaN, a ternary AlGaN, or a quaternary AlGaInN QW active medium represents the options below about 360 nm. Each presents its own set of material challenges, in terms of point defects in the active medium and extended defects emanating from lattice mismatch with the substrate and also within the heterostructure, such as the cladding layer of the SCH structure. To begin with, photopumped laser action from GaN thin films and GaN/AlGaN QWs has been reported under excitation but attaining sufficient electroluminescence efficiency at room temperature has remained elusive [163]. The quaternary involves the issue of simultaneous incorporation of both In and Al and associated phase separation, as discussed in Volume 1, Chapter 3, but offers the added advantage of a tunable bandgap material while maintaining a given lattice constant such as that of GaN. In doing so, cracking and generation of extended defects due to lattice mismatch between the layers constituting the laser structure could be mitigated [164]. The points of attention are associated with the control of electrical properties with the p–n-junction straddling the active emission medium and rapidly increasing difficulty of all aspects of growth with the increasing bandgap. Nagahama et al. [165, 166] demonstrated CW edge-emitting lasers at room temperature near 370 nm with both GaN QW and AlInGaN QWs. In each instance, the template was a thick ( 20 mm) GaN produced by lateral epitaxial overgrowth on sapphire for reducing threading dislocation density. Moreover, a modulation-doped short-period superlattice was incorporated in the cladding layer region for enhanced doping ( 100 periods of 25/25 Å n-Al0.05Ga0.95N/Al0.10Ga0.90N) with Si and correspondingly for the p-doped SL with Mg dopant to enhance particularly, the p-type doping, as discussed in Volume 1, Chapter 4. Finally, these near-UV diode lasers employed only a single quantum well as the

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7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0 0

20

40

60

80

Forward current (mA) Figure 2.58 L–I and I–V characteristics of Al0.03In0.03Ga0.94N SQW laser under CW operation at 25  C [166].

100

Voltage (V)

Output power (mW)

gain medium: a 100 Å thick GaN quantum well with Al0.15Ga0.85N barriers and a 75 Å thick AlxInyGa1
x
yN quantum well with Al0.15In0.01Ga0.84N barriers for the binary and quaternary cases, respectively. This is in contrast to the common violet diode laser active medium that contains about three quantum wells. It should be noted that the issue of the number of well used is merely for technological reasons and will certainly change with the evolution of technology. The predilection for opting for a single quantum well may well be a result of the complexity of having to grow multiple quantum wells of the quaternary material. In principle, the smaller the volume that needs to be pumped, the lower the threshold current, provided that the gain is sufficiently large to overcome the losses, a condition that is met by the conventional III–V semiconductors. The light output–current and current–voltage (L–I and I–V) performance of a quaternary QW device featuring an Al0.03In0.03Ga0.94N QW is shown in Figure 2.58. Note the CW operation at room temperature and the maximum output power reaching several milliwatts. The associated above threshold emission spectrum is shown in Figure 2.59, which is centered near 371.7 nm. This redshifted somewhat, as the center wavelength under pulsed operation is around 366.4 nm. Demonstrating one of the advantages of the quaternary approach, Figure 2.60 presents the wavelength tunability for different Al and In compositions in the quaternary well. Equally noteworthy is the corresponding variation of the threshold current density Jth with composition. Particularly, attention should be paid to the increase in Jth with increasing Al concentrations, up to about 12 kA cm
2 for xAl ¼ 0.08. As expected, Nagahama et al. [166] attributed this increase primarily to the worsening crystal quality of the quaternary AlxInyGa1
x
yN, both in terms of general morphology and defects as the Al content and also In content are increased. Nevertheless, these

Intensity (a.u.)

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

371

371.5

372

Wavelength (nm) Figure 2.59 Emission spectrum of CW Al0.03In0.03Ga0.94N SQW laser operating at output power of 2 mW [166].

attempts expand the basis of knowledge toward developmental efforts in III-nitride materials for light emitters into the deeper ultraviolet. The evolution of the technological developments in light emitters is that laser reports generally follow LED reports. This is also true for VCSEL in that vertically emitting LEDs lead the way to VCSELs. In nitrides and at the time of writing this text, LED developments, presumably some of them will be precursors to laser development, below about 370 nm overlap are in full swing. As already noted in Chapter 1 (dealing with LEDs) and also Chapter 4 (dealing with detectors), the UV range of approximately 350 nm down to 280 nm represents something of an unexplored territory for semiconductor-based light emitters. This wavelength range holds significant potential for a new optoelectronics technology base [81]. For example, a wide range of photochemistry and photobiology applications could benefit from compact and relatively less expensive sources in this region of the spectrum. Even thinking about extending III-nitride diode lasers to this UV range is indicative of how far the technology has come in a short timespan. Suffice it to say that it is simply a major endeavor, one that would require significant research and development efforts. It is for this reason that the UV LED development should be considered as the first step toward laser development. UV LEDs, in their own right, may have applications in some versions of solid-state white-lighting proposals as discussed in detail in Section 1.15.

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

Cleaved and/or etched facets are not applicable to vertical cavities. Instead, reflectors made of either heterostructures composed of materials with similar lattice structure but dissimilar indices of reflection (the more dissimilar they are, the less the number of pairs needed and wider the blocking wavelength band) or dielectric stacks. Two of

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380

370 360

15 Threshold current density (kA cm-2 )

Wavelength (nm)

390

(a)

10

5

0 0

0.02

0.04 0.08 0.06 Al mole fraction, x

0.1

380

370

360

Threshold current density (kA cm-2 )

15

10

5

0 0

0.02

0.04

0.06

In mole fraction , y

0.08

0.1

Wavelength (nm)

390

(b)

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

these stacks are required, one for the bottom and the other for the top. Due to epitaxial growth, requirements the bottom reflector stack is made of the semiconductor variety, while the top can be either, depending on the mode of pumping. The reflectors are also used for microcavity (MC) lasers with small lateral dimension. In what follows, following a succinct discussion of cavity physics, we discuss the activity in nitride-based reflectors and their properties as a segue to vertical cavity lasers, albeit optically pumped at this stage. A typical MC structure is formed by two distributed Bragg reflectors (DBRs), which consist of l/4 wave (l ¼ lair/n is the design wavelength, with n being the refractive index at the wavelength of particular interest) alternating dielectric stacks with lowand high-refractive index layers. This gives rise to a broad stop band in the highreflectivity region centered at lair, with oscillating side lobes on both sides. In the stop band, the mirror reflectivity is given by  R¼

1
(next =nc )(nL =nH )2N 1 þ (next =nc )(nL =nH )2N

2 ;

ð2:229Þ

where nL, nH, nc, and next are the refractive indices of the low- and high-index layers, the cavity material and the external medium, respectively, and N is the number of stack pairs. If the refractive index contrast between the layers of the dielectric stack is relatively small, a large number of pairs are needed for high reflectivity. The required cavity thickness Lc is an integer multiple, m, of l/2 in the medium. The semiconductor MC is very similar to a simple Fabry–Perot resonator with planar mirrors. However, due to the penetration of the cavity field into DBRs, the Fabry–Perot cavity length must be replaced by a larger effective length, Leff ¼ Lc þ LDBR, where LDBR is the penetration depth into the DBRs and given by [167] LDBR ¼

l nL nH : 2nc nH
nL

ð2:230Þ

The cavity mode frequency is given by wm ¼ (Lcwc þ LDBRws)/Leff, where wc is the Fabry–Perot frequency defined by the length of the cavity, and ws is the center frequency of the DBR stop band. For wc 6¼ ws, which may arise from inadequately controlled quarter-wave stack thicknesses, the observed wm is no longer equal to wc and is in fact more sensitive to ws than to wc since LDBR is greater than Lc. Because the mirrors have a finite transmission probability, the cavity mode has a finite width wc (full width at half-maximum), given by, for R ! 1 [167],
hDc ¼


hc(1
R) : nc Leff

3 Figure 2.60 Threshold current density and emission wavelength of AlxIn0.04Ga(0.96
x)N SQW diode lasers as a function of the Al mole fraction x (a) and Al0.03InyGa(0.97
y)N SQW diode lasers as a function of the In mole fraction y (b) [166].

ð2:231Þ

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This width can be considered as homogeneous, or lifetime broadening of the confined cavity mode. A typical width of 1 meV corresponds to a cavity mode lifetime of 4 ps. A planar cavity provides no in-plane confinement, perpendicular to the growth axis, just as for electronic states in a QW, and therefore photon has only in-plane dispersion. Recall that photons are quantized along the cavity as the mirrors force the axial wave vector kz in the medium to be 2p/Lc. Hence, the cavity photon energy is approximately "  #1=2
hc
hc 2p 2 þ k2== : ð2:232Þ E¼ k¼ nc nc Lc For small k//, the in-plane dispersion is parabolic (as depicted in the above equation), and therefore it can be described by a cavity photon effective mass M ¼ 2p2hnc/cLc. This effective mass is very small, 10
5me, and the dispersion can be measured directly in angle-resolved experiments allowed by the introduction of an in-plane component to the photon wave vector. At the cavity mode wavelength, the electric field peaks at the center of the cavity. Experiments involving off-normal incidence can also be modeled by including an appropriate in-plane wave vector for the field. Because the gain region is short, and thus the gain is low, in vertical cavity devices, the reflectivity of the top and bottom reflectors must be high, in the high 90% range, in order to overcome optical losses for lasing. In fact, the bottom one should be as close to 100% as possible while the top one is made slightly less depending on the optical power desired for extraction. A reasonably detailed account of nitride-based optical reflector can be found in Ref. [168]. The thickness control required for highreflectivity DBR scale with the wavelength exacerbating the situation for short wavelengths, the realm of nitride-based lasers. Even small errors in the thickness of each layer in the stack can easily lead to large deviations from the desired central wavelength of the reflection band. It is also imperative that the reflector region be made of materials that have no loss at the operation wavelengths. AlN/GaN alloy has attracted much attention, since this system has the highest refractive index contrast among all the III–V compounds. As a result, a reduced number of periods can accomplish high reflectivity. However, there is a large lattice mismatch between GaN and AlN leading to a tensile strain in vertical structures, which are notorious for crack formation. Unless this issue is addressed, the overall stack thickness is limited. In addition, the lattice mismatch between the two materials causes defects to form and propagate. Despite these difficulties, there have been several reports on single-stack AlN/GaN DBRs [169–172], but reports on microcavity structures inclusive of the bottom AlN/GaN DBR are yet to be available. To mitigate the lattice mismatch and cracking problems, albeit at the expense of increased number of pairs needed and reduced stop bandwidth, AlGaN/GaN DBRs with relatively low Al concentration (99% at 450 nm

99.9% at 347 nm 85% at 358 nm

215 meV

35 nm

35 nm

450 meV

17 nm

17 nm

99.5% at 343 nm

99.4% at 450 nm

99.4% at 450 nm

40

OMVPE

94% at below 360 nm

91% at 353 nm

OMVPE

25

25

7 85% at 358 nm 11(top)/ 14 (bottom) 20 90% at 515 nm

MBE

OMVPE

OMVPE

Lattice-matched DBR

Theoretical

Lattice-matched DBR

Theoretical

Crack-free

Crack-free

[187]

[186]

[185]

[184]

[183]

[184]

[183]

[187]

[182]

[181]

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

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1.0 SiO2/Si3N4 10 pairs

(a)

R = 99.5 %

Reflectivity

0.8

0.6

0.4

0.2

0.0 200

300

400

500

600

Wavelength (nm) 1.0 Al0.50Ga 0.50N / GaN 30 pairs

(b)

R = 99 %

Reflectivity

0.8

0.6

0.4

0.2

0.0 250

300

350

400

450

500

Wavelength (nm) Figure 2.61 Calculated reflection spectra of (a) a 10 pair SiO2/Si3N4 quarter-wave reflector and (b) 36 pair Al0.50Ga0.50N/GaN quarter-wave stack designed for a central wavelength of 380 nm. The dotted line in subpart (b) shows the calculated reflection spectra. Note that the better contrast in

refractive indices provided by the SiO2/Si3N4 pair, even with fewer pairs, produces better reflectivity and wider stop band as compared to the Al0.38Ga0.62N/GaN quarter-wave stack. Courtesy of R. Shimada and J. Xie. (Virginia Commonwealth University).

Figure 2.63 shows a schematic diagram of a VCSEL device structure. The threshold gain gth is given by Equation 2.37. Assuming an optical gain of 103 cm
1 for the active region, a confinement factor G ¼ 1.0, and R1 ¼ R2 ¼ R, the minimum reflectivity R is 0.61, which can be easily provided by thin metallic or multiple-layer Bragg reflectors. To place the maximum of the E field where the gain is generated, the active layer is placed at the antinode of the optical standing wave in a laser cavity. The current confinement structure is necessary to achieve a carrier population inversion state in the active region with low operation current. This is to avoid fruitlessly pumping the regions that do not contribute to the gain of interest. Because the typical size of a VCSEL is only 10 mm in diameter, it is very difficult to

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

n0 n1 n2 n1

0

λ/4n1

1

λ/4n2

2 3

2m-2 n1

2m-1

n2

2m

ns Figure 2.62 Schematic diagram of quarter-wave Bragg reflector incorporating bilayer structure. Each layer has refractive indices of n1 and n2 and thickness of l/4n1 and l/4n2, respectively.

use regrowth methods to form current confinement layers, which are widely employed in edge-emitting laser diodes. There are two typical methods to form current confinement structure in VCSELs. One is the ion implantation technique. Implanted proton and/or other ions such as oxygen have deep levels in the bandgap and create highly resistive regions through carrier trapping, thereby guiding the current away from the implanted regions because the current flow through the least

Light output I

Electrode

Upper reflector, R1 Current confinement layer L

QW

Active region Lower reflector, R2

I

Substrate Electrode

Figure 2.63 Schematic diagram of vertical cavity surface-emitting laser structure along with equicurrent lines and wave pattern.

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path of resistance. The other is the lateral oxidization technique made possible by high mole fraction Al-containing layers, such as AlAs, AlInAs, AlGaAs, or AlGaInP, which are formed near the active region. After etching a mesa structure, the Alcontaining layers are then selectively oxidized by wet-oxidization methods. The current is then funneled from the electrodes through the small unimplanted or unoxidized regions to achieve a high-injection current density the active region, which is what is needed for population inversion and thus gain. As touched upon above, vertical cavity surface-emitting lasers are well suited for potential general-purpose lighting, printer, and others due to their geometry and light emission properties. Even though, very early on, this configuration was considered [190] by some to take advantage of the columnar structure of heteroepitaxial nitrides, technological issues have slowed their developments. From a practical point of view, the challenge presented by the nitride vertical cavity emitters centers around the fabrication of a high-Q optical resonator, and the electrical injection approaches must incorporate lateral current spreading schemes due to primarily the low conductivity of the p-GaN and its alloys. Designs addressing current injection have been considered [191]. Figure 2.64 shows the basic features of a VCSEL. This particular species also sports a SiO2-defined current aperture, which is implemented in conjunction with lateral epitaxial overgrowth. Many dielectric stack applications used for conventional III–V VCSELs are not useful in nitrides because of short wavelengths involved. In principle, the choices are

Top electrode

DBR laser mirrors

p-type nitride Current blocking layer

Active quantum well gain medium Bottom electrode n-type nitride

Buffer layer

Substrate

Figure 2.64 Schematic drawing illustrating a possible blue VCSEL device structure, which features a buried dielectric DBR and current confining aperture. Courtesy of A. V. Nurmikko and Ref. [81].

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

then limited to large-bandgap dielectrics and stacks of in situ AlGaN/GaN DBR. However, due to the small index-of-refraction contrast, the AlGaN/GaN stack requires a large number of pairs. In spite of this, the epitaxial growth of such DBRs has been demonstrated [192–195]. Furthermore, vertical cavity or surface lasing under optical pumping, albeit intense, from GaN, InGaN MQW, or thin-film heterostructures that are encased by in-situ-grown AlGaN/GaN DBR reflectors has been reported [196, 197], but open to argument. Someya et al. [198] have shown stimulated emission employing a hybrid structure comprised of an in-situ-grown AlGaN/GaN DBR and a dielectric DBR. The all-dielectric mirror cavity has been very instrumental in vertical cavity nitride light emitter development. Microcavity resonators with quality factor Q approaching 1000 have been obtained [199]. In this approach, the sapphire substrate was separated by pulsed excimer laser ablation following the method reported by Kelly et al. [124, 125] and Wong et al. [200]. A specific process sequence was used to sandwich the InGaN/GaN/AlGaN heterostructure between two DBR stacks of SiO2/HfO2. It is well-known that smooth interfaces in the dielectric stack and semiconductor surface are required in any VCSEL. The requirement is even more stringent in the GaN system due to shorter wavelengths involved. A mean roughness of 2–3 nm over areas on the order of several hundred square microns is needed. In the absence of good morphology, lasing can be obtained but is readily dominated by in-plane-stimulated emission, which brings about a degree of automatic stringency concerning reports of vertical lasing. Optically pumped quasi-CW VCSEL operation has been achieved by 355 nm excitation to attain a predominant electron–hole pair generation in the InGaN QWs [81, 201]. This wavelength of excitation is just outside the reflectance band of the DBRs and slightly below the bandgap of the AlGaN cladding layer. The upper trace in Figure 2.65 presents the spontaneous emission spectrum at T ¼ 258 K for an average incident power of approximately 17 mW [201]. Several well-defined cavity modes are clearly seen with a typical modal linewidth of approximately 0.6 nm, which is limited by scattering from residual morphological roughness. The bottom trace in Figure 2.65 depicts the onset of stimulated emission at the threshold average input power of Pth ¼ 32 mW. Knowing the excitation density, carrier lifetime (assumed 1 ns), and assuming that the threshold conditions of optically pumped VCSELs, are to a first extent, similar to those in the CW edge-emitting diode lasers, one could estimate the equivalent electrical injection current. A well-defined far-field pattern was obtained, not shown, with a full width at half-maximum radiation angle of approximately 5 , for a linearly polarized nearly Gaussian beam emerging from a 20 mm diameter surface aperture. The large number of AlGaN/GaN layer pairs (>50) required to achieve the high reflectivities present considerable difficulties in that stringent thickness control must be in place in addition to the control of cracks and the DBR morphology, not to mention current conduction-related issues that complicate fabrication. The use of AlGaN interlayers turns out to be effective in controlling mismatch-induced stress and suppressing the formation of cracks that otherwise occur during growth

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Pexc = 0.6 Pth

Intensity (a.u.)

(a)

360

380

400

420

440

460

480

Wavelength (nm) (b)

Intensity (a.u.)

Pexc =1.25 Pth

∆ λ < 1Å

360

380

400

420

440

460

480

Wavelength (nm) Figure 2.65 (a) Spontaneous emission spectra of the optically pumped InGaN MQW VCSEL below threshold. (b) Stimulated emission spectra above threshold under quasi-CW pumping conditions at T ¼ 258 K [201].

of AlGaN directly upon GaN epilayers [202]. This can be accomplished during in-situ monitoring of AlGaN/GaN DBR mirrors and the stress evolution during growth. Incorporation of an AlN interlayer at the beginning of approximately 5 mm thick DBR region causes a sizable modification of the initial stress evolution. For example, tensile stress can be managed or nearly eliminated through multiple insertions of AlN interlayers. Using this technique, crack-free growth of 60 pairs of Al0.20Ga0.20N/GaN DBR mirrors has been achieved over the entire 2 in wafer with a maximum reflectivity of at least 99%. Even so, the associated spectral bandwidths are relatively narrow (10–20 nm). DBRs of the above-mentioned quality with a peak reflectivity R 0.991 have been used to demonstrate room-temperature quasi-continuous-wave operation of an optically pumped InxGa1
xN (x 0.03) MQW VCSEL at near l ¼ 383 nm [203]. The

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

Figure 2.66 Average input versus output power of a violet optically pumped VCSEL device. The inset shows the beam far-field (side) profile captured on a screen [203].

vertical cavity scheme featured a high-reflectivity in-situ-grown multilayer GaN/ Al0.25Ga0.75N and postgrowth dielectric SiO2/HfO2 DBR. Figure 2.66 shows the input/output power characteristics of a VCSEL with lasing threshold at a pump power of 30 mW. Average VCSEL output powers up to 3 mW were obtained. The inset of Figure 2.66 shows an image of the side profile of the low-divergence circular cross section beam. However, finite thickness variations across the wafer led to spectral shifts of the cavity modes relative to InGaN MQW gain spectrum, which in turn led to significant increases in threshold for devices fabricated from near the edge of the wafer. Accounting for the optical excitation volume, the fractional absorption of the pump, and using an electron–hole recombination time of approximately 0.5 ns, the device is estimated to be about 25% efficient with a threshold corresponding approximately to a carrier density of 1019 cm
3. Room-temperature lasing in an optically pumped 4l-InGaN VCSEL structures with both top and bottom dielectic DBRs has been reported at 401 nm by Tawara et al. [204]. The particular structure consisted of an Al0.07Ga0.93N layer with only three periods of a 5 nm thick In0.02Ga0.98N quntum well layer and top and bottom SiO2/ ZrO2 DBRs. This VCSEL structure exhibited a Q factor2) of 460 and a spontaneous

2) 2p times the ratio of the total energy stored divided by the energy lost in a single cycle, or radial frequency times the stored energy over dissipated power. Higher the Q value, lower the loss and narrower the bandwidth.

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Figure 2.67 Schematically cross section of VCSEL structure with much reduced optical threshold pumping energy. Courtesy of R. Butte.

emission factor b of about 10
2. Relatively low threshold of 5.1 mJ cm
2 was achieved by using a high-quality crystalline cavity and quantum well layers without surface roughening or cracking. However, the realization of these structures requires complex, laborious, and advanced processing steps that are not compatible with an electrical injection scheme. To avoid cracking of all-semiconductor DBRs, lattice-matched InAlN can be used in place of AlGaN. In this context, blue-violet lasing action at 422 nm under optical pumping with a low average threshold power density of 50 W cm
2 in a crack-free planar hybrid 5l/2 GaN microcavity containing three InGaN quantum wells with a bottom lattice-matched AlInN/GaN DBR and a top dielectric (SiO2/Si3N4) DBR were reported by Feltin et al. [205]. Figure 2.67 shows a schematic cross section of the VCSEL structure, which was grown by MOCVD on 2 in c-plane sapphire substrate. The cavity region has both n- and p-type regions as well as an AlGaN electronblocking layer on the p-side, making such a planar design in all points identical to that of a real structure ready to be processed for mesas and electrical contacts. A spontaneous emission coupling factor b 2  10
3 is derived from the input–output characteristics for this VCSEL structure. AlInN layer in the n-type cavity region can be subsequently oxidized to form current microapertures allowing access to high current densities (>20 kA cm
2). Figure 2.68a shows the variation of emission spectra for various pump powers at 0 . A clear threshold is observed at an average incident pump power of 1.4 mW. The spectral width of the emission below threshold is about 3.1 nm (Figure 2.68b), which is essentially due to cavity thickness disorder leading to a decrease of the effective Q factor with increasing spot size and with the measured cavity mode revealing the contributions of several narrow modes. Above threshold, the narrowest mode being about 0.37 nm wide is close to the spectral resolution limit of the system. Furthermore, InGaN/GaN LEDs using oxidized

2.14 Reflector Stacks and Vertical Cavity Surface-Emitting Lasers (VCSELs)

2.98

Energy (eV) 2.94 2.92

2.96

2.9

Output intensity (a.u.)

T = 300 K

10 10 10 10 10

6

4

2

0

Increasing pump power

-2

415

420

425

430

λ (nm)

(a)

2.98

2.96

Energy (eV) 2.94 2.92

2.9

T= 300 K

Intensity (×)

P = 1.8 mW

P= 0.3 mW (× 320)

415 (b)

420

425

430

λ (nm)

Figure 2.68 Semilogarithmic plot displaying RT emission spectra at pump powers ranging from 50 to 2 mW at 0 C, shifted for clarity, and linear plot showing two emission spectra (below and above threshold). (a) Variation of emission spectra for various pump powers at 0 C. (b) Spontaneous emission spectrum and VCSEL spectrum [205].

Al0.82In0.12N layer have been reported from the same group [206]. A current density of the order of 20 kA cm
2 has been achieved, a value that should fulfill the injection requirements for nitride-based VCSELs. Figure 2.69 shows the details and the electroluminescence image of the LED structures. A current density on the order of 20 kA cm
2 has been achieved, a value that should fulfill the injection requirements on nitride-based VCSELs.

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Figure 2.69 (a) Schematic cross section of the micro-LED structures, (b) optical microscope image of a 20 mm mesa with a 3 mm nonoxidized aperture (dark area), and (c) electroluminescence from a micro LED under forward bias (250 mA, 5.8 V) [206]. (Please find a color version of this figure on the color tables.)

2.15 Polariton Lasers

During the past decade, planar semiconductor MCs have attracted a good deal of attention owing to their potential to enhance and control the interaction between photons and excitons, which leads to cavity polaritons. The control of the aforementioned interaction is expected to lead to the realization of coherent optical sources such as polariton lasers, which are based on Bose–Einstein condensation (BEC)3) due to collective interaction of cavity polaritons with photon modes. In contrast with the bulk polariton, the cavity polariton has a quasi-two-dimensional nature with a finite energy at zero wave vector, ~ k ¼ 0, and is characterized by a very small

3) Using the atom metaphore, a BEC is a state of matter confined in an external potential and cooled to temperatures very near to absolute zero. Under such conditions, a large fraction of atoms collapse into the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale. This stemed from N. Bose sending a paper in

which he derived the Planck law for black body radiation by treating the photons and a gas of identical particles and Einstein generalizing Bose’s theory and predicting that at sufficiently low temperatures the particles would become locked together in the lowest quantum state of the system, Bose–Einstein condensation.

2.15 Polariton Lasers

in-plane effective mass. These characteristics lead to bosonic effects in MCs that cannot be achieved in bulk material. In particular, the large occupation number and Bose–Einstein [207, 208] condensation, or more strictly nonequilibrium polariton population, at the lower polariton branch can be accessible at densities well below the onset of exciton bleaching. This can potentially pave the way to ultralow-threshold polariton lasers. This feature is markedly different from that governing other lasers. Lasing in conventional lasers is predicated upon population inversion, which requires substantial pumping/carrier injection. In a microcavity system, however, the lasing condition is uniquely dependent only on the lifetime of the lower polariton ground state. This is expected to lead to extremely low threshold lasers, even when compared to the vertical cavity surface-emitting lasers (VCSELs). Planar MCs, also used in VCSELs, whose optical length (ml/2) is a half-integer multiple of the quantum well exciton transition wavelength (l) are highly suited for the manipulation of cavity polaritons [209]. The requisite high-reflectivity mirrors are formed by DBRs composed of alternating l/4 stacks of semiconductor and/or dielectric materials possessing high refractive index contrast. The vertical cavity causes photon quantization in the growth (vertical) direction. However, the in-plane photon states are unaffected by the cavity confinement. As a result, the dispersion of cavity photons is strongly modified relative to that of photons in free space. The exciton states are also quantized parallel to the growth direction with a continuum of the in-plane excitonic wave vector states. In the case of photonic states, the in-plane motion is free. Neglecting the effects of disorder, coupling between exciton and photon states can occur only for the same in-plane wave vector, and the coupled mode eigenstates are termed “cavity polaritons.” The energy splitting between the two coupled modes, called vacuum Rabi splitting (Wi) in analogy to the atom–cavity coupling in atomic physics, is typically of the order of 5 meV. The dispersion of cavity modes and that of the excition polariton that otherwise has no dispersion interact with each other, as shown in Figure 2.70, to form the Rabi splitting. To obtain a significant interaction between the cavity mode and excitonic states, the exciton energy is chosen to be at or close to the resonance frequency of the cavity mode. The coupling is determined by the exciton oscillator strength and the amplitude of the cavity field at the QW position. It is characterized by an energy equal to the vacuum Rabi splitting, which for QWs is close to the electric field antinodes and given by [167]

Wi  2
h

  2G0 cN qw 2 ; nc Leff

ð2:234Þ

hG0 is the radiative where Nqw is the number of quantum wells in the cavity. The term
linewidth of a free exciton, which can be expressed in terms of the exciton oscillator strength per unit area, fex, as
hG0 ¼

p e2
h f : nc 4pe0 me c ex

ð2:235Þ

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Figure 2.70 Schematic showing the interaction of cavity mode dispersion and exciton polariton dispersion leading to Rabi splitting. (Please find a color version of this figure on the color tables.)

In the strong coupling regime, where the vacuum Rabi splitting is greater than the linewidths of the cavity and exciton modes, the splitting can be measured in the optical spectrum where the two modes anticross. At resonance, the two polariton modes arise from symmetric and antisymmetric combinations of the exciton and cavity modes. Only the cavity mode couples directly to the external photons away from resonance, and the excitonic mode becomes weak. Figure 2.71 shows a typical 20

Peak position detuning (meV)

15 10 5 0 -5 -10 -15 -15

-10

-5

0

5

10

15

20

Cavity detuning (meV) Figure 2.71 Reflectivity peak positions as a function of cavity detuning for a MC sample with five QWs at T ¼ 5 K. The theoretical fit is obtained through a standard multiple interference analysis of the DBR Fabry–Perot QW structure [210].

2.15 Polariton Lasers

Ep

Polariton–polariton scattering

∆k

Ωi N0

Thermal lock

Nx k//

Figure 2.72 Dispersion relations of upper and lower polaritons, showing the dominant pair scattering of polaritons feeding energy into the trap at k ¼ 0 [211].

anticrossing behavior [210]. Equation 2.234 is accurate only for the vacuum Rabi splitting much greater than the linewidths of the exciton and cavity modes. The basic principle of a polariton laser is depicted in Figure 2.72 [211], which is based on the dispersion curve of the lower branch exciton polaritons. The strong coupling regime creates a trap containing a small number of polariton states at energies below all the other states in the semiconductor. This polariton trap is sharp with a depth equal to nearly half the splitting Wi between the two polariton modes. Polaritons in the trap have properties suitable for BEC of exciton polaritons. Recombination from this state in the BEC regime is coherent, monochromatic, and sharply directed, representing the characteristics of laser emission. The relaxation of polaritons into the ~ k ¼ 0 state is stimulated if the population of the final state is >1 [212]. This lasing process is fundamentally different from that for conventional lasing. The lasing threshold in conventional lasers depends on population inversion, which balances absorption (loss) by stimulated emission. In polariton lasers, however, the threshold is dependent only on the lifetime of lower polariton ground state. When the relaxation to the ground state of the trap is faster than the radiative recombination from this state, optical amplification occurs. The absorption and reabsorption of light are already taken into account within the polariton picture, and inversion of population is not required for lasing. When Wi is significantly greater than both the exciton and the cavity linewidths, the system is in the strong coupling regime (Figure 2.70). This strong coupling regime is completely different from that for excitons in isolated QWs not embedded in an MC. In the case of QWs without the MC, excitons decay irreversibly into the continuum of photon states along the growth direction due to the discontinuity of translational symmetry along the growth direction. Thus, excitons in QWs would have a finite lifetime ( 20 ps) [213]. The photon states in MCs, which are quantized along the growth direction (along the cavity), and each of the in-plane exciton states can only couple to one external photon state. Quasistationary eigenstates exist as the cavity polaritons, and irreversible exciton decay does not arise. In the time domain,

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excitations oscillate between the two modes on a subpicosecond timescale with a period h/Wi before the leakage of photons from the cavity. Bulk polaritons and cavity polaritons differ drastically. In bulk, photon dispersion is linear (w ¼ kc/n), where n is the refractive index of the material. The splitting between the upper and lower polariton branches at resonance is around 16 meV in GaAs. Although this is larger than that in MCs, it is difficult to probe the properties of exciton–polaritons in bulk materials due to the photon–polariton conversion process at the air–semiconductor interface. Unlike the bulk case, in MCs the polaritons are converted directly into external photons as a result of the finite lifetime of their photon component within the cavity, and propagation to the surface is not required. Moreover, since the wave vector perpendicular to the sample surface is quantized, the complications due to spatial dispersion along the growth direction are eliminated, and there is a one-to-one correspondence between the internal polariton states and the external photons. This allows the polariton dispersion curves to be measured easily in reflectivity or PL experiments. A cavity polariton has a quasi-2D nature with a finite energy at zero wave vector (~ k ¼ 0) and is characterized by a very small in-plane effective mass (of order 10
5 me) that gives rise to narrowing of the linewidth. This feature is due to the half-photon and the half-exciton character of the polaritons, the photon fraction having very strong dispersion. Polariton lasing at room temperature in bulk GaN-based microcavities in the strong coupling regime has been observed under nonresonant pulsed optical pumping [214]. Figure 2.73 shows the integrated output intensity collected at normal incidence for aforementioned condition. A clear nonlinear behavior can be noted for the emission at l  365 nm, with an increase of over 103 at the critical threshold of around Ith ¼ 1.0 mW. This corresponds to a density of N3D 2.2  1018 cm
3, which is an order of magnitude below the Mott density 1–2  1019 cm
3 in GaN at 300 K. The observation of a low-threshold coherent emission, which is an order of magnitude smaller than in previously reported nitride-based VCSELs, and the emision line blueshift due to polariton–polariton interaction demonstrate the first roomtemperature polariton lasing.

2.16 GaInNAs Quaternary Infrared Lasers

Even when a small amount of N (As) is incorporated into GaAs (GaN) lattices, respectively, a large negative bandgap bowing parameter results. Consequently, very small amounts of N in the GaAs or InGaN lattice pave the way for its bandgap to be made very small. The technologically paramount aspect of this is that 1.3 and 1.5 mm lasers can all be built on GaAs substrates, which is more advanced that the InP substrates, which is normally utilized for this wavelength range. In short, the quaternary GaInNAs are emerging as a significant competitor to the more established GaInAsP material system. Anomalously, large bandgap bowing parameters exhibited by GaAsN and GaNAs are caused by large chemical and size differences between As and N as discussed in Volume 1, Chapter 2.

2.16 GaInNAs Quaternary Infrared Lasers L3

10

L2

L1

9

PL (a.u.)

T=300K

10

7

10

5

10

3

10

1

10

-1

3.40

3.46

Energy (eV) Figure 2.73 Emission spectra at pump powers from 20 mW to 2 mW at 0 C. Spectra integrated over 10 ms show multiple emission line [214]. (Please find a color version of this figure on the color tables.)

Semiconductor lasers in the 1.3–1.5 mm range form the optical source backbone in fiber optics-based telecommunication systems. The notable one is based on the GaInAsP/InP quantum well distributed feedback Bragg-grating (DFB) edge emitters with wavelength stability congruous with stringent demands imposed on the carrier. However, there is also a substantial developmental activity in the areas of VCSELs. Gain-switched DFB lasers operate in commercial systems at data rates of 10 GB s
1 and are expected to dominate even the imminent 40 GB s
1 transmission links at least for a while [81]. Among the weaknesses of the GaInAsP QW DFB lasers, albeit few, is their temperature stability, which is attributed in part to the small conduction band offsets in the GaInAsP/InP heterostructures causing carrier leakage. Clever approaches are in progress to circumvent this problem. VCSELs bring into the equation an additional challenge/complication that concerns the small index-ofrefraction contrast for GaInAsP/InP-based DBR mirrors. A lack of such disparity in refractive indices requires a very large number of layer pairs, which inherently introduces excess electrical resistance [81]. The above-mentioned benefits of the GaxIn1
xN1
yAsy quaternary over the more established infrared lasers based on the InGaAsP system have been recognized for

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quite some time [215]. The large electronegativity of nitrogen in the fundamental sense suggests that the conduction band offsets on the order of 500 meV could be achieved for y 0.01 and x 0.7. What is more is that pseudomorphic growth of GaInNAs on GaAs has been shown to be possible in this composition range with critical thickness on the order of 10 nm for the compressively strained QWs, thereby providing for an active medium for an l 1.3 mm laser with strong carrier confinement that can be maintained even at high temperatures [81]. The GaInNAs ternary infrared lasers have been successfully grown by both molecular-beam (MBE) and OMVPE epitaxies on GaAs substrates, in that chronological order. This statement is all relative in that doping considerations require postgrowth anneal, and its both beneficial and adverse impact, the latter regarding the compositional and thus the wavelength change, particularly in the longer wavelength end of the window. Initial growth experiments showed how the wavelength of spontaneous emission could be controlled by changing the concentration of In and N or both. However, with increasing In concentration, the compressive QW strain increases and thus the critical thickness of the QW decreases, creating a special challenge for reaching the 1.5 mm communication wavelengths [81]. The strategy for encountering the increase in compressive strain by increasing the nitrogen content has the problem that the GaInNAs/GaAs QW luminescence intensity decreases and its spectral linewidth increases dramatically, when compared, for example, to GaInAsP. This leads to large increases in the laser threshold current density due to broadening of the gain spectrum, a menace for violet lasers also. Evidently, incorporation of nitrogen into InGaAs leads to pronounced alloy fluctuations beyond the random alloy regime. Depending on growth conditions, such fluctuations may be severe enough to cause pronounced segregation into In- and N-rich regions. The postgrowth thermal annealing can significantly aid in improving the In–Ga interdiffusion and is commonly employed [216], but the wavelength shift and reduced donor activation issues come to being. Edge-emitting lasers including the VCSEL varieties based on the InGaAs system with dilute amounts of N in GaInNAs/GaAs QWs have been demonstrated. The threshold current density has been seeing a continuous improvement as reported by many groups. By the way of an example, the low threshold current density Jth ¼ 270 A cm
2 achieved by solid source MBE growth [217] and the Jth ¼ 450 A cm
2 for approximately 1.3 mm devices grown by OMVPE [218] must be noted here. A large variation in the threshold current and output powers can be found in the literature, depending on the exact wavelength and device configuration, such as the number of quantum wells, optical confinement scheme, and desired output power range. Generally, the longer the wavelengths, the more difficult the task is. Difficulties notwithstanding, edge-emitting lasers up to 1.52 mm have been demonstrated on GaAs substrates. This bodes well for this dilute quaternary system despite the large threshold current density [219]. Compared to the GaInAsP-based VCSELs, the GaInNAs-based system offers an immediate fabrication advantage. The well-established and high index contrast GaAs/AlGaAs multilayer DBR reflectors can be readily incorporated during in-situ growth. Likewise, the resonant cavity photodiodes that have originally been developed

2.17 Laser Degradation

in the GaAs system [220] are actively pursued. Following initial work with optically pumped structures, which gets one around the electrical injection-related complexities that are very beneficial in the early stages of development, electrical injection of a 1.3 mm GaInAsP/GaAs MQW VCSEL was achieved [221]. Reasonably low threshold VCSELs operative at 305 nm have been produced in OMVPE-grown layers [222]. In this particular investigation, the InGaAsN 3QW VCSEL structure was grown with a Sidoped, 40-pair bottom DBR, l-cavity, and a top 28-pair C-doped DBR, including an oxidation layer for current confinement. Room-temperature CW operation of 1.305 mm VCSELs is at a threshold current density of Jth ¼ 5.3 kA cm
2. At higher injection levels, a maximum power of 0.7 mW at room temperature was also obtained. Room-temperature CW lasing was achieved at wavelengths as long as 1312 nm, with somewhat lower output power [81].

2.17 Laser Degradation

A mention was made that with combination of ELO, HVPE, and OMVPE-grown layer structures, the projected laser operation lifetimes are in excess of 15 000 h, albeit at a low power of 30 mW at 60  C. At higher powers such as 60 mW and higher case temperatures, which are imperative for many applications, however, the operating lifetime of the lasers grown on just ELO or combinations of ELO material is not good enough. For these reasons, as should have been expected, the laser community moved aggressively toward the use of what we can term as pseudo-GaN substrates. These are the freestanding GaN wafers grown by a fast growth technique such as HVPE. In time, however, other bulk growth methods such as ammonothermal and growth from liquid may take over. In this section, we will discuss first the degradation of lasers prepared in the prepseudo-GaN sunstrate era followed by the degradation analysis of lasers grown on pseudoGaN wafers. The degradation characteristics of CW InGaN multiple-quantum well laser diodes on sapphire substrates have been reported by Kneissl et al. [223] in structures containing a TEM determined defect density 5  107 cm
2 with longevities nowhere near the record figures mentioned. However, the intent here is to investigate the degradation mechanism. From the temperature dependence of the laser diode lifetimes, activation energy of 0.50  0.05 eV was determined. Similar investigations on SiC substrates have also been conducted [224]. Unlike the sapphire substrates, use of SiC substrates enables a vertical current path, cleaved facets, and excellent heat spreading. The temperature rise during CW operation is measured for different mountings in an effort to determine the thermal resistance. A p-side upmounted diode exhibited a thermal resistance of 18 K W
1 and a 143 h CW lasing at 1 mW optical power (T ¼ 25  C). DC and pulsed aging shows the current flow through the device as being the main degradation source compared to heat for InGaN-LDs on SiC substrates. The same may not be said of sapphire due to low difficulties associated with heat spreading.

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For the 2 mm  800 mm ridge waveguide laser diode on sapphire, the increase in threshold current corresponds to an increase in p–n-junction temperature of about 18  C when operated under CW conditions. This is in good agreement with the temperature increase determined from the shift in emission wavelength. For the total power dissipated under CW operation condition, this change in device temperature corresponds to a thermal resistance of 23 K W
1, which is rather remarkable compared to the above-mentioned SiC substrate case, and speaks to the quality of packaging for heat dissipation. For the lifetime testing, the output optical power was kept at either 2 or 5 mW, and as much current as necessary was allowed to flow to maintain a constant optical power. For the first 13 h, the operating current increased almost linearly and then started to degrade at a faster rate, while the device continued to lase after more than 15 h of operation [223]. The lifetime in the measurement is operationally defined as the elapsed time until the laser operating current for the constant output power has increased by 50% from its initial value. The measured lifetimes for two output powers of 2 and 5 mW versus the p–n-junction temperature are plotted in Figure 2.74 Joule heating of the device was taken into account and the actual p–n-junction temperature was calculated from the thermal resistance, the respective operating current and voltage of the device averaged over the time of operation, and the heat sink

100

Pout = 2 mW Pout = 5 mW

200

Current (mA)

Lifetime (h)

10

1

150 100

2 mW, 20 ºC

50 0

0

5

10

15

Time (h)

0.1 2.7

2.8

2.9

3.0

3.1

3.2

3.3 -1

p–n- junction temperature 1000/T (K ) Figure 2.74 Laser diode lifetimes for constant power output levels of 2 and 5 mW, as defined when the current needed to maintain the constant output power increased by 50%, versus the junction temperature. The junction temperature was determined from the thermal resistance of the packaged device that in turn was determined from the wavelength shift by applied electrical power to the device [223].

3.4

2.17 Laser Degradation

temperature. The degradation rate can be fit exponentially on the reciprocal temperature as. L(T) ¼ eE A =kT ;

ð2:236Þ

where EA is the activation energy, T is the device temperature, and k is the Boltzmann constant. As can be seen from Figure 2.74, this empirical formula describes the measured temperature dependence quite well and through a least square fitting, activation energies of EA ¼ 0.50  0.05 eV and 0.46  0.04 eV can be derived for 2 and 5 mW output powers, respectively. These activation energies are similar to those exhibited by the Nichia chemical laser exhibiting lifetimes of several 1000 h. Kneissl et al. [223] argued that although the activation energy did not seem to be strongly dependent on the light output power of the laser diodes, the overall laser diode lifetime decreased significantly with increasing light output. This reduced lifetime when operated at higher output power levels, however, cannot be merely attributed to a temperature increase. The increase in electric power dissipation, when the laser light output is changed from 2 to 5 mW, results only in a p–n-junction temperature increase of 1 K. This temperature increase is much too small to explain the drastic drop in lifetime for the higher output power. Therefore, the degradation mechanism is not only thermally induced but also photon assisted. Returning to the diodes of K€ ummler et al. [224], which were prepared on SiC substrates without the ELO techniques and consisted of n/p-AlGaN cladding, n/p-GaN waveguide, and three In0.10Ga0.9N/GaN multiple quantum wells with an Al0.2Ga0.8N e-blocking layer, the lifetime testing was very similar to the abovementioned lasers on sapphire substrates except a 1 mW optical output power was maintained. During 143 h of operation, which is the lifetime, the current density rises from 6.5 to about 20 kA cm
2 causing the junction temperature to reach 80  C, a figure calculated using the electrical input power and thermal resistance Rth ¼ 18 K W
1. If one assumes that the degradation of the LDs is caused by a thermally activated mechanism such as Mg diffusion, then the degradation should be sensitive to heat. To distinguish between the heat and the current as to the cause of degradation, K€ ummler et al. [224] investigated several comparable LDs but with a much higher thermal resistance, Rth ¼ 55 K W
1. These LDs were then aged at a constant ambient temperature of T ¼ 25  C. The junction temperature reached more than 100 and 200  C for dc current density of j ¼ 6.7 kA cm
2 and 13.3 kA cm
2 electrical drives, respectively. In addition, pulsed aging with conditions of j ¼ 13.3 kA ¼ cm2; duty cycle 1/30 and 1 ms pulse time were applied. Shorter duty cycles had no effect on the threshold current and slope efficiency. Aging was stopped in certain intervals to measure pulsed I–V and L–I curves. Figure 2.75 shows the change in the threshold current obtained from the aforementioned pulsed measurements. The curve associated with the pulsed aging at 13.3 kA cm
2 follows clearly the CW aging curve with the same current density, even though the temperatures are quite different (TCW > 200  C and Tpuls  25  C).

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Figure 2.75 Change in the threshold current over the period of aging for CW (broken lines) and pulsed operation (solid lines). The time for pulsed operation is corrected for the duty cycle and the threshold current was determined from pulsed measurements [224].

The conclusion that can be drawn is that the junction temperature is not the only parameter governing the degradation process. This means that the thermally activated diffusion is of not major consequence. The junction temperature is only important because higher temperatures require higher threshold currents and cause lower slope efficiencies that culminate in requiring larger operation currents for the same optical output power. Higher currents lead to faster aging as can be seen in the faster degradation at twice the current density. This is consistent with the fact that we measured an exponential increase of LD lifetimes for lower electrical input power (averaged over lifetime). The common thread between the two case studies, albeit not extensive and not performed in state of the art devices, is that at least in these less than par devices, thermal issues are not the main degradation drivers. Let us now turn our attention to the discussion of laser degradation in the era of pseudo-GaN substrates. As a prelude to the section, let us briefly review the saga of GaN-based laser development. Soon after the first report of the blue LD, GaN-based LDs became commercially available and are currently used in Blu-ray DVD technologies for optical storage and recording and reading, in projection TVs, in and laser printers. GaN-based LDs that use InGaN or AlInGaN alloys show good performance with lifetimes in excess of 10 000 h (as long as 15 000 h). However, device reliability remained as one of the main concerns in the industrial circles. The device lifetime depends on the dislocation density. In the early stages, large amounts of threading dislocations (109
10 cm
2) were present due to the lattice mismatch between GaN and sapphire substrates. Having semiconductor lasers lasting even long enough in the laboratory to be measured with this high a dislocation density is a tribute to the low dislocation mobility in GaN, some 14 times smaller than in GaN, for example. Nevertheless, even with relatively small mobility dislocations are a lifetime killer, and efforts have been underway to reduce them. One obvious method, of course, which was not available early on, is to use GaN substrates that are now standard substrates

2.17 Laser Degradation

for GaN-based LDs. As a result, less than 1  10
6 cm
2 dislocation densities were achieved. To realize long-lived and high-power GaN-based LDs, it is also important to reduce planar defects and the associated dislocations that may be formed at the interfaces of InGaN QWs. In the postheteoepitaxy era in which the laser structures were grown on sapphire or SiC substrates, and in the area of pseudo-GaN substrates, the reliability of lasers can be divided into two categories, facet degradation and degradation due to material properties. In terms of the latter, the formation of In–In bonds at the interface between the InGaN QW and the barrier layers, which result in inversion domains (IDs), has to be suppressed using either AlGaN alloy or GaN as barrier layers instead of InGaN [225]. There are still compositional inhomogeneities and the notorious V-shaped defects introduced in the MQW whose main origin lies with dislocations extending to the quantum well region [226]. These V-shaped defects have been reported to take the form of an open hexagonal, inverted pyramid with {1 0 1 1} sidewalls [227]. Other defects not emanating from the dislocations and running upwards toward episurface have been observed in conjunction with quantum well laser structures. These defects have combined the planar defect and the dislocation nature, which provides the impetus for Tomiya et al. [225] to call them “multiple defects.” While the exact size of these multiple defects depends on several parameters including the growth conditions, they can range from 50 to 200 nm, as shown in Figure 2.76. Defects on the order of 4–6 are generated from the boundary of edges of each planar defect. Weak beam (11
2 0) dark field images of these planar defects indicate that they consist of columnar subgrains that are twisted slightly with respect to one another. Furthermore, they are also twisted in relation to the surrounding normal matrix. This twisting creates the a-type dislocations from the boundaries of the planar defects. High-resolution lattice images indicate that these planar defects are formed at the interface between the quantum well region and the barrier layers. Conversion beam electron diffraction studies indicated the polarity to be opposite of the surrounding matrix, which means that they represent inversion domains. Let us now delve into the formation mechanism of these planar defects. The presence of In-rich precipitates observed in the vicinity of multiple defects mitigates the planar defect formation, as shown in Figure 2.77. Recall that the stacking sequence of the closed packed (0 0 0 1) plane of the wurtzite structure is a-A-b-Ba-A-b-B-. If this stacking sequence along the c-axis is a-A-b-B-a-A-b-B-B-b-A-a-B-b, it would represent IDs. Since the In-rich precipitates nucleate near the multiple defects, excess B–B bonds are thought to be In–In. Therefore, Tomiya et al. [225] speculate that the ID defects form due to excess In–In bonds at the interface between the quantum wells and the barrier layers. The need to suppress the formation of metal–metal bonds, particularly at the interface between the quantum well region and the barrier, is obvious. As a result of these measures, such as using GaN substrates and improved growth conditions, the defect levels are reduced and the laser degradation is now taking a route similar to other semiconductors before GaN and focusing on intrinsic degradation of the material such as production of nonradiative recombination centers and dark line defects (DLDs), and laser facet damage. Pertinent to power

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Figure 2.76 Plane view image of multiple defects. (a) Crosssectional TEM h11
0 0i images of the multiple defects with g ¼ 0 0 0 2 and (b) g ¼ 1
1
2 0 where the planar defects observed in the quantum well region are indicated by P and the dislocation nucleating from these planar defects are depicted by D (c). Courtesy of S. Tomiya, Sony Corporation.

2.17 Laser Degradation

Figure 2.77 Cross-sectional bright field TEM image of the multiple defects where the arrows point to the In-rich precipitates. Note that the planar defects and the dislocations are out of contrast. Furthermore, the planar defects are indicated by the dotted lines. Courtesy of S. Tomiya, Sony Corporation.

extraction from edge-emitting lasers one facet, which is called the back facet, is coated with high reflection coating to prevent light emission that is wasted. The other facet, the front facet, might be coated with an antireflection coating to the extent desired for power extraction. To increase power extraction, termed as the edge loss, the laser cavity length can also be made shorter, which requires higher levels of pumping as in the case of partial antireflection coating. The role of dislocations deserves further elaboration in that while their mobility is low, dislocations, even the fully coordinated ones, introduce states within the bandgap. The low dislocation mobility can be used to argue against degradation related to the dislocation multiplication and glide, dubbed DLD [228, 229] in lasers. However, dislocations introduce inhomogeneous local strain that in turn could, under severe demanding device operation, pave the way for generation and then multiplication of nonradiative point defects. Mg dopant used for p-type AlGaN and GaN would not only diffuse, but could also be responsible for nonradiative recombination centers and/or at least making the generation of nonradiative recombination centers more plausible. Although not yet fully identified, diffusion of Mg and native defects could be responsible for laser degradation [230, 231]. There is evidence that degradation in GaN-based lasers is a thermally activated process with a characteristic activation energy of 0.32–0.81 eV [223, 232, 233]. Occurrence of sudden failure with reduction of cavity length in high-power GaNbased laser diodes is a serious issue related to device reliability, as shown in Figure 2.78a. These failures are in part caused by excess heating of the front facet that has its origin in increasing capture cross section of nonradiative recombination centers by the diffusion of point defects/impurities (such as Mg dopants) during laser operation. Employing a current injection-free region near the facet, typically called the noninjected facet (NIF), by as long as 45 mm, formed simply by leaving out of the p-electrode metal near the laser facet, significantly reduces the catastrophic optical damage [225, 234] and increases the lifetime, as shown in Figure 2.78b, which is estimated to be more than 10 000 h under 0.75 W (reduced to 0.65 Wafter 700 h under 6.2 kA cm
2 drive current) CW operation at room temperature [225]. The premature degradation has also been attributed to the photon-mitigated carbon deposition on the dielectric stack on the front facet [233]. The deposited carbon is thought to have its origin in residual organic materials with C–H bonds. When the residual carbon contamination has been successfully removed by plasma

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0.8

25 oC, CW, ACC Output power (W)

Output power (W)

0.8 0.6 0.4 0.2 0

0

200

400

600

800

1000

25 oC, CW, ACC

0.6 0.4 0.2 0

0

Aging duration (h) (a)

200

400

600

800

1000

Aging duration (h) (b)

Figure 2.78 Aging test results of (a) laser diodes with conventional structure and (b) with current injection-free structure. Courtesy of S. Tomiya, Sony Corporation.

cleaning just before cap welding, the improved lifetime of the plasma-cleaned laser diode packed with argon gas exceeded 2000 h under 90 mW CW operation at 60  C [233]. Kim et al. [233] undertook an extensive study of facet damage, as mentioned caused by C contamination, by employing field emission SEM (FESEM) and field emission Auger electron spectroscopy (FEAES) to examine the front facet after aging, as shown in Figure 2.79, for two tested LDs where the dotted lines outline the laser ridge for guide to the eye. During the aging tests, the CW output power of two LDs, LD1 and LD2, was kept at 40 mW and the current increase rate was observed to be much larger in LD1, where the damage was more drastic. The FESEM image of LD1 shows extensive damage to the antireflection (AR) facet, the emitting surface (see top row in Figure 2.79). The oval-shaped damage developed during aging resembles the near-field emission pattern whose center is just below the ridge bottom. The FEAES scans undertaken by Kim et al. [233] indicated that the damage is caused by the presence of C that covered the dielectric facet coating whose shape is similar to that of the near-field pattern as shown in Figure 2.79 (middle row). Therefore, it is fair to conclude that C deposit was induced by the emitted laser light during the aging test and caused increased absorption of photons. Interestingly, C deposited less in the core area of the oval-shaped damage where the optical density of the emitted light is higher than in the areas near the circumference. In addition, the FESEM image of LD2 in Figure 2.79 showed layer peeling, similar to that observed by Marona et al. [234]. On the exposed surfaces, GaN, oxide, and carbon can be seen. Where the GaN is seen, the oxide and the carbon layers had peeled of, and thus C and O elements were not detected in the FEAES analysis. Some areas are covered with C, and in other regions the C layer peeled off. The peeling of the carbon layer is associated with the recovery of the optical properties. LD2 sample showed the partial recovery around 100 h during the lifetime test. The results indicate that the initial degradation due to the carbon deposition is partially recovered in LD2 by peeling off the carbon layer. The results lead to the conclusion that the premature degradation

2.17 Laser Degradation

Figure 2.79 FESEM and FEAES images based on carbon and oxygen mapping for two LD samples. Courtesy of C. C. Kim, L.G. Electronics. (Please find a color version of this figure on the color tables.)

during the early stages of life tests is due to the photon-induced carbon buildup on the light-emitting facet. As already mentioned, the use of the carbon removing process resulted in great improvement in the lifetime of the LDs. The mean time to failure using APC mode life testing at 160 mW under pulsed operation (pulse width ¼ 2 ns, duty cycle ¼ 50%) was estimated as 2240 and 980 h at 60 and 70  C, respectively. The failure here was defined as a 30% increase in operating current. To delineate degradation caused by facet damage and nonradiative processes, Marona et al. [234] undertook a study in which the operation current and emitted optical power, being as the most sensitive from the point of view of the lifetime of the device, were both systematically varied. The threshold current and voltage, differential efficiency, and its characteristic temperature T0 were monitored with

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the conclusion that facet and near-facet degradation, bulk degradation, and quite rarely, contact degradation (attributed to processing flaws) occurred. In a large percentage of devices, high temperature near the facets was noted to be accompanied by the SiO2/TiO2 coating delamination or large catastrophic degradation within the stripe area adjacent to the facet the optical origin of which is now deemed highphoton density related as this type of damage occurs also after testing the device just below the lasing threshold, consistent with other observation [235]. Returning to the NIF, the unpumped region of the waveguide should, in principle, increase the optical loss and thus the threshold current [234]. However, the threshold current was not noted to significantly increase, in part because of the compensating effect of decreasing pumped area and pertinent to reliability possibly reduced heating and resultant minimization of defect generation. Inclusion of a nonpumped region certainly reduces the temperature of the near-facet region. Temperature reduction could aid in facet damage as well as reducing the temperature-induced degradation of the semiconductor. In most cases, the degradation was found to manifest itself mainly by an increase in the threshold current through a square root dependence on aging time, suggestive of the diffusion process being responsible for the damage [225, 234]. The diffusion mitigated failure process may be driven by current itself, current-induced increase of the junction temperature, and optical field or by all these factors together. One of the best methods is to perform the aging test below and above lasing threshold to delineate the effect of optical field from other processes. Doing so led to the conclusion that the optical field does not influence the degradation rate [234]. The optical field being out of the way, performing measurements of the degradation rate as a function of the operating current could shed light on the effect of current and related heat. The time derivative of the evolution of the threshold current density can be used as a measure of the degradation rate. A strong, nearly exponential, dependence of the so defined degradation speed on the operation current has been observed [234]. The results can be interpreted by assuming a normal Joule heating, which can be described as    
E A
E A D ¼ C exp or D ¼ C exp ; ð2:237Þ l(T RT þ a1 I) l(T RT þ a2 P) where D is degradation rate, C the preexponential constant, EA the characteristic activation energy of the degradation process, TRT is room temperature, P is the power dissipation, I is the current, a2 is constant connected with thermal resistance, which is on the order of 12 K W
1 (a1 is similar to a2 in that it is tied to current as opposed to the power). Fitting Equation 2.237 to the experimental data, the value of the a1 coefficient of 200 K A
1 and an activation energy EA of 0.42 eV have been deduced [234]. The good fit lends support that the current-induced increase of the junction temperature is indeed one of the driving forces for the degradation. Let us now turn our attention to aging-related increase of the nonradiative recombination in the active area and increase of the leakage current. Here the word “leakage” is used, as in any optical device, to represent the component of the current flowing through the device but not contributing to radiative recombination. For example, carriers escaping the quantum wells before recombination fall into this

2.17 Laser Degradation

category. To decide which process plays a more important role in degradation, a series of experiments, namely, on cathodoluminescence should be performed for observing spots of nonradiative recombination in the active region and the characteristic temperature T0 in the threshold equation:   T J th ¼ J 0 exp : ð2:238Þ T0 Basically, the CL signal of aged devices showed a decrease of 5–30% as compared to virgin control devices. The obtained results might be susceptible to substantially large errors due to, for example, the compositional inhomogeneities. Nevertheless, the decrease of CL intensity (never increase) statistically confirms the appearance of nonradiative centers in the active area of these devices. The contrast changes in the CL images are typically uniformly distributed over the entire stripe area, which represents a notable deviation from that observed in the GaAs/AlGaAs lasers [228, 229] in which the degradation is manifested through the appearance of black dots or lines (dark lines) in the CL microphotographs. However, in interpreting the aforementioned differences, one should keep in mind the relatively much shorter minority carrier diffusion length in InGaN, which may have a bearing in the resulting CL images. Correlation of the slope efficiency (the slope of the I–L curve) to the threshold current is also illuminating, as is shown in Figure 2.80. A positive correlation would indicate the increase of nonradiative recombination as the main cause of degradation. A strong correlation has been observed in some cases, but in other cases, no such correlation has been noted [234]. A perplexing behavior can be explained by the formation of a parallel path, or leakage current, for the current flow. Due to the formation of a parallel path with the aging time, increasingly larger portion of the 0.30

Slope efficiency (W A-1)

0.28 0.26 0.24 0.22 0.20 600

700

800

900

Ith (mA) Figure 2.80 Correlation between slope efficiency (W A
1) and threshold current (Ith) after aging. Courtesy of P. Perlin.

1000

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current flows without participating in the radiative process meaning without feeding the lasing processes. As mentioned, this effect may be also treated in the context of current leakage (vertical, lateral, or facet related). The fact that the slope efficiency, in some cases, remains constant during aging, and, in other cases, is nearly proportional to the change of the threshold current, indicates that both mechanisms (leakage and nonradiative recombination) indeed contribute to varying degrees to this process. Leakage current is typically reflected in the magnitude of the characteristic temperature
T0 expressed in Equation 2.238. A higher leakage current would imply a lower value of T0. T0 is a measure of the temperature dependence of the threshold current through the exponent (see Equation 2.238) In nitride-based semiconductor lasers, the typical reported values of T0 are between 80 and 235 K [223, 236], but there are also reports of anomalous values of T0 [237], including negative values [238]. The negative value [238] has been attributed either to the anomalous temperature dependence of the carrier capture rates or the increasing temperature facilitating a relatively more homogeneous hole distribution through the multiquantum wells. The holes, having an effective relative mass of two, need additional thermal energy to better overcome the barriers. The characteristic temperature measured before, during, and after the aging procedure [234] is shown in Figure 2.81. The initial value of T0 ¼ 274 K increased by a factor of 4 after 45 h aging. For room-temperature aging the threshold current changed significantly, but for higher temperature aging ( 80  C) the increase was relatively small. The higher temperature behavior can be explained by the formation or increase of a thermal barrier for holes. The barrier can be the quantum barriers or, for example, that introduced by electron-blocking layer in the valence band. It might be prudent to consider that the large value of the T0 parameter does not necessarily imply a good confinement of carriers within QWs. 1200

T0 (K)

900

600

300

0

0

15

30

45

Time (h) Figure 2.81 Characteristic temperature of laser diodes (T0) as a function of aging time. Courtesy of P. Perlin.

2.18 Applications of GaN-Based Lasers to DVDs

To summarize the laser degradation in the era of pseudo-GaN substrates, in terms of the materials issues affecting the laser reliability, it now appears that the degradation is related to multiple defects formed at the interface between the quantum wells and the barrier, and generation of nonradiative recombination centers within the quantum wells. Furthermore, the facet damage that occurs can be mitigated to some extent by introducing a noninjection stripe near the front facet. The material/device designs include mainly the electron leakage (facilitated in part by large effective mass of holes) and processing details to either prevent or remove C deposits on the facets.

2.18 Applications of GaN-Based Lasers to DVDs

As discussed in the introductory section of this chapter, one of the driving forces for the development of what is generically referred to as blue lasers, the majority of which emit in the violet region of the visible spectrum, is the development of optical head for read–write systems in DVD applications. Following respectable CW operation with acceptable longevity, the next stage of effort in the laser development included addressing the requirements by the DVD technology. The imminent DVD system represents the third generation digital video recording (DVR) system after the ones utilizing first IR and later red laser diodes. Suffice it to say, it takes more than just developing the laser and heading to go with it. Issues such as plastic lens development, which are harder to address for shorter wavelengths, the overall systems integration, and error correction schemes must be addressed. As exemplified by the nitride technology as a whole, great progress has been made in a relatively short time in an effort to eventually insert GaN-based violet lasers into the next generation of DVD technology. In this vein, the next generation high-density optical disk systems have seen great progress [239–241]. The system that is under development in many laboratories, including Sony, Philips and Matsushita, goes with the name Blu-ray disk system. This system should be in market, shortly after this book manuscript appears in print, as a digital hi-vision video recorder. The features are a phase-change highdensity optical disk system with a blue-violet LD and a 0.85 numerical-aperture objective lens. High-density capability of this system lends itself to somewhat conventional optical disk systems such as DVD or CD, and new application areas such as various mobile products including camcorders and notebook computers. Even though the imminent optical systems [242] are directed toward about 25 Gb level recording in 120 mm diameter disks, efforts are already underway to develop the necessary optics for larger storage density DVR systems. The possibility of an areal density over 50 Gb in
2 was examined in near-field phase-change recording [243, 244]. The disk structure was optimized to maximize readout signals under the land-andgroove recording condition at a tracking pitch of 160 nm. Near-terms goals include much higher optical recording densities. In an effort to address the cross talk issue, eye diagrams of 50.4 Gb in
2 areal density were demonstrated using 1.5 NA optics and a GaN laser diode. The track pitch and linear bit density employed were 160 nm

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and 80 nm b
1, respectively. The large NA aspherical lens used in the approach (i.e., 1.0 mm-diameter superhemispherical solid immersion lens) was fabricated with plasma etching, which is about three times larger in area than the previous design, by using a novel selectivity control technique during transfer processes [245]. A point of interest is that in the Sony approach, the disk surface is polished by the tape burnishing technique. An eye pattern of (1–7)-coded data at the linear density of 80 nm b
1 was demonstrated on the phase-change disk below a mere 50 nm gap height, which was realized by an air-gap servo mechanism [246]. Let us now take a closer look at optical head of the imminent third generation digital video recording system utilizing an InGaN violet laser with the associated performance data [247]. Keep in mind that data are fluidic with constant improvements. It should not be surprising if the system that makes it to the market place outperforms the parameters listed here. The heart of the third generation digital video recording system that is also relevant to this text is the optical head. While several companies are actively pursuing optical heads for this application, the version reported by researcher at Sony is briefly reviewed. The key device in the optical head is a violet InGaN laser diode and the head itself is the key component of digital recording system, which will likely assume the nomenclature “Blu-ray disk.” There are several critical issues that this system must address. The first among them is the design of a small-sized optical system that has high transmittance to the blue-violet wavelength as well as small aberration. The second is the efficient dissipation of the heat generated in the laser diode. The Sony group used the previously developed In solder to solve these problems and realized this head device, and developed a molded optical element and a package, as well as utilizing the conventional assembly technology developed for CD and mini disk (MD) laser couplers [247]. A schematic diagram of the Sony integrated optical head device is shown in Figure 2.82. It consists of a blue-violet InGaN LD, a 45 mirror, a half-wave plate (HWP), a molded optical element, a laminated prism, a photodiode IC (PDIC), and a molded package. The light emitted from the LD is reflected up by the 45 mirror, and it passes through the HWP, the grating, and a polarizing beam splitter (PBS) film. The HWP rotates the polarization angle of the laser light for it to pass through the PBS film. The light passing through the grating is split into three beams, which are necessary for generating a tracking error signal by the differential push–pull (DPP) detection scheme. Once the light gets through the PBS film, it is focused onto the optical disc, not shown, and traverses through several optical, not shown. The light reflected from the disk is reflected twice by the PBS film and a mirror for it to enter a holographic optical element (HOE) where the side spots for a focusing error signal by the spot size detection (SSD) method are generated. The light that passes through the HOE and the cylindrical lens enters a PDIC detector affixed to the lead frame. The alignment of the overall assembly is as follows: First, the LD, the 45 mirror, and the PDIC on a lead frame are mounted the HWP on the molded package. This is followed by gluing of the laminated prism to a molded optical element. While observing the output signals from the PDIC, the molded optical element is glued to the package after adjusting the X,Y-position and q rotation of the molded element [247]. The

2.18 Applications of GaN-Based Lasers to DVDs

Disc

Mirror Holographic optical element

Polarizing beam splitter

Cylindrical lens

Grating Prism Half wave plate

Molded optical element

Molded package

InGaN LD

45º Mirror

Photodiode IC Lead frame

Figure 2.82 Schematic representation of the Sony integrated optical head element for high-density DVD. The terms used are HWP, PBS, HOE, PDIC, and LD [247].

position accuracy required for the first two steps is about 5 mm, and the angle accuracy required is about 3 mrad. The performance evaluation utilizes a setup that allows the light emitted from the integrated head discussed above to be collimated by a collimator lens. The beam profile of the light is shaped by an anamorphic prism. Using a two-element expander lens, the spherical aberration caused by errors in the cover layer is compensated. After the linearly polarized light is converted to circularly polarized light by the quarter-wave plate (QWP), the beam is focused on the storage disk medium by a two-element objective lens of 0.85 NA (the follow-up version are considering 1.5 NA). The reflected light from the storage disk arrives to the head device following the same light path. Following the experimental conditions listed in Table 2.3, Manoh et al. [247] evaluated the readout performance and tolerances of the optical head for a corresponding disk capacity of 23.3 GB using a 120 mm f (12 cm diameter) disk. Table 2.3 Experimental conditions surrounding the readout performance by laser [247].

LD wavelength Objective lens NA Cover layer thickness Readout power Linear velocity Track pitch Data bit length Groove Equalizer

405 nm 0.85 0.1 mm 0.3 mW 5.28 m s
1 0.32 mm 0.12 mm b
1 No wobble Conventional equalizer

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Table 2.4 Evaluation results of the readout performance [247].

Data-to-clock jitter CNR of 2 T (RBW ¼ 30 kHz) CNR of 8 T (RBW ¼ 30 kHz) Radial tilt tolerance (Jitter < 15%) Tangential tilt tolerance (Jitter < 15%) Defocus tolerance (Jitter < 15%)

7.3% 52.7 dB 58.0 dB 1.4 1.3 0.7 mm

Table 2.5 Write power conditions [247].

Peak power Erase power Cooling power

4.7 mW 2.7 mW 0.3 mW

The eye diagram was clear and a good data-to-clock jitter value of 7.3% was obtained by using a conventional equalizer and a time interval analyzer. The tilt tolerances obtained are sufficient to realize the Blu-ray disk drive. The total readout performance of the device is summarized in Table 2.4. It is almost equal to that of a conventional discrete head. As for the read/write performance tests, Manoh et al. [247] implemented a read/ write performance tests with the write power conditions being listed in Table 2.5 and other parameters being the same as those listed in Table 2.3. The corresponding disk capacity using a 120 mm f disk is also 23.3 GB. The eye diagram tests as well as the equalizer and time interval analyzer used in the readout performance tests indicate a data-to- clock jitter value of 7.9%. This jitter value, which is a little worse than that obtained in the readout test, can be improved by optimizing the write pulse. The results of a direct overwrite (DOW) test and a cross-erase test have also been performed, and the read/write performance results are summarized in Table 2.6.

2.19 A Succinct Review of the Laser Evolution in Nitrides

The semiconductor laser, though it has important applications, is analogous to a race car. As is the case in a race car, the components of an injection laser are put through Table 2.6 Results of read/write test [247].

Data-to-clock jitter (%) Initial recording with cross talk 10 times DOW with cross talk 10 times cross-erase DOW, direct overwrite.

7.9 8.2 8.4

2.19 A Succinct Review of the Laser Evolution in Nitrides

the acid test in that high concentrations of injected carriers, high-photon densities, and large heat generation exist simultaneously, and will challenge the robustness of the device. Since the first report of optically pumped stimulated emission in GaN at 2 K [113], great strides have been made at an astonishing rate, particularly following the report on p-type GaN by the group of Akasaki in 1989 (Volume 1, Chapter 4). Utilizing Al0.1Ga0.9N/GaN separate confinement laser structures, the optical pumping required for lasing at room temperature was reduced to lie well under 100 kW cm
2 from several MW cm
2 in the early versions relying on bulk GaN [163, 248]. Building on a successful commercialization of bright (blue followed by green) LEDs, Nakamura et al. [249] reported the observation of laser oscillations at room temperature in InGaN quantum wells utilizing GaN waveguides and AlGaN cladding layers. Although the initially reported laser utilized 26 periods of In0.2Ga0.9N/ In0.05Ga0.95N MQW structures consisting of 25 Å thick In0.2Ga0.8N well layers and 50 Å thick In0.05Ga0.95N barrier layers, the structures that followed incorporated MQWs with as little as 7 periods [250], which eventually got reduced to 3 periods. A 200 Å thick p-type Al0.2Ga0.8N:Mg layer was employed on top of the InGaN layers to prevent their dissociation during the growth of the subsequent GaN and AlGaN layers, which require much higher substrate temperatures. The 0.1 mm thick layer of n-type In0.1Ga0.9N was imbedded in the buffer layer to prevent cracking. Because it is difficult to cleave the c-face sapphire substrates, reactive ion etching (RIE) was employed to form the cavity mirrors. High-reflectivity facet coatings (60–70%) were used to reduce the threshold current. A Ni/Au contact was evaporated onto the entire area of the p-type GaN layer and a Ti/Al contact onto the n-type GaN layer. The earlier versions of injection lasers fabricated by Nakamura et al. suffered from large forward voltages as high as 30 V, and that is some seven times larger than it should be. This naturally increases the power dissipation by a factor of almost 50. This voltage was later reduced to about 5 V (Figure 2.47), which is reasonable. The threshold current density of the later and more successful varieties of Nichia lasers fabricated in structures grown by the lateral growth technique has been reduced to approximately 1.5 kA cm
2. Akasaki et al. [251] also reported possible lasing action in a single In0.1Ga0.9N well SCH at 376 nm, the shortest of any injection semiconductor laser at the time. A threshold current density of 2.9 kA cm
2 under pulsed operation with a duty cycle of 1% and a pulse width of 0.3 ms has been observed at room temperature. At the time of writing this book, there were some six laboratories reporting lasing action, by current injection, in InGaN DH structures. In the early development of semiconductor lasers, first based on GaAs and later on InP, the mirrors were formed by cleaving. As this process is not a very high-yeilding process, processes were developed to produce etched mirrors. GaN is no exception in that despite difficulties associated with sapphire and GaN, which rotated in-plane with respect to sapphire, cleaved mirrors were employed. Exacerbating the situation is the fact that sapphire has many cleavage planes with approximately equal cleave strengths within a small angular distance of each other. Consequently, the fracture interface can easily be redirected from one cleavage plane to another, even for very small perturbations during the cleaving process. For cleaved mirrors to work, the crystal structure of the epitaxially deposited GaN must align perfectly with a cleavage

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plane in the underlying substrate. This is complicated further in that the GaN columns are rotated with respect to the underlying sapphire, making it impossible to align the cleavage planes of GaN and sapphire. In addition, the method of cleaved facets (mirrors) does not lend itself to achieving cavity lengths below about 500 mm well. To circumvent the problems associated with cleaved mirrors, lasers with etched mirrors on films grown on the c-plane of sapphire have been explored. To go around the problem, Nakamura et al. [252] explored lasers grown on what was then reported to be the a-plane of sapphire (see the details of growth on non-c-plane sapphire discussed in Volume 1, Chapter 3). Even though the materials quality did not compare to that on the c-plane, improved cavity formation along the (1
1 0 0) plane of GaN purportedly outweighed the reduced material quality. Moreover, laser structures on (1 1 1) MgAl204 (spinel substrates), which lead to Wz GaN along the c-plane, have also been explored for optical pumping [253] and injection laser experiments [254]. In this scheme, spinel cleaves along the {1 0 0} planes that are inclined to the surface with cleavage following one of the (1
1 0 0) planes of GaN about where the epilayers are reached. Even though the mirror quality in this scheme was the best among the aforementioned approaches, material quality degradation is too severe to pull it ahead of the other approaches. Nakamura et al. [114] may have found a way to produce etched mirrors on c-plane sapphire substrates, perhaps eliminating the undesired compromise between the best crystalline quality and the best mirrors. During the early exploration of non-c-plane growth and use of other substrates such as spinel in an effort to achieve better quality cleaved mirrors, it became apparent that the materials quality on the c-plane is best, particularly when combined with lateral overgrowth methods and thick HVPE layers. While some manufacturers still prefer cleaved mirrors, albeit with some difficulty, at least some of the efforts shifted toward developing etched mirrors with sufficient quality, mainly smoothness [255]. If it is successful, the etched facets would increase the yield and reduce the cost. Before fully delving into etched facets in GaN, let us briefly discuss the methods that have been developed in the pursuit of GaAs- and InP-based lasers with etched mirrors. In the case of GaAs methods for forming etched mirrors included wet chemical etching [256, 257] and a combination of wet and RIE [258]. The mirror reflectivities, however, attained by these methods were inferior to that of the cleaved varieties until methods such as CAIBE were developed [259]. CAIBE [260] utilizes an inert gas such as Ar in an ion source to generate an ion beam, which is directed toward the sample in the presence a flow of a reactive gas such as Cl2 near the sample. The ion beam provides the physical component of the etching process, which is directed along the ion beam, while the reactive gas provides the chemical component for the etching process. Independent direct control over both the physical and chemical components of the etch is one of the attractive features of the method. This independent control allows the possibility of very high selectivity between the etch mask and the material being etched. As etching takes place only in the direction of the ion beam, the etch is very anisotropic, as required for mirrors. Further development of an appropriate etch mask in conjunction with the unique capabilities of CAIBE paved the way for excellent etched facets for semiconductor lasers [261]. The etched facet (mirror) technology allows lasers to be fabricated on the wafer in much the same

2.19 A Succinct Review of the Laser Evolution in Nitrides

way that integrated circuit chips are fabricated on silicon. Etched facet lasers are monolithically integrable with other photonic devices on a single chip and can be tested inexpensively at wafer level [262]. Facet reflectivity modification can be used to modify the reflectivity of the etched facets through deposition of dielectric coatings with the wafer intact. The CAIBE method was applied to the fabrication of etched facets for blue lasers on sapphire early on, but the etch rates obtained were only double that of Ar ion milling [263]. Kneissl et al. [264] achieved etch rates approaching 100 nm min
1. Unfortunately, the reflectivity obtained from such etched facets was lower than that on cleaved facets. The lower reflectivity was primarily attributed to roughness of the cleaved facet [265]. GaN RIE etched facets having a reflectivity of only 0.05 were improved through the use of focused ion beam (FIB) [266]. The key to obtaining highquality etched facets hinges on the selectivity between the etch mask and the semiconductor material in the etching process. This is, of course, exacerbated by the robustness of GaN in that it is hard to etch GaN, and mask erosion occurs during etching. Behfar et al. [255] overcame this bottleneck and achieved etched facets in GaN with a selectivity of better than 10 : 1 and with an etch rate higher than 0.25 mm min
1. Figure 2.83 shows a scanning electron microscope (SEM) image of an exemplary facet obtained using silicon dioxide as the etch mask in CAIBE, with a section of the facet enhanced through FIB. The smoothness of the facet is clearly noted. Leaving the discussion of etched facets, developments in the materials quality by use of ELO and various liftoff techniques in unison with HVPE templates paved the way to lasers having cleaved facets, and increased power levels and longevity required for digital video write/read applications. The lifetime of 15 000 h under CW operation with output levels of 30 mW at 60  C reported by the researchers at

Figure 2.83 SEM of high-quality etched facets in GaN using CAIBE and FIB etching. Note that FIB is used to etch the active part of the facet for a smooth facet face. Courtesy of A. Befhar, BinOptics Corp.

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Sony and ensuing development of optical heads for digital disks have catapulted the nitride-based lasers to new realms and paved the way for other, but lower volume, applications. What is on the horizon beyond the edge emitters could have a twofold answer. On the one side, the need for efficient pumps for white-light applications may provide the impetus for developing VCSELs. We should be cognizant of the fact that this is not the only application of vertical cavity emitters. On the fold, newer device constructions may be pursued. Among them are the microsized light-emitting devices with the ability to achieve arrays of individually controllable pixels on a single chip. New physical phenomena would emerge as the device size becomes comparable or smaller than the wavelength of light, including modified spontaneous emission, enhanced quantum efficiency, and reduced lasing threshold, all of which warrant fundamental investigations [94]. III-nitride QW microdisk and microring cavities have been fabricated, and enhanced quantum efficiency and optical resonant modes have been observed in these microcavities [267]. Resonant optical modes in microsized GaN pyramids prepared by ELO have also been observed [268]. Precursory in nature to lasers, electrically pumped UV–blue microsized LEDs have been fabricated [269]. These culminated in the demonstration of optically pumped AlIn/GaN lattice-matched bottom DBR optically pumped lasers [205]. Furthermore, InGaN/ GaN LEDs using oxidized Al0.82In0.12N layer have been reported with a current density of the order of 20 kA cm
2 achieved, a value that should fulfill the injection requirements for nitride-based VCSELs [206]. These developments signal the report of GaN-based VCSELs.

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References 217 Livshits, D.A., Egorov, A.Yu. and Riechert, H. (2000) Electronics Letters, 36, 1381. 218 Kawaguchi, M., Miyamoto, T., Gouardes, E., Schlenker, D., Kondo, T., Koyama, F. and Iga, K. (2001) Lasing characteristics of low-threshold GaInNAs lasers grown by metalorganic chemical vapor deposition. Japanese Journal of Applied Physics, 40, L744. 219 Fischer, M., Reinhardt, M. and Forchel, A. (2000) GaInNAs/GaAs laser diodes operating at 1.52 mm. Electronics Letters, 36, 1208. 220 Ünl€ u, M.S., Kishino, K., Chyi, J.I., Arsenault, L., Reed, J., Mohammad, S.N. and Morkoç, H. (1990) Resonant cavity enhanced AlGaAs/GaAs heterojunction phototransistors with an intermediate InGaAs layer in the collector. Applied Physics Letters, 57, 750–752. 221 Choquette, K.D., Klem, J.F., Fischer, A.J., Blum, O., Allerman, A.A., Fritz, I.J., Kurtz, S.R., Breiland, W.G., Sieg, R., Geib, K.M., Scott, J.W. and Naone, R.L. (2000) Room temperature continuous wave InGaAsN quantum well vertical-cavity lasers emitting at 1.3 mm. Electronics Letters, 36, 1388. 222 Takeuchi, T., Chang, Y.-L., Leary, M., Tandon, A., Luan, H.-C., Bour, D., Corzine, S., Twist, R. and Tan, M. (2002) Low threshold 1.3 mm InGaAsN vertical cavity surface emitting lasers grown by metalorganic chemical vapor deposition. SPIE Conference, San Jose, CA. 223 Kneissl, M., Bour, D.P., Romano, L., Van de Walle, C.G., Northrup, J.E., Wong, W.S., Treat, D.W., Teepe, M., Schmidt, T. and Johnson, N.M. (2000) Performance and degradation of continuous-wave InGaN multiple-quantum-well laser diodes on epitaxially laterally overgrown GaN substrates. Applied Physics Letters, 77, 1931. 224 K€ ummler, V., Br€ uderl, G., Bader, S., Miller, S., Weimar, A., Lell, A., H€arle, V., Schwarz, U.T., Gmeinwieser, N. and Wegscheider, W. (2002) Degradation analysis of InGaN laser diodes. Physica Status Solidi a: Applied Research, 194, 419.

225 Tomiya, S., Goto, O. and Ikeda, M. (2008) Structural defects and degradation of high power pure-blue GaN-based laser diodes. SPIE Photonic West, Opto 07: Gallium Nitride Materials and Devices III, January 19–24, 2008, San Jose, CA, Proceedings of SPIE, 6894, art. no. 22. 226 Sharma, N., Thomas, P., Tricker, D. and Humphreys, C. (2000) Applied Physics Letters, 77, 1274. 227 Chen, Y., Takeuchi, T., Amano, H., Akasaki, I., Yamada, N., Kaneko, Y. and Wang, S.Y. (1998) Applied Physics Letters, 72, 710. 228 Fukuda, M. (1991) Reliability and Degradation of Semiconductor Lasers and LEDs, Artech House. 229 Ueda, O. (1996) Reliability and Degradation of III–V Optical Devices, Artech House. 230 Takeya, M., Mizuno, T., Sasaki, T., Ikeda, S., Fujimoto, T., Ohfuji, Y., Oikawa, K., Yabuki, Y., Uchida, S. and Ikeda, M. (2003) Physica Status Solidi c, (7), 2292–2295. 231 Kuroda, N., Sasaoka, Ch., Kimura, A., Usui, A. and Mochizuki, Y. (1998) Journal of Crystal Growth, 189–190, 551–555. 232 Ikeda, M., Mizuno, T., Takeya, M., Goto, S., Ikeda, S., Fujimoto, T. and Ohfuji, Y. (2004) Physica Status Solidi c, 1 (6), 1461–1467. 233 Kim, C.C., Choi, Y., Jang, Y.H., Kang, M.K. and Joo, M. and Noh, M.S. (2008) Degradation modes of high power InGaN/GaN laser diodes on low-defect GaN substrates. SPIE Photonic West, Opto 07: Gallium Nitride Materials and Devices III, January 19–24, 2008, San Jose, CA. Proceedings of SPIE, 6894, art. no. 23. 234 Marona, L., Wis´niewski, P., Leszczyn´ski, L., Grzegory, I., Suski, T., Porowski, S., Czernecki, R., Czerwinski, A., Pluska, M., Ratajczak, J. and Perlin, P. (2008) Why InGaN laser-diode degradation is accompanied by the improvement of its thermal stability. SPIE Photonic West, Opto 07: Gallium Nitride Materials and Devices III, January, 19–24 2008, San

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Jose, CA. Proceedings of SPIE, 6894, art. no. 26. Schoedl, T., Schwartz, U.T., Kummler, V., Furitch, M., Leber, A., Miler, A., Lell, A. and Harle, V. (2005) Journal of Applied Physics, 97, 123102. Uchida, S., Takeya, M., Ikeda, S., Mizuno, T., Fujimoto, T., Matsumoto, O., Goto, S., Tojyo, T. and Ikeda, M. (2003) IEEE Journal of Selected Topics in Quantum Electronics, 9 (5), 1252–1259. Swietlik, T., Franssen, G., Winiewski, P., Krukowski, S., epkowski, S.P., Marona, L., Leszczyski, M., Prystawko, P., Grzegory, I., Suski, T., Porowski, S. and Perlin, P. (2006) Applied Physics Letters, 88, 071121. Ryu, H.Y., Ha, K.H., Lee, S.N., Jang, T., Kim, H.K., Chae, J.H., Kim, K.S., Choi, K.K., Son, J.K., Paek, H.S., Sung, Y.J., Sakong, T., Nam, O.H. and Park, Y.J. (2006) Applied Physics Letters, 89, 031122. Ichimura, I., Maeda, F., Osato, K., Yamamoto, K. and Kasami, Y. (2000) Japanese Journal of Applied Physics, 39, 937. Schep, K., Stek, B., Woudenberg, R., Blum, M., Kobayashi, S., Narahara, T., Yamagami, T. and Ogawa, H. (2001) Japanese Journal of Applied Physics, 40, 1813. Ichimura, I., Masuhara, S., Nakano, J., Kasami, Y., Yasuda, K., Kawakubo, O. and Osato, K. (2002) Proceedings of SPIE, 4342, 168. Schep, K., Stek, B., van Woudenberg, R., Blum, M., Kobayashi, S., Narahara, T., Yamagami, T. and Ogawa, H. (2001) Format description and evaluation of the 22.5 GB digital-video-recording disc. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers, 40 (3B), 1813–1816. Kishima, K., Ichimura, I., Saito, K., Yamamoto, K., Kuroda, Y., Iida, A., Masuhara, S. and Osato, K. (2002) Challenge of near-field recording beyond 50.4 Gbit/in2. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers, 41 (3B), 1894–1897.

244 Kouchiyama, A., Ichimura, I., Kishima, K., Nakao, T., Yamamoto, K., Hashimoto, G., Iida, A. and Osato, K. (2001) Optical recording using high numerical-aperture microlens by plasma etching. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers, 40 (3B), 1792–1793. 245 Kouchiyama, A., Ichimura, I., Kishima, K., Nakao, T., Yamamoto, K., Hashimoto, G., Iida, A. and Osato, K. (2002) Optical recording using high numerical-aperture microlens by plasma etching. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers, 41 (3B), 1825–1828. 246 Kishima, K., Ichimura, I., Yamamoto, K., Osato, K., Kuroda, Y., Iida, A. and Saito, K. (2000) Near-field phase-change recording using a GaN laser diode. Proceedings of SPIE, 4090, 50–55. 247 Manoh, K., Yoshida, H., Kobayashi, T., Takase, M., Yamauchi, K., Fujiwara, S., Ohno, T., Nishi, N., Ozawa, M., Ikeda, M., Tojyo, T. and Taniguchi, T. (2003) Small integrated optical head device using a blue-violet laser diode for Blu-ray Disc system. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers, 42 (2B), 880–884. 248 Amano, H., Watanabe, N., Koike, M. and Akasaki, I. (1993) Japanese Journal of Applied Physics, 32, L1000. 249 Nakamura, S., Senoh, M., Nagahama, S., Iwasa, N., Yamada, T., Matsushita, T., Kiyoku, H. and Sugimoto, Y. (1996) Japanese Journal of Applied Physics, 35, L74. 250 Nakamura, S., Senoh, M., Nagahama, S., Iwasa, N., Yamada, T., Matsushita, T., Sugimoto, Y. and Kiyoku, H. (1996) Applied Physics Letters, 69, 1568. 251 Akasaki, I., Sota, S., Sakai, H., Tanaka, T. and Amano, H. (1996) Electronics Letters, 32, 1105. 252 Nakamura, S., Senoh, M., Nagahama, S., Iwasa, N., Yamada, T., Matsushita, T., Kiyoku, H. and Sugimoto, Y. (1996) Japanese Journal of Applied Physics, 35, L217.

References 253 Kuramata, A., Horino, K., Domen, K., Shmohora, K. and Tanahashi, T. (1995) Applied Physics Letters, 67, 2521. 254 Nakamura, S., Senoh, M., Nagahama, S., Iwasa, N., Yamada, T., Matsushita, T., Kiyoku, H. and Sugimoto, Y. (1996) Applied Physics Letters, 68, 2105. 255 Behfar, A., Schremer, A., Hwang, J., Stagarescu, C., Morrow, A. and Green, M. (2006) Etched facet technology for GaN and blue lasers. Proceedings of SPIE, 6121, 61210. 256 Merz, J. and Logan, R.A. (1976) GaAs double heterostructure lasers fabricated by wet chemical etching. Journal of Applied Physics, 47, 3503–3509. 257 Merz, J., Logan, R.A. and Sergent, A. (1979) GaAs integrated optical circuits by wet chemical etching. IEEE Journal of Quantum Electronics, 15, 72–82. 258 Salzman, J., Venkatesan, T., Margalit, S. and Yariv, A. (1985) Double heterostructure lasers with facets formed by a hybrid wet and reactive-ion-etching technique. Journal of Applied Physics, 57, 2948–2950. 259 Behfar-Rad, A., Wong, S.S., Ballantyne, J.M., Soltz, B.A. and Harding, C.M. (1989) Rectangular and L-shaped GaAs/AlGaAs lasers with very high quality etched facets. Applied Physics Letters, 54, 493–495. 260 Chinn, J.D., Fernandez, A., Adesida, I. and Wolf, W.D. (1983) Chemically assisted ion beam etching of GaAs, Ti, and Mo. Journal of Vacuum Science & Technology: Vacuum Surfaces and Films, 1, 701–704. 261 Behfar-Rad, A., Wong, S.S., Davis, R.J. and Wolfe, E.D. (1989) Masking considerations in chemically assisted ion beam etching of GaAs/AlGaAs laser structures. Journal of the Electrochemical Society, 136, 779–782.

262 Vettiger, P., Benedict, M.K., Bona, G.-L., Buchmann, P., Cahoon, E.C., Datwyler, K., Dietrich, H.-P., Moser, A., Seitz, H.K., Voegeli, O., Webb, D.J. and Wolf, P. (1991) Full-wafer technology – a new approach to large-scale laser fabrication and integration. IEEE Journal of Quantum Electronics, 27, 1319–1331. 263 Adesida, I., Ping, A.T., Youtsey, C., Dowe, T., Khan, M.A., Olson, D.T. and Kuznia, J.N. (1994) Characteristics of chemically assisted ion beam etching of gallium nitride. Applied Physics Letters, 65, 889–891. 264 Kneissl, M., Bour, D.P., Johnson, N.M., Ramano, L.T., Krusor, B.S., Donaldson, R., Walker, J. and Dunnrowicz, C. (1998) Characterization of AlGaInN diode lasers with mirrors from chemically assisted ion beam etching. Applied Physics Letters, 72, 1539–1541. 265 Stocker, D.A., Schubert, E.F., Grieshaber, W., Boutros, K.S. and Redwing, J.M. (1998) Facet roughness analysis for InGaN/GaN lasers with cleaved facets. Applied Physics Letters, 73, 1925–1927. 266 Mack, M.P., Via, G.D., Abare, A.C., Hansen, H., Kozodoy, P., Keller, S., Speck, J.S., Mishra, U.K., Coldren, L.A. and DenBaars, S.P. (1998) Improvement of GaN-based laser diode facets by FIB polishing. Electronics Letters, 34, 1315–1316. 267 Mair, R.A., Zeng, K.C., Lin, J.Y., Jiang, H.X., Zhang, B., Dai, L., Tang, H., Botchkarev, A., Kim, W. and Morkoç, H. (1997) Applied Physics Letters, 71, 2898. 268 Jiang, H.X., Lin, J.Y., Zeng, K.C. and Yang, W. (1999) Applied Physics Letters, 75, 763. 269 Jin, S.X., Li, J., Li, J.Z., Lin, J.Y. and Jiang, H.X. (2000) Applied Physics Letters, 76, 631.

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3 Field Effect Transistors and Heterojunction Bipolar Transistors Introduction

The need for computers to handle large volumes of data for high-speed computing, real-time signal processing, telecommunication, mobile communication, imaging, and low-noise and high-frequency amplification has generated an unequalled interest in advancing the speed of electronic devices and circuits. RF, microwave, and millimeter wave systems for telecommunication and many other traditional uses require devices with ever increasing performance in terms of noise figure (NF) and gain at frequencies exceeding 100 GHz, and also increased power. All of these driving forces have resulted in an intense activity of new device concepts as well as heterostructures based on new semiconductors. In semiconductors, excluding Si, forms other than the ubiquitous metal oxide semiconductor FET (MOSFET) are used for high-performance FETs due to the lack of gate quality dielectrics. Until the advent of modulation-doped FETs (heterojunction FETs (HFETs)), this device was a metal–semiconductor FET (MESFET). The speed of the device depends on carrier transit time under the gate from the source side to the drain side, and the delays inherent in the devices such as those caused by capacitors and resistors. In scaling MESFETs for short channel and fast devices, one encounters limits in terms of doping requirements and the proximity of the gate with respect to the conducting channel. In heterojunction FETs, these limits are alleviated considerably. The semiconductor form that was first used for HFETs was GaAs based. While exploring the properties of quantum wells and superlattices, it was discovered that when the large-bandgap AlGaAs is doped with a donor impurity near the junction it forms with the adjacent small-bandgap GaAs, the electrons donated by donors in the wider bandgap material diffuse to the lower energy conduction band of GaAs where they are confined due to the heterointerfacial potential barrier. The electron gas formed in the process does not get affected, except remotely, by the donors and possesses nearly impurity scattering free transport. Let us now consider briefly the principles of operation of HFETs. For details, the reader is advised to refer to a two-volume text on the topic by Morkoç et al. [1] and references therein, and a review article published in the June 1986 issue of IEEE

Handbook of Nitride Semiconductors and Devices. Vol. 3. Hadis Morkoç Copyright
2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40839-9

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Source contact

Drain contact T-gate

Doped GaAs (InGaAs) layer Doped AlGaAs (InAlAs) layer 2D electron or hole gas

GaAs (or InGaAs) channel layer GaAs (or InAlAs) buffer layer High-resistivity GaAs (1 0 0) (or InP) substrate

Figure 3.1 Cross-sectional view of a pseudomorphic InGaAs channel HFET, used loosely for coherently strained, channel MODFET on GaAs or InP substrates. On GaAs substrates, the mole fraction of InAs of coherent channel is limited to below 35%. On InP, however, a range of 53–80%, later extended to 100%, has been explored.

Proceedings by Drummond et al. [2]. The main visible difference between a MODFET and a MOSFET is that the conduction channel in an HFET is formed in equilibrium by doping the large-bandgap material. In MODFET, the source and drain ohmic contacts are made directly to the twodimensional electron gas (2DEG), while the gate electrode between these two terminals modulates the current. A schematic cross-sectional diagram of a pseudomorphic InGaAs channel HFET on GaAs or InP substrate is shown in Figure 3.1. Large low-field mobilities, relatively large maximum electron velocities, and large electron concentrations provided by many of the compound semiconductors are ideal for high-performance FETs. Proximity of the gate to the conduction channel in MODFETs, and small gate-to-channel distance, among other parameters such as the confinement provided by the field profile in the substrate side of the channel, reduces if not eliminate the enormous increase observed in the output conductance as the gate length is scaled down. In addition, the change in the threshold voltage of this all-epitaxial structure as the gate length is reduced, unlike in conventional metal– semiconductor FETs, is minimal. These last two properties have served well in scalability of these devices so much so that deep submicron (
0.2 mm) small-scale digital circuits have already been fabricated successfully. These same proximity and carrier confinement effects are also responsible for the gains obtained at or over 100 GHz, considered impossible only a few years back. The confinement potential induced by the channel charge on the substrate side alone is not sufficient, as it diminishes near the drain side, particularly in deep saturation. To circumvent this problem, a semiconductor with a larger bandgap than the one forming the channel can be incorporated just below the channel. In GaAs channels, AlGaAs (and in InGaAs channels, InAlAs) can serve this function well. Alternatively, the channel itself can be made out of a smaller bandgap material as in the case of pseudomorphic InGaAs channel devices embedded in an otherwise GaAs/AlGaAs structure. The InAs mole fraction in the pseudomorphic device is limited to a maximum of about 35% to avoid the deleterious crystalline defects that otherwise

Introduction

result and deteriorate the device performance quite substantially. Even though the AlGaAs/InGaAs-based pseudomorphic MODFET became the dominant compound semiconductor FET in the marketplace with its low noise, high gain, and reasonably high-power handling capability, high RF power came at the expense of power combining schemes at a substantial cost. This is where GaN-based HFET came into the picture. GaN’s large bandgap, large dielectric breakdown field, fortuitously good electron transport properties [3–5] (an electron mobility possibly in excess of 2000 cm2 V1 s1 and a predicted peak velocity approaching 3  107 cm s1 at room temperature, although the velocity deduced from device current or speed is much lower and is a topic of raging debate), and good thermal conductivity are trademarks of highpower/temperature electronic devices [6]. To give a flavor, Sheppard et al. [7] have reported that 0.45 mm gate, high-power heterojunction FETs on SiC substrates exhibited a power density of 6.8 W mm1 in a 125 mm wide device and a total power of 4 W (with a power density of 2 W mm1) at 10 GHz. Other groups have also reported on the superior power performance of GaN-based HFETs on SiC and sapphire substrates with respect to competing materials, particularly at X band and higher frequencies [8–11], keeping in mind that record levels belong to structures grown on SiC substrates. This is due to high thermal conductivity of SiC and also enhanced quality of GaN heterostructures on SiC. For starters, GaN grown on SiC does not undergo an in-plane rotation around the c-axis and is thus void of the associated disorder. GaN/AlGaN HFETs prepared at HRL Laboratories by molecular beam epitaxy (MBE) on SiC substrates have exhibited a total power level of 6.3 W at 10 GHz from a 1 mm wide device. A saturated power density of 6.6 W mm1 with a power-added efficiency (PAE) of 35% at 20 GHz also resulted [12]. Credit to the supreme thermal conductivity of SiC, the power level in these devices at the time was not thermally limited as the power density extrapolated from a 0.1 mm device is 6.5 W. When four of these devices are power combined in a single-stage amplifier, an output power of 22.9 W with a power-added efficiency of 37% was obtained at 9 GHz [13]. Equally impressive is the noise figure of 1.0 and 1.75 dB, which was obtained at 10 and 20 GHz, respectively [14], and which improved to 0.85 dB at 10 GHz with an associated gain of 11 dB. In terms of the linearity, a 0.15  100 mm2 device yields an output third-order intercept point (OIP3) of 23 dB m at VDS ¼ 3 V and VGS ¼ 5 V, where a noise figure of 1.0 and 1.75 dB was obtained at 10 and 20 GHz, respectively [15]. The drain breakdown voltages in these quarter-micron gate devices are about 60 V, which are in part responsible for such a record performance [16]. With further optimization, standard devices have a power output up to 9.8 W mm1 at 8 GHz [17] with gradually increasing power densities and total power levels [18]. Polyimide passivated devices with 0.23 mm gate lengths yielded a peak power density of 7.65 W mm1 at 18 GHz [19]. These data suggest that polyimide can be an effective passivation film for reducing surface states. With field plates (FPs) that spread the field between the gate and drain more uniformly (which has also been applied to SiC-based devices [20]), thereby allowing larger drain breakdown voltages [21, 22], power densities up to 30 W mm1 have been reported [23]. On the cutoff frequency front, short channel devices exhibited

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fT ¼ 101 GHz and fmax ¼ 155 GHz [24, 25], fT ¼ 190 GHz and fmax ¼ 241 GHz circa 2007 with actual measurements having been performed up about 50 GHz [26]. Applications of high-power GaN-based HFETs include amplifiers, which are operative at high power levels, high temperatures, and in unfriendly environments. Examples include radar, missiles, and satellites, as well as low-cost compact amplifiers for wireless base stations. A good deal of these applications are currently met by pseudomorphic HFETs developed earlier [27]. The other electronic device, in fact the one that got the electronics industry started, is the bipolar transistor. Unlike FETs, bipolar transistors rely on diffusion of minority carriers across a region called the base, which is straddled by an emitter that injects minority carriers and a collector that collects the injected carriers that have made it across the base. The critical parameters for high performance in terms of speed are the base thickness in relation to the diffusion length in the same, extent of the depletion region in the collector, and emitter injection efficiency. In addition, the base resistance impacts the maximum oscillation frequency very adversely. Efforts to lower the base resistance by increasing the base doping level lead to reduced emitter injection efficiency. With a wide bandgap emitter, the base resistance can be lowered without adversely affecting the emitter injection efficiency because of the added band discontinuity between the emitter and the base, which impedes the reverse injection of minority carriers. Being a vertical device, as opposed to horizontal construction of an FET, one can increase the collector depletion region thickness in a bipolar transistor by reducing the collector doping to get a large breakdown field. In small-bandgap materials, the depletion layer thickness gets to be too large and degrades the device speed. This is where the large-bandgap materials such as GaN come into the picture. With very large breakdown fields, one can keep the collector depletion region small for high speed and still attain large breakdown voltages, which are required by many applications such as smart power supplies, electric motor controllers, and other power electronic applications. Though in its infancy, efforts are underway to exploit nitrides for bipolar transistors. However, the material quality needs to be improved more before the performance expected from GaN can be obtained. Difficulties include the notoriously low p-type doping and low diffusion length in epitaxial layers. In this chapter, operational principles of HFETs, polarization issues pertinent to HFETs (the expanded discussion of polarization can be found in Volume 1, Chapter 2), analytical expressions describing HFET operation, results of HFET simulations, technology, and performance in terms of DC and RF are discussed. This is followed by a discussion of fundamentals of heterojunction bipolar transistor (HBT) operation, and a succinct discussion of nitride-based HBTs closes the discussion.

3.1 Heterojunction Field Effect Transistors

With its reduced impurity scattering and unique gate capacitance–voltage characteristics, the HFET has become the dominant high-frequency device. Among the HFETs

3.1 Heterojunction Field Effect Transistors

most attractive attributes are close proximity of the mobile charge to the gate electrode and high drain efficiency (DE). As in the case of emitters, the GaN-based HFETs have quickly demonstrated record power levels at high frequencies with very respectable noise performance and large drain breakdown voltages. In HFETs, the carriers that form the channel in the smaller bandgap material are donated by the larger bandgap material and ohmic contacts or both. Because the mobile carriers and their parent donors are spatially separated, short-range ion scattering is nearly eliminated, which leads to mobilities that are characteristic of nearly pure semiconductors. A Schottky barrier is then used to modulate the mobile charge that in turn causes a change in the drain current. As a result of this heterolayer construction, the gate can be placed very close to the conducting channel, leading to large transconductances [1]. Figure 3.2 illustrates a schematic representation of a GaN/AlGaN HFET heterostructure in which the donors in the wider bandgap AlGaN or the polarization-induced free carriers when grown on polar surfaces provide the carriers. In an HFETdevice under bias, the carriers can also be provided by the source contact. 3.1.1 Electron Transport Properties in GaN and GaN/AlGaN Heterostructures

Electron mobility is a key parameter in the operation of n-channel FETs as it affects the access resistances as well as the rate with which the carrier velocity increases with electric field. Consequently, we will treat the low-field mobility in GaN and its dependence on various scattering events first. This will be followed by treating the two-dimensional electron gas mobility as it occurs in modulation-doped field effect transistors, which are the types of technological importance hitherto. Ultimately, electron mobility is limited by the interaction of electrons with phonons and, in particular, with optical phonons. This holds for bulk mobility as well as that in AlGaN/GaN modulation-doped field effect transistors. The room-temperature electron mobility values in bulk GaN grown with HVPE to a thickness of 60 mm was reported for GaN to be 950 cm2 V1 s1 [28]. Freestanding GaN templates grown by HVPE exhibited room-temperature electron mobilities approaching 1400 cm2 V1 s1 [29]. Further improved HVPE layers exhibit much higher mobilities on the surface of the layers, which approach those of the freestanding templates. Those reported for organometallic vapor phase epitaxy (OMVPE)-grown layers were in excess of 900 cm2 V1 s1 [30], although the temperature dependence of the mobility in this particular sample was rather unique. Early MBE layers exhibited mobilities as high as 580 cm2 V1 s1 on SiC substrates, which at that time were not as commonly used as in recent times [31]. Typically, however, the MBE-grown films produce much lower mobility values of 100–300 cm2 V1 s1 [32]. The lower mobilities have been attributed to both high dislocation densities [32–34] and elevated levels of point defects [35, 36]. Dislocations are considered by some to be an important scattering mechanism in films having dislocation densities above 1  108 cm2 [32, 33]. We should keep in mind that these are preliminary attributes and more detailed experiments coupled

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T-gate

Si3N4

S o u r ce

D r ain Alx Ga1-x N (e.g., x = 0.25, 30 nm)

2DEG

i-GaN AlN initiation layer Substrate (SI SiC)

(a) di

∆ Ec

Ec

Ec EF

E1

EF2

E0

ED

x =x 2

E Fi

x

x =0

Alx Ga1-x N

GaN

qV 2

qφb

-qV(x)

qVG

EF=0 EFi

(b)

dd Doped

Figure 3.2 (a) Schematic representation of an AlGaN/GaN modulation-doped field effect transistor with a T-gate HFET. (b) Schematic band structure of an AlGaN/GaN modulationdoped heterostructure (some undoped instead relying on the polarization charge only) in which the free carriers are provided to the

E0

E1

di Undoped GaN layer by the dopant impurities placed in the larger bandgap AlGaN barrier layer or by the polarization charge. The band bending in the barrier for the doped barrier case is represented by a solid line whereas that for the literally undoped barrier is shown with a broken line.

3.1 Heterojunction Field Effect Transistors

with detailed analyses are needed to confirm the proposed models. Depending on the particulars of the growth and substrate preparation, GaN films grown by MBE typically have dislocation densities in the range of 5  109–5  1010 cm2 [32]. With refined procedures, however, dislocation densities in the range of 8  108–2  109 cm2 can be obtained when grown directly on sapphire substrates with AlN or GaN buffer layers. Dislocation reduction, and other scattering centers that are inherently related to dislocations, is really the key to achieving high-mobility GaN, which goes to the heart of buffer layer and/or early stages of growth. Based on the premise that the [0 0 2] X-ray diffraction is affected by screw dislocations and the (10–12) peak by edge dislocations and the fact that RF-nitrogen-grown MBE layers produce excellent (0 0 0 2) peaks (in the 40–120 arcs range) while the (10
12) peaks are wider and weaker (in the 360–1200 arcs range on sapphire), one can conclude that the majority of the dislocations in MBE layers are the propagating edge type. The strength of MBE that is producing 2D-growth and low-temperature growth, does not bode well with dislocation reduction as the edge dislocations, which propagate along the c-axis, go right through the active portions of the sample since the dislocation mobility in GaN is small. The details of this simplified picture somewhat depend on the particulars of the growth such as the group V/III ratio and growth temperature, which are discussed in detail in Volume 1, Chapter 3. Some sort of 3D growth at the early stages of the growth, as in the case of growth from vapor, followed by a smoothing layer that also serves as an epitaxial lateral overgrowth, would help reduce dislocation density. The more viable option is to use HVPE or OMVPE buffer layers for MBE growth. This approach in one effort led to a record or near record bulk mobility of 1150 cm2 V1 s1 at room temperature and 53 500 cm2 V1 s1 at 4.2 K in a 2DEG layer [37]. Further work on 2DEG led to increased mobilities [38]. A 2DEG sample prepared by MBE on an HVPE template having an areal carrier density of 2.35  1012 cm2 exhibited a mobility of 75 000 cm2 V1 s1 [39], which was improved later to 167 000 in gated structures [40]. Ammonia MBE has also produced GaN with very high electron mobilities (about 60 100 cm2 V1 s1 at 4 K) when grown on bulk GaN wafers, which in turn were grown under high-pressure and hightemperature conditions [41, 42]. It is clear that the buffer layers grown by the vaporphase epitaxy method helps eliminate a large portion of the problem faced by MBE, that is, the poor quality of the buffer layer with its large edge dislocation content. The other long-standing obstacle for MBE, associated with the preparation of sapphire and SiC substrate prior to the growth, as in situ high-temperature treatment is not available, has been eliminated. In the case of sapphire, a high-temperature anneal in an O2 environment produces atomically smooth and damage-free surfaces [43]. In the case of SiC, some form of H2 etching at elevated temperatures removes the surface damage caused by polishing [44]. Provided that the sample surface preparation prior to growth is done well, controlling the Ga/N ratio and substrate temperature in MBE growth causes the dislocation density across the homoepitaxial interface to remain more or less identical to that in the vapor-phase grown template, allowing the other beneficial attributes of MBE to be brought to bear [45, 46]. Electron mobility is one of the most important parameters associated with the material with great impact particularly on electron devices. The temperature

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dependence of mobility and carrier concentration can be used to extract fundamental information regarding scattering mechanisms [47, 48], which is discussed in great detail in Volume 2, Chapter 3. Compared to the other III–V semiconductors such as GaAs, GaN possesses many unique material and physical properties, see Volume 1, Chapter 1. However, even with improved material, the material quality still remains an obstacle for a thorough investigation of carrier transport. The earlier transport investigations had to cope with poor crystal quality and low carrier mobility, well below predictions [49–52]. We should point out that unintentionally doped GaN exhibits n-type conduction with a typical electron concentration of 1017 cm3, with heavy compensation. Typical compensation ratios observed for OMVPE- and MBE-grown films are about 0.3, though a lower ratio of 0.24 was also reported for HVPE-grown crystals [53–55]. Compensation reduces the electron mobility in GaN for a given electron concentration. Another point that should be kept in mind is that GaN layers are often grown on foreign substrates with very different properties. The degenerate layer at the interface (caused by extended defects and impurities), spontaneous polarization at heterointerfaces, and piezoelectric (PE) effects all should be considered. Experiments show that, even for thick GaN grown by HVPE, the degenerate interfacial layer has an important contribution to the Hall conductivity, especially at low temperatures where the freeze-out occurs for the donors in bulk, leading to the domination by the interfacial layer [28, 56–58]. In these cases, the measured data must be corrected to extract meaningful data [53]. The typical extended defect density of GaN grown by various techniques is 109 cm2 [59]. In many cases, the dislocation and defect scattering may also limit the carrier mobility, especially at low temperatures [60, 61]. Finally, many material and physical parameters of GaN were not available for some of the previous simulations where those parameters were treated as adjustable parameters. Needless to say, reliable parameters are required for accurately calculating the electron mobility and interpreting experimental results. 3.1.2 Heterointerface Charge

A succinct mention of the polarization effect is made here for continuity, as an in-depth discussion has already been provided in Volume 1, Chapter 2. As mentioned in the Section “Introduction,” AlGaN/GaN heterostructures have been the subject of many investigations because of their potential use in high-temperature, high-power devices [6, 62–64]. Owing to large-band discontinuities and polarization-induced screening charge, interface-bound two-dimensional electron gas concentrations exceed 1013 cm2. Although polarization effects cause a redistribution of weakly bound and free charges, unlike implications suggested in the literature, they cannot directly produce free electrons to form a 2DEG [65–68]. In GaN-based systems, issues dealing with heterointerfaces must include a discussion of polarization. Polarization induces a field, which in turn affects the interface charge through screening so that mobile carriers move to where the fixed polarization charge with opposite polarity is. Because nitrides are large-bandgap materials, they tend to be n-type and the hole

3.1 Heterojunction Field Effect Transistors

concentration is extremely low. Consequently, the mobile carriers are normally electrons donated by intentional donors, donor-like defects, or contacts. The AlGaN barrier [69] or the surface [70] have been suggested as the source of electrons. Positive surface charge concept has also been suggested to account for the experimental observations in the form of dependence of the 2DEG density on the thickness and/or alloy composition of the AlGaN barrier [71, 72]. Typically, the AlGaN barrier is grown on a relatively thick GaN layer to form the basis for the 2DEG system in a Ga-polarity sample. The inherent lattice mismatch causes a biaxial tensile strain and the thermal mismatch causes a biaxial compressive strain in the growth plane. The resultant strain induces a macroscopic electric field in the polar material. In addition, due to the particular crystal structure of the wurtzite lattice, a spontaneous polarization field is also found in both materials even in the absence of strain. In most current heterostructures, where the growth takes place along the [0 0 0 1] direction, both spontaneous and induced polarizations are directed opposite to the growth direction. The effect of polarization field on the position of the band edges has been calculated by several groups [65, 69, 73–76]. Polarization-induced fields in Ga-polarity samples, just as any negative gate voltage-induced field in FETs, increase the conduction band edge in AlGaN barrier with distance from the interface. In the presence of free, weakly bound, and surface charges, the internal polarization field is screened by a redistribution of these charges. The surface states may be in the form of donor-like states, which donate their electrons to the lowest unoccupied energy states at the AlGaN/GaN interface. The holy grail of GaN/AlGaN heterostructures is the debate on the origin of the carriers, which end up at the interface. The observed dependence of the 2DEG density on the thickness and composition of the AlGaN barrier has been linked to surface donor states, the binding energy of which is roughly equal to the Schottky barrier height in n-type GaN [70]. Though this may point to the same surface states being responsible for a possible and weak Fermi-level pinning, this is not that notable on the surface of GaN. In addition, pinning of the surface Fermi level in n-type GaN requires electrons to be transferred from bulk donors to surface acceptor states, whereas an excess of surface donors is required to form 2DEG in an AlGaN/GaN heterostructure. This inconsistency can be resolved, however, by assuming that the surface defects are amphoteric (i.e., they can act as either acceptors or donors depending on the circumstances) [77]. The polarization fields present in nitride heterostructures are strong enough to shift the Fermi energy at the surface of the AlGaN barrier below the charge-state level of the surface defects for Ga-face growth. This causes the surface defects on the barrier to transform from being acceptor-like to donor-like surface defects, which can provide the electrons for the 2DEG at the AlGaN/GaN interface [77]. Hsu and Walukiewicz [68, 78] elaborated on the surface donor-like defect that is likely to form at the growth temperature and its manifestation as a source of carriers confined at the underlying interface between the AlGaN top layer and GaN below it. The model calculations appeared to be somewhat insensitive to parameters such as donor formation energy and surface Fermi level. One sensitive parameter is the strength of the polarization field, which will be discussed here.

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The Schr€odinger’s equation and Poisson’s equation can be used self-consistently to study the channel formation and current flow mechanisms in GaN-based HFET [79, 80]. Several approaches have been used to define the system; for example, the Hamiltonian used the Schr€odinger’s equation, namely, effective masses [81], kp expansion [82], and tight-binding expansion [83–86]. The use of sophisticated models such as kp or tight-binding is justified, perhaps even made necessary, by the complex wurtzite band structure, particularly for determining the valence band states. Thus, calculations of optical processes involving band-to-band transitions must consider the details of the band structure beyond the simple effective mass approximation (EMA) [87, 88]. However, when only the conduction band processes are of interest, EMA is still a very accurate means of determining the properties of interest. In fact, nitride-based semiconductors in the wurtzite structure possess a conduction band with a G minimum, which can be described reasonably well within such an approximation. Within the effective mass theory, the Schr€ odinger’s equation takes the form [81, 89, 90]

h2 d 1 d  ð3:1Þ y ðzÞ þ VðzÞym ðzÞ ¼ E m ym ðzÞ; 2 dz m ðzÞ dz m where m(z) is the (position-dependent) effective mass, Em is the eigenenergy for mth subband, ym is the wave function corresponding to this eigenenergy, z is opposite to c-direction, and V(z) is the potential energy that incorporates with conduction band discontinuity and static potential f by VðzÞ ¼ qfðzÞ þ DE C :

ð3:2Þ

Here, D EC is the conduction band discontinuity at the interface. For additional details, see Volume 2, Chapter 3. There are quite a few numerical approaches that can be used to solve the onedimensional Schr€odinger’s equation in a quasitriangular well (see Figure 3.3) among which are finite difference methods, variational methods, the Rayleigh–Ritz method, and moment method [91, 92]. Considering the programming complexity, Rayleigh–Ritz method might be a good choice to attain a good precision with iteration-based solution over moderate depth range. This method gives only finite numbers of the eigen wave functions and eigenenergy levels. In the GaN-based heterojunction system, let z be in the c-direction and z ¼ 0 represent the interface between the channel and the barrier. Then the mth wave function’s Fourier expansion over the range from the top surface (z ¼ d) to the bulk area of GaN (z ¼ a  d, a  d) could be expressed as

Figure 3.3 (a) Schematic band structure of a GaN/AlGaN heterostructure in which the AlGaN may or may not be doped (doped version is shown) with two-dimensional electron gas in the quasitriangular well in GaN; (b) the same showing the surface d charge, AlGaN bulk charge, and interface electron charge in GaN; (c and d) illustration of F(z) and V(z) along with the boundary conditions.

"

3.1 Heterojunction Field Effect Transistors

AlGaN

GaN ∆Ec

+ + + +

+

+

+

- - - - -

+

Ec

-

EF

Ev

(a) ρ (z)

nsf+δ (z)

ns = d . ND+ + nsf+

N D+

n(z)

(b) F(z) I

II

F(∞)=0 0 ε2F 2(0)-ε1F 1(0) ~ ∆ P

(c) V(z)

V(-d) ~ E SD

EF V 1(0)-V 2(0) = ∆ Ec

z=-d

(d)

z=0

z=a-d

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ym ðzÞ ¼


 npz anm sin : a n¼1

N X

ð3:3Þ

To determine coefficients anm, we substitute ym into Equation 3.1 and take the integral after multiplying by each orthogonal basis function. Doing so allows us to rewrite it as N X

Anm anm ¼ E m anm :

ð3:4Þ

n¼1

The matrix A is given by Anm





2 h ¼ 2

þ

np a 2 a

2

2 a

ad ð

d



 1 npz mpz sin sin dz m ðzÞ a a


ad ð

VðzÞsin d


 npz mpz sin dz: a a

ð3:5Þ

If effective mass is constant over the entire range, the first term in Anm will reduce to a d nm function. Equation 3.4 indicates that the expansion coefficients for the mth wave function, anm, are the mth eigenvector of matrix A, and Em is the corresponding eigenvalue. For a two-dimensional system, the density of state of each subband is constant, ignoring the effective mass variation from layer to layer, and it can be written as m ð3:6Þ r2D ¼ 2 : p
h Once the wave function is numerically calculated, the free electron distribution n is then obtained from 1 Xð

nðzÞ ¼

m

r2D f ðEÞjym ðzÞj2 dE:

ð3:7Þ

Em

If we take f as the Fermi–Dirac distribution function, Equation 3.7 will become m X nðzÞ ¼ 2 p
h m dE ¼

1 ð

Em

jym ðzÞj2 1 þ exp½ðEE F Þ=kB T


 m kB T X E F E m  jym ðzÞj2 : ln 1 þ exp kB T p
h2 m

ð3:8Þ

And the total sheet electron concentration will be 1 ð

ns ¼

nðzÞdz ¼ 0


 m kB T X E F E m : ln 1 þ exp kB T p
h2 m

ð3:9Þ

3.1 Heterojunction Field Effect Transistors

The entire discussions above relies on solving the Schr€ odinger’s equation, which depends on the knowledge of potential energy V(z). Considering Equation 3.2, the static potential f(z), electrical field F(z), and charges can be correlated by Poisson’s equation. In a multilayer two-dimensional electron system, if we consider both shallow and deep donors/acceptors in the space charge term, then within each layer the one-dimension Poisson’s equation is given by   X X d½DðzÞ d½FðzÞ d2 fðzÞ þ  ¼ q pðzÞnðzÞ þ N Di  N Ai ; ¼e ¼ e dz dz dz2 i i ð3:10Þ where e is the dielectric constant, n(z) and p(z) are the electron and hole distribution, þ and N  respectively, and N Di Ai are the ionized donors and acceptors density, respectively, with different ionization energy. To solve it, the boundary conditions between each of the layers are of the most important factors. In nitride semiconductors with wurtzite phase, spontaneous and piezoelectric polarization effects are present in the polar direction [93], which must be taken into account in the balance of the boundary condition at the interface. And a sheet charge of donor states is present at the surface, which is assumed to supply electrons in the 2DEG in the absence of contacts. In this respect and for an AlGaN/GaN structure, the schematic band structure, charge distribution, and field and potential distribution are shown in Figure 3.3. The conservation of the normal component of the electrical displacement leads to e2  F 2 ðz ¼ 0Þe1  F 1 ðz ¼ 0Þ ¼ P1 P2 ¼ DP:

ð3:11Þ

In region I characterized by the AlGaN layer, the Poisson’s equation can be written as e1

dF 1 ¼ qnsfþ dðz þ dÞ þ qN Dþ : dz

ð3:12Þ

Here, nsfþ is the density of ionized surface donor states. An integration of which leads to the electric field in the same region as F1 ¼

q þ q n þ N Dþ ðz þ dÞ þ C; e1 sf e1

ð3:13Þ

with C being the integration constant. Doing the same for region II leads to e2

dF 2 ¼ qnðzÞ dz

and F2 ¼ 

ð ð q mC kB Tq X z jyl ðxÞjdx 2 nðzÞdz þ C0 ¼  e2 pe2 0 l


 E F E l  ln 1 þ exp þ C0 : kB T

ð3:14Þ

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The charge neutrality condition requires that nsfþ þ N Dþ d ¼ ns

1 ð

with ns ¼

nðzÞdz:

ð3:15Þ

z¼0

F1 at the interface will be F 1 ðz ¼ 0Þ ¼

q þ q q n þ N Dþ d þ C ¼ ns þ C: e1 sf e1 e1

ð3:16Þ

Noting that F2(z ¼ 0) ¼ C0 , from Equation 3.11 we can determine C0 by C0 ¼

DP þ Ce1 þ qnS : e2

ð3:17Þ

Note Ð z that the electron concentration in the barrier is very small, so for large values of z, 0 jyl ðxÞj2 dx ¼ 1. The constant C can be determined by noting that the electric field in the bulk of GaN vanishes, that is, F2(z ! 1) ¼ 0 as C¼

DP : e1

ð3:18Þ

The electric field in regions I and II can then be expressed as q þ q DP qðN Dþ z þ ns ÞDP nsf þ N Dþ ðz þ dÞ ¼ ; e1 ð e1 e1 e1 q qns : F2 ¼  nðzÞdz þ e2 e2

F1 ¼

ð3:19Þ

For the AlGaN/GaN heterojunction, which has an AlN interfacial layer, there are similar relationships among the electrical fields in each of the three regions, which are described in the Appendix. When a Schottky metal is present at the surface, which is the case in HFETs, then the surface states will be depleted and the potential V(z ¼ d) at the surface will be fixed by the barrier height and applied voltage. Without the Schottky metal, the surface condition is not really well known and is still under debate. One easy approach, regardless the contribution from surface donor states, is to assume that all the electrons are provided from the barrier. Note that this may be valid at high doping or relatively thick barriers. In this case, the surface region is assumed to remain undepleted, and V(z ¼ d), corresponding to EF level at surface, could be determined by simple semiconductor carrier statistics. However, a more prevalent viewpoint is that the surface donor states are the main source of electrons [69, 70], a topic that requires further investigation. At the surface, the total number of surface states could be enormously large, as was noted in the venerable Si MOSET case many decades ago [94]. The ionized donor states at the AlGaN surface have been reported to reach 1013 cm2 eV1 with distributed energy within a range 1–1.8 eV below the conduction band [95]. Including surface-state distribution and the associated statistics will add extra complexity to the self-consistent method. Similar to that proposed in Ref. [96], we can take a simplifying treatment as follows: Neglecting the hole and acceptor concentrations in the

3.1 Heterojunction Field Effect Transistors

semiconductor and considering only the surface donor states, the neutrality of total charges leads to ð Xð þ N Di nsfþ þ dz ¼ ns ¼ nðzÞdz; ð3:20Þ i

nsfþ

is the density of ionized surface donor states, which follows the where Fermi–Dirac distribution given that only one single energy level, ESD, is present: nsfþ ¼

nsf : 1 þ g exp½ðE F E SD Þ=kB T

ð3:21Þ

The degeneracy factor g is 2 and surface donor state’s energy level is approximately 1.2 eV below the conduction band according to scanning Kelvin probe microscopy (SKPM) measurement [97]. The density of total surface donor state Nsuf can be assumed to be up to 1015 cm2, sufficient to pin EF at the surface donor energy level. The boundary condition for V(z ¼ d) (EF at the surface) must be solved by combining Equations 3.20 and 3.21. In short, the carrier, potential energy, and built-in field distributions are interrelated through the Schr€odinger’s and Poisson’s equations. A self-consistent procedure is therefore required to solve the problem. The basic flow chart of the numerical calculation procedure is shown in Figure 3.4. Starting with a trial potential energy V1(z), the Schr€odinger’s equation is solved first to obtain the wave functions and eigenenergy levels. Then, the carrier distribution n(z) and sheet density ns can be calculated as the input to the Poisson’s equation. The built-in electric field F(z) is now determined from the Poisson’s equation incorporating the polarization charge in the boundary conditions. The newly obtained potential energy function V2(z) is then compared with the starting value. If the difference between the two is within a prespecified acceptance level, one then ends the iteration. Otherwise, V1(z) is updated with the new one and the iteration is resumed. The selection of trial potential energy and updating algorithm are tricky in order to get a fast convergence. Figure 3.5 shows a self-consistently calculated potential energy (conduction band diagram) and free electron distribution for an Al0.3Ga0.7N/GaN heterojunction. During the simulation, the surface-state density was assumed to be extremely large, 1015 cm2, with single-donor energy level located at 1.2 eV to make sure a sufficient number of electrons are available from this source alone for the system to reach equilibrium. In addition, assumed shallow and deep donors in the barrier layer have concentrations of 8  1017 and 5  1017 cm3, respectively. The detailed spontaneous and piezoelectric polarization calculations for AlGaN and GaN could be found in Section 3.1.4. In short, the roomtemperature total sheet electron density is approximately 1  1013 cm2, depending on the particulars of the structure, and the 2DEG is confined within a 5 nm channel under the interface. The eigenenergy and wave function of first three subbands are also calculated. Experiments also revealed the similar range for ground energy (100 meV) of 2DEG by Shubnikov-de Haas (SdH) measurements [98, 99].

