Handbook of Nitride Semiconductors and Devices, Materials Properties, Physics and Growth (Handbook of Nitride Semiconductors and Devices (VCH)) (Volume 1)

Hadis Morkoc¸ Handbook of Nitride Semiconductors and Devices

Related Titles Piprek, J. (ed.)

Nitride Semiconductor Devices: Principles and Simulation 2007 ISBN: 978-3-527-40667-8

Adachi, S.

Properties of Group-IV, III-V and II-VI Semiconductors 2005 ISBN: 978-0-470-09032-9

Ruterana, P., Albrecht, M., Neugebauer, J. (eds.)

Nitride Semiconductors Handbook on Materials and Devices 2003 ISBN: 978-3-527-40387-5

Ng, K. K.

Complete Guide to Semiconductor Devices 2002 ISBN: 978-0-471-20240-0

Hadis Morkoç

Handbook of Nitride Semiconductors and Devices Vol. 1: Materials Properties, Physics and Growth

The Author Prof. Dr. Hadis Morkoç Virginia Commonwealth University Dept. of Chemical Enigineering Richmond, VA USA Cover SPIESZDESIGN GbR, Neu-Ulm, Germany

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 . # 2008 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 betz-druck GmbH, Darmstadt Binding Litges & Dopf GmbH, Heppenheim Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-40837-5

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Contents Preface

XIII

Color Tables XXI

1 1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5

General Properties of Nitrides 1 Introduction 1 Crystal Structure of Nitrides 1 Gallium Nitride 30 Chemical Properties of GaN 35 Mechanical Properties of GaN 35 Thermal Properties of GaN 47 Aluminum Nitride 62 Mechanical Properties of AlN 62 Thermal and Chemical Properties of AlN Electrical Properties of AlN 69 Brief Optical Properties of AlN 71 Indium Nitride 75 Crystal Structure of InN 77 Mechanical Properties of InN 77 Thermal Properties of InN 79 Brief Electrical Properties of InN 81 Brief Optical Properties of InN 84 Ternary and Quaternary Alloys 89 AlGaN Alloy 90 InGaN Alloy 92 InAlN Alloy 97 InAlGaN Quaternary Alloy 99 Dilute GaAs(N) 105 References 110

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoç Copyright # 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.8.1 2.8.2 2.9 2.9.1 2.9.2 2.10 2.10.1 2.10.2 2.10.3 2.10.4 2.10.5 2.10.6 2.10.7 2.10.8 2.11 2.11.1 2.11.2 2.11.3 2.12 2.12.1 2.12.2 2.12.3 2.12.3.1 2.12.3.2 2.12.3.3 2.12.4 2.12.4.1 2.12.4.2 2.12.5

Electronic Band Structure and Polarization Effects 131 Introduction 131 Band Structure Calculations 132 Plane Wave Expansion Method 134 Orthogonalized Plane Wave (OPW) Method 134 Pseudopotential Method 135 Augmented Plane Wave (APW) Method 135 Other Methods and a Review Pertinent to GaN 136 General Strain Considerations 154 Effect of Strain on the Band Structure of GaN 159 k
p Theory and the Quasi-Cubic Model 160 Quasi-Cubic Approximation 167 Temperature Dependence of Wurtzite GaN Bandgap 169 Sphalerite (Zinc blende) GaN 172 AlN 176 Wurtzite AlN 177 Zinc Blende AlN 183 InN 184 Wurtzitic InN 185 Zinc Blende InN 200 Band Parameters for Dilute Nitrides 202 GaAsN 205 InAsN 208 InPN 209 InSbN 209 GaPN 210 GaInAsN 210 GaInPN 212 GaAsSbN 212 Confined States 212 Conduction Band 216 Valence Band 224 Exciton Binding Energy in Quantum Wells 227 Polarization Effects 230 Piezoelectric Polarization 236 Spontaneous Polarization 241 Nonlinearity of Polarization 245 Origin of the Nonlinearity 250 Nonlinearities in Spontaneous Polarization 253 Nonlinearities in Piezoelectric Polarization 256 Polarization in Heterostructures 264 Ga-Polarity Single AlGaN/GaN Interface 272 Ga-Polarity Single AlxIn1xN/GaN Interface 276 Polarization in Quantum Wells 278

Contents

2.12.5.1 2.12.5.2 2.12.6 2.12.7

Nonlinear Polarization in Quantum Wells 280 InGaN/GaN Quantum Wells 286 Effect of Dislocations on Piezoelectric Polarization Thermal Mismatch Induced Strain 290 References 299

3

Growth and Growth Methods for Nitride Semiconductors 323 Introduction 323 Substrates for Nitride Epitaxy 324 Conventional Substrates 326 Compliant Substrates 327 van der Waals Substrates 328 A Primer on Conventional Substrates and their Preparation for Growth 329 GaAs 329 A Primer on GaAs 330 Surface Preparation of GaAs for Epitaxy 331 Si 332 A Primer on Si 332 Surface Preparation of Si for Epitaxy 333 SiC 334 A Primer on SiC 334 Surface Preparation of SiC for Epitaxy 338 Sapphire 342 A Primer on Sapphire 343 Surface Preparation of Sapphire for Epitaxy 346 ZnO 350 A Primer on ZnO 351 Substrate Preparation for Epitaxy 353 LiGaO2 and LiAlO2 355 LiGaO2 Substrates 355 LiAlO2 Substrates 358 AlN and GaN 359 Seedless Growth of GaN 362 Seedless Growth of GaN by High Nitrogen Pressure Solution Growth (HNPSG) for Substrates 362 Seeded Growth of GaN by HNPSG Method for Substrates 363 Pertinent Surfaces of GaN 365 GaN Surface Preparation for Epitaxy 369 Other Substrates 371 GaN Epitaxial Relationship to Substrates 372 Epitaxial Relationship of GaN and AlN with Sapphire 372 Epitaxial Relationship of GaN and AlN with SiC 381 Epitaxial Relationship of GaN and AlN with Si 381 Epitaxial Relationship of GaN with ZnO 381

3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.1.1 3.2.1.2 3.2.2 3.2.2.1 3.2.2.2 3.2.3 3.2.3.1 3.2.3.2 3.2.4 3.2.4.1 3.2.4.2 3.2.5 3.2.5.1 3.2.5.2 3.2.6 3.2.6.1 3.2.6.2 3.2.7 3.2.7.1 3.2.7.1.1 3.2.7.1.2 3.2.7.2 3.2.7.3 3.2.8 3.3 3.3.1 3.3.2 3.3.3 3.3.4

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Contents

3.3.5 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.4.1.3.1 3.4.1.3.2 3.4.1.3.3 3.4.1.3.4 3.4.1.3.5 3.4.1.3.6 3.4.2 3.4.2.1 3.4.2.2 3.4.2.3 3.4.2.4 3.4.2.5 3.4.2.6 3.4.2.7 3.4.2.8 3.4.2.9 3.4.2.9.1 3.4.2.9.2 3.4.2.9.3 3.4.2.9.4 3.5 3.5.1 3.5.1.1 3.5.1.2 3.5.2 3.5.3 3.5.3.1 3.5.3.2 3.5.3.2.1 3.5.3.2.2 3.5.3.2.3 3.5.3.2.4 3.5.3.2.5 3.5.3.3 3.5.3.4

Epitaxial Relationship of GaN with LiGaO2 and LiAlO2 and Perovskites 382 Nitride Growth Techniques 384 Vapor Phase Epitaxy 385 Hydride Vapor Phase Epitaxy 385 Organometalic Vapor Phase Epitaxy 393 Modeling of OMVPE Growth 398 Thermal Decomposition of GaN as it Relates to Growth 398 Ga and N Precursor Adsorption and Decomposition 400 Ga and N2 Desorption from the Surface 401 Ga and N Surface Diffusion 403 Kinetic Model: Balance Between Adsorption and Desorption 405 Comparison of Model with Growth Conditions for Surface Morphology 406 Molecular Beam Epitaxy 409 Adsorption 411 Desorption 412 Surface Diffusion 413 Incorporation 415 Decomposition 416 Reflection High-Energy Electron Diffraction 417 Plasma-Assisted MBE (PAMBE) or RF MBE, Primarily N Source 428 Reactive Ion MBE 435 Principles of RMBE and PAMBE Growth 436 Growth by RMBE 437 Growth by PAMBE 446 Which Species of N is Desirable? 451 The Effect of III/V Ratio and Substrate Temperature on Surface Morphology 455 The Art and Technology of Growth of Nitrides 462 Sources 467 HVPE Buffer Layers and Laser Liftoff 468 Benchmark HVPE Layers/Templates 472 Growth on GaAs Substrates 477 Growth on SiC: Nucleation Layers and GaN 479 Stacking and Interfacial Relationship 480 Nucleation Layers on SiC 482 High-Temperature AlN Nucleation Layers on SiC 486 Low-Temperature GaN Nucleation Layers on SiC 488 High-Temperature GaN Nucleation Layers on SiC 489 Alloy and Multiple Layer Nucleation Layers on SiC 491 Nucleation Layers on SiC by MBE 492 Substrate Misorientation and Domain Boundaries 495 Polarity of GaN on SiC 498

Contents

3.5.3.4.1 3.5.3.5 3.5.3.6 3.5.4 3.5.5 3.5.5.1 3.5.5.1.1 3.5.5.1.2 3.5.5.1.3 3.5.5.2 3.5.5.2.1 3.5.5.2.2 3.5.5.2.3 3.5.5.2.4 3.5.5.2.5 3.5.5.2.6 3.5.5.3 3.5.5.3.1 3.5.5.4 3.5.5.5 3.5.6 3.5.6.1 3.5.7 3.5.8 3.5.9 3.5.10 3.5.11 3.5.11.1 3.5.11.2 3.5.11.3 3.5.12 3.5.13 3.5.14 3.5.14.1 3.5.15 3.5.15.1 3.5.15.1.1 3.5.15.1.2 3.5.15.2 3.5.15.2.1 3.5.15.2.2 3.5.15.2.3 3.5.15.3

GaN Growth on SiC 499 Growth on Porous SiC 503 Zinc Blende Phase Growth 507 Growth on Si 507 Growth on Sapphire 512 OMVPE Low-Temperature Nucleation Buffer Layers 513 The Effect of V/III Ratio on Nucleation Buffer Layer 523 Effect of Epitaxial Growth Temperature 525 Effect of Process Pressure 525 Epitaxial Lateral Overgrowth 528 Selective Epitaxial Growth and Lateral Epitaxial Overgrowth with HVPE 537 Lateral Epitaxial Overgrowth on Si 539 Pendeo-Epitaxy 540 Pendeo-Epitaxy on SiC Substrates 542 Pendeo-Epitaxy on Silicon Substrates 544 Point Defect Distribution in ELO Grown GaN 558 Nanoheteroepitaxy and Nano-ELO 564 SiN and TiN Nanonets 569 Selective Growth Using W Masks 583 Low-Temperature Buffer Interlayer 584 Polarity and Surface Structure of GaN Layers, Particularly on Sapphire 586 MBE Buffer Layers 597 Growth on ZnO Substrates 598 Growth on LiGaO2 and LiAlO2 Substrates 603 Growth on GaN Templates 605 Growth on Spinel (MgAl2O4) 611 Growth on Non c-Plane Substrates 611 The a-Plane GaN Growth 613 Epitaxial Lateral Overgrowth of a-plane GaN 616  00) m-Plane GaN Growth 623 The (11 Growth of p-Type GaN 627 Growth of InN 629 Growth of AlN 638 Surface Reconstruction of AlN 642 Growth of Ternary and Quaternary Alloys 652 Growth of AlGaN 653 Growth of p-Type AlGaN 666 Ordering in AlGaN 668 Growth of InGaN 671 Doping of InGaN 678 Phase Separation in InGaN 679 Surface Reconstruction of InGaN 689 Growth of AlInN 695

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Contents

3.5.15.3.1 3.5.15.4 3.5.16 3.5.16.1 3.5.16.2 3.5.16.3 3.5.16.4 3.5.16.4.1 3.5.16.4.2 3.5.16.5 3.5.16.5.1 3.5.16.5.2 3.6

Miscibility Gap in InAlN 697 InGaAlN Quaternary Alloy 699 Growth of Quantum Dots 706 Quantum Dots by MBE 711 Quantum Dots by OMVPE 719 Quantum Dots by Other Techniques 723 Preparation and Properties of Nanostructures 725 Approaches for Synthesis 726 Vapor Phase Growth 726 Nanowires and Longitudinal Heterostructures 737 Coaxial Heterostructures 755 Nanotubes 756 Concluding Remarks 759 References 760

4

Extended and Point Defects, Doping, and Magnetism 817 Introduction 817 A Primer on Extended Defects 818 Dislocations 819 Misfit Dislocations 822 Threading Dislocations 822 Edge Dislocations 824 Screw Dislocations 828 Mixed Dislocations 836 Nanopipes or Hollow Pipes 840 Planar Defects: Domain Boundaries 844 Stacking Faults 851 Grain Boundaries 862 Electronic Structure of Extended Defects 863 Open Core Versus Close Core in Screw Dislocations 865 Edge and Mixed Dislocations 866 Simple Stacking Faults: Electrical Nature 884 TEM Analysis of High Nitrogen Pressure (HNP) Solution Growth (HNPSG) and HVPE-Grown GaN 886 Extended Defect Characterization 887 Pyramidal Defects 894 V-Shaped Defects (Pits) in InGaN Multiple Quantum Wells (MQWs) 905 Structural Defect Analysis by Chemical Etch Delineation 905 Structural Defect Observations with Surface Probes 910 Point Defects and Autodoping 917 Theoretical Studies of Point Defects in GaN 919 Hydrogen and Impurity Trapping at Extended Defects 924 Vacancies, Antisites, Interstitials, and Complexes 928 Vacancies 929

4.1 4.1.1 4.1.1.1 4.1.1.2 4.1.1.2.1 4.1.1.2.2 4.1.1.2.3 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.1.6.1 4.1.6.2 4.1.6.3 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.3 4.3.1 4.3.1.1 4.3.1.2 4.3.1.2.1

Contents

4.3.1.2.2 4.3.2 4.3.2.1 4.3.2.2 4.3.2.3 4.3.2.4 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.6 4.6.1 4.6.2 4.7 4.8 4.9 4.9.1.1 4.9.1.2 4.9.1.3 4.9.2 4.9.2.1 4.9.2.2 4.9.2.3 4.9.2.4 4.9.3 4.9.3.1 4.9.3.1.1 4.9.3.1.2 4.9.3.1.3 4.9.3.2 4.9.3.3 4.9.3.4 4.9.3.5 4.9.3.6 4.9.3.7 4.9.3.8 4.9.4 4.9.4.1 4.9.4.2 4.9.5 4.9.6

Interstitials and Antisite Defects 934 Complexes 935 Shallow Donor – Gallium Vacancy Complexes 936 Shallow Acceptor – Nitrogen Vacancy Complexes 936 Hydrogen-Related Complexes 937 Other Complexes 938 Defect Analysis by Deep-Level Transient Spectroscopy 938 Basics of DLTS 939 Applications of DLTS to GaN 948 Dispersion in DLTS of GaN 970 Applications of DLTS to AlGaN, In-Doped AlGaN, and InAlN 977 Minority Carrier Lifetime 977 Positron Annihilation 982 Vacancy Defects and Doping in Epitaxial GaN 983 Growth Kinetics and Thermal Behavior of Vacancy Defects in GaN 993 Fourier Transform Infrared (FTIR), Electron Paramagnetic Resonance, and Optical Detection of Magnetic Resonance 998 Role of Hydrogen 1004 Intentional Doping 1006 Shallow Donors 1007 Substitutional Acceptors 1007 Isoelectronic Impurities 1009 n-Type Doping with Silicon, Germanium, Selenium, and Oxygen 1010 Si Doping 1010 Ge Doping 1011 Se Doping 1012 p-Type Doping 1013 p-Type Doping and Codoping with Donors and Acceptors 1014 Magnesium Doping 1014 Codoping for Improving p-Type Conductivity 1019 Use of Superlattices for Improving p-Type Conductivity 1032 Role of Hydrogen and Defects in Mg-Doped GaN 1034 Beryllium Doping 1038 Mercury Doping 1040 Carbon Doping 1040 Zinc Doping 1042 Calcium Doping 1042 Cadmium Doping 1043 Other Acceptors in GaN 1043 Doping with Isoelectronic Impurities 1044 Arsenic Doping 1044 Phosphorus Doping 1045 Doping with Rare Earths 1045 Doping with Transition Metals and Rare Earths 1046

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Contents

4.9.6.1 4.9.6.2 4.9.6.3 4.9.6.4 4.9.6.5 4.9.6.5.1 4.9.6.5.2 4.9.6.5.3 4.9.6.6 4.9.6.6.1 4.9.6.7 4.9.6.7.1 4.9.6.7.2 4.9.7 4.9.7.1 4.9.7.2 4.9.7.3 4.9.8 4.9.9 4.9.10 4.9.11 4.9.12 4.10 4.11

Manganese Doping for Electronic Properties 1046 Other TM Doping for Electronic Properties 1060 General Remarks About Dilute Magnetic Semiconductors 1063 General Remarks About Spintronics 1075 Theoretical Aspects of Dilute Magnetic Semiconductor 1082 Carrier – Single Magnetic Ion Interaction 1084 Interaction Between Magnetic Ions 1086 Zener, Mean Field, RKKY, and Ab Initio Treatments 1089 A Primer to Magnetotransport Measurements 1103 Faraday Rotation, Kerr Effect, and Magnetic Circular Dichroism (MCD) 1104 II–VI and GaAs-Based Dilute Magnetic Semiconductors 1123 II–VI-Based Dilute Magnetic Semiconductors 1124 III–V-Based DMS: (GaMn)As 1133 Experimental Results of TM-Doped GaN 1141 Magnetotransport Properties TM-Doped GaN 1141 Magnetic Properties of Mn-Doped GaN 1143 Magneto-Optical Measurements in TM-Doped GaN 1146 Magnetic, Structural, Optical, and Electrical Properties of Cr-Doped GaN 1156 Other TM and Rare Earth Doped GaN:(Co, Fe, V, Gd, and so on) 1163 TM-Doped Nanostructures 1166 Applications of Ferromagnetism and Representative Devices 1167 Summarizing Comments on Ferromagnetism 1186 Ion Implantation and Diffusion for Doping 1188 Summary 1190 References 1191 Index

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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. 1. Hadis Morkoç Copyright # 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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Preface

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 indepth 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 discussion 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 Hall measurements. Low, medium, and high magnetic field cases, albeit only normal to the surface of the epitaxial layers, determination of which is affected by the value of the mobility and various cases 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 are not treated properly or are treated only in a cursory manner in many texts leading to confusion. After providing the necessary fundamental knowledge, 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 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. Experimental performance of GaNbased 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,

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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. A section devoted to reliability with specifics to GaN based high power HFETs is also provided. 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 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,

Preface

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, Q. Fan, 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 and create the figures; 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. 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, V. Fiorentini, 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, J. Bowers, J. C. Campbell, M. Razhegi, A. Nurmikko, M. A. Khan, J. Speck, S. Denbaars, R. J. Trew, A. Christon, 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

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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 highquality images, but the available space does not allow for individual recognition. They are acknowledged in conjunction with the figures. In a broader sense, 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

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Figure 1.4 A stick-and-ball stacking model of crystals with (a, both top and bottom) 2H wurtzitic and (b, both top and bottom) 3C zinc blende polytypes. The bonds in an A-plane (1 1 2 0) are indicated with heavier lines to accentuate the stacking sequence. The figures on top depict the

three-dimensional view. The figures at the bottom indicate the projections on the (0 0 0 1) and (1 1 1) planes for wurtzitic and cubic phases, respectively. Note the rotation in the zinc blende case along the h1 1 1i direction. (This figure also appears on page 6.)

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoc Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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Figure 1.9 The bandgaps of nitrides, substrates commonly used for nitrides, and other conventional semiconductors versus their lattice constants. (This figure also appears on page 12.)

Figure 1.23 An artist’s view of the scanning thermal microscope. Patterned after D.I. Florescu and F.H. Pollak. (This figure also appears on page 57.)

Color Tables

50

40

q D = 800 K

–1

–1

Molar specific heat, Cp (cal mol K )

q D = 500 K

30

20

Cp data, GaN q D = 500 K q D = 600 K

10

q D = 700 K q D = 800 K

0 0

200

400

600

800

1000

Temperature (K) Figure 1.26 Molar specific heat at constant pressure, Cp (cal mol1 K1), of GaN versus temperature. Open circles represent the experimental data. The solid lines are calculation based on the Debye model for Debye temperatures, yD, of 500, 600, 700, and 800 K. Unfortunately, it is difficult to discern a Debye

temperature that is effective over a wide temperature range because a large concentration of defects and impurities is present in GaN. However, a value of 600 K estimated by Slack is used commonly. The data are taken from Refs [215,216], as compiled in Ref. [88]. (This figure also appears on page 61.)

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50

40 Specific heat, Cp (J mol–1 K–1)

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30

20 Specific heat AlN (J mol–1 K–1)

800 K 850 K 900 K 950 K 1000 K 1050 K 1100 K

10

0

0

200

400

600

800

1000

Temperature (K) Figure 1.29 Molar specific heat at constant pressure, Cp (J mol1 K1, 1 cal ¼ 4.186 J), of AlN versus temperature. Open circles represent the experimental data. The solid lines are calculation based on the Debye model for Debye temperatures, yD, in the range of 800–1100 K

with 50 K increments. The data can be fit with Debye expression for yD ¼ 1000 K, which compares with 950 K reported by Slack et al. The data are taken from Ref. [88]. (This figure also appears on page 68.)

Color Tables

Figure 1.41 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 0.8 eV). (This figure also appears on page 101.)

Figure 1.42 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 1.9 eV). (This figure also appears on page 101.)

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0.02

Spontaneous polarization (C m–2)

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+0.333 InxAl1–xN

InxGa1–xN

0.00

+0.193

–0.02

InxGa1–xN

–0.04

GaN

+0.095 –0.06

InxAl1–xN

InN AIxGa1–xN

+0.037 –0.08

Random alloy CH-like CP-like

AIN

AIN

–0.10 0

0.2

0.4

0.6

1

Molar fraction, x Figure 2.37 Spontaneous polarization versus the molar fraction in all three ternary nitride alloys. Circles, squares, and triangles represent random alloy, CH-/LZ-, and CP-like structures, respectively. The dashed/dotted lines (blue) with solid triangles are for the CP-like alloys, the dashed lines (green) with solid squares are for CH-like alloys, and solid lines (black) with filled

circles are for random alloys. The black dashed lines represent the data calculated using Vegard’s law. Numbers indicated in the figure are for CP and CH-/LZ-like ordered alloy bowing parameters in terms of C m2. Courtesy of F. Bernaridini and V. Fiorentini. (This figure also appears on page 249.)

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25

AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

Curvature (1 m–1)

20

15

10

5

0.0

–5 0.0

20

40

60

Thickness (μm) Figure 2.59 A compilation of the variation of thermal curvature, a measure of strain, in epitaxial GaN layers grown on different substrates with respect to layer thickness [492]. (This figure also appears on page 294.)

80

100

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Figure 2.59 (Continued )

Color Tables

5108

Stress on various substrates (Pa)

0

–5108

–1109 AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

–1.5109

–2109

–2.5109 –3109 0100

210–5

410–5

610–5

810–5

Thickness (m) Figure 2.60 A compilation of residual thermal stresses in epitaxial GaN layer on different substrates with respect to layer thickness [492]. (This figure also appears on page 296.)

110–4

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Stress versus thickness of GaN/potential substrates 5.00E+08

0.00E+00 0

0.00002

0.00004

0.00006

0.00008

0.0001

AlN Si MgO

–5.00E+08

3C-SiC 6H-SiC 4H-SiC

Stress (Pa)

–1.00E+09

ZnO Al2O3 LiGaO2

–1.50E+09

MgAl2O4 GaAs NdGaO3** ScAlMgO**

–2.00E+09

LiAlO2 LSAT –2.50E+09

–3.00E+09

Thickness (m) Figure 2.60 (Continued )

Color Tables

Figure 3.1 The perspective view of the GaAs crystal (a) along [1 0 0] (1 · 1 · 1 unit), (b) [1 1 0] (2 · 2 · 2 units), and (c) [1 1 1] (2 · 2 · 2 units) directions [13]. (This figure also appears on page 330.)

Figure 3.2 The perspective view along (a) the [0 0 1], (b) [0 1 1], and (c) [1 1 1] directions of a Si cell. (This figure also appears on page 333.)

Figure 3.3 Tetragonal bonding of a carbon atom with its four nearest silicon neighbors. The bond lengths depicted with a and C–Si (the nearest neighbor distance) are approximately 3.08 and 1.89 Å, respectively. The right side is the three-dimensional structure of 2H-SiC structure. (This figure also appears on page 334.)

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Figure 3.9 (a) Top view of the oxide structure on SiC ð0 0 0 1Þ. The Si2O3 silicate adlayer consisting of a honeycomb structure with SiOSi bonds. At the center of the hexagons, one carbon atom of the topmost substrate bilayer is visible [the dark shaded area indicates the (1 1; 1) unit cell and light shaded the

pffiffiffi pffiffiffi ð 3  3ÞR30  -unit cell]; (b) side view of the oxide structure on the SiC (0 0 0 1) in ð0 1 1 0Þ SiC projection. Linear SiOSi bonds connect the silicate layer and the underlying SiC substrate. Courtesy of N. Onojima (patterned after Ref. [40]). (This figure also appears on page 342.)

Figure 3.10 (a) Top view (projection on the Si-plane of the basal plane of SiC) and (b) side view of SiC after an in situ Ga exposure indicating of the lack of silicate adlayer. Courtesy of N. Onojima. (This figure also appears on page 342.)

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Figure 3.12 The unit cell of sapphire: (a) rhombohedral unit cell; (b) hexagonal unit cell. Smaller spheres are for O and large ones are for Al [13]. (This figure also appears on page 344.)

Figure 3.13 Perspective views in (2 · 2 · 1) unit cells: (a) along the [0 0 0 1] direction in a rhombohedral unit cell; (b) along the (0 0 0 1) direction in hexagonal unit cell [13]. (This figure also appears on page 344.)

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Side view of sapphire r-plane {1 0 1 2}

Oxygen

(a)

Al

Al2O3 [1 0 1 1]

Color Tables

a

2

m

C

b

m

LiGaO2

Lower O

Li

Upper O

Ga

Figure 3.24 Example of the exact fit of GaN atoms over the LiGaO2 lattice if there is no distortion. Courtesy H. Paul Maruska. (This figure also appears on page 356.)

3

Figure 3.20 (a) Sapphire r-plane stacking sequence showing O atoms in larger clear circles and Al atoms in smaller, filled circles. The salient feature is that each Al layer has an O layer above and below it. (b) The atomic arrangement on three layers (the uppermost one is O,

immediately below is Al and third layer down is another O layer) on the r-plane of sapphire. The lines are there just guides to eye and do not represent bonds. (This figure also appears on page 352.)

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LGO: orthorhombic a = 5.402 Å b = 6.372 Å c = 5.007 Å

c b

a

(a)

LGO: orthorhombic a = 5.402 Å b = 6.372 Å c = 5.007 Å

c b

a

(b)

Projection on c-plane ∆b = –0.19 % aG

aN

=3

.18

9Å

∆a = +1.1%

bLGO = 6.372 Å

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∆a = +1.9 % ∆b = –1.1% (c)

aLGO = 5.402 Å

Figure 3.25 Structure of (a) orthorhombic LiGaO2 (LGO), (b) GaN, and (c) a detailed view of the relative orientation of GaN with respect to LGO. Courtesy of H. Paul Maruska. (This figure also appears on page 357.)

Color Tables

Figure 3.43 A schematic representation of a vertical OMVPE system employed at Virginia Commonwealth University along with a picture of the deposition chamber (a); a photograph of the reactor chamber of the same (b). (This figure also appears on page 394.)

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Figure 3.82 Cartoons illustrating the laser liftoff process sequence of 20 GaN membranes. (a) Laser lift-off of the GaN film coated with silicone elastomer and affixed onto a support template; (b) removal of sapphire following laser scanning; (c) deposition of an approximately

3 mm thick thermoplastic adhesive layer at 120  C; (d) peeling-off of the silicone elastomer. In the last step the GaN film is fully removed by dissolving the thermoplastic adhesive in acetone. Courtesy of M. Stutzmann, Ref. [382]. (This figure also appears on page 471.)

Figure 3.128 SEM and CL wavelength image of a cross section of HVPE ELO sample. Courtesy of J. Christen and A.G. Hoffmann, Ref. [695]. (This figure also appears on page 560.)

Color Tables

Figure 3.129 SEM (a) and CL wavelength (b) images of two different regions, coherently grown above the openings and overgrown above the SiO2 stripes. The growth in the windows (between the SiO2 stripes) and the wing

(coalesced regions over the SiO2 stripes) regions indicated in the schematic drawing (c) are clearly visible in the CL wavelength image. Courtesy of J. Christen and A.G. Hoffmann, Ref. [697]. (This figure also appears on page 561.)

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Figure 3.130 A m-Raman scans along c-axis of overgrowth GaN (in blue) and coherently grown GaN (in red): (a) free carrier density and (b) biaxial compressive stress. Courtesy of J. Christen and A. G. Hoffmann, Ref. [697]. (This figure also appears on page 562.)

Color Tables

Figure 3.160 Schematic representation of m-, a-, and c-planes of GaN. (This figure also appears on page 613.)

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Figure 3.165 Cartoon of epitaxial lateral overgrowth (ELO) on a-plane GaN with growth along the [0 0 0 1] direction, representing the Gapolar growth front and along the ½0 0 0 1 direction, representing the N-polar growth front: (a) top view and (b) side view. The growth along the Ga-polar front is about a factor of g ¼ 3 times faster. Courtesy of VCU students Vishal Kasliwal

and Xianfeng Ni. (c) Left 30 mm · 30 mm AFM image for sample B. Right 4 mm · 4 mm AFM image near the window and N-polar wing boundary of sample B, showing different surface pit densities for the window and the wing. Courtesy of VCU students Vishal Kasliwal and Xianfeng Ni. (This figure also appears on page 618.)

Color Tables

Figure 3.168 (a) AFM and (b) NSOM scans from a 40 mm · 40 mm area of a-GaN ELO sample B [875]. (This figure also appears on page 620.)

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Figure 3.249 (a) I–V characteristics of a core multishell (CMS) nanowire device, the top view of which is shown in the inset in the form of a field emission scanning electron microscopy image. Scale bar is 2 mm. (b) Optical microscopy images collected from around theopaque p-contact ofcore multishell nanowire LEDs with increasing In

concentration in the shell quantum well and in forward bias, showing purple, blue, cyan, green, and near yellow emission, respectively. (c) Normalizedelectroluminescencespectraobtained from five representative multicolor CMS nanowire LEDs. Courtesy of C.M. Lieber. (This figure also appears on page 747.)

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Figure 3.249 (Continued )

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(c)

1.0

0.8 Normalized intensity (au)

XLVI

0.6

0.4

0.2

0.0 300

400

500 Wavelength (nm)

Figure 3.249 (Continued )

600

700

Color Tables

Figure 4.8 Side view (projection onto the ð1 1  2 0Þ plane) of a relaxed and neutral screw dislocation: (a) full-core screw dislocation; (b) Ga-filled screw dislocation. Core of a full-core screw dislocation (discussed in greater detail in Figure 4.10 and the associated text) showing the

double helix of Ga bonds. The supercell is repeated twice in the [0 0 0 1] for clarity. Note that the bonds at the core are heavily distorted. Courtesy of Blumenau et al. (patterned after Ref. [29]). (This figure also appears on page 829.)

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Figure 4.36 Potential profile across the dislocation in n-type GaN deduced from the holographic phase map compared with theoretical profile (a). Courtesy of D. Cherns. False colored map showing phase shifts produced by edge dislocations viewed end-on in nominally undoped GaN. Contour lines emphasize dipole-like phase shifts near

dislocation cores (b). Line profile through indicated dislocation in (b) allows quantification of nominally undoped GaN of electric fields, yielding an estimated bound surface charge of 4 · 1011 e cm2 on either side of the defect (c). Courtesy of M. McCartney. (This figure also appears on page 874.)

Color Tables

2 1.5

Phase (rad)

1 0.5 0 –0.5 –1 –1.5 –2 –150

–100

(c)

–50 0 Distance (nm)

50

100

Figure 4.36 (Continued )

Figure 4.42 Contour plots of dislocation-induced electronic gap states for three edge dislocation configurations, namely, (a) fourcore, (b) full-core, and (c) open-core structures. The plots are obtained by calculating atomic geometries with DFT theory used as input to image simulations. Large (small) balls correspond to Ga (N) atoms [125]. (This figure also appears on page 883.)

150

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Mean implantation depth (µm)

0

0.09

0.26

0.50

0.79

1.13

HVPE GaN

0.50

10–14 µm

Mg-doped reference

0.49

S parameter

L

1 µm

36–39 µm

5 µm

49–68 µm

VGa

0.48

0.47

0.46 Defect free

0.45 0

5

10

15

20

Positron energy (keV) Figure 4.97 Ga vacancies and 1, 5, 10–14, 36–39, 49–68 mm thick HVPE GaN layers indicating increased S parameter, thus increased Ga vacancy concentration toward the GaN/Al2O3 interface in each of the films. A Mg-doped p-type sample with very low, if any, Ga vacancy is shown as the reference. Courtesy of K. Saarinen. (This figure also appears on page 986.)

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Figure 4.118 Hole concentration versus ND where the acceptor–donor–acceptor complex model of Ref. [495] is shown with thin and bold lines for NA ¼ 1 · 1019 cm3 (i) and 1 · 1020 cm3 (ii), respectively. The optimum hole concentration where NA ¼ 2ND, as expected from the complex formation, is shown with diamonds. For comparative purpose, the simple compensation model which assumes a single

donor and (unpaired) acceptor is depicted with thin and thick lines for NA ¼ 1 · 1019 cm3 (iii) and 1 · 1020 (iv), respectively. The random pair model is also plotted with thin and thick lines for NA ¼ 1 · 1019 cm3 (v) and 1 · 1020 cm3 (vi), respectively. Discussions with Dr R. Korotkov are acknowledged. (This figure also appears on page 1024.)

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Figure 4.126 Proposed bond center and antibonding site incorporation of H in GaN and its passivation of Mg during growth (the Mg atom is directly below the H atom). In part courtesy of C. Van de Walle. (This figure also appears on page 1037.)

GaN: Mn Mnd projected PDOS (/eV cell)

LII

t+

e-

t+

e+ t-

GaP: Mn

e-

t+ e+

t+

ee+

t-

t+

tt+

GaSb: Mn

e-

e+ t–3

t-

t+

t-

GaAs: Mn

–4

t-

–2

t-

t+ –1

–F Energy (eV)

1

2

3

Figure 4.150 In Mn d projected partial density of states for a single Mn in GaN, GaP, GaAs, and GaSb, where the symmetry (t2 and e) as well as the spin (þ and ) have been indicated. The shaded region represents the t2þ states (after Ref. [699]). (This figure also appears on page 1097.)

Color Tables

3d ion d n-1 t-(d )

e-(d )

t+(d)

Mn on Ga site t-CFR

Anion dangling bonds V Ga3-

CFR

e-

t+CFR t+(p)

e+(d) VBM

CFR

e+

DBH

t-

t-(p)

DBH

t+

Figure 4.151 A schematic energy-level diagram for the levels (central panel) generated from the interaction between the crystal field and exchange-split levels on the 3d transition metal ion (left panel) with the anion dangling bond levels (right panel), when the TM d levels are energetically shallower than the dangling bond levels (after Ref. [700]). (This figure also appears on page 1098.)

LIII

LIV

Color Tables Figure 4.155 Interband transitions in GaAs selected because of its well-known band structure and also its well-established and wellcharacterized properties in terms of magnetic ion doped diluted magnetic semiconductors: (arrows indicate emission but the concept is just as applicable to transitions from the valence band subband to the conduction band as in absorption). (a) Schematic band structure of GaAs near the G point, the center of the Brillouin zone. As for the terms, Eg is the bandgap and DSO the spin–orbit splitting; CB, conduction band; HH, valence heavy hole; LH, light hole; SO, spin–orbit split-off subbands; G6,7,8 are the corresponding symmetries at the k ¼ 0 point representing conduction, HH, LH, spin–orbit (SO) bands, or, more precisely, the irreducible representations of the tetrahedron group Td (see, e.g., Ref. [721]). The terms s1/2 and p3/2 and p1/2 represent the conduction band (s-like) and valence band (p-like) type of orbitals. (b) Selection rules for interband transitions between the Jz projection of the angular momentum along z-direction, sublevels for circularly polarized light sþ(right-hand circular polarization or positive helicity that results from transitions between the Jz ¼ 1/2 conduction band states and Jz ¼ 3/2 heavy-hole states, and Jz ¼ þ1/2 conduction band and Jz ¼ 1/2 light-hole states), and s (left-hand circular polarization or negative helicity, which results from transitions between the Jz ¼ þ1/2 conduction band states and Jz ¼ þ3/2 heavy-hole states, and Jz ¼ 1/2 conduction band and Jz ¼ þ1/2 light-hole states). The numbers by each transition indicate the relative transition intensities, with respect to the light-hole subband to the conduction band (absorption or excitation of carriers to higher bands), or the conduction band to the light-hole subband transition (emission), which apply to both excitation and radiative recombination (depicted by the arrows). The circular polarization (s polarization) for light energies that would not excite the spin split-off band is ideally 50%, which becomes 0 if the spin–orbit split-off band is also excited. For completeness, the transitions between the Jz ¼ 1/2 conduction band states and Jz ¼ 1/2 light-hole states, and Jz ¼ þ1/2 conduction band states and Jz ¼ þ1/2 light-hole states, which are linearly polarized (p polarization), are also shown as depicted by two-way arrows in the figure. The

" transition probability or the emission intensity normalized to the Jz ¼ 1/2 conduction state to the Jz ¼1/2 state transitions (indicated with 1) are also indicated in numbers for GaAs. The circular polarization resulting from the conduction band to the heavy-hole states are three times more intense than the circular polarization resulting from the conduction band states to the light-hole valence band states. The linearly polarized transitions are twice as intense as the circular polarization involving light-hole states. (c) Removal of the valence band heavyand light-hole degeneracy by, for example, strain inducing either by lattice mismatch or by confinement in a quantum well, which increases the electron polarization to nearly 100%. Note that heavy- and light-hole states are no longer degenerate. Both the tensile (left) and compressive (right) in-plane biaxial strain cases are shown. The respective ratios of various transitions (oscillator strengths) have been assumed to be the same as in the relaxed case. Note that spin is indifferent to strain, which means that spin-up and spin-down states are moved in the same direction by strain, but not to magnetic field, as spin-up and spin-down states in a given band are split and moved in opposite directions as shown in (d). In part courtesy of W. Chen, Linko1ping University. (d) Removal of the valence band heavy- and light-hole degeneracy as well as splitting the spin-up and spin-down states by application of magnetic field. The total splitting is enhanced due to sp-d interaction in DMS materials in the form of xN0a<Sz> for the conduction band states, xN0b<Sz> for the HH and LH valence band states, and (1/3)xN0a<Sz> for the spin–orbit split-off band. Here, N0, x, a, b, <Sz> represent the number of cations per unit volume, mole fraction of magnetic ions, the product of Bohr magneton and the g factor for the respective bands, and average spin for each magnetic ion site, respectively. Note that magnetic field/magnetization causes Zeeman splitting, and direction of splitting either up or down in energy is spin dependent. If the semiconductor is ferromagnetic as is the case of GaMnAs, one can either couple polarized light to the symmetry/splitting allowed bands or cause polarized light emission by tuning the wavelength. (This figure also appears on page 1112.)

Color Tables

LV

LVI

Color Tables

Figure 4.155 (Continued )

Color Tables

(a) x = 0.053 B ⊥ plane

2K 55 K 25 K

100 K 125 K 300 K

0.00

0.5

Rsheet(kΩ)

R Hall (kΩ )

0.03

-5

25 K

2K 0.3 -5

-0.03

100 K 125 K 55 K

0.4

300 K 0

5

B (T)

0 B (T )

5

0.08

1/χ Hall(au)

(RHall R ) sheet s

(b) 0.06 0.04 0.02 0.00 0

100

200

300

T (K) 120 (c)

T c (K)

80

40

0 0.00

0.04 x

Figure 4.170 (a, top) Temperature dependence of the Hall resistance RHall for a 200 nm thick Ga0.947Mn0.053As sample for which direct magnetization measurements have been performed but not shown. The inset shows the temperature dependence of the sheet resistance Rsheet. (b, center) Temperature dependence of the saturation magnetization [RHall/Rsheet]S obtained using Arrott plots (solid circles) and inverse susceptibility 1/wHall (open circles), both

0.08 deduced from the transport data shown in (a). Solid lines depict [RHall/Rsheet]S and (c, bottom) 1/wHall (bottom, c) calculated using the mean field Brillouin theory with S ¼ 5/2 for the Mn spin and the Curie–Weiss law, respectively. The dependence of magnetic transition temperature TC on Mn composition as determined from the transport data. Courtesy of Ohno and Matsukura [777]. (This figure also appears on page 1140.)

LVII

LVIII

Color Tables

+1/2 CB

Jz = −1/2

σ+

σ+

σ− π

σ−

π

σ+ σ−

+3/2 HH

−3/2 −1/2

+1/2 LH +1/2 CR

−1/2 VB;HH Figure 4.172 The GCB , G7VB;LH 7 conduction band and G9 VB;CR (spin–orbit split-off band), and G7 (crystal field split off band) valance bands in wurtzitic GaN at the G point along with polarization (sþ right-hand and s left-hand circular polarizations) of various transitions between the conduction and valence band states in the presence of a magnetic field. (This figure also appears on page 1147.)

0.15

Cr: 1%

5K

0.05

Cr: 3%

0.05

Magnetization (emu g–1)

Magnetization (emu g–1)

0.1

0.2 μ B /Cr

Cr: 5%

Cr: 0.5%

0

–0.05 –0.1 –0.15 –10 000

Cr : 5% Cr :1%

0

H c =100 Oe

–0.05 –1000

0

1000

Magnetic Field (Oe)

–5000

0

5000

10 000

Magnetic Field (Oe) Figure 4.177 Magnetization curves for Cr-doped GaN in atomic concentrations of 0.5, 1, 3, and 5% up to a magnetic field normal to the surface of 10 000 Oe (1 T). Note that the film containing 5% Cr does not show any saturation magnetization in the range measured and

appears to be paramagnetic. The blow-up version near the origin indicates hysteresis for 1% Cr sample and a coercive field of 100 Oe. Courtesy of F. Hasegawa. (This figure also appears on page 1157.)

Color Tables

0.04

Magnetization (emu g–1)

H = 200(Oe)

Ferromagnetic

Cr: 1%

0.02

Paramagnetic + ferromagnetic

Cr: 3 %

Paramagnetic

Cr : 5%

0 0

50

100

150 200 250 Temperature (K)

300

Figure 4.178 Temperature dependence of magnetization for 1, 3, and 5% Cr-containing GaN. As indicated, the film with 1% Cr is consistent with ferromagnetic behavior. The films with 3 and 5% Cr exhibit a combination of ferromagnetic and paramagnetic behavior, and paramagnetic behavior, respectively. Courtesy of F. Hasegawa. (This figure also appears on page 1158.)

350

LIX

LX

Color Tables

VG < 0 R (kΩ) Hall

0.04

1

1.5 K 5K 10 K

0

20 K

0.02

RHall (kΩ)

–1 –0.5

0.0 B(T)

VG > 0

0.5

0.00

22.5 K

VG = 0V

–0.02

+125 V –125 V 0V –0.04 –1.0

–0.5

0.0

0.5

1.0

B (mT) Figure 4.186 Hall resistance RHall of an insulated gate (In,Mn)As field effect transistor at 22.5 K as a function of the magnetic field for three different gate voltages. RHall is proportional to the magnetization of the (In,Mn)As channel. Upper right inset shows the temperature dependence of

RHall. Left inset shows schematically the gate voltage control of the hole concentration and the corresponding change of the magnetic phase. Courtesy of Ohno et al. [855]. (This figure also appears on page 1172.)

Color Tables

Figure 4.187 Injection of spin-polarized holes into a light-emitting p–n diode using a ferromagnetic semiconductor (Ga,Mn)As. (a) Sample structure. Spin-polarized holes hþ travel through the nonmagnetic GaAs and recombine with spin-unpolarized electrons in the (In,Ga)As quantum well. I represents the current, and sþ represents circularly polarized light emitted from the edge of the quantum well. (b) Dependence of the polarization DP of the emitted light on the magnetic field B at temperatures of 6, 31, 52 K, the latter above the Curie value. The solid and hollow symbols represent the degree of polarization when the magnetic field is swept in the positive and negative directions, respectively.

The magnetic field was applied parallel to the surface along the easy axis of magnetization of the (Ga,Mn)As. The temperature dependence of the residual magnetization M in (Ga,Mn)As, where the degree of polarization of the zero magnetic field seen in the emitted light exhibits the same temperature dependence as the magnetization (not shown). Dependence on temperature for B ¼ 0 of the change in the relative remanent polarization, DP, (hollow circles) and magnetic moment measured by a SQUID magnetometer (solid circles). Courtesy of Ohno and coworkers [770]. (This figure also appears on page 1174.)

LXI

Color Tables

∆P 1.00

T=6K

Magnetization 6

16 K 0.75

4 0.50

31 K

2

0.25

52 K 0.00 0

20

40

60

Temperature (K) (c) Figure 4.187 (Continued )

80

100

Magnetization (10-5emu)

Relative polarization,∆P (%)

LXII

Color Tables

High resistance state

Low resistance state

(b) Figure 4.190 (a) Schematic representation of a spin valve, a normal metal straddled by two ferromagnetic metals. When the spins in ferromagnetic metals on either end are aligned parallel to each other, the system is in the lowresistance state top. When, for example, the spin of the FM metal on the right is flipped by a magnetic field, making the spins of the

ferromagnetic metals antiparallel, a high resistance state is attained. (b) Schematic representation of transport that is parallel to the plane of a layered magnetic metal sandwich structure for antialigned (upper figure – high resistance) and aligned (lower figure – low resistance) orientations. (This figure also appears on page 1180.)

LXIII

j1

1 General Properties of Nitrides Introduction

GaN as a representative of its binary cousins, InN and AlN, and their ternaries along with the quaternary, is considered one of the most important semiconductors after Si. It is no wonder that it finds ample applications in lighting and displays of all kinds, lasers, detectors, and high-power amplifiers. These applications stem from the excellent optical and electrical properties of nitride semiconductors. The parameters are imperative in determining the utility and applicability of this class of materials to devices, as will be made evident in this chapter and throughout the book. In this chapter, the structural, mechanical, thermal, chemical, electrical, and optical properties of GaN and its binary cousins as well as the substrates commonly used for nitride epitaxy are treated in a general sense for quick reference. The detailed properties associated with electrical and optical parameters and properties are discussed in chapters dealing with transport and optical processes in GaN and related alloys. Because GaN is used in the form of a thin film deposited on foreign substrates, meaning templates other than GaN, a discussion of heteroepitaxial thin films is of paramount importance. Consequently, the properties of nitride films intricately depend on substrates, inclusive of the inherent properties such as lattice constants and thermal expansion coefficients, and on the process-induced characteristics such as surface preparation and chemical and physical interactions at the surface. These too are discussed in the book.

1.1 Crystal Structure of Nitrides

Group III nitrides can be of crystalline structures: the wurtzite (Wz), zinc blende (ZB), and rock salt. Under ambient conditions, the thermodynamically stable structure is wurtzite for bulk AlN, GaN, and InN. The zinc blende structure for GaN and InN has been stabilized by epitaxial growth of thin films on {0 1 1} crystal planes of cubic substrates such as Si [1], SiC [2], MgO [3], and GaAs [4]. In these cases, the intrinsic tendency to form the Wz structure is overcome by the topological compatibility.

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morko
c Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

j 1 General Properties of Nitrides

2

However, Wz structure could very likely be present at the extended defect sites. The rock salt form is possible only under high pressures and, therefore, is laboratory form of exercise. Let us now discuss the space groups for the various forms of nitrides. The rock salt, or NaCl, structure (with space group Fm3m in the Hermann–Mauguin notation and O5h in the Schoenflies notation) can be induced in AlN, GaN, and InN under very high pressures. The reason for this is that the reduction of the lattice dimensions causes the interionic Coulomb interaction to favor the ionicity over the covalent nature. The structural phase transition to rock salt structure was experimentally observed at the following estimated pressure values: 22.9 GPa (17 GPa in other estimates) for AlN [5], 52.2 GPa for GaN [6], and 12.1 GPa for InN [7]. Rock salt III nitrides cannot be produced by any epitaxial growth. The space grouping for the zinc blende structure is F43m in the Hermann– Mauguin notation and T 2d in the Schoenflies notation. The zinc blende structure has a cubic unit cell, containing four group III elements and four nitrogen elements. (Although the term zinc blende originated in compounds such as ZnS, which could be in cubic or hexagonal phase, it has been used ubiquitously for compound semiconductors with cubic symmetry. The correct term that should be used for the cubic phase of GaN is actually sphalerite. However, to be consistent with the usage throughout the literature, even at the expense of accuracy, the term zinc blende has been used in this book). The position of the atoms within the unit cell is identical to the diamond crystal structure. Both structures consist of two interpenetrating facecentered cubic sublattices, offset by one quarter of the distance along a body diagonal. Each atom in the structure may be viewed as positioned at the center of a tetrahedron, with its four nearest neighbors defining the four corners of the tetrahedron. The stacking sequence for the (1 1 1) close-packed planes in this structure is AaBbCc. Lowercase and uppercase letters stand for the two different kinds of constituents. The wurtzite structure has a hexagonal unit cell and thus two lattice constants, c and a. It contains six atoms of each type. The space grouping for the wurtzite structure is P63mc in the Hermann–Mauguin notation and C46v in the Schoenflies notation. The point group symmetry is 6 mm in the Hermann–Mauguin notation and C6v in the Schoenflies notation. The Wz structure consists of two interpenetrating hexagonal close-packed (hcp) sublattices, each with one type of atom, offset along the c-axis by 5/8 of the cell height (5c/8). The wurtzite and zinc blende structures are somewhat similar and yet different. In both cases, each group III atom is coordinated by four nitrogen atoms. Conversely, each nitrogen atom is coordinated by four group III atoms. The main difference between these two structures lies in the stacking sequence of closest packed diatomic planes. The Wz structure consists of alternating biatomic close-packed (0 0 0 1) planes of Ga and N pairs, thus the stacking sequence of the (0 0 0 1) plane is AaBbAa in the (0 0 0 1) direction. Although the main interest is in Wz GaN as opposed to zinc blende GaN, a description of stacking sequence of both GaN polytypes with the accepted Ramsdel notation is warranted, so is the stacking order of SiC polytypes that are relevant to GaN because they are used for substrates in GaN epitaxy. Therefore, a generic description of stacking in Wz semiconductors is given below. A comprehensive description of the tetrahedrally coordinated structures is imperative for a clear picture

1.1 Crystal Structure of Nitrides

j3

of nitride semiconductors, particularly the extended defects that are discussed in detail in Chapter 4. The bonds describe a tetrahedron denoted by T, which has one atom species at each of the three corners and the other atom species in its center [8,9]. The basal plane of this structure is defined by one face of the tetrahedron and the bond perpendicular to this plane defines the c-axis. A rotation of 180
around the caxis produces a twin variant denoted by T0 as shown in Figure 1.1a (left).

(a)

Atom a Atom b [0001]

T

T'

〈1120〉

{1100} In-plane bonds Out-of-plane bonds

(b)

A

T 3´

b

B

T1

a

A b

B

Figure 1.1 Representation of the tetrahedrally coordinated materials in the Ramsdel notation. (a) The two possible 0 0 tetrahedra. (b) The T1, T3, T1, T3, tetrahedral stacking composing the 2H sequence. Courtesy of Pierre Ruterana [9].

T 3´

T1

j 1 General Properties of Nitrides

4

The two variants (twins T and T0 ) are related to one another by mirror symmetry about one of the {1 1 0 0} m-planes. A tetrahedron can occupy one of the three possible positions in the basal plane. The representation of the tetrahedrally coordinated materials in the Ramsdel notation is shown in Figure 1.1a for two possible tetrahedra, one is the mirror image twin of the other with respect to the (1 1 0 0) m-plane. The single bonds are on the (1 1 2 0) plane, called the a-plane. The 0 0 0 layers of the tetrahedra can then be denoted by T1, T2, T3, and by T1 ; T2 ; T3 for its 0 0 twin. An example of T1 ; T ; T1 ; T3 stacking order representing 2H ordering as in wurtzitic GaN is shown in Figure 1.1b. The structure of nitride semiconductors and most relevant polytypes of SiC can be completely described by a combinatorial stacking of the aforementioned six tetrahedra layers. Naturally, not all the stacking sequences must obey the following two rules to keep a corner sharing structure, as such not all stacking orders are allowed: (i) A tetrahedron T can be followed by another one of the same kind with the 0 0 0 following subscript: T1T2T3, and inversely for the twin variant: T3 T2 T1 . (ii) A tetrahedron T1 must be followed by the twin variant of the preceding subscript: 0 0 T1 T3 , and inversely for its twin variant: T1 T2 . In the Ramsdel notation, the stacking order for the wurtzite structure corresponding to various polytypes can be denoted as . . . .

0

0

0

T1 T3 or T2 T1 or T3 T2 for the 2H polytype, which is also applicable to Wz nitride semiconductors; 0 0 T1T2T1 T3 for the 4H polytype; 0 0 0 T1T2T3T2 T1 T3 for the 6H polytype; 0 0 0 T1T2T3 or T3 T2 T1 for the 3C polytype.

The 3C, 4H, and 6H stacking sequences as well as 2H sequence on 6H sequence are discussed in Chapter 3. Recall that GaN crystallizes in the cubic structure (zinc blende or sphalerite, the latter being the correct term and the former being the one used universally) or in the more stable hexagonal structure (wurtzite). The anions (N3) form an hcp structure in which the cations (Ga3þ) occupy half of the tetrahedral sites. The structure of a unit cell of GaN projected along [0 0 0 1] is depicted schematically in Figure 1.2. The open symbols represent g sites that are occupied by nitrogen atoms; the Ga atoms are in the tetrahedral sites, b. These latter sites can either be at heights (3/8)c above (b1) or below (b2) N site, depending on the crystal polarity. A stick-and-ball representation of Ga-polarity and N-polarity Wz structure is depicted in Figure 1.3. The Wz and zinc blende structures differ only in the bond angle of the second nearest neighbor (Figure 1.4). As clearly shown, the stacking order of the Wz along the [0 0 0 1] c-direction is AaBb, meaning a mirror image but no in-plane rotation with the bond angles. In the zinc blende structure along the [1 1 1] direction, there is a 60
rotation that causes a stacking order of AaBbCc. The point with regard to rotation is illustrated in Figure 1.4b. The nomenclature for various commonly used planes of hexagonal semiconductors in two- and three-dimensional versions is presented in Figures 1.5 and 1.6. The Wz group III nitrides lack an

1.1 Crystal Structure of Nitrides

a γ1

β1

γ1

γ1

* γ1

β2

c

β1

γ2

β2 β1

γ1

u

β2

*

β2 β1

β2 β1

γ1

γ1

γ1

Figure 1.2 Schematic diagram showing the b1 and b2 tetrahedral sites of GaN unit cell. Starting with the assumption that N occupies the g sites, only one family of b sites can be simultaneously occupied by Ga atoms. Courtesy of Pierre Ruterana [9].

inversion plane perpendicular to the c-axis; thus, nitride surfaces have either a group III element (Al, Ga, or In) polarity (referred to as Ga-polarity) with a designation of (0 0 0 1) or (0 0 0 1)A plane or a N-polarity with a designation of (0 0 0 1) or (0 0 0 1)B plane. We will use the former notations for each. The distinction between these two directions is essential in nitrides because of their implications for the polarity of the polarization charge. Three surfaces and directions are of special importance in

Ga-polarity

N-polarity

[0001]

[0001]

Ga

N

Ga

N

Ga

N

Ga N

Ga N

Ga N

Ga N

Ga

N

Ga N

Ga

Ga

Ga

N

N

N

Ga

Ga

Ga

N

N

N

Ga

N

Ga

Ga

N

N

N

N

N

Ga

Ga

Ga

N

N

Ga

Ga

N Ga

Ga

N

N

N

N

Ga

Ga

Ga

N Ga

Figure 1.3 A stick-and-ball diagram of a hexagonal structure.

N Ga

N

N

Ga

Ga

j5

j 1 General Properties of Nitrides

6

C View normal to [0001] and [111]

B

B

A

A

Ga N

Wurtzitic

Zinc blende

View along [0001] and [111]

(a) Figure 1.4 A stick-and-ball stacking model of crystals with (a, both top and bottom) 2H wurtzitic and (b, both top and bottom) 3C zinc blende polytypes. The bonds in an A-plane (1 1 2 0) are indicated with heavier lines to accentuate the stacking sequence. The figures on top depict the three-dimensional view. The

(b) figures at the bottom indicate the projections on the (0 0 0 1) and (1 1 1) planes for wurtzitic and cubic phases, respectively. Note the rotation in the zinc blende case along the h1 1 1i direction. (Please find a color version of this figure on the color tables.)

nitrides, which are (0 0 0 1) c-, (1 1 2 0) a-, and (1 1 0 0) m-planes and the directions associated with them, h0 0 0 1i, h1120i, and h1100i as shown in Figure 1.7. The (0 0 0 1), or the basal plane, is the most commonly used surface for growth. The other two are important in that they represent the primary directions employed in reflection high-energy electron diffraction (RHEED) observations in molecular beam epitaxial growth, apart from being perpendicular to one another. They also represent the direction of stripes employed in the epitaxial lateral overgrowth (ELO), details of which are discussed in Section 3.5.5.2. The cohesive energy per bond in the wurtzite form is 2.88 eV (63.5 kcal mol1), 2.2 eV (48.5 kcal mol1), and 1.93 eV (42.5 kcal mol1) for AlN, GaN, and InN, respectively [10]. The calculated energy difference DEW-ZB between wurtzite and zinc blende lattice is small [11]: DEW-ZB ¼ 18.41 meV/atom for AlN, DEW-ZB ¼ 11.44 meV/atom for InN, and DEW-ZB ¼ 9.88 meV/atom for GaN. Wurtzite form is energetically preferable for all three nitrides compared to zinc blende, although the energy difference is small. The Wz structure can be represented by lattice parameters a in the basal plane and c in the perpendicular direction, and the internal parameter u, as shown in Figure 1.8.

1.1 Crystal Structure of Nitrides 〈1100〉

〈1120〉

(m)

(a )

o

1010

30

(m )

o

30

r

n n

n n

c

1100

0110

r

r n

2110

1120

n 0111 s (c)

1210 a 57

o

0114 d 1213 n

1012 r

2113 n

1101 s

1104 d 0001 c

1210 1213 n

o

.6

61

1123 n

c

(n) r

(r) r

n

n

1102 r

32.4

n

o

1100 m

a

a

a

1014 d

2113 n

0112 r 0110

1123 n

1011 s

(m) 2110

1120 1010

n

n

r

n

Common crystallographic planes in sapphire d Plane Miller name index spacing a m c r n s

(1120) (1010) (0001) (1102) (1123) (1011)

2.379 Å 1.375 Å 2.165 Å 1.740 Å 1.147 Å 1.961 Å

Angles between common planes (0001) ^ (1102) (0001) ^ (1123) (0001) ^ (1011) (0001) ^ (1121) (0001) ^ (1120) (0001) ^ (1010) (1120) ^ (1010)

c^r c^n c^s c^ c^a c^m a^m

57º 35' 61º 11' 72º 23' 79º 37' 90º 00' 90º 00' 30 00'

Figure 1.5 Labeling of planes in hexagonal symmetry ( for sapphire).

The u parameter is defined as the anion–cation bond length (also the nearest neighbor distance) divided by the c lattice parameter. The c parameter depicts the unit cell height. The wurtzite structure is a hexagonal close-packed lattice, comprising vertically oriented X–N units at the lattice sites. The basal plane lattice parameter (the edge length of the basal plane hexagon) is universally depicted by a and the axial lattice parameter, perpendicular to the basal plane, is universally described by c. The interatomic distance in the basic unit is described by the internal parameter u. In an ideal wurtzite structure represented by four touching hard spheres, pffiffiffiffiffiffiffiffi the values of the axial ratio and the internal parameter are c=a ¼ 8=3 ¼ 1:633 and u ¼ 3/8 ¼ 0.375, The crystallographic ! pffiffiffi respectively. pffiffiffi ! vectors of wurtzite are a ¼ að1=2; 3=2; 0Þ, b ¼ að1=2;  3=2; 0Þ, and ! c ¼ að0; the basis atoms are (0, 0, 0), (0, 0, uc), pffiffiffi0; c=aÞ. In Cartesian coordinates, pffiffiffi a(1=2; 3=6; c=2a), and a(1=2; 3=6; ½u þ 1=2c=a). Table 1.1 tabulates the calculated structural parameters a, c/a, and e1 ¼ u  uideal for the III–V nitrides by three different groups [12–14]. In the case of Bernardini et al. [12], they optimized the structure within both the generalized gradient

j7

j 1 General Properties of Nitrides

8

(tuvw) coordinate system v

1010 2110

1120

0110

0111 s

1123 n

1012 r 2113 n

0114 d

1104 d 1210

t

0001 c 0112 r

1102 r 2113 n

1100 m

1100

1213 n

1213 n

1210 a

1101s

1014 d

1123 n

0110

1011 s 1120

2110

u

1010

Figure 1.6 A magnified view of labeling of planes in hexagonal symmetry in the (tuvw) coordinate system with w representing the unit vector in the c-direction. The lines are simply to show the symmetryonly. If the lines connecting m-points among each other and a-points among each other were to be interpreted as the projection of those

planes on the c-plane, the roles would be switched in that the lines connecting the m-points would actually represent the a-planes and lines connecting the a-points would actually represent the m-planes that are normal to the plane of the page.

(1120) a-plane [1120] v

(1100) m-plane [0110]

[1010]

[1210]

[1100]

[2110]

t

[2110]

[1010]

[1100]

Ga N

[0110] u [1210]

m-planes a-planes

[1120]

Figure 1.7 The orientations which are commonly used in nitrides, namely the (1 1 2 0) and (1 1 0 0) planes and associated directions are shown as projections on the (0 0 0 1) basal plane.

1.1 Crystal Structure of Nitrides

a

M

M

α c N M

M

b=u x c M

M

b1

N

[0001]

N

b '2

M

N

b 3' N

β b'

N

1

N

M M

M

Figure 1.8 Schematic representation of a wurtzitic metal nitride structure with lattice constants a in the basal plane and c in the basal direction, u parameter, which is expressed as the bond length or the nearest neighbor distance (b) divided by c (0.375 in ideal crystal), a and b (109.47 in ideal crystal) are the bond 0 0 0 angles, and b 1, b 2, and b 3, represent the three types of second nearest neighbor distances.

approximation (GGA) and local density approximation (LDA). The experimental data are from Leszczynski et al. [15]. In all Wz III nitrides, experimentally observed c/a ratios are smaller than ideal and it has been postulated that not being so would lead to the zinc blende phase [16]. There are two avenues that can lead to a deviation from ideal: changing the c/a ratio or changing the u value. It should be pointed out that a strong correlation exists between the c/a ratio and the u parameter so that when c/a decreases, the u parameter increases in a manner to keep the four tetrahedral distances nearly constant through a distortion of tetrahedral angles. For equal bond length to prevail, the following relation must hold: uð1=3Þða2 =c 2 Þ þ 1=4:

ð1:1Þ

Table 1.1 Structural parameters for GaN reported by Bechstedt,

Großner, and Furthm€ uller (BGF) [13] and by Wei and Zunger (WZ) [14] using the local density approximation (LDA).

BGF WZ BFV (LDA) BFV (GGA) Experimental data

a (Å)

c/a

e1 (103c/a)

3.150 3.189 3.131 3.197 3.1890

1.6310 1.6259 1.6301 1.6297 1.6263

6.5 1.8 1.6 1.9 2.0

However, Bernardini, Fiorentini, and Vanderbilt [12] employed both the LDA and GGA methods. Lattice constant a is given in Å and e1 in 103c/a.

j9

j 1 General Properties of Nitrides

10

The nearest neighbor bond length along the c-direction (expressed as b in Figure 1.8) and off c-axis (expressed as b1 in Figure 1.8) can be calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 1 a þ  u c2 : ð1:2Þ b ¼ cu and b1 ¼ 3 2 In addition to the nearest neighbors, there are three types of second nearest 0 0 neighbors designated in Figure 1.8 as b 1 (one along the c-direction), b 2 (six of them), 0 and b 3 (three of them), which are given as [17] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 1 0 0 0 2 a þ c2 u : b 1 ¼ cð1  uÞ; b 2 ¼ a2 þ ðucÞ ; and b 3 ¼ 3 2 ð1:3Þ The bond angles, a and b, are given by [17] "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1 a ¼ p=2 þ arccos 1 þ 3ðc=aÞ2 ð  u þ 1=2Þ2 ; "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b ¼ 2 arcsin

2

4=3 þ 4ðc=aÞ ð  u þ 1=2Þ

2

1

ð1:4Þ

# :

Table 1.2 tabulates the calculated as well as experimentally observed structural parameters discussed above, inclusive of the lattice parameters, the nearest and second nearest neighbor distances, and the bond angles for three end binaries, GaN, AlN, and InN. The distances are in terms of Å. The lattice parameters are commonly measured at room temperature by X-ray diffraction (XRD), which happens to be the most accurate one, using the Bragg law. In ternary compounds, the technique is also used for determining the composition; however, strain and relevant issues must be accounted for as the samples are in the Table 1.2 Calculated (for ideal crystal) and experimentally

observed structural parameters for GaN, AlN, and InN [17]. GaN

u a (Å) c/a b (Å) b1 (Å) 0 Â b 1 ðeÞ 0 Â b 2 ðeÞ 0 Â b 3 ðeÞ a b

AlN

InN

Ideal

Exp.

Ideal

Exp.

Ideal

Exp.

0.375 3.199 1.633 1.959 1.959 3.265 3.751 3.751 109.47 109.47

0.377 3.199 1.634 1.971 1.955 3.255 3.757 3.749 109.17 109.78

0.375 3.110 1.633 1.904 1.904 3.174 3.646 3.646 109.47 109.47

0.382 3.110 1.606 1.907 1.890 3.087 3.648 3.648 108.19 110.73

0.375 3.585 1.633 2.195 2.195 3.659 4.204 4.204 109.47 109.47

0.379 3.585 1.618 2.200 2.185 3.600 4.206 4.198 108.69 110.24

1.1 Crystal Structure of Nitrides

form of epitaxial layers on foreign substrates. The accuracy of X-ray diffraction and less than accurate knowledge of the elastic parameters together allow determination of the composition to only within about 1% molar fraction. In addition to composition, the lattice parameter can be affected by free charge, impurities, stress (strain), and temperature [18]. Because the c/a ratio correlates with the difference of the electronegativities of the two constituents, components with the greatest differences show the largest departure from the ideal c/a ratio [19]. For GaN, the c/a ratio and the value of u are measured as 1.627 (1.634 in Ref. [17]) and 0.377, respectively, which are close to the ideal values [20]. AlN deviates significantly from the ideal parameters: c/a ¼ 1.601 (1.606 in Ref. [17]) and u ¼ 0.382. Although the data for InN are not as reliable, values of u ¼ 0.379 and c/a ¼ 1.601 have been reported [17]. Inhomogeneities, strain, partial relaxation of strain, and high concentration of structural defects may distort the lattice constants from their intrinsic values and cause a wide dispersion among the reported values. Table 1.3 lists a comparison of measured and calculated lattice parameters reported for AlN, GaN, and InN crystallized in the wurtzite structure in more detail in terms of the specifics of the sample used for measurements and complements. The dispersion is even a greater concern in ternary and quaternaries, as compositional inhomogeneities, in addition to the aforementioned issues, cause an additional dispersion. The particulars of the ternaries are discussed in Section 1.5. The wurtzite polytypes of GaN, AlN, and InN form a continuous alloy system whose direct bandgaps range, according to data that adorned the literature for years, from 1.9 eV for InN, to 3.42 eV for GaN, and to 6.2 eV for AlN. A revisit of the InN bandgap indicates it to be about 0.78 eV [30] and the same for AlN is about 6 eV in which case the energy range covered would be about 0.7–6 eV. Thus, the III–V nitrides could potentially be fabricated into optical devices, which are active at wavelengths ranging from the red well to the ultraviolet. The bandgaps of nitrides, Table 1.3 Measured and calculated lattice constants of wurtzite AlN, GaN, and InN.

Compound

Sample

a (Å)

c (Å)

AlN

Bulk crystal [21] Powder [22] Epitaxial layer on SiC [23] Pseudopotential LDA [24] FP-LMTO LDA [25] Bulk crystal [18] Relaxed layer on sapphire [26] Powder [29] Relaxed layer on sapphire [27] GaN substrate – LEO [28] Pseudopotential LDA [24] FP-LMTO LDA [25] Powder [29] Pseudopotential LDA [24] FP-LMTO LDA [25]

3.1106 3.1130 3.110 3.06 3.084 3.189 3.1892 3.1893 3.1878 3.1896 3.162 3.17 3.538 3.501 3.53

4.9795 4.9816 4.980 4.91 4.948 5.1864 5.1850 5.1851 5.1854 5.1855 5.142 5.13 5.703 5.669 5.54

GaN

InN

LDA: local density approximation; FP-LMTO: full-potential linear muffin–tin orbital.

j11

j 1 General Properties of Nitrides

12

Figure 1.9 The bandgaps of nitrides, substrates commonly used for nitrides, and other conventional semiconductors versus their lattice constants. (Please find a color version of this figure on the color tables.)

substrates commonly used for nitrides, and other conventional semiconductors are shown in Figure 1.9 with respect to their lattice constants. All III nitrides have partially covalent and partially ionic bonds. The concept of fractional ionic character (FIC) is useful in interpreting many physical phenomena in the crystals [31,32]. The FIC may be defined for a binary compound AB as FIC ¼ jQ A  Q B j=jQ A þ Q B j, where Q A and Q B are effective charges on atoms A and B. The FIC values range from zero for a covalent compound (each atom has four electrons) to 1 for an ionic compound (all eight electrons belong to the anion). Figure 1.10 displays the charge distribution along the AB bond for all three compounds. The arrow along the bond charge indicates the atomic boundaries in the crystals that are not always at the minimum of the line charge along the bond AB. This should be expected taking into account the partial covalent bond of the compounds, because only in the ionic crystals, the atomic boundary is clearly defined. Table 1.4 lists the calculated effective radii, rIII and rN, the effective charges, and FIC for AlN, GaN, and InN. The ionicity of AlN is high. This may explain the difficulties with AlN doping. It is well known that only covalent semiconductors or semiconductors with a large covalent component can form hydrogen-like shallow levels in the bandgap by substitution of a host atom with a neighbor with one more or one less electron. GaN and InN have a smaller than AlN but nearly equal ionicity. GaN was doped both p- and n-type. Thus, one can expect that InN can also be doped n- and p-type. To date, only n-type InN has been obtained because of high volatility of nitrogen and easiness of nitrogen vacancy formation that acts as a donor in this compound.

1.1 Crystal Structure of Nitrides 2.0

AlN

1.5 1.0 0.5 0.0 0.0

0.5

1.0 r(Al-N)(λ)

1.5

2.0

1.5

2.0

2.0

ρ(r) (electrons/a.u.3)

GaN 1.5 1.0 0.5 0.0 0.0

0.5

1.0 r(Ga-N)(λ)

2.0 InN 1.5 1.0 0.5 0.0 0.0

0.4

0.8

1.2

1.6

2.0

r(In-N)(λ) Figure 1.10 Charge density along the III–N bond in III nitride semiconductors.

The III-nitrides are commonly grown on mismatched substrates because of the lack of suitable native substrates. Thus, the epitaxial layers are strained during cool down, if they are sufficiently thick for them to relax at the growth temperature. The mechanical forces related to strain dramatically change the band structure of the epitaxial layers. The pressure dependence of the bandgap energy Eg can be expressed as Eg ¼ Eg(0) gP þ dP2, where Eg(0) is the bandgap of stress-free semiconductor, g and d are the Table 1.4 Calculated ionic radii (Å), effective charges (electrons), and the fractional ionic character (FIC) for III nitrides [32].

Compound

rIII (Å)

rN (Å)

QIII (e)

QN (e)

FIC

AlN GaN InN

0.8505 0.9340 1.0673

1.0523 1.0119 1.0673

1.12 1.98 1.83

6.88 6.02 6.17

0.72 0.51 0.54

j13

j 1 General Properties of Nitrides

14

3.8

Energy gap (eV)

3.7

3.6

3.5

3.4

3.3

-2

0

2

4

6

8

10

Pressure (GPa) Figure 1.11 Pressure dependence of the GaN energy gap, showing the typical sublinear character. Solid line represents the calculations of Christiansen and Gorczyca [35], which have been rigidly upshifted by 0.82 eV for a better fit with experiments, and the squares represent experimental results [6].

pressure coefficients, and P is the pressure. For GaN, g and d parameters are 4.2 · 103 and 1.8 · 105, respectively [33,34]. The bandgap is in terms of eVand the pressure is in terms of kbar. The pressure dependence has, in general, a sublinear character. The variation of the GaN energy gap with pressure, both theoretical [35] and experimental [6], is shown in Figure 1.11. The calculated pressure coefficients for III nitrides are given in Table 1.5. Parameters associated with mechanical properties of GaN in wurtzitic phase are tabulated in Tables 1.6 and 1.7, the latter dealing with the sound wave velocity. The same parameters for the zinc blende phase of GaN are tabulated in Tables 1.8 and 1.9.

Table 1.5 Calculated pressure coefficients for III nitrides including wurtzitic, zinc blende, and rock salt phases (g in units of meV GPa1 and d in units of meV GPa2) [35].

Zinc blende polytype

Wurtzite polytype

Rock salt polytype

Compound

G

d

g

d

g

d

AlN GaN InN

40 39 33

0.32 0.32 0.55

42 40 16

0.34 0.38 0.02

43 39 41

0.18 0.32 0.08

It should be noted that rock salt phases cannot be synthesized and exist only under high pressures beyond the phase transition point. g and d parameters with values of Eg ¼ Eg(0) þ gP þ dP2.

1.1 Crystal Structure of Nitrides Table 1.6 Parameters related to mechanical properties of wurtzitic GaN (in part after Ref. [36]).

Wurtzite polytype GaN

Parameter value/comments

Group of symmetry Molar volume, Vc (cm3 mol1) Molecular mass (g mol1) Density (g cm3)

C 46v (P63mc) 13.61 83.7267 6.11 or 6.15

Number of atoms in 1 cm3 Lattice constants

8.9 · 1022 a ¼ 3.1893 Å for powder, c ¼ 5.1851 Å for powdera 210 [38]b or 20.4 · 1011 dyn cm2 (204 GPa) 4 150 0.23  0.06 (0.198–0.37) from C parameters 15.5 1200–1700

Bulk modulus B (GPa) (compressibility1) dB/dP Young’s modulus (GPa) Poisson’s ratio, n or s0 (n ¼ C13/(C11 þ C12)) Knoop’s hardness (GPa) Surface microhardness (kg mm2) Nanoindentation hardness (GPa) Yield strength (GPa) Deformation potential, Eds C11 (GPa) C12 (GPa) C13 (GPa)

C33 (GPa) C44 (GPa)

10.8 at 300 K 0.1 at 1000 K 8.54 eV unscreened, 12 eV screened 390  15, 29.6 · 1011 dyn cm2 (296 GPa) 145  20, 13.0 · 1011 dyn cm2 (130 GPa) 106  20, 15.8 · 1011 dyn cm2 (158 GPa)

Comments/ references

The latter by Bougrov et al. [37]

The latter by Bougrov et al. [37]

[39] At 300 K 300 K, using Knoop’s pyramid test [36,40,41]

[38,42] The second set is from Ref. [43] The second set is calculated from the mean square displacement of the lattice atoms measured by X-ray diffraction

398  20, 26.7 · 1011 dyn cm2 (267 GPa) 105  10, 2.41 · 1011 dyn cm2 (241 GPa)

nh0 0 0 1i ¼ (Da/arelaz)/(Dc/crelax) or nh0 0 0 1i ¼ (Da/a0)/(Dc/c0) with Da ¼ ameas  arelax and Dcmeas  crelax. Conversion: 1 dyn cm2 ¼ 0.1 Pa (i.e., 1 GPa ¼ 1010 dyn cm2). For details of elastic constants and piezoelectric constants, see Tables 2.27 and 2.28, and at 300 K Bs ¼ 210  10 GPa, Bs ¼ [C33(C11 þ C12)  2(C13)2]/[C11 þ C12 þ 2C33  4C13]. a See Section 1.2.2 for details and lattice parameter for GaN on different substrates b Average of Voigt and Reuss bulk modulus.

j15

j 1 General Properties of Nitrides

16

Table 1.7 Wave propagation properties in wurtzitic GaN [36].

Wave propagation direction [1 0 0]

[0 0 1]

Wave character VL (longitudinal) VT (transverse, polarization along [0 0 1]) VT (transverse, polarization along [0 1 0]) VL (longitudinal) VT (transverse)

Expression for wave velocity

Wave velocity (in units of 105 cm s1)

(C11/r)1/2 (C44/r)1/2

7.96 4.13

(C11  C12)/2r)1/2

6.31

(C33/r)1/2 (C44/r)1/2

8.04 4.13

Parameters associated with thermal properties of GaN in wurtzitic and zinc blende phases (expected to be identical or nearly identical – treated to be identical here) are tabulated in Table 1.10. The parameters associated with electrical and optical properties of wurtzitic GaN are tabulated in Table 1.11. The same parameters associated with the zinc blende phase of GaN are tabulated in Table 1.12. Table 1.8 Parameters related to mechanical properties of zinc blende GaN (in part after Ref. [36]).

Zinc blende polytype GaN

Parameter value/comments

Group of symmetry Molar volume, Vc, na, or O (cm3 mol1) Molecular mass (g mol1) Density (g cm3) Number of atoms in 1 cm3 Lattice constant (Å) Bulk modulus, B (GPa)

2 Tp d ðF43mÞ ffiffiffi ð 3a2 cÞ=4 ¼ 2:28310  23 cm3

dB/dP Young’s modulus (GPa)

1.936 · 1023 6.15 8.9 · 1022 a ¼ 4.511  4.52 Bs ¼ 204 [36], 201 (theory) [45], 237 [46], 200 [47] 3.9, 4.3 181 [36]

Shear modulus, C 0 (GPa) Poisson’s ratio, n or s0

67 [36] 0.352 [36]

Knoop’s hardness Surface microhardness Nanoindentation hardness Yield strength Deformation potential, Eds C11 (GPa) C12 (GPa) C44 (GPa)

293 159 155

Bs ¼ [C33(C11 þ C12)  2(C13)2]/[C11 þ C12 þ 2C33  4C13] or Bs ¼

Comments/references

2

C12 Þ  2ðC 13 Þ Bs ¼ CC3311ðCþ11C12þ þ 2C 33  4C13

Y0 ¼ (C11 þ 2C12) (C11  C12)/(C11 þ C12) C0 ¼ (C11  C12)/2 n or s0 ¼ C13/ (C11 þ C12)

[42]

C 33 ðC 11 þ C 12 Þ  2ðC13 Þ2 . C11 þ C 12 þ 2C33  4C 13

1.1 Crystal Structure of Nitrides Table 1.9 Wave propagation properties in zinc blende GaN (after Ref. [36]).

Wave propagation direction

Wave character

Expression for wave velocity

Wave velocity (in units of 105 cm s1)

[1 0 0]

VL (longitudinal) VT (transverse)

(C11/r)1/2 (C44/r)1/2

6.9 5.02

[1 1 0]

VL (longitudinal) Vt//(transverse) V t? (transverse)

[(C11þCl2þ2C44)/2r]1/2 Vt// ¼ VT ¼ (C44/r)1/2 [(C11  C12)/2r]1/2

7.87 5.02 3.3

[1 1 1]

V l 0 V l

[(C11 þ 2C12 þ 4C44)/3r]1/2 [(C11  C12 þ C44)/3r]1/2

8.17 3.96

0

For the crystallographic directions, see Ref. [44].

Table 1.10 Parameters related to thermal properties of GaN, wurtzitic, and zinc blende phases are expected to be the in this respect with the exception of the first two parameters, which are for the wurtzitic phase (in part Ref. [36]).

GaN

Parameter value/comments

Comments/references

Temperature coefficient (eV K1) Thermal expansion (K1)

dEg/dT ¼ 6.0 · 104

Wurtzite structure only

Da/a ¼ 5.59 · 106, a|| ¼ aa ¼ 5.59 · 106 (wurtzite structure) [48]

Dc/c ¼ 3.17 · 106; for a plot versus temperature, see Ref. [49] (wurtzite structure only) For low dissociation material (106 cm2)

Thermal conductivity k (W cm1 K1) Debye temperature (K) Melting point (
C) Specific heat (J g1
C1) Thermal diffusivity (cm2 s1) Heat of formation, DH298 (kcal mol1) Heat of atomization, DH298 (kcal mol1) Heat of sublimation (kcal mol1) Heat capacity (J mol1 K1) Specific heat (J mol1 K1) (298 K < T < 1773 K) Enthalpy, DH0 (kcal mol1) Standard entropy of formation, DS0 (cal mol1 K1)

11.9 at 77 K, 2.3 at 300 K, 1.5 at 400 K 600 >1700 (at 2 kbar), 2500 (at tens of kbar) 0.49 0.43 26.4

[50] [37] [37]

203 72.4  0.5 35.4 at 300 K Cp ¼ 38.1 þ 8.96 · 103T

[51]

37.7 32.43

The specific heat Cp of Wz GaN at constant pressure for 298 K < T < 1773 K is Cp ¼ 38.1 þ 8.96 · 103T (J mol1 K1) [51].

j17

j 1 General Properties of Nitrides

18

Table 1.11 Parameters related to electrical and optical properties

of Wz GaN (in part after Refs [36,44]).