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Start with a trial potential V1 (z)

Solve Schrödinger equation, findΨ (z), Ei

Calculate electron distribution n(z), ns

Update V1(z)

Determine boundary conditions, ∆P and potential at surface

Solve Poisson equation to get F(z) and Φ (z)

Calculate new potential V2(z), compare the difference e(z) = V2(z)-V1(z) N e(z) < a Y End Figure 3.4 The flow chart of the self-consistent calculation procedure.

19

5 ×10

2.0 0.06

E = -89.45 meV 1

2

Ψ1

19

4 × 10

2

Ψ2

E = 49.37 meV 3

0.02

2 ψ3

19

3 × 10 0

s

Ec (eV)

1.0

-3

0.04

2

n (cm )

E =12.11 meV

1.5

-0.02 -10

0.5

E 0.0

0

z (nm)

10

20

F

19

2 × 10

19

1 × 10

-0.5

0 -20

-10

0

10

20

30

40

z (nm) Figure 3.5 Calculated conduction band profile, electron distribution, and eigenenergy/wave function for the first three subbands in an Al0.3Ga0.7N/GaN heterojunction.

3.1 Heterojunction Field Effect Transistors

From the band structure across heterojunction obtained from numerical calculation, the potential energy V(z) in the channel area can be approximated as a triangular potential well, so that analytical solution to Schr€ odinger’s equation could be formulated. In terms of the Hamiltonian, the Schr€ odinger’s equation can be expressed as H ¼ H0 þ eF z z ¼ 


2 q2 h þ eF z z: 2mz qz2

ð3:22Þ

Nonparabolicity may induce deviations from the simple parabolic band model. However, this will not substantially change the results that are presented here. For a triangular potential barrier, which can approximately represent the potential distribution at the heterointerface with a 2D charge being present, we can use the following boundary conditions:  DP qF z z; z > 0; VðzÞ ¼ with F z ¼ ; ð3:23Þ 1; z
0; e where Fz is the electric field and z is the distance along the growth direction, normal to the interface, being 0 at the interface. The solution of the wave function is given by [100, 101]  
 2mz eF S Ei yðzÞ ¼ Ai z ; ð3:24Þ eF z
h2 with Ai() being the Airy function given by 1 AiðuÞ ¼ p

1 ð


 1 cos t3 þ ut dt: 3

ð3:25Þ

0

The eigenvalues for energy is
Ei


2 h 2mz


 1=3  3peF z 3 2=3 iþ 2 4

ð3:26Þ

with the (i þ 3/4) replaced by 0.7587, 1.7540, and 2.7575 for the first three lowest subbands, respectively, for the exact eigenvalues. The parameter i takes values of i ¼ 0,1,2,3 with 0 representing the ground state and the rest the excited states. The average value for z, where the 2DEG can be approximated to be, for one subband occupation can be found from ð 2E i < z> ¼ y zy dz : ð3:27Þ 3eF z A variational method introduced by Fang and Howard [102] can be used as well in which case y(z) ¼ Az exp[az], where A is a normalization constant and a is a variationalÐ parameter, as a wave function of ground state. From the normalization condition y ðzÞyðzÞdz ¼ 1, the normalization constant A can be found as A ¼ 2a3/2.

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The expectation value for the total energy is given by Ð hEi ¼ y ðzÞHyðzÞdz
 Ð 
h2 q2 h2 a2 3 eF z
¼ y ðzÞ   2 þ eF z z yðzÞdz ¼ : þ 2mz qz 2mz 2 a

ð3:28Þ

d The variational parameter can be found from da hEi ¼ 0, that is, 2
h2 a3 3mz eF z ¼ 0. Then wave function of the ground state is
 1=3 3 mz eF z 3=2 yðzÞ ¼ 2ðaÞ z exp½az with a ¼ : ð3:29Þ 2
h2

The average position of electrons in the ground state is ð 3 3 3
h2=3 ¼ ¼  hzi ¼ y ðzÞzyðzÞdz ¼   1=3 : 1=3 2a 2 3 m eF z =
12m eF z h2 2

z

ð3:30Þ

z

(See Volume 2, Chapter 3 for a more in depth treatment.) Assuming that only the ground state is occupied at an Al0.2Ga0.6N/In0.1Ga0.9N interface with 0.0055 C m2 interfacial charge, the electric field is Fz ¼ 5.7068  107 V m1, the variational parameter a ¼ 5.9819  108 m1, and the average position of electrons is hzi ¼ 2.5 nm from the interface. Here, effective masses used for GaN and InN are 0.2 and 0.11, respectively, and a linear interpolation is used to find the effective mass of In0.1Ga0.9N. To reiterate, in the nitride semiconductors with wurtzite phase, spontaneous and piezoelectric polarization effects are present [93], which necessitate that the Poisson’s equation be solved for the displacement field, D(z), of Equation 3.10. In the selfconsistent procedure, potential V is obtained by using Equation 3.10 from an initial guess of the mobile charge concentration and then inserted into the Schr€ odinger’s equation, Equation 3.1, which is solved to get the energy levels and wave functions of the systems. The new electron charge density is obtained by applying Fermi statistics as described by Equation 3.9. The calculated density is then plugged into Poisson’s equation (Equation 3.10) and the iteration repeated until convergence is achieved. Convergence of the self-consistent algorithm can be improved adopting special relaxation techniques. Here, a first-order expansion of the model reported in Ref. [103] is used. 3.1.3 Electromechanical Coupling

In the above treatment, we considered the effect of strain in the AlGaN barrier due to lattice mismatch with the underlying and presumed to be strain-free GaN. However, we did not include the effect of electric field induced-strain due to the piezoelectricity of the AlGaN, often referred to as the electromechanical coupling. Therefore, the Hooke’s law and displacement used in the uncoupled case must be replaced with the coupled pair. The coupled formulation is based on the linear piezoelectric constitutive equations for stress and electric displacement [104],

3.1 Heterojunction Field Effect Transistors

sij ¼ Cijkl ekl ekij F k ; Di ¼ eijk ejk k ij F j þ PSi ;

ð3:31Þ

where sij is the stress tensor, Cijkl is the fourth-rank elastic stiffness tensor, ekl is the strain tensor, eijk is the third-rank piezoelectric coefficient tensor, k ij is the secondrank permittivity tensor, Di is the electric displacement, Fk is the electric field, and PSi is the spontaneous polarization. The indices i, j, k, and l run over the Cartesian coordinates 1, 2, and 3 (x, y, and z). Einstein’s rule of summation over repeated indices is implied. The symmetry of the wurtzite crystal structure of GaN and AlGaN reduces the number of independent elastic and piezoelectric moduli. In the devices considered here, the crystals are grown with the c-axis normal to the surface in the z-direction, we make the common assumption that the thick GaN layer is unstrained, and the biaxial strain of the thin AlGaN layer satisfies exx ¼ eyy ¼ (aGaN  aAlGaN)/ aAlGaN, where aGaN and aAlGaN are the c-plane lattice constants of each material constituting the heterojunction. The absence of stress along the growth direction (z-direction) in the barrier AlGaN layer, allows us to express the strain along the growth direction as [105] ez ¼ 2

C13 e33 AlGaN exx þ E ; C33 C33 z

ð3:32Þ

where Ez AlGaN is the z-directed electric field in the barrier AlGaN layer and we have expressed the elastic and piezoelectric moduli in matrix notation [106]. The equivalent sheet charge density due to PE polarization in AlGaN is given by [105]
 C13 e2 ¼ 2exx e31  e33 þ 33 F AlGaN : ð3:33Þ PAlGaN PE C33 C33 z The electric field on both sides of interface follows the boundary condition described by Equation 3.11, while the field on the channel side can be deduced from the Poisson’s equation by neglecting the impurity and hole concentration as follows: 1 ð

1 ð

dFðzÞ ¼ z¼0

z¼0

q h kGaN

dz 

qðzÞnðzÞ þ

X

þ N Di 

X

i

q k GaN

1 ð

nðzÞdz ¼  z¼0

i

i

s2 DEG k GaN

with Fð1ÞFð0Þ ¼ 0Fð0Þ ¼ F GaN ¼ z ; s2 DEG ¼ k GaN F GaN z

N Ai

s2 DEG k GaN

or ð3:34Þ

where k is used as the permittivity of material, instead of the usual e, to avoid confusion with strain, and s2DEG is the two-dimensional electron gas charge density. Therefore, Equation 3.11 will become

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s2 DEG k AlGaN F AlGaN ¼ PAlGaN P GaN ¼ P AlGaN DPSP ; z PE

ð3:35Þ

AlGaN where DPSP ¼ P GaN represents the differential spontaneous polarization SP P SP between the AlGaN barrier and the GaN channel. When we put the modified term of AlGaN piezoelectric polarization into Equation 3.35, we can get the electrical field at AlGaN side:


 1 C 13 s þ DP 2e e  e : 2 DEG SP xx 31 33 C 33 kAlGaN þ e233 =C33

F AlGaN ¼ z

ð3:36Þ

Furthermore, the piezoelectric polarization in AlGaN can be rewritten as
PAlGaN PE

¼ 2exx

 C13 e31  e33 ð1aÞ þ aðs2 DEG þ DPSP Þ; C33

ð3:37Þ

where a ¼ ðe233 =C33 Þ=ðk AlGaN þ e233 =C33 Þ and represents the electromechanical coupling. Setting a ¼ 0 brings us back to the polarization charge with no coupling that can be obtained from Hooke’s law and piezoelectric polarization in an uncoupled fashion. Taking the electromechanical coupling effect into account, the band structure of AlGaN/GaN is shown in Figure 3.6. Also the total sheet carrier concentration will change from 1.19  1013 to 1.17  1013 cm2. There is only a slight difference between simulations with and without the coupling effect. This is due to the small built-in electrical field in the barrier layer ðF AlGaN Þ in the absence of Schottky metal z and applied voltage on the surface.

1.5

1.0

w coupling

0.5

c

E (eV)

w coupling

E 0.0

-0.5 -20

F

-10

0

10

20

z (nm) Figure 3.6 Conduction band profile with and without the electromechanical coupling in effect in an Al0.3Ga0.7N/GaN heterojunction.

3.1 Heterojunction Field Effect Transistors

3.1.4 Analytical Description of HFETs

To qualitatively demonstrate the effect of charge stored at the heterointerface on mobility and carrier velocity, we present in the following an analytical description of the operation of HFET. For the sake of simplicity, we avoid considering the quantitative structural and device analyses and present a model that accounts primarily for the basic and important features of HFETs [2]. The model, which is developed for the GaAs/AlGaAs system and does not account for polarization charge is based on the concept that the amount of charge, which is depleted from the barrier donor layer, is accumulated at the interface, while the Fermi level is kept constant across the heterointerface. Specific issues arising from the use of GaN, particularly the polarization issues, will later be added to the model. The electron sheet charge with no external gate bias (or hole charge in the case of p-channel HFET, which is very unlikely in the case of GaN) provided by the donors in the barrier layer may be given by ns0 ¼

 1=2 2e2 N d

DE C E F2 E Fi þ N 2d d2i N d di ; q

ð3:38Þ

where EF2 is the separation between the conduction band in the barrier layer and the Fermi level, Nd is the donor concentration in the barrier layer, e2 is the dielectric constant of the barrier layer, q is the electronic charge, DEc is the conduction band discontinuity, EFi is the Fermi level with respect to the conduction band edge in the channel layer, and di is the thickness of the undoped layer in the barrier layer at the heterointerface. A graphical description of the aforementioned parameters and the band-edge profile for an AlGaN/GaN heterostructure with gate bias on the surface is shown in Figure 3.2. The electron charge stored at the heterointerface is given by n on o r ns ¼ ln 1 þ ebðE Fi E 0 Þ 1 þ ebðE Fi E 1 Þ ; ð3:39Þ b 2=3

2=3

where b ¼ q/kT and E 0 ¼ g 0 ns and E 1 ¼ g 1 ns are the positions of the first and the second quantum (potential) states at the interface, respectively. These states correspond to a triangular well formed by the interfacial stored charge. The energy reference is the bottom of the conduction band edge in GaAs. It is assumed here that these lowest energy states are the only ones that are either filled or partially filled. The constants g 0 and g 1, which depend on the effective mass of the channel material used, and the density of states r (r ¼ q m/ph2, where m is the effective mass of electrons and h is the Planck’s constant) are derived on the basis that the quantum well may reasonably be triangular in shape. Depending on the value of the applied voltage, the gate on the surface of the barrier layer depletes some or all of the stored charge at the interface. Thus, only a simultaneous solution of Equations 3.38 and 3.39 can result in the determination of the Fermi level provided the interface sheet charge concentration is known. The determination of the sheet charge concentration can similarly be carried out from the same

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equations if the Fermi level is known. With a gate voltage present, Equation 3.38, which depicts the equilibrium situation, must be replaced by 
 e2 1 1 V G  fb V p2 þ E Fi  DE C ; ns ¼ ð3:40Þ qd q q where fb is the Schottky barrier height of the gate metal deposited on the barrier layer, VG is the applied gate-to-source bias voltage, dd is the thickness of the doped barrier layer, d ¼ dd þ di and V p2 ¼ qN d d2d =2e2 , and e2 represents the dielectric constants in the barrier (AlGaN). Similar treatments that have their roots in the models developed for AlGaAs/GaAs HFETs but are modified to take the polarization charge into account can be found in the chapter by Karmalkar et al. [107] and a paper by Rashmi et al. [108]. In addition, a MESFET model, which is somewhat applicable to parts of this treatment, can be found in Ref. [109]. The interface charge concentration in the presence of a gate bias may be expressed by

e2 ns ¼ V g V off ; ð3:41Þ qðd þ DdÞ where Dd ¼ e2aF/q and for the case without polarization-induced interfacial charge, 1 1 V off ¼ fb  DE C V p2 þ DE F0 ; q q

ð3:42Þ

Dd is typically about 2–4 nm for GaN and 8 nm for GaAs and represents in pictorial terms the location of the peak density of the 2DEG from the heterointerface. In addition, the terms DEF0 and aF are determined from the extrapolations. For example, DEF0, which is a temperature-dependent quantity, is the residual value of the Fermi level for zero interface sheet density obtained by extrapolating linearly from the linear region of the curve, as shown by dashed lines in Figure 3.7 for two temperatures, 77 and 300 K for a GaAs/AlGaAs heterointerface. Under ideal conditions, Voff would represent the threshold voltage. However, experimentally, the current and through it the free sheet charge in the channel is plotted against the gate voltage and the extrapolation of the linear region to zero charge would result in the threshold voltage. Owing to substantial subthreshold voltage, the gate voltage leading to nearly zero current flow in the channel would be smaller (in absolute value, it is larger for an n-channel FET). The parameter aF represents the slope of the curve, which is reasonably linear for a wide range of sheet charge except near the vanishing values, relating the Fermi level to the sheet charge. Utilization of this displaced linear approximation leads to the interface Fermi level to be expressed as E Fi ¼ DE F0 ðTÞ þ aF ns

ð3:43Þ

For example, for the GaAs/AlGaAs system, aF 0.125  1016 V m2 and DEF0 0 at 300 K and 0.025 meV at T
77 K. Similar figures can be obtained for the GaN/ AlGaN system neglecting, for the time being, the effect of the polarization charge at the interface.

3.1 Heterojunction Field Effect Transistors

Fermi potential (V)

0.3

0.2 4K 0.1

77 K

0.0

300 K -0.1 0

5

10

15

20

Areal carrier density (10 11 cm -2) Figure 3.7 Interface Fermi potential (EFi) versus the sheet carrier concentration for a GaAs/AlGaAs heterointerface with 2DEG shown for a pictorial view of the parameters used in derivations.

In a field-effect transistor, the drain bias produces a lateral field. For long-channel devices and/or for very small-drain biases, it is generally assumed that the channel voltage, which varies along the channel between the source and the drain and finally reaches a value equal to the drain voltage, is added to the gate potential. When it is done, Equation 3.41 becomes a function of the distance, x, along the channel: ns ¼

e2 fV G V off VðxÞg; qðd þ DdÞ

ð3:44Þ

with V(x) as the channel potential. For a GaN-based modulation-doped FET including the variety dependent only on the polarization-induced charge, the terms Dd and DEF0 will be neglected in which case one can relate the sheet carrier concentration to the gate voltage above threshold as

e2 e2 ns ¼ V g V off or the interface charge Q s ¼ ðV g V off Þ ð3:45Þ qd d and with interfacial polarization charge of np and again neglecting the effect of the Dd and DEF0 on the threshold voltage, but including the charge due to bulk doping NBWd (donor level and depletion layer thickness in the channel layer), one gets 1 qd V off ¼ fb  DE C V p2  ðnp þ N B W d Þ: q e2

ð3:46Þ

It is assumed that the doped layer thickness is very small compared to that of the AlGaN barrier because Equation 3.46 lumps the bulk charge as interfacial charge of equal amount without considering its distributed nature and assumes its distance to the gate metal to be the same as the thickness of the barrier layer. The polarization induced charge can be determined using the methodology discussed in Volume 1, Chapter 2, but suffice it to say that it is about 1013 cm2 under equilibrium conditions.

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For a long-channel MESFET, the case is similar in HFETs in terms of the expressions that follow. The current–voltage relationships can be found in many elementary device texts. Suffice it to say that the drain current in an n-type longchannel device before saturation, where constant mobility can be assumed, is given by "

 # V D 2 V G 3=2 2 V D V G 3=2 ID ¼ G0 V p ; ð3:47Þ þ  Vp Vp 3 Vp 3 where G0 and Vp represent the full channel conductance (assuming no depletion at all) and total gate voltage inclusive of the gate built-in voltage, required to pinch-off the channel, respectively. They are given by G0 ¼

qN D maZ L

and V 2p ¼

qN D 2 a ; 2e

ð3:48Þ

where ND is the channel-doping level assumed to be equal to the electron concentration (should be the electron concentration when these two parameters deviate substantially), Z is the width of the gate, L is the gate length, a is the total channel thickness, and e is the dielectric constant of the channel. For an n-channel device, the applied gate voltage VG < 0. In current saturation regime, as defined by pinch-off due to the reverse-biased gate–drain junction in which case VD  VG ¼ Vp, the drain current is given by " #
 V G 2 V G 3=2 1 þ þ : ð3:49Þ IDS ¼ G0 V p Vp 3 Vp 3 Equation 3.47 is applicable below saturation and Equation 3.49 is applicable in saturation. In Section 3.5.5, the “anomalies section,” the expressions will be modified to include any depletion caused from the substrate or the buffer layer side due to carrier trapping in those regions. Returning to modulation-doped FETs, for small values of V(z), it may be assumed that the constant mobility regime is inapplicable and that ID ¼ qns mZ

dVðxÞ mZe2 dVðxÞ fV G V off VðxÞg ¼ ; d dx dx

ð3:50Þ

where m is the charge carrier mobility and Z is the width of the gate. By integrating Equation 3.50 from the source to the drain while keeping in mind that the drain current remains constant throughout the channel and V (x ¼ 0) ¼ 0; V (x ¼ L) ¼ VDS, one obtains   mZ e2 V2 ðV G V off ÞV DS  DS ¼ bd ½V Geff V DS V 2DS =2; ð3:51Þ IDS ¼ 2 L d where VGeff ¼ (VG  Voff ), and VDS is the drain–source voltage, bd ¼ mZe21/dL, and L is the intrinsic channel length or, in practical terms, the gate length. The current reaches saturation when the drain voltage is increased to the point where the field in the channel exceeds its critical value thereby causing the velocity to saturate.

3.1 Heterojunction Field Effect Transistors

Under these circumstances and utilizing Equation 3.44, the drain current may be calculated by following the steps IDS ¼ qZvsat ns ¼

e2 Zvsat fV G V off V DSS g ¼ bd V 0 ½V Geff V DSS ; d

ð3:52Þ

where VDSS is the saturation drain voltage, IDS is the saturation current, V0 ¼ vsatL/m, and vsat is the saturation velocity. For GaN devices, Dd can be neglected. Because the maximum of the 2DEG is only about 4 nm from the interface that is smaller than the thickness of the barrier layer, which is typically greater than 20 nm. Equating Equations 3.51 and 3.52 at the saturation point, one can solve for saturation drain voltage as V DSS ¼ V G þ V 0 ðV 20 þ V 2Geff Þ1=2 ; which when substituted back into Equation 3.52 yields 2( 3
2 )1=2 V Geff 15: IDS ¼ bd V 20 4 1 þ V0

ð3:53Þ

ð3:54Þ

The treatment presented above is known as the two-piece model, implying that an abrupt transition takes place from the constant mobility regime to the constant velocity regime. A more accurate picture is one in which this transition is smoother allowing the use of a phenomenological velocity–field relationship for a more accurate description of the HFET operation. The simplest of all these pictures is one that neglects the peak in the velocity–field curve and assumes Si-like velocity–field characteristics. One such characteristic may be expressed by v¼

mFðxÞ m0 FðxÞ ; ¼ 1 þ mFðxÞ=vsat 1 þ ðFðxÞ=F C Þ

ð3:55Þ

where F(x) represents the electric field in the channel and is equal to (F(x) ¼ dV(x)/ dx), m is low field mobility, and FC ¼ vsat/m is the electric field at the saturation point. It may be noted that this field is not constant throughout the channel. Recognizing that e2 ð3:56Þ ID ¼ Q total ðxÞZvðxÞ; where Q total ¼ ðV G V off VðxÞÞ; d we get for the drain current ID: ID ¼

Ze2 vðxÞ ½V G V off VðxÞ: d

ð3:57Þ

Using the expression for v(x) given in Equation 3.55 above, Equation 3.57 may be simplified as   Ze2 mvsat dVðxÞ=dx ID ¼ fV G V off VðxÞg ; ð3:58Þ d vsat þ mdVðxÞ=dx where vsat ¼ mFC, FC being the field where the velocity assumes its saturation value. By integrating Equation 3.58 from the source end (x ¼ 0) of the channel to the drain

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end of the channel (x ¼ L), while keeping in mind that the drain current must be constant throughout the channel, one may obtain an expression for the drain current in the framework of a procedure elucidated by Lehovec and Zuleeg [110]. One, thus, obtains   Ze2 mvsat V Geff V D V 2D =2 ID ¼ d vsat L þ mV D    1 mZ e2 V 2D ¼ ðV G V off ÞV D  : ð3:59Þ 2 1 þ mV D =vsat L L d Note that when the saturation velocity, vsat, approaches infinity, Equation 3.59 reduces to Equation 3.51, which is valid for the constant mobility case, and corroborates the gradual channel approximation, which is valid for long-channel HFETs. Following the procedure of Lehovec and Zuleeg [110], using Equations 3.58 and 3.59, and assuming velocity saturation the drain saturation current, IDS may be determined as IDSS ¼

2 2ðV G V off Þ2 Zem  1 þ f1 þ xd g1=2 ; Lðd þ DdÞ

ð3:60Þ

where xd ¼

2mðV G V off Þ : vsat L

ð3:61Þ

Note that when V0  (VG  Voff), drain current saturation occurs due to pinch-off, but not due to velocity saturation as is expected for the long-channel devices. The transconductance is an important parameter in HFETs and is defined by
 qI D gm ¼ : ð3:62Þ qV G V D ¼ const For the saturation regime, the transconductance may be expressed by
 1 qI DSS 2ZemðV G V off Þ sat gm ¼ ¼ fð1 þ xd Þ þ ðð1 þ xd Þ1=2 Þg : qV G V D ¼ const ðd þ DdÞL ð3:63Þ The maximum transconductance is obtained when the sheet charge density is fully undepleted under the gate, which leads to g max m

"   #1=2 Zqmns qmns ðd þ DdÞ 2 1þ ¼ : L evsat L

ð3:64Þ

For very short gate lengths, which occur essentially in all modern HFETs, the second term in the bracket dominates, and Equation 3.64 reduces to g max m

Zevsat : d þ Dd

ð3:65Þ

3.1 Heterojunction Field Effect Transistors

The measured transconductance is actually smaller than that given by Equation 3.65 in that the source resistance, which will be defined shortly, acts as a negative feedback. Taking the circuit effects into account, the measured extrinsic transconductance may be given by 

g max m

 ext

¼

g max m : 1 þ RS g max m

ð3:66Þ

3.1.4.1 Examples for GaN and InGaN Channel HFETs Let us now consider cases with GaN and InGaN channels having either AlGaN or InAlN (can be lattice-matched to GaN) barrier layers. If the GaN channel is grown on a thick GaN buffer layer, it can be assumed relaxed. Therefore, no piezoelectric charge would be induced by GaN. If the InGaN channel is made thin enough for it to be coherently strained, it would be under compressive strain and there would be a piezoelectric polarization. The AlGaN layer on the GaN channel layer would be under tensile strain leading to piezo polarization charge. If an InAlN barrier is used, it can be made lattice matched to GaN in which case the piezo component of the polarization would not exist. However, it can be grown with smaller In concentration with the roughly 18% matching composition for it to be under tensile in-plane strain or with larger In concentration for it to be under compressive in-plane strain. For simplicity, let us assume that in the case of InGaN channel, both AlGaN and InGaN layers assume the in-plane lattice constant of GaN. This example is depicted in Figure 3.8, which also shows the compositional gradient induced spontaneous polarization. Briefly, tensile-strained AlGaN and compressively strained InGaN would have piezoelectric polarization as indicated in Figure 3.8. In addition, all three layers would have spontaneous polarization at each compositional gradient, again as shown in Figure 3.8 where the length of the arrows represents the relative values. We should point out that the spontaneous polarization is negatively larger in InGaN than GaN, which means that InGaN on GaN interface would attract holes that are not assumed present in the system and therefore neglected. However, that at the AlGaN/GaN interface and also that at the AlGaN/InGaN interface would attract electrons. Let us calculate the output I–V characteristics for an AlGaN/InGaN HFET and AlGaN/GaN HFET. The layer structure consists of a GaN buffer that is relatively thick and relaxed, the InGaN channel that is 20 nm thick, and the top layer is AlGaN, part of which next to the interface is undoped. Let us neglect the effect of strain on the band structure while using the appropriate bowing parameters for bandgap calculations. Let us use the linear interpolation for determining the spontaneous and piezoelectric polarization charge, although not strictly correct (see Volume 1, Chapter 2 for an accurate treatment). The mole fraction in AlxGa1xN is 20% (x ¼ 0.2), the doping level in AlxGa1xN is 1018 cm3 (ND), and the thickness of the doped AlxGa1xN is 20 nm. The thickness of the undoped AlxGa1xN is 2 nm. In the case of InyGa1yN channel, the composition y ¼ 0.10 and its thickness is 20 nm. The gate length is 1 mm and centered in a 3 mm channel. The gate width is 100 mm and the Schottky built-in voltage is 1 V. Let us

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P sp/AlGaN

P pe/AlGaN

AlGaN

P sp/InGaN

P pe/InGaN

InGaN

P sp/GaN

[0001]

P sp/AlGaN

GaN(relaxed)

P pe/AlGaN

P sp/GaN

AlGaN GaN channel GaN(relaxed)

Figure 3.8 Schematic representation of an AlGaN/GaN and an AlGaN/InGaN channel structure indicating the piezo and spontaneous polarization under the assumption that both InGaN and AlGaN assume the in-plane lattice constant of GaN in which case InGaN would be under compressive and AlGaN would be under tensile strain.

assume saturated velocity of 1  107 cm s1 and to make the picture simpler, let us also assume that the access resistances are zero. Using the values reproduced from Volume 1, Chapter 2 in Table 3.1, let us calculate the polarization induced charge in Al0.2Ga0.8N/In0.1Ga0.9N/GaN and Al0.2Ga0.8N/ GaN systems. Using a linear interpolation for the lattice constants, piezoelectric constants (which is not strictly true as there is bowing, see Volume 1, Chapter 2 for a more accurate treatment), and elastic constants, one gets (for x ¼ 0.2 and y ¼ 0.1) Table 3.1 Piezoelectric constants, elastic constants, and

spontaneous polarization in nitride-based binaries.

e33 (C m2) e31 (C m2) p e31 , GGA C33 (GPa) C31 (GPa) P0 (C m2) [e31  (C31/C33)e33] a-lattice parameter (Å) See Volume 1, Chapter 2 for details.

AlN

GaN

InN

1.5 0.53 0.62 377 94 0.090 0.86 3.11

0.67 0.34 0.37 354 68 0.034 0.68 3.199

0.81 0.41 0.45 205 70 0.042 0.90 3.585

3.1 Heterojunction Field Effect Transistors

j377

Lattice constants: aAlN ¼ 3:11 Å; aGaN ¼ 3:199 Å; aInN ¼ 3:585 Å aAl0:2 Ga0:8 N ¼ 0:2ð3:11 ÅÞ þ 0:8ð3:199 ÅÞ ¼ 3:1812 Å aIn0:1 Ga0:9 N ¼ 0:1ð3:585 ÅÞ þ 0:9ð3:199 ÅÞ ¼ 3:2376 Å Elastic constants: C 31 ðAlNÞ ¼ 94 C m2 ; C31 ðGaNÞ ¼ 68 C m2 ; C31 ðInNÞ ¼ 70 C m2 C31 ðAl0:2 Ga0:8 NÞ ¼ 0:2ð94 C m2 Þ þ 0:8ð68 C m2 Þ ¼ 73:2 C m2 C31 ðIn0:1 Ga0:9 NÞ ¼ 0:1ð70 C m2 Þ þ 0:9ð68 C m2 Þ ¼ 68:2 C m2 C33 ðAlNÞ ¼ 377 C m2 ; C33 ðGaNÞ ¼ 354 C m2 C33 ðInNÞ ¼ 205 C m2 C33 ðAl0:2 Ga0:8 NÞ ¼ 0:2ð377 C m2 Þ þ 0:8ð354 C m2 Þ ¼ 358:6 C m2 C33 ðIn0:1 Ga0:9 NÞ ¼ 0:1ð205 C m2 Þ þ 0:9ð354 C m2 Þ ¼ 339:1 C m2 Piezoelectric constants: e31 ðAlNÞ ¼ 0:53 C m2 ; e31 ðGaNÞ ¼ 0:34 C m2 ; e31 ðInNÞ ¼ 0:41 C m2 e31 ðAl0:2 Ga0:8 NÞ ¼ 0:2ð0:53 C m2 Þ þ0:8ð0:34 C m2 Þ ¼ 0:378 C m2 e31 ðIn0:1 Ga0:9 NÞ ¼ 0:1ð0:41 C m2 Þ þ0:9ð0:34 C m2 Þ ¼ 0:347 C m2 e33 ðAlNÞ ¼ 1:5 C m2 ; e33 ðGaNÞ ¼ 0:67 C m2 ; e33 ðInNÞ ¼ 0:81 C m2 e33 ðAl0:2 Ga0:8 NÞ ¼ 0:2ð1:5 C m2 Þ þ 0:8ð0:67 C m2 Þ ¼ 0:836 C m2 e33 ðIn0:1 Ga0:9 NÞ ¼ 0:1ð0:81 C m2 Þ þ 0:9ð0:67 C m2 Þ ¼ 0:684 C m2 : The in-plane strain can be calculated by e? ¼ 2

aepi abuffer aa0 ¼2 abuffer a0

with the assumption of GaN as a buffer for AlGaN and InGaN. The piezoelectric polarization is given by

  C13 PPE ¼ e31  e33 e? : C33 The bandgap energy for AlxGa1xN, EG(x) ¼ xEG(AlN) þ (1  x)EG(GaN)  bx(1  x) with b ¼ 1.0 eV, is equal to 3.78 eV, and for GaN EG ¼ 3.4 eV. Using a conduction bandgap discontinuity of 70%, the conduction band discontinuity between Al0.2Ga0.8N and GaN is DEC ¼ 0.7DEG ¼ 0.266 eV. The bandgap energy for InxGa1  xN, EG(x) ¼ xEG(InN) þ (1  x)EG(GaN)  bx(1  x) with b ¼ 1.43 eV, is equal to 3.0113 eV. Using a conduction bandgap discontinuity of 70%, the conduction band discontinuity between Al0.2Ga0.8N and In0.1Ga0.9N is DEC ¼ 0.5381 eV. Let us now calculate the charge induced at the interface due to dopant donors in the Al0.2Ga0.8N layer. Taking the composition dependence of the dielectric constant in AlxGa1xN e(x) ¼ (8.9  1.9x)e0, the dielectric constant or Al0.2Ga0.8N is e2 ¼ 7.5438  1011F m1. The voltage required to deplete the entire Al0.2Ga0.8N layers is

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Vp ¼

qN D 2 ð1:6  1019 CÞð1  1024 m3 Þ d ¼ ð20  109 mÞ2 ¼ 0:4242 V 2e2 d 2ð7:5438  1011 F m1 Þ

3.1.4.1.1 The Case of Al0.2Ga0.8N/GaN For Al0.2Ga0.8N, EG ¼ 3.78 eV; for GaN, EG ¼ 3.4 eV; and the conduction band discontinuity is DEC ¼ 0.266 eV. With the aid of Equation 3.42 and assuming that DEF0 ¼ 0,Voff ¼ 1 V  0.266 V  0.4242 V þ 0 ¼ 0.3098 V and the charge at the interface due to dopants in the barrier is given by

Q¼

7:5438  1011 F m1 ðV G 0:3098 VÞ; 20  109 m

which for VG ¼ 0 (no applied voltage to the gate) is Q ¼ 0.0012 C m2 corresponding to nso ¼ Qs/q ¼ 7.5  1015 electrons m2. Spontaneous polarization charge: PSP Alx Ga1x N ¼ 0:09x0:034ð1xÞ; P SP Al0:2 Ga0:8 N

2

¼ 0:0452 C m

that for x ¼ 0:2 is 2 PSP and GaN ¼ 0:034 C m

The net charge at the AlGaN/GaN interface due to spontaneous polarization is DPSP ¼ 0:0452ð0:034Þ ¼ 0:0112 C m2 : The piezoelectric polarization is contributed only from the AlGaN layer: e? ¼ 2

abuffer alayer 3:1993:1812 ¼2 ¼ 0:0112; alayer 3:1812



 aGaN aAlGaN C 13 ¼ 2 e e ¼ 0:0112ð0:548Þ ¼ 0:0061Cm2 : PPE 31 33 AlGaN aAlGaN C 33 The total charge at the interface due to polarization is a sum of piezo SP SP SP PE ¼ and spontaneous charges, Ptotal ¼ P PE AlGaN þ ðP Al0:2 Ga0:8 N P GaN Þ ¼ P AlGaN þ D 2 0:00173 C m , which in terms of the electron concentration is np ¼ Qp/q ¼ 1.08  1016 electrons m2 with no applied gate voltage. The total charge at the interface is the sum of doping-induced and polarizationinduced charges: Q total ¼ Q þ Q SP þ Q PE ¼ ð0:0012 C m2 Þ þ ð0:0061 C m2 Þ þ ð0:0112 C m2 Þ ¼ 0:0185 C m2 : The total charge is also equal to Q total ¼ CV off ¼

e2 V off ; d

which gives rise to a Voff value of V loff ¼ 4:905 V: The current–voltage or the output characteristics of this HFET can be calculated with the aid of Equation 3.59, which is repeated here for convenience:

3.1 Heterojunction Field Effect Transistors 0.140 VG= 0 V VG= -1.0V

0.120

VG= -2.0 V VG= -3.0 V

Drain current (A)

0.100

VG= -4.0 V

0.080 0.060 0.040 0.020 0.000 0

1

2

3

4

5

Drain voltage (V) Figure 3.9 Calculated output characteristic for the Al0.2Ga0.8N/ GaN HFET with parameters described in the text with low field mobility of 1000 cm2 V1 s1 and a saturation velocity of 107 cm s1 (Voff ¼ 4.905 V).

ID ¼

   1 mZ e2 V2 ðV G þ jV off jÞV D  D ; 2 1 þ mV D =vs L L d

where, Z ¼ 100 mm, L ¼ 1 mm, vs ¼ 1  107cm s1, d ¼ 20 nm, mobility for GaN is 1000 cm2 V 1 s1, and e2 is the dielectric constant for Al0.2Ga0.8N layer e2 ¼ eAlGaN ¼ (8.9  1.9x)e0 ¼ 8.52e0. The calculated output characteristic for the Al0.2Ga0.8N/GaN HFET under consideration is shown in Figure 3.9. If we now include the electromechanical coupling discussed in Section 3.1.3 and take s2D ¼ 0.5  1013 cm2 and C33(Al0.2Ga0.8N) ¼ 0.2(377 GPa) þ 0.8(354 GPa) ¼ 358.6 GPa e33(Al0.po2Ga0.8N) ¼ 0.2(1.5 C m2) þ 0.8(0.67 C m2) ¼ 0.836 C m2, e ¼ 7.5438  1011 F m1, one can determine the electromechanical coupling coefficient to be a ¼ 0.0252. The new PE charge using Equation 3.37 can be found as PPE ¼ 0:0061 C m2 ð1 þ 0:0252Þ0:0252ð0:008 C m2 þ ð0:0112 C m2 ÞÞ ¼ 0:0058 C m2 ; Q total ¼ Q þ Q SP þ Q PE ¼ ð0:0012 C m2 Þ þ ð0:0112 C m2 Þ þ ð0:0058 C m2 Þ ¼ 0:0182 C m2 : V total off ¼ 4:83V. The resulting FET I–V characteristic is then shown in Figure 3.10. We should note that the gate potential-induced vertical field across the AlGaN varies with the gate bias, which causes the electromechanical coupling to vary with gate bias. However, because the effect of coupling is small, not doing so would not lead to a large error.

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380

0.14 0.12

Drain current (A)

V =0 V G

Coupled Uncoupled

0.10

V = -1.0 V

0.08

G

0.06

V = -2.0 V

0.04

V = -3.0 V

G

G

0.02

V = -4.0 V G

0.00 0

1

2

3

4

5

Drain voltage (V)

Figure 3.10 Calculated output characteristic for the Al0.2Ga0.8N/ GaN HFET with parameters described in the text with low field mobility of 1000 cm2 V1 s1 and a saturation velocity of 107 cm s1, (Voff ¼ 4.9 V without the electromechanical coupling and 4.83 V with mechanical coupling). (Please find a color version of this figure on the color tables.)

3.1.4.1.2 The Case of Al0.2Ga0.8N/In0.1Ga0.9N Assuming 70% of the bandgap discontinuity at the interface AlGaN/InGaN being at the conduction band and using the bandgap dependence on composition for AlxGa1  xN EG(x) ¼ xEG(AlN) þ (1  x) EG(GaN)  bx(1  x) with b ¼ 1.0 eV, for InxGa1xN EG(x) ¼ xEG(InN) þ (1  x)EG(GaN)  bx(1  x) with b ¼ 1.43 eV, one gets for Al0.2Ga0.8N a bandgap of EG ¼ 3.78 eV, and for In0.1Ga0.9N a bandgap of EG ¼ 3.0113 eV with a conduction band discontinuity of DEC ¼ 0.5381 eV. Using e(x) ¼ (8.9  1.9x)e0 for the dielectric of the AlxGa1xN alloy, the dielectric constant for Al0.2Ga0.8N becomes e2 ¼ 7.5438  1011 F m1. The voltage required to deplete the dopant charge in Al0.2Ga0.8N (the same as in the case of Al0.2Ga0.8N/GaN FET) is

Vp ¼

qN D 2 ð1:6  1019 CÞð1  1024 m3 Þ d ¼ ð20  109 mÞ2 ¼ 0:4242 V: 2e2 d 2ð7:5438  1011 F m1 Þ

With the aid of Equation 3.42 and assuming that DEF0 ¼ 0, Voff ¼ 1 V  0.5381 V  0.4242 V þ 0 ¼ 0.038 V. The charge at the interface due to dopants in the Al0.2Ga0.8N barrier is given by Q doping ¼

7:5438  1011 F m1 ðV G 0:038 VÞ; 20  109 m

which for VG ¼ 0 (no applied voltage at the gate) becomes Qdoping ¼ 1.43  104 C m2. Spontaneous polarization charge PSP Alx Ga1x N ¼ 0:09x0:034ð1xÞ, that for SP 2 x ¼ 0.2 is P SP Al0:2 Ga0:8 N ¼ 0:0452 C m , P Inx Ga1x N ¼ 0:042x0:034ð1xÞ, that for

3.1 Heterojunction Field Effect Transistors 2 x ¼ 0.1 is P SP In0:1 Ga0:9 N ¼ 0:0348 C m , and the net spontaneous polarization charge at the Al0.2Ga0.8N/In0.1Ga0.9N interface is QSP ¼ DPSP ¼ 0.0104 C m2. As far as the piezoelectric polarization-induced charge is concerned, we need to calculate the in-plane strain in the In0.1Ga0.9N channel, assuming that it assumes the lattice constant of the GaN buffer and it is coherently strained, which can be done as follows:

e? ¼ 2

aGaN aInGaN 3:1993:2376 ¼2 ¼ 0:0238: aInGaN 3:2376

The InGaN channel is under compressive strain while the AlGaN barrier is under tensile strain. Therefore, the piezo charges in InGaN and AlGaN are opposite in sign.

 aGaN aInGaN C13 ¼ 2 e e PPE 31 33 In0:1 Ga0:9 N aInGaN C33 
  68:2 C m2 2 ¼ ð0:347 C m2 Þ 0:648 C m 339:1 C m2 ð0:0238Þ ¼ 0:0115 C m2 : Assuming that AlGaN also takes the in-plane lattice constant of GaN,



 aGaN aAlGaN e e C13 ¼ 0:01112  ð0:548Þ ¼ 0:0061 C m2 : P PE ¼ 2 31 33 AlGaN aAlGaN C33 If we assume that AlGaN assumes the lattice constant of relaxed InGaN, the PE charge is

 aInGaN aAlGaN C13 ¼ 2 e e ¼ 0:0355  ð0:548Þ PPE 31 33 AlGaN aAlGaN C33 ¼ 0:020 C m2 : And in the case of AlGaN and InGaN in-plane lattice constants being the same as PE 2 that for GaN, the charge is Q PE ¼ P PE AlGaN þ P InGaN ¼ 0:0054 C m . In the case PE where InGaN is relaxed, for the sake of discussion, Q PE ¼ PPE AlGaN þ P InGaN ¼ 0:01150:02 ¼ 0:0085 C m2 . Neglecting the spontaneous polarization between the In0.1Ga0.9N channel and the GaN buffer, the total charge at the interface is the sum of calculated charges thus far: Q total ¼ Q doping þ Q SP þ Q PE ¼ ð1:43  104 C m2 Þ þ ð0:0104 C m2 Þ þ ð0:0054 C m2 Þ ¼ 0:0051 C m2 : total Utilization of Q total ¼ ðeAlGaN =dÞðV G V C ðxÞV total off Þ leads to V off ¼ 1:352 V. If one assumes that the InGaN is relaxed and the AlGaN barrier layer assumes the in-plane lattice constant of the relaxed InGaN channel, the total charge is 0.0190 C m2. In this case, V total off ¼ 5:037 V. It should be pointed out that the effect of spontaneous polarization between the InGaN channel and the GaN buffer is neglected as that contribution is small compared to that of the strain-induced piezoelectric polarization. The electron mobility of In0.1Ga0.9N is determined by relating it to GaN using the effective masses in the following form:

j381

j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

382

0.020

Drain current (A)

0.015 VG= 0 V VG= -0.5 V VG= -1.0 V

0.010

0.005

0.000 0

0.5

1

1.5

2

2.5

3

Drain voltage (V) Figure 3.11 Calculated output characteristics for theAl0.2Ga0.8N/ In0.1Ga0.9N HFET with parameters described in the text. The effect of spontaneous polarization between InGaN and the GaN buffer is neglected as that contribution is small compared to that of the strain-induced piezoelectric polarization (Voff ¼ 1.352 V).

mIn0:1 Ga0:9 N ¼

me ðGaNÞ 0:2m0 0:1 m2 V1 s1 mðGaNÞ ¼ 0:191m0 me ðIn0:1 Ga0:9 NÞ

¼ 0:1047 m2 V1 s1 assuming that the relaxation time in both semiconductors is the same. The current–voltage or the output characteristics of this HFET can be calculated with the aid of Equation 3.59, which is repeated here for convenience:    1 mZ e2 V 2D ðV G þ jV off jÞV D  ; ID ¼ 2 1 þ mV D =vs L L d where Z ¼ 100 mm, L ¼ 1 mm, vs ¼ 1  107 cm s1, and d ¼ 20 nm, mobility for GaN is 1000 cm2 V1 s1 e2 is the dielectric constant for Al0.2Ga0.8N layer e2(x) ¼ (8.9  1.9x)e0 ¼ 7.5438  1011 F m1. The calculated output characteristics for the Al0.2Ga0.8N/In0.1Ga0.9N HFET under consideration are shown in Figure 3.11. 3.1.4.1.3 The Case of InAlN/GaN Following a procedure similar to the AlGaN/ GaN and AlGaN/InGaN cases and considering the lattice-matched composition In0.19Al0.81N, we have for the lattice constants as

aGaN ¼ 0:31986 nm; aAlx Inx1 N ¼ aInN xðaInN aAlN Þ ¼ 0:358480:04753x nm:

3.1 Heterojunction Field Effect Transistors

j383

The composition of In should be 18.75% (composition for lattice matched is 17–18% with some uncertainty). The bandgap of InAlN with lattice-matched composition g

E GaN ¼ 3:42 eV; g

g

g

E Alx InN ¼ xE AlN þ ð1xÞE InN bAlInN xð1xÞ ¼ 6:x þ 0:7ð1xÞ3:1xð1xÞ: g

Using x ¼ 0.8125 E Alx InN ¼ 4:62 eV and assuming 70% bandgap discontinuity for the conduction band, the conduction band discontinuity is DEC ¼ 0.84 eV. The dielectric constants can be determined as eðGaNÞ ¼ 8:9e0 ¼ 7:88  1011 F m1 eðAlx InNÞ ¼ ð15:34:9xÞe0 ¼ 10:02  1011 F m1

for x ¼ 0:8125:

The interface charge due to barrier doping (assumed to be 1017 cm3 in this example) Vp ¼

qN D 2 ð1:6  1019 CÞð1  1023 m3 Þ ð20  109 mÞ2 ¼ 0:03194 eV: d ¼ 2e d 2ð10:02  1011 F m1 Þ

Finally Voff ¼ 1 V  0.84 V  0.0319 V þ 0 ¼ 0.1281 V, where the Schottky barrier built-in potential is assumed to be 1 V. The charge at the interface with its gate voltage dependence is given by Q¼

10:02  1011 F m1 ðV G þ 0:1281 VÞ; 20  109 m

which for VG ¼ 0 (no applied voltage to the gate) gives Q ¼ 6.4178  104 C m2. The spontaneous polarization charge first for GaN is (repeated for convenience): 2 PSP GaN ¼ 0:034 C m :

Using the linear model, the spontaneous polarization associated with the barrier SP PSP Alx In1x N ¼ 0:09x0:042ð1xÞ, which for x ¼ 0.8125 reduces to P Alx In1x N ¼ 2 0:081 C m . The net spontaneous polarization charge (the difference of the two) is QSP ¼ DP ¼ 0.047 C m2. In the case of the nonlinear model, PSP Alx In1x N ¼ 0:09x0:042ð1xÞ þ 0:071xð1xÞ, that for x ¼ 0.8125 is P SP ¼ 0:0702 C m2 . Alx In1x N And the charge due to the difference in spontaneous polarization is QSP ¼ DP ¼ 0.0362 C m2. As far as the piezoelectric charge is concerned, there is no piezoelectric component in AlInN for the lattice. The total charge in different cases are as follows: In the case of the linear model, Qtotal ¼ Q þ QSP ¼ (6.42  104 C m2) þ (0.047 C m2) ¼ 0.046 C m2. This value of total charge leads to V total off ¼ 9:2 V. In the case of the nonlinear model, Qtotal ¼ Q þ QSP ¼ (6.42  104 C m2) þ (0.036 C m2) ¼0.035 C m2. This value of total charge leads to V total off ¼ 7:0 V.

j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

384

2.00

VG = 0 V VG= -2.0 V

Drain current (A mm-1)

VG= -4.0 V

1.50

VG= -6.0 V

1.00

0.50

0.00 0

1

2 3 Drain voltage (V)

4

5

Figure 3.12 Calculated output characteristics (normalized drain current versus drain voltage with gate bias being a parameter) for an In0.19Al0.81N/GaN HFET with parameters described above and in the text using the nonlinear polarization charge model. Owing to lattice matching, the piezo component of the polarization charge is nil.

Using the approach used for the two previous cases, the intrinsic current–voltage characteristics can be calculated with the aid of the following parameters and equation (the results are shown in Figure 3.12 for the nonlinear case). mGaN ¼ 0:1 m2 V1 s1 ;  me ðGaNÞ ¼ 0:2m0 ¼ 1:82  1031 kg; ns ðGaNÞ ¼ 1  105 m s1 ; I¼

h i 1 2 ðeAlInN Zm=LdÞ ðV G V total ÞV  V D off 2 D 1 þ ðmV D =vs LÞ

:

3.1.4.1.4 The Case of InAlN/InGaN Following a procedure similar to the AlGaN/ GaN and AlGaN/InGaN cases, and considering two compositions, one latticematched (In0.19Al0.81N) and the other slightly Al-rich (In0.15Al0.85N) composition for the barrier and In0.08Al0.92N and In0.03Al0.97N for the channel, we have for the lattice constants

aGaN ¼ 0:31986 nm; aAlx Inx1 N ¼ aInN xðaInN aAlN Þ ¼ 0:358480:04753x nm; aIny Ga1y N ¼ aGaN þ yðaInN aGaN Þ ¼ 0:31986 þ 0:03862y nm; Iny Ga1y N with In 3% gives aIn0:03 GaN ¼ 0:32102 nm; Alx In1x N with Al 82% has aAl0:82 InN ¼ 0:31951 nm; Alx In1x N with Al 85% has aAl0:85 InN ¼ 0:31808 nm:

3.1 Heterojunction Field Effect Transistors

The bandgap of InAlN barrier for the two compositions under consideration and the channel is g

E GaN ¼ 3:42 eV; g g g E Alx InN ¼ xE AlN þ ð1xÞE InN bAlInN xð1xÞ ¼ 6:1x þ 0:7ð1xÞ3:1xð1xÞ; g g g E Iny GaN ¼ yE InN þ ð1yÞE GaN bInGaN yð1yÞ ¼ 0:7y þ 3:4ð1yÞ2:5yð1yÞ; g

for x ¼ 0:82; E Al0:82 InN ¼ 4:67 eV; g for x ¼ 0:85; E Al0:85 InN ¼ 4:90 eV; g for y ¼ 0:03; E In0:03 GaN ¼ 3:25 eV: Assuming the conduction band discontinuity as 70%, we have Al0:82 InN=In0:03 GaN=GaN;

DE C ¼ 0:99 eV;

Al0:85 InN=In0:03 GaN=GaN;

DE C ¼ 1:12 eV:

The dielectric constants can be determined as eðAlx In1x NÞ ¼ ð15:34:9xÞe0 ; eðIny Ga1y NÞ ¼ ð10:4 þ 4:9yÞe0 ; eðAl0:82 InNÞ ¼ 9:98  1011 F m1 ; eðIn0:03 GaNÞ ¼ 9:33  1011 F m1 ; eðAl0:85 InNÞ ¼ 9:85  1011 F m1 : The interface charge due to barrier doping (assumed to be 1017 cm3 in this example) for the Al0.82InN/In0.03Ga0.97N/GaN case Vp ¼

qN D 2 ð1:6  1019 CÞð5  1023 m3 Þ d ¼ ð20  109 mÞ2 ¼ 0:16032 eV: 2e d 2ð9:98  1011 F m1 Þ

Voff ¼ 1 V  0.99 V  0.16 032 V þ 0 ¼ 0.15 032 V, where the built-in potential is again taken as 1 V. The charge at the interface with its gate voltage dependence is given by Q¼

9:98  1011 F m1 ðV G 0:15032 VÞ; 12  109 m

which for VG ¼ 0 (no applied voltage to the gate) gives Q ¼ 0.00 125 C m2. For Al0.85InN/In0.03Ga0.97N/GaN, we have Vp ¼

qN D 2 ð1:6  1019 CÞð5  1023 m3 Þ d ¼ ð20  109 mÞ2 ¼ 0:16244 eV: 2e d 2ð9:85  1011 F m1 Þ

Voff ¼ 1 V  1.12 V  0.16 244 V þ 0 ¼ 0.28 244 V, where the built-in potential is assumed to be 1 V. The charge at the interface with gate potential dependence is given by Q¼

9:85  1011 F m1 ðV G 0:28244 VÞ; 12  109 m

which for VG ¼ 0 (no voltage applied to the gate) gives Q ¼ 0.00 232 C m2.

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The spontaneous polarization charge first for GaN is (repeated for convenience) PSP Alx In1x N ¼ 0:09x0:042ð1xÞ; PAlx In1x N ¼ 0:09x0:042ð1xÞ þ 0:071xð1xÞ; P SP Iny Ga1y N ¼ 0:042y0:034ð1yÞ; and for Al0.82InN/In0.03GaN/GaN is 2 linear model; PSP Al0:82 InN ¼ 0:0814 C m SP PAl0:82 InN ¼ 0:0709 C m2 nonlinear model; 2 PSP In0:03 GaN ¼ 0:0342 C m :

And the charge due to the difference in spontaneous polarization is QSP ¼ DP ¼ 0.0472 C m2 in the case of linear model and QSP ¼ DP ¼ 0.0367 C m2 for nonlinear model. In the case of Al0.85InN/In0.03GaN/GaN, we have 2 linear model; PSP Al0:85 InN ¼ 0:0828 C m 2 ¼ 0:0738 C m nonlinear model; PSP Al0:82 InN SP 2 PIn0:03 GaN ¼ 0:0342 C m :

And the charge due to the difference in spontaneous polarization is QSP ¼ DP ¼ 0.0486 C m2 in the case of linear model and QSP ¼ DP ¼ 0.0396 C m2 for nonliner model. Because of lattice mismatch, we have to consider the piezoelectric charge. The elastic constants are C31 ðAlNÞ ¼ 94 C m2 ; C31 ðGaNÞ ¼ 68 C m2 ; C31 ðInNÞ ¼ 70 C m2 C31 ðAl0:82 InNÞ ¼ 0:82ð94 C m2 Þ þ 0:18ð70 C m2 Þ ¼ 89:7 C m2 ; C31 ðIn0:03 GaNÞ ¼ 0:03ð70 C m2 Þ þ 0:97ð68 C m2 Þ ¼ 68:1 C m2 ; C31 ðAl0:85 GaNÞ ¼ 0:85ð94 C m2 Þ þ 0:15ð70 C m2 Þ ¼ 90:4 C m2 ; C33 ðAlNÞ ¼ 377 C m2 ; C33 ðGaNÞ ¼ 354 C m2 ; C33 ðInNÞ ¼ 205 C m2 ; C33 ðAl0:82 InNÞ ¼ 0:82ð377 C m2 Þ þ 0:18ð205 C m2 Þ ¼ 346:0 C m2 ; C33 ðIn0:03 GaNÞ ¼ 0:03ð205 C m2 Þ þ 0:97ð354 C m2 Þ ¼ 349:5 C m2 ; C33 ðAl0:85 GaNÞ ¼ 0:85ð377 C m2 Þ þ 0:15ð205 C m2 Þ ¼ 351:2 C m2 : The piezoelectric constants are e31 ðAlNÞ ¼ 0:53 C m2 ; e31 ðGaNÞ ¼ 0:34 Cm2 ; e31 ðInNÞ ¼ 0:41 Cm2 ; e31 ðAl0:82 InNÞ ¼ 0:82ð0:53 Cm2 Þþ0:18ð0:41 Cm2 Þ ¼ 0:508C m2 ; e31 ðIn0:03 GaNÞ ¼ 0:03ð0:41 C m2 Þþ0:97ð0:34 Cm2 Þ ¼ 0:342C m2 ; e31 ðAl0:85 GaNÞ ¼ 0:85ð0:53C m2 Þþ0:15ð0:41 C m2 Þ ¼ 0:512Cm2 ; e33 ðAlNÞ ¼ 1:5C m2 ; e33 ðGaNÞ ¼ 0:67 Cm2 ; e33 ðInNÞ ¼ 0:81 Cm2 ; e33 ðAl0:82 InNÞ ¼ 0:82ð1:5C m2 Þþ0:18ð0:81 Cm2 Þ ¼ 1:376C m2 ; e33 ðIn0:03 GaNÞ ¼ 0:03ð0:81 Cm2 Þþ0:97ð0:67C m2 Þ ¼ 0:674Cm2 ; e33 ðAl0:85 GaNÞ ¼ 0:85ð1:5 C m2 Þþ0:15ð0:81C m2 Þ ¼ 1:397Cm2 : The in-plane strain can be calculated by e? ¼ 2(alayer  abuf/afuf ). The piezoelectric polarization is given by

3.1 Heterojunction Field Effect Transistors


PPE ¼

e31 


  C31 e33 e? : C33

For the case of Al082In0.18N/In0.03Ga0.97N/GaN, let us consider the channel buffer mismatch-induced polarization for In0.03Ga0.97N/GaN and the resultant piezo polarization e? ¼ 2

3:2102 Å3:1986 Å ¼ 0:0073; 3:1986 Å 


  68:1 C m2 2 P ðIn0:03 GaNÞ ¼ ð0:342 C m Þ  0:674 C m 349:5 C m2 PE

2

ð0:0073Þ ¼ 0:0035 C m2 : For the barrier Al082In0.18N and GaN (we assume the InGaN to be coherent on GaN buffer), that is, In0.18Al0.82N, the resultant mismatch and piezo polarization 3:1951 Å3:1986 Å ¼ 0:0022; 3:1986 Å 
  89:7 C m2 2  1:376 C m P PE ðAl0:82 InNÞ ¼ ð0:508 C m2 Þ 346:0 C m2

e? ¼ 2

 ð0:0022Þ ¼ 0:0019 C m2 : PE 2 The total charge is Q PE ¼ P PE AlGaN þ P InGaN ¼ 0:0016 C m : For the case of Al0.85InN/In0.03Ga0.97N/Ga, let us consider the channel buffer mismatch-induced polarization for In0.03Ga0.97N/GaN and the resultant piezo polarization

e? ¼ 2

3:2102 Å3:1986 Å ¼ 0:0073; 3:1986 Å


  68:1 C m2 2 P PE ðIn0:03 GaNÞ ¼ ð0:342 C m2 Þ  0:674 C m 349:5 C m2 ð0:0073Þ ¼ 0:0035 C m2 : For the barrier Al0.85In0.15N and GaN (we assume the InGaN to be coherent on GaN buffer), the resultant mismatch and piezo polarization 3:1808 Å3:1986 Å ¼ 0:0111; 3:1986 Å 
  90:4 C m2 2 P PE ðAl0:85 InNÞ ¼ ð0:512 C m2 Þ   1:397 C m 351:2 C m2 e? ¼ 2

ð0:0111Þ ¼ 0:0097 C m2 : PE 2 The total piezo-induced charge is Q PE ¼ P PE AlGaN þ P InGaN ¼ 0:0062 C m :

j387

j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

388

The total charge including the contributions by barrier doping and spontaneous polarization for the Al0.82InN/In0.03Ga0.97N/GaN case, we have Q total ¼ Q þ Q SP þ Q PE ¼ ð12:5  104 C m2 Þ þ ð0:0472 C m2 Þ þ ð0:0016 C m2 Þ ¼ 0:0501 C m2 : This value of total charge leads to V total off ¼ 6:1 V. In the case of nonlinear model for spontaneous polarization V total off ¼ 4:8 V. For the case of Al0.85InN/In0.03Ga0.97N/GaN, the total charge inclusive of polarization- and doping-induced components Q total ¼ Q þ Q SP þ Q PE ¼ ð23:2  104 C m2 Þ þ ð0:0486 C m2 Þ þ ð0:0062 C m2 Þ ¼ 0:0357 C m2 : This value of total charge leads to V total off ¼ 7:0 V. In the case of nonlinear model for spontaneous polarization V total off ¼ 4:4 V. Using the approach used for the two previous cases, the intrinsic current–voltage characteristics for the In0.15Al0.85N/In0.03Ga0.97N case can be calculated with the aid of the following parameters and equation (the results are shown in Figure 3.13 for the nonlinear model). me ðGaNÞ ¼ 0:2m0 ;

me ðInNÞ ¼ 0:11m0 ;

1.50 VG = 0 V VG = -1.5 V VG = -2.5 V

Drain current (A mm-1)

VG = -3.5 V

1.00

0.50

0.00 0

1

2 3 Drain voltage (V)

Figure 3.13 Calculated output characteristics (normalized drain current versus drain voltage with gate bias being a parameter) for an In0.15Al0.85N/In0.03Ga0.97N HFET with parameters described above and in the text using the nonlinear polarization charge model.

4

5

3.1 Heterojunction Field Effect Transistors

me ðIn0:03 GaNÞ ¼ 0:03me ðInNÞ þ 0:97me ðGaNÞ ¼ 0:197m0 ¼ 1:8  1031 kg; mðIn0:1 GaNÞ ¼

me ðGaNÞ mðGaNÞ  me ðIn0:1 GaNÞ

¼

0:2m0 0:1 m2 V1 s1 0:197m0

¼ 0:102 m2 V1 s1 ; ns ðInGaNÞ ¼ 1  105 m s1 ;  2 1 ðeInGaN Zm=LdÞ ðV G V total off ÞV D 2V D : I¼ 1 þ ðmV D =vs LÞ

3.1.5 Numerical Modeling of Sheet Charge and Current

After analyzing GaN and InGaN channel HFETs, numerical results of the electron density and distribution as well as the band-edge profile for both normal and inverted modulation-doped FETs including the polarization charge are discussed. The presentation follows that reported by Sacconi et al. [79]. The range of parameter values and structural design has been chosen to represent not only the most commonly used structures but also those that would demonstrate how the important properties would change with structural design changes. This also applies to N-polarity samples discussed here not because they are technologically important, but because of the need for a full understanding of how polarization charge affects the parameters important for FETs. Let us consider two HFETstructures, namely, a single-heterojunction AlGaN/GaN normal heterojunction FET (N-HFET), where the AlGaN donor layer is grown on top of the GaN channel layer, and an “inverted” GaN/AlGaN/GaN HFET (I-HFET), where the channel layer is grown on top of the AlGaN donor layer; the latter is very uncommon in experimental GaN HFETs. It should be noted that for the inverted n-channel device to function requires N-polarity samples that are not yet competitive with Ga-polarity samples in devices. The N-HFETstructure consists (from the gate to the substrate) of a 150 Å n-doped (n ¼ 1018 cm3) AlGaN, 50 Å unintentionally doped AlGaN layer, and a thick GaN buffer. The I-HFET consists (from the gate to the substrate) of a 300 Å unintentionally doped GaN layer, 50 Å unintentionally doped AlGaN, 150 Å n-doped (n ¼ 1018 cm3) AlGaN, 300 Å unintentionally doped AlGaN layer, and a thick GaN layer. A residual doping of 1017 cm3 in both GaN and AlGaN layers is assumed, a figure that would be updated downward as the deposition technologies are improved. A Schottky barrier height (FB) of 1.1 eV for the metal–GaN interface and a FB ¼ 1.2 eV for the metal–AlGaN interface is used. Calculations have been performed for AlxGa1xN with Al concentrations of x ¼ 0.1, 0.2, 0.3, and 0.4. Both [0 0 0 1] and ½0 0 0 1 growth directions are considered. In the simulations, an effective mass of 0.19, which is somewhat smaller than the commonly used 0.22m0, for electrons and 1.8 for holes, the midrange among the reported values, in both GaN and AlGaN layers

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Table 3.2 Bandgap and conduction band discontinuities used for AlxGa1xN and AlxGa1xN/GaN heterointerface, respectively.

x(Al)

EG (eV)

DEC (eV)

0.1 0.2 0.3 0.4

3.62 3.85 4.09 4.35

0.17 0.33 0.51 0.69

have been used. The bandgaps and band discontinuities of the AlGaN layers used are tabulated in Table 3.2. As discussed as part of analytical modeling, and in Volume 1, Chapter 2 in great detail, the presence of polarization is quite important in nitride-based HFETs. The conduction band-edge profile for an N-HFETgrown in [0 0 0 1] direction is depicted in Figure 3.14 for the cases (i) with both spontaneous and piezoelectric polarization, (ii) without the polarization, and (iii) with only the piezoelectric polarization. The difference in piezoelectric and spontaneous polarization between AlGaN and GaN layers manifests itself by inducing a strong built-in electrical field across the interface. For the [0 0 0 1] growth direction considered in Figure 3.14, the difference in polarization between the two materials induces a positive charge (s ¼ þ 1.12  1013 cm2) at the Al0.2Ga0.8N/GaN interface. Free electrons are then attracted by this positive bound polarization charge, tending to accumulate at the interface and thus forming a conductive channel. Moreover, the high electric field, due to strong polarization, favors the formation of a high-density two-dimensional charge, which is strongly confined. Within the AlGaN layer, the electric field compensates the space charge contribution due to the ionized donors. Consequently, it prevents the formation of a parasitic channel that would otherwise form in the doped AlGaN layer [82, 111]. 1.5 w all polarizations w/ any polarizations w only piezo polarization

c

E (eV)

1.0

0.5

E 0.0

F

-0.5 -20

-10

0

10

20

z (nm) Figure 3.14 Calculated conduction band edge for the N-HFET structure grown in the [0 0 0 1] direction for VG ¼ 0 with and without polarization fields.

3.1 Heterojunction Field Effect Transistors 8 × 1019 Al = 10% Al = 20% Al = 30% Al = 40%

19

n(z) (cm-3)

6 × 10

4 × 1019

2 × 1019

0 -5

0

5

10

z (nm) Figure 3.15 Electron density distribution in the channel of the [0 0 0 1] grown N-HFET for VG ¼ 0 for several Al fractions of the AlxGa1xN layer.

The comparison illustrated in Figure 3.14 between the three cases with different contributions of the polarization field shows the importance of considering both spontaneous and piezoelectric polarizations in GaN-based device modeling. In fact, by neglecting the spontaneous polarization, as was done in some reports [112–115], the channel electron density is underestimated [82]. Moreover, the sign of the polarization charge is crucial and must be tracked. The depth distribution of the free electron charge in the channel is shown in Figure 3.15 for 10, 20, 30, and 40% Al mole fractions in the AlGaN barrier layer. Increasing the Al content induces a larger polarization charge at the GaN/AlGaN interface and consequently a higher channel electron concentration at equilibrium. The distribution of free electrons in the channel with varying doping (1017–1019 cm3) in the barrier is shown in Figure 3.16. Increasing the doping in the barrier supplies more electrons to the channel and thus increases the total sheet density. However, it comes with side effects such as the decreased mobility and increased gate leakage current [116]. The calculations so far are obtained by considering only the polarization and barrier doping-induced charge at the AlGaN/GaN interface. In theory, however, polarization charge that is called for at the metal–AlGaN and at the bottom end of the GaN buffer region, depending on the composition of the nucleation layer, could be considered. The metal–AlGaN charge is completely screened by the charge induced on the metal surface and can therefore be neglected. However, the charge at the bottom end of the buffer region may induce a large deviation with respect to the situation depicted above. Oberhuber et al. [74] have considered a s/2 charge at the interface between the GaN and the AlN nucleation region. The exact amount of such charge depends, however, on the morphology of the heterojunction and may differ from the theoretical value s ¼ DP/q. The situation is less critical if the bottom interface is far away from the main AlGaN/GaN heterojunction. In this case, the polarization charge that arises can be completely screened by the residual doping in the GaN buffer layer. On the contrary, if such interface is close to the AlGaN/GaN

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1.5

Nd = 1e17 (cm-3)

1.0

Nd = 1e18 (cm-3)

Ec (eV)

Nd = 1e19 (cm-3) 0.5

0.0

-0.5 -20

-10

0

10

20

z (nm)

(a) 19

6 ×10

Nd = 1e17 (cm-3) Nd = 1e18 (cm-3) 19

n(z) (cm-3)

4 × 10

19

2 ×10

0 -10

(b)

Nd = 1e19 (cm-3)

-5

0

5

10

z (nm)

Figure 3.16 (a) Band structure for varying donor density in the [0 0 0 1] grown AlGaN/GaN HFET structure for VG ¼ 0 at several doping levels in AlxGa1xN layer. (b) Electron density distribution in the channel of the same.

heterojunction, the polarization charge can completely deplete the channel. In the simulations considered here, a thick GaN substrate is assumed. Thus, the effect of the polarization charge at the bottom end of the GaN substrate is completely screened. The spacer layer inserted between the channel and the barrier region in a twodimensional system, usually employs an undoped and wide-bandgap material, tends to isolate the source of electrons away from the channel. The insertion of an ultrathin AlN spacer layer between AlGaN/GaN heterojunction has been reported to increase the electron concentration and mobility of 2DEG [117, 118]. The improvement in electron transport is ascribed to the better confinement of 2DEG and thus the associated suppression of alloy scattering [119–121] and possibly scattering by defects/ionized donors in the barrier. The conduction band-edge diagram and electron distribution for an AlGaN/GaN system with an AlN spacer layer is shown in Figure 3.17 and the inset of Figure 3.17, respectively. Compared to the normal AlGaN/GaN heterojunction with the same Al mole fraction and doping

3.1 Heterojunction Field Effect Transistors 3.0 n(z) (cm-3) 8 × 1019 6 × 1019

2.0

Ec (eV)

4 × 1019 2 × 1019

1.0 0 -5

E 0.0

0

z (nm)

5

10

F

Without spacer With spacer

-1.0 -20

-10

0

10

20

30

z (nm) Figure 3.17 Band structure in the channel of a [0 0 0 1] grown AlGaN/GaN HFET structure with and without an AlN spacer for VG ¼ 0. (Inset) Electron density distributions in the channel of a [0 0 0 1] grown AlGaN/GaN HFET structure with and without an AlN spacer for VG ¼ 0.

concentration, the AlN spacer provides a better confinement with smaller proportion of the electron wave function spilling into the barrier. Experiments have been reported in which the thickness of the AlGaN barrier in N-HFET structures has been shown to have a significant influence on the channel carrier density and device performance [120, 122], especially the reliability [123]. The associated interpretation assumes the source of electrons to be the surface states. Once formed the channel charge density is controlled by the gate bias as in traditional N-HFET device. The equilibrium density itself, as we have been discussing all along, depends on the particulars of the structure such as the Al content of the AlGaN layer, AlN spacer thickness if present, and the doping level in the barrier if used. Charge tuning in nitride-based devices can be achieved by adjusting these independent parameters and thus a wide degree of flexibility with respect to traditional devices is available. This can be seen from the sheet charge concentration in the channel as obtained by integrating the electron density distribution along the z-direction. Considering the explicit dependence of the sheet charge density on the gate voltage VG, we have ð ð3:67Þ ns ðV G Þ ¼ nðV G ; zÞdz: We plot the dependence of the total sheet carrier concentration, ns, as a function of the AlGaN barrier thickness in Figure 3.18, for both with and without 1 nm AlN spacer layer cases. The increase of ns with thicker barrier is because when the AlGaN thickness increases, the potential well at the interface becomes deeper due to the stronger polarization-induced electrical field on the channel side, which is proportional to the 2DEG sheet density. In addition, insertion of an AlN barrier increases the piezoelectric-induced polarization (the spontaneous polarization is not influenced in net because the composition returns to that of the AlGaN barrier when traversing

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Sheet carrier density (cm-2)

2.0 × 1013

1.5 × 1013

1.0 × 1013

5.0 × 10

Without AlN

13

With AlN

0.0 0

5

10

Al Ga 0.3

15

20

25

N thickness (nm)

0.7

Figure 3.18 Dependence of the total sheet carrier density on different AlGaN barrier thickness with and without a 1 nm AlN spacer.

toward the surface. Moreover, with the help of AlN spacer, ns has a weakened dependence on the barrier thickness. The calculated data are in good agreement with those reported in Ref. [117]. The reason of this phenomenon is that the large bandgap discontinuity between AlN and GaN serves to “buffer” the change in potential energy caused by the variation of barrier thickness, thus lessening the impact on the potential well profile in the channel. Because of the strong polarization-induced charge by large lattice mismatch between AlN and GaN and the associated compositional gradient at the interface, the insertion of the AlN spacer layer increases the sheet electron density. Very high carrier density (5  1013 cm2) and strong internal electrical fields (109 V m1) at the AlN/GaN interface have been observed or calculated, whichever is applicable [124–126]. Owing to the specifics of the conduction band edge throughout the structure, the sheet density has been calculated to increase with increasing AlN thickness. We plot the dependence of total sheet carrier density on the AlN spacer thickness in Figure 3.19, along with the electrical field distribution in all layers. However, this is not as yet supported fully with experimental data. As part of a study to investigate the effect of the AlN spacer layer on the mobility, heterostructures with and without the AlN have been prepared [127]. One structure without the 1 nm AlN spacer is listed to have roughly the same sheet density as the one without it. In another investigation, some of the same authors [128] presented the Hall mobility and only the resistance for varied AlN spacer thickness, and thus the actual experimental sheet carrier densities are not available. The resistance, which is governed by the mobility and the electron density product, varied with the thickness of the AlN layer. Experimental data in Al0.845In0.155N/AlN/GaN heterostructures show decreasing sheet electron density with increasing AlN spacer thickness, which is counter to calculations in the AlGaN barrier case [129]. While the AlN spacer layer helps to improve the low-field transport properties by providing higher electron concentration and mobility, and lower leakage current, the

3.1 Heterojunction Field Effect Transistors

Sheet carrier density (cm-2)

2.0 × 1013

1.6 × 1013 6

4 × 10

1.2 × 1013

Electrical field (V cm-1 )

0 6

-4 × 10 12

8.0 × 10

6

-8 × 10

z (nm)

7

-1.2 × 10

4.0 × 1012 -0.5

0.0

0.5

1.0

-20

-10

1.5

0

2.0

10

2.5

20

3.0

AlN thickness (nm) Figure 3.19 Dependence of the total sheet carrier density on the AlN spacer thickness ranging from 0 to.2.5 nm and (inset) the electrical field distribution in 1 nm AlN case.

strong built-in electrical field may cause breakdown to occur in the spacer and therefore reduce the long-term reliability of the device (see Section 3.11, for a discussion of reliability). Experimental data involving a large number of HFETs with and without the AlN layers seem to unequivocally indicate that the presence of AlN causes more rapid degradation [130] in terms of the leakage current and output power under RF stress. As always the case, the infrared images also revealed the high-temperature spots corresponding to the local breakdown at the gate region. Therefore, the role of AlN spacer in HFET applications is still in question. Development of inverted HFET is experimentally difficult, despite some progress, as it either involves N-polarity growth for electron channel or Ga-polarity growth for hole channel in addition to the fact that embedded barrier layer below the channel also calls for a parasitic interface channel of the opposite polarity to form below the barrier layer. From a technological point of view, the different growth kinetics and high growth temperature requirements make it difficult to obtain high-quality N-face material [131, 132]. However, reports [133, 134] have revealed some progress in N-polar GaN/AlGaN/GaN structure with 1013 cm2 sheet carrier density and 1400 cm2 V1 s1 mobility, and good DC and small-signal RF performance on the I-HFETs fabricated. Therefore, I-HFET might become a promising candidate for high-power application, which works under the enhancement mode to raise the dynamic scale of the amplifier. Nevertheless, the band-edge profile and electron densities for the I-HFETgrown in the ½0 0 0 1 direction are shown in Figures 3.20 and 3.21, respectively. Specifically, the conduction band edges with piezo and compositional gradient-induced polarization, with piezoelectric polarization only, and without any polarization charge are plotted. Again, the source of electrons is assumed to be the surface states. As in the case of the N-HFET, the presence of the fixed and positive polarization charge at the GaN/AlGaN interface induces the formation of a channel which would not be present in the absence of the polarization charge. For the I-HFET, a s polarization charge is also present at the bottom end of the AlGaN region (i.e., at the AlGaN/GaN interface).

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w/ any polarizations w all polarizations w only piezo polarization

5.0

Ec (eV)

4.0

3.0

2.0

1.0

E

0.0 0

40

80

F

120

z (nm) Figure 3.20 Conduction band edge for the I-HFET structure grown in the ½0 0 0 1 direction for VG ¼ 0 with and without the polarization field.

Similar to the [0 0 0 1] grown N-HFET, a larger Al content of the AlGaN layer induces a larger polarization charge at the GaN/AlGaN interface and consequently causes an increase of the electron concentration in the channel. Naturally, for the [0 0 0 1] orientation, a parasitic interface charge would form below the buried AlGaN layer, which is not desirable for an I-HFET. What is desired is the formation of a sheet electron layer on top of the AlGaN layer, which is possible when the ½0 0 0 1 orientation is employed. The structure in its present shape, that is, the [0 0 0 1] polarity would show FET performance provided the AlGaN layer is completely depleted but with small transconductance. If the AlGaN is not depleted, then the device would function with a parasitic MESFET dominating in part by transport in the AlGaN layer unless the gate potential is large enough to deplete it. To eliminate the formation of a piezo-induced interface electron charge at the bottom of the AlGaN 8 × 1019

n (z) (cm-3)

6 × 1019

Al = 10% Al = 20% Al = 30% Al = 40%

4 ×1019

2 ×10

19

0 -10

-5

0

5

z (nm) Figure 3.21 Electron density distribution in the channel of the ½0 0 0 1 grown I-HFET for VG ¼ 0 for several Al contents of the AlxGa1xN layer.

3.1 Heterojunction Field Effect Transistors

layer, the bottom heterointerface should be graded substantially. In that case, the [0 0 0 1] polarity would cause the band diagram to accumulate holes at the top interface if they are present toward the equilibrium. That top interface would accumulate electrons in the ½0 0 0 1 polarity. Still, the compositional gradientinduced polarization charge would be present at the bottom interface. Our discussion so far in this section centered on determining the total sheet density for various HFET structures. The next step is actually to calculate the current–voltage characteristics of HFETs numerically as we have already covered the analytical calculations. Sacconi et al. [79] have implemented a quasi-2D [135–137] model for the calculation of the current–voltage characteristics of the nitride HFETs. This model uses the exact value of the sheet charge density in an HFET device channel, obtained from the self-consistent Schr€odinger–Poisson solution presented above. We assume an FET model shown in Figure 3.22 where the x-axis is along the channel and the z-axis is along the growth direction. The model also considers the presence of a drain (RD) and source (RS) resistance. When a drain bias (VD) is applied, the potential along the channel may be considered as varying gradually from the source bias (VS) to VD. In this situation, it is still possible to calculate the sheet charge density, ns, at every section grid, provided that one considers the V(x) potential (on the top surface) for each point of the channel. Because for n-channel devices, VD is positive and VS is zero, V(x) contributes to the channel depletion and the sheet charge density, ns, for the generic x section of the FET will be therefor. ns ðxÞ ¼ ns ðV G VðxÞÞ:

ð3:68Þ

By neglecting diffusion contributions, the source to drain current IDS is given by IDS ¼ qZvðxÞns ðxÞ;

Figure 3.22 Schematic representation of the quasi-2D FET model used.

ð3:69Þ

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where Z is the gate width and v(x) the electron mean velocity, supposed to be independent from transverse coordinate. The dependence of the drift velocity on the longitudinal electric field is empirically given by Equation 3.56. Parasitic components are included explicitly through the drain and source resistances (RD, RS) V eS ¼ V S þ I DS RS ; V eD ¼ V D I DS RD ;

ð3:70Þ

where V eD and V eS represent the effective bias boundaries of the gate region on the drain and source sides, respectively. For a certain value of IDS, we can calculate the VD by solving Equation 3.69. The explicit equation for the current is IDS ¼ qZ

m0 FðxÞ n ðV G VðxÞÞ: 1 þ ðFðxÞ=F C Þ

ð3:71Þ

The numerical solution is based on the discretization of this expression into N sections, each with amplitude h, so that Nh ¼ L, where L is the gate length. Given the (i  1)th section potential, the ith potential Vi ¼ Vi1 þ Fih, where Fi is the ith section electric field. We have then the N relations: IDS ¼ qZ

m0 F i n ðV G V i1 F i hÞ: 1 þ ðF i =F C Þ

ð3:72Þ

Because the (i  1)th section potential is known from the previous step, this is a nonlinear equation in the unknown Fi. Solving iteratively for all the N sections, one obtains the value of the drain voltage VD consistent with the assumed current. Repeating this procedure for a suitable range of values of IDS, one obtains the set of corresponding values of VDS and thus the HFET I–V characteristics, which are elaborated on below. 3.1.6 Numerically Calculated I–V Characteristics

In this section, we discuss the simulated I–V characteristic of the normal and inverted HFET, obtained for a gate length of L ¼ 0.3 mm. We have chosen a drain and source contact resistivity of about 1 W mm, which is consistent with experimentally measured values on these types of devices [138]. We use a saturation velocity of 2.5  107 cm s1 [4], while for the low field mobility we choose a value of m0 ¼ 1100 cm2 V 1 s1, slighter higher than the GaN bulk value, according to the experimental and theoretical results for similar devices [4, 82, 139, 140]. In Figure 3.23, we show the IDS versus VDS for the [0 0 0 1] polarity HFET for several gate (VGS) voltages. The results are presented for both x ¼ 0.2 (Figure 3.23a) and x ¼ 0.4 (Figure 3.23b), Al concentration of the top layer. For x ¼ 0.2, the HFET reaches pinch-off for a bias voltage of VGS ¼ 4.4 V while for x ¼ 0.4 the pinch-off is reached at VGS ¼ 9.5 V. However, the saturation drain current for x ¼ 0.2 is

3.1 Heterojunction Field Effect Transistors 3.0

V GS = +0.5 V

0.0 V

2.5

IDS (A mm-1)

2.0

-1.0 V 1.5

-2.0 V 1.0

-3.0 V

0.5

-4.0 V 0.0

0

2

4

6

8

10

12

14

16

V DS (V)

(a) 7

V GS = +0.5 V 6

0.0 V

IDS (mA mm-1)

5

-2.0 V 4 3

-4.0 V

2

-6.0 V 1

-8.0 V 0

(b)

0

2

4

6

8

10

12

14

16

V DS (V)

Figure 3.23 (a) IDS versus VDS for the [0 0 0 1] polarity normal HFET for several gate (VGS) voltages and for x ¼ 0.2. (b) IDS versus VDS for the [0 0 0 1] polarity normal HFET for several gate (VGS) voltages and for x ¼ 0.4.

IDS ¼ 2.4 A mm1 at VGS ¼ 0 and it increase up to 5.76 A mm1 for an Al content of x ¼ 0.4. Thus, the current flowing in the devices depends strongly on the Al contents of the top layer. This is essentially due to the increase of the channel electron density induced by the increasing polarization charge going from an Al content of 0.2 up to 0.4. This peculiarity of the HFET should be considered in the design of these devices because fluctuation of the alloy composition of the top layer may induce large variations with respect to nominal electrical values of the device. It should also be pointed out that the gate leakage would determine the extent of gate voltage that can be applied to the gate. For a gate bias of 9.5 V and AlGaN layer thickness of 20 nm, the vertical field under the gate near the source can reach 4.75 MV cm1. This means that

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HFETs utilizing large mole fractions of Al may require thin AlGaN layers or recessed gates to keep the gate voltage smaller. A similar situation is obtained for the I-HFET with the GaN/AlGaN/GaN structure grown in the ½0 0 0 1 direction, meaning with N-polarity. The calculated IDS versus VDS characteristics are reported in Figure 3.24a and b for x ¼ 0.2 and x ¼ 0.4 Al composition of the AlGaN layer, respectively. Also, in this case, the pinch-off bias depends critically on the Al composition and varies from 3.9 V for x ¼ 0.2 to 9.0 for x ¼ 0.4. Saturation currents are lower for the I-HFETat x ¼ 0.2 with respect to the equivalent HFET structure. Such difference, however, is negligible for the case with x ¼ 0.4.

IDS (mA mm-1)

3.0

2.5

V GS = +0.5 V

2.0

0.0 V

1.5

-1.0 V

1.0

-2.0 V 0.5

-3.0 V 0.0

0

2

4

6

8

10

V DS (V)

(a) 7

V GS = +0.5 V 6

0.0 V

IDS (mA mm-1)

5

-2.0 V

4 3

-4.0 V

2

-6.0 V 1

-8.0 V 0 0

(b)

2

4

6

8

10

12

V DS (V)

Figure 3.24 (a) IDS versus VDS for the ½0 0 0 1 polarity inverted HFET for several gate (VGS) voltages and for x ¼ 0.2. (b) IDS versus VDS for the ½0 0 0 1 polarity inverted HFET for several gate (VGS) voltages and for x ¼ 0.4.

14

16

3.2 The s-Parameters and Gain

3.2 The s-Parameters and Gain

For diagnostics and circuit simulation purposes, it is customary to perform scattering parameter, or the s-parameter configuration for short, measurements. The s-parameters are then converted to y-parameters from which equivalent circuit parameters of varying complexity and accuracy are extracted. In the simplest sense, with years of experience and knowledge of intrinsic and extrinsic device, parameters as well as the feedback components are obtained. In this section, a review of various circuit parameters commonly used to analyze two-port networks, which represent FETs and bipolar transistors. This will segue into the equivalent circuits including the parameters of varying complexity for small-signal and largesignal environments and methodologies used for their determination. All the while, we should recognize the lumped circuit elements, which are well applicable at low frequencies, are more convenient to use as opposed to distributed elements, which are naturally more representative of microwave circuits. To begin the process, we refer to the two-port network of Figure 3.25, which is good for characterization of FETs. A two-port network or device (Figure 3.25) can be described by a number of parameter sets. Among the most popular ones, particularly for relatively low frequencies, are the h-, y-, and z-parameter sets. They are, in order, defined as follows: 

v1 i2





h ¼ 11 h21

h12 h22



 i1 ; v2

 

i1 i2 v1 v2





y ¼ 11 y21



 ¼

z11 z21

y12 y22 z12 z22



 v1 ; v2



 i1 : i2

ð3:73Þ

In the impedance matrix above, relating input and output voltages to the input and output currents, we have taken the output current i2 to be flowing into the two-port network. In some cases, the opposite direction is assumed in which case the i2 ! i2 conversion must be made. To complete the treatment, let us also define the transfer matrix description, which relates the input parameters, voltage and current, to the output parameters as 

v1 i1



 ¼

t11 t21



 v2 : i2

ð3:74Þ

i2

i1 + v1 _

t12 t22

Two-port network

+ v2

_

Figure 3.25 A two-port network showing input and output currents and voltages and their polarities.

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The selection of which one to use is mainly determined by the convenience for the problem in hand. The only difference among the parameter sets is the choice of independent and dependent variables. Let us use the h-parameters to show how the parameter sets of this type can be determined. The parameter h11 is determined by setting v2 equal to zero, meaning applying a short circuit to the output port of the network shown in Figure 3.25. The parameter h11 is then the ratio of v1 to i1, which has the units of impedance, and represents the input impedance of the resulting network ðh11 ¼ v1 =i1 jv2¼0 Þ. The parameter h12 is determined by constructing the ratio of v1 to v2, which represents the reverse voltage gain with the input port open circuited ðh12 ¼ v1 =v2 ji1¼0 Þ. Very importantly, the parameter h21 is determined by constructing the ratio of i2 to i1, which represents the reverse voltage gain with the output port short circuited ðh21 ¼ i2 =i1 jv2¼0 Þ. Note that we use open and short circuits to determine the impedance and admittance parameters, a method which is not necessarily available at high frequencies in part because active devices are not necessarily stable under short- and/or opencircuit conditions. In these circumstances, a new set of network representation parameters, called the s-parameters, are used that do not rely on short or open circuits in parameter definition, but the measurement apparatus, which is a vector analyzer, relies very heavily on calibration procedures and accurate knowledge of reference planes for relative phase determination. The s-parameters are defined as (see Figure 3.26).      b1 ¼ s11 a1 þ s12 a2 b1 s a1 s or ¼ 11 12 ; ð3:75Þ b2 ¼ s21 a1 þ s22 a2 b2 s21 s22 a2 where the parameters a1 and b1 represent the incident and reflected power at the input port of the device under test (DUT) with their amplitude and phase information. Similarly, a2 and b2 represent the incident and reflected power at the output port

i

v2

i

v1 ZS

Two-port network

a1 +

vS

+

S 21 i

r

v1= v1+ v1

S 11

a2

S 22

S 11

i

r

v2= v2+ v2

S 12

_ b1

S 22

_ b2

v1r

r

v2

ΓS

Γin

Γout

Figure 3.26 A two-port network indicating the incident and reflected voltages at the input and output, reflection coefficients at the input and output port, and s-parameters.

ΓL

ZL

3.2 The s-Parameters and Gain

vs

+ v1 _

+ _

i2

i1

Zs Zin a1 b1

Two-port network s

+ v2 _

Zout

ZL

b2 a2

Figure 3.27 A two-port network connected to generator (source) and load.

of the device under test with their amplitude and phase information, as shown in Figure 3.26. In electromagnetic textbooks, the a1 and b1 parameters are also depicted r  by b1 vr1 v 1 and b2 v2 v2 . Similarly, the a2 and b2 parameters are depicted by þ þ i i a1 v1 v1 and a2 v2 v2 , again as shown in Figure 3.26. This means that the RF signal is applied to the input and then an RF signal is applied to the output again for the measurements to be complete, in a sense reversing the roles of the input and output. Having displayed the incident and reflected voltages at the input and output of the two-port network as well as the reflection coefficient, we can from now on refer to a simplified illustration shown in Figure 3.27. As in the example of the h-parameters, the s-parameters can be determined from specific measurements. When the output port of the network is terminated with a matched load that is the characteristic impedance, Z0, s11 is the ratio of b1 to a1 as defined in Equation 3.75. Terminating the output port with the characteristic impedance of the transmission line is equivalent to setting a2 ¼ 0, because a traveling wave incident on this load will be totally absorbed and no reflection would take place. The parameter s11 is the input reflection coefficient of the network. Similarly, s21, the forward transmission through the network, can be found, see Equation 3.75. This is the ratio of b2 to a1. This could either be the gain of an amplifier or the attenuation of a passive network. As for the parameters as seen from the output, the input terminal is terminated to set a1 ¼ 0. The s22, which constitutes the output reflection coefficient, and s12, the reverse transmission coefficient, can then be measured as outlined above for s11 and s21 and/or set up in Equation 3.75. As long as the transmission line is terminated in the characteristic impedance of the line, the network port does not have to be matched to that impedance. To reiterate, to measure the s-parameters one does follow the procedure outlined below: Apply input to port 1; terminate port 2 with a matched load;

s11

s21

Z0 ; making a2 ¼ 0:  b1  ¼  ¼ Reflection coefficient at port 1 a1 a2 ¼0 ðmatched load; Z0 ; at port 2Þ  b2  ¼  ¼ Forward voltage transfer ratio from port 1 to port 2 a1 a2 ¼0 ðmatched load; Z0 ; at port 2Þ;

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which is the gain coefficient of the network in an active circuit or the loss: Apply input to port 2; terminate at port 1 with a matched load;

s22

s12

Z0 ; making a1 ¼ 0:  b2  ¼  ¼ Reflection coefficient at port 2 a2 a1 ¼0 ðmatched load; Z0 ; at port 1Þ  b1  ¼  ¼ Reverse transfer ratio from port 2 to port 1 a2 a1 ¼0 ðmatched load; Z0 ; at port 1Þ; or the reverse transmission coefficient with the input matched:

ð3:76Þ

The above expression is in terms of voltage. To do the same in terms of power, one has to multiply each of the scattering parameter with its complex conjugate or simply take the amplitude square. In this case, |s11|2, |s21|2, |s22|2, and |s12|2 represent the power reflected at port 1, power transmitted from port 1 to port 2, power reflected at port 2, power transmitted from port 2 to port 1, respectively. We should note that the scattering parameters, |sij|exp( jqij), are complex with |sij| representing the amplitude ratio (in the case of s11, it represents the amplitude ratio of b1 over a1, etc.) and qij representing the phase difference. The parameter s11 represents the input reflection coefficient with the output matched to the characteristic impedance, Z0; s21 represents forward transmission gain or loss; s12 represents reverse transmission or isolation; and s22 represents output reflection coefficient with the input matched to the characteristic impedance, Z0. We should mention that in sij designation, i and j represent where the RF power enters and emerges, respectively, and all the s-parameters can be plotted on a Smith chart, which will be shown in Section 3.3. In practice, it is not possible to connect the network analyzer port to the device under test without input and output extension lines. At high frequencies, even an impedance matched line does in fact introduce phase shift, which must be accounted for. This is done by defining a reference plane and keeping track of the phase shifts. The problem is depicted in Figure 3.28 where the two line segments of lengths l1, l2 with their characteristic impedances equal to the reference impedance Z0 are incorporated. In this extended line configuration, the network analyzer measures the waves a0 1 ; b0 1 and a0 2 ; b0 2 , the generator and load ends of the line segments, as shown in Figure 3.28. To handle this problem, one must move the reference plane. From these, the waves at the inputs of the two-port network can be determined rather easily when one assumes lossless transmission line segments, l1 and l2, in the following manner:     0      0  a1 a2 a2 a1 ejbl1 0 ejbl2 0 ¼ and ¼ ; 0 0 b1 b2 0 ejbl1 b 1 0 ejbl2 b 2 ð3:77Þ

3.2 The s-Parameters and Gain

l1 i1

Zs vs

l2 i1

+ v1 _

+ _

+ v1 _

Z0

a1

Two - port network s

+ v2 _

a1 b1

b1

Z0

+ v2 _

t2

ZL b2 a2

b2 a2 t1

t1

i2

i2

t2

Figure 3.28 Two-port network under test with input and output line segments l1 and l2, which are necessary for measurements and whose contributions must be accounted for.

where bl1 and bl2 are the phase lengths of line segments. Equation 3.77 can be rearranged into the forms:     0   0    b1 a1 a1 b1 e jbl1 0 e jbl1 0 ¼ and ¼ : ð3:78Þ a02 b2 0 e jbl2 b02 0 e jbl2 a2 The network analyzer measures the corresponding s-parameters of the primed variables, which represent a combination of the line segments and the DUT, as  0  0    0  s11 s012 b1 s11 s012 a01 0 ¼ 0 ; where s ¼ 0 b02 s21 s022 a02 s21 s022 is the measured scattering matrix:

ð3:79Þ

The s-matrix of the two-port can be obtained then from 

b1 b2





 0    0   b1 s11 s012 a01 e jbl1 0 e jbl2 0 ¼ ¼ b02 e jbl2  0 e jbl2 s021 s022  a01 0 0 0 jbl1 s11 s12 e jbl1 0 a1 0 e ¼ 0 0 jbl2 jbl2 s s a 0 e 0 e 2 21 22    e j2bl1 s011 e j2bðl1 þ l2 Þ s012 a1 ¼ : a2 e j2bðl1 þ l2 Þ s021 e j2bl2 s022

ð3:80Þ

Thus, changing the points along the transmission lines at which the s-parameters are measured introduces only phase changes in the parameters. The s-parameters of the DUT can thus be related to the measured s-parameters as: s11 ¼ s011 e j2bl1 s21 ¼ s021 e j2bðl1 þ l2 Þ

s12 ¼ s012 e j2bðl1 þ l2 Þ s22 ¼ s022 e j2bl2

ð3:81Þ

Without loss of generality, we may replace the extended circuit of Figure 3.28 with that shown in Figure 3.29. This means that we can use the extended two-port parameters s0 . Or equivalently, we can use the one shown in Figure 3.29 where in the generator and segment l1 have been replaced by their Thevenin equivalents, and the load impedance has been replaced by its propagated version to distance l2.

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vs

i2

i1

Zs + v1 _

+ _

Zin

vth

+ _

Zout

+ v2 _ a2 b2

a1 b1 (a)

ZL

(b)

Figure 3.29 Input (a) and output (b) equivalent circuits of a two-port network.

As in the case where we neglected the line segments that are physically necessary, the actual measurements of the s-parameters are made by connecting to a matched load, ZL ¼ Z0. Then, there will be no reflected waves from the load, a2 ¼ 0, and the s-matrix equations will lead to Equation 3.76 for the input reflection and forward transmission coefficient (generally the gain term) and reversing the roles of the generator and the load will lead to the output reflection coefficient and reverse transmission coefficient. Measurement of s-parameter representative of the DUT is not a trivial undertaking. To deembed the contributions by the network analyzer and the measurement test bed, a series of meaningful calibrations must be performed. The lofty goal of being representative of the DUT must be discussed in the realm of systematic errors inherent to the system and random errors. Although random errors will not be removable, those systematic errors arising from an imperfect switch in the test set, impedance mismatches at the test set ports, frequency response of the transmission and reflection paths, and coupling of fields at input and output (cross talk) can be removed. Considering the signal flow in the forward and reverse directions, their combination yields the error adapter, yielding 12 error terms elaborated in Figure 3.30. By measuring sufficient number of “standards” (physical devices: shorts, opens, thrus, and loads) with known physical and electrical properties, equations describing the scattering matrices of the entire system (error adapter plus standard) can be simultaneously solved, resulting in a known error adapter, which can then be mathematically removed from further measurements of devices that one wishes to measure, as well as define a “reference plane.” The reference plane is the plane where the analyzer, cables, probes, connecters, and so on end and the DUT begins. Although several methods exist for the full 12-term error correction, only two are directly implementable in the Agilent 8510: short, open, load, and thru (SOLT) and through, reflect, and line (TRL). The SOLTmethod is frustrated by the need for a good load standard. The TRL approach is somewhat better as it does not rely on known standard loads and allows for simple error boxes to be characterized fully. It should be noted that only two-port test sets (s-parameter) can fully determine all of the sources of error (as opposed to one with two-port and only reflection/transmission capabilities). The block diagram of such a test set is shown in Figure 3.31. The through (thru for short) is a pass between the input and output ports at the desired reference plane. The “reflect” as the name implies utilizes a mismatch load

3.2 The s-Parameters and Gain

Port 2

Port 1

Reverse ELR s 12M ETR

s 21A

ERR

s 22A s 11A

ESR

s 12A

1

s 22M EDR RF in

EXR

DUT

EXF

Forward s 21A

RF in 1 EDF

ESF

s 11M ERF

ETF

s 22A s 11A

s21M

ELF

s 12A Port 1

Port 2

Figure 3.30 EDF, EDR: directivity (cross talk within a port from source directly through the directional coupler); ESF, ESR: source match (closeness of Rl to 50 ohm); ELF, ELR: load match (quality of load path’s Rl termination); ERF, ERR: reflection tracking (frequency response tracking, how well the magnitude and phase of a signal

track the reference signal); ETF, ETR: transmission tracking (transmission mismatch due to impedance mismatch between test systems ports and DUTs ports); EXF, EXR: isolation (leakage of energy between test and reference channels in VNA). Figure from Ref. [141].

with a large reflection coefficient, such as short or open. It is not imperative to know the reflection coefficient a priori since it is determined during the TRL calibration procedure. In the “line” configuration a matched transmission line of a length is connected between the two ports. The length of the line does not have to be known a priori and it does not need to be lossless as the pertinent parameters are determined during the TRL measurements. The length of the line, however, goes toward determining the frequency band within which the TRL calibration procedure is validated. Somewhat counterintuitive at first, a lossy line is better as it leads to a usable calibration over a wider band. To span a wider band, a multiline method can be employed in which shorter lines would be for higher frequencies and the longer counterparts for the lower frequencies. SOLT is the most popular method of calibration [142], due in large part to the facility of use and the dedication on the part of HP in making precision coaxial standards. It is well known, however, that small deviations from ideal in the definitions of the standards can result in large errors [143]; in addition, the characterization of standards above 20 GHz becomes quite difficult [144]. For this reason, standards embedded in coplanar waveguide [145] are difficult to realize for this technique due to the fact that the positioning of the probes is not precise. There is therefore ambiguity in the position of the reference plane. TRL calibration process does not determine the characteristic impedance that is the reference for the subsequent s-parameters. Quite often assumptions and

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Source

Transfer switch

R1

A

B

R2

Port 1

Port 2

Figure 3.31 The block diagram of a test set for measuring s-parameters.

measurements must be made to find Z0, the characteristic impedance of the line, and renormalize (to say 50 W). If this is not done, and the imaginary component of the characteristic impedance and the frequency dependence are ignored, at low frequencies (up to 1 GHz) the ohmic loss dominates the inductive reactance (per unit length) and the imaginary part of Z0 can be quite large. This can also cause some ambiguity in the position of the reference plane. The TRL method is suitable for on-wafer characterization of devices (meaning the calibration standards must be fabricated alongside the DUT on the same substrate) and problems regarding the reference plane position can be circumvented if the DUT is effectively placed behind a transmission line half the length of the thru. At low frequencies, however, the TRL calibration still often fails. TRL can be much more accurate than SOLT at high frequencies, and the standards are simpler to fabricate since there is no need for a precise, well-defined line. However, it consumes valuable space on the wafer (long lines, up to 40 mm, which are required to calibrate down to low frequencies of 1 GHz) and the bandwidth (BW) of the standards only spans a 8 : 1 range of frequency in the traditional TRL [146]. As such, discontinuities at these frequencies often result in discontinuities in the measured s-parameters of devices, which has no physical meaning. Moreover, the reference impedance is set by the characteristic impedance of the line (it must therefore be known precisely), and the reference plane is set at the center of the thru, which can cause a problem in nonzero length thrus (as in all coplanar waveguides (CPW) must be); applying an offset delay is possible in the 8510, but this assumes linear phase of the line, which is only

3.2 The s-Parameters and Gain

rigorously valid in coaxial media (again the moding effects of the CPW are the source of the problem). More advanced calibration techniques exist, but are not directly able to be automated in the 8510. These techniques require either WinCal software available from Cascade Microtek (the now popular and very accurate line, reflect, reflect, match (LRRM) technique [147], which happens to be patented thus LRRM can be implemented using WinCal only) or Multical software available for free from the National Institute of Standards and Technology (NIST) (which runs in the HP Basic (interpreted language of the HP9000 OS environment and can calibrate the NISTTRL or LRM calibration schemes). Another issue in CPW is that the quasi-TEM (transverse electromagnetic) mode assumed to be dominant in CPW is not true as frequency increases (microstrip, parallel-plate modes begin to ruin things). This is especially true for conductorbacked CPW (CBCPW) or at high frequencies on thin substrates. Let us now succinctly lay out the step-by-step approach to measurements and the pros and cons of the measurement method we discussed briefly above. Note that for calibration checks, benchmark or Golden Standards would be ideal. Measure S11 with port 1 open should be 0  1 dB Connect a load to port 1, S11 should be less than “the specified calibrated directivity of the analyzer” (usually less than 30 dB) Connect a short or open to port 1, S11 should be around 0 dB. Connect a thru between ports 1 and 2, S21 should be around 0 dB Connect two loads on each port, see that S21 is lower than specified isolation (usually 80 dB). Pros and cons of various (popular) calibration methods: SOLT (SOLR): Commonly used, standards ideal, known C(open), L(short), nonself-consistency, sensitive to probe placement, Z0 reference: trimmed resistor, fair accuracy. TRL (LRL): Easy to fabricate standards, self-consistent, 8 : 1 frequency range, standards must be on wafer, Z0 reference: transmission line, propagation constant determined (shift reference plane) “poor–fair accuracy.” TRM (LRM): Standards easier than SOLT to fabricate, open/short C/L unknown, Z0 reference: trimmed resistor, better frequency range than TRL, propagation constant unknown (cannot shift reference plane). LRRM: Must have Cascade Microtek software, very good accuracy. NIST TRL: Multiline removes 8 : 1 issue, best accuracy. Let us now continue with our discussion of s-parameters and their use in our characterization of the device. Because s11 and s22 represent the reflection coefficient at the input and output ports, it is instructive to also mention the reflection

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coefficient, G, from and impedance discontinuity such as that in a transmission line with a characteristic impedance of Z0 (can be viewed as the generator impedance) looking into an input or load impedance ZL. G is defined as the reflected voltage divided by the incident voltage at the point of discontinuity, which when done at the load and looking into the load takes on the form GL ¼

ZL Z 0 ZL þ Z0

and

GL ¼

Y 0 Y L ; Y0 þ YL

in terms of impedance and admittances or in terms of reduced parameters (dividing with the characteristic impedance of admittance) GL ¼

zL 1 zL þ 1

or

GL ¼

1yL ; 1 þ yL

in terms of the reduced impedance; ð3:82Þ

respectively, and is a complex number. Similarly, the reflection coefficient at the input looking into the source is given by Z S Z 0 ZS þ Z0

GS ¼

or

GS ¼

Y 0 Y S ; Y0 þ YS

in terms of impedance

and admittances; respectively:

ð3:83Þ

The reflection coefficient at the input looking into the input of the two-port network is given by Gin ¼

vr1 s12 s21 GL Z in Z0 ¼ s11 þ ¼ : 1s22 GL Zin þ Z0 vi1

ð3:84Þ

The reflection coefficient at the load looking into the output of the two-port network is given by Gout ¼

vr2 s12 s21 GS Zout Z0 ¼ s22 þ ¼ : 1s11 GS Zout þ Z0 vi2

ð3:85Þ

By voltage division at the input port of the network, shown in Figure 3.26, we obtain v1 ¼

Zin vS ¼ vi1 þ vr1 ¼ vi1 ð1 þ Gin Þ: Zin þ Z S

ð3:86Þ

Employing Zin ¼ Z0

1 þ Gin 1Gin

ð3:87Þ

and Equation 3.84 and solving for vi1, we obtain vi1 ¼

1GS vS : 1Gin GS 2

ð3:88Þ

3.2 The s-Parameters and Gain

As they are commonly used, let us now relate the incident and reflected voltage terms to a and b parameters. Referring to Figure 3.26, the input and output voltages and currents of the two-port network can be written as v1 ¼ vi1 þ vr1 ; v2 ¼

vi2

i1 ¼

vi1 vr1 ; Z0

vi vr i2 ¼ 2 2 : Z0

þ vr2 ;

ð3:89Þ

pffiffiffiffiffiffi If we divide both sides of these equations by Z 0, the relationship will not change, aij and bij parameters can then be defined as vi1 ffi; a1 ¼ pffiffiffiffiffi Z0

vi2 ffi; a2 ¼ pffiffiffiffiffi Z0

vr1 ffi; b1 ¼ pffiffiffiffiffi Z0

vr2 ffi; b2 ¼ pffiffiffiffiffi Z0

ð3:90Þ

which when constructed in the form of Equation 3.75, the resultant coefficients become the s-parameters. At high frequencies, although used sometimes, lumped element configuration is not as suited as distributed element representation to represent the traveling wave nature of the signal. In the traveling wave case, the relationship will not change, and a1 and b1 at the input port, port 1, and a2 and b2 at the output port, port 2, can be expressed in terms of input and output voltages and currents as a1 ¼

v 1 þ Z 0 i1 v2 Z0 i2 pffiffiffiffiffiffi ; a2 ¼ pffiffiffiffiffiffi ; 2 Z0 2 Z0

v1 Z 0 i1 b1 ¼ pffiffiffiffiffiffi ; 2 Z0

v 2 þ Z 0 i2 b2 ¼ pffiffiffiffiffiffi : 2 Z0

ð3:91Þ

The descriptions of port 1 and port 2 parameters are the same if we recognize that i2 can be represented by incoming current i2. Equation 3.91 can be rewritten to isolate the input and output voltages and currents as pffiffiffiffiffiffi pffiffiffiffiffiffi v1 ¼ Z0 ða1 þ b1 Þ; v2 ¼ Z 0 ða2 þ b2 Þ; ð3:92Þ 1 1 i1 ¼ pffiffiffiffiffiffi ða1 b1 Þ; i2 ¼ pffiffiffiffiffiffi ðb2 a2 Þ: Z0 Z0 In practice, the reference impedance is chosen to be Z0 ¼ 50 W. As mentioned earlier, at lower frequencies the transfer and impedance matrices are commonly used, but at microwave frequencies they become difficult to measure, and thus the scattering matrix description is the preferred method. The s-parameters can be measured by embedding the two-port network (the device under test in our case or DUT) in a transmission line whose ends are connected to a network analyzer.

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Now that we are somewhat familiar with s-parameters and reflection coefficients, we can now turn our attention to the derivation of the power gain equations. The power into a two-port network is expressed in terms of input voltage and current, and input resistance (recalling v1 ¼ Z in i1 ; Rin ¼ ReðZin Þ) and utilizing the input and output equivalent circuits of a two-port network shown in Figure 3.27. Referring to the two-port network shown in Figure 3.27, we can write i1 and i2 in terms of the generator (source) voltage to reexpress the input and output powers as Pin ¼

1 Rin jvS j2 2 jZS þ Z in j2

and

Pout ¼

1 RL jvth j2 : 2 jZL þ Z out j2

ð3:93Þ

The power out of the two-port network similarly is (recalling v2 ¼ ZLi2, RL ¼ Re(ZL)): 1 1 Pout ¼ Reðv2 i2 Þ ¼ RL ji2 j2 : 2 2

ð3:94Þ

Alternatively, we can utilize what is commonly used in transmission lines (and referring to Figure 3.26) and express the power delivered to the network as  i 2   v    jvS j2 1jGin j2 j1GS j2 Pin ¼ 1 1jGin j2 ¼ ; ð3:95Þ 2Z0 8Z0 j1GS Gin j2 where Equation 3.88 was utilized. We should be reminded that peak values are assumed for voltages and currents, and the average power equations are being developed. Following a similar procedure, the average power delivered to the load can be expressed as  r 2 v    ð3:96Þ PL ¼ 2 1jGL j2 : 2Z0 Noting that vr1 ¼ s11 vi1 þ s12 vi2 ¼ s11 vi1 þ s12 GL vr2

ð3:97Þ

and by substituting it into Equation 3.96 and using Equation 3.88, we can write for the average power delivered to the load:  i 2 2     v  js j 1jGL j2 jvi1 j2 js221 j 1jGL j2 j1GS j2 1 21 ¼ : ð3:98Þ PL ¼ 8Z0 j1s22 GL j2 j1Gin GS j2 2Z0 j1s22 GL j2 The average power delivered to the load given by the above equation can also be related to the source voltage by relating vi1 to the source voltage by using Equation 3.88, which when performed leads to PL ¼

  jvS j2 js221 j 1jGL j2 j1GS j2 8Z0 j1s22 GL j2 j1GS Gin j2

:

ð3:99Þ

There are three widely used definitions for the power gain of a two-port network: the power gain G, also called the operating gain, the available power gain GA, and the transducer (amplifier specifically) power gain GT. They are defined as follows:

3.2 The s-Parameters and Gain

Power gain, G, or just the operating power gain, is the ratio of power out of the network into the load (power dissipated in the load ZL) to the power into the twoport network. This gain is independent of ZS. Available gain, GA, is the ratio of power available from the two-port network to the power available from the source. This automatically assumes conjugate matching of both the source and the load. It depends on ZS but not ZL. Transducer gain, GT (amplifier in our case strictly speaking), is the ratio of the power delivered to the load to the power available from the source, and it depends on both ZS and ZL. The operating power gain, or just the power gain, defined as the power out of the network into the load divided by the power into the network is expressed as   1jGL j2 PL 2 G

¼ js21 j  :  Pin 1jGin j2 j1s22 GL j2

ð3:100Þ

The power available from the source is the maximum power that can be delivered by the source (generator) to a load, which is called the available power of the source, PavS. This takes place when the input impedance of the two-port network is conjugately matched to the generator, in which case, PavS ¼ Pin when Zin ¼ Z S . Using Equation 3.95, PavS can be expressed as PavS ¼ P in jGin ¼G ¼ S

jvS j2 j1GS j2   8Z 0 1jGS j2

ðpower available from the sourceÞ: ð3:101Þ

Similarly, the power available from a two-port network, PavN, is the maximum power that can be delivered by the Thevenin-equivalent circuit of Figure 3.29 to a load, that is, PavN ¼ PL when ZL ¼ Zout . It follows then from Equation 3.99 that the available powers are    jvS j2 js221 j 1jGout j2 j1GS j2  Pavn ¼ PL jGL ¼Gout ¼    8Z0 j1s22 Gout j2 j1GS Gin j2   GL ¼Gout

ðpower available from the networkÞ:

ð3:102Þ

In Equation 3.102 Gin must be evaluated for GL ¼ Gout. One can see that, PavS, and Pavn can be obtained from Pin and PL by setting Gin ¼ GS and GL ¼ Gout , which represent the conjugate matching conditions. Utilizing Equation 3.84 we can show that  2  1jGout j2 j1s11 GS j2 1GS Gin j2 GL ¼G ¼ ; ð3:103Þ out j1s22 Gout j2 which simplifies Equation 3.102 to

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Pavn ¼

jvS j2 js221 jj1GS j2 8Z0 j1s11 GS j2 ð1jGout j2 Þ

:

ð3:104Þ

Turning our attention back to the power gain, using Equations 3.101 and 3.104, the available power gain, power available from the two-port network to the power available from the source, can be expressed as GA

P avn 1jGS j2 ¼ js21 j2 : P avS j1s11 GS j2 ð1jGout j2 Þ

ð3:105Þ

The transducer (amplifier in our case strictly speaking) power gain is defined as the ratio of the power delivered to the load to the power available from the source. Therefore, from Equations 3.99 and 3.101 we obtain GT

   1jGS j2 1jGL j2 PL ¼ js221 j P avS j1GS Gin j2 j1s22 GL j2

or



  1jGS j2 1jGL j2 : GT ¼ js21 j    j 1s11 GS 1s22 GL s12 s21 GS GL j2 2

ð3:106Þ

We should note that a special case of the transducer gain, GT, occurs when GL ¼ GS ¼ 0, in which case Equation 3.106 reduces to GT ¼ js221 j:

ð3:107Þ

When the reverse transmission term S12 ¼ 0 or negligibly small, the transducer gain, GT, becomes the unilateral gain, GTU. This nonreciprocal (which is why the term unilateral is used) is the goal of device and amplifier designers with good amplifiers coming close to achieving this. From Equation 3.84, we see that when S12 ¼ 0, the input reflection coefficient Gin ¼ S11. Substituting this now into Equation 3.106 reduces to, we obtain for the unilateral transducer gain, GTU,    js21 j2 1jGS j2 1jGL j2 GTU ¼ : ð3:108Þ j1s22 GL j2 j1s11 GS j2 The transducer gain, GT , is the best representative of gain as it features the effects of both load and generator (source) impedances. For comparison, G depends only on the load impedance and Ga depends only on the generator impedance. If the generator and load impedances are matched to the characteristic impedance Z0, making ZS ¼ ZL ¼ Z0, GS ¼ GL ¼ 0 and Gin ¼ S11, Gout ¼ S22, the power gain then simplifies to GT ¼ js21 j2 ;

GA ¼

js21 j2 1js22 j

2

;

G¼

js21 j2 1js11 j2

:

ð3:109Þ

By definition when the reverse transmission coefficient S12 ¼ 0, which makes S21S12GSGL ! 0, the resultant gains are termed as the unilateral gains. The reverse transmission coefficient being zero provides that Gin ¼ S11, Gout ¼ S22, in which case

3.2 The s-Parameters and Gain

the gain expressions of the three gain equations become (the unilateral transducer gain given in Equation 3.108 is repeated below for completeness) GTU ¼ js21 j2 GAU ¼ js21 j2 GU ¼ js21 j2

1jGS j2

1jGL j2

j1s11 GS j2 j1s22 GL j2 1jGS j2

1

j1s11 GS j2 1js22 j2 1jGL j2

1 2

j1s22 GL j 1js11 j2

; ð3:110Þ

;

:

As can be gleaned from both the bilateral (the reverse transmission coefficient is nonzero) and the unilateral (the reverse transmission coefficient is zero) cases, the gains G and GA are obtained from GT by setting Gin ¼ GS and GL ¼ Gout as we did in the case of PavN and PavS above. The amplifier’s gain is maximum when both the input and the output ports are matched simultaneously with conjugate loads meaning Gin ¼ GS and GL ¼ Gout in which case all three gains become equal to each other and each is called the maximum available gain (also called the matched gain). These conjugate matching conditions for maximum power reduce the gain equation given by Equation 3.108 to GTmax ¼ js21 j2

1

1jGL j2

1jGS j2 j1s22 GL j2

:

ð3:111Þ

Using Equations 3.84 and 3.85 to satisfy the maximum power conditions Gin ¼ GS and GL ¼ Gout leads to GS ¼ s11 þ

s12 s21 GL 1s22 GL

and

GL ¼ s22 þ

s12 s21 GS : 1s11 GS

ð3:112Þ

For a unitary amplifier, that is, S12 ¼ 0, GTU max GMAG ¼ js21 j2

1

1 2

1js11 j 1js22 j2

:

ð3:113Þ

Using GS ¼ Gin ¼ s11 þ

s12 s21 GL 1s22 GL

and GL ¼ Gout ¼ s22 þ

s12 s21 GS : 1s11 GS

ð3:114Þ

substituting the above GS and GL into Equation 3.111 and simplifying leads to pffiffiffiffiffiffiffiffiffiffiffiffi js21 j ð3:115Þ ðK K 2 1Þ: GMAG ¼ js12 j If the FET is unconditionally stable (K > 1) with K¼

1js11 j2 js22 j2 þ jðs11 s22 s12 s21 Þj2 ; 2js12 s21 j

the parameter K is called the Rollett stability factor.

ð3:116Þ

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The maximum available gain is not very useful if the device is not only conditionally stable, see below since in that simultaneous conjugate matching of the source and load is not possible if K < 1. In this case, the maximum stable gain (MSG), which is attained for K ¼ 1, is more meaningful. Using the condition in Equation 3.115 leads to K ¼ 1 GMSG ¼

js21 j : js12 j

ð3:117Þ

As done previously in the unilateral case, MAG is obtained by setting Gin ¼ GS ¼ s11 and GL ¼ Gout ¼ s22 in Equation 3.110, which leads to GMAG; U U ¼ js21 j2 

1 1js11 j2

1  : 1js22 j2

ð3:118Þ

The first term in Equation 3.118 is related to the transistor forward transfer ratio and represents the power gain provided by the device. The second term is the power gain provided by the input circuit (input gain factor) and represents the degree of mismatch between the characteristic impedance of the generator (source) and the input reflection coefficient. The third term represents the power gain provided by the output circuit (output gain factor serves the same function as the second except it is for the output in that it represents the output matching). Again, the maximum unilateral gain can be achieved when GS ¼ s11 and GL ¼ s22 . If the source and load reflection coefficients GS and GL are made zero by matching them, the unilateral gain is reduced to |S21|2. Having mentioned stability warrants some discussion of this important topic. Oscillations in an otherwise amplifier circuit are possible if the real part of the input or the output impedance goes negative, which would imply |Gin| > 1 or |Gout| > 1. The dependence of Gin and Gout on the source and load-matching networks makes the stability of the amplifier to depend on GS and GL. There are two types of stability criteria: Unconditional stability is attained when both |Gin| < 1 and |Gout| < 1 for all passive source and load impedances, which in other words can be stated as |GS| < 1 or |GL| < 1. Conditional stability is attained if |Gin| < 1 and |Gout| < 1 for only a certain range of passive source and load impedances, and thus the nomenclature potentially unstable. Because the reflection coefficients depend on frequency, the stability of the amplifiers also depends on frequency. The amplifier can be stable at the designed frequency but unstable at other frequencies. The stability condition, |Gin| < 1 and |Gout| < 1, can be represented with what is called the stability circles in the Smith chart, drawn for |Gin| ¼ 1 and |Gout| ¼ 1, with the aid of Equations 3.84 and 3.85. The amplifier is stable in the overlapping region of the stability circles. Unlike the conditional stability, meaning finding input and output matching conditions that allow stable operation of the amplifier, there are simpler stability criteria that can be applied. One among them is the K  D test with K given by Equation 3.116 and being the determinant D ¼ s11s22  s12s21 of the scattering matrix.

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

As indicated in the context of conditional stability, the unconditional stability is achieved when K > 1 and |D| < 1 conditions are simultaneously satisfied. Note that the above unconditional stability criterion depends only on the device s-parameters and does not involve the input and output matching networks. If the s-parameters of the device do not satisfy the K > 1 and |D| < 1 conditions, then the stability circles mentioned above must be used to find the values of GS and GL for which the device would be conditionally stable. Another test utilizing only one parameter, again defined by the s-parameters of the device, has been introduced also for stability criterion. This involves a m-parameter given by m¼

1js11 j2 >1: js22 Ds11 j þ js12 s21 j

ð3:119Þ

The greater the m above 1, the more stable the device is. Switching back to the device-level discussion, we can revisit the current gain and determine it in terms of the s-parameters. The short-circuited current gain is obtained by short circuiting the output (v2 ¼ 0). The current gain h21 : hfe can be obtained from Equation 3.73 by h21 ¼ ði2 =i1 Þjv2 ¼0, which when converted to s-parameters takes the form h21 ¼

2s21 : ð1s11 Þð1 þ s22 Þ þ s21 s12

ð3:120Þ

Extrapolation of |h21|2 versus frequency to 0 dB leads to the current gain cutoff frequency fT. Sometimes, another frequency, fs, is also defined, which is the frequency at which |s21|2 (the unilateral gain provided by the device alone with no input and output reflections) goes to unity or 0 dB. We should mention that the slope is 6 dB per octave for one pole circuit. One such exercise for a hypothetical FET is shown in Figure 3.32a with the slope being 6 dB per octave. The plot of |h21|2, GMAG, and GMAG,U for a fabricated short-channel GaN HFET [148] as a function of frequency is shown in Figure 3.32b. We should mention that the unilateral gain (transducer gain when the reverse transmission term s12 ¼ 0 or negligibly small) for a conjugately matched FET amplifier can be obtained by using the equivalent circuit (to be discussed later in conjunction with Figure 3.34b) while neglecting all but the very basic circuit components of the intrinsic device is given by.
2 PL 1 fT ¼ : ð3:121Þ GTU ¼ PavS GDS Ri f This means that the gain drops with the square of the operating frequency with a slope of 6 dB per octave.

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

Modeling of FETs is essential from the engineering point of view, not only because it can provide an accurate prediction of circuit performance under the operating

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20

Gmax

16

s21 2

Gain (dB)

hfe

2

12

8

4

0

fT 7 8

20

10

30

40

fS

50 60 70 80

fmax 100

Frequency (GHz) (a)

60

V DS=10 V for GMSG 4 V for h21(hfe)

50

Gain (dB)

40 30 20 10 0 10-1 (b)

Unilateral gain (GMAG,U)

f max (GMAG,U )=240 GHz

f max (GMSG)=225 GHz

V GS=-3.8 V for both

Maximum stable gain (GMSG) f T=190 GHz Current gain hfe 2

100

101

102

103

Frequency (GHz)

Figure 3.32 Plot of |h21|2, |s21|2, and GMAG versus frequency indicating the current gain |s21|2 and maximum available gain cutoff frequencies. Plot of |h21|2, GMAG, and GMAG,U as a function of frequency for a fabricated device.

conditions but also because it facilitates the design and integration routines in conjunction with microwave applications such as amplifiers, mixers, oscillators, or filters that are widely used in burgeoning communication systems. In general, the modeling of FETs falls into two categories: physical and empirical models. The former constructs the device model based on the electromagnetic and carrier transport theory. Monte Carlo calculations are often adopted based on the real device

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

geometry [149–151], with some parameters required to be adjusted according to the experiments. Merits of the physical device modeling lie in that the theoretical explanations are attempted to undercover the basic physics behind the device operation, which indeed could help us to better understand device design. Unfortunately, a good deal of simplifications and approximations are required to reduce the complexity of numerical calculations and reduce them into the feasible category, albeit at the expense of some accuracy. Even then the computational burden is already heavy. In addition, the lack of clarity for many mechanisms that cause the degradation of devices needs to be addressed for more precise prediction of device performance. Especially for GaN-based HFETs, the presence of more defects and the applications under high-power conditions bring about extra difficulty in fitting the experiments within the available models. In practice, physical device modeling can provide the preliminary guidelines for device optimization, but the routines used are computationally still highly complex, which must be streamlined further before they are incorporated into the circuit design techniques. Empirical or analytical device modeling in the context of parameter extraction is driven by circuit models consistent with the measurements in fabricated devices. It can also predict the device behavior in a range where the measured data are not available. Empirical models need to construct analytical equations for the description of the device; therefore, the computational efforts are much smaller than those in the physical modeling methods. The traditional empirical device modeling methods are often referred to as the equivalent circuit methods that apply discrete circuit elements to the physical attributes of the transistors. For example, a gate–source capacitor represents the capacitance of the depletion region under the gate; a drain–source resistor represents the output resistance of the channel; or a drain inductor represents the parasitic lead inductance of the metallization running to the device. Relatively speaking, the empirical methods could be much more flexible, despite their somewhat black box nature, especially for a better fitting in large-signal situations or interpolating some special effects such as self-heating, trap-assisted current collapse, or current lag than physical modeling methods. For example, tablebased modeling has been studied by many groups with great interest [152–154]. Instead of using linear circuit components, the equivalent circuit is described as nonlinear transconductances and transcapacitances. Those components are symbolized as lookup tables obtained from measurement data. Another hotly pursued approach uses neural networks to map the nonlinearities of equivalent circuit [155, 156] and train the model through a certain algorithm from the measured data. In short, the modeling approaches that offer the flexibility to describe the device behavior in regions where the physical or predefined equation-based models fail to simulate are valuable. The various empirical methods may vary in data acquisition, extraction procedures, computational complexity, and accuracy, but normally their execution time is much shorter than that of physical modeling. Therefore, when properly defined and extracted, empirical device modeling is more practical for design purposes.

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3.3.1 Small-Signal Equivalent Circuit Modeling

To address the above discussion, we will present the theories for empirical equivalent circuit models on III–V-based HFETs. The small-signal modeling is discussed first followed by the large-signal modeling. In terms of the small-signal configuration, the transistor is biased in the saturation region, and the small-signal nature of input signal will not affect the DC bias conditions of the device. Therefore, a small input voltage will produce approximately a linear response in the drain–source current. However, if the amplitude of input signal is increased considerably, the extent of which depends on the device linearity, at certain point, the drain–source current no longer changes linearly. Also, the large signal will bring about a shift in the quiescent operating point of the transistor and strong nonlinearity sets in. In the frequency domain, no additional frequency components are created in the case of small signal, only phase and amplitude of the output signal are changed. However, in large-signal operation, the introduction of multiple harmonics will make the circuit analysis different since the Kirchhoff’s law is not satisfied anymore. Shown in Figure 3.33 is a schematic representation indicating the equivalent circuit elements overlaid onto a cross-sectional FET diagram. Here, we are considering only the lumped rather than the distributed nature of the elements. For smallsignal analysis, all components are assumed to have frequency-independent values. It is customary to establish an equivalent circuit for FETs whose complexity is determined by the depth to which one wants to treat the device in a circuit environment or undertake diagnosis [157–159].

Figure 3.33 Equivalent circuit of a FET overlaid on top of a cross-sectional diagram of the same.

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

For an AlGaN/GaN HFET, a slightly more complicated equivalent circuit including the parasitic capacitances between the terminals and the ground, shown in Figure 3.34, can be developed [160]. The model can be divided into two subparts: (1) a nonlinear and bias-dependent intrinsic part corresponding to the inner device that excludes the contribution from the access regions as well as the stray capacitances, which is outlined with the dashed

G

LG

RG R fgs

Cgsp

R fdg

Cgip

Cdgi R DG

CDG

Vi

CGS Ri

RD Ids Cds Gds RS

LD

Cdip

D

Cdsp

LS S

IDS=Vigmoe-jωτ

S

(b) Figure 3.34 (a) High-frequency equivalent circuit of an FET superimposed on a 3D schematic of the device wherein the stray parasitic capacitances are also included. The equivalent circuit contained in the dashed area represents the intrinsic device without contributions from access regions and stray capacitances; (b) the equivalent circuit gleaned from (a); (c) a somewhat simplified version of the equivalent circuit.

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R fdg LG

G

Cgsp

RG

C GS Cgsi

RD

CDG R DG

R fgs

Vi

IDS GDS

Ri

CDS

LD

Cdsi

Rs

D

Cdsp

Ls S

IDS =V igmoe-jωτ

(c)

S

Figure 3.34 (Continued).

box; and (2) a linear bias-independent extrinsic one corresponding to parasitic access elements. Regarding the extrinsic part, Cgsp, Cdgp, and Cdsp represent the parasitic capacitances introduced by the pad connection and probe contacts, and Cgsi, Cdgi, and Cdsi account for the interelectrode capacitances. If we neglect the extrinsic parasitic capacitances and change to intrinsic differential resistance of the gate–source and gate–drain diodes (Rfgs and Rfgd), the model can be further simplified to that shown in Figure 3.34c, which is more applicable to GaAs MODFET [157]. It is worthwhile to mention that charging resistance, Ri, in the intrinsic part, represents the electron recombination with the depleted donors under the gate depletion region; the delay time t is the time it takes for the depletion region to respond to the change in the gate signal. By matching the two-terminal y-parameters calculated from the s-parameters and the equivalent circuit of Figure 3.34, one can determine the values of the elements shown in the same. The crux of the problem, however, is to converge on a solution set that is unique and representative of the FET under consideration. To ensure that this is the case, many types of measurements including the frequency and bias dependent are made to attain as much confidence as possible on a given parameter or a set of parameters, which is a very cumbersome task to rely on this matching only. It is more convenient to work with y-parameters, in which case it is necessary to convert the measured s-parameters to y-parameters. This is done as follows:    1s11 1 þ s22 þ s12 s21    y11 ¼  ; 1 þ s11 1 þ s22 s12 s21 y12

¼

y21

¼

y22

1 þ s11

2s12    ; 1 þ s22 s12 s21

2s21    ; 1 þ s11 1 þ s22 s12 s21    1 þ s11 1s22 þ s12 s21    ¼ : 1 þ s11 1 þ s22 s12 s21

ð3:122Þ

The network analyzer software generally are able to make the conversion from the y-parameters to s-parameters, which is very convenient.

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

Let us first focus on the intrinsic part and deduce the relationships between y-parameters and intrinsic parameters. The device is assumed to be biased under a given normal operating conditions, which is why this form of modeling is called the “hot modeling” followed by circuit model shown within the dashed box in Figure 3.34c, the y-matrix of the intrinsic part can be written as 2 3    1 1 þ jwCDG jjR1 jwCGS jjR1 i GD þRfdg þRfgs 6 7   1 7  jwC GD jjR1   6 DG Rfdg 6 7 y11 y12 6 7 yint ¼ ¼ 6 g ejwt 7   1 1 6 m 7 y21 y22 R  jwC jjR DG DG fdg 6 1þjwCGS Ri 7 4 5   1 1 R1 DS þRfdg þjwC DS þ jwC GD jjRDG AjjB ¼

with

AB : AþB

ð3:123Þ

The real and imaginary parts of Equation 3.123 can be equated and rearranged to obtain the intrinsic elements shown in the equivalent circuit of Figure 3.34c (getting the y-parameters of the extrinsic part from the measurements requires additional deembedding routines will be done shortly). At low frequencies (in the megahertz range), the capacitance terms in y11 and y12 of Equation 3.123 approach zero so that Rfgs and Rfdg can be easily determined at the following condition: 1 R1 fdg ¼ g fdg Reðy 12 Þ and Rfgs ¼ g fgs Reðy12 þy11 Þ:

ð3:124Þ

Separation of the elements in Equation 3.123 to their real and imaginary parts paves the way to the determination of the rest of the eight circuit elements (t included) as follows: " !2 # Reðy12 ÞþR1 Imðy12 Þ fdg ; ð3:125Þ CDG ¼  1þ Imðy12 Þ w RDG ¼

Reðy12 ÞþR1 fdg wCDG Imðy12 Þ

;

Imðy11 ÞþImðy12 Þ CGS ¼ w

Ri ¼

ð3:126Þ (" 1þ

Reðy11 ÞþReðy12 ÞR1 fgs wCGS ½Imðy11 ÞþImðy12 Þ

gm ¼



Reðy11 ÞþReðy12 ÞR1 gfs ½Imðy11 ÞþImðy12 Þ2

2 #) ;

;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   f½Reðy21 ÞReðy12 Þ2 þ½Imðy21 ÞImðy12 Þ2 g 1þw2 C2GS R2i ;

  Imðy12 ÞImðy21 ÞwC GS Ri ½Reðy21 ÞReðy12 Þ t ¼ w arcsin ; gm 1

ð3:127Þ

ð3:128Þ

ð3:129Þ ð3:130Þ

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CDS ¼

 Imðy22 ÞþImðy12 Þ ; w

R1 DS ¼ Reðy 22 ÞþReðy 12 Þ:

ð3:131Þ ð3:132Þ

It is sometimes necessary to sacrifice accuracy for expediency particularly when the first-order processes are of interest, in which case one takes advantage of the fact that for relatively low frequencies, typically below 5 GHz, the term w2 C2GS R2i V B ; V DG < V B ;

V B ¼ V B0 þ IDS R2 ;

with

ð3:179Þ

where R1 and R2 are approximate breakdown resistances. Similarly, an effective forward resistance is taken into account for the gate–source current source that incorporates the forward gate bias as ( V GS V bi ; V DS >V bi ; IGS ¼ ð3:180Þ RF 0; V DS < V bi : The gate capacitance determined by small-signal s-parameter measurements at different bias conditions follows a relationship similar to that given in Equation 3.176, although inclusion of this nonlinearity will only produce a small influence on the large-signal saturation characteristics, shown in Figure 3.44 as the

200

Voltage variable capacitance

VDS=8.0 V

Output power

80 0.6

Average gate current

40

0.4

Reflection coefficient 0

0.2 0.0

0

10

20

30

40

50

60

Incident input power (mW) Figure 3.44 Output RF power, average drain current, average gate current, and reflection coefficient as a function of RF input power at 12 GHz with VD = 8V, VG = 1 V. The output is tuned for high gain.

70

6 4 2 0

Gate current (mA)

Average drain current

120

Reflection coefficient

Output power (mW) Output current (mA)

160

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

dashed line. Note that all the current dependencies are expressed in the time domain, and the Fourier analysis is necessary to simulate the output power properties for different harmonic components. Good agreement was found between the calculated and experimental load-pull contours under various RF frequencies and input power levels. A shown in Figure 3.44, strong output power saturation occurs as the input power is increased. However, consideration of the second and third harmonics will significantly change this saturation trend. Particularly, the third harmonic dominates the effects of the nonfundamental frequency components, which has also been confirmed by experiments [177], and causes serious third-order intermodulation distortion. Moreover, as the input power is increased, the device performance degrades gradually owing to the increase of gate leakage current, reflection coefficient, and the drop in drain current. The above-mentioned model is very similar to that of Materka and Kacprzak mentioned above in conjunction with Figure 3.42. The Materka and Kacprzak model also provides a good fit to GaAs power FETs [174, 182]. In this model, the bias-dependent drain current source is expressed in a hyperbolic tangent form  
 V GS ðttÞ 2 aV DS IDS ¼ IDSS 1 tanh ; V GS V p Vp

ð3:181Þ

V p ¼ V P0 þ gV DS ;

ð3:182Þ

with

where IDSS, VP0, a, g, and t represent the parameters to be fitted. The diodes Df and Dr in Figure 3.42 appearing in the Materka–Kacprzak model correspond to the gate–source and gate–drain current sources in Curtice-cubic model and represent the metal-to-channel junction and gate–drain breakdown effects. The currents through those diodes are given by If ¼ IS ½expðaS V GS Þ1;

ð3:183Þ

Ir ¼ I sr ½expðasr V DG Þ1;

ð3:184Þ

where IS, Isr, aS, and asr are the parameters to be fitted. The gate–source capacitance still follows the relation in Equation 3.176 within a certain gate voltage range. By utilizing the data obtained from both DC (include forward and breakdown conditions) and RF measurements, the above-mentioned parameters can be fitted through an iteration-based optimization routine. Both the Curtice-cubic model and the Materka–Kacprzak model can provide fair accuracy for GaAs MESFET performance within affordable computational complexity and both are good at predicting the amplifier power saturation characteristics with the aid of HB analysis technique. Statz model [173], which we touched upon above briefly, was proposed in 1987 and utilized by then the already developed SPICE model for conventional JFET or MOSFET. However, instead of taking the hyperbolic tangent function, this model used a segmented polynomial form to describe the current source:

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IDS

"
8  # > bðV GS V T Þ aV DS 3 > > ð1 þ lV DS Þ; 1 1 < 3 1 þ bðV GS V T Þ ¼ > > bðV GS V T Þ > : ð1 þ lV DS Þ; 1 þ bðV GS V T Þ

0 < V DS < 3=a; V DS >3=a: ð3:185Þ

We should mention that the contribution of the Statz model [173] is an attempt to perform the underlying analysis of the gate capacitance components based on the change of charges. An accurate capacitance modeling becomes increasingly important as the frequency increases. The nonlinear capacitances not only determine the frequency response of the device but also introduce distortions and intermodulations into the circuits. Therefore, the simple treatment for the metal–semiconductor junction used in the early models is far from being adequate. In the Statz model, the total charge under the gate is composed of QGD and QGS. In the normal drain voltage operating range (VDS > 0), the variation of QGD is small, which can be expressed as Q GD ¼ CGD0 V eff 2

ð3:186Þ

and QGS is truncated into two parts as 8
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V new > > > ; V new < V max ; < 2C GS0 V B 1 1 V B 
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q GS ¼ V new V new V max > > > þ CGS0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; V new >V max ; : 2C GS0 V B 1 1 V 1V max =V B B ð3:187Þ V eff 1:2 with V new

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1h V GS þ V GD  ðV GS V GD Þ2 þ D2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1h ¼ V eff 1 þ V T þ ðV eff 1 V T Þ2 þ d2 : 2 ¼

ð3:188Þ

The first derivative of charge on the voltage describes the dependence of the gate capacitance on voltage. The capacitance modeling used here, based on the standard metal–semiconductor physics scenario, extended the validity of Equation 3.176 to the entire operating bias range by means of pure mathematical smoothing. With the rapid growth on computational capacities, more and more complicated current and capacitance expressions were proposed in modeling the III–V FETs later on. The model developed by Angelov et al. [176] in 1992, which is more empirical in nature, has been widely accepted because of its ability to handle MODFETs. The model cleverly utilizes the power series expansion near the peak transconductance region that can be easily determined from DC measurements. In this case, the current source takes a modified hyperbolic tangent function form as IDS ¼ Ipk ½1 þ tanhðyÞð1 þ lV DS ÞtanhðaV DS Þ:

ð3:189Þ

Here, y is the power series concentrated at Vpk, namely, the gate voltage corresponding to the peak transconductance, taking VGS as a variable:

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

y ¼ P 1 ðV GS V pk Þ þ P2 ðV GS V pk Þ2 þ P 3 ðV GS V pk Þ3 þ . . .:

ð3:190Þ

The peak transconductance (gmpk) is given by g mpk

qI DS ¼ j ¼ qV GS V GS ¼V pk




 qIDS qy :  j qy qV GS V GS ¼V pk

ð3:191Þ

Together Equations 3.189 and 3.190 lead to P1 ¼

g mpk Ipk0

:

ð3:192Þ

Here Ipk0 is the drain current corresponding to the peak transconductance position. Other coefficients can be determined similarly by taking higher orders of derivatives. The same type of modeling functions can be used to fit the gate capacitance: CGS ¼ C GS0 ½1 þ tanhðy1 Þ½1 þ tanhðy2 Þ; CGD ¼ CGD0 ½1 þ tanhðy3 Þ½1tanhðy4 Þ;

ð3:193Þ

where y1 to y4 are the power series expansion taking VGS or VGD as variables. Several strengths have lately been carried out to this model [183, 184]. The coefficients in Equation 3.190 have been modified taking even the more complicated forms, and y variable used in the capacitance model takes the constant and first order of power series. Although the number of parameters in this model to be fitted is large, the extraction routine is quite straightforward. Moreover, the successful simulation on GaAs p-MODFETand submicron gate-length MODFETsuggests its applicability also to GaN HFETs. There are other analytical large-signal models that utilize different expressions in describing the voltage dependence of the current source and gate capacitance, including the Triquint’s Own Model III (TOM3) [185] and those reported by Wei et al. [186] and Parker and Skellern [187]. They all have both advantages and shortcomings. A specified model might do a good job under certain circumstances but might be far from agreement with experiments under other conditions. The charge and current conservation principles, which are important for the nonlinear HB analysis, have been tested and concluded to bring mathematical guidelines to the large-signal simulations by Root et al. [188]. The idea for the conservation principles can be simply illustrated in Figure 3.45. Considering only the gate voltage, if it is swept in a loop from 1 to 2, then 3, and back to 1, the depleted charges beneath the gate and drain–source current will correspondingly change from a starting position and back to the original position. In another words, mapping between the charge or current and voltage is unique, continuous, and derivable. Therefore, the charge and the current are conserved if the start and stop voltages are the same regardless of the path they follow. For an ideal FETwith a universal intrinsic circuit shown in Figure 3.46, the charges under both the gate and the drain are controlled by VGS and VDS and are the current sources. The y-parameter of the intrinsic part of the circuit can be expressed as.

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Figure 3.45 Charge and current conservation principles in an FET structure.

2

qI G 6 qV GS Y ¼ G þ jwC ¼ 6 4 qI D qV GS

2 3 qIG qQ G 6 qV GS qV DS 7 7 þ jw6 4 qQ qID 5 D qV DS qV GS

3 qQ G qV DS 7 7: qQ D 5 qV DS

ð3:194Þ

Consider the following vectors Fi that are projected along the intrinsic orthogonal voltage axes V^GS and V^DS by charge or current components: Im½Y 11 ðVGS ;VDS ;wÞ ^ Im½Y 12 ðVGS ;VDS ;wÞ ^ ~ F 1 ¼ rQ G ðVGS ;VDS Þ ¼ VGS þ VDS ; w w ð3:195Þ Im½Y 21 ðVGS ;VDS ;wÞ ^ Im½Y 22 ðVGS ;VDS ;wÞ ^ ~ F 2 ¼ rQ D ðVGS ;VDS Þ ¼ VGS þ VDS ; w w ð3:196Þ ~ F 3 ¼ rID ðVGS ;VDS Þ ¼ Re½Y 21 ðVGS ;VDS ;wÞV^ GS þRe½Y 22 ðVGS ;VDS ;wÞV^ DS ; ð3:197Þ

D ID(V GS,V DS)

QD(V GS,V DS)

V GS

QG(V GS,V DS)

IG(V GS,V DS)

G

S Figure 3.46 Intrinsic charge/current source representation for an ideal FET.

V DS

S

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

~ F 4 ¼ rIG ðV GS ;V DS Þ ¼ Re½Y 11 ðV GS ;V DS ;wÞV^ GS þRe½Y 12 ðV GS ;V DS ;wÞV^ DS with r¼

q ^ q ^ V þ V : qV GS GS qV DS DS

The circulatory integrals to these vectors over voltage space are zero, according to the conservation principle: { ~ F i  dV^ ¼ 0; ð3:198Þ or it will be equivalent to r ~ F i ¼ 0:

ð3:199Þ

Root et al. [152] verified the validity of Equation 3.198 from the experimental results, which provides a high degree of accuracy on the gate charge, drain charge, and drain current. Nowadays, those who undertake large-signal extraction with different models will more or less impose conservation laws to constrain the mathematical fitting procedures, such as those reported in Refs [189, 190]. Based on the charge conservation principles, some more flexible large-signal extraction routines regardless of the analytical expressions, the so-called table-based modeling method, have emerged since 1990s helped by improved RF measurements. Taking Figure 3.46 as an example, the characterization to sources in the intrinsic part of the equivalent circuit is no longer centered on finding out the specified expressions but on building mapping tables to reveal such kind of dependencies from the limited measurements. In principle, the varnish of circulatory integrals in Equation 3.199 implies that the quantitative determination of the charge or current source can be obtained by path-independent integrals. Therefore, by performing the following integrals onto Equation 3.194, we can construct relationships as VðGS

I G ðV GS ; V DS Þ ¼ I G ðV GS0 ;V DS0 Þ þ

VðDS

G11 ðV;V DS0 ÞdV þ V GS0

G12 ðV GS ;VÞdV; V DS0

ð3:200Þ VðGS

I D ðV GS ; V DS Þ ¼ I D ðV GS0 ;V DS0 Þ þ

VðDS

G21 ðV;V DS0 ÞdV þ V GS0

G22 ðV GS ;VÞdV; V DS0

ð3:201Þ VðGS

Q G ðV GS ;V DS Þ ¼

VðDS

C11 ðV;V DS0 ÞdV þ V GS0

V DS0

VðGS

VðDS

Q D ðV GS ;V DS Þ ¼

C21 ðV;V DS0 ÞdV þ V GS0

V DS0

C12 ðV GS ;VÞdV;

ð3:202Þ

C22 ðV GS ;VÞdV:

ð3:203Þ

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The constant terms of QD and QG can be assumed to be zero because the current contributed by them is the time derivative. Depending on details of the equivalent circuit one chooses, the C and G quantities can be related to the circuit components extracted from the small-signal measurements under different bias points [191]. The most significant advantage of the table-based large-signal analysis is its accuracy. Careful choice of the spline function allows to minimize the error between experimental and simulated data to the greatest extent possible. However, the tablebased method is somewhat inconvenient in terms of the HB analysis and the circuit design that follows. Additional programming effort has to be carried out to allow the CAD simulator to handle the tabulated data. Moreover, one must pay careful attention to the interpolation algorithm that is used because it always produces discontinuities on the higher order of derivatives on the interpolated data, which will cause fake distortion in the power harmonics analysis. In addition, the table-based method has little ability to predict the performance beyond the frequencies where the measurements are conducted, which limits its application on the devices capable of operating at ultrahigh frequencies. So far we have reviewed many of the theoretical models that are widely used in today’s large-signal analysis techniques involving III–V FETs. It is time to take a closer look at the practical issues related to a typical large-signal modeling scheme, such as the one shown in Figure 3.47, which involves the circuit design application. Besides the software and models that are chosen, systematic and carefully Model selection Load-pull measurement Network analyzer Modeling

CAD

software

simulator

I–V measurement

Mutual operation Figure 3.47 A typical large signal modeling flow diagram.

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

constructed set of measurements are essential because the simulation accuracy depends mainly on the quality of the measurements, which include the DC I–V characteristics, pulsed I–V characteristics, single- and multibias s-parameter measurements, pulsed s-parameter measurements, noise measurements, and temperature-dependent measurements of as many parameters as the effort allows. In many models that do not handle the dispersion effect caused by traps or defects, the choice of the pulsed measurements data can always provide a higher degree of agreement on the current properties. Concerning the modeling, parameter extraction depends on measurements, but measurements are always imperfect. Thus, it makes no sense to force a model to agree with measurements more closely over efforts directed toward improving the measurement accuracy. This point seems obvious, but it is frequently overlooked or ignored. Having said that, any model must be validated so that it can successfully reproduce the phenomenon it is intended to model. It proves little or next to nothing to show that it reproduces only the data used to generate it [192]. There are a number of modeling software available, and their compatibility varies considerably, among which IC-CAP from Agilent LINMIC Design Suite from CST, and LASIMO from Optotek are the most widely used. Specifics being dependent on the selected model, the software can always automatically collect the data and perform the fittings. The CAD simulator would then conduct the power analysis and circuit-level design with the extracted device parameters. Advanced design system (ADS) from Agilent and the pSpice (ORCAD) from Cadence are the typical CAD software used in the industry. One can also verify the simulated power performance with the load-pull measurements that directly characterize device’s large-signal properties. There is also the scaling issue for a modeling that goes to the heart of the frequency range, which is connected to the device dimension, for a reliable extraction from a particular model. The RF measurements may not always be accurate enough at all frequency ranges, especially at low frequencies, so the extraction procedure must select the data that are most reliable. In a multifinger FET device, the increase in device dimension will always result in the reduction of its optimum frequency range. If the device size is large enough compared to the signal wavelength, the elements in equivalent circuit can no longer be viewed as lumped nodes. Some of the extracted parameters have obvious dependence on device dimensions while others do not. The scaling coefficient is often added to the model, which can take the form of area (number of fingers by the width of fingers) ratio between the new device and the reference one [193]. 3.3.2.2 Dispersion and Temperature Effects Anomalies in current and dispersion phenomenon are very common in GaN-based FETs ranging from DC to RF frequencies as touched upon in Section 3.5.5. These anomalies are not unique to GaN in that self-heating and the traps at the surface are widely accepted to contribute to the dispersion effects [194, 195]. For example, physical Monte Carlo calculations [196, 197] indicate that the self-heating at large drain biases increases the channel temperature due to the power dissipation inside the device and decreases the carrier mobility and saturation velocity by increasing the

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electron–phonon scattering. As a result, the transconductance and output current are reduced. Moreover, at frequencies above the detrapping time constant of the defects, electrons that are trapped in the deep levels are unable to respond to the change of the field and therefore cannot participate in the RF current transport process. The dispersion frequency due to the shallow traps in GaN increases exponentially with temperature and can be of the order of megahertz. In general, the self-heating plays a dominant role in current dispersion at low frequencies, while at relatively high frequencies it is mainly caused by trapping effect. Although a relatively slow process, particularly compared to the microwave input signal, the dispersion mechanisms significantly affect the device’s RF performance by producing intermodulations in the amplifier [198, 199]. Therefore, the inclusion of dispersion effect into an HFET large-signal model is critical to accurately predict the RF power characteristics. Self-heating originates from the power dissipated in a transistor, which is a function of the drain voltage and current. However, the current is in turn influenced by temperature. Obviously, the self-heating process is a feedback mechanism that can be modeled in the context of a possible low-pass frequency response [200]. The most straightforward means to address this problem is to strengthen the large-signal models by introducing temperature-dependent terms into the analytical expressions describing the drain current: IDS ¼ f ðV DS ; V GS ; TÞ:

ð3:204Þ

Taking Curtice-cubic model, for example, [201] and temperature-dependent coefficients Ai, the temperature dependence can be modeled as IðV GS ; V DS Þ ¼ ½A0 ðTÞ þ A1 ðTÞV 1 þ A2 ðTÞV 21 þ A3 ðTÞV 31 tanhðaV DS Þ: ð3:205Þ From the pulsed I–V measurements at two different case temperatures, T0 and T1 (the implicit assumption is that the case and the device temperatures are equal), the Ai(T) parameters at these two temperatures can be determined by fitting the current–voltage characteristic to Equation 3.205. Because Ai(T) can be modeled as a simple linear function of temperature, a model expression for its temperature dependence can be developed as follows:
 TT 0 Ai ðTÞ ¼ Ai ðT 0 Þ þ ½Ai ðT 1 ÞAi ðT 0 Þ : ð3:206Þ T 1 T 0 Naturally, the temperature (T ) of the device must be determined under operating conditions. Since the self-heating is caused by the power dissipated within the system, the total power in the time domain can be expressed, if we neglect the contribution from gate side, as follows: pðtÞ ¼ I DS ðtÞ  V DS ðtÞ:

ð3:207Þ

For electrothermal simulations, we make note of the fact that the relation between the power and the temperature is analogous to that between the current and the voltage. This is often implemented in the form of a thermal subcircuit [202]. For example, like shown in Figure 3.48, a thermal resistance Rth denotes the temperature

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

Pdiss(ω)

R th

Cth

∆T

Figure 3.48 Electrothermal subcircuit accounting for self-heating effect.

rise per power applied and its unit is  C W1. Then, the static channel temperature would be (heat dissipation discussed in Section 3.8 is in part repeated here for convenience with the specific intent of device characteristic simulations). T ¼ Rth P þ T 0 :

ð3:208Þ

Here, T0 is the environment temperature and P is the DC component of p(t). More generally, in the time domain, the temperature is determined from the following convolution: TðtÞ ¼ pðtÞ  hth ðtÞ þ T 0 ;

ð3:209Þ

where hth(t) is the impulse response of the electrothermal subcircuit. We should note that the self-heating effect can be eliminated under pulsed measurements when the pulse width is short enough, in which case the channel temperature to a first extent would be equal to the ambient temperature. By using a pulsed system to measure the RF isothermal I–V characteristics of a device at different ambient temperatures, then performing the model-based large-signal extraction on the pulse measurement data, one can find out the dependence of the pertinent coefficients on temperature. Finally, the question reduces to the determination of the response or the parameters of electrothermal subcircuit. The thermal resistance depends on the substrate material and device geometry as indicated by the varying values reported in Table 3.3. A general approach to determine the thermal response function can be found in Refs [206, 207]. It is possible to derive expressions from the extrinsic (measured) small-signal parameters as a function of the intrinsic parameters and the thermal transfer function Hth(w), which is the frequency domain conversion of hth(t), by

Table 3.3 Comparison of the thermal resistance on substrate type and gate dimension.

Rth ( C W1 or K W1)

Material/substrate

Gate dimension

Reference

70 250 45.7 123 135

AlGaN/GaN on Si GaAs p-MODFET AlGaN/GaN on SiC GaAs MESFET AlGaN/GaN on sapphire

0.5 · 150 mm2 4 · 0.15 · 50 mm3 0.35 · 250 mm2 6 · 0.5 · 300 mm3 2 · 0.8 · 94 mm3

[203] [206] [201] [204] [205]

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yext 21

¼

yint 21 ; 1aH th ðwÞV DS

yext 22

¼

yint 22 þ aH th ðwÞI DS ; 1aHth ðwÞV DS

ð3:210Þ

where a¼

qiDS ðtÞ diDS =dT 0 : ¼ 1 þ ðdiDS =dT 0 ÞV DS Hth ð0Þ qT

ð3:211Þ

Here, diDS/dT0 is determined by taking temperature-dependent I–V measurements. By substituting a in Equation 3.210, one can solve Hth(0), which apparently represents the thermal resistance at DC condition. Sequentially, the transfer function Hth(w) will be determined given the measured y-parameters. Other methods include the transient thermal model [203, 215] and Laplace heat spread model [205, 208]. The former is based on the analysis of the transient response of output current when stimulated by a pulsed signal. We must keep in mind that the thermally induced and the time-dependent output envelop waveforms are related to the thermal time constant. Therefore, by analyzing this transient response in the output signal, the thermal resistance parameter can be obtained. The latter method is a physical-based thermal model that is based on the solution of three-dimensional heat spreading (Laplace) equations, which also take device geometry and substrate thermal conductivity into account. The dependencies between the thermal resistance and the gate length, pitch, width, and substrate thickness obtained from this method are consistent with the experimental results. An alternative treatment attempting to separate the self-heating part from the isothermal model is as follows [209]: IDS ðtÞ ¼ I DSO ðt; T 0 Þ  ð1dDTÞ ¼ I DSO ðt; T 0 Þ½1d  pðtÞ  hth ðtÞ:

ð3:212Þ

The thermal response is chosen in the following form HðwÞ ¼

1 n

ð1 þ jw=wo Þ ð1 þ jw=wc Þ1n

;

ð3:213Þ

where wo and wc are the upper and lower roll-off frequencies and n is a fitted parameter to experimental data that is less than unity. The step response function of H(w) in the time domain can be determined from the drain current response after applying a step stimulus and by comparing it with the isothermal current obtained from the pulsed measurements. A comprehensive study involving FET large-signal modeling, which includes the trap-induced dispersion effects, has been reported by Golio et al. [210]. The observed dispersive characteristics can be described in terms of a bias-dependent injection of free carriers into the trap states and the capacitive coupling of this charge-carrier movement into the device channel. The injection on the drain side is assumed to be more pronounced and directly affects the output resistance of the device. To account for these processes, bias-controlled elements (Css, gm2, and Rss) need to be embedded into the existing large-signal models, as shown in Figure 3.49. After the analysis of

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

V1 Ri

V2

RD

+ gm2V 2

RG

CDG + C - GS

gm1V 1

Gate L G

R DS

CDS

LD

Drain

R SS

CSS

RS LS

Source Figure 3.49 The large-signal equivalent circuit considering trap induced dispersion effects.

this equivalent circuit, one can find out the analytical expressions from each element in terms of the y- or z-parameters. A physics-based frequency dispersion model for GaN MESFETs adopting almost the same subcircuit as discussed above to account for the dispersion effect while at the same time analyzing the scope of it in physical means has been proposed by Islam et al. [211]. One of the salient features of the trap-related effects is that the initiation of electron emission (detrapping) at low frequencies would increase drain current. The drain voltage at the onset of the detrapping process can be obtained by solving qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V GS þ V bi þ V ONSET = 1 þ N D =N t0 ¼ qN D d2 =ð2eÞ: V bi þ V ONSET DS DS ð3:214Þ The calculation is based on the assumption that the trapped carrier concentration can be modeled as    : ð3:215Þ N t ¼ N t0 exp a V DS V ONSET DS Here, Nt0 is the total trap concentration in the channel. Therefore, the current due to detrapping of the carriers can be modeled as    : ð3:216Þ IDetrap ¼ qvsat WdN t0 1exp a V DS V ONSET DS Similarly, the transconductance and the resistance are expressed as      a1 þ 2a2 V GS g m2 ¼ qvsat WdN t0 a exp a V DS V ONSET ; DS a1 þ 2a2 V GS þ 1 RDS2

td ; CSS ð1 þ g m2 RDS1 Þ

ð3:217Þ ð3:218Þ

where td is the time corresponding to the frequency at which overall output resistance has decreased to the average of its low-frequency and the high-frequency values. And Css is assumed to be comparable to CGS. Among other methods are the transient time analysis [212] and the optimal method used in table-based large-signal modeling [213]. In the transient analysis,

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a relaxation time differential equation is set up for the nonquasistatic large-signal FET model as d iD ðtÞ iDC ðt; VÞ dV iD ðtÞ þ ¼ D þ g AC ðt; VÞ ; dt t t dt

ð3:219Þ

where g AC ðt; VÞ

dV dV G dV D þ g AC : ¼ g AC m ðV G ; V D Þ D ðV G ; V D Þ dt dt dt

ð3:220Þ

By Fourier transformation of Equation 3.219 into the frequency domain, one can get Y 2i ðwÞ ¼

1 jwt AC g DC ðw; VÞ þ g ðw; VÞ; 1 þ jwt i 1 þ jwt i

i ¼ 1; 2:

ð3:221Þ

This means that the characterization of current sources on both the gate and the drain sides is actually averaged between the DC and AC components weighted by a singlepole transfer function. The transient time (relaxation time) t, corresponding to the time constant of a trapping process, can be determined from the pulsed measurements. However, the single-pole function may not be accurate enough in helping Equation 3.221 to cover the entire range of frequencies, especially between DC and RF range [214]. In the table-based large-signal modeling method, it is assumed that for frequencies well above the dispersion cutoff frequency, the trapping mechanism is dependent on the rate of dynamic change of the applied intrinsic voltages VGS and VDS with respect to the average values, VGS0 and VDS0. Thus, the expression for the current can be set up as follows: IDS ðV DS ; V GS ; V DS0 ; V GS0 ; P diss Þ ¼ IDC DS;IS0 ðV DS ; V GS Þ þ aG ðV DS ; V GS ÞðV GS V GS0 Þ þ aD ðV DS ; V GS ÞðV DS V DS0 Þ þ aT ðV DS ; V GS ÞP diss :

ð3:222Þ

Here, the isothermal DC current is measured in pulsed mode, and aG and aT are the optimized coefficients that model the deviation in the drain current due to the surface- and buffer-trapping-induced dispersions. The terms aG and Pdiss account for the self-heating effects. With the interpolation techniques, all the coefficients are calculated in tabular form from the extracted small-signal data. In the end, during the designing phase of amplifiers for narrow-band input signals, the thermal behavior of the amplifier must be taken into account. In contrast, the trap-induced dispersion effects need to be well represented in the model for a wide band of signals. A dispersion-compensated predistortion network can be used to delineate these effects from memory effects [215]. From a thermal management point of view, the difference that the substrate makes is significant. The selection of SiC, GaN, or Si substrate, as opposed to sapphire with poor thermal conductivity, has been stated to reduce the channel temperature by over 50% [216]. However, the thermal conductivity of the substrate is not the only property that should be considered. From a global point of view, the high-resistivity GaN substrates would

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

be ideal as they would provide a very respectable thermal conductivity (2.3 W cm1 K1 for the lightly doped material at room temperature) as well as nearly defect-free channel layers. In terms of the device layout, a large number of shorter gate fingers as opposed to fewer but longer gate fingers are desirable. The gate finger pitch will impact the horizontal heat flow in the structure as the sparsely placed gate fingers are more efficient for lateral heat flow as they diffuse to a relatively cold body compared to densely spaced gate fingers. Flip-chip mounting on high thermal conductivity materials such as AlN also results in a smaller thermal resistance and reduction in the channel temperature but at considerable complication. The main objectives of large-signal modeling are to simulate the performances of devices under large input signal conditions and facilitate the design procedures for circuit-level applications. The existing large-signal analysis methods have already done an excellent job in modeling the GaAs-based heterojunction transistors. Owing in part to developments in GaAs-based heterostructure device design, we have seen a fast development of GaN HFETs capable of providing competitive performance figures in terms of frequency of operation, output power, and PAE due to a combination of very wide bandgaps afforded by GaN coupled with high electron mobility and velocity. The bandgap is a determining factor for the breakdown voltage and thus the available output power capability. The velocity and the mobility in addition to the reduced gate length of the device is fundamental to getting reasonable amounts of gain at very high frequencies. However, very precise nonlinear models for GaN HFETs are still unavailable to a first extent because of the tremendous thermal issue generated by the large drain voltage and reasonably high drain current, which the device structure must dissipate. Refer to Sections 3.9 and 3.11 for its ramification on the power dissipation. The temperature effects cannot be treated as static parameters as they depend on the dynamic operating conditions. In addition, owing to the imperfection of the semiconductor, the defect-induced electron trapping phenomenon will cause considerable dispersion that is not included in the conventional models. Moreover, the complexity of the input signal in modern digital communication systems will render both RF and quasi-DC operations a dynamic one, with mutual interactive participation by both thermal and trapping effects taking place [217]. Therefore, much effort has been expended to improve the large-signal models, which would account for the dispersion effects, and in the process make them applicable for GaN-based HFETs. In summary, the analysis commences with an appropriate equivalent circuit topology considering the encapsulation and packaging forms as well. Then, the elemental values of the small-signal equivalent circuit parameters are extracted for the fine quiescent points. Also, the device’s DC characteristics are often identified by using pulsed excitations to circumvent the longtime constant thermal effects. The pulsed s-parameter measurements at different ambient temperatures reveal the electrothermal properties of the device. These pulsed DC tests are also complemented by large-signal AC data to produce convenient nonlinear descriptions of the FETs dispersion at different frequency ranges. At last, the device nonlinear model is verified by pull-load and large-signal wave form measurements. There are already some reports on GaN HFET large-signal models being applied successfully to power

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amplifier design [218–220]. And commercial GaN-based MMIC circuits have been available for some time for broadband RFapplications such as WinMAX or WCDMA. But further improvements on GaN HFET large-signal modeling hinge on advances in the understanding to the physical mechanisms within the material during device operation. 3.3.3 Cutoff Frequency

The maximum speed of an FET operation is ultimately limited by the transit time of carriers under the gate together with charging times of the input and feedback capacitances. In addition, extrinsic resistive elements such as the output conductance contribute adversely to lowering the gain. It is, therefore, warranted to present a succinct discussion of the speed of the device in terms of current gain and power gain cutoff frequencies. An examination of Equations 3.227–3.229 indicates that, if larger fmax is desired, it can be achieved by reducing r1. Unlike the GaAs system, the strain-induced polarization due to the compressively strained InGaN and the spontaneous as well as the strain-induced polarization due to tensile-strained AlGaN are against each other and smaller interface charge thus results. However, InGaN as a channel may have benefits in terms of reduced current lag and reduced hot phonon effects, which are discussed in Section 3.9. The transit time under the gate of a submicron HFET is on the order of a few picoseconds. In view of this, the charging time of the input (CGS and CDG) and the feedback capacitance through the input resistance Rin in the equivalent circuit, shown in Figure 3.34, determine the speed of response. Generally, two parameters, the current gain cutoff frequency and maximum oscillation frequency, are figures of merit to gauge the expected high-frequency performance of an HFET. Among them, the current gain cutoff frequency is defined as the frequency at which the current gain goes to unity. fT ¼

gm vsat ; 2pðCGS þ CGD Þ 2pL

ð3:223Þ

where CGS and CGD are the gate–source capacitance and the gate–drain feedback capacitance, respectively. The feedback capacitance (CGD) is smaller than the gate–source capacitance (CGS) and is typically neglected, which when utilized leads to the current gain cutoff frequency in its simplest form fT ¼

gm vsat 1 ¼ ¼ ; 2pCGS 2pL 2ptint

ð3:224Þ

where tint is the intrinsic delay time, which represents the time required for electrons to traverse the physical length of the gate. When the delays due to the extension of the depletion region under the gate because of the drain bias and the RC time constant

3.3 Equivalent Circuit Models, Deembedding, and Cutoff Frequency

are included, Equation 3.66 must be modified as fT ¼

1 ; 2pttotal

ð3:225Þ

Total delay time, τtotal

with ttotal representing the total delay time, which is defined as ttotal ¼ tint þ tD þ tRC with tD representing added drain delay due to the extension of the gate depletion region because of the drain bias and tRC representing the RC delay time constant [221]. To elaborate further on tD and tRC, as the drain bias is increased further into the saturation regime, the depletion under the gate is no longer limited to the physical dimension of the gate. Rather, it extends toward the drain contact, effectively increasing the effective gate length, and the additional time required to traverse the added effective length is represented by the drain time constant, tD. As far as the RC time constant is concerned, the modulated gate voltage changes the charge distribution, and to move charge in and out of a given element requires an additional RC time constant. The RC time constant would approach zero if the drain current approaches infinity. Therefore, the total delay time determined from Equation 3.225 with the measured current cutoff frequency in hand for each of the drain current values, one can plot the total delay time versus the inverse of the drain current (by changing the gate bias) for a constant drain voltage bias. Extrapolating the linear portion of the data to zero would allow the determination of tRC and tD þ tint as shown in Figure 3.50. In short, the delay time for zero inverse current represents the sum of the intrinsic and drain delay times. Isolation of the drain delay can be accomplished by plotting the total delay time versus the channel voltage defined as Vch ¼ VDS  ID(RS þ RD), as shown in Figure 3.51. Here, VDS, ID, RS, and RD represent the applied drain–source voltage, drain current, source resistance, and drain resistance, respectively. Extrapolation from the linear region to vanishing channel voltage will lead

τRC τint+τD

Inverse drain current, 1/IDS Figure 3.50 Charging delay plot demonstrating the extrapolation method for determining tRC and td þ tint. The total delay time is determined from the measured current cutoff frequency and Equation 3.225. The experimental total delay time data are patterned after GaAs-based MODFETs reported in Ref. [221].

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456

τD

τRC+τint

Channel voltage, V ch =V DS-IDS (R S +RD) Figure 3.51 Drain delay plot demonstrating the extrapolation method for determining td and tRC þ tint. The total delay time is determined from the measured current cutoff frequency and Equation 3.224. The experimental total delay time data are patterned after GaAs-based MODFETs reported in Ref. [221].

to the sum of intrinsic and RC time constants. For a given channel voltage then, the difference between the total delay and the sum of the RC and intrinsic delays would lead to the drain delay. Having measured the total delay from the cutoff frequency and isolated the drain and RC delay would allow one to determine the intrinsic delay time. Experimentally, the current gain cutoff frequency (unity current gain frequency output under short-circuit conditions) is determined by extrapolating |h21|2 to 0 dB. The term h21 is defined in terms of the s-parameters (see Equation 3.120, repeated below for convenience), which can be measured, as (see Section 3.1.5 for details) h21 ¼

2s21 : ð1s11 Þð1 þ s22 Þ þ s12 s21

ð3:226Þ

Similarly, the unity power gain frequency is determined by extrapolating the unilateral power gain, U, to 0 dB, (see Section 3.1.5 for details). From Equation 3.225, it may be noted that the higher the saturation velocity and the smaller the gate length, the higher the value of fT. The maximum oscillation frequency, defined as the frequency at which the power gain goes to unity, may be given by [222] f max ¼

fT 2ðr 1 þ f T t3 Þ1=2

:

ð3:227Þ

If Rs is the series resistance, CDG is the drain–gate capacitance, and GD is the differential drain conductance, then the parameter r1 is r 1 ¼ ðRG þ Ri þ RS ÞGD ;

ð3:228Þ

and the feedback time constant t3 is t3 ¼ 2pRG CDG :

ð3:229Þ

3.4 HFET Amplifier Classification and Efficiency

If we consider yet more of the extrinsic elements, the fmax term can be expressed as [107] f max ¼

fT 2f1 þ ½RS þ RG GD þ 2ðCDG =CGS Þ½ðCDG =C GS Þ þ g m ðRS þ G1 D Þg

1=2

:

ð3:230Þ The parameters have their usual meanings in that RS and RG represent the gate and source extrinsic resistances. The terms CDG and CGS represent the drain-to-gate feedback capacitance and gate-to-source capacitance, respectively.

3.4 HFET Amplifier Classification and Efficiency

GaN-based FETs are intended primarily for power application at high frequencies. Consequently, traditional small-signal considerations have to be augmented by largesignal specific issues. The main parameter facing a power device is the maximum power level that can be obtained along with the associated efficiency and gain. In many applications, the noise figure of the device must also be considered. In simple terms, if the device has large drain breakdown voltage, high gain at high frequencies, and high drain efficiency, the stage is set for a desirable device. Even in a welldesigned semiconductor device, the thermal wall is a very formidable one. Thus, it is imperative that the effect of temperature, and therefore the power dissipation, on device performance is accounted for accurately. As in small-signal modeling, the first step in power modeling is to establish the basic device geometrical factors that are needed to calculate the current–voltage characteristics. Once these are known, the output characteristics superimposed with the load line can be used to estimate the power level that can be obtained from the device provided that it is not limited by the input drive details, such as the amplifier classification determined by biasing point must be settled first. To this extent, we give a short description of various types of amplifier configurations first. Depending on the operating point and the load line in the DC characteristic of the transistor, amplifiers can be classified as class A, B, AB, and C where the transistor is used as a controlled current source in an amplifier, as schematically shown in Figure 3.52. For even higher efficiency, there are class E and class F amplifiers in which the transistor is used as a switch [223, 224]. In a class A amplifier, the transistor is biased and driven by RF in a manner in which the complete swing of the output current is in the saturation region of the transistor and the transistor obviously is on for 360 phase angle of the RF swing. As a first approximation for the small-signal case, the output current is proportional to the input voltage with the proportionality constant being the transconductance, gm, of the device at the RF frequency in use. In large-signal operation, however, gm is not constant. Even then the linearity in class A amplifier is the highest and the bandwidth is the widest in comparison to other classes mentioned above. The major drawback, however, is the low efficiency due to high power consumption. The is due to the

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Slope =1/R on

Load lines RF load line

V in,th

V in,sat

V in

V knee

Output current waveforms and on angles

Cl Clas as s sA AB

sB as Cl C ss

Class AB

C la

Class A

Class E, F

Class B Class C

ID

V DD

I2R loss

V br

Class A φ=360o

Class AB 180ο30 V, which in turn get trapped and subsequently detrapped at progressively higher drain biases. The detrapping causes recovery from current anomaly, liberally termed as current collapse. The calculations have been applied to the experimental data of Binari et al. [11], which pertain to a GaN MESFETs with 1.5  150 mm gate device measured in dark and under illumination. The results of such simulations are shown in Figure 3.88. The experimental and calculated data are shown for two different gate biases, namely, 1 and 3 V, in dark and under illumination. In addition, the density of occupied traps is shown in the inset. In dark, meaning without photoexcitation, the I–V curves show significant current anomalies, termed by the authors as “current collapse.” The so-called current collapse was observed only after the drain bias exceeded 30 V in the first measurement, plausibly indicating that the carriers trapped by defect centers in the buffer are generated by impact ionization or the high field is somehow involved in facilitating the trapping mechanism. In the presence of light, the current does not collapse. The solid lines atop the experimental data, which are shown by symbols, represent the calculations as outlined above. In much the same vein, transient simulation of a GaN MESFET has been performed by Horio [313] also, in which deep traps in a semi-insulating buffer layer

3.5 AlGaN/GaN HFETs 20 Light on

Light off

18

V GS = 0 V

16 V GS = -1 V

Drain current (mA)

14 12 V GS = -3 V

10 8 6 N 1(10 16 cm -3)

12

4 2 0

8 4 0 0

0

20

20 30 10 40 Drain voltage (V)

40

60

50

80

Drain voltage (V) Figure 3.88 Experimental (taken from Ref. [11]) and calculated output I–V characteristics for VG ¼ 1 and 3 V for a 1.5  150 mm2 GaN MESFET considering trapping effects. Calculated and measured data are shown by solid lines and

symbols, respectively. The inset shows the variation of occupied trap concentration (NT) with applied drain-to-source voltage. The wavelength of illumination used is 470 nm. Courtesy of A. F. M. Anwar [312].

was considered. It was shown that when the drain voltage VD is raised, the drain current overshoots and when VD is lowered, the drain currents remain at low values for some period of time following the gate bias, signifying drain current lag. Therefore, the drain lag was argued to be responsible for power compression in the GaN MESFET. 3.5.5.2 Effect of Barrier States The above discussion focused only on defects near the buffer layer–channel interface. However, surface states in the case of modulation-doped structure defects in the barrier layer are also capable of causing anomalies in the current–voltage characteristics. In this vein, Mitrofanov and Manfra [286], among others, considered localized trapping centers within the bandgap in the vicinity of the gate where the gate potential defines the energy position of the trap level with respect to the Fermi level. Consequently, any variation in the gate potential causes a change in the trap occupation factor. And occupied traps by electrons can then cause a depletion of the channel and thus the current. The drain current would depend on the charge trapping and detrapping dynamics in addition to an AC gate bias. For example, if the detrapping process is slow compared to the gate AC voltage, the drain current may not

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j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

510

Energy

AlGaN barrier qE, field lowering

∆φ FP

Trap potential without filed

Gate

Frenkel–Poole

ET

Phonon-assisted tunneling

With filed Direct tunneling

Distance, x Figure 3.89 Energy diagram of the trapping center, for example, in the AlGaN barrier under the gate, in the presence of the electric field. Also shown in dashed lines is the potential around a trap in the absence of electric field. Contrasting the ones with and without field schematically

shows the field lowering of the barrier for emission. Possible mechanisms of electron emission such as thermal ionization over the field-lowered barrier (Poole–Frenkel effect), phonon-assisted tunneling, and direct tunneling in case the barrier width is sufficiently thin.

follow the gate bias when that bias calls for an increase in the drain current. In FETs, current transients as opposed to capacitance transients in DLTS are used to gain an understanding of the traps in terms of ionization energy and capture cross section. The measured ionization energies can be skewed by electric field as shown in Figure 3.89, which also embodies the Poole–Frenkel effect on the emission from a trap by phonon-assisted tunneling and direct tunneling processes [314, 315]. There are similarities between this figure and Figure 4 in Vol. 2 Chapter 4, which focused on the Poole– Frenkel effect on the current–voltage characteristics. It should also be pointed out that the thermal ionization energies are different from the optical ones, particularly if lattice defects are involved, as depicted in Figure 3.87. In field effect transistors, it is customary to use current transients to determine the behavior of defects, as opposed to capacitance transients in DLTS, although capacitance transients can also be used. Briefly, in DLTS, traps are filled by applying a forward bias and the dynamics of their emission are monitored during reverse bias. In current transients, the FET is typically held at a constant source–drain bias with source grounded with a gate bias leading to a steady-state drain current of ISS D ðV G Þ. P The gate voltage is then switched from a high level V SS G to a low level V G for a period of time tp during which the channel current drops in response to the gate pulse. The large negative gate bias causes the electrons to tunnel from gate into the semiconductor and filling available trap states. After the time tp, the gate potential is switched back to the initial level. Because the emission process, except in the likelihood of tunneling, is an activated process, the temperature dependence can be very useful. The transient drain current is measured using current probe [286]. The results of one such measurement in unpassivated devices with Si doping are presented in

3.5 AlGaN/GaN HFETs VG = 0 V; VD = 10 V

ID(t) / ID

DC

1.0

0.5

VG = -10 V 0.0 -4

-2

0

2

4

6

t (µs)

8

10

12

14

Figure 3.90 Normalized channel current response to the gate pulse VG(0 < t < 10 ms) ¼ 0 V, after the off-state VG ¼ 10 V. The devices are continuously biased at VD ¼ 10 V. The drain current is measured with the low-insertion impedance current probe. Two traces show devices with and without gate lag. Courtesy of O. Mitrofanov [286].

Figure 3.90 in the form of the channel current response. In this experiment Figure 3.90 maintained the transistor in pinched-off state for t < 0 to allow the entire available trapping center to be filled. At t ¼ 0, the gate potential was switched to the on-state (VG ¼ 0 V) for 10 ms. One of the devices in Figure 3.90 shows an instant recovery but the other exhibits obvious gate lag. Following the initial current switching to 85% of the steady-state level, the drain current in lagging devices slowly completes the full recovery within 50–100 ms. Naturally, the rate of current recovery increases with increasing temperature, as trap emission is an activated process. An illustrative example of the temperature dependence, shown in Figure 3.91, is a series of normalized transients measured at temperatures ranging from 283 to 363 K for a device relying on polarization charge. As in the case of Figure 3.90 and prior to the measurement, the device is held in the pinch-off state with VD ¼ 12 V and VG ¼ 11 V for 10 ms to saturate the traps. As the gate potential changed to VG ¼ 0 V, the captured electrons slowly emit from the traps and the resultant drain current transients exhibit long exponential tails. These can then be used to determine the electron emission rate from the traps. A discussion of the capture–emission dynamics is provided next to extract emission rates and trap thermal activation barriers. Consider a system with traps within the bandgap and in the vicinity of the gate contact. Because the Fermi level is not changed by the gate potential but the conduction band edge is, the potential affects the occupation factor of traps. The source of electrons could be conduction band electrons or those tunneling from the gate. The occupation factor fT of a trap is described by the balance of the capture and emission processes as [286]

df T n n ¼ Ctun ð1f T Þ þ cð1f T Þ ef T 1 ; ð3:251Þ dt N N where the first term represents electron tunneling from the gate into the semiconductor with (Ctun) being the capture coefficient for tunneling electrons, the second

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512

T =10oC T =20oC T =30oC T =40oC T =50oC T =60oC T =70oC T =80oC T =90oC

∆ID(t) / ID

SS

0.1

0.01 0

30

60

90

120

150

t (µs)

(a) 0.5

EA = 0.22 ± 0.01 eV

0.4

-1

-2

A = 410 ± 30 s K

0.2

eT

-2

-1

-2

(s K )

0.3

0.1 0.09 2.6

(b)

2.8

3.0

3.2

3.4

-1

1000/T (K )

Figure 3.91 Temperature dependence of the (b) Experimentally determined values of eT 2 trap emission rate. (a) The difference between versus the inverse temperature. The emission rate is extracted by fitting an exponential decay the steady state and the actual drain current function to the data using Equation 3.254. after switching the gate voltage VG from 11 to 0 V in temperature window ranging from 10 Courtesy of O. Mitrofanov [286]. (Please find a to 90  C and the source–drain bias of 12 V. color version of this figure on the color tables.)

term represents electron capture rate from the conduction band with c being the capture probability, and the third represents the electron emission to the conduction band with e being the emission probability. The term n/N is the occupation factor of the conduction band. Naturally, (1  fT) and (1  n/N) represent the unoccupation factor or the fraction of availability. In the equilibrium, the emission and capture processes balance each other, which leads to a steady-state occupation factor f 0T. When a negative potential is applied to the gate, the additional flow of tunneling electrons results in a rise in occupation factor

3.5 AlGaN/GaN HFETs

and reaches a new equilibrium state f 0T þ f T with temporal dependence obtained from Equation 3.253 as f T ðtÞ ¼ f 0T þ f T ð1eCtun t Þ;

ð3:252Þ

with a characteristic time of (Ctun)1. When the negative gate potential is removed, the electron supply is interrupted and thus the filling process while the emission process continues until equilibrium is reached. However, the nonequilibrium trapped charge temporally remains localized on the defect level. The associated transient dynamics can be approximately expressed f T ðtÞ ¼ f 0T þ f T eet ;

ð3:253Þ

with a characteristic time of e1. The temporal dependence of the occupation factor is shown in Figure 3.92 for a pictorial image. Let us now turn our attention to the emission coefficient e. The electron can acquire sufficient thermal energy to overcome the trap potential barrier and escape from the trap. The temperature dependence of the thermal emission rate can be derived utilizing the principle of the equilibrium, that is, dfT/dt ¼ 0 in the absence of tunneling Ctun and is given in Volume 1, Equations 4.21 and 4.22, which are repeated here for convenience:
 E nA eðTÞ ¼ AT 2 exp  ; ð3:254Þ kT

fT*(1-exp(-Ctunt))

fTo+fT*

ss

ID

Drain current

-V G off

fTo

fT*exp(-et)

-V G on

Occupation factor

fT

∆ID

t1

t2

t

Figure 3.92 A schematic diagram showing the evolution of the trap occupation factor under the gage when the gate is negatively biased at t ¼ t1 and returns to its original bias at t ¼ t2. The constants Ctun and e represent the tunnel electron capture and electron emission

coefficients, respectively. Corresponding current transient is also shown in the constant mobility regime. I SS D represents the steady-state drain current and the amplitude of the current transient is DID.

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j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

514

where

pffiffiffi A ¼ 2sna 3Mc ð2pÞ3=2 k2 m h3 ;

ð3:255Þ

where EnA is the apparent activation energy of the trap, sna is the apparent capture cross section, and A is a constant. The subscript n is used to depict electron traps. The term Mc represents the number of conduction band minima, which in the case of GaN is equal to unity. In general, the apparent values are assumed to be the actual values representing the activation energy and capture cross section of the trap, in which case the activation corresponds to the position of the trap level with respect to the bottom of the conduction band EA ¼ ET for a donor-type trap (with respect to the top of the valence band for an acceptor-type trap). Through Equation 3.255, the constant A can be related to the capture cross section of the trap s in GaN as rffiffiffiffiffiffi 2p Ap
h3 : ð3:256Þ s¼ 3 m k2 The activation energy and the capture cross section of the trap can be found by fitting the temperature dependence of Equation 3.254 to the experimentally measured emission rate. A trapping center can be unambiguously identified through its energy level and the cross section. It should also be noted that Equation 3.254 is very similar to thermionic emission current over a barrier of EA. If the emission rate and T 2 product eT 2 is plotted against 1/T, one can get the activation energy and A constant and thus the capture cross section provided the experimental observations obey the simple exponential decay discussed above. There are cases where the simple exponential decay does not hold. For example, if multiple discrete traps are involved, in which case the transient is a sum of exponential decays with different rates and amplitudes, trap energies are distributed over energy, and emission process is assisted by spatially nonuniform field, simple exponential decay does not hold. In the last case, the overall apparent emission rate is smaller as the electrons first escape from the traps located in the high-field region. All of these three cases require a detailed analysis. As mentioned above, the charge can be trapped on the surface and/or in the barrier or in the traps in the buffer layer. Because the buffer traps has been dealt with earlier in this section, and the focus of this particular section is on the surface of barrier traps, they will be discussed here beginning with the barrier traps. The gate of an FET can be thought of as a capacitor and a negative charge placed between the gate and the channel induces a compensating positive charge in equal quantity at the electrodes, which in this case are the gate metal and metal connected to the 2DEG. The distribution of the positive-induced charge between the electrodes depends on the physical location of the trapped charge. Assuming that the induced charge is much smaller than the total 2DEG density, the induced charge due to electrons trapped at a spatially discrete trap level (such as d function-like oxide charge in an MOS capacitor) can be approximated by
 d1 DQ 2DEG ¼ Q T 1 ; ð3:257Þ d

3.5 AlGaN/GaN HFETs

where QT is the trapped charge and d1 and d are the distances between the trapped charge and the channel and the barrier thickness, respectively. If the charge is trapped by surface traps, the induced charge in the channel equals exactly the trapped charge DQ2DEG ¼ QT. Consequently, the carrier density available for current conduction in the channel is reduced by QT and thus the current will be reduced by a corresponding amount. In the constant mobility regime, the trapped charge is proportional to the difference between the steady-state current and the transient current, that is, Q T ðtÞ / DID ðtÞ ¼ I SS D I D ðtÞ. The amplitude of the current transient, DID(t ¼ 0), corresponds to the total amount of the charge on the traps at the moment of switching the gate voltage [286]. When the gate bias is returned to the steady-state value, the trapped electrons would be emitted and the current approaches to the steady-state level in a fashion similar to that shown in Figure 3.92. The variation of the emission rate with the temperature reported by Mitrofanov and Manfra [286, 316] is consistent with the thermal emission mechanism discussed above. Figure 3.91b shows eT 2 plotted against the inverse temperature in devices relying solely on polarization-induced screening charge. In general, the amplitude of the current transient DID in these undoped devices is larger than that in doped varieties. The activation energy of the process EA is 0.22  0.01 eV and the capture cross section is 6.7  0.7  1019 cm2. It should be reiterated that the measured activation energy is only the apparent activation energy and it does not necessarily correspond to the binding energy of the electron on the trap at all time due to the presence of electric field as will soon be discussed. The shape of the transients does not clearly reflect a simple exponential drop, indicating a more complex dynamics of the trapped charge. The transient shown in Figure 3.91a contains two distinctive stages at different characteristic times, with only the later dynamics adhering to the exponential decay. The utility of the transient spectroscopy critically depends on the accuracy of the emission rate measurement. This requires that effects such as that of the electric field must be considered as well. Mitrofanov and Manfra [286] measured the drain current transient in an unpassivated GaN/AlGaN/GaN HFET. Those authors have observed that as the voltage switches from V PG ¼ 7 V to VG ¼ 0 V, the drain current instantaneously reaches to approximately 95% of the steady-state value. Within a few microseconds, the drain current reaches approximately 99% of its steady-state value. This fast dynamics, which corresponds to the charge emission from the fast state, is followed by a much slower increase that continues for hundreds of microseconds. The relatively fast rise followed by slower transient points to two traps with significantly different emission rates being present. The fast portion of the transient deviates from a simple exponential decay, which prevents a precise value of the emission rate to be deduced. Reducing the gate voltage pulse to 3 V caused the slow portion of the decay to be negligible, and the population of the traps decreases exponentially with a characteristic time of 1 ms, which is easily deduced by fitting DI(t) with an exponential function. If the duration of the filling pulse is extended to 0.1–1 ms, the transient amplitude increases and the character becomes nonexponential. If the distribution of traps in the energy gap is of a few discrete and noninteracting trapping levels, the emission rate for each level can be measured by means of the

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j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

516

selective probing. The probability of trapping an electron under applied negative gate voltage pulse in general varies for different traps. However, by tailoring the depth and width of the gate filling pulse, the single trapping centers can be selectively activated [315]. Selective trap filling by controlling the width and the amplitude of the filling gate pulse provides additional avenues for determining the character of traps. The particulars of the applied gate pulse control the number of trapped electrons or the occupation factor. The emission rate, however, is not affected by the occupation factor associated with that trap, but is affected by the temperature and any local field. During negative swing of the gate voltage, the electrons tunnel into the defects, the likelihood of which is determined by the Schottky barrier height and gate swing, which controls the barrier width. As the negative swing of the gate, in other words, the depth of the filling pulse, is increased, the tunneling probability of electrons from the gate to the trap level is increased. The pulse width, however, determines the amount of charge made available to the defect level for trapping pulses shorter than the characteristic time (Ctun)1. The emission dynamics associated with the trap begins following the removal of the filling pulse. Mitrofanov and Manfra [286] measured the drain current transients in response to a series of gate filling pulses with duration tp ranging from 20 ns to 100 ms, as well as the pulse depth for several sample temperatures. The results are displayed in Figure 3.93a for current transient versus gate pulse width and in Figure 3.93b for current transient versus gate pulse depth. The decay of the current amplitude transient versus gate pulse width practically overlaps for T ¼ 200 K and T ¼ 300 K implying no temperature dependence of the capture process. The amplitude of the transient DI(t ¼ 0), shown in Figure 3.93b as a function of the filling pulse depth, indicates that efficient filling of the traps occurs for sufficiently deep gate pulses only because the large gate-induced electric field causes substantial band bending. As the gate pulse depth is increased, the number of the trapped electrons first rapidly increases and then slows down near the pinch-off voltage. It should be pointed out that the gate channel voltage is actually determined by not just the gate bias but also by the drain bias, namely, the transient amplitude increases with the drain voltage for a given gate pulse depth. As in the case with the duration of the filling pulse, the shape of the transient amplitude shown in Figure 3.93 is independent of the temperature. The lack of notable temperature dependence for both pulse duration and depth suggests that the dominant mechanism for electron migration from the gate to the traps is the direct tunneling. Once again, the vertical electric field assists the tunneling process and results in the large number of the trapped electrons in the vicinity of the gate. Mitrofanov and Manfra [286] concluded that the characteristic time of the process seems to be independent of the applied field. 3.5.5.3 Field-Assisted Emission from the Barrier Traps As mentioned earlier, neglecting the effect of the electric field in determining the activation energy of a given trap would lead to an underestimation of the activation energy along the field, as is the case in GaN/AlGaN HFETs shown in Figure 3.89. This necessitates the consideration of the effect of the electric field on the emission rate and on the apparent activation energy. This effect is known as the Poole–Frenkel effect and,

3.5 AlGaN/GaN HFETs 0.06 T = 200 K T = 300 K

Transient amplitude, ∆I (t = 0) (norm.)

VD = 4.5 V VG = 0 to -6 V

0.05

0.04

0.03

0.02

0.01

0.00

0.01

0.1

(a)

1

10

100

Pulse width, τ p (µs)

Transient amplitude, ∆ I (t = 0) (norm.)

0.04 300 K 200 K 100 K

0.03

0.02

0.01

0.00 -10 -9

(b)

-8

-7

-6

-5

-4

-3

-2

-1

0

Pulse depth (V)

Figure 3.93 Transient current amplitude as a function of the filling pulse (a) with pulse width varying from 20 ns to 100 ms for a filling pulse depth V PG ¼ 6 V and the drain bias of VD ¼ 4.5 V; (b) for depth of the 500 ns filling pulse varying from V PG ¼ 3 V to the pinch-off level V PG ¼ 10 V. The drain bias VD ¼ 4.5 V. Courtesy of O. Mitrofanov [286].

as discussed in Volume 2, Chapter 4, has a unique functional dependence on the field. For a Coulombic-type trap, the trap barrier decreases in the amount DfPF given as [315, 317]
3 1=2 q F ; DfPF ¼ pe

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j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

518

where q is the unit of electron charge and e is the dielectric constant of the material. Equation 3.254 should then be modified to take this change into consideration and doing so leads to


 E A DfPF E AF DfPF ¼ AT 2 exp  ¼ eð0; TÞexp : eðF; TÞ ¼ AT 2 exp  kT kT kT ð3:258Þ The corresponding activation energy of the trap becomes field dependent through pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi E A ðFÞ ¼ E A ð0Þ q3 F=pe, where EA(F ¼ 0) ¼ ET or EA is the binding energy associated with the trap for zero electric field. Consequently, the emission rate is increased as exp(DfPF/kT) leading to a field dependence of the emission rate given as e(F) ¼ e(0) exp (DfPF/kT). The effect is sizable in that for a field strength of 106 V cm1 the apparent activation energy decreases by about 0.25 eV. An example of the Poole–Frenkel emission from the traps in GaN/AlGaN HFET is shown in Figure 3.94a, where the emission rate is plotted as a function of the potential difference between the gate and the drain terminals [315]. The solid line in the plot pffiffiffiffiffiffiffi shows a fit to the data e ¼ eð0Þexpða V D Þ, where the zero-field emission rate e(0) ¼ 0.04  0.03 s1 and the geometrical factor a ¼ 6.4  0.4 V1/2. The characteristic emission time (the inverse of the emission rate) increases from a few milliseconds at low fields corresponding to VD ¼ 2.5 V to submicrosecond at higher fields corresponding to VD ¼ 7–8 V. To verify the functional dependence, Mitrofanov and Manfra [286] fit the measured values of the emission rate with a power law function (ln e ¼ a þ bVp). The best fit resulting in a power factor of p ¼ 0.53, which is very close to 0.5, suggests that the PF mechanism is in effect and that most of the potential drops in the barrier directly under the gate terminal that is assumed to be proportional to the field strength in the barrier. The PF mechanism has a substantial impact on the activation energy of the trap. The apparent activation energy extracted from the thermal dependence of the emission rate at VD ¼ 3 V, using the eT 2 versus 1/T plot (Figure 3.94b), is only 0.11  0.01 eV. However, the measured value differs substantially from the zero-field activation energy, which can be estimated using the fitting parameters of both the field and the temperature dependence. Assuming that the pre-exponential factor A is not modified by the applied field, the zero-field activation energy can be found as EA(0) ¼ kT ln [e(0)/AT2] ¼ 0.39  0.03 eV. However, in the presence of strong electric field, the electrons can escape from the trap via direct or the phonon-assisted tunneling into the conduction band as depicted in Figure 3.89. If the direct or the phononassisted tunneling probability is comparable to the thermal emission, the activation energy EA and the constant A so deduced would be smaller than the actual values. To further verify the applicability of the PF mechanism, Mitrofanov and Manfra [286] measured the temperature dependence of the emission rate for different bias conditions. Figure 3.95 shows the emission rate (for a device different from that shown in Figure 3.94) versus 1000/T for voltages varying from VD ¼ 4.25 V to VD ¼ 5.75 V. The emission rate follows the classical Arrhenius behavior (Equation 3.254) for all bias conditions in the temperature range of 250–360 K, which allows extraction of the activation energy. Doing so shows that the activation energy

3.5 AlGaN/GaN HFETs 6

T = 298 K

-1

Emission rate, e (s )

10

10

10

10

5

4

3

1.5

(a)

2.0 1/2

2.5 1/2

VD ( V ) 0.05 0.04

-2

-1

-2

(s K )

-1

A = 1.4 ± 0.4 s -K EA= 0.11 ± 0.01 eV

eT

-2

0.03

0.02

(b)

2.5

3.0

3.5 -1

1000 / T (K )

pffiffiffiffiffiffiffi Figure 3.94 (a) The emission rate versus V D , the square root of the drain voltage VD for three separate devices represented by three different dots to increase credence in the applicable mechanism. The traps were filled using a 350 ns gate pulse V PG ¼ 3 V, followed by keeping the gate voltage at VG ¼ 0 V. (b) Variation eT 2 versus 1/T, with e being the emission rate, for VD ¼ 3 V. Courtesy of O. Mitrofanov [286].

decreases with the applied field from 0.14  0.005 eV for VD ¼ 4.25 V to 0.089  0.005 eV for VD ¼ 5.75 V (Figure 3.95 inset). The decrease in activation energy is consistent with the PF trap barrier lowering, as observed in Figure 3.94. The pre-exponential factor A ¼ 7  1 s1 K2 remains constant at lower fields and increases slightly to the level of 10  2 s1 K2 at VD ¼ 5.75 V. As the temperature decreases below 200 K, however, the emission rate becomes temperature independent due to either competing emission mechanisms or device self-heating. The results illustrated in Figure 3.95 show that the electron emission from the trap is thermally activated at temperatures above 250 K. The pre-exponential constant A and the apparent activation energy EA in Equation 3.258, which is the difference between the binding energy ET and the PF barrier lowering DfPF(F) can be estimated from the temperature variation of the emission rate at constant bias conditions shown in Figure 3.95 (it should be stated that if the pre-exponential factor A depends on the field F, Equation 3.258 cannot be used for describing the emission process).

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j 3 Field Effect Transistors and Heterojunction Bipolar Transistors

520

5

10

EA (eV)

-1

Emission rate, e (s )

0.14 0.12 0.10

4

10

0.08 4

5 VD (V)

6

3

10

2

10

VD=5.75V VD=5.50V VD=5.25V VD=5.00V VD=4.75V VD=4.50V VD=4.25V

3

4

5

6

7

-1

1000/T (K ) Figure 3.95 The measured emission rate plotted against the inverse temperature for different drain bias conditions. The lines show the fits with the function e ¼ AT2 exp[EA/kT] for T > 250  C. The inset shows the fitted activation energy EA. Courtesy of O. Mitrofanov [286].

The PF barrier lowering DfPF(F) is extracted from the field dependence of the emission rate. Adhering to the PF mechanism, which calls for the emission rate to pffiffiffiffiffiffiffiffiffi exponentially increase with the square root of the applied field e ¼ eð0Þexpða V DG Þ, one can extrapolate DfPF(F) to F ¼ 0 and find the zero-field activation energy ET, which pffiffiffiffiffiffiffiffiffi is a constant. The estimated activation energy E A ðV D Þ ¼ E T kTaðTÞ V DG must be consistent with the measured values (Figure 3.95) at all temperatures. The zero-field binding energy for the device shown in Figure 3.95 is ET ¼ 0.54  0.05 eV. The emission rate is calculated according to Equation 3.258 with A ¼ 7 s1 -K2 and a ¼ 7.5 (V)1/2 and the result overlaps well with the experimentally measured emission rate. One can then conclude, therefore, that above the room temperature, the emission process is thermally activated and is assisted by the electric field due to the gate–drain potential difference via Poole–Frenkel potential barrier lowering. Below 200 K, the emission rate remains constant at the level too high to be explained by the thermal ionization, indicating the increasing relative efficiency of the tunneling effects or the device self-heating. 3.5.5.4 Defect Mapping by Kelvin Probe and Effect of Surface States Transient current measurement discussed above are capable of shedding light on the energetics of the traps responsible for the anomalies observed in AlGaN/GaN HFETs, but they are not applicable to spatial resolution of those traps. Using surface charge sensitive Kelvin probes measurements, Sabuktagin et al. [318] investigated the excess charge between the gate and the drain of an AlGaN/GaN HFETs. Scanning Kelvin probes is capable of mapping the surface potential with nanometer-scale resolution and time response in the range of a few tens of milliseconds. Should there be any charge trapping in the surface states or anywhere in the structure between the

3.5 AlGaN/GaN HFETs

Surface

AlGaN

d1

QT

GaN

Surface

AlGaN

GaN

QT d

d Figure 3.96 Schematic representation of the charge induced on the surface and also at the channel in an AlGaN/GaN modulationdoped structure by spatially discrete AlGaN bulk electron trap density of QT (on the left) and negatively charged surface states of the same density (on the right).

channel and the surface such as the barrier layer, which causes a change in the band bending in the duration of applied bias, scanning Kelvin probe would detect the change in the surface potential, that is, surface band bending. One can consider the 2DEG and gate metal system with the barrier AlGaN in between as a capacitor and any charge on the surface or between the potential extension of the gate and the channel would induce a change in the potential or band bending, which can then be sensed by the Kelvin probe. This is shown graphically in Figure 3.96 for a case where the QT trap charge is on the surface alone and also in the barrier layer. The applicable expressions of charge balance for the two cases are similar to those expressed in Equation 3.257, and the induced differential charge due to a spatially discrete QT trap charge (QT < 0) at a distance d1 from the surface (which is negative) at the surface and at the 2DEG channel can be written as

 d1 d1 DQ channel ¼ Q T 1 and DQ surface ¼ Q T : ð3:259Þ d d This means that the negative delta-like bulk charge in the AlGaN would be balanced by a positive charge a part of which would be at the surface and the other part would be at the AlGaN/GaN interface. The relative values are determined by the ratio of the depth of the charge from the surface to the AlGaN thickness. If the bulk charge is closer to the surface, the positive-induced charge on the surface would be smaller than that at the interface. If the negative charge is entirely on the surface, there would be no positive induced charge on the surface obviously and all the negative charge on the surface would have to be balanced by a positive-induced charge at the interface. In all cases, the band bending would increase. The associated change in the surface potential caused by the induced charge can be found using the Q ¼ CV relationship commonly employed in MOS capacitors, as
 d1 DV surface ¼ Q T : ð3:260Þ eAlGaN If the charge in AlGaN is a distributed charge with a density of r(x), the charge induced at the interface and surface is given by

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ðd DQ surface þ DQ interface ¼ Q T ¼  rðxÞdx 0

and increment in the surface potential ðd 1 xrðxÞdx: DV surface ¼  eAlGaN

ð3:261Þ

0

The charge balance equations indicate that an increase in the surface potential results if negative charge is induced on the surface. Conversely, if positive charge is induced on the surface, there would be a decrease in the surface potential. Because the scanning Kelvin probe is sensitive to the surface charge and naturally any change in it, its measured intensity would allow one to determine if the charge is surface charge or the bulk charge assuming one knows that the charge itself is due to the captured electrons and that it is negative, which is the case in n-channel HFETs. During the turn-on stage of reverse gate bias, the electrons tunnel from the Schottky barrier to the surface where they may be captured by slow traps leading to an increase in the band bending [319–321]. As the negative gate–drain bias is turned off, the excess charge at the surface or at the traps is emitted and surface potential relaxes toward its equilibrium value. If the emission rate is not too fast compared to the time resolution of the probe, one can expect to detect the transient as a change in the surface potential by employing the scanning Kelvin probe. A slow rate of the surface potential transients in the surface photo voltage following laser excitation has been observed [322], which paved the way for applying this technique to sensing charge along a path on the exposed and unpassivated channel between the gate and the drain. In terms of conducting the experiments, the drain current transients were measured by Sabuktagin et al. [318] by applying biases from a pulse generator to the HFETunder test, which every few nanoseconds turn on and turn off the transient response. A digital oscilloscope was used to record the voltage drop across a 100 W resistor connected between the source and the ground. The surface potential was measured between the gate and the drain. Spatial resolution of the probe was about 50 nm, which enabled the detection of surface potential virtually at every point in the 4 mm spacing between the gate and the drain. Two bias conditions were chosen, which represent the gate bias before pinch-off and at pinch-off while keeping the total gate-to-drain bias the same. The difference between these biasing conditions is that in the former case, drain current exists whereas in the latter case, it does not. The purpose is to interrogate any role of the electron injection from the channel onto the surface states or bulk states in the barrier layer. Obviously, in the case of pinch-off, any observed surface potential change would be attributed to charge injected from the gate. If the two biasing conditions were to result in a similar surface potential change, any noticeable electron injection from the channel layer would be ruled out. 3.5.5.4.1 Drain Current Transients After applying a 1 V gate and 7.5 V drain bias, the potential drop across the 100 W resistor decreases slowly as shown in Figure 3.97. The drain current in Figure 3.97a collapses by 20% in approximately 30 s to reach the

3.5 AlGaN/GaN HFETs

Voltage drop across 100 Ω resistor (V)

1.05

0.8

1.00

0.6 0.4

Voltage drop across 100 Ω resistor (V)

0.95

0.2

0.90

0.0

0.85

0

40

80 120 160 200

Time (ms)

(a)

0.80 (b)

0.75

(c)

0.70 0.65 0

2

4

6

8

10

Time (s) Figure 3.97 Drain current transients due to different biases. (a) Current collapse with bias on (gate, 1 V; drain, 7.5 V; source, 0 V). (b) Steady state after 30 s. (c) Drain current recovery after the superimposed pulsed bias is turned off. (Inset) Additional current collapse due to the superimposed pulsed bias (gate, 4 V; 10 ms on, 150 ms off).

steady-state value of Figure 3.97b. Then – 4 V pulses to the gate (10 ms on-time, 150 ms off-time) superimposed on the previous bias (gate 1 V, drain 7.5 V) were applied. Figure 3.97d shows that the drain current during the 150 ms time when the 4 V bias is off is 12% less than the steady-state value of Figure 3.97b, indicating an additional current collapse due to the superimposed pulsed bias. After turning off the 4 V reverse-bias pulses, the drain current recovers (Figure 3.97c) in about 40 s to the level of Figure 3.97b. If we momentarily set both the gate and the drain potential to 0 V and then reapply the bias, we find that the drain current level is almost the same as that in Figure 3.97b. The gate and drain potentials had to be held at 0 V for over 700 s before reapplying the bias to observe a drain current transient that would start at a level similar to the beginning of the transient of Figure 3.97a. 3.5.5.4.2 Surface Potential Transients To understand the effect of bias on the surface potential between the gate and the drain, three different bias conditions have been employed by Sabuktagin et al. [318] in their SKPM investigation, namely, (a) gate: 5 V, drain and source: 0 V; (b) gate: 9 V, drain and source: 0 V; and (c) gate: 1 V, drain: 7.5 V, source: 0 V. Each of these biases was applied for 5 min from the pulse generator. The surface potential traces have been recorded (Figure 3.98) on a straight line from the gate to drain within 1 s of turning off the bias voltages. For about 1 mm from the edge of the gate toward the drain, the surface potential was more negative, albeit with decreasing intensity away from the gate edge, which indicates that the band bending was higher compared to the steady-state case. Figure 3.98a–c shows the changes in surface potential near the gate for the three bias conditions. This figure shows that a higher reverse bias across the gate–drain terminals, as in

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Surface potential after biases (eV)

(a)

0.0

150

(b) (c)

100

-0.5 -1.0 -1.5

0

1

2

50

Drain

Gate

3

4

5

6

Surface height (nm)

200

0.5

0

Distance (µm) Figure 3.98 Surface potential traces between the gate and the drain taken within 1 s of turning off biases of different magnitudes. (a) Gate: 5 V, drain and source: 0 V; (b) gate: 9 V, drain and source: 0 V; and (c) gate: 1 V, drain: 7.5 V, source: 0 V.

Figure 3.98b compared to Figure 3.98a, causes a larger band bending. The basis for this increased band bending is an increase of trapped electrons near the surface. Electrons can be trapped in surface states or in deep levels within the barrier as this measurement method cannot resolve whether surface states or states in the barrier are involved. The source of these electrons can be the gate through tunneling or the channel by hot electron injection. To understand the relative roles of these two processes, the surface potential profiles obtained after applying biases (b) and (c) are compared. For bias (b), no current was flowing through the device so that no hot carrier injection was possible. For bias (c), the current flow through the device was 10 mA. For both biases, the reverse potential between the gate and the drain was similar. As indicated in Figure 3.98, the surface potential profile after bias (c) is similar to the one observed after bias (b). The channel current for bias (c) did not noticeably affect the surface potential, which implies that electron tunneling from the gate [314, 323] is responsible for the observed surface charging phenomenon. To find the rate at which charge accumulates in surface states, 9 V reverse-bias pulses were applied to the gate for 0.4, 1.6,10, 20, 30, 50, and 100 s with the drain and source kept at 0 V. From Figure 3.99, one can see that the surface charging is a rather 200

0.75 0.50

150

0.25 After 0.4 s bias After 1.6 s bias After biases >20 s

0.00 -0.25

100

Gate

-0.50

Drain

50

-0.75 -1.00

0

1

2

3

4

5

6

Surface height (nm)

Surface potential after bias pulses (V)

1.00

0

Distance (µm) Figure 3.99 Surface potential maps on a straight line between gate and drain in 1 sec of turning off biases of different durations.

3.5 AlGaN/GaN HFETs After After After After

0.5

200

1s 100 s 200 s 800 s

160

0.0

120

-0.5

80

Gate -1.0

Drain 40

-1.5

Surface height (nm)

Surface potential after bias off (V)

1.0

0 0

1

2

3

4

5

6

Distance (µm) Figure 3.100 Surface potential maps on a straight line between gate and drain at different instants after turning off a 9 V reverse bias to the gate.

slow process taking place over tens of seconds. The bias duration of 20 s seemed to saturate the surface charging, because for bias durations longer than this, surface potential trace was similar to that of 20 s bias (depicted in Figure 3.99 as >20 s). To find the rate at which the excess surface charge dissipates through emission, Sabuktagin et al. [318] applied a 9 V reverse bias to the gate with the drain and source grounded for 5 min. After turning off the bias, the surface potential was monitored on the same straight line between gate and drain in successive scans by disabling the slow scan axis. One can see in Figure 3.100 that the discharge rate is a rather slow process. It takes about 700 s to reach the steady-sate condition with no excess band bending near the gate edge noted. 3.5.5.4.3 Correlation Between Current Collapse and Surface Charging Drain current lag (collapse) has been attributed to surface states whose thermal distribution has been studied [324]. Figure 3.98a and b shows that a bias applied to the device causes charge to be trapped near the gate. Figure 3.99 indicates a saturation time for this charging of 20 s and Figure 3.100 shows that the discharge of the excess charge is a rather slow process taking hundreds of seconds. These two combined observations help explain current anomaly and recovery of Figure 3.97. The gradual decrease of drain current in Figure 3.97a is caused by a slow accumulation of the charge in the surface states or in the barrier at the gate edge, as shown in Figure 3.101. The time it takes for the drain current in Figure 3.97a to reach its steady-state value is 30 s, which is comparable to the time required for achieving maximum band bending in Figure 3.99 (20 s). As far as the current recovery is concerned, it takes some 700 s for the drain current to recover fully. Similarly, Figure 3.100 shows that excess band bending after the 9 V gate reverse bias takes 800 s to relax. The comparable time scales observed for the drain current behavior and surface charging indicate that the excess band bending near the gate is very possibly related to the drain current level. If we assume that all the excess charge is on the AlGaN surface and use Equation 3.260, a surface charge concentration of about 1.8  1012 cm2 is obtained. This charge should cause a larger drop than the 20% reduction observed. Nevertheless, the increase of band bending definitely indicates depletion of the channel and therefore

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Surface charge Source

Gate

Drain

Depletion profile With surface charge

Without surface charge

Substrate

Figure 3.101 Schematic diagram showing how surface charge can affect drain current in a field effect transistor.

reduction of the drain current. In each of the 10 ms durations of the 4 V bias pulses, more electrons are able to tunnel from the gate and be trapped. Discharge of this excess charge was not complete in the 150 ms time when the 4 V pulse was absent. As a result, there was a net increase of the accumulated charge near the gate edge. This caused the drain current to decrease further, as seen in Figure 3.97c compared to Figure 3.97b. Increase of radio frequency input drive amplitude has been observed to worsen current collapse [325]. During each negative excursion of the gate drive, the reverse bias between gate and drain becomes higher, which is consistent with this model in that a higher reverse bias causes larger number of electrons to tunnel to the surface states. All of this excess charge does not discharge in the duration of positive half cycle. Among other observed anomalies of current–voltage characteristics in nitride-based FETs is a decrease of the drain current with the increase of drain-tosource voltage [323, 325]. As mentioned in the previous section, the drain current decreases with increasing drain-to-source voltage also for the same gate bias because this causes a higher reverse bias between the gate and the drain, which in turn increases tunneling of the electrons from the gate to the surface states near gate edge. Neuburger et al. [326] used an SiN-passivated FET having a second gate between the main one and the drain situated on top of the dielectric layer to sense surface charge by stress experiments. This is somewhat similar to the method of Sabuktagin et al. [318], who used an AFM tip for the same purpose. A simple measurement routine has been developed to identify lateral charge injection from the gate toward the drain. This lateral charge injection, as described in detail in Section 3.5.5.3 is one of the major sources of the virtual gate effect and current and power lag (slump) effects. With the sensing gate, the density and depth of the injected charge centroid at the location of the auxiliary test gate can be determined. An interesting conclusion reached by the authors is that the charge centroid may not be located at the interface between passivation layer and the semiconductor surface but may lie deep within the dielectric where it is trapped. A part of the argument is that injection from the gate is consistent with that reported by Sabuktagin et al. [318]. To add to the plethora of models for current collapse or anomalies, lattice distortions linked to the minimization of the polarization field have been proposed as a likely cause of current lag, in this particular case current collapse [327]. Through an investigation of time decay of current in response to a pulsed voltage applied

3.5 AlGaN/GaN HFETs

between the terminal of the adjacent ohmic contacts, in the form of gated transmission line measurement (GTLM) pattern, authors argue that the time dependence of the current is caused by transient variations of the gate–source and gate–drain resistances, while the channel resistance under the gate remains unaffected. According to these results of the GTLM measurements, the source and drain series resistances are responsible for the current collapse. An increase in the source series resistance should lead to a decrease of current. The same should also cause an increase in the knee voltage, the drain voltage at which the drain current reaches quasisaturation. One plausible explanation for the increase in series resistance during current transient is the change in strain under and around the gate metal. Increased gate bias from its initial value toward a more negative value causes the electric field in the AlGaN barrier layer to increase, which in magnitude is comparable with the built-in piezoelectric field, several MV cm1. If the GaN layer underneath the AlGaN layer is not strained, one then surmises that the AlGaN barrier layer undergoes an in-plane tensile strain to match its lattice constant to the underlying GaN layer. If so, the change in the electric field with gate bias would not affect strain in the GaN channel. However, the surface region of the GaN layer may be somewhat strained. Thus, an increase in the electric field due to increasing gate bias, which is comparable to the piezoelectric field, could increase the tensile strain in AlGaN layer under the gate. This would expand the AlGaN barrier layer laterally. Processes responding to the piezoelectric-induced field, transients of which are slow, could be associated with traps and cause the observed current collapse and lag. In the AlGaN layer under the gate, the gate metal provides a source of electrons in response to the induced piezoelectric charge. Therefore, this region is not expected to contribute to the current collapse [327]. As discussed previously, the current lag can be measured as a function of frequency at RF frequencies with an appropriate load line. Because the drain current does not follow the input stimulus due to surface traps, the term “lag” has been coined to describe the phenomenon. Assuming that the surface is at least in part responsible, it must be appropriately passivated to avoid this degradation. The effective methods so far have been the use of low temperature AlN [328] or Si3N4 [329, 330] postgrowth and fabrication passivation layers. If passivation alone is sufficient to eliminate the current lag, the issue of lattice distortion becomes an interesting one in that it raises the question whether the surface states are involved and, if so, whether passivation layers alter the strain picture also. Placing all the root causes on the surface states does ignore any effect that the barrier states may have, which contradicts the field dependence of the emission rate. At any rate, it is very reasonable to assume that surface states, barrier states, interface states, and also the buffer bulk states are in one way or another responsible for various manifestations of anomalous behavior. As for surface passivation, the surface of any semiconductor, particularly GaN, is sensitive to exposure to various environments and resultant cumulative effect [331–333]. The optical manifestation of surface states and the resultant band bending are discussed in some detail in Volume 2, Chapter 5. It is therefore prudent to passivate the surface. The drain characteristics for a GaN HFET with a 250 Å Si-doped AlGaN layer that exhibit the aforementioned anomalies are shown in Figure 3.102, where two sets of

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Drain current (A mm-1)

1.0 V GS=0

0.8

-1V /step

0.6 0.4 0.2 0.0 0

5

10

15

20

Drain voltage (V) Figure 3.102 Output drain current–voltage characteristics for an AlGaN/GaN HFET with a 250 Å AlGaN layer, gate length of 0.6 mm, and width of 50 mm. The dashed lines are for VDS < 10 V and the solid lines are for VDS up to 20 V. Courtesy of S. Binari [275].

characteristics are included for the same device. The characteristics indicated by the dashed lines are the result when the maximum VDS is limited to 10 V. However, the solid lines are for those measured when the maximum VDS is 20 V. By comparing these characteristics, a reduction in drain current for VDS < 8 V is noted. This reduction in current after the application of a high drain voltage is referred to as current collapse. This effect is similar to that reported for GaN MESFETs and is attributed to hot electron injection and trapping in the GaN buffer layer [279, 334]. As mentioned above, at high drain voltages, the electrons are injected into the GaN buffer layer where they are trapped. This trapped charge depletes the 2DEG from beneath the active channel and results in a reduction in drain current for subsequent VDS traces. The trapped charge can be released through illumination or thermal emission. The gradual reduction in current for VDS > 10 V, as shown in Figure 3.102, is attributed to self-heating due in part to the sapphire substrate with low thermal conductivity. The effect of SiN passivation on the drain characteristics is shown in Figure 3.103 for the same device before and after passivation. The drain current went up as a result of the increase in nsh. It can be seen that the reduction in current associated with the current collapse phenomenon is unaffected. This is consistent with the proposed mechanism for current collapse, that is, hot electron injection and trapping in the buffer layer without surface involvement. The current collapse/lag studied [447] by DC bias-stressing of an Si3N4-insulated device, wherein for a constant drain bias VDS ¼ 10 V, the gate VGS was swept from 5 V to þ 1 V and back to 5 V in 7 s. The results illustrated in Figure 3.104 reveal no discernable degradation for the passivated device but the severe degradation for the standard device. These observations reveal that the bias stress manifested in Figure 3.104 and the concomitant degradation in the standard device, while the passivated device remains unaffected, is an indication that the bulk of the epitaxial

3.5 AlGaN/GaN HFETs

Drain current (mA)

50

w SiN x passivation w /passivation

40

VGS =0

-1 V/step

30 20 10 0.0 0

10

5

20

15

Drain voltage (V) Figure 3.103 Measured drain characteristics of the device of Figure 3.102 before and after SiN passivation. Courtesy of S. Binari [275].

layers is not responsible for this anomaly and the electron injection from the gate to some defects on the surface or near the surface of the semiconductor, extension of the gate termed as “virtual gate”, is responsible as suggested by Sabuktagin et al. [318]. The effects of SiO2 and Si3N4 passivation on DC and RF performance of doped and undoped AlGaN/GaN/Si HFETs have been investigated by comparing the results

10 w passivation w/o passivation

Drain current (mA)

8 Before bias stress w/o passivation

6

4

After bias stress w/o passivation

2

0 -5

-4

-3

-2

-1

0

Gate voltage (V) Figure 3.104 ID-VGS characteristics of a device with (solid line) and a similar device without (broken line) Si3N4 passivation film. Note that the device with passivation remains more or less

the same after 10 V drain stress. However, the device without passivation exhibits much lower drain current after 10 V drain bias stress. Courtesy of T. Mizutani and Y. Ohno [447].

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before and after passivation [448]. Hall effect measurements showed Si3N4 passivation to be more effective than SiO2 passivation in terms of the increase in the carrier concentration at the interface. It should be noted that the dielectric layers were deposited with plasma-enhanced CVD and are therefore assumed to be stoichiometric. The HFET I–V characteristics followed the improvement in the channel charge density as determined by Hall measurements. Small-signal microwave characterization showed a decrease from 18.6 to 9 GHz in fT after SiO2 passivation and an increase from 18.4 to 28.8 GHz after Si3N4 passivation, respectively. Similarly, RF power performance measured at 2 GHz showed similar behavior in that the RF power was halved after SiO2 passivation but more than doubled with Si3N4 passivation. These observations were attributed to high dielectric/semiconductor interface density with SiO2 passivation. In a similar vein, Green et al. [329] applied Si3N4 passivation and noted an improvement in RF power performance. Specifically, RF power measurements on a 0.5 mm gate length and 2  125 mm AlGaN/GaN FETs on sapphire substrates demonstrated an increase in the 4 GHz saturated output power from 1.0 W mm1, with a 36% power-added efficiency, to 2.0 W mm1 with a 46% power-added efficiency peak at a drain bias of 15 V in both before and after passivation cases. Passivation was also noted to increase the drain breakdown voltage by 25% on average. Passivation schemes that are temporary in nature but very effective in terms of passivating surface states, particularly the nonradiative varieties, have been and could be used for investigative purposes while more durable solutions are sought. In addition, surface-sensitive techniques such as photoluminescence can be used as a first-order probe to detect any modification on the surface rather quickly. Even though PL is sensitive only to radiative and due to lack of it to nonradiative recombination centers whether they are charged or not and FETs are sensitive to charged states, PL is still useful in interrogating the surface or near-surface region of the semiconductor. In fact, states that are not charged could actually change their states under bias such as the case in FETs. In this vein, Martinez et al. [335] observed a four to sixfold enhancement in the room temperature PL intensity when the GaN samples were treated with aqueous and alcoholic solutions of inorganic sulfides, such as ammonium or sodium sulfide ((NH4)2Sx and Na2S). It is remarkable that the YL intensity did not change after the treatment. The nonsulfide-based treatments have also been investigated leading to the observations that acidic solutions increased the PL intensity (by up to a factor of 2.5), whereas bases and H2O2 slightly decreased it. Martinez et al. [335] explained the stability of the sulfide passivation by the formation of a strong NS bond instead of the weak NO bond. Similar investigations with Na2S in isopropyl alcohol [336] and CH3CSNH2 solution [337] have been reported with similar trend in that the PL intensity increases when the surface is passivated. Note that along with the enhancement of the PL intensity, the electrical properties of GaN significantly improved after passivation [335, 337]. In particular, the specific contact resistance decreased from 2.4  102 to 3.1  106 W cm2 [337] and the Schottky barrier height increased from 1.18 to 1.63 eV [335] after sulfide passivation.

3.6 Electronic Noise

3.6 Electronic Noise

The low-frequency and high-frequency noise characteristics of GaN HFETs are very important for the microwave applications of these devices. Fluctuations or deviations in the form of irregular changes in parameters of interest such as current, voltage, resistance, frequency, and so on, from long-term time averages or from periodical time dependence are considered noise. The lower limit of any signal intensity that can be measured is set by noise, that is, spontaneous fluctuations in current and voltage of the system under test. In modern devices, nonequilibrium conditions are prevalent for device operation, which exacerbates the situation. Hot carriers prevalent in FETs cause hot (nonequilibrium) phonon population to be generated in addition to impact ionization (inherently noisy), which must be considered [338, 339]. Traditionally, before the advent of high-performance nonequilibrium devices, noise to a first degree was thought of as fluctuations in current flow (shot noise) due to recombination–generation process (generation–recombination noise or simply G–R noise) and due to thermal processes (thermal or Johnson or Nyquist noise). In terms of the thermal noise (Johnson noise [340]), any resistor shows spontaneous fluctuations in current caused by universal thermal motion of carriers (only the term electrons will be used to represent carriers of the type, electrons and holes). The Nyquist theorem [341] treats the thermal noise as the available noise power or a thermal-noise source (such as that due to a resistor) as being a universal function of the absolute temperature. The shot noise is caused by the corpuscular nature of electrical charge and is associated with current flow when it is controlled by a barrier such as in p–njunctions, Schottky barriers across heterojunctions, tunneling structures, barriers induced by nonuniform doping regardless of how small they are, and nonohmic contacts. In a sense, any barrier presented to carrier transport produces shot noise. The noise produced by conductance fluctuations is viewed as 1/f noise when it is dominant at low frequencies decaying as 1/f with frequency. Generation– recombination noise is generated by random transitions from and to the trap levels from the conduction and/or the valence bands as a fluctuation in the number of mobile carriers. In devices such as FETs, the electrons are not in equilibrium with the lattice (hot electrons) and fast kinetic dissipation takes place in the conduction or valence band. Processes such as energy relaxation, intervalley scattering, if applicable, impact ionization and so on, which cause fluctuations resulting in hot electron noise. A fundamental treatment of all of these noise sources is discussed below beginning with shot noise. The low-frequency noise also manifests itself at high frequencies mainly as phase noise [342, 343] with serious implications to oscillators [344]. Investigation of the origin of noise is important for understanding the physical processes taking place in the device. Although GaN HFETs with excellent microwave performance have been fabricated for high-power and high-temperature applications, there is still little investigation on the noise properties. Low-frequency noise, containing fluctuations of currents or voltages with frequency components below 10 KHz, is to a first extent fundamental 1/f noise but to a

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lesser extent also a nonfundamental 1/f noise. In addition, shot noise, GR noise, and thermal noise mentioned above, which are important at higher frequencies, also extend to the low-frequency domain. All these forms of electronic noise are characterized by a mean-squared current fluctuation h(dI)2i (or h(dV )2i : (rms dV )2 for voltage), measured in series with (or across for voltage noise) the device or sample under test, when a constant voltage (or current) is applied, except for thermal noise, which is present even in thermal equilibrium, with no bias applied. All the other forms of noise present in addition to thermal noise are also known as current noise and are absent in thermal equilibrium. Nevertheless, 1/f noise [345] and G–R noise also modulate the root mean square (rms) level of the thermal noise currents (or voltages) in thermal equilibrium, while the available power remains constant at kBT per Hz, where kB ¼ 1.38  1023 J K1 is Boltzmann’s constant and T the absolute temperature [345]. These two forms of current noise are also called modulation noise, because they modulate the resistance. If a bandpass filter is inserted between the measuring device (usually a quadratic meter) and the noise source, the spectral density of the fluctuations h(dI)2if : SI( f ) (or SV( f )) is obtained by dividing the measured mean square by the bandwidth Df of the filter. Let us now briefly discuss the nature of the various noise phenomena mentioned above with the exception of the 1/f noise, which is discussed in Section 3.6.5. 3.6.1 Shot Noise

Statistical fluctuation in current is termed as the shot noise. Owing to quantization of electronic charge e, the electron flow described by number or electrons per unit time is not continuous. It is then possible to separate the current i(t) into DC, I0, and AC, iac(t) components. The latter term is considered as an ergodic fluctuation phenomenon [339]. With the notable exception of 1/f noise, also known as excess noise, the bases for various types of noise phenomena mentioned above were laid by J. B. Johnson, H. Nyquist, and W. Schottky. For example, shot noise is caused by current in devices such as vacuum tubes, electron beams, Schottky diodes, p–n-junctions, and in any other device carrying a current, by the discrete, atomistic nature of electrical conduction. The spectrum is easily described as a Poisson process and is given at low frequencies by the so-called Schottky relation: SI ð f Þ ¼ 2eI 0 ;

ð3:262Þ

where e is the electronic charge of the charge carriers and I0 is the average current in the direction of carrier motion. For electrons, both e and I0 are negative. The mean square shot noise current in a frequency interval Df is thus 2eI0Df. The Schottky relation is valid even at high frequencies. This particular description, Equation 3.262, is suitable for describing the shot noise caused by gate leakage current in FETs in which case the current I0 would be replaced with Ileak. The general formula is given by Carson’s theorem, which gives the spectral density of a random uncorrelated repetition of identical processes with spectrum f( f ) and repetition rate l as

3.6 Electronic Noise

2ljfðf Þj2 :

ð3:263Þ

The case with arbitrary correlations between the moments t0 of passage has been treated by Heiden [346] and is usually not called shot noise. The elementary process in shot noise is the current i(t  t0) caused by the motion, or passage, of a single carrier. Therefore, ð/ iðtt0 Þdt ¼ e ð3:264Þ fð0Þ ¼ /

is the total charge e transported by a single carrier. With el ¼ I0, Carson’s theorem arrives at Equation 3.262. The nomenclature “shot noise” comes from an analogy to a process wherein small shot (or rain drops) fall on a drum. The 1/f noise, however, remained enigmatic and fundamental 1/f noise was understood only after the advent of the quantum 1/f theory [283, 284, 347, 348]. It turns out, there is always the fundamental 1/f noise [345, 349–351] caused by the quantum 1/f effect. But there is also nonfundamental 1/f noise, characterized by accidental 1/f-like spectra arising from a lucky superposition of G–R noise spectra. Both fundamental and nonfundamental 1/f noise types are important in practice, as we stressed below. 3.6.2 Generation–Recombination Noise

The G–R noise is caused by the random generation and recombination or trapping and detrapping of current carrying charge in semiconductors, which is described by the (always one-sided) spectral density SI ð f Þ ¼ 8hðdIÞ2 ipt=ð1 þ w2 t2 Þ;

ð3:265Þ

where t is the lifetime of the carriers and w ¼ 2pf is the angular frequency. According to the Wiener–Khinchine theorem, the spectral density is the Fourier transformation of the autocorrelation function AðtÞ hIðtÞIðt þ tÞi

ð3:266Þ

and is given by 1 ð

AðtÞcosð2pf tÞdt:

Sðf Þ ¼ 4

ð3:267Þ

0

Equation 3.265 is obtained by Fourier transformation from the exponential autocorrelation function A(t) ¼ h(dI)2iet/t, which describes, for instance, the exponential decay of the number of carriers that have not yet recombined at the time t. There is a term similar to Equation 3.265 present in the spectral density of current noise in semiconductors, for each type of carriers. Let N be the number of carriers of a certain type in a semiconductor sample in stationary conditions. In terms of the generation rate g(N) and of the recombination

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rate r(N), Burgess [352] found general formulae for both the lifetime t and the mean square entering in Equation 3.265: t¼

r 0 ðN

1 ; 0 0 Þg ðN 0 Þ

hðdIÞ2 i

I 20 ; N 20

hðdNÞ2 i

I20 tgðN 0 Þ; N 20

ð3:268Þ

where the prime (0 ) denotes a derivative with reference to N. The derivatives are taken for N ¼ N0 hNi, and we have denoted as I0. The following special cases are highlighted: (i) For an n-type semiconductor with Nd deep donors, the generation rate g(N) ¼ g(Nd þ N) is proportional to the number of neutral donors Nd  N, while r(N) ¼ rN2 with constant g and r because there are N free electrons and N ionized donors. Therefore, one obtains t¼

1 N d N 0 ; ¼ g þ 2rN 0 2rð2N d N 0 Þ

hdN 2 i ¼

N 0 ðN d N 0 Þ : ð2N d N 0 Þ

ð3:269Þ

(ii) For a near-intrinsic n-type semiconductor with N electrons, Nd donors (all ionized) and P ¼ (N  Nd) holes (by charge balance), we write g ¼ constant because the fluctuations are due to the thermal generation of electron–hole pairs. In this case, the recombination rate r ¼ rNP ¼ rN(N  Nd). Therefore, t¼

1 ; rðN 0 P 0 Þ

hdN 2 i ¼ hdP 2 i ¼

N 0 P0 : N 0 þ P0

ð3:270Þ

(iii) For a semiconductor with NT traps and N-trapped electrons, the trapping rate is proportional to the number NT  N of empty traps, while the release rate is proportional to N. Therefore, g(N) ¼ a(NT  N) and r(N) ¼ bN. The constants a and b are determined by the equilibrium condition a(NT  N) ¼ bN0, which yields N0 ¼ [a/(a þ b)]NT. Therefore, t¼

1 ; aþb

hdN 2 i ¼

bN 0 abN t ¼ : a þ b ða þ bÞ2

ð3:271Þ

In this special case the rates g and r are not nonlinear functions of N and are therefore independent of N0 (Equation 3.271). In this case, the fluctuation of N obeys the binomial distribution law. 3.6.3 Thermal Noise

Also known as Johnson (or Nyquist) noise, thermal equilibrium noise has a white (frequency-independent) spectrum at moderate, not too high, frequencies. Assuming a temperature greater than the absolute 0, the electrons move about with their thermal energy and due to scattering mechanisms, with their associated random processes, the kinetic motion is accompanied by statistical fluctuations of the voltage across the terminal of the conductor (resistor).

3.6 Electronic Noise

An equivalent noise temperature can be defined to represent the temperature of a blackbody radiator generating the same power in the same frequency bandwidth. The available power generated by a noise source into a frequency band of Df around a central frequency of f can be expressed by comparing it with an absolute blackbody radiator for the same power in the same frequency band. The temperature of the blackbody then generating the same power in the same frequency band is called the equivalent temperature. Therefore, the product of the equivalent noise temperature Tn( f ) and the Boltzmann constant would be the noise power per unit band around a frequency f dissipated into a matched load [338] kB T n ð f Þ ¼

DPn ð f Þ ; Df

ð3:272Þ

where DPn( f ) is the available noise power. Deviations from equilibrium introduce excess noise, spectral features of which depend on the nonequilibrium processes/ conditions. The available noise power, therefore, depends on bias conditions and frequency, making it necessary to view the noise temperature as being frequency and bias dependent. Of particular importance to FETs is that hot electron noise temperature represents features of, say, an electron gas differing from its energy temperature Te that enters into the average energy for electrons in a nondegenerate 3D electron gas as 3 hEi ¼ kB T e 2

ð3:273Þ

and in nondegenerate 2D gas as hEi ¼ kB T e :

ð3:274Þ

The noise spectrum associated with thermal noise depends on the temperature and the real part of the impedance (or conductance). In general, it is given for a circuit component of impedance Z ¼ 1/Y of conductance G ¼ ReY and resistance R ¼ ReZ around the DC bias by the Planck–Nyquist [341] formula SI ð f Þ ¼ 4G

hf 4kB T n ð f ÞG; exp½ðhf =kB T1

ð3:275Þ

which represents the spectral intensity of the noise source for a current source. The same can be expressed as SV ð f Þ ¼ 4R

hf 4kB T n ð f ÞR exp½ðhf =kB TÞ1

ð3:276Þ

for a voltage source where h ¼ 6.62  1034 J s is Planck’s constant and kB ¼ 1.38  1023 J K1 is Boltzmann’s constant. The approximated expression on the right represents the classical limit where the inequality
hw kB T 0 holds. Under equilibrium, the noise temperature equals the absolute temperature of the source of noise as T eq n ¼ T 0 . This means that equilibrium noise spectrum is white, frequency independent for frequencies w ðkB T 0 Þ=
h, and the available power per unit bandwidth can be related to the energy by the equipartition rule.