Wurtzite polytype GaN

Parameter value/comments

Bandgap energy, Eg (eV), direct Breakdown field (cm1) Electron affinity (eV) Energy separation between G and M–L valleys (eV)

3.42 at 300 K, 3.505 at 1.6 K

Energy separation between M–L valleys degeneracy (eV) Energy separation between G and A valleys (eV) Energy separation between A valley degeneracy (eV) Index of refraction

Dielectric constants (static) Dielectric constants (high frequency)

Optical LO phonon energy (meV) A1-LO, nA1(LO) (cm1)

Comments/ references

3–5 · 106 at 300 K 4.1 1.9 at 300 K

[53] [37] [37]

1 at 300 K 0.6 at 300 K

[52] [37]

0.6 at 300 K 1.3–2.1 at 300 K

[52] [37]

2 at 300 K 1 at 300 K

[52] [37]

0.2 at 300 K n (1 eV) ¼ 2.35 or 2.3 2.29, n (3.42 eV) ¼ 2.85 at 300 K (extrapolated to 0 eV), E?c interference method (the value for E||c is 1.5(2)% lower at 500 nm); also see energy dependence and long-wavelength value [54] 10.4 (E||c) 9.5 (E?c) 8.9 in c-direction (E||c) at 300 K 5.35 5.8 (E||c) at 300 K 5.35 (E?c) at 300 K 5.47 (E||c) 91.2

[52]

[55] [55] [37] [37] [56] [55] [37]

710–735

[57]

744

A1-TO, nA1(TO||) (cm1) E1-LO, nE1(LO? ) (cm  1)

533–534 741–742

[34] [58]

533 746

E1-TO, nE1(TO? ) (cm  1) E2 (low) (cm1) E2 (high) (cm1)

556–559 143–146 560–579

[59]

559

Reflectivity [55] Raman [56] Reflectivity [55] Raman [60]

1.1 Crystal Structure of Nitrides Table 1.11 (Continued)

Wurtzite polytype GaN

Parameter value/comments

Energy of spin–orbital splitting, Eso (meV)

11 (þ5, 2) at 300 K calculated from the values of energy gap Eg,dir (given in this table) 40 at 300 K

Energy of crystal-field splitting, Ecr (meV)

Effective electron mass, == me or me

Effective electron mass, me? or m? e Effective hole mass Effective hole masses (heavy), mhh

Effective hole masses (light)

Effective hole masses (split-off band), ms

Effective mass of density of state, mv Effective conduction band density of states (cm3) Effective valence band density of states (cm3) Electron mobility (cm2 V1 s1) Hole mobility (cm2 V1 s1) n-doping range (cm3)

22 (2), calculated from the values of energy gap Eg,dir (given in this table) 0.20m0 at 300 K

Comments/ references [61]

[37] [61] [37]

0.20m0 0.27m0 by Faraday rotation 0.138–0.2 0.20m0, 300 K; fit of reflectance spectrum

[62] [52]

0.15–0.23m0 0.8m0 at 300 K mhh ¼ 1.4m0 at 300 K

[52] [64] Calculated

==

mhhz ¼ mhh ¼ 1:1m0 at 300 K mhh? ¼ m? hh ¼ 1:6m0 at 300 K == mhh ¼ 1:1  2:007m0 m? hh ¼ 1:61  2:255m0 mlh ¼ 0.3m0 at 300 K ==

mlhz ¼ mlh ¼ 1:1m0 at 300 K m? lh ¼ mlh? ¼ 0:15m0 at 300 K == mlh ¼ 1:1  2:007m0 mlh? ¼ 0:14  0:261m0 msh ¼ 0.6m0 at 300 K ==

mshz ¼ mch ¼ 0:15m0 at 300 K msh? ¼ m? ch ¼ 1:1m0 at 300 K == msh? ¼ mch ¼ 0:12  0:16m0 ¼ 0:252  1:96m0 m? ch 1.4m0

[63]

[15] [70] [52] [52] Calculated [15] [70] [52] [52] Calculated [36] [70] [52] [52] [37]

2.3 · 1018 at 300 K

4.6 · 1019 at 300 K 1400 experimental at 300 K 0.42, which may be caused by erroneous determination of the composition and illustrates well the problem at hand. Even additional investigations did not agree on the exact value of the bowing parameter. For example, a value for bInGaN ¼ 3.9  0.5 eV was reported when 0.9 eV was used for the InN gap, but the bowing parameter had to be increased to 5.1  0.4 eV when 1.9 eV was used for the InN gap [385]. Using the bandgap determined by PL, a bowing parameter of 4.5 eV was also reported [386]. However, when reflectivity measurements together with PL data corrected for Stokes shift were used, bInGaN ¼ 2.5  0.7 eV was obtained for 0.9 eV bandgap of InN and bInGaN ¼ 3.5  0.7 eV for 1.9 eV InN bandgap. Optical transmission measurements led to a bowing parameter of 8.4 eV [387]. At least one theoretical effort resulted in a bowing parameter of 1.2 eV [377]. In fact, linear bandgap dependence on composition with a slope of 3.57 eV for up to 25% InN content has also been reported [388]. Linear dependence with a slope of 4.1 eV for InN mole fraction, x < 0.12, has been reported in another publication as well [389]. Wu et al. [390] visited the bandgap dependence of InGaN on composition by considering 0.8 eV for the bandgap of the end binaryInN. Figure1.39 shows the composition dependence of the bandgap of InGaN, determined by using photomodulated transmission [391] and optical absorption [392] measurements, as a

j95

j 1 General Properties of Nitrides

96

Lattice constant (Å) 3 .2

Energy gap (eV)

4

3.3

3 .4

3 .5

3 .6

abs. 300 K PL PT Old data Calorimetric

GaN

3 2

InN

1 0 0.0

0.2

0.4

0.6

0.8

1.0

Composition (x) Figure 1.39 PL peak energy and bandgap of InGaN determined by optical absorption as a function of composition, as compiled in Ref. [390], including previously reported data for InN. The solid curve shows the fit to the bandgap energies (determined by absorption and phototransmission) using a bowing parameter b ¼ 1.43 eV [390].

function of GaN fraction. The data near the GaN binary end include those reported by Pereria et al. [392], Shan et al. [390], and O’Donnel et al. [393]. Care was taken by observing the dependence of squared absorption coefficient on light probe energy and seeing nearly a linear dependence to gain confidence on the measured bandgap and also confirming the values by bandgaps determined by photomodulated transmission measurements. The slight deviation from linearity near the InN end of the ternary has been attributed to the nonparabolicity of the conduction band caused by the k p interaction between the G6 symmetry conduction band and the G8 symmetry valence bands. As shown by the solid curve in Figure 1.39, the compositional dependence of the bandgap in the entire composition range can be well fit by a bowing parameter of b ¼ 1.43 eV. Shown in Figure 1.39 with dashed line is the fit to the empirical expression using energy of 1.9 eV for InN and bowing parameter of 2.63 eV to demonstrate that it does represent the Ga-rich side of the compositions well. However, the bowing parameter of 1.43 eV that is good for the entire compositional range is the one that utilizes 0.77 eV for the bandgap of InN. In an investigation with a different set of objectives, Yoshimoto et al. [368] studied the effect of growth conditions on the carrier concentration and transport properties of InxGa1xN. They observed that if the deposition temperature is increased from 500 to 900
C, InxGa1xN grown on sapphire with x 0.2 suffers from a reduction in carrier concentration from 1020 to 1018 cm3, but gains from an increase in the carrier mobility from less than 10 to 100 cm2 V1 s1. The same group later noted that this trend does not change if the films are grown on ZnO substrates instead of sapphire [369]. They could achieve good InGaN material with In mole fractions as

1.5 Ternary and Quaternary Alloys

large as 23%. Nakamura and Mukai [394] discovered that the film quality of InxGa1xN could be significantly improved if these films are grown on high-quality GaN films. Thus, from the reports cited above it may be concluded that the major challenge for obtaining high-mobility InGaN is to find a compromise in the growth temperature, because InN is unstable at typical GaN deposition temperatures. This growth temperature would undoubtedly be a function of the dopant atoms, as well as the method (MBE, OMVPE, etc.) used for the growth. This is evident from a study by Nakamura et al., who have since expanded the study of InGaN employing Si [395] and Cd [396] as dopants. A review of various transport properties of GaInN and AlInN by Bryden and Kistenmacher [296] is available but predates the bandgap reconsideration of InN; the growth and mobility of p-GaInN is discussed by Yamasaki et al. [344]. 1.5.3 InAlN Alloy

In1xAlxN is an important compound that can provide a lattice-matched barrier to GaN, low fraction AlGaN, and InGaN, and consequently lattice-matched AlInN/ AlGaN or AlInN/InGaN heterostructures. Although there was some discrepancy as to which composition really lattice-matched GaN, continued improvement in layer quality and persistence narrowed the In composition for matching. Compositions In0.29Al0.71N and In0.17Al0.83N have been reported as matching, but the value around the latter composition is gaining more acceptance [397]. The growth and electrical properties of this semiconductor have not yet been as extensively studied compared to the other two ternaries, particularly AlGaN, as the growth of this ternary is also challenging because of diverse thermal stability, lattice constant, and cohesive energy of AlN and InN. Moreover, thermal instability resulting from, for example, the spinodal phase separation phenomenon, which is more of an issue in Al1xInxN than in InyGa1yN [398], must be considered. Despite the above-mentioned difficulties, lattice matching and the lack of crack formation when AlGaN is replaced with InAlN in distributed Bragg reflectors (DBRs) and other structures requiring relatively thicker layers are more than enough to pursue this material. In fact, light emitters, field effect transistors, and DBRs, as mentioned, using InAlN barriers as opposed to AlGaN are gaining considerable momentum. As in the case of AlGaN and InGaN, the calculated lattice parameter of this alloy follows Vegard’s law [347] (also reviewed in Ref. [17]) as ˚ aAlx In1  x N ¼ 3:58480  4753x A

˚ and c Alx In1  x N ¼ 5:8002  0:8063x A: ð1:29Þ

By utilizing various tools such as HRXRD, the experimental data for various AlInN layers support the applicability of Vegard’s law in that the experimental data ˚ and c ¼ ð5:713  0:014Þ  aAlx In1  x N ¼ ð3:560  0:019Þ  ð0:449  0:019Þx A ˚ ð0:745  0:024Þx A are within about 2% of that predicted by linear interpolation, the Vegard’ law. As in the case of AlGaN and InGaN, the bond lengths exhibit a

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nonlinear behavior, deviating from the virtual crystal approximation. Essentially, the nearest neighbor bond lengths are not as dependent on composition as might be expected from the virtual crystal approximation. Early experimental results [288] for the bandgap of In0.29Al0.71N, which was thought lattice matched to GaN, indicate that this alloy has an energy gap of 3.34 eV at low temperatures (the room-temperature value is actually closer to 4.5 eV) that is even below that for GaN. The estimations by Wright and Nelson [399] that followed pointed to a bandgap of about 5 eV for the zinc blende variety. The accompanying bowing parameter reported by Wright and Nelson is 2.53 eV at the time when the larger InN bandgap was accepted. Naturally, when the bandgap is in question the bowing parameter is even more in question. As in the case of AlGaN and InGaN, the compositional dependence of the bandgap of AlInN can be expressed with the following empirical expression using a bowing parameter, bAlInN, as g

g

g

E Alx InN ¼ xE AlN þ ð1  xÞE InN  bAlInN xð1  xÞ ¼ 6:1x þ 0:7ð1  xÞ  bAlInN xð1  xÞ eV:

ð1:30Þ

In addition to the aforementioned calculations, experimental data for the bowing parameter, b, exist. Using a bandgap of 6.2 eV for AlN (the new figure is closer to 6 eV), the values that have been reported encompass b ¼ 3.1 eV deduced by fitting the bandgap of this alloy determined by PL [400], b ¼ 2.384 eV by absorption measurements but by using 2.0 eV for the bandgap of InN and 5.9 eV for AlN [401], and b ¼ 5.4 eV in a review where 1.95 eV was used for InN bandgap [17]. Scaling the bandgap of AlN to about 6.0 eV would reduce the bowing parameter a little. Despite the scattered data, reasonably useful bandgap variation of AlInN with composition can be obtained as shown in Figure 1.40. Kim et al. [404] deposited thin AlInN films with X-ray rocking curve FWHM values between 10 and 20 arcmin. They observed an increase of In content in AlInN of up to 8% by lowering the substrate temperature to 600
C. A further reduction of substrate temperature during OMVPE is not useful because of the needed efficient pyrolysis of ammonia. Yamaguchi et al. [405] also reported on OMVPE growth of AlInN on GaN templates that were in turn deposited on low-temperature AlN nucleation layers on cplane sapphire. In macroscopic sense, the alloys grown were not phase separated and the bandgap variation followed the compositional variations in the InN composition range of 19–44%. From the square of the absorption coefficient versus E–Eg, the bandgap of the alloy was determined. Starosta [406] and later Kubota et al. [407] grew InAlN alloy by radio frequency (RF) sputtering. Kistenmacher et al. [408], however, used the RF magnetron sputtering (RF MS) from a composite metal target to grow InAlN at 300
C. It was observed that the energy bandgap E of this semiconductor varies between 2.0 eV (this is supposed to represent the InN binary end point, which assumes the old and incorrect value) and 6.20 eV (this too represents the old value for the bandgap of AlN with the new figure being approximately 6 eV) for x between 0 and 1 [407]. The carrier concentration and the mobility of In1xAlxN for x ¼ 0.04 were 2 · 1020 cm3 and 35 cm2 V1 s1, respectively, and for x ¼ 0.25 were 8 · 1019 cm3 and 2 cm2 V1 s1,

1.5 Ternary and Quaternary Alloys

Lattice constant (Å) 7 a b c d e

AlN

6

Energy gap (eV)

5

f g h i j k l

4 3 2

InN

1 Eg=6(1–x)+0.7x–3.1x(1–x)

0 0.0

0.2

0.4

0.6

0.8

1.0

Composition (x) Figure 1.40 Dependence of bandgap of the InAlN alloy on composition. Unless otherwise stated, the measurement temperature is room temperature. The solid line between the 0.8 eV gap of InN and 6 eV of AlN is deemed as being reasonably accurate. (a) Absorption; (b) RT PL;

(c) RT absorption [404]; (d) Ref. [401]; (e) absorption, poly [416]; (f) absorption [402]; (g) a theory [403]; (h) RT PL and CL [400]; (i) 8 K optical reflection [400]; (j) RT absorption [405]; (k) RT PL [405]; (l) fit to Eg ¼ 6(1  x) þ 0.7 x  3.1 x (1  x). In part courtesy of Wladek Walukiewicz.

respectively [296]. Thus, the mobility decreases substantially with an increase in the Al mole fraction because the structure of the InAlN approaches the structure of the insulating AlN. 1.5.4 InAlGaN Quaternary Alloy

By alloying InN together with GaN and AlN, the bandgap of the resulting alloy(s) can be increased from 1.9 eV (or near 0.7 eV if we use the updated InN bandgap) to a value of 6.2 eV (or 6 eV if we use the updated value), which is critical for making highefficiency visible light sources and detectors. In addition, the bandgap of this quaternary can be changed while keeping the lattice constant matched to GaN [409,410]. In quaternary alloys of nitrides, the N atoms constitute anion sublattice, whereas group III elements (In, Ga, Al) constitute the cation sublattice. Use of this quaternary material allows almost independent control of the bandgap and thus the band offset in AlInGaN-based heterostructures. However, among other difficulties brought about by the four-component system, the optimal growth temperature is important to optimize and control, as aluminum-based compounds generally require higher growth temperatures and In-based ones require lower

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temperatures. Higher temperatures are also desirable for reducing the O incorporation in the growing film as oxides of Ga and In desorb from the surface. The growth temperature will therefore govern the limits of In and Al incorporation into the AlGaInN quaternary alloy [409]. The quaternary alloy (Ga1xAlx)In1yN is expected to exist in the entire composition range 0 < x < 1 and 0 < y < 1. Unfortunately, as in the case of the InGaN alloy, incorporation of indium in these quaternary alloys is not easy. To prevent InN dissociation, InGaN crystals were originally grown at low temperatures (about 500
C) [411], which also applies to InGaAlN. The use of a high nitrogen flux rate allowed the high-temperature (800
C) growth of high-quality InGaN and InGaAlN films on (0 0 0 1) sapphire substrates. Note that the incorporation of indium into InGaN film is strongly dependent on the flow rate, N/III ratio, and growth temperature in an OMVPE environment. The incorporation efficiency of indium decreases with increasing growth temperatures. Observations made in the case of InGaN should be applicable to In incorporation in quaternary nitrides. Ryu et al. [412] reported on the optical emission in this quaternary system and AlInGaN/AlInGaN multiple quantum wells grown by pulsed metalorganic chemical vapor deposition. Strong blue shift with excitation intensity was observed in both the quaternary layers and quantum wells that was attributed to localization. This would imply the inhomogeneous nature of the structures and/or presence of band tail states indicative of early stages of material development and/or serious technological problems involved. The relationships between composition and bandgap (or lattice constant) can be predicted by the equation below, which was originally developed for the InGaAsP system [413]. Qðx; y; zÞ ¼

xyT 12 ðð1  x þ yÞ=2Þ þ yzT 23 ðð1  y þ zÞ=2Þ þ zxT 31 ðð1  z þ xÞ=2Þ ; xy þ yz þ zx

T ij ðaÞ ¼ aBj þ ð1  aÞBi þ að1  aÞbij : The parameters x, y, and z represent the composition of GaN, InN, and AlN. If GaN, InN, and AlN are represented by 1, 2, and 3, T12 would represent GaxInyN. Further, the term T12 can be expressed as T 12 ðaÞ ¼ aB2 þ ð1  aÞB1 þ að1  aÞb12 , where b12 is the bowing parameter for the GaxInyN alloy and a ¼ ð1  x þ yÞ=2 or ð1  x þ yÞ=2 or ð1  z þ xÞ=2 is the effective molar fraction for GaInN, InAlN, and AlGaN, respectively, B2 the bandgap of InN, and B1 is the bandgap of GaN. Similar expressions can be constructed for T23 and T31 by appropriate permutations. An alternative approach is discussed in conjunction with Equation 1.31. The results of these calculations for the bandgap and lattice constant dependence on composition are shown in the three-dimensional diagrams of Figures 1.41–1.43. An empirical expression similar to that used for the ternaries can also be constructed for the quaternary as g

g

g

g

E Alx Iny Ga1  x  y N ¼ xE AlN þ yE InN þ ð1  x  yÞE GaN  bAlGaN xð1  xÞ  bInGaN yð1  yÞ;

ð1:31Þ

1.5 Ternary and Quaternary Alloys

Figure 1.41 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 0.8 eV). (Please find a color version of this figure on the color tables.)

Figure 1.42 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 1.9 eV). (Please find a color version of this figure on the color tables.)

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Figure 1.43 Lattice constant a versus composition for quaternary AlxInyGa1xyN.

where the first three parameters on the right-hand side of the equation are contributions by the binaries to the extent of their presence in the lattice, the third term represents the bowing contribution related to Al, and the last term depicts the bowing contribution due to In. The bowing parameters, bAlGaN and bInGaN, indicated in Equation 1.31 are the same as those discussed in conjunction with InGaN and AlInN. As such, the values are the same. The parameters x, y, and z represent the molar fraction of binaries in the quaternary. After discussing all three ternary alloys of the nitride semiconductor family, the bandgap (both in terms of energy and also corresponding air wavelength) versus the lattice parameter is shown in Figure 1.44 for convenience. The discussion of alloys individually up to this point paves the way to a collective discussion of alloys in terms of structural parameters for a rapid observation of trends. This discussion would be of special value particularly for the least discussed of alloys, InAlN. Let us first discuss the structural properties such as the lattice constants and bond lengths, and angles of nitride semiconductor alloys, following the discussion in Sections 1.5.1–1.5.3 and that surrounding Figure 1.8, Equation 1.3, and Table 1.2. Following Ref. [17], the lattice parameter calculated using Equation 1.25 (for AlGaN), Equation 1.27 (for InGaN), and Equation 1.29 (for InAlN) can be used to calculate the lattice constants for the three ternaries for all compositions and compared with experiments for AlGaN [414], InGaN [415], and InAlN [416], as shown in Figure 1.45.

1.5 Ternary and Quaternary Alloys

Figure 1.44 The bandgap versus the lattice parameter for AlGaN, InGaN, and InAlN using bowing parameters in the same order, 1, 1.43, and 3.1 eV, and bandgap values of 6 eV for AlN, 3.4 eV for GaN, and 0.8 eV for InN. The lattice constants used for the binary AlN, GaN, and InN are 3.11, 3.199, and 3.585 Å, respectively.

Following the case for the binaries tabulated in Table 1.2 and discussed from a theoretical point of view in Refs [347,417], and the experimental points of view in Refs [414] (for AlGaN), [415] (for InGaN), and [416] (for AlxIn1xN), the cell parameter, u, has been calculated for randomly distributed A0.5B0.5N (here A and B represent the metal components forming the alloy) alloys by the theoretical approach of Ref. [418], the pertinent parts of which are succinctly discussed in Section 1.1. The internal cell parameter can be approximately expressed by the quadratic equation uAx B1  x N ¼ xuAN þ ð1  xÞuBN  bAB xð1  xÞ;

ð1:32Þ

where bAB is the bowing parameter defined as bAB ¼ 2Y AN þ 2Y BN  4Y A0:5 B0:5 N :

ð1:33Þ

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3.8

(a)

Lattice constant, a (x) (Å)

3.6

Alx In1–xN Inx Ga1–xN

3.4

Alx Ga1–x N

3.2

Theory (T = 0 K) Experiment (T = 300 K) 3.0 0

0.2

0.6 0.4 Molar fraction, x

0.8

6.0

1.0

(b)

5.8

Alx In1–xN

Lattice constant c(x) (Å)

Inx Ga1–xN 5.6

5.4

Alx Ga1–x N

5.2

5.0 Theory (T = 0 K) Experiment (T = 300 K) 4.8 0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x Figure 1.45 (a) The a(x) lattice parameter and (b) the c(x) lattice parameter for random ternary alloys of AlxGa1xN, InxGa1xN, and AlxIn1xN as measured by HRXRD at room temperature (solid lines) and the calculated values using Equation 1.25 (for AlGaN) and Equation 1.27 (for

InGaN) for T ¼ 0 K (dashed lines). The agreement between calculations and measured lattice constants is better than 2% over the entire range of compositions, compiled in Ref. [17] utilizing Refs [414,415]. Courtesy of O. Ambacher.

1.5 Ternary and Quaternary Alloys

The internal cell parameters for each of the three alloys then are uAlx Ga1  x N ¼ 0:3819x þ 0:3772ð1  xÞ  0:0032xð1  xÞ; uInx Ga1  x N ¼ 0:3793x þ 0:3772ð1  xÞ  0:0057xð1  xÞ; uAlx In1  x N ¼ 0:3819x þ 0:3793ð1  xÞ  0:0086xð1  xÞ:

ð1:34Þ

The structural and other polarization related parameters of ternaries do not follow a linear relationship of the composition, as discussed in detail in Section 2.7. The nonlinearity in question for an alloy, AxB1xN, where A and B represent the metal components, is approximated by quadratic equations of the form [418] Y Ax B1  x N ¼ xY AN þ ð1  xÞY BN  bAB xð1  xÞ;

ð1:35Þ

where Y represents any parameter, namely the lattice constant, the u parameter or polarization, and the bowing parameter is defined in Equation 1.33. As in the case of binaries discussed in Section 1.1, the cell parameter, u, and the c/a ratio do not follow the ideal crystal values for the three ternaries of nitride semiconductors. They are shown for the three ternaries for varying composition in Figure 1.46. Similar to the binaries, tabulated in Table 1.2 in conjunction with Figure 1.8, the aforementioned two parameters, the nearest and the second neighbor bond lengths, as well as the bond angles have been calculated for the three ternaries and those associated with 50% alloys are tabulated in Table 1.34. As displayed in Figure 1.46, the cell internal parameter increases as one goes from GaN to InN and, more significantly, to AlN. The nonlinear dependencies on the alloy composition are described by a bowing parameter, bAB, whose values are 0.0032, 0.0057, and 0.0086 for AlxGa1xN, InxGa1xN, and AlxIn1xN, respectively. The bowing parameter increases from AlxGa1xN to InxGa1xN, and continues on to AlxIn1xN. It is worth noting that the bowing parameter is negative for all the three ternaries, the average cell internal parameter of the same alloys is always above the ideal value of 0.375. If the lattice constants scale linearly with the alloy composition but the internal parameter does not, the bond angles and/or the bond lengths of the real and the virtual crystal must depend nonlinearly on the alloy composition. The average nearest neighbor bond lengths (b and b1, see Figure 1.8 for a graphical description) and bond angles (see Figure 1.8 for a graphical description) calculated by using Equations 1.2–1.4 are shown in Figure 1.47a and b and listed in Tables 1.2 and 1.34. The average cation–anion distances to the nearest and second nearest neighbors scale nearly linearly with alloy composition for AlxGa1xN, InxGa1xN, and AlxIn1xN. The average bond length along the c-axis is 0.7–0.9% longer than the nearest neighbor bonds in the direction of the basal plane (Figure 1.47a). 1.5.5 Dilute GaAs(N)

When small amounts of N and As are incorporated into GaAs and GaN lattices, respectively, a large negative bandgap bowing parameter results. Consequently, with

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very small amounts of N in the GaAs lattice, its bandgap can be made very small, to a point where 1.3 mm lasers and 1.5 mm lasers if In and Sb are also added to the lattice can all be built with GaAs technology. Anomalously large bandgap bowing parameters exhibited by GaAsN and GaNAs are caused by large chemical and size 1.64 (a)

Ideal

1.63

c(x)/a(x) ratio

1.62

Inx Ga1–xN

Alx Ga1–xN

1.61

Alx In1–xN

1.60 Theory (T = 0 K) Experiment (T = 300 K)

1.59 0

0.2

0.4

1.0

0.8

0.6

molar fraction, x 0.383 (b)

cell-internal parameter u(x)

0.381

Alx Ga1–x N

b = –0.0032

0.379

Alx In1–xN

b = 0.0086

Inx Ga1–xN

0.377

b = –0.0057

Ideal

0.375

0

0.2

0.6 0.4 molar fraction, x

0.8

1.0

1.5 Ternary and Quaternary Alloys Table 1.34 Calculated cell internal parameter, a lattice parameter,

c/a ratio, cation–anion distance between the nearest and second nearest neighbors, and bond angles (given in degrees) for the three ternary random alloys in the virtual crystal limit with a composition of 50%.

u a (Å) c/a b (Å) b1 (Å) 0 b1 (Å) 0 b2 (Å) 0 b3 (Å) a b

Al0.5Ga0.5N

In0.5Ga0.5N

In0.5Al0.5N

0.379 3.154 1.620 1.935 1.924 3.175 3.701 3.694 108.80 110.14

0.377 3.392 1.625 2.078 2.073 3.436 3.977 3.975 109.13 109.81

0.378 3.347 1.612 2.042 2.041 3.354 3.921 3.920 108.76 110.18

The distance is in Å and the angles are in degrees [17].

~

differences between As and N [419–422]. Dependence of the bandgap energy in GaAsN and InPN on nitrogen content is shown in Figure 1.48. To a first extent, the dashed lines originating from both GaN end (in which case small amounts of As are added to GaN) and GaAs end (in which case small amounts of N are added to GaAs) represent the bandgap dependence of GaNAs. However, one must keep in mind that for both GaAsN and InPN the simple treatment behind the aforementioned statement fails and that the arrows shown in the figure indicate the boundaries of the regions where the gap dependence on composition may be predicted with any accuracy. Also shown is the bandgap variation with composition for other commonly used ternaries. The thicker vertical line through GaAs represents the bandgap attainable with GaInAsN, at least in theory, while maintaining lattice matching to GaAs. The decrease in the lattice constant caused by N can be compensated with In added to the lattice. The potential of covering a large range of bandgap energies on GaAs substrates has attracted a great deal of interest in this material system. In fact, the first laser containing N was an InGaAs(N) active layer one. Owing in part to

Figure 1.46 (a) The c/a ratio for the three random ternary alloys determined by HRXRD at room temperature (solid lines) and calculated using Equation 1.24 for T ¼ 0 K (dashed lines). The measured and calculated data confirm that the c/a ratios of Wz InGaN, AlGaN, and AlInN crystals are always less than the value of 1.633 for ideal hexagonal crystal. (b) The cell internal parameter, u, for three random AlGaN, InGaN, and AlInN alloys calculated using the quadratic

Equation 1.35. The nonlinearity of the internal cell internal parameter in its compositional dependence can be described by a negative bowing parameter b. This bowing parameter is 0.0032, 0.0057, and 0.0086 for AlxGa1xN, InxGa1xN, and AlxIn1xN, respectively, as indicated in the figure as well. The u parameter of the ternaries is always larger than 0.375 that is the value for an ideal hexagonal crystal [17]. Courtesy of O. Ambacher.

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(a) M-Nc1

InN

2.2

InN

Metal-N average bond length (A)

M-Nb1

2.1 Inx Ga1–xN Alx In1–xN

2.0 Alx Ga1–xN

M-Nc1

GaN b=M-Nc1 b1= M-Nb1

1.9

0

0.2

0.4

M-Nb1 AlN 0.6

0.8

1.0

Molar fraction, x

111

T=0K

(b)

AlN

Bond angle of virtual lattice (deg)

InN

Alx In1-x N

110

InN

Alx Ga1-x N GaN

Inx Ga1-x N

ideal: α = β =109.47 ο

Inx Ga1-x N 109

β α

Alx Ga1-x N Alx In1-x N

AlN 108

0

0.2

0.8 0.6 0.4 Molar fraction, x

Figure 1.47 (a) The compositional dependence of the average nearest neighbor bond lengths, b and b1 (see Figure 1.8 for a graphical description) in the virtual crystal limit for the metal–nitrogen bonds along the c-axis (solid line) and off c-axis (dashed line). (b) The compositional dependence of the average bond angles a (dashed lines) and b (solid lines) of random

1.0

AlxGa1xN, InxGa1xN, and AlxIn1xN alloys (see Figure 1.8 for a graphical description). Clearly, the average bond angles deviate noticeably from the ideal hexagonal crystal for which a ¼ b ¼ 109.47. Moreover, the deviation increases from GaN to InN and continues onto AlN in a nonlinear fashion [17]. Courtesy of O. Ambacher.

1.5 Ternary and Quaternary Alloys

Γ valley energy gap (eV)

6

AlN

zinc blende T=0K

5 4

AlP AlAs

3

GaP

GaN

2 InN 1

AlSb

GaAs GaAsN InPN

0 4.5

5.0

5.5

InP GaSb InSb InAs 6.0

6.5

Lattice constant (A) Figure 1.48 Direct G valley energy gap as a function of lattice constant for the zinc blende form of 12 III–V binary compound semiconductors (filled circles) and some of their random ternary alloys (lines connecting the solid circles) at zero temperature. The energy gaps for certain ternaries such as AlAsP, InAsN, GaAsN, InPN, and GaPN are extended into regions where

no experimental data have been reported. For GaAsN and InPN, the arrows indicate the boundaries of the regions where the gap dependence on composition may be predicted with any accuracy, patterned after Ref. [423] with necessary changes, particularly the one reflecting the small bandgap of InN.

extreme nonequilibrium conditions employed for growth, MBE is the dominant growth approach for dilute arsenides with nitrogen. The critical issues are compositional control, incorporation of more than a small percentage of N, doping inefficiency, and layer quality. The situation is exacerbated on all fronts when the N concentration is increased for achieving 1.5 mm wavelength of emission. Postgrowth annealing is often employed to improve the crystal quality and/or to increase Si dopant incorporation, however, at the expense of blue shift in the bandgap. While GaAsN is chosen here for the present discussion, there are many other dilute nitride semiconductors as discussed in Section 2.11 in conjunction with band parameters. As alluded to earlier, the chemical and size differences between the N and As atoms are the challenges facing experimentalists. In addition, the generation of atomic nitrogen, although not that different from the technology required for hexagonal GaN growth [424], deserves some attention. While basic mismatch between N and As can be dealt with by growing the layers under nonequilibrium conditions, the issue of atomic nitrogen can be handled by compact RF sources that have seen a good degree of improvement lately. By adjusting the RF power and pressure in the cell, one can tailor the source to produce mostly the atomic species by optimizing the emission at 745 nm of wavelength. Note that the substrate and most of the structure are zinc blende and, consequently, the dilute material assimilates and assumes the same crystalline structure. The desired nitrogen concentrations are in the range of 1–10% for red shifting the transitions out to as long as 1.55 mm. Larger growth rates lead to a reduced incorporation of N in the lattice. Similarly, higher growth temperatures lead to the same. Consequently, when 1.55 mm wavelength material is desired, lower

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growth rates must be employed as well as lower growth temperatures. At substrate temperatures of 500
C or below, if very large As overpressure is employed, incorporation of N is limited because the flux of atomic nitrogen is small. However, atomic nitrogen is very reactive and, therefore, compositional control should be much better as compared to quaternaries relying on P and As (InGaAsP). As expected, owing to dissimilarities of N and As, the luminescence properties of GaInNAs degrade rapidly with increasing nitrogen concentration. Employing remedies such as postgrowth annealing enhances the luminescence efficiency of GaInNAs. However, this enhancement is accompanied by a blue shift in the transition in bulk and quantum well materials. Nitrogen and possibly In diffusion out of GaInAsN are responsible for the observed luminescence shift to shorter wavelengths. For completeness, a one-paragraph discussion of device issues will be made in conjunction with the edge emitting and vertical cavity lasers operating at 1.3 and 1.5 mm portion of the optical spectrum, although other applications such as heterojunction bipolar transistors are possible. Several groups have reported lasers operating at 1.3 mm region [425–442], where the silica-based fiber dispersion is zero, and 1.5 mm region [443–447] (albeit with addition of Sb to the lattice as the quality required for laser operation for InGaAsN layers cannot be obtained), where the loss is low, again for the silica-based fibers. Both are intended for telecommunication purposes. Even 8 W [448] and 12 W [449] CW operation has been reported. Highspeed testing of these lasers has also been performed [450] with data transmission rates as high as Gbit s1 having been achieved already [451]. For interconnects and high-speed data links, vertical cavity surface emitting lasers (VCSELs) have received a great deal of attention. Now that dilute nitrides are becoming potential candidates for long-wavelength lasers, efforts are under way to explore VCSELs in this material system as well [452,453].

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428 Yang, K., Hains, C.P. and Cheng, J.L. (2000) Efficient continuous-wave lasing operation of a narrow-stripe oxideconfined GaInNAs–GaAs multiquantumwell laser grown by MOCVD. IEEE Photonics Technology Letters, 12 (1), 7–9. 429 Kondow, M., Kitatani, T., Nakahara, K. and Tanaka, T. (1999) A 1.3-mm GaInNAs laser diode with a lifetime of over 1000 hours. Japanese Journal of Applied Physics, Part 2: Letters, 38 (12A), L1355–L1356. 430 Li, N.Y., Hains, C.P.,Yang, K., Lu, J., Cheng, J. and Li, P.W. (1999) Organometallic vapor phase epitaxy growth and optical characteristics of almost 1.2 mm GaInNAs three-quantum-well laser diodes. Applied Physics Letters, 75 (8), 1051–1053. 431 Sato, S. and Satoh, S. (1998) Roomtemperature pulsed operation of strained GaInNAs/GaAs double quantum well laser diode grown by metal organic chemical vapour deposition. Electronics Letters, 34 (15), 1495–1497. 432 Kitatani, T., Kondow, M., Nakahara, K., Larson, M.C. and Uomi, K. (1998) Temperature dependence of the threshold current and the lasing wavelength in 1.3-mm GaInNAs/GaAs single quantum well laser diode. Optical Review, 5 (2), 69–71. 433 Nakatsuka, S., Kondow, M., Kitatani, T., Yazawa, Y. and Okai, M. (1998) Index-guide GaInNAs laser diode for optical communications. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers, 37 (3B), 1380–1382. 434 Ougazzaden, A., Bouchoule, S., Mereuta, A., Rao, E.V.K. and Decobert, J. (1999) Room temperature laser operation of bulk InGaAsN/GaAs structures grown by APMOVPE using N2 as carrier gas. Electronics Letters, 35 (6), 474–475. 435 Fischer, M., Reinhardt, M. and Forchel, A. (2000) High temperature operation of GaInAsN laser diodes in the 1.3 mm regime. 58th Device Research Conference, IEEE, pp. 119–120. 436 Shimizu, H., Kumada, K., Uchiyama, S. and Kasukawa, A. (2000) High

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performance CW 1.26 mm GaInNAsSbSQW and 1.2 mm GaInAsSb-SQW ridge lasers. Electronics Letters, 36 (20), 1701–1703. Setiagung, C., Shimizu, H., Ikenaga, Y., Kumada, K. and Kasukawa, A. (2003) Very low threshold current density of 1.3mm-range GaInNAsSb–GaNAs 3 and 5 QWs lasers. IEEE Journal of Selected Topics in Quantum Electronics, 9 (5), 1209–1213. Bank, S.R., Yuen, H.B., Bae, H., Wistey, M.A. and Harris, J.S. (2006) Overannealing effects in GaInNAs(Sb) alloys and their importance to laser applications. Applied Physics Letters, 88, article 221115. Gollub, D., Moses, S., Fischer, M. and Forchel, A. (2003) 1.42 mm continuouswave operation of GaInNAs laser diodes. Electronics Letters, 39 (10), 777–778. Ikenaga, Y., Miyamoto, T., Makino, S., Kageyama, T., Arai, M., Koyama, F. and Iga, K. (2002) 1.4 mm GaInNAs/GaAs quantum well laser grown by chemical beam epitaxy. Japanese Journal of Applied Physics, 41, 664–665. Tansu, N. and Mawst, L.J. (2002) Lowthreshold strain-compensated InGaAs(N) (l ¼ 1.19–1.31 mm) quantum well lasers. IEEE Photonics Technology Letters, 14 (4), 444–446. Peng, C.S., Jouhti, T., Laukkanen, P., Pavelescu, E.-M., Konttinen, J., Li, W. and Pessa, M. (2002) 1.32-mm GaInNAs-GaAs laser with a low threshold current density. IEEE Photonics Technology Letters, 14 (3), 275–277. Yang, X., Jurkovic, M.J., Heroux, J.B. and Wang, W.I. (1999) Molecular beam epitaxial growth of InGaAsN:Sb/GaAs quantum wells for long-wavelength semiconductor lasers. Applied Physics Letters, 75 (2), 178–180. Yang, X., Heroux, J.B., Mei, L.F. and Wang, W.I. (2001) InGaAsNSb–GaAs quantum wells for 1.55 mm lasers grown by molecular-beam epitaxy. Applied Physics Letters, 78 (26), 4068–4070.

445 Vurgaftman, I., Meyer, J.R., Tansu, N. and Mawst, L.J. (2003) (In)GaAsN–GaAsSb type-II “W” quantum-well lasers for emission at l ¼ 1.55 mm. Applied Physics Letters, 83 (14), 2742–2744. 446 Bank, S.R., Wistey, M.A., Goddard, L.L., Yuen, H.B., Lordi, V. and Harris, J.S., Jr (2004) Low-threshold continuous-wave 1.5-mm GaInNAsSb lasers grown on GaAs. IEEE Journal of Quantum Electronics, 40 (6), 656–664. 447 Fischer, M., Reinhardt, M. and Forchel, A. (2000) Room-temperature operation of GaInAsN/GaAs laser diodes in the 1.5 mm range. Conference Digest, 2000 IEEE 17th International Semiconductor Laser Conference (Cat. No. 00CH37092), IEEE, Piscataway, NJ, pp. 115–116. 448 Livshits, D.A., Egorov, Yu.A. and Riechert, H. (2000) 8 W continuous wave operation of InGaAsN lasers at 1.3 mm. Electronics Letters, 36 (16), 1381–1382. 449 Bugge, F., Erbert, G., Fricke, J., Gramlich, S., Staske, R., Wenzel, H., Zeimer, U. and Weyers, M. (2001) 12 W continuous-wave diode lasers at 1120 nm with InGaAs quantum wells. Applied Physics Letters, 79 (13), 1965–1967. 450 Reinhardt, M., Fischer, M., Kamp, M. and Forchel, A. (2000) 7.8 GHz small-signal modulation bandwidth of 1.3 mm DQW GaInAsN/GaAs laser diodes. Electronics Letters, 36 (12), 1025–1026. 451 Steinle, G., Mederer, F., Kicherer, M., Michalzik, R., Kristen, G., Egorov, A.Y., Riechert, H., Wolf, H.D. and Ebeling, K.J. (2001) Data transmission up to 10 Gbit/s with1.3 mmwavelengthInGaAsNVCSELs. Electronics Letters, 37 (10), 632–634. 452 Fischer, M., Reinhardt, M. and Forchel, A. (2000) A monolithic GaInAsN verticalcavity surface-emitting laser for the 1.3-mm regime. IEEE Photonics Technology Letters, 12 (10), 1313–1315. 453 Schneider, H.C., Fischer, A.J., Chow, W.W. and Klem, J.F. (2001) Temperature dependence of laser threshold in an In GaAsN vertical-cavity surface-emitting laser. Applied Physics Letters, 78 (22), 3391–3393.

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2 Electronic Band Structure and Polarization Effects Introduction

The band structure of a given semiconductor is pivotal in determining its potential utility. Consequently, an accurate knowledge of the band structure is critical if the semiconductor in question is to be incorporated in the family of materials considered for serious investigations and device applications. The group III–V nitrides are no exception, and it is their direct bandgap nature and the size of the energy gap that spurred the interest in nitride semiconductors. Nitride semiconductors can be classified into two groups. One group pertains to stoichiometric systems where N represents 50% of the constituents while the other half is made of metal constituents. These stoichiometric nitrides come in wurtzitic and zinc blende (ZB) forms. The other class of nitrides is the dilute compound semiconductors, wherein very small amounts of N are added to the lattice for remarkably large negative bowing of the bandgap, making these dilute nitride systems compete for longer wavelength applications. For example, the bandgap of GaAs can be extended to 1.3 mm applications. Likewise, the bandgap of InGaAs coherently grown on GaAs can be extended with dilute amounts of N in the lattice to be a contender for 1.5 mm applications, which has been the domain of In0.53Ga0.47As lattice matched to InP. The impact of dilute nitrides is that, in at least the aforementioned example, what used to be the domain of InP-based technology can be met by GaAs technology with untold consequences in terms of not only technology but also the cost of that technology. A number of researchers have published band structure calculations for both wurtzite (Wz) and zinc blende GaN, AlN, and InN. To make matters more interesting, the bandgap of InN transmogrified from 1.9 eV downward to about 0.7 eV between the first edition and the current one. It is argued that the first set of bandgap measurements might have been conducted in films containing a large amount of O, which could have caused an upward shift in the measured data. The initial estimate of the 1.9 eV bandgap of InN, in addition to creating confusion concerning the nature and applications of InN, caused uncertainties in the bandgap of the InGaN ternary as well. The situation is exacerbated by inhomogeneities in composition and strain as well as poor sample quality. The situation in fact transformed into one in which

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoç Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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multiple bowing parameters, depending on the InN composition in the alloy, were proposed. It can now be said that, with the small bandgap of about 0.8 eV for InN as agreed on, only a single bowing parameter can indeed account for the data dating back to 1980s where incompatibilities with the 1.9 eV InN bandgap were reported [1]. The first Wz GaN band structure found through a pseudopotential method led to a 3.5 eV direct bandgap. The band structure for ZB GaN has been obtained by a firstprinciples technique within the local density functional framework with a direct bandgap of 3.40 eV and a lattice constant of 4.50 Å. A treatise of the band structure in bulk and quantum wells (QWs) with and without strain for conventional nitrides, GaN, InN, and AlN, along with their alloys in the wurtzite and zinc blende form, and “dilute” nitrides (all of which are zinc blende ternaries and quaternaries, for example, GaAsN and GaInAsN) is discussed. Complete and consistent parameter sets are provided with tabulations of the direct and indirect energy gaps and spin–orbit and crystal field splitting. The alloy bowing parameters, electron and hole effective masses, deformation potentials, elastic constants, and piezoelectric (PE) and spontaneous polarization coefficients are given in many tables in Chapter 1. However, the basis of spontaneous and piezoelectric polarization effects and their practical impact on single- and dual-interface heterostructures are discussed in this chapter following the band structure discussion. The heterostructure band offsets are discussed in Volume 2, Chapter 4. The temperature dependence of the bandgap parameter of nitrides inclusive of the case of band tailing, primarily GaN, is discussed in this chapter as well as in Volume 2, Chapter 5 in conjunction with its optical properties.

2.1 Band Structure Calculations

A glossary of band structure calculations will be given before delving into the specific calculations employed to determine the band structure of nitride semiconductors. This is not an all-inclusive treatment of the field, but it is a compact treatment of the salient features of methods used to illustrate and/or calculate the band structure. The first exposure of many students to band structure calculations is that of a free-electron or nearly free electron approach in a periodic lattice [2–4]. In this one-electron model, the periodic potential can be thought of as arising from the periodic charge distribution associated with ion cores that are situated at the lattice sites. To expand the picture to include many electrons, an average constant potential contribution is added to account for all the other electrons in the system. This problem was first considered by Felix Bloch [5]. In its simplest form, the wave function representing the electrons in a periodic potential would be composed of the product of a plane wave function representing an electron in free space and a function representing the periodicity of the crystal. In the nearly free electron approach, the effect of the crystalline potential on the electronic structure is considered to be weak, and the energy levels of the electrons have little resemblance with those of the atoms. The allowed energies occur in bands of allowed states separated by forbidden energy regions (gaps). Within an allowed and

2.1 Band Structure Calculations

mainly occupied energy band, the electron motion is in many ways similar to that of free particles with an appropriate charge and effective mass. Whether the crystal is an insulator or conductor depends on whether the states within a band or set of bands are completely filled or partially empty. The crystal is considered a semiconductor if the gap between a filled band and the empty band is small, the exact value of which has changed over the years. For example, a gap of 3 eV was taken to be associated with insulators, which is smaller than the gap of GaN. The one-electron picture is an approximation and does not take into consideration processes such as electron– electron interaction, which is neglected other than what is convoluted in the average potential. To understand the basic aspects of band structure, it is instructive to consider an infinite periodic one-dimensional square well potential, which forms the basis for the Kronig–Penney approach [6]. This approximation leads to an exact solution of the Schr€odinger equation. Even though the square well potential approximation is very crude, it serves to illustrate explicitly many important characteristic features of the electron behavior in periodic lattices. In contrast to the free-electron approximation in which the potential energy of the electron is assumed to be small in comparison to its total energy, the opposite is assumed in another commonly practiced approximation, called the tight binding (TB) approximation. Specifically, it is assumed that the potential energy of the electrons accounts nearly for all of the total energy in the case of which the allowed energy bands are narrow compared to the forbidden ones. Unlike the free-electron model, the electronic wave functions are more or less localized around the atoms. Thus, the interaction between neighboring atoms is relatively weak, and the wave functions and the allowed energy levels of the crystal as a whole resemble the wave function and energy levels of isolated atoms. In a sense, an electron associated with an atom is assumed to remain in an orbit associated with that atom, and these orbitals are combined linearly in a form to be consistent with the Bloch–Floquent theorem to represent a state running throughout the crystal. Again, each orbital is localized on a particular atom. As such the results are very sensitive to overlap integrals, which in turn are sensitive to the details of the orbitals outside the cores and to the lattice spacing. Naturally, in this approach, electrons are not affected by atoms more than a single atomic spacing away. The choice of whether the free electron or the tight binding method is good depends on the particular crystal. In fact, in some crystals neither of these is good. The tight binding approach was assumed by Bloch in his original discussion of energy bands. If there is an appreciable interatomic interaction, the tight binding approach must use a linear combination of atomic orbitals (LCAO), in which a quantum mechanical variational procedure is employed to find the combination of s, p, and d orbitals that correspond to the lowest energy in the system. Not surprisingly, the tight binding methods are successful when the effect of the periodic core potential is quite large. This is true, for example, when the band is derived from the 3d states in the first series of transition elements. In these elements, the 3d states are partially filled and one or two electrons are present in 4s subshells. When atoms of these elements are brought together to make a solid, the interaction between the 4s states is very strong while the overlap between the partially filled inner shells is rather weak.

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There are other band structure methods that provide much improvement over the nearly free electron model and the tight binding model. These methods rely on choosing an appropriate basis for the electrons that represent electron behavior both inside and outside the atomic sphere. For example, while a single plane wave may be adequate to represent electron wave function in the interstitial space, to account for the rapid change in the function near the core region, a combination of a large number of plane waves would be necessary. Because the core functions are described in terms of radial functions and spherical harmonics, all unknown wave functions of crystals can be expanded in a set of known functions such as plane waves, radial functions, and spherical harmonics. The various methods for band structure calculations, therefore, differ in the initial choice of boundary conditions that the wave functions must satisfy. A brief qualitative discussion of these methods is given in the following sections. 2.1.1 Plane Wave Expansion Method

Here, the Bloch wave is expanded as a linear combination of plane waves (LCOPW), namely, ! ! ! X ! ð Þ:r Ck ðKÞei k þ k : ð2:1Þ Fk ð r Þ ¼ !

k

The combination coefficients Ck(K) are determined by solving the determinantal equation " # X ! !
h2 ðk þ KÞ2 0
E K jCk ðKÞj2 dK;K 0 þ Ck ðKÞCk ðK ÞVð k
k 0 Þ ¼ 0; 2m 0 K;K

!

! 0

ð2:2Þ

where EK is the energy eigenvalue and Vð k
k Þ is the crystalline potential. Although this is a simple method, its practical implementation is difficult, as it requires a large number of plane waves to represent the behavior of electrons near a core region. Consequently, the convergence in the eigenvalues is poor and it requires solving a large determinantal equation. 2.1.2 Orthogonalized Plane Wave (OPW) Method

The method originally proposed by Herring and Hill [7] and later discussed by Woodruff [8] considers the Bloch function to be a linear combination of OPW basis. The OPW basis consists of a plane wave orthogonalized to the atomic core functions, such that the electron behaves like a core electron while inside the core and like a plane wave while in the interstitial region. The OPW basis can be written as ! ! X 1 ! ! X k ð r Þ ¼ pffiffiffiffiffiffiffiffi ei k : r
mkj Fkj ð r Þ; ð2:3Þ NW j

2.1 Band Structure Calculations

where Fkj are Ðthe core wave functions for constant j. The term mkj is evaluated by ! ! requiring that Fkj ð r ÞX k ð r Þd3 r ¼ 0mX , where W is the volume of the Wigner–Seitz cell, and N is the number of atoms. The Bloch function is then expressed as a linear combination of the OPWs, and the energy eigenvalues are computed by solving the appropriate determinantal equation. This method has been applied to band structure calculation of metallic and nonmetallic solids. 2.1.3 Pseudopotential Method

Phillips and Kleinman [9] later demonstrated that it is possible to rewrite the crystal potential that includes contributions from the core and valence electrons in such a way that the Bloch function can be written in terms of a linear combination of planes as in the plane wave expansion method without compromising the convergence advantages of the OPW method [10,11]. The crystal potential thus obtained is called the pseudopotential. As mentioned earlier, the orbitals between the cores (outer regions) are smooth where the wave functions are somewhat like plane waves. Near the core regions, however, the wave functions are complicated by the strong and rapidly varying potential. The orthogonality requirement causes nodes (zeros) in the wave function in the core region. The weak potential experienced by the electrons in the outer region can be treated as a perturbation that mixes the plane wave components (strongly only at the Brillouin zone (BZ) boundaries). While the orbitals are not like plane waves near the cores and potentials vary rapidly, it is argued that what goes on near the core is irrelevant to the dependence of the energy on the wave vector. The energy wave vector dependence can be calculated by applying the Hamiltonian operator to an orbital at any point in space, which when applied to the outer regions, the picture would be that of nearly free electron energy. The actual potential energy in the core region can be represented by an effective potential energy called the pseudopotential that gives the same wave functions outside the core regions as the actual potential. Surprisingly, the pseudopotential is nearly zero in the core region, which is arrived at by experience with such potentials as well as theoretical considerations. Although use of these pseudopotentials may lead to incorrect wave functions, doing so can, with acceptable accuracy, indicate how the energy varies. 2.1.4 Augmented Plane Wave (APW) Method

This method proposed by Slater [12] also makes use of the fact that the wave function inside the core behaves like atomic functions and outside the core like plane waves as in the OPW method. The difference lies in how one applies the boundary conditions. Unlike the OPW method where the wave function outside and inside the core are matched by the Schmidt orthogonalization condition, in the APW scheme one expands the wave functions outside the core region (r  ri) by a set of plane waves and

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inside the core region (r  ri) by a sum of spherical waves, namely, !! X ! Fk ð r Þ ¼ a0 Qðr
r i Þei k : r þ alm Qðr i
rÞY lm ðq; fÞRl ðE; rÞ; l;m

where Y is the step function with Y(x) ¼ 1 for x  0 and Y(x) ¼ 0 for x < 0. The ! coefficients alm are chosen in such a way that the wave function Fk ð r Þ is continuous across the sphere of radius ri. The term Rl(E, r) represents the solution of the radial equation. Slater [12] proposed a reasonable approximation to the potential with a constant potential outside the cores and an ordinary atomic potential inside a sphere surrounding each ion core. The Schr€odinger equation is then solved separately for each of the two regions and solutions are matched over the spherical boundaries between them, and within each core the wave function can be expanded in spherical harmonics. The solution outside the cores is a superposition of plane waves. The combination gives this method its name, the augmented plane waves. 2.1.5 Other Methods and a Review Pertinent to GaN

Another class of calculations relies on what is called the all-electron approach. Here, one develops potentials within the sphere around the core region and uses their energy derivatives to expand the wave functions in a manner to match those outside the sphere. Among the methods relying on this premise are linear muffin-tin orbital (LMTO) and linearized augmented plane wave (LAPW) methods. Essentially, in the LMTO method, the crystal is divided into nonoverlapping muffin-tin spheres surrounding the atomic sites and an interstitial region outside of spheres. Inside the muffin-tin sphere, the potential is assumed spherically symmetric while in the interstitial region the potential is assumed constant or slowly varying. The early application of LMTO method made use of what is called the atomic sphere approximation (ASA), in which the crystal potential is treated as a superposition of weakly overlapping spherical potentials centered around lattice sites but in such a way as to fill the space and the total volume of muffin-tin sphere is the same as the atomic volume. The potential is also assumed to be spherically symmetric inside each muffin-tin sphere. Additionally, the kinetic energy of the basis function in the interstitial regions is restricted to be a constant and that constant is typically assumed to be zero in the calculations. In open structures, it is customary to include spheres centered on interstitial sites. It is worth noting that the potential so determined is very close to the true full potential and provide a much better representation of the potentials than the original muffin-tin potentials that are constant in the interstitial regions and spherical within nonoverlapping spheres. The limited precision, 0.1 eV, prevents this method from being applied satisfactorily to the very intricate portions of the band structures. For example, very small differences in the total energy between the zinc blende and wurtzitic forms of GaN are a troublesome point for these ASA-LMTO calculations. Likewise, the crystal field splitting at the valence band maxima presents similar problems.

2.1 Band Structure Calculations

The LMTO method has advantages such as using a minimal basis for computational efficiency and thus allowing large unit cell calculations. It also treats all elements in the same manner and is accurate due to an augmentation procedure, which gives the wave function a reasonably correct shape near the nuclei. The method also uses atom-centered basis functions of well-defined angular momentum. The full potential LMTO (FP-LMTO) calculations are all fully relativistic electrons with the shape approximation to the charge density or the potential. As in the LMTO, the crystal is divided into nonoverlapping muffin-tin spheres and interstitial regions outside the spheres. Then, the wave function is presented differently in those two regions in that inside the muffin-tin spheres, the basis functions are as in the LMTO-ASA method and are of Bloch sum of linear muffin-tin orbitals and the kinetic energy is not restricted to zero in the interstitial regions. All of the other above methods rely on our knowledge of the potential in which the electrons inside a crystal move. This potential results from the interaction between electrons, electrons and nuclei, and between nuclei. This presents a many body problem that cannot be solved exactly. Thus, approximations are needed. One of these techniques developed by Hohenberg and Kohn and Sham, known as the density functional theory (DFT) [13,14], has shown a great promise in treating energy bands of solids. DFT can be construed as a self-consistent field method for attaining the crystal potential. Over the years, many band structure calculations evolved to include not only the electronic properties but also binding energies, through which the lattice constant, bulk modulus, elastic constants, and vibrational properties such as phonon frequencies can be computed. The DFT method is well suited for calculating the aforementioned parameters. The basic quantity is the electronic charge density, and the total energy is expressed as functional of this density. The total energy expression as a function of density includes exchange and correlation effects in an average electron gas like manner [15], which is termed as the local density approximation (LDA). Most common form expresses the charge density in terms of occupied oneelectron wave functions, the corresponding eigenvalues of which constitute the band structure [16]. Calculations of the self-energies are best done by the DFT-LDA method within the GW approximation. The nomenclature is adopted from the original paper of Lars Hedin [17] for one-electron Green’s function and Hedin and Lundqvist [18] for screened Coulomb interaction. This approximation represents the first term in a perturbation expansion of the self-energy, which can also be viewed as the screened Hartree–Fock (H–F) theory. It should be pointed out that in semiconductors the GW method differs from the band structure determined by LDA chiefly due to a shift of the conduction band relative to the valence band. Consequently, the fundamental bandgap is underestimated. The genesis for Kohn–Sham [15] local DFT lies in lowering the conduction band in a manner that is k-value dependent, as has amply been pointed out by Sham and Schl€ uter [19] and Perdew and Levy [20] in back-to-back articles. It is for this reason that DFT-based calculations are supplemented by a shift of the calculated fundamental gap inspired by a body of experimental data. Most band structure calculations for GaN and related materials rely on the DFT-LDA approach. Computational development of the GW approximation was later accomplished by a

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series of authors, namely, Hyberstsen and Louie [21], Godby, Schl€ uter, and Sham [22], and even later on by Aryasetiawan and Gunnarsson [23,24]. Let us now discuss the band structure of some specific nitride semiconductors. This semiconductor family can exist in wurtzite and zinc blende crystal polytypes with the Wz phase being the stable and widely used form. While both the Wz and ZB polytypes of GaN have been given in quite some detail, the ZB varieties of AlN and InN are not thermodynamically stable and reports are very sketchy. In fact, there are predictions that ZB AlN has an indirect bandgap [25]. A review of the band structure calculation and the methodologies used for both polytypes of all group III nitrides, BN, AlN, GaN, and InN, has been given by Lambrecht and Segall [16]. Calculations of electronic and optical properties of Wz GaN and related structures have been undertaken over many years [16,26] with more of them emerging continually, as nitride-based devices become more popular. Methods such as the ab initio, tight binding [27], pseudopotential, ASA LMTO, a general treatment of which has been treated by Andersen [28], LCAO, LAPW, full potential linearized augmented plane wave (FP-LAPW) method such as that reported by Wimmer et al. [29] for calculating the electronic band structure of Wz semiconductors within the LDA [15,16], DFT, and GW methods have been employed to calculate the energy bands for both wurtzite and zinc blende GaN, InN, and AlN bulk materials. Christensen and Gorczyca [30,31] utilized the ASA-LMTO method for group III-nitrides mainly GaN and AlN, albeit in relation to their behavior under pressure. The ASA-LMTO method was also applied by Lambrecht and Segall [32] to contrast and compare the nature of the direct and indirect bandgap of various ZB and Wz nitrides and, in particular, in terms of the directness or indirectness of the bandgap. These methods change in their capabilities to varying degrees. Early versions of pseudopotential calculations did not include the contribution by 3d-electrons because of the difficulty of attaining convergence in-plane wave expansions due to the deep N pseudopotential [33]. A mixed basis set was later used to overcome this apparent shortcoming [34] but with the consequence that ZB GaN would be lower in energy, meaning favored over the Wz phase that is inconsistent with experiments, and also was not confirmed with well-converged plane wave calculations performed for GaN by Yeh et al. [35] and InAlN [36]. Yeh et al. [35] focused on the issue of polymorphism for a large number of semiconductors including AlN, GaN, InN, AlP, AlAs, GaP, GaAs, ZnS, ZnSe, ZnTe, CdS, C, and Si, and using the local density formalism (LDF), developed a simple scaling at T ¼ 0 that systematizes the energy difference (DE LDF W-ZB ) between the ZB and Wz forms. This energy difference was found ~ to be linearly dependent on the atomistic orbital radii coordinate RðA; BÞ that depends only on the properties of the free atoms A and B, making up the binary compound AB. Of special interest for the topic under discussion is that Yeh et al. [35] found that DE LDF W-ZB ðABÞ for GaN is
9.9 meV/atom, which is within 0.7 meV/atom of the calculations by Van Camp et al. [37] and within 0.3 meV/atom of the calculations by Min et al. [34]. The energy difference, DE LDF W-ZB ðABÞ, for all three nitride binaries are negative when this quantity is scaled with differences in tetrahedral radii and Pauling’s electronegativity, implying that the equilibrium state of all three nitride binaries is the Wurtzitic form.

2.1 Band Structure Calculations

The issue of d-electrons is an important one, as in Ga 3d, in relation to pseudopotential and all-electron calculations, specifically, to know whether these calculations can handle these bands. Because both In and Ga are heavy and have d cores, Ga 3d and In 4d states overlap with the deep N 2s states with serious implications about bonding and band structure. The way in which the d-electrons are treated as core orbitals with or without nonlinear core corrections or as valence states in pseudopotential methods caused some confusion, which has been the topic of some discussion [32,38]. In the case of Fiorentini et al. [38], the structural and electronic properties, albeit cubic GaN, were studied within the local density approximation by the full potential linear muffin-tin orbitals method, wherein the Ga 3d-electrons were treated as band states with no shape approximation to the potential and charge density. Owing to the resonance of Ga 3d-states with nitrogen 2s states, the cation d bands were found not to be inert, and features unusual for a III–V compound were found in the lower part of the valence band as well as in the valence charge density and density of states. Additional full and frozen (T ¼ 0) overlapped core calculations performed for GaN, ZnS, GaAs, and Si (all cubic) showed that an explicit description of closed-shell interaction has a noticeable effect on the cohesive properties of GaN. The resulting energy resonance causes the Ga 3d-electrons to strongly hybridize with both the upper and lower valence band s and p levels. Such hybridization is predicted to have a profound influence on the GaN properties, including quantities such as the bandgap, the lattice constant, acceptor levels, and valence band heterojunction offsets. Because Al has no 3d core states, there is no hybridization between the cation d states and the N 2s states. In short, the band structure and cohesive properties of GaN are very sensitive to the cation d bands. On the pseudopotential side, Wright and Nelson [36] provided a framework in which accurate calculations treating the Ga 3d- and In 4d-electrons explicitly as valence states were performed by extending the plane wave cutoff to 240 Ry to ensure convergence. An interesting observation is that while d-electrons are important in bonding, they appear as separate states in considering quasi-particle excitations in photoemission experiments [39]. Somewhat of a side note but with legitimate relevance, it has been predicted in the cases of ZnS and ZnSe that potential acceptors, such as Cu, whose d-electrons are resonant with the lower valence band, are repelled by the d-hybridized upper valence band, resulting in a deep level. Impurities without d-electron resonance form shallow acceptors. Mg has no d-electrons and turns out to be sufficiently shallow for roomtemperature p-type doping of GaN. On the contrary, Zn, Cd, and Hg, which all have d-electrons, form deep levels in GaN [40]. Further insight is warranted before conclusive statements can be made with certainty as, for example, photoemission data show the N 2s to be well below the Ga 3d band. Pseudopotential calculations can also be applied to defects [41] and surfaces [42], as has been done for GaN. The calculations relating to surfaces are discussed in Section 3.2.7.2 in reasonable detail. Likewise, the calculations in relation to defects are treated in Section 4.3.1. Other applications of the calculations discussed above are for the determination of dielectric properties and susceptibility [43] and vibrational properties [44].

j139

j 2 Electronic Band Structure and Polarization Effects

140

Calculations in the Hartree–Fock approximation rather than the local density approximation have also been performed for nitride semiconductors, notably GaN [45] and AlN [46,47]. While LDA calculations underestimate the bandgap, the opposite is the case for Hartree–Fock calculations. In the H–F method, exchange is treated exactly but the correlations are fully ignored. Specific to the case of GaN, its total energy as a function of unit cell volume has been calculated for the wurtzite, zinc blende, and rock salt phases by the ab initio all-electron periodic Hartree–Fock method by Pandey et al. [45]. In this case, the gallium 3d levels were treated as fully relaxed band states, and the internal parameters c/a and u in the wurtzite phase were optimized. The calculated transition pressure between the wurtzite and rock salt phases were found to be about 52 GPa at the Hartree–Fock level and about 35 GPa at the correlated level. The calculated electronic structure shows strong hybridization of Ga 3d and N 2s states with the ordering as Ga 3d–N 2s–N 2p in all the phases. The results indicate the bandgap to be direct at G in the wurtzite and zinc blende phases and indirect in the high-pressure rock salt phase where the valence band maximum is shifted away from the G point. The electronic structure of Wz AlN has been investigated by means of periodic ab initio Hartree–Fock calculations for the purpose of calculating the binding energy, lattice parameters (a, c), and the internal coordinate or parameter (u) [47]. The values of the bulk modulus, its pressure derivative, the optical phonon frequencies at the center of the Brillouin zone, and the full set of elastic constants have been calculated and compared with experimental data. When ab initio Hartree–Fock calculations were used to determine the electronic structure of AlN in high pressure, the rock salt phase resulted [46]. In this phase, the calculated lattice constant is 3.982 Å with the bulk modulus of 329 GPa. As in the case of GaN, the rock salt phase is predicted to be indirect at the X point with a gap of 8.9 eV. Moreover, the bonding is essentially ionic between Al and N. The direct gap shows a stronger linear dependence on pressure with a pressure derivative of 68 meV GPa
1 compared to that of the indirect X-valley gap with a pressure derivative of 31.7 meV GPa
1. It should be emphasized that the rock salt phase is favored to exist under high pressure, and as such throughout this book and literature, nitrides are spoken of as if they are wurtzitic with GaN being cubic also when grown away from thermodynamic equilibrium conditions on cubic substrates along h0 0 1i directions. As mentioned above, the electronic properties of nitride semiconductors can more accurately be calculated using first-principles techniques like density functional theory [25] within the Green’s function theory with the characteristic GW approximation of the exchange correlation self-energy [17]. These calculations have been applied to Wz and ZB GaN and AlN by Rubio et al. [25] and to ZB GaN by Palummo et al. [48]. The computational complexity of the full GW method is prohibitive for applications to complex systems with large number of atoms, such as surfaces, interfaces, and clusters. However, it should be mentioned that efficient simplified version of the GW method has been reported to reduce the central processing unit (CPU) time by a factor of 100 (in conjunction with semiconductors Si, GaAs, AlAs, and ZnSe) [49,50]. Unless the simplified GW method is used [48,49], the full GW method is typically limited to simple systems, for example, elemental or binary semiconductors.

2.1 Band Structure Calculations

The GW calculations are reasonably consistent with each other and also with experiments in many cases. In the calculations of Rubio et al. [25], the ab initio pseudopotential method within the local density approximation and the quasi-particle approach have been employed to determine the electronic properties of both Wz and ZB phases of AlN and GaN. The quasi-particle band structure energies were calculated using a model dielectric matrix for the evaluation of the electron self-energy. In the zinc blende structure, AlN was predicted to be indirect (G to X) with (4.9 eV) and that GaN to be direct with 3.1 eV at the G point, the latter in good agreement with absorption experiments on cubic GaN, showing the bandgap to be 3.2–3.3 eV. In the calculations of Palummo et al. [48], models of diagonal and off-diagonal screening with LDA-RPA full calculations in cubic GaN were considered. Simplified GW calculations relying on these models were also compared with full GW calculations. At the time empirical pseudopotential calculations were not available, necessitating ab initio RPA calculations to be done within the DFT-LDA approach. These calculations have already been used for obtaining the full GW band structure of GaN [51]. It should be mentioned that with respect to pure LDA results, the valence band shifts down and the conduction band shifts up, resulting in larger bandgap estimation. The amount of downward shift of the valence bands increases with the increase in energy below the valence band maximum. The N2s states are about 1.5–2 eV more than the valence band maximum. Moreover, the bottom of the N2p valence band, the character of which is of a mixture of N2p cations, shifts by an amount of about 0.5 eV more than the maximum. The absolute shift of the valence band maximum is a problem in GW theories, which is also the case with this method for very established materials such as Si. This seems to stem from the choice parameterization used for the LDA starting point of the calculations. The GW method changes the gaps of GaN and AlN by 1 and 2 eV, respectively [16]. The conduction band correction is on the order of 0.1 eV across the k-points and specific states. This figure appears to increase with the increase in energy dealing with higher conduction bands, awaiting further refinement following any comparison with experiments when accurate measurements become available. Unlike the full GW calculations, the TB approach provides an attractive possibility for an extension of the system size accessible to electronic structure calculations with atomic resolution. TB calculations have been applied to nitride-based systems [52–54]. Moreover, the Slater–Koster parameters transferable between the ZB and Wz crystal phases have been treated [27]. There has been a plethora of reports regarding band structure calculations in nitride semiconductors. The calculated band structures of Wz GaN, AlN, and InN are exhibited in Figure 2.1. For semblance of completeness, results from empirical pseudopotential method are also included here. Fritsch et al. [55] investigated the electronic band structure of both the wurtzite and zinc blende group III nitride semiconductors GaN, AlN, and InN within the empirical pseudopotential approach. Using ionic model potentials and a static dielectric screening function derived, the cationic and anionic model potential parameters were obtained from the zinc blende GaN, AlN, and InN experimental data. Using these model potentials, Fritsch et al. [55] calculated the band structure of group III nitrides in both the wurtzite and zinc blende

j141

j 2 Electronic Band Structure and Polarization Effects

142

(a)

GaN

12 10

3

1 1

3

6

Energy ( eV )

6

1

8

3 1

2

4

3

2

1 1 6 5

0 –2

2

4 3 2

3

–4

3 1

–6

1 3 1

3

–8 –10

3

–12

3

–14 –16

3

1 3

3

1 3

A

S H

R L T U

P K

M

1 M

K

M U L

T

(b) 6

3

3

10

H

P K

4 1

3

8

3 3 1

1

6

S

A

R

AlN

12

3

2

1

4 Energy (eV)

3

4

3

2 1 5

4 3 1

3

3 1

6

0 3

–2

2 3

–4

1

–6 –8

A

S

Γ

R L T U

–10 3

–12 –14 M

K

T

3

1

1

3

M U

H P K

M

L

Figure 2.1 Calculated band structures of (a) wurtzite GaN, (b) AlN, and (c) InN in the LDA within the FP-LMTO method at the experimental lattice constant and optimized u-value. The first Brillouin zone is also shown for convenience [16].

R

A

S

H P K

2.1 Band Structure Calculations

(c)

8

6

3 1 3 2

6 4

1 1

1

3 1 1

3

2 Energy (eV)

InN 4 3

1

2

10

1 6

0 2

–2

5

4

1

3 2

3 1 3

–4 –6

3

3

1 3 1

A

S R L T U

–8 3

–10

3

1 3

–12 3

–14 M

K

1 3 T

H P K

M

3 1 M U

L

R

A

S

H P K

Figure 2.1 (Continued )

form, recognizing the necessity of including the anisotropy of wurtzite crystals in the screening function. The band structures so calculated for wurtzitic GaN, AlN, and InN are shown in Figure 2.2. The same for the zinc blende variety is shown in Figure 2.3. It should be noted at the outset that all these binary materials, including alloy compounds obtained by combinations of these binaries, are wide direct bandgap semiconductors in both crystal phases, except zinc blende AlN that is expected to have an indirect gap with the conduction band minimum being at the X valley. Due to the lack of reliable experimental data, many details of these studies must be improved to provide an accurate band description. Approaches such as the kp model [56,57] including strain have been employed to calculate the valence band structure of Wz GaN [56,58]. First-principles calculations of effective mass parameters and valence band structures in bulk and confined systems with and without strain, utilizing the FP-LAPW method [59–61,66] and envelope function formalism for valence bands in wurtzite quantum wells, have been undertaken [62]. The wurtzite structure has a hexagonal unit cell and thus two lattice constants, c and a. It contains six atoms of each type. The space group for the wurtzite structure is P63mc (C46v ) [63]. The wurtzite structure consists of two interpenetrating hexagonal close-packed (HCP) sublattices, each with one type of atoms, offset along the c-axis by 5/8 of the cell height (5c/8). The zinc blende structure has a cubic unit cell, containing four group III elements and four nitrogen elements. The space group for the zinc blende structure is T 2d : F
43m. The position of the atoms within the unit cell is identical to the diamond crystal structure. Both structures consist of two

j143

j 2 Electronic Band Structure and Polarization Effects

144

interpenetrating face-centered cubic sublattices, offset by one quarter of the distance along a body diagonal. Each atom in the structure may be viewed as positioned at the center of a tetrahedron with its four nearest neighbors defining the four corners of the tetrahedron. The zinc blende and wurtzite structures are similar. In both cases, each

Energy (eV)

10

5

0

–5

A

R

L U M

Σ

Γ Δ A

S

H P K

T

Γ

(a)

15

Energy (ev)

10

5

0

–5

A

R

L U M

Σ

Γ Δ A

S

(b) Figure 2.2 Band structure of (a) wurtzitic GaN, (b) wurtzitic AlN, and (c) wurtzitic InN along high-symmetry lines in the Brillouin zone calculated within the empirical pseudopotential method (EPM), using ionic model potentials obtained experimentally from zinc blend varieties. Courtesy of Daniel Fritsch et al. [55].

H P K

T

Γ

2.1 Band Structure Calculations

j145

Energy (eV)

10

5

0

–5

A

R

L U M

Σ

Γ Δ A

S

H P K

Γ

T

(c) Figure 2.2 (Continued )

Energy (eV)

15 L3

10

K1 X3

L1

5

X1 L3

0

X5 X3

L2

–5

L

Λ

Γ

Δ

X

(a) Figure 2.3 Band structure of (a) zinc blende GaN, (b) zinc blende AlN, and (c) zinc blende InN along high-symmetry lines in the Brillouin zone calculated within the empirical pseudopotential method (EPM), using ionic model potentials obtained experimentally. Courtesy of Daniel Fritsch et al. [55].

K1

K2 K1 K1 UK

Σ

Γ

j 2 Electronic Band Structure and Polarization Effects

146

20

15 Energy (eV)

L3 10

X1

5

0

L3

–5

L2

10

K1

K2

X5

K1

X3 Λ

L

(b)

K1

X3

L1

Γ

Δ

K1 X

U,K

Σ

Γ

L3

Energy (eV)

K1 X3

5

X1

K1

L1 0

K2

L3

X5

K1

L2

–5 L

K1

X3 Λ

Γ

Δ

X

U, K

Σ

Γ

(c) Figure 2.3 (Continued )

group III atom is coordinated by four nitrogen atoms. Conversely, each nitrogen atom is coordinated by four group III atoms. The main difference between these two structures lies in the stacking sequence of closest packed diatomic planes. For the wurtzite structure, the stacking sequence of the (0 0 0 1) planes is ABABAB in the h0 0 0 1idirection. For the zinc blende structure, the stacking sequence of the (1 1 1) planes is ABCABC in the (1 1 1) direction.

2.1 Band Structure Calculations

The structure and the first Brillouin zone of a wurtzite and zinc blende crystal along with the irreducible wedges, calculated using the LDA within the FP-LMTO method at the experimental lattice constant and optimized u value, are displayed in Figure 2.4a and b, respectively. In a crystal with Wz symmetry, the conduction band kz

So

A o

R

Δo

T o

Γ

H L o

o

S’

oP

U K

o

o

ky

T’

Σ

M

kx (a)

kz

L U Q

S Z

kx

K

X

ky

W

(b) Figure 2.4 (a) Structure and the first Brillouin zone of a wurtzite crystal. Schematics of the irreducible wedges of Wz structure, indicating the high-symmetry points and lines, are also shown. The Umin point of the Wz phase is located on the M–L line at two-thirds a distance away

from the M point. (b) Structure and the first Brillouin zone of a zinc blende crystal. Schematics of the irreducible wedges of ZB structure indicating the high-symmetry points and lines are also shown.

j147

j 2 Electronic Band Structure and Polarization Effects

148

Conduction band

E gA E0

E0

C6v

or

5

9

A

6

7

7

7

Valence band

9

, J = 3/2

Valence band

B cr

15

so 15

1 7

ZB

W Crystal field cr = so

W Spin orbit

7

C

W

8 , J = 1/2

Crystal field

1

=

ZB

ZB Spin orbit cr = 0 so

=

2

=

3

Figure 2.5 Schematic representation of the splitting of the valence band in Wz crystals due to crystal field and spin–obit interaction. From left to right, the crystal field splitting is considered first. From right to left, the spin–orbit splitting is considered first. Regardless of which is considered first, the end result is the same in that there are three valence bands that are sufficiently close to one another for band mixing to be nonnegligible.

wave functions are formed of the atomic s orbitals, which transform the G point congruent with the G7 representation of the space group C 46v , The upper valence band states are constructed out of appropriate linear combinations of products of p3-like (px-, py-, and pz-like) orbitals with spin functions. Under the influence of the crystal field and spin–orbit interactions, the hallmark of the wurtzite structure, the sixfold degenerate G15 level associated with the cubic system, splits into Gv9 , upper Gv7 and lower Gv7 levels (Figure 2.5). Figure 2.6 shows the dispersion of the uppermost valence and conduction band structures in Wz GaN and ZB GaN ((a) near the G band for Wz, (b) inclusive of M, L, and A minima in Wz, and (c) inclusive of G, L, and X minima in ZB GaN). The influence of the crystal field splitting, which is present only in the wurtzite structure, transforms the semiconductor from ZB to Wz, which is represented in the section on the left-hand side in Figure 2.5. The crystal field splits the G15 band of the ZB structure into G5 and G1, states of the wurtzite structure. These two states are further split into Gv9 , upper Gv7, and lower Gv7 levels by spin–orbit interactions. Application of the spin–orbit splitting, from right to left, splits the G15 band of the ZB crystal into G8 and G7 states while the crystal possesses the zinc blende symmetry. Application of a crystal field further splits these states into Gv9 , upper Gv7, and lower Gv7 levels, and the crystal now possesses the wurtzite symmetry.

2.1 Band Structure Calculations

E(k)

c

Γ7

Γ9 :HH Δ1 Γ7:LH Γ7:ΧΗ

kz

(a)

Γ

Figure 2.6 (a) Schematic representation of the G point valence and conduction bands in crystal with wurtzite symmetry, such as GaN, where the spin–orbit splitting leads to the bands labeled as HH and LH. The one caused by splitting due to crystal field is labeled as CH [59,60]. (b) Schematic representation of the band diagram for Wurtzite GaN showing the separation between the G, A, and M–L band symmetry points at 300 K. The values with respect to the top of the valence band are EG ¼ 3.4 eV, EM–L ¼ 4.5–5.3 eV, EA ¼ 4.7–5.5 eV,

kx , ky Eso ¼ 0.008 eV, Ecr ¼ 0.04 eV [56]. The values of EG ¼ 6 eV, EM–L ¼ 7 eV, and EA ¼ 8 eV are given by Fritsch et al. [55]. (c) Schematic representation of the band diagram for zinc blende GaN showing the separation between the G, X, and L band symmetry points at 300 K. The values with respect to the top of the valence band are EG ¼ 3.2 eV, EL ¼ 4.8–5.1 eV, Ex ¼ 4.6 eV, and Eso ¼ 0.02 eV. Note that in the ZB structure, the valence band is degenerate [56]. The values of EG ¼ 3.2 eV, EL ¼ 5.1 eV, EX ¼ 4.3 eV are given by Fritsch et al. [55].

j149

j 2 Electronic Band Structure and Polarization Effects

150

Energy

A valley

M and L valleys

valley

EA

EM–L

E Ecr

HH band LH band

kz

k x,y

Split–off band (b) Energy L valleys X valley

EL

valley

EX E

Eso

HH band



LH band Split–off band (c) Figure 2.6 (Continued )

Literature values of the calculated and experimental critical point transition energies for wurtzitic GaN, AlN, and InN are tabulated in Tables 2.1–2.3. Literature values of the calculated and experimental critical point transition energies for zinc blende GaN, AlN, and InN are tabulated in Tables 2.4–2.6. As shown in Figure 2.6a for wurtzitic GaN, the hole effective masses of the three uppermost valence bands Gv9 , Gv7 , and Gv7 exhibit large k-dependence. The bands are labeled as HH (heavy Gv9 ), LH (light, upper Gv7 spin–orbit split) and CH (Gv7 , crystal field split). The mass of the Gv9 band is heavy in all k-directions, whereas that of the upper Gv7 is relatively light in the x- and y-planes but heavy in the z-direction. That of the lower Gv7 is light in the x- and y-planes, but it is heavy along the z-direction

2.1 Band Structure Calculations Table 2.1 Literature values of calculated and experimental critical point transition energies for wurtzitic GaN [55].

A (eV) Parameter

Anisotropic

Isotropic

B (eV)

M v2
Mc1 Mv4
Mc1 Mv4
Mc3 Gv6
Gc1 Gv3
Gv6 Gv1
Gv6 Gv5
Gv6 Gv5
Gc3 Gv5
Gc6 Hv3
Hc3 K v3
K c2 K v2
K c2

7.67 6.07 7.68 3.47 6.97 0.043 1.00 5.96 10.74 8.06 8.54 8.68

7.67 6.07 7.68 3.47 6.94 0.023 1.00 5.96 10.74 8.07 8.55 8.68

8.26 6.61 7.69 3.50 6.80 0.021

9.0 9.43 10.10

C (eV)

3.0 7.0 0.0 1.0 5.9 11.1 8.3 7.9

D (exp) (eV) 7.05 7.0a 7.05 3.6, 3.44a, 3.50a 7.0a 0.022a 5.3 9.4 7.9a 7.65, 9.0a

The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor. A: empirical pseudopotential calculation from Ref. [55]; B: ab initio pseudopotential calculation within local density approximation from Ref. [25]; C: LCAO within local density approximation from Ref. [64]; D: unless stated otherwise, the experimental values are taken from Ref. [65]. a Experimental values taken from Ref. [66].

(c-direction). Two different definitions are prevalent in the literature. The G6, Gl pair has been used in Refs [63,71] and the G5, G1 pair in Refs [62,72,73]; for background information on group theory and symmetries in physics, see Refs [74,75]. We should mention that a carryover habit from the zinc blende nomenclature is still used for wurtzite symmetry by referring to the crystal field split-off band with the nomenclature “SO” as if it is the spin–orbit split-off band, because it happens to be the farthest from the HH band. In the zinc blende symmetry, the crystal field splitting is nonexistent, making the top of the valence band degenerate, and the spin–orbit splitting is large. Portions of this book, unfortunately, participate in the misuse of this nomenclature. Shown in Figure 2.6b are the most pertinent bands near the zone center and A, M, and L valleys. The same for zinc blende GaN is shown in Figure 2.6c. Without the spin–orbit interaction, the valence band would consist of three doubly degenerate bands: HH, LH, and CH bands. The spin–orbit interaction removes this degeneracy and yields six bands. Some band calculations based on the empirical pseudopotential method (EPM) [66] or the empirical tight binding method (ETBM) [76] have shown this splitting to be about 10 meV near the G point, which is comparable to the energy separation of the split-off band in GaN. In early attempts, the general Hamiltonian in kp theory included the spin–orbit interaction,

j151

j 2 Electronic Band Structure and Polarization Effects

152

Table 2.2 Literature values of calculated and experimental critical point transition energies for wurtzitic AlN [55].

A (eV) Parameter

Anisotropic

Isotropic

B (eV)

M v2
Mc1 M v4
Mc1 M v4
Mc3 Gv6
Gc1 Gv3
Gv6 Gv1
Gv6 Gv5
Gv6 Gv5
Gc3 Gv5
Gc6 Hv3
Hc3 K v3
K c2 K v2
K c2

9.56 7.88 8.81 6.11 6.44 0.13 1.04 8.95 12.99 10.10 9.43 9.59

9.54 7.87 8.83 6.11 6.41 0.16 1.03 8.94 12.97 10.91 9.43 9.57

10.0 8.3 8.5 6.0 6.7 0.2 0.9 9.4 14.0 10.5 9.6 9.7

C (exp) (eV)

6.29

8.02 14.00 10.39

The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor. A: empirical pseudopotential calculation from Ref. [55]; B: ab initio pseudopotential calculation within local density approximation from Ref. [25]; C: experimental values taken from Ref. [67].

but it took a while for it to be applied to the calculations of the band structure of wurtzite materials such as GaN. Naturally, the band structures of wurtzitic and zinc blende polytypes are very distinct due to the differences in underlying symmetries. For the zinc blende case, the three Luttinger parameters and the spin–orbit splitting provide a minimal description of the valence band structure. Moreover, the energy gap and interband coupling strength are also required for complete parameterization of both the conduction and valence bands. The split-off hole mass can be treated as an independent parameter within the commonly used eight-band kp model. The increase in the electron effective mass due to interactions with higher conduction bands can be included via the F parameter (see Section 2.2 for details) [72,77–79]. The set of band parameters needed to describe the wurtzite lattice must be augmented due to its lower symmetry. Neglecting the effect of valence band and upper conduction bands on the electron effective mass allows one to omit the interband matrix element and the F parameter. Owing to the reduced symmetry, the electron mass can display a rather weak anisotropy. In contrast, a full description of the valence band in the wurtzite polytype band structure requires both the spin–orbit splitting, Dso, and the crystal field splitting, Dcr, along with the seven so-called A parameters. Analogous to the Luttinger parameters in zinc blende materials, the latter parameterizes the hole masses along the different directions. Figure 2.6a and b highlights the differences in wurtzite and zinc blende varieties in terms of their band structure.

2.1 Band Structure Calculations Table 2.3 Literature values of calculated and experimental critical point transition energies for wurtzitic InN [55].

A (eV) Parameter

Anisotropic

Isotropic

B (eV)

d (exp) (eV)

M v2
Mc1 Mv4
Mc1 Mv4
Mc3 Gv6
Gc1 Gv3
Gv6 Gv1
Gv6 Gv5
Gv6 Gv5
Gc3 Gv5
Gc6 Hv3
Hc3 K v3
K c2 K v2
K c2

7.30 5.94 6.71 2.58 5.63 0.214 0.90 5.22 10.16 7.34 8.13 8.60

7.23 5.88 6.70 2.59 5.50 0.084 0.89 5.18 10.12 7.36 8.12 8.50

6.65 5.05 5.80 2.04 5.77 0.017 1.05 4.65 8.74 6.51 7.38 7.20

7.3, 4.95a 7.3 2.11, 2.0a;b

5.0, 5.5, 4.7a 8.8, 8.9a 5.4a 7.3, 7.2a

The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor. A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [66]. a Experimental values taken from Ref. [68]. b It should be pointed out that the values of bandgap values listed in this table compares with 1.8–2.1 eV values reported during the early stages of InN development and 0.7–0.8 eV reported later on. For an in-depth discussion of this seemingly controversial bandgap determined experimentally, the reader is referred to Section 2.9.1.

Table 2.4 High-symmetry point energies in zinc blende GaN in

reference to the top of the valence band for cases where spin–orbit effects are neglected (included) [55]. Parameter G15 c Gc1 Gv15 X c3 X c1 X v5 X v3 Lc3 Lc1 Lv3 Lv2

(Gc7 ) (Gc6 ) (Gv8 ) (X c7 ) (X c6 ) (X v7 ) (X v6 ) (Lc4;5 ) (Lc6 ) (Lv4;5 ) (Lv6 )

A (eV)

B (eV)

C (eV)

10.098 3.308 0.000 6.010 4.428
2.459
6.294 10.416 5.149
0.834
6.812

10.300 3.383 0.000 6.805 4.571
2.693
6.149 9.916 5.636
0.931
6.743

10.248 3.213 0.000 6.265 4.585
2.086
5.923 10.606 5.510
0.772
6.644

A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [69]; B: empirical pseudopotential calculation from Ref. [70].

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Table 2.5 High-symmetry point energies in zinc blende AlN in

reference to the top of the valence band for cases where spin–orbit effects are neglected (included) [55]. Parameter G15 c Gc1 Gv15 X c3 X c1 X v5 X v3 Lc3 Lc1 Lv3 Lv2

(Gc7 ) (Gc6 ) (Gv8 ) (X c7 ) (X c6 ) (X v7 ) (X v6 ) (Lc4;5 ) (Lc6 ) (Lv4;5 ) (Lv6 )

A (eV)

B (eV)

12.579 5.840 0.000 8.794 5.346
2.315
5.388 12.202 8.264
0.718
6.251

13.406 5.936 0.000 10.661 5.102
2.337 5.262 12.014 9.423
0.728
6.179

A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [69].

Table 2.6 High-symmetry point energies in zinc blende InN in

reference to the top of the valence band for cases where spin–orbit effects are neglected (included) [55]. Parameter G15 c Gc1 Gv15 X c3 X c1 X v5 X v3 Lc3 Lc1 Lv3 Lv2

(Gc7 ) (Gc6 ) (Gv8 ) (X c7 ) (X c6 ) (X v7 ) (X v6 ) (Lc4;5 ) (Lc6 ) (Lv4;5 ) (Lv6 )

A (eV)

B (eV)

9.722 2.112 0.000 6.416 5.187
1.555
4.303 10.168 4.733
0.480
4.667

10.193 1.939 0.000 7.392 2.509
1.408
4.400 8.060 5.818
0.456
5.200

A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [69].

2.2 General Strain Considerations

Strain–stress relationship or Hooke’s law can be used to describe the deformation of a crystal ekl, due to external or internal forces or stresses sij, X sij ¼ Cijkl ekl ; ð2:4Þ k;l

2.2 General Strain Considerations

where Cijkl is the fourth ranked elastic tensor and represents the elastic stiffness coefficients in different directions in the crystal, which due to the C6v symmetry can be reduced to a 6 · 6 matrix using the Voigt notation: xx ! 1, yy ! 2, zz ! 3, yz, zy ! 4, zx, xz ! 5, xy, yx ! 6. The elements of the elastic tensor can be rewritten as Cijkl ¼ Cmn with i, j, k, l ¼ x, y, z and m, n ¼ 1, . . . , 6. With this notation, Hooke’s law can be reduced to si ¼

X

Cij ej :

ð2:5Þ

j

or as treated in Ref. [80] for C6v symmetry, we have 2

3 2 sxx C11 6 syy 7 6 C12 6 7 6 6 szz 7 6 C13 6 7 6 6 sxy 7 ¼ 6 0 6 7 6 4 syz 5 4 0 szx 0

C12 C22 C13 0 0 0

C13 C13 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

32 3 0 exx 6 7 0 7 76 eyy 7 6 7 0 76 ezz 7 7; 6 7 0 7 76 exy 7 0 54 eyz 5 ezx C66

ð2:6Þ

with C66 ¼ C11
2 C12 . If the crystal is strained in the (0 0 0 1) plane, and allowed to expand and constrict in the [0 0 0 1] direction, the szz ¼ sxy ¼ syz ¼ szx ¼ 0, sxx 6¼ 0, and syy 6¼ 0, and the strain tensor has only three nonvanishing terms (with C11 ¼ C22), namely, 

sxx ; syy szz



 ¼

C11 þ C12 2C13

C 13 C 33



 exx ; eyy ; ezz

ð2:7Þ

with exx ¼ eyy ¼ a
a0a0 and ezz ¼ c
c0c0 ¼
CC1333 ðexx þ eyy Þ, the latter of which describes the Poisson effect, and a and a0 and c and c0 represent the in-plane and out-ofplane lattice constants of the epitaxial layer and the relaxed buffer (substrate), respectively. The above assumes that the in-plane strain in x- and y-directions is identical, namely, exx ¼ eyy. When the crystal is uniaxially strained in the (0 0 0 1) c-plane and free to expand and constrict in all other directions, szz is the only nonvanishing stress term, and the strain tensor is reduced to     1 eyy C 12 C33
C213 ð2:8Þ ¼ 2 exx : ezz C13
C11 C33 C 11 C13
C12 C13 If the growth is performed on the ð1
1 0 0Þm-plane, meaning the growth plane is the xz-plane with the growth direction being along the y-axis, the in-plane strain anisotropy dictates that exx 6¼ eyy . The out-of-plane stress, syy ¼ 0, which when utilized in the stress–strain relationship of Equation 2.6 leads to eyy ¼


C12 exx þ C13 ezz : C11

ð2:9Þ

If the growth is performed on the ð1 1
2 0Þa-plane, meaning the growth plane is the yz-plane with growth direction being along the x-axis, the in-plane strain anisotropy

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j 2 Electronic Band Structure and Polarization Effects

156

dictates that exx 6¼ eyy. The out-of-plane stress, sxx ¼ 0, which when utilized in the stress–strain relationship of Equation 2.6 leads to exx ¼


C12 eyy þ C13 ezz : C 11

ð2:10Þ

The values of the elastic stiffness coefficients for GaN have been measured by Sheleg and Savastenko [81] and reproduced in Ref. [59]. These data are listed in Table 1.24. The inversion of the 6 · 6 elastic constant matrix in Equation 2.6 leads to the elastic compliance constants as follows: S11 ¼

C11 C 33
C213 ; ðC 11
C12 Þ½C33 ðC 11 þ C12 Þ
2C213 

S12 ¼

C12 C 33
C213 ; ðC 11
C12 Þ½C33 ðC 11 þ C12 Þ
2C213 

S13 ¼

C13 ; ½C33 ðC11 þ C 12 Þ
2C 213 

S33 ¼

C11 þ C12 ; ½C33 ðC11 þ C 12 Þ
2C 213 

S44 ¼

1 : C44

ð2:11Þ

Through the use of the aforementioned compliance constants, a very useful figure of merit can be determined, which in turn would lead to the directional hardness and the reciprocal Young’s modulus as a function of orientation to the crystal axis. For 0 hexagonal symmetry, the reciprocal Young’s modulus S11 along an arbitrary direction at an angle y with respect to the [0 0 0 1] axis is given by [82–84] 0

S11 ¼ S11 sin4 q þ S33 cos4 q þ ðS44 þ 2S13 Þsin2 qcos2 q:

ð2:12Þ

0

In Figure 2.7a and b, polar plots of S 11 as a function of the direction in reference to the [0 0 0 1] axis and for directions along the basal plane are shown for InN, GaN, and AlN binaries. Clearly, AlN and GaN are harder than InN by more than a factor of 2. The hardness of AlN is almost isotropic, whereas that for GaN and InN exhibit some preferential “softness” along the [0 0 0 1] and ½2
1
1 0 axes. Of paramount interest here is that the hardness of all the wurtzitic binaries is isotropic in the basal plane. This takes on a special meaning as the strain, with growth along the c-axis, caused in epitaxial heterostructures by mismatch lattice and thermal mismatch is along the basal plane. Lack of any force in the growth direction and the fact that the crystal can relax freely in this direction leads to a biaxial strain e1 ¼ e2, which in turn causes stresses s1 ¼ s2

with s3 ¼ 0:

The internal strain is defined by the variation of the internal parameter under strain, (u
u0)/u0. In the limit of small deviations from the equilibrium, Hooke’s law

2.2 General Strain Considerations

[1 0 1 0] 8 S 11 [10 –12 m 2 /N–1]

(a)

6 In N 4

GaN

2 AlN –8

–6

–4

–2

0

0

[1 2 1 0] 2

4

6

8

–2 –4 –6 - –8 [0 0 0 1] 8

S 11 [10 –12 m 2 / N–1]

(b)

6 InN

4

GaN

2

AlN –8

–6

–4

–2

[2 1 1 0]

q 0

0

2

4

6

8

–2 –4 –6 - –8 Figure 2.7 (a) The reciprocal Young’s moduli, S0 11, in the basal plane of InN, GaN, and AlN indicating AlN and GaN are harder than InN to be harder by more than a factor of 2. Of paramount importance, the hardness of the wurtzitic crystals is isotropic in the basal plane. (b) The reciprocal

Young’s modulus, S0 11, along an arbitrary direction making an angle y with respect to the c-axis in the [0 0 0 1] direction. In basal plane the stiffness of AlN is isotropic whereas GaN and InN show preferential “softness” along the [0 0 0 1] and [2
1
1 0] directions [84].

gives the corresponding diagonal stress tensor s with the elements [85] sxx ¼ syy ¼ ðC11 þ C12 Þexx þ C13 ezz ; szz ¼ 2C13 exx þ C33 ezz :

ð2:13Þ

In Equation 2.13, four of the five independent stiffness constants Cij of the wurtzite crystal are involved. The modifications of Equation 2.13 by the built-in electric field due to the spontaneous and piezoelectric polarization are neglected, as the effect is small.