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The thermal noise power available (for a matched load) is Sa ðf Þ ¼

hf kB T: expðhf =kB TÞ1

ð3:277Þ

The right-hand term is obtained by noting that hf kT at microwave and millimeter wave frequencies and thus using only the first two terms in the Taylor expansion. With the exception of ultrahigh frequencies at very low temperatures, only the approximate forms are used in practice and are known as equivalent forms of the Nyquist formula. The amplitude distribution of thermal noise is Gaussian, with small deviations of fundamental origin caused by the quantum 1/f effect even in thermal equilibrium [350]. In terms of circuit-friendly description, the measurable mean square value of thermal noise current hi2th ðtÞi and the thermal noise voltage per unit bandwidth hv2th ðtÞi are given as   2  ð3:278Þ ith ðtÞ ¼ 4pkB TGDf and v2th ðtÞ ¼ 4pkB TRDf ; respectively: Therefore, one can represent a noisy resistor R with a noiseless resistor of value R and a voltage generator whose value is given by Equation 3.278. If one were to connect a resistor R to this noisy resistor (impedance matching for maximum power transmission to the load resistor), the power delivered to the load in a bandwidth Df becomes Pn ¼

hv2th ðtÞi 4pkB TRDf R¼ R ¼ 2pkB TDf ¼ kB TDw: 2R2 2R2

ð3:279Þ

Several observations can be made. As the bandwidth Df ! 0, the noise power goes to 0. As T ! 0, the noise power goes to 0. If Df ! 1, the noise power ! 1, called the ultraviolet catastrophe, which does not really happen because taking the first two terms of the Taylor approximation does not hold anymore. 3.6.4 Avalanche Noise

Even though FETs are not willfully driven into avalanche, high electric fields might be sufficiently large to cause carrier multiplication. There are devices, however, such as avalanche photodiodes, avalanche bipolar transistors, impact ionization and transit time (IMPATT) devices that actually harness impact ionization. If the mean free length of carriers is sufficiently long, carriers can achieve sufficient energy (see Volume 2, Chapter 4) to cause electron–hole pair generation through collisions with lattice atoms. The secondaries created by primary carriers would also in turn impact ionization. The process, once started, continues causing a large increase in both types of carriers, a process known as avalanche breakdown. The noise associated with fluctuations in this process is called the avalanche noise, which is characterized by two ionizations a and b coefficients, a for electrons and b for holes (see Volume 2, Chapter 4 for details). The ionization coefficients are a measure of the number of carriers generated per unit length. Assuming that the initial current, input current,

3.6 Electronic Noise

Iin is of shot noise character and for simplicity a ¼ b, the noise spectrum of the output current can be expressed as SIout ¼ 2eI in M3 ;

ð3:280Þ

where M is the multiplication factor. If one assumes the avalanche coefficients to be related by a ¼ cba (c being a constant) and input current is due to electrons only, the output current is given by [353, 354] "
 # M1 2 3 : ð3:281Þ SIout ¼ 2eI in M 1ð1cÞ M In the case where the input current is purely a hole current, the parameter c should be replaced with 1/c. Before delving into the low- and high-frequency noise, it is helpful to discuss the description of noise in two-port circuits by defining equivalent circuits for noise circuits. Starting with noise impedances and admittances, because ideal inductive and capacitive components do not contribute to noise, real parts of complex impedance or admittance would be considered. The measurable and frequencydependent mean square values of the thermal noise current hi2th i and thermal noise voltage hv2th i per unit bandwidth, Df, are given by Equation 3.278. The thermal spectral noise densities can be expressed as in Equations 3.275 and 3.276 1 ¼ 4kB T Re Y; Df 1 Sth;v ð f Þ ¼ hv2th i ¼ 4kB T Re Z: Df Sth;i ð f Þ ¼ hi2th i

ð3:282Þ

3.6.5 Low-Frequency Noise (1/f Noise)

Low-frequency noise, generically referred to as the flicker noise, is brought on by fluctuations of currents or voltages with frequency components below approximately 10 KHz. To a casual eye, noise appearing at these low frequencies is construed as 1/f noise. However, some G–R and slow surface state-related noise with superposition of many Lorentzian spectra could give the appearance of 1/f noise. This contribution, if any, is termed the nonfundamental 1/f noise, whereas that which is truly 1/f noise in origin is called the fundamental 1/f noise as we are about to discuss. The G–R noise-related component can also be used to monitor G–R center generation during life testing in that hot phonons present in GaN in conjunction with existing defects would be sufficient to generate new ones. This method has been applied to GaAs- as well as GaN-based FETs successfully [355–358]. 3.6.5.1 Nonfundamental 1/f Noise As briefly mentioned above, some G–R noise and noise due to slow surface states have the appearance of 1/f noise, which does not really represent fundamental

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1/f noise based on quantum 1/f effect. In fact, in the experimental description of the low-frequency noise, we gravitated to the common terminology of referring to the low-frequency noise regardless of its origin as 1/f noise. A distinction, however, must be made here. To distinguish the low-frequency noise from the fundamental 1/f noise, the nomenclature “nonfundamental 1/f noise” is used. McWhorter [359] suggested that 1/f-like noise in semiconductor samples and devices may arise from transitions of electrons to and from traps in the oxide at the surface. A superposition of many Lorentzian spectra (Equation 3.265) resulting from traps with different exponential relaxation times t in the interval t1 < t < t2 can yield a 1/f-like spectrum, causing a nonfundamental, or accidental, 1/f noise, in a limited frequency domain if the following two conditions are satisfied: (i) an electron is not allowed to interact with many traps at the same time. (ii) The distribution of the characteristic times must have a probability density c/t. Indeed, one expects then an addition of power spectra tð2 

SI ðf Þ ¼ 8hdI 2 ipc t1

 t dt 1 þ w2 t2 t

pc ¼ 8hdI i ½arctanðwt2 Þarctanðwt1 Þ w 2 p c 8hdI 2 i : w 2

ð3:282Þ

The last approximation is valid only for (1/t2)(1/t2) w (1/t1). There is strong evidence favoring a major contribution of this mechanism in MOSFETs from studies of the relaxation of slow surface states, particularly because the observed spectrum often differs slightly from 1/f. The slow states are distributed uniformly in the oxide volume that serves as gate insulation at the surface of the semiconductor. This nonfundamental contribution is usually larger in MOSFETs than the fundamental 1/f noise. The constant c is proportional to the superficial density of slow surface states, which can in principle be determined from the slow relaxation of the surface charges, but is hard to determine in practice without measuring the 1/f noise. Therefore, Equation 3.283 is difficult to apply in practice. Investigations by Hooge [360] have proven, however, that this McWhorter [359] model is basically inconsistent and that the actual cause of these apparently nonfundamental noise contributions may be fundamental. It is always clear from quantum 1/f noise theory [361] that at least part of it is present in the matrix elements and process rates controlling the current in the MOSFET at the surface. Following the first universal dynamical theory of 1/f noise [362, 363], Hooge [364] has experimentally noted a certain universality of the well-known 1/f noise coefficient. This universality was short-lived, because it did not hold for smaller devices. 3.6.5.2 Fundamental 1/f Noise Let us now turn our attention to fundamental 1/f noise. The conventional quantum 1/f effect [283, 339, 345] is a fundamental fluctuation of physical cross sections and process rates, caused by the infrared-divergent coupling of current carrying charge to

3.6 Electronic Noise

low-frequency photons and other infraquanta, for example, transversal phonons with piezoelectric coupling or electron–hole pairs on the Fermi surface of metals. Even if this noise mechanism were of no major concern in FETs, which is unlikely, the subject must be treated for completeness and informational reasons. For the physical origin of quantum 1/f noise [345], consider for example Coulomb scattering of current-carrying charge, for example, electrons on a center of force. The scattered electrons reaching a detector at a given angle away from the direction of the incident beam are described by de Broglie waves of a frequency corresponding to their energy. However, some of the electrons lose energy in the scattering process due to the emission of bremsstrahlung (this term in German means braking radiation and stands in general to describe electromagnetic radiation emitted (as photons) when a fast-moving charged particle (usually an electron) loses energy upon being accelerated and deflected by the electric field surrounding a positively charged atomic nucleus. X-rays produced in X-ray machines are bremsstrahlung. In particular, the term is used for radiation caused by decelerations when passing through the field of atomic nuclei (external bremsstrahlung) [365]. Therefore, part of the outgoing de Broglie waves is shifted to slightly lower frequencies. When one calculates the probability density in the scattered beam, one obtains also sinusoidal cross terms, linear in the part scattered with and without bremsstrahlung. These cross terms oscillate sinusoidally in time at the same frequency as the emitted bremsstrahlung photons. Therefore, emission of photons at all frequencies results in quantum probability density fluctuations at all frequencies. The corresponding current density fluctuations are obtained by multiplying the probability density fluctuations by the velocity of the scattered current carriers. Finally, these quantum current fluctuations dJ present in the scattered beam are noticed at the detector as low-frequency current fluctuations and are interpreted as fundamental cross-sectional fluctuations in the scattering cross section of the scatterer. The quantum expectation of dJ is zero. However, its power spectral density is not zero, and is proportional to 1/f. Therefore, h J2i is larger than h Ji2. By the way, if J would designate the angular momentum, this would no longer have surprised anyone since 1925, because J( J þ 1) > J2. One may ask why is the spectrum of this conventional fundamental 1/f noise proportional to 1/f ? The energy spectrum of the spontaneously emitted bremsstrahlung radiation is known to be constant at low frequencies. Dividing this by the energy hf of a bremsstrahlung photon [345], we obtain a photon number spectrum proportional to 1/f (known as infrared catastrophe). Any matrix element for bremsstrahlung emission therefore ought to be proportional to (1/f )1/2. The above-mentioned cross terms are linear in this matrix element as was noted above, and therefore are also proportional to (1/f )1/2. The power spectrum of these sinusoidal cross terms present in the expression of the current is thus [283, 345] proportional to their squared amplitude, that is, 1/f. Although the wave function y of each carrier is split into a bremsstrahlung part and a nonbremsstrahlung part, no quantum 1/f noise can be observed from a single carrier [349]. A single carrier will only provide a pulse in the detector. Many carriers are needed to produce the quantum 1/f noise effect, just as in the case of electron diffraction patterns [349], where each individual particle is diffracted, but unless the

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experiment is repeated many times, or one uses many particles, no diffraction pattern can be seen. A single particle only yields a point of impact on the photographic plate in diffraction or a pulse in the detector in the 1/f noise case. While incoming carriers may have been Poisson distributed, the scattered beam will exhibit super-Poissonian statistics or bunching, due to this new effect, which we may call quantum 1/f effect. The quantum 1/f effect is thus a many-body or collective effect present even in the absence of interaction forces between the carriers. It is at least a two-particle effect, best described through the two-particle wave function and two-particle correlation function. Finally, quantum 1/f noise was shown to arise only from spontaneous bremsstrahlung and not to be affected by the statistically independent induced bremsstrahlung, caused by the thermal radiation background [366–368]. The latter just adds a modest uncorrelated white noise contribution. Let us now discuss the coherent quantum 1/f effect [347, 361, 369]. An electrically charged particle embodies the bare particle and its field. The field associated with an electrically charged particle has been shown to be in a coherent state, which is not an eigenstate of the Hamiltonian. Consequently, the physical particle is not described by an energy eigenstate and is therefore not in a stationary state. The fluctuations arising from this nonstationary state have a 1/f spectral density and affect the ordered, collective, or translational motion of the current carriers. This “coherent” quantum 1/f effect should be presented in the observed quantum 1/f noise along with the familiar conventional quantum 1/f effect of elementary cross sections and process rates just as the magnetic energy of a biased semiconductor sample coexists with the mechanical kinetic energy of the individual, randomly moving, current carriers. In the language of coherent state vectors in quantum electrodynamics, the amplitude of the conventional quantum 1/f effect is always the difference of the coherent quantum 1/f noise amplitudes in the “out” and “in” states of the process under consideration and dominates in very small samples, while large samples should exhibit the larger coherent quantum 1/f noise effect. Its power spectrum of fractional current fluctuations is given by the simple formula 2a/pf, where a ¼ e2 =
hc ¼ 1=137 is the fine structure constant. The conventional quantum 1/f effect has an additional factor 2(Dv)2/3c3 where Dv is the change in vector velocity. What is it in the foundations of quantum mechanics and quantum electrodynamics that compels us absolutely to introduce quantum 1/f noise? It turns out that a physical, electrically charged particle must be described in terms of coherent states of the electromagnetic field rather than in terms of an eigenstate of the Hamiltonian. This is the conclusion obtained from calculations [370, 371] of the infrared radiative corrections to any process performed both in Fock space (where the energy eigenstates are taken as the basis and the particle is considered to have a well-defined energy) and on the basis of coherent states. Indeed, with coherent states, all infrared divergences drop out already in the calculation of the matrix element of the process considered, as it should be according to the postulates of quantum mechanics. However, in the Fock space calculation, they drop out only a posteriori, in the calculation of the corresponding cross section or process rate. From a more fundamental mathematical point of view [371], both the description of charged particles, in terms of coherent states of the field, and the undetermined energy are the

3.6 Electronic Noise

consequence of the infinite range of the Coulomb potential. Both the amplitude and the phase of the physical particle’s electromagnetic field are well defined, but the energy, that is, the number of photons associated with this field, is not well defined. The indefinite energy is required by Heisenberg’s uncertainty relations because the coherent states are eigenstates of the annihilation operators, and these do not commute with the Hamiltonian. A state that is not an eigenstate of the Hamiltonian is nonstationary. This means that we should expect fluctuations in addition to the (Poissonian) shot noise to be present, the details of which can be found in Ref. [372]. The additional fluctuations were derived and identified there as 1/f noise with a spectral density of 2a/pf arising from each electron independently as we just mentioned briefly above, where a ¼ e2 =
hc ¼ 1=137 is the fine structure constant and f is the frequency (see Section 3.6.8 for details). 3.6.6 High-Frequency Noise

Many of the noise processes dealing with fluctuations and scattering events that respond to high frequencies manifest themselves as high-frequency noise. Among them of course are thermal noise, shot noise, and avalanche noise. Here, we will also use this nomenclature as a prelude to discuss noise-producing processes specific to high-frequency FETs brought about by high fields, which are present in the channel of a FET, and processes that become imperative at high frequencies. Thermal noise (Johnson noise) comes into play through contact metal resistance and also through the metal–semiconductor contact resistance (metal to semiconductor resistance at the source and drain electrodes) and semiconductor resistance where the electric field is low. Better device designs, much of which are technologically driven, can be employed to lower these resistances and thus the thermal noise. Because the gate provides a barrier to carrier flow and in cases of gate leakage, the fluctuations associated with this barrier activated current flow induce shot noise. The gate leakage problem is somewhat an issue of technology and can be alleviated to some extent, which in turn would eliminate the shot noise. The central issue with FETs, which have high electric fields in the channel, has to do with the random processes involving hot electrons (intervalley scattering, energy gain and loss through interaction with optical phonons, and impact ionization whichever is applicable) that cause fluctuations in the channel current, which constitute the basis of hot electron noise. This noise source is inevitable. The fundamentals of noise theory can be used to reduce the various noise contributions to current sources, which then are added to the equivalent circuit of the device to get a full picture of the noise contribution to the device operation. Linear resistors at low fields produce frequency-independent Johnson noise. Ideal capacitors and inductors do not produce any noise. As mentioned, barriers and diodes produce shot noise. However, deviation from ideality is associated with resistive losses, which induce thermal noise. Nonlinear resistor produce hot electron noise at high fields such as the channel of an FET. Various noise sources (signals) act

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collectively with frequency-dependent impedance of the capacitors and inductors causing frequency-dependent noise, which increases with frequency [338]. Regardless of the complexity of the problem, what is certain is that the shorter the transit time, the less the random hot electron processes are and thus lower the noise is. This can be accomplished by reduced channel lengths and use materials with relatively a high carrier velocity, a scheme that is continually exploited and demands short gate lengths. High carrier velocity naturally implies reduced scattering at high fields, thus lowered fluctuations associated with the relevant processes. On the practical side, empirical expressions have been developed, which allow one to bypass the detailed basic physics for a rapid determination of noise with certain materialdependent parameters and knowledge of resistances, and so on, as described in Section 3.6.9. It is clear that high fields and therefore hot electrons are synonymous with FETs. The scattering processes such as longitudinal optical (LO) phonon absorption and generation and interband scattering impact ionization introduce fluctuations in the system, which manifest themselves as noise, and to be specific this brand of noise will be called hot electron noise and the equivalent noise temperature ascribed to it will be called hot electron noise temperature. In MESFET with three degrees of freedom for carriers, the hot electron noise temperature is different in the direction parallel and normal to the steady-state current flow even in the case of spherical bands and isotropic scattering mechanisms. The noise in the transverse direction is caused by only the velocity fluctuations in an isotropic semiconductor, which is white for frequencies below that corresponding to the mean carrier relaxation time in the context of scattering. This means that the energy relaxation time, which depends on electric field, can be estimated from the hot electron transverse noise temperature. Contributions to thermal and intervalley noise by hot electrons depend on the frequency. Because of intervalley scattering, the hot electron noise is not the simple sum of intravalley scattering-induced noise weighted by the appropriate population densities. We should mention that the intervalley scattering noise is due to carrier scattering from one valley to the other. The source that makes the carriers hot and is involved in intervalley scattering can be either thermal, which is not the case in FETs (the process of phonon emission by electrons accelerated by the electric field heating up the semiconductor is not meant here), or electric field, which is the case in FETs. In both cases, the carrier velocity is nonzero. Outside of the parasitic resistances, the high-frequency noise is affected greatly by hot electron fluctuations and associated velocity and other transport-related fluctuations inclusive of intervalley scattering if applicable. To this end, a full treatment of noise at high fields must include momentum and energy relaxation processes [338, 373, 374]. A functional noise theory in the context of GaAs MESFET has been developed early on [375]. There are some differences between MESFETs and those relying on confined 2DEG system. For example, in FETs relying on confined 2DEG transport, one can to a first extent assume that fluctuations in carrier and carrier transport normal to the overall current flow would be negligible, which would simplify the already complex problem to some extent. The scattering processes involve momentum and energy gain and loss, and assuming that the relaxation approximation applies, the governing equations are

3.6 Electronic Noise

d  m ðEÞv ; ½mn ðEÞv ¼ eF n dt tm ðEÞ dE EE 0 ; ¼ eFv tm ðEÞ dt

ð3:284Þ

where F, E, tm(E), te(E), v, and mn represent the field, energy, momentum relaxation time, energy relaxation time, carrier velocity, and the electron effective mass (which is energy-dependent as nonparabolic portions of the conduction band are involved), respectively. E0 represents the energy under thermal equilibrium. The momentum relaxation expression reduces to v ¼ etF=mn ðor m ¼ et=mn Þ under steady-state conditions and incidentally describes the mobility expression in the realm of relaxation time approximation. Relaxation time approximation assumes, for example, that the energy distribution function can be fully characterized by average carrier energy. These energy and momentum relaxation expressions can be applied to the channel of an FET, as has been done for AlGaAs/GaAs modulation-doped FETs [373, 374, 376]. Application of this method and use of Gauss’ law and making use of the continuity equation along with the charge control equation by the gate in small segments along the channel with the appropriate boundary conditions lead to the determination of important parameters, which are also used to construct the equivalent circuit for noise analysis. Assuming that the noise at high fields is caused mainly by velocity and carrier fluctuations, the segmented approach also can be applied to noise analysis, and noise can be assumed to be represented by uncorrelated noise sources distributed along the channel [374]. The mean square value of the noise current related to each segment j represented by Dx along the channel can be expressed as [374] hi2 ij ¼ e2 hDv2== iN jð2DEGÞ Z=Dx;

ð3:285Þ

where hDv2== i is the mean square velocity fluctuation along the channel and Nj(2DEG) Z/Dx is the number of carriers in that section. As usual, Z represents the width of the device. Diffusion of carriers along the channel is also subjected to fluctuations. As in the case of carrier velocity, in the 2DEG system, one can assume that the component of diffusion normal to the general current flow is suppressed. Consequently, considering only the parallel component, as in the case of velocity, would represent the diffusion fluctuation picture with reasonable accuracy. Considering the diffusion constant as a function of only the average carrier energy in Equation 3.285, the spectral density of the noise current source in the jth section can be written as Sj ¼ 4e2 D== N jð2DEGÞ Z=Dx;

ð3:286Þ

where D== is the longitudinal diffusion constant, which can be determined by, for example, Monte Carlo simulations along with its energy dependence. The physical noise parameters can be translated to noise sources in the equivalent noise sources in the gate and drain circuits. For this, the drain–source voltage DVDS| j and the gate charge DQ | j fluctuations due to fluctuations in the current DI| j in the jth segment

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must be determined. To a first extent, the gate charge fluctuations due to the drain–source voltage variations, which are in turn caused by small current fluctuations DI|j in the jth segment, are given by CDGDVDS|j with CDG being the drain–source capacitance. The gate charge fluctuation can be expressed as DQ 0 j ¼ DQ j CDG DV DS jj :

ð3:287Þ

The spectral densities of the noise sources can be described as summation of the sources due to velocity fluctuations. In this premise, the open-circuit noise voltage hv2DS0 i and short-circuit gate noise current hi2G0 i can be determined. In the special case of carrier transit time being smaller than the period of the signal, the correlation coefficient between the open-circuit drain noise voltage and the short-circuit gate noise current, an imaginary quantity, can also be determined. The drain voltage can be converted to an equivalent noise current source at the drain and gate circuits as     2 ð3:288Þ iG ¼ hðiG0 jwCDG vDS0 Þ2 i and i2D ¼ jy22 j2 v2DS0 ; with the correlation coefficient C ¼

hðiG0 jwCDG vDS0 ÞvDS0 iy22 ;  2  2 1=2 i G iD

ð3:289Þ

where y22 ¼ gD þ jwCDG is the output admittance and y22 is its complex conjugate. Although it is very clear that resistances must be lowered to reduce the noise, the effect of intrinsic material and device properties may not be as obvious. After treating the high-frequency noise, we are now in a position to make some practical statements about the effect of the intrinsic device and materials properties on noise. The noise figure increases with increasing gate length. Noise figure, particularly at higher frequency of operation, decreases with increasing electron mobility and with increasing electron velocity. The issue of velocity is of paramount interest in GaN-based devices due to increased electron–phonon coupling and the resultant high density of hot phonons, as discussed in Section 3.9. Impediments such as these that prevent one from attaining the velocity expected of GaN would prevent achievement of the lowest possible noise in GaN devices. Although not finalized, if ballistic or even quasiballistic transport could be achieved, it would lead to reduced noise. A fully simplified noise treatment, one relating the noise figure to easily discernable device parameters, is given in the next section. 3.6.7 Treatment of Noise with FET Equivalent Circuit

Noise associated with FETs is best described in the context of an equivalent circuit wherein the contribution of noise is added to the equivalent circuit of a noiseless FETs in the form of standard circuit elements. This is accomplished by adding voltage and current sources with appropriate noise resistances added to the equivalent circuit of Figure 3.34. The equivalent circuit-based noise treatment differs from the physicsbased treatment, which utilizes physical processes such as transport, scattering,

3.6 Electronic Noise

ith Y( f) T=0K

Y( f) Tn = 0 K

(a)

Z(f) Tn = 0 K

Z(f) T=0K vth

(b) Figure 3.105 Equivalent circuit of (a) a noisy admittance Y represented by a parallel combination of a noise current source ith and noise-free admittance Y( f ); (b) a noisy impedance represented by a serial combination of a noise voltage source vth and noise-free impedance.

materials properties, and device dimensions. As an introductory material to the equivalent circuit model, let us first discuss the circuit description of noise. Electronic noise can be represented by equivalent current or voltage sources as shown in Figure 3.105. For example, the contribution by an admittance Y( f ) and the impedance Z( f ), which are at a certain nonzero temperature, can be modeled by a combination of a noise source and a noiseless passive element. Specifically, electronic noise can be described by a parallel combination of noise current source ith( f ) and a noise-free admittance Y( f ) (T ¼ 0). Similarly, noise can also be described by a series combination of a noise voltage source vth( f ) and a noise-free impedance Z( f ) (T ¼ 0). The statement T ¼ 0 is to underscore the fact that both the admittance Y( f ) and the impedance Z( f ) are noiseless. The noise present is represented by a noise equivalent temperature indicated as Tn. The aforementioned definition of the equivalent circuit and noise temperature paves the way for calculating the noise behavior of the circuit. In addition, any possible correlation between noise sources can be considered. Proceeding along with the circuit description, Figure 3.106 shows a descriptive circuit representation of a two-port network, which can be adopted to describe an FET

Figure 3.106 Equivalent circuit for the description of noise figure.

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for the purpose of defining the noise figure. The circuit consists of a noisy two-port network (equivalent circuit of an FETcontaining noise sources as well as the extrinsic elements of Figure 3.110) connected to a load impedance assumed to be noiseless (underscored by T ¼ 0 K) and noisy input impedance, which can be described by a noise voltage source and noiseless source impedance as shown in the figure. We are now ready to define the figure of merit universally used to describe the noise performance of a device, the noise figure. To express the impact of noise in a meaningful manner, the noise contribution by a noise source is described in terms of the noise-to-signal ratio at the output port and the noise-to-signal ratio at the input port. The ratio of the two quantities in the form of a logarithmic function in decibels is the noise figure and can be written as NFð f Þ ¼ 10 log10

Sn2v Sn2v0

or NFð f Þ ¼ 10 log10

Tn0 ð f Þ dB; T0

ð3:290Þ

where Sn2v represents the output noise spectrum of the noisy two-port circuit and Sn2v0 represents the output noise spectrum of the noiseless two-port circuit. For the second definition, T 0 n ð f Þ represents the output noise power reduced back to the input of the circuit (meaning it is divided by the amplifier gain in the form T 0 n ð f Þ ¼ T n =Gain) and T0 represents the input noise power generated by a fictitious input matched resistor, which is kept at room temperature (T0 assumed to be 290 K). Using the above definition, the noise figure can also be defined as the output noise equivalent noise temperature divided by a fictitious output resistor, which is kept at room temperature. A noise-free (noiseless) circuit is then described as one having an NF ¼ 0. The noise figure in a device can be minimized under optimized load and bias, and the term NFmin is used for it. The knowledge of NFmin is insufficient to describe the behavior of a two-terminal circuit. Four independent parameters, the minimum noise figure NFmin, the equivalent noise resistance Rn, and the real part GS and the imaginary part BS of the optimal generator (Zs) impedance, are necessary to describe the noise behavior of a linear two-port circuit fully. The noise figure of a noisy linear two-port circuit can be calculated utilizing an applicable equivalent circuit with one current noise source or one voltage noise source at the input circuit as shown in Figure 3.107.

Figure 3.107 A noise equivalent circuit with a noise current and a noise voltage source chained at the input terminal of a noiseless two-port circuit.

3.6 Electronic Noise

Using the equivalent circuit of Figure 3.107, one can calculate the noise figure as NF ¼ 1 þ

Snv þ jZS j2 Sni þ 2Re½ZS Svi  ; 4kB T 0 RS

ð3:291Þ

where Snv and Sni represent the voltage noise spectrum and current noise spectrum, respectively, and Svi is the cross-correlation spectrum between the noise sources vn and in, see Ref. [339] for an expanded treatment. As can be noted, the noise figure is independent of the input impedance Zin and the load impedance ZL. The minimum noise figure NFmin can be obtained by optimizing the input source impedance, which is called noise matching. It should be mentioned that noise-matching conditions are different from power-matching conditions. In addition, the noise-matching conditions, similar to the power-matching conditions, depend on the bias. Generally, the lower noise figures are achieved for lower drain currents up to the point beyond which the gain, transconductance, and possible velocity degradation may occur. The latter could be due to carrier injection into traps in buffer layers, which is enhanced as the channel is depleted by gate bias. In short, the maximum available gain may not be available. Rather an associated gain may be in effect for matching conditions leading to the minimum noise figure. The dependence of NF on matching conditions, that is, deviation from NFmin, can be described by [339, 377] NF ¼ NFmin þ

Rn jY S Y S;opt j2 GS

with Y S ¼ GS þ jBS in terms of the Y-parameters;

ð3:292Þ

and Gn jZ S ZS;opt j2 with ZS ¼ RS þ jX S and the equivalent RS noise conductance Gn ¼ Rn jY S;opt j2 in terms of the Zparameters:

NF ¼ NFmin þ

ð3:293Þ Therefore, to reiterate just four parameters, the minimum noise figure NFmin, the equivalent noise conductance Gn (or the equivalent noise resistance Rn of the transistor), and the real and the imaginary parts of the optimal source admittance, GS,opt and BS,opt, are required to fully describe the noise behavior of a linear two-port circuit. The equivalent noise resistance Rn (or the equivalent noise conductance Gn) is a measure of the sensitivity of the noise figure on deviations from the optimum source admittance or impedance leading to the minimum noise figure and can be construed as a noise figure of merit. In practice, one might have to deviate from optimum matching for noise in favor of matching for power or gain. In the microwave world, the admittance or impedance is normalized with respect to the characteristic impedance Z0 (typically 50 W) and reflection coefficients are determined, which in turn are used to determine impedance/admittance parameters on the source or the load side. In this context, the source admittance can be expressed, using Equation 3.83, as

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YS ¼

1 1GS ; Z0 1 þ GS

ð3:294Þ

where GS is the generator reflection coefficient. Similarly, YS;opt ¼

1 1GS;opt ; Z 0 1 þ GS;opt

ð3:295Þ

where GS,opt is the optimum generator reflection coefficient at the noise-matching conditions for minimum noise figure (both reflection coefficients are complex numbers). With the help of Equations 3.294 and 3.295, we can construct |YS  YS,opt|2 as jY S Y S;opt j2 ¼

jGS GS;opt j2 4 ; Z 20 j1 þ GS j2 j1 þ GS;opt j2

ð3:296Þ

with which Equation 3.292 can be rewritten as NF ¼ NFmin þ 4r n

jGS GS;opt j2 ð1jGS j2 Þj1 þ GS;opt j2

;

ð3:297Þ

where rn ¼ Rn/Z0 is the normalized equivalent noise resistance. Expanding further, the contribution to noise at the input side by the FET can be treated as voltage and current sources with an appropriate noiseless resistance added to the circuit, as shown in Figure 3.108. Gradually easing into relating the noise to the FET parameters, the thermal noise in an intrinsic FET due only to its channel can be described as 2  1 ith ðtÞ ¼ 4pkB T eff Df ; Rch

ð3:298Þ

where Teff represents the effective temperature in the channel of an FET and Rch represents the channel resistance. V DG C DG RDG,T=0 K

D′

G′ vGn IDS =gm e-jωτVGS

n

n R GS

iG

T=0 K

CGS

R GS T=0 K V GS

CDS

R DS T=0 K

S′τ Figure 3.108 High-frequency equivalent circuit of an FET including the noise sources associated with the input and the output circuits.

n

iD

S′

3.6 Electronic Noise

vDG C DG R DG:T=0K

Cimp

G′

n iimp

vGn n RGS

iG

imp

gm vDG

iDS= gm e-jωτvGS

n

D′

n

T=0K

CGS

R GS T=0 K vGS

iD

CDS R DS T=0 K

S′

R imp T=0 K S′

Figure 3.109 The equivalent circuit of an FET with the noise source depicted in Figure 3.108 and the additional noise component by any impact ionization. The gain component and the additional capacitive component due to impact ionization are also indicated.

The noise equivalent circuit of Figure 3.108 can be used to discern RF noise of an FET, which is based on a temperature noise model (TNM) [378, 379] inclusive of gate leakage, which is modeled as RDG and RGS in parallel to the intrinsic gate–drain and gate–source elements, respectively. If needed, the impact ionization effects discussed in the next section can also be added. The equivalent circuit of an FET with an additional noise source brought about by impact ionization in addition to the noise sources depicted in Figure 3.108 is shown in Figure 3.109. The impact ionization components are modeled as an additional imp voltage-controlled current source g m vDG and an RC circuit parallel to the output resistance. As noted, the current source associated with impact ionization is controlled by the voltage drop across the high-field region at the drain side of the gate and can be described by the drain–gate voltage vDG. The characteristic frequency dependence of impact ionization effects is modeled by a combination of Rim and Cim. However, for noise modeling purposes, an additional white noise source iim is added imp parallel to the current noise source g m vDG . The current associated with impact ionization can be described as [339]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 1 1 hi2im;ext i ¼ hi2im i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with w0 ¼ : ð3:299Þ R im C im 1 þ ðw=w0 Þ2 This expression describes the Lorentzian nature of the external noise source iim,ext and it is customary to attribute it to carrier generation processes, which sustain impact ionization. It should be pointed out that thermally activated carrier generation also shows a Lorentzian shape spectrum, particularly the one from deep levels, which can be neglected at high frequencies. The equivalent circuit of an FET with extrinsic elements can be obtained by adding the intrinsic elements such as gate, source, and drain resistances and inductances to the equivalent circuits shown in Figure 3.108 (without impact ionization) and in

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G

vG

LG

R G, T=0 K

D' R D, T=0 K L D

G' Intrinsic FET

vG= 4kBRGTa

S'

vD

D

vD= 4kBRDTa

R S, T=0 K C in

C out

LS vS

vS= 4kBRSTa

S

S Equivalent channel noise voltage: vG= 4kBRGSTG Equivalent output noise current:

iD= 4kBTD/RDS

Equivalent gate leakage current noise: iP = 4kBTP /RPGS Figure 3.110 The equivalent circuit of an intrinsic FET including extrinsic contributions in the input and output circuits. The Johnson noise sources associated with input, source, and drain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kB RG T a, resistances are ffidescribed bypvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vS ¼ 4kB RS T a , and vD ¼ 4kB RD T a (where Ta is the applicable active device temperature as far as the thermal noise is concerned). The equivalent channel noise voltage, equivalent

output noise current, and equivalent gate leakage current (if the gate leakage is modeled with RPGS, meaning parallel resistance between the source and the gate) can be described as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kB RGS T G , iD ¼ 4kB RDS T D , and vG ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iP ¼ 4kB RPGS T G , respectively. Various Tx depict the applicable temperatures associated with Johnson thermal noise. The gate leakage resistance is shown in Figure 3.109 as RnGS .

Figure 3.109 (with impact ionization). Doing so yields the equivalent circuit shown in Figure 3.110. 3.6.8 1/f Noise in Conjunction with GaN FETs

As mentioned previously, the low frequency noise can be upconverted to high frequencies, limiting the performance of these transistors even in the microwave range. Low-noise electronics for communications necessitates some level of knowledge with respect to the origin of the processes responsible for low-frequency noise in GaN FETs. In what follows, we discuss the theory developed by Handel with regard to quantum 1/f noise in GaN/AlGaN HFETs. First, it should be mentioned that the conventional quantum 1/f noise parameter depends on the scattering mechanisms that are in effect, such as impurity scattering and phonon scattering, both acoustic and optical. Moreover, because the gate of an FET intended for RF applications is much wider than its length, it can be construed as pseudo one-dimensional and density N can be described by N0 in terms of number of carriers per unit gate width. To give a flavor of what to expect of GaN, the aforementioned parameters have been calculated for both GaN and another well-established III–V compound semiconduc-

3.6 Electronic Noise

tor, GaAs. As will be shown subsequently, for impurity scattering in small n-type samples, the conventional quantum 1/f formula predicts a quantum 1/f parameter of 6  109 compared to 1.8  108 for impurity scattering in n-GaAs. In p-type samples, the figure for GaN is 1.5  109, to be compared with 2.4  109 in GaAs. For phonon scattering, the figures are 7.5  107 and 4.7  108 for GaN and 6.5  106 and 1.2  107 for GaAs. The conventional quantum 1/f noise spectrum in a scattering cross section s or recombination rate G is given by Sds=s ð f Þ ¼ SdG=G ð f Þ ¼ 2aA=f N Sdj=j ðf ÞM:

ð3:300Þ

Here, S( f ) is the spectral density of fractional (quantum) fluctuations in the current dj/j for a scattering or recombination cross section ds/s or in any other process rate dG/G. The coherent quantum 1/f noise spectrum in a current j is given by Sdj=j ð f Þ ¼ 2a=pf N:

ð3:301Þ

Note that a ¼ e2 =
hc ¼ 1=137 is Sommerfeld’s fine structure constant (here e,
h, and c are the electronic charge, Planck’s constant, and velocity of light in vacuum, respectively). A ¼ 2(Dv/c)2/3p is essentially the square of the vector velocity change dv, normalized to the speed of light in vacuum, of the scattered particles in the scattering, recombination, or tunneling process whose rate fluctuations are under consideration. Finally, N is the number of particles used to define the notion of current j of cross section s or of process rate G. In general, Handel’s Hooge parameter aH has both conventional and coherent components, the former of which gains prominence in small devices with carriers having higher kinetic energy p2/2m than magnetic energy LI2/2. aH

1 s aconv þ acoher 1þs 1þs

 1 4a s 2a ¼ 2ðDv=cÞ2 þ ; 1 þ s 3p 1þs p ¼

ð3:302Þ

where s ¼ 2e2/mc2N0 ¼ 2r0N0 is the coherence parameter defined below following Equation 3.310. Here, r0 ¼ e2/mc2 ¼ 2.84  1013 cm is the classical radius of the electron, n is the concentration of carriers, and N0 the number of carriers per unit length. The treatment holds for a width Z, corresponding to a cross section perpendicular to the flow of the current much smaller than r0/N0 , s 1, and for a cross section much larger than r0/N0 , s 1, provided the length L exceeds the width Z. Therefore, in very small devices, the coherent contribution would be smaller, compared to the conventional or incoherent term. For impurity or charged defect-caused scattering, we obtain therefore the conventional quantum 1/f coefficient as 

2

4a Dv 4a Dp 2 2aA ¼ ¼ ; ð3:303Þ 3p c 3p m c

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which is evaluated assuming a thermal energy 3kT/2 for the motion of the current carrying carriers. Note that (4a/3p) ¼ 3.1  103. Thus,

2

 4a Dv 4a 6kT 2aA ¼ ¼ 3p c 3p m c 2 ð3:304Þ
  T m0 ; ¼ 1:2  109 300 m and using an effective mass m ¼ 0.22m0 for electrons in n-type GaN, with m0 representing the free electron mass, we obtain 2aA|e ¼ 5.45  109. For holes with effective mass m ¼ 2m0 in p-type GaN, we obtain 2aA|h ¼ 0.6  109. For comparison, for n-type GaAs with m ¼ 0.068 m0, 2aA|e ¼ 1.8  108. For p-type GaAs with m ¼ 0.5m0, 2aA|h ¼ 2.4  109. In the case of scattering by acoustic phonons, we enter in Equation 3.303 |Dp| ¼ DE/vs, where hDEi kT is the change in energy of the acoustic phonon scattering and vs is the speed of sound, and obtain


2   4a kT 2 T m0 2 8 ¼ 3  10 : ð3:305Þ 2aA ¼  m 3p m cvs 300 K Therefore, for electrons with effective mass m ¼ 0.22m0 in n-type GaN, we obtain 2aA|e ¼ 6.74  107. For holes with effective mass m ¼ 2m0 in p-type GaN, we obtain 2aA|h ¼ 7.52  109. In the case of polar optical phonon scattering, the fractional quantum 1/f fluctuation dGof the electron scattering rate G on a polar optical phonon of momentum
hq and energy
hwq
hw0 is described by the quantum 1/f coefficient  +
*
 +
*
4a
hq 2 4a
hk 2 ¼ : ð3:306Þ 3p m c 3p m c Here, we have estimated the average quadratic momentum change of the electrons, resulting from the momentum
hq of the optical phonon to be 2ð
hkÞ2 þ 2m W 0 , because the momentum transfer to electrons of momentum
hk is known from energy and momentum conservation to be between the limits
hqmin ¼
h½k þ ðk2 þ 2m W 0 =
hÞ1=2  and
hqmax ¼
h½k þ ðk2 þ 2m W 0 =
hÞ1=2 . If we neglect the term containing W0, Equation 3.306 yields an approximate spectral density SdG=G ð f Þ ¼ Sdm=m ð f Þ ¼ SdI=I ð f Þ ¼ 108 =f N;

ð3:307Þ

where I is the source-to-drain current and N the number of carriers in the FETs channel, in the uniform channel approximation, for the conventional quantum 1/f noise part. The expected rms drain current fluctuation amplitude is thus about 108ID/fN. Both through N and through the drain current ID, this simple result depends on the applied gate voltage VG in FETs. Slightly more exact results are obtained for homogeneous samples by integrating the expression Handel’s quantum 1/f cross-correlations over the states in the conduction band [347]. The general case of the nonuniform channel is treated below in this section.

3.6 Electronic Noise

As in the case of impurity or charge-defect scattering, for electrons with effective mass m ¼ 0.22m0 in n-type GaN and using Equations 3.303 and 3.304 we obtain 2aA|e ¼ 5.45  109. Similarly, for holes with effective mass m ¼ 2m0 in p-type GaN, we obtain 2aA|h ¼ 0.6  109. For comparison, we note that for n-type GaAs with m ¼ 0.068m0 and thus 2aA|e ¼ 1.8  108 and for p-type GaAs with m ¼ 0.5m0 and 2aA|h ¼ 2.4  109. The cumulative conventional quantum 1/f coefficient and spectral density due to mobility, l fluctuations associated with electrons and holes are given by the relation X 2aAje;h ¼ ðm=mi Þ2 2aAi je;h ; ð3:308Þ i

where 2aA|e,h is the quantum 1/f coefficient for electrons or holes calculated for the scattering process i, which contributes to limiting the observed mobility according to P the relation 1=m ¼ i ð1=mi Þ. If both electrons and holes contribute to the conductivity l ¼ e(nmn þ pmp), the spectral density of conventional quantum 1/f fractional fluctuations in l is 2

Sdl=l ð f Þ ¼ 2ðe2 a=f l Þ½ðpmh Þ2 Ajh =N h þ ðnme Þ2 Ajn =N e :

ð3:309Þ

This defines the 1/f noise in any small sample, giving smaller noise than that in GaAs due to larger effective masses. Let us now include the coherent quantum 1/f noise in Equation 3.309, which doing so leads to 2

Sdl=l ð f Þ ¼ 2ðe2 a=f l Þ½ðpmh Þ2 Bjh =N h þ ðnme Þ2 Bjn =N e 

ð3:310Þ

with Bje;h ¼ ½Aje;h þ sje;h =p=½1 þ sje;h , where s ¼ N0  5.7  1013 cm is the quantum 1/f coherence parameter if the length exceeds the width, N 0 is the number of electrons per unit length (or holes, depending on the |e,h subscript) along the current path. In HFETs, the channel length LDS is much smaller than the width Z of the device. In this case, the width Z must be replaced by LDS in the calculation of the coherence parameter s to get a good approximation. In fact, for W  LDS > t, the coherence parameter is given by s pnr0tLDSln(W/2LDS), where t is the channel thickness. This explains why Z must be replaced by Z  (Z  LDS)u[Z  LDS] in general, to get a good approximation for S. Here, the unity step function (heavy side function) u[Z  LDS] ¼ 0 for Z  LDS < 0 and u[Z  LDS] ¼ 1 for Z  LDS > 0. For FETs, Z  LDS > 0 and therefore u[Z  LDS] ¼ 1. However, the case of Z  LDS < 0 is also interesting to consider for a wire or a whisker in which case the unity function is zero and Z  (Z  LDS)u[Z  LDS] ¼ Z indicating the dominance of Z. In short, the general approximate formula for the coherence parameter S, which can be applied for both Z < LDS and Z > LDS is s ¼ r0n(y)tLDS[Z  (Z  LDS)u(Z  LDS)]. Since S depends on the local concentration n, both s and aH will vary along the channel in the direction of current flow. For example, for a GaN/AlxGa1xN-doped n-channel HFET, this yields the spectral density of VDS in terms of device width Z, current Ich, thermal voltage Vth,

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Xð1 2

hðdV DS Þ if ¼ ðjIch j=f Z Deff qV th Þ ðaH =mn Þ½dX =ðX 1Þln2 X  2


 V GS V T V ; X 1 þ exp V th

X0

ð3:311Þ

where X0 and X1 are the values of X at the source (V ¼ 0) and at the drain (V ¼ VDS), while the Hooge parameter aK ¼ 2aBje, Deff ¼ qm/p is q times the effective density of states per unit area for one subband in a 2D system (it should not be confused with the diffusion constant), VT is the threshold value of VGS, and Vth ¼ kT/q is the thermal voltage. By dividing Equation 3.311 through V 2DS , we note that the denominator Z2 in Equation 3.311 allows a fractional noise power reduction proportional to 1/Z with impunity, because the q-function present in s keeps aH from increasing proportional to Z in the transition between conventional and coherent quantum 1/f noise. Finally, we mention that the kinetic energy associated with the flow of current in a sample is an incoherent sum of mv2d =2, proportional to N, for s 1. However, it is a coherent sum of LI2/2 magnetic field energies from each carrier, proportional to N2, for s  1. The quantum 1/f noise contributions behave similarly as a function of s, being proportional to N in the conventional case and to N2 in the transition region between conventional and coherent regimes. Outside the transition region, the s-parameter drops out of Equation 3.302, being either too large or too small compared to unity. Thus, incoherent addition of noise power for each carrier is observed outside the transition region. More details on the calculation of the quantum 1/f noise figure for FET-based amplifiers, which also includes the quantum 1/f noise contribution from the gate, are provided in Ref. [348]. There the reader can also find the simple expression of phase noise for a linear amplifier-based oscillator, as well as the phase noise contribution of the resonator. From the above discussion of 1/f noise based on new engineering formulae, albeit mainly theoretical considerations, we can conclude that depending on dimensions and dominant scattering mechanism(s), the 1/f noise in GaN can be an order of magnitude smaller than that for GaAs. Specifically, this difference is more accentuated in small samples where the conventional quantum 1/f dominates in which case the difference is roughly a factor of 9 in the lattice scattering-limited case and 3 in the impurity scattering-limited case. This clearly indicates that devices in which the impurity scattering is reduced such as the case in 2DEG system, the larger factor would separate the GaN-based devices from GaAs-based devices. Also, the advantage of the GaN-based system is larger in FETs than in BJT and HBT platforms. From a practitioner’s point of view, the low-frequency noise of GaN HFETs usually exhibits flicker characteristics in the form of 1/f g dependence with g, a fudging parameter, being close to 1. The dimensionless Hooge parameter aH, discussed above, as shown below, is commonly used to describe the 1/f noise. aH ¼

SI g f N; I2

ð3:312Þ

where f is the frequency, N is the number of carriers, and SI/I2 is the relative spectral density of noise.

3.6 Electronic Noise

On the experimental side, in an early report, the Hooge parameter aH has been determined for the GaN HFETs. In early investigations, this parameter was found to be approximately 102 [380], which is consistent with a piezoelectric quantum 1/f noise contribution, in which transversal phonons take the role of the soft photons as infraquanta, and the speed of light is replaced by the speed of sound in the piezoelectric analogue of Sommerfeld’s fine structure constant a. Later on, transistors with lower values of aH 104  105 have been reported [381–383] consistent with coherent and conventional quantum 1/f noise contributions. Despite the theory predicting GaN to be better than GaAs, these latter aH parameters are comparable to those for commercial GaAs FETs. Even then much larger values such as aH 150 in p-type GaN [384] and aH 57 in n-type GaN [385] films have been reported, consistent with piezoelectric quantum 1/f noise. However, an n-type thick sample with improved structural quality has shown aH 5  102 [386], which is similar to piezoelectric quantum 1/f noise results [387, 388] that depend on the spontaneous polarization, on sample size, on stress, and on the resulting changes in the speed of transversal sound waves. These figures are substantially larger than those expected of GaN and are not consistent with the low 1/f noise in GaN HFET channels that may be too small for piezoelectric quantum 1/f contributions. To account for the low 1/f noise observed in GaN HFETs in the light of measured large Hooge parameters, additional mechanisms have been proposed [389]. Levinshtein et al. [382] and Dmitriev et al. [390] suggested that the high degree of degeneracy in GaN HFETs results in the reduction of noise level where the noise is caused by the occupancy fluctuations of the localized states with an energydependent capture cross section. As far as the source of noise and its suppression are concerned, Garrido et al. [381] considered the velocity fluctuations in the channel that might be caused by the dislocations present. The very effective screening of the dislocations by the 2DEG in HFET was assumed to suppress the noise level, albeit without the benefit of a stringent treatment. It was shown that the dependence of the Hooge parameter aH on the number of carriers (ns from 5  1012 to 1  1013 cm2) in the channel follows a power law dependence where the exponent is 1 for all samples, which is relevant to dislocations. Other processes contributing to the low-frequency noise have also been suggested, among which is the one by Rumyantsev et al. [391], who reported that aH increases as the carrier concentration increases for highly doped channels (about 6  1013 cm2), while the relationship between aH and the carrier concentration obeys the same power law as above for lightly doped channels. The minimum Hooge parameter was noted to correspond to nearly the same carrier concentration where the maximum mobility is achieved. The increase of the Hooge parameter at higher concentrations has been suggested to be a result of the electron spillover from the 2DEG channel to a low-mobility parallel conducting channel [391]. In addition, the noise contribution from contacts may still be very important. It was found that the total noise power density could be reduced by one order of magnitude (at VGS ¼ 0 V) by improving the contact technology [381]. It should also be mentioned that the FET channels relying on polarization-induced mobile charge may lead to a reduced Hooge parameter, as the doped cousins would suffer from doping fluctuations and impurity scattering.

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Analyzing the magnetic field dependence of the 1/f noise measured at 300 K in magnetic fields up to 10 T and noting a strong magnetoresistance, Rumyantsev et al. [402] concluded that the dominant mechanism for the 1/f noise in HFETs is the fluctuations in the number of electrons [392]. Yet other processes relevant to slow surface states have been attributed as likely contributors to the low-frequency noise. Kochelap et al. [393] presented a theory germane to a nonhomogeneous conducting channel juxtaposed to an insulating layer together with Langevin approach wherein the fluctuations due to processes associated with trapping and detrapping of carriers are accounted for [394, 395]. Deriving general expressions for the fluctuations in the electron concentration, electric field, and the spectral density of the noise current, the authors suggested that due to the self-consistent electrostatic interaction, the noise caused by different regions of the conductive channel is spatially correlated on the length scale to the Debye screening length in the channel. The authors also modified the expression for the Hooge parameter for 1/f noise to take the presence of Coulomb interactions into account. In the vein of suggesting additional likely mechanisms for low-frequency noise, Dmitriev et al. [396] proposed a model for the 1/f noise relating it to tunneling from the two-dimensional electron gas to the tail states near the conduction band of the GaN layer. It should be pointed out that this will be less of an issue as the quality of GaN improves further. A fairly weak temperature dependence of the 1/f noise in the temperature interval between 50 and 600 K and the Hooge parameter aH in the range of 103–105 have been derived, both of which have been noted to be in agreement with experiments. As would be expected, it was noted that the noise level in GaN films depends on the structural perfection or the imperfection as the case may be [384, 386], which is also true for noise performance in GaN HFETs. The noise level for at least the early HFETs grown on sapphire substrates is one order of magnitude larger than those grown on SiC substrates [397, 398]. Figure 3.111 displays the typical frequency dependences of the relative spectral noise density of the drain current fluctuations, SI/I2, for the transistors grown on sapphire and SiC substrates [397]. Either due to the improved technology or aberrations in the early studies, Vitusevich et al. [399] later reported Hooge parameters in HFET devices on sapphire substrates that are as low as those attained on SiC substrates. The Hooge parameters of aH 103 and aH 2  104 for the ungated and gated regions of the transistor channel have been obtained by analyzing the dependence of the noise spectra–gate bias. The noise spectrum for sample Sp2, shown in Figure 3.111, has a plateau at low frequencies, which is a characteristic feature of the generation–recombination noise. Only at very low noise levels (the Hooge parameter on the order of 104) does the contribution of generation–recombination noise become significant. It was found that G–R noise for devices grown on sapphire has an activation energy DE 0.42 eV [398]. On the contrary, the temperature sensitivity for the noise level of devices grown on SiC is very weak, which is essential for high-temperature applications. Balandin et al. [400] reported DE  0.85 eV for transistors grown on sapphire and DE  0.20–0.36 eV for those grown on SiC. The trap densities for undoped and doped devices are 1.1  1016 cm3 and 7.1  1017 cm3, respectively [401]. Rumyantsev

Hooge parameter, αH

Spectral density, S II -2(dB H z-1)

3.6 Electronic Noise

-50

-70

100 Sp3

10-1

Sp2 Sp1

10-2 10-3

SiC1 SiC2

Sapphire

SiC3

10-4

SiC

10-5

-90

-110

Sp3

1/f 2 Sp2

-130

1/f

1/f SiC1

-150 100

101

102

103

104

Sp1

105

Frequency (Hz) Figure 3.111 Typical frequency dependence of the relative spectral noise density of drain current fluctuations SI/I2 for devices fabricated in wafers grown on sapphire substrates (Sp1 to Sp3) and from the wafers grown on SiC substrate (SC1), 300 K. The inset shows the Hooge parameter aH for the samples from different wafers. Wafers

Sp1 to Sp3 were grown on sapphire substrates and wafers SC1 to SC3 were grown on SiC substrates. It should be pointed out that later data indicated that the devices on sapphire and SiC were not as divergent as this [399]. Courtesy of M. S. Shur [397].

et al. [402] reported a large activation energy DE  0.8–1.0 eV for GaN HFET and MOSHFET (HFET with SiO2 whose role is prematurely likened to the gate dielectric in Si MOSFETS) grown on insulating 4H-SiC. The analysis indicates that the traps responsible for the observed G–R noise could originate in the AlGaN barrier layer, which has an estimated trap density of about 5  1016 cm3. Gate voltage dependence of the relative drain current noise spectral density can be utilized to distinguish different noise sources. Balandin [403] reported that the noise spectral density in GaN HFET is inversely proportional to the effective gate voltage in a wide range of biases, which means that the measured noise comes from the channel of the device. It was also found that the slope g of the 1/f g noise density spectrum is in the 1.0–1.3 range for all devices and decreases with the decreasing (i.e., more negative) gate bias [404]. The linear dependence of the exponent g on the gate bias can be explained by the fact that the band bending due to bias increase will change the number of effective traps and, thus, the time constants contributing to the 1/f g noise. These experimental data support the carrier–density fluctuation model for lowfrequency noise rather than mobility fluctuation model. In some studies, the experimental results seem to suggest that the 1/f noise is dominated by the channel noise rather than that originated at contacts or on the surface of the structures [381, 403, 405]. In addition, the noise spectra displayed drastically different behavior in terms of its dependence on the gate voltage in the range of low (VGt
VG
0) and high biases (VG VGt), where VGt is threshold voltage [399]. Rumyantsev et al. [406] also noted a dependence on the gate bias, reported Hooge

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parameters in the range of 2  103 to 3  103, and argued that the noise might be of bulk GaN origin and that the temperature dependence of the noise point to a weak contribution from generation–recombination centers at elevated temperatures. To make matters a little more complex, investigating power and temperature behavior of transmission line model structures, Vitusevich et al. [407] reported deviation from a classical 1/f noise and noted a correlation with nonlinearities in the current–voltage characteristics. The authors proceeded to introduce the notion of nonequilibrium 1/f noise to account for their observations. Obviously, further investigations are required to distinguish the spatial redistribution of effective noise sources in the transistor channel. Vitusevich et al. [408] extended their investigations to transmission line pattern model structures with gap spacings of 5, 10, 15, 20, and 25 mm. This analysis led the authors to conclude that the thermal noise and hot carrier noise play a minor role in the 10–100 MHz range. In the process, the authors uncovered a dominant generation–recombination noise. A correlation as to the genesis of low-frequency noise to the fluctuations of electron concentration on the first-quantum level of the triangle quantum well and the scattering of the electrons in the barrier layer was drawn. Rumyantsev et al. [405] studied the effect of gate leakage current on noise properties for two types of devices, that is, regular HFETs and MOS-HFETs. It was found that the effect depends on the level of noise in the device. At a relatively high level of 1/f noise (aH ¼ 103), the gate leakage current does not contribute to the output noise. In contrast, however, for a low level of 1/f noise (aH ¼ 105  104), it has a profound effect. The gate leakage is related to the material quality of the AlGaN and GaN epilayers in different types of HFETs [405]. As discussed earlier in this chapter and also in Volume 1, Chapter 3, the strain due to the lattice mismatch between GaN and AlxGa1xN induces an electric field and significantly changes the carrier distribution near the interface resulting from its highly piezoelectric nature. Balandin et al. [400, 409] designed two GaN/AlxGa1xN transistors grown on SiC substrates. The same sheet density in the 2D channel is obtained by different doping and Al content, that is, one has external doping and low Al content whereas the other has high Al content without external doping. The results indicate that the noise of undoped channel is much smaller (up to two orders of magnitude) than that of the doped channel device. This increase was attributed to additional defects generated due to doping, which was suggested as leading to increased G–R and flicker noise. When the dust settles, this may be related to doping fluctuations as moderate doping levels, which is certainly the case here, and lower defect concentration. 3.6.9 High-Frequency Noise in Conjunction with GaN FETs

The fundamental discussion with respect to the high-frequency noise in FETs covered in Section 3.6.6 can now be extended to the discussion in the framework of circuit representation of a noisy FET circuit. This step also involves the introduction of contributions by the extrinsic elements, mainly the resistances at the gate,

3.6 Electronic Noise

source, and drain circuit, in the form of thermal noise. However, because the device has a considerable gain, the drain resistance noise source may be neglected. Although they have been defined in the thermal (Johnson) noise section, for continuity they are given as hv2G i ¼ 4kB TRG Df and hv2S i ¼ 4kB TRS Df , see Figures 3.109 and 3.110 in the context of circuit descriptions. The noise figure, associated gain, and other noise parameters can then be calculated using the method of Rothe and Dahlke [410], which transforms a fourpole with internal noise sources to a noise-free fourpole cascaded with a preceding noise network in which all noise sources are concentrated into one noise current source and one noise voltage source, as schematically shown in Figure 3.107. The fourpole contains an equivalent noise conductance Gn, an equivalent noise resistance Rn, and a complex correlation impedance Zc, all of which can be expressed in the following expressions as Gn ¼

RS þ Rnd jZ21 j2

;

ð3:313Þ

  Z11 Zc 2  ðRS þ Rnd Þ; Rn ¼ RS þ RG þ Rng  Z21  Z c ¼ Z11 Z12

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! RS þ C Rnd Rng ; RS þ Rnd

ð3:314Þ

ð3:315Þ

where Rnd ¼ hv2D i=ð4kB TDf Þ; Rng ¼ hv2G i=ð4kB TDf Þ; C ¼ hvG vD i=½hv2G ihv2D i1=2 , and Zij are the extrinsic impedance parameters. Finally, the minimum noise figure is calculated using "
 # Rn 1=2 2 ; ð3:316Þ NFmin ¼ 1 þ 2Gn Rc þ Rc þ Gn where Rc is the real part of Zc. The optimum source resistance for minimum noise figure is Zopt ¼ ½R2c þ Rn =Gn 1=2 jX c ðagain the complexcorrelation impedance Z c ¼ Rc þ jX c Þ:

ð3:317Þ

The functional dependence of hi2D ihi2G i; C and the noise figure calculated using these parameters as outlined in Section 3.6.7 on the drain current is shown in Figure 3.112 to give the reader a feel as to how the noise figure depends on bias conditions, which govern the drain current. The drain current leading to NFmin is indicated. To reiterate, noise performance of linear two-ports can be described in terms of four parameters – the minimum noise figure, equivalent noise resistance, optimum source resistance, and optimum source reactance. Each of these noise parameters can be expressed in terms of intrinsic device parameters and extrinsic parasitics.

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j 3 Field Effect Transistors and Heterojunction Bipolar Transistors Noise figure, (dB)

NF 2

〈i D 〉

NFmin 2

〈i G 〉 C*

Drain and gate noise current (A)

560

Drain current (A) Figure 3.112 The functional dependence of hi2D ihi2G i, C, and the noise figure calculated using these parameters as outlined in this section. The drain current leading to minimum noise figure is also indicated.

The noise and gain performance of the HFET depend critically upon the device parameters (gate length, carrier mobility, carrier velocity, barrier thickness, etc.). The device optimization for noise is somewhat empirical in nature underscoring the complexity of the problem and many sources of noise. Suffice it to say that reduction of resistances, utilizing layered structures that could minimize fluctuations of any kind, such as 2DEG, which would hinder velocity fluctuations perpendicular to the general direction of current flow, reducing the transit time by scaling dimensions, and also using materials with high velocity are among the approaches that can be and have been used to achieve low-noise devices. Let us now attempt to reduce the discussion of Section 3.6.6, which focused on determining the small signal elements as well as the noise sources and any applicable correlation constants by complex simulations. Having given that analysis, it is useful to provide a practical treatment of noise, which is somewhat simplified but easier to grasp in the context of device and materials parameters. To this end, we began with expansion of the analytical treatment of Pucel et al. [375] in the determination of the open-circuit voltage hV 2DS0 i and therefore short-circuit current hi2D i ¼ hV 2DS0 ijy22 j2 sources for the drain-circuit and short-circuit noise source for the gate circuit hi2G i as [373] hi2D i ¼ 4kB TDf LG ðaZ þ bIDS Þ

hi2G i ¼

g 2D þ w2 C 2DG g m ; CGS g 2D

2kB TDf w2 C 2GS ; gm

ð3:318Þ

ð3:319Þ

The correlation coefficient can be expressed as

   iG vDS0 y22 0 : with purely imaginary component C ¼ 2 C ¼C jy22 j ðhiG ihv2DS0 iÞ1=2 

0

ð3:320Þ

3.6 Electronic Noise

The parameters have their usual meaning: Z, LG gm, gD, CGS, and CDG being the gate width, gate length, transconductance, drain conductance, gate–source capacitance, and drain–gate capacitance, respectively, and IDS is the DC drain current. Except the geometrical parameters, the rest represent the small-signal parameters. The parameters a and b are fitting parameters, which are nearly independent of geometrical and materials parameters and assume the values of a ¼ 2  105 pF cm2 and b ¼ 1.25  102 pF mA1. This would imply that these parameters remain the same for GaN-based MESFETs and HFETs. The main factor influencing the factor C0 is the gate aspect ratio meaning the gate length divided by the gate-to-channel separation of HFETs and gate length to epilayer thickness in MESFETs. Cappy et al. [373] reported this parameter to be close to 0.8 for an aspect ratio of greater than or equal to 5 and 0.7 for a ratio of about 3. By the nature of the device geometry, this factor is larger in HFETs leadings to lower noise figures. In the low-frequency regime where wCDG < gD, Equations 3.318 and 3.319 are consistent with those provided earlier by Pucel et al. [375]. To reiterate, the analytical treatment is very beneficial in that a correlation between the noise performance and device and materials parameters can be made, which is helpful for device design for noise. Let us retouch on the effect of various parameters on the noise. Beginning with the small-signal drain–gate capacitance CDG and the drain conductance gD, these two parameters come into play in determining the drain short-circuit noise current in Equation 3.318 through the term dðwÞ ¼ ðg 2D þ w2 C2DG Þ=g 2D . For frequencies less than f0 ¼ gD/(2pCDG), the d(w) term is close to unity but increases as w2 for frequencies f  f0. Recall that the goal here is to have a small drain short-circuit noise current source and therefore keeping the frequency of operation below f0 would lead to lower noise. This means that the drain feedback capacitance CDG must be minimized by device design. Increasing the output conductance gD would achieve the same end in terms of the noise figure, but is deleterious because it reduces the gain and output impedance of the device making it more difficult to impedance match. Going nearly fully practical, an empirical expression for the minimum noise figure in FETs at room temperature provided by Fukui [411], which is generally referred to as the Fukui equation, describes the dependence of noise on device and materials parameters, shown as equation (Equation 3.321) 

NFmin

 kF f 0:5 ¼ 10 log 1 þ ½g ðRG þ RS Þ dB; f t m0

ð3:321Þ

where gm0 is the intrinsic transconductance (mS), RG is the gate resistance function of device geometry (W), RS is the source resistance function of device geometry (W), and kF is an empirical fitting factor that depends on the semiconductor material used. This fitting factor can be expressed as [375]  kF ¼ 2

hi2D i 4kB Tg m Df

0:5 :

ð3:322Þ

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It would be interesting to calculate this factor using the analytical expression of Equation 3.318 with dðwÞ ¼ ðg 2D þ w2 C2DG Þ=g 2D ¼ 1, chosen for simplicity. Doing so leads to  0:5 LG kF ¼ 2 ðaZ þ bIDS Þ : ð3:323Þ CGS Recognizing the gate capacitance CGS ¼ ðeZLG Þ=t þ Cp , where e is the dielectric constant of the semiconductor under the gate (AlGaN in the case of AlGaN/GaN FET), t is the AlGaN thickness, and Cp is the gate fringe capacitance, Equation 3.323 can be rewritten as   t ðaZ þ bIDS ÞLG 0:5 : kF ¼ 2 eZ 1 þ ðC p tÞ=eZLG

ð3:324Þ

As in the case of a large aspect ratio leading to lower noise, a small gate channel distance t, which leads to large aspect ratios, is beneficial in reducing the fitting factor, therefore the noise figure of the device. In this respect also, the HFET is better than MESFET keeping other parameters the same. Substituting Equation 3.323 into Equation 3.321 leads to (  0:5 ) pffiffiffiffiffiffi LG dB: ð3:325Þ ðaZ þ bIDS ÞðRG þ RS Þ NFmin ¼ 10 log 1 þ 8pf fc Taking advantage of the average velocity in the channel through f c ð1=2pÞðv
=LG Þ, Equation 3.325 can be rewritten as (  0:5 ) 1 NFmin ¼ 10 log 1 þ 4pLG f
ðaZ þ bIDS ÞðRG þ RS Þ dB: ð3:326Þ v This expression is a qualitative one linking the minimum noise figure to device parameter where the effect of gate and source resistance is prominently displayed. In another approach, being relevant to the Fukui equation, Delegebeaudeauf et al. [412] developed a relationship of kF as a function of the optimum bias current Iopt.  0:5 I opt kF ¼ 2 ; ð3:327Þ E C LG g m with the cutoff frequency estimated as fT ¼vsat/2pL, the NFmin can be expressed as (  0:5 ) 4pf LG I opt ðRG þ RS Þ dB: ð3:328Þ NFmin ¼ 10 log 1 þ
v E C LG The work by Oxley [413] indicates that the above model predictions for minimum noise figure agree well with experimental data up to about 26 GHz in GaN HFETs. However, the discrepancy between predictions and experiment increases with increasing frequency because Fukui model does not take into account the

3.7 Dielectrics for Passivation Purposes or Gate Leakage Reduction Table 3.6 High-frequency noise characterization of GaN HFETs.

Reference

Gate length (mm)

Drain bias (V)

Gate bias (V)

f (GHz)

Fmin (dB)

Associated gain (dB)

Nguyen et al. [415] Ping et al. [414] Deng et al. [416] Lu et al. [417]

0.15 0.25 2 0.12

6 10 10 10

3 4 1.5 4.8

10 5/10 2 8

0.60 0.77/1.06 0.58 0.53

13.5 14/12 14.13 12.1

gate-to-drain capacitances and the higher order frequency terms. It was estimated that a very low-noise figures of 0.4 dB at 12 GHz might be feasible for a GaN HFET with an improved ft, which depends on material quality. Continuing on with the experimental side further, preliminary investigations of the high-frequency noise performance have shown that GaN HFETs have respectable microwave noise properties that are nearly comparable to those of AlGaAs/GaAs HFETs. Table 3.6 displays recent studies of the microwave noise performance for GaN HFETs. These encouraging results serve to motivate further investigations. Figure 3.113 shows dependence of the minimum noise figure and associated power gain on (a) frequency, (b) temperature, and (c) drain current [414]. Lu et al. [418] also studied the effect of surface passivation on high-frequency GaN HFET with 0.25 mm gate length. Although the gate leakage current was smaller after passivation, the noise measurements after passivation showed that the devices exhibited about 0.2–0.25 dB increase in Fmin. This is mainly due to the 1–1.5 dB decrease of associated power gain attributed to the increase of CGS and CGD. It was concluded that the effect on microwave noise performance is a combination of effects of lower gate leakage current and higher surface dielectric constant.

3.7 Dielectrics for Passivation Purposes or Gate Leakage Reduction

Owing to somewhat defective nature of GaN and surface states, Schottky barriers with very low leakage, particularly at high junction temperatures, are difficult to attain on the one hand and drain current lag, discussed in Section 3.5.5, is problematic on the other. The beneficial effects of surface passivation particularly with Si3N4 on the elimination of drain current lag, caused by charge injection from the gate to the surface or the near surface region of the AlGaN barrier, are treated in some detail in Section 3.5.5.4.3 already. We expand the discussion here to consider various oxides and their effect on FET properties. In addition, any enhancement of gate turn-on voltage, on top of leakage current reduction with dielectric insertion, would translate to increased gate RF voltage swings that can be applied to the gate as well as enhancing reliability. Dielectric insertion under the gate would also allow an FET-based amplifier gate to be driven fully that would lead to larger output power levels when and if RF drain efficiency is

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18

2.5

15

2.0

12

1.5

9

1.0

6

0.5

3

0.0

0 4

6

8

10

12

16

14

Associated gain (dB)

Fmin(dB)

564

18

Frequency (GHz)

(a) 10

Fmin(dB)

4

8 3 6 2 4 1

2

Associated gain (dB)

12

5

0

0

-75

-25

25

75

125

175

225

Temperature (oC) 6

12

5

10

4

8

3

6

2

4

1

2

0

0

0.15

0.3

0.35

0.6

Associated gain (dB)

Fmin(dB)

(b)

0 0.75 900

Normalized drain current ( A mm-1)

(c) Figure 3.113 Dependence of the minimum noise figure (Fmin) and associated power gain for a 0.25 mm gate GaN-based device on (a) frequency, (b) temperature (at 10 GHz, for drain and gate biases of 10 and 4 V, respectively), and (c) drain current (at 10 GHz, for a drain bias of 10 V). Courtesy of M. S. Shur [414].

not limited by anomalous behaviors such as the drain current, which lags behind the input frequency. Furthermore, dielectric insertion would also limit electron tunneling onto the surface state when the gate is reverse-biased, again lowering current lag. Overall, any reduction in gate current, particularly for high-power operation, would lead to better reliability as well [318, 319, 324]. Dielectrics, of which there are many candidates, can be used to mitigate some if not all of these issues including addressing gate leakage and stability issue. Among the dielectric candidates AlN, if a high-resistivity and extremely low-defect variety can be obtained, Ga2O3 (Gd2O3), MgO, Sc2O3, and SiO2 are a few that have been attempted for gate dielectric, passivation, or both. In a similar vein, to overcome thermal stability issues associated

3.7 Dielectrics for Passivation Purposes or Gate Leakage Reduction

with Pt gates, IrO2 gate contact have been employed to suppress Pt diffusion into the underlying AlGaN layer [419]. A depletion-mode GaN MOSFETusing Ga2O3(Gd2O3) gate dielectrics has been reported, with a maximum extrinsic transconductance of 15 mS mm1 and fmax of 10.3 GHz [420]. Although interface trap/defect density values, DIT < 1011 cm2 have been achieved for SiO2/GaN MOS diodes, the measurement method of the interface state density is just as important as the density itself and care must be taken to ensure that interface state density is assessed as accurately as possible. In addition, not only the minimum of the interface state density but also its energy dispersion is important. SiO2 has also been used for an AlGaN/GaN heterostructure FET [421]. As alluded to above, one persistently reported problem for GaN HFETs is that the RF power obtained is much lower than expected from the DC drain characteristics. In other words, RF drain efficiencies are much lower than what one would extrapolate from DC drain efficiency. This is often due to anomalies such as drain current collapse, kinks, and current lag. As mentioned in Section 3.5.5, several mechanisms have been put forth. Among them are surface states between the gate and drain, which deplete the channel in this region with time constants that are longer compared to the speed of signal or traps in the buffer layer. Several studies have shown that the use of SiNx passivation layers can be effective to some extent in reducing the effects of surface states and perhaps the associated current lag, as well as increasing the expected breakdown voltage by passivating the surface states [422]. One purported drawback of typical plasma enhanced chemical vapor deposited SiNx is the high hydrogen content, which could migrate into the GaN or the gate metallization. Alternative candidates for HFET passivation are MgO [423] and Sc2O3 [424, 425], which have received some attention and are also applicable as gate dielectrics for depletion mode GaN-based HFETs. These materials have larger bandgaps [426], as shown in Table 3.7, than, for example, Gd2O3 (5.3 eV) and smaller lattice mismatches to GaN (6.5% for MgO, 9.2% for Sc2O3 versus 20% for Gd2O3). It has been reported that layers of these dielectrics, which can be deposited by oxide molecular beam epitaxy (OMBE) and laser-pulsed deposition (LPD) techniques on AlGaN/GaN HFETs can be effective in mitigating the gate lag response encountered in unpassivated devices. Despite some degree of success in reducing the drain current lag or perhaps also other anomalies, the crystalline varieties of these oxides do not structurally match GaN, not only in terms of the lattice parameter but also in terms of lattice structure. However, a ZnO derivative containing Mg merely to limit the wurtzitic to cubic transition would be much better suited. The Mg concentration can be in the vicinity of about 40% while still maintaining the wurtzitic structure with a bandgap near 5 eV. Drawbacks are that the bandgap is not as large as for some of the others listed in Table 3.7 and that very high-quality films are required to provide an interface with low density of interface defects and high-resistivity bulk to avoid a parallel conduction path shunting the device in addition to complications with oxidation of GaN or AlGaN surface layer. The latter can be avoided with device topology in that a thin gap can be left between the gate metal and oxide and the repeat of the same with the drain contact. However, doing so leaves a thin sliver of the semiconductor unpassivated.

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Table 3.7 Properties of various passivation and gate dielectrics

used in GaN-based HFETs (in part based on Ref. [426]). GaNa Structure Lattice constant (Å) Atomic spacing in the (1 1 1) plane Mismatch to GaN (%) TMP (K) Bandgap (eV) Electron affinity (eV) Work function (eV) Dielectric constant

SiO2b-d Si3N4d AlNe-g

W A 3.1893

A

2800 3.4 3.4

1900 9 0.9

2173 5

9.5

3.9

7.5

GGGh Gd2O3i,j Sc2O3k

MgOl,m

W or A A 3.113

B 0.813 3.828

B 9.845 3.4807

N 4.2112 2.978

2.3 3500 2023 6.2 4.7 0–2.9 0.9–1.2 8.5 14.2

20.1 2668 5.3 0.6 2.1–3.3 11.4

9.2 2678 6.3 3 4 14–14.5

6.5 3073 8 0.7 3.1–4.4 9.8

GGG, gadolinium gallium garnet; W, wurtzite; A, amorphous; B, bixbyite; N, NaCl (rock salt). a Refs [427, 428]. b Ref. [429]. c Ref. [430]. d Ref. [431]. e Ref. [432]. f Ref. [433]. g Ref. [434]. h Ref. [435]. i Ref. [436]. j Ref. [437]. k Ref. [438]. l Ref. [439]. m Ref. [440].

With the exception of the amorphous dielectrics, namely, SiO2 and Si3N4, the oxides listed in Table 3.7 can be grown with OMBE. While the cells’ temperatures vary depending on the partial pressure of the metal involved, the substrate temperature is generally low, typically in the vicinity of 100  C. While the metal can simply be evaporated and or provided by organometallic sources when available, oxygen is provided by some plasma source such as RF or ECR, as molecular oxygen does not lend itself to high-quality oxides. The SiO2 and Si3N4 films can be deposited by plasma enhanced UHVCVD, as the deposition temperatures can be kept low enough to avoid surface decomposition of GaN while providing high-quality dielectrics. MIS structures have been prepared in an effort to characterize the electrical properties of the oxide semiconductor complex. The AC conductance measurements on the MIS structures reported by Pearton et al. [441] showed breakdown fields of 1.2 and 1.5 MV cm1 with MgO and Sc2O3, respectively, at a current density of 1 mA cm2. The capacitance–voltage traces went from accumulation to deep depletion at 25  C for both types of diodes. The relative dielectric constants from the accumulation capacitance were deducted to be 10.5 and 14.2 for MgO and Sc2O3, which are in reasonable agreement with tabulated values of 9.8 for MgO and 14–14.5 for Sc2O3. The conductance method yielded interface state densities of about 2  1011 cm2 eV1 and 5  1011 cm2 eV1, respectively. Another benefit of these oxides is that by

3.7 Dielectrics for Passivation Purposes or Gate Leakage Reduction (a)

Normalized, IDS(%)

100 80 DC mode Pulse mode

60 40

VDS = VDS DC VG switches from -5 to 0 V frequency = 0.1 MHz Duty cycle = 10%

20 0

0

2

4

6

8

10

VDS (V) (b)

Normalized, IDS (A)

100 80 60

VDS = 3 V DC VG switches from -5 to VG frequency = 0.1MHz Duty cycle = 10% DC mode Pulse mode

40 20 0 -5

-4

-3

-2

-1

0

1

V G (V) Figure 3.114 Gate lag measurements on an unpassivated AlGaN/ GaN HFET. (a) VG is switched from 5 to 0 V at 0.1 MHz, while shown in (b) is that for the case when the gate bias is scanned in the reverse direction from 5 V to the value shown on the x-axis. Courtesy of S. J. Pearton [441].

the nature of the deposition technique employed, they are to a first extent free of H, which is known to migrate under field or temperature stress leading to instabilities [441]. The dielectrics deposited using plasma-enhanced processes do contain H as it is an integral part of the gas chemistry employed. The drain current lag measurements on HFETs, that is, the lack of ability for the drain current to follow the gate voltage at high frequencies, are prevalent in unpassivated devices as shown in Figure 3.114, which shows the normalized DC drain current as well as the drain current in response to a gate pulse of 5 V to 0 at 0.1 MHz with 10% duty cycle and also from 0 to 5 V with VDS held constant. The differences in drain current lag between DC and pulsed drain currents are consistent with the premise that surface traps deplete the channel in the region between the gate and the drain terminals. HFETs with Sc2O3 passivation showed strikingly different results to the same gate pulse sweep, as shown in Figure 3.115. A comparison of the data from MgO passivation shows the latter was somewhat more effective in increasing IDS, which we suggest is due to an increase in positive charge at the MgO/GaN (or MgO/AlGaN) interface resulting in an increase in effective sheet carrier density in the channel.

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(a) 120

Normalized, IDS(%)

100 80 60 40 DC mode Pulse mode

20 0

0

2

VDS = VDS DC VG switches from -5 to 0 V frequency = 0.1 MHz Duty cycle = 10%

4

6

8

10

V DS (V)

Normalized, IDS(%)

100 80

VDS = 3 V DC VG switches from -5 to VG frequency = 0.1 MHz Duty cycle = 10%

60 40

DC mode Pulse mode

20 0

-4

-2

0

V G (V) Figure 3.115 (a) Gate lag measurements on MgO-passivated AlGaN/GaN HFETs. At the top, VG is switched from 5 to 0 V, while at the bottom it is switched from 5 V to the value shown on the x-axis. The degradation of the current at high voltages, negative slope, is generally attributed to thermal issues. The positive slope observed after

the oxide passivation is a result of additional current path, which apparently is induced by the oxide if one assumes that oxide itself is not conductive; (b) is the same for Sc2O3 passivation. The negative output resistance seen in the case of MgO passivation is not present in this case. Courtesy of S. J. Pearton [441].

Future work should focus on large signal measurements in which we would expect to observe large increases in saturated power and power-added efficiency. Let us now turn our attention to gate dielectrics and issues related to gate leakage and thermal stability. For a variety of reasons, including reduced gate leakage, improved drain breakdown voltage, increased gate bias that can be applied and therefore increased drain current, and enhanced device linearity, dielectrics have been placed underneath the gate metallization as in MOS-like devices [442, 443]. The use of any dielectric and at least in theory large bandgap AlN with insulating properties is construed here as MIS- or MOS-like device and a brief discussion is given. The SiO2 MIS-like devices have not been as successful as those using Si3N4 as the gate dielectric, an exception being a series of reports from one group [444]. AlN insertion layers under the gate have been proposed along with the some variants in vertical structure of the heterolayers and simulated for a comparative analysis of various designs [445, 446]. Somewhat of a deviation is the use of photoanodically grown Ga2O3 as gate dielectrics for GaN-based MOSFETs [442]. The Ga2O3 layers were produced using aqueous solutions of KOH. The I–V characteristics of MOS capacitors showed insulating behavior of the oxide layers and C–V measurements

3.7 Dielectrics for Passivation Purposes or Gate Leakage Reduction (b)

Normalized, I DS (%)

100 80 60 DC mode Pulse mode

40

VDS = VDS DC VG switches from -5 to 0V frequency = 0.1 MHz Duty cycle = 10%

20 0

0

2

4

6

8

10

V DS (V)

Normalized, I DS (%)

100 80 60

VDS = 3 V DC VG switches from -5 to V G frequency = 0.1 MHz Duty cycle = 10%

DC mode Pulse mode

40 20 0

-4

-2

0

V G (V) Figure 3.115 (Continued).

indicated a small density of states at the oxide/GaN interface. We should also make it clear that there is no comparison to Si-based MOS as the attempts in MIS-like nitride devices have focused normally on devices with no attention paid as to what portion of the bandgap the Fermi level can be made to sweep. A side benefit of Si3N4 dielectric is that it also passivates the gate–drain region of the FET, which has been shown to reduce the current lag after a bias stress [447]. Collapse-free operation of a double-heterostructure MIS-like FET with SiO2 gate isolation has been reported [444], inconsistent with other reports, which indicate Si3N4 being beneficial but not SiO2 [448]. The current collapse/lag studied by bias stressing a Si3N4 insulated device, wherein for a constant drain bias VDS ¼ 10 V the gate VGS was swept from 5 V to þ 1 V and back to 5 V in 7 s, revealed no discernable degradation whereas the standard one showed a considerable amount of degradation [449]. These results indicate that Si3N4 film is effective not only in suppressing the gate leakage current but also in passivating the device surface, which contributes to the suppression of the current collapse [450]. As discussed in Section 3.5.5.4.3, this is most likely due to a reduction if not elimination of the surface states, which capture electrons injected from the gate electrode and are passivated by the Si3N4 film. While the characteristics of Si3N4 for passivating the exposed gate–drain and also the gate–source regions, using it as a dielectric below the gate metal for reduction of gate current and electron injection onto the surface, its relatively small dielectric constant reduces the transconductance. In addition, the thermal mismatch between

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Si3N4 and GaN could be problematic in high-power operation, exacerbated by the piezoelectric nature of GaN. Another oxide such as Ta2O5 with its large dielectric constant (er ¼ 25) and nearly thermal match with GaN is an attractive dielectric to be used between the gate metal and the semiconductors. Kikkawa et al. [451] employed a scheme whereby Si3N4 is used as surface passivant while Ta2O5 is used for the gate dielectric. To accomplish this, one can deposit Si3N4 between the source and the drain region and open a gate window down to the semiconductors by etching. The Ta2O5 can then be deposited between the source and drain followed by gate metal deposition on Ta2O5 in the recessed region. Being a high dielectric constant material, Ta2O5 does not adversely affect the transconductance all that much while providing the insulating properties for low gate leakage and also handling positive RF swings of the gate bias better. The Ta2O5 MIS HFETexhibited small current drop, a transconductance of 200 mS mm1, and very low gate leakage current with a breakdown voltage of 400 V. On-wafer load-pull measurements for the 1 mm devices show an output power of 9.4 W mm1 and a linear gain of 23.5 dB with a PAE of 62% at VDS of 70 Vat 2 GHz. In addition, both a high output power over 100 W and high gain of 16 dB were attained at 2.5 GHz for a large periphery MIS HFET amplifier. As expected, the insulating gate HFETs have been reported to exhibit gate leakage currents some four–six orders of magnitude lower than standard FETs [444]. Another side benefit of these devices is that the drain current versus gate bias characteristics are made more linear. Thus, higher order harmonic generation is hampered somewhat. One would also expect that the intermodulation cross talk would also be reduced in multitone measurements. A comparison of the second harmonic power versus the input RF power measured at 2 GHz (CW) for an MIS-like and conventional HFET fabricated on the same wafer showed the relative power of the second harmonic to be approximately 8 dB lower in the MIS-like device [444]. Yet another benefit of Si3N4 insulated gate device is the reduction in the low-frequency noise compared to that of the conventional devices, which may be attributable to passivating the gate–drain region [449].

3.8 Heat Dissipation and Junction Temperature

Even after many years of efforts in the materials and device arena, which produced high-performance devices, one once again becomes painfully aware of the thermal wall. Nowhere else is this drama played as ostentatiously as in the mobile technology where cooling (heat dissipation option) is limited. Ideally, a material that can handle very high temperatures and one which has a high thermal conductivity is well suited. The junction temperature can then be allowed to be set high compared to the case of temperature-facilitating heat removal, again with the help of good thermal conductivity material between the junction and the case. The nature of high-power devices is such that the absolute maximum performance extraction is coveted and as such one needs to have some fundamental understanding of heat dissipation. Owing to its enormous importance, four-dimensional models (x, y, z for spatial and T for

3.8 Heat Dissipation and Junction Temperature

temperature) have been developed to cope with power devices, mainly in Si devices and mainly for BJTs as they represent the early power devices. The above discussion, particularly in relation to FETs, assumes that heat, whose genesis lies in high electric field and hot electrons, is dissipated through LA phonons without considering the hot electron to LA phonon conversion. While in materials such as Si and to a lesser extent GaAs, this pathway is descriptive, in GaN strong electron–phonon coupling causes the power dissipation to take the route of hot electrons to LO phonons to LA phonons and then to the case. The thermal conductivity as measured and reported of any material is a measure of the efficiency of the last leg of the aforementioned journey. In GaN, the LO phonon to LA phonon decay could be the limiting factor (see Section 3.9 for a detailed discussion). Therefore, power dissipation without consideration of the process prior to LA phonon generation is not representative of the actual picture. The junction temperature plays a role in device characteristics, particularly in large-signal operation, which is discussed in Section 3.3.2.2. In FETs, the heat can be removed from the junction vertically to the case or also through a combination of lateral and vertical heat diffusion, the former needing only a one-dimensional treatment, whereas the latter requiring full three-dimensional spatial treatment. The proximity of gate fingers to each other, the length of gate finger, and their distance from the case are important device parameters to contend with. For example, gates near the center of the chip run higher in temperature compared to those near the periphery. Luckily, negative temperature coefficients associated with mobility and velocity help prevent thermal runaway, but still the thermal wall must be dealt with. Serendipitously, while the parameters change, the treatment of thermal management is transferable from one material to the next. Generally, treatment begins with a region generating the heat, and models are developed for its dissipation. The temperature of the heat source is determined by the dissipated DC power minus the RF power removed. This brings the device efficiency to the forefront as higher the efficiency, the less the heat that must be dissipated. Specific to GaN HFETs, measurements to assess the temperature of a device may be predictors of lifetimes. Infrared (IR) and micro-Raman thermal mapping techniques have been used to deduce temperatures of the devices with micrometer spatial resolution. IR measurements on TLM structures have also been used to measure temperature but micro-Raman showed even higher temperatures. Finite element analysis (FEA) method was then used to model the FET (using a pure thermal as opposed to an electrothermal model), and the linewidth and shift of the micro-Raman signal correlated to FEA results but underestimated the measurements (stokes– antistokes method also correlates with the linewidth method). Two-dimensional electric field corrections have been shown to improve fits [452]. Finally, an Arrhenius plot yielded the mean time to failure and the authors’ claim that by putting the same power into a device at DC, RF failure can be “measured” using DC biases. These techniques yield lifetimes more than 1000 h. The fundamental questions regarding the failure mechanism are discussed in detail in Section 3.11. Bandic et al. [256] applied what is known about heat dissipation to nitride-based structures. It should be mentioned that the models are not FET or HBT sensitive for the same kind of device layout in terms of the gate fingers dimensions and density for

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the former versus the base finger dimensions and density in the latter. The maximum current density or power density is determined by the ability of the device together with its housing/case to dissipate the heat while keeping the junction temperature somewhat lower than the maximum allowed temperature. In a one-dimensional problem where the temperature is the only variable, the difference between the device and the ambient temperature under steady-state conditions is determined by the product of the total device to ambient thermal resistivity and the total power that must be dissipated by the device.
 W DT ¼ T j T ca ¼ ð3:329Þ þ qca PD ; kA where DT is the temperature difference between the maximum junction temperature Tj and the case temperature Tca, W is the distance from the junction to the case, k is the thermal conductivity of the semiconductor material, A is the area of hot source, assumed to be spatially uniform to allow the use of 1D models, qca is thermal resistance of the case to the ambient, and finally PD is the dissipated electrical power, which is represented by the DC power-in and RF power-out. Note that in terms of electrical circuit equivalency, temperature is analogous to voltage and power is analogous to current. Therefore, the (W/kA, with A being the area) term represents the thermal resistance of semiconductor to the case and qca represents the thermal resistance of the case to the ambient. Implicit in Equation 3.329 is that heat dissipation is only from the junction to the case followed by radiation to the ambient unless other cooling mechanisms are employed. A cartoon representing a 1D thermal heat dissipation with its electrical equivalent circuits as constructed by Bandic et al. [256] is shown in Figure 3.116. To reiterate, a convenient technique for measuring not only the average temperature but also the spatial distribution of temperature is micro-Raman spectroscopy. This technique utilizes the change in vibrational mode frequency due to the temperature. In GaN FETs, the heat is generated by electric field and ensuing hot E1 LO phonons, which in turn decay to acoustic phonons that help diffuse the heat. Hot phonon and related decay issues are discussed in Section 3.9 in sufficient Tj A=area PD

W/ κA W

Tj

θca T0

P0 T0

Figure 3.116 A schematic representation along with the electrical circuit equivalent for thermal dissipation described in Equation 3.329. Temperature could be construed as potential (voltage) and the dissipated power as a current in the realm of electrical circuits. A typical value for the case to ambient thermal resistivity is 1 K cm2 W1.

3.8 Heat Dissipation and Junction Temperature

detail. This method has been employed by Ohno et al. [453, 454] to measure the spatial (lateral on the surface) distribution of temperature. The temperature was estimated from the Raman shift of E2 phonons of GaN. The spatial and temperature resolutions of the micro-Raman system were 1 mm and 10 K, respectively. For a DC power dissipation of 248 mW at a drain voltage of 40 V, a peak temperature of 428 K was observed at the gate edge on the drain side at the center of the channel. As expected, the position of the highest temperature is at the high-field region at the gate edge on the drain side. The Raman shift measured on the device was converted to temperature using the following equation proposed by Cui et al. [455]: wðTÞ ¼ w0 

A ; expðB
hw0 =kTÞ1

ð3:330Þ

where w0 ¼ 571.89  0.01 cm1, A ¼ 20.52  0.47 cm1, and B ¼ 1.04  0.01 cm1 for the HFET investigated by Ohno et al. [453]. The temperature error is less than 10 K at 300 K. Figure 3.117 illustrates the surface temperature distribution in an AlGaN/GaN HFET under static bias characterized by VDS, VGS, and ID values of 40 V, 0 V, and 6.2 mA, respectively, corresponding to a power dissipation of 248 mW. An image of the top view of the device is superimposed on the temperature distribution to relate the temperature data to the spatial features of the device. As expected, the hottest section of the device is at the gate edge toward the drain where the electric field is the highest as manifested by the bright area. Figure 3.118 shows the temperature distribution between the source and the drain electrodes (x-direction in Figure 3.117) at the channel center (y ¼ 0) at various VDS values from 0 to 40 V. Naturally, the temperature no matter where it is measured increases with dissipated power governed by VDS. The peak temperature of 428 K was observed at the gate edge on

Figure 3.117 Temperature distribution in an AlGaN/GaN HFET at VDS ¼ 40 V and VGS ¼ 0 V. The region at the edge of the drain side of the gate metal is at the highest temperature. Courtesy of T. Mizutani and Y. Ohno [453].

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S

D

G

440 V DS(V)

420

Temperature (K)

40 400 380

30

360 340

20

320

10

300

0

280 -4

-3

-2

-1

0

1

2

3

4

x (µm) Figure 3.118 Temperature distribution between the source and the drain electrodes taken at the center of the channel in reference to the direction along the longer dimension of the channel. Note that the gate metal blocks the phonons emitted preventing data collection. Note the region at the edge of the drain side of the gate metal is at the highest temperature. Courtesy of T. Mizutani and Y. Ohno [453].

the drain side at VDS ¼ 40 V. This is where the field, therefore the heat dissipation, is the largest as commonly known in FETs of any kind [456]. Figure 3.119 illustrates the temperature distribution along the direction perpendicular to the channel (y-direction), shown in Figure 3.118, at the region between the drain–gate electrodes (x ¼ 2 mm). The peak temperature was observed at the channel center (y ¼ 0), owing to heat dissipation laterally at the ends of the gate/channel. The figure indicates the temperature at the channel edges to be lower by about 30 K than the center of the channel corresponding to y ¼ 0. At a distance some 40 mm away from the edge of 440

Temperature (K)

420 400 380 360

Channel

340 320 300

-20

-10

0

10

20

30

40

50

y (µm) Figure 3.119 Temperature distribution along the gate perpendicular to the source–drain direction indicating that the temperature is lower near the edges of the device because of lateral heat dissipation. Courtesy of T. Mizutani and Y. Ohno [453].

3.8 Heat Dissipation and Junction Temperature 500

Temperature (K)

(a) 450

G D–

(3 5

0K

400

)

D–

G(

RT

)

S–G (

350

RT )

n Simulatio 300 0

50

100

150

200

250

Power dissipation (mW) Thermal resistance (K W-1)

700

(b) 35 0

600 500

K R oo

m m te

t ur pera

e

400 300

Simulation 200 0

50

100

150

200

250

Power dissipation (mW) Figure 3.120 Measured temperature for drain–gate (D–G) region, depicted by filled circles through the solid line, and the source and gate (S–G) region, depicted by open circles through the broken line for a room-temperature heat sink temperature versus the dissipated power. The measured junction temperature at the drain edge of the gate for a case temperature

of 350 K versus the dissipated power is also shown. (b) Thermal resistance calculated from the measured junction temperature at the edge of the gate on the drain side for room temperature and 350 K heat sink as a function of power dissipation. Courtesy of T. Mizutani and Y. Ohno [453].

the channel (y ¼ 50 mm as y ¼ 0 represents the center), the temperature dropped to nearly the room temperature. The dependence of the near-surface temperature, which can be construed as the junction temperature considering the laser absorption depth, on the power dissipation is shown in Figure 3.120 by filled circles through the solid line for the drain and gate (D–G) region and by crosses through the broken line for the source and gate (S–G) region. Also shown is the measured junction temperature at the drain edge of the gate for a case temperature of 350 K versus the dissipated power. A complete simulation would take into account the accurate thermal conductivities of all the layers between the substrate and the junction as well as an accurate modeling of the interfaces between the various layers and that between the buffer layer and the substrate. To simplify the picture so that it can be manageable, Ohno et al. [453] used a two-layer model wherein all the nitride layers were lumped into one 3 mm thick GaN layer and a 430 mm thick sapphire substrate, as shown in Figure 3.121. The calculated

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Hot region

z y

GaN

x Sapphire substrate

Figure 3.121 A two-layer model wherein all the nitride layers were lumped into one 3 mm thick GaN layer and a 430 mm thick sapphire substrate used for the 3D spatial thermal simulations. The volumetric region of 0.1  20  0.1 mm3 on the surface of the GaN layer represents the heat source. Courtesy of T. Mizutani and Y. Ohno [453].

junction temperature using a three-dimensional (plus temperature) thermal simulator is also shown by broken lines in Figure 3.120. In addition, constant thermal conductivities of 130 W m1 K1, which had been the long-standing figure with the updated figure for high-quality GaN being 230, and 24 W m1 K1were used for GaN and sapphire, respectively. Furthermore, a volumetric region of 0.1  20  0.1 mm3 was introduced on the surface of the GaN layer as the heat source. The simulated temperature was averaged within a volume of 1 mm3, governed by the resolution of the Raman apparatus used. The measured temperatures in both the source–gate and the gate–drain regions are higher than the simulations, and the dispersion gets larger with increasing dissipated power. Moreover, the measured temperature increased superlinearly with power dissipation, whereas the calculated temperature was proportional to the power dissipation as expected from the use of a constant thermal resistance of 215 K W1 (dominated by sapphire considering its absolute thickness and relative thickness in relation to the composite GaN layer). Because the thermal resistance is dominated by the sapphire substrate in this model, the use of the updated GaN thermal conductivity would not represent any consequential change. However, using SiC substrate would reduce the thermal resistance by more than an order of magnitude to near 10 K W1. For the small dissipated power of 250 mW, the first term on the right-hand side of Equation 3.329 would be reduced by nearly a factor of 20, perhaps leaving the radiation component, which is the second term on the right-hand side, as the dominant figure. Using the measured temperature adjacent to the drain side of the gate electrode for a case (heat sink) at room temperature, Ohno et al. [453] calculated the thermal resistance versus dissipated power and noted an increase from 200 to 500 K W1 with increasing power dissipation to 250 mW, as shown by filled circles in Figure 3.120b. When a case (heat sink) temperature of 350 K was used, the thermal resistance increased from 350 K W1 for nearly no dissipation to 700 K W1 for 250 mW power dissipation. The increase in measured thermal resistance for high-power dissipations cannot be attributed to an increase in the heat sink temperature [457] because the temperature outside the channel region dropped to room temperature some

3.9 Hot Phonon Effects

50 mm away from the channel center, as shown in Figure 3.119. The case temperature dependence and the thermal resistance change, as shown in Figure 3.120b, and the superlinearly increasing measured junction temperature with power dissipation are all a clear sign of increasing thermal resistance with temperature. Equation 3.232 can be used to estimate the thermal conductivity of sapphire versus temperature. For example at 600 K, the thermal conductivity of sapphire would reduce by about 50%. This would reduce the discrepancy between the predictions and the experimental values. Another practical issue that should be considered is that closer the heat source to the heat sink, the better the heat dissipation. To this end, the sapphire substrate thickness can be reduced from 430 to some 200 mm, halving the thermal resistance and thus the temperature differential between the heat source and the heat sink. Another option that is available is flip-chip mount in which case the device is mounted upside down on, for example, ceramic-grade AlN, which has thermal conductivity three times better than that of the sapphire. To accomplish this, patterns that match the pad patterns of the device are produced on the ceramic mount with additional extension for accessing the device, which is then mounted upside down. As discussed many times already, the layers can be grown on high-resistivity SiC substrates or better yet GaN substrates which are becoming available, which have high thermal conductivity with the added benefit of improved layer quality. Heat removal is a critical process in power FETs (it is also so in high-density smallsized devices that are pushed to higher current and/or voltage levels). In GaN, hot spots take on a special meaning because of its pyroelectric nature as it would cause dynamic polarization. The hot strip by the gate in the gate–drain region is particularly susceptible in that as the device current/voltage is increased for power performance, an accompanying degradation due to high field coupled with local defects as well as pyro- and piezoelectric effects (additional strain induced due to dissimilarities between the gate metal and the semiconductor) would limit the device performance and degrade longevity. We should also note that the above heat dissipation is at the device level and skips to a large extent the fundamentals of how the electric power is converted to heat via hot electrons and hot phonons and how they in turn decay to LA phonons that in turn transmit the dissipated power to the thermal bath. The fundamental treatment of this process can be found in Section 3.11, which deals with reliability where the junction temperature is a critical parameter.

3.9 Hot Phonon Effects

Despite the blazing experimental achievements in GaN-based HFETs, what could be construed is that an insufficient amount of progress has been made in understanding the physical properties of these structures in terms of high-field transport and lack of attaining the electron velocity predicted early on. This is in part due to the poor material quality but mainly, one could argue also, due to the unbalanced overemphasis on bare device power performance at the expense of solid fundamental work. HFETs, as is the case for any FET, operate at high electric fields, and the carrier

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concentrations are high. As such, the carriers are heated by the field and carrier interaction with lattice at high fields assumes central importance. The electrons gain energy from the field and the energy of hot electrons are relaxed by emitting LO phonons, through their decay into acoustic and other phonons, and energy transfer toward a heat sink away from the channel. For GaN, being a polar semiconductor, the dominant scattering mechanism at high temperatures is polar LO phonons. As tabulated in Volume 1, Table 1.26, there are nine optical and three acoustical phonon branches in wurtzitic GaN. The acoustical modes are simple translational modes and optical modes are more complex, and the nature of their mechanical vibrations is shown in Volume 1, Figure 1.14. The modes with the strongest interaction with electrons are the E1 and A1 LO phonons with an energy of 92 meV in GaN. The E1 and A1 modes are very close in energy (about 8 cm1) and there are three LO phonon modes in w€ urtzitic GaN. The E1 mode couples perpendicular to the c-axis, which is to a first extent represented by the case of FETs built-in layers on c-plane, and the A1 mode couples along the c-direction, which is applicable to vertical p–n structures built on the c-plane. However, if for example there exist scattering events that cause non-cplane transport, the phonon wave vector acquires an orthogonal component, making both phonons to be involved. If the angle between the c-axis and the phonon wave vector (often designated by q) is q, the angular dependence of the zone-center LO phonon frequency can be expressed as wLO ðqÞ ¼



2 2  1=2 wALO1 cos2 q þ wELO1 sin2 q :

ð3:331Þ

This means that the LO phonon frequency depends on the scattering geometry and has a frequency lying between that of the A1(LO) mode (wave vector along the c-axis or q ¼ 0 ) and that of the E1(LO) mode (q ¼ 90 ), with the maximum difference being about 8 cm1. Depending on the scattering angle, both modes might be present. One can then argue that hot electrons relax primarily through E1 and/or A1 LO phonon emission. In steady state, energy loss takes place by LO phonon emission and electron energy gain stems from the electric field balance, so the electrons propagate at a saturated velocity, which precedes the velocity limited by the intervalley scattering in a multiconduction valley system such as GaN [458]. Experimental and theoretical investigations, particularly in GaAs, pointed out that hot carriers (with a density approximately greater than 1017 cm3) can produce a nonequilibrium distribution of phonons, or hot optical phonon, which slows down the rate of energy relaxation. The extent of the effect depends on the discrepancy between the phonon generation time and the decay time or the lifetime of these hot phonons. The phonon generation time is about 10 fs for GaN (about 100 fs in GaAs), but decay time could be 2 ps or longer, as discussed in Section 3.9.1. Essentially, hot electrons and hot phonons are in equilibrium in terms of their temperature. Moreover, hot phonons tend to change the mobility as the reduction of energy relaxation at low field implies steeper rise in electron temperature (or energy) resulting in an increase in mobility. However, at high fields, the presence of large concentrations of hot phonons reduces mobility through phonon scattering. At high

3.9 Hot Phonon Effects Table 3.8 Fr€ ohlich interaction constants for several wurtzite materials and GaAs.

Parameter

AlN

GaAs

GaN

InN

ZnO

a

0.74

0.075

0.41

0.22

1.04

a

For the wurtzite materials, the values of a are those associated with interactions along the c-axis. Courtesy of M. Stroscio.

fields where scattering from ionized impurities is unimportant, the dominant main effects appear to be the enhancement of electron temperature for a given field, which causes a reduction in carrier velocity. Conventional treatment of hot phonons involves a forward displaced distribution of the nonequilibrium phonons in momentum space that is caused by the drift of hot electrons. The emission and reabsorption of nonequilibrium phonons have been argued to reduce the overall energy relaxation rates for electrons. The highly polar nature of GaN leads to a strong Fr€ohlich interaction, as tabulated in Table 3.8, which likely needs more than the common perturbative treatment used in the typical Monte Carlo calculations to predict the velocity field characteristics of nitride binaries and ternaries, as treated in Volume 2, Chapter 3. High-field effects in terms of hot phonons have been treated in GaAs. Timeresolved luminescence and Raman measurements performed in GaAs pointed to the presence of nonequilibrium phonon distribution due to cooling of photogenerated electrons and holes [459–462]. Analytical [463–466] and numerical (Monte Carlo) [467] tools have been developed in consideration of this problem in conjunction with not only photoexcited carriers but also high-field transport encountered in FETs early on, particularly in conjunction with GaAs. The Monte Carlo method [468] has the advantage in that an accurate description of the physical processes is possible without assumptions on the phonon distribution functions provided the phonons treated represent the system, that is, bulk, interface, and so on. Characteristic times for various scattering mechanisms relevant to cooling also come out. During the time-resolved PL and Raman processes, which are assumed to apply to high-field transport in GaN and relevant to FET operation, a strongly perturbed phonon distribution has been noted in the first few picoseconds following the laser pulse, which was attributed to the reduction of the cooling rate of the photoexcited carriers in GaAs [468]. The best available predictions for high-field transport properties in GaN, albeit without the treatment of hot phonons, are those of Brennan and collaborators [469– 471], which predict a peak velocity of more than 3  107 cm s1 at 140 kV cm1 at room temperature and valley velocity above 2  107 cm s1 over a wide range of fields, as discussed in Volume 2, Chapter 3. In contrast, estimates from the current gain cutoff frequency of AlGaN/GaN HFETs place the effective velocity to relatively lower values, as shown in Figure 3.122. To a first extent, the effective electron drift velocity (vd) can be related to the microwave current gain cutoff frequency through vd = 2pfTLG, where LG is the gate length and fT is the cutoff frequency, the use of which leads to the effective velocities of less than about 1  107 cm s1, see, for example, Refs [472, 473]. The effective velocity deduced experimentally is not the peak velocity

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103 fT ,InGaAs HFET fmax, InGaAs HFET fT , GaN HFET

Frequency (GHz)

fmax, GaN HFET

102

v=107 cm s-1

101 10-2

10-1

100

Gate length (µm) Figure 3.122 Effective carrier velocity, representing GaN, determined from the dependence of current cutoff frequency on gate length (squares). Also shown are the values of maximum oscillation frequency versus the gate length (circles). It should be noted that there is a considerable variation in the published data of both cutoff frequencies.

but is assumed to closely relate to the valley velocity encountered at very high fields. But the valley velocity argument requires that electrons scatter to the upper valley which is not trivial giving the high density of hot phonons due to strong electron phonon coupling. To be more focused and to a first extent, one might argue that this velocity may represent the average under the velocity–field curve accounting for the fraction of time the carriers are under the influence of the high field. Using pulsed I–V measurements [474] (300 ns pulses), a velocity of 1.0  107 cm s1 has been deduced at a lattice temperature of 77 K, which is comparable to that determined from FETs at room temperature and higher fields, undoubtedly. It should be mentioned that in all I–V measurements, the carrier concentration must be known to deduce the velocity. At high fields, needed to determine the saturated velocity, one must make certain that any injection from contacts does not enter into the equation. In addition, the heating effects must be reduced to a point where they are not applicable by using short pulses. A saturated velocity of 2  107 cm s1 obtained at 130 kVcm1, but by using 3 ns pulses to avoid heating, has been reported by Ardaravicius et al. [475]. In the same vein [476, 477], velocity measurements deduced from pulsed I–V measurements with short 10–25 ns voltage pulses in a 50 W environment have shown that the velocity in the GaN/AlGaN heterostructure does indeed reach 3.1  107 cm s1 at 140 kVcm1 at room temperature, see Figure 3.123. This is consistent with the early Monte Carlo calculations of Brennan et al. [469] and Yu and Brennan [478], which did not take hot phonon and interface phonon effects into consideration. However, when the pulse width is increased to 200 ns, the velocity reaches only about 107 cm s1 at a field of 190 kV cm1 [479], most likely due to hot phonon effects although heating effects should also be considered. This value is consistent with those deduced from FET cutoff frequency data. To shed some light on the issue, Ardaravicius et al. [475]

3.9 Hot Phonon Effects 3.5

Electron velocity (×10 7 cm s-1)

3.0 2.5 2.0 1.5 1.0 2

-1 -1

Mobility = 435 cm V s

0.5

2

8 × 2µm 2 8 × 2µm 2 12 × 3µm

0 0

50

100

150

Electric field (kV cm-1) Figure 3.123 Experimentally determined velocity field characteristics using pulsed I–V measurement using 10 ns input pulses in three different AlGaN/GaN samples. The inset shows the top view of the pattern used for the measurements whose active areas (indicated

by the arrow) were 8  2 mm2 (open circles), 8  2 mm2 (closed diamonds), and 13  3 mm2 (open diamonds). When the pulse width is increased to 200 ns, the velocity reaches only 1  107 cm s1 due possibly to hot phonon effects. Courtesy of J. M. Barker and Dave K. Ferry.

found a good agreement with Monte Carlo calculations if the hot phonon effects with a hot phonon decay time of 1 ps are taken into consideration, although heating effects have also been proposed. A hot phonon decay time of 0.35 ps overestimated the velocity while 3 ps underestimated it in comparison to experiments [480, 481]. In contrast, the phonon population relaxation time for A1 (LO) phonons was directly measured to be tph ¼ ffi 5  1 ps at 25 K along the c-direction in early samples prepared by MBE having an electron concentrations of n 5 1017 cm 3 [482]. Repeating the measurements at additional lattice temperatures of 150 and 300 K resulted in a phonon decay time of 3 ps at 300 K [483]. Phonon decay time (lifetime) depends on the available channels for optical phonon decay to lower phonons. 3.9.1 Phonon Decay Channels and Decay Time

Energy gained by electrons from the field (or optical excitation) is given off by phonon emission with an activation barrier equal to the LO phonon energy governed by Fr€ohlich interaction (coupling). Some of the emitted phonons are reabsorbed, but by and large the LO phonons, both nonequilibrium and equilibrium varieties, decay by various pathways to acoustic phonons, which can propagate through the sample to the heat sink, thus dissipating the energy given off by hot electrons as heat. The pathways for optical phonons to acoustical phonons are therefore an important issue. Many pathways such as impurity scattering, carrier scattering, and anharmonic

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interaction are available for phonon dephasing in solids. In high-quality samples, phonon dephasing due to carrier–phonon scattering and impurity scattering is negligible. Moreover, the depopulation of hot phonons takes place on a timescale on the order of 1 ps (the figures for InAs, InP, GaAs [484], and GaN are in the same ballpark), which is much longer than the carrier–phonon interaction of 10 fs in GaN and about 100 fs in GaAs [485]. These arguments as a whole justify the conclusion that the lifetime of the hot phonons is almost exclusively determined by anharmonic interactions in the form of decay into phonons of lower energies, that is, acoustical phonons. The phonon decay in semiconductors has been theoretically treated by Cowley [486] in Ge who considered the anharmonic interaction as an axially symmetric force between the nearest neighbors and summed all the possible decay processes. Later Klemens [487] introduced the concept of optical phonons decaying into two acoustic phonon modes with opposite wave vectors. However, the available theoretical data had a large dispersion and did not agree with experiments well. Debernardi et al. [488] were able to develop a detailed theory that provided a good insight into the anharmonic phonon decay with a clear picture of various decay pathways. This gave way to the calculation of anharmonic decay of phonons based on the electronic density functional theory, which culminated in the determination of the lifetime of the zone-center phonon in diamond, Si, and Ge [488] and of the zone-center LO and transverse optical (TO) phonons in GaAs, GaP, AlAs, and InP by Debernardi [489]. The phonon decay issue goes directly to the heart of hot carrier or energy relaxation of carriers in semiconductors. When the hot carrier density is high, the LO phonon emission time is short (10 fs in GaN) and emission rate is high (1014 s1 in GaN, consistent with 10 fs generation time), which gives rise to nonequilibrium population of LO phonons and increases the phonon reabsorption rate of carriers reducing the energy loss rate adversely affecting optical and electrical properties germane to devices. Not only the velocity discussed at length above but also the thermal management is adversely affected. These phonons occupy a limited k-space near the zone center (center of the Brillouin zone). In polar semiconductors, such as GaN, it is the LO phonons that take the center stage. The LO phonon decay in polar semiconductors, mainly GaAs, has been investigated both in the frequency domain using spontaneous Raman spectroscopy [490, 491] and time-resolved incoherent [492, 493] and coherent [494, 495] anti-Stokes Raman scattering. Time domain [483, 504] and frequency domain (Shi et al. [528]) Raman spectroscopy for phonon decay time investigations have been conducted in GaN. An optical phonon can decay into lower energy optical and/or acoustic modes determined by energy and momentum conservation conditions. In high-quality samples, phonon relaxation occurs through anharmonic interactions between three phonons. These three phonon interactions are governed by third-order anharmonicity and represent the most plausible processes for phonon decay and thus phonon lifetime. Four possible channels for such a reduction have been proposed. (i) Klemens proposed that the optical phonon decays into two acoustic phonons with opposite momenta (known as the Klemens channel) [487]. (ii) Ridley [270] suggested the possibility of a zone-center longitudinal optical mode decaying into a transverse optical mode and a longitudinal acoustic mode. This has been generalized

3.9 Hot Phonon Effects

to involve a process whereby the optical mode decays into a lower branch optical mode and an acoustic mode, and this has been termed as the generalized Ridley channel. (iii) Vallee–Bogani [496] and Vallee [495] proposed that an optical mode may decay into a lower mode of the same branch and an acoustic mode, and this channel is referred to as the Vallee–Bogani channel. (iv) Furthermore, in wurtzite materials a zone-center optical mode may also decay into two lower branch optical modes [497]. To compare and contrast these four different possibilities and their applicability to the semiconductor of interest, knowledge of the phonon dispersion curves would be needed. Phonon dispersion curves for GaN and related binaries have been reported for the wurtzite [498, 499] and the zinc blende phases [500]. In the long wavelength case, applicable to acoustic phonons, the associated dispersion curves can be approximately represented by straight lines. However, the shorter wavelength optical branch for a diatomic linear chain can be approximated by a quadratic dispersion relation. The dispersion relations for three-dimensional crystals are in general quite complicated and a simple approximation is not necessarily possible. To derive analytical expressions for the decay rate of a zone-center optical mode due to cubic anharmonicity in the crystal potential, Barman and Srivastava [497] made appropriate approximations for phonon dispersion curves for zinc blende and wurtzite materials. Using the Debye model within the continuum approximation for the acoustic branches, and considering optical branches as flat or linearly (rather than quadratically) dispersive, Barman and Srivastava [497] determined the dispersion curves for zinc blende materials as shown in Figure 3.124. For the wurtzite structure, the Debye model within a semi-isotropic model was applied with calculations having been performed along the three principal symmetry directions G–K (in basal plane), G–M (in basal plane), and G–A (c-direction), the results of which are illustrated in Figure 3.125. These are not realistic phonon dispersion curves, but descriptive of the first-order phenomenon near the zone center in a small section of the k-space. Therefore, the quadratic variation with frequency of the density of states obtained from the application of these linear dispersion relations would be noticeably different when realistic phonon dispersion relations are used. The energy and momentum conservation considerations are such that the zonecenter LO and TO phonon modes in zinc blende materials decay through one or many of the Klemens, Ridley, and Vallee–Bogani channels. Specifically, the following combinations of daughter modes have been argued as being allowed [501]: LO mode decay in zinc blende GaN: (i) Klemens channel: LO ! LA þ LA; LA þ TA; TA þ TA, (ii) Ridley channel: LO ! TO þ LA; TO þ TA, (iii) Vallee–Bogani channel: LO ! LO (zone edge) þ LA; LO(zone edge) þ TA. TO mode decay Klemens channel: TO ! LA þ LA; LA þ TA; TA þ TA. The details of the zinc blende material are not included here in favor of the wurtzitic phase, which are technologically more important.

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(a)

ω LO phonon

LA TA q Schematic representation of Vallee–Bodani decay

ω LO phonon

TO phonon

LA TA q

Schematic representation of Ridley decay with dispersive TO phonon Figure 3.124 Schematic representation of Klemens, Ridley, generalized Ridley, Vallee–Bogani, and additional possible phonon decay channels in a zinc blende semiconductor such as cubic nitrides. Because the schematic diagram is limited to a small section of the k-space in the zone center, the dispersion curves are approximately represented by straight lines. Patterned after Barman and Srivastava [497].

3.9.1.1 LO Phonon Decay Channels in Wurtzitic GaN In wurtzite materials A1(LO), E 22 , E1(TO), A1(TO), and E 12 constitute the zone-center optical modes. Barman and Srivastava [497] stated that the decay of the A1(LO) mode is allowed via channels (a) (Klemens), (b) (Ridley), and (c) modes involving daughter modes B11 ; B21 ; E 22 ; E 12 ; E 1 ; ðTOÞ; A1 ðTOÞ, and acoustic modes. However, the experimental data of the temperature dependence of the A1(LO) lifetime for GaAs [495] and GaN [483] are not consistent with it reducing to two LA modes with half the LO phonon energy and opposing wave vector. The experimentally measured temperature dependence of the lifetime varies with temperature much faster than this phonon decay channel predicts. Modes E1(TO), E 22 , and A1(TO are also allowed to decay via channels (a), (b), and (c). According to Barman and Srivastava [497], only the Klemens channel is allowed for the decay of E 12 .

3.9 Hot Phonon Effects

ω

(b)

LO or TO phonon

LA TA q Schematic representation of Klemens phonon decay

ω LO phonon TO phonon

LA TA q Schematic representation of Ridley decay with flat TO phonon Figure 3.124 (Continued)

ω

E1(LO)

A1(LO) E 22 E1(TO) B11 E 12

LA TA q

Figure 3.125 Phonon dispersion curves for a wurtzitic semiconductor near the zone center calculated along the three principal G–K, G–M (in the basal plane), and G–A (along the c-direction) symmetry directions and averaged. Because the

schematic diagram is limited to a small section of the k-space in the zone center, the dispersion curves are approximately represented by straight lines Patterned after Barman and Srivastava [497].

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7

1:γ = 0.57

1

6

Phonon lifetime (ps)

High-quality GaN pin

Wurtzitic GaN A1(LO)

2:ω LO 3:ω LO

5

ωLA,TA ωTO,LA(TA)

4: γ = 0.8 3

4

2

3

4

2

1 0

50

100

150

200

250

300

Lattice temperature (K) Figure 3.126 The phonon lifetimes deduced from time-resolved Raman spectra such as those shown in Figure 3.127 [483], as a function of the lattice temperature for a wurtzite GaN sample (full circles). The one depicted with solid square is for a high-quality GaN sample. The solid and dashed lines represent the decay channels of

wLO ! 2wLA,TA and wLO ! TO þ LA(TA), respectively, from Equation 3.332 [483]. The lines marked as 1 and 4 represent calculations for the mode average Gr€ uneisen’s constants of g ¼ 0.57 and 0.8, respectively, as discussed in some detail in Section 3.9.1.

Let us now examine the time-resolved Raman investigations and relate those studies to the phonon decay pathways as carried out by Tsen et al. [483]. Using the decaying part of the Raman signal such as the one shown in Figure 3.127, the population relaxation time of the LO phonons as a function of the lattice temperature ranging from 10 to 300 K can be obtained, which is shown in Figure 3.126. Klemens [487] and Ridley [270] used perturbation theory to demonstrate that the temperature-dependence part of the decay of the LO phonon population ni(w,T ) for phonon i in semiconductors can be expressed as. dnðw; TÞ nðw; TÞ½1 þ n1 ðw1 ; TÞ þ n2 ðw2 ; TÞ : ¼ dt t0

ð3:332Þ

Experimental results shown in Figure 3.126 can be used to gain some understanding into the decay channels of zone-center LO phonons in wurtzite GaN. The decay of zone-center LO phonons into a small wave vector LO and a small wave vector LA or TA phonon does not seem to be likely because this channel predicts a much larger temperature dependence of the lifetime of LO phonons than that supported by the experimental data shown in Figure 3.126 [483, 497]. The solid line, designated as 2 in Figure 3.126, represents the decay channel represented by wLO ! 2wLA,TA with
hwTA;LA ¼ 370 cm1 , which is half the energy of the LO phonon, as predicted by Equation 3.332. We should hasten to remind, however, that wLO

3.9 Hot Phonon Effects

2wLA,TA path is forbidden in GaN by energy conservation violation in that the LA phonon energy is 29 meV whereas the LO phonon energy is 92 meV, necessitating participation by not two but by four LA phonons. Within the experimental uncertainty, the zone-center LO phonons cannot decay into two large wave vector, equal-energy but opposite-momentum LA or TA phonons, which are usually assumed in the decay of LO phonons in other III–V compound semiconductors [502]. However, if one assumes that the TO phonon dispersion curve is relatively flat across the Brillouin zone, the decay channel of zone-center LO phonons in wurtzite GaN into a large wave vector TO phonon and a large wave vector LA or TA phonon is consistent with the experimental data. The dashed line designated as 3 in Figure 3.126 [483, 497] corresponds to such a decay channel represented by wLO ! TO þ LA(TA), as predicted by Equation 3.332. The energies of the TO and LA or TA phonons involved in the fitting process are w1 ¼ 540 cm1 and w2 ¼ 200 cm1, respectively. The lines designated as 1 and 4 represent the calculations performed by Barman and Srivastava [497] for the Ridley decay channel using the mode average Gr€ uneisen’s constant of g ¼ 0.57 and g ¼ 0.8, respectively. The calculated decay time value at room temperature with g ¼ 0.8 agrees reasonably well with experiments performed in the high-quality sample. Pending more detailed experimental investigations coupled with theory, the available experimental data show that among the various possible decay channels, the LO phonons in wurtzite GaN decay primarily into a large wave vector TO and a large wave vector LA or TA phonon. This preliminary suggestion is consistent with the theoretical calculations of the phonon dispersion curves for wurtzite of Azuhata et al. [503]. The lifetimes of A1(LO), E1(TO), A1(TO), E 22 , and E 12 modes in wurtzite semiconductors have been calculated by employing Debye’s scheme within a semiisotropic continuum model for acoustic modes [497]. In addition, lifetimes of the zone-center LO and TO phonons in cubic materials, specifically cubic phases of nitrides, have been performed by modeling the acoustic phonon modes within Debye’s isotropic continuum scheme. More applicable to this particular section are the pertinent phonon decay paths that have also been investigated [497]. Phonon decay channels for a given semiconductor determine the decay speed of nonequilibrium phonons. The intrinsic channels have to do with phonon–lower phonon anharmonic interactions as discussed above. The lifetime of LO phonons is an important parameter. Following up on earlier measurements, Tsen et al. [504] performed more refined measurements along the c-direction, as shown in Figure 3.127, which indicate this figure for the A1(LO) phonon lifetime to be as tph ¼ 2  0.5 ps and electron drift velocity to be vd ffi (2.5  0.5)  107 cm s1 when the injected electron–hole pair density is about 1  1017 cm3 and the electric field intensity is about 30 kV s1. As a side note, Ridley et al. [458] indicated that the optical measurements and many of the theories undertaken treat the case with implicitly small carrier concentrations. A pair concentration of 1  1017 cm3 does not probably fit into that description. These direct decay time measurements should be compared with 2 ps obtained through a convoluted fitting process in a Monte Carlo environment with many assumptions [475, 480]. Also of interest is that in the experimental conditions employed in Refs [475, 480], the well-defined excitation may not

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Wurtzitic GaN A1(LO) Nonequilibrium phonon population (a.u.)

10

T=300 K, high-quality GaN pin T=25 K

8

T=150 K T=300 K Best fit

6

4

2

0 -2

0

2

4

6

8

10

12

Time delay (ps) Figure 3.127 Nonequilibrium A1(LO) phonon populations for a wurtzite GaN sample as a function of time delay between the pump and the probe pulses for the lattice temperatures of T ¼ 25 (decay time is 5 ps), 150, and 300 K (decay time is 3 ps). The decay parts of the signal are

used to deduce the population relaxation time of A1(LO) phonons. Also shown is A1(LO) phonon relaxation in a high-quality pin GaN at 30 kV cm1 and 300 K indicating an LO phonon relaxation time of tph ¼ 2  0.5 ps. Courtesy of K. T. Tsen.

necessarily be the LO phonons but perhaps plasmon–LO phonon-coupled mode as discussed later in this section in conjunction with Figure 3.133. A reasonably full picture can be gained if GaAs and GaN data are used together as the data for GaN alone are scarce. The data obtained in GaN from time-resolved Raman measurements indicate that the hot phonon lifetime decreases from about 10 ps at 1016 cm3 electron concentration to about 6 ps at 1017 cm3 electron concentration [483] and then to 0.3 ps at an electron concentration of 1019 cm3, see the text in conjunction with Figure 3.133 and the description therein. The data obtained from noise measurements in 2DEG GaN indicate the hot phonon lifetime to be about 0.9 ps at 3  1018 cm3 and 0.2–0.3 ps at 1019 cm3 or slightly higher electron concentrations, as reviewed by Matulionis [505]. The dependence of the hot phonon lifetime on the electron concentration helps clarify to some extent the longer hot phonon lifetimes measured by time-resolved Raman measurements in bulk GaN samples and shorter hot phonon lifetimes deduced from noise measurement in FETs with high density of 2DEG. Although the follow-up Raman experiments with high photogenerated electron concentrations indicate shorter hot phonon lifetimes (see the text in conjunction with Figure 3.133), there is a difference between electrical and Raman measurements in the sense that the Raman measurements are for very small k-vectors limiting the picture to about the zone center. All these issues are pertinent with considerable merit and the problem can benefit from more refined Monte Carlo calculations, which treat the bulk and interface hot

3.9 Hot Phonon Effects

phonons properly inclusive any of their directional dependence as well as careful experiments to help guide the predictions for a clearer picture. The phonon lifetimes in wurtzitic GaN and AlN have also been determined experimentally [506] by using the simple (first-order t ¼
h=DE relation, where DE is the Raman linewidth in units of cm1 and
h ¼ 5:3  1012 cm1 s) energy–time uncertainty relation [507]. This exercise led to the observation that the phonon lifetimes in GaN and AlN fall into two main time regimes describable by a relatively long time of the E 12 mode and much shorter times of the E 22 , E1(TO), and A1(TO) modes. The lifetimes of E 12 , E 22 , E1(TO), and A1(TO) of a GaN crystallite were found to be 10.1, 1.4, 0.95, and 0.46 ps, respectively. Unfortunately, the A1(LO) mode in the GaN was not observed in this particular experimental study, the absence of which was attributed to plasmon damping. However, the carrier concentration either by doping (intentional or unintentional) and/or induced by any other means has not been reported. The lifetimes associated with the E 12 , E 22 , E1(TO), A1(TO), and A1(LO) modes in an what was termed as a high-quality AlN crystallite were reported to be 4.4, 0.83, 0.91, 0.76, and 0.45 ps, respectively. This particular A1(LO) lifetime is generally consistent with the phonon decay model of that mode in wurtzite structure materials. The lifetimes of the Raman modes in an AlN crystallite containing about two orders of magnitude more impurities than the high-quality crystallite AlN were found to be 50% shorter. The experimental decay times in GaN compare with 1 ps used in Ref. [475], which resulted in the best agreement between the Monte Carlo simulations inclusive of hot phonons and the pulsed current–voltage measurements (with a drift velocity of 2  107 cm s1, 130 kV cm1). These inconsistencies are indicative of a problem that is not well understood and will remain so until Monte Carlo calculations, which treat the phonons accurately (inclusive of bulk and interface phonons on the c-plane and at the heterointerface), reliable experimental data of phonon decay time and good understanding of the hot phonon decay mechanism are available. The role of a reliable theory and experiments supporting the development of one another cannot be overstated. What is almost certain is that the phonon emission time on the order of 10 fs or a high scattering rate on the order of 1014 s1 is fairly consistent with a large body of reports. Irrespective of the controversy as outlined above, the decay time is much longer than the emission time, which makes the phonon generation and decay issue very complex and raises the question whether many different types of phonons are involved and whether the anharmonicity is treated properly, and the validity of Fermi’s golden rule. Returning to the theory, Barman and Srivastava [497] calculated phonon lifetimes for various modes at low temperatures and room temperature. The results of their calculations along with the available experimental data from Raman scattering measurements and the time-resolved spontaneous Raman scattering technique are tabulated in Table 3.9 for the three nitride binaries. However, it should be mentioned that the effective anharmonic constant (Gr€ uneisen’s parameter that is defined as the negative logarithmic derivative of the frequency of the mode with respect to volume [508]) g for different modes in the various nitrides is not really all that well

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Table 3.9 Calculated lifetime (ps) of the zone-center optical

phonons in wurtzite GaN at 6 K (LT) and 300 K (RT) using a mode average (Gr€ uneisen’s parameter) g ¼ 0.8 for the mode-averaged Gr€ uneisen’s constant. Low temperature Mode

Material

A1(LO)

InN GaN AlN InN GaN AlN InN GaN AlN InN GaN AlN InN GaN AlN

E 22

E1(TO)

A1(TO)

E 12

Theory 0.19 3.0 0.27 30.4 2.2 1.52

Experiment

5.0a, 0.35f, 2e 0.75d

2.9d

Room temperature Theory 0.08 1.5 0.16 3.3 1.3 0.89

Experiment

3.0a, 0.62–1.16b 0.57d, 0.45c, 0.72g 1.4c 0.83c, 1.7g

2.3 0.8

1.4 0.46

0.95c 0.91c, 1.8g

1.2 0.73 252.2 507 44.0

0.7 0.41 25.21 81.7 11.2

0.4c 0.76c, 1.2g 10.1c 4.4c, 5.3g

For comparison, available experimental results are also given. The inverse of the lifetime would represent the scattering rate. Tabulated in Ref. [497]. a Ref. [483]. b Ref. [510]. c Ref. [506]. d Ref. [511]. e Ref. [504]. f Ref. [480]. g Ref. [512], calculated from half-width of Raman peaks.

known. The work by Bruls et al. [509] suggests that for AlN the value of g between 300 and 1600 K lies in the range of 0.70–0.95 (Bruls et al. treated the g parameter in AlN in a temperature range of 90–1600 K using thermal expansion, elastic constants, and heat capacity data of these materials). Barman and Srivastava [497] used g ¼ 0.8, which is the average of the results reported in Bruls et al. [509] for 300 and 1600 K) for all modes in all the nitride materials discussed here. This makes rescaling of g to fit the theory with experiment necessary when reliable experimental data for different modes are available. An additional issue is the simplified dispersion relation, perhaps oversimplified, employed in the model. The decay of the zone-center A1(LO) mode in GaN can take place only via Ridley’s channel. The Klemens channel decay mechanism is not consistent with GaN and experiments in that the LO phonon energy is more than twice the LA phonon energy or any available acoustic phonon energy for that matter. The various combinations of daughter phonon modes involved in Ridley’s decay processes are (i) A1(LO) ! E 22 þ LA,

3.9 Hot Phonon Effects

(ii) A1(LO) ! E1(TO) þ LA, (iii) A1(LO) ! E 22 þ TA, (iv) A1(LO) ! E1(TO) þ TA, (v) A1(LO) ! B11 þ LA, (vi) A1(LO) ! B11 þ TA. However, an inspection of the phonon dispersion curves [500] reveals that the processes (i)–(iv) are not allowed along G–A (c-direction), process (ii) is not allowed along G–M (in basal plane), and process (v) is not allowed along G–K (in basal plane). Computed results of Barman and Srivastava [497] give a very slow decay rate for processes (v) and (vi) in the three symmetry directions. Similarly, along G–K and G–M directions (both in basal plane) processes (iii) and (iv) give much smaller lifetime than the other processes, thus providing the main contribution with process (iii) providing 60% of the decay. The details of the other phonon mode decay are beyond the scope of this text and interested reader is referred to Barman and Srivastava [497] and references therein. The temperature dependence of the A1(LO) phonon decays calculated with g ¼ 0.57 and g ¼ 0.8 by Barman and Srivastava [497] and by Tsen et al. [483] using Klemen’s and Ridley’s channels along with the experimental data are shown in Figure 3.126. The lines designated as 1 and 4 represent calculations of Barman and Srivastava [497] for mode average Gr€ uneisen’s constants g ¼ 0.57 and g ¼ 0.8, respectively. The calculations for g ¼ 0.8 appear to agree reasonably at room temperature with the data obtained in the high-quality sample. The net rate of LO phonon annihilation via Ridley’s anharmonic process in GaN is given by [270, 513] 1 G2
hwTO ðwLO wTO Þ3 ¼ ½nðwTO Þ þ nðwLO wTO Þ þ 1; t0 2prLO wLO n3LA

ð3:333Þ

where rLO is the reduced mass density, uLA is the LA phonon group velocity, G is the linewidth broadening, and n(wTO) and n(wLO–wTO) are the occupation numbers for TO and the mixed TO–LO phonons. AlN is a pivotal binary in that its ternary with GaN is used in every modulationdoped GaN FET. In addition, the same ternary is crucial in optoelectronic devices. In this vein, it is imperative that phonon decay processes in AlN are discussed. The decay channels available for the A1(LO) phonon, which couples along the c-direction, are [497] (i) A1(LO) ! B21 þ TA, (ii) A1(LO) ! B21 þ LA, (iii) A1(LO) ! E 22 þ TA, (iv) A1(LO) ! E 22 þ LA, (v) A1(LO) ! E1(TO) þ LA,

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(vi) A1(LO) ! A1(TO) þ TA, (vii) A1(LO) ! A1(TO) þ LA, (viii) A1(LO) ! E 22 þ E 12 . However, along the c-axis only the channels (i), (ii), and (viii) contribute. The decay mechanisms channels (viii) and (v) are absent along the G–M and G–K directions (both in the c-plane), respectively. The relative main contributions of various processes are found as follows [497]: Process (vi) 35%, (iv) 23%, (i) 15%, and (viii) 16% over the entire temperature range. The rest of the total contribution comes from other Ridley-like processes. Kuball et al. [511] measured A1(LO) phonon lifetimes in AlN of 0.75 and 0.57 ps at 10 K and room temperature, respectively, which compares with 0.45 ps, deduced from the linewidth analysis of micro-Raman signal, by Bergman et al. [506]. The figure reported by those authors might have been affected by impurities and other peculiarities of the samples used. Bergman et al. [506] utilized the decay of the A1(LO) mode occurring via Ridley’s channel into a large wave vector TO mode and an LA mode. On the contrary, Kuball et al. [511] assumed Klemens’ channels in analyzing their data. An inspection of the phonon dispersion curves reported in Ref. [500] and shown in Volume 1, Figure 1.27 for AlN illustrates clearly that owing to substantial mass difference between the Al and the N atoms, there is a large gap between the acoustic and optical phonon branches, with the A1(LO) frequency at G point being 910 cm1 and at the zone-edge LA frequency being 314 cm1. Because 2wLA < wLO, Klemens’s model is not applicable for the decay of A1(LO) mode in AlN, similar to the case of the A1(LO) decay in GaN. Instead, the Ridley-type processes contribute 84% and with further channels with two optical daughter modes contributing the rest, approximately 16%. For further details on temperature dependence, both calculated and experimental, as well the discussion of the channels for the other phonons, the interested reader is referred to Barman and Srivastava [497]. Owing to their larger LO phonon energy, smaller Fr€ ohlich coupling (electron– phonon coupling) and shorter LO phonon lifetime compared to those in GaN, InNcontainingchannelsforFETsmightjustgaintremendouspopularity.Allthesefavorable parameters imply that it takes more energy to create LO phonons in InN, it takes more electrons to create them, and once they are created, they are eliminated by reducing them to other lower phonon relatively quickly. All this means that nonequilibrium phonon density in InN would be much lower compared to GaN, everything else being similar. This would also imply that the saturation velocity perhaps would not be limited byphononemissionasreadilyasitisthecasewithGaN. Inthisvein,letusdiscuss theLO phonon decay channels for InN as we have done for GaN. With the help of Figure 3.125 and available phonon dispersion curves, one concludes that the A1(LO) mode can decay via the Ridley and “further” channels. In particular, Barman and Srivastava [497] reported the following Ridley decay processes as being allowed in wurtzitic InN: (i) A1(LO) ! E 22 þ TA, (ii) A1(LO) ! E 22 þ LA,

3.9 Hot Phonon Effects

(iii) A1(LO) ! E1(TO) þ LA, (iv) A1(LO) ! E1(TO) þ TA, (v) A1(LO) ! A1(TO) þ LA, (vi) A1(LO) ! A1(TO) þ TA, (vii) A1(LO) ! E 22 þ E 12 , (viii) A1(LO) ! E1(TO) þ E 12 , (ix) A1(LO) ! A1(TO) þ E 12 . Barman and Srivastava [497] reported that along G–K (in basal plane) all these decay mechanisms are allowed, along G–M (in basal plane) only the second of the Ridley processes and (vii), (viii), and (ix) processes are allowed, and along G–A (the c-direction) only (vii), (viii), and (ix) channels are allowed. Barman and Srivastava [497] calculated lifetime values as 0.19 ps at 5 K (low temperature) and 0.08 ps at 300 K. It is found that channels (vii), (viii), and (ix) contribute very strongly, accounting for about 80%, toward the decay process. In particular, channels (viii) and (ix) provide a contribution of about 60%. The LO phonon decay mechanism is proving to be a rather complex problem now that its dependence on the electron concentration is established experimenpffiffiffi tally as illustrated by Matulionis [505]. The empirical 1= n dependence of the hot phonon lifetime, where n is the electron concentration, is beginning to shed some much needed light on the possible (or improbable) LO phonon decay mechanisms. Because the Joule heat can only be removed by acoustic phonons, the acoustic phonons must be involved in the process and in the phonon decay mechanisms. Considering energy conservation alone allows the conversion of an LO phonon into four acoustic phonons, but this process is quite improbable as five particles must partake in this event, which is unlikely. Other phonon conversion schemes such as LO ) TO þ TA/LA do not seem to explain fully the experimentally observed dependence of the hot phonon lifetime on the electron density. Screening of the conversion potential by electrons would lead to an increase in the lifetime with increasing electron density, which goes counter to the experimental observations. The data showing the hot phonon lifetime to be decreasing with electron pffiffiffi concentration as 1= n suggest a plasmon-assisted decay of LO phonons into acoustic phonons. The obvious conclusion is that more refined investigations are needed to clarify the hot phonon decay channels involved in the LO phonon decay processes. 3.9.1.2 Implications for FETs Relevant is the fact that transport in FETs occurs in the c-plane and at the heterointerface (representing the case for velocity deduced from I–V measurements and associated fitting of the velocity field characteristics with Monte Carlo calculations of Refs [475, 480] presumably using bulk phonons only), which could very well be different from the bulk and also along the c-direction (representing the reported

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phonon lifetime measured directly). It should, however, be pointed out that the early predictions (see Volume 2, Chapter 3) ignored the role of nonequilibrium and interface phonons. Later theoretical studies, using ensemble Monte Carlo techniques, which take into account the nonequilibrium phonons explicitly, predict peak velocities below 2  107 cm s1 at room temperature [480, 514, 515]. It has once again been demonstrated that transmission line patterns used for measuring the current– voltage characteristics and thus the velocity depend on lattice heating and thus the pulse width of the voltage applied for such measurements [475]. The delineation of Joule heating from hot phonon effects has been the topic of another study [516]. It should be pointed out that in all of these experiments where the velocity is deduced from the current–voltage characteristics is the implicit assumption that the carrier concentration determined by Hall measurement at low fields hold under pulsed and high-field current–voltage measurements. Another issue that must be kept in mind is that the velocities determined from the current cutoff frequencies represent the steady-state or CW conditions. It appears that one group of researchers appears to report an agreement between the velocities determined by pulsed current–voltage measurements (1  107 cm s1, heating is always an issue and measurements depend on the pulse width used) and analytical theory treating hot phonons as the velocity saturation mechanism (predicting the velocity to be approximately between 0.9  107 and 1.2  107 cm s1 for carrier concentration levels of 5  1017 and 5  1018 cm3) [458, 474]. Another group appears to report an agreement also, but at 2  1017 cm s1, between the Monte Carlo simulations, taking hot phonons into account and the pulsed current–voltage measurements [475]. None of these results is consistent with the values determined from the time-resolved Raman measurements. On a fundamental level, one may argue that these inconsistencies between the experimental measurements of peak velocity (with short pulses) and the predictions, which take into account the nonequilibrium phonons, may have their genesis in an inappropriate treatment of phonons in calculations. Another imbricated issue is the hot phonon decay mechanism and thus energy relaxation and heat removal. Experimental [517] and theoretical investigations [463, 467, 518] show that hot phonons (nonequilibrium phonons) decrease the energy loss rate at high electric fields and also the electron drift velocity, both of which degrade the performance of FETs. Conventional treatments of hot phonon effects allow a forward displacement of the nonequilibrium phonon distribution in momentum space owing to drifting hot electrons [463]. The emission and reabsorption associated with drifting nonequilibrium phonons reduce the overall energy relaxation rates for electrons [519]. A highly relevant and very important question has to do with the pathways for hot LO phonon decay and whether they involve decay to acoustic phonons in any way. As alluded to above, adding the hot phonon effects to the Monte Carlo calculations in GaN led to a predicted saturated velocity of 2  107 cm s1 [475, 480, 516]. An analytical model [458] predicted the velocity to depend on carrier concentration and in the range of 0.9  107–1.2  107 cm s1. At high electric fields, both electrons and phonons are not in equilibrium with the lattice. This is characterized as nonequilib-

3.9 Hot Phonon Effects

rium electrons and phonons or hot electrons and hot phonons. The electron temperature under the influence of electric field depends on the rate of energy gain from the field and the loss of energy given off to the lattice by a combination of phonon emission (relaxes energy) and reabsorption (increases energy). High electron concentrations and strong electron–phonon coupling in GaN lead to hot phonons in a limited k-space whose net generation rate is determined by a balance between phonon emission (depends on electron–phonon coupling and phonon energy) and absorption. The phonon lifetime is determined by the rate at which those phonons are converted to acoustic phonons. As such, this decay time or lifetime has an impact on the energy relaxation of carriers and thus the electron temperature, and velocity in that phonon emission reduces carrier energy and lingering phonons cause scattering and degrade velocity. Therefore, once generated it is best to remove the phonons as efficiently as possible. However, the decay pathways of LO phonons to acoustic phonons have not received sufficient theoretical and experimental attention. The decay pathways for all nitrides are discussed at the end of this section. The electron temperature is typically handled by Monte Carlo simulations. The experience with GaN with relatively large phonon population has provided the impetus to include hot phonon issues as well [520]. In the work of Matulionis et al. [480], the electron temperature was experimentally deduced from microwave noise measurements performed at sufficiently high frequencies (10 GHz) for the 1/f noise contribution to be negligible. The details of the method can be found in Ref. [521]. The contribution due to parasitic elements was also accounted for. The noise intrinsic to the device is caused by hot electron fluctuations associated with hot electron temperature and fluctuations in the same quantity. Moreover, the Monte Carlo simulations [520] also show that electron–gas degeneracy influences the velocity, particularly at low fields. If the degeneracy is neglected, the velocity, as shown in Figure 3.128 [520] with dashed line, exceeds that obtained with the model including both hot phonon and carrier degeneracy (solid line) effects. The carrier degeneracy within the context of Monte Carlo simulation has been treated by Lugli and Ferry [522]. It should be pointed out that because the electron energy increases with electric field, the degeneracy effects become less important. Consequently, the drift velocities calculated with and without the degeneracy tend to converge at high fields. However, when the hot phonon effect is neglected, the calculated drift velocity, shown with dotted line in Figure 3.128 [520], exceeds the experimental data noticeably (indicated by squares) and the simulations including all the aforementioned effects (solid line). When an LO phonon relaxation time of tph ¼ 1 ps is used in the simulation including all the aforementioned contributions, the best fit to the experimental data was obtained. The phonon lifetime is a critical and hotly debated issue and will be elaborated on next. An understanding of the phonon decay rates is closely connected to power dissipation through phonon emission. In steady state, the power supplied to electrons and power dissipated by them must balance. The power supplied to electrons increases the electron temperature and the dissipated power reduces the electron temperature and consequently the dissipated power is a function of electron temperature. We should recall that there is an activation energy for dissipation that is

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Velocity (106 cm s-1)

14 12 10 8 6 4 2 0

0

2

4

6

8

10

12

14

Electric field (kV cm-1) Figure 3.128 Electron velocity–field characteristics in GaN two-dimensional channel determined by Monte Carlo simulations for 300 K. The solid line represents the case with electron–gas degeneracy and hot phonon effects included using a phonon lifetime of 1 ps as a fitting parameter, the dashed line represents the

case without degeneracy but with hot phonon effects, and the dotted line represents the case with degeneracy included but without hot phonons. Open squares stand for experimental results reported in Ref. [481]. Courtesy of A. Matulionis [520].

roughly equal to the LO phonon energy in GaN, which is 92 meV. In the absence of hot phonons at low lattice temperatures satisfying
hwLO =kB T L  1 (wLO is the LO phonon frequency), the mean power dissipated by an electron through LO phonon emission is estimated, assuming Boltzmann statistics [480],

hwLO
hwLO Pd ¼ exp  ; ð3:334Þ t0 kB T e where t0 is the spontaneous LO phonon emission time constant (t1 0 represents the phonon emission rate, spontaneous emission rate). The exponential term accounts for the number of electrons that are able to emit optical phonons. Equation 3.334 leads to a 30 times higher dissipated power than that observed experimentally (see Figure 4 in Ref. [480]). This suggests that hot phonons must be taken into account. Hot carrier relaxation, which is achieved by LO phonon emission in this case, can best be investigated by optical means with one caveat that has to do with carrier concentration. For a meaningful comparison, the electron concentration must be high. In addition to the discussion of time-resolved Raman and band-to-band transient excitation-related investigations covered in Vol. 2, Sections 3.9 and 3.10, pump and probe measurements have also been used. A technique wherein carriers in the conduction band are heated by an infrared laser pulse (coupled through free carrier absorption if the electron concentration is sufficiently high, i.e., 1018 cm3) together with a tunable UV low-power probe pulse to monitor the carrier distribution in the conduction band has been used to determine hot carrier relaxation in GaN [523]. The hot carriers so generated would in general have an energy distribution but in cases where the carrier–carrier scattering is efficient and IR excitation laser pulse is relatively

3.9 Hot Phonon Effects

long, a common temperature greater than the lattice temperature can be approximately assigned to the hot electrons to make model calculations a little easier. The hot carriers, electrons, then transfer energy to the lattice through LO phonon emission (actually goes by the difference in emission and reabsorption rates). The exponential decay in transmitted signal following the excitation laser pulse, as determined by the probe pulse, would represent this cooling or the energy relaxation mechanism with time. Temporal behavior of the relative optical transmission at wavelengths corresponding above and below the bandgap, coupled with model calculations, has been used to determine the energy loss rate or relaxation rate by fitting the experiment to the model calculations [523]. Such fitting resulted in phonon emission time of 0.2 ps and phonon relaxation time constant of 0.68 ps. However, questions would arise as to how one avoids hot phonons in the hot electron experiment and why one even would expect Equation 3.334 to be valid if hot phonons are present. To my opinion, interpretation of experimental data [523] in terms of Equation 3.334 may not be well taken as evident by fitting deduced time constant of 0.2 ps that exceeds the spontaneous emission time constant of 10 fs by a factor of 20. The discrepancy would be reduced substantially if hot phonons are taken into consideration. The time required for an electron to emit an LO phonon in a semiconductor with low carrier concentration is determined by t0, which is given by the expression [524] 2 h t1 0 ¼ ðe wLO Þ=2p


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m =ð2
hwLO Þð1=eopt 1=es Þ;

ð3:335Þ

where wLO is the frequency of the LO phonon, eopt is the optical constant, and es is the static dielectric constant. This points to a phonon emission time near 10 fs, which is much shorter than the above-mentioned 0.2 ps resulting from the fitting process. As mentioned above, the discrepancy may have its roots in neglecting hot phonon effects in determining the 0.2 ps figure. It should be mentioned that the calculated 10 fs figure is consistent with a later publication (5–10 fs in samples with carrier concentrations in the mid-to-high 1018 cm3 range) [525]. At room lattice temperature or higher, the equilibrium phonons are present necessitating the inclusion of the phonon occupation factor fph which requires modification of Equation 3.334 in the form, again within the Boltzmann statistics,
  
hwLO
hwLO
hwLO Pd ¼ ð1 þ f ph Þ exp  f ph : ð3:336Þ t0 kB T e t0 The first term in Equation 3.336 represents the power given off by the system in terms of phonon emission (spontaneous and stimulated). The power returned to the system by phonon reabsorption is represented by the second term. The power Pd can be obtained by dividing the intrinsic power given to the electron system as a whole divided by the number of electrons. At low temperatures, the discrepancy between the lattice and electron temperatures is greater and the equilibrium phonon occupancy factor is negligible in which case Equation 3.334 can be applied. In dealing with hot phonons, again within the Boltzmann distribution framework, Equation 3.336 must be modified wherein the lattice temperature T is replaced with the electron

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temperature Te. Doing so leads to 

hwLO hwLO
IV Pd ¼ ð1 þ f ph Þ 1 þ exp ; ¼ t0 kB T e Ns

ð3:337Þ

where I and V represent the current through the sample, V is the internal voltage, excluding the voltage drop across external resistances, and Ns is the total number of electrons. For these expressions to be used, one has to ensure that a full and accurate accounting of power applied to the system and what fraction of it really is consumed heating the carriers must be undertaken. As far as the nonequilibrium or hot phonons are concerned, the phonon occupancy number can be treated within the Bose–Einstein distribution as [480] 
 1
hwLO f ph ¼ exp 1 ; kB T ph

ð3:338Þ

hwLO ¼ 92 meV, and t0 ¼ 10 fs in where Tph is the phonon temperature. Using
Equation 3.337, the dependence of the phonon occupancy factor fph versus electron temperature can be solved, the results of which are displayed in Figure 3.129 for an Al0.15Ga0.85N/GaN system with 5  1012 cm2 carriers in the system. Knowledge of the phonon occupancy fph and use of Equation 3.338 leads to the determination of the phonon temperature. The hot phonon temperature so determined as a function of the inverse hot electron temperature is shown in Figure 3.130, where symbols represent the data using Equation 3.338 and the experimental results of Figure 3.129. For comparison, the dashed line in Figure 3.130 represents the hot electron temperature. The lattice temperature measured at an ambient temperature of 293 K, which is strong function of any heat sinking scheme used, is also shown in Figure 3.130 as a solid line.

10 0 Phonon state occupancy

Al0.15Ga0.85N/GaN ns=5×1012cm -3 10 -1

10 -2

293K 80 K

10 -3 0

1

2

4 3 1000/T e (K-1)

5

6

Figure 3.129 Dependence of the LO phonon occupancy factor on the inverse hot electron temperature at 293 K (circles) and 80 K (squares). The occupancy factor is solved using the balance Equation 3.337. The solid line represents Equation 3.338, where Te ¼ Tph is assumed. Courtesy of A. Matulionis [480].

Al0.15Ga0.85N/GaN

Hot phonon temperature

500

500

ns =5×1012cm-3 400

400

300

300

200

200

100

2

3

4 1000/T e(K-1)

5

6

Electron temperature (K)

3.9 Hot Phonon Effects

100

Figure 3.130 Hot phonon temperature (symbols) as a function of the inverse electron temperature at two ambient temperatures: 80 K (squares) and 293 K (circles). The dashed line is the electron temperature. The solid line is the lattice temperature at room temperature. Courtesy of A. Matulionis [480].

A striking feature of Figure 3.130 is that the hot phonon temperature is comparable to the hot electron temperature (represented by the dotted line), which can be read from the horizontal axis or the right-hand scale. This allows one to assume Te ffi Tph as was previously done by Artaki and Price [518]. This approximation together with Equation 3.338 is used to plot the dependence of fph on the inverse temperature (Figure 3.129, solid line). The experimental data points in Figure 3.129 can be represented by a straight line, which validates Boltzmann statistics. Under the steady state, the power Ps supplied equals the power (Pd)ph dissipated into the lattice. The dissipated power is lower than the power exchanged inside the electron–LO phonon subsystem. Different time constants are involved with phonon emission followed by reabsorption and emission, which in turn are followed by decay into most likely acoustic phonons. The decay time is of consequence here. If we designate phonon decay time as tph and use a concept similar to the relaxation time approximation (meaning exponential decay), the hot phonon decay rate can be expressed as [480]  qf ph   qt 

¼

decay

f ph ð f ph Þeq tph

:

ð3:339Þ

The power dissipated by the LO phonons to the lattice can be found by multiplying the rate of [ fph( fph)eq]/tph with the LO phonon energy
hwLO. Doing so leads to Pd j ph¼
hwLO

f ph ð f ph Þeq tph

:

ð3:340Þ

Use of Ps ¼ IV/Ns ¼ Pd|ph together with Equation 3.340 leads to the determination of phonon decay time tph ¼ 350  100 fs [480]. What also results is that the phonon decay time is weakly, if at all, dependent on the hot phonon temperature Tph below

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500 K. This compares with tph ¼ 2  0.5 ps measured directly for LO phonons by time-resolved Raman measurements, but in samples where the electric field is along the c-direction, 0.68 ps in Ref. [523] and 0.35 or 1 ps, depending on the interpretation [480], giving the best fit between the experimentally determined electron velocity and Monte Carlo calculations. Matulionis et al. [480] in a series of papers undertook an effort to determine the electron velocity by bringing to bear experimental data involving current–voltage measurements for the determination of electron velocity and high-frequency noise data for the determination of electron temperature. They also coupled the abovementioned work with semiclassical Monte Carlo simulation of electron motion to find the best phonon relaxation time describing best the experimental velocity field dependence. As touched upon above on several occasions, the Monte Carlo simulation takes into account electron–phonon scattering, 2DEG degeneracy, and the nonequilibrium phonon distribution. In addition, in treating the optical phonon scattering, screening of the phonon potential is taken into account. Furthermore, the inelastic electron–acoustic phonon interaction is treated within the framework of piezoelectric and deformation potential approaches. The LO phonon lifetime is introduced in conjunction with reaching the steady-state distribution of hot phonons. Culmination of all the above leads to velocity field curves shown in Figure 3.131 for three values of the LO phonon lifetime, 0.35 (arrived at from the analytical treatment presented above), 1, and 3 ps. As expected, the calculated electron drift velocity is higher for shorter hot phonon lifetimes, which reduce the hot phonon density and therefore scattering by reabsorption. Open circles stand for the drift velocity estimated according to vd ¼ I/(qnsW) where the current I is

Velocity (106 cm s-1)

10

AlGaN/GaN

8 6 4 2 0

0

5 10 Electric field (kV cm-1)

Figure 3.131 Calculated (Monte Carlo inclusive of hot phonon effects) and experimental (determined from pulsed current–voltage measurements) of electron drift velocity versus electric field. Experimental data represent those reported in Ref. [475] and are obtained at approximately 293 K (open circles). Monte Carlo results correspond to the lattice temperature of 300 K (symbols) for three hot phonon lifetimes of

15 350 fs (triangles), 1 ps (squares), and 3 ps (diamonds). The solid line and squares cross at 7.5 kV cm1 and diverge at lower and at higher fields. At low fields (E < 5 kV cm1), neither 1 ps nor 350 fs fit. At very low fields, there is no difference between 1 ps and 350 fs indicating that hot phonons play a minor role in this regime as expected. The solid line is simply a guide to the eye. Courtesy of A. Matulionis [480].

3.9 Hot Phonon Effects

measured for 3 ns voltage pulses for a fixed lattice temperature of the channel, W represents the sample width and ns represents the sheet electron density at the interface of the modulation doped structure, and q (e also is used interchangeably) is the electronic charge. The experimental drift velocity is close to the calculated one for tph ¼ 350 fs. As pointed out by Matulionis [526], Monte Carlo model does not take into account the impurity scattering and other mechanisms of elastic scattering. Consequently, drift velocity deduced as such cannot be below the experimental points. However, one can see that some data points denoted by squares are below the experimental circles. Consequently, 1 ps lifetime (squares) does not fit. If elastic scattering mechanisms (impurities, dislocations, alloy, etc.) are added to fit the experimental data at low fields (e.g., E ¼ 1 kV cm1) where the hot phonons play no role, no fitting of squares and the experimental data (circles) can be achieved at E > 7.5 kV cm1 for tph ¼ 1 ps. Therefore, a fitted value less than 1 ps is appropriate. The dependence of the phonon lifetime in random samples indicates an n1/2 dependence of electron concentration in the channel [526]. The square root dependence suggests plasmon involvement in phonon decay. This dependence is inconsistent with LO phonons decaying to TO and AC phonons. It should be pointed out that these results are preliminary and a more definitive work is needed to be certain. It is instructive and helpful in developing intuition if analytical treatments are made available for any problem in general and hot phonon effect in relation to the saturated velocity in GaN in particular, as has been done by Ridley et al. [458]. Even though the heterointerface nature of the AlGaN/GaN interface would require a full treatment of phonons including half-space bulk modes and interface modes, scattering rates were obtained using bulk-like phonon spectrum assuming that would provide a good representation of the system at hand for simplification necessary for the analytical model developed. In addition to the practical reason of simplifying the problem to be tackled by analytical means, a combination of the high carrier concentrations and high electron temperature would pave the way for the scattering to involve on average over many subbands, allowing the use of bulk-like scattering processes. High fields and the ensuing hot carriers would also make the effect of degeneracy negligible and allow the use of a Maxwell–Boltzmann distribution characterized by an electron temperature to describe the occupancy of states. The analytical calculation begins with the estimation of the power dissipated per electron by emitting LO polar phonons. To complete this step, knowledge of the phonon energy and phonon lifetime, which were taken as 91 meV and 3 ps, respectively, by Ridley et al. [458] is needed. An expression for the momentum relaxation rate associated with polar optical phonon (POP) scattering in a nonparabolic band has been used to obtain the momentum relaxation time averaged over the electron distribution to determine the POP mobility as a function of electron temperature. For simplicity, the contribution from other scattering mechanisms has been configured in the form of a temperature-independent mobility so chosen to result in a low-field mobility of 1200 cm2 V1 s1. From the dissipated power, assumed to be equal to that gained from the field, which is valid for steady state, one can find the electric field using P ¼ qmF2 and the drift velocity from v ¼ mF.

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(a) 0.5×1018 cm-3

Velocity (107 cm s-1)

1.4

1×1018 cm -3 2×10 18 cm -3 3×10 18 cm -3

1.2 1.0 0.8

5×1018 cm-3

0.6 0.4 0.2 0.0 0

50

0

1

150 200 100 Electric field (kV cm -1)

250

Velocity (107 cm s-1)

(b)

1.4 1.2 1.0 0.8 0.6 2

3

4

5

Electron density (1018 cm-3) Figure 3.132 (a) Velocity–field curves for several electron densities in the range of 0.5  1018 and 5  1018 cm3. (b) Saturation velocity as a function of carrier density at 250 kV cm1. Courtesy of B. K. Ridley [458].

Shown in Figure 3.132 is the velocity field plots for several carrier densities between 0.5  1018 and 5  1018 cm3, which represent a range of modulation-doped FETstructures. This analytical treatment predicts a velocity well above 2  107 cm s1 at fields beyond 50 kV cm1 if the nonequilibrium hot phonon effects were neglected. Hot phonon velocity saturation appears to be unavoidable at high densities. These calculations have been limited to electron temperatures below 5000 K, corresponding to an average energy of 0.65 eV, which is well below energies at which intervalley processes can occur, assuming the lowest energy predicted for the U valley is 1.34 eV. As alluded to earlier, despite the recognition of hot phonon effects and their inclusion in Monte Carlo calculations [475, 527], there remain unanswered questions. A velocity of 2  107 cm s1 determined from pulsed current–voltage measurement appears to be consistent with Monte Carlo method that includes hot phonon effects (with a hot phonon lifetime of 0.35–1 ps). However, the experimental data [475] so obtained are not consistent with 1  107 cm s1 determined from FET current gain cutoff frequency and also other pulsed current–voltage

3.9 Hot Phonon Effects

characteristics [474]. To make matters more puzzling, the carrier velocity determined from time-resolved Raman investigations, shown in Figure 3.122, appears to agree with the early Monte Carlo calculations, which did not take hot phonons into account. Note, however, the configuration of the TR Raman measurements probed the carrier velocity in c-direction in samples with no heterointerfaces. It should be pointed out again that the value tph ¼ 350 fs reported in Ref. [480] is much lower than the anharmonic lifetime values reported by Tsen et al. [483] of approximately 5 and 3 ps at 20 and 300 K, respectively, measured by time-resolved Raman measurements. A follow-up measurement by Tsen et al. [504] of the A1 LO phonon decay time with time-resolved Raman measurements utilizing 10 ns pulses with a photogenerated electron–hole pair density of about 1  1017 cm3 resulted in t ¼ 2  0.5 ps at room temperature in very high-quality samples. We should be reminded that the A1 LO phonon couples with electrons along the c-direction. When the Raman FWHM of the A1 LO phonon is converted to decay constant, through a linewidth analysis, Shi et al. [528] obtained approximately 2.1 and 0.9 ps at 20 and 300 K, respectively, but this method is very indirect and does not address the various sources that could contribute to linewidth broadening. An analytical treatment of the dependence of hot phonon-induced friction or reduction of velocity on the carrier concentration has been provided by Ridley et al. [458] for bulk GaN. Essentially, the saturated velocity decreases with increasing carrier concentration with best saturation velocity of slightly over 107 cm s1 for 5  1017 cm3. A similar trend has been observed experimentally in that the transistor cutoff frequency (attributed to an increase in the drift velocity) increases when the electron density decreases as the channel pinch-off is approached [242]. Because of its importance, we reiterate that Matulionis et al. [505, 529] tabulated hot phonon lifetimes deduced from time-resolved Raman measurements for bulk GaN (for the lower doping range) and from microwave noise measurements in a series of 2DEG channels of FETs for the higher doping range corresponding to electron concentrapffiffiffi tions of 1018 cm3 to slightly over 1019 cm3. A 1= n dependence of the hot phonon lifetime, where n is the electron concentration, was empirically deduced. In short, the 2DEG channels exhibited shorter phonon lifetimes than the bulk samples with much lower doping levels pointing to the possibility of plasmon involvement in the LO phonon decay mechanisms. Phonon lifetime measurements deduced by timeresolved Raman measurements versus the carrier concentration between 1016 and 2  1019 cm3 also indicate decreasing lifetime with increasing carrier concentration, which also include fluctuation technique, as shown in Figure 3.133. Essentially, the lifetime of A1(LO) phonon mode decreases from 2.5 ps for 1016 cm3 to about 0.35 ps for 1016 cm3, the latter being consistent with that deduced from 2DEG [530]. The question arises whether the trend of reducing hot phonon lifetime with increasing electron concentration would continue. The answer is not likely as the coupling with plasmons becomes more pronounced for carrier concentrations larger than 1017–1018 cm3 when the plasmon energy exceeds the acoustic phonon energy as shown in Figure 3.134. Increasing the electron density further to well above 1019 cm3 causes the plasmon energy to increase past the LO phonon energy. Therefore, the reduction in the hot phonon lifetime observed with increasing

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GaN@ 300 K

Hot phonon lifetime (ps)

10

1

0.1

16

10

17

10

18

10

19

10

10

20

Plasma density (cm-3) Figure 3.133 Phonon lifetime as a function of the photoexcited electron density or the plasma density in the range of slightly under 1016 cm3 to slightly over 1019 cm3. The solid stars are for data obtained in bulk GaN by time-resolved Raman scattering [530], the pentagon symbol also represents the Raman data [483], the filled circles represent Raman data [492], the upward

pointing triangle, diamond, and square symbols represent the data obtained by fluctuation technique in the laboratory of Prof. A. Matulionis between 2003 and 2006, the down-pointing triangle is for an AlGaN/GaN heterostructure obtained by Wang et al. [531]. Courtesy of A. Matulionis.

electron density would most likely reverse the course and begin to increase again above 1019 cm3. Having discussed the case of AlGaN/GaN FET structures and implications of hot phonons on device performance, let us turn our attention to other designs. On the device design side, a large LO phonon energy, a small Fr€ ohlich coupling (small

140

GaN

Energy (meV)

120 100

LO phonon

80 60

Bulk plasmon

40 20 0 16 10

Acoustic phonons 17

18

10 10 10 -3 Electron density (cm )

19

Figure 3.134 Representation of LO phonon, acoustic phonon, and plasmon energies versus the electron concentration in GaN indicating that plasmons would begin to play a role in determining the hot phonon lifetime when their energy exceeds that of the LA phonon energy and

20

10

would be a dominant mechanism for electron concentrations above 1019 cm3 where a reduction in lifetime with increasing electron concentration would reverse its course and begin to increase with further increases in electron concentration. Courtesy of A. Matulionis.

3.9 Hot Phonon Effects

electron phonon coupling), and if hot phonons are to be formed, then a short hot phonon lifetime are desirable. The overwhelming data generated so far appear to indicate that the AlGaN/GaN system is heading to a limit in terms of the hot phonon lifetime, which would cause limitation in power dissipation as well as the electron velocity as outlined above. One fine point, however, did not receive much attention and that is the effect of strain and also any effect that the barrier may play. Although preliminary in nature and the extent of the effect is somewhat not obvious, the use of lattice-matched InAlN barriers for GaN channel appears to lower the LO phonon lifetime and therefore allows better power dissipation and attainment of relatively high electron temperature and carrier velocity. Let us now discuss the hot phonon relaxation and carrier velocity in InAlN barrier GaN-based HFET structure even though the field is evolving. Under steady-state conditions, the supplied power must be equal to the power dissipated by hot electrons. In the electron temperature approximation, the dissipated power per electron Pd is expressed in terms of the hot electron energy relaxation time ten, which is defined as ten ¼ k

dT e ; dP d

ð3:341Þ

with Te being the electron temperature, which can be obtained from the measured noise temperature in that they are nearly the same within several percentile points. Knowledge of the noise temperature together with Equation 3.341 allows us to determine the dependence of the energy relaxation time on the supplied electric power as illustrated in Figure 3.135 with the solid line [532].

Noise temperature (K)

10

4

AlInN/AlN/GaN 7 µm 10 GHz 291 K 10

3

10

2

100 ns 2 µs 0.01

0.1

1

Supplied power (nW

10

100

electron-1)

Figure 3.135 Dependence of the noise temperature on the supplied power per electron Ps ¼ VI/N (V, I, and N represent the voltage, current, and the number of electrons, respectively) for the Al0.82In0.18N/AlN/GaN 2DEG channel studied at room temperature.

Voltage pulse duration: 100 ns (triangles), 2 ms (circles). The solid curve is a fit to symbols, and the dash–dotted line depicts the linear dependence A þ BPs line. The thin dashed lines represent an estimate of the limits of accuracy. Courtesy of A. Matulionis.

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10 AlInN/AlN/GaN AlGaN/AlN/GaN,simulation AlGaN/AlN/GaN,experiment GaN, simulation

1

Energy relaxation time (ps)

606

5 4 3 2 1 0 0 1 2 -1 Supplied power (nW electron )

0.1

0.01

0

50

100

150

200

Supplied power (nW electron-1) Figure 3.136 Dependence of hot electron energy relaxation time on supplied power per electron for the Al0.82In0.18N/AlN/GaN 2DEG channel investigated at room temperature (solid line). The dashed lines around the solid line represent the estimated accuracy. Noise experiment on

AlGaN/AlN/GaN (open squares). Monte Carlo simulation (solid symbols): AlGaN/AlN/GaN (solid squares) and GaN (solid circles). The inset shows the data in the power range of 0 < P < 2 nW electron1. Courtesy of A. Matulionis.

Let us now attempt to determine the energy relaxation time in the AlInN barrier system and compare with the more conventional AlGaN/GaN system. Recall that the hot electron energy relaxation time depends on the interaction by the electron with both acoustic and LO phonons. The acoustic phonons prevail at low electric fields, but the electron–electron collisions play an important role as well. According to the experimental data shown in Figure 3.136 with the solid curve, the associated energy relaxation time is 6 ps. As discussed on many occasions, the LO phonons come into play at moderate-to-high fields, particularly in GaN as the LO phonon energy is relatively high at about 92 meV. The transition from one mechanism to the other is illustrated by a steep decrease of the relaxation time (Figure 3.136, solid curve), which takes place at hot electron temperatures below 500 K (Ps < 2 nW electron1). In this range, the results for the Al0.82In0.18N/AlN/GaN 2DEG channel (solid curve) are close to the experimental data for AlGaN/AlN/GaN (open squares) and those of the Monte Carlo simulation (closed squares). However, at a higher power levels, that is, Ps > 2 nW electron1, the electron energy relaxation is primarily determined by the electron–LO phonon interaction. The energy relaxation time changes slowly in the range 2 < Ps < 12 nW electron1 and is close to the dash–dot line as depicted with the experimental data of Figure 3.135. The latter leads to a constant value of ten ¼ 650 fs. Furthermore, according to the fitted solid curve in Figure 3.135, the energy relaxation time does not remain constant (Figure 3.136, solid curve) in that it continues to decrease and reaches the values of 400 fs and 75 fs at power of Ps 15 nW electron1 and 200 nW electron1, respectively. The shortest obtained energy relaxation time of 75  20 fs is several times less than the average experimental value of 360 fs reported for AlGaN/AlN/GaN (Figure 3.136, open squares).

3.9 Hot Phonon Effects

In the region where the electron–phonon interaction is responsible for power dissipation, Ps > 2 nW electron1 and Te > 500 K, the hot electron energy relaxation time for the Al0.82In0.18N/AlN/GaN 2DEG channel is longer than that predicted by Monte Carlo simulations for an electron concentration of 1017 cm3 (Figure 3.136 solid line) when the hot phonons play a negligible role. However, hot phonons play a major role for a density of 1019 cm3, which is representative of the measured Al0.82In0.18N/AlN/GaN sample. Monte Carlo simulations for the AlGaN/GaN structures under the assumption of constant energy relaxation time predict a linear increase of the electron temperature with the supplied power. The data of Figure 3.136 (solid line) for the Al0.82In0.18N/AlN/GaN sample show decrease of the energy relaxation time with increase of power supplied, which implies that the effect of hot phonons is not as important in this system as it is in the AlGaN/GaN system. Let us now estimate the hot phonon lifetime for a range of supplied power wherein electron–hot phonon interaction is the dominant power dissipation route and the hot electron and hot phonon system forms an isolated hot subsystem due to slow decay of hot phonons to acoustic phonons and thus weak coupling to the thermal bath. In this scenario, the hot electron and hot phonon temperature and lifetime can be assumed to be nearly the same (equivalent hot phonon temperature being several percentile point lower). This allows the interpretation of the data, Figure 3.136 (solid line), in terms of the dependence of the hot phonon lifetime on the hot phonon temperature, as reconfigured in Figure 3.137 (dashed line). In short, the hot electron lifetime in the AlGaN/GaN system, which is a few percentile points higher than the hot phonon temperature, is about 360 fs. This compares with 75  20 fs for the Al0.82In0.18N/AlN/GaN 2DEG channel, which implies that much higher electron temperatures and also much higher electron velocities are possible

Hot phon on life tim e (fs)

1000

100

291 K AlInN/AlN/GaN AlGaN/GaN

10

0

500

1000

1500

2000

2500

Hot phonon temperature(K) Figure 3.137 Effective hot phonon lifetime as a function of equivalent hot phonon temperature for Al0.82In0.18N/AlN/GaN 2DEG channels (solid curve) and AlGaN/AlN/GaN (squares). Courtesy of A. Matulionis.

3000

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608

3×107

2×107

1×107

AI0.82In0.18N/AIN/GaN AIGaN/GaN AIGaN/GaN GaN AIGaN/GaN/AIN/GaN

0 0

50

100 150 200 Electric field (kv cm-1)

Figure 3.138 Dependence on electric field of the estimated electron drift velocity for an Al0.82In0.18N/AlN/GaN channel at room temperature (filled circles). The duration of voltage pulses used to extract the data is 3 ns and the channel length of the fabricated sample is 7 mm (filled circles). The line represents the

250

velocity field characteristics assuming v ¼ m0E. Monte Carlo simulations for an AlGaN/GaN calculated with hot phonons taken into account are shown with square symbols. Experimental data for an AlGaN/GaN are indicated with diamonds and those for GaN with stars. Courtesy of A. Matulionis.

with this system as compared to the AlGaN/GaN system, which is the next topic of discussion. The carrier velocity in the nearly lattice-matched Al0.82In0.18N/AlN/GaN heterostructure has been obtained at room temperature from pulsed current–voltage measurements [533]. It should be pointed out that in these measurements no current saturation was observed even at fields up to 150 kV cm1 because of the minimization of Joule heating, which bodes well for the accuracy of the measurements. Assuming a uniform electric field and field-independent electron density, the maximum electron drift velocity has been estimated at 3.2  107 cm s1 at 150 kV cm1, as shown in Figure 3.138. The velocity value was deduced from the current–voltage measurements conducted by using nanosecond pulses to avoid heating. The absence of dependence on the channel length indicates the lack of injection of carriers from contacts. A straightforward calculation indicates that charge that can be injected into the dielectric body between the source and the drain electrons is some three orders of magnitude smaller than the available charge in the 2DEG. Any negative differential mobility is not an issue in these experiments conducted in the high density of electron and LO phonon as the electron runaway due to electron–electron and electron–LO phonon interaction tends to be excluded, which increases the threshold for intervalley scattering, and thus the drift velocity continues to increase with field. Voltage pulse widths of 1 and 3 ns were used with only 5% decrease in current for the longer pulse indicating the absence of heating. Therefore, the 3 ns pulses have been relied on more in the experiments as they lead to more accurate results. The experimental results for the investigated 2DEG channel formed in an Al0.82In0.18N/AlN/GaN structure (filled circles in Figure 3.138) are compared with

3.10 InGaN Channel and/or InAlN Barrier HFETs

those for AlGaN/GaN (diamonds) [534] and doped GaN (stars) [535]. A reasonably good agreement of the bullets and the diamonds is obtained at fields below 20 kV cm1. At high electric fields, no tendency for velocity saturation is observed (bullets and stars). The experimental results (filled circles, stars, and diamonds) also show no negative differential mobility. The latter observation contradicts the results of Monte Carlo simulation: the negative differential mobility due to intervalley transfer has been predicted for AlGaN/GaN 2DEG channel (squares) [476]. The highest electron drift velocity 3.2  107 cm s1 (bullets) is reached at 150 kV cm1 (filled circles). This value is comparable to 3  107 cm s1 at 215 kV cm1 reported for low-mobilitydoped GaN (stars). Monte Carlo results for AlGaN/GaN 2DEG channel (squares) stay above the experimental data (filled circles) at fields 20 < E < 60 kV cm1 and are below the bullets at 70 < E < 150 kV cm1. In addition to exploiting the possible effect of strain, more likely much more is unknown at this time, the other channel materials with somewhat weaker electron–phonon coupling can be explored. Among the three binaries available in the GaN system, InN is inherently better in all of the aforementioned aspects than GaN. It is therefore desirable to develop InN-containing channels, such as InGaN. In fact, if the technological developments permit, even InN channels should be considered provided that one does not compromise on the drain breakdown strength. The critical issue here is the technology of depositing high-quality InGaN on GaN and capping it with a barrier. The available experimental data in this regard are summarized below.

3.10 InGaN Channel and/or InAlN Barrier HFETs

The role of an InAlN barrier in an otherwise conventional GaN-based HFET in terms of hot electron and phonon lifetimes and electron velocity is discussed in the previous section. Motivation for exploring InGaN channels is multifaceted and includes potentially increased mobility and velocity and intriguingly reduced hot phonon generation. As in the case of the GaAs wherein InGaAs channel HFEToutperformed other GaAs-based FETs, InGaN at least in theory is expected to provide enhancement of the device performance owing to intrinsically higher mobility and velocity. This has not yet proven itself to be the case experimentally because of in part technological impediments such as phase separation into higher and lower InN fraction regions, which cause carrier localization at low fields, and local piezoelectric-induced electric field beyond that which is applied externally. The need to grow InGaN at relatively low temperatures and juxtaposed AlGaN at relatively higher temperatures aggravates the experimental methodology in preparing these structures. As mentioned, the case of InGaN is of special importance because GaN has a much larger phonon–carrier coupling coefficient (Fr€ohlich coupling) than that of InN, thus InGaN, as tabulated in Table 3.8. This takes on a special meaning in GaN-based FETs because of high hot LO phonon densities, which in turn limit carrier velocity and heat removal causing havoc to device reliability. Despite technological impediments, it is deemed beneficial to

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discuss first FETs with InGaN channels and AlGaN barriers followed by a discussion of FETs utilizing InAlN barriers, which are lattice matched to GaN to avoid misfit strain, as any strain in general particularly in a semiconductor such as GaN where it is coupled to polarization does not bode well for prolonged device operation. High-quality InGaN has been difficult to implement due to the large differences in parameters of the constituents, leading to, among other detrimental effects, phase segregation. The use of the one-mode behavior of selected w€ urtzite ternaries to reduce phonon emission as well as designing device channels and interfaces to produce phonon envelopes that result in minimal carrier energy loss due to interface, confined, and half-space modes, would lead to reduced velocity degradation [513, 536]. In addition, the related dimensional confinement of phonons can be exploited to realize reduction of the phase space and the consequent decrease in carrier energy losses due to scattering with phonons in an effort to increase carrier velocity. Other mechanisms to enhance carrier mobilities such as coherent phonon effects and related Cerenkov processes [537–539], hot phonon effects, and runaway transport can be investigated. These latter processes have been considered primarily for the case of bulk phonons, and there is a need to extend these investigations to interface phonons. As mentioned, experimental investigations of InGaN channel FETs are dwarfed by the focused work on GaN channel varieties. As the GaN channel once became mature and limitations noted, forays into InGaN began to take place. In this vein, Maeda et al. [540] prepared In0.06Ga0.94N channel AlGaN/GaN heterostructures for FETs and reported room-temperature and 77 K mobilities of 850 and 7600 (sheet carrier concentration ¼ 7.6  1012 cm2) cm2 V 1 s1, respectively, but with no device results. Simin et al. [541] investigated similar structures with the room-temperature Hall mobility and sheet carrier concentration of 730 cm2 V 1 s1 and 1.1  1013 cm3, respectively. Owing to the difficulty of growing InGaN channel HFETs with AlGaN barriers, the reports in the literature are limited and typical Hall mobilities for InGaN channels are in the range of 200–850 cm2 V 1 s1 at room temperature [541–543]. Using 1.5  200 mm2 devices, Simin et al. [541] obtained a CW power level of 4.3 W mm1. Hsin et al. [544] obtained 2DEG carrier concentration of 3.4  1013 cm2 and a Hall mobility of 174 cm2 V 1 s1 at 300 K. A transconductance of around 65 mS mm1, a current gain cutoff frequency of fT ¼ 8 GHz, and an fmax ¼ 20 GHz were 1 mm gate length, respectively. Lanford et al. [542] reported InGaN channel devices, grown by MBE, with in relative terms better saturation characteristics compared to GaN channel ones and most importantly no dispersion between the pulsed and CW I–V measurements, indicating little or no current lag that besets the GaN variety. On the contrary, doped-InGaN channel InGaN/GaN devices reported by Hsin et al. [544] showed DC output characteristics without any anomalies when the drain voltage was limited to below 15 V. When the drain voltage exceeded 15 V, kinks appeared when the data were collected in dark. Those kinks were not present when the measurements were conducted in light. Eventually, channels made of InN might be explored because the transport characteristics associated with this material are very conducive for FETs in terms of relatively higher low-field mobilities and reduced electron phonon coupling, but pending to be fully investigated are the high-field strength, material quality and robustness, and thermal conductivity issues

3.10 InGaN Channel and/or InAlN Barrier HFETs

among others, not to mention technological difficulties in producing high-quality layers and inevitably large lattice mismatch and associated strain that would be encountered. It may be that the benefits might be outweighed by technological limitations. To avoid strain between GaN and AlGaN in an HFET structure, lattice-matched In0.18Al0.82N barriers on GaN channels have been explored in an effort to enhance FET performance. In addition, a relatively large band discontinuity ensues, which is comparable to that with 40% AlN mole fraction in AlGaN, but of course without the associated strain. Perhaps more importantly, defects in InAlN as compared to AlGaN even with much smaller bandgap are shallower. Furthermore, doping AlGaN with In, perhaps 1% or so, reduces the defect concentration substantially but without changing the energy levels [545]. These types of structures sometimes [546–553] associated with InGaN channel layers [554, 555] have been investigated reasonably well. High electron mobilities by using AlN spacer layers have been obtained as well, but with caution that the presence of AlN degrades device reliability [130] perhaps due to high fields present in AlN coupled with hot electron injection and predilection of Al to O. Electron mobilities of 1600 and 17 600 cm2 V 1 s1 for the nearly lattice-matched Al0.82In0.18N/AlN/GaN heterostructure with a sheet carrier density of 9.6  1012 cm2 have been obtained at room temperature and 10 K, respectively [129]. Kuzmik [554] calculated the current–voltage characteristics for AlInN barrier HFETs with various barriers and InGaN and GaN channel layers using the available analytical formulae. He suggested that 0.08
x
0.27 for a 15 nm thick InxAl1xN barrier or 0
y
0.18 for a 5–10 nm thick InyGa1yN channel could be implemented for strain without layer relaxation while the quantum well free electron densities up to 4.6  1013 cm2 and the transistor open-channel drain currents up to 4.5 A mm1 might be expected. Kuzmik et al. [555] undertook a comparison of HFETs fabricated using Al0.2Ga0.8N/GaN, In0.17Al0.83N/GaN, and In0.17Al0.83N/In0.10Ga0.90N channels. These authors showed that as compared to the AlGaN/GaN HFETs, HFETs based on InAlN/(In)GaN exhibit two to three times higher quantum well polarization-induced charge. Use of an analytical model to calculate InAlN/(In)GaN HFET drain currents and transconductances led to 3.3 and 2.2 A mm1 drain current for In0.17Al0 83N/In0.10Ga0.90N and In0.17Al083N/GaN HFETs, respectively. This represents up to about 200% current enhancement as compared to AlGaN/GaN HFETs, which was argued to potentially lead to high power performance. Pursuing the matter experimentally with 1 mm gate-length HFET, the same group [546] achieved 0.64 A mm1 drain current at VGS ¼ 3 V and 122 mS mm1 transconductance, respectively, falling quite short of expectations, most likely due to technological ailments such as the low mobility obtained as compared to 1600 cm2 V 1 s1 reported by Xie et al. [129]. HFETs reported by Xie et al. [129] having gate dimensions of 1.5  40 m2 and 5 mm source–drain separation exhibited a maximum transconductance of 200 mS mm1, with good pinch-off characteristics, and over 10 GHz current gain cutoff frequency. Somewhat similarly, Medjdoub et al. [552] obtained current gain cutoff and maximum oscillation frequencies about 26 and 40 GHz in HFETs with 0.25  50 mm2 dimensions, respectively at a drain bias of VDS ¼ 10 V. A maximum drain current density of more than 1.3 A mm1 with a pinch-off breakdown voltage

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about 40 V without any passivation was also noted. It should be mentioned that InAlN, although not to the same extent as InGaN, suffers from phase separation and thus localization effects. In addition, growth of InAlN on InGaN has the added advantage in that both layers can be grown at the same temperature or similar temperatures avoiding the conflict of an otherwise inevitable compromise that results in quality degradation. It is also noteworthy that high mole fraction AlGaN barriers with 40% AlN mole but doped with In have also been used coupled with very short channel length to obtain very high cutoff frequencies in an effort to explore the potential cutoff frequencies in HFETs based on GaN. To reiterate, owing to catalytic effect of In at the AlGaN growth temperatures (hardly if any In would incorporate), defect concentration is reduced and very likely smoother heterointerface is obtained. Pursuing the high-frequency performance potential, by using a 60 nm gate In-doped AlGaN/GaN heterostructure field effect transistors featuring a 6 nm thick InAlN barrier with an Al composition of 0.86, Higashiwaki et al. [556, 557] achieved a maximum drain current density of 1.34 A mm1 and a maximum extrinsic transconductance of 389 mS mm1. The smallsignal performance of the devices showed a current gain cutoff frequency of 172 GHz and a maximum oscillation frequency of 206 GHz, indicating the superior nature of InAlN barriers over the AlGaN (8 nm)/GaN HFETs with the same gate length.

3.11 FET Degradation

Despite great progress and strides made over the years, the RF high power operation of AlGaN/GaN HFETs (mainly owing to high voltage capability) is unfortunately accompanied by a reliability problem manifesting itself as degradation in the DC current and RF output power continually as a function of time, as depicted in Figure 3.142. Generally, GaN HFETs, due to the high electric fields involved and coupling of thermal, polarization, and mechanical properties, exacerbated by large density of structural and point defects, have unique reliability problems, which have not yet been addressed fully. The field distribution in the channel can be calculated as well as measured. The calculated electric field distribution in a GaN/AlGaN FET with Si3N4 passivation and gate dielectric without and with a field plate is shown in Figure 3.139. The field plate extension is 0.4 mm. The calculations have been performed for a gate–source voltage and the drain–gate voltage at 4 and 50 V, respectively. The field/potential distribution has been measured by cleaving the HFETs and measuring the potential distribution using Kelvin probe [558]. The results of such an effort are shown in Figure 3.140 for cases without and with a Si3N4 dielectric passivation for us to gain an appreciation of the extent of the potential in the channel. It is too early at this time to reconcile the differences between the calculations and the measurements. The field distribution can also be altered by using a doped layer on the surface such as an n-GaN cap layer. This layer would also have the added advantage of mitigating the virtual gate extension due to electron injection from the gate metal. Figure 3.141 shows simulations for the field

3.11 FET Degradation

Figure 3.139 The electric field distribution between the gate and the drain of an HFET having a dielectric surface layer, including between the gate metal and the semiconductor, for cases with (a) and without the field plate (b). The gate–source voltage and the drain–gate voltage are 4 and 50 V, respectively. The field plate extension is 0.4 mm. Courtesy of R. Trew. (Please find a color version of this figure on the color tables.)

distribution for two cases: (a) for an undoped AlGaN-cap case and (b) for an n-GaN cap case [451]. In the case of the undoped AlGaN-cap layer, the electric field reaches nearly the breakdown electric field strength, degrading reliability. On the other hand, when an n-GaN cap layer is employed, the electric field can be suppressed by piezoelectric charge between the n-GaN cap layer and the underlying n-AlGaN layer, which forms the barrier. Attempts to go around the problem by employing field plates naturally hurt the high-frequency performance. The high electric field and the coupled phenomena paving the way for defect generation could have been the cause of sudden failure and must be considered together with the ubiquitous wear-related reliability problems. The reliability in regard to electronic devices was recognized by the military personnel before electronics became such an intricate and important part of the society. The issues faced by compound semiconductors and remedies, along with a historical look at the memory lane, have been provided by Roesch [559]. Represented in Figure 3.142a is a typical longevity performance observed in early 2003, and Figure 3.142b shows some of the best performance in 2006. Consistent with problems uniquely associated with technologically available GaN, the best performance is not consistently obtained and there is significant variation from device to device and from wafer to wafer. Although the degree of performance degradation varies with the device design, processing steps, and device manufacturer, all high-voltage AlGaN/GaN HFETs are affected and the problem becomes more acute when the device is scaled to reduced dimensions and the operating frequency is increased. The time-dependent reduction in current and RF power due to virtual gate extension is generally a reversible process, and removal of the drive source and a period of inactivity generally cause the device to return to its original or nearly original state. However, it is observed that reliability is a strong function of gate leakage current and that a “sudden reliability” problem [560] also exists when the electric field achieves a certain critical magnitude. Under these conditions, permanent damage is

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Figure 3.140 Variation of cross-sectional electric field distribution in an AlGaN/GaN HFET without an SiNx passivation layer under the gate (a) and with an SiNx passivation layer (b) for gate bias voltage of 5 V and drain bias voltage of 40 V. The unit of color bar is MV cm1. Courtesy of Y. Nanishi. (Please find a color version of this figure on the color tables.)

done to the semiconductor, with resulting performance degradation that does not recover with any length of recovery time. Depending on the failure rate versus operating time, there are three regimes of failure, which are depicted schematically in Figure 3.143 in the form of failure rate versus operating time. Some devices suffer from what is called the early (infant) failure followed by random failure, which then enters into the realm of wearout failure, altogether leading to a bathtub-shaped dependence on time. Ideally, it would be desirable to eliminate the infant and random failures and reduce the wearout failure rate to the extent possible.

3.11 FET Degradation

Change in Pout, I DS (%)

Figure 3.141 Calculated electric field distribution with undoped AlGaN surface layer with SiN passivation (a) and with doped GaN surface layer with SiN passivation. Courtesy of T. Kikkawa, Fujitsu Laboratories [451]. (Please find a color version of this figure on the color tables.)

0

IDS (self -bias current) during operation

-10 -20 ∆Pout(%) ∆IDS(%)

-30

Output power during operation

-40 0

5

0

400

(a)

10 15 20 RF stress time (h)

25

Pout, (dB m)

45

40

35

(b)

800

1200

1600

RF stress time (h)

Figure 3.142 RF output power and IDS versus RF stress time for AlGaN/GaN HFETs (a) circa 2003 in terms of % change in output current and power; (b) circa 2006 for five devices in terms of output power. Courtesy of R. Trew.

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Failure rate, λ (t)

Early (infant) failure period Random failure period

Wearout failure period Total failure rate

Wearout failure rate

Random failure rate

Time, t Figure 3.143 A schematic representation of three failure regions, namely, the infant failure experience very early on, the random failure period, and the wearout failure. Courtesy of A. Christou.

The genesis of the reliability problem is rather difficult to know unless the specific failure mechanisms are illuminated. Typically, there are both reversible and sudden or gradual catastrophic (nonreversible) failure mechanisms simultaneously at play. Reversible degradation, such as the one caused by nonregenerative point defects capturing electrons temporarily, is characterized by recoveries with time constants congruent with the dominating trap/defect. The catastrophic failure mechanisms can be attributed to defect generation and metallurgy/mechanical (strain-induced) failure. Possible mechanisms contributing to HFET failure are schematically depicted in Figure 3.144.

Figure 3.144 Effects that might contribute to HFET instability and failure. In part courtesy of R. Trew and G. Bilbro.

3.11 FET Degradation

3.11.1 Reliability Measurements

A typical procedure to ascertain lifetimes of various types of devices is the accelerated life test to attain a quick assessment of projected lifetime. Thus, critical issues for accelerated life testing are that they must provide estimates of device reliability in a time much shorter than that required to produce a significant number of failures under normal operating conditions. To be successful, the tests must stress the largest possible number of devices in a controlled manner without introducing artifacts (i.e., unrealistic failure modes). The accelerated tests normally adopted, which must definitely be augmented by RF and pulsed IV tests, and the various combinations thereof as they, among others, stress the gate by pushing forward operating mode during part of the RF swing, in particular for power HFETs, are [561] (a) High-temperature storage test (HTS): Unbiased samples are stored at different temperatures to accelerate thermally activated failure mechanisms, such as interdiffusion processes, occurring at the metal/semiconductor and semiconductor/semiconductor interfaces. (b) High-temperature operating life test (HTOL or HTOT): Samples are stored at different temperatures and biased in conditions similar to those experienced by the device during normal operations, aiming at studying the combined effects of thermal and electrical stresses. This approach in the form of the three-temperature plot is used to determine the MTTF Arrhenius plot of Figure 3.146, which is very limited and does not really represent the real operating conditions of the device. (c) High-forward gate current test (HFGC): The Gate Schottky diode is forward biased to investigate the effects of high-current densities (at high temperatures). (d) High-temperature reverse-bias test (HTRB): The Gate Schottky diode is reverse biased close to breakdown voltage to observe the cumulative effects of high electric fields and temperatures. (e) Temperature–humidity bias test (THB) and highly accelerated stress test (HAST): Samples are biased in a high-temperature and high-relative humidity environment. To increase both temperature and relative humidity, a pressure cooker is employed in the HAST test. Usually, the gate diode is reverse biased to analyze the effects of humidity directly on the chip and the protection capability of the passivation layers and of plastic packages. Sometimes the accelerating factors are not kept constant during the test but are increased at defined times, giving rise to the so-called “step stress” test, which can be either thermal or electrical. The purpose of this test is twofold: (i) it can give very quick information on the limiting conditions for the investigated technology and (ii) step stresses are used (with carefully defined time intervals) to obtain reliability predictions.

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Accelerated DC testing including tests at elevated temperatures do not gage the device under operating conditions representative of the real operating environment. A clear advantage of RF life tests over the above-mentioned tests is that the devices operate very close to the actual system working conditions, so that reliability predictions can be more accurate and representative. In particular, for high power devices, large RF signals can drive the devices in electrical conditions not experienced during DC life testing. However, it is not easy to control RF working conditions during life testing, so that it is possible to introduce spurious failure mechanisms due to overstress, input/output impedance mismatching in particular at high temperature, and so on. Moreover, the same failure mechanisms enhanced by RF life testing, such as electromigration caused by forward gate conduction or degradation due to operation close to gate–drain breakdown, can be induced by proper DC tests such as HFGC or HTRB, which can be performed under wellcontrolled conditions. This thermal method, however, is a limited test and does not consider RF biasing and pulsed operation, wherein the gate voltage is swung in the forward direction among others. Another aggravating factor is that it is difficult to measure the peak junction temperature, particularly when a field plate is used that obstructs optical access to the semiconductor. In addition, the junction temperature at the heterointerface should be measured as opposed to the surface temperature. For the abovementioned reasons, it is prudent to perform a series of tests as outlined in the following paragraph augmented by various RF, pulsed measurements, and their combinations thereof. The infant mortality and failures of similar manifestation must also be investigated, which is discussed later on in this section. For parameter monitoring, it is generally preferable to measure both DC and RF parameters in spite of the fact that in some cases a clear correlation was found between DC and RF degradations. In the case of DC parameters, it is, however, recommended not only to limit the characterization to classical transistor parameters (drain saturation current (Idss), pinch-off voltage (Vp), transconductance (gm), and so on) but also to include the measurement of other parameters, such as parasitic resistances, gate diode characteristics, and so on, which can help correctly identify the actual failure mechanisms. In the thermal characterization methods, accurate evaluation of the channel temperature of devices under test is needed to correctly accelerate the different failure mechanisms and to evaluate their activation energy (see Section 3.8 for a partial discussion). The channel temperature (Tch) of an electronic device is conventionally described as the sum of the case temperature (Tcase) and the product of the power dissipation (PD) by the thermal resistance (Rth), that is, Tch ¼ Tcase þ PDRth. To evaluate Rth of microwave MESFETs, the electrical method based on the current–voltage forward I(V) characteristics of the gate Schottky barrier diode is widely used because it enables Tch (DVgs) to be evaluated through a calibration curve, so that Rth is then obtained from the knowledge of Tch (DVgs), Tcase, and PD. It should be noted that Rth is itself a function of the temperature. In operating conditions or during accelerated life tests, however, the power dissipated in the active device areas leads to a nonuniform increase of the device

3.11 FET Degradation

temperature. Tch (DVgs) is therefore an unknown weighted average of the temperature distribution on the device and can be very inaccurate, in particular if a small area of high temperature exists within the structure. The actual temperature distribution on the chip can be measured by liquid crystal techniques or directly observed by means of high lateral resolution infrared (IR) thermography (IR near-field optical microscopy and m-Raman spectroscopy mapping can also be used), which allows one to detect the thermal gradients caused by local differences in the heat dissipation or by structural inhomogeneities. 3.11.2 GaN HFET Reliability

The defect-related anomalies surrounding GaN HFETs are treated in great detail in Section 3.5.5. All of these are a form of degradation, including the reversible ones, and therefore must be addressed if the goal is to attain long operating lifetimes under realistic conditions, which are typically simulated in the laboratory through a series of tests, mainly accelerated life testing. Accelerated life testing is typically performed at three different but elevated temperatures in which cumulative failure rates are recorded and the mean time to failure (MTTF) times for each of the three temperatures are established. The failure is defined as, for example, reduction in the drain current by 10%. RF testing can and must also be used. The MTTF figures so determined are then converted to a single Arrhenius plot wherein the MTTF values are plotted in a log scale as a function of inverted temperature. Extrapolation to a given temperature then determines the expected MTTF for an HFET operating at that temperature, more applicably the channel temperature, which is always higher than the case temperature. For example, the cumulative failure rates measured at 260, 285, and 310  C in a lot of GaN power HFETs are plotted in log–log scale versus the temperature as shown in Figure 3.145 what is typically referred to as the three-temperature life test. The Arrhenius plot of activation energy obtained is shown in Figure 3.146, indicating an extrapolated operating lifetime greater than 107 h at an operating temperature of 150  C which when solely relied upon looks very good. The reliability problem in GaN HFETs has been generally attributed by the practitioners mainly to surface processing and passivation because significant performance variations are observed as these steps are altered. For example, it has been shown that a basic mechanism involving quantum mechanical tunneling of electrons from the gate electrode on the drain side to the surface of the semiconductor exists, particularly when subjected to high terminal voltages, as discussed in detail in Section 3.5.5.4. Electrons leaking from the gate electrode to the surface of the semiconductor have been shown to affect the reliability [562]. The effect of hot electrons and hot phonons, discussed from the basic physics point of view in Section 3.9, on potential longevity will be discussed after the discussion of the gate current. In addition, traps that are located in the bulk AlGaN and GaN, as well as any at the heterointerface including those that might be generated during operation/ testing would not bode well for reliability, particularly in the presence of hot electrons

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Cumulative fail (%)

99

310°C

50

10

285°C 260°C

1 1

10

1000

100

5000

Time (h) Figure 3.145 Three temperature life test data showing the cumulative failure rates versus time obtained at three different temperatures, namely, 260, 285, and 310  C. The s values for 180 and 285  C are approximately 1, but the s value at 310  C is 1.5, which might imply contribution by the infant failure. Courtesy of Dr A. Hanson of Nitronex.

8

10

Operating at 150ºC 7 leads to MTTF>10

7

10

6

MTTF (h)

10

5

10

4

10

3

260ºC

10

2

285ºC

10

1

E a = 2.0 eV

310ºC

10

0

10

1.6

1.8

2.0

2.2

2.4

Temperature (1000 K -1 )

Figure 3.146 Arrhenius plot of the mean time to failure determined at three different temperatures 260, 285, and 310  C, which lead to an activation energy of about 2 eV and an extrapolated MMTF value greater than 107 h at 150  C. Courtesy of Dr A. Hanson of Nitronex.

and hot phonons. Furthermore, metallurgy-related reliability problems such as phase change and electromigration would have to be considered. When a high drain bias voltage is applied, imperative for high power operation, and the HFET is driven with a large RF input signal, the peak voltage at the drain can attain a magnitude essentially twice the magnitude of the DC bias voltage in class A operation. Simulations indicate that the magnitude of the electric field at the edge of the gate electrode on the drain side can easily exceed 6–8 MV cm1, which is

3.11 FET Degradation

sufficient to produce quantum mechanical electron tunneling. The electrons that tunnel from the gate electrode can (i) accumulate on the surface of the semiconductor next to the gate (can be modified by surface passivation), (ii) move along the surface by a trap-to-trap hopping mechanism, creating a gate-to-drain leakage current, or (iii) if the energy of the electrons is sufficiently high, avalanche ionization could occur on the surface. If ionization does occur, light should be observed emanating from the gate edge. Light emission from the gate edge is often observed in the large-signal operation of GaAs MESFETs, but so far has not been observed with AlGaN/GaN HFETs. To gain an insight on the effect of gate leakage current on the stability of HFETs, the gate I–V characteristics for a device before and after RF stress are shown in Figure 3.147. Note that after the RF stress the I–V characteristics are essentially those of a tunnel diode with conduction in both the forward and the reverse direction. When the electrons accumulate on the surface of the semiconductor near the gate between it and the drain, a “virtual gate” effect is created causing an effective lateral extension of the gate toward the drain. As the electron tunneling proceeds, the density of electrons on the semiconductor surface increases. The electrons so accumulated on the surface create an electrostatic charge that causes a partial depletion of the conducting channel electrons, thereby causing a reduction in the channel current, and a corresponding decrease in the RF output power. Electron tunneling continues as long as the device is operating. Therefore, the degradation is a function of time, which creates a reliability problem. The shift in the forward turnon voltage and the magnitude of the reverse current are affected by the degree of electron tunneling experienced by the gate. The current conduction characteristics evolve as improved device design, surface processing, and passivation are implemented. It is possible to modify and reduce the tunnel leakage, for example, by using an optimized field-plate HFET design. However, the field plate, while effective in reducing the electric field by spreading it out somewhat degrades highfrequency performance by increasing the gate–drain capacitance. A better solution seems to lie in reducing if not eliminating the current conduction paths such as surface state and bulk defects in the barrier as well as improving the layer quality and optimizing device design for a better robustness. Typical performance degradation is shown in Figures 3.142 and 148. Specifically, Figure 3.148 shows the drain–source current and gate current as a function of gate voltage for increasing RF stress time. As can be seen, the gate diode reverse-bias conduction can vary significantly and by an order of magnitude or more as a function of time. The shift in the forward turn-on voltage and the magnitude of the reverse current is affected by the degree of electron tunneling experienced by the gate. The measured current conduction results, shown in Figures 3.142, 3.147, and 3.148, correlate with degradation in RF power output and are typical. The current conduction characteristics evolve as improved device design, surface processing, and passivation are implemented. It is possible to modify and reduce the tunnel leakage, for example, by using an optimized field-plate HFET design. Doing so provides the results shown in Figure 3.142b. As indicated, this device demonstrates minimal

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After RF stress

Gate–drain diode

IGD (mA mm-1)

2.0

1.0

0.0

-1.0 -4 -3

-2

-1

0 1 VGS (V)

3

2

4 Before RF stress

(a) Gate–source diode 1.00

IGS (mA mm-1)

0.75

Before RF stress

0.50 0.25 0.00 -0.25 -0.50 -0.75

After RF stress

0

2

4

6 VDS (V)

8

10

(b) Figure 3.147 Gate DC I–V characteristics for an AlGaN/GaN HFET before and after RF stress, (a) for the gate–drain diode and (b) gate–source diode. Courtesy of R. Trew.

degradation as a function of time. However, as indicated earlier, the field plate, while effective in reducing the electric field by spreading it out, degrades high-frequency performance. A better solution seems to lie in reducing, if not eliminating, the current conduction paths such as surface state and bulk defects in the barrier as well as improving the layer quality and optimizing device design for a better robustness. There are two current paths for gate leakage, as shown in Figure 3.149. The main path is established by an electron tunnel leakage from the gate with electrons flowing along the AlGaN surface to the drain contact. The electron conduction occurs by a trap-to-trap hopping mechanism, which is inherently very slow, where both thermionic emission and tunneling are likely involved for the source of electrons, as illustrated in Figure 3.150. It has been reported that there are five factors (i) the gate edge roughness, (ii) net charge at the AlGaN/passivation interface, (iii) dielectric constant of the medium

3.11 FET Degradation

ID (A mm-1)

0.90

t=0 h ∆t/step=25 h

0.85

t=100 h

0.80

∆t/step=50 h 0.75

t=600 h

0.70 4.0

4.2

4.4

5.0

4.8

4.6 VGS (V)

(a)

100

IG(A mm-1)

10-1 25 h

0h

10-2 10-3 600 h 10-4 -8

-6

75 h -4

-2

0

2

4

VGD (V)

(b) Figure 3.148 (a) ID and (b) IG versus VG for AlGaN/GaN HFETs as a function of RF stress time. Courtesy of R. Trew.

Figure 3.149 Possible gate leakage current paths, which contribute to device instability. “Normal” gate leakage produces surface conduction, and at a critical electric field conducting path to the channel is formed. Courtesy of R. Trew.

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Strained energy band Electron accumulation forms a “virtual gate”

Thermionic emission

E Tunnel emission

VDG =0 EF Critical parameters N ss: Surface state density m* tun VDG=V

Gate metal

AlGaN surface

Figure 3.150 Proposed gate–surface leakage current mechanisms. Courtesy of R. Trew.

above the HFET surface, (iv) nontriangular potential barrier, and (v) negligible angular tunneling from the gate edge [563]. However, there have been no experimental results proving these attributions. In one simulation effort that was undertaken to better understand the surface effects of GaN-based devices, devices utilizing a surface into which electrons can tunnel from the gate were used to generate good agreements with experimental values of drain current (VDS/IDS family of curves) as well as gate leakage (VDS/IG family of curves). The model is physically understood as electrons tunneling from the gate to the surface, and then hopping along surface states. This phenomenon results in leakage current (along the surface), although at some critical field a conducting channel into the AlGaN barrier layer is formed [564]. Both of these phenomena would give rise to a virtual gate, resulting in reduced current and transient analysis would give rise to lags mentioned above. Considering electrostatic feedback (electrons returning to the gate due to electronic attraction to image charges), transient responses of IDS and IGS could also be predicted [564]. 3.11.2.1 Gate Current At least some of the degradations mechanisms appear to have their roots in the gate current, particularly under RF stress. Preliminary simulations undertaken indicate that it is likely that the exact conduction mechanism changes as the electric field increases due to high DC and RF terminal voltages. This performance degradation mechanism is entirely reversible and nondestructive. Therefore, removal of the bias and drive signals, with a period of device inactivity, causes the device to return to its initial state. The second current path illustrated in Figure 3.149 consists of electron tunneling from the gate with electron flow through the AlGaN layer to the 2DEG conducting channel. This current path requires a higher electric field and often produces permanent damage to the AlGaN lattice with resulting increased gate leakage. The lattice damage is observed in TEM images as well.

3.11 FET Degradation

By using the gate tunnel leakage and surface conduction model, it is possible to simulate the drain and gate current characteristics with excellent accuracy in comparison to measured data. The gate tunnel leakage and surface conduction model accurately simulates both the drain and gate current characteristics, as shown in Figure 3.151. In addition, the model can be extended to simulate the timedependent drain and gate current performance, as shown in Figure 3.152.

0.20 Measured VGS=0V Modeled

IDS (A mm-1)

0.16

0.12

0.08 VGS=-2.5V 0.04 VGS=-5V 0.00 0

1

2

3

4

(a)

5 6 VDS (V)

7

8

9

10

0 ×100 VGS =-1 V

-2×10-5

VGS =-3 V

IGS (A)

-4 ×10 -5 -6×10 -5

VGS =-5 V -8 ×10 -5 Measured -1 ×10 -4

-1.2 ×10 -4 0×100 (b)

Modeled

2×10 -5

VGS =-7 V 4×10 -5

6×10 -5

8×10 -5

VDS (V)

Figure 3.151 Simulated and measured (a) IDS–VDS characteristics and (b) IGS versus VDS characteristics for an AlGaN/GaN HFET (red lines, measured data; blue, modeled data) Courtesy of R. Trew. (Please find a color version of this figure on the color tables.)

10×10 -5

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I D (mA)

48 47 46 45 0

100

300

200

400

500

Time (s) (a)

I G (mA)

-1×10 -3 -3×10-3 -5×10 -3 -7×10-3 0

100

200

300

400

500

Time (s) (b) Figure 3.152 Simulated and measured (a) stress time-dependent IDS–VDS characteristics and (b) stress time-dependent IGS versus VDS characteristics for an AlGaN/GaN HFET (lines: measured data; solid circles: modeled data). Courtesy of R. Trew.

The simulations are performed with a model that includes the effects of electrostatic feedback from the electrons that tunnel to the surface of the AlGaN layer adjacent to the gate electrode, as illustrated in Figure 3.149, which are in excellent agreement with measurements. The electrostatic feedback reduces the electric field at the edge of the gate electrode, thereby reducing the electron tunnel leakage. As electrons accumulate at the gate edge toward the drain as a function of stress time, the feedback produces reduced gate leakage current. Also, the increased electron density on the AlGaN surface partially depletes the 2DEG electrons, and a reduction in gate current occurs. Although the gate leakage can be attributed to both the current conduction along the surface and the vertical transport through the barrier to the channel at the heterointerface, it is still not well understood which mechanism is the most important. For GaN Schottky contacts, screw dislocations have been shown to increase the reverse current by 300–500 times while only a sixfold increase results from edge types [565]. However, in these experiments, the spacing between Schottky and ohmic contacts is sufficiently large so that the lateral field (along the surface) is not nearly as large as it is in an HFETdevice where the gate–drain separation is much

3.11 FET Degradation

shorter. More studies are necessary to determine whether the dislocations are really that important in the HFET devices. A complete model for GaN HFETs encompassing the effects of bulk, surface and interface defects, hot electrons and phonons together with electromechanical coupling, and current paths that may be in effect in high-voltage operation is missing. A model describing GaN HFETreliability may be built on existing GaAs HFETs models, expanding them to include the events unique to GaN as outlined above. In this vein, a model for gate tunnel leakage in GaAs MESFETs has previously been reported [566] and has been modified for use with AlGaN/GaN HFETs [567]. A modified model can be used in a harmonic-balance simulator to investigate the gate tunnel mechanism as a function of DC and large-signal RF operation conditions. The gate tunnel leakage can be represented as a current generator between the gate and drain electrodes. The model also includes RF breakdown within the conducting channel, which is represented by a current generator between the drain and source terminals. This particular model accurately simulates the DC and RF performance of AlGaN/GaN HFETs. Full RF test data should in time be available. However, suffice it to say, that RF tests show degradation to varying degrees of devices with no apparent reason for the observed dispersion even for devices from the same wafer. In one study in which the devices were subjected to a 24 h RF stress with VDS ¼ 30 V, IDQ ¼ 100 mA mm1(here Q stands for quiescent operating point) with RF input at the 3 dB compression point, and maximum power-added efficiency, the drain current in response to a gate-tosource voltage increased most likely due to increased gate leakage current, the gate lag increased by about 20%, and RF power output dropped between 0 and 2 dB. There also exist other cases wherein the devices seem to have operated with little to no change in RF performance at channel temperatures as high as 260  C. In another experiment, fixtured FETs were tested at two different temperatures of 135 and 260  C over nearly 6000 h with modest degradation at 260  C with a total output power loss of 0.6 dB over that period, which leads to a projected MTTF of 104 h. Degradation and false recovery observed is indicative of defect generation during RF testing. When the RF testing is resumed, rapid degradation ensues. Attempts have also been made to find a correlation between DC and RF stress in devices that showed rapid degradation. In one set of tests, the RF performance of the as-received and DC-stressed (12 h at 28 V drain bias and 100 mA mm1 drain current at a measured channel temperature of 110  C) devices remained nearly identical. However, subjecting the device to RF stress for 12 h caused 0.3 dB decrease in output power, while subjecting it to DC stress for another 12 h caused an additional 0.05 dB power loss. Returning to a 12 h RF stress caused an additional 0.1 dB power loss. The decrease in output power might be related to increased gate leakage current. Despite the observation of some long lifetimes, as mentioned previously, the device failure is not predictable with some devices failing quickly and some lasting a while. The consensus is that any permanent device failure does begin at the drain side of the gate where the electric field is the highest, so are the electron and phonon temperatures. The damage after a catastrophic failure done at the drain side of the gate is even notable in a few TEM images taken.

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A complete understanding of the “sudden failure” problem [560] requires the development of a model that can simulate the mechanical stress, coupled with pyroelectric and polarization effects, on the lattice resulting from high electric fields. Defects consisting of screw dislocations, misfit defects, cracks, and point defects that are certain to be generated under high field and elevated temperatures can be introduced. The lifetime of semiconductor devices is strongly related to the densities of dislocations or point defects. It is known that in GaAs-based devices the moderate dislocation density (104 cm2) can affect the operating life of the devices. Since in GaN-based structures the dislocation density is several orders of magnitude higher (108 cm2 or higher), the degradation rate can be related to the presence of extremely high dislocation density and simple and complex point defects associated with them. After discussing the heat and thus power dissipation exacerbated by high fields and associated hot electrons and phonons, let us now examine the possible effects of hot phonons on power dissipation and possible sources of device degradation. While the physics of hot phonon is discussed in Section 3.9, the particulars dealing with power dissipation and possible failure mechanisms are discussed in this section. 3.11.2.2 Metallurgical Issues In addition to degradation inherent to semiconductor, be it permanent or temporary, the metal–semiconductor diffusion, phase change in metal stacks, and electromigration within the metal are sources of potential degradation. Electromigration of the gate electrode metal, impurity activation, and contact diffusion effects are reasonably well understood particularly in conjunction with, for example, GaAs-based devices. However, GaN-based devices push the metallization technology to its limit causing some metallurgical change, particularly under prolonged high-current operation. Exacerbating the Schottky barriers is the presence of electric high field and gate leakage current (small gate cross section leads to high current densities). The Ti/Albased contacts on GaN are reasonably stable against oxidation and cracking when Al is sealed with Ni/Au [568] but not sufficiently stable for prolonged operation. However, Ti/Au seal over Ti/Al becomes rough after annealing with bad line definition, which leads to current filamentation and therefore premature breakdown. Mo/Au and Pt/Au seals have also been used with limited or no success leaving Ni/Au seal as the preferred choice. Ultimately, though, an n þ layer atop the AlGaN barrier would need to be used to not only reduce the contact resistance but also to ensure uniform current distribution. This would necessitate selective dry etching methods for gate recessing, which have already been explored for this purpose to some extent [569, 570]. The reliability investigations of GaN FETs have been relatively less developed in comparison to the GaAs counterparts, in fact to a larger extent due to large current levels/fields involved. As a result, both the gate and drain/source metallurgy undergoes change including phase change. Degradation of this kind is progressive and not reversible. Therefore, this process must be understood well and incorporated into the physics-based reliability model. As in the case of GaAs-based FETs, the extent of electromigration depends on factors such as conductor-line properties and any

3.11 FET Degradation

inhomogeneities as well as structural features of the conductor layout. Naturally, electromigration must be studied noting that the magnitude of defect transfer depends superlinearly on current density, which poses formidable challenge for GaN-based FETs. High temperatures, particularly applicable to gate and drain metallization, cause mass transport facilitated by short-distance diffusion associated with defects, such as dislocations, grain boundaries, interphase boundaries, and/or external surfaces. It is therefore imperative that we study and understand fully the mechanisms involved in electromigration and apply that knowledge to the reliability model. 3.11.2.3 Hot Electron and Hot Phonon Issues As mentioned on several occasions, already hot electrons and hot phonons due to high electric field could cause new defects to form, a process that is exacerbated by inhomogeneities and electromechanical coupling in a piezoelectric material such as AlGaN and GaN. Hot phonons are intimately related to reliability of the GaN-based FET devices in that the generation of large quantities of hot phonons will inevitably lead to the generation of defects. Obviously, one would like to minimize, if not fully eliminate, the quantity of defects in devices and when defects are being generated during the operation of a device, the reliability of the device is clearly in question. Existing defects and those so generated due to hot phonons/electrons have a G–R noise signature that can be monitored rather well with low-frequency noise measurements as has been applied to GaAs-based and GaN-based FETs previously [571–573]. Strong electron–phonon coupling in GaN afforded by the ionic nature leads to efficient phonon generation (perhaps as high as some 30 times stronger compared to that of GaAs), which in turn leads to hot phonon population as the LO phonon decay mechanism cannot keep up with generation. The hot phonons are capable of generating point defects particularly in the presence of point defects formed during growth and high local field (piezo nature coupled with inhomogeneities). This causes drift and degradation in device performance, which is not incorporated in many garden variety statistical models that are often applied to GaN. The hot phonon decay mechanisms are not well developed in GaN. Noncentrosymmetric materials such as GaN and AlN can be represented with the well-known triangle wherein stress, electric field, and temperature representing the corners of the triangle. The branch between the stress and the electric field represents the piezoelectricity, the branch between the electrical field and the temperature represents the electrothermal effect, and the branch between the thermal and the mechanical stimuli represent the thermoelastic effect, as shown in Figure 3.153. The detailed descriptions of connective processes that take place are also indicated between the properties such as strain, displacement, and entropy. This clearly means that field, temperature, strain are intricately interconnected to one another, all of which exist in GaN FETs by the drain edge of the gate with grave consequences to reliability. It is simply imperative to uncover the physics in effect, not just hot carrierinduced effects, for a comprehensive model that is realistic. What might not meet the eye is the double blow delivered by hot phonons. Hot LO phonon population occurs because of the very efficient electron–phonon coupling in GaN and relatively inefficient decay of LO phonon to LA phonon. High electric fields

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Electrical

ect

effec t Permittivity[2]

e pie zoele ctric

ce ffe ct Conv ers

f ic ef

of a pol at riz ion

i re ct p

calor

ct ffe ct al e 1] fe erm ) [ ef ic oth iciy ctr ctr ctr Ele roele ele ro (Py Py

ro Elect

Displace ment (0)

at He

Ele (Pi ctrom ezo ele echan ie c tr zo icit ical e el y)[ ffe ec ct 3] tri

Field (1)

D

Strain Heat of deformation Entropy 4] (2) (0) Heat ty[ i cap c Therm t i c t fe f s aci e al exp ty aloric E la ansion [0] Piezoc

Stress (2)

Mechanical

Thermal pressure

Tempe rature

Thermoelastic effect [2]

Figure 3.153 The well-known triangle used to describe the pathways between mechanical, electrical, and thermal energies in a class of noncentrosymmetric materials such as GaN and particularly AlN exhibiting piezoelectric effect,

Thermal pyroelectric effect (converse of electrocaloric effect), and piezo caloric effect (converse of thermal expansion). Tensor rank of the variable is shown in parenthesis and the tensor rank of the property is shown in brackets.

not only generate these phonons, which limit the velocity but also hinder heat dissipation, as they are not able to efficiently decay to LA phonons. The thermal conductivity of GaN is about 2.3 W cm1 K1 but only for LA phonons to the thermal bath. If the heat dissipation path is through the LO phonons to LA phonons to the thermal bath, which is the case here as shown in Figure 3.154, the LO phonon decay would be a limiting factor in heat removal also. We should note that a rudimentary device-level treatment of power dissipation and junction temperature measurements are detailed somewhat in Section 3.8. It is, therefore, imperative that the hot phonons are characterized unearthing their generation and decay to LA phonons culminating in determination of phonon lifetime. Naturally, as short an LO phonon lifetime as possible is preferred. In addition, the effect of hot phonons through localized high heat generation on defect generation must be considered. As mentioned above, strong hot phonon coupling along with inefficient hot phonon decay to LA phonons has an adverse effect on carrier velocity, which does not bode well for high-frequency operation. Therefore, structural modification that can be made to the GaN-based HFETs to prevent velocity degradation would be very helpful to the device operation. These include the use of InGaN channels and short-period superlattices, albeit with technological complications, as outlined in Section 3.10. The effect of hot phonons, degeneracy, and self-heating on electron velocity in GaN based devices on Monte Carlo simulations and experiments is shown

3.11 FET Degradation

Electric power

UI

Hot electrons LO phonons Negligible at high bias

Acoustic phonons

Thermal bath

Figure 3.154 A schematic representation of electric power dissipation paths in effect in GaN. Clearly, the LO phonon to LA phonon decay must be efficient before the LA phonon-to-thermal path heat transfer, which defines the thermal conductivity, can take place with tremendous

implications to heat removal from the junction. The hot electron–LA phonon path at high electric fields in FET channels with high hot electron concentrations, which would be inefficient for low hot electron concentrations, is negligible Courtesy of Prof. A. Matulionis.

Drift velocity (107cm s-1)

schematically in Figure 3.155. The top line depicts the velocity–field relationship that can be expected when hot phonon, self-heating, and degeneracy effects are neglected altogether. The second line from top is for the case where only the hot phonon-induced velocity degradation is considered. The third line from top considers degeneracy in addition to hot phonons, and the red line denotes hot phonons, degeneracy, and self-heating. The circles are experimentally determined values. It should be noted that when investigating systems with high electron densities, it is imperative that one accounts for the degeneracy of the electron gas, which can be handled by the rejection technique proposed by Lugli and Ferry [522]. We also should 1.5

No degeneracy No hot phonon No self heating

AlGaN/GaN 300 K Hot phonons

1.0

Degeneracy

No degeneracy No self heating No self heating

0.5

0.0

Self heating

0

+Hot phonons +Degeneracy +Self heating

5 10 15 Electric field (kV cm-1)

Figure 3.155 The effect of hot phonons and selfheating on the carrier velocity versus electric field as determined by Monte Carlo simulations. The top line depicts the velocity–field relationship that can be expected when hot phonon, selfheating, and degeneracy effects are neglected altogether. The second line from top is for the

case where only the hot phonon induced velocity degradation is considered. The third line from top considers degeneracy in addition to hot phonons and the bottom line considers hot phonons, degeneracy, and self-heating. Courtesy of A. Matulionis. (Please find a color version of this figure on the color tables.)

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point out here that the trend is the main focus rather than pinpoint accuracy of the values themselves. Impediment caused by inefficient hot LO phonon decay presents a formidable problem for AlGaN/GaN FETs, which is not sufficiently recognized and appreciated. Power dissipation by phonons is similar to that by electrons assuming that hot electron–LA phonon path is in effect, which is not for high electron concentration. Nevertheless, the physical expressions are similar. Power dissipation by a hot electron is given by Pd ¼ kðT e T 0 Þ=te ;

ð3:342Þ

where Pd is the dissipated power per electron, Te is the hot electron temperature, T0 is the equilibrium lattice temperature, and te is the hot electron energy relaxation time. As Equation 3.342 indicates, the smaller the hot electron energy relaxation time, the larger the power that can be dissipated per electron. Hot electron temperature and hot LO phonon temperature are comparable to one another. This follows then that the smaller the hot phonon energy relaxation time, the larger the power dissipation per phonon when, of course, the LO phonon to LA phonon path is the dominant power dissipation route, the case for large electron concentration, which in turn is the case in HFETs. The power dissipated per hot phonon is similarly given by Pd ¼


hwLO ðN ph N ph0 Þ ; tph

ð3:343Þ

where
hwLO is the LO phonon energy, Nph is the hot phonon population, Nph0 is the equilibrium phonon population, Tph is the hot phonon temperature, and tph is the hot phonon relaxation time. Utilizing the hot phonon population N ph ¼ ½expð
hwLO =kT ph Þ11 (note that 1 is the Bose correction to Boltzmann distribution) and the fact that hot electron and hot phonon temperature can be construed as the same, Te ¼ Tph, Equation 3.343 can be written as
wLO h ½expð
hwLO =kT ph Þ11 N ph0 g tph
hwLO expð
hwLO =kT e Þ: tph

Pd ¼

ð3:344Þ

The above expression clearly indicates that the power dissipation in the mode wherein it is dominated by hot phonons (high electron concentrations) is inversely proportional to the hot phonon lifetime. The shorter that lifetime the more efficient the power dissipation is. The fact that electron energy relaxation time is hampered by hot phonons is illustrated in Figure 3.156, which shows a lifetime within 200–400 fs for supplied power levels of about 5 nW electron1 or larger, which is reasonable for GaN HFETs. The bottom line in Figure 3.156 depicts Monte Carlo calculations excluding the hot phonon effects. The squares display the Monte Carlo calculations including the hot phonon effects (the top line is a guide to the eye). The circles indicate the experimental data (the second line from top is a guide to the eye). The Monte Carlo calculations and experiments (deduced from power dissipation measurements) are consistent and also agree with data obtained by femtosecond laser experiments.

3.11 FET Degradation

Energy relaxation time (ps)

AlGaN/AlN/GaN at 293 K

101

Experiment Monte Carlo with hot phonons Monte Carlo w/hot phonons

100

10-1

10-2

0

20 40 60 80 Supplied power (nW electron-1)

Figure 3.156 Hot electron energy relaxation time versus power supplied to an electron. The green line depicts Monte Carlo calculations excluding the hot phonon effects. The squares display the Monte Carlo calculations including the hot phonon effects (the top line is guide to the eye). The circles indicate the experimental data (the second line from top is guide to the eye). The

100 bottom line depicts Monte Carlo calculations excluding the hot phonon effects. Optical measurements with femtosecond pulsed lasers indicate the relaxation time between 300 and 500 fs that are in agreement with Monte Carlo and measurement data. Courtesy of A. Matulionis.

Hot phonons with temperatures approaching 2000 K along with the large electric field, particularly localized relatively high electric fields coupled with piezoelectricity/ strain are certain to generate defects, made easier with the starting material being defective already. It is also likely that nonuniform sheet electron density would ensue, which would also cause local heating under operation. Monitoring noise at low frequencies, sensitive to defects, which are generally referred to as generation recombination centers, could shed some light on the defect generation of the type mentioned during operation. The traps also affect the low-noise performance and the phase noise of devices. The upper limit of the hot phonon temperature is really determined by device failure. Much higher hot phonon temperatures are attained in modified heterojunction designs. These designs eventually should provide high phonon and electron temperatures to be reached before failure as well as short phonon lifetimes. However, the hot phonon and hot electron (particularly the former)-induced generation of point defects is a problem that can be mitigated by high-quality layers, which eventually would have to be on high-resistivity freestanding GaN or high-resistivity GaN when it is available. Fe doping is capable of providing stable high resistivity. It should also be stated that use of expensive SiC substrates for the high thermal conductivity may not even be well justified over using GaN substrates as the bottleneck for power dissipation appears to be LO phonon to LA phonon decay anyway. Once such decay occurs then the thermal conductivity comes into the picture. 3.11.2.4 Other Reliability Issues Various methods and materials for passivating FETdevices have been employed with varying degrees of success. While the reliability of the passivant itself is an issue (as,

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for example, a Si3N4 passivation layer deteriorates with time) that is still to be resolved, different passivants and surface treatments will have different results on the performance of the device. Typically, a passivant tends to increase the sheet carrier density as the (negative) charges on the surface tending to deplete the underlying channel are essentially moved away from the surface, or as the AlGaN surface tends to relatively become positively charged [574]. Moreover, passivation has been shown to reduce the RF current slump as well as increase the breakdown voltages of devices [289]. Although the use of a passivant would tend to point the problem of poor performance at the surfaces of the devices, one must be careful and consider that, for example, in any deposition scheme that utilizes hydrogen either as a carrier or a precursor (such as SiH4 in Si3N4 deposition), H can diffuse into the AlGaN and even into the GaN and “passivate” the deep levels in the underlying epitaxial layers [575]. In short, all aspects of processing must be considered when attempting to describe the physical phenomena occurring in the devices, and everything must be optimized simultaneously to achieve reliable device operation. Though not a fundamental issue like material quality or metallurgy, the issue of long-term stability warrants further attention since it is particularly evident in the GaN devices. In one study, after 1 month, in a device with Al concentration of 29%, the sheet resistance increased by 7 times, sheet density decreased by 33%, and the mobility decreased by 5 times. AFM imaging showed cracks on the AlGaN surface that were not present in the as-grown samples (acute problem in GaN on SiC, particularly with further processing and stress); however, by storing the devices in N2 instead of air, resistance and carrier density slightly increased and mobility decreased by 33%. This degradation was attributed to O-diffusion into surface (25 nm by SIMS) [576]. Similar results were found by another group when studying other samples grown by both HVPE and MBE with varying growth conditions, concentrations [577]. Although a cap layer and gate recess would likely alleviate some of this degradation, considering the depth to which the O diffuses as evidenced by the SIMS analysis, the development and application of some sort of hermetic barrier in packaging will obviously be desirable for long-term usage of GaN-based devices. In summary to the reliability section, a complete understanding of the “sudden failure” problem requires the development of a model that can simulate the mechanical stress, coupled with pyroelectric and polarization effects, on the lattice resulting from high electric fields. Defects consisting of screw dislocations, misfit defects, cracks (particularly on Si and SiC substrates), and point defects that are certain to be generated under high field, high local temperature, and elevated temperatures can be introduced. The lifetime of semiconductor devices is strongly related to the densities of dislocations or point defects. It is known that in GaAs-based devices the moderate dislocation density (104 cm2) can affect the operating life of the devices. Since in GaN-based structures the dislocation density is several orders of magnitude higher (108 cm2 or higher), the degradation rate can be related to the presence of extremely high dislocation densities and simple and complex point defects associated with them.

3.12 Heterojunction Bipolar Transistors

In short, there is still work to be done. Although some lifetimes look reasonable at 28 V, the research now needs to focus on gate leakage, and high fields potentially causing metal diffusion, affecting AlGaN crystal quality (particularly at the AlGaN/ GaN interface due to hot electrons and phonons to the issue of hot phonons) and defect generation. Research on methods to suppress the hot phonon phenomenon by developing structures to enhance the power dissipation per electron and finally pushing forward toward an understanding of the mechanisms of failure through design of test structures that allow us to tease the intertwined effects of high field, hot electrons, and hot phonons and piezoelectric and spontaneous strains from one another in an effort to determine the real physics of failures is also needed.

3.12 Heterojunction Bipolar Transistors

The state of heterojunction bipolar transistors based on the GaN material system is in the very early stages of development with not much to show for. There is actually a good technological reason for this state, as it is very difficult to grow very high-quality layers on top of p-type GaN. It is the need for very high doping levels in the base, upward of 1019 cm3, which exacerbates the problem further in npn varieties where the highly doped base has to be p-type. In LEDs and lasers, the function of the top p-layer is to inject holes. In bipolars, E–B junction needs to be of high quality with as little as possible interface recombination, and the base should be of high quality for minority carrier transport with as little as possible recombination. The collector–emitter (C–E) junction needs to have large breakdown voltage. Without the large breakdown voltage, the GaN technology is left to compete with SiGe and conventional compound semiconductor technologies. Although the intrinsic breakdown voltage of GaN can be high, the quality of available layers is not at a state where breakdown voltages approaching the intrinsic values have not been possible. To obtain large breakdown voltages, the doping level in the collector depletion region must be very low. In epitaxial GaN, the doping level is in mid-1016 cm3, a figure that is destined to improve some in time. Regardless, one needs to bring this figure to 1015 cm3 range. In summary, the E–B and also C–B junctions fabricated in the GaN system have fallen short of critical performance for HBTs based on this material system to be viable. Because the goal of this book is twofold, to discuss principles and the technology, fundamentals of operation of HBTs are treated. This is followed by what could be expected from GaN if the technological issues are somehow taken care of and what little experimental data available in GaN-based bipolars. Heterojunction bipolar transistors, based on the traditional compound semiconductors, such as GaAs, InP, and SiGe/Si systems, have progressed to the point where their speed and power performance are very attractive for many system applications requiring high performance. Compared to FETs, higher linearity, high transconductance, more uniform turn-on voltage than that can be obtained in FETs, and larger power densities per unit wafer area can be obtained in bipolar transistors. In addition, being a vertical device, larger breakdown voltages can be obtained allowing larger load

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resistances to be used, which is easier to impedance match. Unlike field effect transistors, their closest competitors, bipolar devices rely on minority carrier transport and as such their critical dimension is a vertical one, which is determined by deposition as opposed to lithography. In the silicon world, because of their large current handling capability, bipolar transistors (BJTs) are even used in unison with field effect devices, CMOS, to drive large capacitive loads for even faster performance (BiCMOS). As alluded to earlier, the basic operation of a bipolar transistor involves minority carrier injection in the forward biased emitter–base junction, minority carrier transfer through the base, and the collection of minority carriers in the reverse-biased collector junction. Increasing the minority carrier injection efficiency while maintaining a high base doping is a basic requirement for transistors designed for high frequency and high speed applications. These design criteria are difficult to achieve using a homojunction emitter, but they may be realized through the use of a heterostructure. The concept of the heterojunction bipolar transistor was first proposed by Shockley [578] followed by other reports [579, 580] pointing out potential advantages over conventional homojunction devices (BJTs). Because the semiconductor material with a wider bandgap is used for the emitter, this transistor concept was initially called the wide bandgap emitter transistor. A lightly doped n-GaN collector, a p þ -GaN (or a graded AlGaN for field-aided transport across the base), and an n-AlGaN emitter layer capped with a n þ -GaN contact layer complete the structure. A schematic diagram of what a fabricated device would look like is shown in Figure 3.157. Shown in Figure 3.158 is the schematic band diagram of an abrupt npn HBT under normal bias conditions (forward-biased emitter–base junction and reverse-biased collector–base junction) and that for an abrupt pnp HBT. In each case, the base doping is assumed to be much larger than the emitter doping, forcing the bulk of the depletion region at the forward-biased EB junction into the emitter region. The arrows indicate the carrier motion. For the npn case, while the forward injected electron does not really experience any barrier, the reverse injected hole from the base does experience a large barrier. Consequently, the emitter injection efficiency is high. The same is valid for holes in a pnp HBT. Here, the uppercase letter depicts the larger

Emitter n + -GaN Base Collector

n -AlGaN p + -Al,GaN n - -GaN n+ -GaN

Base

Substrate Figure 3.157 Schematic diagram of an AlGaN/GaN heterojunction bipolar transistor with or without a compositionally graded base.

Collector

3.12 Heterojunction Bipolar Transistors Emitter

Base Collector

∆V n

∆V p

(a)

Collector

Emitter

Base

∆V n

∆V p

(b) Figure 3.158 Schematic band diagram of (a) an npn HBT under normal bias conditions (forwardbiased emitter–base junction and reverse-biased collector–base junction) and (b) of a pnp HBT under normal bias conditions. The arrows

indicate the carrier motion. The base doping is assumed to be relatively much larger than the emitter doping to the point that the depletion region into the base from the emitter side under forward bias condition is very small.

bandgap material. In HBTs, two types of emitter–base (E–B) and base–collector (B–C) heterojunctions, graded and abrupt, are used. The abrupt junction gives rise to discontinuities in the energies of the conduction band minima and valence band maxima. On the other hand, the graded junction presents a continuity of both the conduction band minima and valence band maxima. The valence and conduction band discontinuity in npn and pnp HBTs play a very important role in reducing the back injection current, thus giving rise to larger current gains than otherwise can be obtained. A good quality base coupled with a very small thickness increases the base transport factor. In cases where the diffusion length is small and/or the surface recombination is severe, grading Al mole fraction in an AlGaN base down toward

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the collector causes an electric field in the base helping the electron motion in an effort to increase the base transport factor and thus the overall current gain. Although the HBT concept was proposed in 1950s, only in 1980s did HBTs record dramatic advances. These advances were to a large extent fueled by improved crystal growth methods, such as MBE, OMVPE and ultrahigh vacuum chemical vapor deposition (UHVCVD). These technologies provided atomic-level precision in layer thickness and doping concentrations with unprecedented control ensuring improvements in material quality. Physicists and engineers were thus able to explore new device structures and to verify nonequilibrium transport mechanisms such as ballistic transport in heterostructures. The current gain, hFE, of an HBT is sensitive to the material quality and as the quality improves, the HBT current gain increases. The highest current gain current gain cutoff frequencies reported for various conventional compound semiconductors are in the range of about 100–200 GHz [581]. In fact, the technology in the conventional semiconductors is so well advanced that GaAs/AlGaAs layers grown by MBE and OMVPE can be commercially obtained for HBT fabrication. GaN-based electronic devices such as high-power and heat-tolerant HBTs can be important components of integrated systems designed for high-frequency and highspeed applications, for example, in satellites and all-electric aircraft. However, this hinges on some technologically intractable problems being conquered. As mentioned earlier, GaN HBTs could lend themselves to high-power operation with larger operating voltages and better linearity than those that can be attained by FETs, again pending improvements in epitaxial technologies. The basic operations of HBTs, involving minority carrier injection in the forward-biased emitter–base junction, minority carrier transfer through the base, and minority carrier collection in the reverse-biased collector junction, also lead to high-speed performance, which is imperative. This high-speed performance is represented by the current-gain cutoff frequency fT and the maximum oscillation frequency fmax [582]. The latter parameter critically depends on base resistance, which is a problem for GaN, known to suffer from low hole concentration in p-type layers, about 1018 cm3. In addition, the deep nature of the most commonly and successfully used dopant, Mg, causes temperaturedependent ionization. This leads to temperature-dependent base resistance, which may actually drop with increasing temperature even though the hole mobility would decrease somewhat. In addition, p-type GaN layers are known to contain domains, getting worse with an increased doping level, which could adversely affect the diffusion constant, definitely the lateral component. To reiterate, a succinct description of HBT fundamentals is given for the reader to be familiar with the fundamentals of this important device. A good deal of the discussion in conjunction with the fundamentals is based on GaAs and InP systems because of the substantial amount of prior data that is available in comparison GaN system for which there is very little data available. 3.12.1 HBT Fundamentals

The structure of an HBT is almost the same as that of a BJT described in textbooks, with the exception that a wide bandgap emitter is used. Moreover, a compositionally

Doping Concentration (cm-3)

3.12 Heterojunction Bipolar Transistors 10

21

10

20

10

19

10

18

10

17

10

16

10

15

p

n

n

0

0.2

0.4

0.6

0.8

1

Depth ( µm ) Figure 3.159 Dopant profile in an npn heterojunction bipolar transistors with heavily doped emitter contact layer and subcollector.

graded base or a doping graded base is used to add a drift component to the carrier transport for further improvement. A wider emitter layer allows the use of a lower doping concentration in the emitter, leading to a low emitter depletion region capacitance. Moreover, the emitter–base band discontinuity partially blocks the back injection of base majority carriers into the emitter. If kB is the Boltzmann constant and T is the absolute temperature, then the reduction in the reverse injection is, in the first approximation, on the order of the exponential of the ratio of the appropriate band discontinuity (valence band discontinuity DEV, for npn and the conduction band discontinuity DEC for pnp transistors) to kBT. The reduction of emitter capacitance reduces the emitter junction charging time. In HBTs with a graded base, a drift field is present, which accelerates injected carriers from emitter to collector, reducing the base transit time. All of the effects mentioned above enhance the current gain, hFE, and the cutoff frequency, fT, beyond those achievable with BJTs. Wide-gap emitters also allow very high base doping levels without degrading the emitter injection efficiency, so that a small base resistance can be obtained [583]. The maximum oscillation frequency, fmax, is thus increased without degrading the current gain, hFE. Figure 3.159 shows the typical impurity profiles for HBT structures and highlights the larger base doping compared to the emitter in HBT. 3.12.1.1 Current Transport Mechanism Across the Heterojunction In a homojunction, the semiconductor material, and therefore the energy bands, is the same on both sides of the junction so that the conduction and valence bands coincide at the junction. In a heterojunction, however, two different semiconductor materials are joined together at the junction and the energy bands on each side do not generally coincide. Discontinuities in both the conduction band minima and the valence band maxima are formed at the junction. The formation involves complicated participation of the intrinsic material properties and the microscopic properties at the junction.

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IB

I3e IE

p

E-field

n E-field

I3d

Collector

Emitter

Base n

I4g I4f IC

I2c

I5h

I2b

I5i

I1a

I1a

Figure 3.160 Various current components in an npn bipolar transistor operating under normal conditions wherein the emitter–base junction is forward biased for electron injection and the collector–base junction is reverse biased for electron collection. Carrier motion is indicated with light arrows while the current direction is indicated with dark arrows. The electric field in both emitter and collector depletion regions is also shown.

In a bipolar transistor inclusive of heterojunction varieties, there are multiple current flow channels resulting from forward and reverse carrier injections, thermal generation of electron–hole pairs and subsequent carrier motion, carrier diffusion, and in cases of high collector voltages carrier generation due to impact ionization and motion in the high-field region. Let us now specify the sources of current flow in bipolar and assign nomenclature to each component. In an npn bipolar variety, electrons are injected from the emitter, some of which traverse to the collector edge and are collected, defined as I1a in Figure 3.160. Some recombine with holes in the base, as indicated with symbol X in the base region and vanish, the associated current for which is indicated with I2b in Figure 3.160. The current associated with hole injection from the base into the emitter, reverse injection albeit negligibly small in heterojunction bipolar transistors, is depicted in Figure 3.160 by I2c. Following reverse injection into the emitter, some of these holes recombine with majority electrons, indicated with symbol X in the emitter region. Some of them diffuse to the emitter metal contact and get collected with increasing rate as the emitter thickness is made thinner. Thermally generated electrons in the base (holes so generated diffuse away from the depletion layer’s edge and are small in concentration with respect to the majority carrier concentration) are swept across the emitter–base junction depletion region due to the electric field giving rise to a current component designated by I3d. In addition, thermally generated holes (electrons so generated diffuse away from the edge of the depletion region in the emitter and are small in

3.12 Heterojunction Bipolar Transistors

concentration with respect to the majority carrier concentration) in the emitter region but on average within a diffusion length of the edge of the depletion region are swept across the emitter–base junction giving rise to a current component designated by I3e. Both of these current components, I3d and I3f, oppose the prevalent current due to electron injection from the emitter to the base. See Figure 3.160 for a graphical presentation. Focusing our attention to the base for now, electrons that recombine with holes, leading to the I2b current, and the resultant deficit of holes sets up hole diffusion, the holes for which are provided by the base contact. Similarly, the holes that are reverse injected by the forward-biased emitter–base junction are also provided by the base contact, giving rise to I2c current component. Thermally generated holes in the emitter within, on average a diffusion length, are swept across the emitter–base junction and end up at the base contact after giving rise to the I3e current component. Thermally generated electron–hole pairs in the base within, on average, a diffusion length of the edge of the base–collector depletion region in the base have an impact on the carrier transport in the base. While electrons are swept across the depletion region by the electric field giving rise to a current component of I4f, the holes in the base diffuse away. See Figure 3.160 for a graphical presentation. Finally, turning our attention to the collector, the electrons injected from the emitter that did not get annihilated in the base through recombination get collected at the collector junction (assuming 100% collection efficiency, which does not hold if the junction interface is of low quality and leads to carrier loss and/or if there is heterojunction barrier as in double-heterojunction bipolar transistors) and constitute the current component I1, which is the current component giving rise to the forward gain. Again thermally generated electron–hole pairs within, on average a diffusion length, contribute to the current. The holes are swept across the collector–base junction and are collected by the base contact giving rise to a current component with a designation of I4g, as shown in Figure 3.160. Finally, in cases of large collector biases and other conditions permitting, electrons entering the collector depletion region could cause avalanching and the resulting electrons and holes are swept in opposite direction giving rise to current components I5h due to electron motion and I5i due to hole motion as depicted in Figure 3.160. Among the current components, the dominant current is that associated with injection from the emitter followed by transport through the base before being collected at the collector terminal I1a. However, if the device is pushed into avalanche, the impact ionization current can be sizeable. Assuming this not to be the case, the terminal currents, in the order of emitter, collector, and base currents, can be expressed as IE ¼ I1 þ I2 þ I3 ;

ð3:345aÞ

IC ¼ I1 þ I4 ;

ð3:345bÞ

IB ¼ I 2 I 3 I 4 :

ð3:345cÞ

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The current transport mechanisms in heterojunctions have been discussed by Lundstrom [584]. Marty et al. [585] first presented a detailed diffusion model describing the current transit mechanism across a heterojunction bipolar transistor. In fact, the model applies only to the graded HBTs and is not much different from models of homojunction bipolar transistors. A number of authors have pointed out that the correct mechanism of current transport across an abrupt heterointerface is thermionic emission [586–590]. To take this into account, the thermionic emission current from the emitter should, for example, be balanced by the conventional Shockley diffusion current in the base. For an npn HBT with E–B heterojunction at z ¼ 0 and the B–C junction at z ¼ WB along the z-direction, the excess electron concentration Dn(z) ¼ DnE(z) þ DnC(z) in the base should satisfy the following boundary conditions: DnE ð0Þ ¼ np0 ½expðqV BE =kB TÞ1;

ð3:346aÞ

DnC ðW B Þ ¼ np0 ½expðqV BC =kB TÞ1;

ð3:346bÞ

DnE ðW B Þ ¼ 0;

ð3:346cÞ

DnC ð0Þ ¼ 0;

ð3:346dÞ


 
 qV BE qvn DnE ð0Þ qDnE 1  J nE0 exp ; ¼ qDB kB T qx x¼0 4

ð3:347Þ

where DnE(z) and DnC(z) are the excess electron concentrations due to the injection fluxes from the emitter and the base, respectively, DB is the diffusion coefficient for electrons in the base, m is the effective mass for electrons in the base, and vn is the electron mean thermal velocity, vn ¼ (8kBT/pm)1/2. Also, q is the electronic charge, VBE is the emitter–base applied bias, VBC is the base–collector applied bias, and JnE0 is the thermionic emission current. If An is the Richardson constant, FBn is the barrier height, and np0 is the thermal equilibrium value of the electron concentration in the base, then
 qvn np0 qFBn J nE0 ¼ An T 2 exp  :

kB T 4

ð3:348Þ

Note that the potential barrier varies slowly within the de Broglie wavelength of those electrons that are responsible for electrical transport. Because of this, in Equation 3.347, the quantum mechanical reflection QR at the E–B interface has been ignored. The term qvnDnE(0)/4 corresponds to the reflection QR of those injected electrons from within the base. The turn-on voltage of the abrupt emitter–base junction of an HBT is modified by the conduction potential spike. A model developed for this turn-on voltage must treat the current transport mechanism across the E–B heterojunction as a balanced two-

3.12 Heterojunction Bipolar Transistors

step process of thermionic emission followed by Shockley diffusion, instead of the conventional one-step diffusion process. The Ebers–Moll equations would consequently be modified to IE ¼ 

IF þ aR I R ; g E12

ð3:349Þ

IC ¼ 

IR þ aF I F ; g C12

ð3:350Þ

where aF ¼ aEg Epn and aR ¼ aCg Cpn are the forward and reverse common-base current gains, respectively, g E12 and g C12 are the respective electron injection efficiencies at the emitter and collector junctions by considering injection currents only, aE and aC are the forward and reverse transport factors, respectively, IF is the current due to electron injection from the forward direction (e.g., from emitter to base), and IR is the current due to electron injection from the reverse direction (e.g., from collector to base). These are given by 
  qV BE 1 ; IF ¼ IE0 exp kB T

ð3:351Þ


  qV BC 1 : IR ¼ I C0 exp kB T

ð3:352Þ

The new parameters g E12 and g C12 are introduced to take into account the effect of recombination. Owing to the recombination components, the total emitter current is not only the emitter injection current but also the sum of emitter injection current and the emitter recombination current. Similarly, the total collector current is not the collector injection current, but the sum of collector injection current and the collector recombination current. The term g E12 is the ratio of the emitter injection current and the total emitter current. Similarly, g C12 is the ratio of the collector injection current and the total collector current. The band diagram of an npn HBT under normal operation and labels of several common properties of this HBTare depicted in Figure 3.161. The saturation currents IE0 and IC0 at the respective E–B and C–B junctions of this HBT may be given by. IE0 ¼

IC0


 AE An T 2 qFBn þ DE C exp  ; g Epn kB T


 AC An T 2 qFBn ¼ exp  ; g Cpn kB T

ð3:353Þ

ð3:354Þ

where AE and AC are the emitter and collector area, respectively. The conduction band discontinuity, DEC, the valence band discontinuity, DEV, and other related parameters of a typical HBT that contribute to the currents, IR and IF, are shown in Figure 3.162. If instead of a single-heterojunction bipolar transistor, we consider a

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(p) GaN

(n) Al x Ga1-x N InE ∆EI

(n) GaN I nC

qf Bn

E F IE

I C

I B I PE

V BE

Normal operation

V

CB

Figure 3.161 Band diagram, biasing conditions, and current flow in an npn HBT under normal operation.

double-heterojunction bipolar transistor (DHBT), in the same way as Equations 3.353 and 3.354, it would involve the conduction band discontinuity, DEC. The Ebers–Moll model is a very simple and useful model that takes the carrier injection and extraction into account to determine the total emitter and collector currents in a transistor. Equations 3.349 and 3.350 differ from the conventional Ebers–Moll equations by the presence of the g parameters. This differs from the conventional Ebers–Moll model also by the presence of conduction band discontinuity, which is absent in a homojunction. In the absence of recombination, the g parameters reduce to unity, and for a homojunction Equations 3.349 and 3.350 reduce to the conventional form for the Ebers–Moll model. The common-emitter current gain may be derived from Equations 3.349 and 3.350 as bF ¼

aF g E12 ; 1aF g E12

ð3:355Þ

bR ¼

aR g C12 : 1aR g C12

ð3:356Þ

Note that the current gain, bF, corresponding to the forward injection takes the effects of recombination and conduction band discontinuity into consideration. However, bR takes only the effect of recombination into account.

3.12 Heterojunction Bipolar Transistors

(a)

∆ δEFn

qV EB

- qV CB

(b) n

n

(+)

n

0

(-)

W

z

Figure 3.162 Quasi-Fermi levels in the base of an abrupt and forward-biased npn HBT under forward-biased emitter and reverse-biased collector (a); and a schematic diagram of an electron concentration profile (b) [615].

The C–E offset voltage DVCE under forward operation can be derived as ! DE C kB T R ; ð3:357Þ þ ln DV CE ¼ q q aF g C12 g Cpn where R ¼ AC/AE is usually larger than 1, and aF the base forward transport factor, which is close to unity for high-gain bipolar transistors. The height of the potential spike DEb of the E–B junction (see Figure 3.162) at IC ¼ 0 is a function of the E–B applied voltage. Only at a high level of injection in which most of the applied voltage drops across the bulk or the contact resistance, instead of the junction itself, does DEb approach a constant value. With the aid of this model, the potential spike height DEb has been determined from the measurement of the DC current and is around 0.12–0.18 eV for Al0.5Ga0.5As/GaAs material.