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j 2 Electronic Band Structure and Polarization Effects

158

In the case of uniaxial stress, for example, along the c-direction, there is an elastic relaxation of the lattice in the c-plane. The ratio of the resulting in-plane strain to the deformation along the stress direction is expressed by the Poisson’s ratio, which in general can be anisotropic. For the wurtzite lattice subjected to a uniaxial stress szz parallel to the c-axis, sxx ¼ syy ¼ 0 holds. Then Equation 2.13 gives the relation exx ¼
½C13 =ðC 11 þ C12 Þezz ¼
nezz ;

ð2:14Þ

with n ¼ C13 =ðC 11 þ C12 Þ being the Poisson’s ratio. The uniaxial stress correlates with strain along the direction of the stress by the Young’s modulus E as szz ¼ Eezz and thus E ¼ C33


2C 213 : C11 þ C 12

ð2:15Þ

A homogeneous biaxial stress in the basal plane is described by a constant force in the plane with sxx ¼ syy and vanishing force in the c-direction szz ¼ 0. The Hooke’s law of Equation 2.13 leads to a relationship between axial and basal plane strain components as ezz ¼
RBexx, which reproduces Equation 2.7 with the biaxial relaxation coefficient being RB ¼

2C13 : C33

ð2:16Þ

The biaxial relaxation coefficient is also referred to as simplylater in this chapter in the polarization section. The in-plane stress–strain relationship using the biaxial modulus is sxx ¼ Yexx, which leads to Y ¼ C11 þ C12


2C 213 : C33

ð2:17Þ

The strain–stress relationship along the c-axis is sxx ¼
ðY=RB Þezz :

ð2:18Þ

We can then relate the Young modules E (is also commonly described by nomenclature as Yo) to biaxial modulus Y as E¼

C33 Y 2n or E ¼ B Y: C11 þ C12 R

ð2:19Þ

Equation 2.13 can now be expressed as sxx ¼


E ezz : 2n

ð2:20Þ

In the case of hydrostatic pressure sxx ¼ syy ¼ szz ;

ð2:21Þ

and from the Hooke’s law ezz ¼ RH exx ;

ð2:22Þ

2.3 Effect of Strain on the Band Structure of GaN

with RH expressed as RH ¼

C11 þ C12
2C13 : C33
C 13

To calculate elastic stiffness constants, Wagner and Bechstedt [85] considered C11 þ C12 as an independent quantity and made use of the relation between the elastic constants and isothermal bulk modulus, B0 (nomenclature Bs is also used): B0 ¼

ðC11 þ C12 ÞC13
2C213 : C11 þ C12 þ 2C 33
4C13

ð2:23Þ

Equation 2.22 can be obtained from Equation 2.13 with the aid of linearized relation DP/B0 ¼
DV/V0 ¼
2(e11 þ e33) or Dp/B0 ¼
DV/V0 ¼
2(e11 þ e33), where DV is the variation of volume with pressure and V0 is the static volume. The values of the bulk modulus B0 have been obtained by fitting the Vinet equation of state to the calculated dependence of energy on volume [86,87], using     1 Y ¼ 2 þ 1=2 ð2:24Þ
4 RB B 0 ; v and C13 ¼ Y=½ð1=vÞ
RB :

ð2:25Þ

Wagner and Bechstedt [85] obtained the absolute values of the elastic stiffness coefficients. The strain- and stress-related issues represent the cornerstone of the discussion on piezoelectric polarization, and the above discussed section provides sufficient material to embark on the discussion of polarization issues in nitride semiconductor heterostructures.

2.3 Effect of Strain on the Band Structure of GaN

The strain in conventional group III–V semiconductors has been a much desired feature for its beneficial effects [88]. In the world of GaN, however, it is not necessarily a desirable commodity but could be construed as a nemesis brought upon by the lack of lattice- and thermal-matched substrates and uncomfortably large lattice and thermal mismatch with its ternaries. It is therefore imperative that strain effects be considered. Figure 2.8 exhibits the valence band structure of GaN in the x- and y-planes under biaxial compressive strain and uniaxial strain in the c-plane with the direction of strain as in Figure 2.8c. There are no major changes in the HH, LH, and CH bands, other than crystal splitting becoming larger, with the hole effective mass remaining heavy, the density of states staying high, and the crystal symmetry remaining the same, C46v . In contrast, the uniaxial strain in the c-plane causes an anisotropic energy splitting in the x- and y-planes, which leads to a symmetry lowering from C 46v to C2v . When a compressive uniaxial strain is induced along the y-direction, the HH band in the x-direction and the LH band in the y-direction move to higher energies. This causes a

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j 2 Electronic Band Structure and Polarization Effects

160

(a)

(b)

E

E

(c) Biaxial eff so

eff cr

Uniaxial z

kx

ky

kx

ky

y x

Figure 2.8 The valence band structure of GaN under (a) biaxial strain in the c-plane, (b) uniaxial strain in the c-plane, and (c) schematic of the particulars of the strain [59,89].

reduction in the density of states. A tensile uniaxial strain along the x-direction has the same effect. On the contrary, when a tensile uniaxial strain is induced along the ydirection, the HH band in the x-direction and the LH bands along the y-direction move to lower energies. This causes a reduction in the density of states. A compressive strain along the x-direction has the same effect [89].

2.4 kp Theory and the Quasi-Cubic Model

The conduction and valence bands of nitride semiconductors are comprised of s- and p-like states, respectively. Unlike the conventional ZB III-N semiconductors and the lack of a high degree of symmetry, the crystal field present removes the degeneracy at the top of the conduction band. Moreover, unlike the ZB case, the spin–orbit splitting is very small and makes all three bands in the valence band closely situated in energy. Consequently, the three valence bands and the conduction band must be considered in unison and this makes the use of an 8 · 8 kp Hamiltonian imperative. Because the bandgaps of nitrides are very large, the coupling between the conduction and valence bands can be treated as a second-order perturbation, which allows the 8 · 8 Hamiltonian to be split into one 6 · 6 Hamiltonian dealing with the valence band and another 2 · 2 dealing with the conduction band [59]. As indicated above, the conduction band is made of s-like states, which means that it can be treated as parabolic with the dispersion relation EðkÞ ¼ E c0 þ


2 k2z h

==

2mc

þ


2 ðk2x þ k2y Þ h 2m? c

== þ a? c ðexx þ eyy Þ þ ac ðezz Þ;

ð2:26Þ

==

where a? c and ac represent the in-plane and out-of-plane deformation potentials, respectively. For an isotropic parabolic conduction band, Equation 2.26 reduces to EðkÞ ¼ E c0 þ


2 k2 h þ ac e: 2mc

ð2:27Þ

2.4 kp Theory and the Quasi-Cubic Model

Ec0 is the conduction band energy at the k ¼ 0 point, e is the strain, and ac is the deformation potential for the conduction band. The other terms have their usual meanings. It should be pointed out that we are dealing with a linear system. Using the basis jY 11 ">; jY 11 #>; jY 10 ">; jY 10 #>; jY 1
1 ">; jY 1
1 #>, the 6  6 Hamiltonians can be expressed as 0 1 F 0
H 0 K 0  B 0 G D
H 0 K C B C B
H D l 0 I 0 C B C; ð2:28Þ B 0
H 0 l D I C B C @ K 0 I D G 0 A 0 K 0 I 0 F where F, G, l, D, H, I, and K are defined as (two forms are given by Ren et al. [90] and Albrecht et al. [91]) F ¼ D1 þ D2 þ l þ q; G ¼ D1
D2 þ l þ q; l ¼ A1 k2z þ A2 ðk2x þ k2y Þ þ D1 ezz þ D2 ðexx þ eyy Þ; q ¼ A3 k2z þ A4 ðk2x þ k2y Þ þ D3 ezz þ D4 ðexx þ eyy Þ; H ¼ iA6 kz ðk2x þ k2y Þ1=2
A7 ðk2x þ k2y Þ1=2 ; H ¼ ðiA6 kz
A7 Þðkx þ iky Þ þ iD6 ðexz þ ieyz Þ; I ¼ iA6 kz ðk2x þ k2y Þ1=2 þ A7 ðk2x þ k2y Þ1=2 ; I ¼ ðiA6 kz þ A7 Þðkx þ iky Þ þ iD6 ðexz þ ieyz Þ; K ¼ A5 ðk2x þ k2y Þ; K ¼p A5ffiffiffiðkx þ iky Þ2 þ D5 ðexx
eyy þ i2exy Þ; D ¼ 2D3 ;

½91 ½92 ½91 ½92 ½91 ½92

ð2:29Þ

where Ai is the valence band parameter corresponding to the Luttinger parameters in the ZB system, D1 and D2,3 are the crystal field and spin–orbit splitting energies (D2, ¼ D3 ¼ Dso), Di parameters represent the deformation potentials for the valence band, and exx, eyy, and ezz are the strain tensors (also referred to as e11, e22, and e33 in this book and many other publications). Both forms used by Albrecht et al. [91] and Ren et al. [90] for H, I, and K parameters are given. The former uses the basis and Hamiltonian in the (kx þ iky) form and contains all of the shear strain information. The latter does the phase rotation to get two 3 · 3 and the relevant matrix elements are then functions only of k-transverse and all of the directional phase information has been compiled onto the basis states by a unitary transformation. For presentation of the pseudomorphic strain, either form would be just as easy to use. Another important point is that in the case where the shear strain is nonzero, such as when uniaxial in-plane strain, the D5 and D6 terms must be included as represented by the Albrecht et al. [91] notation above. If only biaxial strain is considered, the shear terms of the strain tensor are zero and thus the D5 and D6 terms vanish. In the latter case, the representation by Ren et al. [90] and others similar to it hold. A good description of pertinent issues is discussed in Ref. [62]. Similar to the cubic system, the 6 · 6 matrix can be block diagonalized into two 3 · 3 matrices, and this can considerably simplify the band structure calculation. It should

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j 2 Electronic Band Structure and Polarization Effects

162

be mentioned that A7 can be assumed nearly zero due to symmetry considerations. Chuang and Chang [58] derived the two 3 · 3 Hamiltonians when A7 is neglected and obtained three doubly degenerate bands. Ren et al. [90] investigated the effect of A7 parameter on the valence band dispersion in wurtzitic crystals such as GaN. In fact, Ren et al. [90] argued that theories forwarded by Chuang and Chang [58] replicates that of Bir and Pikus [72] reported decades earlier. Choosing A7 ¼ 0 reduces the results to that of Chuang and Chang [58] and Sirenko et al. [62]. To underscore the effect of A7 parameter, Ren et al. [90] compiled data from empirical pseudopotential method (EPM) [67]. The same method was also applied to wurtzitic and zinc blende phases of all the three binaries of nitrides by Fritsch et al. [55]. Ren et al. [90] calculated the band structure for GaN for values of A7 ¼ 93.7 meV Å and 0 as, shown in Figure 2.9. Clearly, inclusion of the A7 parameter results in a much better fitting between the kp theory and EPM calculations. From the (a) 0.00

Wz GaN

–0.01

HH

Energy (eV)

–0.02 LH –0.03

–0.04 CH

–0.05 0.0

0.050

0.100

0.15

Wave vector, kx,( k// ) (1 Å–1) Figure 2.9 (a) Valence band structures of wurtzite GaN with the kp theory fitting including the spin–orbit interaction with A7 ¼ 93.7 meVÅ in the solid line. The dash-dotted line is the result of fitting with A7 ¼ 0. The empirical pseudopotential method (EPM) calculation data (o) are from Ref. [67] ([90]). (b) Valence band

structure of wurtzite GaN, using the parameters recommended by Ren et al. Courtesy of I. Vurgaftman and J. Meyer and Ref. [90]. (c) The same using the parameters recommended in Ref. [152]. The dashed lines represent the case for A7 ¼ 0. Courtesy of I. Vurgaftman and J. Meyer and Ref. [90].

2.4 kp Theory and the Quasi-Cubic Model

(b) 0.00

Wz GaN

HH

–0.01

Energy(eV)

LH –0.02

–0.03

–0.04 CH –0.05

–0.06 –0.15

–0.10 k z , (k )

–0.050

0.0

0.050

0.100

0.15

-1 Wave vector (1 Å ) kx, (k//)

(c) 0.00

Wz GaN (c)

HH

–0.01 LH

Energy(eV)

–0.02

–0.03

–0.04 CH –0.05

–0.06 –0.10

–0.15

k z , (k ) Figure 2.9 (Continued )

–0.050

0.0

0.050

Wave vector (1 Å-1)

kx, (k//)

0.100

0.15

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j 2 Electronic Band Structure and Polarization Effects

164

Table 2.7 Fitted splitting energies and Luttinger-like parameters

for the valence band of the wurtzite GaN: Di parameters are in unit of meV, and the Rydberg terms are in units of
h2 =2m0 except for A7, which is in unit of meV Å. D1

D2

D3

A1

A2

A3

A4

A5

A6

A7

21.1

3.61

3.61


7.21


0.440

6.68


3.46


3.40


4.9

93.7

quality of the fit, one can argue that the 93.7 value for the Luttinger-like parameter A7 is a good one. The other parameters giving this fit are listed in Table 2.7. In addition to the cycled parametric values of A parameters by kp theorists, a good extraction of A1–A7 parameters from empirical pseudopotential method band structures of AlN, GaN, and InN can be found in Ref. [55]. Vurgaftman and Meyer [152] followed a nearly identical path to that of Ren et al. [90] but calculated the band structure in both kx and ky crystal directions, as shown in Figure 2.9. The dashed lines, as in the case of the paper by Ren et al. [90], correspond to the case representing the case of A7 ¼ 0. In addition, Vurgaftman and Meyer [152] and Vurgaftman et al. [92] mentioned the calculation of the valence band structure for GaN using parameters representing the properties of GaN that they prefer, the results of which are also shown in Figure 2.9 for comparison. The effective masses are calculated using the parallel and perpendicular hole masses m//and m? that can be expressed in terms of their dependence on the Luttinger-like parameters Ai as follows: ==

m0 =mhh ¼
ðA1 þ A3 Þ; ==

m0 =mlh ¼
ðA1 þ A3 Þ; == m0 =mso

ð2:30Þ

¼
A1 :

?

Here, m represents the mass in the (kx, ky) plane, which means kz ¼ 0 and m0 =m? hh ¼
ðA2 þ A4
A5 Þ; 2 m0 =m? lh ¼
ðA2 þ A4
A5 Þ
2A7 =jD1 j ;

m0 =mh? so

¼

ð2:31Þ


A2 þ 2A27 =jD1 j;

where m//is along the kz-direction (kx ¼ ky ¼ 0) and m? in the (kx, ky) plane, which means kz ¼ 0. The effective masses are calculated using the parallel and perpendicular hole masses m//and m? together with Luttinger-like parameters using Equations 2.30 and 2.31. As indicated in the schematic of Figure 2.5, both the spin–orbit and the crystal field splitting affect the structure of the valence band in wurtzitic crystals [93]. Typically, the relevant parameters are correlated to one another as Dso ¼ 3D2 ¼ 3D3, in spite of the fact that a small D2/D3 anisotropy has sometimes been reported [94,95] and Dcr ¼ D1. Experimentally, the splitting parameters are obtained from the energy differences of the A, B, and C free excitons, which have nonlinear dependencies on the various splittings [96]. It should be pointed out that the nomenclature for the three valence

2.4 kp Theory and the Quasi-Cubic Model

bands for hexagonal system is A, B, and C for HH, LH, and SO (CH) bands when including A7 terms, because spin splitting and strain can significantly alter as to which band of eigenstates is “heavy” or “light” at various k-values, particularly in the c-plane. An early experimental undertaking by Dingle et al. [97] led to Dcr ¼ 22 meV and Dso ¼ 11 meV. An analysis by Gil et al. [98] yielded Dcr ¼ 10 meV and Dso ¼ 18 meV. Chuang and Chang [58] attempting to rederive these parameter from the same data but with what was termed as a more precise description of the effect of strain on the valence band edge energies arrived at values of Dcr ¼ 16 meV and Dso ¼ 12 meV. Reynolds et al. [99] obtained Dcr ¼ 25 meV and Dso ¼ 17 meV from a fit to exciton energies, with A and B determined by photoluminescence (PL) and C determined by reflection but with a geometry not fully ideal in terms in that some error is introduced in the value of C exciton energy. Again, using exciton energies values of Dcr ¼ 22 meV and Dso ¼ 15 meV were obtained by Shikanai et al. [100], Dcr ¼ 37.5 meV and Dso ¼ 12 meV by Chen et al. [101], Dcr ¼ 9 meV and Dso ¼ 20 meV by Korona et al. [102], and Dcr ¼ 9–13 meV and Dso ¼ 17–18 meV by Campo et al. [103] and Julier et al. [104]. The values of Dcr ¼ 10 meV and Dso ¼ 17 meV were determined by both Edwards et al. [105] and Yamaguchi et al. [96]. Noticeable is one of the smallest reported crystal field splittings to date, Dcr ¼ 9 meV, along with Dso ffi 18 meV reported by Rodina et al. [106] based on detailed experimental investigation. On the theoretical side, an ab initio calculation by Wei and Zunger [107] overestimates the crystal field splitting Dcr ¼ 42 meV, but arrives at a Dso ¼ 13 meV that agrees well with the experimental data. Suzuki et al. [56] reported Dcr ¼ 40 meV and Dso ¼ 8 meV or Dso ¼ 3D2 ¼ 3D3 1.16 mRy and Dcr ¼ D1 ¼ 5.36 mRy for these splittings. Many firstprinciples calculations focusing on the valence band splitting are available in the literature [58,90,67,154,108]. The experimental data, however, appear to converge on the splittings Dcr ffi 10 meV and Dso ffi 17 meV, as suggested by Vurgaftman and Meyer and tabulated in Table 2.8 [152]. The spin splitting of the valence band of wurtzitic GaN can be determined via the A7 parameter, as shown in Figure 2.9a, which is derived by Vurgaftman and Meyer [152] assuming the parameters of Ren et al. [90]. On the contrary, Figure 2.9b is derived by Vurgaftman and Meyer [152] using A parameters from Ren et al. [90] combined with what is believed to be the more representative spin–orbit and crystal field splittings. Modification of the A parameters alone, but with the corrected values of the splitting energies, does not allow the recovery of the band structure resembling that shown in Figure 2.9a. This simply implies that the field is not yet settled on a set of reliable parameters and more refinement is needed for the most appropriate values of the A parameters to be arrived. To be sure six distinct valence band deformation potentials, as well as the strain tensor and the overall hydrostatic deformation potential, are necessary to describe the band structure of GaN under strain. In the cubic approximation, these can be expressed in terms of the more familiar av, b, and d potentials [93]. In terms of the calculation and combined calculation and measurement efforts, Christensen and Gorczyca [31] reported a hydrostatic deformation potential a ¼
7.8 eV, which has been shown to agree well with
8.16 eV obtained by Gil et al. [98]. On the contrary, a somewhat lower value of a ¼
6.9 eV was attained through an ab initio calculation by

j165

j 2 Electronic Band Structure and Polarization Effects

166

Table 2.8 Recommended band structure parameters for wurtzitic GaN from Ref. [152].

Parameter

Value

Parameter

Value

Parameter

Value

Eg (eV, low temperature) a (meV K
1) b (K)

3.510

A1


7.21

D1 (eV)


3.7

0.909 (1 in [119]) 830 (1100 in [119]) 10 17 0.20 0.20
4.9
11.3

A2 A3


0.44 6.68

D2 (eV) D3 (eV)

A4 A5 A6 A7 (meV Å) d13 (pm V
1) d33 (pm V
1) d15 (pm V
1) Psp (C m
2)


3.46
3.40
4.90 93.7
1.6b 3.1b 3.1b
0.034

D4 (eV) D5 (eV) D6 (eV) c11 (GPa) c12 (GPa) c13 (GPa) c33 (GPa) c44 (GPa)

Dcr (meV) Dso (meV) == me =m0 m? e =m0 a1 (eV) a2 (eV)

4.5 8.2
4.1
4.0
5.5 390 145 106 398 105

367a 135a 103a 405a 95a

See Tables 2.27 and 2.28 for details related to the elastic constants, piezoelectric constants, and spontaneous polarization charge. Any dispersion among the tables is a reflection of the uncertainty in the available parameters. See Volume 2, Chapter 5 for an extended discussion of Varshni parameters. a The second column figures for the Cii parameters are from Table 2.28 where a more expanded list of elastic coefficients is given. b Table 2.28 provides additional data on d-parameters.

Kim et al. [109]. Noting that the hydrostatic potential is anisotropic, owing to the reduced symmetry of the wurtzite crystal, Wagner and Bechstedt [85] calculated values of
4.09 and
8.87 eV for the two hydrostatic interband deformation potentials. On the transport side, fits to the experimental mobility data [110,111] yielded a conduction band deformation potential approaching
9 eV, a topic discussed in some detail in Volume 2, Chapter 3. Using pressure-dependent optical transition energies with pressure, Shan et al. [112] reported a1 ffi
6.5 eV and a2 ffi
11.8 eV and uniaxial deformation potentials b1 ffi
5.3 eV and b2 ffi 2.7 eV for the two hydrostatic interband components. Employing photoreflectance measurements on compressively strained M-plane GaN films (grown along the h1 0
1 0i direction) grown on g-LiAlO2 (1 0 0), Ghosh et al. [113] reported a1 ¼
3.1 eV with a2 ¼
11.2 eV. Again, using M-planeoriented GaN layers, Gil and Alemu [114] obtained a1 ¼
5.22 eV with a2 ¼
10.8 eV. Vurgaftman and Meyer [152] recommend a set of a1 ¼
4.9 eV and a2 ¼
11.3 eV, which represents an average of all the measured values. Numerous sets of valence band deformation potentials have been derived from both first-principles calculations [58,60,61,115,116] and fits to experimental data [94,98,100,96,112–114,117]. The dispersion among the reported data is unacceptable, which calls for further work to resolve the discrepancies and converge on accurate parameters. If one were to average the deformation potentials that are most widely quoted, values of D1 ¼
3.7 eV, D2 ¼ 4.5 eV, D3 ¼ 8.2 eV, D4 ¼
4.1 eV, D5 ¼
4.0 eV, and D6 ¼
5.5 eV, which satisfy the quasi-cubic approximation that is about to be discussed [58], are obtained. The deformation potential values for GaN are

2.5 Quasi-Cubic Approximation

tabulated in Table 2.8, and their effect on optical transitions are discussed in detail in Volume 2, Chapter 5.

2.5 Quasi-Cubic Approximation

The genesis of the quasi-cubic approximation relies on the fact that the Wz and ZB structures are both tetrahedrally coordinated and hence are closely related. The nearest neighbor coordination is the same for Wz and ZB structures but differs at the next nearest neighbor positions. The basal plane (0 0 0 1) of the Wz structure corresponds to one of the (1 1 1) planes of the ZB. When the in-plane hexagons are lined up in Wz and ZB structures, the Wz [0 0 0 1], ½1 1
2 0, and ½1
1 0 0 planes are

parallel to the ZB [1 1 1], ½1 0 1, and ½1 2 1 planes, respectively. This, in turn, leads to correlations between the symmetry direction and the k-points for the two polytypes. There are, however, twice as many atoms in the Wz unit cell as there are in the ZB one. In addition to the band structure similarities between the doubled ZB and Wz structures, one can establish a correlation between the Luttinger parameters in the ZB system and parameters of interest in the Wz system by taking the z-axis along the [1 1 1] direction and the x- and y-axes along the ½1 1
2 and ½
1 1 0 directions. For details regarding the symmetry relations between the ZB and Wz polytypes, refer to Refs [32,118]. Doing so leads to D2 ¼ D3 ; A1 ¼ A2 þ 2A4 ;pffiffiffi A3 ¼
2A4 ¼ 2A6
4A5 ; A7 ¼ 0; D1 ¼ D2 þ 2D4 ;pffiffiffi D3 ¼
2D4 ¼ 2D6
4D5 :

ð2:32Þ

The A parameters can be related to the classical Luttinger parameters gi through A1 A2 A3 A4 A5 A6

¼
ðg 1 þ 4g 3 Þ; ¼
ðg 1
2g 3 Þ; ¼ 6g 3 ; ¼
2g 3 ; ¼
p ðg ffiffi2ffi þ 2g 3 Þ; ¼
2ð2g 2 þ g 3 Þ:

ð2:33Þ

The calculated values of the spin–orbit and crystal field splitting parameters with those deduced from the observation of A, B, and C excitons are listed in Tables 2.8 and 2.9, which will be presented shortly. The calculations agree well in terms of the spin–orbit splitting, but the theoretical crystal field splitting is much too large compared to experimental data. The discrepancy may be due to the unaccounted residual strain and strain inhomogeneities present in GaN films, as well as the inaccuracy of the parameter values. The debate will probably continue until strainfree or homogeneously strained films can be prepared.

j167

j 2 Electronic Band Structure and Polarization Effects

168

Table 2.9 Effective masses and band parameters for wurtzitic GaN.

Parameter Aniso ==

me m? e == mhh == mlh == mch m? hh m? lh m? ch A1 A2 A3 A4 A5 A6 A7 D1

0.138 0.151 2.000 2.000 0.130 2.255 0.191 0.567
7.692
0.575 7.192
2.855
2.986
3.360 0.160 0.043

Iso

A

B

0.138 0.151 2.007 2.007 0.130 2.249 0.261 0.317
7.698
0.600 7.200
2.816
2.971
3.312 0.171 0.023

0.20 0.18 1.10 1.10 0.15 1.65 0.15 1.10
6.56
0.91 5.65
2.83
3.13
4.86

0.20 0.18 1.76 1.76 0.16 1.61 0.14 1.04
6.27
0.96 5.70
2.84
3.18

a

0.039

C

0.14 0.15 1.479 1.479 0.130 1.592 0.299 0.252
7.706
0.597 7.030
3.076
3.045
4.000 a 0.194 0.038 0.022

D

E

0.14 0.15 1.453 1.453 0.125 1.595 0.236 0.289
7.979
0.581 7.291
3.289
3.243
4.281 0.179 0.022

0.19 0.17 1.76 1.76 0.14 1.69 0.14 1.76
7.14
0.57 6.57
3.30
3.28

F

0.19 0.17 1.96 1.96 0.14 1.87 0.14 1.96
7.24
0.51 6.73
3.36
3.35
4.72 0 0 0.021 0.021

G

H

I

0.19 0.19 0.23 0.19 1.89 2.00 1.96 1.89 2.00 1.96 0.12 0.16 0.16 2.00 2.04 1.20 0.15 0.18 0.16 0.59 1.49 1.96
7.21
6.4
6.36
0.44
0.50
0.51 6.68 5.9 5.85
3.46
2.55
2.92
3.40 2.56 2.60
4.9 3.06 3.21 0.094 0.108 0 0.021 0.036

Effective masses in units of free-electron mass m0, Luttinger-like parameters Ai (i ¼ 1, . . ., 6) in units of h2 =2m0 , and A7 in units of eV Å . The crystal field splitting energy D1 is given in units of meV. The term
aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric == tensor. Here, me and m? e represent the effective electron masses along and perpendicular to the c-axis [55]. Anisotropically screened and isotropically screened values are from Ref. [55]. A: FP-LAPW band structure calculations are from Ref. [56], and effective mass parameters are obtained through a 3D fitting procedure within cubic approximation; B: FP-LAPW band structure calculations are from Ref. [56], and effective mass parameters are obtained by direct line fit; C: Ai from Ref. [153] obtained through a Monte Carlo fitting procedure to the band structure and effective masses calculated using Equations 2.30 and 2.31; D: direct kp calculations for Ai from Ref. [153] and effective masses calculated using Equations 2.30 and 2.31; E: effective masses and Ai from Ref. [67] obtained through a line fit to the band structure; F: direct kp calculation in a 3D fit from Ref. [67]; G: Ai obtained through a direct fit from Ref. [90] and effective masses calculated using Equations 2.30 and 2.31; H: direct fit of Ai to firstprinciples band structure calculations from Ref. [154]; I: Ai and effective masses obtained in the quasicubic model from zinc blende parameters from Ref. [154]. a A7 in the range of 0.136 eV Å has been set to zero.

The parameters mentioned in bandgap-related discussion for wurtzitic GaN are tabulated in Table 2.8 [152] for the wurtzitic phase GaN. All conventional nitridesinthe wurtzite phase exhibit a directenergy gap, and the next satellite conduction valley, which is the M valley, is some 2 [31]to 5 eV [56]higher than the G valley. In addition to the one made available here, the wurtzite indirect gap related issues have been amply discussed in the literature, some of which can be found in Refs [31,53,75,120–122]. Sufficing it to state the bottom of the conduction band in GaN can be well approximated by a parabolic dispersion relation, although a slight anisotropy is expected due to the reduced lattice symmetry [56]. In many devices, the pertinent property of the band structure is the region near the bottom of the conduction band, which can be represented to a great deal with effective mass. In relatively early

2.6 Temperature Dependence of Wurtzite GaN Bandgap

experimental studies in GaN grown by hydride VPE, Barker and Ilegems [123] obtained an electron effective mass of mn ¼ 0.20m0 from reflectivity measurements. Again early on, Rheinlander and Neumann [124] inferred 0.24–0.29m0 for the effective mass from a Faraday-rotation investigation of heavily n-doped GaN. Using heavily doped samples, which was the norm then, and fits to the thermoelectric power, Sidorov et al. [125] obtainedelectroneffective masses of0.1–0.28m0, dependingon whatprimary scattering channel was assumed. For a review of early investigations of these and other properties, thereaderisreferredtoreviewsfromthe1970s,suchastheonebyPankoveetal.[126]and that by Kesamanly [127]. Congruent with the increased activity in GaN, fuelled by the device demonstrations, particularly LEDs and later on lasers, a substantial body of work has since produced more precise estimations of the electron mass. Among them are the works by Meyer et al. [128] and Witowski et al. [129] who obtained masses of 0.236m0 and 0.222m0, respectively, utilizing shallow donor transition energies; the latter is with the smallest error bars quoted in the literature (0.2%). Underscoring the importance of the polaron correction, which is about 8% in GaN and comes about because of the strong polar nature of GaN, Drechsler et al. [130] derived a bare mass of 0.20m0 from cyclotron resonance data. A similar result was obtained by Perlin et al. [131] using infrared reflectivityandHalleffectmeasurements,whichalsoledtoananisotropyoflessthan1%. For comparison, a slightly larger dressed mass of 0.23m0 has been obtained by Wang et al. [132] and Knap et al. [133]. A small downward correction may be necessary in the former,astheelectronswereconfinedataninterface.Thelatterauthors,however,appear to have corrected for that effect. Using n-type bulk GaN, which does not require an appreciable correction that is needed in confined systems and employing infrared ellipsometry measurements, Kasic et al. [134] reported slightly anisotropic electron masses of 0.237 0.006m0 and 0.228 0.008m0 along the two axes. Again, using modulation-doped structures, a series of authors, Elhamri et al. [135], Saxler et al. [136], Wong et al. [137], Wang et al. [138], and Hang et al. [139], also reported on the effective mass, the values of masses for which ranged from 0.18m0 to 0.23m0 from Shubnikov–de Haas data. Elhamri et al. [135] suggested that strain effects, which are somewhat difficult to be certain of, could have compromised somewhat the masses reported in some of these reports. A value of 0.20m0 is very commonly used for the bare electron effective mass and 0.22m0 for the experimentally relevant dressed mass. A more in-depth discussion of the cyclotron and Shubnikov–de Haas measurements can be found in Volume 2, Chapter 3. This bare mass figure of 0.20m0 agrees reasonably well with a number of estimates based on theory, as outlined in a list in Ref. [140]. Owing to the large uncertainty, no attempt is made to specify an F parameter for wurtzite GaN. However, the interband matrix element may be obtained from the relation between the electron mass and the relevant zone center energies [92].

2.6 Temperature Dependence of Wurtzite GaN Bandgap

The temperature dependence of the bandgap in semiconductors is often described by an imperial expression (assuming no localization)

j169

j 2 Electronic Band Structure and Polarization Effects

170

EðTÞ ¼ Eð0Þ
aT 2 =ðb þ TÞ:

ð2:34Þ

In the case of localization, which can also be construed as band tail effect, the temperature dependence deviates from the above equation. In the framework of the band tail model and Gaussian-like distribution of the density of states for the conduction and valence band, the temperature-dependent emission energy could be described by the following modified expression [141], which is based on a model developed for Stokes shift in GaAs/AlGaAs quantum wells [142]. EðTÞ ¼ Eð0Þ
½aT 2 =ðb þ TÞ
½s2 =ðkTÞ;

ð2:35Þ

where the last term represents the localization component with s indicating the extent of localization or band tailing, which is nearly imperative for In-containing alloys. The values of the parameters a (in units of energy over temperature) and b (in units of temperature), for wurtzitic GaN, are listed in Volume 2, Table 5.1. Although a detailed discussion of these parameters in very high-quality samples is deferred to Volume 2, Chapter 5, the evolution of them is discussed here to give the reader a flavor that when the sample quality is under question, the fits to experiments could lead to varying if not erroneous parameters. In concert with this approach, the spread in the values of a and b for A exciton is also discussed in the text surrounding Volume 2, Table 5.3. Varshni parameters have been deduced from the measured variation of the A, B, and C excitonic energies with temperature early on by Monemar [143] with Varshni parameters of a ¼ 5.08 · 10
4 meV T
1 and b ¼
996 T, the sign for the latter of which is contradictory to the agreed upon values deduced from high-quality samples. In chronological order of the reports, using optical absorption measurements on bulk single crystals and also epitaxial layers grown on sapphire, Teisseyre et al. [144] reported a ¼ 0.939–1.08 meV K
1 and b ¼ 745–772 K; note the positive sign of b. Shan et al. [145] reported a ¼ 0.832 meV K
1 and b ¼ 836 K deduced from the temperature variation of the A exciton resonance. Petalas et al. [146] determined the Varshni parameters to be a ¼ 0.858 meV K
1 and b ¼ 700 K using spectroscopic ellipsometry. Relying on PL measurements, Salvador et al. [147] obtained a ¼ 0.732 meV K
1 and b ¼ 700 K. Using absorption measurements, Manasreh [148] reported a ¼ 0.566–1.156 meV K
1 and b ¼ 738–1187 K on samples grown by MBE and OMVPE. Using a variation of electroreflectance, the contactless electroreflectance, Li et al. [149] led to a ¼ 1.28 meV K
1 and b ¼ 1190 K for the A exciton transition energy. Utilizing PL spectra of excitonic transitions, Zubrilov et al. [150] reported values of a ¼ 0.74 meV K
1 and b ¼ 600 K based on exciton luminescence spectra. PL data of free and bound excitons were fitted by Reynolds et al. [151] to a modified Varshni-like form that resulted in a ¼ 0.5 meV K
1 and b ¼ 1060. Some of the dispersion in the reported values can be attributed to the difficulty in identifying and resolving various excitonic transitions. Vurgaftman and Meyer [152], averaging what they term as more credible results, recommend a ¼ 0.909 meV K
1 and b ¼ 830 K. A detailed discussion of the temperature dependence of the bandgap of GaN along with the Varshni parameters for all three excitons (A, B, and C) are listed in Volume 2, Table 5.3, where A exciton related parameters are a ¼ 1 meV K
1 and b ¼ 1100 K.

2.6 Temperature Dependence of Wurtzite GaN Bandgap

The complete Ai parameters calculated by Fritsch et al. [55] as well as others deduced from alternative methods are tabulated in Table 2.9. In addition, a compilation of the dispersion in the effective mass for both the conduction band and various valence bands as obtained by various computational methods as well as parameters used in the description of the bandgap for wurtzitic GaN, particularly, in the context of empirical pseudopotential method, as described in Ref. [55], are also included in Table 2.9. Suzuki and Uenoyama [59] have determined the deformation potentials by the fullpotential linearized augmented plane wave (FLAPW) calculations. The values recommended for GaN by Vurgaftman and Meyer [152] are tabulated in Table 2.8. In the calculations of Suzuki and Uenoyama [59], biaxial and uniaxial strains have been introduced and reduced shifts in the G point energy to which a linear fit in terms of strain was obtained. From the linearfit,the deformation potential values for thevalence band were deduced. The figures obtained from the quasi-cubic approximation are listed too. The good agreement between the calculated values and those determined from the quasi-cubic model is strikingly obvious, which is indicative of the excellence of the quasi-cubic approximation. Assuming an approximately spherical potential in the neighborhood of the N atoms, of the two spin states, the higher energy is in the one in which the electron spin and the orbital angular momentum are parallel. This result is also anticipated on the basis of the atomic spin–orbit splitting in which the P1/2 state is known to have energy higher than the P3/2 state. The contributions of spin–orbit interaction and the crystal field perturbation to the experimentally observed splittings (E1,2 and E2,3) have been calculated with different linear combination of atomic orbitals (LCAO) approximations [155,156]. The large effective mass and the small dielectric constant of GaN, relative to more conventional group III–V semiconductors, lead to relatively large exciton binding energies and make excitons, together with large exciton recombination rates, clearly observable even at room temperature. The bottom of the conduction band of GaN is predominantly formed from the s levels of Ga, and the upper valence band states from the p levels of N. Even though sophisticated methods have been introduced and discussed on these pages, the method of Hopfield and Thomas [157], which treats the wurtzite energy levels as a perturbation to the zinc blende structure is discussed briefly, as it provides a physical picture of band splitting in the valence band. Using the quasi-cubic model of Hopfield [98], one obtains E 1 ¼ 0; v" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # u   dþD u dþD 2 2 t E2 ¼
dD ; þ 2 2 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v" # u  2 dþD u d þ D 2
dD ;
t E3 ¼ 2 2 3

ð2:36Þ

ð2:37Þ

ð2:38Þ

j171

j 2 Electronic Band Structure and Polarization Effects

172

where Dcr and Dso represent the contributions of uniaxial field and spin–orbit interactions, respectively, to the splittings E1,2 and E2,3.

2.7 Sphalerite (Zinc blende) GaN

The zinc blende GaN crystal consists of two interpenetrating face-centered cubic lattices, one having a group III element atom, for example, Ga, and the other a group V element atom, for example, N. The matrix element of the momentum operator between the conduction and valence bands has been expressed by Kane [92,158] in terms of a single parameter P whose value is termed as EP, which is an important parameter. The other parameter of importance is the F parameter, again defined by Kane, which comes about from a second-order perturbation theory and takes into account the higher order band contributions to the conduction band. Their importance aside [92] EP and F are inherently difficult to determine accurately, due to the fact that the remote band effects can be calculated but are not measurable quantities. One experimental technique relies on measuring the effective g factor, which is not as influenced by remote bands as the effective mass. A number of experimental and theoretical studies have determined energy gaps for zinc blende GaN [69,70,146,159–163]. The term “gaps” is used, as there seems to be a dispersion in the reported values. Typically, the excitonic transitions [164–166] observed in low-temperature PL is used to infer the bandgap, provided the exciton binding energy is known, which in this case is 26.5 meV. Although, low-temperature bandgaps ranging from 3.2 to 3.5 eV have been measured, most of them tend to be between 3.29 and 3.35 eV. It is therefore reasonable to use a low-temperature bandgap of 3.3 eV for zinc blende GaN. A reasonable figure for the room-temperature fundamental bandgap is 3.2 eV, although the range of 3.2–3.3 eV stated in Chapter 1 remains. As in the case of wurtzitic GaN, the temperature dependence of the energy gap was also studied for zinc blende GaN, examples of which can be found in the works of Petalas et al. [146] and Ramirez-Flores et al. [167]. Both group of authors found b ¼ 600 K (using the more reliable model 1 in Ref. [146]), but the a parameters differed somewhat. Not having any real basis for selecting one or the other, the average value of 0.593 meV K
1 is considered the default value. Although the indirect gap energies have not been measured, for a calculation by Fan et al. [69], the X-valley and L-valley minima had been put at 1.19 and 2.26 eV above the G valley, respectively. These compare with an earlier prediction by Suzuki et al. [56] of about 1.4 and 1.6 eV, respectively, as shown in Figure 2.6c. Ramirez-Flores et al. [167] measured the spin–orbit splitting in zinc blende GaN to be 17 meV. Electron spin resonance measurements on zinc blende GaN determined an electron effective mass of 0.15m0 [168], which may represent the only experimental results, and the value is similar to the G-valley masses derived from first-principles calculations by Chow et al. [169] and Fan et al. [69]. Effective masses of ml ¼ 0:5m0 and mt ¼ 0:3m0 have been calculated for the X valleys in GaN [70], which are similar to the theoretical results of Fan et al. [69].

2.7 Sphalerite (Zinc blende) GaN

0.00

HH

ZB GaN

LH

–0.01

Energy (eV)

j173

–0.02

SO

–0.03

–0.04

–0.05

–0.06 –0.15

–0.10 [1 1 1]

–0.050

0.0

0.050

Wave vector (1 Å-1)

0.100 [0 0 1]

Figure 2.10 Valence band structure of zinc blende GaN [152].

The valence band of zinc blende GaN has been the topic of various theoretical efforts, and the E–k diagram by Vurgaftman and Meyer [152] is show in Figure 2.10. Although the hole effective masses in zinc blende GaN have apparently not been measured, a number of theoretical predictions of Luttinger parameters are available in the literature [69,70,164–166,168,169,425,170,171]. Once the Luttinger parameters are known, the full picture in terms of the hole effective masses can be determined. First, it should be pointed out that in polar semiconductors such as the III–Vcompounds in general and GaN in particular, it is the nonresonant polaron [172] mass that is actually measured. The polaron mass exceeds the bare electron mass by about 1–2%, the exact value of which depends on the strength of the electron–phonon interaction. Because the band structure is governed by the bare electron mass, this is the quantity that is typically reported whenever available. At the valence band edge, the heavy hole (hh) effective masses in the different crystallographic directions are related to the free mass by the Luttinger parameters in the following manner [92]: mzhh ¼

m0 2m0 m0 ½110 ½111 ; mlh ¼ ; mlh ¼ : g 1
2g 2 2g 1
g 2
3g 3 g 1
2g 3

ð2:39Þ

0.15

j 2 Electronic Band Structure and Polarization Effects

174

Here, the z-direction is perpendicular to the growth plane of (0 0 1). These expressions described by Equation 2.39 show the relationship of the Luttinger parameters to the hh effective masses that can typically be measured in a more direct manner. The light hole (lh) and so hole effective masses are given by mzlh ¼

m0 2m0 m0 ½110 ½111 ; mlh ¼ ; mlh ¼ ; g 1 þ 2g 2 2g 1 þ g 2 þ 3g 3 g 1 þ 2g 3

m0 E P Dso ¼ g1
: mso 3E g ðE g þ Dso Þ

ð2:40Þ

ð2:41Þ

Equation 2.41, which relates the split-off hole mass to the Luttinger parameters, should in principle contain an additional parameter to account for the effects of remote bands that is analogous to the F parameter [92], but the remote bands are not necessarily the same ones that cause the largest correction to the electron mass. Due to the dominance of the wurtzitic GaN, insufficient effort and thus data exist for zinc blende GaN, which is also true for even the well-investigated III–V materials to describe the effect of the interaction with remote bands on the split-off hole mass quantitatively. To restate, although the hole effective masses in zinc blende GaN have apparently not been measured, a number of theoretical predictions of Luttinger parameters are available in the literature. The values are based on averages of the heavy-hole and lighthole masses along [0 0 1], as well as the degree of anisotropy in g3–g2. Doing so leads to the parameter set as g1 ¼ 2.70, g2 ¼ 0.76, and g3 ¼ 1.11. Similarly, averaging all the reported split-off masses [69,70,163,171,173] leads to mso ¼ 0:29m0 . In its simplest form, the Luttinger parameters can be used to quickly determine the effective masses in various valence bands both in equilibrium and also under biaxial strain. In fact, with biaxial strain, the valence band degeneracy can be removed, and most strikingly the heavy-hole in-plane mass can be made smaller by compressing strain, a notion that has been exploited in the InGaAs/GaAs system very successfully. An average of the two theoretical values for EP in zinc blende GaN [163,173] yields EP ¼ 25.0 eV, which in turn implies F ¼
0.95. Caution is advised because these values have not been verified experimentally. In conjunction with calculations of the electronic band structure of binary nitrides and specifically effective masses in the valence band, Fritsch et al. [55] also arrive at the Luttinger-like kp parameters by empirical fits for the effective masses at the G point. These Luttinger-like parameters for the valence band of zinc blende GaN are listed in Table 2.10. Those for zinc blende AlN and InN will be given in Sections 2.8.2 Sections 2.9.2. The Ai parameters transformed from the Luttinger parameters obtained with the help of the quasi-cubic approximation can be found in Tables 2.8 and 2.9. It has been argued that the Ai parameters calculated in this manner are in good agreement with the calculated values that have been used to support the value and validity of the quasi-cubic approximation, which greatly simplifies the calculations. Fritsch et al. [55] obtained the valence band effective masses for zinc blende binaries by solving the eigenvalues of the kp matrix while taking the spin–orbit interaction into account. The effective masses so calculated for the conduction and valence bands, the latter involving the light and heavy holes, as well as the spin–orbit

2.7 Sphalerite (Zinc blende) GaN Table 2.10 Luttinger parameters g1, g2, and g3 for zinc blende GaN obtained from a fit along the [1 1 0] direction along with those available in the literature, as compiled in Ref. [55].

Parameter

A

B

C

D

E

c1 c2 c3

2.89 0.85 1.20

2.96 0.90 1.20

2.70 0.76 1.07

3.07 0.86 1.26

2.67 0.75 1.10

A: empirical pseudopotential calculation by Fritsch et al. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: first-principles band structure calculations from Ref. [170]; D: empirical pseudopotential calculation from Ref. [174]; E: recommended values taken from Ref. [152].

split-off mass, and the anisotropy taken into account are listed in Table 2.11 for zinc blende GaN. The data contain those obtained by full potential linearized plane waves (FP-LAPW), empirical pseudopotential method (EPM) calculations and those calculated with Luttinger parameters employing ðm0 =mhh=lh Þ½100 ¼ g 1 2g 2 ; ðm0 =mhh=lh Þ½111 ¼ g 1 2g 3 ; 2g g 2 g 3 ðm0 =mhh=lh Þ½1100 ¼ 1 ; 2

ð2:42Þ

where subscripts hh and lh represent the heavy-hole and light-hole effective masses, respectively. The spin–orbit split-off-hole effective mass, mso, is isotropic in all the three directions and is given by ðm0 =mso Þ ¼ g 1 :

ð2:43Þ

Table 2.11 Effective masses for electrons (e), heavy holes (hh), light holes (lh), and spin–orbit split-off (so) holes in units of the free-electron mass m0 along the [1 0 0], [1 1 1], and [1 1 0] directions for zinc blende GaN. [100]

[100]

[111]

[111]

[110]

[110]

Reference

me

mhh

mlh

mhh

mlh

mhh

mlh

mso

A B C D E F

0.14 0.14 0.13 0.17 0.15 0.12

0.84 0.86 0.76 0.85 0.85 1.34

0.22 0.21 0.21 0.24 0.24 0.70

2.07 2.09 1.93 1.79 2.13 1.06

0.19 0.19 0.18 0.21 0.21 0.63

1.52 1.65 1.51 1.40 1.55 1.44

0.20 0.19 0.19 0.21 0.21 0.58

0.35 0.30 0.32 0.37 0.29 0.20

Compiled by Fritsch et al. [55]. A: after Ref. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: empirical pseudopotential calculation from Ref. [69]; D: calculated from Luttinger parameters from Ref. [170], using Equations 2.42 and 2.43; E: calculated from recommended Luttinger parameters from Ref. [152], using Equations 2.42 and 2.43; F: empirical pseudopotential calculation from Ref. [70].

j175

j 2 Electronic Band Structure and Polarization Effects

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Because various calculations place the hydrostatic deformation potential for zinc blende GaN in the range
6.4 to
8.5 eV [31,69,70,104,169,175], for the lack of a better choice, the average value of the hydrostatic deformation potential a ¼
7.4 eV is chosen. It should be mentioned that due to the nature of the atomic bonding in III–V materials, the bandgap increases for a compressive strain. Under positive hydrostatic pressure that produces negative strain because the lattice constant gets smaller, the change in the bandgap corresponding to a change in volume, DV/V ¼ (exx þ exx þ ezz), given by DEg ¼ a(exx þ exx þ ezz) is positive, which necessitates the sign of the deformation potential to be negative. The deformation potential is the sum of the conduction and valence band deformation potentials, a ¼ ac þ av. Under pressure, the conduction band edge is believed to move upward in energy while the valence band moves downward, with most of the change being in the conduction band edge. Similar to the total bandgap change, the change in valence band under pressure can be described as DEg ¼ av(exx þ exx þ ezz). Naturally, it is much easier to measure the change in the total bandgap due to strain than its component effects on the conduction and valence bands. The valence band deformation potential value reported by Wei and Zunger [176], namely, av ¼
0.69 eV, is suggested. The sign convention for av is adopted from Vurgaftman et al. [92]. To restate, the shrinking volume and negative strain cause the valence band to move down. The values reported in the literature are in the range of
0.69 to
13.6 eV. The suggested value is consistent with the expectation that most of the strain shift should occur in the conduction band. As for the shear deformation potentials b and d, the same is applied, which yields a suggestion that b ¼
2.0 eV with the full range of reported values being
1.6 to
3.6 eV. The recommendation d ¼
3.7 eV is an average of the published results from Ohtoshi et al. [175], Van de Walle and Neugebauer [177], and Binggeli et al. [178]. No experimental confirmations of any of these deformation potentials for zinc blende GaN appear to exist. Turning to elastic constants, the values of C11 ¼ 293 GPa, C12 ¼ 159 GPa, and C44 ¼ 155 GPa have been taken from the theoretical calculations of Wright [179]. Very similar sets have also been calculated by Kim et al. [109,180] and Bechstedt et al. [181]. For more details of elastic constants for all three binaries, refer to Tables 2.25–2.27 and 2.28 that will follow later on in this chapter. The parameters in conjunction with the band structure for zinc blende GaN are compiled in Table 2.12.

2.8 AlN

AlN forms the larger bandgap binary used in conjunction with GaN for increasing the bandgap for heterostructures. As in the case of GaN, AlN also has wurtzitic and zinc blende polytypes, the latter being very unstable and hard to synthesize. Owing to increasing interest in solar blind devices and expectations that larger AlGaN with large mole fractions of AlN would have large breakdown properties, this material has been steadily gaining interest. It should also be mentioned that the N overpressure on Al is the smallest among those over Ga and In, paving the way for equilibrium growth of AlN bulk crystals, albeit not without O contamination.

2.8 AlN Table 2.12 Parameters associated with the band structure for zinc blende GaN.

Parameter alc (Å) at T ¼ 300 K E Gg (eV) a (G) (meV K
1) b (G) (K) E Xg ðeVÞ a (X) (K) b (X) (meV K
1) E Lg ðeVÞ a (L) (meV K
1) b (L) (K)

Value 4.50 3.3 0.593 600 4.52 0.593 600 5.59 0.593 600

Parameter

Value

Parameter

Value

Dso (eV) me ðGÞ ml ðXÞ mt ðXÞ c1 c2 c3 mso

0.017 0.15 0.5 0.3 2.70 0.76 1.11 0.29

EP (eV) F VBO (eV) ac (eV) av (eV) b (eV) d (eV) c11 (GPa) c12 (GPa) c44 (GPa)

25.0
0.95
2.64
6.71
0.69
2.0
3.7 293 159 155

Bandgaps are for low temperature [152].

2.8.1 Wurtzite AlN

Wurtzite AlN is a direct bandgap semiconductor with a bandgap near 6.1 eV and still considered to be semiconductor. The zinc blende polytype is not stable with a predicted indirect bandgap, as will be discussed in the next section. The AlN derives its technological importance from providing the large bandgap binary component of the AlGaN alloy, which is commonly employed both in optoelectronic and electronic devices based on the GaN semiconductor system. Early on, the absorption measurements carried out by Yim et al. [182] and later by Perry and Rutz [183] indicated a large energy gap for wurtzite AlN of 6.28 eVat 5 K to 6.2 eVat room temperature. The actual figures are converging at values about 0.1 eV below those contained in those early reports. As for the dependence of the bandgap on temperature, the Varshni parameters of a ¼ 1.799 meV K
1 and b ¼ 1462 K were reported by Guo and Yoshida [184], who also found the low-temperature gap to be 6.13 eV, which is similar to that reported by Vispute et al. [185], a value closer to the values observed of late. Tang et al. [186] resolved what they believed to be the free or shallow impurity bound exciton in their cathodoluminescence (CL) data, at an energy of 6.11 eV at 300 K. Brunner et al. reported a variation from 6.19 eV at 7 K to 6.13 eV at 300 K [187]. A group of same authors Wethkamp et al. [188] used spectroscopic ellipsometry and determined that the energy gap varies from 6.20 eV at 120 K to 6.13 eV at 300 K. Kuokstis et al. [189] resolved a low-temperature free-exciton transition at 6.07 eV. Guo et al. [190] reported the temperature dependence of the reflectance spectra, while fitting it to the Bose–Einstein expression. Using the low-temperature data in conjunction with the Varshni parameters of Guo and Yoshida [184], leads to an intermediate value of 6.23 eV for the lowtemperature bandgap. The Varshni parameters reported by Brunner et al. [187] indicate no significant divergence from GaN for the entire AlGaN alloy composition range, which may bring the accuracy of the data into question. As in any semiconductor, the quality and strain nature of the films can alter the results. The availability

j177

j 2 Electronic Band Structure and Polarization Effects

178

of high-quality homoepitaxial AlN with presumably no strain has shed the much needed light onto the issues surrounding the actual bandgap of AlN [191]. However, even then O contamination could cause the near-band emission peak observed for shift, as it has done so for high-quality bulk substrates reported by Slack et al. [192]. In the experiments of Silviera et al. [191], the epitaxial layer was 0.5 mm, plenty considering the small penetration depth of 10 keV electrons used for the CL experiments, and the substrate was 287 mm thick with O concentrations of about 5 · 1019 cm
3 as measured by Neutron activation. The homoepitaxial AlN films have been grown by organometallic vapor-phase epitaxy on the single-crystal AlN substrates and efforts were undertaken to reduce the oxygen content of the film. The lowtemperature near band edge CL spectrum of the AlN film is shown in Figure 2.11. The open squares correspond to the experimental data obtained at 5 K, while the full line representing the best fit to the experimental data using Corinthian line shapes with transitions are indicated with dashed lines. The assignments shown are a result of thermal quenching behavior. The full widths at half maxima (FWHM) of the narrowest emission line at 6.023 eV is about 1.0 meV and was reported to be perhaps limited by the slit size used during the experiment. A measurement of the 253.65 nm emission line of a low-pressure Hg lamp using the same slits size set resulted in a FWHM of about 0.7 meV, as tabulated in Table 2.13. Figure 2.12 shows the temperature-dependent CL spectra for the AlN film. A rapid decrease in the intensities of the four peaks initially observed between 5.98 and 6.01 eV with the increase in temperature is evident, which is consistent with recombination processes involving excitons bound to shallow neutral centers. The peaks at 6.023 and 6.036 eV remain intense with increasing temperature, which lets them gain the free exciton A (FXA) and free exciton B (FXB) assignments, respectively,

Figure 2.11 High-resolution CL spectrum of an AlN homoepitaxial film. The full line represents our best fit using Lorentzian line shapes, and the dashed lines are the transitions composing the full line [191].

2.8 AlN Table 2.13 Energy positions, full widths at half maxima (FWHM),

and preliminary assignments associated with the transitions shown in Figure 2.11 [191]. Energy (eV)

FWHM (meV)

Assignment

5.98 6.000 6.008 6.01 6.023 6.036

49.4 11.0 1.5 44.0 1.0 8.0

A07 X A D027 X A D017 X A D017 X B FXA FXB

due to their large binding energies. The line around 6.07 eV, shown as FX2A , is some 2 orders of magnitude weaker than the most intense bands observed in the spectrum. On the basis of the similarity in the luminescence spectra of both GaN and AlN, the peak at 6.07 eV is attributed to the first excited state of the FXA. This assignment allows an estimation of the FXA binding energy as 63 meV, which is about twice the value for GaN. This leads to an estimated low-temperature bandgap of AlN of 6.086 eV (the sum of the FXA energy and its binding energy). Returning to the band structure of AlN, of considerable significance, the crystal field splitting in wurtzitic AlN is believed to be negative. The ramification of this is that the topmost valence band is the crystal hole band. Calculations have yielded a range of crystal field splittings, namely, Dcr ¼
58 meV by Suzuki et al. [56], Dcr ¼
217 meV by Wei and Zunger [107],Dcr ¼
176 meV by Shimada et al. [193], and Dcr ¼
244 meV by Wagner and Bechstedt [85]. Moreover, splittings of Dcr ¼
104 and
169 meV were obtained from first-principles and semiempirical pseudopotential calculations, respectively, by Pugh et al. [163] and Dcr ¼
215 meV by Kim et al. [154]. Averaging all of the available theoretical crystal field splittings, one obtains a value of Dcr ¼
169 meV. Silveira et al. [194], using optical reflectance data performed on a- and c-plane bulk AlN and a quasi-cubic model developed for the wurzite crystal structure, determined the crystal field splitting to be D ¼
225 meV. Note that the negative sign for the crystal

Figure 2.12 Temperature-dependent CL spectra of 0.5 mm thick AlN film deposited on 287 mm bulk AlN substrate [191].

j179

j 2 Electronic Band Structure and Polarization Effects

180

field splitting has important implications, namely, that the G7 valence band is on the top of G9 valence band, which is opposite of that in GaN. As for the spin–orbit splitting, the literature values range from 11 [163] to 20 meV [124]. Silveira et al. [194] again using the optical reflectance spectra in bulk AlN determined the spin–orbit splitting energy to be d ¼ 36 meV. In view of the experiments in high-quality bulk AlN, the value of 36 meV is recommended even though it is much larger than the calculated value of 19 meV recommended by Wei and Zunger [107]. As in the case of GaN, the region of the energy band near the bottom of the conduction band, as it manifests itself in devices, can be represented by the effective mass. The same of course applies for the top of the valence band. A number of investigators have calculated the AlN electron effective mass [154,124,163,195,196], with the prediction that it displays a greater anisotropy than that for wurtzitic == GaN [56]. The bare mass values of m? e ¼ 0:30m0 and m e ¼ 0:32m0 obtained by averaging the available theoretical masses may represent a good set of default values as this stage. It should again be underscored that further experimental studies are needed to verify the calculations. As for the valence band, a number of theoretical sets of valence band parameters are available [56,58,154]. There is an apparent disagreement in the signs for A5 and A6 among these reports, which may be irrelevant, because only absolute values of those parameters enter the Hamiltonian [154,163]. The A parameters given by Kim et al. [154] are suggested because the crystal field and spin–orbit splittings reported by these authors are closest to the ones suggested here. The hydrostatic deformation potential for wurtzite AlN has been reported to be in the range of
7.1 and
9.5 eV [31,180], which is consistent with the observation that the bandgap pressure coefficients for AlGaN alloys have little dependence on composition, as reported by Shan et al. [197]. The calculated values of a1 ¼
3.4 eV and a2 ¼
11.8 meV reported by Wagner and Bechstedt [85] are assumed to represent the material well. Theoretical values are also available for a few of the valence band deformation potentials such as D3 ¼ 9.6 eV, and D4 ¼
4.8 eV [180]. However, the complete set D1 ¼
17.1 eV, D2 ¼ 7.9 eV, D3 ¼ 8.8 eV, D4 ¼
3.9 eV, D5 ¼
3.4 eV, and D6 ¼
3.4 eV with the last value derived using the quasi-cubic approximation presented by Shimada et al. [193] can be used in the absence of any other reliable data. For mere availability reasons and few other issues, the mechanical properties of AlN have seen a good deal of experimental activity very early on, which later was followed by theory. Tsubouchi et al. [198], McNeil et al. [199], and Deger et al. [200] measured the elastic constants of wurtzitic AlN. A good many theoretical papers have also been reported [154,193,201–204]. The values suggested by Wright [179], who also provided a detailed discussion of their expected accuracy, namely, C11 ¼ 396 GPa, C12 ¼ 137 GPa, C13 ¼ 108 GPa, C33 ¼ 373 GPa, and C44 116 GPa, are recommended. Several piezoelectric coefficients [205,206] for early AlN at least in part can be found in Ref. [207]. The result for d33 ¼ 5.6 pm V
1 reported in Ref. [207] is in reasonably good agreement with the previous determinations but differs somewhat from d33 ¼ 5.1 pm V
1 measured by Lueng et al. [208]. While these experiments focused on only d33, both d33 and d13 can be determined from first-principles

2.8 AlN Table 2.14 Recommended band structure parameters for wurtzitic AlN [152].

Parameter

Value

Parameter

Value

Parameter

Value

Eg (eV, low temperature) a (meV K
1) b (K) Dcr (meV) Dso (meV) == me =m0 m? e =m0 a1 (eV) a2 (eV)

6.077 [194] 1.799, 0.9a 1462, 1000a
225 [194] 36 [194] 0.32 0.30
3.4
11.8

A1 A2 A3 A4 A5 A6 A7 (meV Å) d13 (pm V
1) d33 (pm V
1) d15 (pm V
1) Psp (C m
2)


3.86
0.25 3.58
1.32
1.47
1.64 0 (default)
2.1 5.4 3.6
0.090

D1 (eV) D2 (eV) D3 (eV) D4 (eV) D5 (eV) D6 (eV) C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa)


17.1 7.9 8.8
3.9
3.4
3.4 396 137 108 373 116

See Tables 2.27 and 2.28 for details related to the elastic constants, piezoelectric constants, and spontaneous polarization charge. Any dispersion among the tables is a reflection of the uncertainty in the available parameters. Note that the G7 valence band is above the G9 valence band, which is opposite of GaN. It is also similar to that in ZnO, which is somewhat controversial. See Zinc Oxide: Materials Preparation, Properties, and Devices, by H. Morkoç and Ü. Özg€ ur, Wiley (2008) regarding the valence band ordering in ZnO. a Obtained using the temperature dependence of the A exciton energy reported in Ref. [194].

calculation [193,209–212]. The recent theoretical values of Bernardini and Fiorentini, d33 ¼ 5.4 pm V
1 and d13 ¼
2.1 pm V
1 [210], are suggested although the elastic coefficients given in that reference are somewhat larger than the recommended values. On the basis of recent measurements [206,207] and a calculation [210] of the shear piezoelectric coefficient, Vurgaftman and Meyer [152] recommend d15 ¼ 3.6 pm V
1. The parameters concerning the bandgap-related issues for wurtzitic AlN recommended by Vurgaftman and Meyer [152] are tabulated in Table 2.14. A compilation of the dispersion in the effective mass for both the conduction band and various valence bands as obtained by various computational methods as well as parameters used in the description of the bandgap for wurtzitic InN, particularly, in the context of empirical pseudopotential method, as described in Ref. [55], are tabulated in Table 2.15. Even though a detailed discussion of polarization is reserved (Section 2.12), a succinct treatment of the topic is given here as it is relevant to the topic under discussion. The difference between the GaN and AlN spontaneous polarizations strongly causes a net polarization at the interface between the two materials that extends to the GaN/AlGaN interfaces as well. This charge, which is bound, influences the band profiles and energy levels in GaN/AlN quantum heterostructures. Although rigorous calculations [84,181,209,213] of the spontaneous polarization Psp(AlN) have produced results spanning a fairly broad range, from
0.09 to
0.12 C m
2, values for the difference Psp(AlN)
Psp(GaN) have tended to be more consistent, with most falling between 0.046 and 0.056 C m
2. Experimentally, for some time the majority of workers on the GaN/AlGaN system reported somewhat smaller Psp(AlN)
Psp(GaN). For example, Leroux et al. [214,215] derived 0.051 < Psp <
0.036 C m
2 for AlN. A study of the charging of GaN/AlGaN field effect transistors led to a similar

j181

j 2 Electronic Band Structure and Polarization Effects

182

Table 2.15 Effective masses and band parameters for wurtzitic AlN.

Parameter ==

aniso

iso

A

B

C

D

E

F

me m? e

0.231 0.242

0.232 0.242

0.33 0.25

0.33 0.25

0.24 0.25

0.24 0.25

0.35

0.33

mhh

2.370

2.382

3.68

3.53

1.949

1.869

3.53

4.41

2.370

2.382

3.68

3.53

1.949

1.869

3.53

4.41

0.209 3.058 0.285 1.204
4.789
0.550 4.368
1.511 1.734
1.816 0.134
0.128

0.209 3.040 0.287 1.157
4.794
0.571 4.374
1.484 1.726
1.788 0.153
0.160

0.25 10.42 0.24 3.81
4.06
0.26 3.78
1.86 2.02

0.229 2.584 0.350 0.709
4.367
0.518 3.854
1.549 1.680
2.103 0.204
0.093

0.212 2.421 0.252 1.484
4.711
0.476 4.176
1.816 1.879
2.355 0.096
0.093

0.26 11.14 0.33 4.05
3.86
0.25 3.58
1.32 1.47 1.64

0.27 2.18 0.29 4.41
3.74
0.23 3.51
1.76 1.52 1.83 0

==

== mlh == mch m? hh m? lh m? ch

A1 A2 A3 A4 A5 A6 A7 D1

0.25 6.33 0.25 3.68
3.95
0.27 3.68
1.84 1.95
2.91 0
0.059

0
0.059


0.215

Effective masses in units of free-electron mass m0, Luttinger-like parameters Ai (i ¼ 1, . . ., 6) in units of
h2 =2m0 , and A7 in units of eV Å. The crystal field splitting energy D1 is given in units of meV. The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor [55]. Anisotropically screened and isotropically screened values are from Ref. [55]. A: FP-LAPW band structure calculations are from Ref. [170], and effective mass parameters are obtained through a 3D fitting procedure within cubic approximation; B: FP-LAPW band structure calculations are from Ref. [170], and effective mass parameters are obtained by a direct line fit; C: Ai from Ref. [153] obtained through a Monte Carlo fitting procedure to the band structure and effective masses calculated using Equations 2.30 and 2.31; D: direct kp calculations for Ai from Ref. [153] and effective masses obtained from Ai using Equations 2.30 and 2.31; E: direct fit of Ai to first-principles band structures from Ref. [154]; F: Ai and effective masses obtained in the quasi-cubic model from zinc blende parameters from Ref. [154].

conclusion [216], and Hogg et al. [217] were able to fit their luminescence data by assuming negligible spontaneous polarization. Park and Chuang [218] required Psp ¼
0.040 C m
2 to reproduce their GaN/AlGaN quantum well data. On the contrary, Cingolani et al. [219] reported good agreement with experiment using a higher value derived from the original Bernardini et al. [209] calculations. A significant step toward resolving this discrepancy has been the realization that the AlGaN spontaneous polarization cannot be linearly interpolated between the values at the binary end points [220–222]. In combination with an improved nonlinear strain treatment of the piezoelectric effect, the discrepancy between theory and experiment for GaN/AlGaN quantum wells has been largely eliminated [85]. For additional details, see Section 3.14 and the text dealing with Tables 2.25, 2.27 and 2.28. We adopt Psp ¼
0.090 C m
2 as the recommended value for AlN, in conjunction with Psp(GaN) ¼
0.034 C m
2. The recommended band structure parameters for wurtzite AlN are compiled in Table 2.14.

2.8 AlN

2.8.2 Zinc Blende AlN

Only a handful of purportedly successful growths of zinc blende AlN on zinc blend substrates, such as GaAs and 3-C SiC, and Si substrates following low-temperature zinc blende GaN buffer layers have been reported [223–226]. Consequently, much of the discussion here relies primarily on theoretical projections and so do the parameter set recommended for this polytype. The only quantitative experimental study of the bandgap indicated a G-valley indirect gap of 5.34 eV at room temperature [224]. Assuming that the Varshni parameters for the wurtzitic AlN hold for the zinc blende polytype, the aforementioned room-temperature bandgap translates to a low-temperature gap of 5.4 eV. Vurgaftman and Meyer [152] recommend 4.9 and 9.3 eV for the X- and L-valley gaps, respectively [31,69,163]. The spin–orbit splitting is expected to be nearly the same as in wurtzite AlN at 19 meV [154,107,171,227]. Averaging the theoretical results from a number of different publications [69,154,163,170,173], one arrives at a G-valley effective mass of 0.25m0. The longitudinal and transverse masses for the X valley have been predicted to be 0.53m0 and 0.31m0, respectively [69]. If the method used previously for the GaN is applied to zinc blende AlN, one arrives at recommended Luttinger parameters of g1 ¼ 1.92, g2 ¼ 0.47, and g3 ¼ 0.85, and mso 0.47m0 [69,154,170,173]. These as well as the other literature values of the Luttinger parameters are listed in Table 2.16. Fritsch et al. [55] calculated the effective masses for conduction and valence bands, the latter involving the light and heavy holes, as well as the spin–orbit split-off mass, which with the anisotropy taken into account are listed in Table 2.17 for zinc blende AlN. The data contain those obtained by FP-LAPW, EPM calculations and those calculated with Luttinger parameters (Equations 2.42 and 2.43). When the calculated [163,173] values for the momentum matrix are averaged, a value of EP ¼ 27.1 eV (with F ¼
101) is obtained. Hydrostatic deformation potentials of
9.0 eV [31] and
9.8 eV [69] have been reported. The deformation potential values, a ¼
9.4 eV, av ¼
4.9 eV [69,107], b ¼
1.7 eV [69,177,178], and d ¼
5.5 eV [180,177,178], have been suggested [152]. The elastic constants of C11 ¼ 304 GPa, C12 ¼ 160 GPa, and C44 ¼ 193 GPa calculated by Wright [179] are Table 2.16 Luttinger parameters g1, g2, and g3 for zinc blende AlN obtained from a fit along the [1 1 0] direction along with those available in the literature, as compiled in Ref. [55].

Parameter

A

B

C

D

E

c1 c2 c3

1.85 0.43 0.74

1.54 0.42 0.64

1.50 0.39 0.62

1.91 0.48 0.74

1.92 0.47 0.85

A: empirical pseudopotential calculation by Ref. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: first principles band structure calculations from Ref. [170]; D: empirical pseudopotential calculation from Ref. [174]; E: recommended values taken from Ref. [152].

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Table 2.17 Effective masses for electrons (e), heavy holes (hh), light holes (lh), and spin–orbit split-off (so) holes in units of the free-electron mass m0 along the [1 0 0], [1 1 1], and [1 1 0] directions for zinc blende AlN. [100]

[100]

[111]

[111]

[110]

[110]

Reference

me

mhh

mlh

mhh

mlh

mhh

mlh

mso

A B C D E

0.23 0.28 0.21 0.30 0.25

1.02 1.44 1.05 1.39 1.02

0.37 0.42 0.35 0.44 0.35

2.64 4.24 2.73 3.85 4.55

0.30 0.36 0.30 0.36 0.28

1.89 3.03 2.16 2.67 2.44

0.32 0.37 0.31 0.38 0.29

0.54 0.63 0.51 0.67 0.47

Compiled by Fritsch et al. [55]. A: after Ref. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: empirical pseudopotential calculation from Ref. [69]; D: calculated from Luttinger parameters from Ref. [170], using Equations 2.42 and 2.43; E: calculated from recommended Luttinger parameters from Ref. [152], using Equations 2.42 and 2.43.

similar to the sets quoted in other theoretical works [180,181,228] and are therefore suggested. These and other band structure parameters recommended for zinc blende AlN are tabulated in Table 2.18 [152].

2.9 InN

As in the case of AlN, the interest in InN has so far been not necessary because of its properties, but because of the InGaN alloy that is used in lasers and LEDs operative in the visible and violet regions of the optical spectrum. In fact, if and when the technological issues are overcome, the InGaN channel FETs may also be superior to the GaN channel varieties [229], some details of which are discussed in Volume 3,

Table 2.18 Parameters associated with the band structure for zinc blende AlN.

Parameter

Value

Parameter

Value

Parameter

Value

alc (Å) at T ¼ 300 K E Gg ðeVÞ a (G) (meV K
1) b (G) (K) E Xg ðeVÞ a (X) (K) b (X) (meV K
1) E Lg ðeVÞ a (L) (meV K
1) b (L) (K)

4.38 5.4 0.593 600 4.9 0.593 600 9.3 0.593 600

Dso (eV) me ðGÞ ml ðXÞ mt ðXÞ c1 c2 c3 mso

0.019 0.25 0.53 0.31 1.92 0.47 0.85 0.47

EP (eV) F VBO (eV) ac (eV) av (eV) b (eV) d (eV) c11 (GPa) c12 (GPa) c44 (GPa)

27.1
1.01
3.44
4.5
4.9
1.7
5.5 304 160 193

Bandgaps are for low temperature [152].

2.9 InN

Chapter 3. The properties, particularly the fundamental parameters of InGaN for a given composition, depend very much on the InN parameters, particularly its bandgap. It is therefore important to understand the properties of bulk InN in its wurtzitic form. InN, however, is not all that easy, even considering the general difficulties encountered in the nitride semiconductor system, to synthesize. The somewhat intractable problem with InN is the enormous difference in the ionic size of its constituent atoms in that the atomic radii for In and N are largely different, which leads to highly distorted interatomic distances, interatomic bonding charges, tendency to form metallic clusters of the group III constituent, and inhomogeneous strain. All of these could, in principle, lead to pronounced anomalies in all the properties of InN, inclusive of measured bandgap and nature of defects. To make matters worse, the InN layers are grown at best on GaN epitaxial layer with large lattice mismatch (lm), aggravating many of the aforementioned problems. In spite of all these, progress is steadily made. It should be added that after having been accepted as the bandgap of InN, the 1.98 eV figure came under new scrutiny in that a plethora of reports concluded the actual bandgap to be 0.7–0.8 eV. Just at a time when a good many got convinced of the newer data, questions have been raised about the models along with heightened level of scrutiny of the new low-bandgap data. Assuming that interpretation of experiments pointing to the small bandgap figures are impeccable, theories also begin to be developed, even though long-standing understanding such as cation rule would be broken by the small bandgap figure. 2.9.1 Wurtzitic InN

The bandgap of InN has been a point of controversy dating back to the early days of InGaN development [1,230,231]. Early absorption studies on sputtered thin films concluded that the InN bandgap is in the 1.7–2.2 eV range [232–236]. However, no band-to-band PL could be observed in the samples prepared by sputtering in early developmental stages or later on in films grown by OMVPE and MBE. A review [237] of various crystal growth related issues and resultant properties as well as a proceedings [238] of a meeting devoted to debating these issues is available in the literature. In contrast to earlier reports of no near band edge emission, Davydov et al. [239–242] and others [243–245] reported near band edge emission but at much lower energies near 0.6–0.8 eV, depending on the report. Also see the comment, Ref. [246], on Ref. [239] and reply to that comment [247]. These reports relied to various extents on absorption, photoluminescence, and photoluminescence excitation experiments that showed the experimental evidence for the bandgap of InN to be overwhelmingly in the range of 0.7–0.8 eV and recommended a zero-temperature gap of 0.78 eV [248] and 692 2 meV [249], and in fact values as low as 0.65 eV [239] and 0.67 eV [250] have been reported as well. InN films of N-polarity grown by RF molecular beam epitaxy exhibited a bandgap value of about 0.7–0.75 eV as measured by optical transmission and reflection measurements [251–256]. The PL emission

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line appeared at 0.7 eV with no shift in energy between 300 and 77 K. These particular samples also exhibited room-temperature electron mobilities in the range of m ¼ 1750–2000 cm2 V
1 s
1 and an electron concentration in the range of n ¼ 2–3 · 1018 cm
3 at room temperature. Some details about the growth of these films are provided in Section 3.5.13. The polycrystalline or nanocrystalline nature of those early thin films associated with high electron densities and low mobilities led the proponents of the small bandgap for InN to suggest that those early films most likely contained a good deal of oxygen coupled with Moss–Burstein effect that could push the apparent bandgap upward. Another possible explanation for the dispersion in the reported values of the bandgap and also in support of the smaller bandgap may have to do with blue shift caused by any quantum size effects. At least the correct trend has been established by Lan et al. [257] who reported PL emission at 1.9 eV in nanorods of diameter between 30 and 50 nm (dubbed the brown InN), whereas 0.766 eV emission (measured at 20 K) was observed in rods with a diameter in the range of 50–100 nm (dubbed the black InN). Further refinement of the work led to the observation that the samples with fine (10 nm) or containing very high carrier concentrations exhibit the visible emission. In fact, both IR and visible peaks could be observed in the same sample when the samples show bimodal distribution of grain size [258]. The InN nanorods catalytically formed in the upstream portion of the substrate were of the brown type while those downstream were of the black variety. The Au catalyst that floats on the top of the InN nanorods as the growth progresses were shown to be encapsulated with In2O3, which is somewhat unexpected and may be made possible owing to close epitaxial relationship between wurtzitic InN and In2O3, indicating the participation of O in the catalytic process. The source of O, in this case, was attributed to residual O in the reactor as well as the quartz tube used. Furthermore, the black InN photon emission in a PL experiment quenched above 150 K while that from the brown InN exhibited strong emission at room temperature, albeit broad. The variation in the observed emission wavelength has been attributed to a variety of sources including O incorporation, Moss–Burstein shift due to high electron concentration, and quantum size effect. The quantum size effect would require diameters of less than 5 nm that is much larger than the 20–50 nm brown InN rods, which cannot therefore explain the bandgap shift to 1.9 eV from about 0.7 eV. It should be stated that the characteristic PL peak in O-implanted InN is different from the broad visible peak. The possibility of InN quantum dot formation in InGaN as being responsible for bandgap variations has not been of as much use in that even intentional dot formation, albeit limited to possibly only one report, did not lead to any blue shift [259]. The results are not precise but speak to the trend and demonstrate that great care must be taken to make sure that all the samples are grown under identical conditions, and doping levels are the same and not very large. Another issue of paramount importance that comes into the equation is the degree of crystallinity of the InN films. For example, Anderson et al. [260] produced polycrystalline InN with 0.8 eV luminescence present but did not identify the size of polycrystalline grains.