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The model for the collector–emitter offset voltage DVCE suggested by Mazhari et al. [591] is the most general. The DVCE resulting from this model for singleheterojunction bipolar transistors is identical to that of Lee et al. [586]. However, for double-heterojunction bipolar transistors it provides significant improvement over that of Lee et al. [586]. This improvement is based on the argument that, at zero collector current, the minority carriers injected into the base from the emitter and the collector should have different functional relationships to VBE and VBC, respectively. This difference is due to the different directions and magnitudes of the currents that flow across the two junctions. Thus, even when emitter–base and base–collector junctions are identical in all respects, they may have a substantial offset voltage with a magnitude that depends on the current gain of the transistors. A notable difference between the model of Mazhari et al. [591] and the model of Lee et al. [586] is that, unlike the latter, the former takes the emitter resistance explicitly into account, while unlike the former the latter takes the thermionic emission explicitly into account. Lundstrom [584] offered an explanation of the influence of the potential spike on the electron injection from the emitter to the base. Current transport across the emitter–base junction is a two-step process. Electrons are first injected over or through the emitter–base barrier and then diffuse across the quasineutral base. When the slowest process is diffusion across the base, the conventional law of the junction applies, but when the slowest process is injection across the barrier, the current–voltage (I–V) characteristics resemble more closely that of a metal–semiconductor junction. A thermionic diffusion theory of minority transport in HBTs was developed by Grinberg and Luryi [590] with a particular emphasis on the differences between abrupt and graded emitter–base junctions, as well as on the role of the quasi-Fermilevel discontinuity at the interface in the abrupt junction. The starting point was the following two basic relations of Marty et al. [585] for homojunctions: C ð2




C1

 Jn dz ¼ qV A ; mB n

ð3:358Þ


 qV A ; kB T

ð3:359Þ

np ¼

n2i exp

where Jn is the electron current density, VA is the applied voltage, mB ¼ (qDB/kBT) is the electron mobility, and n, p, and ni are the electron, hole, and intrinsic carrier concentrations in the base near the emitter junction, respectively,. The integral in Equation 3.358 is taken between the emitter contact C1 and an auxiliary contact C2 deep in the base. For heterojunctions, Equations 3.358 and 3.359 are respectively modified to C ð2

C1




 Jn n þ N EC dz ¼ qV A DE C þ kB T ln ; mB n n N BC

ð3:360Þ

3.12 Heterojunction Bipolar Transistors


 
N BC n DE C qV A 2 exp ¼ n ; exp i kB T kB T N EC n þ

 nþ pþ

ð3:361Þ

where n þ and p þ are, respectively, the electron and hole concentrations in the base near the emitter junction, n is the electron concentration on the emitter side of the abrupt boundary (see Figure 3.162), and N EC and N BC are the conduction band densities of states for electrons in the emitter and base, respectively, in the vicinity of the discontinuity. Equations 3.360 and 3.361 take into consideration the fact that the electron concentration n þ in the base side of the E–B junction is not the same as the electron concentration n in the emitter side of the E–B junction and that the electron current density across the E–B junction is reduced by the conduction band discontinuity, DEC. The presence of a notch DEn (Figure 3.162) in the base side of the AlGaAs/GaAs E–B heterojunction of an AlGaAs/GaAs HBT creates a quantum well, and the electron concentration n þ in that well may essentially be two dimensional. There is no reason for the electron concentration n to be different from that of the three dimensional. The product of electron and hole densities on the base side of the E–B junction depends on not only n þ and n but also the height of the conduction band discontinuity, DEC. While it is increased by the applied bias, it is decreased by the conduction band potential spike, DEb. Assuming that the quasi-Fermi level for electrons is continuous in the emitter and base side of the E–B junction, for example, d EFn ¼ 0, and that   E
n NC DE C ¼ : exp  nþ kB T N BC

ð3:362Þ

Equations 3.360 and 3.361 reduce to Equations 3.358 and 3.359, respectively. This indicates that the consideration of thermionic emission and the discontinuity of the quasi-Fermi levels for electrons on both sides of the E–B junction can be implemented within the framework of drift-diffusion model. There is no need for losing the generality established for HBTs by the approach of Marty et al. [585]. The actual value of the ratio n/n þ may be determined from the boundary condition on the electronic flux

B NC DE C þ

J nl J nr ; J n ¼ qvR n qvR n exp  kB T N EC 

ð3:363Þ

where vR is the Richardson velocity, vR vn/4 ¼ (kBT/2pm)1/2. Equation 3.363 is based on the consideration that the net electron current density in the base must be equal to the difference between current density Jnl due to electron flow from the emitter to the base (e.g., left to the right) and the current density Jnr due to electron flow from the base to the emitter (e.g., right to the left). The density of states factor enters this equation because noncontinuity of electron quasi-Fermi levels on the emitter and the base sides of the E–B heterojunction causes the effective density of states N BC and N EC to be different. According to the thermionic emission model, the ideality factor of the collector current should be unity if the entire forward bias VBE drops in the E–B junction. But,

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in fact, it is slightly larger than 1. Tunneling through the potential spike DEb in the emitter conduction band is shown to be important for a heavily doped emitter. This affects the initial energy of carriers that are injected into the base, their effective mass distribution, and the collector turn-on voltage. Injection of electrons through and over a barrier leads to carriers that are not in equilibrium [592] as determined from the electroluminescence spectra obtained in abrupt HBTs with emitter AlAs mole fraction in the range of 28–55%. The height of the potential spike, DEb, influences the electron injection from the emitter to the base and in turn the collector current ideality factor and the C–E offset voltage. The ideality factor is around 1.05 for graded HBTs and 1.1–1.3 for abrupt HBTs. Equations 3.346–3.350 show that for abrupt junction HBTs the current is extremely sensitive to the height of the spike. Slight variations in composition or displacements in the doping can have large effects on the current. For this reason, the turn-on voltage for an abrupt junction HBT is much more difficult to control than for a graded junction HBT. Under these circumstances, a significant nonuniformity of E–B turn-on voltage has been found on a local scale (between adjacent devices within a differential pair) as well as on a wafer scale in millimeter wave ICs (MMICs). Therefore, a graded E–B junction may be used. 3.12.1.2 Current Transport Mechanism Across Base Understanding the minority carrier transport across the base with width from 100 to 1000 Å is an important issue in BJTs in general and HBTs in particular because the high-frequency performance is partially dictated by the base design. Even though, delay occurs not only across the base but also in the emitter and collector depletion regions, the base transit time is often the most talked about due to its dominating influence on the speed of BJTs. In short, in conventional bipolar transistors, the total collector–emitter delay time, tCE, is often dominated by the base transit time, tB, because the base must be lightly doped to maintain a certain current gain and must not be too narrow or else the base resistance increases. In addition, an effect called the induced base at high injection levels adds one more component to the total device delay. Various device delays are schematically shown in Figure 3.163. As in the case of BJTs, the base transit time, tB, together with emitter transit time and the transit time at the junction, provides the total transit time in an HBT. The transit time has been widely investigated for well-known and well-established bipolar transistors, including Si homojunction bipolar transistors [593–596], heterojunction SiGe HBTs [597, 598], and AlGaAs/GaAs HBTs [599]. These studies suggested that, for the sake of lower base resistance, the base doping concentration of an HBT must be increased. However, such an increase accompanies very minimal increase in the built-in potential and a decrease in the carrier mobility in the base. The increased built-in potential lowers the base transit time, tB. However, the decreased mobility increases the tB. Nonuniform doping and the compositional grading of the base also influence the tB. Numerical simulations to determine the room-temperature dependence of the base transit time on various parameters, such as base doping and base compositional grading of npn GaN/InxGa1xN HBTs (x is the In mole fraction in InGaN) were undertaken some years ago [600]. Additional simulations using a

3.12 Heterojunction Bipolar Transistors

E

B

C

τe

SC

τc τb

τ SCR

(a)

E

B

C

τe

SC

τc τb

τCIB

τSCR

(b) Figure 3.163 Schematic representation of various delay times in an HBT under low and high injection conditions, for small collector bias (a) and for large collector bias (b). SC is for subcollector. The capacitor symbols show the delay to be due to charging time of capacitors across the emitter and collector junctions. The rest are due to transit time of carriers.

compact simulator of the DC and cutoff frequency performance of GaN/AlGaN have also been performed [601, 602]. As in any calculations, the results depend on the parameters used. In HBT based on GaN, the major problem is one of a good estimate for the base diffusion length, which critically depends on the layer quality. Other issues such as collector breakdown voltage, heterojunction band discontinuities, and surface state depend on the extended regions of the base (beyond the emitter region). Generally, the minority carrier–hole diffusion length in GaN would be smaller than that of the conventional compound semiconductors. In very high-quality GaN templates, the diffusion length of holes in n-type material was measured to be about 1 mm on the nitrogen face, which represents the early stages of growth on sapphire and is more defective, and 4 mm on the Ga-face, which is some 200 mm away from the initial substrate epitaxial layer interface. Likewise, the minority carrier lifetime ranged from 50 ns near the N-face to 800 ns near the Ga-face [603]. These numbers are really sensationally outstanding, which calls for extreme caution, as additional measurements must be made to gain confidence in the results. Most of the conclusions of Refs [601, 602] are complementary and predictable. Basically, the current gain is limited by carrier transport across the base, the cutoff frequency is limited by the base transit time, which can be enhanced by using a graded junction, and graded base would reduce the transit time. The use of pnp

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HBTs, to circumvent the high base resistance plaguing the npn device, runs into the hole transport limitations across the n-type base. As expected, the authors of Ref. [601] also argue that the polarization effects are unlikely to adversely affect the performance of HBTs, current gains in the range of 200–2000 may be possible at room temperature, and the cutoff frequency appears to be around 30 GHz. Despite the odds, experimentalists have been charging ahead. First reports indicated current gains of about 3 [604, 605] and 6 with an early voltage of about 400 V [606]. An issue of special interest in HBTs is that in these devices a much higher base doping can be employed, provided it is attainable, and the base can be reduced to several hundred angstroms, to even several tens of angstroms. Therefore, it becomes necessary to examine the validity of using standard drift-diffusion analysis for the thin base and to explore the possibility of ballistic transport in the base. This is particularly so in semiconductors with highly mobile carriers. From the classical drift-diffusion equation, the base transit time, tB, and the average velocity, vAV, in the neutral base region can be drawn as tB ¼

W 2B ; lDB

vAV ¼

W B lDB ¼ : tB WB

ð3:364Þ

ð3:365Þ

It is easy to see from Equation 3.365 that with a decrease in the base width, the average base velocity increases without bound. For a GaAs-based HBT with the base width WB ¼ 100 Å, base diffusion coefficient DB ¼ 25 cm2 s1, and the adjustable parameter l ¼ 2, the average carrier velocity in the base is vAV ¼ 5  107 cm s1. This value is unreasonably high. In fact, the drift-diffusion equation is derived from the Boltzmann transport equation by using the relaxation time approximation. This assumes that the carriers have a quasithermal equilibrium distribution and that scattering may be described in terms of a relaxation process. When the neutral base width becomes comparable with the carrier mean free path, the carriers may cross the base without losing velocity and without randomizing collisions. In this case, the transit time, Equation 3.364, based on the drift-diffusion model is no longer accurate. Using Monte Carlo techniques, Maziar and Lundstrom [607] estimated the transit time across a field-free uniform GaAs base. Levi et al. [608] found a departure from the conventional dependence of the common-emitter current gain b (note that bF for the forward injection and bR for the reverse injection) on the base width WB in abrupt AlInAs/InGaAs HBTs with base width of 200–4000 Å. Figure 3.164 shows the current gain as a function of base width at IC ¼ 100 mA and VBC ¼ 0 V. The broken line 1 represents the behavior expected for b / W 2 B and the solid line for b / W B . It is clear from the data that for WB < 1000 Å the current gain b scales as 1/WB, whereas for WB > 1000 Å varies approximately as W 2 B . The latter shows that the current gain is limited by base recombination and the electron transport in the base is controlled by diffusion. The former case derives from extreme nonequilibrium electron motion in the base. The presence of nonequilibrium electron transport at the base–collector junction should enhance avalanche multiplication in the collector. This has been

3.12 Heterojunction Bipolar Transistors 10

3

Current gain (β)

1/W B 10

2

10

1

10

0

10

1/(W B)2

100

1000

Base thickness, WB (nm) Figure 3.164 Variation of the current gain b of AlInAs/InGaAs HBTs as a function of base width WB at IC ¼ 100 mA and VBC ¼ 0 V [608].

verified by the measured avalanche multiplication constant GAM as a function of base width [608] involving nonequilibrium transport in the base. Ritter et al. [609] found that the current gain of InP/InGaAs HBTs varies as W 2 B for the investigated base thickness range of 200–1000 Å, as is the case in diffusive base transport. As electrons are injected into the InGaAs base from the InP emitter with energy of 0.25 eV, the appropriate value of the diffusion coefficient is determined not by the equilibrium diffusivity but by the hot electron transport, which may be nonequilibrium. Harmon et al. [610] described thermal velocity limits to diffusive electron transport in thin-base npn GaAs HBTs. Based on experimental analyses on transistors with base width 1000, 2000, and 4000 Å, these authors concluded that the diffusive transport mechanism is invalid in thin-base HBTs. They noted that heavy impurity doping of thin p-base suppresses electron–hole scattering and leads to large mean free paths. Thin-base effect thus becomes strong for thermal electrons in the p-GaAs base, and the measured collector current shows sign of saturation. This occurs because the collection velocity of electrons at the base–collector junction is restricted to the kinetic limit. It seems plausible that semiballistic transport and finite collection velocity together with diffusion model control electron flow in thin-base HBTs. Researchers have employed abrupt heterojunctions in the hope of achieving ballistic base transport in high-frequency applications [580, 611–614]. A case in point is that by Ankri and Eastman [611] who used the conduction band energy difference between the emitter and base as a carrier-launching ramp into the low-field base region. The ramp energy was selected such that upon launching, the energy acquired by the carrier was less than the threshold energy for transfer into the L-valley (about 0.3 eV) of GaAs with its heavier electron mass. Because the base is heavily doped and ungraded, the field throughout the base is small. Thus, electrons gain little energy as they cross the base and their motion approaches that of the ballistic

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transport. However, Lundstrom et al. [584] argued that the advantages of ballistic base HBTs (BBHBTs) are limited due to the intervalley scattering in the GaAs base. For example, in an Al0.27Ga0.73As/GaAs HBT with doping of 5  1017 cm3, a 9kBT ramp at the emitter can be obtained. In this case, 65% of electrons occupy the G-valley, 30% are found in the L-valley, and the remaining 5% are in the X-valley. Because a significant fraction of electrons injected into the base reside in the upper valleys, intervalley scattering contributes an important velocity-randomizing component to the velocity distribution. Slow electrons spend large periods of time in the base, lowering the steady-state ensemble velocity significantly. This may explain the lack of clear experimental evidence for nonequilibrium transport in the base of AlGaAs/ GaAs HBTs. Consequently, to avoid intervalley scattering, a reduction of the ballistic ramp height, for example, in the AlGaAs/GaAs HBTs is necessary. The situation in InP/InGaAs or AlInAs/InGaAs HBTs is different because of the larger G–L intervalley separation (0.55 eV for In0.53Ga0.47As) and electron scattering rate in InGaAs material. While the conduction band discontinuity DEC ¼ 0.50 eV for InP/InGaAs, DEC ¼ 0.52 eV for InAlAs/InGaAs. Using Monte Carlo simulations, Dodd and Lundstrom [613] examined the base transit time and electron distribution function of InP/InGaAs HBTs with base width from 10 to 10 000 Å. A clear ballistic behavior was observed only for extremely thin bases less than 100 Å. In the thin bases, the ballistic fraction exceeds 80% and the base transit time is proportional to the base width, WB. Over the range of base widths of interest, suppose larger than 200 Å, base transport appears to be diffusive, but the electrons are very far from thermal equilibrium. The diffusive behavior may have arisen from the sensitivity of the steady-state carrier population to large-angle scattering. Grinberg and Luryi [615] carried out a theoretical work on this subject. The timedependent Boltzmann equation was used to calculate the small signal complex base transport factor a(w) for different ratios of the base width WB and the scattering mean free path lSC. It was shown that the phase trajectory (Re(a), Im(a)) has a universal character both in the diffusion limit (WB  lSC) and in the ballistic limit (lSC  WB). The mechanism of a high-frequency current gain roll-off in an abrupt HBT with ballistic propagation of minority carriers across the base has also been treated. The band structure of an abrupt HBT can be engineered in such a way that the minority carriers are injected into the base over the potential spike DEb and carry electron kinetic energy DEb in the base. The ballistic or coherent regime arises at temperatures low enough compared to the injection energy so that the injected minority carriers form a nearly collimated and monoenergetic beam. The coherent transistor with base transport can have both current gain and power gain at frequencies far above the limit wherein scattering occurs. As indicated earlier, the minority carrier transport in the base of a bipolar transistor is adequately described by the drift-diffusion equation, provided the base width WB is sufficiently large so that WB  lSC, where lSC  DB/vT is the characteristic scattering length in the base, and vT is the thermal velocity of minority carriers in the base. Based on a Maxwellian ensemble, vT ¼ (3kBT/m)1/2. For narrower bases, where WB < lSC (lSC ¼ 250 Å, if DB ¼ 25 cm2 s1 and vT ¼ 1.0  107 cm s1), the drift-diffusion equation breaks down and a more refined Boltzmann transport model is required. For the state-of-the-art AlGaAs/GaAs HBTs, the base

3.12 Heterojunction Bipolar Transistors n + -GaN cap n + -AlGaN emitter

E p + -GaN or graded AlGaN base n - -GaN collector n + -GaN subcollector Sapphire or SiC substrate

Figure 3.165 Cross-sectional view of an HBT showing the lateral flow of injected electrons in the base region. In the case of SiC substrates, because they can be conductive, the collector contact can be made to the substrate if desired and the interface between SiC and GaN is designed for low-resistance current conduction.

width can be wide enough for the drift-diffusion equation to describe the behavior of minority carriers in the base. An alternative means of imparting energy to electrons in the base is through a built-in field in the base. Both compositional grading and doping induced grading can cause a strong quasielectric field in the base. This field greatly reduces the base transit time and in turn increases the current gain cutoff frequency. Employing compositional grading of the base from Al0.1Ga0.9As at the emitter edge to GaAs at the collector edge, Ito et al. [616] reported current gain enhancement due to an electron drift motion from the built-in field. In addition to the improved current gain, at low current densities, the built-in field due to compositional grading of the base was found to suppress the current gain reduction accompanied by scaling of AlGaAs/GaAs HBTs [617, 618]. A lateral diffusion of injected carriers at the extrinsic base surface between the emitter mesa and the base ohmic contact of a scaled down HBT leads to recombination of electrons and holes and a capture of electrons by surface traps. This lateral flow of carriers is schematically illustrated by the solid lines in the cross section of the HBTshown in Figure 3.165. For HBTs with a built-in field in the base, Figure 3.165 also shows the direction of the electric field, E, and the dashed lines illustrate the bending of the lateral flow of injected electrons toward the collector under the influence of built-in field. The disadvantages of compositional grading of the base result from the reduction of minority carrier mobility and the high-base ohmic contact resistivity caused by the AlGaAs base material as opposed to GaAs. The design of a doping concentration gradient in place of a compositional gradient presents the potential advantage of superior high-frequency performance. However, this advantage may be compensated by the base bandgap shrinkage effect [619, 620], if the base doping is excessively high. Let us now consider a device with both doping grading and compositional grading. If the base doping is varied exponentially, the hole concentration p(z) along the z-direction in the base, recognizing that at low injection levels (for npn HBT), it is equal to the base acceptor concentration, can be written as pðzÞ ¼ N B ð0ÞexpðbzÞ:

ð3:366Þ

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The compositional grading, however, may cause the energy bandgap along z-direction in the base to vary linearly with position as E G ¼ E G0 ðakB TÞz:

ð3:367Þ

This leads to a simplified model for the base transit time, tB, at low-level injection as [624] tB ¼

exp½ða þ bÞW B  þ ða þ bÞW B 1 DB ða þ bÞ2

;

ð3:368Þ

where a and b are constants related to the base built-in field and DB is an average value of the electron diffusion constant in the base valid only at low injection levels. A more general expression for the tB assuming only diffusion transport is given by [621] 2W 3 W ðB 2 ðB nie ðzÞ 4 pðyÞ tB ¼ dy5dz; ð3:369Þ pðzÞ Dneff n2ie ðyÞ 0

z

where nie is the effective intrinsic carrier concentration, p is the nonuniform carrier concentration in the base, and WB is the metallurgical base width, which is the width of the neutral base region between the emitter and the collector. Taking the compensation of the free carrier concentration into account, the effective diffusion coefficient, Dneff, is defined as Dneff ¼ fcom Dn with fcom being the base doping compensation factor. The effective intrinsic carrier concentration nie was calculated from [622] n2i ¼ n2i0 expðHi0 Gi0 Þ;

ð3:370Þ

where Hi0 represents the effect of bandgap narrowing and Gi0 is the effect of the Fermi–Dirac statistics. Because simulations are available for HBTs having InGaN base layers, those results in conjunction with the base transit time are discussed here. The base transit time so calculated with varying compensation factor, doping nonuniformity in the base (typically exponential in the form of Na(z) ¼ NaE exp(bz)), mole fraction of InGaN in the base (although the technological difficulties associated with this material limited reported efforts to either GaN or AlGaN base varieties), and compositional gradient expressed as x ¼ xE þ (xE  xC) z with xE and xC indicating the composition at the emitter and collector ends of the base, respectively, indicates the typical dependence. Simply, doping and compositional gradients help reduce the transit time whereas the increased compensation ratio increases it. The calculated dependence of tB on the neutral base width WB both with and without compositional grading is depicted in Figure 3.166. For these calculations, fcom ¼ 0.1, NaE ¼ 1.0  1019 cm3, and NaC ¼ 9.0  1017 cm3. While for results in Figure 3.166a, xE ¼ 0.20 at the emitter edge of the base and xC ¼ 0.20 at the collector edge of the base (e.g., no compositional grading), for results in Figure 3.166b, xE ¼ 0.20 at the emitter edge of the base and xC ¼ 0.35 at the collector edge of the base, and Na(z) ¼ NaE þ az. An enlarged view of the tB versus WB variation for WB ¼ 0.1–0.5 mm is shown in the inset of Figure 3.166b. Suzuki [593] suggested a doping profile.

3.12 Heterojunction Bipolar Transistors

Base transit time ( × 10 2), ps

3

1: Exponential I 2: Linear 3: Exponential II 4: Suzuki (s =10)

2

1 1

2

4

3

0 0.0

0.2

0 .4

0.8

0.6

1.0

Base width, W B ( µm)

(a) 100

1.8 4

60

τB (ps)

Base t r ansit time, τB (ps)

80

1.6

2

1.2

1

1 3

0.8 3

0.4

40

0.0 0.1

0.2

20

0.3 W B ( µm)

0.4

0.5 4 2

0.0 0.0 (b)

0.2

0.4

0.6

0.8

1.0

Base width, W B ( µm)

Figure 3.166 Variation of the base transit time tB with the base width WB for the InGaN base having different types of doping profile. The In mole fraction in the InGaN (nongraded) base in (a) is x ¼ 0.20; the linear compositional grading of the InGaN base in (b) is achieved with xC ¼ 0.20 and xC ¼ 0.35. The inset in (b) is a blown-up section for base widths of less than 0.5 mm. The curves 1, 2, 3, and 4 correspond

respectively to exponential I, linear, exponential II, and the Suzuki (s ¼ 10) type of doping profile. Exponential I corresponds to Na(z) ¼ NaEexp (bz), exponential II corresponds to Na(z) ¼ NaE 1=2 exp(gz1/2), where g ¼ logðN aC =N aE Þ=W B , Suzuki profile corresponds to that given in Equation 3.371 and linear corresponds to Na(z) ¼ NaE þ az, where a ¼ (NaC  NaE)/ WB) [623].

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 x s N aE N a ðzÞ ¼ N ae exp  ln N aC WB

ð3:371Þ

with an adjustable parameter s. It was noted that when s ¼ 10, the doping profile is essentially linear over most of the neutral base region and that it falls off very rapidly near the collector edge of the base. The variations of tB with WB for this doping profile are also included in Figure 3.166a. Figure 3.166b shows that the base transit time increases almost exponentially with the base width. This increase is most pronounced for the Suzuki type of doping profile and least pronounced for the exponential II doping profile. The variation of tB with WB was less sensitive for the linear doping profile than for the Suzuki type of the doping profile. Figure 3.166b shows that the variation of tB with WB remains essentially linear (see the inset) until WB ¼ 0.6 mm and that at WB 0.9 mm tB increases almost vertically upward. In short, although the above discussions are only for AlGaN/InGaN HBTs, their implications are valid for all group-III nitride HBTs that involve heavy doping and/or compositional grading in the base. These HBTs include AlGaN/GaN and AlGaN/ InGaN HBTs. For all these HBTs, the above discussions suggest that compositional grading along with an exponential base doping profile with a higher value at the emitter edge and a lower value at the collector edge should yield a lower value of the base transit time. In an HBT design, one must make sufficient effort to ensure that the base width of the transistor is as small as possible. Additional details can be found in the reference by Mohammad and Morkoç [623]. The transit time through the base is reduced by electric field induced by doping nonuniformity, typically exponential and/or compositional grading in such a way as to support the carrier motion from the emitter end of the base to the collector end of the base. On the intrinsic side, the higher the mobility in the base, which is related to the minority carrier diffusion constant through the Einstein relation, the shorter the transit time. This does not bode well for npn GaN-based HBTs, as the hole mobility is low in the base, thus so is the diffusion constant. These coupled with low Mg dopant activation, which causes large base resistance and the Early effect, reduce the output impedance. This led some to explore pnp HBTs in GaN-based bipolar transistors so that high base doping levels could be obtained in the base. This variety of devices will be discussed below. Another point to remember is that the minority carrier motion in GaN may be dominated by boundary recombination as many EBIC studies noted, and very little if any difference was noted between the minority carrier diffusion length in AlGaN and in GaN, contrary to the expectations that the former would be inferior to the latter. To reiterate the nascent nature of GaN-based HBTs does not lend itself to discussion of basic issues within the realm of GaN. For a better understanding the fundamental issues, we instead rely on the work carried out in the most heavily researched HBT, which is based on GaAs. A transmission line matrix (TLM) model accounting for diffusion, drift, and recombination was used [624] to calculate the base transit time of AlGaAs/GaAs HBTs, which show that the base transit time is reduced by 40–60% for base widths of 50–100 nm when the field introduced by compositional grading increases from 0 to 10 kV cm1. The base built-in field factor (lBI) is found to

3.12 Heterojunction Bipolar Transistors

be not only a function of the base field strength but also of the base width, and the typical value for this ranges between 2 and 5.5. The diffusion-drift effects in the base were characterized according to the relation tB / W nB (1
n
2), with lower n representing transport by drift and higher n representing transport by diffusion. It is worthwhile to note that when doping gradients are employed in the bases of HBTs, the base bandgap shrinkage has to be taken into account. The effective field Eeff in the quasineutral base is then composed of an electric field due to the positiondependent band structure. The effective field is [625]
   kB T rN A rðDE GN Þ E eff ¼  ; ð3:372Þ NA q q where DEGN is the effective bandgap shrinkage in the base. The first term in Equation 3.372, the impurity drift field, is negative and tends to accelerate electrons across the base. However, the second term, the quasielectric field, is a retarding field. As a result, the net accelerating field in the base is reduced. 3.12.1.3 Electron Velocity Overshoot in the Collector Space Charge Region The collector transit time through the collector space charge region (SCR) tSCR constitutes a large part of the total transit time from emitter to collector tBC. Electron velocity overshoot with electron velocity temporarily exceeding the nonequilibrium saturation velocity improves device performance by providing a high electron velocity over a wide collector region allowing the reduction of the tSCR. Electron velocity overshoot has been explored in Si, GaAs, and InP during the latter part of the twentieth century. Specifically, Shockley [626] reported that electrons escaping collisions could attain high kinetic energy from the electric field in the semiconductor. Later, Ruch [627] calculated electron velocity overshoot in Si and GaAs using a Monte Carlo method. It is very important for device designers to know whether or not velocity overshoot in the collector SCR significantly reduces the collector transit time. In this vein, Rockett [628], using a Monte Carlo simulation, investigated electron velocity overshoot in the collector SCR region and its influence on the collector transit time of AlGaAs/GaAs HBTs with collector doping of 5  1016 cm3. Similar concepts would be applicable to GaN-based devices as well, but with different parameters as the mean free length between collisions is material dependent. According to Rockett, velocity overshoot has only a marginal effect on reducing the total collector transit time in GaAs-based HBTs. In other words, the collector transit time is not appreciably different from those predicted with a constant saturation drift velocity vs of 1  107 cm s1 throughout the entire collector SCR region. By using transient Monte Carlo simulations, Das and Lundstrom [629] presented a new impulse response technique, which rigorously evaluates the total transit time delay, including base transit time and the collector transit time for AlGaAs/GaAs HBTs. The result shows that velocity overshoot reduces the collector delay by about 40% at VBC ¼ 0 V with an effective electron velocity veff of 1.35  107 cm s1. The benefits of velocity overshoot, however, are reduced as the base–collector bias increases. At VBC ¼ 3.0 V, velocity overshoot reduces the collector delay only about 10%. The effects diminish at higher biases because the electric field profile changes due to the increase

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in the space charge region width, which reduces the importance of the initial velocity overshoot. Of course, the doping profile changes the field profile and in turn the extent of the velocity overshoot. For example, an inverted field profile showed a decrease in the base–collector delay by more than 60%, which will be discussed below. An early attempt was made by Asbeck et al. [630] to model HBTs and address the issue of overshoot in the collector with a uniform 75 kV cm1 field. Following Asbeck et al., Maziar et al. [631] applied a Monte Carlo technique to simulate the impact of velocity overshoot on the collector SCR with inverted field on the transit time. An inverted field collector structure or p-type collector (np þ pn þ ) HBT structure was purported to provide a 25% improvement in the transit time over the conventional structure. Hu et al. [632] carried out Monte Carlo simulations to demonstrate that the inverted field HBT has the shortest fall time and overall switching time, although the storage time tends to be longer than that for the conventional collector HBT. The velocity overshoot effect and other particulars of the SCR region necessitate a correction for the collector transit time tSCR [633, 634]. Such a correction leads to a model [633] zðm

tSCR ¼


1

0

 z dz ; zm vðzÞ

ð3:373Þ

where zm is the width of the B–C depletion region, and v(z) is the position-dependent velocity of electrons in the B–C depletion region. To minimize tSCF, the weighting function (1  z/zm) dictates that v(z) should be the largest near z ¼ 0, with smaller values of v(z) resulting in progressively less penalty near z ¼ zm. The collector transit time in the Ishibashi model [634] is     1 W0 W0 Ws Ws þ þ tSCR ¼ ; ð3:374Þ v0 vs W 0 þ W s 2 v0 where v0 is the overshoot velocity in the B–C depletion region, W0 is the overshoot width and Ws is the width where the saturation velocity vs is maintained, and W0 þ Ws ¼ WC, the collector width. Simply, electron velocity overshoot in the collector region of HBTs in general plays an important role in high-frequency performance as illustrated here with AlGaAs/ GaAs HBTs. Electron velocity overshoot is most important when the B–C junction is forward or zero biased and at room temperature or less, and the influence on the high-frequency performance decreases at large reverse bias and at high temperatures. The collector velocity attains its maximum very near the base end of the depletion region and decays toward the neutral collector. The overshoot distance is about 300–500 Å for GaAs, could be as long as 1000 Å for InGaAs, but much shorter in GaN. The reduction in the collector transit time is beneficial for reducing the highfrequency saturation voltage RFVCE, which in turn increases the power-added efficiency in power applications. In short, although the above discussions are only for AlGaN/InGaN HBTs, their implications are valid for all group-III nitride HBTs that involve heavy doping and/or

3.12 Heterojunction Bipolar Transistors

compositional grading in the base. These HBTs include AlGaN/GaN and AlGaN/ InGaN HBTs. For all these HBTs, the above discussions suggest that compositional grading along with an exponential base doping profile with a higher value at the emitter edge and a lower value at the collector edge should yield a lower value of the base transit time. In an HBT design, one must make sufficient effort to ensure that the base width of the transistor is as small as possible. 3.12.1.4 Current Gain and Recombination Current of HBTs The salient feature of an npn HBT, as compared to BJT, lies in the energy barrier in the valence band at the emitter–base junction. This barrier resulting from the valence band discontinuity, DEV, suppresses the injection of holes into the emitter, thus leading to larger current gain and smaller minority carrier charge storage in the emitter. High injection efficiency can be traded somewhat for reduction of the base resistance, which in turn would increase the maximum oscillation frequency fmax. The current gain is adversely affected by the high surface recombination velocity of GaAs at low collector currents and by the Kirk effect at high collector currents. Therefore, the current gain, for example, of an npn HBT is a function of surface recombination current IS, bulk recombination current IR, collector current IC, and the base current IB. The value of current gain can be evaluated by

hFE

hFEmax 1 þ ðIS þ I R Þ=I P

ð3:375Þ

where IB ¼ IS þ IR þ IP is the base current, IP is the hole injection current, and hFEmax is the peak value of the current gain hFE. The bulk recombination current results from the presence of traps and other scattering parameters, and is negligibly small in high-quality materials. Ryum and Abdel-Motaleb [588] have demonstrated that both IR and IS depend, among others, on the applied bias, carrier lifetime, carrier diffusion length, and the width of the emitter (base) region. The mathematical formulae derived by these authors for IR and IS appear to be elegant enough for reasonable estimation of bulk and surface recombination currents. Surface state density, which can be higher for higher lattice mismatch at the heterojunction, is an important factor for IS. A comparison of the current gain expressions of Equations 3.355 and 3.356 with that of Equation 3.375 would indicate that unlike bF and bR, hFE involves the suppression of hole injection into the emitter from the base by the valence band discontinuity and that by virtue of treating both bulk and surface recombination effects, it is quite general. The peak value of the current gain hFEmax for an HBT with the uniform emitter doping density NE, and uniform base doping density NB, may be given by
 N E DB W E DE V hFEmax ¼ : ð3:376Þ exp N B DE W B kB T Depending on the material system, the ratio of DB, the carrier diffusivity in the base, and the ratio of DE, the carrier diffusivity in the emitter, may vary between 5 and 50. The ratio of the emitter width WE and of the base width WB may in general be between 2 and 5.

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In the case of a Si bipolar transistor, for which DEV ¼ 0 and (DBWE/DEWB) varies between 1 and 5, a larger hFEmax must require a large value of DB/DE. From the viewpoint of material quality, a high current gain means low defect density in the layer. For a circuit application, however, a uniform current gain hFE over a large current regime is more important than the peak value hFEmax. Surface recombination in HBTs, particularly in AlGaAs/GaAs varieties, is a crucial issue with serious implications for low-frequency noise, current gain, and device stability. The elusive nature of surfaces in general has led to scores of proposals as to the origin and nature of the surface recombination current IS, with a good deal of controversy. Henry et al. [635] have studied the effect of surface recombination on current in AlxGa1xAs/GaAs heterojunctions in detail, showing that the 2kBTcurrent in double-heterostructure AlxGa1xAs/GaAs/AlxGa1xAs p–n-junctions is primarily due to surface recombination at the junction perimeter. The 2kBT surface recombination behavior occurs when the recombination is limited by the surface recombination velocity and when the quasi-Fermi levels between the surface and bulk are flat. In fact, for a high surface recombination velocity, carriers are depleted by recombination rapidly, making recombination limited by the availability of minority carriers, which are supplied by diffusion [636]. Consequently, the quasi-Fermi levels do not remain flat and 2kBT current becomes 1kBT current. In the case of AlGaAs/GaAs HBTs, the extrinsic base surface is depleted and the recombination current changes from a kBT dependence at low currents to a 2kBT dependence at high currents, due to the two-dimensional flow of carriers into the extrinsic base surface depletion region. The ideality factor of the base current hB is used to account for various recombination mechanisms and a measure of quality for a bipolar transistor. Generally, hB is 1 for Si bipolar transistors and 1.5–2.0 for GaAs bipolar devices due to the high surface recombination velocity. Schemes such as the emitter edge-thinning design [636], the emitter guard-ring structure [637], and p-AlGaAs passivation layer design [637] have led to dramatic improvements in hB and reduced current gain degradation at low current levels. The overriding principle governing these designs is to prevent electron injection into the extrinsic base surface. Despite these efforts, the best reports of hB still remains high, being 1.5 for AlGaAs/GaAs HBTs with emitter edge-thinning design [638], 1.68 for GaInP/GaAs with the same design [638], and 1.4 for AlGaAs/GaAs HBTs with carbon-doped base [639]. For AlxGa1xAs/AlyGa1yAs DHBTs the best report of hB is 1.37 [640] and for InGaAs/InP DHBTs the best report is 1.3 [641]. Experimental results demonstrate that the extrinsic base surface recombination current increases with the base–emitter voltage with an ideality factor between 1.00 and 1.33, which is closer to unity than to 2 [642]. The origin of the 2kBTcomponent base current is a critical issue facing the AlGaAs/ GaAs HBT technology. Despite considerable research on this topic, confusion regarding its origin and nature still remains. The base current is composed of (i) recombination current in the space charge region (SCR) of the emitter–base junction, (ii) recombination current at the emitter sidewall, (iii) recombination current in the bulk base, and (iv) recombination current in the exposed extrinsic base surface between the emitter mesa and the base ohmic contacts. It is recognized that recombination of electrons and holes at the extrinsic base surface, forming the

3.12 Heterojunction Bipolar Transistors

base current component (iv), results in a size-dependent current gain in AlGaAs/ GaAs HBTs [617, 643, 644]. Only a few studies have dealt with the recombination current at the emitter sidewall [645, 646]. Gao et al. [646] proposed an N-i-p þ emitter sidewall model to interpret an ideality factor greater than 1 for base currents. As is well known, the high density of surface defects causes band bending at both AlGaAs and GaAs surfaces. As a result, the sidewall is depleted forming a thin surface depletion region, and the Fermi level at the sidewall surface is just at the middle of the bandgap. We refer to the surface depletion region as the emitter sidewall depletion region (ESDR). The recombination in the emitter sidewall depletion region may be the main contribution to the 2kBT base current of realistic AlGaAs/GaAs HBTs. For AlGaAs/GaAs HBTs on Si substrates, the carrier lifetime is reduced by about a factor of 10 or more, making the recombination current in the E–B SCR all the more important, which explains the measured hB ¼ 2.0. Reliable, small-sized, and highspeed AlGaAs/GaAs HBTs are required for circuit development. So far, the base recombination current remains high due to the recombination current in the ESDR. Results of Ref. [646] show that the use of AlGaAs at the air-exposed part of the lateral base is effective in reducing the recombination current in the extrinsic base substantially, down to about 0.2 mA mm1 around the emitter stripe perimeter, 40 times less than that for an unpassivated device. This particular layer, however, has no effect on the recombination current at the emitter sidewall, which is equal to 12 mA mm1, some 60 times larger than the recombination current at the extrinsic base. The importance of recombination current at the emitter sidewall is also apparent in the experiments by Nittono et al. [647]. Using O þ -implant E–B junction isolation technology, the device fabrication exhibited a sidewall recombination current of 4 mA mm1 at collector current density of 2.5  104 A cm2, compared to 20 mA mm1 recombination current at the same collector current density for H þ implant isolation devices. This particular effect suggests that O þ -implant E–B junction isolation technology is much better than H þ -implant isolation for reducing sidewall recombination. 3.12.1.5 Current Gain at High Currents The generally accepted explanations for the fall-off of current gain hFE at high currents are the Kirk effect and the temperature effect. It is well known that the power output of a power amplifier depends on the maximum operating current and voltage. Therefore, it is important to design a power HBT with high current capability and high breakdown voltage. To increase the collector current without seriously compromising the breakdown voltage requires a careful design of the collector region. This design may involve construction of a double-layer collector structure [648] with higher doping and thinner width near the B–C junction, which improves the current while preserving large breakdown voltages. If Wc1 and Nc1 are, respectively, the width and the doping density of the collector region 1, Wc2 and Nc2 are, respectively, the width and the doping density of the collector region 2, eS is the dielectric constant of the semiconductor concerned, and e0 is the absolute permittivity of the vacuum level, then the breakdown voltage for the double-layer collector structure may be given by

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qN c1 qN c2 BV CB0 ¼ E cri ðW c1 þ W c2 Þ W 2c1 þ 2W c1 W c2 W 2c2 V bi ; 2eS e0 2es e0 ð3:377Þ where Ecri is the critical field at breakdown and Vbi is the built-in voltage of the junction. For a double-collector layer, the critical current density JC required for the Kirk effect to occur will increase, and it is approximately given by   2eS e0 V BC J 0 ¼ qveff þ N ð3:378Þ c2 ; qW 2c2 where veff is an electron effective velocity taking the overshoot effect into account. Because veff is higher than the drift velocity vd, and Wc1 is less than (Wc1 þ Wc2), JC should be reasonably high. Consequently the cutoff frequency fT and the current gain hFE at high current density will be improved. 3.12.1.6 Emitter Current Crowding Effect At high current densities, the performance of bipolar transistors is limited by parasitic effects, such as the emitter current crowding effect. This arises when there occurs a lateral base distribution of the base resistance across the E–B junction and tends to pose important limitations, particularly in cases where the base resistance is relatively high (such as in Si bipolar transistors in which the base sheet resistance is usually high at 10 kW square1), and to a large extent in GaN-based HBTs (where the base doping is low and mobility is low). Combination of low mobility and low doping level creates an enormous challenge for GaN-based HBTs intended for high-current applications. As far as the AlGaAs/GaAs-based HBTs are concerned, it was generally assumed that because of lower sheet resistance, the current crowding effect is a minor issue. However, Liu and Harris [649] noted that this is true only when the base doping level is not excessively high. They noted that when the base doping approaches 1020 cm3, the current gain of HBTs drops dramatically and is accompanied by an overall increase in the voltage drop across the lateral base resistance. It was ascribed to the emitter crowding effect. A calculation indicated that the emitter widths above which this effect is significant is about 2 mm for npn HBTs and 4 mm for pnp HBTs. While seeking evidence for this, Fournier et al. [650] carried out some experimental investigations and concluded that there may be no emitter crowding effect even in 3 mm large HBTs under very high bias conditions. The problem of emitter current crowding is similar to the current crowding issues in LEDs and lasers that are grown on high-resistivity sapphire substrates where the n-contact is placed lateral to the anode. In fact, the treatment of current crowding in LEDs and lasers is based on the current crowding treatment in bipolar transistors developed much earlier. 3.12.1.7 Noise in Bipolar Transistors Similar in some ways to FETs, heterojunction bipolar transistors too involve noise sources that would need to be defined. Deviating from FETs, the most important noise source in HBTs (BJTs as well) is the shot noise. Of course, the thermal noise due to resistive components is also present. Succinctly, an HBT consists of two

3.12 Heterojunction Bipolar Transistors

i1b i4 rµ

B

rπ ia

1

i2

i3

C ra

v gmv

E

i1 E

Figure 3.167 Noise equivalent circuit for a BJT with the associated noise sources representing the shot noise components. The noise sources ia1 and ib1 can be lumped together and represented by the source i1 shown in broken lines across the emitter–collector terminals. All the thermal noise sources associated with resistors are not shown here but are included in the small signal equivalent circuit of Figure 3.168 [339].

p–n-junctions, the emitter–base junction of which is forward biased and the base–collector one is reverse biased. Consequently shot noise associated with various current components comes in the picture [339]. The various current paths in a bipolar transistor biased under normal operating conditions are shown in Figure 3.160. The terminal currents are given in Equations 3.345a–3.345c. In terms implications to noise, the four current components, Ii (i ¼ 1,2, . . ., 4) can be modeled by uncorrelated shot noise sources, which are indicated in Figure 3.167 in a simplified equivalent circuit form where the shot noise associated with the current component I1 is represented with two correlated noise sources ia1 and ib1 for that associated with the emitter–base and collector–base junctions. These two components are related by a phase shift of a(f )/ a0, where a( f ) represents the current gain at frequency f and a0 is the DC current gain. The amplitudes of the noise current sources are related to DC current through [339]. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ia1 ¼ 2qI1 Df exp½jðwt þ f1 Þ; p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ib1 ¼ 2qI1 Df exp½jðwt þ f1 Þ; a0  2 ð3:379Þ i ¼ 2qI Df ; 2

 2 i3  2 i4

2

¼ 2qI 3 Df ; ¼ 2qI 4 Df :

For cases where a/a0 1, such as at moderate frequencies, ia1 and ib1 noise sources can be combined into one noise source i1 across the collector–emitter junction, shown with broken lines in Figure 3.167 with the magnitude of  2 i1 ¼ 2qI 1 Df : ð3:380Þ In addition to the shot noise, there is the contribution from the thermal noise. The small-signal equivalent circuit inclusive of both shot and thermal noise sources for a

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CpBC CxBC inBB B

LB RBB

LC

Rπ C π

gmv

C

RCC

Cµ v

inB

inCC Rµ

inC

CpBE

R0 CpCE

inEE

REE

LE E Figure 3.168 Small-signal equivalent circuit of a bipolar transistor inclusive of thermal and shot noise sources.

bipolar transistor is shown in Figure 3.168. The circuit can be used to determine the two-port network parameters in the s-parameter or the y-parameter format. Focusing on the shot noise contribution first, their generators at the input and output of the intrinsic bipolar transistor can be expressed as [339]. 2  ð3:381Þ inB ¼ 2qInB Df with InB being the DC base current causing the shot noise. Similarly, the shot noise current source for the collector, output, with a DC current of InC can be expressed as 2  ð3:382Þ inC ¼ 2qInC Df : These two noise sources are correlated to each other through qffiffiffiffiffiffiffiffiffiffiffiffi hinB inC i ¼ C i2nB i2nC ;

ð3:383Þ

where C ¼ CR þ jCI is the complex correlation coefficient [339]. The thermal noise caused by resistances RBB, REE, and RCC (see Figure 3.168 for description) is the source for three uncorrelated noise sources termed inBB, inEE, and associated thermal Johnson noiseffi current sources are defined, in order, as inCC. The pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi inBB ¼ 4kB T B =RBB ; inEE ¼ 4kB T E =REE ; and inCC ¼ 4kB T C =RCC . For a complete description of noise behavior of the circuit, the equivalent noise temperatures, TB, TE, and TC, of the associated resistances, InB, InC, and C must be extracted. In addition, the parasitics due to wire bonds to the terminals, such as the inductances,

3.12 Heterojunction Bipolar Transistors

and the parasitic capacitances, such as those caused by pads, are modeled by LB, LE, and LC, and CpBE and CpCE, respectively, as shown in Figure 3.168. It should be mentioned that the large number of model parameters renders it impractical to extract the pertinent parameters using analytical means, making numerical algorithms imperative [339]. 3.12.2 Nitride-Based HBTs

One of the major attractions of GaN-based bipolar transistors is the potential for large collector breakdown voltages and concomitant high-power applications, which are yet to be realized. Considering the low diffusion constant and low in-plane conductivity in p-type GaN coupled with relatively low p-type doping, and the inferior emitter–base n–p-junction quality, it is very unlikely that GaN bipolar transistors would be competitive. Nevertheless, a discussion of the issue is warranted. Actually, this matter is discussed in some detail in Section 3.5.3. To gain an appreciation of the ultimate breakdown voltage based on calculated ionization coefficients [262, 264], see Volume 2, Chapter 4. After estimating the ionization coefficients with the following power law expressions [651], the depletion layer width Wbr at breakdown and breakdown voltage Vbr against the background doping concentration can be obtained as shown in Figure 3.169. 23=25

W br ¼ 8:84  1010 N D

and

21=25

V br ¼ 7:43  1016 N D

;

ð3:384Þ

where ND is the doping concentration. As a precursor to bona fide high-voltage GaNbased power devices, breakdown of linearly graded p–n-junctions has been examined to have a critical field of 1.5–3 MV cm1 [652]. The breakdown voltage measured was in the range of 40–150 V determined at a reverse leakage current of 10 mA cm2. The p–n-junction breakdown exhibited a positive temperature coefficient of 0.02 V K1 reflecting the temperature coefficient of ionization coefficients. The breakdown was reported to be of a localized mesoplasmic nature, implying that it is expected that these results do not represent the upper limit for avalanche breakdown in GaN. The carrier lifetime in the base, together with the base width, determines the base transport ratio. While shorter lifetimes lead to smaller transport or transfer ratios, the excess charge in the base can be removed faster thus making the device a faster one. In this regard, the direct bandgap semiconductors are good in terms of reduced charge storage. The issue of carrier transport factor across the base associated with short minority carrier lifetimes can be addressed by making the base thinner. However, doing so increases the base resistance unless the doping level in the base is increased. This can be achieved in GaAs-based HBTand C doping is able to produce hole concentrations in the 1020 cm3 range. Unfortunately, this is the crux of the problem in GaN as p-type doping cannot be made sufficiently high to have the freedom desired in the choice of the base thickness. The collector breakdown voltage issue can be shifted mostly to the lightly doped collector region provided the base doping is sufficiently high, except for conductivity modulation discussed in the next paragraph. Otherwise, the depletion region can extend into the base region and may even reach

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(a)

Depletion layer thickness (µm)

10 3

Wz GaN ZB GaN

10 2

10 1

10 0

10-1

10-2 14 10

15

10

16

10

17

10

18

19

10

10

Doping level (cm-3) (b)

Breakdown voltage (V)

105

Wz GaN ZB GaN

104

103

102

101 14 10

15

10

16

10

17

10

18

10

19

10

Doping level (cm -3) Figure 3.169 (a) Depletion layer thickness at breakdown Wpp and (b) parallel plane breakdown voltage BVpp both versus doping concentration for a one-sided GaN junction. Patterned after Ref. [651] using the ionization coefficients reported in Refs [262, 264].

the emitter junction, which is called the reach through, at high collector voltages. Again, the base doping is the single most important parameter in any HBT. To have conductivity modulation in the collector of a GaN-based HBT in saturation, the collector width should be about five times the diffusion length or less. Taking the high number for the device to sustain a large voltage and using Figure 3.169 with the assumption of a 1 mm diffusion length, one arrives at a 5 mm collector width, which could support about 1000 V [651]. With a wider collector, the conductivity modulated would not be available and a resistive drop due to the unmodulated collector region

3.12 Heterojunction Bipolar Transistors

would increase the voltage required for saturation. For further increases in the collector voltage to more than 1000 V, diffusion lengths longer than 1 mm are necessary. Let us now turn our attention to the experimental observations, which are few, in GaN-based bipolar transistors. The first heterojunction bipolar transistor utilizing nitride semiconductors was a hybrid as both GaN and SiC technologies were used to construct it. The energy bandgap of GaN and SiC are 3.4 and 2.9 eV, respectively, and the great majority of the band discontinuity occurs at the valence band, which bodes well for npn HBTs. In addition, both GaN and SiC have high thermal conductivities, 2.3 W cm1 K1 for GaN, and 4.0 W cm1 K1 for SiC. A GaN/SiC HBT with high current gain has been reported by Pankove et al. [653] but measured in the common-base configuration where the collector–base junction leakage current must be accounted for. The n-GaN emitter, 0.57 mm thick, had an unintentional doping level 1  1018 cm3 (grown by MBE), and the 6-H p-SiC base, 0.2 mm thick, had a doping level 9  1018 cm3. The SiC substrate with n-type doping of 1.8  1018 cm3 formed the collector. The GaN layer was etched in CCl2F2 plasma in unwanted regions during device fabrication. High doping concentrations in the base as well as in the collector led to negligible early voltage and very small breakdown voltage. The common-collector configuration used to get at the current gain in the light of a leaky collector junction relied on differential current gain determination, which led to extremely high current gains. Using appropriate parameters, an emitter injection efficiency of 0.999 999 was deduced. Moreover, using a mobility of 110 cm2 V1 s1 and a lifetime of 5 ms, the diffusion length and the base transport factor were calculated to be 37.7 mm and 0.999 987, respectively, for SiC base [653]. The calculated parameters led to a current gain of 80 409, which is very close to experimental observations. Again, it should be pointed out that the measurements are not based on common-emitter configuration. All GaN-based HBTs with AlGaN emitter with common-emitter characteristics have been demonstrated [654]. One technological issue with p-type base is that large Mg concentrations are required, which exacerbate the already problematic Mg memory effect/segregation and displace the p–n-junction from the AlGaN/GaN heterojunction. Depending on the displacement, the p–n-junction could be in all AlGaN in that case. Employment of regrown emitter, in other words, the growth is terminated after collector and base growth and the sample after some processing is placed in the growth vessel, can mitigate the Mg memory and outdiffusion effects. Doing so, common-emitter characteristics with collector breakdown voltages >80 V have been obtained [655]. The structure for this particular work was grown on c-plane sapphire with OMVPE, which commenced with n-type subcollector (1  1019 cm3 Si doped), followed by an unintentionally doped n-type GaN (5  1016 cm3 donors) collector and a 100 nm p-type GaN base (on the order of 1019–1020 cm3 Mg doped). The sample was then removed from the growth vessel and a dielectric regrowth mask was deposited and patterned. The sample was then returned to the growth chamber and a 400 nm thick n-type (1  1019 cm3 Si doped) GaN emitter was grown selectively on the base. The dimension of the emitter was 17  350 mm2. Owing to the deep nature of Mg in GaN, the activation is expected on the order of a small

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percentage. The measured room-temperature current gain of 3 corresponds to a current transfer ratio of 0.75, which results in a calculated minority carrier lifetime of about 80 ps in the base. The high base resistance in npn heterojunction transistors (due to low efficiency of p-type doping) has compelled some to explore the pnp varieties, because it would be easier to achieve high doping in an n-type GaN base, the relatively higher carrier mobility in n-type material would reduce base resistance. In HBT structures, there would also be larger emitter–base energy bandgap differences and band discontinuities than in the npn configuration for base minority carriers back injected into the emitter [656]. Consequently, the current gain degradation due to back injection would be extremely small, leaving the base recombination, and possibly the E–B interface recombination as the dominant current gain reduction mechanism. In the case of the pnp HBT investigated by Zhang et al. [656], the device structure was grown by OMVPE at approximately 1050  C on (0 0 0 1) c-plane sapphire following the typical OMVPE practices that included a low-temperature initiation layer at 500  C followed by ramping to the final growth temperature where an Mg-doped (p-type) subcollector was grown. This was then followed by a GaN buffer layer to reduce the defect density, a 1 mm thick Mg-doped GaN subcollector with hole concentration more than 2  1017 cm3, followed by a 0.5 mm thick Mg-doped collector with hole concentration 2  1016 cm3, a 0.1 mm Si-doped (n-type 1018 cm3) base, an Mg-doped (p-type, 0.2 mm thick) Al0.1Ga0.9N emitter with hole concentration 2  1017 cm3, and an Mg-doped GaN (0.5 mm thick) emitter contact layer with hole concentration more than 2  1017 cm3. The resistivity of the base layer is estimated from similar bulk layers at 2.5  102 W m, corresponding to a sheet resistivity of 2.5 k W q1 for a 100 nm thick base. This is some two orders of magnitude lower than for a p-type GaN base with typical hole concentration 2  1017 cm3 and hole mobility of 10 cm2 V1 s1. Device fabrication proceeded by exposing the base and subcollector layers for base and collector ohmic contacts by using inductively coupled plasma (ICP) etching in a Cl2/Ar discharge at 300 W source power and 85 V DC self-bias, which led to an etch rate of about 100 nm min1. The etch damage was removed by annealing at 750  C for 30 s under N2, and the device isolation was also achieved by dry etching to the substrate. The fabrication process was completed by Ni/Pt/Au metallization and annealing for the emitter and subcollector and Ti/Al/Pt/Au for the base in the form of circular geometry emitters in the range of 50–100 mm. The specific contact resistivities were 102 W for p-type and 105 W for n-type, the former representing a problem in that it increases the emitter resistance. Testing was performed up to 250  C using a high-temperature probe station and a parameter analyzer. Common-base characteristics from a 50 mm diameter HBT are shown in Figure 3.170 in the range of 25–250  C because the GaN material is a robust one and intended for high-power/temperature applications. It is presumed that common-emitter characteristics were not observed. The collector emitter offset voltage, discussed in Section 3.12.1.1, is on the order of 2 V, which is similar to that for a GaN npn BJT, but lower than the 5 V for a GaN npn HBT. Note that the breakdown voltage decreases with increasing temperature. Previous reports of temperature-dependent

Collector current (mA)

3.12 Heterojunction Bipolar Transistors

-1.5

25 °C IE step=200 µA

-1.0

-0.5

0.0

Collector current (mA)

0 -1.5

-5

-10

-15

-20

-25

150 °C IE step=200 µA

-1.0

-0.5

Collector current (mA)

0.0 -1.5

0

-5

-10

-15

-20

-25

0

-5

-10

-15

-20

-25

250 °C IE step=200 µA

-1.0

-0.5

0.0

Collector–base voltage (V) Figure 3.170 Common-base I–V characteristics at 25, 150, and 250  C for AlGaN/GaN/GaN pnp HBTs. The emitter current was stepped in 200 steps from 200 mA [656].

performance of pnp and npn GaN BJTs have shown little change in the common-base current–voltage characteristics at elevated temperatures, which imply technological problems with the heterojunction and/or the emitter. Unfortunately, due to excessive leakage in the collector–base junction, the common-emitter characteristics were not reliable [656]. However, in the best devices Gummel plots showed DC current gains of 20–25 at room temperature. A plot from one of these devices is shown in Figure 3.171. The offset is still quite high at 4–4.5 V, while collector currents up to several milliamperes were achieved. At room temperature, the maximum current densities are near 2.5 kA cm2 at 10 V, which correspond to a power density of 20.4 kW cm2. The current densities in these large area devices are nowhere near what they are for GaAs- and SiGe-based devices, where these numbers run well above 105 A cm2. If the breakdown voltage can be increased while providing comparable current densities, the GaN devices would have applications.

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102

AlGaN/GaN pnp HBT 50 oC

Base current

100

Current (mA)

Colector current

10-2

10-4 2

3

4

5

6

7

8

9

10

Base–emitter voltage (V) Figure 3.171 Gummel plot of HBT with VCB ¼ 0 [656].

Extending the GaN base to InGaN base, Makimoto et al. [657] reported on NpN DHBT with GaN emitter and collector and InGaN base layers. The particular DHBT structure was grown on SiC with OMVPE and the emitter and base junction areas, the latter provides access to subcollector for contacts, were defined by electron cyclotron resonance plasma etching. In the common-emitter configuration, the maximum current gain of 20 was obtained at room temperature. The In composition used in the base was 6% and the base thickness was 100 nm. To avoid the adverse effects of conduction band discontinuity at the base–collector junction, such as charge buildup and the associated time delay, the In mole fraction was graded up to GaN. It was reported that a chemical Mg doping concentration in the base layer of 1  1019 cm3 led to a hole concentration of 5  1018 cm3 at room temperature. The common-emitter collector output current–voltage characteristics are presented in Figure 3.172 showing a GaN/p-InGaN/GaN DHBT with base current increments of 0.1 mA. The maximum common-emitter current gain attained is 20 at a collector current of 3 mA, which compares with 10 reported for AlGaN/p-GaN/GaN variety. A very large offset voltage is noted, which is due to the asymmetry between the E–B and B–C junctions including the asymmetry in junction areas and resistances inclusive of the ohmic contacts [591], as discussed in Section 3.12.1.1. In this particular case, the E–B junction area was 60  350 mm2. A carrier diffusion model with a constant lifetime and a unity emitter injection efficiency [616] led to a minority carrier diffusion length of 0.32 mm in the p-InGaN base layer, compared to 0.2 mm reported for p-GaN [658]. Moreover, with an assumed electron mobility of 100 cm2 V1 s1 in p-type In0.06Ga0.94N base, one obtains a minority carrier lifetime of 400 ps.

Collector current (mA)

3.13 Concluding Comments Base current 0.1 mA/step

8 6 4 2 0 0

2

4

6

8

10

12

14

Collector–emitter voltage (V) Figure 3.172 Common-emitter output I–V characteristics of a GaNp-InGaN/GaN DHBT [657].

The purpose of discussing above-mentioned results is to show that the state of the art of this material system is such that transistor action can be obtained. The road to functional devices in systems, however, is a long way away.

3.13 Concluding Comments

GaN-based heterojunction field effect transistors, in the form of doped or undoped varieties, have great potential due to large RF power densities and absolute power levels that can be obtained with them when used as amplifiers. Unlike the GaAsbased HFETs on (1 0 0) surfaces, polarization-induced charge in GaN varieties on the polar (0 0 0 1) surfaces is quite large. Consequently, even undoped structures contain sheet electron concentrations in the 1013 cm2 range. In unintentionally doped structures, donor-like surface defects most likely provide the electrons except when fabricated into FETs in which case the source of electrons would most likely be the source contact as in the case of Si-based MOSFETs. Nonpolar orientations of GaN can be used to eliminate polarization charges, but the layer quality lags well behind that on c-plane. Experimental data and calculation results have been provided on the particulars of the interface charge in relation to parameters such as the AlGaN mole fraction and thickness. In addition, calculation results for electron distribution and current–voltage characteristics of HFETs have been presented. On the experimental side of HFET performance, CW power levels of about 22.9 W at 9 GHz have been reported in devices with four 1 mm gate periphery devices in a single-stage power-combining scheme with an associated power-added efficiency of 37%. A minimum noise figure of 0.85 dB with an associated gain of 11 dB at 10 GHz has been obtained. These power figures continually changed upward to over 400 W at 2 GHz, some 100 W at 10 GHz, and nearly 8 W at 30 GHz, all of which are remarkable. A discussion of the current kinks, collapse, and current lag occurring in GaN-based HFETs is provided. In short, GaN-based HFETs have made great strides and are continuing to do so despite the less than ideal material properties. Anomalies in the current–voltage characteristics at low and higher frequencies observed in these devices are

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attributed to traps in the structure, surface states, and slow trapping processes associated with the field-induced lateral extension of the strain near the gate. It may be only a matter of time for inclusion of these devices in systems, but not before the reliability issues are fully addressed. Using field plate to extend the field and thus reduce its peak values increased the power levels at low frequencies at the expense of precluding the use of these devices at millimeter-wave applications. The efficient electron–phonon coupling in highly ionic GaN and resulting hot phonons and the less than ideal decay process to LA phonons are of concern in terms of performance and heat dissipation. In addition, the gate leakage, particularly under RF stress and coupling of field, heat, polarization, and strain along with hot electron issues, is a source of device failure, details of which are also discussed in this chapter. Moreover, both the low-frequency and high-frequency noise fundamentals along with available data are provided. Because this chapter is on FETs and HBTs more than just GaN varieties, fundamentals of HBTs are treated in detail even though GaN-based HBT performance is really very dismal and may remain so. The lackluster HBT performance in GaN, in part due to the low quality of p-type base (localization effects and associated low carrier diffusion length) and Mg memory and segregation effects that the emitter layer grown on that base, leads to low-quality layers and junctions. In addition, growth of what would be the emitter atop a poor quality p-type GaN also leads to inferior quality emitter layers. The fundamentals of HBTs provided in this chapter, however, are applicable to any HBT.

3.14 Appendix: Sheet Charge Calculation in AlGaN/GaN Structures with AlN Interface Layer (AlGaN/AlN/GaN)

In nitride semiconductors with wurtzite phase, spontaneous and piezoelectric polarization effects are present [659], which must be taken into account in the balance of boundary condition at the interface (z ¼ d). In this respect, the conservation of the normal component of the electrical displacement leads to (see Figure 3.173). e1  Fðz ¼ d þ Þe2  Fðz ¼ d Þ ¼ P2 P 1 ¼ DP:

ð3:385Þ

In region I characterized by the AlGaN layer, the Poisson’s equation can be written as e1

dF 1 ¼ qnsfþ dðz þ t þ wÞ þ qN Dþ : dz

ð3:386Þ

Here, nsfþ is the density of ionized surface donor states. An integration of which leads to the electric field in the same region as F1 ¼

e þ q n þ N Dþ  ðz þ t þ wÞ þ C e1 sf e1

ð3:387Þ

with C being the integration constant. Doing the same for region II leads to

3.14 Appendix: Sheet Charge Calculation in AlGaN/GaN Structures with AlN Interface

Figure 3.173 (a) Charge distribution, (b) electric field distribution in a c-plane Ga-polarity AlGaN/AlN/GaN heterostructure.

e2

and

dF 2 ¼0 dz

and

F2 ¼ C

e1 q e1 q þ ðn þ þ N Dþ wÞ ¼ C þ ns e2 e2 sf e2 e2

ð q nðzÞdz e2  
ð mC kB Tq X z E F E l þ C0 : ¼ jyl ðzÞj2 dz ln 1 þ exp k T pe2 B d l

F2 ¼ 

ð3:388Þ According to charge neutrality condition: nsfþ

þ N Dþ d

1 ð

¼ ns

with ns ¼

nðzÞdz;

ð3:389Þ

z¼0

which is a constant as the doping level in AlN is taken to be zero. Again doing likewise

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674

for region III leads to e3

dF 3 ¼ qnðzÞ dz

and F3 ¼ 


 ð ðz q mc kTq X E F E l 2 þC 0 : j y ðxÞ j dx ln 1þexp nðzÞdzþC 0 ¼  l kT e3 p
h2 e3 l 0

F1 at the interface will be F 1 ðz ¼ dÞ ¼

q þ q þ q n þ N d þC ¼ ns þC: e1 sf e1 D e1

ð3:390Þ

The constant C0 can determined noting that F3(z ¼ 0)C0 as C0 ¼

Ce1 þqns DP : e3

Noting that F2(z ¼ d) ¼ C0 , from Equation 3.385, we could determine C0 by C0 ¼

DP þCe1 þqns : e2

ð3:391Þ

Ðz Note that for large values of z, 0 jyl ðxÞj2 dx ¼ 1. The constant C can be determined noting that the electric field in the bulk of GaN vanishes, that is, F 3 ðz!1Þ ¼ 0 as C¼

DP : e1

ð3:392Þ

The electric field in regions I, II, and III can then be expressed as q DP q þ DP e ðns þN Dþ tÞþ þ N z; F 2 ¼ þ ns ; and e1 e1 e1 D e2 e2 ð q qns F3 ¼  nðzÞdzþ e3 e3

F1 ¼

ð3:393Þ

with 1 ð

ns ¼

nðzÞdz: z¼0

Summary

The chapter treats field effect transistor fundamentals that are applicable to any semiconductor materials with points specific to GaN. The discussion primarily focuses on 2DEG channels formed at heterointerfaces and their use for FETs, including polarization effects. A succinct analytical model is provided for

References

calculating the carrier densities at the interfaces for various scenarios and current voltage characteristics of FETs with several examples. The 2-port network analysis, s-parameters, various gain expressions, circuit parameter extraction of equivalent circuit parameters, for both low and high rf power cases, temperature and dispersion effects are discussed in detail. Experimental performance of GaN-based FETs and amplifiers is then discussed followed by an in-depth analysis of anomalies in the current voltage characteristics owing to bulk and barrier states, including experimental methods and probes used for cataloging these anomalies. This is followed by the employment of field spreading gate plates and associated performance improvements. This segues into the discussion of noise both at the low-frequency end and high-frequency end with sufficient physics and practical approaches employed. The combined treatment of various low-frequency noise contributions as well as those at high frequencies along with their physical origin makes this treatment unique and provides an opportunity for those who are not specialists in noise to actually grasp the fundamentals and implications of low- and high-frequency noise. Discussion of high-power FETs would not be complete without a good discussion of heat dissipation and its physical pathways, which is made available. Unique to GaN is the awareness of the shortfall in the measured electron velocity as compared to the Monte Carlo simulation. Hot phonon effects responsible for this shortfall are uniquely discussed with sufficient theory and experimental data. Power dissipation pathways from hot electrons to hot LO phonons followed by decay to LA phonons and in turn to heat transfer to the bath are treated with sufficient physics. In particular, the dependence of the hot phonon lifetime on the carrier concentration and its implications to carrier velocity is treated. The effect of lattice matched AlInN Barrier layers vis a vis AlGaN barrier layers on the hot LO phonon lifetime and carrier velocity is treated. A section devoted to reliability with specifics to GaN based high power HFETs is also provided. Such effects as surfaces, carrier injection by the gate to the surface states and the resultant virtual extended gate, surface passivation, interplay of temperature, strain, and electric field and their combined effect on reliability are treated in detail. Finally, although GaN-based bipolar transistors are not all that attractive at this time, for completeness and the benefit of graduate students and others who are interested in such devices, the theory, mainly analytical, of the operation of heterojunction bipolar transistors is discussed along with available GaN based HBT data.

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heterostructures. Japanese Journal of Applied Physics, 46, 547. Hierro, A., Ringel, S.A., Hansen, M., Speck, J.S., Mishra, U.K. and DenBaars, S.P. (2000) Hydrogen passivation of deep levels in n-GaN. Applied Physics Letters, 77, 1499. Kikawa, J., Yamada, T., Tsuchiya, T., Kamiya, S., Kosaka, K., Hinoki, A., Araki, T., Suzuki, A. and Nanishi, Y. (2007) An analysis of the increase in sheet resistance with the lapse of time of AlGaN/GaN HEMT structure wafer. Physica Status Solidi c, 4, 2678. Elhamri, S., Saxler, A., Mitchel, W.C. et al. (2003) Study of deleterious aging effects in GaN/AlGaN heterostructures. Journal of Applied Physics, 93, 1079. Shockley, W. (1951) US Patent No. 2,569,347. Kroemer, H. (1957) Theory of a wide-gap emitter for transistors. Proceedings of IRE, 45, 1535–1537. Kroemer, H. (1982) Heterostructure bipolar transistors and integrated circuits. Proceedings of IRE, 70, 13–25. Gao, G.B., Mohammad, S.N., Martin, G.M. and Morkoç, H. (1996) III–V compound semiconductor heterojunction bipolar transistors, in Compound Semiconductor Electronics: The Age of Maturity (ed. M.S. Shur), World Scientific, pp. 85–174. Mohammad, S.N. and Morkoç, H. (1996) Progress and prospects of group III–V nitride semiconductors. Progress in Quantum Electronics, 20 (5–6), 361–525. Chang, M.F. and Asbeck, P.M. (1990) III–V heterojunction bipolar transistors for high-speed applications. International Journal of High Speed Electronics, 1, 245–301. Lundstrom, M. (1991) III–V heterojunction bipolar transistors, in Heterojunction Transistors and Small Size Effects in Devices (ed. M. Willander), Krieger, California (Chapter 1). Marty, A., Rey, G.E. and Bailbe, J.P. (1979) Electrical behavior of an npn GaAlAs/

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GaAs heterojunction transistor. SolidState Electronics, 22, 549–557. Lee, S.C., Kau, J.N. and Lin, H.H. (1985) Current transport across the emitter–base potential spike in AlGaAs HBTs. Journal of Applied Physics, 58, 890–901. Lundstrom, M.S. (1986) An Ebers–Moll model for the heterostructure bipolar transistors. Solid-State Electronics, 29, 1173–1179. Ryum, B.R. and Abdel-Motaleb, I.M. (1990) A Gummel–Poon model for abrupt and graded heterojunction bipolar transistors (HBTs). Solid-State Electronics, 33, 869–880. Parikh, C.D. and Lindholm, F.A. (1992) A new charge-control model for single-and double-heterojunction bipolar transistors (HBTs). IEEE Transactions on Electron Devices, 39, 1303–1311. Grinberg, A.A. and Luryi, S. (1992) On the thermionic-diffusion theory of minority transport in heterostructure bipolar transistors. IEEE Transactions on Electron Devices, 40, 859–866. Mazhari, B., Gao, G.B. and Morkoç, H. (1991) Collector-emitter offset voltage in heterojunction bipolar transistors. SolidState Electronics, 34, 315–321. Ramberg, L.P. and Ishibashi, T. (1988) Abrupt interface AlGaAs/GaAs heterojunction bipolar transistors: carrier heating and junction characteristics. Journal of Applied Physics, 63, 809–820. Suzuki, K. (1991) IEEE Transactions on Electron Devices, 38, 2128–2133. Rosenfeld, D. and Alterovitz, S.A. (1994) IEEE Transactions on Electron Devices, 41, 848. Rosenfeld, D. Alterovitz, S.A. (1994) Solid-State Electronics, 37, 119–126. Lu, T.C. and Kuo, J.B. (1993) IEEE Transactions on Electron Devices, 40, 766–772. Kuo, J.B. and Lu, T.C. (1993) Solid-State Electronics, 36, 917–921. Winterton, S.S., Peters, C.J. and Tarr, N.G. (1993) Solid-State Electronics, 36, 11611164.

References 598 See Ref. [624]. 599 Mazier, C.M. and Lundstrom, M.S. (1987) IEEE Electron Device Letters, 8, 90–92. See also Ritter, D. Hamm, R.A. Feygenson, A. Smith, P.R. (1994) Applied Physics Letters, 64, 2988–2990. 600 Mohammad, S.N. and Morkoç, H. (1995) Base transit time in GaN/InGaN heterojunction bipolar transistors. Journal of Applied Physics, 78, 4200–4205. 601 Pulfrey, D.L. and Fathpour, S. (2001) IEEE Transactions on Education, 48 (3), 597–602. 602 Monier, C., Ren, F., Han, J., Chang, P.-C., Shul, R.J., Lee, K.-P., Bacca, A.G. and Pearton, S. (2001) IEEE Transactions on Education, 48 (3), 597–602. 603 Chernyak, L., Park, Y.J. and Morkoç, H., unpublished data. 604 Ren, F., Abernathy, C.R., Van Hove, J.M., Chow, P.P., Hickman, R., Klaassen, J.J., Kopf, R.F., Cho, H., Jung, K.B., LaRoche, J.R., Wilson, R.G., Han, J., Shul, R.J., Baca, A.G. and Pearton, S.J. (1998) 300  C GaN/AlGaN heterojunction bipolar transistor. MRS Internet Journal of Nitride Semiconductor Research, 3, 41. 605 McCharty, L.S. et al. (1999) IEEE Electron Device Letters, 20, 277–279. 606 McCharty, L.S., Smorchkova, I.S., Xing, H., Kozodoy, P., Fini, P., Limb, J., Pulfrey, D.L., Speck, J.S., Rodwell, M.J.W., DenBaars, S.P. and Mishra, U.K. (2001) IEEE Transactions on Education, 48 (3), 543–549. 607 Maziar, C.M. and Lundstrom, M.S. (1987) On the estimation of base transit time in AlGaAs/GaAs bipolar transistors. IEEE Electron Device Letters, 8, 90–92. 608 Levi, A.F.J., Jalali, B., Nottenburg, R.N. and Cho, A.Y. (1992) Vertical scaling in heterojunction bipolar transistor with nonequilibrium base transport. Applied Physics Letters, 60, 460–462. 609 Ritter, D., Hamm, R.A., Feygenson, A. and Panish, M.B. (1991) Diffusive base transport in narrow base InP/Ga0.47In0.53 as heterojunction bipolar transistors. IEEE IEDM 91 Technical Digest, pp. 967–969.

610 Harmon, E.S., Melloch, M.R., Lundstrom, M.S. and Cardone, F. (1994) Thermal velocity limits to diffusive electron transport in thin-base np þ n GaAs bipolar transistors. Applied Physics Letters, 64, 205–207. 611 Ankri, D. and Eastman, L.F. (1982) GaAlAs–GaAs ballistic heterojunction bipolar transistor. Electronics Letters, 18, 750–751. 612 Lundstrom, M.S., Datta, S., Bandyopadhyay, S., Cahay, M., Das, A., Dungan, T.E., Klausmeier-Brown, M.E., Maziar, C.M. and McLennan, M.J. Physics and modeling of heterostructure semiconductor devices. Technical Report TR-EE 86-31, School of Electrical Engineering, Purdue University, West Lafayette, IN. 613 Dodd, P. and Lundstrom, M. (1992) Minority electron transport in InP/ InGaAs heterojunction bipolar transistors. Applied Physics Letters, 61, 465–467. 614 Fukano, H., Nakajima, H., Ishibashi, T., Takanashi, Y. and Fujimoto, M. (1992) Effect of hot-electron injection on highfrequency characteristics of abrupt In0.52(Ga1xAlx)0.48As/InGaAs HBTs. IEEE Transactions on Electron Devices, 39, 500–506. 615 Grinberg, A.A. and Luryi, S. (1992) Ballistic versus diffusive base transport in the high-frequency characteristics of bipolar transistors. Applied Physics Letters, 60, 2770–2772. 616 Ito, H., Ishibashi, T. and Sugeta, T. (1985) Current gain enhancement in graded base AlGaAs/GaAs HBTs associated with electron drift motion. Japanese Journal of Applied Physics, 24, L241– L243. 617 Nakajima, O., Nagata, K., Ito, H., Ishibashi, T. and Sugeta, T. (1985) Emitter–base junction size effect on current gain hFE of AlGaAs/GaAs heterojunction bipolar transistors. Japanese Journal of Applied Physics, 24, L596–598.

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618 Nakajima, O., Nagata, K., Ito, H., Ishibashi, T. and Sugeta, T. (1985) Suppression of emitter size effect on current gain in AlGaAs/GaAs HBTs. Japanese Journal of Applied Physics, 24, 1368–1369. 619 Jain, S.C. and Rouslton, D.J. (1991) A simple expression for bandgap narrowing (BGN) in heavily doped Si, Ge, GaAs and GexSi1x strained layers. Solid-State Electronics, 34, 453. 620 Van Vliet, C.M. (1993) Bandgap narrowing and emitter efficiency in heavily doped emitter structures revisited. IEEE Transactions on Electron Devices, 40, 1140–1147. 621 Kroemer, H. (1985) Solid-State Electronics, 28, 1101–1103. 622 Mohammad, S.N. (1988) Journal of Applied Physics, 63, 1614–1627. 623 See Ref. [600]. 624 Gao, G.B. and Morkoç, H. (1991) Base transit time for SiGe-base heterojunction bipolar transistors. Electronics Letters, 27, 1408–1409. 625 Klausmeier-Brown, M.E., Lundstrom, M.S. and Melloch, M.R. (1989) The effects of heavy impurity doping on AlGaAs/ GaAs bipolar transistors. IEEE Transactions on Electron Devices, 36, 2146–2155. 626 Shockley, W. (1961) Problems related to p–n junctions in silicon. Solid-State Electronics, 2, 35–37. 627 Ruch, J.G. (1972) Electron dynamics in short channel field effect transistors. IEEE Transactions on Electron Devices, 19, 652–659. 628 Rockett, P.I. (1988) Monte Carlo study of influence of collector region velocity overshoot on the high-frequency performance of AlGaAs/GaAs heterojunction bipolar transistors. IEEE Transactions on Electron Devices, 35, 1573–1579. 629 Das, A. and Lundstrom, M. (1991) Does velocity overshoot reduce collector delay time in AlGaAs/GaAs HBTs? IEEE Electron Device Letters, 12, 335–337.

630 Asbeck, P.M., Miller, D.L., Asatourian, P. and Kirkpatrick, C.G. (1982) Numerical simulation of GaAs/AlGaAs heterojunction bipolar transistors. IEEE Electron Device Letters, 3, 403–406. 631 Maziar, C.M., Klarsmeier, M.E. and Lundstrom, M.S. (1986) A proposed structure for collector transit time reduction in AlGaAs/GaAs bipolar transistors. IEEE Electron Device Letters, 7, 483–485. 632 Hu, J., Tomizawa, K. and Pavlidis, D. (1989) Monte Carlo approach to transient analysis of HBTs with different collector designs. IEEE Electron Device Letters, 10, 55–57. 633 Laux, S.E. and Lee, W. (1990) Collector signal delay in the presence of velocity overshoot. IEEE Electron Device Letters, 11, 174–176. 634 Ishibashi, T. (1990) Influence of electron velocity overshoot on collector transit times of HBTs. IEEE Transactions on Electron Devices, 37, 2103–2105. 635 Henry, C.H., Logan, R.A. and Merritt, F.R. (1978) The effect of surface recombination on current in AlxGa1xAs heterojunctions. Journal of Applied Physics, 49, 3530–3542. 636 Lin, H.H. and Lee, S.C. (1985) Super-gain AlGaAs/GaAs heterojunction bipolar transistors using an emitter edgethinning design. Applied Physics Letters, 47, 839–841. 637 Zhu, E.J., Fischer, R., Henderson, T. and Morkoç, H. (1985) An emitter guard ring structure for GaAs high-gain heterojunction bipolar transistors. IEEE Electron Device Letters, 6, 91–93. 638 Lu, S.S. and Huang, C.C. (1992) Highcurrent-gain Ga0.51In0.49P/GaAs heterojunction bipolar transistor grown by gas-source molecular beam epitaxy. IEEE Electron Device Letters, 13, 214–216. 639 Hobson, W.S., Ren, F., Abernathy, C.R., Pearton, S.J., Fullowan, T.R., Lothian, J., Jordan, A.S. and Lunardi, L.M. (1990) IEEE Electron Device Letters, 11, 241–243.

References 640 Berger, P.R., Chand, N. and Dutta, N.K. (1991) An AlGaAs double-heterojunction bipolar transistor grown by molecularbeam epitaxy. Applied Physics Letters, 59, 1099–1102. 641 Nottenburg, R.N., Chen, Y.K., Panish, M.B., Humphrey, D.A. and Hamm, R. (1989) Hot-electron InGaAs/InP heterostructure bipolar transistors. IEEE Electron Device Letters, 10, 30–32. 642 Liu, W. and Harris, J.S., Jr (1992) Diode ideality factor for surface recombination current in AlGaAs/GaAs heterojunction bipolar transistors. IEEE Transactions on Electron Devices, 39, 2726–2732. 643 See Ref. [617]. 644 Hayama, N. and Honjo, K. (1990) Emitter size effect on current gain in fully selfaligned AlGaAs/GaAs HBTs with AlGaAs surface passivation layer. IEEE Electron Device Letters, 11, 388–390. 645 deLyon, T.J. and Casey, H.C., Jr (1989) Surface recombination current and emitter compositional grading in npn and pnp GaAs/AlxGa1xAs heterojunction bipolar transistors. Applied Physics Letters, 54, 641. 646 Gao, G.B., Fan, Z.F. and Morkoç, H.,Edge injection model for heterojunction bipolar transistors. Unpublished. 647 Nittono, T., Nagata, K., Yamauchi, Y., Makimura, T., Ito, H. and Nakajima, O. (1991) Advanced IC fabrication technology using reliable, small-size, and high-speed AlGaAs/GaAs HBTs. IEEE IEDM 91 Technical Digest, pp. 931–934. 648 Stanchina, W.E., Metzger, R.A., Pierce, M.W., Jensen, J.F., McCray, L.G., WongQuen, R. and Williams, F. (1993) Monolithic fabrication of npn and pnp AlInAs/GaInAs HBTs. 5th International Conference on InP and Related Materials, pp. 569–571. 649 Liu, W. and Harris, J.S. (1991) Dependence of base crowding effect on base doping and thickness for npn

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AlGaAs/GaAs HBTs. Electronics Letters, 27, 1048–1050. Fournier, V., Dangla, J. and DubonChevallier, C. (1993) Investigation of emitter current crowding effect in heterojunction bipolar transistors. Electronics Letters, 29, 1799–1800. Chow, T.P., Khemka, V., Fedison, J., Ramungul, N., Matocha, K., Tang, Y. and Gutmann, R.J. (2000) SiC and GaN bipolar power devices. Solid-State Electronics, 44, 277–301. Dmitriev, V.A., Irvine, K., Carter, C., Jr, Kuznetsov, N. and Kalinina, E. (1996) Applied Physics Letters, 68, 229–231. Pankove, J.I., Chang, S.S., Lee, H.C., Molnar, R.J., Moustakas, T.D. and Van Zeghbroeck, B. (1994) IEDM Technical Digest, pp. 389–392. McCarthy, L., Kozodoy, P., DenBaars, S.P., Rodwell, M. and Mishra, U.K. (1999) IEEE Electron Device Letters, 20, 277. Limb, J.B., Xing, H., Moran, B., McCarthy, L., DenBaars, S.P. and Mishra, U.K. (2000) High voltage operation >80 V of GaN bipolar junction transistors with low leakage. Applied Physics Letters, 76 (17), 2457–2459. Zhang, A.P., Ren, F., Anderson, T.J., Abernathy, C.R., Singh, R.K., Holloway, P.H., Pearton, S.J., Palmer, D. and McGuire, G.E. (2002) High power GaN electronic devices. Critical Reviews in Solid State and Materials Sciences, 27, 1–71 and references therein. Makimoto, T., Kumakura, K. and Kobayashi, N. (2001) High current gains obtained by InGaN/GaN double heterojunction bipolar transistors with p-InGaN base. Applied Physics Letters, 79 (3), 380–382. Bandic, Z.Z., Bridger, P.M., Piquette, E.C. and McGill, T.C. (1998) Applied Physics Letters, 73, 3276. See Ref. [93].

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4 Ultraviolet Detectors Introduction

The direct bandgap nitride semiconductors, ranging from about 0.8–6.2 eV, are ideally suited for a large variety of detectors, especially in the ultraviolet (UV) range as there are currently no serious semiconductor-based competing technologies. A large absorption coefficient (a result of the direct bandgap) and the ability to detect the UV- and solar-blind regions of the spectrum make them very attractive detector candidates. The ozone layer around the Earth absorbs nearly all the solar radiation in the band near 280 nm, so the background seen by a detector operating at this wavelength would be dark. This would eliminate cosmic radiation noise, and so the only noise that one is left to contend with the one generated by the device itself. However, many applications near this wavelength require detection of just a few photons, putting tremendously stringent requirements on the detector. Truly, solarblind detectors and detector arrays have not yet been demonstrated in the AlGaN system. However, the wavelength of operation has been blueshifted very close to that particular wavelength. Before the advent of wide-bandgap nitrides, detectors in that spectral range were based solely on photomultiplier tube (PMT) imagers with very high gain. PMT detectors are classified into Generation I (Gen I), Generation II (GenII), and Generation III (Gen III) UV sensors. The wavelength range of interest can be divided into IR, visible, and UV, each in turn being further divided into subranges. For UV, these regions are “near-ultraviolet” (400–300 nm), “mid-ultraviolet” (300–200 nm), “far-ultraviolet” (200–100 nm), and “extreme-ultraviolet” (100–10 nm). Another scheme classifies these regions into ultraviolet-A (400–320 nm), ultraviolet-B (320–280 nm), deep ultraviolet (350–190 nm), and vacuum ultraviolet (200–10 nm). Ultraviolet light was first mentioned by J. W. Ritter in 1801. He stated that some chemical reactions were influenced by an invisible light with a wavelength that was smaller than violet. Ultraviolet light has a wide spectral range spanning from visible light (400 nm, 3.1 eV) to the low-energy X-ray spectral region (10 nm, 124 eV). Efforts to uncover the properties of UV radiation began in the later half of the nineteenth century with the realization that the radiation beyond the blue end of the visible spectrum was very important. Spectrographic records obtained at high altitudes Handbook of Nitride Semiconductors and Devices. Vol. 3. Hadis Morkoç Copyright
2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40839-9

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underscored the premise that the stellar and solar radiation indeed existed in the UV range. However, the enthusiasm was soon ebbed with the observation that the terrestrial atmosphere blocks radiation shorter than about 300 nm from reaching the surface of the Earth. UV radiation of 200 nm
l
300 nm is absorbed by the ozone layer enveloping the Earth and radiation of 110 nm
l
250 nm is absorbed by the molecular oxygen. Ozone and molecular oxygen together form the terrestrial atmosphere. UV radiation with l < 110 nm is absorbed by almost all types of atomic and molecular gases in the atmosphere rendering it nonexistent on the surface of the Earth. For these reasons, the UV region with l < 200 nm is generally called the vacuum region. A detector operating in the solar-blind region sees a dark background, which allows the detection of minute emissions. Solar-blind UV detectors are imperative, particularly for the military, because they enable air, sea, and ground equipment to detect and warn against the ground-to-air, air-to-air, air-to-ground, and ground-to-ground missile threats. The system, at the heart of which is a UV detector, must be able to detect and track extremely weak signals from rapidly moving threats, which necessitates operation in the solar-blind region of the spectrum, 260–290 nm. There are also commercial applications such as in boilers and jet engines, though they may not require operation in the solar-blind region of the spectrum. The continuing proliferation and increasing lethality of surface-to-air and air-to-air missiles pose serious threats to aircraft. Countering these threats successfully requires an early detection of the missile launch. One promising method for detecting incoming missiles is sensing the UV emissions from the rocket plume. Particularly at altitudes lower than 20 000 ft, the solar-blind region is ideal for this purpose because the solar background radiation is almost entirely absorbed by atmospheric ozone. Thus, the 3000 K blackbody emission from a missile plume stands out prominently against a dark background, minimizing the number of false alarms. Such a missile-threat warning system would therefore require two-dimensional arrays of highly sensitive photodetectors to image the solar-blind UV spectral range. This problem has been solved by using PMT-type detectors and optical filters to block out all lights except that between 260 and 290 nm. However, this approach suffers from a limited sensitivity due to low transmission through the filters and low quantum efficiency of the photocathodes, detection of wavelengths outside the solarblind region caused by filter roll-off at wavelengths longer than 290 nm, and large size and weight of the photomultiplier assembly and associated high-voltage power supply. By comparison, solid-state detectors offer the advantages of being compact and rugged, having near-unity quantum efficiency and better rejection of long wavelength radiation due to their sharp UV band edge. Currently, however, no solid-state UV photodetector has the sensitivity required for this application. UV detectors have more applications including biological and chemical sensors (ozone detection, determination of air pollution levels, and biological agent detection), flame sensors (fire alarm systems and combustion engine control), spatial optical communications (intrasatellite and intersatellite secured communications),

Introduction

Normalized responsivity (a.u.)

emitter calibration, and UV imaging such as solar UV measurements and astronomical studies [1]. In terms of biological applications, there is a good deal of areas involving detection of wavelengths or ranges of wavelengths specific to certain biological processes where GaN-based UV detectors could be applied [2–5]. Among these are pigmentation (maximum wavelength at l ¼ 360–440 nm), solar erythema (maximum sensitivity at l < 297 nm), D2 and D3 vitamin synthesis (l ¼ 249–315 nm with maximum yield at 290 nm), damage to plants (l < 317 nm), bactericidal action (l ¼ 210–310 nm with maximum yield at 254 nm), carcinogenic effects of UV B and UV C bands (peaks at 310 nm), and ADN damage (occurs at l < 320 nm with damage increasing as wavelength decreases) [6]. Simple, accurate, reliable, and low-cost instruments are therefore necessary to evaluate the biological effects of the UV radiation. A wide spectral range of UV detectors have also been developed to monitor UV A and UV B erythema action [4]. Muñoz et al. [7] have demonstrated that using a special mole fraction of Al and suitable growth conditions for AlGaN Schottky photodiodes makes it possible to precisely fit the UV A and UV B erythema action curve (Figure 4.1). The spectral response of these devices thus provides direct information on the biological effects of the UV light. In this application, the absorption tails of state-ofthe-art AlGaN-based detectors make it possible to correctly weigh the UV A and UV B spectra so that they accurately fit the erythema reference functions. In general, UVdetectors may be categorized into photon detectors (photodetectors) and thermal detectors. The photodetectors are highly sensitive to radiation and are used very widely to measure the rate of photon arrival. Photon detection in semiconductors takes place through absorption and ensuing creation of free electrons and holes, which lead to a current flow. The change in electrical energy distribution resulting from this absorption of photons gives rise to an observed electrical signal. If they are designed appropriately in a suitable material, they would respond only to UV radiation of certain selected wavelengths. Unlike photodetectors, the thermal detectors exhibit increased temperature caused by photon absorption.

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Al0.32Ga0.68N 10-1

10-2

10-3

10-4 250

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Wavelength (nm) Figure 4.1 Normalized spectral response of an AlGaN Schottky photodiode, compared to the erythema standard action curve (CIE) [6].

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Consequently, these detectors measure the photon flux and yield (for photons of certain selected wavelengths) in the form of a change of certain temperaturedependent parameter of the material. Essentially, the radiation can be absorbed even on a black surface coating, demonstrating that the thermal effects in a thermal detector are generally wavelength independent. These detectors often find their uses as absolute radiometric standards at UV wavelengths. Among the other types of detectors, the pyroelectric detectors rely on the change of the internal electrical polarization and the bolometers rely on the change of electrical resistance. The photoelectric detectors are known to yield a greater stability of response and greater linearity characteristics. This has been achieved by a considerable progress in image recording and processing as evidenced by the recently developed photovoltaic and photoemissive array detectors. Interestingly, these two detectors can offer, for the first time, the sensitivity and radiometric stability of photomultipliers, paving the way for high-resolution image capability. Figure 4.2 illustrates the specific detectivity (a figure of merit inversely proportional to the noise level with higher numbers indicating better performance, further discussion is provided in Section 4.1.5) of photodetectors covering various parts of the spectrum, namely, the ultraviolet (250–400 nm), the visible (400–750 nm), and the near-infrared (750 nm and beyond) [8]. For reference, a line denoting the background-limited detectivity at 293 K, expected from an ideal photodetector, is also shown. Results for solar-blind AlGaN detectors are also presented with depictions of a star [9, 10], open circle [11], open square [12], full circle [13], full square [14], and asterisks. The detector shown with star has 53% quantum efficiency and 3.2  1014 cm Hz1/2 W1 specific detectivity. The PMT-based detectors, having traditionally shown the highest specific detectivities, are used as benchmark. Compound 1015 1014 1013 10

D (cm Hz 1/2 W-1) *

10 10

10

PMT Si diode InGaAs diode

13 12

10

10

15

14

11

Ideal photovoltaic detector limited by 293 K background

CCD PbS at 243 K PbSe at 243 K

10

10 10

8

10 10

HgCdZnTe at 243 K

S1 PMT

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Pyroelectric PDA

HgCdZnTe at 293 K

Thermopile

7

6

2

4

6

8

10

12

Wavelength (µm) Figure 4.2 Specific detectivity D plotted as a function of wavelength for a number of photodetectors, from the ultraviolet to the nearinfrared. Detectivity of an ideal detector, limited by room-temperature background radiation, is

plotted for reference. The boxed area is the region of interest for nitride semiconductorbased photodetectors, which is also shown with nitride detector detectivities indicated with circles Courtesy of H. Temkin and J. C. Campbell.

4.1 Principles of Photodetectors

semiconductors, mainly operating in the near-IR region for telecommunications applications (see Ref. [15]) and many kinds of detectors for detection and imaging in the medium wavelength IR region of the spectrum (see Ref. [16]) have been described well in reference books and will not be treated here. After many decades of research and development, nearly ideal semiconductorbased detectors are available in the visible and near-infrared spectral regions. The same, however, cannot be said of the UV region of the spectrum. UV-enhanced Si-based ultraviolet detectors suffer from low responsivity, in part due to visible rejection filters employed. Until the advent of nitride semiconductors, only the bulky photomultiplier detectors offered high ultraviolet detectivities. In contrast, the visible and infrared regions offer a number of excellent detector choices. In the near-infrared, direct bandgap InGaAs grown on InP forms excellent detectors, and further into the infrared, other direct bandgap semiconductors from the family of IV–VI lead salts such as PbS offer excellent room-temperature detectivities. Principles of operation of photodetectors followed by noise in semiconductor detectors and detectivity will be given. This will be followed by a primer on solar radiation and specifics dealing with GaN-based detectors.

4.1 Principles of Photodetectors

To develop insight into the operation of the device, we must follow the trail of photons and created electrons. In a photocathode device, the photons are incident on the active region material that forms the detector. When these photons collide with the solid surface of the photodetector material, electrons are emitted into a vacuum, provided that the barrier to electron emission has been eliminated, preferably by a surface treatment, which leads to the generation of a current. A voltage is applied between the photocathode surface and the positively biased anode, collecting the electrons and measuring the ensuing current. The magnitude of the photocurrent will depend on the quality of the photocathode surface. Nevertheless, the photoelectric current flowing through an appropriate circuit is, in general, proportional to the intensity of the incident photon radiation. Because the surface material plays a key role in the functioning of a photoemissive UV detector, the wavelength range of the detector is dictated primarily by the absorption band of the cathode material and, to a lesser extent, the work function of its surface. In all semiconductor photodetectors, one or more semiconductor layers are typically grown on a suitable substrate to form a semiconductor photodetector. The simplest of semiconductor photodetectors is of the photoconductive type as shown in Figure 4.3. The other types require a junction, either a p–n-junction variety or a Schottky barrier variety as shown in Figure 4.4a–e. In both cases, the light must be able to penetrate the absorption region. This is accomplished by either providing a window in the metal (ohmic contact in the case of the p–n-junction or Schottky barrier) or making the metal sufficiently thin to be semitransparent. The latter is preferred due to series resistance considerations. In the back-illuminated case

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h ν P opt

-

Ohmic contact D L

R

Si

lo

gn

ad

al

W Ohmic contact

Current Figure 4.3 A schematic representation of a photoconductive detector with the critical dimensions and external circuit indicated.

through the polished sapphire substrate, the metal on the top surface can be opaque. Generally, the absorption layer is made so that it is depleted at the operating voltage (generally close to zero bias although reverse biases can increase the speed of the device) to avoid diffusion effects by allowing the carriers to be swept by the electric field. Heterojunctions can be employed in such a way so as to reduce the dark current of the device as well as to get the photons of the desired wavelengths to the absorptive layer with minimal photon loss. The Schottky barrier or the p–n-junction then collects the minority carriers. In the nitride world, the silver lining among all the problems induced by the lattice-mismatched sapphire substrates is that sapphire is transparent to the wavelengths of interest. This lends itself nicely to the backillumination geometry, which can be used very conveniently with Si readout circuitry for imaging as in focal plane arrays (FPAs), is shown in Figure 4.5a and b [17]. Unlike photoemissive detectors (photocathode), photons incident on semiconductor photodetectors are absorbed in the bulk of the semiconductor material. When the energy of the absorbed photon is large enough to raise the electrons to the conduction band from the valence band, they create holes in the valence band. The electric field present, either due to the built-in potential or the applied voltage, separates electrons and holes causing the terminal current to flow, which is proportional to the photon flux. In the case of photovoltaic detectors, the electric field generated within the depletion region of the device is caused by the differences in Fermi levels on either side of the junction. Among the varieties of photodetectors, photoconductive types are simple to fabricate, provide gain and large Johnson noise and low-frequency noise, but they suffer from persistent photoconductivity (PPC) making them inherently slow. On the other hand, the p-i-n varieties operated near zero bias offer high quantum efficiency,

4.1 Principles of Photodetectors

UV radiation

Semitransparent Schottky barrier Ohmic contact

Ohmic contact

n-GaN n+-GaN Substrate

Ti/Al/Ti/Au Bondpad to Schottky contact

10 nm Pd

Silicon dioxide

Bondpad to ohmic contact

n-GaN Substrate

(a)

UV radiation

p-contact

p-contact

p-GaN n-contact

n-GaN

n-contact

n+-GaN Substrate

AlN initiation layer

(b) Figure 4.4 Schematic cross-sectional views of photodiode structures implemented in the GaNbased semiconductor system: (a) GaN homojunction, (b) AlGaN/GaN single heterojunction, and (c) double heterojunction of AlGaN applicable to bottom illuminated solarblind applications. (d) Schematic representations of an AlGaN/AlGaN

homojunction detector, the type used for backilluminated solar-blind detectors. (e) Backilluminated AlGaN/GaN photodiode used in the context of an array with In bumps shown. The boxed schematic diagram is a display of the cuton and cutoff wavelengths due to the window and absorber layers, respectively. Courtesy of M. Reine of BAE systems [17].

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UV radiation

p-AlGaN window Ohmic contact

Ohmic contact

n-GaN n+-GaN Substrate

(c) p-contact

p-AlGaN n --AlGaN

n-contact

n-contact

n+-AlGaN AlN initiation layer

Substrate

UV radiation

(d)

Relative response (per photon)

1.0 0.8 0.6 0.4 0.2 0.0

Cut-on due to Al0.6Ga0.4N

Cutoff due to Al0.45Ga0.55N

Wavelength Common contact

Unit cell

Unit cell

p-GaN p-Al0.45Ga 0.55N n --Al0.45Ga 0.55N n-Al0.6Ga 0.4N

Contact layer p-layer Absorber Window and common contact

Sapphire substrate

UV radiation (e)

Figure 4.4 (Continued )

4.1 Principles of Photodetectors

ROIC

p-contact p-GaN or p-AlGaN n-contact

n-absorbing AlGaN

n-transparent AlGaN Sapphire substrate

(a) UV illumination

Detector mesas

Sapphire substrate AlGaN/GaN heterojuction

Bump bond Electronics boards

Unit cell

Si readout chip (CMOS multiplexer with N×N units)

(b)

Figure 4.5 (a) Schematic diagram of a back-illuminated nitridebased detector butted to an Si readout circuit by In bumping. (b) A more detailed view of (a). Courtesy of M. Reine of BAE systems.

low-frequency noise, and high detectivity, but have a gain of 1. Schottky barrier types operated near zero bias have low-frequency noise, a gain of unity, and offer simple device structure, but are surface sensitive. The metal–semiconductor–metal (MSM) varieties share much of the features of the Schottky barrier types, but with lowfrequency noise, and depending on the bias might offer gain. Avalanche photodiodes (APDs) offer gain and low dark current, but with low-frequency and also highfrequency noise as the avalanching process adds on additional statistical variations. Phototransistors offer high gain, large visible rejection, and low-frequency noise, and the persistent photoconductivity can also be eliminated with pulse bias. As discussed in Chapter 3, the low-frequency noise exists in all types of semiconductor devices. Here, we mention a device if it is relatively large compared to the varieties supporting low-frequency noise. The fundamentals of detectors as discussed below follow to a great extent from Ref. [18] and to some extent from Refs [19, 20]. The performance of a photodetector

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can be described by an electron Jn and hole Jp current density equations, two continuity equations for electrons and holes, and one Poisson’s equation for electrons and holes as follows:

 dn dV J n ¼ qDn qnmn ; ð4:1Þ dx dx
J p ¼ qDp


 dp dV qnmp ; dx dx

ð4:2Þ


 1 dJ n þ ðGRÞ ¼ 0; q dx

ð4:3Þ


 1 dJ p ðGRÞ ¼ 0; q dx

ð4:4Þ

d2 V q ¼  fN d N a þ png; dx 2 e

ð4:5Þ

where V is the electrostatic potential, Nd is the donor concentration, Na is the acceptor concentration, Dn and Dp are the electron and hole diffusion coefficients, respectively, n and p are the electron and hole concentrations, respectively, and e is the dielectric constant of the semiconductor material. The generation–recombination (G–R) term in Equations 4.3 and 4.4 accounts for all possible types of generation and recombination components, which are very important, particularly in large-bandgap semiconductors where the minority carrier concentration in the quasineutral regions is negligible. Examples of sources for the G–R current include Shockley–Read–Hall (SRH) recombination, radiative recombination, and Auger recombination. Lattice defects and impurity energy levels within the energy bandgap are the primary causes of Shockley–Read–Hall recombination. When Equations 4.1–4.5 are solved by a selfconsistent iterative procedure, they yield the electrostatic potential V. A reformulation of this V in terms of integral equations incorporating the necessary boundary conditions and eliminating the current densities Jn and Jp gives the carrier densities from the potential distribution. 4.1.1 Current and Voltage Response to Incident Radiation

The signal current, photocurrent Ip generated in a photodetector under steady-state illumination with monochromatic light of wavelength l can be written as Iph ¼ qhext gAFs

or

I ph ¼ qhext g

Pl ; hn

ð4:6Þ

where Iph is the short-circuit photocurrent at zero frequency (DC condition), Fs is the photon flux density on the detector with an area of A ¼ wL (w is the width and L is the length of the detector), Pl is the incident power, and hv is the photon energy. The

4.1 Principles of Photodetectors

parameter hext is the external quantum efficiency and is affected by the internal quantum efficiency, absorption depth, and surface reflectivity of the device. The photoconductive gain “g” is another parameter of a detector (the photoconductive variety) and is defined as the number of carriers passing through the contact per generated electron–hole pair. This gain describes how efficiently the generated electron–hole pairs are used to create electric current. 4.1.1.1 Photoconductive Detectors As the photoconductivity is a two-carrier phenomenon, it is composed of the current generated by electrons and holes. Thus, the photocurrent in photoconductive detector can be expressed as

Iph ¼

wtqðmn Dn þ mp DpÞV a ; L

ð4:7Þ

where t is the detector thickness, which is small compared to the minority carrier diffusion length, Va is the applied bias, mn is the electron mobility, mp is the hole mobility, Dn ¼ n  n0 and Dp ¼ p  p0, where n and n0 are the instantaneous and thermal equilibrium values of the electron concentrations, respectively, and p and p0 are the instantaneous and thermal equilibrium hole concentrations, respectively. Consequently, Dn and Dp are the excess electron and hole concentrations, respectively. When a photoconductor (PC) operates under constant current conditions, a voltage across the load resistance is essentially the open-circuit voltage: V s ¼ I ph Rd ¼ Iph

L ; qwtnmn

ð4:8Þ

where Rd is the detector resistance. Nearly all photoconductors are designed so that the conductivity is dominated by electrons and holes play a secondary role. Consequently, if the absorption is uniform and all the photons are absorbed, the rate equation for the excess electron concentration is given by d h Fs Dn ðDnÞ ¼ ext  ; t dt t

ð4:9Þ

where t is the excess carrier lifetime and under steady state is given by t¼

tDn ; hext Fs

ð4:10Þ

which can be obtained from the generation rate expression of Dn hext Fs : ¼ t t

ð4:11Þ

Equations 4.6 and 4.7 yield the photoelectric (or photoconductive) current gain, g, as g¼

mn tDnV a : hext L2 Fs

ð4:12Þ

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When expressed in terms of the carrier lifetime t, this photoelectric current gain is obtained from Equations 4.10–4.12 as g¼

tme V a t t ¼ 2 ; ¼ 2 t L L =ðme V a Þ tm

ð4:13Þ

where ttm is the transit time of electrons between ohmic contacts. Equation 4.13 suggests that the photoconductive gain of a photoconductor is determined by the ratio of the carrier lifetime t to the carrier transit time ttm of the sample electrons. When t > ttm, an internal photoconductive gain results. Because, in general, mn > mp, the photogenerated electrons traverse faster through the semiconductor than holes. To preserve charge neutrality and current continuity, the external electrode must supply electrons from the opposite side. These new electrons may move so fast across the detector that the original holes cannot recombine with them. If this happens, a gain is produced that is proportional to the number of times an electron can transit the detector electrode within its lifetime. This internal photoconductive gain is lower than unity if the recombination lifetime is short because the electron cannot always reach the electrode before recombining with a hole. The transit time for a typical 10 mm detector is on the order of 1010 s for an average velocity of about 107 cm s1. Depending on the detector size, material, and doping, the photoconductive gain can vary from less than 1–105. The upper limit is restricted, however, by parameters such as space charge effects, dielectric breakdown, and ionization effects. 4.1.1.2 p–n-Junction Photovoltaic Detectors Photodetectors employing p–n-junctions with built-in potential barriers at the junction interface are essentially photovoltaic in nature if they are used at or near zero bias. The operating principle is that optically generated excess carriers are injected into the vicinity of the barriers. The built-in electric field then drifts the carriers in the appropriate direction, depending on the external circuit. The photovoltaic effect can be created in a number of ways. These include p–n-homojunctions, p–n-heterojunctions, Schottky barriers, and metal–insulator–semiconductor (MIS) photocapacitors. Each of these devices has its own advantages and disadvantages, and the intended application plays a large role in determining which one to choose. Abrupt homojunction is the most common type of photovoltaic detector. When photons with energy greater than the energy bandgap are incident on the front or back surface of this semiconductor, they create electron–hole pairs in the vicinity of the junction space charge region (SCR). Minority carriers generated within a diffusion length of the junction enter the space charge region where they are pulled by the strong electric field, as shown in Figure 4.6, generating a terminal current as shown in Figure 4.7. The equivalent circuit of a p–n-junction photodiode is shown in Figure 4.8. The total current density of a p–n-junction photodetector is given by

JðV a ; Fs Þ ¼ J d ðV a ÞJ ph ðFs Þ;

ð4:14Þ

where Jd is the dark current density and Jph is the photocurrent density. The dark current density Jd depends mainly on the applied voltage Va, and the photocurrent

4.1 Principles of Photodetectors

p-type

n-type W E-field

+ p=Na

++ n=Nd

n=Nd

pn0 np0

x=-xp x=0

x=-t

x=d

x=xn

x

(a) p-type

n-type

+ p=Na

W

++ n=Nd

n=Nd

E-field

pn0

np0

x=-t

x=-xp

x=0

x=xn

x=d

x

(b) Figure 4.6 Schematic representation of a p–n-junction photodetector indicating the depletion region and the electric field therein for a junction with neutral regions much longer than the respective diffusion lengths (a) and for a junction with neutral regions comparable or shorter than the respective diffusion lengths (b).

density Jph depends mainly on the photon flux density Fs. The current gain in a photovoltaic detector (with the exception of avalanche photodiodes) is almost equal to 1, and following Equation 4.6 gives the magnitude of the photocurrent as Iph ¼ hext qAFs :

ð4:15Þ

Under the open-circuit condition of a p–n-junction photodiode, the electrons and holes are accumulated on the sides of the junction producing an open-circuit photovoltage (see Figure 4.7). This photovoltage causes a current flow through the circuit when a load is connected across the diode. The highest value of this current is obtained if an electrical short is placed across the diode terminals. This is called the short-circuit current. The open-circuit voltage Vp is obtained by multiplying the short-circuit current by the incremental (differential) diode resistance R ¼ (qI/qV )1 at V ¼ Va:

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Photocurrent

Voltage

Dark

Vs

With radiation

Figure 4.7 Terminal current through a p–n-junction photodiode in dark and under illumination.

V p ¼ hext qAFs R;

ð4:16Þ

where I ¼ I(Va) is the voltage-dependent current through the photodetector. For many applications, the photodetector is operated at zero-bias voltage yielding
1 qI : ð4:17Þ R0 ¼ qV V a ¼0 A figure of merit for a photodiode is the R0A product and is given by
1 qJ ; R0 A ¼ qV V a ¼0 where J ¼ I/A is the current density.

Rs

iph

is2

Cj

Output voltage Rj

Figure 4.8 Equivalent circuit of a p–n-junction diode where iph and is represent the photocurrent and noise current components, Rj and Rs the junction and series resistances, and Cj the junction capacitance.

ð4:18Þ

4.1 Principles of Photodetectors

The photodiode can be operated at any point along the current–voltage (I–V ) characteristic. The diodes are usually reverse biased for very high-frequency applications, which assists in reducing the transit time of the device (through increased carrier velocity) and allows the transit region to be increased, to reduce the diode capacitance and thus the RC time constant. The speed is determined by a combination of the transit time and RC time constant, and a global optimization approach is often employed. For p–n-junction photodiodes made of wide-bandgap nitrides, the minority carrier concentrations are very small and thus the diffusion of carriers is not the dominant mechanism for dark current. Several other mechanisms, which are more important, must therefore be involved. In general, though, the dark current is contributed by the bulk of the semiconductor, the space charge region, and the surface region. The thermally generated current in the bulk and the depletion region arises from (a) diffusion of carriers in the bulk p- and n-regions (negligible in wide-bandgap semiconductors), (b) generation–recombination in the depletion region, (c) bandto-band tunneling, (d) trap-to-trap and trap-to-band tunneling, (e) anomalous avalanching of carriers, and (f ) ohmic leakage across the space charge region. However, surface leakage arises from (a) surface generation current from surface states, (b) generation current in the field-induced surface depletion region, (c) tunneling induced near the surface, (d) ohmic and nonohmic shunt leakage, and (e) avalanche multiplication in a field-induced surface region. 4.1.1.2.1 Diffusion Current for a p–n-Junction Detector Diffusion plays a fundamental role in current conduction in a p–n-junction photodetector. In Figure 4.6, which shows a schematic diagram of a p–n abrupt homojunction, the spatial charge of width w ¼ xp þ xn surrounds the junction boundary, x ¼ 0. Moreover, there are two quasineutral–neutral regions of several diffusion lengths from the edges of the depletion region at xp and xn, which are assumed to be uniformly doped. The dark current is caused by electrons injected from the n-side over the potential barrier into the p-side and a similar current due to holes injected from the p-side into the n-side region. The diffusion-limited minority carrier current is important in small-bandgap semiconductors, but not as important in large-bandgap semiconductors because the minority carrier concentration is negligibly small under ideal conditions. The current–voltage characteristics of an ideal diffusion-limited diode can be calculated beginning with the diffusion equation (consider for now that for minority holes on the n-side and assume the origin to be at the edge of depletion region) as follows:

Dp

d2 Dpn Dpn  ¼ 0 with dx2 tp

Dpn ðxÞ ¼ pn ðxÞpn0 ;

ð4:19Þ

where tp, Dpn, and Dp are the hole minority carrier lifetime, the excess hole concentration in the quasineutral region on the n-side and hole diffusion length, respectively. The general solution of Equation 4.19 is

 x x þ B exp ; ð4:20Þ Dpn ðxÞ ¼ A exp Lp Lp

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pffiffiffiffiffiffiffiffiffiffi where Lp ¼ Dp tp , Dp, and tp are the diffusion length, diffusion constants, and minority carrier lifetime, respectively. Considering the long junction case depicted in Figure 4.6a, the coefficient B must be zero to allow the excess hole concentration Dp to drop to zero for large values of x. Using the boundary condition at x ¼ xn0, which states that 
  qV Dpn ðx ¼ x n Þ ¼ pn0 exp 1 ; ð4:21Þ kT we obtain for the A coefficient from Equation 4.20 as

  xn qV exp 1 : A ¼ pn0 exp Lp kT Turning our attention to the hole diffusion current on the n-side,  
  qDp pn0 dp  qV J p ¼ qDp n  ¼ exp 1 : dx x¼xn Lp kT

ð4:22Þ

ð4:23Þ

Similarly, the minority electron current on the p-side (reversing the direction of the coordinate system and taking the edge of the depletion region as the origin for the sake of simplifying the equations) can be expressed as  
  dnp  qDn np0 qV J n ¼ qDn ¼ exp 1 : ð4:24Þ dx x¼xp Ln kT Sum of the hole and electron currents leads to the total current: 
    qV ID ¼ A J p þ J n ¼ AJ S exp 1 ; kT

ð4:25Þ

where JS is the saturation current density, which, for a thick quasineutral regions (dimensions on n- sand p-sides are longer than the pertinent diffusion lengths) is given by   Dn np0 Dp pn0 þ : ð4:26Þ JS ¼ q Ln Lp Let us now turn our attention to the case of short p–n-junction depicted in Figure 4.6b. Now we need to solve the general solution of Equation 4.19 given by Equation 4.20 with the boundary condition given by Equation 4.21 and the additional one, which states that  h i dp  Dp n  ð4:27Þ ¼ s1 pn ðx ¼ dÞpn0 ; dx x¼d where s1 stands for the surface or interface recombination velocity at the edge of the quasineutral region away from the junction (at x ¼ d). In the case of very high interface recombination velocity or assumption that the excess hole concentration must tend to be zero at the edge of quasineutral region, the term in brackets on the right side of Equation 4.27 must also tend to be zero. To facilitate the determination of

4.1 Principles of Photodetectors

coefficients in the solution of differential equation given by Equation 4.20, we can rewrite the general solution in the following form (assume the origin to be at the edge of depletion region):

 x x Dpn ¼ A cosh þ B sinh : ð4:28Þ Lp Lp Restating the boundary conditions of Equations 4.21 and 4.27:

 
  x x qV þ B sinh ¼ pn0 exp 1 ; A cosh Lp Lp kT 



 d d d d g p A cosh þ B sinh ¼ A sinh þ B cosh ; Lp Lp Lp Lp

ð4:29aÞ

ð4:29bÞ

where g p ¼ s1Lp/Dp. Solving for A and B coefficients using Equations 4.29a and 4.29b, we obtain: 
  coshðd=Lp Þg p sinhðd=Lp Þ qV exp A ¼ pn0 1 ; ð4:30aÞ g p sinh½ðdx n Þ=Lp  þ cosh½ðdx n =Lp Þ kT B ¼ pn0


  g p coshðd=Lp Þcoshðd=Lp Þ qV exp 1 : g p sinh½ðdx n Þ=Lp  þ cosh½ðdx n Þ=Lp  kT

ð4:30bÞ

By substituting Equations 4.30a and 4.30b, the hole current with the aid of Equations 4.28 and 4.23 can now be expressed as  

 qDp dp  xn xn J p ¼ qDp n  ¼ A sinh þ B cosh dx x¼xn Lp Lp Lp 
  qDp pn0 U p qV exp ¼ 1 ; ð4:31Þ Lp kT where Up 

U p1 g p cosh½ðdx n Þ=Lp  þ sinh½ðdx n Þ=Lp  : ¼ U p2 g p sinh½ðdx n Þ=Lp  þ cosh½ðdx n Þ=Lp 

ð4:32Þ

Similarly again reversing the direction of the coordinate system and taking the edge of the depletion region as the origin to simplify the equations, we can express the electron current on the p-side as Jn ¼


  qDn np0 U n qV exp 1 ; Ln kT

ð4:33Þ

where Un 

U n1 g n cosh½ðtxp Þ=Ln  þ sinh½ðtxp Þ=Ln  ¼ ; U n2 g n sin h½ðtx p Þ=Ln  þ cosh½ðtx p Þ=Ln 

ð4:34Þ

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where g n ¼ s2Ln/Dn with s2 being the recombination velocity at the edge of the quasineutral region away from the junction on the p-side (at x ¼ t). The total junction current due to diffusion for the short neutral region case is the sum of Equations 4.31 and 4.33, which doing so leads to [21]: JS ¼

qDp pn0 U p qDn np0 U n þ ; Lp Ln

ð4:35Þ

where g p  s1Lp/Dp, g n  s2Ln/Dn, xn is the position with respect to the junction in n-type semiconductor, xp is the position with respect to the junction in the p-type semiconductor, Ln is the electron diffusion length, and Lp is the hole diffusion length. Among other parameters, pn0 is the minority hole concentration on the n-side and np0 is the minority electron concentration on the p-side of the photodetector under equilibrium conditions; s1 is the surface recombination velocity at the front (illuminated) surface and s2 is the surface recombination velocity at the back surface of the top illuminated photodetector. Thus, it is apparent from Equations 4.26 and 4.35 that the saturation current density JS depends on minority carrier diffusion lengths, minority carrier diffusion coefficients, minority carrier concentrations, surface recombination velocities, and junction dimensional parameters (e.g., xn, t, W, and d). For nondegenerate semiconductors, the intrinsic carrier concentration is given by ni ¼ (np0pn0)1/2, diffusivity by D ¼ (kT/q)m, and diffusion length by L ¼ (Dt)1/2. They can be used to reduce Equation 4.26 to " JS ¼

ðkTqÞ1=2 n2i



 # 1 mn 1=2 1 mp 1=2 þ pp0 tn nn0 tp

ð4:36aÞ



 # U n mn 1=2 U p mp 1=2 : þ pp0 tn nn0 tp

ð4:36bÞ

and Equation 4.35 to " JS ¼

ðkTqÞ1=2 n2i

Once again, here, pp0 and nn0 are the majority hole and electron concentrations, respectively, and tn and tp are the electron and hole minority carrier lifetimes, respectively. As clearly seen, the temperature dependence of the diffusion current is dominated by the temperature dependence of n2i . The resistance at zero bias of an ideal diode, whose current conduction is solely diffusion-based, can be obtained by the differentiation of I–V characteristics, with In being the diode forward current: R0 ¼

kB T : qIn

ð4:37Þ

The R0A product, a figure of merit determined by the diffusion current, is obtained as
 dJ D 1 kB T ¼ : ð4:38Þ ðR0 AÞD ¼ dV V a ¼0 q J S

4.1 Principles of Photodetectors

4.1.1.2.2 Space Charge Current In a p–n-Junction Detector Generally, the space charge-limited current is composed of the generation–recombination current and the tunneling current within the space charge region, surface current, and impact ionization current. Each of the components can be characterized by its own I–V and temperature–voltage (T–V ) dependence. Because of this, an accurate solution to the problem is attained by numerically fitting the sum of all the current components to experimental data over a range of both applied voltages and temperatures from which one can determine all the pertinent parameters. Below, we will examine these components of current. 4.1.1.2.3 Generation–Recombination Current Even when the width of the space charge region is much smaller than the minority carrier diffusion length (which is usually the case), the generation–recombination current in this region can be more important than the diffusion current, especially at low temperatures and for largebandgap semiconductors such as GaN and AlGaN. This is because the carriers are generated in the space charge region by traps, which causes the generation rate to be much higher than that in the bulk region of the material. Under reverse bias, the generation–recombination current may be given by

Igr ¼ qGscr V scr ;

ð4:39Þ

where Gscr is the generation rate and Vscr is the volume of the space charge region. If the generation rate is dominated by the generation rate from traps in the space charge region, then it may be given more correctly by the SRH formula as Gscr ¼

n2i ; n1 te0 þ p1 th0

ð4:40Þ

where n1 and p1 are the electron and hole concentrations, respectively (corresponding to the Fermi energy being equal to the trap energy), and te0 and th0 are the carrier lifetimes in the strongly n-type and p-type materials, respectively. If n1  p1 or vice versa and the trap level coincides with the intrinsic Fermi level, then Gscr ¼

ni ; t0

ð4:41Þ

which leads the generation–recombination current density to be J gr ¼

qwn i : t0

ð4:42Þ

A comparison of Equations 4.41 and 4.104 indicates that the generation rate Gb in the bulk is a function of n2i , while the generation rate Gscr for mid-gap states in the space charge region is a function of ni. Furthermore, unlike the diffusion-limited reverse current, which is independent of the applied bias above a few kBT/q, the generation–recombination current is weakly dependent on the applied bias. For an abrupt junction where the depletion depth is proportional to V1/2, it varies as the square root of the applied bias and for a graded junction (W  V1/3), it varies as the cube root of the applied bias.

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4.1.1.2.4 Tunneling Current Tunneling current is often the dominant component of the dark current. It can be both direct and indirect. When direct, it arises from the tunneling of electrons across the junction from the valence band to the conduction band without the assistance of the intermediate traps. When indirect, it arises from the tunneling of electrons across the junction from the valence band to the conduction band with the assistance of impurities or defects within the space charge region. More specifically, the carriers in this process first undergo thermal transitions from one band to the trap states, and then tunnel from the trap state to the other band. Because of this, indirect tunneling requires lower electric fields than the direct tunneling where carriers need to traverse a longer tunneling distance. Assuming a triangular potential under the influence of an applied bias Va, the current JT resulting from the direct tunneling of electrons of effective mass m is " #
 4ð2m E 3g Þ1=2 q2 EV a 2m 1=2 JT ¼ ; ð4:43Þ exp  Eg 3qE
h h2

where E is the electric field and h is the Planck’s constant. For an abrupt p–n-junction, it is given by  
 2q E g pn 1=2 E¼ Va : ð4:44Þ es q pþn An inspection of Equations 4.43 and 4.44 suggests that the tunneling current is strongly sensitive to the energy bandgap Eg, applied bias Va, and the effective doping density Nd ¼ np/(n þ p). It is, however, relatively insensitive to the variation of temperature and the shape of the junction barrier. For a parabolic barrier, the tunneling current JT is 2 J T / exp4

pm E 3g 8

!1=2

3 1 5 : q
hE

ð4:45Þ

4.1.1.2.5 Surface Leakage Current Though the surfaces of GaN-based wide-bandgap semiconductors are more stable than other compound semiconductors, they too exhibit traps and surface states due to the formation of native oxides [22], predominantly Ga2O3. As revealed by ultraviolet and X-ray photoelectron spectroscopy and Auger electron spectroscopy, the surface of, for example, GaN grown on sapphire is contaminated with carbon and oxygen. These results must be treated with great precaution to ensure that what is measured is not caused by the environment in which experiments are carried out. Assuming care was taken, native oxides such as Ga2O3 together with some components of a mixed oxynitride of Ga also form an overlayer. It is suspected that due to the growth of GaN on sapphire substrate, the lattice of which is highly mismatched to the epilayer, dangling bonds are present on the GaN surface and are actually responsible for the formation of the Ga2O3 native oxide layer. Moreover, these defects are present in the bulk of the films, which is inevitably a good source of increased dark current. While the exact mechanisms of the defects that produce dark

4.1 Principles of Photodetectors

current are still not clear, all of the aforementioned phenomena including Poole–Frenkel current must be considered. In addition, a part of the dark current results from recombination at traps and surface states existing on the surface of a GaN-based p–n-junction, particularly at low temperatures. The SRH recombination mechanism dominates this process creating minority carriers. These minority carriers along with the additional charges existing on the surface affect the position of the space charge region at the surface and participate in the current conduction. Both diffusion and drift are responsible for this current conduction. Fast surface states also act as generation–recombination centers yielding a leakage current given by Igrs ¼

qni wc Ac ; t0

ð4:46Þ

where wc is the channel width and Ac is the channel area. The surface breakdown mechanism in areas of high electric field also produces leakage current at the surface. Unlike the generation–recombination processes occurring at the surface and within surface channels, this mechanism is largely temperature independent. 4.1.2 Noise in Detectors

Detectors and noise are synonymous in that the minimum detectable radiation power is determined by the noise floor of the detector. The noise sources can be categorized into two very important groups: noise internal to the detector (resulting from one or more sources) and noise due to the environment (background radiation). At the limit of the minimum detectable radiant power, all detectors experience a noise wall. The potential sources of noise are fluctuations in the detector itself, in the radiant energy to which the detector responds, and in the electrical system accompanying the detector. Usually, signal fluctuations and background fluctuations are considered responsible for the radiant noise. However, under most operating conditions in the ultraviolet, the background fluctuation is minimal and the radiant noise is dominated by the signal fluctuations. In a photodetector, the free carriers always exhibit random thermal motion and because of this, fluctuations occur in the velocity of these carriers creating internal noise. Fluctuations in the free carrier density due to varying rates of thermal generation and recombination also contribute to the internal noise. All noise in a photoconductor falls under four different types: Johnson noise, 1/f noise, G–R noise, and preamplifier noise. All of these noise components manifest themselves as current between detector terminals, though some types of noise are more pronounced in certain types of devices. The physics of noise in semiconductor detectors has been treated thoroughly in reference books [23] that the interested reader is encouraged to read. This book presents a short primer. 4.1.2.1 Thermal Noise Thermal noise (also known as Johnson or Nyquist noise) is caused by fluctuations in the velocity of carriers due to random thermal motion and creates an instantaneous

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current flow. This noise is common to all conducting elements. In a photodiode detector operating at zero bias, the thermal noise current arising from random thermal motion of charged carriers (Johnson–Nyquist noise) dominates the dark current, which can be expressed as hI2n i ¼

4kB T B; R0

ð4:47Þ

where kB is the Boltzmann constant, T is the temperature, R0 is the zero-bias resistance (also called the differential resistance), and B is the response bandwidth of the postdetector filter or amplifier. The zero-bias resistance should be maximized to minimize dark current. It is proportional to minority carrier diffusion lengths on both sides of the p–n-junctions and increases with decreasing density of defects in materials. Noise current depends on frequency and because detectors are sensitive within a bandwidth, it is essential to discuss the power spectrum of noise here. In this context, a power spectrum of noise hi2n ð f Þi in any frequency interval df can be defined. It is then convenient to discuss noise components in terms of spectral density of the power spectrum SI( f ), which is a measurable parameter and is defined as the derivative of the mean square value of the noise current with respect to frequency [8]. SI ð f Þ ¼

dhi2n ð f Þi : df

ð4:48Þ

Spectral density can be also defined in terms of voltage noisehV 2n ð f Þi or resistance fluctuations hdR2n ð f Þi in the form SI ð f Þ SV ð f Þ SR ð f Þ ¼ ¼ : I2 V2 R2

ð4:49Þ

The usual units of SI( f ) and SV( f ) are A2 Hz1 and V2 Hz1, respectively. Because S( f ) describes the noise for a bandwidth of 1 Hz, owing to the derivation with respect to frequency, the total device noise is now obtained by integrating the spectral density function over the spectral band that is sensed by the detector. Applying the concept of spectral density, the Johnson noise can be expressed as SI ð f Þ ¼

4kB T : R0

ð4:50Þ

If the zero-bias resistance of the photodiode is high (possiblely using high-quality materials and junctions), the Johnson noise can be limited to a low value, see Chapter 3 for a discussion of the fundamentals of noise. Ultimately, however, the thermal noise component imposes a floor on photodetectors that can be lowered only by reducing the ambient temperature. 4.1.2.2 Shot Noise The other noise source within the photodiode is the shot noise. Shot noise is generated by a series of discrete random events called shots that are independent of each other. Its manifestation in semiconductors is usually made possible by a

4.1 Principles of Photodetectors

potential barrier. Phenomena such as thermionic emission or diffusive motion of carriers across a p–n-junction create shot noise. Absorption of a photon in a photodiode is also a random process. When the detector is followed by a filter with a bandwidth B, the resulting mean square photocurrent due to shot noise is hI2n i ¼ 2qI p B;

ð4:51Þ

where I is the average current. Similar to the thermal noise case, the spectral density function for shot noise is given by SI ð f Þ ¼ 2qI p ;

ð4:52Þ

which is a characteristic of shot noise. Neither thermal nor shot noise is frequency dependent, both types of noise are known as “white.” 4.1.2.3 Generation–Recombination Noise The process of generation and recombination of free carriers in a semiconductor gives rise to what is termed as generation–recombination noise. These processes are random in nature and could be considered a form of shot noise. However, the distinguishing characteristic of the G–R noise is the presence of a well-defined time constant t. Hence, the carrier density changes with time as dN/dt ¼ N/t. Assuming that the current gain for the photocurrent and the noise current is the same, then the resulting generation–recombination noise is

In ¼ qg½2Wgr ðDnÞAe 1=2 ;

ð4:53Þ

where Wgr is the areal thermal generation–recombination, with (Wgr ¼ [te(G þ R)], G and R are the generation and recombination rates, respectively, te is the thickness of the detector, and Ae is the electron area of the detector. The generation rate G is a combination of the optical generation rate Gop and the thermal generation rate Gth, so that G ¼ Gop þ Gth. Under a constant applied bias, variations in the carrier density result in resistance fluctuations dR, which manifest themselves as current fluctuations dI ¼ V/dR. Thus, it is convenient to describe G–R noise by [8] SR SN hðDNÞ2 i 4t ¼ ¼ : R2 N 2 N 2 1 þ w2 t2

ð4:54Þ

The spectral density function SN( f ) is of the type characteristic of a low-pass filter. It is frequency dependent up to the frequency that corresponds to the characteristic time constant(s) t. Beyond that point, it decreases rapidly with a rate of 20 dB decade1. Experimentally, carrier trapping and recombination at deep levels can be observed through measurements of SN( f ). 4.1.2.4 1/f Noise Another important noise component is the 1/f noise, which dominates low frequencies. The power spectrum of the 1/f noise, as the name suggests, depends on

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frequency as 1/f a, where a is a constant close to unity [8]. This noise has a few names such as flicker noise [24] and excess noise [25]. To make matters worse, there are many distinct physical phenomena that produce nearly the same 1/f characteristics. For example, generation–recombination noise associated with a large number of different recombination time constants would produce the 1/f signature [26]. Devices as distinctly different from photodetectors as bipolar junction and field effect transistors (FETs) also exhibit this type of noise. The spectral purity of an oscillator critically depends on the 1/f noise as well. The 1/f noise is described by a characteristic spectral density function Sn( f ): S n ¼ S0

I 2d ; fg

ð4:55Þ

where Id represents the detector dark current, and S0 and g (typically close to unity) are fitting parameters. Two models of the 1/f noise in hopping conductivity have been proposed. Both rely on localized states separated from each other by distances greater than the characteristic electron jump length. These sites are not incorporated into the critical current-carrying circuit, but function as “dead ends,” slowly exchanging electrons with other sites within the circuit. In the number fluctuation theory [27], this leads to slow fluctuations in the total number of electrons taking part in hopping conduction. In the mobility fluctuation theory [28], trapping an electron at a dead end is assumed to change the distribution of charge and shift the energy levels of neighboring traps. This leads to slow fluctuations in the effective mobility of hopping electrons. The mobility fluctuation theory predicts a very weak temperature dependence of the low-frequency hopping noise and it does not seem to apply to GaN homojunctions well. The number fluctuation theory results in a frequency dependence of the noise spectral density given by a specialized form of Equation 4.55 proposed by Hooge as [29] I2 Sn ¼ a d
; fN

ð4:56Þ

where N
is the average number of electrons taking part in the conduction process and a (the Hooge parameter) is expected to be on the order of unity. The Hooge parameter can be estimated from Equation 4.55 as a ¼ S0 N
assuming g ¼ 1. At room temperature and bias of 10 V, the parameter S0 is approximately 1.3  105 and the dark current of the photodetector is approximately 1  109 A, resulting in a diode resistance of R  1  1010 W. The number of electrons N
can then be found from R ¼ L2 =ðqmN
Þ, where the length L is assumed equal to the depletion layer width W, which can be obtained from the C–V data. For W ¼ 0.25 mm, the Hooge parameter would be close to unity and for W ¼ 0.3 and 0.4 mm, the Hooge parameter is 1.5 and 2.6, respectively. The effective mobility m is estimated from the theory of hopping conduction [30] using parameters deduced from the fit to the I–V data at room temperature. This analysis yields m  5  106 cm2 V1 s1, N
 2:3  105 , and a  3, consistent with expectations of the number fluctuation theory [8]. Both the

4.1 Principles of Photodetectors

bias and temperature dependencies of S0 can be represented by the change in the number of hopping electrons N
. With increasing bias, N
goes up due to increasing width of the depletion layer and increasing temperature. There are also noise sources introduced by the circuit, which will not be discussed here, and background radiation. The extremely low radiation noise in the solar-blind region is the reason why AlGaN-based detectors operating near 280 nm have been catapulted to prominence. A succinct discussion of the fundamentals behind the Hooge parameter is provided in Chapter 3. 4.1.3 Quantum Efficiency

The quantum efficiency is a very important parameter because it is a measurement of how efficiently the photons are converted to current. This parameter hext is defined as the number of electron–hole pairs generated and collected per incident photon in a photodetector. In a well-designed photodetector, nearly all photons penetrating the semiconductor are absorbed and result in carrier generation followed by efficient carrier collection. If all of these processes were done perfectly, the internal quantum efficiency h0 for this photodetector would be unity. Usually, however, the external quantum efficiency also includes reflections and other external losses so that hext is lower than the internal quantum efficiency h0. 4.1.3.1 Quantum Efficiency in Photoconductors In a photoconductor, there are slabs of material(s) as well as the top and the bottom surfaces all of which exhibit finite reflection coefficients (r1, r2, r3, etc.). The absorption coefficient a of the slab is also finite. All the slabs absorbing optical radiation lead to the internal photogenerated charge profile in the y-direction (see Figures 4.3 and 4.9). This profile can be expressed by [31]

SðyÞ ¼ S1 ðyÞS2 ðyÞ; S1 ðyÞ ¼

h0 ð1r 1 Þa ; 1r 1 r 2 expð2atÞ

ð4:57Þ ð4:58Þ

and S2 ðyÞ ¼ expðayÞ þ r 2 exp½að2t þ yÞ:

ð4:59Þ

The external quantum efficiency hext is simply the integral of S(y) over the photodetector thickness t: ðt hext ¼ SðyÞ dt 0

h ð1r 1 Þ½1 þ r 2 expðatÞ½1expðatÞ : ¼ 0 1r 1 r 2 expð2atÞ

ð4:60Þ

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S0 r1 S0

(1-r1) S0

(1-r1 r2 )r1 S0 e -2 α t

(1-r1 )r2 S0 e -2 α t

r1 (1-r1 )r1 r2 S0 e -2 α t

.....

t (1-r1 )S0 e -α t

(1-r1 )r2 S0 e -α t

x

(1-r1 )r1 r2 S0 e -3 α t

y r2

(1-r1 )(1-r2 )S0 e -α t

(1-r1 )(1-r2 )S0 e -α t 1-r1 r2 e -2 α t Figure 4.9 Surface and internal photoreflections indicating the intensity of the optical signal within the detector. The optical profile must be multiplied with the absorption coefficient to arrive at the internal charge profile.

If r1 ¼ r2 ¼ r, the external quantum efficiency is reduced to hext ¼

h0 ð1rÞ½1expðatÞ : 1r expðatÞ

ð4:61Þ

In a real photodetector with a well-designed assembly, the reflection capability of only the top surface is very important because the intrinsic detector materials tend to be highly absorptive. This allows a simplification of Equation 4.61 to hext h0 ð1rÞ ð1rÞ:

ð4:62Þ

4.1.3.2 Quantum Efficiency in a p–n-Junction Detector Referring to the p–n-junction detector structure shown in Figure 4.6, where the light is incident on the detector from the left side (must have at least partially transparent contacts), the light is first absorbed in the p-layer, what is not absorbed is assumed to be partially absorbed in the depletion region, and what is left is fully absorbed in the n-layer. Based on this premise, we can calculate the quantum efficiency in such a detector. Ideally, the detector structure is designed in such a way that the top p-layer would be transparent and the thickness of the depletion region is made equal to the absorption depth for the central wavelength of operation. Three different regions contribute to the quantum efficiency of a p–n-junction photodetector: two neutral regions (n-type and p-type) and one space charge region. Thus, h ¼ hn þ hp þ hscr : ð4:63Þ

The quantum efficiency is defined as

4.1 Principles of Photodetectors

h¼

j735

number of photon-generated charges contributing to current J ph ; ¼ qf number of incident photons ð4:64Þ

where Jph is the photocurrent in certain layer and f is the incident photon flux. The generation rate of electron–hole pairs at distance x from the surface (with x increasing away from the surface) can be expressed as G ¼ afð1rÞ expðaxÞ;

ð4:65Þ

where r is the reflection coefficient from the front (top p-layer) surface. Considering the p-type region first, the change in the electron density due to photon-generated carriers satisfies the continuity equation: De

d2 Dnp Dnp  þ G ¼ 0: dx 2 te

ð4:66Þ

Substituting Equation 4.65 into Equation 4.66 leads to De

d2 Dnp Dnp  þ afð1rÞexp½aðx þ tÞ ¼ 0: dx 2 te

ð4:67Þ

The general solution for Equation 4.67 is given by (with x increasing away from the surface)

 xþt xþt afð1rÞtn Dnp ¼ A cosh exp½aðx þ tÞ; þ B sinh  2 2 Ln Ln a Ln 1 ð4:68Þ where A and B are the integration constants, which can be determined using the boundary conditions as follows: at

x ¼ x p ;

at

x ¼ t;

np ðx ¼ x p Þ ¼ np0 Dn

ðwith no biasÞ;

 dnp  ¼ s2 ½np ðx ¼ tÞnp0 : dx x¼t

ð4:69aÞ ð4:69bÞ

After applying the boundary conditions to Equation 4.68, we have for Dnp, aið1rÞtn a2 L2n 1 ðg n þaLn Þsinh½ðxxp Þ=Ln þexp½aðtx p Þ½g n sinh½ðtþxÞ=Ln þcosh½ðtþxÞ=Ln   g e sinh½ðtx p Þ=Ln þcosh½ðtx p Þ=Ln 

exp½aðtþxÞ : ð4:70Þ Dnp ¼

The photogenerated current density can be expressed as   dnp  dDnp  ¼qD ; J ph ¼ qDn n dx  dx  x¼xp

x¼xp

ð4:71Þ

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When Equation 4.64 is used with the help of Equations 4.70 and 4.71, we can find quantum efficiency as

að1rÞLn ðg n þaLn Þexp½aðtxp ÞU n1 hp ¼ ¼ 2 2 aLn exp½aðtx p Þ : qf a Ln 1 U n2 J ph

ð4:72Þ Similarly, for the n-type region, the change of hole density satisfies the continuity equation Dp

d2 Dpn Dpn  þafð1rÞexp½aðxþtÞ ¼0: dx 2 tp

ð4:73Þ

The boundary conditions for solving the above differential equation are as follows: at pn ðx ¼ xn Þ ¼pn0 ; at x ¼d;

Dn

x ¼x n ;

ð4:74aÞ

 dnp  ¼s1 ½np ðx ¼dÞnp0 : dx x¼d

ð4:74bÞ

The photon-generated current density is J ph ¼ qDp

  dDnp  dpn  ¼qD : p dx x¼xn dx x¼xn

ð4:75Þ

Equation 4.75 can be solved with the help of boundary conditions given by Equation 4.74. Then Equation 4.64, describing the efficiency, can be used with the help of Equation 4.75 to determine the efficiency for the n-layer as hn ¼



U p1 ðg p aLp Þexp½aðdx n Þ ð1rÞaLp Þ aL  : exp½aðtþx n p U p2 a2 L2p 1 ð4:76Þ

For the space charge region, the photon-generated carriers are all extracted by the built-in field and contribute to the current. The continuity equation becomes Dn

d2 Dnp ¼G¼fð1rÞa exp½aðtþxÞ; dx2

where

x p
x
x n :

ð4:77Þ

Therefore, the photocurrent density is J ph ¼qðDn ;Dp Þ

xðn ð dðDnp ;Dpn Þ ¼q Gdx ¼qfð1rÞa exp½aðtþxÞdx dx x p

¼qfð1rÞfexp½aðtx p Þexp½aðtþx n Þg: ð4:78Þ

4.1 Principles of Photodetectors

The efficiency, therefore, in the space charge region is hd ¼

J ph qf

 ¼ð1rÞ exp½aðtxp Þexp½aðtþx n Þ :

ð4:79Þ

If the loss associated with the optical reflection at the semiconductor surface is neglected, the external quantum efficiency h and the internal quantum efficiency h0 are the same. To obtain a high value for h in a p–n-junction photodiode, the illumination region of the junction must be sufficiently thin for the carriers generated within that region to make it to the edge of the depletion region. The excess minority carriers in that case would be swept across the depletion region. If the device is operated with no external bias voltage, the excess majority carriers making it to the edge of the depletion region would be injected across leading to a terminal current as well. Generally, p–n-homojunction photodiodes (supplanted by heterojunction varieties really) are so designed that the impinging radiation is absorbed on only one side (e.g., the p-side) of the junction (see Figure 4.4b). This is achieved in practice by making the top layer thin compared to the diffusion length. Depending on the material system and layer thicknesses, the photons canbe absorbed in the p-layer, the depletion region, and/ or the n-layer below. The minority carriers generated would diffuse to the edge of the depletion region and be swept across resulting in terminal current. The carriers generated inthe depletionregion are then swept across by the electricfield givingriseto a terminal current. The optimum device design is one wherein the carriers are generated in the depletion region only. The meaning of thin needs further elaboration. If this layer is much thicker than both the absorption depth and the minority carrier diffusion length, all of the photons will be absorbed, but the minority carriers in whole cannot diffuse to the junction and create a terminal current. However, if this region is made very thin relative to the absorption depth (assuming that the diffusion length is not the limiting factor), some of the photons will be absorbed on the other side (n-side) of the junction. If the diffusion length is the limiting factor then some of the minority carriers cannot make it across the junction and will not create a terminal current. The net effect is a reduction of the internal quantum efficiency. If we assume the electrons to be the minority carriers (top layer is n-type) and the back contact is several minority diffusion lengths Ln away from the junction, then the quantum efficiency is given by hðlÞ ¼ ð1rÞ

aðlÞLn : 1 þ aðlÞLn

ð4:80Þ

If the back contact is less than a diffusion length away from the junction and no radiation is reflected from the back surface, the quantum efficiency would reduce to hðlÞ ¼ ð1rÞf1eaðlÞd g;

ð4:81Þ

where d is the thickness of the p-type region assuming the top layer is n-type. If the front surface reflection is minimized by incorporating antireflection coating, and the device is made thicker than the absorption length (for full absorption of radiation), the reflection coefficient r would be negligibly small and nearly all the photons

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would be absorbed. Combining these two features would maximize the quantum efficiency. 4.1.4 Responsivity

The spectral current responsivity of a photodetector is expressed, as the current in amperes divided by the incident power in watts required to generate that current. The expression for the responsivity can be constructed by recognizing that for each photon, hextg electrons are generated. Consequently, the responsivity can be expressed as

 Ip device current lhext q ¼ gq ð4:82Þ Rl ¼ ¼ g lhext ¼ Pl hc optical power hc where h is the Planck’s constant, c is the velocity of light, and q is the electric charge (instead of device current you can use photocurrent). In the ultraviolet spectral region of interest, the responsivity varies from 0.161 A W1 at 200 nm to 0.294 A W1 at 365 nm for ideal hext ¼ 1 and g ¼ 1. Assuming that the change in conductivity under illumination is small compared to the dark conductivity, the voltage responsivity can be expressed as
 open-circuit voltage V s hext lt V a RV ðlÞ ¼ ¼ ; ð4:83Þ ¼ Pl lwt hc n0 optical power where the absorbed monochromatic power is Pl ¼ Fs Ahn. For high responsivity at a given wavelength l, device should have high quantum efficiency hext, long excess carrier lifetime t, the smallest possible dimension, low thermal equilibrium carrier concentration n0, and the highest possible applied voltage Va. The frequency-dependent responsivity can be determined by the following equation: RV ðlÞ ¼

hext lteff V a 1 ; wt hc n0 ð1 þ w2 t2 Þ1=2 eff

ð4:84Þ

where teff is the effective carrier lifetime. 4.1.5 Signal-to-Noise Ratio, Noise Equivalent Power, and Detectivity

The above discussion neglects any current in the absence of incident optical signal (known as the dark current and caused by sources of noise), which is the crux of any detector. Under illumination, one can define the signal-to-noise ratio, that is, the ratio of the photocurrent to the noise current In (which is random by definition) as Ip S R Pl ¼ qffiffiffiffiffiffiffiffi ¼ qlffiffiffiffiffiffiffiffi : N hI 2 i hI2 i n

n

ð4:85Þ

4.1 Principles of Photodetectors

Because the photocurrent is proportional to the responsivity for a given optical signal power, the signal-to-noise ratio is proportional to responsivity as well. Enhancement of the signal-to-noise ratio then hinges on lowering noise whenever possible and enhancing the responsivity. Noise current is measured over a certain frequency range, at some frequency resolution, and analyzed with a Fast Fourier transform (FFT) spectrum analyzer. Advances made in measurement systems allow very reliable noise measurements; it is currently possible for the noise floor of the entire experimental apparatus to be lower than 1025 A2 Hz1. Noise in the detector determines the detectivity. Another term used to account for noise in a detector is the noise equivalent power (NEP). The NEP of a photodetector is defined as the incident optical power required to produce a signal current (photocurrent) that is equal to the noise current at a given wavelength. It can also be described as the signal power required to yield S/N ¼ 1 and is a measure of the minimum detectable signal. In other words, it is defined as the noise current divided by the responsivity of the detector. For several sources of noise current such as thermal, shot, generation–recombination, 1/f, and background radiation, NEP is given by qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hI 2th i þ hI 2shot i þ hI2gr i þ hI 21=f i þ hI2bk i hI 2n i noise current NEP ¼  : ¼ Rl responsivity Rl ð4:86Þ We should mention that the maximum responsivity (corresponding to an external quantum efficiency of 100%) reduces with increasing photon energy (decreasing photon flux for a given optical power). The NEP for various noise sources is discussed further down the section, but for now, in terms of the spectral power density function, the NEP can be defined as NEP ¼

pffiffiffiffiffiffiffiffiffiffiffiffi Sn ð f Þ ; Rl

ð4:87Þ

where Rl is the responsivity measured in A W1 and NEP in terms of W. Some define NEP as the optical signal power required to produce a photocurrent that is equal to the noise current at a given wavelength and within a bandwidth of 1 Hz. In this case, the detectivity for a unit area would just simply be the inverse of NEP with NEP having unit as W Hz1/2. We can now define two figures of merit that are commonly used to describe photodetector performance, namely, detectivity (D) and specific detectivity (D ) [32]. The detectivity D is simply the inverse of NEP specified for a bandwidth of 1 Hz, and D increases with improved detector performance. Specific detectivity D is the value of D normalized to a detector area of 1 cm2 and to a response bandwidth B of 1 Hz. D therefore represents the signal-to-noise ratio normalized to a standard size. If A0 is the photon area of the detector, then the normalized signal-to-noise ratio of this detector can be determined by the specific detectivity D and is given by

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D ¼

ðA0 BÞ1=2 Rl ðA0 BÞ1=2  qffiffiffiffiffiffiffiffi ; NEP hI2 i

ð4:88Þ

n

where the unit of D is cm Hz1/2 W1. For a detector with an area of 1 cm2 and the response bandwidth of 1 Hz, D equal to D. It is common to make a distinction between the noise and the background radiationlimited performance [32]. When the thermal noise (Johnson noise), shot noise, generation–recombination, or a combination thereof is dominant over the radiation noise, the detectivity is limited by noise internal to the device. Likewise, when the background radiation noise is dominant over the internal noise, the detectivity is said to be limited by the background radiation. The thermal noise, shot noise, and generation–recombination noise are represented by Equations 4.47, 4.51, and 4.53, respectively. After defining NEP and D , let us now determine them for thermal, shot current, generation–recombination, and background radiation-limited cases. 4.1.5.1 Thermal Limited When the thermal noise of the photodiode detector is larger than the noise induced by fluctuations in the background radiation, the thermal-limited detectivity is given by

DT

rffiffiffiffiffiffiffiffiffiffiffi R0 A 0 ; ¼ Rl 4kB T

ð4:89Þ

where Rl is the responsivity and R0 is the differential resistance of the photodetector. 4.1.5.2 Shot Current Limited In this shot noise-limited case, using Equations 4.82, 4.85, and 4.51, the signal-tonoise ratio for shot noise-dominated photodetector can be expressed as

S P ¼ ghext l : 2hnB N

ð4:90Þ

Using Equation 4.90 with the aforementioned conditions, NEP for a shot noisedominated photodetector can be written as NEP ¼

2hn B; ghext

gh D ¼ ext 2hn 

rffiffiffiffiffiffi A0 : B

ð4:91Þ

ð4:92Þ

4.1.5.3 Generation–Recombination Limited Equations 4.53, 4.82, and 4.88 yield the generation–recombination-limited specific detectivity as

1=2 h A0 D ¼ ext : ð4:93Þ hv 2Wgr Ae

4.1 Principles of Photodetectors

The photon and electron areas of a photodetector are generally different because they have to make contact with the semiconductor, blocking the radiation from reaching the semiconductor. An examination of Equation 4.93 indicates that, for a given wavelength l and operating temperature T, the highest performance of the photodetector can be obtained by maximizing h/(Wgr)1/2. If the recombination process takes place in a region of the device where the photoelectric gain is marginally small (e.g., at the contacts in sweep-out photoconductors or in the neutral regions of the diode types), the effect of fluctuating recombination is quite negligible. In this manner, the noise can be lowered by as much as a factor 2.5 times, increasing the detectivity by the same factor. The generation process and its associated fluctuation cannot, however, be avoided completely. The generation and recombination rates are equal at equilibrium. By assuming A0 ¼ Ae, we obtain
 h 1 1=2 D ¼ ext : ð4:94Þ hv 4Gte 4.1.5.4 Background Radiation Limited In background-limited detectors, the NEP is given by   2AFB B 1=2 ; NEP ¼ hn hext

ð4:95Þ

where FB is the total background radiation impinging on the detector (expressed as background radiation flux density (ph s1 cm2 nm1)) and the detectivity in the photodetector (operating at zero bias) is given by DBL ¼

Rl qð2hext FB Þ1=2

:

ð4:96Þ

If more than one noise source is in effect, the detectivity expressions given above must be added appropriately. For example, if both thermal and background radiation noise are to be dealt with, then the detectivity is given by  1=2 4kB T þ 2qhext FB ; ð4:97Þ D ¼ Rl R0 A 0 where the first term in the bracket corresponds to thermal noise and the second to radiation noise. 4.1.5.5 Noise in a p–n-Junction Detector The two fundamental processes – fluctuations in the velocity of free carriers due to random motion and fluctuations due to randomness of thermal generation and recombination – are less readily identifiable in p–n-junction photodiodes than in photoconductive detectors. The combined effect of these two processes is measurable as noise for the minority carriers, yielding the net junction current. This shot noise is the result of the random motion of the minority carriers, which creates fluctuations in the diffusion rates (in the neutral regions of the devices) and in the

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generation–recombination processes (in the neutral regions as well as the space charge region). A general theory of noise in p–n-junction photodiodes in terms of all possible sources of leakage and arbitrary applied bias is not yet available. An approximate form of the model developed for an ideal diode is I2n ¼ ½2qðId þ 2I s Þ þ 4kB TðGj G0 ÞB;

ð4:98Þ

where Id ¼ Is[exp(qV/kT )  1], B is the bandwidth, Gj is the conductance of the junction, and G0 is the low-frequency value of Gj. In the absence of applied bias and external photon flux, the first term on the right-hand side of Equation 4.98 is negligibly small and thus for a diode in thermal equilibrium the mean square noise current is equal to the Johnson–Nyquist noise of Equation 4.47. Note that the mean square shot noise in reverse bias is half the mean square Johnson–Nyquist noise at zero bias. However, this predicted improvement in the noise level of reverse-biased diode is not normally observed due to increased 1/f noise. 4.1.5.6 Detectivity for a p–n-Junction Detector Typically, the photoelectric gain for a p–n-junction photodiode is unity, and Equation 4.82 for responsivity is given by

Rl ¼

qhext qlhext ¼ : hn hc

Similarly, following Equation 4.93, the detectivity or the D is  1=2 qlh A D ¼ : hc 2qðI d þ 2Is Þ

ð4:99Þ

ð4:100Þ

Under reverse-bias conditions and negligible radiation noise, Id tends to be Is and the expression in the parenthesis of Equation 4.100 reduces to Is. This suggests that the performance of an ideal diffusion-limited photodiode can be optimized by maximizing the quantum efficiency h and minimizing the reverse saturation current Is. For a photodiode, the general expression for the saturation current due to electrons on the p-side can be obtained from Equations 4.34 and 4.35 as
 qADn np0 sinh q þ g n cosh q ; ð4:101Þ Ips ¼ Ln cosh q þ g n sinh q where g n ¼ s2Ln/Dn and q ¼ (t  xp)/Ln The leakage current can be more easily brought to a manageable level by properly treating the inactive side of the junction, which does not contribute to the postsignal. For example, if the doping or the energy bandgap of the inactive side is increased, the minority carrier generation rate and hence the diffusion current will greatly decrease. If the back contact is several diffusion lengths away from the junction, Equation 4.101 reduces to Is ¼

qADn np0 : Ln

ð4:102Þ

This means that as the back contact is brought closer to the junction, the leakage current will either increase or decrease, depending on whether the surface recombination

4.1 Principles of Photodetectors

velocity is greater than the diffusion velocity Dn/Ln. In the limiting case where d  Ln, the saturation current is reduced by a factor of d/Ln relative to that in Equation 4.101 for s ¼ 0 and increased by a factor of Ln/d for s ¼ /. If the surface recombination velocity is small, then Equation 4.101 is simplified to I s ¼ qGV diff ;

ð4:103Þ

where G is the bulk minority carrier generation rate per unit volume and Vdiff is the effective volume of material from which minority carriers diffuse into the junction. The effective volume is ALn for Ln  d and tends to be Ad for Ln  d. Because of this, the generation rate for p-type material may be given by Gb ¼

np0 n2 ¼ i : tn tN d

ð4:104Þ

It is thus apparent that the device performance of a photodetector is strongly dependent on the quality of its back contact. To alleviate the problems caused by this contact, it is advisable to move it many diffusion lengths away to one side and to ensure that the backside surface is passivated properly. An alternative solution is to design the back contact to produce a low surface recombination velocity. This can be achieved by introducing a barrier for minority carriers between the metal contact and the rest of the device. Increasing doping concentration and the energy bandgap are good options for this barrier. Such a barrier, if realized, will effectively isolate the minority carriers from regions with high recombination rates (such as at the contacts). 4.1.6 Surface and Bulk Recombination in Detectors

In a typical photoconductor, there is a finite probability of electron–hole recombination at the crystal surface. Because of this recombination, the total number of steadystate excess carriers is reduced and the photoconductive lifetime teff then represents the lower limit of the bulk lifetime t of a carrier. The two lifetimes are related by teff c ¼ 2 2 ; t a LD 1 where c ¼ aLD ½zA þ zB eat ð1eat Þ; ðaDD þ s1 Þfs2 xA þ xB g ; xC þ xD ðaDD s2 Þfs1 xA þ xB g ; zB ¼ x þx
C D t xA ¼ cosh 1; LD

 DD t sinh ; xB ¼ LD LD zA ¼

ð4:105Þ

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 DD t ðs1 þ s2 Þcosh ; LD LD
2
 DD t þ s1 s2 sinh xD ¼ ; LD LD

xC ¼

DD ¼

Dn p0 mp þ Dp n0 mn : n0 mn þ p0 mp

ð4:106Þ

Note that DD is the ambipolar diffusion coefficient, Dn and Dp are the diffusion coefficients of the electrons and holes, respectively, s1 and s2 are the surface recombination velocities at the front and rear surfaces of the photoconductor, respectively, and LD ¼ (DDt)1/2. If the absorption coefficient a is large, exp(at) 0 and s1  aDD. Under this condition, Equation 4.105 is simplified to teff ¼ t




DD LD



s 2 xA þ xB : LD xC þ xD

ð4:107Þ

Further simplification based on the assumption that s1 ¼ s2 ¼ s leads to 1 1 2s ¼ þ : teff t t

ð4:108Þ

For some nitrides, the predominant recombination mechanism is indirect and mediated by imperfection centers in the crystal. These centers act as recombination centers if carriers reaching them have a high probability of recombining with carriers of opposite polarity and essentially no probability of returning to a nearby band. In contrast, these centers act as traps if the probability of these carriers returning to the band is high. When these centers act as traps, the total time between the excitation and the recombination is effectively identical for electrons and holes. The total time spent out of the traps is, however, different for electrons and holes. It is only during this time that the carriers contribute to the photoconductive process. Analysis of the decay characteristics of responsivity over time and the power density for GaN photoconductive detectors strongly suggests that the carrier dynamics is affected by recombination centers rather than trapping centers. Had it been affected by traps, there would have been a re-emission of captured carriers with a strong temperature variation of the emission rate.

4.2 Particulars of Deep UV Radiation and Detection

The Sun is the most important natural source of terrestrial UV light, because 9% of its overall light power is UV. Until the advent of wide-bandgap nitrides, most of the applications of UV detectors involved solar measurements. UV photodetectors therefore require, in general, visible and infrared blindness to have a sufficient level of UV selectivity. UV detectors mainly work as high-pass filters and are classified as “visible-blind” detectors when their cutoff wavelengths are in the 400–280 nm range

4.2 Particulars of Deep UV Radiation and Detection

Wavelength (nm) 300

400

b=1

1.0

Al mole fraction

200

0.8 0.6 0.4 0.2 0.0 -0.2 3

3.5

4

4.5

5

5.5

6

6.5

Photon energy (eV) Figure 4.10 Bandgap of AlGaN, both in terms of energy gap and corresponding wavelength of AlGaN, as a function of AlN molar fraction in the lattice.

and “solar-blind” detectors when their cutoff wavelengths are shorter than 280 nm. In the latter case, solar-blind detectors are blind to the whole solar UV light spectrum coming to the ground on Earth, the UV C band is normally absorbed completely in the upper layers of the atmosphere. Visible-blind detectors are widely used in flame detection applications, and they are also potentially of a high interest in space applications such as secured optical telecommunications between satellites, using UV wavelengths lower than 280 nm that can neither be seen nor detected on Earth. The naturally large bandgap of AlGaN lends itself to absorb the wavelengths of 280 nm and below while not responding to the longer wavelengths. Shown in Figure 4.10 is the bandgap in terms of both the energy gap and the corresponding wavelength of AlGaN as a function of the AlN molar fraction in the lattice. To classify a photodetector fabricated in AlGaN as solar blind, the AlN molar fraction must be 45%. This is beyond the efficient activation of the only successful p-type impurity, Mg, as discussed in Volume 1, Chapter 4. The objective here is to highlight the stringent requirements that must be met in the design of AlGaN/GaN solar-blind UV photodetector arrays that will have responsivities, detectivities, gain, speed, and low noise levels comparable to and preferably better than those of high-gain PMT detectors [33]. To highlight these objectives, the following topics will be considered: solar UV radiation, stratospheric ozone and absorption of solar UV radiation, computational methods such as the code called “Plexus,” the UV transmission of the atmosphere (the number of solar photons reaching lower altitudes), UV photon range estimates, and unavoidable atmospheric losses of UV photons from emission to detection. Inevitable is a comparison with Generation I (Gen I), Generation II (Gen II), and Generation III (Gen III) UV sensors.

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4.2.1 Solar UV Radiation

A discussion of detectors collecting radiated signals from the environment is intricately connected to a discussion of solar radiation [33]. The topic of nitride semiconductors focuses the discussion on solar UV radiation emanating from the Sun. The Sun is a star with a very complex layered structure composed primarily of hydrogen and helium along with traces of other atomic elements [34]. The core of the Sun transforms hydrogen nuclei into helium in a set of physical processes that lead to fusion of hydrogen nuclei yielding great amounts of energy in the form of gamma radiation. Then through repeated absorption and emission processes from the core of the Sun to the outer chromosphere and photosphere solar layers, the shorter wavelength gamma radiation emanating from the core is downconverted into relatively longer wavelength ultraviolet, visible, and infrared radiation. This UV, visible, and infrared radiation is then emitted by the surface of the Sun and propagates in all directions throughout and away from our solar system. From Earth, the Sun appears to be radiating as a blackbody at a temperature of 5770 K based upon the measured value of the total amount of solar power reaching the Earth and the use of Stefan’s law and the Sun to Earth distance. The total amount of solar power reaching the Earth is also known as the solar constant and has a value of approximately 1373 W m2 based on the experimental measurement [35]. In general, the amount of radiation from the Sun in any wavelength interval is in accordance with Planck’s blackbody radiation law, which can be used to estimate the approximate number of UV photons reaching the Earth’s upper atmosphere. In UV detectors, the number of solar UV photons that are transmitted from the stratosphere, which is wavelength dependent, down to lower altitudes becomes the UV sensor’s background. In the solar-blind region of the spectrum, the background imaged by the UV sensor appears black as if solar illumination were not present. Thus, the background noise is very limited even at high noon and when viewing the Sun directly. The number of solar UV photons that reach an altitude of 41.4 km at the solar zenith angle of 29.6 was measured by Altrock [36] on April 19, 1978 at 10:22 a.m. local time using a spectrometer carried aloft by a balloon that lifted off from Holloman Air Force Base. Hall’s UV measurements (discussed in reference [36] by Altrock et al.) have been conveniently tabulated in 1 A wide bins [37] in the form of the number of photons in each angstrom-wide bin. Ten of these angstrom-wide photon bins are added together to give the number of photons in 1 nm wide spectral bins beginning with the number of photons in the 260–261 nm bin. Then, the number of photons for each of the 1 nm wide bins from the 261–262 nm bin to the 289–290 nm bin are added to arrive at the number of photons in the spectral range from 261 to 290 nm. Ultimately, these nanometer wide photon bins, which are based on Hall’s balloon measurements, are used to determine the number of solar photons that are transmitted down to lower altitudes using the computational program known as “Plexus,” discussed in Section 4.2.3.

4.2 Particulars of Deep UV Radiation and Detection

Figure 4.11 Schematic diagram of various atmospheric layers including the O3 layer, which absorbs the band around 280 nm.

4.2.2 Stratospheric Ozone and UV Absorption

The stratosphere is the layer of the atmosphere between the troposphere and the mesosphere. The top of the troposphere is approximately at an altitude of 18 km and the bottom of the mesosphere is at an altitude of 50 km, so that the thickness of the stratosphere is roughly 32 km, as shown in Figure 4.11. The stratosphere contains nearly 90% of the ozone in any vertical column of the atmosphere. Stratospheric ozone densities vary from mid-1012 cm3 at an altitude of 18 km to mid-1011 cm3 at 50 km. The ozone density peaks at a value of 6  1012 cm3 at an altitude of approximately 23 km. As is well known today, it is the ozone content of the stratosphere that absorbs the solar UV photons to make the Earth habitable to animal and plant life because mid- and short-wavelength UV photons have enough energy to break and mutate biological molecules. What is germane to this chapter is the solar UVabsorption by ozone that helps to create a dark background for UV sensor imaging, as shown in the cartoon of Figure 4.12. Beginning at 310 nm, solar UV photons are absorbed by ozone, which dissociates into atomic and molecular oxygen. As the solar UV photon energy increases, ozone continues to absorb but also begins to dissociate into excited atomic and molecular oxygen. However, excited atomic oxygen and ground-state molecular oxygen recombine exothermally, producing more ozone. Protection of this naturally occurring physical process is of concern today because reduction of stratospheric ozone levels has been shown to lead to decreased absorption of solar UV photons. 4.2.3 Computational Method Called Plexus

Plexus (Phillips Laboratory Expert-Assisted User Software) [38] is a Windows-based computer program. Plexus acts as a driver to run two atmospheric optical radiation

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Figure 4.12 Cartoon showing how the O3 ozone layer prevents the solar radiation to reach the sensor. Courtesy of R. J. Manning of BAE systems. (Please find a color version of this figure on the color tables.)

propagation models known as MODTRAN and HITRAN, which are based on field and laboratory measurements. These models evaluate the optical transmission of the atmosphere in the UV, visible, and infrared portions of the electromagnetic radiation spectrum. In calculating atmospheric transmission, Plexus takes into account the geographic location, altitude, weather, air transparency, and solar and geomagnetic activity. The calculations of atmospheric transmission were performed in accordance with Hall’s balloon flight data as discussed in Section 4.2.1. Thus, the UV transmission of the atmosphere was obtained for a slant range beginning at a higher initial altitude down to a lower final altitude along a slant range parallel to the direction of the Sun’s rays for a solar zenith angle of 29.6 in accordance with Hall’s balloon flight measurements discussed above. 4.2.4 UV Transmission of the Atmosphere

The measured spectrum of the Sun’s radiation at different altitude/atmospheric pressure levels in terms of photon flux for a unity bandwidth is shown in Figure 4.13. The transmission of the Earth’s atmosphere was calculated for ultraviolet photons with wavelengths ranging from 260 to 290 nm in 1 nm wide spectral bins and tabulated from 270 to 290 nm [39]. These atmospheric transmission numbers were obtained at a solar zenith angle of 29.6 as was the position of the Sun on the day Hall carried out his balloon flight measurements of solar photons at Holloman AFB at 10:22 h local time. The UV atmospheric transmission data for the 260–290 nm spectral region was obtained with Plexus as discussed in Section 4.2.3. Specifically, the number of solar UV photons reaching and entering a UV sensor is calculated

4.2 Particulars of Deep UV Radiation and Detection

Solar irradiance (photons/s-1 cm-2 nm-1)

1015 1014 1013 1012 1011 1010 109 108 107 106

0.24 atm cm

105 104

0.30 atm cm

103

0.36 atm cm

102

280

285

290

295

300

305

310

315

Wavelength (nm) Figure 4.13 The spectrum of the Sun’s radiation in terms of photon flux per unity bandwidth for three different atmospheric masses (atm cm) ranges. (Please find a color version of this figure on the color tables.)

when the sensor is located at the altitudes of 9.656 km (6 miles/31 680 ft), 6 km (3.728 miles/19 684 ft), 4 km (2.5 miles/13 200 ft), 2 km (1.242 miles/6561 ft), and 1 km (0.625 miles/3280 ft). The sensor is assumed to be looking directly at the Sun for these calculations. The results of these evaluations will be shown below and will establish the solar-blind region considering the wavelength region from 260 to 290 nm. The spectrum of the Sun’s radiation in terms of photon flux or I could also use the one from Manning of BAE systems, but the wavelength scale is classified that shows only the flux as irradiance. 4.2.5 Number of Solar UV Photons Reaching Lower Altitudes

In this section, the number of solar UV photons propagating down to lower altitudes and into a UV sensor pointed directly at the Sun is calculated. Thus, the next step in the evaluation process is to multiply the photon flux density per bin (photons cm2 sec1 nm1) obtained from Hall’s data times the atmospheric transmissions. The spectral region covered in Tables 4.1 and 4.2 is 260–290 nm in 1 nm wide wavelength bins. Figure 4.14a and b shows the number of solar photons that are transmitted down to an altitude of 9.656 km from 41.4 km. Figure 4.14a covers the spectral range from 260 to 270 nm. Figure 4.14b covers the spectral range from 270 to 286 nm. Similar results were also obtained, which show the number of solar photons transmitted down to 6, 4, 2, and 1 km. The results for 6, 4, 2, and 1 km are the

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Table 4.1 The number of photons intercepted per second by a 2 in.

diameter aperture at a slant range distance of 3 miles and an altitude of 2 miles. UV photon 2500 K wavelength blackbody bins (nm) emission; emissivity ¼ 1.0 (ph cm2 s1) 260–261 261–262 262–263 263–264 264–265 265–266 266–267 267–268 268–269 269–270 270–271 271–272 272–273 273–274 274–275 275–276 276–277 277–278 278–279 279–280 280–281 281–282 282–283 283–284 284–285 285–286 286–287 287–288 288–289 289–290

1.04 · 1013 1.12 · 1013 1.20 · 1013 1.28 · 1013 1.37 · 1013 1.46 · 1013 1.56 · 1013 1.67 · 1013 1.78 · 1013 1.90 · 1013 2.04 · 1013 2.17 · 1013 2.31 · 1013 2.46 · 1013 2.62 · 1013 2.87 · 1013 2.96 · 1013 3.14 · 1013 3.34 · 1013 3.54 · 1013 3.76 · 1013 3.99 · 1013 4.22 · 1013 4.48 · 1013 4.74 · 1013 5.02 · 1013 5.31 · 1013 5.61 · 1013 5.93 · 1013 6.27 · 1013

Annulus; 6 in. outer diameter; 3 in. inner diameter (cm2)

Slant range, 3 miles; altitude, 2 miles; transmission

Solid angle, 2 in. diameter Number of aperture, range photons per ¼ 3 miles second

136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8 136.8

3.80 · 104 5.03 · 104 5.49 · 104 7.72 · 104 8.11 · 104 1.05 · 103 1.37 · 103 1.68 · 103 2.24 · 103 2.65 · 103 3.54 · 103 4.44 · 103 5.56 · 103 7.11 · 103 9.32 · 103 1.14 · 102 1.42 · 102 1.73 · 102 2.16 · 102 2.62 · 102 3.21 · 102 3.92 · 102 4.64 · 102 5.24 · 102 6.32 · 102 7.42 · 102 8.32 · 102 9.43 · 102 1.07 · 101 1.19 · 101

8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011 8.70 · 1011

47.03 67.04 78.80 117.60 132.23 182.45 254.36 333.91 474.54 599.20 8.59 · 102 1.15 · 103 1.53 · 103 2.08 · 103 2.91 · 103 3.89 · 103 5.00 · 103 6.48 · 103 8.59 · 103 1.10 · 104 1.86 · 104 9.89 · 104 2.33 · 105 2.79 · 105 3.57 · 105 4.43 · 105 5.26 · 105 6.30 · 105 7.55 · 105 8.00 · 105

These photons have been emitted by a blackbody annulus at a temperature of 2500 K and emissivity ¼ 1. Atmospheric UV absorption has been taken into account. Courtesy of Drs P. J. Schreiber and C. W. Litton.

same as for 9.656 km, which are shown in Figure 4.14a and b. These results will now be discussed in more detail. Inspection of the 280–281 nm spectral bin and beyond, see Figure 4.14a and b, indicates that the number of photons arriving at an altitude of 9.656 km cm2 s1 has begun to increase substantially. Thus, it may be concluded, with the aid of Figure 4.14a and b, that the solar-blind UV region is shorter than the 280–281 nm spectral bin.

4.2 Particulars of Deep UV Radiation and Detection Table 4.2 Number of photons intercepted per second at slant

ranges of 1–5 miles and at corresponding altitudes of 0.5–4 miles.

Number of photons intercepted per second

UV photon wavelength bins (nm)

Slant range, 1 mile; altitude, 0.5 mile

Slant range, 2 mile; altitude, 1 mile

Slant range, 3 mile; altitude, 2 mile

Slant range, 4 mile; altitude, 3 mile

Slant range, 5 mile; altitude, 4 mile

260–270

134.93 · 104

339.24 · 102

599

268

48

270–280 Totals 2 in. aperture Reduction factor Totals 1 in. aperture

55.84 · 105 6.933 · 106

31.495 · 104 3.48 874 · 105

43 479 44 078

9651 9919

2863 2911

0.0 133 875

0.0 133 875

0.0 133 875

0.0 133 875

0.0 133 875

92 820

4670

590

133

39

Annulus temperature ¼ 2500 K and emissivity ¼ 1. Courtesy of Drs P. J. Schreiber and C. W. Litton.

Consequently, if a UV sensor is pointed directly at the Sun, the stratospheric ozone absorption of solar UV radiation makes the Sun dark in the wavelength bins shorter than 280–281 nm. The results obtained for transmission of solar photons from an altitude of 41.4 km down to 6, 4, 2, and 1 km are same as those shown in Figure 4.14a and b. 4.2.6 Atmospheric Detection Range

To develop a suitable solar-blind UV detector array, it is necessary to develop an AlGaN/GaN photodiode UV detector array with a very sharp spectral cutoff longer than 280 nm. The spectral cut-on wavelength of an ideal UV detector array is determined by the spectral energy distribution of hot UV-emitting sources, chosen to be 260 nm here. Because the long wavelength spectral responsivity cutoff of a solar-blind UV photodiode detector array is established above, it would be worthwhile now to examine how sensitive such a UV detector array must be in order for the UV sensor to demonstrate sufficient detection range to meet sensor requirements. Thus, the purpose of this section is to examine the approximate number of photons that arrive at a solar-blind UV sensor after they have been emitted by a hot UV-emitting object. The actual temperatures and temperature profiles of hot UV-emitting objects can be determined experimentally. However, a conservative estimate can be made of the number of UV photons emitted from a hot UV-emitting object based on the use of a

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(a)

ph cm-2 s-1 nm-1

The number of solar photons in 1 nm wide wavelength bins transmitted from 41.4 to 9.656 km at a solar zenith angle of 29.6° .

nanometers 260 261

262

263

264

265

266

267

268

269

271

270

1.68 1.49 2.15 1.76 7.99

4.95 8.58

×

×

10 -25

×

10 -25

1.56 ×

×

2.02 5.46

×

×

×

10 -19

10 -19

×

10 -16

8.38 ×

×

10 -13

10 -16

10-21

10 -22

10-22

10 -27

(b)

ph cm-2 s-1 nm-1

The number of solar photons in 1 nm wide wavelength bins transmitted from 41.4 to 9.656 km at a solar zenith angle of 29.6° .

Nanometers 270 271

272

273

.

5.2 × 2.32 10-11 × 1.68 -12 × 10 10-13

274

275

276

277

278

279

3.23 × 2.89 10 6 × 5 5.92 10 × 5.08 10 4 × 9.5 10 3 × 7.44 10 2 × 1.82 10 1 × 10 0

6.0 280 7.77 × 4.59 × 10 -2 1.25 2.99 × 10 -3 × 10 -4 4.72 × × 10-6 10 -5 1.85 10 -8 × 10 -9

281

282

283

284

285

286

4.2 Particulars of Deep UV Radiation and Detection

blackbody to represent a hot UV-emitting object. From time to time, educational television channels present home construction shows. During the presentation of these shows, visits are made to modern-day factories, which manufacture home construction materials such as tiles, bricks, and bathtubs. Often the furnaces used to make these commercial products are featured. The fires in these furnaces are white hot. Usually the furnace temperature is stated to be 1700–2700 C without reference to the temperature scale being used. However, commercial product manufacturers often use the centigrade temperature scale. Therefore, the temperature range of 1700–2700 C translates into approximately 2000–3000 K. Although the sensor itself would not actually be at these high temperatures, such temperatures in the proximity of the sensor still place stringent demands about the thermal stability of the material used for this application. At this point, it will be assumed that a hot UV-emitting object can be represented by a blackbody radiative surface and that Planck’s radiation law can be used to calculate the number of photons emitted by the blackbody surface in 1 nm wide spectral bins beginning at 260 nm. It will be assumed that the UV photons emitted by a hot UV-emitting object can be represented by a blackbody radiative surface with an emissivity ¼ 1, at surface temperatures of 2000, 2500, and 3000 K. Later on, the emissivity value can be adjusted to represent a typical gray body. Atmospheric UV transmission measured with very simple UV sensor geometries will be considered in the evaluations and have already been tabulated elsewhere [39] for the spectral bins 270–280 nm. The atmospheric transmission values for 260–270 nm are not shown here but have also been tabulated elsewhere [40]. Essentially, the UV transmission numbers of the atmosphere were obtained using Plexus and are already described at slant ranging from 1 to 5 miles for the spectral bins of 260–290 nm. The solid angles subtended by a 2 in. diameter UV sensor optical aperture with a detection range of 1–5 miles with respect to the hot UV-emitting object have also been previously reported [39]. One of the numerical evaluations described above will be presented here as representative of the many evaluations performed. To simplify these numerical evaluations, it is imagined that the UV sensor is looking directly at the hot UVemitting object, or very close to a near-zero aspect angle. Imagine that the UV sensor focal plane array detector has an individual pixel field of view (FOV) large enough to encompass the entire hot UV-emitting object at a detection range of 1–5 miles. It is

3 Figure 4.14 (a) The number of solar photons (cm2 s1 nm1) that have been transmitted from an altitude of 41.4 km down to an altitude of 9.656 km at a solar zenith angle of 29.6 in 1 nm wide wavelength bins. This evaluation is based upon the use of the Plexus computer program. The spectral region covered in this figure is 260–270 nm. Courtesy of Drs P. J. Schreiber and C. W. Litton. (b) This diagram shows the number of solar photons

(cm2 s1 nm1) that have been transmitted from an altitude of 41.4 km down to an altitude of 9.656 km (6 miles/31 680 ft) along a pathway at the solar zenith angle of 29.6 in 1 nm wide wavelength bins. This evaluation is based on the use of the Plexus computer program. Notice the paucity of photons shortward of 280 nm. This break defines the long wavelength end of the solar-blind region. Courtesy of Drs P. J. Schreiber and C. W. Litton.

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assumed that the UV sensor’s optical aperture is 2 in. in diameter and has a 2 in. focal length. For the cases considered here, the blackbody surface representing the hot UV-emitting object is taken to be a flat circular annulus with an outer diameter of 6 in. and an inner diameter of 3 in. The annulus area is then 136.811 cm2. Under these very simplified conditions, the annulus looks much like a point source to the UV sensor. The numerical calculation presented here was carried out as follows. The photon emissions per unit area were multiplied by the area of the annulus, by the corresponding UV atmospheric transmission, and by the solid angle for each detection range. This evaluation was performed in each spectral bin from 260 to 290 nm. By staying shorter than 280 nm, the results remain in the solar-blind UV region. The results of the calculations are then tabulated and discussed below. Table 4.1 presents the results of our evaluations for a blackbody annulus at a surface temperature of 2500 K, emissivity equal to 1, and a slant range of 3 miles. Atmospheric transmission along the slant range is taken into account in the evaluations. Complete emission tables have been made for annulus temperatures of 2000, 2500, and 3000 K as well as atmospheric transmission tables for the 260–280 nm spectral region along with the corresponding tables that show the number of photons available at detection range of 1–5 miles [40]. Some conclusions can be drawn from the tabulated results described above. If a small but hot UV-emitting object is represented by a flat blackbody annulus at a temperature of 2000, 2500, or 3000 K, a detector would require close to single photon sensitivity for any detection range from 1 to 5 miles in the solar-blind UV spectral region to be of much use. This is especially true when unavoidable photon losses are taken into account as described below. There will be further inevitable and unavoidable losses of photons due to relative misalignment of the sensor aperture and the hot UV-emitting object, optical filter losses, and other factors not considered yet in arriving at the numbers shown in Table 4.1. These inevitable and unavoidable losses of photons are discussed in Section 4.2.7. These photon losses further reinforce the need that a UV sensor AlGaN/GaN photodiode detector array have almost single photon sensitivity to have an appreciable detection range especially in comparison to Gen II type UV sensors. 4.2.7 Inevitable and Unavoidable Losses of Photons

The numbers shown in Table 4.1 are based on ideal detection conditions. However, it is recognized there will be unavoidable losses of photons not yet considered. The temperature profiles of hot UV-emitting objects can be measured and these UV-emitting objects will be gray bodies rather than ideal blackbodies. In addition, atmospheric UV transmission numbers may be much less than those used in the calculations to obtain Table 4.1. Consideration should be given to the effects of weather conditions on the atmospheric transmission of UV photons especially in the solar-blind spectral region.

4.2 Particulars of Deep UV Radiation and Detection

In these calculations, photons absorbed by ground-hugging ozone were ignored. Under certain conditions, ground-hugging ozone can absorb almost every UV photon emitted by a hot UV-emitting object. In Addition, UV photons lost to atmospheric haze were not taken into account. Even though a spherical arrangement of UV sensors could be imagined, there would be unavoidable losses of UV photons due to relative misalignments between the hot UV-emitting object and the spherically arranged UV sensors. However, the blackbody model considered here comes very close to physical reality as the hot UV-emitting object and UV sensor separation distance decreases. The computer program Plexus can be used to evaluate the atmospheric UV transmission under various weather conditions and groundhugging ozone layers. One very important factor we have ignored is the use of a filter in the optical system of the UV sensor to limit both the longward and shortward solar-blind band edges. Filters may very well be required to accomplish spectral cutoffs to meet specialized requirements. These same filters will also reduce the number of signal photons reaching the UV sensor. All of these factors should be taken into account to more accurately determine the number of UV photons that reach the photodiode array detectors per second. The purpose of this section is to evaluate more realistic counts of photons intercepted per second. Thus, an effort will be made below to consider the various factors that contribute to the reduction of photons: at emission, during atmospheric transmission and during interception and detection in the UV sensor. An overall photon reduction factor will be obtained by multiplying these factors together. This overall reduction factor will be applied to the photon numbers shown in Table 4.1. This table shows the number of photons intercepted by a 2 in. aperture of a UV sensor located at a slant range of 3 miles distance away from the blackbody annulus and emitted within the solar-blind spectral range. Specific photon reduction factors will now be discussed. Blackbody Emissivity Hot UV-emitting objects are most likely to be gray bodies and not blackbodies. Thus, a gray body emissivity equal to 0.6 would be an excellent initial value to represent a hot UV-emitting object. UV Sensor Optical Aperture From a practical point of view, UV sensors should be as small as possible and thus as low cost as possible. A 1 in. diameter f/2 UV sensor optical aperture would help to ensure the size of future UV sensors is quite small. However, the solid angle presented by a 1 in. optical aperture is one-fourth the value of the solid angle presented by a 2 in. aperture at the same range and orientation. Filter Loss To assure that the UV sensor operates exclusively within the solar-blind spectral region and meets specific UV sensor requirements, a filter set may be required. It is assumed that the UV sensor filter set transmissivity is within 25% of the solar-blind spectral region. The UV sensor filter set’s out-of-band rejection ratio will be assumed to be 104.

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Weather, Haze, and Ozone Local atmospheric UV spectral transmission is likely to be different than used previously in this chapter. Local weather, haze, and ozone can each easily account for 20, 20, and 20% of the photon losses, respectively. Misalignment of UV Sensor and Hot UV-Emitting Object The UV sensor and the hot UVemitting objects will likely pitch and yaw with respect to each other resulting in a repetition of on–off signals at the UV sensor. Thus, cosine losses due to misalignment of the UV sensor and the hot UV-emitting object could be quite large. Here, we will assume cosine losses to be 30% or larger. Thus, we will assume only 70% of the photons emitted within the solid angle defined by the UV sensor and the hot UV-emitting object will reach the UV sensor focal plane array of photodiodes. Total Photon Losses Some of the more important factors have been described above that contribute to the loss of photons in their emission, transmission, and detection beginning at the hot UV-emitting object and ending at the UV sensor focal plane array. This has the overall effect of reducing the range of detection by the UV optical filters, and a host of other factors not considered yet in arriving at the numbers are shown in Table 4.2. In summary, the above-described reduction factors are . . . . .

Gray body emittance ¼ 0.6. 1 in. aperture diameter of UV sensor instead of 2 in. aperture ¼ 0.25. Solar-blind filter reduction ¼ 0.25. Weather, haze, and ozone ¼ 0.8  0.8  0.8 ¼ 0.51. Misalignment ¼ 0.7.

The overall reduction factor is the product of all the aforementioned factors: 0.6  0.25  0.25  0.51  0.7 ¼ 0.013 387 5. This overall reduction factor will be applied to the number of UV solar-blind photons reaching the 3 mile range and the other ranges of 1–5 miles in Section 4.2.8. 4.2.8 Practical UV Sensor Detection Ranges

Practical UV detection ranges will be evaluated in this section by applying the reduction factor obtained in the previous section to all the ranges considered from 1 to 5 miles. The first step will be to calculate the number of 260–280 nm photons intercepted per second by a 2 in. sensor aperture at the blackbody source temperatures and 1–5 mile ranges. The next step will be to apply the overall reduction factor equal to 0.013 387 5. The results for a hot UV object at a temperature of 2500 K for the five different aforementioned ranges are compiled in Table 4.2 shown below. This table represents the number of photons most likely to be intercepted by a UV sensor at the ranges indicated. Thus, it can be seen now how sensitive a UV sensor must be to be able to detect photons emitted by a small, hot UV-emitting source.

4.2 Particulars of Deep UV Radiation and Detection

The overall reduction factor that has been applied to Table 4.1 (2500 K and 2 mile range) has also been applied to the tables not shown here for the temperature of 2500 K and the range from 1 to 5 miles. These results are tabulated and shown in Table 4.2. Not shown here are similar results for hot UV-emitting objects at temperatures of 2000 and 3000 K. Table 4.2 clearly shows how sensitive a UV sensor must be to have sufficient UV solar-blind photon detection range to be considered for most practical applications. 4.2.9 Available UV Sensors

Prior to the advent of nitride semiconductors, other semiconductors such as conventional compound semiconductor and Si-based detectors were used for the UV region. The most notable drawback of silicon and other small-bandgap semiconductor-based photodetectors is that they cannot be directly exposed to the incident radiation. First, these detectors are sensitive to visible light and, depending on their bandgap, to a portion of the IR spectrum as well. Moreover, the quantum efficiency of a semiconductor-based photodetector is at its maximum for photon energies just above the bandgap. When the photon energies are much larger than the bandgap (which is the case for small-bandgap semiconductors used for UV light detection), part of the light energy is lost as heat in the semiconductor lattice where phonon modes are excited. UV detection with small-bandgap semiconductors requires highpass filters. Sometimes, a phosphor-based filter is used, which absorbs the UV light and reemits longer wavelengths closer to the edge of bandgap of the semiconductor [41]. However, this makes the detection system complicated and expensive. In addition, aging of the filters is a concern. The 6H-silicon carbide with a bandgap of 2.86 eV at room temperature paved the way for semiconductor-based detectors to operate in the UV portion of the spectrum. Commercial solar-blind flame detectors based on SiC in the UV C range are already available. However, insertion of high-pass optical filters is still necessary, and the indirect bandgap of SiC produces low quantum efficiency. Fortunately, the direct bandgap of AlxGa1xN increases with the Al mole fraction x and ranges from 3.42 eV (l ¼ 362 nm) (x ¼ 0) to almost 6.2 eV (l ¼ 200 nm). This spectral range notably includes the UV B and UV C cutoffs, which correspond to 3.87 eV (or l ¼ 320 nm) and 4.43 eV (or l ¼ 280 nm), respectively. Therefore, filter-free UV photodetectors are possible and would considerably simplify the UV photodetection systems. For the solar-blind and nearly solar-blind high-detectivity detectors, three types of UV sensors have been developed over the past two decades. The first type of UV sensor combines a PMT, a filter, an optical aperture, a high-voltage power supply, and a signal processor. The signal processor has evolved into a preamplifier, an amplifier, a signal conditioner, a computer, and a detection algorithm. This type of UV sensor is referred to as a Generation I (Gen I) type of UV sensor. The use of a fish-eye-type optical lens gives the Gen I sensor the ability to “stare” and cover a wide field of view. An optical filter set is required to make this UV sensor solar blind.

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Electrons UV photons

UV photons

Visible photons

Electrons

Photocathode Filter/ optics

Visible photons

CCD focal plane array

Phosphor screen

Microchannel plate electron multiplier

Tapered fiber optic coupler

Figure 4.15 Component of a Gen II UV sensor. Courtesy of BAE systems.

Many of these Gen I type UV sensors have been manufactured and used for UV applications. But only Gen II and Gen III type UV sensors are discussed here to some extent. Huffman [42] has previously outlined the concept of an imaging type of UV sensor, referred to in this chapter as a Generation II (Gen II) type of UV sensor, a diagram of which is shown in Figure 4.15 [43]. As sketched by Huffman, a Gen II UV sensor consists of an optical aperture, a photocathode, a microchannel plate (MCP), a phosphor, a CCD, and a video display. Like Gen I type UV sensors, a signal processor, a computer, and a detection algorithm can be used in a Gen II type system to give the system flexibility in detecting a wide variety of hot UV-emitting objects. A UV filter set would have to be added to the Gen II system to make it solar blind. Brown et al. [13, 44] have discussed the development of an imaging-type AlGaN/ GaN photodiode array and described combining the array with a UV-transmitting lens. In this book, the combination of an AlGaN/GaN array with a UV lens, a readout integrated circuit (ROIC), a computer, and a display is referred to as Generation III or Gen III type UV sensor. A Gen III type UV sensor is shown in Figure 4.16. Gen I, Gen II, and Gen III type UV systems are compared in Table 4.3, which shows the advantages and disadvantages of these three generations of UV sensors. Table 4.4 shows a comparison of the performance of Gen II and Gen III type UV sensors. This table is important for the future design and fabrication of Gen III type UV sensors as the third column clearly shows the performance enhancement factors that Gen III type UV sensors can attain over Gen II type UV sensors. There is another UVdetector structure under development, which promises close to single photon detection sensitivity for Gen III type UV sensors. This type of detector structure is known as the APD operating in the Geiger mode. GaN APDs operating in the Geiger mode with single photoelectron gains have been demonstrated, and AlGaN

4.2 Particulars of Deep UV Radiation and Detection

• Ohmic contacts

Generation III

• Good high-quality i-layer for better absorption, QE

• Low trap p-doping

Interface electronics

AlGaN device structure

128 × 128 pixel data

Photons Photons

ROIC Optics Transparent large area lattice-matched substrate to reduce dislocation density (106 makes this device worthy for advanced development to determine its experimental capability and limitations. This APD offers to be solar blind with single photon counting capability giving Gen III type UV sensors the potential for maximum possible detection range. In conclusion, this section when coupled with the prior nine sections of this book outlines in detail the performance characteristics that both Gen II and Gen III type UV sensors must attain in order to be considered for many UV applications. 4.2.10 Design Requirements for UV Solar-Blind Imaging Detectors

The object of the work in the preceding sections was to determine the design objectives to follow to develop a Gen III AlGaN/GaN UV photodetector array that performs as well as a Gen II UV type sensor. In order for a UV sensor to be solar blind, as shown in the preceding section, the spectral wavelength cutoff of a UV photodetector should be shorter than 280 nm. The number of photons available in the spectral bins from 260 to 280 nm from an ideal blackbody at the temperatures considered in this chapter is quite limited. Therefore, the spectral responsivity of a solar-blind UV detector should have close to unity response from 260 to 280 nm and a drop-off of four orders of magnitude or more between 280 and 290 nm and between 250 and 260 nm. The solar-blind UV sensor detector should have zero responsivity longer than 290 nm and shorter than 250 nm. Gen II sensor detectors use filter sets to achieve this objective. Gen III photodiode arrays will have to be tailored to achieve this stringent requirement.

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Table 4.3 Detector type, period of development, period of use, and

advantages and disadvantages of Gen I, Gen II, and Gen III sensors. Gen I

Gen II

Gen III

Detector type

PMT point detector

GaN/AlGaN pixel array

Period of development Period of use Advantages

1980–2002

MCP PMT imaging detector 1995–present Prototype in testing Imaging High sensitivity Photon counting Moderate FOV

Under development Imaging High sensitivity Photon counting Large field of view

Fast speed of response Large dynamic range Higher cost than Gen I

Fast speed of response

1985–present Point detection High sensitivity Photon counting Fast speed of response Large dynamic range Low noise

Disadvantages

PMT ¼photomultiplier tube Large size and weight

MCP ¼ microchannel plate Large size and weight

High-voltage operation High power dissipation Costly filter/optics High maintenance Nonimaging Quadrant field of view

High-voltage operation High power dissipation Costly filter/optics High maintenance

1996 and —

Very small size and weight Low maintenance Low-power operation Low power dissipation Fewer optical components Device material reduces optics/filter requirements Lower cost than Gen I/Gen II Still to be developed, but faces the typical material problems such as point defects and complexities involving p-type doping

Courtesy of Drs P. J. Schreiber and C. W. Litton.

Based on the number of photons intercepted at the range of 1–5 miles, shown in Table 4.2, the UV Gen III solar-blind photodetector should have the capability to detect a few photons per second. This is comparable to sensitivity achieved by Gen II UV sensors of a few photons per second beyond a 5-mile range. In Gen II type UV sensors, the photocathodes, microchannel plates, and phosphors are very fast, but the time response of the CCD is approximately 1 kHz. Thus,

4.2 Particulars of Deep UV Radiation and Detection Table 4.4 Performance comparison of Gen II and Gen III UV

sensors, and Gen III enhancement factors. Type

Imaging point PMT (includes phosphors, MCP and CCD) GaN/AlGaN pixel array

Si ROIC

260–285 nm Spectral region Edge rejection 106

260–285 nm

260–285 nm

106

Responsivity

Moderate

Potentially high

Quantum efficiency

Low to moderate (photocath- Needs improvement, Z  0.75 ode þ MCP þ phosphor þ CCD) (0.3 · 0.10 · 0.30 · 0.95) Required 106 in MCP

Sharper cut-on and cutoff Higher resp. required Higher QE required

Gain

Go to APD/ Geiger mode Higher sensitivity required

Sensitivity

Individual photon counting capability

Dynamic range Electrical bandwidth Bias

106

High sensitivity not demonstrated yet (104 ph pixel1), NEPth  1018 W, NEPdemo  1015 W 103

1 kHz

1 kHz

—

0 V Geiger 200 V

Geiger  200 V

103

2 noise e APD or Geiger mode APD Improved fill factor Improved detector active area Improved wide-field optics Improved wide-field optics

1–2 kV (photocathode þ MCP 9 kV total) Readout noise 106 Detector size

100 · 100 mm2

30 · 30 to 50 · 50 mm2

Individual FOV

>1 mrad

0.6 mrad

Total FOV

>30 · 30

>90 · 90

Number of elements

525

735

103

Courtesy of Drs P. J. Schreiber and C. W. Litton.

the CCD response limits the Gen II UV sensor time response. The solar-blind UV AlGaN/GaN photodetector array should have as short a time response as physically possible to follow the microsecond time changes of hot UV-emitting objects. Fast response would be a valuable tool in a target recognition algorithm embedded in the UV sensor’s computer software program. Though for practical purpose, the speed of a Gen III type UV sensor photodiode array must be only as fast as a CCD.

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It would be very helpful to have a spatial detection capability in the Gen III UV sensor. If optical systems with a 2 in aperture and a 2 in focal length were used, a detector of size 100  100 mm2 would give each detector in the focal plane a field of view of approximately 10  10 ft2 at 1 mile. Accordingly, at 2 miles, each individual detector would cover a field of view approximately 20  20 and 30  30 ft2 field of view at 3 miles detection range. As indicated above, a 100  100 mm2 detector size in a f/1 optical system of 2 in. diameter and 2 in. focal length was chosen for consideration. The solid angle covered by one UV sensor would cover roughly 2.094 sr2. Under these conditions, the required array size would be approximately 735  735 pixels2. Thus, the total array size would be approximately 7.35  7.35 cm2. A smaller detector size would require more individual detectors in the array. Table 4.2 shows the number of photons in the spectral bins from 260 to 280 nm, at detection ranges from 1 to 5 miles. However, as the UV sensor approaches the hot UVemitting object, the number of UV photons impinging on the UV sensor increases about six orders of magnitude. This suggests the dynamic range of the Gen III photodetectors should be at least six orders of magnitude. The technical objectives required to develop an AlGaN/GaN photodetector array to meet or exceed Gen II performance are summarized below: Design requirements Spectral response Spectral responsivity Reduction factor Spectral responsivity Beyond 290 nm Beyond 250 nm Photon sensitivity Electrical frequency response Detector size Array number Array dimension Dynamic range

100% shortward of 280 nm and longward of 260 nm. 104 longward of 280 nm Zero Zero 3 ph s1 in the 260–280 nm spectral region >103 100  100 mm2 735  735 for a field of regard >100 deg2 7.35  7.35 cm2 >106

The analysis shown in this chapter has developed stringent design requirements that should be met for a Gen III AlGaN/GaN photodetector array to achieve comparable performance to that of a Gen II type UV sensor.

4.3 Si and SiC-Based UV Photodetectors

The most common commercial solutions for UV photodetection are based on Si or SiC. Some less common applications use compound semiconductors such as GaAs,

4.3 Si and SiC-Based UV Photodetectors

GaP, and GaAsP. These materials are mainly used to fabricate Schottky barrier photodiodes. This special range of devices notably fits very well to UV detection applications, because they offer a remarkable operating stability. The recent emergence of diamond-based photodetector technologies has enabled the creation of short cutoff wavelength photoconductors (lC ¼ 225 nm) with a UV/visible contrast of about six orders of magnitude [46, 47]. Such devices fit very well to UV photodetector applications and high-energy particle detection because of the strong radiation hardness of diamond. However, the development of this special kind of photodetector is hindered by the technical difficulty in getting high-quality monocrystalline epitaxial films. The final material is, in the best cases, a polycrystal whose adjacent grains are significantly misoriented. This results in a large density of electrically and optically active defects that significantly hinder the operation of the photoconductors in terms of responsivity and UV/visible contrast. Despite this, diamond-based UV photodetectors are now commercially available, which is the clear indicator of the growing interest in these devices for UV photodetection [48]. 4.3.1 Silicon-Based UV Photodetectors

The UV-enhanced silicon-based p–n-junction photodiodes are commonly available and are applicable to a very wide spectrum ranging from the near-UV to the VUV, and even to the threshold of the soft X-rays. This together with Si fabrication technology makes UV-enhanced Si photodiodes very inexpensive. However, the spectral response is not uniform and reaches the infrared region, as can be expected from the 1.1 eV at room temperature. To narrow the spectral response to the intended range only (the UV in this case), an absorbent filter that passes only the desired spectral range is placed in front of the detector. Another approach is to take advantage of the large absorption coefficient at UV wavelengths and employ a very thin Si layer representing the only absorber layer in the structure. As in the case of any semiconductor-based detector, two families of UV-enhanced Si detectors (p–n-junction photodiodes and metal-oxide-semiconductor photodiodes) are available [49, 50]. P–n-junction Si UV photodetectors include shallow p–n-junction photodiodes, where the p–n-junction is typically located 0.2 mm below the surface and coated with a thin SiO2 surface layer. This insulating oxide surface layer plays the double role of a surface passivation layer and an antireflection coating. Junction depth is a very important parameter because light absorption is bound increasingly near the surface as photon energy increases. The goal is to optimize the photocurrent by increasing the number of photogenerated carriers near the junction before recombination. Moreover, the reduction of surface recombination is important for higher efficiency, which requires that special care be taken when the wafer surface is prepared. In particular, optimization of the Si/SiO2 interface properties is pivotal to lowering the surface trap density, which includes production of an internal electric field on the silicon surface to sweep the carriers and subsequently reducing carrier recombination at the surface.

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P–n-junction Si-based UV photodiodes of the n- on p-type with nearly perfect internal quantum efficiencies have been reported [51, 52]. The devices were fabricated by diffusing phosphorus (P) into a dislocation-free Si(p) substrate followed by the deposition of a 60 nm thick SiO2 layer. This thickness is good for the spectral window of 350–600 nm and photodiode has 100% internal quantum efficiency. For wavelengths shorter than 350 nm, the internal quantum efficiency is larger than unity due to secondary impact ionization phenomena. When the wavelength is larger than 600 nm, the quantum efficiency drops below unity, because the light is absorbed far deeper than the p–n-junction. The photogenerated electron–hole pairs cannot diffuse to the junction because their diffusion lengths are too short and simply recombine in the bulk of the semiconductor, not contributing to the photocurrent. Turning our attention to the Si-based p–n-junction photodiodes designed for the VUV (up to 124 eV), the basic design of the p–n-junction remains nearly identical. However, the surface oxide layer is much thinner, about 4.5 nm instead of 60 nm. Such a thin oxide layer is transparent to all short wavelength radiation, eliminating the possibility for the UV light and humidity to generate traps in the oxide. In these structures, the electron–hole pair photogeneration paves the way for carriers with a large kinetic energy to multiply by secondary impact ionization. In the high photon energy range, the internal quantum efficiency h of the VUV Si photodiodes typically increases linearly with photon energy E, as h ¼ E/3.63 eV [50]. The typical value of the internal quantum efficiency for a VUV Si photodiode is about 30 for photon energy of 124 eV. These photodiodes also employ a bandpass filter in the 10–50 nm spectral range, which is produced by thin metallic layers such as Al/C, Al/C/Sc, Ti, Sn, and Ag [50]. Si-based charge inversion-type photodiodes [41, 50] are in some ways similar to the metal-oxide-semiconductor (MSM) structures and utilize strong electric fields at the Si/SiO2 interface. A strong electric field is established close to the surface precisely where the high-energy photons are absorbed. Doing so maximizes the internal quantum efficiency, which is enhanced further by the elimination of any dead zone recombination region near the surface layer, a problem that plagued the earlygeneration Si UV junction photodiodes. Charge inversion UV photodetectors exhibit high internal quantum efficiencies in the 250–500 nm spectral range. The 120 nm cutoff wavelength is mainly due to the high-energy photon absorption by the surface oxide layer. However, the performance of these photodetectors degrades in time under UV illumination, as the UV light tends to degrade the properties of the oxide layer. Moreover, charge inversion UV photodetectors have smaller linear range than p–n-junction photodiodes because of the high electrical resistance of the twodimensional charge layer at the SiO2/Si interface. 4.3.2 SiC-Based UV Photodetectors

Wide-bandgap semiconductors are desirable for UV optical detection applications because of their insensitivity to longer wavelengths and their very small dark currents, even at elevated temperatures [53]. Low dark current enables SiC photodiodes to

4.3 Si and SiC-Based UV Photodetectors

exceed Si UV detector sensitivity by four orders of magnitude [54]. Many of the applications for UV detection involve hostile environments such as in-situ combustion monitoring and satellite-based missile plume detection, where the ruggedness of SiC is an important advantage. Other applications capitalize on the sensitivity of widebandgap semiconductor detectors, such as air quality monitoring, gas sensing, and personal UVexposure dosimetry. The advent of AlGaN-based UV detectors limited the activity in SiC detector development to the period when the GaN-based technology was getting off the ground. In the early stages, before the advances in the GaN technology, detectors based on 6H-SiC p–n-junction photodiodes exhibited relatively better performance [50], albeit short-lived. The optimized n-on-p structures employ a 0.2–0.3 mm thick n þ - type SiC layer grown on top of a p-type SiC substrate. The doping levels are about 5  1018 cm3 for the n þ -type SiC layer and 5–8  1017 cm3 for the p-type SiC substrate, respectively. As in the case of Si detectors, the surface is passivated by a SiO2 layer. A nickel layer forms the ohmic contact for the n þ side. Figure 4.17 shows a typical SiC UV photodiode device structure fabricated from commercial 6H substrates on which a p–n-junction was grown epitaxially [55]. Photodiodes reported by Edmond et al. [56] have demonstrated extremely small dark currents, as little as 1011 A at 1 V and 200 C (Figure 4.18), representing a considerable improvement over previous efforts. The same devices exhibited nearunity peak responsivities between 268 and 299 nm at temperatures as high as 350 C (Figure 4.19). The typical responsivity values are about 0.15–0.175 A W1 at 270 nm, which corresponds to 70–85% internal quantum efficiency. The spectral response displays peak responsivity in the 268–299 nm range. However, as is apparent in Figure 4.19, the responsivity falls off quickly with decreasing wavelength. Brown et al. [55] have attempted to push the operation of UV SiC photodiodes to still shorter wavelengths. As the detection wavelength is decreased, a larger proportion of the absorption occurs near the semiconductor surface and surface recombination becomes more important. By thinning the top n þ layer outside of the mesa contact, a responsivity of 0.05 A W1 was achieved at 200 nm. Further improvement might be realized by adding a SiO2 cap layer, which is transparent down to nearly 125 nm, to serve as a UV window while simultaneously passivating the SiC surface.

Contact 0.2 µm 1-5 µm

n+-SiC p-SiC

p-SiC substrate

Figure 4.17 SiC photodiode device cross section [55].

0.1 µm

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Current density (A cm-2)

10 -7

10 -8

10 -9 473 K 523 K 573 K 623 K

10 -10

10 -11

0

2

4

6

8

10

12

Reverse voltage (V) Figure 4.18 SiC photodiode dark current density as a function of temperature and reverse bias [56].

6H-SiC-based Schottky photodiodes that operate in the 200–400 nm spectral range are also available [50]. The best performances are obtained for p-type SiC-based Schottky devices because the Schottky barrier height on this material is twice that on the n-type SiC. Furthermore, the diffusion length is much larger for electrons in p-type material than for holes in n-type material. Current leakage is very low for Schottky barriers formed on p-type SiC. Leakage currents less than 1012 A are common reverse-bias voltages of 10 V. In spite of the argument favoring p-type SiC, Anikin et al. [57] experimented high-quality Schottky junctions on n-type SiC reported surprising results. The gold Schottky contact gave rise to a Schottky barrier height in the 1.4–1.63 eV range. These devices exhibit a very low leakage current, on the order of 1010 A for reverse-bias voltages in the range of 100 to 170 V, and a high responsivity of 0.15 A W1 at 215 nm [50]. Gas sensing is also of interest regarding SiC photodetectors. Typical MOS Si-based sensor performance is temperature limited, which severely restricts the range of

Responsivity (mA W-1)

250

90%

223 K 300 K 498 K 623 K

200 150 100 50

QE = 10 %

0 200

250

300

350

400

450

Wavelength (nm) Figure 4.19 Temperature-dependent responsivity of a SiC photodiode [56].

4.4 Nitride-Based Detectors

detectable chemical reactions [58]. SiC with its large bandgap and reasonably good oxide can perform the same type of functions but at much higher temperatures. For example, hydrogen sensors are fabricated from MOS-based structures with Pd gates. In a typical process, hydrogen molecules or compounds are dissociated in a chemical reaction and rapidly diffuse through the Pd and oxide causing a detectable shift in the flat band voltage, or at a fixed voltage, a change in the device capacitance. As such, this scheme is limited to reactions occurring at and below 280 C (the limit for Si-based device operation). Important processes, such as the dissociation of hydrocarbons on catalytic metals, typically occur at 500 C or higher [59]. By using SiC MOS devices with Pt gates, hydrogen at 800 C and methane at 700 C at the ppm level, have been detected.

4.4 Nitride-Based Detectors

As mentioned previously, the recent surge of interest in nitride semiconductors is due in part to unique nitride-based detectors that cover the wavelength band much shorter than 400 nm. The Sun’s radiation in the range of 280–300 nm is absorbed by the ozone layer leading to a dark background as shown in Figure 4.13. Because the UV photons of interest for detection are very few in number, it is imperative that any UV or, for that matter, any solar-blind detector has a very sharp cutoff on the long-wavelength side. This is typically accomplished by using the absorption edge of the semiconductor to absorb the photons of interest. For a sharp cutoff, the absorption edge must be sharp, which requires high-quality material free of band-tailing states. The term “UV-to-visible contrast” is used to describe how sharp the detector cuts off long wavelengths. The spectral response measurements are typically obtained by a white light source (Xe lamp) with a strong UV emission coupled to a monochromator for selecting the wavelength incident on the detector, as shown in Figure 4.20. The photocurrent of the device is measured as a function of the wavelength, which is normalized with respect to the optical power P(l) of the lamp DC Bias source

Computer

Monochromator

λ

Probe station

Chopper

UV-fiber

DUT

Xenon lamp ω

Lock in amplifier

Figure 4.20 A schematic diagram showing the spectral response measurements of a UV detector.

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determined with a precisely calibrated detector, often an Si-based one. To increase the sensitivity of the measurement and eliminate the DC parasitic light coming from the environment, mode-locking techniques are employed. Sharpness of the cutoff depends on the sharpness of the absorption edge of the semiconductor. In this context, direct bandgap semiconductors lead to sharper cutoffs because of sharper absorption edges compared to indirect varieties. In the case of UV photodetectors, the UV–visible contrast is defined as Rlmax ðl ¼ lG Þ=Rlðl ¼ 400 nmÞ;

ð4:109Þ

where Rlmax is the maximum value of the responsivity at the gap wavelength lG and Rl(l ¼ 400 nm) is the responsivity value at the edge of the visible spectrum [6]. The UV–visible contrast is primarily affected by the presence of electrically and optically active levels deep in the gap of the semiconductor. These deep levels are caused by point defects in the semiconductor, some of which may have their genesis in extended defects. They can also be due to deep impurities and states caused by surface imperfections. Fabrication is important in that photoemission by the metal Schottky in front-side-illuminated detectors with a semitransparent contact can affect this ratio. The III-nitride semiconductor family represents a great opportunity and platform to attain UV detectors with desired characteristics. The extent of success critically depends on the quality of the AlGaN with a large molar fraction of AlN, particularly p-type for reduced resistance. The potential of nitride semiconductors was recognized very early in the studies of polycrystalline gallium nitride (GaN) in the 1970s for detector applications. The development of the epitaxial techniques since then has led to higher quality material and heterostructures with both n- and p-type doping, paving the way for the development of competitive detectors and imagers. This is in spite of the lack of lattice-matched substrates, let alone native ones. Numerous studies concerning GaN and AlxGa1xN UV detectors have appeared in the literature. The detector structures reported include PCs, MSMs, and photovoltaics (PVs, including Schottkys, p–n-junctions, and p-i-n’s). These detector structures were typically grown on sapphire substrates by variations of MBE and organometallic vapor-phase epitaxy (OMVPE) methods. Improvements in III-nitride materials growth and detector processing have resulted in progressively higher responsivity PV and PC ultraviolet detectors in the last few years [50, 60–77]. In the past two years, AlxGa1xN/GaN-based detector response times have decreased dramatically from the milliseconds to the nanoseconds regime. To illustrate the range of wavelengths that can be detected by the alloy AlGaN including the end binary compounds [61, 78], we present Figure 4.21. Note that to detect wavelengths of 280 nm or shorter, an AlN mole fraction of at least 50% must be employed. 4.4.1 Photoconductive Detectors

The photoconductive detectors are essentially radiation-sensitive resistors. They respond to radiation by absorbing a photon of energy E ¼ hn, which must be greater

4.4 Nitride-Based Detectors

1

AlxGa1-xN

Photoresponse (a.u.)

300 K

x=0.0 (GaN) x=0.21

x=0.64 x=0.75

0.1

x=0.47

x=1.0 (AlN)

200

x=0.34

250

300 Wavelength (nm)

350

400

Figure 4.21 Room-temperature spectral response of AlGaN, including the end binaries, UV photoconductive detectors showing a tunable cutoff wavelength from 200 to 365 nm. Courtesy of M. Razeghi [62, 78].

than the bandgap energy Eg of the active region, and yielding an electron–hole pair. Production of this type of device in (e.g., AlGaN) consists of a 1 mm thick AlGaN (Si) epitaxial layer with two ohmic contacts. A DC bias is applied to the photoconductor, which is connected in series with a small resistance. The induced photocurrent is simply obtained from the voltage drop in the load resistance when the photoconductor is illuminated. Under an externally applied electric field, the photogenerated electrons and holes are separated from each other and cause a change in the electrical conductivity of the semiconductor, which is measured in an external circuit, see Figure 4.3. The signal is detected by a change in the voltage developed across the sample. In a series of experiments, the photoconductive behavior of some of the III–V nitrides has been investigated in some detail. In one of these experiments, visibleblind UV photodetectors were fabricated using single-crystal AlxGa1xN with varying AlN mole fractions [79]. The active region of the photodetector was a 0.8 mm thick insulating GaN layer deposited on a 0.1 mm thick AlN buffer layer. Sapphire was used as the substrate for the structure and Au, about 500 nm thick, was used as the contact metal to form an interdigitized contact on the epilayer. Using a standard photolithographic procedure and a liftoff technique, the interdigitized electrode (about 3 mm wide, 1 mm long, and having 10 mm spacing) was formed on the epilayer. The purpose was to ease the measurements of photoconductive response. When the spectral responsivity Rl as a function of wavelength l of the photodetectors was measured, the peak responsivity obtained was about 100 A W1 at about 365 mm. The variation of detector responsivity Rl as a function of current I under 254 nm excitation is essentially linear over five orders of incident power radiation. In another experiment [80], a GaN film of about 2 mm thick, grown by OMVPE on a sapphire substrate via an AlN buffer layer, was used to measure photodetective nature of GaN photodetectors. The spectral steady-state responsivity of the detector was measured under an electric field of 3 kV cm1 across the GaN region. For photon energies higher than the GaN energy bandgap Eg, the spectral responsivity of the

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10 7

λ= 325 nm

Gain

10 6

Gain

3

10

λ = 488 nm λ = 514 nm

2

10

1

10 5

10

10 4

10

0

T = 300 K 10

10 3

0

10

1

Optical

10

2

10

3

power (W m-2)

10 2 T = 120 K; k= 0.90

10 1

T = 300 K; k= 0.88 T = 370 K; k= 0.85

10 0

T = 460 K; k= 0.80

10 -2

10 -1

10 0

10 1

Optical power (W

10 2

10 3

m-2)

Figure 4.22 Gain variation of GaN photoconductors as a function of the incident light power, for different values of the device temperature [82].

detector was rather flat with a value of about 4  103 A W1. Band-edge response dominated by excitonic effect was purportedly responsible for the high value of the spectral responsivity, which exhibited an abrupt exponential cutoff at energy E < Eg. As a result, at 0.5 mm, the responsivity died down to 4  103 A W1, demonstrating an impressive wavelength selectivity of the detector for E > Eg and E < Eg. A testament to the relatively low quality of the material, photoconductors based on GaN and AlGaN display a responsivity that is highly dependent on the incident optical power. The dependence can be characterized as Pg with 0.5 < g < 0.95 over more than five decades, as shown in Figure 4.22 [81–83]. This behavior is independent of the excitation wavelength as shown in the inset of the figure. The value of g is sample dependent and is in general a decreasing function of the electrical resistivity of the layer. It has also been observed that the value of g is a decreasing function of the temperature. The variation of the photoconductive responsivity as a function of incident optical power P at two different temperatures, 80 and 300 K, is shown in Figure 4.23. For this experiment, the detector was excited by an He–Cd laser (0.325 mm) and the attenuation of the incident power was obtained using optical densities. It was observed that, at low optical densities, the linear increase in photocurrent with optical power leads almost to a constant responsivity having a value of 4.5  103 A W1. However, in the high-power regime (P 40 W m2), the responsivity decreased with increasing power, and in this particular case following a power law of P0.5. Extremely high gains observed in photoconductive detectors are attributed to slow trapping phenomenon, as will be discussed shortly. The evolution of photoresponsivity both at low

4.4 Nitride-Based Detectors

10 4

Responsivity (A W-1)

λ =325 nm V= 5 V

10 3

10 2 T= 300 K T= 80 K

10 1 10 0

10 1

10 2

10 3

10 4

10 5

Power density (W m-2) Figure 4.23 Responsivity of a GaN photoconductor versus incident optical power density measured at 325 nm, He–Cd laser wavelength, and 300 and 80 K [84].

Responsivity (a.u.)

Responsivity (a.u.)

and high power densities is displayed in Figure 4.24. Note that, at low power, the responsivity decays almost exponentially with time. However, at high power, the decay is almost hyperbolic with less sensitive exponential nature. The variation of the response time with power density is also exponential but in the high power density regime ( 3  102 W m2), response times at various temperatures are essentially identical (not shown). Very high values of responsivity at lower power densities obtained for GaN photoconductors [84] suggest that the photoconductive gain for 100

(a) Low power density τ = 3.9 ms

10-1

10-2 100

(b) High power density τ = 2.6 ms

10-2

10-4

0

1

2

3

4

5

6

7

8

9

Time (ms) Figure 4.24 Evolution of the photoresponse of a GaN photoconductor for low (a) and high (b) incident optical power levels [84].

10

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these detectors was also very high (see Equation 4.82) and that the quantum efficiency was high as well. Nitride semiconductor-based PV and PC ultraviolet detectors have been prepared by MBE [60, 85] and OMVPE [50]. As in the case of OMVPE-grown materials, ultraviolet detectors fabricated in all MBE-grown materials showed good quantum efficiencies, noise equivalent power, and speed of response. We should mention that standard photoconducting detectors suffer from large dark current and memory effects. An analysis of decaying characteristics of responsivity as a function of time as well as power density for GaN photoconductive detectors (see Figure 4.24a and b) strongly suggests that responsivity is affected by recombination centers rather than trapping centers. Had it been affected by traps, there could be a re-emission of captured carriers with a strong temperature variation in the emission rates. It is known that, at room temperature, deep recombination centers located several kT from the conduction band exist in GaN grown both by OMVPE and MBE. The photodetector’s long response time signifies that these centers may indeed be responsible for controlling the photoresponse. The long delay in turnoff response and the low UV/visible contrast compared to that expected from the absorption coefficient of the material have been attributed by many to relatively shallow majority carrier traps, a small electron capture cross section coupled with a large hole capture rate [67, 86] and unique potential barriers [87, 88]. Muñoz et al. [87] and Garrido et al. [88] argued that the high gain in PC detectors is caused by a modulation mechanism of the conductive volume in the layer. As the carriers are photogenerated, they are separated spatially by potential barriers generated by band bending associated with surface and bulk dislocations. Carrier recombination and capture are controlled by these potential barriers and an intrinsic nonexponential recovery process, leading to long delay times and large gains. These together with the sublinear dependence of the photocurrent on optical power appear to support the notion of a photoconductive mechanism based on potential barriers caused by localized charge at defect sites. This model assumes that the current responsivity Rl consists of two terms: Dn, the photogenerated free carriers, and DS, the light-induced modulation of the effective conduction cross section. Employing this assumption in Equation 4.7, and limiting the carriers to electrons only, as illustrated in Figure 4.25, one can arrive at the following expression: Rl ¼

DIp qV a mn ¼ ðSDn þ nDSÞ: P0 LP 0

ð4:110Þ

Although already defined, q is the electron charge, me is the electron mobility, L is the distance between contacts, S is the conductive area (device area), Va is the bias voltage, n is the free carrier concentration, and P0 ¼ Pl is the incident power of wavelength l. The conduction cross section does not correspond to the geometrical cross section of the devices but is based on the presence of depletion regions around lattice discontinuities (threading dislocations, grain boundaries, and interfaces), similar to those shown in the diagram of Figure 4.25.

4.4 Nitride-Based Detectors

Figure 4.25 Energy bands diagram in a photoconductor section perpendicular to the electrical current [88].

By recognizing that the current density is simply the charge flow per unit time t and Dn is simply the photogenerated excess electron concentration Dn (assuming electron charge only as in p-type semiconductor), the current is given by Ip ¼

qDn : t

ð4:111Þ

With the help of Equation 4.6, the photogenerated excess electron concentration Dn (assuming electron charge only as in p-type semiconductor) can be expressed as Dn ¼ hext gt

P0 : hv

ð4:112Þ

Substituting Equation 4.112 into Equation 4.110, we get

 DI qV a mn hext gtS n qV a mn hext gtS n Rl ¼ ¼ DS ¼ DS : þ þ L L P0 hn P0 hl=c P0 ð4:113Þ The spectral dependence of the first term in the parentheses of Equation 4.113 is given by lhext. A notable rejection to the response to visible light would be expected if the first term in Equation 4.113 were dominant, because the quantum efficiency is a direct function of the absorbance. However, the dependence of photoconductive responsivity, in the form of gain, on optical power shown in Figure 4.22 suggests that the second term in Equation 4.113 is dominant. That is, the modulation of the effective area between the depleted regions by the potential barrier, which in turn is

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caused by the fixed sheet charge localized at defects. This mechanism also explains the high response to visible light below the bandgap due to either homogeneously distributed defects in the semiconductor, such as dopants or vacancies, or defects localized at extended defect sites (dislocations, grain boundaries). If charged, defects cause a depletion region around them, reducing the effective conduction cross section of the device (Figure 4.25). Light absorption by those defects in aggregate may be negligible, so they will hardly affect the absorption coefficient and hence the photovoltaic detection. However, their effect on the responsivity is large because the charge localized at these discontinuities changes and thus modulates the length of the effective conduction region. Other suggestions have also been forwarded to account for the long decay, also called PPC, in GaN. For example, Si or Mg impurities have been proposed as the source of PPC in doped samples [89], as has been oxygen contamination [90]. PPC has also been attributed to intrinsic material defects, such as Ga vacancies [91, 92], or defects responsible for the yellow photoluminescence emission [93]. Kung et al. [94] and Binet et al. [80] noted the nonexponential shape of the photocurrent decay and reported a model that relies on a trap level in the bandgap that gets ionized by the incident light. When the light is removed, these ionized traps behave as recombination centers with decreasing efficiency in time. This model fits the first few milliseconds of the photocurrent decay and its dependence on optical power [80] precisely, but it fails to account for the long nonexponential tail observed in most AlGaN photoconductors (Figure 4.26). The slow photocurrent decay can be accounted for by the recombination of electrons at deep levels localized at extended defects or dislocations [82]. The band bending around these defects causes a spatial separation of electrons and holes that

Al0.22Ga0.78N T=300 K

Normalized photoresponse (a.u.)

10 0

10 -1

C1 C3

Undoped n=6.1 ×1017 cm -3

C4

n=1.7 ×1018 cm-3 10 -2 0

2 ×10 3

4 ×10 3

6 ×10 3

8 ×10 3

Time (s) Figure 4.26 Normalized photocurrent decay of Al0.22Ga0.78N photoconductors for different values of the doping level, after a short He–Cd laser pulse illumination [6].

1 ×10 4

4.4 Nitride-Based Detectors

may be responsible for the PPC effect. When the incident light is turned off, electrons will recombine but not before they cross the potential barrier that separates them from the recombination centers. The height of this barrier depends on the localized charge at the defect and evolves as recombination proceeds. In this process, the SCR slowly recovers its width in dark and thus the resistance of the device also evolves nonexponentially toward its dark value. This model can be applied either when the dominating defect is at dislocation sites or when it is on the surface of AlGaN or AlGaN–substrate interface. However, it is difficult to delineate them from one another, and their relative importance may depend on the crystalline quality of the material. The contribution of the AlGaN–substrate interface to responsivity and PPC has been reported by Seifert et al. [95] who observed an increase in the persistence of photocurrent when the sample was illuminated through the polished sapphire substrate. However, when the device is front-side illuminated with above-gap radiation, the light does not reach the substrate, so that the PPC in this case would be due to dislocations or the AlGaN surface. Again, the difficulty here is that the region of the nitride layer closer to the substrate is inferior in quality and has a high defect concentration. Increased persistence in photoconductivity simply means that defects have something to do with it, which is what one suspects to begin with anyway. In short, despite many theoretical and experimental efforts, the local nature of a metastable defect or defects causing the PPC in GaN and AlGaN is not understood at this time. In the absence of reducing the defects causing the long decay, a lock-in detection scheme could be used to eliminate all frequencies lower than the optical chopper frequency. Chopping frequencies even as low as 700 Hz nearly eliminate the incident power dependence of the responsivity in photoconductive detectors [6]. Better still, p–n-junction or Schottky barrier varieties can be employed that are not affected by these defects in the same manner as the photoconductive devices are, and their responsivities are independent of optical excitation intensity. 4.4.2 Photovoltaic Detectors and Junction Detectors

As the discussion in Section 4.4 clearly indicates, the performance of photodetectors is inextricably connected to the quality of the semiconductor materials. Of particular interest are minority carrier diffusion lengths Lp(n) for holes (electrons) and surface recombination velocity sp(n). Hole diffusion length in n-type GaN increases linearly from 0.25 mm at the GaN/sapphire interface up to 0.63 mm at the surface of a 36 mm thick HVPE-grown GaN [96], and it reaches several micrometers in freestanding GaN templates, details of which are discussed in Volume 1, Chapter 4 (see Volume 1, Figure 4.93. Likewise, the diffusion length increases from 1.2 to 3.4 mm, as the carrier concentration decreases from 2  1018 to 5  1015 cm3 [97]. Diffusion length of minority electrons in p-type GaN is considerably shorter, down to 0.2 mm in the material with hole concentration of (1–4)  1017 cm3, reflecting the lower quality of p-type layers [98]. Even though a clear picture is not available, the diffusion length scales with dislocations as discussed in Volume 1, Chapter 4.

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4.4.2.1 GaN and AlGaN-Based Schottky Barrier Photodiodes As elaborated by Temkin [8], the lure of Schottky barrier photodetectors is that they are simpler to grow, because they do not require p-type doping (very problematic process, particularly for high mole fraction AlGaN). Both planar and vertical structures are used for Schottky barrier photodiodes. The vertical structures are good for attaining fast switching speed and having a high responsivity. However, current limitations of III-nitride device fabrication technology (i.e., damage caused during mesa formation by RIE etching) degrade the performance, which affects the bandwidth and noise level. A typical cross-sectional view of a Schottky barrier detector is shown in Figure 4.4a. Following growth, the mesas are defined by dry etching methods down to the n þ -layer for ohmic contact formation. In some varieties, the ohmic contact is formed on the top surface without mesa etching. This device structure is called the planar or lateral device owing to the fact that both ohmic and Schottky contacts are placed side by side. Once the ohmic contact is deposited and annealed, the Schottky contact that typically uses Pd or Pt is deposited. The Schottky metal thickness is kept below 10 nm for transparency to light. The shorter the wavelength desired for detection, the thinner this layer needs to be. As shown in Figure 4.4a, the diode can be illuminated either through the Schottky contact or, if grown on sapphire, through the transparent substrate. In the latter case, performance is degraded due to the lower quality of the layer next to the substrate. It is also possible to form two back-to-back Schottky contacts in the form of interdigitated metal fingers separated from each other by a few micrometers. This goes by the nomenclature of MSM detectors and will be discussed separately. The advantage of this process is its simplicity, because it requires only one deposition step, but the device geometry is not well defined. These detectors have the potential for high quantum efficiency and high speed. Often, this variety is used to evaluate the absorber material and other pertinent device issues, as opposed to becoming captive to the difficulties involved in p-type doping of AlGaN. Noteworthy progress has already been made in the development of GaNbased Schottky barrier detectors with near-unity gain responsivities and response bandwidth in megahertz range [99–115]. Performance of these devices increases with reduction in leakage current, which leads to lower noise. This is a result of improved materials quality and more sophisticated processing methods that cause less damage. GaN Schottky barrier photodetectors with quantum efficiencies of h  50% and responsivities of about 0.18 A W1 can be fabricated by paying careful attention to layer growth and processing [100, 109, 116]. The photoresponse of these detectors is fairly flat above the band edge (350 nm), and it drops by as much as three orders of magnitude below the band edge because of increasing photon energy and absorbance of the semitransparent metal. After pointing out the importance of traps and other generation–recombination processes involving multiple levels, the noise performance of detectors investigated by Chen et al. [103] will be discussed. The device structure consisted of a 40 nm thick buffer layer of AlN, followed by 1 mm thick n þ -GaN layer doped n-type to 3  1018 cm3 and a 0.4 mm thick n-GaN layer doped to 3  1016 cm3. Vertical geometry diodes with semitransparent Pd Schottky barrier contacts were fabricated

4.4 Nitride-Based Detectors

by defining 4 4 mm2 mesas using reactive ion etching. The ohmic contact was made to the n þ layer and annealed at 800 C for 30 s by a rapid thermal annealing process. At a reverse bias of 5 V, these diodes showed dark current density of 3.8  107 A cm2. Following Equation 4.48, Chen et al. [103] obtained the mean square noise current by integrating the spectral density over the frequency range from 0 to the bandwidth B, which was limited to 1 Hz in the experiment, as given here: ðB hi2n i

ð1

ðB

¼ Sn d f ¼ S0 d f þ 0

0

1


  S0 B d f ¼ S0 ln þ1 : f 1 Hz

ð4:114Þ

Assuming a load resistor of RL ¼ 600 W, noise power Pn ¼ hi2n iRL ffi 2:9  1015 W was obtained [8]. The NEP is obtained by dividing the mean square noise current by the responsivity of the device, as defined in Equations 4.85 and 4.87. In these particular detectors, the noise is predominantly of the 1/f nature with the noise spectral density given by Equation 4.55 with the coefficient a close to unity. The characteristic value S0 was proportional to the square of the average dark current, once again underscoring the importance of minimizing such currents. For frequencies up to the photodetector bandwidth (B) of about 3 MHz, the 1/f noise power density Sn was much higher than the shot noise density. Utilizing the measured responsivity of Rl ffi 0.18 A W1 at 360 nm, the NEP was calculated to be 1.2  108 W. The NEP normalized by the square root of the bandwidth, the inverse of which is indicative of the specific detectivity, was calculated as approximately 7  1012 W Hz1/2. As discussed shortly, similar NEP values with specific detectivity of 6.1  109 cm Hz1/2 W1 were also obtained by Monroy et al. [117] in lateral Schottky barrier detectors. The dark current in detectors is a binary process in which both the material quality and the device fabrication process must be acceptable. Addressing the latter, Adivarahan et al. [118] and Deelman et al. [119] focused on the importance of device fabrication processes in achieving low dark current densities. Devices fabricated with SiO2 surface passivation layers along with 0.1 mm layer of SiO2 underneath the 5.0–7.5 nm thick semitransparent Pt Schottky barrier exhibited reduction in dark current levels in the range of 102–104 beyond a bias of 5 V compared to unpassivated devices. At 5 V, the dark current as small as 1 pA was measured in a 200 mm diameter device, corresponding to a current density of about 3  107 A cm2. The plasma-enhanced CVD (PECVD) process used for SiO2 deposition is characterized by low deposition temperature and results in low density of interfacial steps on GaN [120]. Schottky barrier diodes fabricated on GaN grown on Si (1 1 1) substrates by gas source molecular beam epitaxy were investigated by Deelman et al. [119]. The GaN layers were grown on AlN grown on Si, mesas were etched, and ohmic contact metallization was applied, leading to the formation of Schottky barriers (10 nm thick Pd), which were all followed by the PECVD deposition of SiO2 surface passivation layer. The quality of the Schottky barriers was demonstrated when the current was dominated by thermionic emission, which led to a Richardson constant of A ¼ 24

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A cm2 K2 that is nearly consistent with the accepted electron effective mass of 0.22, but a high ideality factor of 14 was a problem. Measurements of 86  86 mm2 devices show dark currents of 4 pA at a reverse bias of 5 V, resulting in the very low dark current density of 5.4  108 A cm2. The noise power density, as a function of diode current, was measured and as is common for Schottky diodes prepared on GaN, the noise was 1/f limited at low frequencies [103, 104]. The noise spectral density measured at 1 Hz was 9  1029 A2 Hz1 for zero bias. For the lowest dark currents, noise power density of these Schottky diodes approached the shot noise limit Ishot ¼ (2qI)1/2, as indicated in Equation 4.51 for 1 Hz bandwidth. Further reduction of low bias dark currents and noise power requires better insight into the defect-assisted tunneling frequently encountered in Schottky diodes on GaN. The tunneling current also coincides with rapid degradation of the current–voltage characteristics under voltage stress. The strong voltage dependence of the dark current at low bias is indicative of tunneling and was investigated by Carrano et al. [101, 116] who suggested a sequential set of deep-level assisted processes. A schematic band diagram of the metal–semiconductor interface with defects likely to be responsible for tunneling is shown in Figure 4.27. Accordingly, following the initial application of a small external bias, electrons tunnel through the Schottky barrier to an interfacial state labeled 1 in Figure 4.27. At this stage, at least two paths become available – namely, tunneling through the remaining barrier (depicted as process 2) and thermal excitation through a set of deep states (depicted as process 3). The completion of the first voltage sweep would leave most of the available defect states filled. This leaves only a few empty states available for deep-level assisted tunneling, causing the second voltage sweep to show a degraded current–voltage

4 3 1 2

Figure 4.27 Band diagram of a Schottky diode interface illustrating specific defect-assisted tunneling processes proposed for explaining the large leakage at low bias voltages [101].

4.4 Nitride-Based Detectors

characteristic with lower, albeit unstable, dark current. However, once the filled state population reaches equilibrium, the current–voltage characteristic becomes stable. It is also plausible to release the trapped electrons, depicted as process 4, by tunneling back to the Schottky metal. This gives rise to excess dark current at low bias voltages. Some of the detrapping processes appear to be caused by illumination with white light. This frees defect states and results in increased leakage current when dark current measurements are made following illumination. Refer to Volume 2, Chapter 1 for details regarding current conduction, inclusive of excess current, in Schottky barriers. Another sign of interfacial trapping in Schottky barrier detectors is the appearance of gains as large as 50 at a forward bias of 0.7 V [109]. This is typically attributed to the trapping of photogenerated holes at the metal–semiconductor interface. Under reverse bias, the induced field separates electrons and holes and gain disappears. As discussed in Section 4.4.1, large gains observed in GaN photoconductive detectors are also attributed to carrier trapping at deep levels. This results in a highly nonlinear response to the incident light, a significant photoresponse to subbandgap light degrading the visible-to-UV ratio, and very long response times. AlGaN-based Schottky barrier detectors are, at this point, somewhat less advanced than their GaN cousins. In spite of this, detectors fabricated with AlGaN (with the Al content up to 0.35) exhibited ultraviolet spectral response peaking at 280–290 nm with the responsivity of 0.1 A W1 [121, 122]. A very large drop in the below the gap response (four orders of magnitude) has been demonstrated, even in the absence of sophisticated postgrowth processing. This high UV-to-visible ratio was maintained even for AlGaN-based detectors on Si(1 1 1) substrates [86]. Figure 4.28 shows I–V characteristics of Al0.35Ga0.65N and GaN Schottky photodiodes [6]. In GaN devices, the ideality factor is about 1.2, the series resistance is in the range of 20–50 W, and the leakage resistances are higher than 1 G W. Semitransparent Schottky contacts in this particular case were either Ni or Au, generally 10 nm or less,

10 -2

Current (A)

10

Alx Ga1-x N/Au Diameter=200 µm

-4

10 -6

10 -8 x=0

10 -10

-5

x = 0.35

-4

-3

-2

-1

0

1

2

3

4

Bias voltage (V) Figure 4.28 Typical dark I–V characteristics of GaN and Al0.35Ga0.65N Schottky barrier photodiodes on sapphire [6].

5

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with barrier heights of 0.86 and 1.2 eV for Ni and Au, respectively. The latter is consistent with the Schottky–Mott theory, but that the barrier height for Ni is lower than the expected value. The deviation in the case of Ni can be attributed to the interface chemistry such as Ga4Ni3 formation as reported by Guo et al. [123]. As the mole fraction Al increases, the quality of AlGaN decreases, increasing the leakage current and the ideality factor (which reach a value of about 4). Consequently, reliable information on the Schottky barrier height cannot be obtained from the I–V curve for the AlGaN material. Noise characteristics of AlGaN Schottky barrier detectors with the Al content up to x  0.22–0.26 were investigated by Osinsky et al. [107] and Monroy et al. [117] who extended the range to x ¼ 0.35 as discussed below. Sapphire substrates with structures similar to those for GaN were used, and the detectors exhibited cutoff wavelengths as short as 290 nm. The detectors showed leakage current densities of 2.5  103 A cm2, considerably larger than those of GaN devices, underscoring the defective nature of AlGaN. Variable size detectors (from 50  50 to 500  500 mm2) were fabricated in an effort to shed light on the source of the excess current. The dark current scaled as 1.3 power of the detector size, about halfway between bulk leakage (power  2) and surface leakage (power  1), which suggests that both surface and bulk contribute. The normalized responsivity of AlxGa1xN Schottky barriers with AlN molar fraction from the binary (representing no AlN) to 0.35 is shown in Figure 4.29. As desired, Schottky barrier devices show a linear dependence of the photocurrent on the incident power over five orders of magnitude (from 10 mW m2 to 2 kW m2) as

Alx Ga1-x N

10

Photocurrent (A)

Normalized responsivity (a.u.)

10 0

-1

10

10

10

-6

x x x x

=0 = 0.19 = 0.26 = 0.35

-8

-10

λ=325 nm

10 -2 -1

1

10 10 10 Irradiance (W m-2)

10 -3

x x x x 280

=0 = 0.19 = 0.26 = 0.35 320

360

400

440

Wavelength (nm) Figure 4.29 Normalized room-temperature spectral response of AlxGa1xN Schottky barrier photodiodes for different values of the Al mole fraction. The inset shows the dependence of the photocurrent on the incident light power [6].

480

3

4.4 Nitride-Based Detectors

shown in the inset of Figure 4.29, which represents a welcome deviation from photoconductive varieties. Figure 4.29 also plots the spectral responsivity of AlxGa1xN Schottky diodes with different Al molar fractions, specifically x ¼ 0, 0.19, 0.26, and 0.35. A UV/visible contrast ratio of more than three decades is noted. The cutoff wavelength shifts from 362 to 293 nm as one traverses from GaN to Al0.35Ga0.65N. While the normalized responsivity is flat for wavelengths below the bandgap, the absolute responsivity decreases with increasing AlN molar fraction, namely, 0.054, 0.045, 0.030, and 0.010 A W1 for x ¼ 0, 0.19, 0.26, and 0.35, respectively. The rapid drop in the responsivity to radiation below the bandgap is indicative of the effect of the defects in the absorbing material only, in contrast with the results obtained in photoconductive detectors. It should be noted that these values for NEP and specific detectivity D are approaching specification of typical commercial Si ultraviolet detectors. Unlike the p-i-n where the delay due to the RC time constant (where C is the sum of junction capacitance and the load capacitance, and RL is the sum of the load resistance and the series resistance of the device) and the transit time through the i-region is made nearly equal, the AlxGa1xN photodetector time response is limited by just the RC constant. Consequently, the photocurrent decay time constant, defined as the time required for the photocurrent to fall from its maximum value to 1/e of that value, increases linearly with the load resistance, as expected from RC time limitation. A minimum time constant of 69 ns has been obtained in diodes with a diameter f ¼ 1 mm and 15 ns for smaller devices with a diameter f ¼ 240 mm [117]. An NEP normalized to the square root of the bandwidth of 8  1012 W Hz1/2 (specific detectivity ¼ 6.1  109 W1 Hz1/2 cm) and 41 pW Hz1/2 (specific detectivity ¼ 1.2  109 W1 Hz1/2 cm) has been obtained for GaN/Au and Al0.22Ga0.78N/Au Schottky photodiodes, respectively, at 2 V bias [117]. As iterated several times, both the detectivity and the UV-to-visible ratio of detectors are affected adversely by defects. It is reasonable to argue that if the extended defect concentration can be reduced, so can the point defects and complexes, paving the way for performance improvements. As discussed in Volume 1, Chapter 3, epitaxial lateral overgrowth (ELO) and other defect reduction techniques have reduced the dislocation density by some two orders of magnitude. It is then natural to investigate detector performance on material prepared with ELO templates. To this end, semitransparent Au Schottky barrier photodiodes have been fabricated on two-step ELO GaN templates [124], displaying a responsivity of 0.130 A W1. An improvement of one order of magnitude in the UV/visible contrast has been observed, in comparison with devices fabricated with standard GaN on sapphire, as shown in Figure 4.30. The 1  109 A cm2 dark current at 1 V bias is significantly less than that obtained in conventional GaN/sapphire Schottky barrier photodiodes. The time response of the device is again RC limited, and the reduced residual doping level in these epitaxial layers, which reduces the junction capacitance, leads to bandwidth values of over 30, 12, and 8 MHz for devices with a diameter of 200, 400, and 600 mm, respectively. A detectivity of 5  1011 W1 Hz1/2 cm has been determined for ELO GaN photodiodes with a diameter of 400 mm at 3.4 V bias voltage.

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10

0

ELO GaN GaN/Sapphire

-1

-5

10

10

-2

10

-3

10

-4

10

-5

Photocurrent (A)

Normalized responsivity (a.u.)

10

λ= 325 nm

10-7

-9

10

ELO GaN GaN/sapphire -2

10

-1

10

0

10

10

1

2

10

3

10

Irradiance (W m-2)

250

300

350

400

450

Wavelength (nm)

Figure 4.30 Normalized responsivity of ELO GaN-based Schottky photodiodes compared to the responsivity of similar GaN/ sapphire devices. The inset shows the variation of the photocurrent versus incident light power [6].

4.4.2.2 Metal–Semiconductor–Metal Detectors Because of geometric convenience, speed, and not requiring p–n-junctions, MSM detectors have been explored in many of the detector materials, and the GaN-based system is no exception. A schematic diagram of a typical MSM detector structure with biasing, field lines, depletion front profile, and band structure is shown in Figure 4.31. Back-to-back Schottky barriers pave the way for very large dark resistance [125], unlike

Figure 4.31 A schematic representation of a typical MSM detector structure with biasing, field lines, depletion front profile, and band structure.

4.4 Nitride-Based Detectors

Current (A)

10-8

10-10

10-12

10-14 0

5

10

15

20

Reverse bias (V) Figure 4.32 Current–voltage characteristics of Schottky diodes fabricated on GaN epitaxial layers of different thicknesses. The upper curve is for the case of 1.5 mm thick GaN while the lower is for 4.0 mm thick GaN. The dashed lines represent calculations [100].

the traditional photoconductor detectors. This leads to low dark current [126], an imperative in high-sensitivity detectors with low noise. A detailed study of mechanisms leading to excess leakage currents was conducted by Carrano et al. [127]. Backto-back Schottky diodes used in their study were fabricated as interdigitated metal–semiconductor–metal structures consisting of 2 mm wide fingers separated by varying gaps of 2, 5, and 10 mm in various samples. The metal contact started with the deposition of 5 nm of Ti, to ensure good adhesion to the GaN, followed by 80 nm of Pt, the Schottky metal with a relatively high work function (5.6 eV). Contact fingers were deposited into openings in a SiO2 passivation layer, used also as an antireflection coating, deposited on the surface of GaN. Figure 4.32 shows current–voltage characteristics of Schottky diodes obtained on two GaN MSM detectors, one having been fabricated by using a 1.5 mm thick and the other by using 4.0 mm thick GaN grown by OMVPE on sapphire substrates, which exhibited Hall measured n-type carrier concentrations of about 6  1016 cm3. At low voltages, Schottky diodes exhibited current densities in the mid-106 A cm2 range [100], with detectors fabricated on 4.0 mm thick GaN showing markedly improved I–V characteristics. The dashed line in Figure 4.32 shows calculations that assume the current transport is due to either thermionic emission [128] or thermionic field emission including tunneling currents [129], see Volume 2, Chapter 1 for details. Good agreement with the thermionic-emission-only model for detectors on 4.0 mm of GaN is indicative of negligible tunneling current. However, the thermionic field emission with tunneling model must be used for detectors on 1.5 mm thick GaN, with higher defect concentrations pointing to defect-assisted tunneling when the barriers become narrow enough by the application of a sufficient reverse bias. The MSM detector arena has been expanded to include AlGaN-based photodiodes with two interdigitated Ni/Au Schottky contacts deposited on 1 mm thick AlxGa1xN with unintentionally doped epitaxial layers, and 2, 4, and 7 mm finger widths and spacing [130]. The optical area of the devices investigated ranged from 250  250 mm2 to 1  3 mm2. In Section 4.4.2.1, the point was made that thinner epitaxial layers lead

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10-6

Current density (A cm-2)

T=300 K

Tunnel current

10-7

2

A opt= 1×3 mm

10-8 A opt= 250 × 250 µm

2

10-9 Thermoionic current 16

N d =6 × 10 cm-3

10-10

qφ0= 1.04 eV

0

20

40

60

80

Voltage (V) Figure 4.33 Current-to-voltage characteristics of an MSM photodetector for two different optical areas, Aopt, 250  250 mm2 and 1  3 mm2. Dotted lines are for fits assuming thermionic emission transport for the smaller detectors and tunneling dominated current transport in large detectors [130].

to Schottky barriers where defect-assisted tunneling had to be invoked to represent the current flow, while the thicker layers led to thermionic emission current (including the Schottky barrier lowering due to image force effects [129]) being sufficient to fit the data. Even when the defect concentration is relatively low, increased device area necessitates the use of tunneling current as shown in Figure 4.33. In the 250  250 mm2 detector, the ideal GaN effective Richardson constant of A ¼ 26 A cm2 K2 was obtained with the associated Schottky parameters of a barrier height of 1.04 eV and a doping concentration Nd ¼ 6  1016 cm3. The normalized spectral response of AlGaN MSM photodiodes for various AlN molar fractions and bias voltages is shown in Figure 4.34. Notice that the responsivity is relatively flat over the bandgap and the cutoff wavelength shifts to shorter wavelengths with increasing AlN molar fraction [131]. A UV/visible rejection ratio of four to five orders of magnitude is observed. However, the contrast decreases by one decade when the bias is reduced from 5 to 1 V and remains at approximately 103 for lower biases [132]. The photocurrent scales nearly linearly with optical power for wavelengths both above and below the bandgap. In another report, a dark current of 50 pA at a bias voltage of 10 V and a responsivity of 0.3 A W1 have been reported [125]. The rejection ratio is determined by the sharpness of the absorption edge, as is the case in p–n-junction and Schottky barrier vertical detectors. In another report, several nanoamperes of dark current at bias voltages of about 5 V have been reported along with responsivities of about 0.05 A W1 [133]. The variation of the responsivity with bias has also been investigated for diodes of different sizes for excitation above the bandgap, and the results are shown in Figure 4.35 [132]. For VB < 2 V, the responsivity scales sublinearly with bias (Rl / V 0:7 B ), which fits the theoretical behavior expected for an MSM photodiode

Normalized responsivity (a.u.)

4.4 Nitride-Based Detectors

AlGa N x 1-x

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

x = 0; V = 1 V x = 0; V = 5 V x = 0.25; V = 10 V

T = 300 K 250

300

350 400 Wavelength (nm)

450

500

Figure 4.34 Spectral response of AlGaN MSM photodiodes, measured under different bias. Triangles were obtained with the 458 and 488 nm lines of an Ar þ laser [131].

Responsivity (A W-1)

in the absence of the gain calculated from a one-dimensional model (solid line). An abrupt increase of the responsivity is observed between 2 and 5 V, indicative of a biasactivated gain mechanism that saturates for higher bias voltages. This behavior is independent of the finger width and the gap spacing, as also shown in Figure 4.35. The gain mechanism at high bias was also wavelength dependent [6]. For wavelengths corresponding to below bandgap response, the detectors follow the trend expected for an MSM photodiode in the absence of gain. The deviation from this behavior appears only for wavelengths shorter than 370 nm, so that the enhancement

10

0

10

-1

10

-2

10

-3

10

-4

(2 × 2) (4 × 4) (7 × 7)

model (4 × 4)

GaN T = 300 K λ = 325 nm 0.1

1

10

100

Voltage (V) Figure 4.35 Responsivity dependence on bias for GaN MSM photodiodes, measured with an excitation wavelength of 325 nm, which is above the bandgap. Performances of three sizes of devices with finger widths and spacings of 2  2, 4  4, and 7  7 mm2 in an active area of 250  250 mm2 are shown. The model calculations of the responsivity for the 4  4 mm2 active area are also shown with the solid line [6, 132].