2.9 InN

Proponents of the earlier and larger bandgap for InN bring on the table several arguments undermining the validity of the low-bandgap figure. Among theme is that the absorption squared versus energy plots used to obtain the apparent bandgap in a semiconductor with very high carrier concentration underestimates the bandgap due to band tailing [261]. In addition, the 0.7–0.8 eV peak ascribed by some to the bandgap of InN is attributed to defects caused by nonstoichiometry of the films, which are grown under extremely In reach conditions and far away from equilibrium condition [261]. Moreover, The Moss–Burstein blue shift used by proponents of the small bandgap to account for the large bandgap reported earlier is not consistent among all the samples in that the sample with low carrier concentrations in the past confirmed the large bandgap of InN, albeit they were most likely heavily compensated. Finally, the opponents of the larger bandgap argue that the observed transition at 0.7–0.8 eV is due to Mie resonance that in turn is due to scattering or absorption of light in InNcontaining clusters of metallic In, which may have been mistaken for the low bandgap, as only the In cluster containing samples do show the 0.7–0.8 eV peak [262]. Let us now discuss the evolution of InN, and in particular, its perceived and admittedly controversial perceived bandgap. The early attempts to produce InN relied on not well-developed methods, as compared to modern crystal growth techniques, and as such produced mostly powder and nanopowders [263–265]. Naturally, the early work on thermodynamic decomposition studies and X-ray diffraction (advantageous for power diffractometry due to the nature of the films) were performed on powder films. Even in these very powder samples, there was some observed variation in the color of the material in that some have been noted to be black or blackbrown [264–266] as opposed to the deep red that one would expect for the 1.8 eV bandgap originally estimated [265]. For a witty historical account of these early efforts through the controversy, the reader is referred to Refs [267,268]. Below a discussion of the role of oxygen, possible defects that might be responsible for the 0.7 eV emission, the Moss–Burstein shift, particulars of carrier absorption, and Mie resonance is given in an effort to give an overall appreciation of issues causing the controversy in determining the bandgap parameter of InN. Focusing on the issue of O, because In2O3 has a bandgap of 3.75 eV [269], the argument goes that if InN films are heavily O contaminated, the bandgap would be pushed upward. This is one of the points used by the proponents of the small bandgap. However, as shown in Figure 1.34, to bring the bandgap from 0.7 to 1.9 eV would require some 30% O in the film assuming that the Vegard’s law holds. It should also be noted that the stated O must form an alloy for increasing the bandgap not just as surface contamination or inclusion of O at grain boundaries. One of the samples used to support the InN–In2O3 alloy formation was an RF-sputtered film with poor mobility and high carrier concentration [270]. Auger analysis used to determine the O content in the aforementioned layer is sensitive to only O and cannot determine whether it is alloyed in the semiconductor InN. In addition, any depth profiling accompanied by that technique requires argon ion etching, which is known to result in severe nitrogen loss and lead to overestimation of the atomic oxygen concentration owing to the recycling of sputtered oxygen on the film surface coupled with the strong bond between oxygen and surface indium [271]. An alternative method, Rutherford backscattering (RBS), was

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188

also used by Davydov et al. [270] with the sample mounted on glass, which in consideration of the sample thickness may not have allowed a definitive determination whether the O signal is due to O in InN or the glass substrate used. Nevertheless, the O content so measured was 20%, which is to be contrasted to 10% obtained [272] from measurements performed in similar layers using elastic recoil, which is accurate for elemental analysis. It should again be mentioned that elemental O in the film, even if present in stated quantities, is not the same as InN:In2O3 alloy, which would increase the bandgap, not to mention the fact that such alloying is not favored by temperatures employed in RF sputter deposited InN. One can surmise that amorphous InON, NO2 and surface hydroxide species are implicated as being responsible for the 1.8–2 eV bandgap reported. These species are only as surface species, and there has not as yet been any evidence reported of any other form of InN–In2O3 alloy species during the growth of InN, as can be discerned from Figure 1.34 that 10% or even 20% oxygen could not account for the bandgap of 1.9 eV if a bandgap of 0.7 eV is assumed for InN. Figure 1.34 clearly indicates that an alloy with about 37 at.% oxygen, which translates to 44% In2O3, would be needed to provide a bandgap of 2.0 eV if the bandgap of pure InN is 0.7 eV. Consequently, 10% (at.) oxygen, even if all were alloyed with InN, would account for a blue shift of about only 0.3 eV, much less than some 1 eV for consistency. If the bandgap of the alloy follows a bowing parameter, unless it is positive, which is unlikely, the blue shift caused by alloyed O would even be smaller. This view is supported by the results of Yoshimoto et al. [273] that reported a bandgap value of 1.8–2.0 eV for MBE InN grown on quartz with 3% atomic oxygen present in the film. Evidently, this level of O contamination does not prevent one from arriving at the longstanding bandgap of InN, as any blue shift caused by this level of elemental O even if all is in the alloy form does amount to much. The role of O in InN had been investigated early on in sputtered films. Among them is the work of Westra et al. [274] who produced InN with carrier concentrations between 7 · 1019 and 2 · l020 cm
3 and mobility of 4–10 cm2 V
1 s
1. Rutherford backscattering data indicated 11% atomic oxygen, and indium-to-nitrogen ratios slightly above a value of 1. No evidence of oxygen or oxynitride phases was observed in the X-ray diffraction spectra, which led those authors to propose that oxygen is in the form of an amorphous indium oxynitride, similar to the observations of Foley and Lyngdal [275]. In this case the structure would maintain the stoichiometry, while the oxygen would not be detected by X-ray diffraction. Contribution from NO2 has been observed for samples containing higher oxygen content in a polycrystalline InN matrix with no evidence for O in X-ray diffraction or infrared absorption spectra [271], wherein grain boundaries were proposed to be the host for O. Raman data for sputtered InN with 10% atomic oxygen concentration reveals only InN-related phonon peaks, implying the lack of alloy formation [272,276]. The moral of the aforementioned discussion is that any O present in at least polycrystalline InN that has been examined with the role of O in mind did not seem to be incorporated as an alloy, which explicitly leads to the conclusion that could not contribute to the bandgap. An in-depth discussion of the topic can be found in Ref. [267]. Moss–Burstein effect has been forwarded as a plausible argument for observing 1.8–2 eV absorption if the 0.7 eV bandgap is assumed. This shift occurs when carrier

2.9 InN

concentration is above the Mott critical density, meaning larger than the conduction band density, that is, the Fermi level lies in the conduction band. In such a case, electrons fill the bottom of the conduction band so that the apparent bandgap measured by optical absorption is increased by an amount of extension of the Fermi level. As a result, extra photon energy is required to excite electrons from the top of the valence band to the Fermi-level position within the conduction band suggested by Trainor and Rose [277]. But the required large electron concentrations produces bandgap renormalization, which is in competition with the Moss–Burstein effect. High doping concentrations cause band tailing effect, which acts to reduce the apparent bandgap. Layers of InN with large variation in electron concentration have been reported along with the measured bandgap, as compiled in Figure 1.35, as function of electron concentration. The measured bandgap is the convolution of Moss–Burstein shift and bandgap renormalization culminating in the measurement of EG þ EF, the EF being measured with respect to the conduction band edge. If the plot is limited to MBE films only, the data appear to support the argument for small bandgap and variation attributed to Moss–Burstein shift. However, when all the data in films prepared by any growth method in aggregate are plotted, there seems to be quite a scatter, including large bandgap associated with relatively lower impurity levels. The data basically do not show any trend in that a good many of the samples with high and low electron concentration both do exhibit the large bandgap value. The critical data having to do with the samples of relatively lower electron concentrations are associated with compensated samples. This is an important issue in that material compositional, or stain-related nonuniformities may also hamper efforts to determine the bandgap accurately. A critical view may raise question about the quality of the samples. What is clear is that the bandgap of the nondegenerate InN is still unknown and will require considerably more investigation [272]. What about some yet unknown defect being responsible for or the source of the 0.7 eV emission and absorption? Ask Butcher [267] who considered this possibility and provided arguments backed by experimental data in support of it. Butcher [267] suggested that the evidence for a 0.7 eV bandgap is also consistent with the presence of a 0.7 eV deep-level trap in that the available absorption data for the best published apparently 0.7 eV material (without the Moss–Burstein shift due to low carrier concentrations) exhibit an energy dependence consistent with a deep-level trap of |si-like orbital symmetry and was inconsistent with direct band-to-band transitions. Butcher [267] also made the case that the slope dependence of absorption coefficient plots is consistent with variations in the density of such a deep-level trap. Further, the 0.7 eV material showed that emission to be emanating from regions of indium-rich aggregates [270]. If one wants to be skeptical, one could argue that it is likely that the said emission could be a result of surface states at the metal–semiconductor interface rather than being associated with the InN band edge. It was also shown that material grown without such aggregates had an absorption edge nearer 1.4 eV [278]. The early InN data were produced by N-rich material, whereas the 0.7 eV material is grown under In-rich conditions and the same probably holds for OMVPE-grown layers. The early samples grown under N-rich conditions exhibited a higher than usual unit cell

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volume [279]. Such variation in the unit cell volume with stoichiometry would be consistent with that observed for GaN [280]. Being forced to grow InN at very low temperatures, typically below 550  C, with the caveat that the temperature measurements are not absolute, the driving force for desorption of excess In is not all that potent. The potency of In droplet formation has been noted by Yamaguchi et al. [281] who indicated that as little as a 20  C increase in temperature from optimum growth conditions can result in indium droplets. Clearly the stoichiometry, specifically In inclusion issue, is something that would need to be dealt with. With the evidence for indium-rich aggregates in the InN matrix and suggestion that they are responsible for the 0.7 eV luminescence properties of InN, the data of Wu et al. [282] has been discussed under a different light [267]. The samples investigated by Wu et al. had undergone an irradiation with 2 MeV protons to a dose of 2.23 · 1014 protons cm
2 that resulted in a factor of 2 increase in the intensity of 0.7 eV PL peak, which was interpreted as radiation hardness. This increase in PL peak intensity can be ascribed to increased density of radiative defects, as has been well documented for III–V materials by Lang [283]. This is typically accompanied by band-edge photoluminescence that is either quenched or left unperturbed by radiation damage. Another interpretation of the observations of Wu et al. [282] might therefore be an increase in radiation-induced defects although not being promoted [267]. An issue is the fact that InN is particularly susceptible to nitrogen loss when bombarded with ions [284–287], leading to possible indium-rich aggregates. The aforementioned discussion lays the groundwork for a plausible connection between the 0.7 eV peak and In-rich aggregates, but does not quite attempt to discuss the nature of that emission. Shubina et al. [288] and also Ivanov et al. [278] attempted to do just the same, meaning shed some light into the nature of the 0.7 eV emission. The efforts of those authors utilizing microcathodoluminescence studies coupled with imaging of metallic In have shown that bright infrared emission at 0.7–0.8 eV arises in the close vicinity of In inclusions and is likely associated with surface states at the metal–InN interfaces. Employing thermally detected optical absorption (TDOA) measurements, Shubina et al. [288] suggested a bandgap near 1.5 eV, reserving a more definitive judgment until after more accurate measurements could be performed. Shubina et al. [288] have actually broadened the range of samples, examining various substrates including sapphire by choosing two representative sets of InN epilayers grown by both plasma-assisted molecular beam epitaxy and organometallic VPE methods. The dominant IR emission in these samples were observed to be in the range of 0.7–0.8 eV, independent of the growth technique used to prepare them and of excitation, such as optical with different laser lines, or by an electron beam in conjunction with a CL performed at 5 K in an analytical scanning electron microscope. No correlation between the IR emission and the electron concentration, which ranged from 2.1 up to 8 : 4 · 1019 cm
3 (determined from IR ellipsometry measurements using an effective electron mass of me ¼ 0 : 11m0) [234] and measurement temperature was discernable, consistent with other reports [289]. Not all the 18 samples studied emitted light, and all the samples emitting IR PL emission did contain In-rich aggregates. Analytical microscopy sensitive to atomic weight in

2.9 InN

backscattered electrons (BSE) geometry, and energy dispersive X-ray (EDX) analysis along with CL were employed to establish a definite correlation between the In-rich aggregates and IR PL emission. Again, the bright 0.7–0.8 eV IR emission in both MBE and OMVPE sets of the samples was found to be associated with the In aggregates. Total optical extinction losses in a semiconductor matrix with metallic clusters have been established. In addition to the interband absorption in the matrix, those losses contain two additional components, namely, a bipolar absorption of radiation energy and its conversion into heat in small particles, and resonant scattering on plasmon excitations, which is important for larger particles [290]. Both characteristic components have been observed in the optical spectra obtained by Shubina et al. [288]. Those authors also observed emission in the range of 0.8–1.4 eV that was attributed to the scattered background signal, the root cause of which was most likely associated with nonchromatic spontaneous emission of the semiconductor laser used or the fluorescence of all optical components at high excitation power levels. Owing to electron beam excitation, the scattered signal was absent in the CL spectrum. Unless the aforementioned spurious signals are accounted for, erroneous conclusions could be drawn. Citing the absence of such signals, Shubina et al. [288] employed thermally detected optical absorption technique performed at 0.35 K. The method is based on the detection of a small increase in the sample temperature caused by phonons produced by nonradiative recombination processes as a result of optical absorption and bipolar absorption of light in In-rich aggregates. The TDOA spectra contained additional peaks below the principle absorption edge in films containing In-rich aggregates but not others, owing to absorption within those aggregates. The sharpness of the observed feature is related to the resonance at the extremely low measurement temperature that prevents thermal broadening induced by electron acoustic phonon scattering [288]. A natural conclusion of the aforementioned observations is that the strong IR absorbance is most likely associated with Mie [291] resonances due to scattering or absorption of light in InN-containing In aggregates or metallic In clusters. In the Mie theory [291], the extinction losses for a metallic sphere depend on the complex dielectric functions of both the matrix material, e, which is InN in this case, and metal, em, which is In in this case. Consequently, the resonance energy of the In clusters in InN matrix with a high-frequency dielectric constant of e0 ¼ 8.4 (based on stoichiometric InN, which is the case here as the InN films studied were In-rich) is considerably smaller than that in vacuum. For a treatment of the dielectric constant in InN, the reader is referred to Ref. [292]. Shubina et al. [288] argued that the In inclusions are predominantly formed either between columns or initiated at the interface with the substrate. Not knowing the shape and density of those clusters accurately, the authors employed a model developed for nonspherical metal particles in an absorbing matrix [293], which is based on the Maxwell–Garnett approximation for an effective medium [294], to demonstrate that the resonance absorption energy in the InN–In composite can shift down to the IR range depending on the In content and the shape of the In clusters. A lack of accurate knowledge of the InN complex dielectric constant and the specifics of In clusters prevented the authors to arrive at a

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shape and density of In clusters responsible for the IR absorption. Suffice it to say that given the available parametric data, it is very plausible that the observations are related to Mie resonance absorption caused by In-rich aggregates in the InN matrix. Clearly, additional experiments are imperative and will surely be available in due time. Even with the reported available data and analysis, Shubina et al. [288] argue that it is unlikely that the bandgap of InN is at 0.7–0.8 eV range. Confirming to a large extent the results obtained in InN reported in 1980s, using epitaxially grown wurtzite InN, Guo and Yoshida [184] measured low-temperature and room-temperature gaps of 1.994 and 1.97 eV, respectively, along with Varshni parameters of a ¼ 0.245 meV K
1 and b ¼ 624 K. Estimates of the crystal field splitting in wurtzite InN range from 17 to 301 meV [66,107,163]. A value of 40 meV can be adopted. On the basis of the calculation, spin–orbit splittings vary from 1 to 13 meV [107,163], but a value of Dso ¼ 5 meV is recommended by Vurgaftman and Meyer [152]. Ironically, the small bandgap of InN actually goes against the long held cation rule, in which for isovalent, meaning common-cation semiconductors, the direct gap at the G point increases as the anion atomic number decreases. This implies, for example, that the bandgap of InN should be larger than that of InP, which is 1.4 eV, which is consistent with the values of 1.5 eV and higher reported for InN. It should, however, be stated that the breakdown of the common-cation rule is not unusual in ionic semiconductors. This is articulated in an interesting observation where in Nag [246] pointed out that this gap is unusually low in the context of trends exhibited by other semiconductor materials. Fore example, the bandgap of ZnO is also smaller than that of ZnS. This unexpected behavior has been attributed to two effects. Wei et al. [295] argued that a much lower 2s atomic orbital energy of N (
18.49 eV) compared with P (
14.09 eV) and other group V elements lowers the conduction band minimum at the G point. Moreover, the smaller bandgap deformation potential of InN (
3.7 eV) in comparison to InP (
5.9 eV) weakens the atomic size effect. The atomic size effect is the one that forms the basis of the common-cation rule. Similarly, the effect of the lower orbital energy of O as compared to the other group VI elements such as S is responsible for ZnO bandgap being smaller than that of ZnS. The controversy on the theory side matches that of the experiments. It could be argued that in the end carefully thought of experimental evidence will carry the day and theories with appropriate approximations will be developed to support the general direction of experiments. Tsen et al. [296] studying nonequilibrium optical phonons in a high-quality singlecrystal MBE-grown InN with picosecond Raman spectroscopy reached the conclusion that their results are not consistent with the large bandgap of InN. The basis for this argument is as follows: above gap photons cause creation of electron and hole pairs that very quickly relax toward the bottom/top of conduction/valence band by emitting phonons. For wurtzitic InN, using me ¼ 0:14 me [297], mh ¼ 1:63 me [298],
hwL ¼ 2:34 eV, and Eg ¼ 1.89 eV gives an index of refraction n ¼ 3.0 (measured by Tsen et al. [296]) and
hwLO ¼ 73:4 meV (corresponding to A1(LO) phonon mode energy); the phonon wave vector in the experiments of Tsen et al. [296] is q ¼ 2nkL 7.08 · 107 m
1, where kL is wave vector of the excitation laser with photon energy

2.9 InN

2.34 eV. Owing to a much larger associated hole mass, the phonons emitted by holes can be ignored. In a parabolic band, excess electron energy is given by   mh DE ¼ ð
hwL
E g Þx ffi 0:41 eV; ð2:44Þ me þ mh which is about five times the LO phonon energy
hwLO ¼ 73:4 meV, which means that the energetic electrons are capable of emitting five LO phonons during their thermalization to the bottom of the conduction band. However, because of energy and momentum conservation during the electron–LO phonon interaction, there exists a range of LO phonon wave vectors that electrons can emit. As depicted in Volume 2, Figure 3.7, for an electron with wave vector ~ ke and excess energy DE, the minimum and maximum LO phonon wave vectors it can emit are given by [299] pffiffiffiffiffiffiffiffiffi 2me pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð DE
DE

hwLO Þ; kmin ðor qmin Þ ¼
h and kmax ðor qmax Þ ¼

pffiffiffiffiffiffiffiffiffi 2me pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð DE þ DE

hwLO Þ:
h

ð2:45Þ

Because of the nature of E–k relationship of the electron, the lower the electron energy, the larger kmin (also referred to as qmin) and the smaller kmax (also referred to as qmax) are. Therefore, at some electron energy, kmin of the LO phonon will be larger than the wave vector probed by the Raman scattering experiment, which is q ¼ 7.08 · 107 m
1. When this occurs, the energetic electrons can no longer emit LO phonons with wave vectors detectable in the Raman scattering experiments. These conditions corresponds to kmin ffi 1.1 · 108 m
1 and kmax ffi 2.4 · 109 m
1 in the experiments of Tsen et al. [296]. Although the effect of nonparabolicity of the conduction band for InN is currently not known, the general trend is that the effective electron mass increases with the increase in energy in the conduction band for a typical semiconductor. By taking the nonparabolicity into account, the detectable kmin can be revised as being larger than 1.1 · 108 m
1. Consequently, no nonequilibrium A1(LO) phonon population should be detected in the Raman experiments of Tsen et al. [296] when the effect of nonparabolicity of the conduction band is considered. This means that if the bandgap energy of InN were 1.89 eV, there would be no detectable nonequilibrium A1(LO) phonon population with the excitation laser photon energy of 2.34 eV. This contradicts the fact that nonequilibrium A1(LO) phonons have been observed and argues against the large InN bandgap. A similar argument can be applied to the scenario that the bandgap of InN is ffi0.8 eV, in the case of which the same laser photon energy (2.34 eV) can excite the electrons up in the conduction band so that it would take 6 A1(LO) phonons to be emitted for full thermalization. No A1(LO) phonons should be detected with 1.96 eV laser excitation. Both of the above arguments are consistent with the experimental observations. Earlier, it was stated that the proponents of the larger bandgap InN argue that the small bandgap reported for InN could be due to incorrect attribution of the deep-level emission to the band-edge emission as defects capture electrons or

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j 2 Electronic Band Structure and Polarization Effects

194

holes and emit photons (through a radiative relaxation process) of lower energy than the bandgap. This capture process can also emit phonons through a nonradiative relaxation process. Let us for a moment suppose that the 0.8 eV luminescence reported in the literature is due to such a capture process by some unknown deep-level defect(s) in InN. If so, the observations of Tsen et al. [296] of nonequilibrium A1(LO) phonons in InN with excitation laser having photon energy 2.34 eV suggests that nonradiative relaxation processes also play a role in the capture. However, due to deep-level defects having very localized wave functions, their momenta are widely spread due to the uncertainty principle. This suggests that electron–phonon interaction during the capture process does not need to conserve momentum. Therefore, phonon wave vectors of almost every magnitude can be emitted. In other words, the defect-model predicts that if one detects nonequilibrium A1(LO) phonons with 532 nm excitation then one should also detect nonequilibrium A1(LO) phonons with 634 nm excitation. This is in contradiction with the experimental observation of Tsen et al. [296]. In short, the nonequilibrium phonon experiments are consistent with the 0.8 eV bandgap. In a renewed effort to determine the fundamental gap of InN, Arnaudov et al. [249] studied the shape and energy position of near band edge photoluminescence spectra of InN epitaxial layers with different doping levels. They implied that samples with high doping concentration have been used to infer the bandgap of InN without properly accounting for the effects of band filling, band nonparabolicity, and electron–electron and electron–impurity interactions. In addition to usual difficulties associated with highly doped samples, another aggravating factor is a clear lack of convergence in the value of the electron effective mass. Properly accounting for the aforementioned effects, Arnaudov et al. suggest a bandgap of Eg ¼ 692 2 meV for an effective mass at the conduction band minimum mn0 ¼ 0.042m0. They also argue that the value of Eg ¼ 0.69 eV reported in Ref. [270] extracted from the absorption and photoluminescence spectra of samples with carrier concentration n > 6 · 1018 cm
3 within the band-to-band recombination model while taking into account the Burstein–Moss shift and bandgap renormalization due to many-body effects agrees with the data; the value used for the effective mass at the bottom of the conduction band, mn0 ¼ 0.1m0, is inconsistent with the universal Kane’s relation (mn0; Eg). This relation predicts a bandgap energy of 1.7 eV for mn0 ¼ 0.1m0, and to obtain near to 0.7 eV bandgap one must use a much smaller effective mass of mn0 ¼ 0.042m0. Interestingly, the same value of Eg ¼ 0.69 eV was obtained by fitting the absorption spectra, this time in a lightly doped sample with n ¼ 3.5 · 1017 cm
3, with a sigmoidal function that includes only the band tailing effect and does not involve any value for the effective mass [300]. Taking advantage of improved crystalline quality and utilizing samples with electron concentrations in the range of 7.7 · 1017–6 · 1018 cm
3 Arnaudov et al. [249] undertook the task of determining the bandgap on InN from optical data but with the interpretation of the emission spectra in such highly conducting layers in terms of the free-electron recombination band (FERB) model, which has been previously reported in the context of GaAs [301,302], InP [303], InSb [304], and GaN [305]. By analyzing not only the emission energy position but also the shape of the spectra simultaneously and taking into consideration the specifics associated with both high and low electron concentrations, Arnaudov

2.9 InN

E

1.0

Ec

0.8 PL intensity (au)

n = 1.7 × 1018cm–3

E F = E Fn Degenerate band tails G* = E

0.6

Ev

0.4

n = 6 × 1018 cm–3

Fp

gn ,gp

0.2

Eg -G2*Eg-G1* Eg EF2

0.0 0.5

0.7

0.6

EF 3 0.8

Energy (eV) Figure 2.13 Experimental, depicted with symbols, and calculated, depicted with solid lines, PL spectra of samples with 1.7 · 1018 and 6.0 · 1018 electron concentrations as measured by Hall effect. The inset schematically depicts the recombination mechanism between the

degenerate electrons in the conduction band DOS to the level G* in the valence band tails as relied on in modeling. The energy positions representing the best fits for Eg, EF, and the intrinsic, or the unperturbed, bottom of the conduction band Eg–G* are also indicated [249].

et al. [249] are able to determine the fundamental bandgap for the electron effective mass in InN as Eg ¼ 692 2 meV for an effective mass at the bottom of the conduction band mn0 ¼ 0.042m0, which is consistent with Kane’s relation. PL spectra of two samples with electron concentrations of 1.7 · 1018 cm
3 6 · 1018 cm
3, as determined by Hall measurements, by Arnodov et al. [249] are shown in Figure 2.13 (points)). Both samples exhibit a broad emission band with a maximum at 685 meV and 705 meV, respectively, the former for the lower doped and the latter for the higher doped sample due to a larger Burstein–Moss shift. Noteworthy, however, is that the emission band of the sample with a higher Hall concentration is broader and more asymmetric, and its PL peak is at a higher energy due to the larger Burstein–Moss shift. The low-energy side of the spectral band of the sample with higher electron concentration shifts to lower energy compared to the sample with lower Hall concentration, which tends to narrow the apparent optical bandgap, consistent with emission spectra from highly doped semiconductors. In addition, these observations are characteristic of free to bound recombination of degenerate conduction electrons with nonequilibrium valence holes in the valence band tail, as shown in the inset of Figure 2.13 [301–305]. Moreover, the shape of the emission bands follows the energy distribution of electrons in the conduction band, all the while their energy positions are determined by the interplay of the equilibrium Burstein–Moss shift (blue shift) and the effective bandgap renormalization (red shift).

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j 2 Electronic Band Structure and Polarization Effects

196

To deduce the bandgap energy Arnaudov et al. [249] fit the experimental emission spectra with that obtained from the general expression for the intensity versus the photon energy I(hn) given by (neglecting the energy dependence of the probability for radiative transitions, as was done for heavily doped GaN [305]) ð¥ ð¥ IðhnÞ g n ðE Fn Þf n ðE n
E Fn Þg p ðE p Þf p ðE p
E Fp ÞdðE n
E p
hnÞdE n dE p ; 00

ð2:46Þ where gn(En) and gp(Ep) are the density of states in the conduction and valence bands at electron and hole energies En and Ep, respectively. The terms fn and fp represent the associated nonequilibrium Fermi–Dirac functions, and EFn and EFp are the quasiFermi levels for electrons and holes. The conduction band density of states gn(En), as well as the electron effective mass mn(En), can be calculated in the framework of Kane’s two-band kp model as described in Ref. [300]. Although the bandgap information is implicit in the En
Ep term of the delta Dirac function, the bandgap term Eg is sometimes explicitly subtracted from En
Ep for emphasis, as done by Arnaudov et al. [249]. The reader is referred to Volume 3, Chapter 2 for a detailed discussion of spontaneous emission intensity calculations. Because the nonequilibrium holes are situated in a relatively narrow energy window deep in the band tails, they do not significantly affect the spectral distribution of the emitted light. Therefore, the quantity gn(En) fn(En
EFn) in Equation 2.46 roughly reproduces the shape of the FERB shown in the upper part of the inset in Figure 2.13. Likewise, the term gp(Ep) fp(Ep
EFp) in Equation 2.46 determines primarily the energy position of the emission band associated with the unperturbed fundamental bandgap Eg via Equation 2.47, as depicted in the lower part of the inset in Figure 2.13. It should be pointed out that the FERB model includes a calculation of the spectral shape as well as analytical renormalization of the bandgap due to the presence of ionized impurities. In this case, the energy positions of both the low- and high-energy slopes of the emission band are sensitive to the electron concentration induced by ionized impurities. In conjunction with the nonequilibrium Fermi–Dirac function of electrons, fn, Arnodov et al. [249] used the Fermi level EFn ffi EF corrected for the temperature of electrons y, which can differ from the lattice temperature T, electron–electron and electron–impurity interactions [305], and the nonparabolicity of the conduction band density of states (DOS), which can be determined in framework of the two-band kp model. The valence band DOS gp(Ep) is replaced by a Gaussian, determining the tails deep in the bandgap through the root mean square (rms) impurity potential G as detailed in Ref [305]: pffiffiffi 4pe2 ðN i R3s Þ1=2 ; G¼2 p eRs with Rs ffi

ð2:47Þ

aBe e h2 ðna3Be Þ
1=6 and aBe ffi ; 2 4pe2 4pmn0

where the terms e represent in order the electric permittivity, Rs the Thomas–Fermi screening length, Ni ¼ [(1 þ K)/(1
K)]n the total ionized impurity concentration,

2.9 InN

K the compensation ratio, n the extrinsic electron concentration, and aBe the effective Bohr radius of electrons. The value of Rs is smaller than aBe and thus the equilibrium and nonequilibrium degenerate electrons are free above the bottom of the conduction band [305]. The situation with the nonequilibrium holes is quite different in that in III–V materials the effective Bohr radius aBh is much smaller than Rs due to the relatively large hole effective mass mp. Moreover, holes are classically localized, at least for not extremely high impurity concentrations and high temperatures [306], at the potential minima of the valence band tails near the thermal equilibrium level pffiffiffi G ¼
E v þ 2G
kT=2. As shown for heavily doped GaN [305], the level G* plays the role of the quasi-Fermi level in the nonquasi-equilibrium recombination FERB model. Thus, we can replace the value of EFp in the Fermi–Dirac function for holes fp can be replaced by G*, meaning one case set EFp ¼ G*. To comment on the electron effective mass, the FERB emission spectra of samples with 1.7 · 1018 and 6.0 · 1018 cm
3 Hall electron concentration can be calculated with varying n, y, and K and using the value of mn0 ¼ 0.042m0 suggested in Ref. [307]. For the relative static permittivity, a value of e ¼ 14.61 was used by Arnaudov et al. [249]. Assuming a zero-compensation ratio, which is a good first-order approximation, the best fits of the spectra are obtained with bandgap values of 690 and 692 meV, for the sample with 1.7 · 1018 cm
3 electron concentration and the sample with 6.0 · 1018 cm
3 electron concentration, respectively. To reconcile the small difference in the aforementioned bandgap values, a small compensation ratio of K ¼ 0.06 for sample with 1.7 · 1018 cm
3 electron concentration and K-value of 0.01 for sample with 6.0 · 1018 cm
3 electron concentration was introduced. Doing so resulted in a bandgap valued of Eg ¼ 692 meV for both samples. The best fit values for the two samples are shown in Figure 2.13 (solid lines). It should be noted that best fit values of the electron concentration nopt are noticeably lower than those deduced by the measured Hall effect. This dispersion may perhaps be related to the inhomogeneities in the films. In spite of this both represent degenerate cases because the Mott’s transition concentration is estimated to be about nMott ¼ 5 · 1016 cm
3 for mn0 ¼ 0.042 m0. The calculated curves agree very well with the experimental spectra, with the exception for the low-energy range. In this region, an additional contribution from a deeper emission center could in principle be possible, which is not included in the model. In the high-energy portion of the spectrum, the accuracy is more reliable. To summarize the above discussion, using the shape as well as the energy position of the near band edge PL spectra of InN epitaxial layers with different doping levels, Arnaudov et al. [249] concluded that the radiative transition is between the degenerate electrons in the conduction band and nonequilibrium holes in the valence band tails and that the fundamental bandgap of InN is Eg ¼ 692 2 meV for an effective mass at the conduction band minimum of 0.042m0, which is consistent with the Kane model. The optical transmission and reflection data obtained from a high-quality InN film grown by MBE with N-polarity support the smaller bandgap figures of InN as shown in Figure 2.14. The growth details and transport properties of InN layers similar to the one that led to the data presented in Figure 2.14 are discussed in Section 3.5.13.

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j 2 Electronic Band Structure and Polarization Effects

198

70

100

80

65

60

60

40

55

20

Reflection (%)

Transmission (%)

E g = 0.7– 0.75 eV

50

0 45 1.0

1.5

2.0

2.5

3.0

3.5

Wavelength (μm) Figure 2.14 Optical reflection and transmission data obtained in an N-polarity InN film grown by MBE indicative of a bandgap between 0.7 and 0.75 eV. Courtesy of A. Yoshikawa.

As for the true value of bandgap of wurtzitic InN, although the data for small bandgap data are convincing and the arguments for the small bandgap are compelling, for some reason, some controversy remains. This controversy is expected to evaporate when and if the large bandgap observed is earlier and some InN samples are explained satisfactorily. We should reiterate that the data in high-quality samples converge on the small bandgap. However, there is still some dispersion in the exact bandgap value that seems to be between 0.65 and 0.8 eV. In this chapter, the arguments for both the large and small bandgap as well as the pitfalls for each in terms of the reliability of the data and a historical review are provided for the reader to be abreast with the conflicting issues surrounding the matter. Turning our attention momentarily to other electronic properties affected by the band structure, measurements of the electron effective mass in wurtzitic InN produced values of 0.11m0 [234], 0.12m0 [308], and 0.14m0 [297], as well as 0.24m0, for the mass perpendicular to the c-axis [309]. Kasic et al. [297] used infrared spectroscopic ellipsometry and micro-Raman scattering to study vibrational and electronic properties of wurtzitic in 0.22 mm thick InN layers grown by RF MBE, as well as Hall effect measurements, and arrived at the isotropically averaged effective electron mass of 0.14m0. The mass value of 0.14m0 closely matches at least one theoretical projection [53]. It should be mentioned that all the InN films used for these investigations featured very high electron concentrations, which are endemic to InN, in the 1018 cm
3 or higher, and causes the Fermi level to degenerate well in the conduction. Consequently, any nonparabolicity in the conduction band would affect the effective mass measurements. However, the realization that the InN bandgap is narrower than that previously thought prompted a reexamination of the effective mass issue also [310]. Accounting for the substantial nonparabolicity that can cause

2.9 InN

an overestimate of the mass because high doping leads to a band-edge effective mass of 0.07m0, which is what is recommended here as was done by Vurgaftman and Meyer [152]. Turning our attention to holes, valence band mass parameters have been calculated by Yeo et al. [67] using the empirical pseudopotential method, and also by Pugh et al. [163] and Dugdale et al. [153] using more or less the same technique. The results of the first two investigations are quite similar, Pugh et al. [163] employed three different levels of computation comprising first-principles total energy calculations, semiempirical pseudopotential calculations and kp calculations. Band structures were obtained from each method in a consistent manner and were used to provide effective masses and kp parameters. These parameters are useful in investigating the electronic structure of alloys and quantum well heterostructures. These valence band parameters are the recommended values with the caveat in that the lower InN energy gap may require a downward revision of the light-hole mass. The parameters concerning the bandgap-related issues for wurtzitic InN recommended by Vurgaftman and Meyer [152] are tabulated in Table 2.19. A compilation of the dispersion in the effective mass for both the conduction band and various valence bands as obtained by various computational methods, as well as parameters used in the description of the bandgap for wurtzitic InN, particularly, in the context of empirical pseudopotential method, as described in Ref. [55], are tabulated in Table 2.20. Christensen and Gorczyca [31] predicted a hydrostatic deformation potential of
4.1 eV for wurtzite InN, which compares to a smaller value of
2.8 eV calculated by Kim etal. [180].Inthe absence of any predilection forany of the tworeports, averaging the two leads to a ¼
3.5 eV. In the absence of any calculations of the valence band deformation potentials, appropriating the parameter set specified above for GaN could be a good default at this point. The elastic constants measured by Sheleg and Savastenko [81], early on more refined values arrivedby calculations suchasthe set reported by Wright [179], are available, which are C11 ¼ 223 GPa, C12 ¼ 115 GPa, C13 ¼ 92 GPa, Table 2.19 Recommended band structure parameters for wurtzitic InN [152].

Parameter

Value

Parameter

Value

Parameter

Value

Eg (eV, low temperature) a (meV K
1) b (K) Dcr (meV) Dso (meV) == me =m0 m? e =m0 a1 (eV) a2 (eV)

1.5–1.8 0.245 624 0.040 0.005 0.07 0.07
3.5
3.5

A1 A2 A3 A4 A5 A6 A7 (meV Å) d13 (pm V
1) d33 (pm V
1) d15 (pm V
1) Psp(C m
2)


8.21
0.68 7.57
5.23
5.11
5.96 0 (default)
3.5 7.6 5.5
0.042 (
0.041)

D1 (eV) D2 (eV) D3 (eV) D4 (eV) D5 (eV) D6 (eV) c11 (GPa) c12 (GPa) c13 (GPa) c33 (GPa) c44 (GPa)


3.7 4.5 8.2
4.1
4.0
5.5 223 115 92 224 48

See Tables 2.27 and 2.28 for details related to the elastic constants, piezoelectric constants, and spontaneous polarization charge. Any dispersion among the tables is a reflection of the uncertainty in the available parameters.

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200

Table 2.20 Effective masses and band parameters for wurtzitic InN.

Parameter ==

me m? e == mhh == mlh == mch m? hh m? lh m? ch A1 A2 A3 A4 A5 A6 A7 D1

anisotropic

isotropic

A

B

C

D

0.138 0.141 2.438 2.438 0.140 2.661 0.148 3.422
7.156
0.244 6.746 3.340
3.208
4.303 0.072 0.214

0.137 0.140 2.493 2.493 0.137 2.599 0.157 1.446
7.298
0.441 6.896 3.064
3.120
3.948 0.103 0.084

0.11 0.10 1.56 1.56 0.10 1.68 0.11 1.39
9.62
0.72 8.97 4.22
4.35

0.11 0.10 1.67 1.67 0.10 1.61 0.11 1.67
9.28
0.60 8.68 4.34
4.32
6.08 0

0.10 0.10 1.431 1.431 0.106 1.410 0.196 0.209
9.470
0.641 8.771 4.332
4.264
5.546 0.278 0.0375

0.10 0.10 1.350 1.350 0.092 1.449 0.165 0.202
10.841
0.651 10.100
4.864
4.825
6.556 0.283 0.0375

0

Effective masses in units of free-electron mass m0, Luttinger-like parameters Ai (i ¼ 1, . . ., 6) in units of
h2 =2m0 , and A7 in units of eV Å. The crystal field splitting energy D1 is given in units of meV. The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor [55]. Anisotropically screened and isotropically screened values are from Ref. [55]. A: effective masses and Ai are from Ref. [67] obtained through a line fit to the band structure; B: direct kp calculation in a 3D fit from Ref. [67]; C: Ai from Ref. [153] obtained through a Monte Carlo fitting procedure to the band structure and effective masses calculated using Equations 2.30 and 2.31; D: direct kp calculations for Ai from Ref. [153] and effective masses calculated using Equations 2.30 and 2.31.

C33 ¼ 224 GPa, and C44 ¼ 48 GPa. Other sets of parameters calculated by Kim et al. [180] and Davydov [201] are also available in the literature. For a more detailed discussion of elastic and piezoelectric coefficients as well as polarization issue, refer to Section 2.12 and Tables 2.25–2.27 and 2.28. Owing to the fact that the piezoelectric coefficients in InN have apparently not been measured, for consistency the theoretical values of Bernardini and Fiorentini [210], d33 ¼ 7.6 pm V
1, d13 ¼
3.5 pm V
1, and d15 ¼ 5.5 pm V
1, are suggested. In spite of the fact that the spontaneous polarization data for GaN/GaInN structures are not as conclusive as one would like at this point, most likely owing to relatively poor material quality, the value Psp(InN) ¼
0.042 C m
2 is consistent with a thorough comparison of experiment and theory [84]. Recommended band structure parameters for wurtzite InN are compiled in Table 2.19 [152]. 2.9.2 Zinc Blende InN

Experiments on zinc blend InN are very rare although this polytype has been reported [311]. The bulk of the reports comprise theoretical estimates of its band

2.9 InN Table 2.21 Luttinger parameters g1, g2, and g3 for zinc blende InN

obtained from a fit along the [1 1 0] direction along with those available in the literature, as compiled in Ref. [55]. Parameter

A

B

c1 c2 c3

7 0.97 1.22

3.27 1.26 1.63

A: empirical pseudopotential calculation by Fritsch et al. [55]; B: recommended values taken from Ref. [152].

parameters. The zinc blende variety has been projected to have a direct band alignment, with G-, X-, and L-valley gaps of 1.94, 2.51, and 5.82 eV, respectively [70]. However, this particular calculation was performed before the controversy in the bandgap of wurtzitic InN. If the arguments available in the literature and presented in the Section 2.9.1 dealing with wurtzitic InN were to hold in favor of the large bandgap, then the aforementioned bandgap values will be more credible. For the valence band, spin–orbit splittings in the range 3–13 meV have been projected [107,171,227]. Vurgaftman and Meyer [152] recommends the 5 meV value among them. As in the case of the bandgap, the effective mass value for the wurtzitic polytype is recommended for the zinc blende variety, that is, 0.07m0. As mentioned in the previous section, this value is arrived at after the nonparabolicity effects are accounted for. To reiterate, the range of values reported for the Wz InN is 0.10–0.14m0 [70,163,173]. The longitudinal and transverse masses for the X valley have been calculated to be 0.48m0 and 0.27m0, respectively [70]. The recommended Luttinger parameter set by Vurgaftman and Meyer [152] is g1 ¼ 3.72, g2 ¼ 1.26, and g3 ¼ 1.63, which is derived from the work of Pugh et al. [163], and the split-off mass is chosen to be mso ¼ 0.3m0 [70,163]. These as well as the other literature values of the Luttinger parameters for zinc blende InN are listed in Table 2.21. Fritsch et al. [55] calculated the effective masses for conduction and valence bands, the latter involving the light and heavy holes, as well as the spin–orbit split-off mass, which along with the anisotropy taken into account are listed in Table 2.22 for zinc blende InN. The data contain those obtained by FP-LAPW, EPM calculations, and those calculated with Luttinger parameters Equations 2.42 and 2.43. The EP parameter value given by Meney et al. [173] is more likely because the alternative value by Pugh et al. [163] implies too large a value for F. The resulting parameter set is EP ¼ 17.2 eV and F ¼
4.36. For the hydrostatic deformation potential, an average value of
3.35 eV from the theoretical [31,180] candidates of
2.2 to
4.85 eV has been recommended [152]. The valence band deformation potentials listed in Table 2.19 are compiled from the calculations of Wei and Zunger [107], Kim et al. [180], Tadjer et al. [70], and Van de Walle and Neugebauer [177] are av ¼
0.7 eV, b ¼
1.2 eV, and d ¼
9.3 eV. Elastic constants of C11 ¼ 187 GPa, C12 ¼ 125 GPa, and C44 ¼ 86 GPa are assumed from the calculations of Wright [179],

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202

Table 2.22 Effective masses for electrons (e), heavy holes (hh), light holes (lh), and spin–orbit split-off holes (so) in units of the free-electron mass m0 along the [1 0 0], [1 1 1], and [1 1 0] directions for zinc blende InN. [100]

[100]

[111]

[111]

[110]

[110]

Reference

me

mhh

mlh

mhh

mlh

mhh

mlh

mso

A B C

0.13 0.12 0.10

1.18 0.83 2.18

0.21 0.16 0.89

2.89 0.83 2.29

0.19 0.16 0.93

2.12 1.55 3.10

0.20 0.15 0.79

0.36 0.30 0.30

Compiled by Fritsch et al. [55]. A: after Ref. [55]; B: calculated from recommended Luttinger parameters from Ref. [152], using Equations 2.42 and 2.43; C: empirical pseudopotential calculation from Ref. [70].

Table 2.23 Parameters associated with the band structure for zinc

blende InN bandgaps are for low temperature [152]. Parameter

Value

Parameter

Value

Parameter

Value

alc (Å) at T ¼ 300 K E Gg ðeVÞ a (G) (meV K
1) b (G) (K) E Xg ðeVÞ a (X) (K) b (X) (meV K
1) E Lg ðeVÞ a (L) (meV K
1) b (L) (K)

4.98 0.78 1.5–1.8 0.245 624 2.51 0.245 624 5.82 0.245 624

Dso (eV) me ðGÞ ml ðX Þ mt ðX Þ c1 c2 c3 mso

0.005 0.07 0.48 0.27 3.72 1.26 1.63 0.3

EP (eV) F VBO (eV) ac (eV) av (eV) b (eV) d (eV) c11 (GPa) c12 (GPa) c44 (GPa)

17.2
4.36
2.34
2.65
0.7
1.2
9.3 187 125 86

which are similar to other calculated sets [180,181]. The recommended band structure parameters for zinc blende InN are compiled in Table 2.23.

2.10 Band Parameters for Dilute Nitrides

Dilute nitrides can be described as conventional III–V compound semiconductors, wherein a small N fraction on the order of a small percentage is added. Addition of even very small amounts of N causes substantial changes to the bandgap and the lattice contact of the host compound to the point that standard bowing parameters for the bandgap variation linear interpolation of lattice constants between the host material and zinc blende GaN or GaInN do not apply. A single bowing parameter is inadequate even if the goal is only to describe the energy gap for a relatively wide range of compositions [312]. The technological driving force is really compelling in

2.10 Band Parameters for Dilute Nitrides

that wavelengths of interest for short and long haul communications systems can be obtained on GaAs technology by adding small fractions of N into the GaAs or InGaAs lattice. The motivation here is in materials incorporating only a small percentage of nitrogen [313], because it is highly questionable whether more than 10–16% N can be incorporated stably. Even in cases when one can, the quality is very inferior. Annealing techniques employed to improve layer quality end up causing N segregation, reducing the fraction in the bulk. Having mentioned this, it should also be pointed out that N isoelectronic doping of compound GaP, which has an indirect bandgap, for LEDs [314] predates the flurry of activity aimed at imparting substantial changes in the band structure, which is the topic of this section. The band structure of dilute nitride compound semiconductors has been reviewed in the literature by Vurgaftman and Meyer [152]. The treatise here follows the same philosophy in that the N-containing GaAs followed InP is treated before segueing onto other compound semiconductors such as those based on Sb anion. The properties of the conduction band in dilute nitride semiconductors can be described in terms of the band anticrossing (BAC) model [315]. It should be mentioned that a small percentage of N usually has little effect on the valence bands. This twoparameter anticrossing model can be spun formally in terms of the many-impurity Anderson model within the coherent potential approximation. It can also be thought of as the interaction between a single, spatially localized N level and the conduction band of the underlying traditional As-, P-, Sb-based compound semiconductors. Alternatively, Lindsay et al. [316] predicted the identical fundamental bandgap by assuming that the interaction involves a weighted average of perturbed upper states as opposed to a single N level. If one neglects the effect on the valence bands completely, the energy dispersion relation for the two-coupled bands within the BAC model can be expressed as   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  C E ðkÞ ¼ ð2:48Þ E ðkÞ þ E N ½E C ðkÞ þ E N 2 þ 4xV 2 : 2 Here, EC(k) is the conduction band dispersion of the nominal nonnitride semiconductor (e.g., GaAs in the case of GaAsN), EN is the position of the nitrogen isoelectronic impurity level in the nonnitride semiconductor, V is the interaction potential between the two bands, and x is the N mole fraction. The energy dispersion relation for the twocoupled conduction bands in GaAs0.99N0.01 showing the characteristic anticrossing is plotted in Figure 2.15. As can be gleaned from the band structure treatment for standard nitride semiconductors, any temperature dependence arises from the shift of the conduction band dispersion EC(k) not that of the valence band. In the dilute semiconductor case also, the bandgap is assumed to follow the Varshni formula of Equations 2.34 and 2.35, but with EN taken temperature independent. One consequence of this assumption is a considerable weakening of the variation of the fundamental energy with temperature [317]. The weak shift of EN with applied pressure necessitates a fresh look at the deformation theory. The strain dependence of the E transitions can be determined by substituting the applicable deformation potentials of the host nonnitride semiconductor to obtain EC(k), followed by deriving E (k) from Equation 2.48. The implementation of Equation 2.48 requires input of

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2.4

T = 300 K

GaAs0.99N0.01

Energy (eV)

2.2

E+

2.0

1.8 EN 1.6

EC E–

1.4

1.2 –0.15

–0.10

–0.050

0.0

0.050

0.100

0.15

Wave vector (1 Å–1) Figure 2.15 Conduction band dispersion relations for GaAs0.99N0.01 at 300 K from the band anticrossing model (BAC) [319] (solid curves). For comparison, the unperturbed GaAs conduction band and the position of the nitrogen level are shown as the dotted and dashed curves, respectively [152].

band parameters of the host semiconductor. Although, we do not reproduce the nonnitride parameters in this work to preserve space, all of the required values are tabulated in a review [92]. The more detailed kp theory can also be applied. The BAC model can be extended to treat spin-doubled conduction bands, valence band, and nitrogen impurity bands by modifying the eight-band kp theory [318–321]. Coupling of the nitrogen band to the X and L valleys have also been introduced [322,323] but at the expense of additional complexity, which may not be warranted. Unless compelled, it is preferable to stick to the simple two-parameter fit of Equation 2.48. It has been pointed out that the fixed position of the nitrogen level with respect to the vacuum level implies a tandem shift with the valence band maximum of the host nonnitride material. To account for the experimental observation of nonnegligible deviations from referencing to the valence band offset, a separate nitrogen level for each host material is typically specified. Vurgaftman and Meyer [152] suggest that the valence band offset for an unstrained dilute nitride be set equal to that of the host semiconductor. As mentioned above, within the realm of the BAC model, the primary effects of the nitrogen are on the conduction band. Further,

2.10 Band Parameters for Dilute Nitrides

even the 10-band model does not shift the valence band maximum in the absence of strain or quantum confinement, while of course influencing the hole dispersion relations. Although a finite type-I or type-II offset in strained structures have been reported on the basis of experiments, they are not sufficiently compelling to deviate from a null offset relative to the host in the absence of strain. Despite its simplicity, Equation 2.48 provides a basis for describing material properties, such as the fundamental energy gap, which are governed by the transition from E
to the top of the valence band, the temperature dependence of the gap, the electron effective mass, and the characteristics of the upper band Eþ. It must be kept in mind that within the theory of Lindsay et al. [316], there is not necessarily a single well-defined Eþ band. Somewhat in similar vein, the extent to which the BAC representation may be considered fundamentally realistic is still a matter of active discussion [315,324]. Clearly, the BAC model considers only a single nitrogen level on a substitutional lattice site or a narrow impurity band formed from such levels. In the process, it neglects mixing with the L and X valleys, and more complex nitrogen behavior in the semiconductor such as the formation of nitrogen pairs and clusters. In contrast, the more complicated pseudopotential calculations that consider some of these issues are computationally demanding [325–327]. In addition, the numerical results do not lend themselves to simple formalism such as Equation 2.48. Buttressing the simpler BAC approach is the discovery by Lindsay et al. [316] that the verifiable prediction of the dependence of the bandgap on the N content may be unaffected by generalizing to a multiplicity of higher lying states. Let us now turn our attention to specific dilute nitride semiconductor systems. 2.10.1 GaAsN

In conjunction with early investigations motivated by visible LED development, it has been known that small quantities of nitrogen in GaAs and GaP form deep-level impurities [314]. However, it has only been after the advent of nitride semiconductors that also paved the way for investigating traditional compound semiconductors, such as GaAs with N content definitely beyond the quantities ( 1% or more) used for doping experiments [328,329]. On the flip side of this, reports [330] of the incorporation of small amounts of As into GaN are neither common nor easy. The presence of As in GaN has been stated to cause modified surface reconstruction and/or act as surfactant or be a source of dopant impurities. Even though care must be exercised, PL measurements can be used to determine the bandgap, provided the relative position of the particular transition energy is known, and have been applied to GaAsN with N fractions up to 1.5% in an effort to determine the dependence of the energy gap on N composition [331]. As compared to other ternaries, a large bandgap bowing parameter of 18 eV was found, which for small compositions is equivalent to a linear model [332]. Early theoretical studies projected bandgap bowing parameters based primarily on the dilute nitride semiconductor with large N content [333–341]. Although the large bowing parameter was originally supposed to produce a semimetallic overlap at intermediate

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compositions [336], more detailed investigations that followed led to reduced bowing parameter with increasing composition [338,339], with experimental backing based on investigations of Bi and Tu [342], who studied N compositions as large as 15%. In discord with the above-mentioned linear dependence of the bandgap for small N fractions, subsequent investigations pointed to a highly nonlinear reduction in the energy gap for small N compositions [343–349]. Another noteworthy discovery was the realization of a significant weakening of the temperature and pressure dependencies of the bandgap for GaAsN and also GaInAsN with small In fractions [346,350,351]. In aggregate, it is clear that a simple bowing approximation could not adequately describe the GaAsN alloy. In this vein, Shan et al. [352] proposed the band anticrossing model and confirmed a weak pressure dependence for the nitrogen band transitions with a deduced deformation potential of
1.2 eV. To put matters in context, the density functional calculation of Jones et al. [353,354] also predicted reduced pressure dependence without invoking the BAC model. Additional report supporting the BAC model has been the finding by Skierbiszewski et al. and others pointing to a significantly heavy electron mass in GaInAsN [355–361]. On the anticorrelation side is another set of measurements by Young et al. [362] who found a reduction in the effective mass with increasing N content, in direct conflict with the BAC model that predicts an increase even at the zone center. Increasing mass with increasing N content, however, appears to hold. Consistent with the overall expected behavior of dilute nitride semiconductors based on the BAC model, the temperature dependence of the bandgap has been confirmed to be notably weaker in GaAsN than in GaAs [317,363]. In a similar vein, GaAsN film electroreflectance [364] resolved both the E
and Eþ transitions, the band description of which can be seen in Figure 2.15. Extending to a ternary, the bandgap reduction was also observed in nitrogen-implanted Al0.27Ga0.73As samples [365]. The transition between the doped and alloyed materials was studied by Zhang et al. [366,367]. Here, doped implies quantities manifesting themselves as dopants without radical changes incurring on the host material. The alloy material implies that the N concentration is sufficiently high to cause substantial changes in the host material. Zhang et al. observed evidence for impurity banding at N concentrations as small as 0.1% N. Zhang et al. also proposed an alternative method for the characterization of the bandgap energy, which is not based on the BAC model [368]. This transition point was quantified as 0.2% by Klar et al. [369]. Figure 2.16 depicts the fundamental energy bandgap that is between the valence band maximum and the E
conduction band minimum, as a function of N fraction, x, for GaAs1
xNx at 300 K. Also shown is a curve with a constant bowing parameter of 18 eV (dotted line) along with another incorporating a variable bowing parameter of (20.4–100x) eV, as suggested in a review by Vurgaftman et al. [92] (dashed line). Ostentatiously, the BAC model predicts a substantially higher energy gap beyond the N fraction of 1.5%. The available experimental data (points in Figure 2.16), compiled in Ref. [315], show much better consistency with the BAC model than with either of the two utilizing bowing parameters. It should be reiterated that GaAs1
xNx alloys with x > 5% become increasingly difficult to grow while retaining quality. As such, compositions with large N fraction

2.10 Band Parameters for Dilute Nitrides

GaAs 1.4

Bandgapenergy (eV)

GaAs1–xNx 1.2

1.0

BAC 0.8

C = 20.4 –100 xeV C = 18 eV Experimental data

0.6 0.00

0.01

0.02

0.03

0.04

0.05

N mole fraction, x Figure 2.16 Energy of the fundamental bandgap in GaAsN as a function of nitrogen concentration x (a) from the BAC model (solid curve), (b) using the variable bowing parameter from the review of Vurgaftman et al. [92] (dashed curve), and (c) using a constant bowing parameter (dotted curve). For comparison, the available experimental data as compiled in Ref. [315] are also plotted (circles) [92,152].

may not have the technological significance [313]. On the theory side, no upper limit on the N compositions beyond which the BAC model becomes invalid has been reported. Despite some spread in the reported values for the two primary parameters EN, 1.65–1.71 eV, referenced with respect to the GaAs valence band maximum, and V (2.3–2.7 eV), the values reported by Shan et al. [352], namely, EN ¼ 1.65 eV, V ¼ 2.7 eV, are suggested. In discord with the conventional BAC model, which assumes that the addition of N has little effect on the valence bands, two reports noted a larger than expected heavy-/ light-hole splitting in GaAsN containing a small percentage nitrogen [370,371]. This implies a strong bowing in the valence band shear deformation potential b, although the increase in the deformation potential is inconsistent in these two reports. Expanding their earlier work, Egorov et al. [372] again observed strain-induced splitting of light-hole and heavy-hole bands of tensile-strained GaAsN. The observed dependence of the bandgap in unstrained GaAsN on the nitrogen content differed substantially from that predicted by the theory assuming that the bandgap in GaAsN

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can be reduced to zero. In cases like this when there is discrepancy, the default position is to underscore the importance of additional investigations. An issue of importance in device applications is the nature and magnitude of the band alignment at a GaAs/GaAsN heterojunction. While the BAC model implies that the GaAsN conduction band minimum must lie below that in GaAs, it is not clear whether the valence band maximum in GaAsN should exhibit any relative shift. Various predictions disagree on this issue [334,341]. On the experimental side, X-ray photoelectron spectroscopy data suggest a type-II band alignment [373] but with quite large error bars on the valence band offset (VBO). This type of band alignment was supported by PL measurements [374]. Follow-up investigations utilizing not just optical but electrical characterizations as well have concluded a definite type-I alignment [375–379]. A point of discord may lie in the fact that the built-in strain was not completely relaxed in any of the heterostructures covered in the aforementioned investigations. Egorov et al. [377] accounted for strain effects and deduced that the band offset for unstrained GaAs0.98N0.02 with respect to GaAs. The value of the valence band discontinuity between GaAs and GaAsN0.02 at 18 K was evaluated to be 15 5 meV, taking into account the bandgap of GaAsN alloy [372]. On the contrary Egorov et al. [379] reported the band alignment of InxGa1
xAs/GaAsN heterojunctions to be type I or type II, depending on the In content x, a point that needs to be confirmed by additional measurements. The BAC model parameters recommended by Vurgaftman and Meyer [152] for GaAsN and all of the other dilute nitrides for which information is available are summarized in Table 2.24. 2.10.2 InAsN

Another binary compounded with N is InAsN, which garnered early theoretical interest [334,380], followed by tight binding calculation focused on the effects of nitrogen clustering in the alloy [381]. Experimental investigations of this dilute nitride have been reported [382–386]. Measurements of the electron effective mass in this alloy indicated a large increase [380,390], analogous to that in GaInAsN [355].

Table 2.24 Band anticrossing (BAC) model parameters for some

of the dilute nitride semiconductors [152]. Parameters

EN w.r.t. VBM (eV)

V (eV)

GaAsN InAsN Ga1
xInxAsN GaPN InPN Ga1
xInxPN InSbN

1.65 1.44 1.65(1
x) þ 1.44x
0.38x(1
x) 2.18 1.79 2.18(1
x) þ 1.79x 0.65

2.7 2.0 2.7(1
x) þ 2.0x
3.5x(1
x) 3.05 3.0 3.05(1
x) þ 3.0x
3.3x(1
x) 3.0

2.10 Band Parameters for Dilute Nitrides

While the authors [385,386] appear to argue that the BAC model could not account for any increase greater than doubling of the mass in the nitrogen-free host material, upon closer examination of the model it becomes evident that their view is contradicted, and also by the results available for GaInAsN [355], clearly displaying a similarly large increase in mass. Consequently, it is reasonable to assume that the BAC model is applicable to InAsN. At the same time, however, it should be stated that the available information is somewhat incomplete and future investigations may alter this assertion. Extracting the position of the nitrogen level with respect to the valence band maximum in the host InAs from the valence band offsets tabulated in a review article [92], a value for EN ¼ 1.44 eV is recommended. The measurements of Naoi et al. [382] are consistent with values for the potential V ranging between 1.9 and 2.3 eV. A round figure of V ¼ 2.0 eV could be a good default. 2.10.3 InPN

A few experimental studies of InP1
xNx are available in the literature [387,388]. Because the bandgaps and valence band offsets of GaAs and InP differ only slightly, it would follow that the BAC model would apply equally well to InPN. Yu et al. [388] derived band structure parameters in which the GaAs/InP VBO was assumed to be 0.35 eV. However, Vurgaftman et al. [92] recommended a value of 0.14 eV, but considering the result EN ¼ 2.0 eV by Yu et al. led them to update their recommended value to EN ¼ 1.79 eV, both being with respect to the valence band maximum of InP. [92] A reexamination of the coupling potential that is most consistent with the data of Yu et al. [388] led Vurgaftman et al. [92] to recommend V ¼ 3.0 eV. 2.10.4 InSbN

Even though InSb has the smallest bandgap for the conventional III–V binaries, addition of N has attracted some attention because of the potential that even longer wavelengths, beyond those accessible by InSb, can be accessed. Murdin et al. [389– 391] made experimental observations that the effective mass in InSb1
xNx increases despite a considerable reduction in the bandgap, which is consistent with all the other dilute nitride semiconductors discussed above. The authors derived EN ¼ 0.65 eV. This compares with EN ¼ 0.85 eV above the top of the valence band. At this stage, the experimental values are recommended as default. The same authors also reported V ¼ 2.2 eV and supplemented the minimal BAC dispersion relation of Equation 2.48 with an additional shift of the level of nitrogen position with increasing N fraction [389,390]. It should, however, be pointed out that it appears that their value for V underestimates the observed bandgap decrease [389]. This led Vurgaftman et al. [152] to recommend V ¼ 3.0 eV, which should yield better consistency with the available data, albeit little. Because the effect of nitrogen on the band structure of InSbN is significant in terms of the light-hole dispersion, the full 10-band kp model may be required for any realistic calculations.

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2.10.5 GaPN

The early LED [314] developments utilized N-doped GaP because of the unique band structure of this binary. It is an indirect gap semiconductor with both X and L valleys lower in energy than the G point. Nitrogen acts as an isoelectronic impurity in GaP and has been employed as the active material of visible LEDs [314] until the advent of other conventional direct bandgap and III–V and nitride-based varieties. Initial studies of GaP with an alloy-like concentration of N were reported by Baillargeon et al. [392,393]. Miyoshi et al. [394] investigated the transmogrification of the GaPN luminescence spectrum with increasing N content and were able to observe the emission from excitons bound to nitrogen pairs for x < 0.5%. Bi and Tu [395] reported GaP1
xNx with x as large as 16% using gas-source molecular beam epitaxy. A number of theoretical reports predicted that GaPN retains its indirect gap nature up to relatively large (beyond dilute) N concentrations [334,339,341]. Shan et al. [396,397] reexamined this view of indirect bandgap within the BAC model, and in the process found out that the anticrossing between G-valley states and the N impurity band moves E
below the X valley for arbitrarily small values of x. They derived V ¼ 3.05 eV employing the well-established value of EN ¼ 2.18 eV, relative to the valence band maximum. This places the nitrogen level slightly below the conduction X valley. As in the case of GaAsN whose band structure is less dependent on pressure and temperature than its host material GaAs, GaPN too sports a decrease in the pressure dependence of the fundamental transition with a deformation potential of
1.2 eV [396]. The temperature dependence of the fundamental gap also sees a reduction [398,399]. As in the case of all the dilute nitrides, a large increase in the electron effective mass is observed [400]. Congruous with popular LED materials for decades, strong luminescence for small N fractions [401,402], occurring despite the indirect gap of the GaP host, is another feature of this material. The evolution of transitions due to isolated N centers, N pairs, and N clusters that hasbeenobservedforx < 1%,andthemixingwithX-andL-valleystates,however,cannot be described by the simple BAC model [403–405]. For example, Buyanova et al. observed a sudden reduction in the radiative lifetime of the fundamental transition for x > 0.5%, which they attributed to an effective indirect-to-direct crossover [406]. The wealth of phenomena reported for GaPN can perhaps beexplainedwith a more flexible theoretical approach such as the supercell pseudopotential formulations of Kent and Zunger [324– 326]. Even though the accuracy of the BAC model is more limited in GaPN than in the other dilute nitride alloys due to the proximity of the X valley and the plethora of complex experimental observations for intermediate compositions, the parameter set EN ¼ 2.18 eV and V ¼ 3.05 eV of Shan et al. [398] can be used as default values. 2.10.6 GaInAsN

Let us now extend our treatment of the effect of N to ternary alloys. Having established BAC parameters for GaAs1
xNx and InAs1
xNx, they need to be determined for the

2.10 Band Parameters for Dilute Nitrides

Ga1
yInyAs1
xNx alloy whose host material is InGaAs. Owing to applications in long haul communications systems, most of the technological interest among all the dilute nitrides has so far focused on this quaternary. Additional impetus can be found in its applications to solar cells and photovoltaics that can be grown on GaAs substrates rather than InP, the former having a more established and less expensive technology base. Addition of N in InGaN narrows the bandgap for In concentrations, while at the same time providing tensile strain compensation far less than those required in conventional GaInAs quantum wells that are under compressive strain, countering the composition-induced reduction in the bandgap. Moreover, the wavelengths near 1.55 mm are not accessible with standard InGaAs-based quantum wells on GaAs substrates, which necessitates the use of InP substrates, which allow employment of larger concentrations of In. Incorporation of N in InGaAs can mitigate the situation and allow the use of GaAs substrates even for the extended wavelength for long haul fiber-based communications systems. Specific to the discussion of the bandgap, the applicability of the BAC model to the case of GaInAsN is quite well established [320,321,351,352,355,357] in that much of what is reported for the dilute nitrated binaries discussed above has also been found to be applicable to this quaternary. Investigations employing very low In fractions [351,352,355], for example, on the order of 10%, typically found no differences of significance from GaAsN except of course the decrease in the InGaAs energy gap [92]. For materials with larger In fractions, prepared on both GaAs [407,408] and InP [409] substrates, Zhukov et al. [410] proposed an alternative model. Pan et al. [359] took EN to be independent of the In concentration and employed V ¼ 2.5 eV. Although Choulis et al. [320,321,411] utilized the same assumption with respect to EN, their value for the coupling potential was considerably lower: V ¼ 1.675 eV. A similar value (V ¼ 1.7 eV) was independently deduced by Polimeni et al. [408] for In compositions ranging from 25 to 41%. In contrast, Sun et al. [412] found that, depending on the particular transition between the conduction and valence subbands, V in the range of 2.8–3.0 eV was imperative to account for the luminescence data for InGaAsN/GaAs quantum wells having an In content of 27.2%. The finding of a smaller bandgap reduction in GaInAsN than in GaAsN with N fraction is in fact expected and is due to the ordering of the nitrogen atoms in the InGaAs matrix [413,414]. There is some experimental evidence for carrier localization in the presence of both In and N in the quaternary alloy [415,416]. In one investigation [416] a series of five distinct transitions, which were attributed to five different environments for the N atom in the alloy, have been reported. Vurgaftman et al. [92] attempted to provide the best available parameters for the InGaAsN quaternary, while maintaining consistency with the parameters recommended above for GaAsN and InAsN. Accordingly, the position of the nitrogen band EN should be determined from the shift of the valence band offset in GaInAs [92]. This should include the small, yet nonnegligible, bandgap bowing with composition. This leads to a smooth variation of EN between 1.65 eV (for GaAsN) and 1.44 eV (for InAsN) and is consistent with the intuitive expectation, the basis of which is in no contradiction with any definitive experiments in that the position of the nitrogen level should not vary with respect to vacuum. Within this framework, Vurgaftman et al. [92] proposed a

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bowing of the coupling potential V in the Ga1
yInyAsN alloy: V ¼ 2.7(1
y) þ 2.0y
3.5y(1
y) eV. Admittedly, this parameterization does not fully agree with the experimental results for this quaternary [320,321,351,352,355,359,408]. But, it is nearly consistent with the median values and the recommended binary end points. As in the case of the ternaries, any strain must be added to the host semiconductor properties in the basic BAC model while employing these parameters. 2.10.7 GaInPN

It must be mentioned at the get go that there are not as much data available in this quaternary system. One report of the energy gap in bulk Ga0.46In0.54PN is available in the literature [417]. According to estimates based on the VBO dependence in GaInP, the quaternary alloy under discussion represents a special case in that one would expect a proximity of the nitrogen energy level and the host conduction band edge, perhaps within as little as 10–20 meV [92]. If one presumes, for the sake of discussion, that the BAC model remains valid in this limit, the reported energy gaps allow one to obtain a coupling potential in the 2.1–2.3 eV range. Considering that this is considerably smaller than the recommended values for both InPN and GaPN, by analogy with GaInAsN one could surmise a bowing of the Ga1
yInyPN interaction potential: V ¼ 3.05(1
x) þ 3.0x–3.3x(1
x) eV. It is noteworthy that the bowing parameters derived for the two quaternaries are quite similar. It should again be underscored that further studies are needed to confirm and/or update this value. 2.10.8 GaAsSbN

In addition to the mixed cation ternary host materials, mixed anion host materials also present opportunities. Among them is the alloy host GaAsSb that produces the quaternary alloy GaAs1
x
ySbyNx [418–420]. This quaternary has the potential for reaching long wavelengths also on GaAs substrates. Unfortunately, because of the sparse nature of the data for this material, and the total lack of any reports on GaSb1
xNx, one can only conjecture that the procedure recommended above for GaInAsN be followed. The material-specific parameters, that is, the constant V ¼ 2.7 eV at least for Sb fractions 20% should be assumed.

2.11 Confined States

If the physical size of the semiconductor in any direction is comparable to the de Broglie wavelength for electrons in electronic processes and exciton diameter in lowtemperature optical processes, the size effects become important. The constriction can form one side, two sides, and three sides in the case of which the system represents three-dimensional, two-dimensional, one-dimensional, and zero-dimensional, as

2.11 Confined States

Figure 2.17 Schematic representation of three-dimensional, twodimensional, one-dimensional, and zero-dimensional systems in real space.

shown in Figure 2.17. The two-dimensional, one-dimensional, and zero-dimensional correspond to quantum wells, wires, and dots, respectively. From the transport point of view, when the size of the well in a quantum well structure is comparable to the de Broglie wavelength in the semiconductor forming the well, the conduction and valence bands are modified noticeably in that the density of states in both the conduction and valence bands are discretized. From the excitonic optical transitions point of view, the length scale is the Bohr radius. This picture is depicted schematically in Figure 2.17a and b for a wurtzite semiconductor bulk (3D system), quantum well (2D system), quantum wire (1D system), and quantum dot (0D system). Moreover, equally important is the modification of these energies in the semiconductor caused by strain effects. In short, eigenstates of strained-layer superlattices require the consideration of both strain and quantum size effects. Eigenstates applicable to the early versions of compound semiconductor based quantum wells without strain were modeled by the envelope function formalism of Bastard [421]. If this method is to be used, one has to calculate the strain-dependent bandgap first. Bastard’s formalism can then be utilized to determine the transition energies in quantum wells, which must be added to the strain component, as shown by Marzin [422]. Marzin developed a method for calculating the bandgap of a strained cubic semiconductor that, when employed in conjunction with the envelope function approximation, leads to eigenstates in quantized structures of strained systems (Figure 2.18). In Bastard’s method revised by Marzin, an 8 · 8 Kane Hamiltonian matrix [423] is employed to give an accurate description of a strained quantum structure. The

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Figure 2.18 Schematic representation of (a) a GaN/AlGaN multiple quantum well structure and the conduction and valence band edges at the G point, and (b) valence and conduction band diagrams of the same with the confined states indicated.

size of the Hamiltonian is justified because there are three valence bands, in addition to the conduction band, with spin-up and spin-down for each band. Similar to cubic semiconductors, the wurtzite phase also has one conduction band and three valence bands, heavy- and light-hole states as well as spin–orbit bands, each with spin-up and spin-down, necessitating the use of an 8 · 8 Hamiltonian. The strain effects in the GaN system have been treated by Gil et al. [98,424] and others, as reviewed in Ref. [59]. However, until the time when uniformly strained films can be grown, correlations to experiments will remain weak and reduce the level of confidence in the predictions especially and the analysis as a whole. This is particularly true for InGaN wells, as there is also phase separation to deal with, in addition to nonuniform strain. We shall once again emphasize that there exist many discrepancies concerning the determination of some important parameters for the relevant bulk material properties such as hole masses, bandgap energy bowing parameters, shear and deformation potentials, and band offsets that are necessary for determining and understanding the properties of confined structures. Calculations, using ab initio methods, have been applied to estimate the unknown but necessary parameters for the band structure calculations. Later on, Sirenko et al. [62] have performed envelope function calculations of the valence band in wurtzite quantum wells following the formalism of Rashba– Sheka–Pikus (RSP) developed for bulk wurtzite semiconductors. Employing a 6 · 6 Luttinger–Kohn model, Ahn [425] studied the effect of a very strong spin–orbit split-off band coupling on the valence band structure of GaN-based materials. Considering that the spin–orbit band is extremely important for GaN because of its very narrow

2.11 Confined States

spin–orbit splitting (10 meV), the spin–orbit split-off coupling was taken into consideration in the calculations. In addition, it was assumed that the electrons in QW are confined by the conduction band offset (DEc) and the holes by the valence band offset (DEv), the values of which are also of some controversy. When the size of the well is comparable to the Bohr radius of an exciton in the semiconductor forming the well, the exciton transition energies are modified. The Bohr radius is given by a0 ¼

4pes
h2 ; m  q2

ð2:49Þ

where es is the dielectric constant of the semiconductor. In GaN, due primarily to its large effective mass, this radius is about 28 Å, necessitating very small physical dimensions before noticeable quantization can occur. In an experimental quantum well, the wave function is constricted along the growth direction, which we shall term as the z-direction. In the orthogonal directions, the system is free. The problem is similar to that of a vibrating string with the ends held stationary. The vibration wavelengths are given by ln ¼

2L n

with n ¼ 1; 2; 3;

ð2:50Þ

where L represents the length of the string. In a quantum well, the rapidly varying Bloch waves will be affected by the barriers in the z-direction, and the effect can be lumped into an envelope function that is slowly varying. The wave function can be expressed as [49,426] Cn ðk? ; zÞ ¼

X n

f n ðk? ; zÞu n expðjk?  r ? Þ;

ð2:51Þ

where fn(k?, z) and uv represent the envelope and Bloch functions, respectively, and n is the subband index. The summation is performed for spin-up and spin-down of the conduction band where the value of six is assumed for the three valence bands. The wave function expression must be solved for the conduction band and the valence band with the envelope function satisfying Schr€ odinger’s equation for the particular potential barrier height [59]. For the case where the conduction band is almost s-like, the G7 state suffices. However, for the valence band, due to band mixing, a 6 · 6 Hamiltonian including all the three uppermost valence bands must be used. Moreover, if strain is present, which is the case in almost all nitride-based structures, the Hamiltonian must include the effect of strain as well. To complicate matters further, the piezoelectric effect induces large electric fields at the heterointerfaces, particularly, in samples utilizing InGaN wells. The envelope function must satisfy    X
h q Hnn0 k? ; þ V n ðzÞdnn þ Hvn0 ðeÞ yv;n0 ðk? ; zÞ j qz ð2:52Þ v0 ¼ E n ðk? Þyv;n0 ðk? ; zÞ;

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j 2 Electronic Band Structure and Polarization Effects

216

where m ¼ 1, 2 for the conduction band state and n ¼ 1, . . . , 6 for the valence band state. The details of how the dispersion of the conduction and valence band states can be found in Ref. [59]. Suffice it to say that the conduction band is formed of nearly slike states and can be considered nearly parabolic, ameliorating the confined state calculations, as will be discussed in the next section. In finding the total energy between the confined conduction and valence band states, one may assume that the majority of the contribution is due to the confinement energy in the conduction band because of the large disparity between the electron and hole effective masses in favor of the conduction band. However, when gain in semiconductor lasers is considered, the dispersion of valence band states must be taken into consideration. 2.11.1 Conduction Band

If the potential barrier is infinitely high, the wave vector in the z-direction will be quantized and assumes the discrete values of   2p np ; n ¼ 1; 2; 3; ð2:53Þ ¼ kzn ¼ ln L where L is the thickness of the quantum well. Assuming a parabolic band structure that satisfactorily describes the s-like conduction band, the confinement energy can be expressed as
h2 k2zn
h2 np2 ¼ n ¼ 1; 2; 3: ð2:54Þ DE conf ¼ 2m 2m L Taking into account the energy dispersion relationship in the x- and y-directions for a parabolic conduction band, we have  
h2 np2 þ k2x þ k2y n ¼ 1; 2; 3; ð2:55Þ E ¼ Ec þ 2m L with n ¼ 1, 2, 3, and Ec representing the conduction band edge. For a true one-dimensional wire along the x-direction, discretization along the ydirection would also occur in addition to the z-direction, giving rise to confined energy states (with complete confinement) of  
h2 np2 E ¼ Ec þ þ k2x þ k2y ; n; m ¼ 1; 2; 3 . . . : ð2:56Þ 2m L If we consider a semiconductor whose constant energy surface for conduction band in k-space is a sphere, such as the case in GaN, the volume of that sphere and the number of available states in k-space are proportional to k3 in terms of momentum and E3/2 in terms of energy, as shown in Figure 2.19a. The density of states per unit energy associated with that system is proportional to E1/2, again as shown in Figure 2.19a. The area and the number of available states in k-space in an ideal system confined in one direction only (representing quantum wells), which is often the z or the growth direction, is proportional to k2 or E, as shown in Figure 2.19b. The

2.11 Confined States

density of states in this case is given by m  =p
h2 and forms a staircase as shown in Figure 2.19b. If we continue and place a confinement in the x-direction in addition to the z-direction, which represents quantum wires, the line length and the number of available states in k-space is proportional to k in terms of momentum and E1/2 in terms of energy as shown in Figure 2.19c. The corresponding density of states takes the dependence of E
1/2, again as shown in Figure 2.19c. If confinement is imposed in all three directions, which represent the pseudoatomic or quantum dot state, the energy is discretized in all directions and the resultant density of states takes a deltalike function in energy, as shown in Figure 2.19d. The case for quantum dots where

(a)

kz

dN/dE ~E 1/2 3

N ~ k (N-E3/2)

ky

E

kx

(b) kz dN/dE N ~ k 2(N-E)

~Constant

ky E kx Figure 2.19 Constant energy surfaces and density of states for 3D in (a) , 2D in (b) , 1D in (c) , and 0D in (d) systems, respectively. The constant energy surface are represented by a sphere, circle, a line, and a point in 3D, 2D, 1D, and 0D systems, respectively, in semiconductor such as GaN conduction band that has a spherical constant energy surface in 3D.

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j 2 Electronic Band Structure and Polarization Effects

218

(c)

kz Bulk dN/dE ~ E–1/2 N ~ k(N-E 1/2 )

n=4

n=3 n=2 n=1

ky

E

kx

(d)

dN/dE ~ δ (E)

kz

Bulk n=4

n=3 n=2 n=1

ky E

kx Figure 2.19 (Continued )

physical dimension in all directions are scaled to the size of the wave function or smaller is treated in Volume 2, Chapter 5. Considering quantum wells formed in growth along the z-direction, the confinement occurs in the same direction and thus the wave vector is quantized. The component of the wave vector experiencing confinement is typically referred to as the z-component, out-of-plane component, or simply kz. In-plane components of the wave vector kx and ky, however, are not quantized and the usual energy momentum dispersion (E–k) diagram would apply. A pictorial description of the E–k diagram in a two-dimensional system with confinement along the z-direction for two quantum energy levels is shown in Figure 2.20. If the barrier potential is large but not infinite, the wave function outside the well decays exponentially, which is called the evanescent wave. No analytical solution exits for the subband energies when the potential barrier is not infinitely high. Graphical solutions treated in many textbooks on quantum mechanics and numerical solutions as well do exist. The problem is made even more complicated in semiconductors in

2.11 Confined States

E

E

E2 E1 π/Lz 2π /L z

kx ,k y

kz

kx Figure 2.20 Energy momentum dispersion relation in a twodimensional system with confinement along the z-direction. Shown on the left is the three-dimensional view while that on the right represent two slices depicting the E–kin-plane relationships for quantum levels.

that not only is the barrier not infinite but also the barrier and well materials do not have the same carrier mass. In this case, the boundary condition must be changed from the continuity of the derivative of the wave function in the z-direction to the continuity of the particle flux in the z-direction, that is,     1 qyB 1 qyW ¼  ; ð2:57Þ mB qz mW qz at Z L/2, assuming the origin of the z-axis to be in the middle of the well. The terms mB and mW represent the effective masses in the barrier and well materials, respectively. Likewise, fnB and fnW depict the envelope wave functions [427] in the barrier and well materials, respectively. The solution for the subband energies can be computed numerically. The conduction band minimum for GaN as well as AlN is at the zone center and twofold degenerate. The confinement energies for the GaN/AlGaN quantum wells can reasonably be estimated by means of the envelope function approximation in the same manner as that extensively used for the GaAs/AlGaAs material system [427,428]. Following the Weisbuch and Vinter notation, the low-lying conduction electron state can be represented by [427,428] X CðrÞ ¼ ½expðik? ðrÞucj ðrÞf n ðzÞ; ð2:58Þ j¼W;B j

where W and B represent the well and barrier materials, uc ðrÞ is the conduction band zone center Bloch wave function of GaN or AlGaN, fn(z) is a slowly varying envelope function, k? is the transverse (in-plane) wave vector, is the envelop wave function j [427], and the growth direction is along the z-axis. Because uc ðrÞ is the same for GaN and Al(Ga)N, Schr€odinger’s equation reduces to  
h2 q2
þ VðzÞ f n ðzÞ ¼ E n f n ðzÞ: ð2:59Þ 2m0 qz2

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j 2 Electronic Band Structure and Polarization Effects

220

In the above equation, m*(z) is the corresponding effective mass of the conduction electron, V(z) represents the profile of the minimum of the conduction band along the growth direction, and E(z) is the confinement energy. Assuming no doping in either regions, V(z) has a rectangular well-like profile. The solution of the confinement energies is similar to a particle in a box problem. The boundary conditions are 1 df n ðzÞ that fn(z) and mðzÞ dz be continuous across the interface. The latter is necessary to ensure the conservation of the particle current.

Al x Ga1–x N/GaN, x = 0.2, L B = L W = 2 nm

0.25

Energy (eV)

0.20 0.15 0.10 0.05 0.00

–2

–1

0

1

2

3

4

Distance (nm) (a) Figure 2.21 (a) The conduction band edge potential profile for a representative Al0.2Ga0.8N/ GaN single quantum well with a barrier and well thickness of 2 nm (or 20 Å) each for a Ga polar sample with the [0 0 0 1] direction pointing to the left. Polarization charge causes deviation from the square well and no screening due to free carriers is accounted for. The scales for the vertical and horizontal axes are in terms of eV and nm, respectively. Courtesy of V. Litvinov. (b) The conduction band edge potential profile for a representative Al0.3Ga0.7N/GaN single quantum well with a barrier and well thickness of 2 and 4 nm, respectively for a Ga-polar sample with the [0 0 0 1] direction pointing to the right.

Polarization charge causes deviation from the square well and no screening due to free carriers is accounted for r. The scales for the vertical and horizontal axes are in terms of eV and nm, respectively. Courtesy of V. Litvinov. (c) The conduction band edge potential profile for a representative In0.3Ga0.7N/GaN single quantum well with a barrier and well thickness of 2 and 4 nm, respectively, for a Ga-polar sample with the [0 0 0 1] direction pointing to the right. Polarization charge causes deviation from the square well and no screening due to free carriers is accounted for. The scales for the vertical and horizontal axes are in terms of eV and nm, respectively. Courtesy of V. Litvinov.

2.11 Confined States

Al x Ga1-x N/GaN, x = 0.3, LB = 4 nm, L W = 2 nm

0.5

Energy (eV)

0.4 0.3 0.2 0.1 0.0 –4

–2

0

2

4

Distance (nm)

(b)

In x Ga

1–x

N/GaN, x = 0.3, LB = 4 nm, LW = 2 nm

1.4 1.2

Energy (eV)

1.0 0.8 0.6 0.4 0.2 0.0 –4 (c)

–2

0

2

4

Distance (nm)

Figure 2.21 (Continued )

The case of quantum wells in the nitride system requires not just confinement effects due to barriers but also polarization-induced charge. The latter tends to distort the band profile due to the induced electric field. The details of the polarization charge and its effect on the quantum wells can be found in Section 2.12.5 and Volume 2,

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j 2 Electronic Band Structure and Polarization Effects

222

Chapter 5. Essentially, the conduction and valence band edges get skewed due to the field induced by polarization, as shown in Figure 2.21a and b for representative Al0.2Ga0.8N/GaN (Ga-polar) and Al0.3Ga0.7N/GaN (Ga-polar) single quantum well structures, the former with barrier and well thicknesses of 2 nm (or 20 Å) each, with [0 0 0 1] direction pointing to the left, and the latter with barrier and well thickness of 2 and 4 nm and [0 0 0 1] direction pointing to the right. The same for a representative In0.3Ga0.7N/GaN single quantum well with a barrier and well thickness of 2 and 4 nm, respectively, for a Ga-polar sample with the [0 0 0 1] direction pointing to the right is shown in Figure 2.21c. The band profile depicts the case with no screening. Free carrier induced screening would alter the profile. The calculated eigenstates, only the ground levels exist, for Al0.1Ga0.9N/GaN and Al0.2Ga0.8N/GaN as a function of quantum well thickness for a barrier thickness of 2 nm are shown in Figure 2.22. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. The calculated eigenstates for Al0.3Ga0.7N/GaN ground and the first excited states (only two states are available) for a barrier thickness of 4 nm as a function of quantum well thickness are shown in Figure 2.23. For increased activity in deeper UV devices, both

0.30

x = 0.1, level 1 x = 0.2, level 1 x = 0.2, level 2 LB = 2 nm

Alx Ga1–x N/GaN Dashed lines: without polarization

0.25

Energy (eV)

0.20

0.15

0.10

0.05

0.00 1

2

3

4

5

LW (nm) Figure 2.22 Calculated eigenstates, only the ground levels exist, for Al0.1Ga0.9N/GaN (lower curve) and Al0.2Ga0.8N/GaN (upper curve) for a barrier width of 2 nm as a function of quantum well thickness. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

6

7

2.11 Confined States

in terms of emitters and detectors, it is necessary to consider structures where AlxGa1
xN is used as the active part of the device in the form of a quantum well. Choosing AlN, arbitrarily, as the barrier for AlxGa1
xN, the eigenstates for two representative cases are presented. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. Shown in Figure 2.24 are the ground and first excited states as a function of well width for an AlN/Al0.4Ga0.6N quantum having a barrier width of 2 nm. It should be noted that only the ground and first excited states are available. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. The same structure but with AlN/Al0.5Ga0.5N, representing nearly the solar blind region of the Sun’s spectrum, is shown in Figure 2.25. Also shown in dashed lines are the energy levels without polarization, such as the case of a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. For the visible or nearly visible part of the spectrum, InGaN quantum wells typically with GaN barriers are employed. This is in part due to not only benefits gained by heterostructure devices but also technological reasons, the latter due to the decomposition of thick InGaN layers and inordinate amounts of ammonia required

0.45 0.40 0.35

AlxGa1–x N/GaN

level 1 level 2 L B = 4 nm, x = 0.3 Dashed lines: without polarization

Energy (eV)

0.30 0.25 0.20 0.15 0.10 0.05 0.00 1

2

3

4 LW ( nm )

Figure 2.23 Calculated eigenstates for Al0.3Ga0.7N/GaN for ground and the first excited states (only two states are available) for a barrier thickness of 4 nm as a function of quantum well thickness. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

5

6

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j 2 Electronic Band Structure and Polarization Effects

224

1.0

level 1 level 2 x = 0.4 L B = 2 nm

AlN/Al x Ga 1–x N Dashed lines: without polarization

Energy (eV)

0.8

0.6

0.4

0.2

0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

L W (nm) Figure 2.24 Calculated ground and first excited states (only two states are available) as a function of well width for an AlN/ Al0.4Ga0.6N quantum having a barrier width of 2 nm. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

to grow thick layers. In this vein, the calculated ground state (for x ¼ 0.1) and ground and excited states (for x ¼ 0.2) as a function of well width for an InxGa1
xN/GaN quantum well having a barrier thickness of 2 nm are presented in Figure 2.26. 2.11.2 Valence Band

In an attempt to determine the valence band subband structure, Suzuki and Uenoyama [59] calculated the band discontinuities from first-principles calculations and found them to be 0.11 and 0.43 eV for the valence and conduction bands of GaN/ Al0.2Ga0.8N. The elastic stiffness constants, taken from prior experimental data, employed for GaN were, in units of 1011 dyn cm
2, 29.6, 13.0, 15.8, 26.7, and 2.41 for C11, C12, C13, C33, and C44, respectively (see Table 1.24). For illustrative purposes, the valence band structure in unstrained 30 and 50 Å GaN/Al0.2Ga0.8N quantum wells is exhibited in Figures 2.27a and 2.28 where the strain due to the lattice and thermal mismatch are neglected. Bandgap discontinuities of 0.11 and 0.43 eV were adopted for the valence and conduction bands, respectively. The confinement energies in deep wells are inversely proportional to the effective mass in the growth direction and

2.11 Confined States

1.4 AlN/Al x Ga1–xN 1.2

Dashed lines: without polarization

level 1 level 2 x = 0.5 LB = 2 nm

Energy (eV)

1.0

0.8

0.6

0.4

0.2

0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

LW (nm) Figure 2.25 Calculated ground and first excited states (only two states are available) as a function of well width for an AlN/ Al0.5Ga0.5N quantum well having a barrier width of 2 nm, representing nearly the solar blind region of the Sun’s spectrum. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

directly proportional to the square of the well length. The HH and LH bands are not coupled. Consequently, the HH band can be construed as parabolic with little, if any, change in strain. This is because the C6v crystal symmetry of the bulk remains. The upper pffiffiffi (LH) and lower (CH) bands are coupled with a constant coupling coefficient 2D3 , which means that coupling of these bands is independent of kz and strain. The effective masses of the HH and LH bands are too heavy to cause substantial confinement energy, whereas the CH band with a lighter mass causes more split due to quantization and, in a sense, makes the crystal splitting (Dcr) larger. For comparison, in ZB structures the coupling between the light hole and the spin–orbit band is dependent on kZ and thus the bands change considerably with strain, and the in-plane heavy-hole mass becomes light and the in-plane light-hole mass becomes heavy. The effect of strain on the quantum well was also considered by Suzuki and Uenoyama [60], assuming coherently strained quantum wells. Consequently, the inplane lattice constant of the layers is assumed to be that of the substrate with a substantial effect on the subband structure of the quantum wells. The beneficial effects of strain in electronic and optoelectronic devices based on ZB crystals have

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j 2 Electronic Band Structure and Polarization Effects

226

0.8 0.7

Dashed lines: without polarization

In x Ga1–x N/GaN

x = 0.1 x = 0.2, first x = 0.2, second L B = 2 nm

Energy (eV)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

2

3

4

5

6

LW (nm) Figure 2.26 Calculated ground state (for x ¼ 0.1) and ground and excited states (for x ¼ 0.2) as a function of well width for an InxGa1
xN/GaN quantum wells having a barrier width of 2 nm. For x ¼ 0.2, the ground and the first excited states are available. Also shown in dashed lines are the energy levels without polarization, such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

been documented well [88]. This and the fact that there is some lattice mismatch between GaN and its ternaries make it imperative that the effect of strain on the properties of Wz quantum wells be considered. Figures 2.27b and 2.28b exhibit the valence band subband structure for 30 and 50 Å GaN/Al0.2Ga0.8N quantum wells. The biaxial strain was assumed to be 0.5% in the (0 0 0 1) c-plane. Superimposed is the valence band structure for 30 and 50 Å Wz GaN/Al0.2Ga0.8N quantum wells with 0.5% tensile strain in the c-plane. Results indicate that the wells get deeper for compressive strain and shallower for tensile strain. The density of states at the valence band maximum gets smaller for the compressive and larger for tensile strain. However, the change is very small as the symmetry remains as in the bulk with no further removal of the degeneracy. It must be mentioned though that uniaxial strain along the x- or the y-direction only causes the HH band to move to a higher energy and leads to a reduced density of states. Consequently, the effects of this type of strain resemble those in bulk GaN. In a sense, the Q well alone does not result in any special characteristic that would lead to much improved results for lasers [429]. Valence band confinement energies for relaxed and 0.5% compressively strained GaN/Al0.2Ga0.8N quantum wells in the c-plane are depicted in Figure 2.29. The data

2.11 Confined States

50

L = 30 Å

100

No strain

50

0

–50

0 Energy (meV)

Energy (meV)

HH1 LH1 LH2 –100 HH2 –150

–200 0

–50

L = 30 Å

j227

With 0.5 % compressive strain HH1

LH1

LH2 HH2

–100 –150

20 10 15 5 kx ,ky ,Wave number (× 106 cm –1)

–200 0

(a)

20 10 15 5 kx ,ky ,Wave number (× 10 6 cm –1) (b)

Figure 2.27 Plot of the upper valence band structure (HH and LH bands) in a 30 Å GaN/Al0.2Ga0.8N quantum well with bandgap discontinuities of 0.11 and 0.43 eV for the valence and conduction bands, respectively; (a) without strain and (b) with 0.5 compressive strain in the c-plane [60].

have been deduced from the calculations of Suzuki and Uenoyama [59] for 30, 40, 50, and 60 Å well thicknesses and band discontinuities of 0.11 and 0.43 eV for the valence and conduction bands, respectively. The other parameters utilized can be found in the tables presented in Chapter 1. 2.11.3 Exciton Binding Energy in Quantum Wells

Exciton binding energies in reduced dimensional systems vary from those of the bulk [430]. In the GaN/AlGaN system, one would expect the binding energy to go up, as the confinement gets stronger. If the well thickness is continually reduced, at some point the overlap with AlGaN becomes very noticeable. Toward zero well thickness, the binding energy should approach the value of AlGaN. Considering the strong localization and possible role of excitons in optical processes even at room temperature, it is imperative that the binding energy be known. Bigenwald et al. [431] considered the very problem of exciton binding energy in a GaN/Al0.2Ga0.8N system with results leading to the obvious conclusion that the effect of confinement is large and cannot be ignored. They calculated the exciton binding energies and oscillator strengths with the formalism developed [432] prior to the aforementioned investigation by applying a two-parameter trial function. Due to the anisotropy of the structure, the dielectric constant was globalized and the particle masses were weighted with the

j 2 Electronic Band Structure and Polarization Effects

228

50

100

L = 50 Å

Strain free

L = 50 Å HH1

50

0

0.5 % Compressive strain LH1

E n e r g y (m e V )

HH1 –50 HH2 –100 LH1

0

LH2

HH2

–50 –100

HH3

LH2

–150

–150 –200 0

–200 0

5

10

15

20

5

10

15

20

k x ,k y ,Wave number (× 106 cm–1)

kx,ky,Wave number (× 106 cm–1) (a)

(b)

Figure 2.28 Plot of the upper valence band structure (HH and LH bands) in a 50 Å GaN/Al0.2Ga0.8N quantum well with bandgap discontinuities of 0.11 and 0.43 eV for the valence and conduction bands, respectively; (a) without strain and (b) with a compressive strain of 0.5 in the c-plane [60].

0

HH LH CH

100

CH 1

Strained b ul k GaN

50

LH 2 Bulk Al 0.2Ga 0.8N

C onfin em ent en ergy (m eV)

LH 1

HH 2 CH 2

( 9 )1 HH 1 ( 9 )2 HH 2

HH 1

150 HH LH CH

200 0

50 Well width (Å)

Figure 2.29 Valence band confinement energies versus the thickness of the well in GaN/Al0.2Ga0.8N quantum wells. Courtesy of Professor Bernard Gil.

100

(

) 71 1

LH 1

(

71 )2

LH 2

( (

) CH 1 72 1 ) 72 2 CH 2

2.11 Confined States

j229

40

Exciton binding energy (meV)

35 B Exciton A Exciton 30

25

C Exciton 20

15 0

20

40

60

80

GaN Q Well thickness (Å) Figure 2.30 A, B, and C exciton binding energies as a function of the well width in a GaN/ Al0.2Ga0.8N system. Note that the exciton binding energies reported for GaN range from about 20 to 30 meV. The likely value of A exciton

binding energy in GaN is 21 meV. If so, the absolute values of the binding energies shown should be treated with caution. However, the trend with quantum well thickness holds, which is the reason for inclusion of this figure [431].

probability densities. The A, B, and C exciton binding energies computed as a function of GaN well thickness are shown in Figure 2.30. Two essential points stand out. The first point is that the G9v and G17v states are confined in the well as is the electron state. The largest binding energy corresponds to a well thickness of L  15 Å, which is when the spreading of both electron and hole functions in the barrier area minimal. The second point is that the relatively light G27v hole state leaks out of the well (for L < 100 Å) so that the electron–hole pair has a small binding energy and is almost constant for 20 < L < 100 Å. Caution should be exercised in applying the calculations of Ref. [432] for well thicknesses larger than about 100 Å, three times the Bohr radius. In GaN-based systems, the terms quantum well and superlattices have been used very liberally in that structures with well thicknesses well in access of the Bohr radius are referred to as quantum wells. The term superlattice requires that barriers are penetrable by the wave function, and further, the wave functions in adjoining wells overlap and form the superlattice minibands. If these standards are strictly applied at this point in time, there may not be much to discuss. Consequently, a conscious decision was made to treat many heterostructures with reduced dimensions, in at least one direction, as “quantum-confined structures.”

100

j 2 Electronic Band Structure and Polarization Effects

230

2.12 Polarization Effects

Solids are different from vacuum in that they respond to electric fields present or applied. There are three forms of polarization that are present at the atomic level [2]. One is due to partial or complete alignment of dipole moments of polar molecules with the electric field. When atoms forming the solid are different, as in binary, ternary, and quaternary semiconductors, and they have different electronegativities, any asymmetrical molecule has a permanent dipole moment, a process referred to as dipole orientation or paraelectric response. This component cannot keep pace with varying electric field above about 1010 Hz, causing a drop in the real part of the dielectric constant and a jump in the imaginary part (the loss part – appreciable loss factor). The loss is caused by dipoles attempting to respond to the field but seriously lagging in phase. At higher frequencies, the dipoles cannot follow the field and the effect is negligible including the loss factor. Electric field paves the way for rotation of dipole to participate in the electric displacement and align the dipoles collectively, barring thermal disturbances. In a completely or partially ionic solid, dipoles can be induced by relative motion of positive and negative ions under the influence of electric field, causing what is termed as the ionic polarization. Similar to the case of dipole orientation, this process cannot respond to frequencies above 1013 Hz (Reststrahlen frequency), causing yet an additional drop in the dielectric constant and a surge in the loss factor due to the phase lag. Again, as the frequency is increased further, above the Reststrahlen frequency, ionic motion cannot respond to the field and the loss factor due to this process diminishes. The third kind, which occurs in every dielectric, is called the electronic polarization. This is caused by displacement of electrons in an atom relative to the nucleus under the influence of electric field, in a sense deforming the electron shells. The electronic polarization remains at frequencies above Reststrahlen frequency, making the real part of the dielectric constant larger than unity. This process too cannot follow the field above 1015 Hz, above which the dielectric constant of the solid becomes very close to unity. Group III–V nitride semiconductors exhibit highly pronounced polarization effects. Semiconductor nitrides lack center of inversion symmetry and exhibit piezoelectric effects [209] when strained along h0 0 0 1i. Piezoelectric coefficients in nitrides are almost an order of magnitude larger than in many of the traditional group III–V semiconductors [209,433–437]. The strain-induced piezoelectric and spontaneous polarization charges have profound effects on device structures. The piezoelectric effect has two components. One is due to lattice mismatch (misfit) strain while the other is due to thermal strain (ts) caused by the thermal expansion coefficient difference between the substrate and the epitaxial layers. The low symmetry in nitrides, specifically, the lack of center of inversion symmetry present in zinc blende structure, may be interpreted as some sort of nonideality, which is not the case. Nonvanishing spontaneous polarization is allowed in an ideal wurtzite structure [212,438]. This spontaneous polarization is noteworthy, particularly when heterointerfaces between two nitride semiconductors with varying electronegativity are involved. This manifests itself as a polarization charge at heterointerfaces.

2.12 Polarization Effects

Spontaneous polarization was only understood fully not too long ago by King-Smith and Vanderbilt [439] and Resta et al. [440]. In heterojunction devices such as modulation-doped field effect transistors (MODFETs) where strain and heterointerfaces are present, the polarization charge is present and is inextricably connected to free carriers, which are indeed present. As such, polarization charge affects device operation in all nitride-based devices, particularly HFETs, and thus must be taken into consideration in device design unless nonpolar surfaces such as the a-plane are used. The quality of films on nonpolar surface has not kept pace with those on polar basal plane, which make the topic of discussion quite relevant. As mentioned above, polarization charge arises from two sources: piezoelectric effects and the difference in spontaneous polarization between AlGaN and GaN, even in the absence of strain. These charges exit in all compound semiconductors to varying degrees unless self-cancelled by the symmetry of the particular orientation under consideration such as the nonpolar surfaces/interfaces. In relative terms, spontaneous polarization is larger than the piezoelectric polarization in AlGaN/GaN-based structures. In the case of InGaN/GaN structures, spontaneous polarization is relatively small but not as small as the earlier predictions called for, but still noteworthy, as spontaneous polarizations in GaN and InN are not as different from one another. However, the strain-induced piezoelectric polarization can be sizable. If and when defect-associated relaxation occurs reducing the strain in the films, the strength of the piezoelectric polarization is lowered. Spontaneous polarization and piezoelectric polarization affect the band diagram of heterostructures. The effects are very large and can easily obscure the engineered designs. Polarization is dependent on the polarity of the crystal, namely, whether the bonds along the c-direction are from cation sites to anion sites or visa versa. The convention is that the [0 0 0 1] axis points from the face of the N-plane to the Ga-plane and marks the positive z-direction. In other words, when the bonds along the c-direction (single bonds) are from cation (Ga) to anion (N) atoms, the polarity is said to be the Ga-polarity, and the direction of the bonds from Ga to N along the c-direction marks the [0 0 0 1] direction, which is generally taken to be the þz-direction. By a similar argument, when the bonds along the c-direction (single bonds) are from anion (N) to cation (Ga) atoms, the polarity is said to be the N-polarity, and the direction of the bonds from N to Ga along the c-direction marks the direction, which is generally taken to be the
z-direction. To shed further light, the Ga-polarity means that if one were to cut the perfect solid along the c-plane where one breaks only a single bond, one would end up with a Ga-terminated surface. The Ga- and N-polarity of a model GaN crystal is shown in Figure 1.3. A schematic representation of the spontaneous polarization in a model GaN/AlN/GaN wurtzitic crystal is shown in Figure 2.31. The spontaneous polarization Pspont (also commonly referred to as P0) in a solid has not always been well defined, although much better understanding of it has been emerging. Only those differences in P between two phases that can be linked by an adiabatic transformation that maintains the insulating nature of the system throughout are well defined. For example, one phase can be considered unstrained and the other strained. Vanderbilt proved that the polarization difference DP between the

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j 2 Electronic Band Structure and Polarization Effects

232

[0 0 0 1] Axis

P0AlN

P0GaN N

N

Ga

Ga

Al

Al

N

N

N

Al

Al

N

N

Ga

Ga

N

P0

P0 GaN

P0GaN

P0AlN

AlN

Figure 2.31 Schematic depicting the convention used for determining the polarity and crystalline direction in wurtzitic nitride films. The diagram shows the case for a Ga-polarity film with its characteristic bonds parallel to the c-axis (horizontal in the figure) going from the cation (Ga or Al) to the anion (N). The spontaneous polarization components P0Ga and P0AlN for a periodic GaN/AlN structure are also indicated with that for AlN having a larger magnitude. The

GaN

spontaneous polarization is negative and thus points in the [0 0 0
1] direction. Caution must be exercised here as there is no long-range polarization field, just that it is limited to the interface. The polarization in AlN is larger in magnitude than in GaN. There exists a difference in polarization at the interface, DP0 pointing in
direction for both GaN/AlN the [0 0 0 1] interfaces. The Born factor is defined in Equation 2.82 [433].

wurtzite and zinc blende phases could be calculated by considering an interface between the two phases and by defining Pspont to be zero in the zinc blende phase. In short, by calculating the integral of a quantum mechanical Berry phase along a line in the Brillouin zone from one end to the other in the bulk wurtzite symmetry leads to polarization P with respect to that in zinc blende (which is zero by definition because zinc blende is cubic and cannot have a spontaneous polarization in an infinite bulk periodic crystal). The Berry phase actually represents an overlap integral between the periodic part of the Bloch function at k and a neighboring k-point, k0 . Zorroddu et al. [212] and Bernardini et al. [433] showed that the charges accumulating at each interface in a self-consistent calculation can be obtained from the DP of the two bulk layers forming the heterointerface. The relation between the charge and P follows basically from Gauss’s law. The bound charge density rb ¼
ÏP. This means that across an abrupt interface with P1 on one side and P2 on the other side, one gets P P2
P 1 ¼ i rs (surface charge density at the interface with the appropriate signs). Even though it is overly simplistic, a graphical picture of polarization due to strain (piezo component) and heterointerfaces (spontaneous component), the latter is in the case of different ionicity, can be obtained, which is helpful. Shown in Figure 2.32 is a ball-and-stick diagram of a tetrahedral bond between Ga and N in Ga- and N-polarity configurations, showing the polarization vector due to the electron cloud being closer to the N atoms. Actually, the cumulative polarization due to the triply bonded atoms is along the direction of the single bond. The in-plane and vertical components of polarization due to pairs of atoms cancel one another if the tetrahedron is ideal.

2.12 Polarization Effects

Ga Polarity

[0 0 0 1]

N Polarity

N

[0 0 0 1]

P0 P0 P0

P0zr

P0

P0

Ga

Ga P0

P0zr

P0

P0

N

Ga

P0

N

j233

Ga

P0

N

Ga

Figure 2.32 Ball-and-stick configuration of an ideal GaN tetrahedron with proper c/a ratio and internal parameter u for both Ga and N polarity in a relaxed state.

However, when a Ga-polarity film is under homogeneous in-plane tensile strain, the cumulative z-component, [0 0 0 1] direction, of the polarization associated with the triple bonds decreases, causing a net polarization that would be along the ½0 0 0
1 direction, as shown Figure 2.33. In a nitrogen-polarity film, the same occurs except that the net polarization would be in the opposite, [0 0 0 1], direction. When an inplane and homogeneous compressive strain is present, the net polarization would be in the [0 0 0 1] direction in the Ga-polarity case and ½0 0 0
1 direction in the N-polarity case, as shown in Figure 2.34. Ga Polarity

[0 0 0 1]

N Polarity

N

P0

P0 P0 P0

P0zr

P0

P0 Ga

N

[0 0 0 1]

F

P0

P0

N

N

Ga

Ga P0 Ga

Figure 2.33 Ball-and-stick configuration of a GaN tetrahedron for both Ga and N polarity with a homogeneous in-plane tensile strain showing a net polarization in the [0 0 0 1] direction for Ga-polarity and [0 0 0
1] polarization for N polarity.

P0

P0

P0zr

P0

Ga

F

j 2 Electronic Band Structure and Polarization Effects

234

Ga Polarity

[0 0 0 1]

N Polarity

N

[0 0 0 1]

Ga

P0 P0 P0

P0zr

P0

Ga P0

N N

P0

P0 P0zr

P0

P0

N

Ga Ga

P0

Ga Figure 2.34 Ball-and-stick configuration of a GaN tetrahedron for both Ga and N polarity with a homogeneous in-plane compressive
direction for Ga strain showing a net polarization in the [0 0 0 1] polarity and [0 0 0 1] polarization for N polarity.

The same graphical argument can be used to attain a mental image of spontaneous polarization at heterointerfaces as well. For example, if we were to construct two tetrahedra, one representing a GaN bilayer and another on top of it representing an AlN bilayer, the top N atom shown in Figure 2.32 for the Ga-polarity configuration would make triple bonds with it. Because AlN is more electronegative than GaN, the net component of the polarization vector in the [0 0 0 1] direction in triply bonded N with Al is larger in amplitude than in the GaN tetrahedron, and there would be a net interfacial polarization in the ½0 0 0
1 direction even without strain. In short, the source of piezoelectric polarization is strain in an electronegative binary. That for spontaneous polarization is the change in electronegativity across an interface such as the AlN and GaN interface. Substrates upon which nitride films are grown lack the wurtzitic symmetry of nitrides. Consequently, the polarity of the films may not be uniform, as schematically depicted in Figure 4.21 where the section on the left is of Ga-polarity and the section on the right is of N-polarity, representing a Holt-type inversion domain. In this type of inversion domain, the wrong type of bonds, for example, Ga
Ga and N
N, are formed at the boundary projected on (1 1
2 0) plane. The structural and electrical details of the inversion domains observed and investigated in GaN can be found in Section 4.1.3. Inversion domains combined with any strain in nitride-based films lead to flipping piezoelectric fields with untold adverse effects on the characterization of nitride films in general and the polarization effect in particular, and on the exploitation of nitride semiconductors for devices. Such flipping fields would also cause much increased scattering of carriers, as they traverse in the c-plane. Having made the case, it should be mentioned that if proper measures are taken, the Gapolarity films grown by OMVPE and to a lesser extent by MBE are nearly or completely inversion domain boundary free even on sapphire substrates. However,

2.12 Polarization Effects

in the case of MBE, unless optimum AlN buffer layers are employed or N-polarity films are grown by incorporation of GaN initiation layers, inversion domain boundaries do occur, sometimes in high concentrations, see Section 3.5.6. The magnitude of the polarization charge converted into number of electrons can be in the mid-1013 cm
2 level for AlN/GaN heterointerfaces, which is huge by any standard. For comparison, the interface charge in the GaAs/AlGaAs system is used for MODFETs in less than 10% of this figure. An excellent review of the polarization effects can be found in Ref. [440]. The magnitude of the polarization charge is tabulated in Table 2.25 along with elastic coefficients taken from a series of publications by Bernardini, Fiorentini, and Vanderbilt. The data in bold are those reported in an earlier publication [433] and the remaining data points are taken from a later publication [212]. Following the initial reports of piezoelectric and spontaneous polarization [441], the authors returned to the topic [212] as the values of the initial parameters were not consistent with other reports [181]. Bernardini et al. [442] reanalyzed the polarization as obtained using the Berry phase method within two different DFT exchange correlation schemes. Specifically, the authors used the Vienna ab initio simulation package (VASP) and the pseudopotentials provided therewith, as in Ref. [180]. The newer calculations were carried out using both the generalized gradient corrected local density approximation (GGA) to density Table 2.25 Elastic constants and spontaneous polarization charge

in nitride semiconductors.


2

e33 (C m ) LDA GGA e31 (C m
2) LDA GGA p e31 LDA GGA C33 (GPa) GGA C31 (GPa) GGA P0 (C m
2) LDA GGA P0 (C m
2), ideal wurtzite structure R ¼
C31/C33 LDA GGA e31
(C13/C33)e33 (C m
2)

AlN

GaN

InN

1.46 1.8 1.5
0.60
0.64
0.53
0.74
0.62 377 94
0.081
0.10
0.090
0.032
0.578
0.499
0.86

0.73 0.86 0.67
0.49
0.44
0.34
0.47
0.37 354 68
0.029
0.032
0.034
0.018
0.40
0.384
0.68

0.97 1.09 0.81
0.57
0.52
0.41
0.56
0.45 205 70
0.032
0.041
0.042
0.017
0.755
0.783
0.90

The data in bold are associated with DFT in the generalized gradient approximation (GGA) that are more accurate than others reported prior. Moreover, the resultant predictions are in relatively better agreement with experimental data as well as the bowing parameters observed in polarization charge in alloys [212,442]. e31 and e33 are piezoelectric constants. C m
2 is Coulomb p per square meter. C31 and C33 are elastic stiffness coefficient or elastic constants. e31 is the proper piezoelectric constant. The data in bold, which are from Refs [212,442], are recommended. The remaining data are from Refs [212,433,442].

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j 2 Electronic Band Structure and Polarization Effects

236

functional theory in the Perdew–Wang PW91 version, and the LDA in the Ceperley– Alder–Perdew–Zunger form. For a glossary of these approximations within the DFT method, the reader is referred to Section 2.15. Ultrasoft potentials were used treating Ga and In d-electrons as valence at a conservative cutoff of 325 eV. Finally, the reciprocal space summation was done on a (8 8 8) Monkhorst–Pack mesh. The results of the refined GGA calculations in terms of spontaneous polarization, piezoelectric, and elastic constants so calculated are tabulated in Table 2.25 along with earlier calculations. The two sets of data, the earlier and refined, are within 10% agreement. The results of more refined LDA calculations are also provided. For deference to earlier work, the piezoelectric constants, albeit an incomplete list dealing with e14, for GaN were estimated theoretically for cubic GaN [443] and used in early investigation of piezoelectric effects in GaN [444] and deduced from the mobility data [445], which is indirect particularly in samples containing many scatterers. For comparison, the interface charge in the GaAs/AlGaAs system is used for MODFETs (or HFET) in less than 10% of this figure. An excellent review of the polarization effects can be found in Ref. [446]. 2.12.1 Piezoelectric Polarization

In a polarizable medium, the displacement vector can be expressed in terms of two components due to the dielectric nature of the medium and the polarizability nature of the medium as [447] !

!

!

!

!

!

D ¼ eE þ 4pP in cgs and D ¼ eE þ P in mks units; !

!

!

$ !

$

!

ð2:60Þ

where E and P represent the electric field and polarization vectors. Considering only the piezoelectric component, the piezoelectric polarization vector is given by [448] PPE ¼ e e ;

ð2:61Þ

where e and e are the piezoelectric and the stress tensors. To gain a quantitative understanding of the piezoelectric polarization, the piezoelectric tensor, which is defined as the derivative of the polarization with respect to strain, must be considered. In hexagonal P63mc symmetry, piezoelectric polarization is related to strain through the piezoelectric tensor (ei,j) as [449] 2 3 exx 36 eyy 7 2 3 2 7 Px 0 0 0 0 e15 0 6 6 ezz 7 5 4 Py 5 ¼ 4 0 6 7: 0 0 e24 0 0 6 ð2:62Þ eyz 7 7 Pz 0 0 6 e31 e31 e33 0 4 exz 5 exy Note that e24 ¼ e15 for hexagonal symmetry that reduces to 2 3 2 3 e15 exz Px 4 Py 5 ¼ 4 ðe24 ¼ e15 Þeyz 5; e31 ðexx þ eyy Þ þ e33 ezz Pz

ð2:63Þ

2.12 Polarization Effects

where Pi, eij, and eij represent the electric polarization, electric piezoelectric coefficient, and strain, respectively. It is clear from Equation 2.63 that the piezoelectric polarization along the ½1
100mdirection Px ¼ e15exz ¼ 0, along the ½11
20a-direction Py ¼ e15eyz ¼ 0, and along the [0 0 0 1] c-direction Pz ¼ e31(exx þ eyy) þ e33ezz. The strain components given in Equation 2.7 are repeated here for convenience, without shear and eyz ¼ ezx ¼ exy ¼ 0. If only a biaxial strain is present, the in-plain strain can be calculated by relative difference in the in-plane lattice constants of the epitaxial layer and template (buffer or substrate) through a
a0 : ð2:64Þ exx eyy e11 e22 ¼ a0 Here, abuffer a0 and aepi a represent the relaxed (equilibrium) in-plane lattice constants of the buffer layer or the substrate, depending on layers and their thicknesses, and of the epitaxial layer of interest, the strained epitaxial layer, respectively. The nomenclature as or a0 for the substrate (buffer) and ae or a for the epitaxial layer is also used. The expression for the out-of-plane strain is e33 ezz ¼

c
c0 : c0

ð2:65Þ

Similarly, c0 and c represent the relaxed and the out-of-plane lattice parameters, which would correspond to the buffer layer and epitaxial layer, respectively. In case the in-plane strain is anisotropic, e11 6¼ e22. As it may have already become obvious, the nomenclature used in the literatures for parameters surrounding strain and piezoelectric and elastic constants vary in that, for example, exx and e11 are interchangeably used. The two-way transformation of x 1, y 2, z 3 can be used to convert from one nomenclature to the other. Likewise, PPE, pz pz PPE, PPE are commonly used in literature interchangeably to depict 3 , P 3 , and P piezoelectric polarization. If the subscript 3 is also used as in P PE 3 , it specifically indicates that for biaxial in-plane strain, the only nonvanishing polarization is along the c-axis. Even in cases where the subscript is not employed, the underlying assumption is that the polarization is along the c-direction, as growth of nitride semiconductor structures is performed predominantly on the basal plane. In the same vein, P0, Psp, and Psp indicate spontaneous polarization along the c-axis. DPsp and DP0 both represent differential spontaneous polarization at a heterointerface such as an AlN/GaN interface, a topic which follows the discussion of piezoelectric polarization. The components of the piezoelectric polarization tensor given by Equation 2.62 pz can be expressed in terms of a summation, using Pi instead of PPE, as X pz Pi ¼ eij ej with i ¼ 1; 2; 3 and j ¼ 1; . . . ; 6; ð2:66Þ j pz

where P i is the ith component of the piezoelectric polarization. The wurtzite symmetry reduces the number of independent components of the elastic tensor e to three, namely, e15, e31, and e33. The third independent component of the piezoelectric tensor, e15, is related to the polarization induced by a shear strain that is not applicable to the epitaxial growth schemes employed. The index 3 corresponds

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j 2 Electronic Band Structure and Polarization Effects

238

to the direction of the c-axis. It is clear that the piezoelectric properties of the Wz structures are somewhat more complicated. If we restrict ourselves to structures with growth along the [0 0 0 1] direction or z-direction, or along the c-axis, only the e31 and e33 components need to be considered. The piezoelectric polarization in the [0 0 0 1] direction can be obtained by setting i ¼ 3. The electric polarization component in the c-direction, which is designated by z- in the above nomenclature, is given by Pz ¼ e31 exx þ e31 exx þ e33 ezz ¼ 2e31 exx þ e33 ezz :

ð2:67Þ

For isotropic basal plane strain, the strain components exx e?/2 and thus Equation 2.67 can be written as pe

Pz P 3 ¼ e31 e? þ e33 ezz :

ð2:68Þ

In hexagonal symmetry, strain in the z-direction can be expressed in terms of the basal plane strain e? through the use of Poisson’s ratio, which is expressed in terms of the elastic coefficients Cij as ezz ¼
2(C13/C33)exx ¼
(C13/C33)e? (see Equations 2.7, 2.14–2.16 for specifics). In the case of externally applied pressure in addition to mismatch strain, the out-of-plane strain can be related to the in-plane strain through ezz ¼
[(p þ 2C13exx)]/C33, where p is the magnitude of compressive pressure (in the same unit as the elastic coefficients). In terms of the nomenclature again, it should also be noted that e1 e11 exx and e3 e33 ezz in the other notation used in the literature and also in this text.   C13 Pz ¼ e31
e33 ð2:69Þ e? : C33 pz

3 The z-component of the electric polarization is also referred to as P PE 3 , P 3 , and P pz . Piezoelectric polarization is also described in terms of piezoelectric moduli in the literature, which is treated here for completeness. In terms of piezoelectric moduli, dij, which are related to the piezoelectric constants by X eij ¼ dik Ckj with i ¼ 1; 2; 3 and j ¼ 1; . . . ; 6 and k ¼ 1; . . . ; 6: ð2:70Þ k

Using Equation 2.70 in Equation 2.66 and strain–stress relationship (stress is equal to the product of elastic constant and strain in a tensor form), the piezoelectric polarization can be expressed in terms of piezoelectric moduli as X pz dij sj with i ¼ 1; 2; 3 and j ¼ 1; . . . ; 6: ð2:71Þ Pi ¼ j

Symmetry considerations lead to d31 ¼ d32, d15 ¼ d24 and all other components dij ¼ 0, and thus Equation 2.71 reduces to a set of three equations: 1 1 pz pz P1 ¼ d15 s5 ; P2 ¼ d15 s4 ; 2 2

pz

and P 3 ¼ d31 ðs1 þ s2 Þ þ d33 s3 :

ð2:72Þ

For biaxial strain, which is the case with epitaxial layers, additional conditions are imposed in that s1 ¼ s2, s3 ¼ 0. Moreover, the shear stresses are negligible, which leads to s4 ¼ s5 ¼ 0. Consequently, in cases primarily applicable to epitaxial layers grown along the c-direction, the piezoelectric polarization is left with only one

2.12 Polarization Effects

nonvanishing component, which is in the growth direction and is given by, using Equation 2.72, pz

P3 ¼ 2d31 s1 :

ð2:73Þ

Utilizing stress–strain relationship   C2 s1 ¼ e1 C 11 þ C12
2 13 ; C33

ð2:74Þ

we obtain   C2 pz P3 ¼ 2d31 e1 C 11 þ C12
2 13 ; C33

ð2:75Þ

pz

where P 3 represents the piezoelectric polarization along the c-direction and similar to Equation 2.69, which expresses the same in terms of piezoelectric constants as opposed to piezoelectric moduli. For hexagonal crystals, the relations between piezoelectric constants and piezoelectric moduli expressed in Equation 2.70 can be reduced to e31 ¼ e32 ¼ C11 d31 þ C12 d32 þ C13 d33 ¼ ðC11 þ C 12 Þd31 þ C13 d33 ; e33 ¼ 2C13 d31 þ C33 d33 ; e15 ¼ e24 ¼
C 44 d15 ; eij ¼ 0 for all other components:

ð2:76Þ

The nonvanishing component of the piezoelectric polarization of Equation 2.66 due to only the biaxial strain is for i ¼ 3 and is given by, which recovers Equation 2.67, pz

P3 ¼ e1 e31 þ e2 e32 þ e3 e33 ¼ 2e1 e31 þ e3 e33 ¼ e? e31 þ e3 e33 because e31 ¼ e32 : ð2:77Þ Steps can be taken to express Equation 2.77 in the form of Equations 2.68 and 2.69, which are straightforward and therefore not presented. We have so far focused on lattice mismatch induced strain. However, the thermal expansion coefficients of the layers used to compose many of the heterostructures are different, which upon cooling from growth temperature could lead to thermalinduced strain. A larger effect in this vein, however, is that caused by differences in the thermal expansion coefficient between the substrate and epitaxial stack used. In that case, the piezo component would have two parts, namely, the lm or misfit strain and by the thermal strain leading to Ppe ¼ Plm þ Pts. Another issue that must be considered is that the electric field induced due to strain (piezoelectric field) in adjacent layers of a heterostructure comprised of A and B (i.e., A for AlGaN, B for GaN, and A,B for AlGaN/GaN) is sp

pe

E A;B ¼ E A;B þ E A;B :

ð2:78Þ

If, for example, A is composed of a ternary, then the linear interpolation for both sp and pe polarizations can be used to a first extent. Again, the nonlinearities are discussed later on in Section 2.12.3.

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j 2 Electronic Band Structure and Polarization Effects

240

Another relevant but not discussed nearly as much topic is that the properties that cause the piezoelectric polarization can also lead to pyroelectric effects. Such phenomena can be rather important in nitride-based devices, as the junction temperature is high by the nature of the applications such as lasers and high-power amplifiers. Consequently, the thermally induced electric field, pyroelectric effect, would most likely be present [450] with consequences similar to those ascribed to polarization effects. If one considers the distortion of the u parameter as well, the piezoelectric polarization can be expanded as qP 3 ¼

qP3 qP3 qP3 ða
a0 Þ þ ðc
c 0 Þ þ ðu
u0 Þ: qa qc qu

ð2:79Þ

The internal u parameter is defined as the average value of the projection of the connecting vector of a nitrogen atom with its first neighbor in the ð0 0 0
1Þ direction along this same direction. The three parameters, a, c, and u, are not independent of each other. If the partial derivatives in Equation 2.79 are known, following Ref. [209], one can write for the two piezoelectric constants qP3 4qc 0 qu þ pffiffiffi 2 Z  ; qc qc 3a0

ð2:80Þ

e31 ¼

a0 qP 3 2q qu þ pffiffiffi Z ; 2 qa qa 3a0

ð2:81Þ

Z ¼

pffiffiffi 2 3a0 qP3 ¼ ZT3 ; 4q qu

ð2:82Þ

e33 ¼ c 0

where

is the axial component of the Born, or the transverse component of the charge tensor ZT3 . Various structural parameters that are useful in treating the polarization issue in nitride semiconductors are tabulated in Table 2.26.

Table 2.26 Structural parameters of wurtzitic AlN, GaN, and InN

reported in Ref. [209] and updated with DFT in the GGA approximation in Ref. [212]. c0/a0

a0 (Å) Ref. [209] GGA [212] Exp. AlN 5.814 GaN 6.040 InN 6.660

3.1095 3.1986 3.5848

u0

Ref. [209] GGA [212] Exp.

3.1106 1.6190 3.1890 1.6336 3.538 1.6270

1.6060 1.632 1.6180

Ref. [209] GGA [212] Exp.

1.6008 0.380 1.6263 0.376 1.6119 0.377

0.3798 0.3762 0.377

0.3821 0.377

It should be mentioned that GGA produced data for structural as well as polarization-related parameters, see Table 2.25, are in better agreement with refined experimental data.

2.12 Polarization Effects

In Equation 2.80, it is implicit that the vector connecting the cation with the anion has a modulus uc associated with the internal cell parameter and points in the direction of the c-axis. The first term in Equations 2.80 and 2.81 signifies the term called the clamped-ion term, and represents the effect of the strain on the electronic structure. The second term represents the effect of internal strain on the polarization. The derivatives of u with respect to c and a in Equations 2.80 and 2.81 are related to the strain derivatives of u through c0du/dc ¼ du/de3 and a0du/da ¼ 2du/de1. In addition to binaries, the nitride heterojunction system utilizes ternary and to a lesser extent quaternary alloys as well. Knowing the piezoelectric parameters of the end binary points is generally sufficient, to a first order, to discern parameters for more complex alloys. For example, in the case of AlxGa1
xN, the piezoelectric polarization vector expression, using linear interpolation within the framework of Vegard’s law, can be described as [448] $

$

!

Ppe ¼ ½x e AlN þ ð1
xÞ e GaN e ðxÞ:

ð2:83Þ

The same argument can be extended to piezoelectric polarization in quaternary alloys such as AlxInyGa1
x
yN in a similar fashion as $

$

$

$

!

Ppe ¼ ½x e AlN þ ð1
xÞ e InN þ ð1
x
yÞ e GaN e ðxÞ:

ð2:84Þ

The linear interpolation is very convenient and does give reasonably accurate values. However, as will be discussed later in this section, while the Vegard’s law applies to the alloys, the polarization charge itself is not a linear function of composition [220,221]. 2.12.2 Spontaneous Polarization

Spontaneous polarization calculated for the binary nitride semiconductors are tabulated in Table 2.25. For ternary and quaternary alloys, the simplest approach is to use a linear combination of the binary end points, taking into account that the mole fraction can be used under the auspices of the Vegard’s law. However, this linear interpolation falls short of agreeing with the experimental variation of spontaneous polarization with respect to the mole fraction. Consequently, nonlinear models have been developed that are discussed later in Section 2.12.3. For now, the linear interpolation is applied for simplicity in which the spontaneous polarization in quaternary alloys such as AlxInyGa1
x
yN can be expressed as
sp þ yP
sp
sp
sp ðx; yÞ ¼ x P P InN þ ð1
x
yÞP GaN : AlN

ð2:85Þ

The ternary cases can be obtained by simply setting either x or y to zero. This is again predicated on the assumption that polarization charge obeys Vegard’s law, as shown below. Later in this section, the nonlinearity involved is discussed. Using the GGA calculation results for the spontaneous polarization and linear interpolation, as in Equation 2.95, for ternaries one gets

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j 2 Electronic Band Structure and Polarization Effects

242

sp

PAlx Ga1
x N ¼
0:09x
0:034ð1
xÞ; sp PInx Ga1
x N ¼
0:042x
0:034ð1
xÞ; sp PAlx In1
x N ¼
0:09x
0:042ð1
xÞ:

ð2:86Þ

The total polarization charge inclusive of spontaneous and piezoelectric must be considered in dealing with heterojunctions such as quantum wells, quantum wires, and quantum dots. The case of the quantum wells and dots are treated in Volume 2, Chapter 5. The case of the modulation-doped structures is treated in Volume 3, Chapter 3. Therefore, the total polarization involving layers A (AlxGa1
xN) and B (GaN) in contact is Ptotal ¼ P sp þ Ppe :

ð2:87Þ 


2

Spontaneous polarization P (C m ), Born Z or the transverse component of the charge tensor ZT3, piezoelectric constants (C m
2), elastic constants (GPa), and the ratio R ¼
2C31/C33 of wurtzitic nitrides, as obtained in the LDA and GGA approximation [212] are tabulated in Table 2.27. Also, tabulated following the elastic ðpÞ constants e33 and e31 is e31 , which is the applicable piezoelectric constant in the context of experiments dealing with current flow across the sample. The constant e31 is relevant to systems in depolarizing fields such as nitride nanostructures [448]. A comprehensive table including experimental and calculated values of elastic compliance, elastic constants, and piezoelectric constants as well as the Poisson number of wurtzite binary group III nitrides at room temperature are shown in Table 2.28. Table 2.27 Spontaneous polarization, Born effective charges, Z*

(in units of e), piezoelectric constants, dynamical charges, elastic constants (GPa), and the ratio R ¼
2C31/C33 of wurtzitic nitrides, as obtained with DFT calculations in the LDA and GGA approximation.

AlN LDA LDA [179] GGA GaN LDA [179] LDA GGA InN LDA LDA [179] GGA

P (C m
2)

Z

e33 (C m
2)

e31 (C m
2)

«(p) 31 (C m
2)

C33 (GPa)


0.100

2.652

1.80


0.64


0.74

384 373

111 108


0.578
0.579


0.090

2.653

1.50


0.53


0.62

377

94


0.499


0.032

2.51

0.86


0.44


0.47

415

83


0.400


0.034

2.67

0.67


0.34


0.37

405 354

103 68


0.508
0.384


0.041

3.045

1.09


0.52


0.56

233 224

88 92


0.755
0.821


0.042

3.105

0.81


0.41


0.45

205

70


0.683

The last column reports the proper e31 piezoelectric constant [212].

C31 (GPa)

R¼
2C31/C33

0.22c

367 135 103 405 95 0.52 3.267 1.043 0.566 2.757 10.53

Theory [179]


1.253 2.291
1.579


0.34 0.67

0.38

68 354

Theory [209,212]

GaN

396 137 108 373 116 0.58 2.993 0.868 0.615 3.037 8.621

370a 145a 110a 390a 90a 0.56 3.326 1.118 0.623 2.915 11.11

0.30d

Theory [179]

Experiment


2.298 5.352
2.069


0.53 1.50

0.50

94 377

Theory [212,433]

AlN

a

For an expanded list of elastic constants, see tables in Chapter 1 under mechanical properties [84]. Ref. [200]. b Ref. [198]. c Shur, M.S., Bykhovski, A.D. and Gaska, R. (1999) MRS Internet Journal of Nitride Semiconductor Research, S41, G16. d Ref. [437].

C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa) n(0001) S11 (10
12 N m
2) S12 (10
12 N m
2) S13 (10
12 N m
2) S33 (10
12 N m
2) S44 (10
12 N m
2) e31 (C m
2) e33 (C m
2) e24 ¼ e15 (C m
2) d31 (10
12 C m
2 Pa) d33 (10
12 C m
2 Pa) d15 (10
12 C m
2 Pa)

Parameter

of wurtzite binary group III nitrides at room temperature (theory [179,212,433]).

410a 140a 100a 390a 120a 0.51 2.854 0.849 0.514 2.828 8.333 0.58b 1.55b 0.48b
2.65 5.53
4.08

Experiment 223 115 92 224 48 0.82 6.535 2.724 1.565 5.750 20.83

Theory [179]

Table 2.28 Experimental and predicted elastic compliance, elastic constants, and piezoelectric constants as well as the Poisson number


3.147 6.201
2.292

0.41 0.81

0.68

70 205

Theory [212,433]

InN

2.12 Polarization Effects

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j 2 Electronic Band Structure and Polarization Effects

244

In heterojunctions containing donors and acceptors and shallow defects, the associated free carriers within the Fermi statistics diffuse to the semiconductor with the smaller bandgap where they are confined, due to potential barriers, to potential minima. The resulting charge separation due to free carriers causes an internal electric field, screening field, which is represented by the first term in Equation 2.60. In addition, an electric field can also be induced by the application of an external voltage such as done through the use of Schottky barriers, metal oxide semiconductor structures, and p–n junctions. Spontaneous and strain-induced piezoelectric polarization can influence the final status of the interfacial free-charge density in these heterostructures. Any shallow defects (induced fields would change the ionization ratio), free carriers, and surface contacts must be included for a complete treatment. Let us now calculate the polarization charge for model heterojunctions using the linear interpolation method. For AlxGa1
xN coherently strained on a relaxed GaN substrate, the strain e? is expected to be proportional to x and given by e? ¼ 2 (aGaN
aAlGaN)/aAlGaN, which is 0.051x and is tensile. The piezoelectric polarization is then Ppiezo ¼
0.0464x, that is, pointing in the ½0 0 0
1 direction. The corresponding difference in spontaneous polarization between AlxGa1
xN and GaN is also expected to be proportional to x, the AlN mole fraction, and is given by DPspon ¼
0.056x. Consequently, the two are in the same direction for this particular orientation and are comparable in magnitude. This treatment assumes that the polarization charge scales linearly with alloy composition, which does not necessarily hold but is used for simplicity. The matter is discussed below in more detail. The total polarization for AlN–GaN interface, which is defined in this case as the sum of the piezoelectric polarization and the differential polarization charge is
0.102x. Note that these are all in C m
2 and that 1 C m
2 ¼ 0.624 · 1015 electrons cm
2. Thus, for x on the order of 0.1, we are dealing with total polarization charge of the order of mid-1012 cm
2. In case the ternary AlxIn1
xN is used for the barrier, the composition of Al0.82In0.18N can be grown lattice matched to GaN [451] and the piezoelectric polarization vanishes. For lower Al concentrations, that is, x < 0.82, the piezoelectric polarization increases due to the increase in biaxial compressive strain. For higher Al concentrations, that is, x > 0.82, the layer is under tensile strain and the piezoelectric polarization becomes negative. For a coherently strained InxGa1
xN layer on relaxed GaN, the difference in spontaneous polarization is much smaller, DPspon ¼
0.012x. Furthermore, the InxGa1
xN layer on GaN would be under compressive strain e? ¼
0.203x and Ppiezo ¼ þ0.139x. Here, the piezoelectric polarization dominates and is opposite in direction to the spontaneous polarization charge but even larger in absolute magnitude. Unlike for tensile-strained AlxGa1
xN or AlxIn1
xN (for large x-values) on GaN layer, wherein the piezoelectric and the spontaneous polarization are negative and point in the same direction, thus they add up, the spontaneous and piezoelectric polarizations oppose one another for compressively strained InxGa1
xN or AlxIn1
xN (for small x-values) layers. To calculate the differential spontaneous and piezoelectric polarization associated with alloys, one can to a first order employ a

2.12 Polarization Effects

linear interpolation for the spontaneous polarization and piezoelectric and elastic constants from the binary compounds [452]. In addition to nonlinear behavior of polarization that is discussed in Section 3.14.3, there are technological issues that must be considered. Some further words of caution about the above estimates based on linear interpolation are needed. If the AlGaN layers are not pseudomorphic but partially relaxed (by misfit dislocations for example), then the piezoelectric effect would be reduced but the spontaneous polarization would still be present. Finally, if domains with inverted polarity exist, the overall polarization effects may be washed out. Also note that in an inverted structure with nitrogen (N) polarity toward the surface, it may be possible to create a two-dimensional hole gas (2DHG) at the AlGaN/substrate GaN interface, provided that free holes are available. However, if an n-type GaN layer is placed on top, a 2DEG may form on top of the AlGaN layer. Polarization effects and devices are inextricable. In devices with a large concentration of free carriers, the polarization charge would be screened. In devices where the modulation of charge is the basis of operation such as the MODFETs, a detailed accounting of all polarization charge must be undertaken. In considering a Normal MODFET (N-MODFET) structure where the larger bandgap AlGaN donor layer is deposited on top of a GaN channel layer, both the spontaneous polarization and the piezoelectric polarization must be accounted for. For an N-MODFET structure with Ga-polarity, the potential will slope down from the surface of the AlGaN layer toward the AlGaN–GaN interface. The topic is discussed in some detail for both AlGaN/GaN and AlxIn1
xN/GaN structures in Section 2.12.4.1 and for the InGaN/GaN heterojunction system in Section 2.12.4.2. 2.12.3 Nonlinearity of Polarization

In the treatment above and in the literature for quite sometime, a linear interpolation from the binary end points were used to deal with both piezoelectric and spontaneous components of polarization. Despite the linear interpolation method being reasonably successful in getting the figures to a first extent, as done in conjunction with Equation 2.86, discrepancies with experiments were noted. Thus, efforts continued to refine the polarization figures and, while in the process, entertain whether a bowing parameter such as the case in bandgap of ternaries, while the lattice parameter obeys the Vegard’s law, that is, linear interpolation, could also be considered for polarization. This forms the basis of the treatment of the problem by Bernardini and Fiorentini [220,221,453] and others [454,455]. The nonlinearity is quite pronounced in AlInN and InGaN alloys for which the binary constituents are very largely lattice mismatched. Spontaneous polarization bowing strongly depends on the microscopic nature of the alloy. What is more is that chemical ordering in the form of short period superlattices may increase the bowing up to a factor of five. Similarly, the piezoelectric polarization is also nonlinear. However, in random alloys, this nonlinearity is entirely due to a nonlinear strain dependence on piezoelectric polarization in pure end binary compounds while the

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j 2 Electronic Band Structure and Polarization Effects

246

piezoelectric coefficients follow Vegard’s law. On the contrary, in chemically ordered InGaN and AlInN alloys piezoelectric coefficients deviate from Vegard’s law, this effect reduces the strength of the piezoelectric polarization up to 38% of its value in AlInN alloy. However, linear extrapolation is simple and gets reasonably accurate figures. The details of this nonlinearity are discussed below. To reiterate, discrepancies between the experimental data and the theory using linear interpolation led Bernardini and Fiorentini to consider a bowing parameter for polarization in conjunction with ternaries, and it follows that this applies to quaternaries as well. For freestanding alloys, meaning relaxed for this purpose, and assuming that the ternary nitride alloys have random microscopic structure with no strain, the values of the lattice constants within the realm of Vegard’s law are given by aAlx Ga1
x N ¼ xaAlN þ ð1
xÞaGaN ; c Alx Ga1
x N ¼ xc AlN þ ð1
xÞcGaN :

ð2:88Þ

The spontaneous polarization Psp of an alloy, for example, AlxGa1
xN, can be expressed in terms of the polarization values of the binary constituents, AlN and GaN, in the realm of nonlinearities (i.e., non-Vegard behavior). The polarization in an alloy can be described in a generic measurable quantity to a first approximation by a parabolic model involving a bowing parameter, similar to that used for the bandgap of alloys. In conjunction with nonlinear polarization effects in alloys, Bernardini and Fiorentini [453] considered ordered structures to get at the spontaneous polarization across the entire composition range by calculating it for compositions of 0.25, 0.5, and 0.75 in addition to the binary end points. The chalcopyrite-like (CH) structure, used for the 0.5 composition, is formed by each anion site being surrounded by two cations of one species and two of the other, with the overall condition of conforming to periodical (2 · 2 · 2) wurtzite supercell. This structure is highly symmetric, as there are only two kinds of inequivalent anion sites, differing in the orientation of the neighbors but not in their chemical identity. Among the possible ordered structures, CH is in a sense the most homogeneous for a given composition. A further useful ordered structure considered is a luzonite-like (LZ) structure, used for the 0.25 and 0.75 points, resembling zinc blende based alloys. In this structure, each nitrogen atom is surrounded by three cations of one species and one of the other: in a sense, this is the analog to the CH structure for molar fractions x ¼ 0.25 and 0.75. In the comparison between CH and random structures, the latter used for the 0.5 composition could provide insight into the effect of randomness versus ordering without the biases due to specific superlattice ordering, as in the CuPt (CP) structure used for the 0.5 composition point. Luzonite-like structure can provide values of the polarization in the intermediate molar fractions. Shown in Figure 2.35 is a comparison of calculated equilibrium basal and axial lattice parameters a and c for three binaries and their alloys with those determined in the realm of Vegard’s law (dashed lines). The agreement between the equilibrium a and c lattice parameters so calculated and those determined using Vegard’s law is quite good. The dependence of the polarization on composition is then the same as that on the lattice parameter(s), modulo a multiplicative factor.

2.12 Polarization Effects

5.9 InN

c-Lattice constant (Å)

5.7 In x G

a

1–x

N

5.5

In x

A l 1 –x

N

GaN 5.3

5.1

A lx G a

AlN

1 –x N

AlN

4.9 0

0.2

0.4

0.6

1.0

0.8

Molar fraction, x InN

a-Lattice constant (Å)

3.6 In x G

a 1 –x

N In x

3.4

A l 1 –x

N

GaN

A lx G a

3.2

1 –x

N

AlN

AlN 3.0 0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x

Random alloy CH-like CP-like Figure 2.35 The basal plane lattice parameter a and axial lattice parameter c of wurtzite nitride binaries and alloys directly calculated versus those determined by Vegard’s law. The open circles denote the random alloy with 0.5 molar fraction. The dashed lines are Vegard’s law. Courtesy of F. Bernaridini and V. Fiorentini.

Shown in Figure 2.36 are again the spontaneous polarization values for the aforementioned ternaries in the freestanding strain-free form. The solid lines represent interpolations utilizing Equation 2.95 for AlInN, the binary points of which were determined using 32-supercell calculations. The dashed lines represent the simple Vegard’s law based interpolations and the numbers indicate the bowing parameters in terms of C m
2.

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j 2 Electronic Band Structure and Polarization Effects

248

–0.02 In x Ga 1–x N

GaN

Spontaneous polarization (C m–2)

+0.038 InN –0.04 +0.071

+0.019 –0.06

In x Al1–x N Alx Ga1–xN

–0.08

AlN

AlN

–0.10 0

0.2

0.4

0.6

0.8

1

Molar fraction, x Figure 2.36 Spontaneous polarization versus molar fraction in freestanding (strain free) random nitride alloys (solid circles). The solid lines represent the results of the bowing model of Equation 2.95. The dashed lines are determined by Vegard’s law. The numbers shown in the figure are the bowing parameters expressed in units of C m
2. Courtesy of F. Bernaridini and V. Fiorentini.

Depicted in Figure 2.37 are the spontaneous polarization values versus the molar fraction. The solid circles, squares, and triangles represent the values for the random alloy, CH-/LZ-, and CP-like alloys, respectively. The dashed lines represent the values calculated using Vegard’s law. The numbers shown in the figure represent the bowing parameters for the CH-/LZ- and CP-like ordered alloys. The CH and LZ calculations can be used to verify efficacy of the interpolation model inclusive of the bowing parameter. The ordered LZ structure is analogous to CH for molar fractions of 0.25 and 0.75. The extent to which the LZ values deviate from those calculated by Equation 2.95 and the CH (x ¼ 0.5) value indicates whether or not nonparabolicity occurs in the P(x) relation for CH-like order. Because the polarization of the CH structure behaves qualitatively as that of the random structure (Figure 2.37), the conclusions drawn for CH are applicable to the random phase. Fortuitously, the values of the polarization calculated for the LZ structures at molar fractions 0.25 and 0.75, shown in Figure 2.37, are very close to the parabolic curve for InGaN and AlGaN calculated using the quadratic expression depicted in Equation 2.95, paving the way for the use of the analytical (quadratic) expression for polarization calculations. However, for AlInN, the calculated values are somewhat

2.12 Polarization Effects

Spontaneous polarization (C m–2)

0.02

+0.333 InxAl1–xN

InxGa1–xN

0.00

+0.193

–0.02

InxGa1–xN

–0.04

GaN

+0.095 –0.06

InxAl1–xN

InN AIxGa1–xN

+0.037 –0.08

Random alloy CH-like CP-like

AIN

AIN

–0.10 0

0.2

0.4

0.6

1

Molar fraction, x Figure 2.37 Spontaneous polarization versus the molar fraction in all three ternary nitride alloys. Circles, squares, and triangles represent random alloy, CH-/LZ-, and CP-like structures, respectively. The dashed/dotted lines (blue) with solid triangles are for the CP-like alloys, the dashed lines (green) with solid squares are for CH-like alloys, and solid lines (black) with filled

circles are for random alloys. The black dashed lines represent the data calculated using Vegard’s law. Numbers indicated in the figure are for CP and CH-/LZ-like ordered alloy bowing parameters in terms of C m
2. Courtesy of F. Bernaridini and V. Fiorentini. (Please find a color version of this figure on the color tables.)

above the quadratic relation for x ¼ 0.25 and below it for x ¼ 0.75, indicating some nonparabolicity in polarization. Specifically, the bowing is higher for low In concentration in AlInN. This nonparabolicity is relatively modest, of order 10%, as compared to the quadratic nonlinearity Equation 2.95 for AlInN. One can then conclude that the analytical (quadratic) expression would predict polarization in AlGaN and InGaN fairly accurately and also for AlInN but with about 10% accuracy. To understand the physical origin of the spontaneous polarization bowing, Bernardini and Fiorentini [453] decomposed the spontaneous polarization into three distinct components on the basis of their genesis, namely, the internal structural and bond alternation, volume deformation, and disorder. The internal structural and bond alternation (strain) can be caused by varying cation–anion bond lengths. The volume deformation can be due to compression or dilation of the bulk binaries from their original equilibrium lattice constants to the alloy values. The disorder effect is due to the random distribution of the chemical elements on the cation sites. Bernardini and Fiorentini [453] showed that in ordered alloys the structural

j249

j 2 Electronic Band Structure and Polarization Effects

250

contribution is dominant, the volume deformation accounts for one-third of the bowing found in random alloys, and the effect of disorder appears insignificant in terms of its effect on the bowing of spontaneous polarization. 2.12.3.1 Origin of the Nonlinearity To understand the origin of the structural contribution to the nonlinear behavior, Bernardini and Fiorentini [453] considered that while the bandgap is a scalar, the polarization is a vector of defined direction. The basal plane normal in wurtzite structures is the [0 0 0 1] direction. Thus, the bond length and angle alternation will affect the polarization bowing only if it changes the projection of the bond length along the c-axis, which is the [0 0 0 1] axis. This is consistent with the notion that the polarization in pure binaries is strongly affected by the relative displacement of the cation and anion sublattice sites in the [0 0 0 1] direction [441]. Also consistent is that there is a clear correlation between the u parameter of the wurtzite structure, which is the bond length along the singular polar (or pyroelectric) axis, and the value of the polarization. Shown in Figure 2.38 is the calculated spontaneous polarization of freestanding x ¼ 0.5 alloys of AlGaN, InGaN, and AlInN versus the average internal 0.02 Random CH – like CP – like Spontaneous polarization (C m–2)

0.00 InGaN AlInN – 0.02

– 0.04

AlGaN – 0.06

0.380

0.378

0.376

0.374

Average u lattice parameter Figure 2.38 Spontaneous polarization versus the average internal parameter u in AlGaN, InGaN, and InAlN ternary alloys. Open circles, squares, and triangles, refer to random, CH-like, and CPlike structures, respectively. Courtesy of F. Bernaridini and V. Fiorentini.

0.372

0.370

2.12 Polarization Effects

parameter u. The internal u parameter is defined as the average value of the projection of the connecting vector of a nitrogen atom with its first neighbor in the ð0 0 0
1Þ direction along this same direction. This is in the realm of the wurtzite structure convention in which each anion is situated at the (0, 0, u) Cartesian point from the cations and all the vertical bonds between Ga and N point along [0 0 0 1] direction. This definition can also be used for random phase alloys in spite of the displacement from the ideal sites. Figure 2.38 shows that for a given alloy composition, the spontaneous polarization of relaxed (freestanding) nitride alloys of different microscopic structure depends linearly on the average internal u parameter of the alloy structure. This indicates that spontaneous polarization differences between alloys of the same composition are primarily due to structural and bond alternation effects; disorder appears to have a negligible influence. The structural and bond alternation effects discussed above can also shed light on the random and CP phases in AlGaN having almost the same average u, hence, nearly the same polarization. In InGaN, the random alloy has a larger u than the CH phase, while the opposite is true for AlInN and AlGaN accounting for the CH versus random bowing behavior in InGaN being opposite of that in AlInN and InGaN. Moreover, the large bowing of CP-ordered AlInN and InGaN is consistent with the very large deviation of the average u as compared to the random and CH-like structures. If the internal strain were the only source of polarization bowing, all of the points in Figure 2.38 would fall on the same straight line. This not being the case suggests that another factor related to the chemical identity of the constituents plays a role of some importance and brings to the role of volume deformation. To investigate this role, Bernardini and Fiorentini [453] set up a model based on the polarization in a constrained ideal wurtzite structure in which only the a parameter is a variable, whereas c and u are fixed atp the determined by maximal sphere packing, ffiffiffiffiffiffiffivalues ffi namely, u ¼ 0.375 and c=a ¼ 8=3. Each nitrogen atom is then surrounded by four equidistant cations, meaning all bonds have the same length for a given lattice constant a. By design, the bond alternation caused by perturbation in the internal parameter u would not play any role, and the effects of chemical identity of the constituents can be easily distinguished. To continue to tackle the aspect of volume deformation, Bernardini and Fiorentini [453] assumed that Vegard’s law holds for the lattice constant a. Naturally, this establishes a linear relationship between the composition and the lattice constant. This segues into the calculation of the polarization in each of the binary nitrides in their ideal structure as a function of the lattice constant a(x). Finally, they could express the alloy polarization as a composition-weighted Vegard-like average of the polarizations of the binary end points (through a reduction of Equation 2.85) as aðxÞ Psp ðAlx Ga1
x NÞ ¼ xPaðxÞ sp ðAlNÞ þ ð1
xÞP sp ðGaNÞ:

ð2:89Þ

In this approach, any nonlinearity must have its origin in the different response of polarization to perturbations in a(x), hence, to hydrostatic compression. To illustrate this point, shown in Figure 2.39 are calculated polarizations in the ideal wurtzite structure for the three ternary alloys, polarization of the binaries in the ideal

j251

j 2 Electronic Band Structure and Polarization Effects

252

InGaN

Spontaneous polarization (C m–2)

–0.015 GaN

InN

–0.020 AlInN –0.025

AlGaN

AlN GaN InN

–0.030 AlN –0.035 3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

Equilibrium lattice constant (Å) Figure 2.39 Spontaneous polarization versus the lattice constant in ideal wurtzite structures. Solid circles depict the values of binary compounds and random ternary alloys. Open circles, squares, and triangles represent the polarization calculated as a function of the lattice constant in bulk AlN, GaN, and InN. Solid lines correspond to Vegard interpolations based on

the ideal binaries under hydrostatic pressure. The dashed lines represent the Vegard interpolation of the polarization using the values for the binaries at equilibrium. The data show that the polarization in the alloys is a direct result of the hydrostatic pressure and thus volume deformation. Courtesy of F. Bernaridini and V. Fiorentini.

structure, and polarization interpolated by the Vegard interpolation of Equation 2.89. It can be seen that the calculated values of polarization and those interpolated by Vegard’s prediction agree well. Because the Vegard interpolation intrinsically account for the volume deformation, the origin of the volume deformation component of the nonlinearity and its large values in In-containing alloys become clear. Essentially, this has its genesis in the fact that polarization decreases with hydrostatic pressure in AlN and GaN, while it increases in InN. Also to be noted, polarizations in the ideal structure are between 35 and 50% of their values in freestanding (relaxed) alloys, and despite the absence of bond alternation, which is designed for the purpose separating the components in effect, the bowing is still very large. The above model dealing with the effect of strain on polarization provides the basis for developing an expression of the bowing parameter bmodel for an ideal wurtzite structure alloy as a function of the polarization response to hydrostatic pressure (for the model case of AlGaN): 0 1 qP qP GaN AlN A AlGaN @
bmodel ¼ ðaGaN
aAlN Þ qa qa a¼að1=2Þ 0 1 ð2:90Þ 2 2 1 q P q P GaN AlN A 2@ þ ðaGaN
aAlN Þ
: qa2 qa2 4 a¼að1=2Þ

2.12 Polarization Effects

The agreement of the latter expression with the b resulting from a fit to the calculated values is very good (e.g., for the extreme case of AlInN, bmodel
20.0225 C m
2, while from direct calculation, we get
20.0208 C m
2). On the basis of the model, it is now understandable that the AlGaN bowing is pretty moderate because the region of interest is small (3.1–3.2 Å) and the responses to hydrostatic pressure of AlN and GaN are similar. On the contrary, in the large range 3.1–3.6 Å, AlN and InN have opposite behavior, when the huge bowing is found in AlInN alloys. The same goes, although to a lesser extent, for the InGaN alloys. 2.12.3.2 Nonlinearities in Spontaneous Polarization Any nonlinearity in the spontaneous polarization can be treated by using a bowing parameter as commonly employed in interpolating the bandgap of an alloy from the binary point with the help of a bowing parameter. In this vein, the spontaneous polarization for a ternary Psp (AxB1
xN) with A and B representing the metal components and N representing nitrogen is given by [84,220,221] sp

sp

sp

PAx B1
x N ¼ xP AN þ ð1
xÞP BN
bAB xð1
xÞ: sp

ð2:91Þ

sp

PAN and PBN are the spontaneous polarization terms for the end binaries forming the alloy. The bowing parameter is as defined bAB ¼ 2PAN þ 2P BN
4P A0:5 B0:5 N ;

ð2:92Þ

which requires only the knowledge of the polarization of the ternary alloy at the midpoint, that is, molar fraction x ¼ 0.5. Knowledge of the bowing parameter from Equation 2.97 would lead to the determination of the spontaneous polarization at any composition. For AlxGa1
xN, Equations 2.91 and 2.92 take the form sp

sp

sp

PAlx Ga1
x N ¼ xPAlN þ ð1
xÞPGaN
bAlx Ga1
x N xð1
xÞ;

ð2:93Þ

bAlx Ga1
x N ¼ 2P AlN þ 2P GaN
4P Al0:5 Ga0:5 N :

ð2:94Þ

with

The first two terms in Equation 2.93 are the usual linear interpolation terms between the binary constituents. The third term, quadratic, represents the nonlinearity. Higher order terms are neglected because their contribution is estimated to be less than 10%. Using the numerical GGA values in Table 2.25 for the spontaneous polarization in AlN and GaN and the bowing parameter for random alloy AlGaN given in Refs [220,221] leads to sp

PAlx Ga1
x N ¼
0:09x
0:034ð1
xÞ þ 0:0191xð1
xÞ; and sp

sp

sp

PInx Ga1
x N ¼ xPInN þ ð1
xÞPGaN
bInx Ga1
x N xð1
xÞ;

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j 2 Electronic Band Structure and Polarization Effects

254

with bInx Ga1
x N ¼ 2P InN þ 2PGaN
4PIn0:5 Ga0:5 N : Again, using the numerical GGA values in Table 2.25 and the bowing parameter for random alloy InxGa1
xN given in Refs [220,221] leads to sp

PInx Ga1
x N ¼
0:042x
0:034ð1
xÞ þ 0:0378xð1
xÞ; and sp

sp

sp

PAlx In1
x N ¼ xP AlN þ ð1
xÞP InN
bAlx In1
x N xð1
xÞ; with bAlx In1
x N ¼ 2PAlN þ 2PInN
4P Al0:5 In0:5 N: Using the numerical GGA values in Table 2.25 and the bowing parameter for random alloy InxAl1
xN given in Refs [220,221] leads to sp

PAlx In1
x N ¼
0:090x
0:042ð1
xÞ þ 0:0709xð1
xÞ:

ð2:95Þ

In calculating the bowing parameters for the three ternaries mentioned above, Bernardini and Fiorentini [453] used a 32-atom supercell for both alloys and binary nitrides for spontaneous polarization for the end binaries and ternaries. These calculations, while being more efficient in terms of computer time, are not as accurate as those reported in Refs [220,221,441]. In the above treatment, the bowing parameters are taken from Ref. [453], while the spontaneous polarization for the end binaries is taken from Ref. [441]. Relying simply on the 32-atom supercell calculations, the bowing parameters and related spontaneous polarization figures for ordered alloys such as CuPt-like ordered alloy (CP-like), chalcopyrite-like ordered alloy (CH-like), luzonite-like ordered alloy (LZ-like) have also been obtained by Bernardini and Fiorentini [453]. In the series of tables below, the results of such calculations for the aforementioned ordered alloys are given for completeness. The binary figures in terms of spontaneous polarization and the lattice parameter resulting from the 32-atom supercell calculations are also given for consistency. Again, more accurate binary data exist in Refs [441]. For convenience, the lattice parameter (a), spontaneous polarization (Psp), and bowing parameter (bAB) for the three ternary nitride alloys in the form of random, ordered chalcopyrite (CH-like), ordered luzonite (LZ-like), and CuPt-ordered alloy (CP-like) are tabulated in Tables 2.29–2.35, in addition to the data shown in Figures 2.35–2.37.

Table 2.29 The lattice parameter (a) and spontaneous polarization (Psp) for AlN, GaN, and InN determined by 32-atom supercell calculations by Bernardini and Fiorentini.

AlN a (Å) Psp (C m
2)

3.1058
0.0897

GaN 3.1956
0.0336

InN 3.5802
0.0434

2.12 Polarization Effects Table 2.30 The lattice parameter (a), spontaneous polarization (Psp), and the bowing parameter (bAB) for random alloy ternaries with a molar fraction of x ¼ 0.5 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453].

R 50%

Al0.5Ga0.5N

In0.5Ga0.5N

Al0.5In0.5N

a (Å) Psp (C m
2) bAB (C m
2)

3.1500
0.0569 þ0.0191

3.3872
0.0290 þ0.0378

3.3352
0.0488 þ0.0709

Table 2.31 The lattice parameter (a), spontaneous polarization (Psp), and the bowing parameter (bAB) for CuPt ordered alloy (CPlike) with a molar fraction of x ¼ 0.5 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453].

CP 50%

Al0.5Ga0.5N

In0.5Ga0.5N

Al0.5In0.5N

a (Å) Psp (C m
2) bAB (C m
2)

3.1489
0.0573 þ0.0176

3.3884 þ0.0098 þ0.1934

3.3222 þ0.0168 þ0.3336

Table 2.32 The lattice parameter (a) and spontaneous polarization (Psp) for the luzonite (LZ-like) and chalcopyrite (CH-like) alloy with a molar fraction of x ¼ 0.25 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453].

LZ 25%

Al0.25Ga0.75N

In0.25Ga0.75N

Al0.25In0.75N

a (Å) Psp (C m
2)

3.1724
0.0413

3.2920
0.0323

3.4510
0.0385

Table 2.33 The lattice parameter (a) and spontaneous polarization

(Psp) for the luzonite (LZ-like) and chalcopyrite (CH-like) alloy with a molar fraction of x ¼ 0.5 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453]. CH 50%

Al0.5Ga0.5N

In0.5Ga0.5N

Al0.5In0.5N

a (Å) Psp (C m
2) bAB (C m
2)

3.1474
0.0523 þ0.0374

3.3949
0.0328 þ0.0226

3.3369
0.0427 þ0.0952

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j 2 Electronic Band Structure and Polarization Effects

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Table 2.34 The lattice parameter (a) and spontaneous polarization

(Psp) for the luzonite (LZ-like) and chalcopyrite (CH-like) alloy with a molar fraction of x ¼ 0.75 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453]. LZ 75%

Al0.75Ga0.25N

In0.75GaN0.25

Al0.75In0.25N

a (Å) Psp (C m
2)

3.1276
0.0690

3.4828
0.0366

3.2146
0.0564

Table 2.35 The bowing parameter (a) for the random alloy and luzonite (LZ-like), chalcopyrite (CH-like), and CuPt ordered alloys (CP-like) calculated by Bernardini and Fiorentini [453].

Random CH and LZ CP

AlGaN

InGaN

AlInN

þ0.019 þ0.037 þ0.018

þ0.038 þ0.023 þ0.193

þ0.071 þ0.095 þ0.333

2.12.3.3 Nonlinearities in Piezoelectric Polarization The total polarization at heterointerfaces is the sum of spontaneous and piezoelectric polarization. Having treated the nonlinearity in spontaneous polarization, attention must be turned to piezoelectric polarization, specifically, its nonlinear dependence on composition in alloys. Some if not all of the components of the heterojunctions are grown pseudomorphically and are, therefore, under strain on the (0 0 0 1) axis. The ensuing symmetry-conserving strain causes a change in polarization that amounts to a piezoelectric polarization. The aim here is to show that piezoelectricity in nitride alloys is nonlinear, and that this nonlinearity is due to a pure bulk effect with its nonlinear behavior of bulk binary piezoelectric constants versus symmetry-conserving strain. The model heterostructure considered by Bernardini and Fiorentini [453] is a coherently strained alloy grown on a relaxed binary buffer layer (bulk for this purpose) in which the in-plane lattice parameter is aalloy ¼ aGaN. The piezoelectric component is the difference between the total polarization to be obtained and the spontaneous polarization discussed above. Shown in Figure 2.40 is the piezoelectric polarization as a function of the alloy composition. Symbols, which have similar designation as in the spontaneous polarization, represent the calculated polarizations for AlGaN, InGaN, and InAlN alloys as a function of compositions. After the publication of Ref. [453], to reconcile a discrepancy with a paper by Al-Yacoub and Bellaiche [454] who showed that CuPt-like ordering in In0.5Ga0.5N wurtzite-structure alloys causes a sizable deviation of the piezoelectric constants from Vegard’s like behavior. Bernardini and Fiorentini [220,221] revisited the piezoelectric polarization in nitride alloys. In fact, Figure 2.40 contained here represents the updated calculations. The error in question stems from the fact that the polarization values for the strained and unstrained alloys were subtracted

Piezoelectric polarization (C m–2)

2.12 Polarization Effects

InN

Vegard's Random alloy CH,LZ CP

0.2

0.1

InGaN AlInN GaN

0.0 AlGaN AlN

AlN – 0.1 0

0.2

0.4

0.6

0.8

1

Alloy molar fraction Figure 2.40 Piezoelectric component of the macroscopic polarization in ternary nitride alloys epitaxially strained on a relaxed GaN layer (template). Open symbols represent the directly calculated values for random alloy (circles), CHlike and LZ-like (squares), and CP-like (triangles)

structures, respectively. Dashed lines represent the prediction of linear piezoelectricity, while the solid lines are the prediction of Equation 2.96 using the nonlinear bulk polarization as shown in Figure 2.41. Courtesy of F. Bernaridini and V. Fiorentini.

correctly, but values calculated with different k-point meshes were used instead of those for the same k-point [454,455]. It is clear that, contrary to the spontaneous component of polarization discussed above, the piezoelectric polarization component hardly depends on the microscopic structure of the alloy. One might then ask whether the piezoelectric polarization of the alloy can be reproduced by a Vegard-like model interpolated from the binaries in the form pe

pe

pe

PAlGaN ðxÞ ¼ xPAlN ½eðxÞ þ ð1
xÞP GaN ½eðxÞ; pe P AlN ½eðxÞ

ð2:96Þ

pe PGaN ½eðxÞ

and represent the strain-dependent bulk piezoelectric where polarization of the binary end points. With obvious permutations, this expression can be constructed for InGaN and InAlN ternaries as well. To a first approximation, one may calculate the piezoelectric polarization of the binary compounds for symmetry conserving in-plane and axial strains as pe

PAlN ¼ e33 e3 þ 2e31 e1 :

ð2:97Þ

The piezoelectric constants e can be calculated for the equilibrium state of the binary, AN, and as such they do not depend on strain. The dashed lines in Figure 2.40 represent the piezoelectric term as computed from the above relations using the piezoelectric constants computed for the binaries [433]. The Vegard’s law of Equation 2.96 when combined with Equation 2.97 clearly fails to reproduce the calculated polarization and misses the strong nonlinearity of the piezoelectric term evident in

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j 2 Electronic Band Structure and Polarization Effects

258

Figure 2.40. This is due to a valid nonlinearity of the bulk piezoelectricity of the binary constituents, which is of nonstructural origin. It should be stated that bowing due to the microscopic structure of the alloys is negligible. The argument forwarded by Bernardini and Fiorentini [453] is that they calculate the piezoelectric polarization as a function of the basal strain for AlN, GaN, and InN while optimizing all structural parameters. The results depicted in Figure 2.41 clearly indicate that the piezoelectric polarization of the binaries is an appreciably nonlinear function of the lattice parameter a, which is related to basal strain. Because all lattice parameters closely follow Vegard’s law, the nonlinearity cannot be related to deviations from linearity in the structure. Bernardini and Fiorentini [453] substitute the nonlinear piezoelectric polarization computed for the binaries into the Vegard interpolation, Equation 2.96 In doing so, they obtain excellent agreement with the polarization calculated directly for the alloys as shown with solid lines in Figure 2.40. This led them to conclude that the nonlinearity in bulk piezoelectricity dominates over any effects related to disorder, structure, bond length and angle alternation, and so on. Importantly, they concluded that Vegard’s law still holds in calculating the piezoelectric polarization of the III–V nitrides alloys, provided that the nonlinearity of the bulk piezoelectric of the constituents is accounted for. Serendipitously, this means that the piezoelectric polarization of any nitride alloy at any strain can be found by noting the value for x (the composition), followed by calculating the basal plane strain, e(x) from Vegard’s law, and Ppe from Equation 2.96 using the nonlinear piezoelectric polarization of the binaries (Figure 2.41). This approach is of paramount value in the modeling of nitride heterostructures, especially those with high In content, and AlInN alloys. It should be noted that nonlinearities in the calculated piezoelectric constants of AlN and GaN have also been reported by Shimada et al. [456] but in the realm of

Piezoelectric polarization (C m–2)

0.4

AlN GaN InN

0.3

AlN InN

0.2

GaN 0.1

0.0 0.00

– 0.05

– 0.10

– 0.15

Basal strain Figure 2.41 Piezoelectric polarization in binary nitrides as a function of basal strain (symbols and solid lines) compared to linear piezoelectricity prediction (dashed lines). The c- and u-lattice parameters are optimized for each strain. Courtesy of F. Bernaridini and V. Fiorentini.

2.12 Polarization Effects

volume-conserving strain. A direct quantitative comparison with the results of Shimada et al. and Bernardini and Fiorentini is not possible, because the latter seeks to optimize the volume of the cell so that the stress component along [0 0 0 1] vanishes, which is more appropriate for epitaxial structures. In spite of this substantial difference, there is a common trend between the derivative of the piezoelectric polarization shown in Figure 2.41 (i.e., an effective piezoconstant) and the values of the piezoelectric constants reported in Ref. [456]. The nonlinear piezoelectricity of the binaries can be described by the relations (in C m
2) [84] pz

PAlN ¼
1:808e þ 5:624e2 for e < 0;

pz

P AlN ¼
1:808e
7:888e2 for e>0;

pz

PGaN ¼
0:918e þ 9:541e2 ; pz

PInN ¼
1:373e þ 7:559e2 : ð2:98Þ It is important again to note that the nonlinearity in the bulk piezoelectricity exceeds any effects related to disorder or bond alternation, which have been taken into account [220,221]. The calculation of the piezoelectric polarization of an AxB1
xN alloy for any level of strain would proceed with first calculating the strain e ¼ e(x) for a given molar fraction, x, using the Vegard’s law, and the piezoelectric polarization by pz

pz

pz

PAx B1
x N ¼ xP AN ðeÞ þ ð1
xÞPBN ðeÞ; pz

ð2:99Þ

pz

where PAN ðeÞ and PBN ðeÞ are the end binary strain dependent piezoelectric polarizations that can be calculated for a given strain, eðxÞ, using Equation 2.98. Application of this process to each of the three ternaries for all the possible cases of ternaries is as follows. Using Equation 2.96 and linear interpolation for the elastic constants that are tabulated in Table 2.28 and strain determined from Equation 2.64, the piezoelectric polarization between a given ternary and binary can be calculated, as represented below (in C m
2) [84]. pz

PAlx Ga1
x N=GaN ¼
0:0525x þ 0:0282xð1
xÞ; pz P Alx Ga1
x N=AlN ¼
0:026x þ 0:0282ð1
xÞ; pz PAlx Ga1
x N=InN ¼
0:28x
0:113ð1
xÞ þ 0:042xð1
xÞ:

ð2:100Þ

pz

PInx Ga1
x N=GaN ¼
0:148x þ 0:0424xð1
xÞ; pz PInx Ga1
x N=AlN ¼
0:182x þ 0:026ð1
xÞ
0:0456xð1
xÞ; pz P Inx Ga1
x N=InN ¼ 0:113ð1
xÞ þ 0:0276xð1
xÞ:

ð2:101Þ

pz

P Alx In1
x N=GaN ¼
0:0525x þ 0:148ð1
xÞ þ 0:0938xð1
xÞ; pz PAlx In1
x N=AlN ¼
0:182ð1
xÞ þ 0:092xð1
xÞ; pz PAlx In1
x N=InN ¼
0:028x þ 0:104xð1
xÞ:

ð2:102Þ

The calculated values are a function of molar fraction for the ternaries for the epitaxial layer and template combinations represented in Figure 2.42, assuming that the template is fully relaxed and the epitaxial layer is completely coherently strained

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j 2 Electronic Band Structure and Polarization Effects

260

with strain relaxation. For partial relaxation, unless the degree to which the relaxation that occurs is known, the calculations cannot be made. The extent of relaxation just depends on whether strain-relieving defects propagate from the template to the epitaxial layers. In addition, the effect of cooldown-induced thermal mismatch strain due to cooling from the growth temperature down to the operating temperature of the structure must also be considered. As stated, the results shown are for a perfect system with fully relaxed template and fully strained epitaxial layer on top of it. We should point out that for a particular pair with a particular composition, Al0.82In0.18N/ GaN heterostructure, there is a perfect lattice match; thus, the misfit-induced piezoelectric polarization is equal to zero. It should also be pointed out that other

0.2

Al x In1–x N/AlN In x Ga1-x N/AlN

b = 0.092

Piezoelectric polarization (C m–2)

b = – 0.046

0.1

A1 x Ga 1–x N/AlN b = 0.025 0.0

–0.1 0 (a)

0.2

0.4

0.6

0.8

1.0

Molar fraction, x

Figure 2.42 (a) Piezoelectric polarization of fully and coherently strained ternary alloys on fully relaxed AlN templates in the case of which the ensuing positive piezoelectric polarization and the negative spontaneous polarization are antiparallel. The bowing parameters describing the nonlinearity in the compositional dependence of Ppz are also indicated. (b) The piezoelectric polarization of coherently strained ternary alloys on fully relaxed GaN template. Note that Al0.82In0.18N/GaN heterojunction (indicated with an arrow) is lattice matched, thus

strain and piezoelectric polarization vanishes. (c) Also note that other experiments indicate the lattice matching composition for AlInN on GaN where the piezoelectric polarization charge vanishes to be different. The piezoelectric polarization of coherently strained ternary alloys on fully relaxed InN template. For ternary alloys grown on InN, the negative piezoelectric polarization and the spontaneous polarization are parallel and oriented to along the c-axis. Courtesy of O. Ambacher.

2.12 Polarization Effects 0.2

Piezoelectric polarization (C m–2)

Al x In1–x N/AlN b = 0.094

In x Ga 1–x N/GaN b = – 0.042

0.1

0.0

A1 x Ga 1–x N/GaN b = 0.028

–0.1

0

0.2

(b)

0.8

0.6

0.4

1.0

Molar fraction, x

0.0

Al x In 1–x N/InN b = 0.104

Piezoelectric polarization (C m–2)

In x Ga 1–x N/InN b = – 0.028 –0.1

–0.2

Al x Ga1–xN/InN b = 0.042

–0.3 0

(c) Figure 2.42 (Continued )

0.2

0.4

0.6

Molar fraction, x

0.8

1.0

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j 2 Electronic Band Structure and Polarization Effects

262

Nonlinear

InN 0.4

Piezoelectric polarization, P3 (C m–2)

AlN GaN 0.2

0.0

AlN

–0.2

–0.4

–0.2

–0.1

0.0

0.1

0.2

Strain, ε pz

Figure 2.43 The piezoelectric polarization (P 3 ) of wurtzitic binary group III nitrides under biaxial tensile or compressive strain calculated using Equations 2.98 and 2.99, the premise of which relies on nonlinearity as depicted in the figure. Only the AlN case is shown for positive strain consistent with Equation 2.98. In addition, the values of polarization exceed those of the linear model expressed in Equation 2.69. Courtesy of O. Ambacher.

experiments indicate the lattice matching composition for AlInN on GaN, where the piezoelectric polarization charge vanishes, to be different. The piezoelectric polarization versus strain, ignoring nonlinearity, can be calculated using Equation 2.69 for both compressive and tensile biaxial strain for all the epitaxial layer/template configurations. Use of Equations 2.98 and 2.99, however, takes the nonlinearity discussed above into consideration, as shown in Figure 2.43. As expected, the piezoelectric polarization for a given strain increases from GaN to InN and AlN. Significantly, higher piezoelectric polarizations, especially in cases of high strains (high In concentrations), result when nonlinearities in the piezoelectric charge are considered. The piezoelectric polarization associated with coherently strained random ternary AlGaN, InGaN, and AlInN alloys grown on GaN templates is shown in Figure 2.44. To reiterate, the piezoelectric polarization is nonlinear with respect to the alloy

2.12 Polarization Effects

Piezoelectric polarization, P3 (C m–2)

0.2

Al x In1–x N

In x Ga 1–x N

0.1

0.0

Al xGa1–x N/

0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x Figure 2.44 The piezoelectric polarization of random coherent ternary alloys on relaxed GaN templates. Those calculated using Equation 2.69 and linear interpolations of the physical properties such as the piezoelectric constants and elastic coefficients (exy , Cxy) for the relevant binaries are shown in dashed lines. Calculations considering the nonlinearity in the piezoelectric polarization in terms of strain reflected in

Equations 2.98 and 2.99 are shown in solid lines. For alloys under high biaxial strain such as those containing high In concentration, the piezoelectric polarization is underestimated by the linear approach. It should be pointed out that the lattice matching composition for AlInN on GaN where the piezoelectric polarization charge vanishes is converging on 16–18% In in the InxAl1
xN lattice. Courtesy of O. Ambacher.

composition, spanning a range of
0.461 to 0.315 C m
2, the end points corresponding to coherently strained AlN on InN and InN on AlN, respectively. A point of side interest, albeit not practical, is that in alloys with high indium concentration, leading to high strain, the piezoelectric polarization is larger than that of the spontaneous polarization. As mentioned earlier in conjunction with Figure 2.42 that Al0.82In0.18N can be grown on GaN in a lattice-matched form making the piezoelectric polarization for this heterostructure vanish. To underline the point, the nonlinear model described by Equations 2.98 and 2.99 leads to larger piezoelectric polarization charge than linear interpolation where strain, elastic constants, and piezoelectric constants are assumed to vary linearly with composition.

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2.12.4 Polarization in Heterostructures

Investigation of heterostructures such as quantum well and single heterointerfaces such as those present in modulation-doped structures are ideal platforms for putting to test the knowledge base of spontaneous and piezoelectric polarization. Depending on the polarity of the sample, that is, Ga or N, the order of growth, meaning GaN on AlGaN or the other way around, and the buffer layer that determines to a first extent whether the barrier, the well, or both are in strain, the piezoelectric and spontaneous polarization charge add or subtract. Regardless of this, the resultant band bending leads to a red shift in energy due to the presence of electric field, commonly referred to as quantumconfined Stark effect (QCSE), and the ensuing deformation of the wave functions being pushed to opposite ends of the interface leads to reduction of radiative efficiency, increase in lifetime, and also blue shift with injection that is screening. In parallel, if the well size is small enough, carrier confinement induces a blue shift. Consequently, the red shift induced by polarization and the blue shift induced by localization compete in determining the transition energy. The optical transitions both in CW and timeresolved forms are discussed in detail in Volume 2, Chapter 5. Here, the manifestation of polarization charge in heterostructures is treated. We should mention that the electric field resulting from the polarization in GaN would cause a large Stark shift in optical measurements and reduce the effective bandgap in, for example, quantum wells (see Volume 2, Chapter 5), which is of paramount importance in optoelectronic devices. The field induced has been calculated in the context of the GaAs-based superlattices, which show notable Stark shift when grown along the [1 1 1] direction [457]. Stark shift can be screened on a length scale of the order of the Debye length ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pThe ekT=q2 n by injecting free carriers in the GaN layer(s), as is the case in LEDs, lasers, and PL experiments with large excitation intensities. In such a case, the strain-induced field causes carrier separation that, in turn, causes a field that opposes the strain field. Strain-induced polarization in such a heterostructure can also lead to a net electric field that can be measured as a voltage drop across the sample. The impact of the Stark shift and concomitant screening has been considered for laser gain also [458]. Fiorentini et al. [448] and Bernardini and Fiorentini [459] demonstrated that for an alternating sequence of wells (w) and barriers (b), the total electric field in the well can be calculated by recognizing that the normal components of the displacement vector D of Equation 2.60 are continuous (provided that there is no interface charge). total eW E W þ 4pPtotal W ¼ DW ¼ DB ¼ eB E B þ 4pP B

ð2:103Þ

in cgs units (for MKS remove the 4p terms). Utilizing Equation 2.103 together with the periodicity-imposed equality of voltage drops being equal but with opposite sign LW E W ¼
LB E B : One arrives at

ð2:104Þ

2.12 Polarization Effects total E W ¼
4pLB ðPtotal W
P B Þ=ðLW eB þ LB eW Þ;

or

total E W ¼
4pðPtotal W
P B Þ=ðLW =LB eB þ eW Þ;

total E B ¼
4pLW ðPtotal B
P W Þ=ðLB eW þ LW eB Þ; 0 1 L B total @ or E B ¼
4pðPtotal þ eB A: B
P W Þ= LW eW

ð2:105Þ

where eW and eB are the dielectric constant of the well and barrier layers, respectively. Likewise, LW and LB represent the well and barrier thicknesses. The same notation is used for polarization also. Both W and B superscripts and subscripts relate to wells and barriers, respectively. The total polarization term can be changed to piezoelectric term and spontaneous polarization term for cases when only the former or the latter is in effect, respectively. total Explicit in Equation 2.105 is that whenever Ptotal W „P B there will be an electric field. Due to strain or screening, the electric field will be present in both the well and the barrier. If we limit ourselves to piezoelectric polarization and somehow achieve relaxed heterostructures, the piezoelectric polarization charge and therefore the field will be zero. The electric field in wells and barriers in a quantum well has two components, one from spontaneous and the other from piezoelectric polarization. If the thickness of the wells and barrier are the same, the field in wells and barriers are related to each other as sp

pe

sp

pe

E W þ E W ¼ E W ¼
E B
ðE B þ E B Þ;

ð2:106Þ

where EW and EB represent the electric field in the well and barrier material. The superscripts indicate the field due to spontaneous and the other from piezoelectric polarization. Additional comments that can be made are that if lattice-matched AlInGaN alloy is used, the piezoelectric component in that material is eliminated. However, the spontaneous component would still be present. Another point to be recognized is that the piezoelectric field induced in InGaN and AlGaN layers of the same composition of In in the former and Al in latter, if grown on relaxed GaN, is larger in the former because of the larger lattice mismatch between GaN and InN as compared to that between GaN and AlN. The spontaneous polarization in alloys can be found using Equations 2.85 and 2.86 together with values listed for the binary end points in Table 2.25. The spontaneous polarization so formulated together with piezoelectric polarization of Equation 2.77 (and also Equation 2.83) would allow the computation of total polarization charge. This must be done for both the barrier and well material. The electric field can then be calculated using Equation 2.105. Any free carrier present in the well and barrier regions as well as those injected by optical and/or electrical means tend to screen the polarization-induced field. A complete picture can be obtained by solving self-consistently a set of effective mass theory or tight binding theory and simultaneously the Schr€ odinger–Poisson equation [460], which is discussed in Section 2.1.14. Combining the ab initio calculations

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j 2 Electronic Band Structure and Polarization Effects

266

of Fiorentini and Schr€odinger–Poisson solver of Di Carlo allowed the authors to calculate the charge distribution and field profiles in quantum wells in the presence of free carriers [448,461]. Poisson’s equation is solved using the boundary condition that the electric field is zero at the ends of the simulated regions. This corresponds to LW reaching the infinity limit in Equation 2.105. The potential thus obtained across the structure is plugged into the tight binding Schr€ odinger equation, which is solved to obtain energies and wave functions [462]. The new quasi-Fermi levels are then calculated, which then lead to carrier concentrations, and the procedure is iterated until self-consistency is obtained. A case of interest for FETs and parts of quantum wells is one that features a ternary on the surface where polarization charge would exist. The same is true for the interface between an AxB1
xN and a GaN heterostructure. The characteristic of this charge is that it changes abruptly, leading to a fixed two-dimensional polarization charge density s on the surface and at the interface, which is given by sp

pz

sAx B1
x N ¼ P Ax B1
x N þ P Ax B1
x N sp

pz

on the surface and sp

pz

sAx B1
x N=GaN ¼ ðPGaN þ P GaN Þ
ðP Ax B1
x N þ P Ax B1
x N Þ at the interface; ð2:107Þ respectively. Figure 2.45 shows the polarization-induced surface and interface sheet density s/q for relaxed and coherently strained binary nitrides and ternary–binary AxB1
xN/GaN interfaces, such as AlxGa1
xN/GaN. The spontaneous polarization surface charge density on relaxed InN, GaN, and AlN layers is 2.62 · 1013 cm
2, 2.12 · 1013 cm
2, and 5.62 · 1013 cm
2, respectively (there is no piezoelectric charge in this case). If a biaxial compressive strain of e ¼
0.002 is present, the surface charge is reduced to 0.72 · 1013 cm
2 · 1013, 0.74 · 1013, and 3.22 · 1013 cm
2, as can be deduced from Figure 2.45a. For compressively strained InN, GaN, and AlN at the level of e ¼
0.025,
0.030, and
0.045, respectively, the piezoelectric polarization fully compensates the spontaneous polarization and thus no polarization-induced field should exist. As can be seen from Figure 2.45a, polarization-induced surface charge is reduced by compressive and increased by tensile strain. For fully relaxed layers grown on the c-axis on a substrate, the spontaneous polarization-induced surface charge is negative for Ga-polarity and positive for N-polarity samples. The bound surface charge can be screened by oppositely charged surface defects and charges adsorbed on the surface. If this screening were incomplete, the carrier concentration profiles in the crystals would be different. Consequently, surface band bending, as well as that of interface, would be affected by polarization, and further such band bending would be dependent of the polarity of the sample. Experimental results are not sufficiently complete to draw a definitive conclusion. For coherently strained Ga-face AlxGa1
xN/GaN (for 0 < x  1) and AlxIn1
xN/ GaN (for 0.71 < x  1 interfaces), the polarization-induced interface charge is calculated to be positive as shown in Figure 2.45b. For both ternary and binary interfaces, the bound polarization charge increases nonlinearly with composition, x,

2.12 Polarization Effects

up to 7.06 · 1013 cm
2. However, for Ga-face InxGa1
xN/GaN (for the entire compositional range, 0 < x  1) and AlxIn1
xN/GaN (or 0 < x  0.71), the polarization charge is calculated to be negative and nonlinear with respect to composition. At the limit for the former, InN/GaN structure, a bound sheet density of 14.4 · 1013 cm
2 is calculated. Upon screening in n-type heterostructure, a positive polarization sheet 4

3 AlN Bound surface density s /q (1014 cm–2)

2

1

0

AlN

–1 InN –2

–3 GaN –4 –0.2

–0.1

0.0

0.1

0.2

Strain, ε (a) Figure 2.45 (a) In-plane biaxial strain dependence of surface polarization (piezoelectric plus spontaneous) charge for wurtzitic GaN, InN, and AlN. As in the case of Figure 2.43, the AlN case is with positive strain. (b) Bound interface charge density for coherent AlxGa1
xN, InxGa1
xN, and AlxIn1
xN grown on relaxed Ga-polarity GaN. Polarization-induced

bound interface charge (positive in n-type samples and negative in p-type samples) is screened by free electrons in n-type and holes in p-type samples, respectively, leading to twodimensional gas. Note that the interface charge can be converted into carrier concentration by dividing it with the electronic charge of q ¼ 1.602 · 10
19 C. Courtesy of O. Ambacher.