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of the UV/visible ratio with bias and the enhancement of gain are brought about by the same mechanism. In the device investigated by Monroy et al. [131], the variation of photocurrent response time with load resistance did not show saturation for low resistances, implying the response speed to be below 10 ns [131]. The intrinsic bandwidth of the MSM photodiodes could be limited either by the RC product or by the carrier transit time of the detector, both of which are estimated in the picosecond range. Considering the doping concentration and Schottky barrier height (which represents a small part of the voltage drop across the device), the reach-through voltage of these MSM structures is >200 V. Thus, most of the applied voltage drops in the reverse-biased cathode contact, and the photocurrent is produced by absorption in the cathode space charge region, with a small negative contribution by the anode. The higher responsivity observed in devices with a shorter gap spacing is not due to a more intense electric field, but due to a higher number of fingers in the illuminated region. Therefore, responsivity saturation at high bias voltages is not due to a full depletion of the device, but due to gain saturation. The gain mechanism observed is only active at bias higher than 2 V and only for excitation over the bandgap. This mechanism is responsible for the superlinear increase of the responsivity with bias [131, 133] and for the enhancement of the UV/visible contrast that is observed in AlGaN-based MSM photodiodes. Gain in interdigitated MSM photodiodes has been attributed to different mechanisms [134, 135]. The most often suggested mechanism consists of electron tunneling enhanced by trap-mitigated hole accumulation at the cathode. Anomalous gain beyond t/ttr (the minority carrier lifetime divided by the transit time) in photoconductive-like detectors and electron tunneling enhanced by trap-induced hole accumulation at the cathode (see Equation 4.13 for photoconductive devices) is typically due to slow trapping and detrapping phenomena. Trapping effects manifest themselves as slow exponential decay in photocurrent, if one simple trapping level is involved. On the practical side, however, the decay rate can be limited by the RC product of the measurement system, independent of bias. Obviously, the experimental setup must not be the limiting factor in measuring the time-resolved phenomena. Trapping at surface states or dislocations produces persistent photoconductivity effects and a sublinear behavior with optical power, degrading the spectral response. Both the linearity and the fast response of MSM photodiodes make the case for the absence of this gain mechanism that is dominant in GaN and AlGaN photoconductors [83]. The presence of deep hole traps in the unintentionally doped bulk GaN is responsible for the photoresponse to below bandgap radiation, but because the gain is not observed for l > 370 nm, trapping at these levels as the origin of this gain is not likely. The increase in hole density near the cathode can be attributed to the difference in electron and hole transit times [134], but this is a self-limiting process because electron–hole separation sets up a field opposing the original accelerating field. Moreover, the electron–hole separation cannot explain a gain that only occurs for above bandgap photon excitation. Another process, which could give rise to an anomalous gain, is avalanching in the valence band. At high electric fields, holes

4.4 Nitride-Based Detectors

generated in the valence band near the cathode move toward the contact under the influence of a large electric field and might have sufficient energy to cause impact ionization. However, recent calculations of the multiplication coefficients in GaN [136] suggest that higher fields than those in these devices would be required (greater than 20 V for the device geometries employed). In short, the gain mechanism is not yet clearly understood in AlGaN MSM photodiodes. Despite the trapping phenomenon discussed above, the MSM detectors are characterized by their high speed [137] as shown in Figures 4.37 and 4.38. The time response t of MSM photodiodes is affected by the carrier transit time ttr and the device capacitance C: t2 ¼ ðRL CÞ2 þ t2tr ;

ð4:115Þ

where RL is the load resistance. The transit time is the time it takes for a carrier to travel across the depletion region and is approximated by the transit length divided by saturation velocity. ttr ¼

L ; vsat

ð4:116Þ

where L is the pitch and vsat is the hole saturation velocity, whereas the device capacitance approximately corresponds to C¼

Ae0 ðeGaN þ 1Þ p ; LþW 4 ln½ð8=pÞ þ L=WÞ

ð4:117Þ

where A is the detector area, e0 is the vacuum dielectric constant, eGaN is the relative dielectric constant of GaN, and W is the finger width. It is then clear that for a given detector area, there is an optimum finger separation for the maximum speed of response, as reported by Monroy et al. [138] and shown in Figure 4.36. The estimates agree well with experimental data of Carrano et al. [139] for 50  50 mm2 devices with

ttr -limited region

Response time (s)

10-9 R LC-limited region

10-10 Active area 250 × 250 µm 2 100 × 100 µm 2

10-11

50 × 50 µm 2

0.1

1

10

100

Finger width and pitch (µm) Figure 4.36 Calculated time response of GaN MSM photodiodes as a function of finger separation for devices with various active areas. A saturation velocity vsat of 107 cm s1 is assumed [138]. The experimental data points are from Ref. [139].

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1.0

Fit

2 µ m gap M SM E ≈ 0.1 W cm -2

Normalized amplitude

Electrons

0.5 5V

25 V

15 V

Holes

0.0 43.4

43.5

43.6

43.7

43.8

43.9

44.0

44.1

Time (ns) Figure 4.37 Time domain response of an MSM detector at various bias voltages for a 2 mm gap spacing. For a bias voltage of 25 V and without the hole response, 90–10% decay time is about 200 ps. Courtesy of J. C. Campbell.

gap spacings of 2 and 10 mm. The time response of Equation 4.115 is minimized when the transit time and RC time constant are about the same. Shown in Figure 4.37 is the time domain response of an MSM detector at various bias voltages for 2 mm gap spacing. For a bias voltage of 25 V and without the response 10

E≈0.1 W cm -2 25 V, 2 µm gap

8 6

Power (dB)

4 2

f 3 dB =3.8 GHz

0 -2 -4 -6 -8 0

2

4

6

8

10

12

14

Frequency (GHz)

Figure 4.38 Frequency domain response of an MSM detector at 25 V bias voltages for 2 mm gap spacing. For a bias voltage of 25 V, the 3 dB frequency is 3.8 GHz. Courtesy of J. C. Campbell.

4.4 Nitride-Based Detectors

by holes, 90–10% decay time is about 200 ps. Shown in Figure 4.38 is the frequencydomain response of an MSM detector at 25 V bias voltages for 2 mm gap spacing. For a bias voltage of 25 V, the 3 dB frequency is 3.8 GHz. The MSM detectors have been differentiated by their noise characteristics, which can be done with Stanford Research tools such as the SR530 lock-in amplifier or the Stanford Research Model 7870 network signal analyzer. The noise performance of the MSM devices has been measured for biases up to 28 V. At this bias, the spectral noise power remains below the background noise level of the system, implying a normalized noise equivalent power (NEP ) below 2  1012 W Hz1/2 in devices with finger width and gap spacing of 2 mm (2  2). At 28 V bias, the measured NEP was approximately 2.4  1011 W Hz1/2 in Al0.25Ga0.75N photodiodes [130]. 4.4.2.3 p–n- and p-i-n-Junction Detectors There are basically three types of GaN-based p–n-junction detectors that have been fabricated. One is strictly a GaN homojunction. The second employs a p-type AlGaN top window layer. The third, developed for bottom-illuminated solar-blind applications, relies on a window buffer layer, an n-type AlGaN absorber, and a p-type top AlGaN layer, as depicted in Figure 4.4b–d. In GaN homojunction devices, above-gap radiation, is absorbed in the top layer (Figure 4.4b). In order for the minority carrier to make it across the junction, the thickness of the top layer must be comparable, if not smaller than, the diffusion length. If the top layer is p-type (which is inferior in quality to n-type), the electron diffusion length could be small, in which case the p-layer must be made even thinner. The depletion depth determines the lower limit for the thickness because this depth should not be allowed to reach the surface. To eliminate the problem of absorption in the top layer, a wider gap window layer is used for p-type material (Figure 4.4c). The short wavelength cutoff for the detector is then determined by the absorption edge of this window layer. For solar-blind devices, the difficulty in achieving p-type conduction in high mole fraction AlGaN and the ease of coupling the detector to an Si readout circuit (ROC), generally for imaging purposes, make the back-illuminated device more desirable. In that case, the window layer is below the absorber layer for the radiation of interest and it again determines the short wavelength cutoff. The long wavelength cutoff is determined by the absorbing layer. Surface-illuminated and back-illuminated GaN and AlGaN/GaN p-i-n UV photodetectors have been prepared by reactive or gas source molecular beam epitaxy (RMBE) and organometallic vapor deposition sapphire substrates with excellent performance. The p-i-n structures are generally grown on c-plane sapphire substrates. Figure 4.4b shows the typical structure of a GaN detector. In the configuration shown, the light is coupled from the top. The thickness of the GaN layer can be adjusted for the light to be absorbed entirely in the top p-type GaN layer, the lightly and unintentionally n-doped GaN layer (referred to as the i-layer), or both. AlGaN (AlGaN homojunction), GaN (GaN homojunction), and GaN with AlGaN window varieties have been explored. The spectral response of GaN UV detectors with an Al0.12Ga0.88N window layer has been obtained in a top illumination configuration at normal incidence with a

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light intensity of approximately 0.06 mW cm2 for zero bias and 10 V bias. The responsivity dropped by more than three orders of magnitude as the incident light’s wavelength increases from 360 to 390 nm. Moreover, the reverse bias enhances the response (0.15 A W1 at 10 V), while maintaining the sharp cutoff edge and low-noise characteristics of the zero-bias responsivity. The spectral response of p–n-junction photovoltaic detectors has been obtained yielding peak responsivities of 0.21 A W1 at 356 nm with an internal quantum efficiency of 82%. As in the case of Schottky barrier and p–n-junction detectors, the dark current (leakage current) and its anomalous behavior is inextricable from the noise performance of the detector. As discussed in Volume 2, Chapters 1 and 4, the saturation current expected from thermal excitation of carriers in a wide-bandgap material such as GaN is expected to be in the 1018 A range or less at room temperature. Specifically, due to the large bandgap of GaN and extremely small thermal excitation rate, diffusion currents in the neutral region and generation–recombination currents in the depletion region should be negligibly small (see Volume 2, Chapter 4 for numerical simulations of the reverse current for various generation–recombination rates in GaN). This is some six orders of magnitude smaller, meaning the experimental saturation current is six orders of magnitude larger than expected. Even though etching the surface of GaN in KOH prior to Schottky barrier deposition [140] reduced the leakage current down to 4.5  1014 A with an associated ideality factor of 1.01, the case is still valid. In addition, the exponential dependence of the current on both the voltage and the temperature is difficult to explain using conventional models of reverse-bias conductivity [50]. Measurements on devices with different sizes, in an effort to discern the surface and bulk components, showed that the current scaled with the device area, rather than the perimeter of the active region, thus ruling out any significant contribution of surface leakage currents at the device mesa sidewalls, although some reports indicate a mixture of bulk and surface current. The onset of impact ionization in typical photodiodes that have been reported is normally observed only at a bias greater than 40 V, which should not have any impact on the current–voltage characteristics for smaller bias voltages. Therefore, impact ionization is not expected to contribute significantly to the measured I–V characteristics. Finally, direct band-to-band or trap-assisted tunneling current has exponential voltage dependence but should be only weakly dependent on temperature, which is a visceral characteristic of tunneling. It does not take much of an interpolation to realize that the anomalous current in GaN is a result of the highly defective nature of this material. Detailed analysis of the I–V characteristics and their correlation to dislocation density utilizing ELO material [141] led to the realization (one that is not new because other semiconductors such as GaAs and InP went through the same steps) that dislocations affect the dark conductivity of diodes (see Volume 1, Chapter 3 for details of ELO). The reasons why this happens are not fully established. Models developed for systems containing large concentrations of point defects could be applied for gaining insight into the current conduction mechanisms. One such model assumes that the current is due to hopping of charged carriers via localized defect-related states (traps) in the depletion region, which explains well the field dependence of the reverse current [8], details of which are discussed in Volume 2,

4.4 Nitride-Based Detectors

Chapter 4 dealing with the I–V characteristics in p–n-junctions and Volume 2, Chapter 1 dealing with Schottky barriers. Spectral response of semiconductor photodetectors can be described as a function of material parameters based on a one-dimensional model shown in a schematic cross section in Figure 4.6. In this type of a diffusion-limited model, the diode structure is divided into n- and p-type and depletion regions. A similar model can be used to describe Schottky barrier detectors. The spatial charge of width w extends from the junction at x ¼ 0 to the n-type semiconductor, and the quasineutral region (xn, wn) is uniformly doped. The depletion width w depends on the doping levels. The dark current consists of electrons injected from the n-side and holes injected from the p-side. The saturation current density can then be calculated from standard expressions [50]. A finite surface recombination velocity is normally assumed, unless determined independently for each surface and interface of the device. The saturation current density is a function of minority carrier diffusion lengths, minority carrier diffusion coefficients, minority carrier concentration, detailed junction design, and surface recombination velocity. The total current in the photodetector is the difference between the dark current and photocurrent, the latter given by Equation 4.6, and the two current components being assumed independent of each other. The diode current (which takes into account the diffusion length and the surface recombination velocity) can be calculated using Equation 4.14 and related equations that follow. The wavelength dependence of responsivity can be entered through the wavelength dependence of the absorption coefficient [142]. The responsivity of a p–n-junction detector can be calculated with the help of Equations 4.63, 4.72, 4.76, and 4.79 with appropriate materials parameters. The results of such a calculation are shown in Figure 4.39a and b. Figure 4.39a shows the calculated spectra for a p–n-junction detector illuminated from the n-side for a constant surface recombination velocity S ¼ 100 cm s1 and different values for the minority hole diffusion length (0.5, 1.0, and 1.5 mm). Figure 4.39b plots the responsivity for a fixed value of the minority hole diffusion length Lp ¼ 2.0 mm and different values of s1 ¼ 100, 1000, and 4000 cm s1 with no external bias. For the calculations an electron diffusion length of 1 mm, hole concentration of 1017 cm3, electron concentration of 1  1018 cm3, electron mobility of 300 cm2 V1 s1, hole mobility of 20 cm2 V1 s1, and n-and p-side thicknesses of 0.4 and 0.1 mm, respectively, were used. As can be seen in Figure 4.39a, longer diffusion lengths result in higher responsivity and less peaking at the band edge. As expected, lower surface recombination velocities also result in improved responsivity and less of a decrease above the band edge. Diffusion lengths increase in material with lower defect density, while the effective surface recombination velocity can be controlled through passivation layers. Already, freestanding GaN templates with extended defect concentrations below 106 cm2 exhibit a preliminary hole diffusion length of about 4.5 mm and a minority carrier lifetime of about 800 ns at 175 mm away from the substrate side of the template. The minority carriers generated in the n-layer (illuminated side) and those that diffuse to the junction depletion edge are collected and contribute to the photocurrent. Therefore, the responsivity would depend on the layer thickness. Also, the responsivity would depend on the wavelength through the wavelength

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0.25

µ

Responsivity (A W-1)

0.2

µ

0.15

e

p

-2

-1 -1

-2

-1 -1

300 cm V s 20 cm V s

Lp=1.5 µm

S

100 cm s

t

0.4 µm

L =1.0 µm

d

L =0.5 µm

N

p

0.1 µm

p

0.1

N L

0.05

0 200

250

(a)

n-type

t

p-type

d

300

350

-1

18

-3

17

-3

10 cm

d

10 cm

a

3.0 µm

e

400

450

500

Wavelength (nm) 0.25

Responsivity (A W-1)

0.2

µe µp

0.15

S =100 cm s

-1

-1 -1

-2

300 cm V s -2

-1 -1

20 cm V s

Lp

1.0 µm

-1

t

0.4 µm

-1

d

S =1000 cm s S= 4000 cm s

0.1

10 cm

18

-3

Na

17

-3

Le

0.05

n-type

t

p-type

d

0.1 µm

Nd

10 cm

3.0 µm

0 200

(b)

250

300

350

400

450

500

Wavelength (nm)

Figure 4.39 Comparison of measured spectral response of a GaN-based photodetector and simulated spectra plotted for (a) constant surface recombination velocity S of 100 cm s1 and different values of the minority hole diffusion length Lp of 0.5, 1.0, and 1.5 mm, and (b) constant Lp of 1.5 mm and different values of S, namely, 100, 1000, and 4000 cm s1. Courtesy of F. Qian.

dependence of the absorption coefficient and photon absorption profile in the n-layer in addition to the fact that as the wavelength is reduced, the number of minority carriers generated per optical power would diminish because of the increasing photon energy, but electron–hole pairs generated for each photon would remain the same. This is one reason why in some UV detector lexicon, the responsivity is defined not in terms of A W1, but A ph1 to remove the penalty due to increasing power with increasing photon energy with no increase in photon count. The dependence of responsivity of the photodiode, depicted in Figure 4.39, on the n-layer (absorbing

4.4 Nitride-Based Detectors 0.2 µe

λ=350 nm

Responsivity (A W-1)

0.18

λ=300 nm

0.16

λ=250 nm

0.14 0.12

-2

-1 -1

300 cm V s

µp

20 cm V s

S

100 cm s

Lp

1.0 µm

-2

-1 -1 -1

d

0.1 µm

Nd

10 cm

18

-3

Na

17

-3

10 cm

3.0 µm

Le

0.1 0.08 0.06 0.04

0

n-type

t

p-type

d

0.5

1

1.5

2

Top layer thickness (µm) Figure 4.40 The dependence of responsivity on the n-layer (absorbing layer) thickness of the photodiode for wavelengths of 350, 300, and 250 nm assuming a minority carrier diffusion length, Lp ¼ 1 mm. It is clear that the responsivity is peaked at a certain thickness, which depends on wavelength, and degrades for shorter wavelengths due to wasted photon energy above the bandgap. Courtesy of F. Qian.

layer) thickness is shown in Figure 4.40 for wavelengths of 350, 300, and 250 nm, assuming a minority carrier diffusion length Lp ¼ 1 mm and with no external bias. As expected, the responsivity peaks at a certain thickness, which depends on wavelength and degrades for shorter wavelengths due to the wasted photon energy above the bandgap. In well-designed photodetectors, the absorption is made to take place in the depletion region where the electric field separates all the photogenerated carriers. This is accomplished by using window layers and optimal thickness and bias conditions. 4.4.2.3.1 Noise Under Reverse Bias Measurements of noise power density for GaN homojunction diodes as a function of frequency and temperature were taken in the reverse bias ranging from 5 to 30 V and in a temperature range of 298–523 K. The data at 10 V are shown in Figure 4.41 [143]. The equipment was calibrated by measuring the room-temperature noise current of a conventional 600 W resistor. At elevated temperatures and/or high values of the reverse bias, the dark current of the photodiode increased above 1  109 A, dominated by 1/f noise. All of the measured noise spectra satisfy the standard expression for the 1/f noise given by Equation 4.55 with the fitting parameter g in the range 1.0
g
1.1. The fitting parameter S0 (S0 ¼ Sn f g =I 2d ), which relates the noise power to the dark current, is plotted in Figure 4.42 as a function of bias voltage and sample temperature. The S0 values decreased slightly with increasing bias and rapidly with increasing temperature. The dark current noise spectra are near the noise floor of the measurement apparatus for GaN photodetectors at room temperature below a reverse-bias voltages

j793

j 4 Ultraviolet Detectors

794

Noise current density2(A2Hz-1)

10-21

-10V bias 250oC

10-22

Thermal noise current of 600 Ω load resistor at 25oC

200oC 150oC

10-23

100oC 10-24 25oC 10-25 102

103

104

Frequency (Hz) Figure 4.41 Dark current noise spectra (squared) and corresponding fits at five different temperatures. Noise current density of a 600 W resistor is shown for comparison as a horizontal line [143].

Noise coefficient, α(SI ( f )/Id2)

of 10 V [143–145], often making it necessary to perform the measurements either at larger reverse biases or at higher sample temperatures. Noise characteristics of the three types of UV detectors (GaN homojunction, Al0.03Ga0.97N homojunction, and Al0.1Ga0.9N/GaN heterojunction) fabricated in the author’s laboratory and measured in the laboratory of Prof. H. Temkin at Texas Technical University are tabulated in Table 4.5. Because the detector noise for all three devices at small bias values was below the noise floor of the measurement setup, the noise characteristics were measured at a reverse bias of 28 V. The corner frequency fc (the frequency where the 1/f noise density is equal to the shot noise density) was also noted. The NEP is then found as usual by calculating the total rms noise current and dividing it by the responsivity of the device. In the photodetectors reported by Kuksenkov et al. [143], the dark current at 3 V was 2.7 pA, resulting in Sn (at 1 Hz) ¼ 7.3  1029 A2 Hz1. For this bias, the 10 -5

25 ºC 100 ºC

150 ºC

10-6

200 ºC 250 ºC

0

5

10

15

20

25

30

Bias voltage (V) Figure 4.42 Bias and temperature dependence of the proportionality coefficient between dark current and noise power. Courtesy of H. Temkin.

4.4 Nitride-Based Detectors Table 4.5 Noise data for GaN homojunction, Al0.03Ga0.97N

homojunction, and Al0.1Ga0.9N/GaN heterojunction devices. Parameter

GaN diode

AlGaN diode

AlGaN/GaN diode

Id @ 28 V Id @ 2 V fc NEP (total) NEP @10 kHz

5 nA 7 pA 158 Hz 1.18 · 1011 W 2.14 · 1014 W Hz1/2

7 nA 7 pA 1683 Hz 9.1 · 1012 W 1.25 · 1014 W Hz1/2

7 nA 7 pA 1683 Hz 2.06 · 1011 W 2.49 · 1014 W Hz1/2

Bandwidth was assumed to be 30 MHz for GaN and 50 MHz for AlGaN. Responsivity was assumed to be 0.07 A W1 for GaN and 0.12 A W1 for AlGaN. 1/f noise power density at 1 Hz is 8  1018 A2 Hz1 for the Al0.1Ga0.9N/GaN diode, 3.8  1021 A2 Hz1 for the Al0.03Ga0.97N diode, and 1.8  1022 A2 Hz1 for the GaN diode.

amplitude of the 1/f noise for frequencies above approximately 85 Hz is lower than that of the shot noise Sshot 8.6  1031 A2 Hz1. The corresponding NEP ( f 100 Hz) was 6.6  1015 W Hz1/2, comparable to the best results reported for ultraviolet-enhanced silicon photodiodes. 4.4.2.3.2 Noise Measurements Under Forward Bias Let us now turn our attention to forward biasing where the built-in electric field in the depletion region is lower and the current is dominated by diffusion in the neutral regions. The hopping conductivity is no longer the dominant dark current mechanism [143], the 1/f current noise disappears, and most devices show a very clear G–R noise component with a Lorentzian spectrum of the form (see Equation 4.54) [8].

Sn ðwÞ ¼ A

t0 1 þ ðwt0 Þ2

where A is a constant and t0 is the characteristic time constant of the generation–recombination process. For an accurate determination of t0 from experimental data, it is convenient to multiply the measured noise density Sn =I 2d by w ¼ 2pf and plot it in linear scale as a function of frequency f [146]. The Lorentzian spectrum then appears as a symmetrical peak about a frequency f0, and t0 is then found from t0 ¼ 1/2pf0. Monitoring the frequency dependence of the normalized noise density under forward bias, at three different temperatures, one notes that f0 increases (and therefore t0 decreases) with temperature. For a thermally activated G–R process, the characteristic time t0 follows an Arrhenius form e t0 ¼ t00 exp ; ð4:118Þ kT where t00 is a constant and e is the activation energy. The value of e can be estimated from the slope of log(t0T2) plotted against the inverse temperature [147]. A value for the activation energy e of approximately 0.49 eV was obtained, [143] which is in agreement with the activation energy for point defect centers associated with Ga antisite defects [148]. Refer to Volume 1, Chapter 4 for details regarding antisites.

j795

j 4 Ultraviolet Detectors Noise current density (A Hz -1/2)

796

3.2×10-13 3.0×10-13 2.8×10-13 2.6×10-13

Shot noise

2.4×10-13 2.2×10-13 2.0×10-13

0.15

0.18

0.21

0.24

0.27

0.30

Photocurrent (µA) Figure 4.43 Measured spectral noise density at 20 kHz, under illumination, as a function of photocurrent. Solid line shows the calculated shot noise level. Courtesy of H. Temkin [8].

4.4.2.3.3 Noise Measurements Under Illumination Noise measurements on illuminated GaN photodiodes were performed by Kuksenkov et al. [143]. The photodiode bias was kept at 5 V, and the intensity of light produced by Xe arc lamp was adjusted by changing the current applied to the lamp. At low frequencies, the dominant noise source was that of the lamp itself. However, above approximately 10 kHz, the measured noise spectrum was completely flat. The corresponding noise current density is plotted in Figure 4.43 as a function of the photocurrent Ip. The noise pffiffiffiffiffiffiffiffiffi changes with illumination as 2eI p , indicative of domination by shot noise [26]. 4.4.2.4 AlGaN/GaN Heterojunction Detectors GaN homojunction photodiodes frequently suffer from relatively low external quantum efficiencies and nonuniform spectral response that shows a peak at the band edge, 360 nm, and a falloff at shorter wavelengths because of absorption in the surface layer [61]. The carriers that are not within a diffusion length of the junction do not usually contribute to the quantum efficiency. To exacerbate matters, the absorption coefficient of GaN increases with decreasing wavelength, and consequently the UV light is absorbed very close to the surface. The small diffusion constant of electrons in p-type GaN due to the poor quality of the materials makes matters worse. Specifically, most of the electrons and holes that are absorbed within 0.14 mm of the surface will recombine before they can diffuse to the depletion region, degrading the efficiency at these shorter wavelengths. Addition of a p-AlGaN window layer allows more of the incident light to propagate to the intrinsic absorption layer, which results in improved quantum efficiency [149]. However, the higher resistivity of AlGaN results in a current crowding effect because of the large lateral resistance that leads to spatial variations in the responsivity and the time response. To reduce the field crowding, GaN/AlGaN heterojunction p-i-n photodiodes with a semitransparent p-contact are employed [60, 150]. A schematic cross section of such a singleheterojunction device is illustrated in Figure 4.4c. Noise measurements of recessed-window Al0.13Ga0.87N/GaN p-i-n photodiodes with semitransparent p-contacts were carried out by Kuryatkov et al. [151]. The diode

4.4 Nitride-Based Detectors

structure was grown on sapphire substrate by metalorganic vapor-phase epitaxy. A thin p-metal of 50 A Ni/100 A Au covered the entire mesa, resulting in a uniform distribution of the electric field throughout the device active volume. Quantum efficiency of these photodiodes reached 38% at 15 V. The response remained relatively flat in the spectral region from 360 down to 330 nm. The 300 nm efficiency was lower, about 22% at 15 V. Room-temperature I–V characteristics typical of AlGaN/GaN single-heterojunction [8] photodiodes have been measured for two devices, with the device having a mesa diameter of 50 mm exhibiting a dark current below 1014 A at 5 V. The larger devices with a mesa diameter of 250 mm exhibited a dark current below 1010 A at 5 V. Because the current for the small-area device near zero bias is below measurement capability, the zero resistance (used to calculate the detectivity) must be deduced via extrapolation. Substantial photoresponse was noted up to 360 nm light excitation, indicating flat photoresponse down to zero bias. I–V measurements at temperatures up to 250 C lead to the deduction of a 0.4 eV dark current activation energy, which did not vary with the bias ranging from 1 to20 V. This is similar to the activation energy previously obtained in gate leakage measurements of AlGaN/GaN field effect transistors [152]. As indicated above, the zero-bias detector resistance R0 is estimated from the temperature-dependent I–V characteristics [8]. In thermally limited case, the quantity R0A0 (the product of the zero-bias resistance and the area of the diode) is a commonly used figure of merit (Equations 4.47 and 4.89). The R0 values are 2.5  1014 and 4  1015 W for the large and small devices, respectively, which results in a R0A0 product of 1.13  1011 W cm2 for the large device. Because the dark current of the small device cannot be measured, all that can be said about the R0A0 product is that it is larger than that for the large device. Inserting the measured responsivity Rl of about 0.1 A W1 and the RA values discussed above into Equation 4.89, a roomtemperature thermal noise-limited detectivity of D  3.2  1014 cm Hz1/2 W1 is obtained at 360 nm. This detectivity, which is illustrated in Figure 4.2 as a solid square, is close to the performance of a UV photomultiplier discussed in Section 4.2.9. High values of R0A  1.5  109 W cm2 were reported for similar single-heterostructure AlGaN/GaN diodes by Brown et al. [13] resulting in roomtemperature detectivities of D  6.3  1013 cm Hz1/2 W1. Figure 4.44 shows the noise power density measured for an AlGaN/GaN heterojunction photodiode [8] as a function of reverse bias at 1 Hz. A 1/f-type noise was noted for reverse-bias voltages below 10 V. For higher bias voltages, though, the dependence changed to a 1/f g dependence with g ¼ 3. This suggests that the noise is due to a number of generation–recombination centers [153]. Increasing the frequency above 1 kHz changed the noise characteristics in that shot noise was detected, reducing the primacy of the thermal noise. The overall noise levels were still low in that the signal was below the experimental floor (2  1029 A2 Hz1) at room temperature for frequencies greater than 200 Hz. The temperature dependence of the noise power density showed an activated behavior with energy of 0.37–0.42 eV, which is in the same ballpark with 0.4 eV determined from I–V data. The data of Figure 4.44 shows a welldefined exponential dependence of spectral noise power density Sn with bias. It

j797

j 4 Ultraviolet Detectors Noise power density (A2 Hz-1)

798

1E-18 o

T=22 C, f =1 Hz

1E-20

250 µm

1E-22 1E-24 50 µm

1E-26 1E-28 1E-30 1E-32

0

5

10

15

20

25

30

35

Voltage (V) Figure 4.44 Noise spectral density of a GaN/AlGaN singleheterojunction diode as a function of the bias voltage. Measurements were carried out at room temperature at 1 Hz. The zero-bias noise spectral density is extrapolated from higher bias measurements. Also indicated is the experimental floor of the measuring apparatus [151].

changed by some five to eight orders of magnitude as the bias was changed from 5 to 30 V for larger and smaller devices, respectively. This well-behaved bias dependence paved the way for extrapolating the values of Sn at zero bias, which is the bias most imaging applications require. The extrapolation exercise led to zero-bias spectral noise densities (Sn) of 1.5  1030 and 3.6  1032 A2 Hz1 for 250 and 50 mm diameter photodiodes, respectively [8]. The 1 Hz data of Figure 4.44 replotted as a functionp offfiffiffiffiffiffiffiffiffiffiffiffiffiffi dark current could be fitted ffi with a phenomenological expression Sn ¼ ðI2d =f g Þ A0 =A00 ; where A00 is the characteristic area parameter and A0 is the area of the diode (both measured in cm2) and Id is the dark current of the photodiode. The area dependence of the noise power density indicates a bulk, not surface, scaling of the noise, indicative of noise caused by conduction and threading dislocations, which is in good agreement with GaN homojunction photodiodes [154]. With the help of Equation 4.87 and spectral noise power in hand, one can calculate the NEP for these photodiodes. The NEP values are 1.2  1014 and 1.9  1015 W for 250 and 50 mm diameter photodiodes, respectively [8]. As defined in Equation 4.87, the NEP is the noise current density divided by the responsivity and Df, Dv, or B is the electrical bandwidth of the receiver, included because only the noise seen by the receiver is important. Using Equation 4.89, specific detectivities of D of 3.5  1013 and 5.2  1013 cm Hz1/2 W1 were calculated for device diameters of 50 and 250 mm, respectively [8]. An equally important figure of merit for a photodetector is its speed, though certain applications do not really challenge the speed limits of either the material system or the device structure. The time responses for the devices incorporating several design features such as all GaN, all AlxGa1xN, and heterojunction types at zero bias have been measured at 355 nm using a pulsed nitrogen laser by monitoring the decay of the photocurrent. The exponential decay times of the photocurrent of the GaN, Al0.05Ga0.95N/GaN, and Al0.1Ga0.9N p–n were found to be about 29, 22, and 12 ns,

4.4 Nitride-Based Detectors

10

1

Relative response

Relative response

10 0

-1

0 0

10

10

10

20

30

40

50

-2

GaN p-i-n, τ=29 ns Al0.1Ga 0.9N p-i-n, τ=12 ns Al0.05Ga0.95N p-i-n, τ=22 ns

-3

0

200

400

600

800

1000

Time (ns) Figure 4.45 The time decay of the photocurrent for GaN, Al0.1Ga0.9N/GaN, Al0.05Ga0.95N p-i-n detectors. The inset shows the same over an expanded time scale below 50 ns.

respectively, as shown in Figure 4.45. These particular data are very important because they demonstrate how defects can impact the speed of response in p–njunction varieties as well. Fall times as low as 6 ns have been measured [155]. These detectors exhibited two regimes in the photocurrent time decay. At high excitation intensities, trap states in the GaN bandgap presumably become saturated, and the detectors are then capable of resolving the 8 ns excitation pulse. At lower excitation intensities, a longer time response tail dominates the decay. In this regime, the detectors have a rise time of less than 5 ns and a fall time (1/e) of approximately 31 ns. The response speed data in more advanced and developed devices utilizing improved materials and in p–n-like structures were near or under 1 ns at a reverse-bias voltage of 10 V, which is strongly influenced by the series resistance through the player [156]. When the device is illuminated near the center of the window and away from the contact area, the decay time increased. The 3 dB point is nearly 1 GHz. A speed of response as fast as 90 ps has been reported [157]. We have to reiterate that the speed is determined by convolution of the RC time constant and the transit time. In this regard, p–n varieties provide low capacitance due to a large depletion region, and so the transit time can be reduced by the application of a reverse-bias voltage, which increases the electric field. The caveat is that the dark current will increase with the application of a bias voltage if the diode is leaky. Back illuminated, through the substrate side (as shown in Figure 4.4e), solar-blind detectors and arrays with Al0.45Ga0.65N absorber layer and Al0.6Ga0.4N window layer have been operated at or near zero bias [17]. The choice of zero bias is motivated by negligible current flow through the device, which renders the 1/f noise a nonissue. In addition, at least ideally, the doping level in the absorber layer (for a given thickness) is chosen so as to cause it fully depleted at zero bias and allowing the photogenerated carriers to be swept by the electric field. This has the advantage that carrier transport is

j799

j 4 Ultraviolet Detectors

800

Probe data 250 µm diameter 13 diode with R 0 >10 Ω

10-3 10-4

and R 0 A >109 Ω cm2

Current,I(V)-I0(A)

10-5

1014 1013 1012 1011

n = 2.3

10-6

1010

10-7

109

10-8

108

10-9

107

10-10

106

10-11

105

10-12

104

10-13

103

10-14 -10

-8

-6

-4

-2

0

2

4

6

8

Dynamic resistance (Ω)

10-2

102 10

Voltage (V) Figure 4.46 Current–voltage characteristics and dynamic resistance versus voltage for the best regions of the best solarblind films measured in dark at room temperature. Courtesy of M. Reine BAE systems [17].

not dependent on the diffusion process in high Al concentration AlGaN with a small diffusion length, which leads to high quantum efficiencies. The back-illuminated detector has the advantage that the entire top (front) GaN surface can be covered with p-type ohmic contact alleviating somewhat the series resistance-related effects through this layer. Current–voltage characteristics of such solar-blind detector with 250 mm diameter were measured under both dark and illumination conditions using a parameter analyzer with low-noise triax probes, leading to a current floor of about 1  1013 A, which can be measured. Extremely low leakage currents have been achieved in the solar-blind detectors, a representative best result of which is shown in Figure 4.46, where the current is shown on the left axis and dynamic resistance on the right. The straight line in the semilogarithmic curve indicates the ideality factor, which ranged between 2 and 5. Near zero bias, the current is too small to measure on the probe station. True solar-blind performance can be obtained only from heterojunctions of AlGaN/GaN, and the type of device structure illustrated in Figure 4.4e. Preparation of solar-blind photodetectors and optimization of their performance are thus of considerable interest. A representative photoresponse in the form of wavelength dependence of the external quantum efficiency for back-illuminated solar-blind detector on sapphire obtained at zero bias is illustrated in Figure 4.47. A schematic cross section of this device is shown in Figure 4.4e. The device is designed for illumination through the sapphire substrate, which is highly transparent to the radiation wavelength of interest. It consists of three layers. The middle layer (absorbing layer) has an AlN molar fraction of about 0.45 to produce peak responsivity

4.4 Nitride-Based Detectors 60%

Cut-on = 262.6 nm Cutoff = 279.3 nm FWHM = 16.8 nm QE (272 nm, V=0) = 51.5%

Quantum efficiency (external)

50% 40% 30% 20% 10%

0% 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380

Wavelength (nm) Figure 4.47 Spectral response at V ¼ 0 for back-illuminated solarblind AlGaN-based photodiode. Courtesy of M. Reine BAE systems [17].

near 280 nm. The composition of this layer also determines the cutoff response on the long wavelength side. The lower, n-type AlGaN layer, which needs to be transparent to the wavelength of interest, generally contains an AlN molar fraction of about 0.6, which corresponds to the band edge of 4.85 eV or 255 nm, see Figure 4.10. This layer determines the cutoff wavelength response on the short wavelength side. The top AlGaN layers together provide a bandpass filter that is responsible for the spectral responsivity curve shown in Figure 4.47. Because the detector is bottom illuminated, the top p-layer can be made of GaN. This design results in photodetector response limited to the 260–280 nm range and high quantum efficiency [14]. However, it is necessary to achieve ohmic contacts on this layer. If not, the carriers can be collected by the Schottky barrier (formed by nonohmic contacts), which thus formed is then responsible for out-of-phase photoresponse shown in Figure 4.48. The performance of the arrays made of these solar-blind detectors is discussed later in Section 4.5. 4.4.2.5 AlGaN Detectors Including the Solar-Blind Variety Solar-blind detectors of the p–n-junction variety require high mole fraction p-type AlGaN for which OMVPE has so far been more conducive. The layers are grown on sapphire substrates because of their transparency to the wavelength of interest, 280 nm, in a back-illuminated configuration. The growth on substrates begins with a thin nucleation layer of AlN followed by a 1 mm thick n-type layer of AlGaN (an undoped absorbing layer of AlGaN with a somewhat lower Al content) and a 0.5 mm thick p-type layer of AlGaN. To produce peak responsivity near 280 nm, the AlN molar fraction in the absorbing layer should be about 0.5, as mentioned previously and shown in Figure 4.10. If back illumination through the substrate is desired, the bottom AlGaN layer (buffer layer) should be transparent to the wavelength of interest. This leads to an Al content of about 0.55–0.6, considering the bandtailing. With the Al

j801

j 4 Ultraviolet Detectors

802

100

Quantun efficiency (external)

Wafer probe data taken on 250 µm diameter diodes at 500 Hz, V = 0 10-1

10-2

10-3

10-4 250

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

Wavelength (nm) Figure 4.48 Spectral response of three different solar-blind AlGaN detectors inclusive of the one shown in Figure 4.47 in logarithmic scale showing an additional response band between 300 and 380 nm due to nonohmic top contact. The carriers generated in the spectral response in question are collected by this undesirable Schottky barrier, which can be eliminated by increased p-doping. Courtesy of M. Renie BAE systems [17].

content of the p-type layer fixed at 0.45–0.5, the photodetector response in the range of 250–280 nm can be obtained. Though, top illumination is not really feasible because of the difficulty in obtaining p-type AlGaN with molar fractions above that of the absorber layer. With precise fabrication of the n- and p-type layers and a high-quality epitaxial growth, this design yields high peak responsivities and quantum efficiencies, as discussed below. For instance, a responsivity Rl  0.051 A W1 has been obtained at 273 nm, corresponding to an internal quantum efficiency of 27% [158]. A zero-bias responsivity of 0.041 A W1 with quantum efficiency of 23% has also been obtained at 243 nm by using double-heterostructure photodetectors optimized for either top or through-the-substrate illumination [159]. The solar-blind rejection ratio was four orders of magnitude. The highest detectivity in an AlGaN photodetector is of the type illustrated in Figure 4.4d, D ¼ 3.2  1014 cm Hz1/2 W1 at 275 nm and 1.9  1014 cm Hz1/2 W1 at 269 nm, and zero-bias differential resistance is R0 ¼ 1.47  1014 W, which was described by Collins et al. [10, 160]. This device achieved a remarkable combination of excellent optical and electrical characteristics, a zero-bias external quantum efficiency of 53% (Rl  0.12 A W1), and low leakage current density of 8.5  1011 A cm2 at a reverse bias of 5 V.

4.4 Nitride-Based Detectors 100

Current (fA)

0 -100 -200 -300 -0.2

-0.1

0.0

-0.1

0.2

Bias voltage (V)

(a) 13

1.6 × 10

13

dV/dI (Ω)

1.2 ×10

12

8.0 ×10

4.0 × 10

12

0.0 -0.2

(b)

-0.1

0.0

0.1

0.2

Voltage (V)

Figure 4.49 (a) Current–voltage characteristics and (b) differential resistance near zero bias of an AlGaN heterostructure photodiode. Patterned after Ref. [158].

The I–V characteristics and the zero-bias resistance of a double-heterostructure photodetector with a 200  200 mm2 square mesa are shown in Figure 4.49 [158]. The dynamic resistance dV/dI peaks near zero-bias point for R0 ¼ 2  1011 W, another device with square mesa of 200  200 mm2, corresponding to the product R0A  8  107 W cm2 [158]. This yields room-temperature thermal noise-limited detectivity of D  3.3  1012 cm Hz1/2 W1 at a wavelength of 273 nm. Figure 4.50 plots the zero-bias resistance of a 50 mm diameter double-heterostructure photodiode as a function of temperature [8]. The responsivity Rl  0.03 A W1 and the zero-bias resistance R0 is 1  1011 W at 520 K. R0 increases rapidly with decreasing temperature, reaching 3  1014 W at 350 K. The extrapolated value of R0 at 300 K is 2.5  1016 W, which is not as reliable as that determined by direct measurements. The product R0A0 is 6.1  1012 W cm2, which results in a thermal noise-limited detectivity of D  3.8  1014 cm Hz1/2 W1 at 280 nm. The datum point is shown in Figure 4.2 as an open circle [11]. As discussed in detail in Volume 2, Chapter 4, hopping conductivity through dislocations in GaN p–n-junctions gives rise to exponential dependence of current on temperature at low electric fields. The data shown in Figure 4.50 suggest a similar mechanism in AlGaN. The field dependence of the dark current of AlGaN heterojunctions, the fundamentals of which are discussed in detail in Volume 2,

j803

j 4 Ultraviolet Detectors

804

Photoresponse (a.u.)

1E19 1E18 1E17

(R 0 )D(Ω)

1E16 1E15

240

1E14

260 280 300 Wavelength (nm)

320

1E13 1E12 1E11 1E10 1E9

(R 0 )300 K=2.5 × 1016 Ω (R 0)diff =kT/ql sV=0 300

350

400

450

500

550

Temperature (K) Figure 4.50 Zero-bias resistance R0 of an AlGaN heterostructure photodiode measured as a function of temperature. The inset shows spectral response of the photodetector [8].

Chapter 4, is in agreement with the Poole–Frenkel conduction mechanism with a Poole–Frenkel constant of bPF ¼ 3.3  104 eV V1/2 cm1/2, very similar to that obtained for GaN [8]. The importance of dislocations in dark conductivity of AlGaN was addressed by epitaxial lateral overgrowth experiments, similar to those illustrated in Volume 2, Figure 4.31, and applied to GaN devices [161]. Lateral growth of GaN from the seed layer was continued until the adjacent wings coalesced, resulting in a planar wafer. Homojunction-type photodetectors based on Al0.33Ga0.67N were formed on this template wafer. Diodes were fabricated on dislocated areas, directly over the seed layer, on low dislocation density wing areas, and on coalesced areas that contain some defects and dislocations. Diodes fabricated over the wing areas had very low leakage currents, 10 nA cm2, at a reverse bias of 5 V. Leakage currents in diodes formed over coalesced areas were about an order of magnitude higher. Leakage currents in dislocated diodes were larger again, by as much as six orders of magnitude. The discussions so far clearly indicate that extended defect concentrations correlate to dark current in photodiodes. Naturally, material with lower defect concentration would lead to smaller dark currents. In this respect, the growth of device structures on ELO GaN templates, as was done for GaN homojunction devices, is very promising. When ELO templates are used, the devices can be placed on regions in the windows with high defect concentration and on the wings with lower defect concentration. They can also be placed on the coalescent boundaries where the defect concentration is also high. In this way, Al0.33Ga0.77N-based p-i-n photodiodes have been fabricated on ELO GaN [161]. In this particular ELO process, the windows formed by a SiO2 mask were 5 mm and spacing between the windows was 35 mm. The photodiodes of 10  100 mm2 were formed on the wings. Diodes of 30  100 mm2 active area were fabricated on the coalesced regions, and photodiodes of 300  300 mm2 size were fabricated on standard GaN. Leakage current densities as low as 10 nA cm2 have been measured for the smallest devices fabricated on the wing regions. This leakage current density was one

4.4 Nitride-Based Detectors

Leakage current density, JL (A cm -2 )

10

-10

10

10

10

10

Wing

-8

-6

Coalesced -4

-2

10

Dislocated 0

-20

-15

-10

-5

0

Reverse bias, V R(V) Figure 4.51 Typical leakage current densities on p–n photodiodes fabricated from the epitaxial lateral overgrown region of ELO GaN, and from high dislocation density GaN [161].

order of magnitude higher in devices on the coalescence boundaries, and the leakage current was more than six orders of magnitude higher in devices fabricated in the standard GaN on sapphire (depicted as “dislocated” in Figure 4.51). As is the case in previously discussed devices, the low value of the dark current is correlated to reduction of the dislocation density and thus the defect-assisted tunneling [162]. Another important benefit of reduced dislocation density is improved spectral response, specifically a relatively sharp roll-off of the responsivity below the band edge [161]. In this vein, the spectral response curves obtained in AlGaN homojunction photodiodes prepared on ELO templates are illustrated in Figure 4.52. This structure provides an ideal platform in that the window regions contain the standard density of dislocation expected of GaN grown directly on sapphire. The wing regions represent a much lower dislocation density. Alas, the coalesced region boundaries again exhibit larger dislocation density in part because of wing sagging, lattice index disorder, and so on. It should again be pointed out that the implication here is that the higher dislocation material would have higher point defect density. As discussed in Volume 1, Chapter 4, the dislocations in GaN are to a large extent either electrically active due to local strain-induced trapping of impurities and point defects or directly active due to incomplete coordination of the lattice at those sites. Comparing devices prepared in these three different regions of the same wafer, and thus attempting to establish a correlation with the dislocation density, the peak responsivity of Rl  0.057 A W1 at 287 nm did not appear to be affected by the dislocation density. However, a significant difference was observed in responsivity curves below the band edge, as shown in Figure 4.52. In low dislocation density devices, the responsivity falls off rapidly at a rate of three orders of magnitude within 54 nm, which bodes well for solar-blind operation. It should be pointed out that ELO can be accomplished only with GaN, and thus the template on which the solar-blind section of the device is grown must be GaN. In top-illuminated detectors, the longer

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10 -1

Wing Coalesced Dislocated

Responsivity (A W-1)

10 -2

10 -3

10-4

10 -5 200

250

300

350

400

Wavelength (nm) Figure 4.52 Spectral response of an AlGaN homojunction p-i-n photodiodes fabricated in the epitaxial lateral overgrown region of ELO GaN, in the lateral growth region including the coalescence boundary, and on high dislocation density GaN [161].

wavelength light is not absorbed in the AlGaN layer grown on the GaN template and reaches this underlying GaN template layer where it is absorbed. In ideal material, the bandgap discontinuity between AlGaN and GaN is large and thus holes photogenerated in GaN do not contribute to the photocurrent. However, dislocations in GaN/ AlGaN structures act as leakage paths for the photogenerated holes, resulting in excess photoresponse at longer wavelengths. These observations are consistent with the Schottky photodiodes fabricated in the ELO material [124] as discussed earlier in Section 4.4.2.1. 4.4.2.6 AlGaN/GaN MQW Photodetectors MQWs in the active region of the devices are expected to enhance the quantum efficiency due to the high absorption coefficient. In a 2D system, a high joint density of states enhances the oscillator strength (due to confinement) for upward carrier transition, which can enhance optical absorption, leading to higher quantum efficiency in the photodetectors [128]. Strong piezoelectric fields, such as those existing in GaN/AlGaN quantum wells, are also expected to improve the carrier transport. MBE-grown MQW detectors prepared on c-plane sapphire substrates and fabricated in the back-illuminated vertical Schottky geometry, as well as photoconductive varieties, have been reported [163, 164]. The growth of the MQW structures was initiated by a thin AlN buffer layer (50 nm) on top of a c-plane sapphire substrate followed by an Si-doped AlGaN layer of 0.5 mm thickness (which acts as a short wavelength cutoff filter on top of the AlN layer). The active layer for all three detector structures investigated consisted of 20 periods of GaN/AlGaN MQWs with 4 nm well

4.4 Nitride-Based Detectors

Figure 4.53 (a) Schematic structure of a back-illuminated Schottky barrier multiquantum well UV photodetector. Growth parameters are given in the text. (b) Schematic conduction band profiles of MQW Schottky detector together with carrier transport [164].

thickness. The thicknesses of barriers were 7, 5, and 3 nm for samples A, B, and C, respectively. The growth sequence was finished by growing a thin AlGaN cap layer. The schematic of the device structure and potential profile are shown in Figure 4.53. A nearly flat spectral responsivity between 325 and 350 nm with a 0.054 A W1 peak responsivity was achieved from the single-side rough-polished backsideilluminated GaN/AlGaN MQW devices. The photocurrent exhibited a nearly linear increase with incident optical power up to 1 kW m2. The cutoff wavelength of the MQW photodetector can be tuned by adjusting the well width, well composition, and barrier height. A model has been developed to gain insight into the operation principles of MQWs photodiodes. The peak responsivity increased with decreasing barrier thickness due to enhanced tunneling of photogenerated carriers. The bias dependence of the responsivity for all three types of detectors has also been studied up to 5 V. Figure 4.54 shows the spectral responsivity taken at different bias voltages for sample C. As seen from Figure 4.54, the spectral responsivity curve does not show any significant difference as far as its shape is concerned, but the noise level increases with applied bias due to the noise of the voltage source. The inset shows the peak responsivity at 350 nm as a function of applied bias. Below 2 V, the responsivity increases linearly with applied voltage. Beyond 2 V, the responsivity saturates at 0.19 A W1. The saturated responsivity values for samples A and B were obtained as 0.052 and 0.12 A W1, respectively. The ratio between the saturated peak responsivity and zero-bias peak responsivity is about 3.6 times and the same for all devices, supporting the effect of the barrier width. It has been shown that the photocurrent was limited by tunneling through the barriers, which upon further optimization can lead to improved device performance. There are five optimization criteria. First, for a given Al composition and hence a given band offset, the lower limit on the well width is determined by the necessity of having at least two subbands inside the well, and the upper limit should be set to a value that facilitates an efficient intersubband energy relaxation through emission of optical phonons, DE
hwLO ; where
hwLO 92 meV is the optical phonon energy for GaN. Second, the thickness of the barrier should be neither very large (because of a

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808

Responsivity (A W-1)

0.2

0.15

0.25

A B C

0.2 0.15 0.1 0.05 0 0

1

2

0.1

4

5

6

0V -0.5 V -1 V -2.5 V -5 V

0.05

0 280

3

Applied voltage (V)

300

320

340

360

380

400

420

440

Wavelength (nm) Figure 4.54 Spectral responsivity at different bias voltages for the device fabricated using sample C. (Inset) The peak responsivity versus the applied voltage for all three devices fabricated using A, B, and C [164].

low tunneling probability) nor very small (due to low absorption efficiency at shorter wavelengths). Third, to provide resonant tunneling, the average electric field should be about equal to the energy separation of the first excited state and the ground states (to do so for a given Al content, the thickness of the well, the barrier, and the ratio between, as well as the period of the superlattices and the number of periods have to be taken into account). Fourth, quality of the layers must be considered. Finally, to obtain a complete picture of device operation for comparison with the experiment results, calculation of the current densities because of tunneling and thermionic emission needs to be included in the proposed model. 4.4.2.7 Heterojunction Phototransistors A bipolar junction transistor, which already has two back p–n-junctions, can also be used as a photodetector. With lattice-matched semiconductors whose bandgaps can be adjusted, the designer could choose to have the light absorbed in the emitter, within a diffusion length of the emitter–base junction, in the base, or in the depletion region at the base–collector junction. If the photons are absorbed in the emitter, the minority carriers diffuse across the junction, are injected into the base, diffuse through the base, and are collected by the collector junction. If the emitter is of a larger bandgap material than that of the base and the photogenerated carriers are in the base, the holes get confined to the emitter–base interface, assuming npn transistor, because that junction would be forward biased, causing electron injection from the emitter into the base. Electrons then diffuse across the base before being collected at the collector junction. Because of the nature of heterojunction, electron injection is much greater than the hole back injection into the emitter, giving rise to gain. If photogeneration takes place in the depletion region of the base–collector

4.4 Nitride-Based Detectors

Cap

Base

Emitter

∆EC

e-current Collector

EC qVBE qVCB

∆E V

EV

EFC

h-current WB

WE

xne

xpe

WBC

xpc

WC

xnc

Figure 4.55 Schematic representation of the band diagram of an npn floating base phototransistor with electron and hole currents indicated assuming that carriers are generated only in the base and collector depletion regions that can be accomplished for the photon energies below the bandgap of the emitter.

junction, the carriers are swept away by the field, electrons to the collector and holes to the base in the npn variety. The holes in base diffuse toward the emitter junction, and so the transistor acts as though the photons were absorbed in the base. Because the base is floating, the minority carriers generated by the incident light control the total current flow. A schematic band diagram of an npn heterojunction transistor is shown in Figure 4.55. Very high gains in the tens of thousands can be obtained. In the case of FETs, the channel conductivity through photogeneration of carriers is modified. In an n-channel device, the electrons would be accelerated toward the drain and holes toward the source. Except for being used side by side with electrical FETs or for taking advantage of an established fabrication methodology, the motivation to specifically fabricate FET-based phototransistors is not really that strong. As in other semiconductors, GaN-based BJTs [165] and FETs have been reported [166]. Yang et al. [165] fabricated a GaN(n)/GaN(p)/GaN(i)/Al0.20Ga0.80N(n) heterojunction phototransistor. The larger bandgap material Al0.20Ga0.80N is near the substrate, and the light is shone through the transparent sapphire substrate. The UV light is thus absorbed in the GaN(i) layer. The photogenerated electron–hole pairs are separated by the electric field in the i-region, and the electrons and holes are drifting toward the base and the collector, respectively. The accumulation of holes in the floating base increases the injection of electrons from the emitter, resulting in a current gain. A gain larger than 105 has been demonstrated. The photocurrent exhibited a sublinear behavior as a function of incident light power, but the phototransistor showed PPC in much the same manner as photoconductors. In normal operation, the holes would recombine in the base with the electrons injected from the emitter. However, the holes are trapped in defects, which reduce their recombination rate, causing persistent charge effects. Yang et al. [165] have demonstrated that the recombination is enhanced when a bias is applied to the detector to reduce the barrier for hole back injection to emitter. A bias voltage pulse thus acts as an electrical flush. To avoid the persistent effects, the phototransistor can be subjected

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104

Responsivity (A W-1)

103 102 101 100 10-1 10-2 10-1 10-4 10-5 200

240

280

320

360

400

Wavelength (nm) Figure 4.56 Spectral response of a GaN-based phototransistor [165].

to a voltage flush pulse before each measurement (7–10 V were used for the purpose), and the photocurrent can be measured at a 3–4 V bias voltage. In these operating conditions, the responsivity still decreases as a function of the optical power and the gain is very dependent on the frequency, so that the gain bandwidth product remains at a constant value. The spectral response of the device to a 10 Hz frequencymodulated input optical signal is displayed in Figure 4.56. A UV/visible contrast over eight orders of magnitude is obtained [165]. AlGaN(n)/GaN(n)/GaN(i) heterostructure field effect transistors used in the photodetection mode (illuminated through the sapphire substrate) have also been reported [166]. The photogenerated holes in the channel rapidly drift to the source under the lateral electric field giving rise to drain current. The photogenerated electrons would do the reverse journey adding to the drain current. As in the photoconductive devices, the device would have a gain if the transit time is smaller than the recombination lifetime. These devices display high responsivity values that reach 3000 A W1 with a sharp cutoff edge and a time response on the order of 200 ms. However, due to the residual carriers in the channel, the dark current in FET-based phototransistors tends to be high for many detector applications. 4.4.2.8 GaN Avalanche Photodetectors Avalanche photodiodes offer the combined advantages of a fast operation, high gain, and high sensitivity [167]. The diode structure is basically a p–n type, and the device is operated in the reverse-bias regime at breakdown. The high field present in the iregion accelerates the photogenerated carriers until they reach a certain threshold energy (depending on the semiconductor material used), at which point they generate secondary electron–hole pairs across the bandgap. These in turn gain energy from the field and create their own secondary pairs and so on. This gives rise to large currents flowing in the reverse-bias direction. This carrier generation process is known as the impact ionization process and is responsible for carrier multiplication, generating gain. This is the reason why the process is called avalanching. The speed is

4.4 Nitride-Based Detectors

determined by the RC time constant and transit time constants. The gain bandwidth product is set by the materials and device design. The higher the gain, the lower the bandwidth. In addition, various material limitations (i.e., the electron and hole impact ionization coefficients are comparable, which appears to be the case in GaN) [168] give rise to noise, so despite the gain, these photodiodes are not necessarily automatically desirable. Calculations of the ionization parameters of electrons and holes (impact ionization rates) in GaN indicate that the electric field necessary for impact ionization is very large, on the order of 1.6 MV cm1 (values in the range of 2–3 MV cm1 have been predicted [169]). However, from the practical point of view, it is most important to note that the high defect density in the GaN material makes it very difficult to achieve homogeneous carrier multiplication over the entire optical area of the device because of possible microplasma formation. Photodiodes based on GaN and AlGaN have a reduced density of dislocations, resulting in improved performance; but surprisingly, devices are operable even with large numbers of dislocations. In the case of avalanche photodiodes, a minimum dislocation density is required before gain can be obtained. The high electric fields needed to initiate avalanche multiplication (greater than 1.6 MV cm1) simply cannot be reached in material with high dislocation density. A competing process of microplasma breakdown occurs at much lower electric fields. This is illustrated in the micrograph of Figure 4.57, which shows a GaN p-i-n diode, 200  200 mm2 in size, under a pulsed reverse bias of 80 V, corresponding to a field of 1.2 MV cm2 [8, 170]. The bright points correspond to regions of microplasma breakdown, each of which is most likely associated with a cluster of dislocations. Stringent processing technology is needed to avoid high field spikes, premature breakdown at or near the edges of mesas, and spatially localized high gain, which often require stringent passivation processes and high-quality materials.

Figure 4.57 A top view of GaN avalanche photodetector with 200  200 mm2 in size under a probe station and under bias, left. The magnified image of the active area of the device under reverse bias shows regions of microplasma breakdown, assumed to be associated with highly dislocated regions, right. The reverse bias equal to 80 V. Courtesy of Dr. A. Osinsky. (Please find a color version of this figure on the color tables.)

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In spite of the difficulties with the material quality, McIntosh et al. [171, 172] reported GaN avalanche photodiodes prepared by hydride vapor-phase epitaxy (HVPE). This method is capable of growing GaN at very high rates, enabling the production of thick layers with low dislocation density, details of which are discussed in Volume 1, Chapter 3. The device structure was placed on top of a 10 mm thick GaN buffer layer. Dry etched mesa diodes with diameters between 30 and 60 mm were prepared. Resulting devices were characterized by low dark current (100 fA at an unspecified low bias), but this value was measurement limited. With a reverse bias as large as 150 V, the dark current was still below 200 nA. A maximum gain of 10, corresponding to an external quantum efficiency of 350%, was measured just below the reverse breakdown bias of 220 V. This gain was spatially uniform (as demonstrated by two-dimensional raster images of the photocurrent) and not high by avalanche photodiode standards. Much larger avalanche gains could be realized in the photon counting or Geiger mode [172, 173]. In this operation regime, a reverse bias greater than the breakdown voltage is applied for just a few microseconds. If photon is absorbed during this short pulse, the voltage across the device is reduced by a few volts (about 1–2 V). The voltage drop is monitored by a decision circuit connected to the diode that counts the number of pulses above a preset threshold. Using this mode, effective gains as large as 107 have been reached with GaN avalanche photodiodes, which some consider a semiconductor answer to PMT-based detectors. GaN-based avalanche photodiodes have also been reported using material grown by OMVPE [174, 175]. Figure 4.58 shows the room-temperature I–V characteristics of a GaN avalanche photodiode. The device diameter was 24 mm, and a low leakage current of about 2  1011 A at a reverse bias of 5 V was attained [174]. For low bias voltages, down to zero bias, the responsivity of the device was independent of the voltage. The dependence of gain on the applied voltage is also shown. A clear onset of gain with respect to voltage is noted near 35 V, and the maximum gain of approximately 25 was achieved near the breakdown of 48–50 V, which led to the estimation

40 10-4

35

Photocurrent

30 25 Dark current

10 -8

20

Gain

Current (A)

10 -6

15 10 -10

10

Gain

5

10 -12

0 0

10

20

30

40

50

Reverse bias (V) Figure 4.58 Current–voltage and gain–voltage plots for a GaN avalanche photodetector [174].

4.4 Nitride-Based Detectors

that the breakdown field is about 3.5 MV cm1, consistent with devices prepared with HVPE. The temperature dependence of the avalanche breakdown process in GaN diodes has been experimentally analyzed in some detail [173, 176]. As mentioned above, each photogenerated carrier in the depletion region generates, on average, one electron–hole pair at the breakdown voltage. This corresponds to the condition aW ¼ 1, where a is the ionization coefficient and W is the depletion width [167, 177]. As mentioned earlier, the electron and hole ionization coefficients in GaN are expected to be comparable to a ¼ 5  104 cm1 and b ¼ 4  104 cm1 for electrons and holes, respectively, at 4 MeV cm1 [168]. This is in excellent agreement with the experimentally determined electron impact ionization coefficient a ¼ 2.9  108 exp (3.4  107/E ) cm1, where E is the electric field [178]. The depletion width in a GaN avalanche photodiode at breakdown can then be written as W  (ab)1/2  0.22 mm, which is in agreement with the measured value W  0.26 mm. The depletion width condition is not satisfied in devices with defect-induced premature breakdown. In well-established semiconductors, the breakdown voltage has a positive temperature coefficient making the breakdown voltage to increase with temperature. In early developmental stages of relatively new semiconductors, however, poor quality of the material led to erroneous conclusions indicating otherwise, as was the case with SiC. In GaN between 100 and 250 K, the avalanche breakdown voltage has been reported to increase from about 72 to 92 V, which follows the tradition of the other conventional compound semiconductors. Above 200 K, the temperature dependence of the breakdown was linear and could be described by a coefficient dVb/dT 0.2 V K1. The temperature dependence of the breakdown could be modeled by assuming that avalanche gain was limited by carrier–phonon scattering [176]. In this model, the energy gained by an accelerating charge is proportional to the applied field eVb/W, where Vb is the breakdown voltage and W is the depletion width at breakdown. Knowing that carriers generated by excitation of valence electrons to the conduction band limit the energy gain at most to the bandgap, one gets lqVb/W ¼ Eg, where l is the mean free path between collisions, and W is the depletion width. For l ¼ l0(2N þ 1)1, where l0 is the T ¼ 0 K mean free path and N is the phonon occupation number, the breakdown voltage can be written as VB ¼

EgW ð2N þ 1Þ; ql

where the phonon occupation number is 
 1 hvq 1 ; N ¼ exp kT

ð4:119Þ

ð4:120Þ

where hvq is the effective energy of the phonons involved in the scattering process. With the measured dEg/dT ¼ 6.7  104 eV K1 (see Volume 1, Chapter 1 for details of the thermal properties of nitrides) and Eg ¼ 3.4 eV, an excellent fit was obtained to the measured temperature dependence of the breakdown voltage by assuming l0 ¼ 13 nm, the phonon energy of 42.3 meV. The effective phonon energy is within the range of active Raman phonon modes in GaN, which are listed in

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Volume 1, Chapter 1. The agreement between the measurement and the model confirms the avalanche nature of breakdown and imposes a limit on the temperature dependence of Vb. For comparison, defect-induced premature breakdown tends to have much weaker temperature dependence with opposite sign. These reports of avalanche photodiodes in GaN are really simply first steps toward a very long developmental journey and serve only as feasibility demonstrations.

4.5 UV Imagers

The single-pixel detector development has begun to give way to arrays for imaging purposes. By combining the spectral signature and image of the source, one can more definitively identify the origin. For example, an 8  8 GaN Schottky barrier photodiode array for UV imaging with a pixel size of 200  200 mm2, responsivity of 0.06 A W1, and an RC time-limited response of 50 ns has been reported [179]. In another effort, a 1  18 GaN MSM linear array with a responsivity of 3250 A W1 at a bias of 10 V and a response time of 0.5 0.2 ms was also reported [180]. Large, nearly solar-blind 32  32 array detector imagers with mesa etching, ohmic contact formation, In bump bonds over p-contacts, and flip-chip mounting on an Si readout circuit chip have been developed jointly by Prof. Schetzina and his colleagues at North Carolina State University, Honeywell, Inc., and Night Vision Laboratories [14]. The Si ROC was already in place because of developments of other imagers operating in the IR spectrum. Photodiodes were illuminated through the sapphire substrate polished on both sides. Specifically, a visible-blind UV camera based on a 32  32 array of backilluminated GaN/AlGaN p-i-n photodiodes has been demonstrated. Each photodiode in the array consisted of an n-type base layer of AlGaN (20%) followed by an undoped GaN layer topped by a p-type GaN layer, all deposited by MOVPE. The photodiode array was connected to a silicon readout integrated circuit using In bump bonds. This visible-blind digital camera is sensitive to radiation from 320 to 365 nm in the UV spectral region. UV images from a III–V nitride-based photodiode arrays in the visible-blind region of the spectrum were obtained using a FPA through a collaborative effort at Night Vision Laboratories on July 28, 1999. Following the preliminary developments of 32  32 pixels2 imagers, arrays as large as 128  128, consisting of 16 384 individual photodiodes, have been prepared and used in a focal plane array camera [44]. After the initial demonstration of UV focal plane arrays, Lamarre et al. [181] and Long et al. [182] reported nearly solar-blind AlGaN UV imagers with 128  128 and 256  320 FPAs using back-illuminated p-i-n structures. These developments preceded the report of true solar-blind full frame image with a 256  320 AlGaN FPA [183]. These authors similarly used a back-illuminated AlGaN p-i-n structure with Si (n-type) doped Al0.5Ga0.5N layer, undoped Al0.32Ga0.68N absorption layer, and Mg (p-type) doped GaN top layer. A 256  256 pixels2 truly solar-blind imager whose device-level performance is shown in Figures 4.46–4.48 has been fabricated and tested [17]. The cross-sectional schematic view showing the layer structure and some fabrication details as well as the

4.5 UV Imagers

Figure 4.59 Image of a 2 in. diameter AlGaN p-i-n wafer processed into 256  256 arrays with close-up views of the waferprobe diagnostic features located in the dicing streets and test circuitry containing photodiodes with 250, 200, 150, and 100 mm diameters. Courtesy of M. Reine BAE systems [17].

scheme used for packaging are shown in Figures 4.4e and 4.5a and b. A four-mask process was used to fabricate back-illuminated AlGaN p-i-n’s 256  256 arrays [17] with Ni–Au was the p-side contact and Ti-Al-Ni-Au the n-side contact. Mesa etching was performed using an inductively coupled plasma (ICP) etcher. Figure 4.59 shows a 2 in. diameter AlGaN p-i-n wafer containing many 256  256 pixels2 focal plane array imagers, each of which is 8  8 mm2 in area, with test circuitry after ICP mesa etching and after n-side and p-side contacts. Test circuitry can be diced out and hybridized by bump mounting to fan-out boards for probe testing. Wafer-probe diagnostic features are located in the dicing streets between adjacent 256  256 arrays. Diagnostic features included circular transmission line method (CTLM) features for both n-side and p-side contacts, circular photodiodes with four different areas (250, 200, 150, and 100 mm diameter), a step feature for measurement of mesa etch depth, and a clear area for optical transmission [17]. Promising 256  256 AlGaN p-i-n photodiode arrays, following the wafer-level probe testing of diagnostic photodiodes in the dicing streets, were selected and hybridized to low-noise 256  256 ROIC chips at BAE Systems. The ROIC chips were designed specifically to match the ultrahigh-resistance, low-noise AlGaN UV p-i-n photodiodes [17]. The input circuit uses an amplifier with a 16 fF integration capacitor and a correlated double sampler and is capable of measuring diode impedances as high as 1017 W. Frame rates that can be attained are >400 Hz. Also important to note is that the ROIC layout has features to shield the ROIC input circuits from UV

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Table 4.6 Performance summary of the best solar-blind 256  256 AlGaN FPAs.

R0 A FPA Cut-on Cutoff Response Response R0 wavelength wavelength Dk QE (%) QE (%) operability uniformity (median) (median) (O) (O cm2) (nm) (nm) (nm) (V ¼ 0) (V ¼ 5) (%) (r/l) a b c d e f

262 261 264 265 261 265

278 278 282 282 281 284

16 17 18 17 20 19

44.5 47.5 46.6 58.1 43.4 38.2

54.7

64.5 56.5

99.8 95 99.6 85 99.4 99.4

6 3.1 2.5 5.5

4 · 1015 6 · 1012 1.5 · 1015 >1 · 1016 5 · 1015 1 · 1016

3 · 1010 4 · 107 1.1 · 1010 >7 · 1010 4 · 1010 7 · 1010

Courtesy of M. Reine.

radiation. The effectiveness of this UV shielding was verified by testing an ROIC chip with no AlGaN array. UV response, dynamic resistance, and noise of the promising 256  256 arrays were measured for each of the 65 536 pixels in each FPA and organized as both spatial maps and histograms. The excitation source was broadband unfocused UV radiation at normal incidence from an Oriel 63 165 high-irradiance high-stability ozone-free 30 W deuterium lamp, located about 30 cm from the FPA. The FPA was shielded against stray radiation. The relative response was then converted to quantum efficiency by normalizing the FPA data to the average quantum efficiency measured by wafer-level probe on the 250 mm diameter diagnostic photodiodes located nearest to each array. AlGaN photodiode resistances were measured in dark with a 1 s integration time and external correlated double sampling at a 100 Hz frame rate. The performance of a few solar-blind arrays is summarized in Table 4.6. In terms of imagery, a UV reflection image of a US dollar coin taken with solarblind FPA (f ) in Table 4.6 is shown in Figure 4.60. The UV response operability of this FPA is 99.4%, and the image quality is very good despite a few dark pixels, as response operability is less than 100%. Here the response operability is defined as that fraction of the 65 536 pixels with response more than half the average response value. The noise equivalent irradiance (NEI) for solar-blind FPA (a) in Table 4.6 has been measured. The median NEI is about 1150 ph pixel1 s1 at 12 Hz. When the NEI is computed at 1 Hz, the median NEI becomes 332 ph pixel1 s1. The best pixels exhibited NEI values around 144 ph pixel1 s1 at 1 Hz. These NEI values correspond to Dl values at l ¼ 272 nm of 8.7  1012 cm Hz1/2 W1 (median) and 2.0  1013 cm Hz1/2 W1 (best pixels). NEI data (median and the “best element” values) measured for four 256  256 UV AlGaN FPAs plotted versus dynamic resistance (median values) are shown in Figure 4.61. Also shown in Figure 4.61 are the NEI curves plotted versus R0 calculated for three values of preamp noise using Equations 4.121–4.124 given below. The signal photocurrent, we also referred to this as Ip, through the diode is expressed in terms of amperes as Isig ¼ qhQ sig A;

ð4:121Þ

4.5 UV Imagers

Figure 4.60 UV reflection image of a US dollar coin taken with 256  256 AlGaN solar-blind FPA( f ) in Table 4.6. Courtesy of M. Reine BAE systems [17]. (Please find a color version of this figure on the color tables.)

where A is the device area, Qsig is the signal photon flux in ph cm2 s1, incident on the device active area, and h is the quantum efficiency. The squared rms noise current in terms of amperes squared is given by the sum of Johnson noise due to the device and a term associated with the noise of the input preamplifier circuit as
 4kT 1 qN N 2 þ ; ð4:122Þ I2N ¼ tint R0 2tint where R0 is the dynamic resistance at zero-bias voltage, in ohm, tint is the integration time, in s, NN is the rms noise of the input circuit (preamp), in electrons. The NEI in terms of ph cm2 s1 (can be converted to units of ph pixel1 s1 by multiplying by the pixel area A) is expressed as NEI ¼

IN : qhA

ð4:123Þ

The detectivity at wavelength l in terms of cm Hz1/2 W1 can be calculated using pffiffiffiffiffiffi Df l  pffiffiffiffi ; ð4:124Þ Dl ¼ hc NEI A where Df ¼ noise bandwidth in hertz.

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103 Visible-blind FPAs a, e, and f solar blind

N N=100 electrons NEI (ph/pixel)

102

NN = 10 electrons Goal

10

NN =1 electron

Curves: QE=0.9 t1N=1 s A TD=30 × 30 µm2 T= 300 K

1 1012

1013

1014 10 15 Zero-bias resistance, R0 (W)

1016

1017

Figure 4.61 NEI in terms of photons per pixel at 1 Hz for a visibleblind 256  256 AlGaN UV FPA and three solar-blind AlGaN UV FPAs (a), (e), and (f ), the latter group is featured in Table 4.6, versus median zero-bias resistance R0. Also shown are NEI curves versus R0, calculated from the above equations, for three values of preamp noise NN. Courtesy of M. Reine BAE systems [17].

Despite the remarkable developments detailed throughout this chapter, it would be desirable to attain ternary p-type AlGaN with large AlN molefractions with high-quality material. The p-doping possibly limited to about 30% AlN mole fraction necessitates modifications in the device design such as that done for the solar-blind detectors in the form of back illumination. Back illumination forces one to use high mole fraction AlGaN buffer layer, which is known to produce as high-quality material as that grown on GaN buffer layers. In addition, the ELO process, which has been demonstrated with AlGaN, cannot be used to improve materials quality. If top illumination were to be used, the large lateral resistance in the p-type top layer(s) would cause the device to respond nonuniformly. This is, in a way, similar to the well-known emitter crowding effect in bipolar transistors. Schottky barrier varieties [184–186] may alleviate the problems associated with p-doping, but it is difficult to get light through even very thin Schottky barriers at solar-blind wavelengths, leaving the bottom-illuminated heterojunction devices as the only plausible option. In addition, if avalanche photodiodes are pursued for their much-needed gain (required by some systems applications), the Schottky barrier varieties may not be as successful as the p–n-junction varieties in avalanche photodiodes based on conventional compound semiconductors. Moreover, very little is really known about the ionization coefficients of electrons and holes in GaN and its alloys. The calculations are frustrated by the lack of reliable data on the

Summary

properties (such as complete band structure) of these materials. These difficulties should, however, be viewed as challenges rather than obstacles. And they are being met in the sense that avalanche photodiodes based on the GaN system have already been reported. A reasonable set of midway performance goals would be to achieve near 80–90% internal efficiency, 1014 W NEP, higher than 100 MHz frequency response, 260–290 nm wavelength response, and 104 of peak response at 300 nm for AlGaN detectors. These types of electrical characteristics should yield specific detectivities in the 1015 cm Hz1/2 W1 range.

4.6 Concluding Comments

UV and solar-blind detectors, the latter operative around 280 nm (where the solar radiation is absorbed by the ozone layer), with attractive performance have been demonstrated. One GaN photodiode exhibited a zero-bias responsivity of about 0.21 A W1 at 356 nm, which corresponds to an internal quantum efficiency of 82%, and other exhibited dark current levels as low as 1014 A at 5 V for 50 mm diameter devices. The NEP in photodetectors is 1.9  1015 W for 250 and 50 mm diameter photodiodes. Specific detectivities of approximately 3  1014 cm Hz1/2 W1 have been obtained. Finally, the GaN-based detectors with AlN mole fractions corresponding to the solar-blind region of the spectrum have been fabricated into arrays for imaging. Solar-blind detector arrays with pixel counts of 256  256 and 256  320 images have been fabricated and tested already. Despite tremendous progress, it is still desirable to increase specific detectivity for the solar-blind imagers given the abyssal number of photons available from the targets of interest.

Summary

This chapter discusses optical detectors with special orientation toward UV and solarblind detectors. Following a discussion of the fundamentals of photoconductive and photovoltaic detectors in terms of their photo response properties, a detailed discussion of the current voltage characteristic of the same, including all the possible current conduction mechanisms, is provided. Because noise and detectors are synonymous with each other, sources of the noise are discussed, followed by a discussion of quantum efficiency in photoconductors and p-n junction detectors. This is then followed by the discussion of vital characteristics such as responsivity and detectivity with an all too important treatment of the cases where the detectivity is limited by thermal noise, shot current noise, generation-recombination current noise, and background radiation limited noise (this is practically nonexistent in the solar-blind region except the man-made noise sources). A unique treatment of particulars associated with the detection in the UV and solar-blind region and requirements that must be satisfied by UV and solar-blind detectors, particularly, for the latter, is then provided. This leads the discussion to various UV detectors based

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on the GaN system, including the Si- and SiC-based ones for comparison. Among the nitride-based photodetectors, photoconductive variety as well as the metal-semiconductor, Schottky barrier, and homo- and heterojunction photodetectors are discussed along with their noise performance. Nearly solar-blind and truly solar-blind detectors including their design and performance are then discussed, which paves the way for the discussion of avalanche photodiodes

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168 Oguzman, I.H., Bellotti, E., Brennan, K., Kolnik, J., Wang, R. and Ruden, P.P. (1997) Theory of hole initiated impact ionization in bulk zincblende and wurtzite GaN. Journal of Applied Physics, 81 (12), 7827. 169 Shur, M.S. (1998) GaN based transistors for high power applications. Solid-State Electronics, 42, 2131–2138. 170 Osinsky, A., Osinsky, A., Shur, M.S., Gaska, R. and Chen, Q. (1998) Avalanche breakdown and breakdown luminescence in p-n-n GaN diodes. Electronics Letters, 34 (7), 691–692. 171 McIntosh, K.A., Molnar, R.J., Mahoney, L.J., Lightfoot, A., Geis, M.W., Molvar, K.M., Melngailis, I., Aggarwal, R.L., Goodhue, W.D., Choi, S.S., Spears, D.L. and Verghese, S. (1999) GaN avalanche photodiodes grown by hydride vaporphase epitaxy. Applied Physics Letters, 75 (22), 3485. 172 McIntosh, K.A., Molnar, R.J., Mahoney, L.J., Molvar, K.M., Melngailis, I., Efremov, N., Jr and Verghese, S. (2000) Ultraviolet photon counting with GaN avalanche photodiodes. Applied Physics Letters, 76 (26), 3938. 173 Verghese, S., McIntosh, K.A., Molnar, R.J., Mahoney, L.J., Aggarwal, R.L., Geis, M.W., Molvar, K.M., Duerr, E.K. and Melngailis, I. (2001) GaN avalanche photodiodes operating in linear gain mode and Geiger mode. IEEE Transactions on Electron Devices, 48 (3), 502. 174 Carrano, J.C., Lambert, D.J.H., Eiting, C.J., Collins, C.J., Li, T., Wang, S., Yang, B., Beck, A.L., Dupuis, R.D. and Campbell, J.C. (2000) GaN avalanche photodiodes. Applied Physics Letters, 76 (7), 924. 175 Yang, B., Li, T., Heng, K., Collins, C., Wang, S., Carrano, J.C., Dupuis, R.D., Campbell, J.C., Schurman, M.J. and Ferguson, I.T. (2000) Low dark current GaN avalanche photodiodes. IEEE Journal of Quantum Electronics, 36 (12), 1389. 176 Aggarwal, R.L., Melngailis, I., Verghese, S., Molnar, R.J., Geis, M.W. and

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Mahoney, L.J. (2001) Temperature dependence of the breakdown voltage for reverse-biased GaN p-n-n þ diodes. Solid State Communications, 117, 549. Stillman, G.E. and Wolfe, C.M. (1973) Semiconductors and Semimetals, Vol. 12 (series eds R.K. Willardson and A.C. Beer), Academic Press, New York. Kunihiro, K., Kasahara, K., Takahashi, Y. and Ohno, Y. (1999) Experimental evaluation of impact ionization coefficients in GaN. IEEE Electron Device Letters, 20 (12), 608. Lim, W., Gangopadhyay, S., Yang, J.W., Osinsky, A., Chen, Q., Anwar, M.Z. and Khan, M.A. (1997) 8  8 GaN Schottky barrier photodiode array for visible-blind imaging. Electronics Letters, 33 (7), 633–634. Huang, Z.C., Chen, J.C., Mott, D.B. and Shu, P.K. (1996) High performance GaN linear array. Electronics Letters, 32 (14), 1324–1325. Lamarre, P., Hairston, A., Tobin, S.P., Wong, K.K., Sood, A.K., Reine, M.B., Pophristic, M., Birkham, R., Ferguson, I.T., Singh, R., Eddy, C.R., Jr, Chowdhury, U., Wong, M.M., Dupuis, R.D., Kozodoy, P. and Tarsa, E.J. (2001) AlGaN UV focal plane arrays. Physica

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Index 1 dB compression point 462 1/f current noise 795 1/f noise 531, 729, 731 – GaN FETs 550 – density spectrum 557 – slope g of 557 1kBT current 660 2DEG, average position of 365 2kBT current 660 3 dB compression point 462

a abrupt heterojunctions 651 absorption 8 – coefficient 7, 10, 205, 216, 217, 224, 733, 792 – process 8, 172, 197, 204, 206 – rate 216 access regions 421 activation energy 317, 514, 795 – characteristic 324 – degradation 317 advanced design system (ADS) 447 aggregation 144 aging test results 322 Airy function 365 AlGaN detectors 801 – heterojunction 796 AlGaN/GaN – HFETs 378, 463 – interface charge 378 – modulation-doped field effect transistor 354 – MQW photodetectors 806 – sheet charge calculation 672 AlGaN/GaN interface charge 378 AlGaN Schottky barrier photodiodes 780 – spectral response of 780

AlInN/AlN/GaN 2DEG channel 606, 607, 608 AlInN barrier 606 all-dielectric mirror cavity 302 ambipolar diffusion coefficient 744 amplifier classification 458 analytical device modeling 419 angled sidewall 22 anharmonic interaction(s) 581, 582, 587 anharmonic lifetime 603 anharmonic phonon decay 582 anomalies 497 anomalous gain 786 – characteristics 498 apparent activation energy 514 apparent capture cross section 514 applied bias 719 areal thermal generation-recombination 731 areal trap density 506 Arrhenius plot 619, 620 – mean time failure 620 atmospheric detection range 751 atmospheric layers 747 atmospheric transmission 748 – UV transmission 748 atomic nuclei 539 Auger recombination processes 76, 78, 79, 249 – coefficient 76, 77 – rate 80 available noise power 535 available power gain (GA) 412, 413 avalanche breakdown voltage 486 avalanche coefficients 537 avalanche ionization 621 avalanche multiplication 811 avalanche noise 536

Handbook of Nitride Semiconductors and Devices. Vol. 3. Hadis Morkoç Copyright Ó 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40839-9

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832

avalanche photodetectors 810, 811 avalanche photodiodes (APDs) 717 average velocity 650

b backlight configuration 117 bactericidal action 711 ballistic transport 651 bandgap narrowing 654 bandgap reduction 238 bandgap renormalization 237, 238, 261 band structure 392, 393 band tails 68 band-to-band radiative recombination 80 barrier height 642 barrier lowering process – Poole–Frenkel effect 519, 520 barrier states – current anomalies effect 509 base-collector 637 base transit time 648, 650, 654, 655 beam intensity 202 biaxial strain 242, 243 bipolar transistors 662 – noise 662 blackbody emissivity 755 blackbody radiator 535 black dots 325 Bloch functions 221 blue phosphors 110 Bohr radius 238 Bose–Einstein condensation 308, 309 boundary conditions – HBTs 642 Bragg reflectors 55, 295 breakdown voltage 487, 494, 665 – gate length dependency 491 bremsstrahlung 539 brightness 28 broadening factor 228, 260 broadening process 237, 238 bulk current 790 bulk degradation 324 bulk polaritons 312 BYY approach 101

c capture cross section of the trap 514 capture/emission processes 511 – balance of 511 capture of electrons 507 carbon contamination 321 carbon layers 322 carbon mapping 323

carrier confinement 213, 215 carrier degeneracy 595 carrier-density fluctuation model 557 carrier diffusion model 670 carrier-launching ramp 651 carrier leakage 76 carrier overflow 265 carrier recombination 253 – lifetime 263, 720 carrier transit time 787 case temperature 572 catastrophic failure 616 cavity length 280 cavity losses 278 cavity mode(s) 194, 199, 201 – dispersion 310 cavity photon effective mass 296 cavity polaritons 312 cavity thickness 295 CCD 760, 761 – CD-Rom (CD read only memory) 170 CD-Rom player 171 Cerenkov processes 610 channel potential 371 channel resistance (RC) 424 – delay time constant 455 – subcircuit 432 – time constant 723 channel temperature 618 characteristic temperature 273, 275, 323, 325, 326 characteristic time 731, 795 – constant(s) 731 charge-transfer excitons (CTE) 143, 145 charge transport 132 charged trap 503 charging delay plot 455 chromaticity color coordinates 30, 31 chromaticity diagram 31, 90 cladding layers 11 classical radius 551 coherence parameter 551 coherent quantum 1/f effect 540 coherent quantum 1/f noise 554 – spectrum 551 coherent regime 652 cold cathode fluorescent light bulbs 120 cold conditions 424 collapse-free operation 569 collector-common-emitter output I-V characteristics 671 collector-emitter offset voltage 646 collector transit time 658 colloids 112

Index color matching functions 31 color-mixing efficiency 29 color rendering index (CRI) 42, 88, 89, 95, 98 – general 88, 89 – special 88 color temperature 30, 31 collisions mean free path 813 Combining LEDs and Phosphor(s) 100 Commission Internationale de l’Éclairage (CIE) 27 common-base I-V characteristics 669 common-emitter current gain 644 compact disk (CD) 170 compact disk read only memory (CD-ROM), player 170, 171 compositional fluctuations 253 compositional gradient 654 compositional grading 653, 654 compositional strain inhomogeneities 252 computational method 747 – plexus 747 conditional stability 416 conduction band discontinuity 639 conduction band edge 390, 396 conduction band profile 364 Cones 23, 24 confinement factor 186, 190, 194, 195, 260, 301 contact degradation 324 continuity equation 4, 718, 736 contrast parameter 256 conventional quantum 1/f 554 – coefficient 551 – effect 538 – noise 552 – noise parameter 550 – noise spectrum 551 coordinate diagram configuration 507 coplanar waveguides (CPW) 408 – conductor-backed 409 correlated color temperature (CCT) 32, 88, 89, 95, 104 correlation coefficient 544, 560, 664 Coulomb effect(s) 235, 250 Coulomb hole contribution 237, 238 Coulombic interaction 237, 248, 252 Coulombic processes 247 Coulombic-type trap 517 Coulomb interaction 249 Coulomb potential 541 critical angle 8, 181 – loss 8 critical loss efficiency 8 cross-correlation spectrum 547

cross talk 327 cumulative conventional quantum 1/f coefficient 553 cumulative failure rates 619 current collapse 497, 498, 508, 525, 526, 528 current confinement structure 302 current crowding 11 current density equations 718 current dependent spectra 67 current gain cutoff frequency 417, 454, 456, 473, 638 current gain(s) 643, 659, 661, 663 – DC current gain 663 – forward/reverse 643 – high currents 661 current lag 498, 499, 500, 528, 565 – buffer traps effect 500 current noise spectrum 547 current/power gains 471 current response 718 current-spreading length 13 current transport mechanism 639, 648 – across base 648 – across heterojunction 639 Curtice-cubic model 439, 441, 448 Cutoff frequency 454

d dark conductivity 738 dark current(s) 778, 790, 793, 812 – density 766 dark lines 325 – defects (DLDs) 319 dark resistance 782 data transfer rate 172 Davydov splitting 136, 144 de Broglie wavelength 642 Debye model 583 decorative lighting 83 deembedding 417 deep UV radiation/detection 744 defect-assisted tunneling processes 778, 784 defect mapping 520 degeneracy effects 631 degradation 70 degradation rate 317, 324 delay time 265 density-of-states 200, 222, 223, 225 – 1D 223 – 2D 223, 232 – joint 224, 225 depletion layer 640 – width 732

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834

detectivity 738, 739, 741, 817 – background radiation limited case 741 – thermal noise 803 detector dark current 732 detector resistance 719 determinant 416 dielectric function, complex 181 differential gain 263 differential push-pull 328 differential quantum efficiency 267 differential resistance 730, 803 diffusion coefficients 718 diffusion constants 724 diffusion current 723 diffusion fluctuation 543 diffusion length 724 diffusion limit 652 diffusion-limited diode 723 diffusion model 642 direct-bandgap semiconductor – optical transitions 209 direct tunneling 510 dispersion relation 199 – upper/lower polaritons 311 dispersion/temperature effects 447 distributed Bragg reflectors (DBRs) 55, 295, 297 – crack-free 297 – mirror 297, 300 – penetration depth 296 – reflectivity 297, 300 distributed losses 196 doping graded base 639 doping induced grading 653 double-heterojunction bipolar transistors 641 downward transitions rate 197, 203, 211 drain breakdown voltage 459, 483, 491 drain current 372, 373, 397 – fluctuation 552 – lag 567 – transient(s) 515, 516, 522, 523 drain delay time 455 drain delay plot 456 drain efficiency 480, 498 drain feedback capacitance 561 drain saturation current 374 drain short-circuit noise current 561 drain-to-gate feedback capacitance 457 drain-voltage 483 drifting nonequilibrium phonons 594 drift velocity 600 DVD 170 DVD-ROM drives 170

DX centers 498 dynamic polarization 577 dynamic resistance 800, 803

e early (infant) failure 614 Ebers-Moll equations 644 Ebers-Moll model 644 effect of threading Dislocation 57 effective carrier velocity 580 effective diffusion coefficient 654 effective electron drift velocity 579 effective intrinsic carrier concentration 654 effective length 296 effective mass 209, 222 – approximation 358 – density-of-states 222 – electron 209 – hole 209 effective temperature 548 effective transit time 429 efficiency 7, 460, 736, 737 – degradation 76 – droop 76, 81 eigenenergy/wave function 364 eigenvalues 365 Einstein’s coefficient 197, 203, 211, 218, 221 – A coefficient 203, 211, 221 – B coefficient 198, 220 elastic constants 377 elastic stiffness tensor 367 electrical displacement 361, 367 electric field 362, 365, 373 – saturation point 373 electric field distribution 614, 615 electric power dissipation paths 631 electrical efficiency 28, 104 electroluminescent devices 62 electromagnetic energy 198 – energy density 200 – volume density 198 electromagnetic radiation 539 electromagnetic wave, intensity of 204 electromechanical coupling 366 electron-blocking layer 326 electron concentration 232 electron density distribution(s) 391, 392, 393, 396 electron diffusion current 724, 725 electron distribution 360, 364 electron-electron interaction 249 electron emission 519 electron energy relaxation time 633

Index electron-hole pairs 641 electronic noise 531, 545 electron mobility 353, 791 electron occupancy 229 – factor 226 electron-phonon coupling 592, 629 electron sheet charge 369 electron temperature 595 electron transport 353 electron transport layer (ETL) 126, 127, 131 electron tunneling, gate to surface 624 electron velocity-field characteristics 596 electron velocity overshoot 657, 658 electrothermal simulations 448 electrothermal subcircuit 449 emission coefficient 513 emission processes 204, 205 emission rate 217, 520 emitter–base junction 637 emitter current crowding effect 662 emitter injection efficiency 667 empirical methods 419 end loss 267 energy balance 202 energy-dependent capture cross section 555 energy relaxation time 543, 632 energy temperature 535 energy transport fundamentals 132 epitaxial lateral overgrowth (ELO) 270, 273 equilibrium phonon population 632 equivalent circuit models 417 – cold condition at low frequency 426 – zero gate bias cold condition at high frequency 426 equivalent noise conductance 559 equivalent noise current source 544 equivalent noise resistance 546, 559 equivalent noise temperature 535, 542 etched facet (mirror) technology 332, 333 – lasers 333 excess carrier lifetime 719 excess electron concentration 773 excess hole concentration 723 excess noise 732 exciton oscillator strength 309 exciton phase-space filling density 247 exciton polariton dispersion 310 excitons pathways 246 expectation value, energy 366 expected two dimensional electron gas (2DEG) position 366 extended line configuration 404 extended states 255 external noise source 549

external quantum efficiency (EQE) 29, 44, 81, 267, 733 extraction efficiency 28, 106 extraordinary refractive index 176 extreme-ultraviolet 709 extrinsic capacitance elements 425 extrinsic parameters 424 extrinsic resistances 428 extrinsic transconductance 375 eye diagrams 327

f Fabry–Perot frequency 296 facet damage 319 facets coating delamination 324 far-field pattern 191, 193, 195 far-ultraviolet 709 fast Fourier transform (FFT) 739 Fe doping 633 feedback capacitance 454 feedback resistances 432 Fermi’s golden rule 218 Fermi-Dirac distribution function 228, 360 Fermi-Dirac statistics 210 fiber-optic communications 3 field effect transistors (FET) degradation 612 field-assisted emission 516 field distribution 192 field emission Auger electron spectroscopy 322 field emission scanning electron microscope (FESEM) 322 field plates (FPs) 351, 464, 475, 485, 494, 495, 613, 621 figure of merit 722 filter loss 755 fine structure constant 540 finesse 306 finite element analysis (FEA) 571 flat panel displays 62 flicker characteristics 554 – external quantum efficiency 45 flicker noise 732 flip-chip LED external quantum efficiency 45 flip chip LED (FCLED) 17, 44 flip-chip mount LEd 46 flip chip-mounted heterojunction field effect transistors (HFET) 465 floating base phototransistor 809 focal plane arrays (FPAs) 714, 816 Fock space 540 focused ion beam 333 forward voltage transfer ratio 403 four-LED chips 91

j835

j Index

836

– approach 98 four-level system 208 Fröhlich coupling 592, 604, 609 fractional quantum 1/f fluctuation 552 Franck-Condon (FC) shift 501, 506 Franck-Condon correction 502 Franck-Condon principle 141 Frenkel excitons 136 frequency-dependent noise 542 Fresnel loss 8 fudging parameter 554 full channel conductance 372 fundamental 1/f noise 538

g gain 175, 203, 205, 211, 236, 262, 263, 264, 401, 480 – absorption spectrum 285 – avalanche photodiode 812 – broadening 238 – expression 229 – spectra 280, 284, 285 gain calculations 241 – Wz GaN Q wells with strain 241 – Wz Q wells without strain 241 gain coefficient 204, 233, 263, 284, 285 Gain in GaN photoconductive detector 770 gain in ZB Q wells, with strain 245 gain in ZB Q wells, without strain 244 gain measurement 257, 261 – electrical injection 261 – optical pumping 258 GaInNAs quaternary infrared lasers 312 GaN-based lasers 327 – applications of 327 GaN HFET reliability 619 GaN HFETs performance 465 GaN MSM photodiodes 785, 787 – responsivity dependence on bias 785 – time response of 787 GaN photoconductive detector gain 770 GaN photoconductor 771, 792 – responsivity of 771 – spectral response of 792 GaN quantum wells gain 241 gated transmission line measurement (GTLM) 527 gate filling pulses 516 gate lag 499 gate leakage current 488, 490, 623 – dependence on gate length 492 – paths 623 gate leakage reduction 563 gate-source capacitance 454, 457

Geiger mode 758, 812 gen III photodiode arrays 759 generation rate expression 719, 727, 731, 743 generation-recombination (G-R) 718 – centers 729 – current 727 – noise 531, 533, 729, 731 generator reflection coefficient 548 graded base 639 green gap 86 green phosphors 110 group refractive index 201 Grüneisen’s constant 587 Grüneisen’s parameter 589 guidance condition 183

h H-aggregate 144 h-parameters 402 Hakki–Paoli method 281 half-wave plate 328 Hamiltonian 365 – operator 218 – interaction 218, 220 Handel’s quantum 1/f cross-correlations 552 harmonic condition 220 heterojunction bipolar transistor (HBT) 352, 637 649, 658 – delay time 649 – fundamentals 638 – Gummel plot 670 – nitride-based 665 heat dissipation 570 heat removal 269, 577 heat sink 576 Henry approach 284 heterointerface charge 356 heterojunction bipolar transistor 635, 636 heterojunction field effect transistors (HFETs) 349, 352, 369 – AlGaN/InGaN 380 – analytical description 369 – GaN/InGaN channel 375 – InAlN barrier 609 – InAlN/GaN 382 – InAlN/InGaN 384 – InGaN channel 609 – insulating gate 570 – large-signal 464 – small-signal 464 heterojunction field effect transistors (HFET) amplifier 457 – class A amplifier 457

Index – class AB amplifier 458 – class B amplifier 458 – class C amplifier 459 – class E amplifier 459 – efficiency 457 heterojunction phototransistor(s), 808, 809 higher lying triplet state 138 highest occupied molecular orbital (HOMO) 126, 136 high-forward gate current test (HFGC) 617 high-frequency equivalent circuit 421 high-frequency noise 541, 558 high-resistivity GaN 633 high-resolution lattice images 319 high-temperature operating life test (HTOL) 617 highly accelerated stress test (HAST) 617 High-temperature reverse-bias test (HTRB) 617 High-temperature storage test (HTS) 617 hole concentration 233 hole diffusion current 724, 725 hole injection 640 hole mobility 791 hole occupancy 229 – factor 226 hole-transporting molecules (HTMs) 131 hole transport layer (HTL) 125, 127 holographic optical element 328 Hooge parameter 554, 556, 732 – aH 551 – aK 554 hopping conductivity 732 hot carrier relaxation 596 hot electron energy relaxation time 605, 606 hot electron fluctuations 542 hot electron noise 541, 542 – temperature 535, 542 hot electrons 531, 578 hot field effect transistors operating conditions 429 hot modeling 423 hot optical phonon 578 hot phonon lifetime 588, 607 – dependence on electron concentration 593 hot phonons 578, 579, 629, 631 – depopulation 582 – effects 577 – implications for FETs 593 – induced friction 603 – phonon temperature 598, 599, 607, 632 – population 632 hot UV-emitting object 756

i IC-CAP 447 ideality factor 660 IG vs.VG 623 impact ionization 508 – current 641 impedance matrix 401 impulse response technique 657 InAlGaNS SQW laser 292, 293 – L-I curve 293 – spectrum for 293 In/AlGaN 53 InAlN barriers 611 In in AlGaN 53 incident power 718 incremental (differential) diode resistance 721 inductances 428 infant failure 616 infrared catastrophe 539 inhomogeneities 286 injection efficiency 265 injection lasers analysis 263 in-plane strain 377, 381, 386 in-plane wave vector 296 interface charge 383, 385, 370 – due to dopants 380 interfacial trapping 779 internal conversion, OLEDs 143 internal loss 194, 267 internal photogenerated charge profile 733 internal quantum efficiency (IQE) 28, 81, 104, 267 International Commission on Illumination (CIE) 27 intervalley scattering 542, 652 intrinsic capacitances 429 intrinsic delay time 454 intrinsic part 421 intrinsic resistances 429 ionization coefficient(s) 492, 665, 813 ionized donor states 499 isothermal current 450 isothermal model 450 I-V characteristics numerical calculations 398

j J-aggregates 144 Johnson-Nyquist noise 531, 534, 664, 714, 729, 730, 817 joint density of states (JDOS) 175, 233 junction temperature 460, 570, 571, 575

j837

j Index

838

k Kelvin probe 520, 521 kink, I-V characterstics 498 Kirk effect 661 Klemens’ channel(s) 582, 583, 590, 592 Klemens phonon decay 585 knee voltage 459

l Lambartian design 19 landscape lighting 83 large signal characteristics 473, 475 large-signal equivalent circuit model 434, 437, 438, 453 – HFET 438 laser annealing(LA) phonon group velocity 591 laser degradation 315 laser evolution 330 laser holography 21 lasers principles 172 LASIMO 447 lattice coordination diagram 498 lattice distorting defects 498 layer laser structure 272 leakage current 326, 729, 742 leakage resistances 779 lifetime 265, 279, 288 – anharmonic 603 – broadening 296 – excess carrier 719 – laser diode 316 – minority carrier 724, 791 – recombination 263, 264, 266, 720 light box 97 light emission 621 – gate edge 621 – OLEDs 139 light-emitting diodes (LEDs) 2, 3 – Amber 47 – blue 40 – degradation 70 – efficiency 76 – fiber-optic 3 – four-chip white light 17, 46, 97 – green 40 – laser 114 – monochrome applications 2, 82 – sapphire substrates 38 – Si substrates 60, 61 – SiC substrates 60 – organic/polymeric 122 – phosphor conversion 33

– photon-recycling semiconductor (PRS-LEDs) 111, 113 – resonant cavity-enhanced 55 – superradiant devices 3 – surface emitters 3 – thin-film 19 thin-film flip-chip (TFFC) 84 – three-chip white-light 91, 94 – two 93 – utilizing rare earth transitions 61 – UV 48, 53 white light from Four-Chip 97 light-emitting polymers (LEPs) 52 light receptors 22 light-voltage vs. current characteristics 274 lightly doped collector 636 line broadening 236 line, reflect, reflect, match (LRRM) 409 linewidth broadening 228, 249, 591 LINMIC design suite 447 liquid crystal display (LCD) 116, 117 – TVs 119 load line 462 load resistance 787 localization effects 66, 67 localized states 246, 251, 252, 253, 255, 732 localized states pathways 246 longevity performance 613 longitudinal-mode spacing 196 longitudinal optical (LO) 542 – phonon absorption 542 – phonon decay channels 584 – phonon decay mechanism 593 – phonon population 586 – phonons 578 – phonon to LA phonon decay 571 Lorentzian spectrum 795 losses 7 loss modes 194 low-energy X-ray spectral region 709 lowest unoccupied molecular orbital (LUMO) 126, 136 low-frequency noise (1/f noise) 531, 537, 714 low-noise amplifiers (LNAs) 478 low-pass filter 731 luminance 28 luminescence conversion 85 luminous efficacy 26, 29, 98, 104 luminous efficiency 28, 45 luminous energy 28 luminous flux 28 luminous intensity 28

Index

m matched gain 415 Materka–Kacprzak model 441 matrix element 218 Maury load-pull system 496 maximum available gain 415 maximum electron drift velocity 608 maximum gain 231 maximum luminous efficacy of radiation 29 maximum oscillation frequency 456, 638 maximum power conditions 415 maximum stable gain (MSG) 416 maximum transconductance 374 mean power dissipation 596 mean square photocurrent 731 mean thermal velocity 642 mean time to failure (MTTF) 288, 619 mechanical, electrical, and thermal energies pathways 630 metal-semiconductor-metal (MSM) 717, 782, 788 – AlGaN photodiodes 785 – detector 783 – frequency domain response 788 – GaN detectors 783 – photodetector 784 – time domain response of 788 metal-oxide semiconductor FET (MOSFET) 349 metal-semiconductor field effect transistors (MESFET) 349 microcavity (MC) lasers 295 microchannel plate (MCP) 758, 761 microplasma breakdown 811 mid-ultraviolet 709 mini disk 328 minimum noise figure 476, 548, 559, 562 minority carrier injection 638 mirror reflectivity 295 mobility fluctuation model 557 mobility fluctuation theory 732 modal gain 281–283 modes 200, 201 – density 200 – hopping 287, 288 – number of 200, 201 – spacing 280 MODFET 350 momentum matrix element 231 momentum operator 230 momentum relaxation time 543 momentum vector 200 Monte Carlo calculation(s) 580, 593, 594, 600, 602, 603, 632, 633

Monte Carlo environment 587 Monte Carlo method 579, 594 Monte Carlo model 601 Monte Carlo simulation 543, 589, 594, 595, 596, 606, 607, 608, 609, 631, 657 multichip SSL-LEDs 92 multiple defects 319, 320, 321 multiplication factor 537 Munsell samples spectra 89

n National Institute of Standards and Technology (NIST) 409 near/far-field patterns 22 near-ultraviolet 709 net modal gain 263 neurotransmitter 25 NIST TRL 409 nitride-based detectors 767 – back-illuminated 717 nitride LED light emission, its genesis 64 nitride LED performance 37 noise bandwidth 817 noise, circuit description of 545 noise current source 545, 663 noise detectors 729 noise equivalent circuit 544, 546, 663 – BJT 663 noise equivalent irradiance (NEI) 816, 817 – solar blind dedetector 818 noise equivalent power (NEP) 738, 739, 777, 798 – background radiation limited case 741 – in shot noise dominated detector 740 noise figure 482, 546, 547 noise-matching conditions 547, 548 noise measurements under forward bias 795 noise measurements under illumination 796 noise power 793 – unit band 535 noise spectra 537, 793 noise spectral density 798 noise temperature 605 noise-to-signal ratio 546 noise under reverse bias 793 noise voltage source 545 noisy admittance 545 noisy impedance 545 nonequilibrium phonons 587 – distribution 594 – populations 588 nonequilibrium trapped charge 513 nonfundamental 1/f noise 537, 538 non injected facet (NIF) 321, 324

j839

j Index

840

nonlinearities 436 nonradiative multiphonon (NMP) 507 nonradiative recombination 143 nonradiative recombination time 6 normalized responsivity 781, 782 npn bipolar transistor 640 numerical-aperture (NA) 172, 327 numerical gain calculations 239 Nyquist noise 531, 534, 729

o

occupancy fluctuations 555 occupation probability 198, 210 offset voltage 645 organic light emitting diodes (OLEDs) 124, 156 – degradation 156 – devices 146 displays 151 – lighting 155 – transparent (TOLEDs) 128 – white 147 open channel conductance 505 open-circuit noise voltage 544 open-circuit photovoltage 719, 721 open-loop configuration 117 operating gain 412 optical field (E field) distribution 186, 187, 188, 189, 191, 193 optical gain 196, 205, 260 – bulk GaN 239 – semiconductor approach 215 – spectra 260 optical generation rate (Gop) 731 optical head element 329 optical length 309 optical losses 279 optical power P(l) 767 optical recording densities 327 optical transition matrix 230 optimum DC load resistance 461 optimum generator reflection coefficient 548 optimum source resistance 559 ordinary refractive index 176 organic crystals 135 organic vapor-phase deposition 122 out-of-band rejection ratio 755 output I-V characteristics 375, 488 output noise equivalent noise temperature 546 output noise spectrum 546 output power 474, 480 oxygen mapping 323 ozone layer 748

p packaging 15, 110 parasitic capacitances 421, 428 parasitic resistances 426 passivation 529, 530, 563 passive display 153 pcLEDs 101, 103, 104, 107 peak gain 231 peak junction temperature 574, 618 peak voltage 621 Perception of Visible Light and Color 22 permittivity, tensor 367 persistent photoconductivity (PPC) 714, 774, 809 phase-change 327 phase trajectory 652 Phillips laboratory expert-assisted user software (Plexus) 747 phonon-assisted tunneling 510 phonon decay 582 – channels 581 – time 581 phonon dephasing 582 phonon dispersion curves 583 phonon emission time 589, 597 phonon lifetimes 586, 589, 601 – dependence on electron concentration 601 phonon lifetime vs. electron density 604 phonon occupancy factor 597 phonon occupancy number 598, 813 phonon population 595 phonon relaxation time 597 phosphor conversion LED (pcLED) 33 photocathode 713 photoconductive 713 – detector 714, 719, 768 – gain 719 – lifetime 743 – response 769 photoconductor (PC) 719, 773 – quantum efficiency 733 – ultraviolet detectors 772 photocurrent 718, 719, 721, 816 – decay 774 – density 736 – time decay 799 photodetector 743 – bulk recombination 743 – surface recombination 743 photodiode arrays 815 photodiode structures 715 photoelectric current gain 719, 720 photogenerated current density 735 photoionization cross section 506

Index photoionization energies 502 photoionization spectrum 502 photometry 27 photomultiplier tube (PMT) 709 – detectors 745 photon absorption profile 792 photon area 739 photon-assisted tunneling 256 photon conversion schemes 111 photon density 201 – per unit frequency 201 – per unit volume 201 photon energy 296, 718 – density 198, 199 – quanta 201 photon extraction efficiency 104 photon flux density 718 photon-generated carriers 735 photon-generated current density 736 photon-mitigated carbon deposition 321 photon mode density 200, 202 photonic crystal(s) 21, 47 photopic vision 26 photoreceptors 23 photoreflections 734 – internal 734 – surface 734 photoresponse 771 – evolution 771 photovoltage 721 photovoltaic detectors 775 – ultraviolet 772 p-bonds 135 piezoelectric charge(s) 381,386 piezoelectric coefficient(s) 253, 367 – tensor 367 Piezoelectric constants 377, 386 piezoelectric-induced fields 253 piezoelectric polarization 368, 376, 377, 387 pigmentation 711 pinch off voltage 372 p-i-n-junction detectors 789 planar cavity 296 planar defects 319 Planckian locus 31 Planckian spectrum 86 Planck–Nyquist formula 202, 203, 535 Planck’s blackbody radiation distribution law 202 plasmons 603, 604 p-n-junction detector 720, 721, 734, 789 – detectivity 742 – noise 741 – quantum efficiency 734

p-n-junction diode 722 – equivalent circuit 722 Poisson’s equation 358, 361, 363, 504, 718 polar longitudinal optical (LO) phonons 578 polar optical phonon (POP) 601 polariton lasers 308 polarization vector 220 polarizing beam splitter 328 polyethylene terephthalate (PET) 129 poly(para-phenylene vinylene) 127 Poole–Frenkel constant 804 Poole–Frenkel effect 510 Poole–Frenkel emission 518 population inversion 203, 206, 207, 211 potential distribution 365 potentially unstable 416 potential spike 648 power-added efficiency (PAE) 460, 461, 480 power amplifiers 478 power available 413, 414 power dissipation 618 – by a hot electron 632 – by the LO phonons 597, 599 power gain 412, 413, 414 power, input-output 412 power-matching conditions 547 power performance 479 power spectrum 539 power vs. frequency 478 Poynting vector 220 preamplifier circuit noise 729, 817 premature degradation 322 pressure dependence of spectra 64 proposed gate-surface leakage current mechanisms 624 pseudomorphic MODFET 351 pSpice (ORCAD) 447

q quality (Q) factor 302, 306 quantum-confined stark shift 2 quantum deficit 104 quantum dots 66, 67, 112 quantum efficiency 267, 719, 733, 737 – definition 734, 735 – external 719, 734, 737 – internal 719, 737 quantum mechanical electron tunneling 621 quantum 1/f coefficient 552 quantum 1/f noise 555 quantum size effects 111 quantum wells gain 231 quarter-wave Bragg reflector 301

j841

j Index

842

quarter-wave plate 329 quarter-wave stack 297 quasi-Fermi levels 209, 212, 645, 647 quasistationary eigenstates 312 quaternary 52 quiescent operating point 627

r R0 A product 722, 726 Rabi splitting 309 radiative current 262 – density 235 radiative linewidth 309 radiative recombination time 6 radiometry 27 Raman shift 573 random access memory (RAM) 171 random failure 614 – period 616 rare earth transitions 62 rate equation, semiconductors 210 rate of change in population 198, 206 Rayleigh–Ritz method 358 reciprocal (symmetric) nonlinear device 424 recombination coefficient 226 recombination cross section 551 recombination current 659 recombination process 80 recombination rate 264, 731 red-emitting phosphors 110 red phosphor 110 reduced effective DOS 234 reference impedance 411 reference plane 404 reflection coefficient(s) 196, 403, 404, 409, 410, 733 reflections 8 reflection spectra 300 refractive index 176, 178, 180 – AlGaN 176 – dispersion 178, 179, 201 – distributions 188, 189, 193 – GaN 176 – InGaN 180 relaxation rates 579 relaxation time approximation 543 reliability 617, 624, 629, 633 – gate current 624 – hot electron/hot phonon issues 629 – measurements 617 – metallurgical issues 628 – other issues 633 responsivity 738, 739, 793 retina 22, 23

reverse injection 641 reverse transfer ratio 404 reversible failure 616 red, green, and blue (RGB) all-LED array 117 RF input signal 621 RF output power 615 RF stress 621, 622 RGGB approach 100, 101 Richardson constant 642 Richardson velocity 647 ridge waveguide 276 Ridley decay 585 – channel 587 – processes 592, 593 Ridley’s anharmonic process 591 Ridley’s channel 583, 590 rods 23, 24, 26 Rollett stability factor 415 RRGGB approach 101 Rydberg energy 238

s saturation current 742 – density 726 – HBT 643 saturation effects 258 scanning Kelvin probe microscopy (SKPM) 522, 523 scattering angle 578 scattering efficiency 29 scattering matrix 405 scattering parameters 404 Schottky barrier height 370 Schottky photodiodes 776, 779 Schrodinger’s equation 358, 363, 365 scotopic vision 26 screened carrier exchange 238 screening length 238 self-heating 448, 631 semiconductor lasers, glossary of 208 semiconductors statistics 204 semi-isotropic continuum model 587 sheet carrier concentration 371 sheet charge and current numerical modeling 389 sheet charge density 393, 397 sheet electron concentration 360 Shockley–Read–Hall (SRH) recombination 718 short-channel field effect transistors (FETs) 424 short-circuited current gain 417 short-circuit gate noise current 544 short-circuit noise source 560

Index short-circuit photocurrent 718 short p-n-junction 724 short, open, load, and thru (SOLT) 406 shot noise 531, 532, 541, 664, 730 – dominated photodetector 740 SiC-based UV photodetectors 763, 764 – silicon-based 763 SiC photodiode 766 SiC polishing 468 – in H 468 – in HCl 468 sidewall deflector 22 s-bond 135 signal photon flux 817 – SiN passivation 528 signal-to-noise ratio 738 single-particle model 197 singlet fission OLEDs 143 Si read out IC 761 slope efficiency 325 slow electrons 652 small-signal equivalent circuit 420, 664 – HBT 664 – modeling 420 Snell’s law 8 solar-blind detector 800, 801 – array 799 solar-blind films 800 solar-blind region 710 solar-blind UV detectors 710 solar erythema 711 solar photons 753 – inevitable losses 754 – number 753 – unavoidable losses 754 solar radiation 749 – spectrum 749 – UV 746 solar UV photons reaching lower altitudes, number of 749 SOLT (SOLR) 409 space charge current 727 space charge region (SCR) 720 – collector depletion region in spacer layer 392 space layer inserted 392 s-parameters 401, 402, 403, 405, 411 specific detectivity (D*) 712, 739 – generation-recombination-limited case 740 spectral density 201, 552, 730 – conventional quantum 1/f fractional fluctuations 551, 553 – function 731

spectral distribution 202 spectral gain 238, 239 spectral intensity 535 spectral noise densities 537 spectral power distribution 93 spectral response 789, 791 – back illuminated solar blind photodiode 801 – current 505 – measurement setup 767 – phototransistor 810 – solar-blind detectors 802 spectral steady-state responsivity 769 spectrally integrated gain 286 spectrum locus 31 spontaneous emission 172, 173, 197, 199, 204, 206, 210, 216 – factor 258 – lifetime 198 – rate 217, 218, 226, 227, 228, 235, 258 spontaneous polarization 367, 368, 376, 378, 380, 381, 383, 386 stacking sequence 319 state-filling process 255 states density 200, 221 Statz model 441 stimulated absorption 173, 198, 210 stimulated emission 173, 197, 198, 199, 203, 206, 210, 215, 216, 259 – process 202 – rate 217, 227, 228 Stokes shift 85, 86, 104, 110, 252 storage capacity 172 strain, tensor 367 stratospheric ozone 745, 747 stray capacitances 421 stress tensor 367 stress time-dependent Ids -Vds characteristics 626 stress time-dependent IGS vs: Vds characteristics 626 structure constant 540 substrate thermal conductivity 460 sudden failure 321, 613, 616, 634 superluminescent devices 3 superradiant devices 3 surface charging 525, 526 surface conduction model 625 surface current 790 surface donor states 357, 363 – density of 363 surface emitters 3 surface leakage current 728 surface potential 524, 525

j843

j Index

844

– transients 523 surface processing 620 surface recombination 660 – velocity 726, 791 surface states 362

t TE-mode 181 temperature dependence 448 – coefficients 448, 665 – spectra 67 temperature distribution 574 temperature effects 453 temperature-humidity bias test (THB) 617 temperature noise model (TNM) 549 test set 408 T-gate GaN-based FET 470 thermal conductivity 572, 577 thermal dissipation 572 thermal emission rate 513 thermal generation rate (Gth ) 731 thermal ionization 506 thermal management 582 thermal noise 531, 534, 541, 548, 729 – current 536, 537 – limited detectivity 740, 797 – mean square value 536 – power 536 – voltage 536, 537 thermal resistance 315, 575, 618 thermal velocity 651 thermal wall 570 thermionic diffusion theory 646 thermionic emission current 642 thermionic emission model 647 thermoluminescence (TL) 131 thin-film flip-chip 84 thin film LED 19 thin-film transistor (TFT) 123 third-order intermodulation 475 three-color approach 92 three-LED solution 93 three-level system 207 three temperature life test data 619 threshold condition 194 threshold current 262 – density 206, 264 threshold gain 196, 205 threshold modes 194 threshold voltage 371 through, reflect, and line (TRL) 406 time-resolved Raman investigations 586 time response 798 top-emitting diode 46

total delay time 455 total efficacy 30 total junction current 726 total loss 194 total piezo-induced charge 387 total reflection 8 total sheet carrier density 394, 395 traffic light 82 transconductance 374, 429 – dispersion 498 transducer power gain (GT) 412, 413, 414 transfer matrix 401 transfer probability 141 transient current 517 transient simulation 508 transition rates, per unit volume 210 transit time 454, 656 transmission line matrix 656 transparency carrier concentration 260, 263 transparency condition 211 transparency current 264 transparency onset 175 transparent electrode 12 transport factor, forward/reverse 643 trap emission rate 512 trapped charge 515 trapping centers 499 trap-to-trap hopping mechanism 621, 622 triangular potential barrier 365 triangular well 369 trichromatic color-mixing approach 94, 106 tricolor phosphors 86 triphosphor blend 108, 109 – blue emission spectra 109 – UV emission spectra 109 triplet dynamics 140 tris-(8-hydroxyquinoline) aluminum (Alq3) 125, 126 tristimulus values 31 TRL (LRL) 409 TRM (LRM) 409 tunneling current 728 tunnel junction 14 tunnel leakage 625 two-dimensional electron gas 350 two-level system 207 two-piece model 373 two-port network 401, 402, 403, 405, 406

u ultraviolet catastrophe 536 unconditional stability 416 uniaxial tensile strains 242, 243 unilateral gain, GTU 414, 415

Index unitary amplifier 415 unity power gain frequency 456 unity step function (heavy side function) 553 universal intrinsic circuit 443 upward transition rate 197, 203, 211 UV absorption 747 UV imagers 814 UV photodetectors 762 – Si-based 762 – SiC-based 762 UV sensor optical aperture 755 UV sensors 709, 745 – available 757 – generation I (gen I) 709, 745 – generation II (gen II) 709, 745, 758 – generation III (gen III) 709, 745, 758, 759 – misalignment 756 – practical detection ranges 756 UV solar-blind detectors, design requirements 759 UV spectra 711 – UVA spectra 711 – UVB spectra 711 UV-visible contrast 768

vertical cavity surface-emitting laser 301 virtual gate 529, 621, 624 visible-light terminology 27 vitamin synthesis 711 voltage noise spectrum 547 voltage response 718 V-shaped defects 319

v

y

vacuum fluorescent displays 124 valence band discontinuity 639 Vallee–Bogani channel 583 vector potential 220 velocity field characteristics 581 velocity-field curves, several electron densities 602 velocity-field relationship 373 velocity fluctuations 544 velocity measurements 580

w wall-plug efficiency (power efficiency) 28, 106 wave equation 199 wave function 365 wave guiding 175 – analytical treatment 180 – modes 183 – numerical solution 184 – problem 180 wearout failure 614, 616 white-light generation 85, 91 – application 114 – approaches 91 – history 114 white noise 731 Wiener–Khinchine theorem 533

y-matrix 423 y-parameters 422, 423, 424, 425, 432 yttrium aluminum garnet (YAG) 90, 101

z zero-bias resistance 804 – area product 797 zero resistance 726 zone-center LO phonon frequency 578 zone-center optical phonons 590

j845

Appendix

Appendix: Periodic Table of Pertinent Elements

Atomic weight based 69.72 on carbon 12 5.91 Ga Symbol Ar3d 10 4s 2 p1 Electronic configuration r r: Radioactive Solid Gallium Name

Atomic number 31 Density -3

Solid and liquid : g cm Gas:g/ l at 273 K and 1 atm

II A 4

IB

IIIB

IIB

9.0122

5

VB

12 .0 1

B

VIB

7 14. 00 7 8 1.251 N 1.429

16

2.62 O C 1s 2 2s 2p 2 1s2 2s 2 p 1 1s 2 2s 2p 3 1s 22 s2 p 4 1s 22s 2 Gas Solid Solid Solid Gas Boron Nitrogen Be ryllium Carbon Oxygen 13 26 .9 8 14 30.97 16 12 32.06 28.09 15 24.305 1.74 2. 70 P 2.07 S Mg Al 2.33 Si 1.82 2 Ne 3s p 4 Solid Ne3s 2p 1 Ne 3s2 p 3 Ne 3s2 2 2 Solid Solid Solid Ne3s p Solid Ma gnesium Phosphorus Su lfur Silic on Aluminum 69.72 32 65.38 31 72. 59 33 20 40.08 29 63.55 30 7 8.96 74.92 34 As 4. 80 Ca 8.96 Cu 7.14 Zn 5.91 Ga 5. 32 Ge 5.72 1. 55 Se Ar3d 10 4s2 p 1 Ar3d 10 4s 2 p2 Ar3 d 10 4s2 p 3 Ar4s2 Ar3d 10 4s 2 Ar3d 104s 2p 4 Ar3d 10 4s 1 Solid Solid Solid Solid Solid Solid Solid Copper Arsenic Calcium Zinc Selenium Gallium Germanium 118.69 51 112.41 49 47 107.87 48 127.60 114.82 50 121.75 52 7.31 7.30 10. 5 6.88 Sb 6.24 Te Ag 8.65 Cd In 1.85

Be

IVB

10.81 6

VIIIB 2

2.34

Sn

Kr4 d 10 5s 2 Kr4 d 10 5s1 Kr4 d10 5s2 p 1 Kr4d 10 5s 2p 2 Kr4d 105s 2 p 3 Solid Solid Solid Solid Solid Antimony Silve r Cadmium Indium Tin 79 19 6.97 80 200.59 81 204.37 82 207.2 83 208.98 19.3 Au 13.53 Hg 11.85 Tl 11. 4 Pb 9.8 Bi Xe4f 144d 10 6s1

Solid Gold

Xe4f1 45 d106s 2 Xe4f1 45 d106s 2p1 Xe4 f145d 106s2p 2 Xe4f 145d106 s2p3

Solid Mercury

Solid Tallium

Solid Lead

Solid Bismuth

Kr4d 105s 2 p 4 Solid T ellerium

63 .55

0.1787 He 1s 2 Gas Helium 10 20 .18 0.901 Ne 1s2 2s 2p 6 Ga s Neon 18 39.95 1.784

Ar

Ne 3s2 p 6 Gas Argon 36 83.8 0 3.74

Kr

Ar3d 1 04s 2p 6 Gas Krypton 54 131.3 5.89

Xe

Kr4d 10 5s2 p 6 Ga s Xenon

j 847