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j 2 Electronic Band Structure and Polarization Effects

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2.0 Ga-Polarity on GaN

2DEG

Bound interface density s /q (1014cm–2)

1.5

1.0 AlN 0.5

N A l xG a1 –x

GaN 0.0

–0.5 In

x Ga

N

I n 1 –x

Al x

–1.0

1– x

N

InN

–1.5

InN 2DHG

–2.0 0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x (b) Figure 2.45 (Continued )

charge, which is bound, leads to a 2DEG with a sheet carrier concentration close to the concentration of the bound interface density þs/e. Now that a conceptual issues related to polarization has been discussed, the attention can now be turned to calculating the sheet charge distribution and band structure at a single heterointerface typically used for MODFET. The available data indicate that GaN channel devices are the only viable ones, even though In-containing channels may someday work well as well. For the moment, the likely GaN channel MODFET structures utilize AlxGa1
xN, AlxIn1
xN, or perhaps the quaternary. The criterion is that the bandgap of AxB1
xN ternary must be larger than that of GaN. As discussed often, the AlxIn1
xN ternary lattice matches GaN; however, the reported values for lattice matching composition vary. For example, AlN molar fraction of 71% (corresponds to InN molar fraction of 29%) and 82–83% (corresponds to InN molar fraction of 18–17%, which is the more likely lattice matching composition) have been reported to lattice match GaN [451]. To treat the MODFET interface charge problem,

2.12 Polarization Effects

the bandgap of nitride semiconductor alloys, which are discussed in Section 1.5 in detail, must be known. For completeness, the expressions for the compositional dependence of the alloy bandgap are repeated below. It should be pointed out that the bandgap bowing parameter data bAx B1
x N for AlGaN are converging onto a value of nearly 1 eV even though early figures spanned from
0.8 eV (upward bending) to þ 2.6 eV (downward bending). The data for InGaN are fluidic also in part because of reasons having to do with difficulties in determining the composition and bandgap, detailed in Section 1.5.2, and also lack of high-quality samples with composition midway in the alloy range. For example, optical reflectivity measurements together with PL data corrected for Stokes shift led to a bInGaN ¼ 2.5 0.7 eV for 0.9 eV bandgap of InN and bInGaN ¼ 3.5 0.7 eV for 1.9 eV InN bandgap [463]. PL data alone in another report states the bowing parameter to be 4.5 eV [464]. The figure determined by optical transmission measurements is 8.4 eV [465], while the theory indicates 1.2 eV [466]. Other experimental values for lower InN molar fraction end of the ternary are near null, meaning linear variation of the bandgap [467,468]. The reported data in aggregate, taking into consideration the small bandgap of InN, leads to a value of 2.53, as discussed in Section 1.5.2. The value for the bowing parameter in AlInN is somewhat too controversial. For example, one particular theoretical report points the bowing parameter to be 2.53 eV [469] while a value of 3.1 eV was determined by fitting the bandgap of this alloy determined by photoreflection [470], 2.384 eV by absorption measurements but by using 2.0 eV for the bandgap of InN, and 5.9 eV for AlN [471] and 5.4 eV in a review article where 1.95 eV was used for InN bandgap [84]. However, when all the available data are considered in aggregate with 0.7 eV bandgap for InN, a bowing parameter of about 3 eV appears to be a very good value. g

g

g

E Alx Ga1
x N ¼ xE AlN þ ð1
xÞE GaN
bAlGaN xð1
xÞ ¼ 6:1x þ 3:42ð1
xÞ
xð1
xÞ eV; g E Inx Ga1
x N

g

g

¼ xE InN þ ð1
xÞE GaN
bInGaN xð1
xÞ ¼ 6:1x þ 0:7ð1
xÞ
1:43xð1
xÞ eV;

g

ð2:108Þ

g

E Alx InN ¼ xE AlN þ ð1
xÞE EInN
bAlInN xð1
xÞ ¼ 6:1x þ 0:7ð1
xÞ
bAlInN xð1
xÞ eV: The presence of displacement or polarization gradient leads to a volume charge density rv given by !

!

!

r: D ¼ r:ðe E þ P total Þ ¼
rv

!

in MKS units: For cgs; add 4p before P : ð2:109Þ

In a one-dimensional system, which is considered here along the c-axis, and utilizing E ¼
dV/dz, Equation 2.109 can be rewritten as the Poisson’s equation in the form of dD=dz ¼ d=dzð
eðzÞdV=dz þ P total ðzÞÞ ¼
qrz ¼
q½N Dþ ðzÞ þ pðzÞ
nðzÞ
N A
ðzÞ:

ð2:110Þ

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j 2 Electronic Band Structure and Polarization Effects

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The term on the right-hand side of Equation 2.110 represents the volume density of net charge, and the position-dependent quantities D, e, and V are the displacement field, dielectric constant, and potential, respectively. The term Ptotal is the position-dependent total transverse polarization (along the c-axis and perpendicular to the interfaces). N Dþ ðzÞ and N A
ðzÞ represent the ionized donor and acceptor concentrations, and p(z) and n(z) represent the hole and electron concentrations, respectively. The effects of composition, polarization, and free-carrier screening are thus fully included. With the knowledge of the band profile for both the valence band and the conduction band, one can determine the electronic states in the heterostructure by solving the Schr€odinger (Equation 2.59) equation, as a function of the spatial coordinate z. In the effective mass approximation, one needs to solve the following eigenvalues problem:  
2 d 1 dCi ðzÞ h
þ V c ðzÞCi ðzÞ ¼ E i Ci ðzÞ; 2m0 dz mz dz

ð2:111Þ

for which the appropriate choice of the effective masses in the conduction band and in the valence band is done by preliminary calculation using the tight binding approximation. The solution of Equation 2.111 determines the eigenstates Ei in the conduction band and in the valence band, and the corresponding eigenfunctions Ci. It is imperative to note that free carriers cannot eradicate the polarization charge, which is bound and invariable (unless structural changes are made), but one can screen it to a degree determined by carrier concentration. Likewise, polarization charge is bound charge and cannot by itself be the source of free carriers, but it can cause a redistribution of free carriers that would tend to screen the polarization charge. In addition to the polarization charge related parameters that must be known to solve the Schr€odinger and Poisson equations, one must also know the dielectric constants, doping levels, effective masses, and band discontinuities, in addition to compositions and the layer thicknesses used. The band discontinuity issue is discussed in Volume 2, Chapter 1, but suffice it to say that the bandgap of the ternaries is found from Equation 2.108 with the appropriate bowing parameter, and a certain fraction of the total bandgap difference is assigned to the conduction band and the rest to the valence band. To mitigate the process, the band alignment for all the nitride heterostructures is of type I, which means that the larger bandgap straddles the smaller bandgap one. Typically, 60–70% of the total band discontinuity is assigned to the conduction band. A linear interpolation for the effective masses, given in Chapter 1, in ternaries from binary end points can be used. While the tables provided in Chapter 1 list the dielectric constants for binaries, knowledge of them leads to the ternary dielectric constants via linear interpolation. Relative dielectric constants of 10.28, 10.31, and 14.61 (compares with 15.1 presented in Chapter 1) for dielectric constants in GaN, AlN, and InN, respectively, have been calculated [472]. The GaN dielectric constant is nominally larger than that of AlN, but in this particular investigation the data are as listed above. Following the so calculated figures and applying the linear interpolation leads to dielectric constant for ternaries.

2.12 Polarization Effects

eAlx Ga1
x N =e0 ¼ 10:28 þ 0:03x; eInx Ga1
x N =e0 ¼ 10:28 þ 4:33x; eAlx In1
x N =e0 ¼ 14:61
4:33x: Using the values for the binary dielectric constants tabulated in Chapter 1, the linear interpolation scheme for the ternary dielectric constants lead to eAlx Ga1
x N =e0 ¼ 10:4
1:9x; eInx Ga1
x N =e0 ¼ 10:4 þ 4:9x; eAlx In1
x N =e0 ¼ 15:3
4:9x:

ð2:112Þ

In dealing with devices such as MODFETs where a Schottky barrier is placed on top, the Poisson’s equation, through boundary conditions, is affected by the metal barrier height as well as the applied bias on the metal. The Schottky barrier height for the ternaries can also be interpolated from the binary end points. It should also be noted that the barrier height depends on the particular metal used, the details of which is discussed in Volume 2, Chapter 1. Because the Fermi level on the surface of GaN is not fully pinned in that heavier metals with large work functions lead to larger barrier heights, it is plausible and natural to assume that the barrier height would increase with an increase in AlN content and decreases with InN content. However, for InN the Fermi level is most likely in the conduction band already, as in InSb and InAs. Limited studies of metal contact potential on nitride semiconductors make it difficult to state a molar fraction dependence of this parameter. However, Ti and Ni contacts on Al0.15Ga0.85N have been studied and were compared to those on GaN. The barriers heights for Ti and Ni increased on Al0.15Ga0.85N [473]. Comparing the data deduced for Ni using I–V, C–V, and photoemission methods, barrier heights of 0.95 (0.84 eV if not corrected for the nonideal ideality factor), 0.96, and 0.91 eV, respectively, have been obtained. For Al0.15Ga0.85N, the same figures are 1.25 (1.03 eV if not corrected for the nonideal ideality factor), 1.26, and 1.28 in order, using the same methods. These figures represent an increase of about 0.3 eV in barrier height for Al0.15Ga0.85N over GaN. Assuming that Ni is used for Schottky barriers, the barrier height for GaN is 0.95 eV, and more boldly reported figures for one mole fraction do actually represent a figure consistent with a linear interpolation, one can express the molar fraction dependence of the barrier height on AlxGa1
xN as fAlx Ga1
x N ¼ 0:95 þ 2xV:

ð2:113Þ

Following Ref. [84], the linear interpolation for the other two ternaries modified for barrier height on GaN used here are fInx Ga1
x N ¼ 0:95
0:36xV; fAlx In1
x N ¼ 0:59 þ 1:36xV:

ð2:114Þ

These expressions lead to barrier heights (qf or ef) of 0.95, 2.95, and 0.59 eV on GaN, AlN, and InN, respectively. With the exception of the value for GaN, the rest is really speculative at this point.

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j 2 Electronic Band Structure and Polarization Effects

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2.12.4.1 Ga-Polarity Single AlGaN/GaN Interface Returning to single-interface structures, such as those used in MODFETs (or HFETs), the total interface charge, for example, at a gated AlxGa1
xN and GaN interface, rs, for an n-type case would be sum of total polarization charge rp and free-carrier charge ns.    er e0 E Fi
DE C ; ð2:115Þ rs ¼ rp
ns ¼ rp
V G
fB
V p2 þ qdAlGaN q

where VG is the applied gate bias in terms of V, fB is the Schottky barrier height in terms of V, on AlxGa1
xN, EFi is the Fermi level in GaN at the interface with respect to the edge of conduction at the interface, DEC is the conduction band discontinuity between the ternary AlxGa1
xN and GaN, and Vp2 is the voltage drop across the doped AlGaN, which in turn is given by V p2 ¼ qN d d2d =2er e0 where Nd is the donor concentration in AlxGa1
xN, all assumed to be ionized, dd is the thickness of the doped AlGaN, and ere0 is the dielectric constant of AlxGa1
xN. Vp2 is negative for depleting voltage in the case of which it would add to the Schottky barrier height. We should point out at this stage that the form of Equation 2.115 is good for determining the sheet carrier concentration at the interface, but when used for FETs both the polarization and the charge induced by doping must be lumped into the threshold or off voltage. For details see Volume 3, Chapter 3. For a p-type semiconductor, the semiconductor charge as well as the reference for the Fermi level and the band discontinuity should be that at the valence band edges [88,474]. The effect of any undoped AlxGa1
xN layer designated as having a thickness of di is small because its thickness is several nanometers and is neglected in Equation 2.115. This setback layer was originally employed by the author in the GaAs system to further reduce remote Coulomb scattering for increased mobility. A detailed treatment including the effect of di can be found in Ref. [474]. For an undoped AlGaN/GaN heterostructure, the Vp2 term can be set to zero. In this case the boundary conditions for potential or the Fermi level are made consistent with the Schottky barrier height on the Alx Ga1
x N surface. In the bulk of the structure, due the special nature of the quasi-triangular barrier at the interface, shown in Figure 2.46, the Fermi level is generally taken to be near the midgap of the smaller bandgap material, which in this case is GaN. As in the case of quantum wells, the one-dimensional Schr€ odinger–Poisson solver can be iteratively used to determine the band and carrier profile (simultaneous solution of Equations 2.110 and 2.111 in an iterative mode either in the tight binding realm or the effective mass approximation, but in the current example it is in the effective mass realm). The bound charge can be represented by a thin and heavily doped interfacial layer, the thickness of which is about 1 nm or less, keeping in mind that the total charge associated with such a fictitious layer must be equal to the bound charge. In one investigation, a thickness for this fictitious layer of 0.6 nm was used. To reiterate, to solve this pair of equations boundary conditions at the interface and surface as well as the structural parameters must be known. In a typical undoped AlGaN/GaN MODFET, the GaN buffer layer is unintentionally doped with a level of ND  1016 cm
3, and the thickness of this buffer layer spans 1–3 mm. The barrier thickness spans 10–20 nm. Because the structure is not intentionally doped,

2.12 Polarization Effects

Ec

AlGaN

GaN

di

qφB

Ec ΔEc

-qVG

E1 E0

EF

EFi

dd

x=0 Figure 2.46 Conduction band edge of what is generically referred to as a modulation-doped structure based on the AlGaN/GaN system. The origin of the interface charge is aggregate due to polarization and free carriers. In Ga-polarity samples, AlGaN grown on GaN produces polarization charge (due to spontaneous polarization and piezoelectric polarization

x because AlGaN is under tensile strain), causing accumulation of electrons at the interface in addition to any electron donated by the any donor impurities, intentional or unintentional, in AlGaN. The diagram is shown for a doped AlGaN, the doped portion of which is indicated by dd and the undoped part is indicated by di.

it is assumed that unintentional impurities in the form of contamination or native defects, such as O, Si, VN, are responsible for supplying the free carriers, which is measured with the accompanying assumption that it is sufficient to completely screen the positive polarization charge, þr/q, at the interface [476]. The influence of doping specifically in the case of AlGaN/GaN has been treated, for example, by Chu et al. [475]. In short, the sheet charge at the interface of such an undoped structure is dominated by the polarization-induced charge. On Ga-polarity surfaces, this charge increases with increasing AlN molar fraction in the barrier because both the piezo and spontaneous components of the polarization charge increase, assuming of course that the barrier is coherently strained. The conduction band and electron concentration profiles for an undoped Ga-face Al0.3Ga0.7N/GaN (30 nm/ 2 mm) heterostructure with an Ni Schottky contact on top are shown in Figure 2.47a. An electric field strength of about 0.4 MV cm
1 in the barrier and a sheet electron concentration of 1.2 · 1013 cm
2 are induced by polarization. The bound sheet density and 2DEG sheet carrier density induced by polarization in heterostructures (identical to that shown in Figure 2.47a with the exception that the alloy compositions in the barrier has been changed) are shown in Figure 2.47b and c with solid lines for both the linear and nonlinear polarization cases. The underlying assumptions are that the GaN buffer layer is relaxed, the barriers are coherently strained, and the physical parameters of importance (Cij, eij, and Psp) linearly scale from binaries to ternaries.

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j 2 Electronic Band Structure and Polarization Effects

274

The estimated error depicted by the gray area in Figure 2.47c is primarily due to uncertainties in the barrier thickness and the conduction band offsets. The sheet carrier concentrations of 2DEGs confined in an AlxGa1
xN/GaN heterostructure for x ¼ 0.5 have been measured by C–V profiling using Ti/Al ohmic and Ni Schottky contacts [84]. The highest measured and calculated sheet carrier concentration for 2.0

+s E = 0.41 MV cm–1

1.5

2DEG

1.0

Energy (eV)

13

Ns = 1.2 10 cm –2

eΦ

1.23 eV

0.5

CB

0

Δ

0.30 eV

–0.5 (a)

0

10

ΔE

EF

c

0.38 eV

E0 = 0.17 eV

30 40 20 Depth (nm)

50

Figure 2.47 (a) Self-consistent calculation of the Schr€ odinger–Poisson equations for the conduction band edge and the electron density profile for an undoped Ga-polarity Al0.3Ga0.7N (30 nm)/GaN (2 mm) single-interface heterostructure. Also shown are the Schottky metal on the surface and the polarizationinduced surface and interface charges. (b) The polarization-induced bound interface charge density as a function of the alloy composition in the barrier for Ga-polarity AlxGa1xN/GaN heterostructures for the case of relaxed buffer layer and coherently strained barrier layer. The upper solid line for sheet charge corresponds to the case of linear interpolation of physical parameters (Cij, eij, and Psp) from the binary compounds. The lower solid line for the sheet charge, on the contrary, corresponds to the case of nonlinear extension of polarization from binary end points. The dashed lines depict the lagging sheet density with increasing molar fraction, x, due to partial relaxation, which is

60 accounted for by the measured degree of barrier relaxation into account. (c) 2DEG sheet carrier concentrations ns in unintentionally doped Gaface AlxGa1
xN/GaN heterostructures (with 30 nm AlGaN and 2000 nm GaN – the structure referred to in (b)) as obtained by C–V profiling versus alloy composition of the barrier (open symbols), compared with the theoretical predictions for the bound interface charge r/q calculated using (i) a linear interpolation between the macroscopic polarizations of the binary compounds (upper solid line) and (ii) the nonlinear piezoelectric and spontaneous polarization (lower solid line). The sheet charge of the 2DEGs is then calculated as in (ii), considering in addition the depletion by a Ni Schottky contact (dotted line). The dashed lines account for the experimentally observed strain relaxation of the barrier for x > about 0.4. Courtesy of O. Ambacher. The related details can be found in Refs [476,84,479].

2.12 Polarization Effects

1014

Bound interface density σ/q(cm–2)

Linear interpolation

Coherently strained

1013

Nonlinear

1012

0

0.4

0.2

(b)

0.6

0.8

1.0

Molar fraction, x

3.5

Sheet carrier concentration ns (1013 cm–2)

Relaxed

Ga-face Alx Ga1–x N/GaN (30/2000 nm)

3.0

Pseudomorphic

2.5 2.0 Partly relaxed

Linear interpolation

1.5 1.0 0.5

Nonlinear

0 (c)

0

0.1

0.3 0.4 0.2 Molar fraction, x

0.5

0.6

Figure 2.47 (Continued )

AlxGa1
xN/GaN heterostructures is 2 · 1013 cm
2 for x ¼ 0.37, as higher AlN molar fractions in the alloy for 30 nm barrier causes a partial relaxation, lowering the piezo contribution [476]. The extrapolation, indicated by dashed lines in Figure 2.47c, represents the case when the mole fraction is sufficiently large to cause some degree of relaxation. As expected, the sheet density decreases nonlinearly with reducing mole fraction for the 30 nm barrier modeled. Among the two curve that bend over, the upper one is the total calculated polarization charge and the lower one is the

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j 2 Electronic Band Structure and Polarization Effects

276

calculated screening charge using the one-dimensional Schr€ odinger–Poisson equations. When full or partial relaxation occurs, the piezo component of the polarization charge is reduced while the spontaneous component remains unchanged. The experimental data along with the error bars are shown in open circles. Overall, there is then a reduction of the charge. Likewise, the screening sheet carrier concentration is also reduced. The obvious conclusion that can be made is that a much better agreement is attained between calculations and experiments when the nonlinear polarization is used. The case of the nonlinear polarization is stronger in the 2DEG case than it is in the quantum well case due to the direct nature of the measurement. Ridley et al. [477] presented an empirical expression relating the sheet carrier concentration for a nominally undoped AlGaN/GaN single-interface heterostructure having barrier thicknesses of greater than 15 nm and AlN molar fractions over 6%. ns ðxÞ ¼ ½
0:169 þ 2:61x þ 4:50x 2 1013 cm
2 :

ð2:116Þ

Naturally, the sheet charge becomes affected by parameters other than the AlN molar fraction when the barriers are made much thinner, as can be deduced from Equation 2.115. Using an expression similar to Equation 2.115, Ambacher et al. [84] plotted the dependence of the sheet density of barrier thickness for several AlN molar fractions, namely, x ¼ 0.15, 0.30, and 0.45. A priori, it is clear that beyond a certain thickness of the barrier, the density should saturate even if coherent strain prevails, as shown Figure 2.48. Ambacher et al. [84] also measured the sheet density for a set of samples with x ¼ 0.3 by C–V profiling for barrier thicknesses spanning the range of 1–50 nm, and presented the data together with those from other reports. 2.12.4.2 Ga-Polarity Single AlxIn1
xN/GaN Interface This system, if for nothing else, is of importance because the entire structure can be lattice matched, leading to vanishing piezoelectric polarization and thus allowing one to investigate and probe only the spontaneous polarization in undoped structures. Doing so would enhance our confidence in spontaneous polarization calculations, as there are fewer parameters to be determined and thus less uncertainty. For high concentrations of Al (for x > 0.6), the bandgap of AlxIn1
xN is larger than that of GaN, and if used in conjunction with Ga-polarity GaN, a 2DEG would result due to polarization or doping in AlxIn1
xN or both. For lattice-matched conditions where the x-value is about 0.82, the bandgap of the AlxIn1
xN alloy is about 4.7 eV to produce sufficiently large band discontinuity with GaN, imperative for MODFET structures. Ambacher et al. [84] prepared coherently strained AlxIn1
xN (50 nm)/ GaN (540 nm) single-interface heterostructures with Al concentrations between 0.78 and 0.88, and applied X-ray diffraction to determine the structural state of GaN, meaning relaxed or strained (GaN grown on sapphire is typically under compressive strain due to thermal mismatch). Through the in-plane and out-of-plane strain (which can both be measured using reciprocal space X-ray diffraction mapping) or the lattice constants, the mole fraction of the barrier and, through repeated attempts, the mole fraction giving rise to lattice matching conditions can be determined. Ambacher et al. [84] determined the lattice matching composition of AlxIn1
xN to

2.12 Polarization Effects

10

14

Sheet carrier concentration ns (cm–2)

x = 0.45

10

13

= 0.3

= 0.15

1012

Ga-face AlGaN/GaN

10

11

0

10

20

30

40

50

60

d AlGaN (nm) Figure 2.48 Barrier thickness, d, dependence of sheet density in nominally undoped and coherently strained AlxGa1
xN/GaN heterointerfaces for x ¼ 0.15, 0.30, and 0.45 (solid lines). Experimental data available for x ¼ 0.3 measured by C–V profiling for barrier thicknesses spanning 1 and 50 nm, representing an aggregate from several reports, are shown [84].

GaN to be x ¼ (0.83 0.01), as shown in Figure 2.49, which is compared with other experimental values of 0.82–0.83 [451]. The weak compressive strain in GaN was determined to be e ¼
1.9 · 10
3, which would result in a piezoelectric polarization of 1.5 · 10
3 Cm
2. The residual strain in GaN would lead to a bound sheet density of only
1012 cm
2, which is much smaller than the 1013 cm
2 electron sheet density. This implies explicitly that interface charge is dominated by the gradient in the spontaneous polarization across the GaN/AlxIn1
xN interface, as show in Figure 2.50, which shows the calculated bound sheet density induced by a gradient in spontaneous polarization (upper dashed line) and strain-induced piezoelectric (lower solid line) for a range of compositions near the lattice matching conditions and the resultant 2DEG sheet carrier density (upper solid line). Also shown are the experimental 2DEG densities obtained by C–V measurements. The polarization and thus the sheet density have been obtained by a linear interpolation of the physical parameters (Cij, eij, and PSP) of the binary compounds but by taking the nonlinearity of Ppz and PSP into account. The calculations confirm the obvious in that the measured high electron sheet density can be accounted for by the spontaneous polarization charge.

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j 2 Electronic Band Structure and Polarization Effects

278

6

cAlInN (x) 4 aAlInN (x)

Strain, e (10–3)

2 Tensile strain 0 Compressive strain –2

–4

Lattice matched –6 0.75

0.85 0.80 Molar fraction, x

Figure 2.49 Strain in AlxIn1
xN barrier along the c-axis and on the basal plane as determined from the measured lattice constants, cAlxIn1
xN and cAlxIn1
xN by high-resolution X-ray diffraction reciprocal mapping. AlxIn1
xN for x ¼ 0.83 can be

0.90 grown lattice matched to GaN, leading to vanishing piezoelectric polarization due to misfit. However, residual strain, due to thermal mismatch between GaN and the substrate, can still induce piezoelectric polarization [84].

Note that for Ga-polarity samples and AlN compositions less than the lattice match conditions, the AlxIn1
xN barrier layer is under compressive strain in-plane and therefore the piezo-induced polarization is opposite in sign to that for spontaneous polarization. When the AlN molar fraction exceeds that for matching condition, the in-plane strain is tensile in the case of which the two polarization charges are additive. Near the lattice matching conditions, the calculated sheet density is about 2.95 · 1013 cm
2 and falls above the measured values. The reason for this discrepancy could be related to samples themselves, either in the form of defects near the interface, surface adsorbates, and/or inhomogeneities caused by In, and underestimation of the bowing parameters associated with the polarization charge. 2.12.5 Polarization in Quantum Wells

For multiple interface heterostructures, the sheet carrier density and barrier thickness, as well as the width quantum wells, are of interest because the total potential drop across the structure is directly proportional to the product of polarization field and well width in constant field approximation if free-carrier screening is neglected.

2.12 Polarization Effects

4

s sp e (P )

2

n (x) S

1

0

–1

–2 0.75

s z (P ) E

0.80

Lattice matched

Sheet carrier concentration (1013cm–2)

3

0.85

0.90

Molar fraction, x Figure 2.50 Compositional dependence of the spontaneous polarization (dashed line) and piezoelectric polarization (lower solid line) induced charge density reduced to interface sheet charge in an AlxIn1
xN/GaN heterojunction using the nonlinear interpolation for polarization discussed in the text. For AlN molar fractions below the lattice matching

conditions, the two polarization charges oppose and above the lattice matching value, they add. The calculated sheet electron concentration (upper solid line) and the measured sheet electron density are also shown, the latter of which has been obtained by both Hall effect (solid symbols) and C–V profiling (open symbols) [84].

This issue goes to the heart of lasers, particularly in the stages as the gain is built up, in that the well widths greater than about 5 nm are not used. At higher injection levels, the polarization-induced field is screened pretty much [458]. To illustrate the point, the conduction band profile of a 10 nm GaN/In0.2Ga0.8N quantum well as calculated by Della Salla et al. [461] is shown in Figure 2.51 for several sheet densities. Even at a substantial sheet density of n2D ¼ 5 · 1012 cm
2, a nearly uniform electrostatic field of strength 2.5 MV cm
1 is still present in the well. One needs to increase n2D to 5 · 1013 cm
2 before recovering the quasi-field-free shape of the quantum well that is needed for lasers. This is achieved much earlier in thinner quantum wells. In wells, the electrons and holes are indeed spatially separated by the polarization field, but the free carrier induced field acts to cancel the polarization field, which is efficient for high sheet densities. This reestablishes the efficient electron–hole recombination. Time-resolved PL (TRPL) is a wonderful way of seeing the effect of polarization. Due to band bending induced by polarization, when the optical excitation pulse is turned off, the free-carrier density goes down and so does the screening. As a result, a

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j 2 Electronic Band Structure and Polarization Effects

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[0 0 0 1]

Conduction band edge energy (eV)

4.0

GaN/In0.2Ga0.8N

3.5 3.0 2.5 2.0

n2 D = 1013cm–2 0.5 1.0 2.0 5.0

1.5 1.0 0.5

0

10

20 30 Depth (nm)

40

50

Figure 2.51 Conduction band profile of a 10 nm GaN/In0.2Ga0.8N quantum well for various levels of free carriers ranging from 5 · 1012 cm
2 to 5 · 1013 cm
2 either present by doping or injection. The effect of polarization is nearly all but wiped out for the largest sheet electron concentration [461].

red shift accompanied with increased carrier lifetime due to lowering of the overlap between the electron and hole wave function occurs, as they are pushed to the opposing end of each well [478]. For details, the reader is referred to Volume 2, Chapter 5. 2.12.5.1 Nonlinear Polarization in Quantum Wells Asinthecaseofbulk binariesand alloys, the polarization issue inheterostructuresneeds a revisit to consider the nonlinearities discussed above in the context of binary and alloy nitride bulk layers. Assuming that the ternary nitride alloys have random microscopic structure, the spontaneous polarization of random ternary nitride alloys, in unit C m
2, has been expressed by Fiorentini et al. [479] to the second order in the composition parameter x, as expressed in Equation 2.95 but repeated here for convenience. sp

PAlx Ga1
x N ¼
0:09x
0:034ð1
xÞ þ 0:019xð1
xÞ; sp

P Inx Ga1
x N ¼
0:042x
0:034ð1
xÞ þ 0:038xð1
xÞ; sp P Alx In1
x N

ð2:117Þ

¼
0:09x
0:042ð1
xÞ þ 0:071xð1
xÞ:

The first two terms in all three equations indicated in Equation 2.117 are the usual linear interpolation between the binary compounds represented by Equation 2.86. However, the third term is the so-called bowing term encompassing the quadratic nonlinearity as in the case of the bandgap bowing parameter discussed in Section 1.5. The coefficient of the third term is the bowing parameter discussed in conjunction with Equations 2.95 and 2.92. Higher order terms are neglected, but their effect was estimated to be smaller than 10% in the worst case being the AlInN alloy [453].

2.12 Polarization Effects

For piezoelectric polarization, it was shown [453] in conjunction with Equation 2.96 that Vegard’s law holds provided that the appreciable nonlinearity of the bulk piezopolarization of the component binaries as a function of strain is accounted for. Doing so has led, in general, to a good agreement with experimental results [480]. For a model AlxGa1
xN alloy, the piezoelectric polarization can be related to the binary end points using the Vegard’s law, but recognizing that the terms for the bulk binaries must contain the nonlinear terms as indicated in Equation 2.96. Such polarizations can be expressed accurately and compactly (in units of C m
2) as pe

PAlN ¼
1:808e þ 5:624e2 pe

PAlN ¼
1:808e
7:888e2 pe PGaN ¼ pe PInN ¼

for e < 0; for e > 0;


0:918e þ 9:541e ; 2

ð2:118Þ


1:373e þ 7:559e2 ;

as a function of the basal strain of the alloy layer in question, with a(x) and asubs or a0 as the lattice constants of the unstrained alloy at composition x and of the relaxed buffer layer or the substrate. eðxÞ ¼ ½asubst
aðxÞ=aðxÞ:

ð2:119Þ

In the case of pseudomorphic growth on GaN buffer layers, basal strain e can be calculated directly from the lattice constants, which are found to follow Vegard’s law as a function of composition (depending on the lattice constants used): aAlx Ga1
x N ðxÞ ¼ aGaN
xðaGaN
aAlN Þ ¼ 0:31986
0:00891x nm; aInx Ga1
x N ðxÞ ¼ aGaN þ xðaInN
aGaN Þ ¼ 0:31986 þ 0:03862x nm; aAlx In1
x N ðxÞ ¼ aInN
xðaInN
aAlN Þ ¼ 0:35848
0:04753x nm:

ð2:120Þ

The combination of Equations 2.118–2.120 provide a convenient way of determining the polarization dependence on basal strain. The coefficients in Equation 2.118 are related (not equal) to piezoelectric constants and come about from the ab initio calculations [453]. The polarization charge values calculated using Equations 2.118–2.120 for heterostructures can be used together with a self-consistent Schr€ odinger–Poisson solver based, for example, on effective mass theory (not as accurate but efficient) or tight binding (more accurate but computation intensive) to determine field and charge distribution in the entire heterostructure as well as the effect of free carriers [448,460]. Typically, for a given structure or set of structures, two classes of observable interest can be simulated and compared with experiment. Experimental confirmation for the nonlinear theory can be garnered from quantum wells by probing the red shift for a given sized well and its dependence on excitation, and measuring the magnitude and gate voltage dependence of the interface sheet carrier concentration at an AlGaN–GaN interface with the aid of the one-dimensional self-consistent simultaneous solution of Schr€ odinger and Poisson equations in the effective mass approximation. Additional confidence can be estimated by repeating these experiments for these structures where, for example,

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j 2 Electronic Band Structure and Polarization Effects

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only the barrier mole fraction is changed [479]. Fortunately, the results are very sensitive to the values of the polarization in the different layers, and therefore the built-in field or the screening charge. The particular bowing parameters in the polarization expression yielding the best agreement with experiment can be assumed to be valid. The point should be made that the C–V measurements are probably the most direct and optical shifts the least direct. The C–V data and the simulations based on the theories described above agree very well [479]. The attempt here is to show that inclusion of the nonlinear effects in calculations lead to a better agreement between the experiments and theory in quantum wells and AlGaN/GaN heterointerfaces. In terms of the quantum confined stark effect (QCSE), the photoluminescence energy of AlxGa1
xN/GaN QWs with well thicknesses of 1,1.5, 2.5, 4, 6, and 8 nm, and with barrier alloy composition of x ¼ 0.08, 0.13, 0.17, and 0.27, respectively, reported in Ref. [481] were used to determine the polarization-induced electric field. The field determined by measuring red shift from the PL transition energies versus the mole fraction in the barrier of quantum wells as well as calculated values from the polarization charge determined using the linear and the nonlinear approach are shown in Figure 2.52. In both types of calculations, the electric field in the GaN QWs was obtained by solving self-consistently the coupled Schr€ odinger–Poisson equations including polarization-induced interface charges. In the first approach the polarization of the AlGaN barriers that are assumed under tensile coherent strain, as the bulk buffer layers were GaN, is determined by a linear interpolation between the elastic 2.5 Linear interpolation

Electric field (MV cm–1)

2.0

Nonlinear approach

1.5

1.0

Ga-polar AlxGa1–xN/GaN QWs

0.5

0 0

0.1 0.2 0.3 0.4 0.5 0.6

AIxGa1–x N molar fraction, x Figure 2.52 Polarization-induced electric fields in high resistivity and Ga-face AlxGa1
xN/GaN MQWs versus the alloy composition of the barrier. The upper line represents the field predicted by linear interpolation of binary compound polarization. The lower line represents the calculated field using the nonlinear polarization concept. Open circles

are deduced, via self-consistent effective mass calculations, from the polarization-induced Stark shift of excitonic recombination reported in Ref. [481] via a self-consistent effective mass calculation. In addition, the experimental data published by Langer et al. [482] and Kim et al. [483] are also shown in filled circles [479].

2.12 Polarization Effects

and piezoelectric constants and the spontaneous polarizations of the binary compounds AlN and GaN. In the second approach, the nonlinearity of the polarization of AlGaN as described by Equations 2.117 and 2.118 was taken into consideration. Clearly, the electric field calculated including the nonlinearity in polarization versus barrier alloy composition does much better at reproducing the experimental data taken from Ref. [481]. In addition, the experimental data published by Langer et al. [482] and Kim et al. [483] are also shown in filled circles. A more convincing arguments for nonlinearity in polarization in the context of polarization charge and resultant red shift in the spectra is made by pressuredependent measurements of the transition energies as presented by Vaschenko et al. [484]. They considered a quantum well system with background unintentional dopants and excitation-induced carriers in the case of which the field deviates from that indicated in Equation 2.105. total ðPtotal Vs W
PB Þ þ r þ LW þ LB LW eB þ LB eW      1 d
LW 1 d
LB
qN D þ ; 2 2 eW eB

E W ¼
LB

ð2:121Þ

where eW,B is the permittivity of the GaN wells and AlxGa1
xN barriers, respectively, (assumed to be independent of pressure in this work), LW,B are the cumulative thicknesses of the wells and the barriers in the MQW structure, r is the total twodimensional photogenerated charge density in the wells, Vs is the surface barrier potential determined as in Ref. [485], ND ¼ 1017 cm
3 is the assumed background doping concentration based on bulk GaN layers grown by MBE, which was used to produce the structure, and d is the distance from the barrier–buffer interface to the well where the field is calculated. The first term in Equation 2.121 is similar to Equation 2.105 but the former having an additional charge r (the total two-dimensional photogenerated charge) in the first term in addition to second and third terms. The second term represents the field due to the surface barrier potential which neglects the effect of dielectric discontinuity between AlGaN and GaN, and the third terms represents the effect of dopant on the field. If Vs, r, and ND were made to go to zero, Equation 2.121 reduces to Equation 2.105. To find PW
PB as a function of pressure, the experimentally measured PL peak energy variation, which was assumed to be representing the n ¼ 1 electron and heavyhole transitions, with well width was fitted to the calculated dependence of the same transition, The nonlinear behavior in this case is revealed by determining the PW
PB (barrier–well polarization difference) as a function of applied hydrostatic pressure. Figure 2.53 shows the fit to the measured PL peak energies in samples in which the AlN mole fraction varied as 0.2, 0.5, and 0.8 at a pressure of 5 GPa. The solid lines are calculations with the polarization-induced field, whereas the dashed ones are without such field. The experimental data are shown in open symbols. The good agreement between the calculations where PW
PB was treated as fitting parameter and the experiment underscores the crucial nature of the built-in electric field in the determination of the well width dependence of the ground-state energy.

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j 2 Electronic Band Structure and Polarization Effects

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4.2 4.0 3.8

Photon energy (eV)

3.6 3.4 3.2 3.0 2.8 2.6

x = 0.2 x = 0.5

2.4

x = 0.8

2.2 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Well width (nm) Figure 2.53 Well width dependence of the PL peak energy at 5 GPa. Open symbols correspond to the experimental points and the solid lines depict the fits to the experimental data obtained with PW
PB as an adjustable parameter. The dashed lines correspond to the e1
hh1 transitions calculated assuming zero field [484].

The polarization difference PW
PB and resulting electric field in the 2.9 nm wells for the x ¼ 0.5 MQW sample, as determined by the procedure described above, is presented in Figure 2.54. For the x ¼ 05, and the others that are not shown here, PW
PB noticeably increases with pressure, resulting in an increase of the built-in field in accordance with the mole fraction in the barrier reaching 0.76 MV cm
1 in the samples with x ¼ 0.5 at 8 GPa. Figure 2.54 also shows the calculated results of the pressure dependence of PW
PB in that the solid lines correspond to PW
PB calculated with the linear polarization [448]. This linear polarization model overestimates the values of PW
PB at atmospheric pressure and underestimates the pressure dependence as compared to experiments. The dashed-dotted lines represent calculations where only the volume-conserving strain dependence of the GaN and AlN piezoelectric coefficients is taken into account [456]. Clearly, this model agrees better with the experimental data than the linear polarization model that is consistent with the conclusions of Perlin et al. [486], where the pressure dependence of PL in GaN/Al0.13Ga0.87N QWs was found to be adequately described by the volumeconserving strain dependence of the piezoelectric coefficients. Lastly, the dashed line

2.12 Polarization Effects

0.058

3.6

0.056

0.052

3.2

0.050 3.0 0.048 2.8

0.046 0.044

Built in electric field, MV/cm

Polarization difference (Pw-PB), (cm–2)

3.4 0.054

2.6

0.042 2.4

0.040

T = 35 K x = 0.5

0.038 0

2

4

6

8

10

2.2

Pressure (GPa) Figure 2.54 Pressure dependence of PW
PB and corresponding electric field in 2.9 nm AlxGa1
xN/GaN MQWs with x ¼ 0.5. The open circles represent the calculated points obtained from the fit to the PL data as shown in Figure 2.53. The solid line corresponds to PW
PB calculated with the linear polarization that overestimates the data obtained from experiments at zero pressures and underestimates the slope of the pressure dependence of polarization. The dashed-dotted

line shows the calculations, where only the volume-conserving strain dependence of the GaN and AlN piezoelectric coefficients is considered, which overestimates the differential polarization but does very well in terms of the pressure dependence of the differential polarization. Lastly, the dashed line shows the result of calculations using the nonlinear polarization behavior with the distinctly best overall fit to the experimentally determined values. The dotted line is a guide to eye [484].

shows the result of calculations where the nonlinear behavior of both the spontaneous and piezoelectric polarizations had been taken into account using the results of Bernardini and Fiorentini [220,221]. Only the change in piezoelectric polarization due to hydrostatic compression of the ideal crystal and that due to the increase in the internal parameter u with pressure is considered here [86]. The spontaneous polarization bowing was included at p ¼ 0 [220,221]. Although this is only an approximation of the theory developed in Refs [220,221], this model also predicts the slope of the pressure dependence of PW
PB significantly larger than that of the linear model. In short, the added value of pressure dependence of differential polarization causes this parameter to be very sensitive to which polarization picture is employed for the transition energies in AlGaN/GaN MQWs to the point that one can clearly state that the nonlinear dependence of the piezoelectric polarization in GaN and AlN unequivocally predicts the experimental data best.

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j 2 Electronic Band Structure and Polarization Effects

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2.12.5.2 InGaN/GaN Quantum Wells Relatively better transport properties of InN as compared to GaN and possible translation of the same to InxGa1
xN are one of the draws for considering InxGa1
xN channels [229] for MODFETs, discussed briefly in Volume 3, Chapter 3. However, experimental results have so far been disappointing. On the contrary, InxGa1
xN proved to be the magical material for its high radiative recombination efficiency for optical emitters. As detailed in Volume 3, Chapters 1 and 2, all the highperformance optical emitters feature InxGa1
xN in one form or another in their active regions. Unlike the FETcases where GaN is used as the active layer and also the buffer layer where the buffer layer is high resistivity, the conductivity of InxGa1
xN is high, which rules out its use as the buffer layer in an InxGa1
xN channel FET. Consequently, even if InxGa1
xN were to be considered for the active layer, it must be grown on GaN buffer layers. As for the barrier layer, it can be made of GaN or some composition of AlxGa1
xN. In any case, the InxGa1
xN layer would be straddled by large bandgap material on both sides with the resultant single quantum well structure. Consequently, polarization in single-well InxGa1
xN quantum wells must be considered as discussed here. In a single quantum well structure, in addition to interface sheet charge that contains information on polarization and can be measured by electrical means, these structures also offer an additional avenue to probe the polarization-induced charge through quantum-confined Stark shift. While the optical properties of InxGa1
xN quantum wells are discussed in great detail in Volume 2, Chapter 5, including the associated QCSE and any Stokes shift, a discussion of single quantum well for the sole purpose of polarization effects is provided here for completeness. Ambacher et al. [84] reviewed the issue in conjunction with nominally undoped, n-type GaN/In0.13Ga0.87N/GaN structures with Ga-polarity, where the width of the quantum well dInx Ga1
x N varied between sp 0.9 and 54 nm. The spontaneous polarization of the InGaN layer, P In0:13 Ga0:87 N ,
2
is
0.031 Cm and points in the ½0 0 0 1 direction and the piezoelectric polarization is calculated to be 0.016 Cm
2, which is antiparallel to the spontaneous polarization because the InxGa1
xN quantum well is under compressive strain. The bound charge at the GaN–InGaN interface near the surface is positive and that at the lower interface is negative for Ga-polarity sample due the compressive strain that InxGa1
xN is under. This implies that the electron accumulation caused by screening would occur at the interface near the surface in n-type samples. If p-type samples were considered, a hole accumulation would occur at the other interface. The total polarization-induced interface sheet density is then given by sp

pz

sp

pz

ðPGaN þ P GaN Þ
ðPInx Ga1
x N þ PInx Ga1
x N Þ:

ð2:122Þ

Recognizing that GaN is relaxed in this case, the samples are grown on GaN buffer layers that are presumed to be relaxed and any residual strain is neglected. This leads pz to PGaN ¼ 0, and substituting the numerical values, one gets for the total polarization ð
0:034 þ 0Þ
ð
0:031 þ 0:016Þ Cm
2 ¼ q1:18  1013 Ccm
2 ;

or 1:18  1013 electrons cm
2 :

2.12 Polarization Effects

InGaN SQW x = 0.13

d = 54 nm

ns = 5 × 1012 cm–2

22

10

26 nm

Electron concentration (cm–3)

2.0 ×

× 10

1012

10

10 16 12 nm 2.6 × 1011

18

16

10

10

20

1018

2

20

10

10

10

22

1014

Electron concentration (cm-3)

10

10 12

4.3 nm 3.6 × 1011

+s –s

14

12

10

1

10

2

2

× 10

10

3

10 4

Depth (nm) Figure 2.55 Electron concentration profiles unintentionally doped, n-type GaN/In0.13Ga0.87N/GaN QWs having well widths of 4.3, 12, 26, and 54 nm, respectively, as deduced from C–V depth profiling [84].

The GaN/In0.13Ga0.87N/GaN the heterostructures with GaN top layer and In0.13Ga0.87N quantum well with thicknesses of 130 and 20 nm, respectively, have been examined by Ambacher et al. [84], and their electron profiles, as determined by C–V measurements, are shown in Figure 2.55. For well widths less than 4 nm, any electron accumulation was not observed, which implies failure to screen the bound charge fully. In fact, as the quantum well thickness was increased from 4.3 to 54 nm, the sheet carrier concentration increases from ns ¼ 3.6 · 1010 to 5 · 1012 cm
2. The increase in 2DEG sheet carrier concentration with well width, which follows a 2.5 power of well width, is much faster than the well width (the volume), which may be attributed to the system not being in equilibrium in terms of screening for thinner wells. Overall, the sheet density is about half of what is expected even in thick quantum wells. The discrepancy between predictions and experiments raises an interesting question if the polarization-induced charge is fully screened by electrons and ionized donors. It is clear that an electric field causes band bending, the total extent of which scales with thickness, which in turn causes a red shift in radiative recombination transition energy. Assuming that the polarization-induced charge at the free surface and the GaN–substrate interface are fully screened, an electric field in the constant

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j 2 Electronic Band Structure and Polarization Effects

288

field approximation forms, which can be expressed as E well ¼

ptotal e0 ðeInGaN r


1Þ

¼

total P total GaN
P InGaN : InGaN e0 ðer
1Þ

ð2:123Þ

The terms here have their usual meanings. The electric field has been predicted to be 2.2 MV cm
1 for In0.13Ga0.87N QWs, which causes band bending to the extent of turning otherwise square quantum well potential distribution to a triangular distribution in the constant field approximation. The resultant red shift in transition energy, Stark shift, can be related to the polarization charge through the field as  g

E InGaN
E energy ¼ qE well W þ

9phqE W pffiffiffi 8 2

2=3 

1 mnInGaN

þ

1 p

mInGaN

1=3 ; ð2:124Þ

where mnInGaN

p and mInGaN

are the electron and hole effective masses in InGaN. For an InN mole fraction of 0.13, these values will not be much different from those for GaN. W represents the InGaN well width. This would provide an additional means for determining the polarization charge with an all-optical method such as photoluminescence. Extreme care must then be exercised to be certain about the InN composition and its bandgap as well as making sure that the optical transitions observed are accurately related to the band edge, meaning their nature must be known. The experiments must also be conducted at low injection levels as to not screen the charge, and high injection levels to screen the charge. If screening is full and low injection levels is truly low, then the difference in energy between the very low and very high injection levels would represent the red shift due to the polarization charge, as was done by Ambacher et al. [84] who also determined the bandgap of InGaN by reflection measurements such as spectroscopic ellipsometry and room-temperature PL using an excitation energy of 3.41 eV, only absorbed by the InGaN well. The QW region was pumped with high-intensity light to generate a high density of electron–hole pairs to fully screen the polarization-induced charge, thus the field. For In0.13Ga0.87N QWs with widths of greater than 26 nm, a bandgap of 2.902 0.012 eV was measured, which is consistent with the value for bulk (2.946 eV). The spectroscopic ellipsometry and PL data show a monotonic increase in energy, reaching a value of 3.21 eV for a well width of 0.9 nm, as the Stark shift diminishes with decrease in well width while the quantum confinement increases, as depicted in Figure 2.56. The PL spectra measured by 3.81 eVexcitation with a pump power density of 103 Wcm
2 yield a PL peak position that is increasingly shifted to lower energies if the QW width is increased from 0.9 to 5.3 nm (Stark effect). Knowledge of this red shift in the PL peak position together with Equation 2.124 allows one to calculate the electric field strength as 0.83 MV cm
1. This field corresponds to a polarization-induced bound charge density of about 5 · 1012 cm
2 and compares well with the data of 0.62 MV cm
1 and charge of 3.7 · 1012 cm
2 for In0.12Ga0.88N/GaN MQWs by Wetzel et al. [487]. As can be deduced from Figure 2.56, the polarization-induced red shift (Stark shift) is not notable for QWs wider than 26 nm where the bandgap of InGaN and the PL

2.12 Polarization Effects

3.4

3.2 g

Energy (eV)

3.0

2.8

E InGaN

660 meV

E 0 e, 0 h

2.6 5 nm

2.4

2.2 10 –1

10 1 10 0 InGaN SQW thickness (nm)

Figure 2.56 The dependence of the effective bandgap (without confinement effects) of In0.13Ga0.87N used in QWs (solid line) and the energy of the radiative recombination in QWs assumed to be between electron ground state (E0e) and hole ground state (E0h) on well width. The effective bandgap is determined by

10 2 spectroscopic ellipsometry and RT PL with 3.41 eV excitation (high and low intensity for PL to account for the Stark shift). Also noted in the figure is that a change of 5 nm in the well thickness leads to a change of 660 meV change in the QW emission energy [84].

peak position agree within 40 meV. It also appears that polarization-induced charged may either be fully screened, negating the constant field approximation throughout the quantum well, or be the notorious In fluctuations discussed in details in Chapter 3 in terms of growth, Chapter 4 in terms of its effects on optical transitions, and Volume 3, Chapters 1 and 2 in terms of its effects on optical emitters. 2.12.6 Effect of Dislocations on Piezoelectric Polarization

To fully consider the true nature of III-Nitride epilayers, the effect of dislocations on piezoelectric polarization must be included. The case does hardly needs to be made owing to the high density of dislocation (edge, screw, and mixed) that alters the strain distribution and thus the piezoelectric polarization. In this vein, Shi et al. [488] calculated the piezoelectric polarization around c-oriented screw and edge dislocations in Wz GaN and found that polarization around screw dislocations (having Burgers vector h0 0 0 1i) has no z-component, which is similar to a magnetic field around a conducting line, so there is no charge induced either at the core or around the screw dislocation. In the case of edge dislocations (having Burgers vector 1/3h1
2 0i), in which the strain field is compressive on one side and tensile on

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j 2 Electronic Band Structure and Polarization Effects

290

(0 0 0 1) c-plane

-

Qs

+

Pe Ps bs be

+

-

Figure 2.57 Schematic diagram illustrating the dislocation geometry, associated polarizations and charge densities [488].

the other, calculations show that piezoelectric polarization has only the z-component and its divergence vanishes at the core and around the dislocation. But at the interface, it results in an effective surface charge for the difference in polarization across the interface, as shown in Figure 2.57. It was estimated that for a c-oriented edge dislocation the charge density could reach 1011 e cm
2 within 0.1 mm of the core. On the experimental side, electron holography was applied to investigate the built-in field caused by polarization [489] and charge distribution at the dislocation [490]. In undoped GaN, the holography results confirm that all dislocations are negatively charged and the line charge densities are calculated as 1 and 0.3 for screw and edge dislocations, respectively. Cai and Ponce [491] argued that screw dislocations always have relatively higher charge density. The electrical activity associated with extended defects is discussed in considerable detail in Section 4.1.6. We should be cognizant of the fact that not only extended defects in general cause local strain and therefore inhomogeneous strain but they also attract and often trap impurities, point defects, and free charge. Moreover, the strain component affects the charge through the piezoelectric component. 2.12.7 Thermal Mismatch Induced Strain

Having made the case that strain induced by both lattice mismatch and also by thermal mismatch plays a profound role in polarization, let us discuss the thermal mismatch case in some detail because the lattice mismatch component got good deal of coverage already. Within the realm of thermal mismatch, the dominant component is that introduced by the nonnative substrates used. The substrates used are many in number and kind, but the dominant ones are sapphire and SiC both of which introduce sizable thermal mismatch. An incomplete list of

2.12 Polarization Effects

substrates used, in addition to the ones cited, include other substrates, GaN, AlN, g-LiAlO2, b-LiGaO2, NdGaO3, Si, GaAs, MgO, ZnO, ScAlMgO4, MgAl2O4, and (La, Sr)(Al,Ta)O3. For a complete discussion refer to Chapter 4 and for a complete compilation refer to Ref. [492]. In a nut shell, the thermal stress is relatively small if GaN layers are grown on AlN, SiC, ScAlMgO4, Si, GaAs, and ZnO. However, it is relatively larger when all other substrates are used. The stress in GaN is compressive for all the substrates substrate except Si and SiC, which due to their small expansion coefficients give rise to tensile strain, which is notorious for layer cracking. The overall stress remains the same for nominally thick GaN layers when a sapphire substrate is used with or without an AlN buffer layer but reduces by an order when a 6H-SiC substrate is used with an AlN buffer layer. In these pages, the treatment reported in Refs [492,493], which followed the model of Olsen and Ettenberg [494], for an arbitrary stack of epitaxial layers on a substrate is followed. The three-layer heterostructure stack layer model used to study the thermal-induced strain is shown in Figure 2.58 with length L, width W, Young’s modulus Ei, layer thickness ti, moments Mi, coefficient of thermal expansion a, forces Fi, strain e, and curvature k. The index i ¼ 1 represents the substrate and i ¼ 2, 3, 4, . . . represents the epitaxial layers.

Figure 2.58 (a) A set of unstrained platelets of thicknesses ti (i ¼ 1, 2, 3, . . .) used to construct the composite layer. (b) A case depicting the films and the substrate to be in contact, representing the epitaxial growth of two different films on a substrate with a larger thermal

expansion coefficient (positive bending) as compared to the films. The dimensions are characterized by length L (into the page), width W, and thickness ti. The terms Ei, Mi, and Fi represent the Young’s modulus, moments (bending forces), and forces, respectively.

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j 2 Electronic Band Structure and Polarization Effects

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For one epitaxial layer with i ¼ 1 representing the substrate and i ¼ 2 representing the epitaxial layer on it, one can surmise that from equilibrium of force, the total force must vanish as F 1 þ F 2 ¼ 0:

ð2:125Þ

Similarly, equilibrium of moments leads to

Wk t1 t2  ¼ 0: ðE 1 t31 þ E 2 t32 Þ þ F 1 þ F 2 t1 þ 2 2 12

ð2:126Þ

The strain at the interface between the film and the substrate in terms of mechanical parameters can be expressed as e¼

F2 F1 ðt1 þ t2 Þk

: E 2 t2 L E 1 t1 L 2

ð2:127Þ

The strain is also determined from the difference between the coefficients of thermal expansion (CTE) of the substrate and the film multiplied by the difference between growth and room temperature, DT. e ¼ DTða1
a2 Þ:

ð2:128Þ

Considering a negligibly small bending stress, the one-dimensional stress in the ith epitaxial layer is then taken as constant and given by si ð1DÞ ¼

Fi ; ti W

ð2:129Þ

where si(1D) is the one-dimensional stress. By assuming a spherical bending for a square sample (meaning L W), the twodimensional stress can be deduced from the one-dimensional stress expression of Equation 2.129 as follows: si ð2DÞ ¼

si ð1DÞ

ð1
vÞ
1

;

ð2:130Þ

where si(2D) and n are the two-dimensional stress and Poisson’s ratio, respectively. For two epitaxial layers the equilibrium of forces Equation 2.125, the equilibrium of moments Equation 2.126, strain in terms of mechanical parameters Equation 2.127, strain in terms of the difference in CTE, and differential temperature Equation 2.128 take the form of F 1 þ F 2 þ F 3 ¼ 0;

ð2:131Þ



Wk t1 t2  t3  þ F 3 t1 þ t2
¼ 0; ðE 1 t31 þ E 2 t32 þ E 3 t33 Þ þ F 1 þ F 2 t1 þ 2 2 2 12 ð2:132Þ e1 ¼

F2 F1 ðt1 þ t2 Þk

¼ DTða1
a2 Þ; E 2 t2 L E 1 t1 L 2

ð2:133Þ

2.12 Polarization Effects

e2 ¼

F3 F2 ðt2 þ t3 Þk

¼ DTða2
a3 Þ; E 3 t3 L E 2 t2 L 2

ð2:134Þ

where e1 and e2 represent the strain between the first epilayer and substrate and the second and first epilayers, respectively. Then si(2D), the two-dimensional stress in the epilayer i, is given as si ð2DÞ ¼

ðF i =ti WÞ

ð1
nÞ
1

:

ð2:135Þ

To calculate the misfit strain and to some extent CTE mismatch, the epitaxial relationship between the epitaxial layer and the substrate must be known, which is known simply as epitaxial relationship. That relationship between GaN and a variety of other substrates are discussed in Section 3.3. In spite of this, a succinct description is provided here as a part of the present topic on stress and strain for continuity and to help support the discussion on polarization. For sapphire substrates, orientations of (0 0 0 1), ð0 1
1 0Þ, ð2
1
1 0Þ, and ð0 1
1 2Þ (basically c-, m-, a-, and r-planes) have been used, see Table 3.6. The largest lattice mismatch,
33%, is between (0 0 0 1) GaN and ð2
1
1 0Þ sapphire along the ½0 1
1 0==½0 1
1 0 in-plane direction. The smallest mismatch, 1.19%, is between ð2
1
1 0Þ GaN and ð0 1
1 2Þ sapphire (r-plane) along the ½0 0 0 1==½0
1 1 1 in-plane direction. This orientation produces a-plane sapphire on r-plane sapphire and is discussed at some length in Sections 3.3.1 and 3.5.11. From the view point of lattice mismatch alone, the r-plane of sapphire is predicted to be most suitable for GaN growth. A thermal strain of
0.18% exists in GaN (0 0 0 1) when grown on sapphire (0 0 0 1), which is compressive. The next common substrate used to grow GaN is various polytypes of SiC. The epitaxial relationship on SiC substrates of hexagonal symmetry is tabulated in Table 3.7. The lattice parameter misfits between GaN and 6H-SiC, 3C-SiC, and 4H-SiC are very close to each other, namely, 3.48%, 3.46%, and 3.50%, respectively, and the corresponding thermal strains are 0.01, 0.09, and 0.03, respectively. The thermal strain in GaN on SiC is tensile and notorious for causing cracks in the epitaxial layers, beginning at about 2 mm thickness. In case AlN substrates would become available, growth on that substrate is also considered. Lattice misfit between GaN and AlN is only 2.41% and the thermal strain is
0.06% and 4.08% for epitaxial relationships (0 0 0 1)//(0 0 0 1) and ð1 1
2 0Þ==ð1 1
2 0Þ, respectively. From the thermal strain point of view, growth on c-plane AlN is preferred. GaN and ZnO share the stacking order and close lattice parameter, the epitaxial relationship for which is provided in Section 3.3.4. GaN can be grown on (0 0 0 1) ZnO, with the lattice parameter mismatch being only
1.97% and the misfit strain
0.21%. Some oxides have also been explored because of the small lattice misfit with GaN they provide. The epitaxial relationship between GaN and LiAlO2 is ð1
1 0 0Þ==ð1 0 0Þ with a lattice misfit of 0.31% along the [0 0 0 1]//[0 1 0] in-plane direction and a misfit strain of
0.41%. Some details can be found in Section 3.3.5. The structure of LiGaO2 is similar to the wurtzite structure, but because Li and Ga atoms have different ionic radii, the crystal has orthorhombic structure. Figure 3.36 shows the

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j 2 Electronic Band Structure and Polarization Effects

294

transformation of the hexagonal unit cell of GaN to an orthorhombic cell, which has lattice parameters close to that of LiGaO2. Similarly, Table 3.10 shows the corresponding lattice parameters of GaN before and after the transformation to orthorhombic cell as well as that of orthorhombic LiGaO2 unit cell. The misfit along [1 0 0] is 2.13%, with a misfit strain of
0.54%. Hexagonal (0 0 0 1) GaN has been prepared on (0 0 0 1) ScAlMgO4. The lattice parameter mismatch between the ½1 0
1 0 GaN and ½1 0
1 0 ScAlMgO4 is
1.49% and the misfit strain is
0.15%. Wurtzite (0 0 0 1) GaN films have been grown on cubic (1 1 1) spinel (MgAl2O4) substrates, as detailed in Section 3.5.10. The lattice parameter mismatch between the ½1 1
2 0 GaN and ½1
1 0 MgAl2O4 is 11.55% and the misfit strain is 0.06%. Zinc blende GaN was grown on (0 0 1) MgO with an epitaxial misfit of 6.99% and a misfit strain of
1.19%. Zinc blende GaN was grown on (0 0 1) MgO with an epitaxial misfit of 6.99% and a misfit strain of
1.19%. Mixed perovskite (La,Sr)(Al,Ta)O3 (LSAT) that is grown by Czochralski method could also be a promising substrate for GaN epitaxial layers and are reported. The lattice parameter 25

AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

Curvature (1 m–1)

20

15

10

5

0.0

–5 0.0

20

40

60

Thickness (μm) Figure 2.59 A compilation of the variation of thermal curvature, a measure of strain, in epitaxial GaN layers grown on different substrates with respect to layer thickness [492]. (Please find a color version of this figure on the color tables.)

80

100

2.12 Polarization Effects

Figure 2.59 (Continued )

mismatch between ½1 1
2 0 GaN//LSAT ½1
1 0 is 16.64% and the corresponding misfit strain is 0.04%. Cubic GaN films can be epitaxially grown onto (0 0 1) Si and hexagonal polytype on (1 1 1) Si with a lattice misfit of
16.93% and misfit strain of 0.19%. The epitaxial relationship between GaN andSi is discussed in Section 3.3.3 and tabulated in Table 3.8. As for the case on GaAs, GaN with (0 0 0 1) orientation can be grown on (1 1 1) GaAs with an in-plane lattice arrangement of ½1 1
2 0==½1
1 0. The corresponding lattice misfit is
20.19% and the misfit strain 0.07%. Perovskite oxide substrates have also been considered as substrates for GaN and related structures in an effort to perhaps find a better match and/or utilize the

j295

j 2 Electronic Band Structure and Polarization Effects

296

nonlinear optical properties of perovskite oxide along with what GaN has to offer. The lattice mismatch of GaN to NdGaO3 has been calculated by assuming a perovskite cell of NdGaO3 with lattice parameters a, b, and c each being equal to 3.86 Å. Next, a new unit cell is constructed with a0 and b0 , where a0 and b0 are the diagonals of the old perovskite cell, as shown in Figure 3.37. The c0 -axis of the new cell is parallel to the caxis of the perovskite cell, but its length is doubled. In a sense this transforms a perovskite unit cell to a tetragonal unit cell. Accordingly, (1 0 0) plane becomes ð1
1 0Þ plane and (0 0 1) becomes (0 0 1). The corresponding lattice misfit is 1.72% and the misfit strain is
0.66%. Owing to the difference in stress between the thin film and substrate, the composite of film and substrate bends, which is the basis for many predigital thermometers and temperature controllers with spiral elements. The strain can be deduced from bending radius, for example, by optical means even during growth at elevated temperature, to monitor the evolution of stress and in attempts to reduce stress by epitaxial heterojunction layer design as performed on SiC substrates, see Section 3.5.3. It is therefore imperative to establish the relationship between the radius of curvature and the difference in the strain between the film and substrate, as discussed in the model of Olsen and Ettenberg [494]. In the calculation of curvature and stress instead of an average CTE over the entire range of temperature, the variation CTE with temperature has been considered for accuracy [492]. The final 5108

Stress on various substrates (Pa)

0

–5108

–1109 AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

–1.5109

–2109

–2.5109 –3109 0100

210–5

410–5

610–5

810–5

Thickness (m) Figure 2.60 A compilation of residual thermal stresses in epitaxial GaN layer on different substrates with respect to layer thickness [492]. (Please find a color version of this figure on the color tables.)

110–4

2.12 Polarization Effects

j297

Stress versus thickness of GaN/potential substrates 5.00E+08

0.00E+00 0

0.00002

0.00004

0.00006

0.00008

0.0001

AlN Si MgO

–5.00E+08

3C-SiC 6H-SiC 4H-SiC

Stress (Pa)

–1.00E+09

ZnO Al2O3 LiGaO2

–1.50E+09

MgAl2O4 GaAs NdGaO3** ScAlMgO**

–2.00E+09

LiAlO2 LSAT –2.50E+09

–3.00E+09

Thickness (m) Figure 2.60 (Continued )

curvature and stress is then the integrated values from growth temperature to room temperature. Figures 2.59 and 2.60 show the curvature and the residual thermal stresses in epitaxial GaN layers grown on various substrates with respect to layer thickness, respectively. The parameters associated with each of the substrate used as well as the growth temperatures are tabulated in Table 2.36. An inspection of Figure 2.60 leads to the conclusion that the thermal stress is relatively small when GaN is grown on AlN, SiC, ScAlMgO4, Si, GaAs, and ZnO and much higher when NdGaO3 and MgO substrates are used. In all other cases (Al2O3, LiAlO2, LiGaO2, MgAl2O4, LSAT) the values are intermediate. Of paramount importance, the thermal stress is tensile in GaN when grown on Si and SiC, whereas in all other cases it is

j 2 Electronic Band Structure and Polarization Effects

298

Table 2.36 Properties and residual thermal stress of 1 mm epitaxial GaN film with other III-N compounds and substrates [492].

Substrate

Melting point ( C)

GaN AlN a-Al2O3 6H-SiC 3C-SiC 4H-SiC c-LiAlO2 b-LiGaO2 Si GaAs NdGaO3 MgO ZnO ScAlMgO4 MgAl2O4

>1700 at 2 kbar 2400  C at 30 bar 2030 2700 sublimes 1825 sublimes 2797 1700 1595 1415 1238 1600 2852 1975 2130

CTE (·10
6  C) (room temperature)

Growth temperature of GaN by grown OMVPE ( C)

aa ¼ 4.997, ac ¼ 4.481, a ¼ 5.45 aa ¼ 5.411 aa ¼ 8.31, ac ¼ 8.5 aa ¼ 4.76, ac ¼ 4.46 4.5 [76] 4.75 aa ¼ 12.1 aa ¼ 10.1, ab ¼ 21.1, ac ¼ 13.6 3.9 6.7 aa ¼ 11.9, ab ¼ 6.6, ac ¼ 5.8 13.9 aa ¼ 6.9, ac ¼ 4.75 aa ¼ 6.2, ac ¼ 12.2 8.7

950–1050 450–1040 950–1100  1000 600 600–1000 600 700 810 450–800 700 650 1000

compressive in nature, which leads to serious cracking issues. This is caused by CTE in SiC and Si being much smaller than that for GaN. As a result, during cooldown GaN is not freely permitted to reduce its lattice constant to the extent GaN naturally would like to, and consequently, the film remains under tensile strain, which causes cracks when the film thickness is about 2 mm or larger. Focusing on the most commonly used substrates for GaN and related epitaxy, Figures 2.61 and 2.62 show variation of the residual thermal stress in GaN versus the 7 × 107

Stress (Pa)

6 × 107 5 × 107 4 × 107 3 × 107 2 × 107 0

20

40

60

80

Thickness (μm) Figure 2.61 Residual thermal stress in GaN for a dual layer GaN/ AlN (0.1 mm) structure on 6H-SiC with respect to GaN thickness [492].

100

References

–0.2

Stress (GPa)

–0.4 –0.6 –0.8 –1.0 –1.2 0

20

60 40 Thickness (μm)

80

100

Figure 2.62 Residual thermal stress in GaN for dual layer GaN/ AlN (0.1 mm) structure on sapphire (Al2O3) with respect to GaN thickness [492].

thickness of GaN with an AlN buffer layer of thickness of 0.1 mm for dual-layer GaN/ AlN on 6H-SiC and GaN/AlN on Al2O3 heterostructures, respectively. There is very little change in thermal stress when GaN is grown on Al2O3 with or without a buffer layer of AlN. On the contrary, the stress decreases by an order by using a buffer layer of AlN while growing GaN on 6H-SiC. In summary, GaN layers grown on substrates such as AlN, SiC, ScAlMgO4, Si, GaAs, and ZnO has a residual thermal stress that is smaller by a factor of two or more as compared to the cases when the GaN layers are grown on other substrates. Moreover and very pivotally, the thermal stress is tensile in nature when grown on Si and SiC substrates whereas in all other cases it is compressive. The tensile residual strain in GaN grown on SiC and Si is notorious for cracking, and multiheterolayer buffer structures are used to deal with the problem, but not without limitations on layers thicknesses for a given substrate temperature used.

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xInxN epitaxial films on GaN templates grown by metalorganic vapor phase epitaxy. Journal of Applied Physics, 94 (4), 2449. 471 Lukitsch, M.J., Danylyuk, Y.V., Naik, V.M., Huang, C., Auner, G.W., Rimai, L. and Naik, R. (2001) Optical and electrical properties of Al1
xInxN films grown by plasma source molecular-beam epitaxy. Applied Physics Letters, 79 (5), 632. 472 Bernardini, F., Fiorentini, V. and Vanderbilt, D. (1997) Polarization-based calculation of the dielectric tensor of polar crystals. Physical Review Letters, 79 (20), 3958–3958. 473 Yu, L.S., Qiao, D.J., Xing, Q.J., Lau, S.S., Boutros, K.S. and Redwing, J.M. (1998) Ni and Ti Schottky barriers on n-AlGaN grown on SiC substrates. Applied Physics Letters, 73 (2), 238. 474 Morkoç, H., Ünl€ u, H. and Ji, G. (1991) Principles and Technology of MODFETS, vol. II, John Wiley & Sons, Ltd, Chichester, UK, p. 317.

References 475 Chu, R.M., Zhou, Y.G., Zheng, Y.D., Han, P., Shen, B. and Gu, S.L. (2001) Influence of doping on the two-dimensional electron gas distribution in AlGaN/GaN heterostructure transistors. Applied Physics Letters, 79 (14), 2270. 476 Ambacher, O., Foutz, B., Smart, J., Shealy, J.R., Weimann, N.G., Chu, K., Murphy, M., Sierakowski, A.J., Schaff, W.J., Eastman, L.F., Dimitrov, R., Mitchell, A. and Stutzmann, M. (2000) Journal of Applied Physics, 87, 334. 477 Ridley, B.K., Ambacher, O. and Eastman, L.F. (2000) Semiconductor Science and Technology, 15, 270. 478 Reale, A., Massari, G., Di Carlo, A., Lugli, P., Vinattieri, A., Alderighi, D., Colocci, M., Semond, F., Grandjean, N. and Massies, J. (2003) Comprehensive description of the dynamical screening of the internal electric fields of AlGaN/GaN quantum wells in time-resolved photoluminescence experiments. Journal of Applied Physics, 93 (1), 400. 479 Fiorentini, V., Bernardini, F. and Ambacher, O. (2002) Evidence for nonlinear macroscopic polarization in III–V nitride alloy heterostructures. Applied Physics Letters, 80 (7), 1204. 480 G€ orgens, L., Ambacher, O., Stutzmann, M., Miskys, C., Scholz, F. and Off, J. (2000) Applied Physics Letters, 76, 577. 481 Grandjean, N., Damilano, B., Dalmasso, S., Leroux, M., La€ ugt, M. and Massies, J. (1999) Journal of Applied Physics, 86, 3714. 482 Langer, R., Simon, J., Ortiz, V., Pelekanos, N.T., Barski, A., Andre, R. and Godlewski, M. (1999) Applied Physics Letters, 74, 3827. 483 Kim, H.S., Lin, J.Y., Jiang, X.H., Chow, W.W., Botchkarev, A. and Morkoç, H. (1998) Applied Physics Letters, 73, 3426. 484 Vaschenko, G., Patel, D., Menoni, C.S., Ng, H.M. and Cho, A.Y. (2002) Nonlinear

485

486

487

488

489

490

491

492

493

494

macroscopic polarization in GaN/ AlxGa1
xN quantum wells. Applied Physics Letters, 80 (22), 4211. Simon, J., Langer, R., Barski, A., Zervos, M. and Pelekanos, N.T. (2001) Physica Status Solidi a: Applied Research, 188, 867. Perlin, P., Suski, T., Lepkowski, S., Teisseyre, H., Grandjean, N. and Massies, J. (2001) Physica Status Solidi a: Applied Research, 188, 839. Wetzel, C., Takeuchi, T., Amano, H. and Akasaki, I. (1999) Japanese Journal of Applied Physics, 38, L163. Shi, C., Asbeck, P.M. and Yu, E.T. (1999) Piezoelectric polarization associated with dislocations in wurtzite GaN. Applied Physics Letters, 74, 573. Cartney, M.R., Ponce, F.A., Cai, J. and Bour, D.P. (2000) Mapping electrostatic potential across an AlGaN/InGaN/AlGaN diode by electron holography. Applied Physics Letters, 76, 3055. Cherns, D., Jiao, C.G., Mokhtari, H., Cai, J. and Ponce, F.A. (2002) Physica Status Solidi b: Basic Research, 234, 924. Cai, J. and Ponce, F.A. (2002) Determination by electron holography of the electronic charge distribution at the threading dislocation in epitaxial GaN. Physica Status Solidi a: Applied Research, 192, 407. Barghout, K. and Chaudhuri, J. (2004) Calculation of residual thermal stress in GaN epitaxial layers grown on technologically important substrates. Journal of Materials Science, 39 (18), 5817–5823. Morkoç, H., Ünl€ u, H. and Ji, G. (1991) Fundamentals and Technology of MODFETs, vols I and II, John Wiley & Sons, Ltd, Chichester, UK. Olsen, G.H. and Ettenberg, M. (1977) Journal of Applied Physics, 48, 2543.

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3 Growth and Growth Methods for Nitride Semiconductors Introduction

Although the synthesis of GaN goes to back more than a half century, there are several pivotal developments, which, in the opinion of the author, are responsible for laying the technological framework and paving the way for the tremendous commercial and scientific interest in nitrides. They are as follows: synthesis of AlN by Tiede et al. [1], synthesis of GaN through the reaction of Ga, and ammonia by Johnson et al. [2] synthesis of InN by Juza and Hahn [3], epitaxial deposition of GaN using the hydride VPE technique by Maruska and Tietjen [4] employment of nucleation buffer layers by Amano et al. [5] and Yoshida et al. [6] and achievement of p-type GaN by Akasaki et al. [7]. A more recent development that paved the way for all the commercial activity is the preparation of high-quality InGaN by Nakamura et al. [8], which followed the synthesis of InGaN by Osamura et al. [9]. Nearly every crystal-growth technique, substrate-type, and orientation has been tried in an effort to grow high-quality group III–V nitride thin films. Increasingly, researchers have successfully taken advantage of the hydride vapor phase epitaxy (HVPE), organometallic vapor phase epitaxy (OMVPE), and molecular beam epitaxy (MBE) techniques, which have yielded greatly improved film quality. All of these epitaxial methods must contend with two main problems: the lack of native GaN substrates and difficulty with nitrogen incorporation and concomitant high ammonia flow rates needed particularly for In-containing nitride semiconductors. A major drawback of GaN is that native substrates are not yet available in large quantities. This is, in part, owing to the low solubility of nitrogen in bulk Ga and the high vapor pressure of nitrogen over GaN at the growth temperature of bulk crystals. The best alternatives now lie in development of sapphire, SiC, or AlN substrates. Interest in AlN substrates has increased recently owing to the closer lattice match over sapphire, matched stacking order, and high thermal conductivity. These factors make AlN one of the best choices for growth of detectors requiring high AlN content AlGaN and backside illumination. Early work on producing bulk AlN looked promising [10]. The problem of nitrogen is endemic in epitaxial deposition techniques as well. Regardless of the growth method employed, the major difficulty in growing group III Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morko
c Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

j 3 Growth and Growth Methods for Nitride Semiconductors

324

nitrides arises from the need to incorporate stoichiometric quantities of nitrogen into the film. This is accomplished in vapor phase processes at high substrate temperatures by decomposing a nitrogen-containing molecule, such as ammonia, on the substrate surface. It should be noted that as the N vapor pressure increases, going from AlN to GaN and then to InN, the ammonia flow rate must be increased reaching tens of liters per minute for InGaN and V/III ratios of 200 000 for InN. It can also be accomplished at lower temperatures in MBE growth by increasing the reactivity of nitrogen through remote plasma excitation or ionization or using ammonia as the reactive nitrogen source. A great deal of effort has been spent in trying to overcome the problems arising from lack of GaN substrates. Some of the best device results have been achieved through use of buffer layers such as thick HVPE, including freestanding, variety, or lateral epitaxial overgrowth variety.

3.1 Substrates for Nitride Epitaxy

As there is lack of a commercial native substrate, a plethora of substrates have been employed in the growth of GaN films [11–14]. Recognizing the thermodynamic bottleneck with regard to native substrates, unconventional methods employing various implementations of compliant substrates and lateral epitaxy have been explored. The most promising results on more conventional substrates so far have been obtained on sapphire and SiC, with SiC making substantial inroads. Also coming on the scene are thick freestanding GaN templates grown by HVPE and then separated from the sapphire substrates. GaN, AlN, and InN have been grown primarily on (0 0 0 1) sapphire but also on the ð2 1  3 1Þ, ð1 1 0  1Þ, ð1  1 0 2Þ, and  ð1 1 2 0Þ surfaces. In addition, III–V nitrides have been grown on Si, NaCl, GaP, InP, SiC, W, ZnO, MgAl2O4, TiO2, and MgO. Other substrates as well have been used for nitride growth, including Hf and LiAlO2, and LiGaO2. Table 3.1 is a compilation of the lattice parameters and thermal characteristics of a number of prospective substrate materials for nitride growth. Lattice-mismatched substrates lead to a substantial density of misfit and threading dislocations (in the range of 108 and 1010 cm
2), though selective epitaxy followed by coalescence, which goes by many names such as epitaxial lateral overgrowth (ELO), lateral epitaxial overgrowth (LEO), and epitaxial lateral overgrowth (ELOG), is a promising method for reducing dislocations down to 106 cm
2. In comparison, the extended defect densities are essentially zero for silicon homoepitaxy and 102–104 cm
2 for gallium arsenide homoepitaxy [13]. Additional crystalline defects besetting the layers include inversion domain boundaries (IDBs) and stacking faults. Such defects are either directly or indirectly responsible for the creation of nonradiative recombination centers, which manifest themselves as energy states in the forbidden energy bandgap reducing the quantum efficiency as well as producing scattering centers. Other adverse effects of structural and point defects are that impurities diffuse more readily along threading dislocations, and carrier transport is

3.104 3.189 4.758

AlN (hexagonal) GaN (hexagonal) Al2O3 (sapphire) (rhombohedral) 4H-SiC (hexagonal) 6H SiC (hexagonal) ZnO (hexagonal) ScAlMgO4 (hexagonal) c-LiAlO2 (tetragonal) LiGaO2 (orthorhombic) MgAl2O4 (cubic/spinel) Si (cubic) GaAs (cubic) b-SiC (cubic) MgO (cubic/rock salt) 6.372

—

Conventional b (Å)

10.053 15.1123 5.2065 25.195 6.2679 5.407

4.966 5.175 12.991

c (Å)

3.1340 6.372

3.0817 3.2496

3.104 3.19 2.747

Matched a (Å)

1.5 0.5 4.9

4.9 0.3–0.4

3.2 2.3 0.3–0.5

Thermal conductivity, j (W cm
1 K
1)

10.5

4.2, 4.68 4.75, 2.9 6.2, 12.2 7.1, 15 a ¼ 6, b ¼ 9, 7 7.45 3.59 6

4.2, 5.3 5.59, 3.17 7.5, 8.5

Da/a, Dc/c (·10
6 K
1)

In part from Landolt-B€ ornstein, vol. 17, Springer, New York, 1982. ZnO data are from W. Harsch of Eagle Picher.

3.073 3.0817 3.2496 3.246 5.1687 5.402 8.083 5.4301 5.6533 4.36 4.216

a (Å)

Crystal

Table 3.1 Lattice parameters and thermal characteristics of a number of the prospective substrate materials for nitride growth and their lattice mismatch with GaN.

P63mc P63mc R3m P41212 Pna21 Fd3m Fd3m F 43m F 43m Fm3m

P63mc P63mc R3c

Space group


3.63%
3.36% þ1.9% þ1.8%
1.7% 0.18%


2.7% 0% 49% (13%)

Mismatch

3.1 Substrates for Nitride Epitaxy

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j 3 Growth and Growth Methods for Nitride Semiconductors

326

either impeded, as in lateral transport, or aided, as in vertical transport. The high density of defects also leads to boundary-limited transport making the important basic parameters such as diffusion constant and mobility nearly impossible to measure. The extended defects in nitrides also lead to inhomogeneities in electric potential because of the high piezoelectric constants of GaN. Electrically active defects induced either directly or indirectly by extended defects cause excess leakage that is detrimental to both optical devices (in the form of dark current in detectors and reduced quantum efficiency in emitters) and electrical devices (in the form of increased gate current leakage and output conductance in field-effect transistors). Details regarding the piezoelectric properties can be found in Chapter 2. Lattice-mismatched substrates are commonly used at present in spite of efforts to produce GaN [15,16] and AlN [10] bulk materials. Despite the lack of matched substrates, remarkable progress in the growth of high-quality epitaxial III nitride films has been achieved by a variety of methods such as hydride vapor phase epitaxy (inorganic VPE or hydride VPE also goes with the acronym HVPE) [11,17], OMVPE [18], and reactive molecular beam epitaxy (RMBE) [19,20]. Moreover, thick freestanding GaN templates for further epitaxy have been prepared by HVPE [21]. By far the most frequently used methods are the VPE methods with heterojunction capability. The most versatile among the VPE methods is the metalorganic chemical vapor deposition (OMVPE). OMVPE is the primary method employed in the investigation and production of optoelectronic devices, such as LEDs and lasers, albeit the quality of MBE films comes close to that grown by OMVPE. Electronic devices with higher quality interfaces are achieved principally with OMVPE and MBE. Inorganic VPE was the first method used to grow epitaxial III-N semiconductors but was nearly abandoned [22]. The technique, however, got revived recently by growing very high-quality and thick GaN buffer layers and templates [23] for the growth of device structures using MBE or OMVPE [11]. Efforts are underway to expand the method to the growth of AlGaN. Below, a discussion of the class of substrates that have been explored is followed by a discussion of the properties of and processing steps for the conventional substrates before growth for each of these substrates. 3.1.1 Conventional Substrates

GaN, as the most studied member of the semiconducting group III nitrides, has been grown on many substrates. Many of the major problems that have hindered the progress in GaN and related semiconductors can be traced back to the lack of a suitable substrate material that is lattice and thermally matched to GaN. Lattice mismatch is responsible for stacking faults and dislocations. Thermal mismatch causes the epilayer to crack on cooling. Specifically, the semiconductors GaN, AlN, and InN have been grown primarily on sapphire, most commonly in the c (0 0 0 1) orientation but also on the a- ð1 1 2 0Þ and R- ð1  1 0 2Þ planes [14]. Growth on a-plane produced c-plane GaN, but growth on R-plane sapphire produces a nonpolar, a-plane

3.1 Substrates for Nitride Epitaxy

GaN. In addition, the group III–V nitrides have been grown on SiC, ZnO, MgAl2O4, Si, GaAs, MgO, NaCl, W, and TiO2. As high-resistivity SiC substrates became available, it became the proffered substrate for transistor work, in part owing to its high thermal conductivity. In addition, freestanding GaN prepared by HVPE is at or nearing production capacity primarily for low threshold lasers that need low-defect material. Some of the suitable substrate materials have become commercially available only recently. Almost all the group III–V nitride semiconductors have been deposited on sapphire despite its poor structural and thermal match to the nitrides. The preference for sapphire substrates can be ascribed to its wide availability, hexagonal symmetry, and ease of handling and pregrowth cleaning. Sapphire is also stable at high temperatures (1000  C) required for epitaxial growth using the various CVD techniques commonly employed for GaN growth. Owing to thermal and lattice mismatches between sapphire and the group III–V nitrides, it is necessary to grow a thick epilayer to obtain good-quality material. 3.1.2 Compliant Substrates

When large mismatch exists between an epilayer and its substrate, the misfit is typically accommodated by the introduction of misfit dislocations at the interface, which are accompanied by threading dislocation segments in the epilayer. To overcome this problem, a compliant substrate is used in high-misfit systems. The role of the compliant substrate is to accommodate the large mismatch either by plastic deformation of the compliant substrate in a manner that avoids the formation of dislocations in the heteroepitaxial film or by homogeneous elastic strain of the threading dislocation, which also avoids formation of the threading dislocations. Both mechanisms are facilitated by a compliant substrate whose stiffness constants are well below those of the epilayer and the supporting bulk substrate. As for the former mechanism, the soft and thin nature of the compliant substrate energetically favors the capture of dislocations resulting from the mismatch by the substrate rather than by the stiffer epilayer, thus paving the way for predilection toward misfit accommodation (MA) by homogeneous elastic strain rather than misfit dislocations. An attractive approach is to insert a pillarlike interfacial layer that is capable of accommodating thermal strain, the effectiveness of which depends on the height of the pillars and the size of the wafer. An effort has been made to find a universal substrate onto which any epitaxial layer can be grown with a very low density of structural defects. Some experimental success has been achieved, but only in specific cases, since 1991 when Lo and colleagues [24,25] introduced the basic idea of a compliant substrate. Different kinds of epitaxial layers have been grown on compliant substrates of GaAs twist bonded to bulk substrates of GaAs. Among them are InGaP, In0.22Ga0.78As, GaSb, and InSb where the respective misfits with GaAs are 1,
1.5,
8, and
15%, respectively [26,27], with positive figures denoting tensile stresses in the layer on GaAs. The use of a thin compliant substrate is not limited to the epitaxial growth of III–V compounds but may also be found in SiGe grown on a thin compliant

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substrate of Si on a viscous SiO2 layer [28]. Other approaches pursue compliancy by twist bonding, that is, deposition of a low melting temperature interlayer. Twist bonding Si to SOI provides a high density of interlayer dislocations that can elastically accommodate misfit between the compliant substrate and the heteroepitaxial film. Otherwise, if the bonding between the compliant layer and substrate is strong, an array of screw dislocations form, transforming to edge/screw dislocations upon deposition of a lattice-mismatched film by elastic deformation. Various explanations have been offered to understand the mechanisms of misfit accommodation [29]. The term compliant is used liberally to describe an approach or a set of approaches to grow lattice mismatch materials where the substrate or some interface layer accommodates the mismatch by expansion/contraction or generating defects within itself as opposed to epitaxial layers. Specifically, the concept behind the latter case is to force the defects caused by mismatch to propagate into the substrate as opposed to the epilayer, requiring generation of misfit dislocations in the thin, weakly bound template layer rather than the growing epitaxial layer. In case of nitride, the initial approach proposed was that Si on an insulator be utilized for GaN growth. Here, a thin Si layer on silicon dioxide, which, in turn, is on Si would be employed. The downside is that the quality of GaN on compliant Si has been poor at best. The modified approach to overcome this barrier is to carbonize the Si to convert it to SiC. If one utilizes the (1 1 1) orientation, one would then get the wurtzitic phase of GaN. To be specific, Yang et al. [30] suggested that before the beginning of the carbonization process, SiO2, Si, and C need to be deposited successively on a Si substrate. By exposing the new composite substrate to a flux of acetylene or carbon particles at 900  C, a thin layer (less than 50 nm) of Si (on SiO2) will be partially or completelyconverted into SiC. GaNis then grownonthis SiC. Again, the problemhere, setting aside the problems associated with the growth on SiC, is that SiC so formed is not contiguous and is extremely defective both in terms of bulk and surface structural properties. In addition, air gaps form beneath the layer surface. Consequently, this technique has not yet lived up to the original proposal and expectations. Efforts still continue to exploit this approach despite the lack of progress so far. Compliance based on expansion and/or contraction is a very neat idea and may be workable for small-sized wafers. However, it is impractical for larger wafers. For example, if the lattice mismatch between the compliant substrate is 4% (epitaxial layer having the larger lattice constant) and the wafer is 50 mm, an expansion in the substrate required for producing defect-free epitaxial layer is 2 mm, which is substantial and unlikely. 3.1.3 van der Waals Substrates

To get around the lattice-mismatch problem, a new growth method called van der Waals epitaxy has been proposed [31], which delivers strain-free films. In this approach, the substrate and epitaxial film are separated by an intermediate epitaxial two-dimensional (2D) buffer material such as MoS2, WS2, or other materials such as II–VI (ZnTe) or III–VI compounds (GaSe, InSe, etc.) having weak van der Waals

3.2 A Primer on Conventional Substrates and their Preparation for Growth

bonding to the substrate and the film. Strain from lattice mismatch between the epitaxial film and the substrate is completely relieved in the region between the layer and the buffer. As in the case of the compliant substrate scheme, this approach has not been very successfully applied to nitrides.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

A substrate is like the foundation of a building. As such, substrate preparation deserves the mostattention.Though thedetailsof the proceduresemployed vary fromone growth method to the next, a chemical preparation before loading into the growth reactor is common. In the OMVPE technique, this is followed by either a simple heat treatment or a combination of heat treatment with gas-phase etching, where temperatures for heat treatment in the vicinity of 1200  C are possible. In the case of vacuum-deposition techniques where it is not always possible to achieve sufficiently high temperatures, dry processing techniques are employed. One of the dry processing techniques is utilizing ECR remote plasma etching with a mixture of hydrogen and helium, as discussed below. The purpose of the He gas is to take advantage of its energetic metastable states with long mean free paths. In addition to a clean surface, the goal is to get as flat a surface as possible, because the nitride stacking order, AaBbAaBb, is different from the stacking order found in most of the substrates under investigation. The exception is ZnO whose stacking order matches that of the nitrides. Because the atomic steps on the (0 0 0 1) surface are of the bilayer type, the surface terraces would have the same surface polarity so that stacking mismatch boundaries (SMB) can be avoided. The surfaces of the substrates used have to be prepared for epitaxial growth, a process that includes degreasing followed by chemical etching when possible. Surfaces of as-received sapphire and SiC substrates contain mechanical polishing damage that must be removed. Chemical etches are not yet available for this purpose. Consequently, a high-temperature treatment under a controlled environment is employed, as will be discussed below. The degreasing procedure, which is the first step for growth, for Si, sapphire, SiC, ZnO, LiGaO2 and LiAlO2, and GaN and AlN, whose specifics will be discussed below, is the same. The substrate is first dipped in a solution of trichloroethane (TCE) kept at 300  C, for 5 min. It is then rinsed for 3 min each in acetone and methanol. This is followed by a 3-min rinse in deionized (DI) water. The above process is repeated three times to complete the degreasing process. The substrates are then etched, which is a substrate dependent procedure. Following degreasing, a variety of substrate-specific methods are employed to get various substrates growth ready as discussed below. 3.2.1 GaAs

GaAs, as a substrate for GaN epitaxy, is justified on the premise of obtaining pure (wurtzite-free) zinc blende GaN on GaAs(1 0 0), attaining thick wurtzite GaN films on

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GaAs(1 1 1) substrates, and dilute GaAsN films for infrared applications. The bulk of the research on GaAs based nitrides is on (1 0 0) surface for cubic GaN. GaAs(1 0 0) is one of the few semiconductor substrates on which metastable zinc blende GaN epitaxial films readily form, and many researchers have investigated the best ways of avoiding any inclusion of the wurtzite polytype into these films. GaAs is much more readily wet etched than any of the other substrates used for nitride epitaxy, which also makes GaN films easier to separate from GaAs than sapphire. Thus, GaAs(1 1 1) substrates are considered a template for creating freestanding thick GaN films for subsequent epitaxy. Because the decomposition rate of GaAs in NH3 or an ultrahigh vacuum (UHV) rapidly increases at temperatures above 700  C, a deposition process including multiple temperatures is required. Unless the substrate temperature is high, the maximum growth rate (GR) attainable is limited. Moreover, even a small amount of GaAs decomposition could be detrimental, as surface roughening or faceting enhances the onset of mixed polarity growth. Because MBE is capable of depositing epitaxial GaN films at a lower temperature as compared to vapor phase methods, it has been more commonly employed in this respect. The maximum allowed temperature could be increased once the GaAs substrate is completely encased with GaN deposited at a low temperature (LT), thereby making OMVPE and HVPE more viable. 3.2.1.1 A Primer on GaAs GaAs has the zinc blende crystal structure with the symmetry group of F  43m. Figure 3.1 displays the perspective view of the GaAs crystal along [1 0 0], [1 1 0], and [1 1 1] directions. Table 3.2 lists the physical, chemical, thermal, mechanical, and optical properties of GaAs important for the GaN epitaxy. For additional details, see Liu and Edgar [13]. GaAs substrates are grown with either liquid encapsulated Czochralski (LEC) or vertical gradient freeze (VGF) methods. GaAs has seen a spectacular improvement over a period of two decades in that crystal defects, impurities, and micro-inhomogeneities have been reduced. In fact, GaAs wafers with diameters greater than

Figure 3.1 The perspective view of the GaAs crystal (a) along [1 0 0] (1 · 1 · 1 unit), (b) [1 1 0] (2 · 2 · 2 units), and (c) [1 1 1] (2 · 2 · 2 units) directions [13]. (Please find a color version of this figure on the color tables.)

3.2 A Primer on Conventional Substrates and their Preparation for Growth Table 3.2 Properties of GaAs at room temperature (partially after Ref. [13]).

Parameter

Value

Lattice constant (Å) Density (g cm
3) Melting point ( C) Heat capacity (J g
1 K
1) Thermal conductivity (W cm
1 K
1) Thermal diffusivity (cm2 s
1) Thermal expansion (linear) (·10
6 K
1) Percent change in lattice (300–1200 K) Bulk modulus (GPa) Young’s modulus (GPa) Poisson’s ratio Refractive index Relative dielectric constant Electrical resistivity (undoped)

5.6536 5.32 1240 0.327 0.45 0.26 6.03 Da/a0 ¼ 0.5876 75.0 85.5 0.31 3.66 near band edge e0 ¼ 13.1 1.0 · 104 O cm, nonstoichiometric defect compensated

150 mm and with various doping types and concentrations are commercially available. Silicon and tellurium are common n-type dopants with resultant electron concentrations in the range of 1016–1018 cm
3. On the contrary, zinc is the standard p-type dopant with resultant hole concentrations in the range of 1018–1019 cm
3. Commercial LEC GaAs has a typical etch pit density (EPD) of less than 104 cm
2 and an electron mobility greater than 4000 cm2 V
1 s
1. But, VGF grown GaAs offers a lower defect density, with EPD less than 103 cm
2. Both (1 0 0) and (1 1 1) types of GaAs with different vicinal degrees are available. 3.2.1.2 Surface Preparation of GaAs for Epitaxy The substrate preparation, following degreasing, includes etching in acid such as H2SO4 : H2O2 : H2O. After rinsing, the surface can be treated in dilute HF for H passivation of the surface. The H passivation layer can be desorbed in the deposition chamber. However, owing to the well-advanced nature of GaAs technology, epi-ready substrates with protective oxides are commercially available. Once in the deposition reactor, the oxide layer can be removed by thermal desorption. Direct nucleation of GaN on GaAs is difficult owing to great chemical and mechanical mismatch with GaN (the large lattice match of about 20% caused by 5.65 Å vs. 4.51 Å for zinc blende GaN), but it is a task mitigated somewhat with the deposition of a GaAs prelayer. Zinc blende GaN on (2 · 4) GaAs can be grown by first establishing an atomically smooth GaAs surface with minimized step density. This is accomplished by depositing a GaAs prelayer, on the order of 100 nm to as high as 1 mm, and choosing appropriate nucleation conditions. Without a GaAs prelayer, the more stable wurtzite polytype of GaN grows preferentially once (1 1 1) facets of the GaN film are generated on the rough substrates. The content of hexagonal GaN phase

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is drastically reduced with an epitaxial GaAs prelayer in MBE growth. The growth temperature for a GaAs prelayer is typically about 600  C. It is imperative that the GaAs surface is nitridated prior to GaN growth for chemical and mechanical transitioning. Nitridation of the GaAs substrate results in a thin nitride surface layer, which provides a cubic template for growth, improving the quality of the GaN layer and suppressing GaAs decomposition at higher substrate temperatures needed for GaN epitaxy. GaAs substrates are nitridated either by exposure to nitrogen plasma (MBE) or by annealing in ammonia (OMVPE and RMBE). At temperatures below 200  C, the nitridation is hindered by kinetic limitations. At temperatures above 600  C, simultaneous etching of the surface may occur along with the nitridation process. Nitridation of GaAs(0 0 1) does not take place homogeneously but proceeds along {1 1 1} facets into the underlying GaAs layer. Complete nitridation can lead to a highly facetted interface between the GaN layer and GaAs substrate, which, in turn, leads to the nucleation of the wurtzite phase and could lead to polycrystalline GaN. Using the highest growth rate possible to quickly bury the interface or maintaining an As flux during growth of the first few monolayers of GaN helps to prevent the deterioration of the GaAs surface. A buffer layer of GaN at low temperature following the initial nitridation generally improves the eventual quality. AlN is not used as buffer because its zinc blende polytype is extremely difficult to nucleate. 3.2.2 Si

Si is the most perfected and least expensive substrate that is available in sizes up to 300 mm. Unlike GaAs, ZnO, and a few others, silicon has good thermal stability under conditions used for GaN epitaxy. However, Wz GaN and AlN grown on Si(1 1 1) are highly defective. The incentives for using Si substrates remain high, however, and good progress in reducing the defect density by using epitaxial lateral overgrowth or pendeo-epitaxy has been reported. 3.2.2.1 A Primer on Si Si has a diamond-lattice structure with the space group of Fd 3m (No. 227) and can be thought of as two interpenetrating fcc sublattices with one sublattice displaced from the other by one quarter of the distance along a body diagonal of the cube (i.e., the pffiffiffi displacement of a 3=4, where a ¼ 5.43102 Å is the lattice constant). Each atom in the lattice is surrounded by four equidistant nearest neighbors that lie at the corners of a tetrahedron. Figure 3.2 illustrates the perspective view along the [0 0 1], [0 1 1], and [1 1 1] directions of a Si cell. Table 3.3 lists physical, chemical, thermal, mechanical, and optical properties of Si at room temperature. Single crystalline ingots are produced by the Czochralski (CZ) method (over 85% of silicon crystals are grown by this method) or the Float Zone (FZ) method, used mostly for purification. These ingots eventually become thin Si wafers through the processes of shaping, slicing, lapping, etching, polishing, and cleaning. Impurities

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Figure 3.2 The perspective view along (a) the [0 0 1], (b) [0 1 1], and (c) [1 1 1] directions of a Si cell. (Please find a color version of this figure on the color tables.)

can be added directly to the melt to create p-type and n-type silicon. The only discernible half-drawback of the CZ method is that oxygen (typical at a level of 1018 cm
3) and carbon (typical at a level of 1016 cm
3) can be incorporated because of the reduction of the quartz crucible and contamination by graphite fixtures. It should be noted that these impurities are not without benefits in that oxygen increases the yield strength or acts as internal getter to tie up metallic contaminants. The FZ method does not use any crucible, and thus the impurity level is markedly reduced, making it easier to grow high-resistivity material. For more details, refer to Ref. [13]. 3.2.2.2 Surface Preparation of Si for Epitaxy For wurtzitic GaN growth, (1 1 1) plane Si is used. The (0 0 1) surface is also used for cubic GaN growth, albeit in only a few cases. As-received Si surface is already

Table 3.3 Properties of Si at room temperature (partially after Ref. [13]).

Parameter

Value

Lattice constant (Å) Density (g cm
3) Melting point ( C) Heat capacity (J g
1 K
1) Thermal conductivity (W cm
1 K
1) Thermal diffusivity (cm2 s
1) Thermal expansion (linear) Percent change in lattice (298–1311 K) Shear modulus (GPa) Bulk modulus (GPa) Young’s modulus (GPa) Poisson’s ratio Refractive index Relative dielectric constant Electrical resistivity (undoped)

5.43102 2.3290 1410 0.70 1.56 0.86 2.616 · 10
6 K
1 Da/a0 ¼ 0.3995 680 97.74 165.6 0.218 3.42 e0 ¼ 11.8 Up to 50 kO cm

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excellent and removal of only a very thin surface layer, using the RCA etch followed by hydrogenation of surface dangling bonds, is sufficient. This is accomplished by immersing Si for 10 min in a 1 : 1 : 5 solution of HCl : H2O2 : H2O kept at 60  C, which grows a porous oxide, followed by a rinse in deionized water. The resulting oxide layer is then removed by dipping the substrate in a 10 : 1 solution of H2O : HF for 20 s. The hydrogenation process takes place through a short exposure of the wafer to an HF solution. 3.2.3 SiC

The cohesive bond strength of SiC is so large that it was once considered an element under the name of carborendum. Owing to its large thermal conductivity and dearth of defect causing in-plane rotation of GaN with respect to SiC lattice, and availability of high-resistivity substrates, SiC is continually gaining recognition as a very viable substrate for epitaxy for both optical and electronic devices. Much of the early drawbacks having to do with pre-epitaxy surface preparation, micropipes, size, and, to an extent, cost issues have been mitigated to the point that some commercial LEDs and almost all of high power field-effect transistors utilize nitride heterostructures on SiC. 3.2.3.1 A Primer on SiC A basic unit of crystalline SiC is a covalently bonded tetrahedron of C atoms with a Si atom at its center or vice versa, that is, either SiC4 or CSi4, as illustrated in Figure 3.3. Variation in the stacking order of SiC along the c-direction leads to more than 250

Figure 3.3 Tetragonal bonding of a carbon atom with its four nearest silicon neighbors. The bond lengths depicted with a and C–Si (the nearest neighbor distance) are approximately 3.08 and 1.89 Å, respectively. The right side is the three-dimensional structure of 2H-SiC structure. (Please find a color version of this figure on the color tables.)

3.2 A Primer on Conventional Substrates and their Preparation for Growth

A B C

Carbon Si Base

B A

C

A

B

C

A

C

A

A

C

B

B

B

A

A

B A

A 3C

A

2H

4H

6H

Figure 3.4 Stacking sequence of cubic and three polytypes of wurtzitic SiC.

polytypes, of which a few prominent ones are shown in Figure 3.4. (A basic discussion of stacking is given in Section 1.1.) By observing the SiC crystal from the side, the stacking sequence can be projected as in Figure 3.5. The distance a between neighboring silicon or carbon atoms is approximately 3.08 Å for all polytypes. The height of the unit cell c varies with the different polytypes, as tabulated in Table 3.4. Consequently, the c/a ratio varies from polytype to polytype but is always close to the ideal for a close packed structure. This ratio is approximately 1.641, 3.271, and 4.908 for the 2H-, 4H-, and 6H-SiC polytypes, respectively, whereas theffiffiffiffiffiffiffi equivapffiffiffiffiffiffiffiffi p ffi lent ideal ratios for these polytypes are 1.633, 3.266, and 4.899 ( 8=3 , 2 8=3 , and pffiffiffiffiffiffiffiffi 3 8=3), respectively [13]. Each polytype has a unique set of electronic and optical properties. The bandgaps at liquid helium temperature of the different polytypes range between 2.39 eV for 3CSiC and 3.33 eV for the 2H-SiC polytype. The two most important polytypes as substrates for GaN epitaxy, 6H-SiC and 4H-SiC, have bandgaps at liquid helium temperature of 3.02 and 3.27 eV, respectively. The hexagonal polytypes of SiC, such as 4H- and 6H-SiC, belong to the same space group, P63mc (No. 186), as wurtzite GaN. The most studied substrates for GaN epitaxy are the 3C-SiC/Si(1 0 0) and 6H-SiC, as these polytypes have been the most readily prepared or commercially available for the longest time. With 4H-SiC now commercially available, its use will become more common. Table 3.4 shows the physical, chemical, thermal, mechanical, and optical properties of SiC at room temperature. The thermal expansion coefficient of SiC in c- and a-planes as a function of temperature is shown in Figure 3.6. Bulk SiC crystals are produced by sublimation in the modified Lely process, developed by Tairov and Tsvetkov [33], which employs a SiC seed crystal for the control of polytype and orientation. Growth is achieved by the vapor transport of Si, Si2C, and SiC2 driven by a temperature difference in an argon atmosphere in a graphite, tantalum, or tantalum carbide crucible at 20–500 Torr and at about 2200  C. The 4H-SiC and 6H-SiC(0 0 0 1) varieties both on- and off-axis (typically 3.5 for 6HSiC and 8 for 4H-SiC), silicon and carbon face, are available in sizes up to 100 mm in

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Figure 3.5 Views of the ½1 1 2 0 planes for the 3C-, 2H-, 4H-, and 6H-SiC polytypes.

diameter. Screw dislocations occur in high densities and depending on the magnitude of their Burgers vector, the core of a screw dislocation can be hollow (nanopipes or micropipes) or closed and run through the entire wafer. Hollow core screw dislocations take place when the Burgers vector (b) is two or more times the c-lattice constant (c) for 6H-SiC or three times the lattice constant for 4H-SiC. Because the best micropipe density has been reduced to about 1 cm
2, research has shifted on to closed-core screw dislocations, which occur in densities of approximately 103–104 cm
2. Both 4H-SiC and 6H-SiC wafers are available in low resistivity nand p-type forms with concentrations in the range of 1015–1019 cm
3. The resistivities for n- and p-type material for the aforementioned doping range lie in the range 0.01–0.10 and 1–10 O cm, respectively. Interest in semi-insulating SiC is driven by

3.2 A Primer on Conventional Substrates and their Preparation for Growth Table 3.4 Properties of SiC at room temperature (after Ref. [13]).

Parameter

Polytype

Value

Lattice constant (Å)

3C 2H 4H 6H 3C 2H 6H 3C 6H 3C 4H 6H 3C 6H 6H 3C 3C Ceramic 3C 2H 4H 6H 3C 6H

a ¼ 4.3596 a ¼ 3.0753, c ¼ 5.0480 a ¼ 3.0730, c ¼ 10.053 a ¼ 3.0806, c ¼ 15.1173 3.166 3.214 3.211 2793 0.71 3.2 3.7 3.8 3.9 4.46 for a-axis, 4.16 for c-axis Da/a0 ¼ 0.4781, Dc/c0 ¼ 0.4976 Da/a0 ¼ 0.5140 440 0.183–0.192 2.6916 at l ¼ 498 nm 2.6686 at l ¼ 500 nm 2.6980 at l ¼ 498 nm 2.6894 at l ¼ 498 nm e(0) ¼ 9.75, e(1) ¼ 6.52 e(0) ¼ 9.66, e(1) ¼ 6.52 ? c-axis e(0) ¼ 10.3, e(1) ¼ 6.70 || c-axis 102–103, higher in V doped

Density (g cm
3)

Melting point ( C) Heat capacity (J g
1 K
1) Thermal conductivity (W cm
1 K
1) Linear thermal expansion coefficient (·10
6 K
1) Percent change in lattice (300–1400 K) Young’s modulus (GPa) Poisson’s ratio Refractive index (ordinary ray)

Dielectric constant

Electrical resistivity (undoped)

4H

Thermal expansion coefficient (%)

1.8

α−SiC

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2 0

200 400

600

800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800

Temperature (K) Figure 3.6 Thermal expansion coefficient of SiC in c and a planes as a function of temperature [32].

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FETs. The resistivities close to 1010 O cm in both V-doped and intrinsically compensated SiC are possible. 3.2.3.2 Surface Preparation of SiC for Epitaxy The surface preparation of SiC prior to deposition takes on a different meaning in MBE as opposed to OMVPE. Lacking high temperature capability and H, MBE growth relies on ex situ cleaning procedures. In contrast, SiC surfaces can be epi readied in situ during growth by vapor phase. However, one of the best, if not the best, approach involves gas phase preparation of SiC surface in which the surface is exposed to in H and/or HCl gases at high temperatures, the details of which are discussed below. Unless SiC substrates are H polished a priori or in the growth reactor, as can be done in HVPE or OMVPE methods, it is recommended that approximately 3 mm from the surface be removed in a hot KOH solution (300–350  C). If the substrate quality is not high, the etch rate in defective regions is high and smooth surfaces do not follow. Assuming that the previous step is successful, it is followed by a DI rinse for 3 min and the wafer is blow dried by N2. The SiC substrate is then subjected to a series of oxidation and passivation procedure. The substrate is immersed for 5 min in a 5 : 3 : 3 solution of HCl : H2O2 : H2O at 60  C, followed by a 30 s DI rinse. The resulting oxide layer is then removed by dipping the substrate, for 20 s, in a 10 : 1 solution of H2O : HF. This procedure is repeated several times (three to five times) after which the substrate should not be exposed to the atmosphere for longer than 30 min, otherwise another oxidation–passivation procedure would be required. Skipping the chemical etching step leaves the surface with residual damage from mechanical polishing. As mentioned above, techniques have been developed to remove the surface damage by plasma or vapor etching. One such technique is the mechanical chemical polish, which has made substantial progress [34] (commercial service is available from Novasic), and the other is etching in H and/or Cl environments at very high temperatures. The surface morphology of the as-received SiC substrate that underwent a standard mechanical chemical polish (MCP) contains much surface and subsurface damage as characterized by many grit scratches shown in Figure 3.7a. Special MCP procedures have been developed to obtain smoother surfaces (see Ref. [34]), two example of which obtained by Eagle Picher and Novasic are shown in Figure 3.7b and c. Schottky barrier diodes on as-received and Novasic MCP-treated substrates indicated as much as four orders of magnitude reduction in the reverse bias current, albeit with a considerable nonuniformity. Another method that leads to atomically smooth SiC surfaces and much improved GaN overlayers, in terms of reduced extended defect concentration as shown by plan Figure 3.7 (a) AFM image of as-received SiC surface following a standard mechanical chemical polish. Image size is 10 mm · 10 mm and vertical scale is 50 nm. Note the presence of scratches; (b) AFM image of SiC surface after a mechanical chemical polish performed at Eagle

" Picher. Note that scratches are no longer present. Image 10 mm · 10 mm, vertical 5 nm; (c) a 2 mm · 2 mm AFM image of a Cree 6H-SiC wafer MCP polished by Novasic showing a root mean square roughness of 0.134 nm.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

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view transmission electron microscopy (TEM) images, is the high temperature H annealing [35]. Those investigating various issues dealing with SiC have recently developed and exploited the in situ hydrogen etching [36]. H2, H2 þ HCl, or H2 þ C3H8 etching at temperatures between 1300 and 1550  C removes the scratches caused by mechanical polishing [37]. This is more effective than wet HF etching of SiO2 after oxidation. Other wet etching techniques in bases such as molten salts (Na2O2, NaOH, KOH, etc. at temperatures approaching or at 500  C) reveal the defect features of SiC surface and thus are not suitable for surface preparation for epitaxy. Even though nitridation has been shown to improve the smoothness of SiC substrate owing to a combination of nitrogen chemisorption and etching at 1050  C in NH3 flow for 30 min, the utility of long nitridation processes is questioned because of SixNy formation. An atomic force microscopy (AFM) image of a SiC substrate polished by a hightemperature H treatment (1500  C), similar to that reported in Ref. [38], and used for GaN growth in Ref. [35], is shown in Figure 3.8, which clearly shows well-ordered and unbroken atomic terraces indicative of superb surface quality. Similar results can be obtained by a H treatment in a typical SiC growth reactor at about 1500  C. A light follow-up etch in molten KOH solution ensures atomically smooth terraced surfaces if the H polishing steps are not ideal. As previously mentioned, vacuum deposition equipment, such as the one used in MBE, is not compatible with the high-temperature H or HCl treatment, but remote plasma etching techniques can be employed to at least remove the damaged surface layer if H-etched samples are not available. To circumvent the need for high temperatures and exotic treatments incompatible with conventional MBE setups, a preparation procedure adapted from conventional Si technology, and augmented by H plasma cleaning, has been shown to work for SiC. In the first step of this procedure reported by Lin et al. [39], the surface is hydrogen passivated using an HF dip before

Figure 3.8 AFMimageofSiCsurfaceaftera5 min1600  C hydrogen polishing step. Note that scratches are no longer present. The step height seen in a full c-direction lattice parameter for 6H-SiC.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

being introduced into vacuum. In the second step, the substrate is treated with hydrogen plasma, which reduces the CO level (oxygen–carbon bonding) to a value below the X-ray photoemission detection limit. Detailed investigation of SiC surface after H anneal have been undertaken [40] using such techniques as low-energy electron diffraction (LEED) experiments and Auger electron spectroscopy (AES) in addition to reflection high-energy electron diffraction (RHEED). The emphasis is to determine the surface chemical and structural properties taking it beyond what may be needed simply for epitaxy but looking at it with the precision needed for MOS-like structures with AlN/SiC composite. When (0 0 0 1) SiC samples subjected to a H polish at 1500  C for 5 min in flow 3000 scum are introduced into the UHV system, they exhibited a 2 ffiffi pffiffiaffi Hp ffi ( 3  3) R30 LEED pattern with bright and sharp superstructure spots. For comparison on the C-polarity surfaces, ð0 0 0 1Þ, no background was observed, which indicates a high degree of order. On the (0 0 0 1) surfaces, however, always a faint background was visible. The ratio between the average intensities of fractional and integer order beams was above 0.5 for both surfaces, indicating strong surface reconstruction in both cases. The typical Auger spectra displayed a strong OKLL peak in addition to the typical SiLVV and CKLL peaks [40]. The SiLVV signal on the ð0 0 0  1Þ surface showed both bulk-related peak at 90 eV and a feature at around 65 eV attributed to oxygen-bonded silicon. The SiLVV signal of the (0 0 0 1) surface is more complex as expected because of differently coordinated Si for both surfaces. The LEED pattern on the ð0 0 0 1Þ surface changed to (3  3) structure following a 30-min  annealing at 1050 pffiffiffi C.pIn ffiffiffi addition, both the oxygen and the Si–O AES signals vanished. The ( 3  3) R30 structure remained for the (0 0 0 1) surface after a 30-min annealing at 1000  C. However, OKLL peak disappeared and SiLVV signal returned the bulklike shape. The simultaneous structural evolution and oxygen removal from both surfaces are indicative oxide on the surface, which was pffiffiffi of pffiffisilicon ffi confirmed by LEED analysis of the ( 3  3) R30 phase of the presence of Si2O3 overlayer, referred to as the honeycomb silicate adlayer, on the SiC surface. The honeycomb silicate adlayer is formed by two Si atoms per unit cell and each oxygen atom connects two of the Si atoms completing a ring-type structure, as shown in Figure 3.9 pffiffiffi p ffiffiffi depicting the top view or the projection on the c-plane. The silicate-related ( 3  3) R30 structure on the (0 0 0 1) surface, shown as a side view, in Figure 3.9 is identically arranged with Si atoms also oriented toward the substrate. Interestingly, the silicate layer and substrate are linearly bridged by oxygen, Si–O–Si, not connected via Si–Si bonds. This simply implies that the surface as it stands is not optimum for nitride growth. It is possible that in OMVPE environment, the silicate adlayer may be removed, a statement that cannot be unequivocally made for MBE growth. In fact, the growth of AlN on such SiC surface leads to three-dimensional (3D) growth and, as discussed earlier, exposureto Gaathightemperature removes thesilicate layer [41–44]. Treating the surface of ex situ HCl-treated (1300  C) SiC with in situ Ga spray, Onojima et al. [41] obtained SiC surfaces free of the silicate adlayer, as shown schematically in Figure 3.10, and were able to obtain a 2D AlN growth. Owing pffiffiffi topGa ffiffiffi deposition on the SiC surface and subsequent flash-off, an oxygen-free ( 3  3) R30 surface structure was achieved and initial 2D growth with an evident RHEED

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[1 1 0 0]

(a)

[1 1 2 0] [0 0 0 1]

Top Si Second Si (b) Silicate adlayer

C Top O

SiC surface

Second O

Figure 3.9 (a) Top view of the oxide structure on SiC ð0 0 0 1Þ. The Si2O3 silicate adlayer consisting of a honeycomb structure with Si
O
Si bonds. At the center of the hexagons, one carbon atom of the topmost substrate bilayer is visible [the dark shaded area indicates the (1 1; 1) unit cell and light shaded the

pffiffiffi pffiffiffi ð 3  3ÞR30 -unit cell]; (b) side view of the oxide structure on the SiC (0 0 0 1) in ð0 1 1 0Þ SiC projection. Linear Si
O
Si bonds connect the silicate layer and the underlying SiC substrate. Courtesy of N. Onojima (patterned after Ref. [40]). (Please find a color version of this figure on the color tables.)

intensity oscillation was demonstrated. The initial growth mode of AlN closely correlated with the crystalline quality of AlN layer. Figure 3.11 shows the RHEED images of HCl-treated surface with a silicate adlayer, HF treated surface with has residual O, and finally in situ Ga spray treated SiC, of O on the surface for the ½1 1  2 0 and ½1  1 0 0 azimuths. Note the pffiffiffi which pffiffiffi is void  ( 3  3) R30 RHEED surface reconstruction for the HCl-treated surface, 1 · 1 pffiffiffi pffiffiffi RHEED reconstruction, and again ( 3  3) R30 RHEED surface reconstruction for the in situ Ga spray treated surface. 3.2.4 Sapphire

Owing to its relatively low cost, availability in large area, and continual improvement in its quality, both in terms of bulk and surface properties, sapphire has become the [1 1 0 0] [1 1 2 0] [0 0 0 1]

(a) Si

(b) 1/3 ML Si ad atom

ad-Si C

SiC surface Figure 3.10 (a) Top view (projection on the Si-plane of the basal plane of SiC) and (b) side view of SiC after an in situ Ga exposure indicating of the lack of silicate adlayer. Courtesy of N. Onojima. (Please find a color version of this figure on the color tables.)

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Figure 3.11 RHEED images along the ½1120 and ½1100 azimuths for HCl, HF,pand in ffiffisitu pffiffiSi-polarity ffi pffiffiffi SiC surface ffiffiffi p ffi Ga spray treated indicating ð 3  3ÞR30 , 1 · 1, and ð 3  3ÞR30 , respectively. Courtesy of N. Onojima.

dominant substrate material for epitaxy. Although there are other reasons, sapphire is transparent for most of the bandgaps of nitride alloys; thus, it affords certain benefits in detectors, for example, for back illumination and in LEDs, for lack of absorption. 3.2.4.1 A Primer on Sapphire Sapphire has the space group of R3c (No. 167), as provided in the International Tables for Crystallography, and is primarily of ionic bond nature. It can be represented by both rhombohedral unit cells, with volume 84.929 Å3, and hexagonal unit cell, with volume 254.792 Å3, which is displayed in Figure 3.12 [13]. In the rhombohedral unit cell there are 10 ions in total, 4 Al3 þ ions and 6 O2
ions. The hexagonal unit cell has 30 ions in all, 12 Al3 þ ions and 18 O2
ions. Oxygen is located at (x, y, z) ¼ (0.306, 0, 0.25). If this position is approximated to (x, y, z)  (1/3, 0, 1/4), the anion framework forms an hcp lattice with a ¼ 0.476 nm and c ¼ 1.299 nm. The unit cell described by Miller–Bravais indices consists of six close-packed (0 0 0 1) planes of O2
ions sandwiching 12 planes of Al3þ ions that occupy two thirds of the available octahedral voids created by the O2
ions. An Al3þ ion is located at (x, y, z) ¼ (0, 0, 0.352) instead of (0, 0, 1/3), thus the cations are shifted by 0.025 nm along the c-axis from the ideal octahedral sites. The oxygen ion is larger than the aluminum ion by a factor of about 3 in terms of its radius; therefore, the steps on the substrate are limited to those in the oxygen sublattice, leading to step heights in multiples of c/6 (d(0006)  0.216 nm). The (0 0 0 1) Al2O3 surfaces are oxygen terminated and present steps along f1 1 2 0g and f1 1 0 0g planes [45]. Two crystallographically equivalent surfaces are related by a symmetry operation of the space group. Along the [0 0 0 1] direction, A–A or B–B surfaces are separated by c/3, 2c/3, and c steps. Steps separating two “A” surfaces are noted as A–A, and c/3 steps of height c/6, c/2, or 5c/6

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separate the two surfaces related by a glide symmetry operator. Such steps are dubbed demi-steps and are noted as A–B, c/6 [46,47]. The unreconstructed basal c-plane perspective views for both unit cells are given in Figure 3.13 [13], where the cell boxes are polyhedra. A schematic representation of sapphire unit cell indicating the six O layers in the unit cell is shown in Figure 3.14. The oxygen ions form a pseudohexagonal lattice. The small Al ions occupy the octahedral sites. The labeling of planes and directions in the context of sapphire substrates are shown in Figure 1.5. Properties of sapphire are provided in Table 3.5 [13]. All common surfaces employed for GaN epitaxy including the (0 0 0 1) and ð1  1 0 0Þ are nonpolar. Thus, the polarity control on sapphire depends on the particulars of growth conditions employed with the ominous inversion domain formation always a possibility. Because

Figure 3.13 Perspective views in (2 · 2 · 1) unit cells: (a) along the [0 0 0 1] direction in a rhombohedral unit cell; (b) along the

3.2 A Primer on Conventional Substrates and their Preparation for Growth 0.287 nm

0.252 nm

b B

O 2–

Al3+

a A

0.052 nm

b

0.0797 nm

C

0.1358 nm

a B b

0.1441 nm

0.1661 nm [0 0 0 1]

A a

[1 0 1 0]

Figure 3.14 A schematic diagram of the Al2O3 sapphire unit cell, there are six oxygen layers in the unit cell, the distances between the various atomic layers change as shown in the figure. The oxygen ions form a pseudohexagonal lattice. The small Al ions occupy the octahedral sites. Courtesy of P. Ruterana and Ref. [47]. Table 3.5 Properties of sapphire (in part after Ref. [13] and references therein).

Parameter

Value

Condition

Lattice constant (Å) Melting point ( C) Density (g cm
3) Thermal expansion coefficient (K
1)

a ¼ 4.765, c ¼ 10.2982 2030 3.98 6.66 · 10
6 || c-axis 9.03 · 10
6 || c-axis 5.0 · 10
6 ? c-axis a/a0 ¼ 0.83, c/c0 ¼ 0.892

20  C

Percent change in lattice constants with DT Thermal conductivity (W cm
1 K
1) Heat capacity (J K
1 mol
1) Young’s modulus (GPa)

Tensile strength (MPa) Poisson’s ratio Hardness: Knoop nanoindentation (GPa) Energy band gap (eV) Resistivity (O cm)

20  C 20–50  C 20–1000  C 20–1000  C 293–1300 K

0.23 || c-axis 0.25 || a-axis 77.9 452–460 in [0 0 0 1] direction, 352–484 in ½1 1 2 0 direction 190 0.25–0.30 23.9  2.0

296 K 299 K 298 K

8.1–8.6 >1011

Experimental value 300 K

300 K 300 K 300 K

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2.4 2.2 2.0 1.8

Polycrystalline

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2

0

c-axis

Al2O 3

a-axis Si

200

400

600

800

100

1200

1400 1600

1800 2000

Temperature (K) Figure 3.15 Thermal expansion coefficient of sapphire on the cplane (along the a-axis) and along the c-axis, and that of Si as a function of temperature [48].

of heteroepitaxy of GaN on sapphire, it is useful to display the thermal expansion coefficient of sapphire as done in Figure 3.15. 3.2.4.2 Surface Preparation of Sapphire for Epitaxy As-received sapphire substrates contain scratches caused by mechanical polishing with root mean square (RMS) roughness values between 0.8 and 2.1 nm over l mm2 areas. Wet chemical etches such as phosphoric acid (H3PO4), sulfuric–phosphoric acid combination (H2SO4–H3PO4), fluorinated and chlorofluorinated hydrocarbons, tetrafluorosulfur (SF4), and sulfur hexafluoride (SF6) have been employed. None of these techniques, however, produces a surface free of damage and scratches. For MBE growth, which does not allow in situ cleaning of the surface in H at high temperatures, a 3 : 1 solution of H2SO4 : H3PO4 is used as the etchant. The substrate is dipped in this solution and kept at 300  C for 20 min. This is followed by a rinse in DI water for 3 min. Although the hot etching removes some material, the resultant surface still bears the scratches caused by mechanical polishing. However, the surface becomes flatter after etching, with the RMS roughness being reduced from 0.323 nm in image to 0.211 nm in image in one case. A high-temperature annealing technique after wet chemical etching of mechanically polished sapphire substrates has been shown to result in atomically smooth surfaces [49]. Figure 3.16 shows the AFM images of two c-plane sapphire surfaces, (a) before and (b) after the chemical etching. In an OMVPE or HVEP environment, the typical process is to simply heat the sapphire under flowing hydrogen at temperatures between 1000 and 1100  C. This process etches sapphire slightly leading to the formation of hexagonal pits if there are residual amounts of gallium left in the reactor from prior runs. The crystal quality of subsequently deposited GaN films was insensitive to the presence of these pits [13]. To eliminate surface damage altogether, a high-temperature annealing step has been employed, which gives rise to atomically smooth surfaces. A very high

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Figure 3.16 (a) AFM image of an as-received sapphire substrate. Note the scratches caused by mechanical chemical polishing; (b) AFM image of a sapphire substrate after a 180  C etch in sulfuric/ phosphoric acid. Some improvements are apparent, but the scratches remain and are accentuated to some extent. Image size 2 mm · 2 mm.

temperature annealing investigation of sapphire substrates was recently undertaken. Annealing experiments in air at 1000, 1100, 1200, 1300, and 1380  C (the ceiling of the furnace employed) for 30- and 60-min periods were conducted to determine the best conditions with the aid of AFM images of the finished surface. This was followed by observation of RHEED patterns once in an MBE system. A small, but progressive, improvement was observed in the reduction of scratches up to 1300  C. However, annealing at 1380  C for 1 h led to scratch-free and smooth surfaces to the point where the only noticeable feature in AFM images were the atomic steps about 0.15 nm in height. AFM images indicated that annealing at

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Figure 3.17 An AFM image of sapphire following a 1380  C–1 h annealing in atmosphere. Atomically flat surface is clearly visible. Atomic step heights are about 0.15 nm, which represent the only roughness in the image. The diagonal lines, from left to right, are the artifacts of AFM.

1380  C for 1 h leads to atomically smooth surfaces as shown in Figure 3.17. An atomically smooth surface is maintained after nitridation as well. RHEED images typically show extended and bright rods associated with sapphire at temperatures as low as 600  C during the ramp-up as shown in Figure 3.18.

Figure 3.18 A RHEED image at about 800  C of an annealed sapphire at 1380  C for 1 h (½1 1 2 0 azimuth). Clear streaky RHEED pattern observed at temperatures as low as 600  C indicates that the high temperature annealing step produces clean epiready surfaces. Without the annealing procedure, the RHEED images are not as clear and elongated and not reproducible.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Sapphire is nitridated by exposing it to nitrogen plasmas or thermally cracked ammonia (a practice that has been abandoned) in MBE reactors or by to ammonia/ hydrogen gas mixtures in OMVPE reactors. Sapphire substrates that have not undergone a heat treatment in O, as described above, exhibit the polishing damage in the form of random scratches when they undergo MBE-like pregrowth in situ heat treatment followed by an exposure to ammonia [50]. Considering the stacking-order mismatch between sapphire and nitrides, these features are likely to have deleterious effects on growth. A low density (108 cm
2) of surface outgrowths was observed after 30 min of nitridation. The presence of surface damage does not appear to have influenced the formation of protrusions. There is no clear correlation between the positions at which the protrusions have formed and the local surface topography. Uchida et al. [51] observed similar protrusions after 5 min of nitridation at 1050  C in an OMVPE system, but at a much higher density than that observed in the MBE process. It is likely that a combination of a higher substrate temperature and the background ammonia pressure promotes a more rapid nitridation reaction leading to a higher density of protrusions in OMVPE-grown samples. Although what is reported for these particular samples may hold, it should again be noted that atomically smooth surfaces following annealing in air, as described above, do not show discernible change after nitridation. Noting that AlOxN1
x would be unstable at the nitridation temperatures employed, the nitridation of sapphire should result in the formation of AlN. In fact, in the MBE process, this can be observed with RHEED in that the pattern associated with the ½1 1 2 0 azimuth of sapphire gives way to the pattern associated with the ½1 1 2 0 azimuth of AlN but with a 30 rotation to minimize misfit strain. Nitridation has direct consequences in the quality of the low-temperature buffer layer and final layer(s). The ultimate test whether nitridation was done properly requires going through the buffer and final layer growth. The particulars of the final layer are then used to draw conclusions about the nitridation process as was done by Wickenden et al. [52], the details of which are discussed in Section 3.5.5.1. In the study of Uchida et al. [51], nitridation was reported to occur very rapidly for times less than 3 min and then slowed considerably. For short nitridation times, of 650  C)

(0 0 0 1) (0 0 0 1) (0 0 0 1) (0 0 0 1) (0 0 0 1)

½2 1 1 0 jj ½0 1 1 ½1 1 2 0 jj ½1 1 0 ½2 1 1 0jj½0 2 2 ½1 0 1 0jj½0 2 2 ½0 1 1 0jj½1 1 2

(1 1 1) if Wz AlN is produced (Ts ¼ 400–600  C)

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Table 3.9 Epitaxial relationship of GaN grown on Si with the

atomic distance mismatch (ADM) and the extended atomic distance mismatch (EADM).

Epilayer/substrate ð0 0  1Þh-GaN/(1 0 0)Si

(1 0 0)c-GaN/(1 0 0)Si

(0 0 1)h-GaN/(1 1 1)Si

Atomic distance (Å) of epilayer/substrate [0 1 0]Ga–Ga/[0 1 0] Si–Si 3.186 5.431 [2 1 0]Ga–Ga/[1 0 0] Si–Si 5.518 5.431 [0 1 0]Ga–Ga/[0 1 0] Si–Si 4.500 5.431 [0 1 0]Ga–Ga/[1 1 0] Si–Si 3.186 3.84

ADM (%)

EADM (%) of epilayer/substrate


41.3

(Ga–Ga) · 5/(Si–Si) · 3

1.6

2.5 (Ga–Ga) · 1/(Si–Si) · 1


17.1

1.6 (Ga–Ga) · 6/(Si–Si) · 5


17.0


0.571 (Ga–Ga) · 6/(Si–Si) · 5
0.438

For details, refer to Refs [13,100,101].

c-plane is very small, under 2%, no discernible in-plane rotation for strain reduction is expected. However, if ZnO is deposited on sapphire followed by GaN deposition, the sapphire substrate and both epitaxial layers are oriented toward each other by a 30 rotation of the unit cell. That is, the in-plane epitaxial layer and substrate are in the form of ZnO, GaN ½1 0 1 0 k Al2O3½1 1  2 0, GaN ½1  1 0 0k Al2O3½1 1  2 1 0, GaN or    ZnO ½2 1 1 0 k Al2O3½1 1 0 0, all of which indicates a 30 rotation of the epitaxial layer with respect to sapphire substrate. 3.3.5 Epitaxial Relationship of GaN with LiGaO2 and LiAlO2 and Perovskites

Some oxides have also been explored because of the small lattice misfit with GaN they provide, with LiGaO2 and LiAlO2 being the ones, particularly the latter, generating a good deal of interest. The epitaxial relationships between GaN and LiGaO2 are expected to be ½1 1 2 0 GaN || [0 1 0] LiGaO2 and [0 0 0 1] GaN || [0 0 1] LiGaO2. On the contrary, the epitaxial relationships between GaN and LiAlO2 are expected to 1 1 0 GaN//[0 0 1] LiAlO2, ð1  1 0 0Þ GaN// be ð0 1 1 0Þ GaN//(1 0 0) LiAlO2 with ½2   g-LiAlO2 (1 0 0) with ½1 1 0 0GaNjj½1 0 0LiAlO2 and ½1 1  2 0GaNjj½0 0 1LiAlO2 . The lattice misfit strain between GaN and LiAlO2 with ð1  1 0 0Þ=ð1 0 0Þ and along the [0 0 0 1]//[0 1 0] in-plane is
0.41%. The lattice structure of LiGaO2 is similar to the wurtzite structure. However, owing to Li and Ga atoms having different ionic radii, the crystal has orthorhombic structure [102]. Figure 3.37 shows the transformation of the hexagonal unit cell of GaN to an orthorhombic cell that has lattice parameters close to that of LiGaO2 (Table 3.10). Perovskite oxides have also been employed in the growth of GaN in an effort to attain a better match compared to more conventional substrates or for applications

3.3 GaN Epitaxial Relationship to Substrates

1

2

Dangling bonds

Dangling bonds Si top layer Si first underlayer [0 1 0] 2

〈2 1 1 0〉

Si second underlayer Al or N sublattice

[1 0 0] [0 0 1]//[ 0 0 0 1] 〈1 1 1 0〉

Figure 3.36 Atomic arrangement for the heteroepitaxial nucleation of 2H-AlN on the Si(0 0 1) surface. The two AlN domains with a 30 rotation are formed on neighboring terraces (1) and (2), separated by a single atomic step boundary according to the Si dangling bond directions. Refs [13,101].

where the properties of nitride semiconductors and electro-optic and nonlinear optics properties of perovskites could be offered in the same template/stack. Among the perovskite is NdGaO3 with its orthorhombic unit cell. The lattice mismatch of GaN with NdGaO3 has been calculated [102] by assuming a perovskite cell of NdGaO3 with lattice parameters a, b, and c each equal to 3.86 Å. This is followed by creating a new unit cell with a0 and b0 where a0 and b0 are the diagonals of the old perovskite cell, as shown in Figure 3.38. The c0 -axis of the new cell is naturally parallel to the c-axis of the perovskite cell but its length is doubled. Basically, this operation transforms a perovskite unit cell to a tetragonal unit cell representing GaN, as tabulated in

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GaN b a a'

b'

Figure 3.37 Transformation of a hexagonal unit cell to an orthorhombic unit cell. Courtesy of J. Chaudhuri and Ref. [102].

Table 3.11. Accordingly, (1 0 0) plane becomes ð1  1 0Þ plane and (0 0 1) becomes (0 0 1). The corresponding lattice misfit is 1.72% and misfit strain is
0.66%.

3.4 Nitride Growth Techniques

HVPE, OMVPE (inorganic VPE), RMBE, and bulk crystal growth from Ga solution are the main growth methods used for nitrides. By far the most frequently used methods are the variants of VPE methods. Although HVPE is used to produce thick GaN layers, including those thick enough to be self-supporting once peeled from the sapphire substrate, OMVPE produces sharp heterojunctions for devices. With the exception of FETs, OMVPE is the primary method employed in the investigation and production of optoelectronic devices, such as LEDs and lasers, albeit the quality of MBE films grown on HVPE buffers is slightly better than of those grown by OMVPE. Inorganic VPE was the first method used to grow epitaxial III-N semiconductors, but was nearly abandoned. The technique, however, was revived recently by growing very high quality and thick buffer layers and templates for the growth of device structures by MBE and OMVPE.

Table 3.10 Transformation of GaN hexagonal unit cell to an orthorhombic unit cell [102].

GaN original hexagonal cell with a and b lattice parameters

GaN transformed orthorhombic cell with lattice parameters a0 and b0

LiGaO2 orthorhombic cell

a ¼ 3.189 Å b ¼ 3.189 Å c ¼ 5.185 Å

pffiffiffi a ¼ 3a ¼ 5:52 Å 0 b ¼ 2b ¼ 6.38 Å c0 ¼ 5.185 · 1 Å

a ¼ 5.4063 Å b ¼ 6.3786 Å c ¼ 5.0129 Å

0

3.4 Nitride Growth Techniques

a

b'

a'

b

Figure 3.38 Transformation of a perovskite unit cell to a tetragonal unit cell. Courtesy of J. Chaudhuri and Ref. [102].

3.4.1 Vapor Phase Epitaxy

VPE has long been employed for the growth of many semiconductor structures. With ongoing source developments and improved reactor designs, this technique has become very powerful, particularly for GaN and related materials. Growth from the vapor phase is categorized on the basis of the sources used. If the sources are inorganic in nature, the term inorganic vapor phase epitaxy is used. This too can be subdivided on the basis of the sources used. For example, if a hydride source is used for the group V element, the term hydride vapor phase epitaxy is applied. If at least some of the sources are organic in nature, the terms organometallic vapor phase eptixay, organometallic chemical vapor deposition, metalorganic chemical vapor deposition, or metalorganic vapor phase epitaxy are employed. 3.4.1.1 Hydride Vapor Phase Epitaxy The genesis of the HVPE growth method can be traced to its wide use in silicon and conventional III–V semiconductors. In the early 1960s, the development of the halide precursor techniques applied to Si and Ge provided the foundation for their subsequent application to the growth of GaAs, which was coming to eminence. The HVPE method has since played an important role in the growth of III–V

Table 3.11 Transformation of a perovskite unit cell to a tetragonal unit cell.

Lattice parameters a b c

Perovskite unit cell

New unit cell

Final unit cell

3.86 Å 3.86 Å 3.86 Å

pffiffiffi a= ¼ 3:86pffiffi2ffi = b ¼ 3:86 2 c0 ¼ 3.86 · 2

a0 ¼ 5.43 Å b0 ¼ 5.43 Å c0 ¼ 7.72 Å

Courtesy of J. Chaudhuri [102].

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semiconductors. In fact, it has the unique distinction of being the first method to produce AlN [103] and, as reported by Maruska et al. [17], to produce single crystalline GaN with quality sufficient to launch the first stages of GaN technology that gained so much prominence in the 1980s. Typical thicknesses for these deposits were in the range of 50–150 mm. The advantage of this technique is that it is conducive to the growth of thick buffer layers at high growth rates on any available substrates to be used as templates for OMVPE and MBE growth, particularly the latter, of high-quality heterostructures with relatively low defect concentrations. A comprehensive treatment of this method can be found in Ref. [104]. A brief historical treatment and a succinct review will be given. After the report of Maruska et al. [17], Wickenden et al. [105] reported on GaN deposition on a-SiC and a-Al2O3, meaning wurtzitic varieties, in 1971, and Ilegems [106] obtained 100–200 mm thick single-crystalline GaN layers on sapphire substrates in 1972. Continuing on, Shintani et al. [107] investigated in 1974 the effects of the important growth parameters, such as the position of the substrate in view of the gas flow dynamics in the reactor, the reactant gas flow rate, and the substrate temperature, on the epitaxial growth rate of GaN on (0 0 0 1) sapphire substrates. Sano et al. [108] considered the influence of the surface sapphire orientation on the growth rate in 1976. In 1977, Madar et al. [109] and Jacob et al. [110] achieved doped GaN with n-type conductivity on sapphire substrates. A study on the growth rate of GaN in hydrogen as well as in inert gas ambience was undertaken by Seifert et al. [111] in 1981, which resulted in growth rates up to 800 mm h
1. Owing to difficulties associated with uniform seeding of GaN on sapphire, coalescence of islands in a timely manner and resultant high n-type background doping (typically 1019 cm
3), large defect concentrations, and inability to produce p-type GaN for light emitters, this technique was largely abandoned in the early 1980s, although Maruska et al. [112] later showed that Zn and Mg doping could be achieved by the simultaneous evaporation of the dopant source in the HCl stream. Since the first report of Maruska and Tietjen [17] many reports on GaN growth [113–124] became available in the literature. Further, several others [103,125,126] have extended this method to the growth of high-quality AlN. But when low-temperature nucleation buffer layers were employed in the context of OMVPE, as discussed in Section 3.5.5.1, followed shortly thereafter by reports of p-type conductivity in GaN, the HVPE method reappeared [127–129] because, in part, of its improved ability to grow thick GaN films with relatively low defect concentration and new techniques for nucleation layers (NLs). Further improvements resulted from lateral epitaxial overgrowth on patterned SiO2 masks [130–132], the concept of which is discussed in Section 3.5.5.2. Freestanding GaN substrates have been prepared by Kim et al. [133] and Melnik et al. [134], which eventually culminated in the production of very high quality freestanding GaN templates prepared at Samsung Advanced Institute of Technology [135], the detailed characteristics of which are discussed in Section 3.5.1.2. Despite these remarkable achievements, the nitride materials still suffer from a very high defect density because of the lattice mismatch between the nitrides and all the available foreign substrates. Recent twostep processes employing low-temperature GaN buffer layers [136] and techniques

3.4 Nitride Growth Techniques

for substrate removal [135,137] have shown good-quality materials with very promising characteristics. The development of the HVPE technique combined with other techniques for producing GaN templates may be the key to resolve the high density defect issue in the III nitride device technology. In HVPE, the group III precursors are chlorides formed by flowing hydrogen chloride gas over the liquid metal in a quartz tube. The group V precursors are hydrides fed into the reaction chamber by a separate quartz line in order to avoid premature reaction with molten source metal. For GaN growth, the chloride and hydride precursors are GaCl, which is formed by reacting Cl from HCl gas with molten Ga, and NH3, respectively. The GaCl in vapor phase is transported to the deposition zone by a carrier gas, which can be hydrogen, and/or an inert gas. The pressure inside the reaction chamber is kept at the atmospheric pressure. The reactor walls are made of high-purity quartz tube. For high-quality semiconductor films to result, the gases are of electronic quality with purity better than 1 ppm for contaminants. For nitrogen or hydrogen, each impurity is in concentration below 1 ppm. For the other gases, the total concentration of all the impurities is below 1 ppm. The metallic sources employed are of 7N, meaning they have a purity of 99.99999%. A multiple zone furnace is used as the metal source zone and substrate temperatures are different. In particular, the zone containing the metallic gallium, the central zone where the gases are homogeneously mixed, and the zone where the substrate resides and the deposition takes place are all kept at different temperatures, as shown schematically in Figure 3.39. The three-temperature zone allows one to independently set the partial pressure of each species, such as the chlorides and hydrides, and explore optimum growth conditions systematically. The influence of parametric variations in the vapor phase composition on the growth rate becomes relatively easier. The same applies to physical processes taking place during a growth. In brief, the vapor phase composition depends on the metallic source efficiency, the ammonia decomposition, and the flow of various gases introduced into the reactor. The vapor phase composition, the partial pressures of the various reactive gaseous species, and the temperature of the three zones in the reactor determine the growth rate and the solid composition of the epitaxial layer if ternaries are attempted. Obviously, the growth takes place in a thermodynamical equilibrium and a wide range of conditions can be applied. If a large flow of HCl is not introduced into the reactor, the Source zone

Mixing zone

Deposition zone

NH3

NH3

N2/H2+HCl

N2/H2+HCl GaCl

N2/H2+HCl add

Substrate

GaCl HCl H2/N2/NH3

Ga source

Bypass line

Bypass

Three-zone furnace Figure 3.39 Schematic diagram of a three-zone HVPE reactor, which utilizes Ga and ammonia sources, used for nitride growth. Patterned after Ref. [104].

Exhaust

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use of NH3 leads to initial conditions far from thermodynamical equilibrium. The growth rates from a low of 1 mm h
1 to a high of more than 100 mm h
1 can be achieved [104]. In HVPE as employed by Maruska and Tietjen [17] for GaN, HCl vapor flowing over a Ga melt causes the formation of GaCl, which is transported downstream. On the substrate surface, the GaCl reacts with NH3 and leads to GaN through the following chemical reaction: 2GaðlÞ þ 2HClðgÞ ! 2GaClðgÞ þ H2 ðgÞ; GaClðgÞ þ NH3 ðgÞ ! GaNðsÞ þ HClðgÞ þ H2 ðgÞ;

ð3:9Þ

where g, l, and s depict gaseous, liquid, and solid species, respectively. The efficiency of the first reaction had been estimated by Ban [138] to be about 99.5%. Using the GaAs analogy [139], two thermodynamic reaction pathways leading to the deposition of GaN [140] can be forwarded: GaClðgÞ þNH3 ðgÞ()GaN þ HClðgÞ þ H2 ðgÞ; 3GaClðgÞ þ 2NH3 ðgÞ()2GaN þ GaCl3 ðgÞ þ 3H2 ðgÞ:

ð3:10Þ

The gaseous species in the reactor are GaCl, GaCl3, HCl, NH3, H2, and either H2 or some other inert gas is used as carrier gas. Barin [141] used thermodynamical data to calculate the reactions of Equations 3.9 and 3.10. For additional details, see Ref. [104]. The supply of GaCl is controlled by the Ga boat temperature and the flow rates of the HCl gas and the carrier gas. The reactor in this method is heated by a two-zone resistance furnace with the region containing the Ga boat being kept at a different temperature than the region housing the substrate for reaction, as shown in Figure 3.39. The Ga zone temperature impacts the growth rate a great deal. The Ga source is held at a constant temperature between 850 and 900  C. The reaction efficiency of HCl with Ga is near unity. Although dependent on the reactor itself, the typical flow rates are about tens of sccm for HCl, 1 l min
1 for NH3 and 2 l min
1 for the carrier gas. GaN films can be grown with rates up to 1 mm h
1 on Al2O3 substrates at atmospheric pressure. However, those high growth rates deplete the Ga source material rather quickly and lead to very rough surfaces with columnar growth. The substrate zone temperature varies between 1050 and 1200  C. At lower substrate zone temperatures, the growth rate decreases exponentially owing to the decreasing pyrolysis efficiency of the GaCl and NH3. On the contrary, at higher substrate zone temperatures, thermally induced decomposition of reactants reduces the growth rate. The hydrogen ambience also aids this reduction in that competing processes such as GaHx would take place. The main issue of concern associated with HVPE and other nitride growth methods is that the initial nucleation layer on sapphire substrates determines to a larger extent the material properties of the subsequent epitaxial layer than in other methods [142–144]. The nucleation or the prelayer is typically deposited using a GaCl or NH3 pretreatment consisting of flowing GaCl or NH3 over the sapphire surface prior to the initiation of growth at high temperatures. In other cases, a ZnO wetting layer is used [127,145,146]. Nitridation of sapphire has been mentioned as a means to improve HVPE materials by several groups [147,148].

3.4 Nitride Growth Techniques

Thermal dissociation of group V species during HVPE of GaAs leads to the formation of As2 or As4 molecules, which remain volatile and chemically reactive and thus participate in the film growth. However, in the case of GaN, by-products of decomposed NH3 are N2 and H2, and N2 molecules that are stable and unreactive at the temperature of interest. Other forms of N species such as NCl3 are not considered, as they are explosive. Moreover, there is a strong thermodynamic driving force for forming parasitic gas-phase reactions, which cause deposition on the walls, making the growth mechanism difficult to unravel. The process also tends to produce large amounts of NH4Cl, GaCl3, and GaCl3-NH3, which condense and clog the exhaust lines unless they are heated to sufficiently high temperatures (>150  C). Exchange reactions with the hot quartz walls of the reactor make it difficult to use HVPE for aluminum- and magnesium-bearing compounds required for AlGaN growth and efficient p-doping, respectively. Endemic to MBE and not considered that important in the case of vapor phase deposition techniques until the advent of GaN is the concept of kinetics, involving adsorption and desorption discussed in detail for OMVPE in Section 3.4.1.3 and for MBE in Section 3.4.2. Adsorption and desorption processes depend on the kinetics of the gaseous species on the substrate surface and the diffusion kinetics of the adatoms or admolecules before arriving at the incorporation sites dubbed half-crystal or K sites (see Figure 3.40) [104]. To put it simply, kinetics consists of complex but simplified approaches that can be considered to get insight into the growth resulting from two superficial diffusion flows, namely, the NGa admolecules and the other from the NGaCl admolecules. Near the step edges, only the NGa flow directly leads to incorporation through desorption of chlorine from the NGaCl. Because GaN growth relies on GaCl, which is identical to that in GaAs, much of the GaN work benefited from systematic investigations performed for GaAs. Among the studies of GaAs growth were those reported by Shaw [149,150] in a series of articles including a systematic measure of the growth rate of GaAs on {0 0 1}, {1 1 1}A, {1 1 1}B, and {1 1 0} surfaces (using hydrogen as carrier gas and the chloride Vapor phase AB

AB+B Vapor phase followed by diffusion

Adsorption followed by diffusion

B* A

AB*

Dissociation A A*

Su bstra

A

te * Adsorbed species

Surface diffusion Figure 3.40 Kinetics processes occurring from the growth by vapor phase. Patterned after Ref. [104].

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method) as a function of temperature and the GaCl partial pressure. The nitride growth rate in HVPE exhibits an increase followed by a decrease in the growth rate with decrease in substrate temperature. The latter was found consistent with Langmuir GaCl adsorption isotherm together with a kinetic model based on GaCl adsorption [151,152]. In this treatment, the lateral interaction between GaCl adsorbed molecules and the Cl desorption by hydrogen of adsorbed GaCl molecules (dubbed the H2 mechanism) are invoked to account for the slight decrease in the GaAs growth rate at low temperatures when the GaCl partial pressure is increased. Hollan and Schiller [153,154] and Hollan et al. computed surface diffusion, a component of kinetics at play, by fitting the experimental data and computed values as a function of the substrate orientation, and of Gentner [155] on 6 off {0 0 1} GaAs substrates. In the latter study, both atmospheric and reduced pressures in hydrogen and helium carrier gases were considered. These studies were also expanded to include AsCl3 instead of HCl gas. A desorption mechanism for two adsorbed chlorine atoms by GaCl in GaCl3 was, therefore, considered with an intermediate GaCl3 adsorption step [139]. This mechanism is called the GaCl3 mechanism. Cadoret [156] developed a model involving adsorption and desorption of species on the surface of the substrates. Among the adsorbing species considered are NH3 molecules, adsorption of N atoms resulting from NH3 decomposition, and adsorption of GaCl molecules on N atoms. They follow the reactions: V þ NH3 ðgÞ()NH3 ; 3 NH3 ()N þ H2 ðgÞ; 2

ð3:11Þ

N þ GaClðgÞ()NGaCl; where (g) depicts the gas-phase species and V is a vacant site. Cadoret also considered two desorption mechanisms of chlorine: that is, desorption in HCl vapor molecules following a surface reaction with H2 and desorption in GaCl3 vapor molecules following adsorption of a GaCl molecule on two GaCl underlying molecules, which is schematically shown in Figure 3.41. The representation is based on the premise that the substrate surface, which is sapphire, is terminated with Al atoms on which N atoms bond leading to a Ga polarity sample. As in the GaAs model [139], the processes follow the reactions: 2NGaCl þ H2 ðgÞ()2NGa þ 2ClH; NGa
ClH()NGa þ HClðgÞ:

ð3:12Þ

The two mechanisms are labeled H2 and GaCl3 mechanisms. They can be treated by means of a one-monolayer model of adsorption on a (0 0 0 1) Ga or Al surface. The adsorbed species are NH3 molecules, N atoms, NGaCl, NGa–ClH, and 2N Ga–GaCl3 molecules. The one-monolayer adsorption model and the Bragg–Williams approximation are used to simplify the problem. The number of activated molecules involved in the reactions described by Equations 3.11 and 3.12 as well as possible intermediate states of hydrogen desorption from NH3 are neglected. GaCl adsorption on a Ga adatom [157,158] is assumed to be negligible because it would lead to antisite positions following chlorine desorption or would act as a simple inhibitor of the deposition

3.4 Nitride Growth Techniques

Ga N

H

H Cl

Cl

Ga

Ga

N

N

Cl Ga

NH3 N

N

N

V

(a)

Cl Ga

Ga N

N

Cl

Cl

Ga

Ga

N

Cl Ga N

NH 3 N

N

V

(b) Figure 3.41 Schematic steps of adsorption and desorption processes involved in the (a) H2 mechanism and (b) GaCl3 mechanism. Patterned after Ref. [104].

process that is not necessarily observed in GaAs. The two overall reactions corresponding to the H2 and GaCl3 growth mechanisms can be written as V þ NH3 ðgÞ þ GaClðgÞ()NGa þ HClðgÞ þ H2 ðgÞ; 2V þ 2NH3 ðgÞ þ 3GaClðgÞ()2NGa þ GaCl3 ðgÞ þ 3H2 ðgÞ:

ð3:13Þ

The adsorption and desorption processes including their flux, potential barriers associated with their activation barriers, are somewhat involved and are beyond the scope of the present treatment. An in-depth treatment of the topic can be found in Ref. [104]. In addition to adsorption and desorption, quantification dealing with growth by HVPE also involves mass transport, which is then followed by growth. In the mass transport case, partial pressures of reactant species are treated that, depending on local conditions and the substrate area, can lead to production or depletion of species. Coupled with the treatment of mass transport, the growth phase, which involves equilibrium between the vapor and substrate surface, must be treated. To get a handle on the problem, superficial flow of diffusive molecules toward step edges, which are assumed to be monomolecular in height, and surface coverage of vacant sites, their distribution over the substrate surface (assumed uniform), are considered. So is whether the growth is based on GaCl3 or H2 growth mechanisms. When the terrace width is small enough to not hinder surface diffusion, the parameters of importance are the adsorption energies of GaCl(g), HCl(g), and GaCl3(g). Further details can be found in Ref. [104]. Those who are more focused on the growth- and properties-related issues involving GaN by MBE are referred to Ref. [159].

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Summarizing, the growth process of GaN by HVPE could be analyzed by a combination of thermodynamical and kinetic considerations. The ample experimental data available in the literature regarding the growth rate on exact and misoriented {0 0 1} surfaces including those measured by Seifert [111] in He and H2 environments on 3 off (0 0.1), in the hkl configuration and (0 0 0 1) in the hjkl configuration, GaN by HVPE allowed for modeling and understanding the physical processes involved in growth. Relative kinetics considerations would indicate that the mass transfer is larger for GaN than for GaAs. Additionally, the high supersaturation, in effect, generally leads to important parasitic GaN deposition before the substrate zone, or the deposition zone, that reduces and even could potentially negate the relative supersaturation. The accuracy with which the growth rate is measured and controlled is the key to the study of the thermodynamics of the system. The variants of HVPE have also been explored and used for growth of GaN and its ternaries with AlN. Among them is a modified VPE process, dubbed the sublimation sandwich method (SSM), which was reported by Wetzel et al. [160] and Fischer et al. [161] (Figure 3.42). Initially, the GaN films were grown from metallic Ga and ammonia on (0 0 0 1) 6H-SiC, using a modification of the sandwich method described previously by Vodakov et al. [162]. In this approach, the quartz reactor contains a Ga cell for each substrate for multiwafer processing with one ammonia stream only. The gap between the substrate and the Ga source is typically about 5 mm. The ammonia flow rate through the gap is very high, 25–50 l m
1 at atmospheric pressure. Under these conditions, there is an effective mass transport of Ga vapor and nitrogen to the surface of the substrates. At growth temperatures between 1170 and 1270  C, GaN layers were obtained at growth rates of up to 0.3 mm h
1. Many other variations of this approach, including [163–171] gaseous sources such as GaCl, GaCl2, GaCl3, Ga(C2H5)2Cl, GaCl2NH3, AlC3, AlBr3, InCl3, and GaBr3 for group III element(s) as reactants for NH3, have been used. Pastrnak et al. [172] chose to react N2 with GaCl3, AlCl3, and InCl3 in their CVD process. Dryburgh [173] grew RF Coils Water

Graphite Substrate NH3

ΔT

Ga Graphite Water

Figure 3.42 Schematic diagram of a proximity HVPE vessel for the growth of GaN at very high growth rates, approaching 0.3 mm h
1. It is dubbed the SSM.

Quartz

3.4 Nitride Growth Techniques

AlN from AlSe and N2. By introducing PH3, Igarashi et al. [174] achieved several percent P incorporation in GaN. The technique became popular because of the highquality buffer layers on the freestanding GaN templates for epitaxy by heterostructure deposition systems such as OMVPE and MBE [175]. The transport and optical properties of HVPE-grown films are discussed in Volume 2, Chapters 3 and 5, respectively. 3.4.1.2 Organometalic Vapor Phase Epitaxy High-quality epitaxial III-N films and heterostructures for devices have been accomplished by OMVPE technique. Manasevit et al. [176] applied this technique to the deposition of GaN and AlN in 1971. Using triethylgallium (TEG) and ammonia (NH3) as source gases for group III and V species, respectively, the authors obtained c-axis oriented films on sapphire (0 0 0 1) and on 6H-SiC(0 0 0 1) substrates. MIS-like LEDs followed, albeit they relied on deep states induced by Zn and suffered from very low efficiencies because of their poor crystalline quality. The development of LT buffer layers addressed the quality issue some [177]. The technique improved over the years to the point that undoped GaN films with a low background carrier concentration of 5 · 1016 cm
3 and with an X-ray symmetric peak FWHM of 30 arcsec have been grown [178]. The X-ray data should be treated with caution, as the symmetric peak is not as sensitive to the edge dislocation as the asymmetric peak. OMVPE has been used for the development of LEDs [179], lasers [180], transistors [181], and detectors [182]. The best OMVPE reactors for group III nitride film growth incorporate laminar flow at high operating pressures and separate inlets for the nitride precursors and ammonia to minimize predeposition reactions. A successful, two-flow OMVPE reactor is shown in Figure 3.43 [183]. The main flow composed of reactant gases with a high velocity is directed through the nozzle parallel to a rotating substrate. The subflow gas composed of nitrogen and hydrogen is directed perpendicular to the substrate. The purpose of the flow normal to the substrate surface is to bring the reactant gases in contact with the substrate and to suppress thermal convection effects. Hydrogen is the carrier gas of choice. A rotating susceptor was used to enhance uniformity of the deposited films. If one goes with the premise that smallest rocking curve half width implies an all-around good quality, GaN films can claim this quality. These films with one of the narrowest rocking curves with FWHM values of 37 arcsec (values even under 30 arsec have been obtained) were grown with a modified EMCORE GS 3200 UTM reactor. It should be stated, however, that the X-ray data based on the symmetric diffraction peak are not a critical measure of sample quality necessarily. For a more complete analysis, one should also inspect the asymmetric peak, which is sensitive to edge dislocations. This reactor generally incorporates separate inlets for ammonia and the nitride precursor, all are normal to the substrate surface, which rotates at speeds over 1000 rpm, and a laminar flow cell to assure a uniform growth [212]. OMVPE reactors incorporating new concepts have been designed to grow layers at lower temperatures. (It should be mentioned that the motivation for lower temperature growth spawned from the perceived need to minimize the loss of nitrogen from

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Figure 3.43 A schematic representation of a vertical OMVPE system employed at Virginia Commonwealth University along with a picture of the deposition chamber (a); a photograph of the reactor chamber of the same (b). (Please find a color version of this figure on the color tables.)

the surface. However, later it became evident that high temperatures are needed to mobilize treading dislocations, as they are useful for reducing dislocation density and facilitating lateral growth. For the sake of completeness, a discussion of relatively lower temperature growth is provided.) These technologies utilize an activated form of nitrogen to lower deposition temperatures of group III nitrides. That these technologies are interesting is apparent, for example, from the deposition of polycrystalline and amorphous GaN films at temperatures lower than 350  C by plasma-enhanced CVD. Epitaxial GaN and AlN have been grown by variants of methods activating nitrogen, such as laser-assisted CVD, remote plasma enhanced CVD, atomic layer epitaxy with

3.4 Nitride Growth Techniques

NH3 cracked by a hot filament, with ammonia catalytically decomposed, photoassisted CVD, and ECR plasma-assisted CVD. However, none of these approaches has been able to produce material comparable in quality with the standard OMVPE systems and, consequently, they did not really become players in the field. We alluded to the genesis of the lower quality above. As for the mechanism involved, growth of nitride semiconductors by OMVPE relies on the transport of organometallic precursor gases, hydrides for the nitrogen source, and reacting them on or near the surface of a heated substrate. The deposition is through pyrolysis. The underlying chemical mechanisms are complex and involve a set of gas phase and surface reactions. Although OMVPE has long been assumed to be a thermodynamically equilibrium process, nitride OMVPE processes may involve kinetics as well. The fundamental understanding of the processes involved is still evolving and, as such, the reaction mechanism and the related kinetic rate parameters are poorly understood. The deposition of epitaxial nitride layers by OMVPE involves the reaction of metalcontaining In, Ga, or Al gases with ammonia, NH3. Commonly, the metal-containing gases are trimethylgallium ((CH3)3Ga), trimethylindium ((CH3)3In), or trimethylaluminum ((CH3)3Al). Radicals, reactive by most definitions, react in the gas phase with donors containing acidic hydrogen, such as NH3, and form adducts. The key here is to eliminate the unwanted radicals by forming stable molecules followed by their removal from the reaction region. The analysis of the mechanisms involved in the OMVPE process clearly indicates that any precursor must balance the requirements of volatility and stability, which often counter each other, to be transported to the surface and decomposed for deposition. To put it another way, these precursors must have appropriate reactivity to decompose thermally into the desired solid and to generate readily removable gaseous side products. Ideally, the precursors should be nonpyrophoric, water and oxygen insensitive, noncorrosive, and nontoxic. The trialkyls, trimethylgallium (TMG) [184] and triethylgallium (TEG), trimethylaluminum (TMA) [185], trimethylindium (TMI) [186], and others are usually used as III metal precursors. Ammonia (NH3) [187], hydrazine (N2H4) [142,188,189], monomethylhydrazine (CH3)N2H2 [190,191], and dimethylhydrazine (CH3)2N2H2 have all been used as nitrogen precursors with varying degrees of success. Although trialkyl compounds (TMA, TMG, TMI, etc.) are pyrophoric and extremely water and oxygen sensitive, and ammonia is highly corrosive, much of the best material grown today is produced by conventional OMVPE by reacting these compounds with NH3 at substrate temperatures close to 1000  C [192–203]. Investigators have reacted TMG [204,205], TEG, and GaCl [206,207] with NH3 plasma. Sheng et al. [208] reacted TMA and NH3 in the presence of hydrogen plasma. Wakahara et al. [209] grew InN by reacting TMI with microwave-activated N2. Eremin et al. [210] used nitrogen to transport metallic Ga to the reaction zone where it was reacted with active nitrogen. These commonly employed precursors at least satisfy the criteria of sufficient volatility and appropriate reactivity. During the growth of nitrides by employing trialkyl precursors, adduct formation between ammonia and TMA and TMG is well documented. Usually mixing at room temperature, adduct

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j 3 Growth and Growth Methods for Nitride Semiconductors

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formation between TMG or TEG and ammonia is complete in less than 0.2 s. The resulting adduct Ga(CH3)3-NH3 has a vapor pressure of 0.92 Torr at room temperature, while the vapor pressure of Ga(C2H5)3 : NH3 is much lower. To counter early beliefs that stability of ammonia and required relatively high growth temperature the use of other more volatile nitrogen sources were explored. For example, Fujieda et al. [142] replaced NH3 with N2H4 and observed that a significantly smaller amount of N2H4 was required to maintain the same growth rate. However, they also noted that the CVD growth rate was limited by the decomposition of TMG, thus limiting the benefits of N2H4. Matloubian and Gershenzon [211] used TMA and NH3 and a substrate temperature range of 673–1473 K to grow nitrides. Single-crystal AlN films were obtained only at 1473 K. Dupuie and Gulari [212] reported that the presence of a hot filament near the substrate increased the growth rate of AlN grown with TMA and NH3 by two orders of magnitude. However, the use of a hot filament immediately raises concerns about residual contamination, most prominently oxygen, which was not addressed by the author. A case in point illustrating the elimination of unwanted radicals by forming stable molecules followed by their removal from the reaction region is AlN growth from mixtures of methyl alkyls that may proceed by the formation of an intermediate gas phase adduct (CH3AlNH3), followed by the elimination of CH4. The exact path may be that coadsorption of (CH3)3Al and NH3 at room temperature generates surface adduct species such as ((CH3)2AlNH3) and adsorbed NH3 [213]. As the substrate temperature is raised above 320  C, the appearance of vibrational bands corresponding to AlN indicates the formation of extended (Al–N) networks on the surface. These Al(NH2)2Al species finally eliminate H2 at the surface to form AlN [214]. The possible chemical reactions in the process are (a stands for adsorbate on the surface and g stands for gas-phase product):

and

2ðCH3 Þ2 Al:NH3 ðaÞ ! CH3 AlðNH2 Þ2 AlCH3 ðaÞ þ 2CH4 ðgÞ; CH3 AlðNH2 Þ2 AlCH3 ðaÞ ! AlðNHÞ2 Al þ 2CH4 ðgÞ;

ð3:14Þ

AlðNHÞ2 AlðaÞ ! 2AlNðaÞ þ H2 ðgÞ: As for GaN, investigations are relatively limited, but it would be fair to assume that processes similar to that with AlN growth are most likely in place. Adducts of Ga compounds are weaker electron acceptors than the corresponding Al adducts, and therefore these adducts may not be abundant owing to redissociation in the hot zone. It may be because of this that successful GaN growth by OMVPE requires very large V/III ratios, which favor adduct formation. Thermal stability of NH3, although low compared to that of N2, could be partially responsible for the use of high substrate temperatures, typically above 550  C for InN and above 1000  C for GaN and AlN. The high growth temperature necessitated by the process itself, associated with high nitrogen vapor pressure over GaN, lead to the inevitable nitrogen loss from the nitride film. This may also be the path to carbon contamination from the decomposition of the organic radical during metalorganic pyrolysis. The loss of nitrogen can be

3.4 Nitride Growth Techniques

alleviated by using high V/III gas ratios during the deposition, particularly for InGaN (e.g., >2000 : 1). Assuming that high substrate temperatures represent a problem in relation to ammonia, which seems reasonable particularly in early days, various alternative approaches can be and have been taken. One approach is to use alternative nitrogen precursors that are thermally less stable than NH3. Hydrazine (N2H4), which is a larger and less stable molecule, has been used in combination with TMA to deposit AlN at temperatures as low as 220  C [189]. However, hydrazine is toxic, unstable, and not as pure as NH3. Consequently, a compromise between quality and substrate temperature must be made. Researchers took the quality/purity as the primary parameter and stayed with NH3. More recently, other nitrogen sources such as tertbutylamine (t-BuNH2) [215], isopropylamine (i-PrNH2), and trimethylsilylazide (TMeSiN3) have been used with TMA or t-Bu3Al to deposit AlN films at lower substrate temperatures (400–600  C) and reduced V/III gas ratios (5 : 1–70 : 1) [216]. However, the deposited films were invariably contaminated with high levels of residual carbon (up to 11 at.%). Hydrogen and, to a lesser extent, nitrogen are predominantly used as the transport gas. They can influence the chemical reaction mechanism of Et3M or (CH3)3M in the gas phase by changing the reaction temperature of the metalorganic compounds or the concentration of reaction products. Hydrogen at the surface of the growing film can influence the growth rate and the structural properties [217,218]. To obtain a basic understanding of the role of hydrogen in GaN growth, the possible sources of hydrogen and the influence of hydrogen on the chemical reaction mechanism in the gas phase and at the surface are unique and important issues in the context of OMVPE. Pyrolysis of highly concentrated NH3 in the presence of H2 as the carrier gas results in a high concentration of molecular and atomic hydrogen near the substrate surface. Because the growth temperatures above 900  C are employed, which are higher than the decomposition temperature of the metalorganic compounds and their hydrocarbon ligands, the conditions for the desired bond breaking between the metal atom and the methyl or ethyl groups of the precursors are in place. However, the same can also lead to pyrolysis of the hydrocarbons with incorporation of hydrogen and carbon into the films. In this vein, decomposition of (CH3)3Ga (TMG) and Et3Ga in hydrogen and nitrogen atmospheres using a quadrupole mass analyzer has been investigated [219]. The decomposition reaction of the metalorganic precursors was found to be strongly affected by the presence of molecular hydrogen. The decomposition of (CH3)3Ga occurs at 400 and 500  C in H2 and N2, respectively. Similarly, the decomposition of Et3M occurs at 260 and 300  C in H2 and N2, respectively. Clearly, molecular hydrogen reduces the reaction temperature. The reaction mechanisms involve hydrolysis for (CH3)3Ga in H2, homolytic fission for (CH3)3Ga in N2, and b-elimination for Et3M in both H2 and N2. By changing the reaction temperature and the reaction mechanism of the metalorganic precursor, the partial pressure of hydrogen affects the deposition rate of GaN and therefore the structural properties of the resulting film, especially at low growth temperatures [220].

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3.4.1.3 Modeling of OMVPE Growth The GaN kinetic model with the associated details, to the extent necessary, of the GaN thermal decomposition, precursor adsorption, and decomposition, Ga and N2 desorption rates, and surface diffusion rates is discussed next to identify the important chemical steps for GaN growth. This will aid in developing the kinetic model. 3.4.1.3.1 Thermal Decomposition of GaN as it Relates to Growth The rule of the thumb for modeling the growth of conventional compound semiconductors is that MBE is a nonequilibrium growth with several activation barriers in place and, as such, a kinetic model is better suited. On the contrary, OMVPE is an equilibrium process and as such, a thermodynamic model is appropriate. However, nitrides are different in that temperatures employed are high and barriers such as NH3 dissociation are in play. Consequently, a kinetics-based model for OMVPE is appropriate as reported by Koleske et al. [221]. The group III nitrides have strong chemical bonds with the Ga–N bond strength being estimated near 4.2 eV, which is comparable to C–C bond strength of 3.8 eV in diamond. In addition, the group III nitride bonds are more ionic (31–40%) (excluding BN) than those in other III–V semiconductors (3 1.45

V/III desorption ratio 1.93 · 1016 kN/kGa

3.41

RHEED oscillations on vicinal surfaces Ga dosed, RHEED pattern recovery Estimate based on present work RHEED pattern recovery Calculation by Koleske et al. [223]

Note that a GaN decomposition activation energy of 3.6 eV has also been reported in Ref. [222].

3.4.1.3.2 Ga and N Precursor Adsorption and Decomposition Adsorption and decomposition processes are described in detail in conjunction with MBE growth of GaN in Section 3.4.2. However, the particulars of these processes, although the basics are the same, in an OMVPE environment are discussed here. Several factors, such as the precursor flux, the gas-phase diffusion rate, the sticking coefficient, the number of available surface sites, and the precursor decomposition rate, contribute to the concentration of active growth species on the surface. For the particular kinetics-based model, it will be assumed that, once adsorbed, the Ga and N precursors decompose completely to produce adsorbed Ga and N atoms and that precursor sticking coefficients are near unity. The assumption that TMG and TEG undergo full dissociation to Ga atoms is supported by several studies in that nearly complete pyrolysis of TMG for T > 500  C or TEG for T > 300  C in H2 or N2 occurs. In flowing NH3, measurable and complete TMG decompositions are observed for T > 525  C and T > 625  C, respectively. TMG and TEG have been shown to partially decompose in the gas phase but at a reduced pressure (e.g., 0.1 atm), which is parasitic. However, most of the decomposition occurs on the surface rather than in the gas phase. To ensure that the decomposition takes place on the surface, gas manifolds are designed to keep the precursors cool until they come in contact with the surface. The sticking coefficients for TMG and TEG are large because of the accommodation ratio, which increases as the molecular weight increases. TMG forms a strong adduct bond with NH3, which would further

3.4 Nitride Growth Techniques

increase the accommodation ratio and therefore the sticking probability. TMG conversion to metallic Ga has been observed on a sapphire surface at 525  C at atmospheric pressure, suggesting that metallorganic Ga precursors efficiently stick and completely dissociate to metallic Ga on GaN and sapphire surfaces. The ability to grow low temperature nucleation layers at substrate temperatures not much higher than 500  C is also indicative of this conversion. Although not much is known about NH3 adsorption and dissociation, the general assumption is that sticking of NH3 to a Ga-terminated surface should be large. Moreover, the sticking of NH3 may depend on the N surface coverage in a fashion similar to the reduced sticking of other group V hydrides on group V terminated surfaces. Once adsorbed on the surface, NH3 dissociates and forms adsorbed H and NHx720  C), the kinetic order may change and this cannot be ruled out. Of paramount pertinence to OMVPE growth is that the desorption rates of N and Ga become equal at 780  C (one monolayer per second or 1.14 · 1015 atoms cm
2 s
1), and thus, at the growth temperatures commonly used in OMVPE, the N desorption rate is 1000–10 000 times that of Ga. At about 1030  C, a reasonable

3.4 Nitride Growth Techniques

OMVPE growth temperature, the estimated GaN decomposition rate is about 3 · 1017 cm
2 s
1, which coincides well with kGa. Assuming that desorption occurs only from the top layer, the desorption limited surface residence times, tD, can be calculated by dividing the surface coverage (1.14 · 1015 cm
2) by the desorption rates, kGa and kN, for Ga and N, respectively. Most studies indicate tD,Ga ¼ 1 s at temperatures in the range of 750–780  C, which is larger than for Ga on GaAs(1 1 1). 3.4.1.3.4 Ga and N Surface Diffusion At high growth temperatures and during growth where the surface diffusion length is larger than the terrace width, atoms move by surface diffusion from the terrace to the step edge for incorporation into the growing lattice, which is called the step-flow growth. As the diffusion length decreases, the surface morphology changes from step flow to islanded growth on the terraces. These two regimes have been established in MBE growth of GaAs and other conventional compound semiconductors. If thermal desorption is large, the diffusion length may be dramatically shortened. Consequently, diffusion lengths at growth temperatures must be evaluated. This can simply be done by assuming that the surface diffusion follows an Arrhenius form and that the mechanism does not change with temperature. In these descriptions and the following Einstein equation, the diffusion length, lS, is given by [228] pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:19Þ lS ¼ DS tS ¼ DS0 expð
E SD =kTÞtS ;

where DS is the diffusion coefficient,tS is the lifetime of the diffusion event, DS0 is the temperature-independent diffusion coefficient, ESD is the diffusion activation energy, and kB is the Boltzmann’s constant. Table 3.12 lists representative values of DS0 and ESD for GaN and GaAs. The values of tS can be in two limiting values, which are the lifetime before desorption (tD), which depends on the desorption rate and the lifetime before lattice incorporation (tI), which depends on the GR. The estimate lS, Ga for both limiting values of tS will be presented here. Using the DS,Ga reported in the literature [229] and tD,Ga calculated in Section 3.4.1.3.3, Koleske et al. [221] estimated lS,Ga using Equation 3.19. The temperature dependence of lS,Ga is plotted as a solid line in Figure 3.44, which indicates that lS,Ga decreases slightly as the temperature increases. This is unexpected because the temperature increases as lS increases. The decrease in lS,Ga at higher temperatures takes place because the increase in DS,Ga is counterbalanced by the Ga desorption rate, which results in a reduced tD. As shown in Figure 3.44, lS,Ga is less than 1.2 nm for temperatures ranging from 600 to 1200  C. If the Ga coverage exceeds one monolayer, lS,Ga would increase and approach the lS,Ga value for Ga on liquid Ga metal. However, Ga droplet buildup on the surface during growth is not advantageous for maintaining the desired smooth surface morphology (in contrast to MBE). Therefore, lS,Ga is about 1 nm if the Ga desorption rate from the surface is large. It must be remembered that at OMVPE growth pressures (30–770 Torr), Ga desorption appears to be suppressed, suggesting that the Ga surface lifetime, tS,Ga, may be substantially longer than the desorption limited lifetime. The limit, of course, is the incorporation lifetime (tI) into the growing lattice. The value of tI depends inversely on the growth rate, as the slower the growth rate the larger tI. To

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102 3

Diffusion length, (nm)

10 30

λ S,Ga = (D S,Gaτl)1/2

101

100

nm min–1

100 λ S,Ga = (D,S,GaτD)1/2

10–1 600

700

800

900

1000

1100

1200

Temperature (ºC) Figure 3.44 Estimates of the Ga diffusion length, lS,Ga, on GaN. The solid line is a calculation of lS,Ga based on tD (desorption limit) and the dashed lines are calculations of lS,Ga based on tI (growth rate limit) for growth rates of 3, 10, 30, and 100 nm min
1 [223].

show the dependence of tI on the growth rate, lS,Ga is plotted in Figure 3.44 as open segmented lines for four different GR ranging from 30 to 1000 Å min
1. For this calculation, tI was determined by dividing the thickness per monolayer (actually bilayer, which is 2.583 Å and represents one half of the lattice parameter along the c-direction) by the growth rate. Several observations can be gleaned from Figure 3.44. The term lS,Ga increases with temperature and this temperature dependence is solely owing to the temperature dependence of DS,Ga because tI is determined by the GR. As the GR decreases, both tI and lS,Ga increase, and consequently, the Ga diffusivity increases, which should aid in the growth of a more ordered lattice because the number of adatoms that incorporate is increased. Enhancements in GaN crystalline quality have been attributed to reduced growth rates for the migration-enhanced epitaxy (MEE) approach where the surface is dosed with an alternating sequence of N and Ga exposures with dwell times between exposures in an effort to increase lS,Ga. At 1050  C, the estimated values of lS,Ga using tI (no desorption regime or complete incorporation) are about 10–100 times larger than the estimated values of lS,Ga based on tS,Ga (high desorption regime or growth rate limited case). For GaN growth by OMVPE, practical growth rates are typically >150 Å min
1 (about 1 mm h
1). At a growth rate of 300 Å min
1, lS,Ga is about 11.3 nm at 1050  C. Turning our attention to the other species N, its surface diffusion length (lS,N) should be smaller than lS,Ga. Unlike Ga, no evidence for N diffusion was observed in

3.4 Nitride Growth Techniques

the RHEED intensity analysis in MBE when radio frequency (RF) activated nitrogen was used. This is not necessarily applicable to NH3 as RHEED oscillations have been observed. Assuming RF-activated nitrogen and N do not migrate outside the unit cell during the RHEED pattern, the recovery time, DS,N, must be 800  C, once N migrates to next to an adjacent N, N2 forms and desorbs. If the surface migration of atomic Ga and N is limited, partially decomposed reactants may diffuse more readily on the surface. This may be particularly important for OMVPE growth, because partially decomposed precursors that are weakly bound to the surface may more freely diffuse across the surface. Weaker N–N or N–Ga bonds between partially dehydrogenated NH3 and the GaN surface may form with these species diffusing more readily than Ga or N atoms. Therefore, the mean diffusion rates for Ga and N species on the surface may be substantially larger than the values calculated here using diffusion constants measured by MBE in a vacuum environment. 3.4.1.3.5 Kinetic Model: Balance Between Adsorption and Desorption Using the temperature dependence of desorption rates for N and Ga, kN and kGa, reported in Ref. [229] for an MBE environment, the ratio kN/kGa can be plotted and when done, a temperature dependence of kN/kGa ¼ 1.9205 · 1016 exp[
39 563/T (K)] results. Using the kinetic values, kN/kGa is about 1000 at T ¼ 1020  C and about 0.1 at T ¼ 725  C. By choice of temperatures, these values of kN/kGa are comparable with the V/III ratios used for OMVPE and MBE growth, respectively. In addition to MBE analogue, if one also relies on the models developed for GaAs [231], one can then construct a kinetic model for GaN grown by OMVPE where activation barriers play an important role as done by Koleske et al. [221]. In this scenario, the Ga and N surface coverage, yGa and yN can be related to the incident fluxes, F, and desorption rates, k, as follows:

dqGa ¼ sGa F Ga
kGa qGa
vG ðqGa ; qN Þ for Ga; dt

ð3:20Þ

dqN ¼ sN F N
mkN qm N
vG ðqGa ; qN Þ for N; dt

ð3:21Þ

and

where s is the precursor sticking coefficient, m is the kinetic order of desorption, and vG is the growth rate. The Ga and N precursors decompose to atomic species (as discussed in Section 3.4.1.3.1 and SGa ¼ 1. For the NH3 sticking coefficient, SN, we assume that NH3 does not stick to surface sites already occupied by N. The value of SN will therefore depend on both yN and the NH3 decomposition mechanism. If two adjacent sites are necessary for the sticking and dissociation, then SN ¼ (1
yN)2; however, if only one site is required for NH3 adsorption, which is followed by

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dissociation, then SN ¼ (1
yN). The growth rate, vG, is then given by vG ðqGa ; qN Þ ¼ g GaN qGa qN
dGaN ð1
qN Þ;

ð3:22Þ

where g is the incorporation rate of the surface atoms into the crystal and d is the decomposition rate of atoms from the crystal to the surface (see Section 3.4.1.3.1). The incorporation rate (the first term on the right-hand side) depends on the thermal activation of the adatoms to overcome the kinetic barriers for bonding to step edges and surface diffusion. Because both the incorporation rate and the decomposition rate depend on combinations of activation barriers (kinetic factors) [231], both increase exponentially with temperature. For growth, the incorporation rate, the gGaNyGayN term, is slightly larger than the decomposition rate, the dGaN ð1
qN Þ term. For GaAs growth, the incorporation rate is maximum near yGa ¼ yN with yGa and yN both less than 0.25 ML [231]. Above the temperature where the preceding conditions apply, the desorption rate approaches the incorporation rate, resulting in a rapid decrease in vG. Clearly, having these two rates as large as possible will lead to improved ordering of the lattice, resulting in a near equilibrium growth. Under steady state, the change in surface coverage (left sides of Equations 3.20 and 3.21) is zero, which leads to sGa F Ga ¼ kGa qGa

and

sN F N ¼ mkN ðqN Þm :

ð3:23Þ

To achieve the condition where yGa ¼ yN, the ratio of incident flux and desorption flux for the two components must be equal, or sGa F Ga =kGa ¼ ðsN F N =mkN Þ1=m :

ð3:24Þ

As discussed in Section 3.4.1.3.2, N2 desorption from GaN has been observed to be a first-order process, which means that m ¼ 1. Assuming that SGa ¼ 1 and near equilibrium, and yN is small, sN ¼ ð1
qN Þx  1 results. Using these relations, Equation 3.24 becomes, F N =kN ¼ F Ga =kGa

or after rearranging

F N =F Ga ¼ kN =kGa :

ð3:25Þ

Equation 3.25 implies that for near equilibrium growth where yGa ¼ yN, the ratio of input fluxes, FN/FGa, must match the ratio of the desorption fluxes, kN/kGa. Because the N surface diffusion rate is nearly zero (see Section 3.4.2 for details regarding growth under N-rich conditions), the N incorporation rate will depend on the number of times N atoms are adsorbed and desorbed on the surface. For growth to take place, vG > 0, g GaN qGa qN > dGaN ð1
qN Þ, kN > kG, and FGa > kGa, which lead to F N =F Ga > kN =kGa :

ð3:26Þ

Because kN/kGa depends exponentially on the temperature, the choice of FN/FGa also depends exponentially on the temperature. 3.4.1.3.6 Comparison of Model with Growth Conditions for Surface Morphology It is beneficial to relate values of these terms to various growth processes. For MBE growth of GaN, growth temperature, TG ¼ 600–800  C, and the films are grown in either

3.4 Nitride Growth Techniques

Ga-rich conditions, that is,FN/FGa < 1,or nitrogen-rich conditions, that is, FN/FGa about 1–10. For HVPE and OMVPE growth, TG ¼ 800–1100  C and FN/FGa ranges from 100 to 10 000, depending on the growth temperature. It is also useful to relate the predicted condition for optimized growth, that is, FN/FGa ¼ kN/kGa to the growth temperature, TG, V/III ratio, and the eventually resultant material quality. Data for this comparison were taken from the literature from several sources to determine the applicability of the growth model forwarded [221]. Generally, the data support the model and show how the V/III ratio and TG must be selected for the growth of high-quality GaN. For OMVPE growth on sapphire with a nucleation layer, if the V/III ratio is too low or TG is too high, both of which lead to the condition FN/FGa < kN/kGa, Ga droplets form to produce a rough surface morphology. Conversely, when FN/FGa > kN/kGa or quite N rich, the film morphology is smooth. An example of the change from smooth to rough morphology as affected by the V/III ratio for a constant growth temperature is shown in Figure 3.45a, where GaN was grown at a fixed TG using different V/III ratios. The filled circles denote GaN films with a smooth morphology, while the open circles denote a film with a rough morphology. Pictures of the two surface morphologies are also shown in Figure 3.45a, along with the kN/kG line. Note that the kN/kGa line properly depicts growth conditions that result in either smooth, that is, FN/FGa > kN/kG, or rough, that is, FN/FGa < kN/kGa, surface morphology. To further illustrate this, the values of the TG and V/III ratios used by many researchers are plotted in Figure 3.45b for GaN grown on either AlN or GaN nucleation layers. Again the open squares denote growth conditions leading to rough morphology, whereas the filled circles denote growth conditions where either smooth surface morphology was obtained or optimized films, judged by electronic properties, were grown. For each of the data points, the V/III ratio and TG were assumed to be reported correctly. Note that all but three of the filled circles are above the kN/kGa line and that all of the open circles fall below the kN/kGa line. Knowledge of the growth process would pave the way for optimizing growth conditions to obtain lower free carrier concentrations and higher mobilities, narrower XRD peaks, and intense photoluminescence (PL) by Koleske et al. [221]. Optimized growth temperature values and V/III ratios taken from five different references are shown in Figure 3.46. For example, the growth conditions leading to optimum electron mobility (500 cm2 V
1 s
1) occur at TG ¼ 1040  C and a V/III ratio of 1590. This condition is very close to the kN/kGa line shown in Figure 3.46. For minimum symmetric X-ray linewidth, the conditions TG ¼ 1025  C and a V/III ratio ¼ 1240 were used, which is slightly above the kN/kGa line. However, asymmetric diffraction linewidth must be observed to draw conclusions about the quality, as this diffraction is sensitive to in-plane extended defects. If the ratio of the band edge emission and yellow emission is used as a quality indicator in terms of radiative recombination, TG ¼ 1080  C and a V/III ratio of 5130 conditions had the best figures. This too is very close to the kN/kGa line. However, the ratio monitored strongly depends on the excitation intensity and further extrapolations based on this ratio alone should be handled with caution. The kinetic model presented above brings to the foreground the importance of the growth parameters. Specifically, the control of V/III ratio and TG is essential for the

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Figure 3.45 (a) Values of the V/III ratios for a growth temperature of above 1040 C. The inset pictures represent the surface morphology for the growth conditions plotted. The filled circles denote smooth morphology and the open circle denotes a rough morphology. The line is the ratio of kN to kGa. (b) Values of the growth temperature

and V/III ratio for smooth surface morphology films or a film grown with optimized growth temperature and V/III ratio (filled circles) and rough morphology films (open squares). Data were taken from 39 sources, which are listed in Ref. [221] where the details can be found. The line is the ratio of kN to kGa. Courtesy of D. Koleske.

reproducible growth of optimized material. Available mass flow controllers (MFC) used in OMVPE systems control flows to within 0.5–1%, which translates to a V/III ratio control to within 2%. The temperature control near 1000–1100  C is more challenging, and there is a greater uncertainty in both setting TG and measuring it. Temperature nonuniformities across the substrate are common; especially for larger substrates despite great strides made in uniform heat coupling and substrate rotation. This temperature nonuniformity leads to nonuniformities in the composition and the physical properties of alloys in the following manner. For growth at 1000  C, a 1  C temperature difference shifts the kN/kGa ratio by 15 out of a total kN/ kGa of 611, which represents a 2.5% change, and at 1100  C, the kN/kGa shifts by 123 out of the total kN/kGa of 5880, representing a 2.1% change. If the wafer center is 10  C

3.4 Nitride Growth Techniques

V/III ratio

104

Conditions for optimized GaN growth

103

k N/k Ga line

102

900

1000

1100

Temperature (ºC) Figure 3.46 Optimized growth temperature and V/III ratio values taken from five different references (see Ref. [221] for details). The kN/kGa line is also plotted. Note the good correlation between the optimized growth conditions and the kN/kGa line.

cooler than the edge for TG ¼ 1040  C, the kN/kGa ratio will be 25% larger at the edge than at the center. These examples demonstrate that to have uniform material properties over the entire wafer, the TG needs to be accurately and reproducibly controlled [221]. Before we delve into the world of MBE, which is next, a discussion for historical content is provided. In spite of the early assumption that CVD process using ammonia led to N-deficient layers and plasma-activated N would lead to better films, further research and ensuing high-quality layers proved this hypothesis to be invalid. For the sake of completeness, a few words are shared here on the topic. Although trialkyl compounds (TMA, TMG, TMI, etc.) are pyrophoric and extremely water and oxygen sensitive, and ammonia is highly corrosive, much of the best material grown today is produced by conventional OMVPE by reacting these trialkyl compounds with NH3 at substrate temperatures in the vicinity of 1000  C [196,232–238]. Temperatures in excess of 800  C are required to obtain single crystalline high-quality GaN films, and the GaN films with the best electrical and optical properties are grown at or around 1050  C. At substrate temperatures exceeding 1100  C, the dissociation of GaN results in voids in the grown layer. A similar situation is observed also for AlN film growth. 3.4.2 Molecular Beam Epitaxy

In this section, a succinct overall view of growth by MBE is given followed by an indepth discussion of physical processes that take place in MBE growth environments. In MBE technique, thin films are formed in vacuum on a heated substrate through various reactions between thermal molecular beams of the constituent elements and

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the surface species on the substrate [239]. The composition of the epilayer and its doping level depend on the arrival rates of the constituent elements and dopants [240]. Therefore, MBE growth is carried out under conditions governed primarily by the kinetics, rather than by mass transfer [241]. A thorough understanding of the growth kinetics, especially the surface processes of growth, is therefore critical. MBE is an extremely versatile technique for preparing thin semiconductor heterostructures owing to the control over the growth parameters that it offers, and the inherent in-situ monitoring capability. As mentioned earlier, thin films are formed on a heated substrate through various reactions between thermal molecular beams (atomic beams in the case of RF-activated nitrogen) of the constituent elements participated by the surface species on the substrate originating from the substrate itself by surface or bulk contamination. The composition of the epilayer and its doping level depend on the arrival rates of the constituent elements and dopants, respectively. The typical growth rate of 1 mm h
1, or slightly more than one monolayer per second (ML s
1), is sufficiently low to allow for surface migration of the impinging species on the surface. In the case of growth along the h1 1 1i for cubic and c-directions for wurtzitic systems, one monolayer constitutes a bilayer. As MBE growth occurs under conditions that are governed primarily by the kinetics, rather than by mass transfer, this allows the preparation of many different structures that are otherwise not possible to attain. This is in contrast to OMVPE of conventional compound semiconductors, such as GaAs. However, in the case of GaN, even the OMVPE has elements of kinetics [242]. In nitride growth by MBE, the metal species are provided by Ga, In, and Al metal sources, the dopants are provided by pure Si for n-type and Mg for p-type using conventional Knudsen effusion cells that are heated to sufficient temperatures for the desired growth rate, composition, and doping levels. On the contrary, nitrogen gas is one of the least reactive gases because of its large molecular cohesive energy (946.04 kJ mol ¼ 9.8 eV per N2 molecule). Because of the triple bond between the two nitrogen atoms, dissociation of one molecule into two reactive nitrogen atoms requires a large amount of energy, which cannot be provided by thermal means. In a plasma environment, however, and at reduced pressures, a significant dissociation of the nitrogen molecules takes place. Atomic nitrogen is reactive even at room temperature and bonds with many metals. Consequently, group III nitrides can be grown by plasma-assisted molecular beam epitaxy, where the plasma-induced fragmentation of nitrogen molecules is combined with the evaporation of metal atoms from effusion cells. In this vein, MBE growth of GaN has been reported by electron cyclotron resonance microwave plasma assisted molecular beam epitaxy (ECR-MBE). Several laboratories in the past have attempted RMBE growth in which N2 or NH3 was decomposed on the substrate surface [243–247]. The sequence of processes taking place during growth by MBE is adsorption, desorption, surface diffusion, incorporation and decomposition. All of these processes are in effect in many ways compete with each other during growth by MBE. Adsorption can be summed as the atoms or molecules impinging on the substrate surface and sticking by overcoming an activation barrier. Desorption, on the contrary, is a process in which the species that are not incorporated into the crystal lattice leave the substrate surface by thermal vibration. During surface diffusion, imperative for

3.4 Nitride Growth Techniques

Impinging atoms

Lattice site

Desorption

Surface diffusion

Adsorption Surface diffusion Decomposition

Substrate, T s Overgrowth Incorporation into lattice

Surface nucleation (aggregation)

Interdiffusion

Figure 3.47 Surface processes during the MBE growth: adsorption, desorption, surface diffusion, lattice incorporation, and decomposition.

growth, the constituent atoms or molecules diffuse on the substrate surface to find the low-energy crystal sites for incorporation. During the incorporation phase, the constituent atoms or molecules enter the crystal lattice of the substrate or the epilayer already grown by attaching to a dangling bond, vacancy, step edge, and so on. Owing to high temperatures involved, albeit much lower than those employed in OMVPE, during decomposition, the atoms in the crystal lattice leave the surface by breaking the bond. These processes are schematically shown in Figure 3.47. Let us discuss these five processes that govern growth by MBE. 3.4.2.1 Adsorption Impinging gas molecules (atoms) condense on the surface and, depending on the strength of the interaction between the adsorbate and the surface, can be adsorped by physisorption (weak) or chemisorption (strong) [248]. Physisorption represents weak adsorbate–surface interaction because of van der Waals forces with typical binding energies on the order of 10–100 meV. Owing to the weak interaction, physisorbed atoms or molecules do not disturb the structural environment near the adsorption sites to a significant degree. In chemisorption, on the contrary, an adsorbate forms strong chemical bonds with the substrate atoms with typical binding energies on the order of 1–10 eV, thus changing the adsorbate’s chemical state.

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The adsorbate coverage characterizes the surface concentration of adsorbed species expressed in monolayer units. The coverage is a relative value associated with a given substrate. It can be converted to an absolute surface density of atoms. Considering the kinetic approach in the case of a uniform solid surface exposed to an adsorbing gas, the adsorption rate is defined as [248] p r a ¼ sf ðQÞexpð
E act =kB T s Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pmkB T s

ð3:27Þ

where p is the partial pressure of the adsorbing gas; s is the condensation coefficient responsible for the effects of the orientation and the energy accommodation of the adsorbed molecules; f(Y) is a coverage-dependent function that describes the probability of finding an adsorption site; exp(
Eact/kBT) is the temperature-dependent Boltzmann term associated with the energetics of the activated adsorption. Finite equilibrium Ga adlayer coverage has been reported for typical substrate temperatures and Ga fluxes [249]. For large Ga fluxes, up to 2.5  0.2 monolayers of Ga are adsorbed on the GaN surface. For higher Ga fluxes, Ga droplets are formed [250]. At typical growth temperatures used in MBE, Ga adlayer does not condense into a reconstruction but behaves like a liquid film [251]. The reported height of the Ga adlayer is about 3.88 Å, as measured by scanning tunneling microscopy (STM), which corresponds to 1.9 ML [252]. For lower Ga fluxes, a discontinuous transition to Ga monolayer equilibrium coverage is found, followed by a continuous decrease toward zero coverage. The practical implications of this process in terms of GaN growth are discussed in Section 3.4.1.3.1. 3.4.2.2 Desorption Desorption is a process in which the adsorbate species gain sufficient thermal energy to escape from the adsorption well and leave the surface. The probability of desorption depends on the bonding strength of the particular atom to the surface. Bonding energies are different for different materials and the strength of a bond is expressed in terms of the amount of energy needed to break it. In the kinetic approach, desorption is described in terms of a desorption rate, rdes, which represents the number of species desorbed from unit surface area per unit time, and can be expressed as [248]

r des ¼ s f ðQÞexpð
E des =kB T s Þ;

ð3:28Þ

where f (Y) describes the coverage dependence and s is the desorption coefficient standing for steric and surface mobility factors. For desorption to occur, the adsorbed species must overcome a barrier called the desorption energy, Edes. In case of activated chemisorption, the desorption energy is the sum of the binding energy in the chemisorbed state and the activation energy for adsorption, Edes ¼ Eads þ Eact. In case of nonactivated chemisorption, the desorption energy is simply the binding energy in the chemisorbed state, Edes ¼ Eads. The lifetime of an adsorbate as a function of temperature is needed for studying the desorption energy. The lifetime t is defined as the average time spent by the adsorbate on the surface marked from the time of adsorption to the time of desorption and

3.4 Nitride Growth Techniques

obeys an Arrhenius dependence in the form [250]:   E des t ¼ t0 exp : kB T s

ð3:29Þ

Equations 3.28 and 3.29 describe the desorption process to be quite sensitive to temperature. For low Ts, the lifetime of adsorbates is sufficient to point that the desorption process can be neglected. For intermediate temperatures, the growth rate is determined through the competing processes of adsorption and desorption. However, for sufficiently high Ts, the desorption rate can be greater than the deposition rate, in which case evaporation rather than deposition would take place. Specific to the case of GaN, Ga desorption has been investigated by mass spectrometry [250,253] and RHEED [254,255] with consistency lacking. The measured activation energies reported for Ga desorption are in the range of 0.4–5.1 eV [250–255]. For N, which is usually desorbed as a dimer (N2), the desorption rate is limited by the surface diffusion of two N atoms. Some reports indicate this to be a first-order process [256], meaning that the surface diffusion may play less of a role in the desorption mechanism than the surface structure. Another plausible mechanism is that once an N atom diffuses near another N atom, the N2 molecule would immediately form owing in part to the large N–N binding energy and desorbs from the surface because of the highly exothermic nature of the N2 formation [257]. The practical implications of these processes in terms of GaN growth are discussed in Section 3.4.1.3.1. 3.4.2.3 Surface Diffusion As mentioned earlier, surface diffusion describes the motion of adsorbates (atoms or molecules) on the substrate (film) surface. The motion of adsorbates that become mobile because of thermal activation is described as a random walk. When a concentration gradient is present, this random walk motion of many particles results in their net diffusion being opposite to the gradient direction in a macroscopic sense. The microscopic view of surface diffusion is an activated process that is also affected by factors such as interaction between diffusing adsorbates, formation of surface phases, and defects. For an atom on the surface to diffuse to the next lattice position, it must overcome the lattice potential between the two neighboring positions. This activation energy required for diffusion, E0, is the microscopic origin of the lattice potential. The average length of diffusion, ls, in a unit time interval is a very important parameter to characterize the diffusion process and has an exponential temperature dependence given by Equation 3.19, called the Einstein equation [248]. At high growth temperatures where the surface diffusion length is larger than the terrace width, atoms move by surface diffusion from the terrace to the step edge to be incorporated into the growing lattice, which is called the step-flow growth. As the diffusion length decreases, for example, by reduced temperature, atoms meet and nucleate new islands on the terraces before reaching the edge of an existing island or a step edge. Moreover, islands can nucleate on top of the existing islands, a process that leads to rough surface formation and 3D growth. In the intermediate region, the

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(0 0 0 1)

Top view (c-plane)

(0 0 0 1)

hcp

Diffusion path

Bridge On top

N layer

Ga2L fcc Ga1L

Ga layer

Side view (a-plane)

Ga1L N Ga2L (a)

(b)

Figure 3.48 Schematic diagram of diffusion paths on the GaN (0 0 0 1) (a) and (0 0 0 1) (b) surfaces. The side views as in the a-plane for each are also shown [256].

islands do form on terraces, but the diffusion length is sufficiently long for adsorbates to diffuse to the edges of these islands and incorporate there, which leads to smooth surface also. In this mode of growth, the RHEED intensity oscillations are observed in that the intensity is maximum when the islands expand to cover the terrace and the intensity is minimum when the coverage is 50%. Specific to the case of GaN, diffusivity for Ga and N adatoms on GaN surface is different. Zywietz et al. [256] calculated the surface potential energy for Ga and N adatoms on GaN(0 0 0 1) surface. The simulations led to the presence of two transition sites, as shown in Figure 3.48. For Ga diffusion, the lower energy transition site is the bridge position (0.4 eV) and the higher energy site is the “on-top” position (>3 eV). However, for N adatoms, the barrier for bridge diffusion is 1.4 eV and the barrier for the on-top position is similar to Ga. Significantly lower diffusion barrier for Ga relative to N results in Ga being very mobile at typical growth temperatures, whereas the diffusion of N is slower by orders of magnitude. Further, the presence of excess N strongly increases the Ga diffusion barrier from 0.4 to 1.8 eV. The very divergent surface mobilities of Ga and N adatoms have serious consequences. In the Ga-rich regime, the Ga adatoms are highly mobile and a step-flow mode results in 2D growth. Furthermore, if excess Ga adatoms are present on the surface, N adatoms can be efficiently incorporated because the probability of

3.4 Nitride Growth Techniques

fast-moving Ga adatoms capturing N atoms is high. The presence of a Ga bilayer on the surface in Ga-rich growth conditions reduces the lateral diffusion barrier from about 1.3 to about 0.5 eV, paving the way for growth of smooth layers at low temperatures [258]. For details on the Ga-bilayer issue, refer to Section 3.5.6. However, N-rich growth conditions show roughly a five times higher diffusion barrier for Ga. A rough surface can, therefore, be expected for N-rich conditions, consistent with experimental observations. 3.4.2.4 Incorporation In the incorporation process, the molecules or atoms bond to the crystal through various reactions between the constituent elements and the surface species on the substrate. This process is controlled by the interplay of thermodynamics and kinetics. The general trends in film growth are understood within the thermodynamic approach in terms of the relative surface and interface energies. However, film growth by MBE is a nonequilibrium kinetic process in which rate-limiting steps affect the growth mode [241]. There are three modes of growth/incorporation that come to bear in general, namely, Frank–van der Merve (FM), Stranski–Krastanov (SK), and Volmer–Weber (VW). These three processes with graphics are discussed in Section 3.5.16, which pertains to quantum dots. Suffice it to say, the Frank–van der Merve mode is a layer-by-layer process. Each layer is fully completed before the next layer starts to grow, and it is strictly a two-dimensional growth mode. The Volmer–Weber mode is an island growth mode. Three-dimensional islands nucleate and grow directly on the substrate surface, which is typically seen in metals unless very low – even below room temperature – deposition conditions are employed. The Stranski–Krastanov mode is a layer-plus-island growth mode and represents the intermediate case between the FM and VW growth modes. After the formation of a complete two-dimensional layer of a few monolayer thickness (the exact value of which depends on the local strain), the growth of a three-dimensional layer (islands) takes place. The occurrences of growth have their genesis in the competition between the surface desorption and the surface diffusion. Because the desorption and diffusion processes are noticeably affected by the deposition rate, the surface condition and temperature, the growth mode, and the epilayer surface can be controlled by choosing a proper III/V ratio and a substrate temperature. Several studies [259,260] have revealed that not all incident Ga atoms are incorporated into the growing GaN at the usual growth temperature in PAMBE (650–750  C), even when an excess of N flux is present (III/V flux ratio Gc, which paves the way for flattening. In the case of Gs/Gc  2, coalescence boundaries are formed together with voids. The red lines represent the behavior of threading dislocations up to the coalescence boundaries. Figure 3.120 is a typical image of a 2S-ELO sample but at the end of the first step, viewed in cross section along the ½1 0 1 0 zone axis. The second step that is centered around the lateral growth was induced by the introduction of (CH3Cp)2Mg in the vapor phase [604]. The resulting microstructure is similar to the case when the lateral growth is induced by the much higher growth temperature 1120  C. Full coalescence and smoothing of the surface required 1.5 h, resulting in a thickness of 12 mm. Once full coalescence is achieved, the behavior of the dislocations is different than in

Converging (1) followed by diverging trajectories of (0 0 0 1) facet edges

(2)

(1)

Mask GaN template Figure 3.119 Schematic representation of the two stages of the two-step (2S-)-ELO process during the second ELO step performed either at relatively higher temperature or at typical growth temperatures but with introduction of (MeCp)2Mg. The evolution of growth is such that

for GS > GC flattening occurs and for GS/GC  2, coalescence boundaries are formed together with voids. The gray lines represent the threading dislocations up to the coalescence boundaries. Courtesy of P. Gibart and Ref. [575].

3.5 The Art and Technology of Growth of Nitrides

Figure 3.120 (a) Cross section along the ½1 0 1 0 zone axis of a 2SELO film at the end of the first step. Dashed lines join the dislocation bending points. (b) Schematic representation of 2SELO. Dotted lines represent the shape of the ELO material at different stages of the first step of the 2S-ELO process. Broken black lines join the successive edges of the top C facet. Solid black lines represent dislocations. Courtesy of P. Gibart and Ref. [575].

standard one-step ELO. Specifically, the dislocations under the masks are also blocked when they encountered the mask. A close examination of the dislocation propagation behavior shows that the dislocations above the window first propagate vertically, as in the standard ELO. However, they afterwards bend by 90 to assume a direction in the (0 0 0 1) basal plane. It should be mentioned that this behavior is observed regardless of whether the type of the dislocations are, edge, screw, or mixed (a, c, or a þ c. refer to Chapter 4 for description of each of the three dislocations). Some of the dislocations adopt an intermediate inclined direction before bending again to assume a direction lying in the basal plane. Broken lines in Figure 3.120 indicate the points at which the dislocations bend. The dislocations close to the edge of the mask bend first. A

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vertical growth rate higher than the lateral growth rate leads to intermediate overgrowth whose material shapes are represented by dotted lines in Figure 3.120. As soon as a vertical dislocation line gets sufficiently close to the edge between the top and inclined facets, it bends and adopts a new direction in the basal plane. This general picture can be best understood in the frame of image dislocation and image forces (in much the same way as charge and image charge), as already proposed [616] and treated in books [670]. A dislocation generates a stress field, which should be compensated when the dislocation line gets sufficiently close to surfaces for it to leave the free surfaces stress free. An image dislocation is a virtual dislocation located outside the material, which generates a stress field compensating at the free surface the stress field of the actual inner dislocation. These image dislocations induce attractive forces acting on inner dislocations. In the experiments of Gibart et al. [575], owing to symmetry considerations, the image force has no effect on the dislocations when TDs are perpendicular to the (0 0 0 1) basal surface without lateral facets within the interaction range. However, the dislocation lines may lie in the basal plane when the threading dislocations are close enough to an inclined facet. In fact, during the first step of 2S-ELO, the lateral inclined facet progressively moves toward the dislocation lines, reaching first to the dislocation lines that are closest to the edge of the mask followed by reaching to the dislocations situated just in the center of the window. The dislocations situated in the middle of the window may not bend and propagate vertically to the surface because interactions with both the free surfaces on the left and right, ð1 2 1 2Þ and ð1 2 1 2Þ, may compensate each other. The 90 bending of dislocations could be the result of the general principle of minimization of the free enthalpy of the system. As such during growth, dislocations follow a direction leading to minimum enthalpy. As the line energy of a dislocation depends also on its character, the energy of a screw being the lowest, bending at 90 of an edge eventually produces a screw dislocation, or introduces a screw component, thus lowering the enthalpy of the system. Bending of dislocations results in a discernible reduction of their density in the upper part of the film provided that the thickness is greater than the height of the pyramids formed at the end of the first growth step. After bending, most of the dislocations have a line parallel to ½1  2 1 0, which extends to the coalescence boundary with the overgrown GaN advancing from the adjacent stripe. Defects therefore accumulate at the boundaries. Dislocations at the interfaces manifest themselves in several ways as has been reviewed by Vennegues et al. [666]. Voids are clearly visible over the mask, as discussed in the next paragraph, and have a basal triangular shape with an extension over their top apex. Most of the lateral dislocations are blocked when they encounter such a void and therefore do not extend further. Other possible manifestations of these lateral dislocations when they reach the coalescence boundary have been shown to bend down to the void resulting in their termination therein, or bend up in the boundary and thread up to the surface. Single ELO GaN layers with a thickness of 2 mm were obtained using 3 mm wide stripe openings with 10 mm pitches and oriented along the h1  1 0 0i direction (Figure 3.121a). The growth parameters were temperature of 1100  C and a TEG flow rate of 26 mmol min
1. Atomic force microscopy showed that the surfaces of the

3.5 The Art and Technology of Growth of Nitrides

Figure 3.121 Cross-sectional and surface SEM micrographs of the first, (a) and (b), and second, (c) and (d), coalesced GaN layers, respectively, grown on 3 mm wide and 7 mm spaced stripe openings oriented along < 1 1 0 0>. Courtesy of R. Davis.

ELO GaN had a terraced structure with an average step height of 0.32 nm. As in the single ELO, one must deal with the voids formed in two-step ELO at the coalescence interface. If the voids were to form in the dislocated regions and cause the dislocations to move laterally owing to local strain modification, they would be beneficial. However, at the coalesced region, their presence point to problems with proper indexing of the lattice. As an example of these undesirable voids, each black spot in the overgrown double ELO GaN layers shown in Figure 3.121a and c is a subsurface void that forms when two growth fronts coalesce. These voids were most often observed using the lateral growth conditions wherein rectangular stripes having vertical f1 1 2 0g side facets developed. The morphologies of the finished surfaces of single and double ELO layers imaged by SEM are shown in Figure 3.121b and d. Surface morphology of the second overgrown layer was comparable to the first layer. Cracks were occasionally observed along the coalesced interface under selected growth conditions, probably owing to the thermal mismatch between the GaN layers and the SiO2 mask. The cross-sectional TEM image of Figure 3.122 shows a typical laterally overgrown GaN. Threading dislocations, originating from the GaN/AlN buffer layer interface, propagate to the top surface of the regrown GaN layer within the window regions of the mask. The dislocation density within these regions, calculated from the plan view TEM micrograph is approximately 109 cm
2. By contrast, however, additional microstructural studies of the ELO regions showed much fewer dislocations. Cross-sectional TEM observation of the double ELO sample in the micrograph presented in Figure 3.123 shows that a very low density of dislocations parallel to the (0 0 0 1) plane, formed via bending of threading dislocation, exists in the first and second ELO GaN layers on the SiO2 masks. The second SiO2 mask is slightly misaligned relative to the first. These results suggest that very low defect density GaN layers can be fabricated by precise alignment of the mask in the second lithographic process.

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Figure 3.122 Cross-sectional TEM micrograph of a section of a laterally overgrown epitaxial GaN layer on a SiO2 mask region. Courtesy of R. Davis.

The dislocation densities in ELO GaN are low enough so that questionable figures could be arrived at by TEM, which, in this context, led to report of figures [599] in the range of 104–10 5 cm
2, which are very optimistic. The etching methods reported in Section 3.5.1.2 dealing with HVPE samples, and also in Section 4.2.4 are much more reliable, although they too could be incorrect unless extreme care is taken and many calibration samples are utilized. Ultimately, when sufficient progress is made, electrical and optical properties and to some extent electrically active point defect density would take the place of extended defect density. In this particular report, the work was done with stripes oriented in the ½1  1 0 0 direction to yield a large lateral growth rate/vertical growth rate ratio. The stripe spacing was varied to give ratios of open width and patterned period of 0.1–0.5. The ELO GaN was bound by the (0 0 0 1) facet on top and by vertical ½1 1 2 0 sidewalls on the edges, which showed a lateral growth rate of up to 6 mm h
1. Patterns with 10 mm stripes and a ratio of open width and patterned period of 0.5 enabled full coalescence of the overgrown GaN film after 90 min of growth. TEM and AFM (counting the pit density where the bilayer steps are terminated/broken) observations indicated that the density of mixed character dislocations reaching the surface of the ELO GaN and leaving a dark spot in the AFM image is in the 104–105 cm
2 range. To draw the contrast between the overgrown region and the region directly over the template GaN, a cross-sectional TEM image and a plan view TEM image around the vicinity of a stripe are shown in Figures 3.122 and 3.124, respectively. Clearly, the ELO process is effective in reducing the extended defect density. As in the case of one-step ELO, CL has been used copiously to image the radiative recombination on and near the surface. Dislocations with a screw component are expected to cause deep states in the gap and act as nonradiative recombination centers in GaN and therefore lead to contrast in CL images. Assuming a one-to-one

3.5 The Art and Technology of Growth of Nitrides

Figure 3.123 Cross-sectional TEM micrograph of a section of the second lateral grown GaN layers. Courtesy of R. Davis.

correspondence between the CL dark spots and the threading dislocations, the difference in CL images taken at 90 K of one-step and two-step ELOs is striking [575], as illustrated in Figure 3.125. Figure 3.125a shows the top view monochromatic CL mapping at l ¼ 358 nm, which corresponds to the free exciton A of a 2S-ELO GaN sample. Figure 3.125b shows a 1S-ELO GaN sample. To make the comparison more meaningful, the conditions used to grow the top GaN, which is what is probed with CL, have been kept identical. For both samples, full coalescence was achieved and no topography contrast was observed in secondary electron images. Therefore, stark differences noted in the CL images of Figure 3.125 are attributable only to the nature of two-step and one-step ELO. In the images, the dislocations pointing up appear as

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Figure 3.124 Plan view TEM micrograph of a section in the vicinity of the lateral grown GaN and that grown on the GaN template below. Courtesy of R. Davis.

black points, and it can be seen that they are organized along the lines parallel to the stripes, consistent with dislocation bending and pooling in the coalescence plane arguments. Lines with a high density of dislocations (white arrows) alternate with lines with a lower density of dislocations (black arrows) and are separated by 5 mm, which corresponds to half the stripe period. The highly defective and lightly defective lines correspond to the coalescence boundaries and the center of the windows, respectively. Filtering of extended defects through dislocation bending is therefore not completely effective and dislocations located just at the center of the window, which meet the apex of the pyramid at the end of the first stage of the 2S-ELO, may not bend and may propagate up to the surface. Getting back to the CL image, a few dislocations are present between the lines as well. The density of defects average over the entire surface as determined by CL imaging is 1.7 · 107 cm
2. If only the areas between the coalescence boundaries, spaced 10 mm apart, are considered, the density is 107 cm
2. The two-step process reduces the threading dislocation density by at least one order of magnitude compared with the standard non-ELO-grown GaN layers.

Figure 3.125 CL map of GaN layers grown with the two-step (a), and one-step ELO processes (b). The scale bar is 20 mm. Courtesy of P. Gibart and Ref. [575].

3.5 The Art and Technology of Growth of Nitrides

Figure 3.126 UV CL monochromatic image recorded at 90 K on a bevel edge of a 2S-ELO GaN. The luminescent triangles correspond to laterally overgrown material whereas the dark triangle corresponds to the coherent part. Coalescence boundaries and TDs are running parallel to the c-plane after bending. In the top surface, the dark spot corresponds to merging dislocations. Courtesy of P. Gibart and Ref. [575].

To get an insight into the evolution, as layer growth progresses, of dislocations, cross-sectional CL experiments in a two-step ELO sample have been undertaken [671]. Images displayed in Figure 3.126 are for coalescence using (CH3Cp)2Mg, but similar features are obtained when coalescence is performed by the use of increased substrate temperature. In addition to imaging, CL spectra in cross-sectional mode were also obtained for an in-depth analysis [671]. The spectra, which are not shown, indicate that the growth in the windows is defective as judged from a lack of near band edge emission and strong yellow line (YL) emission. At the lateral edges of the stripes (GaN grown on the striped windows) exhibited broad near band edge emission. In the region where the lateral epitaxy is enhanced by addition of (CH3Cp)2Mg, donor– acceptor pair band emission was observed. It should be pointed out that Si and O contamination coupled with Mg contamination from the mask and vapor lead to impurity incorporation, which could adversely affect the devices such as FETs. The unintentionally doped top layer exhibited sharp excitonic transitions with free and bound excitons visible with a line width of 0.7 eV, which degrades to 3–4 meV when ELO is not employed [575]. Even though the research effort on 2S-ELO in HVPE has been less intensive as compared to OMVPE, the 2S-ELO technology is achievable in HVPE, which is motivated by the potential of HVPE providing templates for further growth by other methods such as OMVPE and MBE. The basic mechanisms in effect for HVPE are, however, not completely identical to those for OMVPE and the experimental conditions for achieving the growth anisotropy are significantly different. There are also similarities in that the morphology is controlled by the composition of the carrier gas. Starting with windows aligned in the h1 1 0 0i direction and under N2 at 1050  C, the

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equilibrium shape of the overgrowth leads to trapezoidal stripes with (0 0 0 1) top faces and f1 1 2 2g lateral facets. When the carrier gas is switched to H2, the lateral growth rate is significantly reduced and triangular stripes with f1 1  2 2g lateral facets are obtained. This means that a mixture N2 : H2 carrier gas would lead to pyramidal nucleation islands during the initial states of growth. In a second step, pure N2 carrier gas, which enhances lateral growth, can be employed. With two-step ELO, complete coalescence without pits and planarization have been obtained, which are in Figure 3.127 as cross-sectional SEM images GaN layers grown with one-step and two-step ELOs. The pits present along the coalescence boundaries in one-step ELO are absent in the two-step ELO case. In addition, the two-step ELO process produces planar surfaces [672,673]. Figure 3.127c shows cross-sectional SEM image of a 60 mm 2S-ELO HVPE layer obtained by carrying out the first growth step with a mixture of H2 þ N2 as carrier gas (using 5 mm windows and 10 mm mask stripes): growth proceeds until the growth

Figure 3.127 Comparison of coalescence on onestep (a) and two-step (b) ELO performed by HVPE. Note that the one-step ELO leads to a high number of pits along the coalescence boundary while those pits are absent in the two-step ELO with the additional benefit of good planarization

obtained, (c) Cross-sectional SEM images of a 60 mm thick HVPE-grown GaN layer showing the initial coalescence followed by an enhanced lateral growth by switching to pure N2 carrier gas leading to planar growth (openings, 5 mm, period 15 mm) Courtesy of V. Wagner and M. Ilegems (Ref. [673]).

3.5 The Art and Technology of Growth of Nitrides

front from two adjacent f1 1 2 2g facets meet. In the second growth step, pure nitrogen carrier gas is used as to achieve full coalescence and a flat surface, see Figure 3.127. The nature of threading dislocations in HVPE two-step ELO is somewhat similar to that observed in 2S-ELO in OMVPE in that they first propagate from the template. They then bend 90 and propagate horizontally only to be terminated at the coalescence boundaries. In some cases, the 90 bending occurs in two steps as determined by TEM investigations. Once again the wavelength resolved crosssectional CL allows an in-depth evaluation of the evolution of the crystallographic quality and the density of incorporated impurities. The seed region and the part grown in the windows on the template show about the same optical and crystallographic quality as characterized by relatively narrow near bandgap emission. However, the laterally overgrown region exhibits a broad blue-shifted emission but with a high-integrated intensity, the origin of which may be associated with impurities. Three-Step Growth with ELO Template It has been demonstrated that several steps led to ELO GaN with wider usable surfaces, free of emerging TDs. Even better quality can be foreseen with more than two steps at the expense of higher complexity. Nagahama et al. [674] implemented a three-step ELO process in an effort to further reduce the threading dislocation density. The effort led to densities slightly above the mid 105 cm
2 point, and thereby allowed the fabrication of LDs lasting 15 000 h with an output power of 30 mW. In this 3S-ELO, a standard OMVPE ELO GaN is first grown as discussed. On top of this ELO GaN, a 200 mm thick HVPE layer is grown. The sapphire substrate is then removed by mechanical polishing to get a 150 mm freestanding GaN layer. Finally, in the last step, another ELO GaN is grown on top of this self-supported GaN, leading to a threading dislocation density of 7 · 105 cm
2 in the top layer on which the laser structure is built. 1S-ELO on 2S-ELO with the mask in the second step located exactly above the first mask (to eliminate coalescence ELO boundaries) further reduces the threading dislocation density [675]. The inverted version, the 2S-ELO on 1S-ELO produces similar results [676]. An AFM image of the top surface of a 3S-ELO GaN, where the threading dislocation density over the entire surface is 1 · 107 cm
2, showed very few threading dislocations between the coalescence boundaries. In another approach, starting from grooved stripe substrate as in Ref. [681–683]. Ishida et al. [677] employed two regrowths on periodically grooved structures and in the process reduced the density of threading dislocations down to 6.3 · 106 cm
2. Once again, the mechanism of dislocation reduction is bending. These improvements, however, come at the expense of complicated processes, which increase the cost of templates. Maskless Epitaxial Lateral Overgrowth As discussed, two epitaxial growth steps, one masking step and one photolithography step, are required in one-step ELO. A simpler ELO process that does not significantly degrade the final quality of the ELO GaN is of considerable interest. In a method for which the regrowth step is not necessary, the ELO process is initiated directly on the SiC or sapphire substrates patterned with SiNx mask [678]. The AlGaN nucleation layer is deposited, which wets the exposed substrate

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but not the silicon nitride mask, followed by GaN, which proceeds both vertically and laterally. AFM studies indicate that the threading dislocation densities are in the mid 108 cm
2 in the coherent region and 106 cm
2 in the ELO region. Further effort along these lines led to the use of textured sapphire substrates [679]. In this particular process, mesas are formed on the substrate by etching prior to just one growth sequence. Growth takes place laterally to produce GaN cantilevers that finally coalesce. Cantilever epitaxy (CE) results in a significant reduction of threading dislocations, with densities below 107 cm
2 above the support mesa. Threading dislocations in the edge region bend and further extend laterally into the cantilever. Compared to the conventional ELO, the dislocation density at the coalesced front and at the mesa is lower. To date, this method presents the limitation of the 1S-ELO. Despite a significant decrease in dislocations reached by CE, the wing tilt of approximately 0.8 was observed for CE of GaN on SiC and Si(1 1 1) [680]. With further improvements, this process may produce low dislocation GaN at a lower cost than the conventional ELO. An alternative method, which relies on periodically grooved substrates (sapphire or SiC) with stripes parallel to the h1 1 0 0i GaN direction, was proposed by Sano et al. [681–683]. On these grooved substrates, a fast vertical growth rate produces triangular stripes with f1 1 2 2g facets as in the ELO process that finally coalesce. Low dislocation density GaN was produced with this method, with nearly similar crystallographic quality to that produced by conventional ELO. The coalesced boundaries also produce high densities of threading dislocations. In a slightly modified version, Strittmatter [684] utilized textured Si(1 1 1) substrates with grooves parallel to Sih1 1 0i. In such a case, GaN grows on the bottom of the grooves and then the V/III ratio is increased to enhance the lateral growth rate and obtain full coalescence. As usual, the wing areas exhibit improved optical and crystallographic quality. An ELO process in which nucleation is performed on a sapphire substrate patterned with a SiO2 mask was proposed to further simplify the process, thereby restricting nucleation and further lateral growth within the windows. Direct lateral overgrowth occurs on these SiO2 patterned sapphire substrates. In spite of achieving ELO quality, the random nucleation does not allow smooth coalesced films [685]. 3.5.5.2.6 Point Defect Distribution in ELO Grown GaN In standard ELO, two regions are clearly identified: the overgrown region over the mask and the coherent growth in the windows. The structural data provided clearly show that ELO reduces the threading dislocation density as well as tilting of the c-axis and twisting of the cplane, with improvement being more dramatic in the wing regions as quantified by Xray measurements conducted in ELO GaN grown by both OMVPE and HVPE [686]. In OMVPE, the tilting and twisting strongly depend on the stripe orientation of the pattern (h1 1 2 0i or h1 1 0 0i), which is closely related to the difference in the growth process. However, the extended defect density is much lower in the wing regions than that over the seed template in the windows. Therefore, an assessment of point defects and impurities necessitate position-sensitive tools such as micro-Raman, micro-PL (m-PL) both CW and time-resolved [628], and cathodoluminescence (CL). The CL method is extensively used to map the luminescence efficiency in the layer and thus gain some insight into the spatial distribution of radiative recombination

3.5 The Art and Technology of Growth of Nitrides

efficiency. Higher efficiency is associated with less nonradiative defects and the converse is true for more nonradiative defects. For correlation to structural defects, it is also assumed that structural defects cause nonradiative recombination centers, the exact nature for which is complex and is discussed in Section 4.3. This can be done with emission at a given wavelength (monochromatic) or all wavelengths (panchromatic). In addition, excitation electron energy can be changed to affect the absorption depth, and so on. However, waveguiding of the emitted light as well as secondary excitations both in terms of electron-initiated excitation and photon-initiated excitation are among the processes that cause some uncertainty in the collected data. In one such investigation, the monochromatic CL image at 365 nm reveals that in the coherent region above the windows, the CL shows mottled luminescence contrast as usually observed in uniformly grown OMVPE GaN. On the contrary, the ELO material extending on both sides of the seed region exhibits fewer nonradiative regions, but the coalesced boundaries are characterized by dark CL lines [687]. The vertical and horizontal propagation of threading dislocations were investigated by Rosner et al. [688], who intercalated a GaInN single quantum well (SQW) grown with the ELO method and obtained a 421 nm CL mapping including depth analysis, showing that the lateral defects do not propagate into the upper part of the film. Stripes Along the h1 1 0 0i Direction in HVPE ELO For ELO GaN on AlN/6H-SiC with stripes along the h1 1 0 0i direction, CL imaging of uncoalesced stripes shed light on the threading dislocation propagation and impurity incorporation [689]. When the overgrown GaN has the form of trapezoidal cross section (such as that shown in Figure 3.108 region II), the fast vertical growth rate results in the incorporation of a high density of defects and produces a strong YL emission in the coherent region, whereas in the triangular part, a blue emission is clearly visible. In the region with rectangular cross section (such as that shown in Figure 3.108 region III), most of the YL emission originates from the region between the stripes. PL measurements show an additional band at 3.4643 eV, which are most likely linked to a donor incorporated in the ELO process, and a blue-shifted emission in relation to the underlying GaN. These observations show that the ELO GaN exhibits improved material quality and reduced biaxial strain, as judged by point defect activity as well. On GaN pyramidal structures partially coalesced along ½1 1 0 0 direction, spatially resolved CL demonstrates that the coalescence region exhibits stronger and more uniform luminescence than the pyramidal sidewalls [690]. Moreover, these results are consistent with Raman scattering data where a reduced linewidth and a slight shift of the E2 phonon in the ELO region are observed [689]. It should be mentioned that stress affects Raman frequencies in addition to defect generation and propagation, such as horizontal dislocations [691], which makes stress analysis an integral part of ELO. ELO GaN results from a growth process involving different materials (dielectric mask, substrate) with different lattice parameters and thermal coefficients and involves several thermal cycles. Therefore, such a system generates inhomogeneous stress. The magnitude and spatial distribution of stress have been modeled using finite-element analysis for the system GaN/AlN-6HSiC with SiO2 mask [692]. It was shown that the edge of ELO GaN in the vicinity of the

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Figure 3.128 SEM and CL wavelength image of a cross section of HVPE ELO sample. Courtesy of J. Christen and A.G. Hoffmann, Ref. [695]. (Please find a color version of this figure on the color tables.)

GaN/dielectric interface could be under localized compressive stress fields up to 3 GPa. The stress inhomogeneities are endemic to ELO no matter which growth method is employed. The stress analysis for HVPE ELO of GaN was carried out using simulations based on continuum elastic theory. The lattice mismatch a was adjusted to reproduce the biaxial stress in the GaN template layer and was taken to be zero for the amorphous SiO2 mask. The calculated biaxial stress, sxx, distribution agrees very well with the experimental values deduced from the m-Raman data. The ELO GaN is under biaxially compressing stress; sxx relaxes rapidly through the window to
0.32 GPa and to
0.45 GPa when stripes are along the h1 1  2 0i and h1  1 0 0i directions, respectively. Along the mask, the ELO GaN is also under biaxial stress [693]. Other efforts of stress analysis in the pyramids led to the deduction of compressive stress in the GaN buffer template layer, tensile stress in facets, and full relaxation in the mid to upper parts of the pyramids [694]. Kaschner et al. [695] investigated ELO with stripes along the h1  1 0 0i direction with cross-sectional CL and m-Raman. Cross-sectional CL mapping and SEM images are shown in Figure 3.128. The coherent region grown in the windows exhibits a monochromatically uniform rectangular region dominated by CL emission at 3.467 eV. The CL spectra of this coherent part exhibits sharp D0X recombination at 3.467 eV. As can be deduced from Figure 3.128, the overgrown region is red shifted, which is similar to the case with the stripes along h1 1  2 0i, which will be discussed below. It is postulated that this red shift results from incorporation of impurities in the coalesced region. The TRPL measurements support the basic conclusions deduced from CL and m-Raman. Decay time constants in the coherent region of samples over the windows reach about 220 ps, indicating high-quality material, which is also supported by m-PL measurements, the spectra of which are dominated by two peaks D0X

3.5 The Art and Technology of Growth of Nitrides

Figure 3.129 SEM (a) and CL wavelength (b) images of two different regions, coherently grown above the openings and overgrown above the SiO2 stripes. The growth in the windows (between the SiO2 stripes) and the wing (coalesced regions over the SiO2 stripes) regions

indicated in the schematic drawing (c) are clearly visible in the CL wavelength image. Courtesy of J. Christen and A.G. Hoffmann, Ref. [697]. (Please find a color version of this figure on the color tables.)

and XA. Both peaks show a weak intensity on the coherent region and a high intensity in the wings. From the analysis of the cross-sectional m-PL, it is concluded that the incorporation of donors is enhanced in the wing regions before full coalescence, but once GaN coalesces over the mask, the incorporation of donors is suppressed [696]. Stripes Along the h1 1 2 0i Direction in HVPE ELO In addition to planar CL mapping, cross-sectional mapping of ELO with stripes along the h1 1  2 0i direction has been conducted [697] for a detailed investigation and evolution of nonradiative defects. As can be seen in Figure 3.129, the coherently grown GaN over the windows in the mask exhibits a homogeneous emission at 3.463 eV. On the contrary, the ELO region, the wing region, exhibits an inhomogeneous and blue-shifted emission around 3.483 eV. Conversely, a red-shifted (3.425 eV) CL peak dominates at the coalescence boundaries. A CLwavelength image and local CL spectra from the ELO regions and the coherently grown GaN were also taken by Betram et al. [697]. The spectra in the coherent growth region show sharp excitonic lines associated with the free exciton X, D0X (two donors) and A0X. With increasing distance from the substrate into the layer, an 8 meV blue shift was observed for all lines. In the ELO region, broad and blue-shifted CL emission was observed. To complement the CL measurements, m-Raman scattering experiments were carried out on the same spots [697]. These experiments showed that E2 mode is directly related to the local strain. The free carrier concentration deduced from the LO phonon–plasmon coupled modes (LPP) indicated that the free carrier density in the overgrown wing region increased to about 1019 cm
3 above the buffer and remains constant in the growth direction, as shown in Figure 3.130a. In the coherent region, it starts at a low level ( 0.5) magnitude for the slowdecaying components underscores the increased radiative efficiency in the sample with TiN nanonet formed by 15 min annealing. Although detailed deep level transient spectroscopy (DLTS) analyses (see Section 4.4 for description of the method and its applications to GaN) for samples grown using TiN and SiNx nanonets are not yet available, preliminary data for SiNx (whose deposition time was varied at increases of 0.5 min between 4 and 6 min) have been obtained and compared to standard that for OMVPE samples and standard ELO samples. Specifically, two samples with single (5 and 6 min) and one with double (5 þ 5 min) SiNx nanonetwork interlayers along with a standard ELO sample and a control sample without ELO have been investigated and compared to each other [728]. For each sample, the bias voltage in DLTS measurements was chosen to probe the top 200–300 nm from the surface, and the pulse’s amplitude and width were adjusted to fill electron traps, and the results obtained are shown in Figure 3.143. The dominant trap A for all the layers under study has a peak at 325 K in the DLTS spectra. The activation energy for this trap determined from Arrhenius plots varied from 0.55 to 0.58 eV for different layers with the capture cross sections being in the low 1015 cm2 range. This trap is similar to E2 (0.58 eV) [739], D2 (0.60 eV) [740], and B (0.62 eV) [741] traps commonly observed in HVPE-, OMVPE- and MBE-grown n-GaN layers and has been suggested to relate to substitutional nitrogen atom on the Ga site [739], in addition to being dependent of the type of Ga source used for growth, as it disappeared with TEG [742]. Clearly, these are all indirect deductions and the origin of this trap still remains unclear.

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Figure 3.142 Normalized time-resolved PL spectra for the TiN nanonet grown GaN sample with 15 min nitridation (annealing) time, control sample, and bulk GaN. The solid lines are biexponential fits to the data.

Figure 3.143 Comparison of the DLTS spectra associated with the topmost GaN : Si layers (n  0.5–1 · 1017 cm
3) grown on standard GaN template (reference), templates with single 5 and 6 min SiNx nanonetwork, that with double SiNx nanonetwork (5 min þ 5 min), and standard ELO technique. The rate window for all spectra is 120 s
1 [728].

3.5 The Art and Technology of Growth of Nitrides

Peak B (inset of Figure 3.143) located at 155 K corresponds to a trap with activation energy 0.21–0.28 eV, which has also been observed by others [739,741]. The concentration of this trap was nearly two orders of magnitude lower than that of the dominant trap A for all the samples investigated. Trap B concentration was the highest in the reference sample and the lowest in the GaN layer with 6 min SiNx. It has been suggested [743] that trap B is related to screw dislocations, which is also supported by the trend observed among the samples investigated here. As the data indicate, the lowest trap concentration is seen in the sample with 6 min of SiNx followed by the standard ELO. As expected, the highest trap densities are seen in the sample on the standard GaN template. 3.5.5.4 Selective Growth Using W Masks Selectiveareagrowth(SAG)andlateralepitaxialovergrowth(ELO)ofGaNwithtungsten (W) masks [744] and WNx masks [745] using OMVPE and HVPE have also been investigated. The WNx mask was employed to prevent dissolution of the underlying GaN layer owing to the Wcatalytic effect. The WNx mask was produced by nitridation of the W film using NH3, which is already present in the reactor, at temperatures higher than 600  C. Thermal stability of WNx is good and the WNx/n-GaN contact forms a Schottky type. The selectivity of the GaN growth on the W mask as well as the control SiO2 mask was found to be excellent for both OMVPE and HVPE. The ELO GaN layers were successfully obtained by HVPE on the stripe patterns along the h1  1 0 0i crystal axis with the W mask as well as the SiO2 mask. No voids between the SiO2 mask and the overgrown GaN layer were observed. In contrast, there were triangular voids between the W mask and the overgrown layer. The surface of the ELO GaN layer was quite uniform for both mask materials. In the case of OMVPE, the ELO layers on the W mask and SiO2 masks are similar for stripes oriented along the h1 1  2 0i and h1  1 0 0i directions. In other words, no voids were observed between the W or SiO2 mask and the overgrown GaN layer by using OMVPE. As in the case of SiO2 masks, W and WNx masks led to growth patterns where triangular growth results for stripes along the h1 1 2 0i direction and truncated triangular growth results for stripes along the h1 1 0 0i direction, as shown schematically in Figure 3.144. The details of the experimental methods employed are as follows: [745] The growth experiments were performed using atmospheric pressure HVPE and OMVPE systems on 3.0–4.5 mm thick (0 0 0 1) GaN layers grown on the c-plane (0 0 0 1) of sapphire substrates. As usual, a LT buffer layer was first grown by OMVPE on the substrate. A 120 nm thick W film was deposited on the GaN surface by RF sputtering at room temperature. Stripe windows 10 mm wide with a periodicity of 20 mm were formed on the W film with conventional photolithography and wet chemical etching in H2O2 at room temperature. In the case of HVPE, GaCl and NH3 were used as the source gases and N2 was used as the carrier gas. The flow rates of HCl and NH3 were 10 sccm and 0.5 l min
1, respectively. The growth temperature was 1090  C. In the case of OMVPE, TMG and NH3 were used as the source gases and H2 was used as the carrier gas. The flow rates of TMG and NH3 were 18.7 mmol min
1 and 2.5 l min
1, respectively. The growth temperature was 1060  C. The nitridation of W was

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W or SiO 2

Stripes along

W or SiO 2

Stripes along Figure 3.144 Schematic representation of GaN grown in windows opened in W or SiO2 masks. The triangular growth results for stripes along the < 1 1 2 0> direction and truncated triangular growth results for stripes along the < 1 1 0 0> direction. Patterned after a figure provided by K. Hiramatsu and Ref. [745].

performed at 600–950  C in NH3 in H2 or N2 ambience [745]. To remove any O from the surface of the W film, the substrate was annealed in an H2 ambience at 400  C for 10 min. Then, the substrate was heated to the nitridation temperature in H2 and NH3 ambience. Following the nitridation of W, the ELO of GaN was attempted by a twostep growth at 950 and 1050  C at a low pressure (LP) of 300 Torr. The two-step process was to prevent the dissociation of GaN in contact with W by completing the first stage at a lower temperature, which was 950  C for 30 min in this case. In the second step, to bury the W mask easily, a high-temperature (1050  C) growth was performed for 90 min. 3.5.5.5 Low-Temperature Buffer Interlayer In parallel to ELO, other techniques, such as those that do not require exposure to atmosphere part way through the process and photolithography, have been explored. Among them is low-temperature interlayer [746]. The LT buffer layers inserted periodically a few times reduce the dislocation density in the top layer. The LT buffer layers are grown using growth conditions very similar, if not identical, to the standard LT buffer layers. After each LT buffer layer, the structure is automatically annealed in H2 prior to the growth of the high-temperature layers. In the process, the propagating defects collide with the new interfacial defects generated in LT insertion buffer layer and annihilate one another. Defect concentrations have been lowered to about the same value available by ELO using this method. Recently, this method has been extended to grow AlGaN on GaN layers without the notorious cracking effect observed in tensile strained AlGaN on GaN.[746]. The problem is very serious for nitride lasers as the cladding layer thickness is limited by cracking. The reduced thickness employed to avoid cracking causes leakage of the optical field to the GaN buffer layer with deleterious effects.

3.5 The Art and Technology of Growth of Nitrides

The insertion of a low-temperature GaN or AlN interlayer between high-temperature grown GaN reduces the threading dislocation densities down to the low 108 cm
2 range [747–750].TEM observations showed a significant reduction of dislocations with a screw component [434], which were found to terminate at the interlayer [751,748]. Similar features were observed in GaN growth using dimethylhydrazine as the nitrogen precursor [752,753]. The mechanisms of reduction in dislocation density are attributed to those in effect in Section 3.5.5.2 dealing with ELO technology. Figure 3.145a and b shows images of two GaN layers grown by OMVPE after they have been subjected to a defect staining etching. Corresponding schematic diagrams of the grown layers are also shown. Figure 3.145a shows a sample grown only on a LT buffer layer, which, in turn, is grown on sapphire. While Figure 3.145b shows an additional LT buffer layer inserted in the main GaN layer. As the images indicate, the etch pit density dropped from about 108 to about 5 · 106 cm
2 when an additional LT buffer layer was inserted [754]. LT AlN insertion buffer layers have been examined with a particular attention to electrical transport and optical properties of the resulting GaN layer on top [434]. It was shown that as the number of AlN IL/HT GaN layers increased, the electron mobility increased in the top Si-doped GaN layer, nearly as much as by a factor of two, from 440 to 725 cm2 V
1 s
1. The dependence of the electron mobility on temperature

Figure 3.145 Images of two GaN layers grown by OMVPE after they have been subjected to a defect staining etch, (a) is for a sample grown only on an LT buffer layer, which in turn is grown on sapphire and (b) has an additional LT buffer layer inserted in the main GaN layer. Courtesy of M. Koike.

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had the characteristic peak at about 100 K before falling off toward lower temperatures. This implies that the mobility was not dominated by any two-dimensional electron gas (2DEG), which could form at each of the interfaces between the AlN layer and the underlying GaN layer and that the measured values do indeed show an improvement in the electrical quality of GaN with LT AlN insertion. Cross-sectional transmission electron microscopy images were remarkable in that a significant reduction in the screw dislocation density for GaN films grown on the AlN IL/HT GaN layers was obtained. This is consistent with the assertion that, in GaN, edge dislocation core energy is minimal along the c-direction and it is unlikely that the edge dislocations would be affected by these AlN insertion layers. Of course, the situation is different with screw dislocations. As for the XRD investigation, the symmetric and off-axis linewidths increased as the number of AlN IL/HT GaN layers increased, indicating a greater relative misalignment of the adjacent HT GaN layers. An interesting observation about the difference between LT AlN and GaN insertion layers is the notably different residual strain in GaN [746]. It was shown that the inplane biaxial stress thickness product increased with each successive repetition of the LT GaN insertion. Further, the HT GaN layers were under tensile strain. On the contrary, the picture with LT AlN buffer insertion remained the same with each insertion of LT AlN buffer layers. Threading dislocations (TDs) were also measured using plan view TEM analysis. A reduction of TDs with each insertion of a LT buffer layer is observed, from about mid 109 cm
2 in HT GaN with no LT insertion buffers to about mid to high 107 cm
2 when some six LT buffer layers were inserted. Although there were some variations, the TD density did not seem to be that dependent on the type of LT buffer, meaning GaN or AlN is used [746]. Consistent with the structural properties with reduced screw dislocations, electrical transport properties also could be improved by inserting multiple low-temperature buffer layers of the AlN type [755]. For Si-doped GaN films grown on five AlN interlayers, the room temperature electron mobility, mRT, was 725 cm2 V
1 s
1 and the room temperature electron concentration, nRT, was 1.47 · 1017 cm
3, both of which are similar to the electrical properties for GaN grown on both 6H- and 4H-SiC, but without multiple buffer layer insertion. The dependence of the electron mobility on temperature was such that conduction by any two-dimensional electron gas, which would give erroneously high electron mobilities falsely assumed to be for the bulk, was not in effect, as the mobility decreased when the temperature was lowered below about 70–100 K. An increase in mobility accompanied by a decrease in screw dislocations suggest that screw-type dislocations may cause scatterers as well as compensating n-type donors in GaN. 3.5.6 Polarity and Surface Structure of GaN Layers, Particularly on Sapphire

Because GaN does not share the same atomic stacking order as many of the substrates on which it is grown, the crystal direction [0 0 0 1] of GaN film, the direction of the long bond along the c-axis from Ga to N atoms, can be either parallel or antiparallel to the growth direction [756]. The epilayer in the former case is conventionally referred to as

3.5 The Art and Technology of Growth of Nitrides

having Ga-polarity or Ga-face, whereas the latter has N-polarity or N-face [757]. Investigations have shown that these two polar films have vastly differing growth and surface properties. For examples, a Ga-face is typically smoother than a N-face. For MBE growth near stoichiometric conditions, the growth rate of N-polar domains may be slightly lower than that of Ga-polar matrix, leading to the formation of pits with inversion domains at their centers. A p-type doping by Mg is easier in Ga-polar films, whereas C, O, Si, and other residual impurities are more likely to incorporate into Npolar films. The Ga-face is also more stable than a N-face against wet chemical etching. The photoluminescence spectra, the Pt/GaN Schottky barrier, the band discontinuities, and the two-dimensional charges in GaN/AlGaN heterostructures are all affected by the polarity of the structures. Different polarities can be identified by convergent beam electron diffraction, hemispherically scanned X-ray photoelectron diffraction [758], convergent beam electron diffraction [758,759], coaxial impact collision ion scattering spectroscopy, and X-ray standing wave method [760,761]. The polarity impacts the dopant incorporation also in that the N-face is more amenable to incorporation [762]. Readily available to an MBE grower is the RHEED capability, which allows the determination of the polarity of some of the films by inspecting the surface reconstruction during cooldown [763,764]. With more insight and control over the MBE growth process, reports emerged as to which growth parameters lead to which polarity. It has been implied that GaN buffer layers grown at 700  C lead to N-polarity with poor layer quality and that the N-polarity samples are characterized with much higher background concentration and overall inferior quality [765]. Others reported N-polarity when grown directly on sapphire and Ga-polarity when grown on an AlN buffer layer [766,767]. Surface charge sensitive electric force microscopy for determining the polarity of the sample and its distribution on the surface has also been reported [768]. Furthermore, a close correlation between the morphology and the polarity has been established for both as-grown and etched GaN surfaces [769,770], making it possible to determine the film polarity using more easily available techniques such as AFM. Wet chemical etching has been used to determine the polarity in GaN. Particularly, NaOH-, KOH-, and H3PO4-based solutions at different temperatures have been demonstrated to attack N-polar surface, whereas the morphology of Ga-polar surface remained unchanged [758]. For device applications, an understanding and control of the crystal polarity in the epitaxial growth is essential. For this purpose, the effects of the substrate nitridation [771], buffer layer materials, and growth conditions [772], such as III/V ratio, have been investigated. The change in film polarity by Mg doping was also observed [773,774]. It has been demonstrated that, in the case of OMVPE on sapphire substrates, either GaN or AlN low-temperature buffer layers lead to Ga-polar films [757]. For MBE growth, however, the published results show that AlN buffer layers commonly lead to Ga-polar films, whereas GaN buffer layers lead to N-polar films [772]. Although there are suggestions that low-temperature GaN buffer layers may increase Ga domains or lead to Ga-polar films in some cases [771,775], a correlation between the film polarity and buffer layer growth is still not very well established.

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MBE with its inherent control over the growth parameters can be used to interrogate certain structural and electrical processes in the crystal. One such topic is the polarity of the films because the c-plane of sapphire is a polar surface and GaN does not share the stacking order with sapphire. Consequently, GaN grown on sapphire could either be terminated with Ga ((0 0 0 1) or (0 0 0 1)A-face) or N (ð0 0 0  1Þ or (0 0 0 1)B-face) [776,777]. Being non-centro-symmetric owing to its wurtzitic and ionic structure, nitrides exhibit large piezoelectric effects when under stress along the c-direction. Moreover, spontaneous polarization charges also appear at the heterointerfaces owing to the different degrees of ionicity of the various binary and ternary nitrides [778]. The signs of spontaneous polarization and piezoelectric polarization depend on the polarity of the film. This charge and its sign must be known and controlled in electronic devices, particularly in modulation-doped FETs. For example, electric field caused by polarization effects can increase or decrease interfacial free carrier concentration causing the gate potential needed to vary drastically [779]. The polarity of the film can also have an impact on the effective band discontinuities [780]. GaN layers grown by MBE can be either N- or Ga-polarity, and each can be grown under Ga- and N-rich conditions. In the case of growth with ammonia as the nitrogen source and in conjunction with high-temperature growth >800  C, very nitrogen rich conditions lead to better quality films in all respects, that is, PL, X-ray diffraction, and electron mobility. This is in part because of simultaneous removal of material, which leads to the elimination of regions that are not of as high quality. In addition, the tips of the clusters are partially etched away, which leads to smoother surfaces. Systematic RHEED investigations supported by LEED, scanning tunneling microscopy (STM), and theory [764,763,781,357] mapped the reconstructions of GaN(0 0 0 1) surface that may appear on each GaN{0 0 0 1} surface. The 2 · 2, 5 · 5, 6 · 4, and “1 · 1” reconstructions were identified for the Ga-face and the 1 · 1, 3 · 3, 6 · 6, and c(6 · 12) reconstructions were identified for the N-face [764]. Moreover, a 1 · 2 reconstruction and on occasion 3 · 2 and 2 · 3 RHEED reconstructions were also observed for the Ga-face, which was supported by AES, and first principles total energy calculations were used to identify possible atomic structures [781]. Smith et al. [764] have constructed schematic diagrams for the surface reconstructions observed on the N and Ga faces and corresponding RHEED patterns, as viewed along the ½1 1  2 0 azimuth, which are shown in Figure 3.146a and b, respectively. The N-polarity GaN is prepared by nucleating and growing GaN directly on sapphire using RFMBE. The 1 · 1 reconstruction depicted in Figure 3.146 was produced by heating the as-grown film surface to high temperature (800  C) to remove excess Ga adatoms. The 3 · 3, 6 · 6, and c(6 · 12) reconstructions, which are sustainable only below 300  C (Figure 3.146a), were obtained by depositing submonolayer quantities of Ga onto the surface. When the surface temperature is increased, the structures undergo reversible order–disorder phase transitions and the nonintegral RHEED features disappear. The Ga-polarity GaN is obtained by growth on AlN buffer layers, as will be discussed in the next section. However, the work depicted in Figure 3.146b utilized Ga-polarity OMVPE-grown GaN templates on sapphire [764]. During growth at growth

3.5 The Art and Technology of Growth of Nitrides

Figure 3.146 Schematic diagrams along with RHEED images illustrating the coverage and temperature dependence of the RHEED reconstructions on the: (a) N face and (b) Ga face. Ga coverage increases from left to right in both diagrams. Temperatures given correspond

to either order–disorder transitions or annealing transitions. The crosshatched regions indicate either mixed or intermediate phases. RHEED patterns for both the Ga and N face, as viewed along the ½1 1 2 0 azimuth, are also shown. Courtesy of R.M. Feenstra and Ref. [764].

temperatures, a streaky 1 · 1 RHEED pattern is observed, which upon cooling, transforms to a pseudo “1 · 1” at 350  C. When viewed along the ½1 1  2 0 azimuth, satellite lines adjacent to the integral order lines accompanied this “1  1” structure. Additional RHEED patterns from this structure can be found in Ref. [782]. When the

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surface exhibiting the “1  1” reconstruction is annealed at 750  Cto remove excess Ga atoms, the RHEED pattern changes to a 1  2 (not shown in Figure 3.146a), with a weak half order streak. Deposition of Ga onto the 1  2 surface at room temperature removes the half orderstreak and nofractional orderstreaks appear. However, by depositing half ML of Ga and treating the surface to 700  C followed by cooling, the 5  5 reconstruction is formed. Further deposition of an additional half ML of Ga and annealing to 700  C followed by cooling results in a 6  4 reconstruction. The 6  4 reconstruction undergoes a reversible phase transition at 250  C, whereas the 5  5 structure is stable up to 700  C. Deposition of about one ML of Ga on the surface exhibiting the 6  4 reconstruction, followed by a rapid annealing at 700  C, results also in the “1  1” structure.[763] The 2  2 RHEED pattern results when the 5  5 surface is annealed at 600  C or nitridated at 600  C. The dependence of polarity on growth conditions, particularly the type of buffer layers used, in the context of RF MBE growth on c-plane of sapphire substrates has been investigated [783]. It should be mentioned that a low-temperature nitridation, in combination with low-temperature buffer growth, has been reported to lead to interfaces void of cubic GaN and improved quality [784]. Four groups of samples were grown and investigated. The first and second sets utilized GaN buffer layers grown at about 500 and about 800  C, respectively. The third and fourth groups utilized AlN buffer layers grown at about 500 and 890–920  C, respectively. Following the buffer layers, typically 1 mm thick GaN layers were grown at a substrate temperature between 720 and 850  C with growth rates in the range of 300–1000 nm h
1 under N-limited (Ga-rich) conditions. In addition to the in-situ RHEED images, AFM, X-ray diffraction, and hot H3PO4 at 160  C were employed to confirm the polarity assignment. Layers with high-temperature GaN buffer layers (around 770  C or higher) invariably turned out to be of N-polarity regardless of whether a static or graded substrate temperature was employed during the buffer growth. Upon cooling, the RHEED pattern indicated only the bulk 1 · 1 structure, though others have reported higher order reconstruction [764]. Conversely, layers with AlN buffers grown in the temperature range of 880–960  C (some are not included in the data presented here) with thicknesses in the range of 8–35 nm and growth rates of 40–60 nm h
1 led to Gapolarity. Consequently, a (2 · 2) RHEED pattern was observed upon cooldown at temperatures ranging between 280 and 650  C depending on the V/III ratio employed for the mail layer. The low substrate temperature buffer growth, Ts 650  C but primarily around 500  C, resulted in layers with either polarity with either GaN or AlN buffer layers. The GaN structures with 100–150 nm thick GaN buffer layers at a growth rate of about 600 nm h
1 led to Ga-polarity with the characteristic 2 · 2 pattern upon cooldown after the entire structure was completed. However, when the thickness of the buffer layer was reduced to 30–60 nm, keeping the growth rate constant, the layers turned out to be of mixed polarity with a faint 2 · 2 reconstruction observed upon cooldown. When about 110–220 nm thick buffer layers grown at 500  C with 220 nm h
1 growth rate were used, the resultant layers were of N-polarity with only the 1 · 1 reconstruction observed during cooldown.

3.5 The Art and Technology of Growth of Nitrides

The AlN buffer layers grown at Ts 650  C, but primarily around 500  C, exhibited Ga- or N-polarity depending on the growth conditions. When 10–15 nm thick buffer layers grown at a rate of 60 nm h
1 were employed, Ga-polarity resulted. However, when 2.5–22 nm thick buffer layers were employed with a growth rate of 15–25 nm h
1, N-polarity resulted. The typical surface morphologies of as-grown Ga-polar films with different buffer layers are presented in Figure 3.147. In this case, a high-temperature AlN buffer layer tended to result in a smooth, but pitted layer (Figure 3.147a), consistent with the group III/V ratio employed. Higher group III/V ratios generally led to disappearance of the pits. A low-temperature AlN buffer layer grown at a high rate led to a Ga-polar surface morphology with irregular stepped terraces, often with pits and/or a rough surface (Figure 3.147b). When a low-temperature GaN buffer layer grown at high rate was used, a similar morphology to that shown in Figure 3.147 b with a more drastic variation in terrace height and shape was observed (Figure 3.147c). The surface morphologies of as-grown N-polar films with different buffer layers are presented in Figure 3.148. With a high-temperature GaN buffer layer, the film morphology is that of noncoalesced columns (Figure 3.148a). In general, smoother morphologies with stepped terraces were found when a low-temperature AlN buffer layer grown at a low rate was used (Figure 3.148b). Using low-temperature GaN buffer layers grown at a low rate, the morphologies vary from extremely rough surfaces with noncoalesced columns to a surface shown in Figure 3.148c, where very tall columns and terraces are separated by deep troughs. The simple model that explains the N-polarity with GaN buffer layer calls for Ga to form [0 0 0 1] or the long bond to the O-surface of sapphire. In the same vein, the Gapolarity results with AlN buffer layers when O leaves the surface and N forms the [0 0 0 1] or the long bond with Al of sapphire. This simple model, although consistent with high-temperature buffer layers, does not explain our results with low-temperature AlN and GaN buffer layers, which led to either polarity depending on the growth conditions. Detailed investigations are necessary to gain an insight into the mechanisms involved. High growth rates mainly leading to Ga-polarity and low growth rates to N-polarity would indicate that there must be some atomic exchange or interaction that may be suppressed or promoted by large growth rates, depending on the case. Experimental and theoretical investigations [764,785] led to possible models of the above-mentioned surface structures (see Ref. [83] for a detailed account). It has been pointed out that reconstructions on both Ga- and N-face GaN{0 0 0 1} surfaces consist of metallic layers of Ga, bonded to the GaN, unlike other semiconductor surfaces that prefer to be nonmetallic because the opening of a surface gap provides a mechanism for lowering the energy of the system [785]. Moreover, the separation of Ga atoms in bulk GaN along the [0 0 0 1] direction is typically 2.7–2.8 Å, which is smaller than the lattice constant of GaN, 3.19 Å, and should not favor the metallic surface layers. In spite of this, the ð0 0 0 1Þ N surface with the 1  1 reconstruction was found to consist of one ML of Ga atoms (adlayer) located on sites above the N atoms [786]. Charge transfer from Ga to N atoms has been proposed to stabilize Ga- on N-face because the Ga atoms in the adlayer are slightly positively charged so that there is a Coulombic repulsion between them [785]. RHEED intensity variations in combination with

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Figure 3.147 AFM images of Ga-polarity samples obtained under three different buffer layer and growth conditions: hightemperature AlN buffer layer (a), low-temperature AlN buffer layer (b), and low-temperature GaN buffer layer (c).

3.5 The Art and Technology of Growth of Nitrides

Figure 3.148 AFM imges of N-polarity samples obtained under three different buffer layer and growth conditions: hightemperature GaN buffer layer (a), low-temperature AlN buffer layer (b), and low-temperature GaN buffer layer (c). The rough surface morphology can be improved by growing the top GaN layer at lower temperature such as 720  C instead of 800  C, which was the case here.

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 0i DMS led Held et al. [777] to confirm the stability of Ga adlayer on gallied h0 1 1 surface. The 3  3, 6  6, and c(6  12) reconstructions are formed by additional Ga adatoms weakly bound on top of the 1  1 Ga adlayer. In particular, the 3  3 structure has one additional atom per 3  3 cell compared with the 1  1 [786]. A lateral relaxation of the nearest-neighbor Ga adlayer atoms allows the adatom to move much closer to the adlayer plane, which aids in stabilizing the structure [786]. The 3  3, 6  6, and c(6  12) reconstructions are characterized by reversible order–disorder phase transitions at 300  C, owing to the weak bonding of the Ga adatoms. Turning our attention to the Ga(0 0 0 1) surface of GaN under N-rich conditions, it is terminated with an arrangement of adatoms, with each adatom forming three bonds with the underlying Ga atoms [785]. Most likely N and Ga atoms participate as adatoms, as can be inferred from the relative surface energies calculated by Northrup et al. [357], which are shown in Figure 3.149. The theoretical calculations also point to the stability of the 2 · 2 N and 2 · 2 Ga adatom structures, as described in Refs [787– 790]. STM observations showed that the 2 · 2 reconstructed surface is well ordered, but only in small domains, which is consistent with the half-order diffraction lines in RHEED not being very sharp [791]. Because the 2 · 2 structure is formed by nitridation, it is most plausible that it is associated with N adatoms at the H3 hollow site, supported by theory [763,789]. It is also plausible that nitridation might convert the surface from a Ga-rich structure to a 3.0 GaN(0 0 0 1)

2.5

Energy (eV/1× 1)

2.0 Laterally contracted Ga monolayer

1.5 1.0 0.5 0.0 –0.5

2 × 2 N adatom

2 × 2 Ga adatom

–1.0 Laterally contracted Ga bilayer –1.5 –2.0 –1.2

–1.0

–0.8

–0.6

–0.4

μ Ga - μ Ga(bulk) (eV) Figure 3.149 Relative energies of the surfaces are plotted as a function of the Ga chemical potential. For Ga-rich conditions the most stable structure is the laterally contracted Ga bilayer. (Courtesy of J. Northrup and Ref. [357].

–0.2

0.0

0.2

3.5 The Art and Technology of Growth of Nitrides

less Ga-rich structure, with the 2 · 2 reconstruction being associated with the Ga adatoms. If so, the calculations indicate that the Ga adatoms prefer the T4 site, right above the second layer N atoms. Regarding the 5 · 5 reconstruction, the experimental evidence suggests adatoms being at H3 and T4 sites with three adatoms and a dangling bond site at a Ga rest atom per each 5 · 5 unit cell [763]. However, the 6 · 4 structure was reported to be quite complicated in origin and is likely to involve more than just adatoms and/or vacancies on the surface [763]. In the Ga-rich limit, the pseudo 1 · 1 structure of the (0 0 0 1) surface contains a fluidlike overlayer of Ga [357,763,790]. Motivated by RHEED and LEED diffraction observations and modeling of the AES Ga/N peak intensities, the premise that there are two monolayers of Ga on top of the Ga-terminated GaN bilayer has been forwarded. In this case, the Ga layers assume spacing close to their bulk values so that they form an incommensurate structure on the surface. Moreover, utilizing first principles total energy calculations, Northrup et al. [357] provided the theoretical backing (Figure 3.149) in favor of a laterally contracted mobile bilayer structure p ffiffiffi pffiffiffi containing 2.3 ML of Ga, as shown in Figure 3.150. Modeling utilized a 3  3 cell, which allowed pffiffiffi one to model hexagonal Ga adlayers with a reduced Ga–Ga spacing of ac ¼ ð 3=2Þa1·1 ¼ 2.75 Å (because a1·1 ¼ 3.17 Å is the in-plane spacing of Ga atoms on the 1 · 1 reconstructed ideal surface). The lattice vectors of the overlayer were rotated by 30 with respect to those of the substrate. In the structure depicted in Figure 3.150, layer 1 contains three atoms and layer 0 contains

0.16 Å

ac

a1×1

Ga-layer 0

z

01

z

12

T1 sites Ga-layer 1

a1×1 Ga-layer 2 N-layer 3

N

Ga

Figure 3.150 A schematic representation of a laterally contracted Ga bilayer above a Gaterminated (0 0 0 1) substrate. The average separations between layers are z12 ¼ 2.54 Å and z01 ¼ 2.3 7Å. The filled and open circles in layer 0 represent a time-averaged image of the Ga atoms. The filled circles in layer 0 correspond to

the positions at a particular time. The timeaveraged vertical corrugation of layer 0 is approximately 0.16 Å. Note: In this projection the laterally contracted monolayer (layer 0) has been rotated by 30 for ease of viewing. Courtesy of J. Northrup et al. and Ref. [357].

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pffiffiffi pffiffiffi four atoms in each ( 3  3) cell. In the Ga-rich limit, the energy of this laterally contracted Ga bilayer structure was the lowest among all the structures investigated, as shown in Figure 3.149 inclusive of the special case of the Ga adatom structure by 0.25 eV/1  1. Several inequivalent registries of layer 0 relative to layer 1 were considered pffiffiffi pffiffiffi within a tolerance of about 0.02 eV/1  1. However, the energy of the ( 3  3) laterally contracted bilayer (LCB) structure was independent of the registry with the substrate. This is consistent with sufficiently mobile Ga atoms and also accounts for the observation of a 1  1 corrugation pattern in STM [763]. The Ga-bilayer model associated with the stable structure of the Ga-rich (0 0 0 1) surface has been supported by another research group [350,792]. In these investigations, the amount of excess Ga on the surface was measured by observing the transients in the intensity of the specular RHEED beam. However, confirmation of the incommensurate bilayer structure [785] remains, because STM imaging of the (0 0 0 1) surface did not yield the expected incommensurate pattern (i.e., series of fringes). Moreover, the observed atomic spacing corresponds to the 1 · 1 spacing (3.19 Å) as opposed to a smaller value by 1/6 or 1/12 as implied by the diffraction experiments [782]. There have been several other investigations of the two GaN {0 0 0 1} surfaces inclusive of a discussion of theoretical calculations of Ref. [793]. For a detailed account, the reader is referred to Ref. [79]. studies pointed to additional surface pffiffiffi These pffiffiffiffiffi reconstructions, such as the 10 · 10, 5 3  2 13 and 4  4 for the (0 0 0 1) face and pffiffiffi pffiffiffi 6  8, 7  7 and 2  3 for the N-face. A 2  pffiffiffi pffiffiffi 2 reconstruction has been also reported for a ð0 0 0 1Þ surface [794] and ( 3  3) R30 for a (0 0 0 1) surface [795]. The observation of a 2 · 1 RHEED pattern on a ð0 0 0  1Þ surface has been attributed to a poorly ordered 2  2 reconstructed surface, in which the twofold periodicity is more easily seen with the electron beam along the ½1 1  2 0 azimuth [794]. Another important issue that could have a bearing on the surface reconstruction is that of contaminants. Several reports have indicated the sensitivity of the surface reconstruction on the presence of contaminants. A 4 · 4 reconstruction on the Npolar surface has been attributed to oxygen or arsenic contamination, the latter associated with systems that have previously been used for arsenide semiconductors [795,796]. A body of experimental work attributed the 2 · 2 reconstruction [797,798] to impurities such as As or Mg. It was shown that a partial pressure of 10
9 Torr of arsenic modified the growth kinetics significantly (As behaves as a surfactant) and induced a 2 · 2 surface reconstruction during growth of GaN (0 0 0 1) [799]. The proposed model of this 2 · 2 structure consists of one As adatom per 2 · 2 unit cell [799,89]. Moreover, adding small amounts of Ga to the arsenicinduced 2 · 2 surface resulted in the formation of 4 · 4 and 5 · 5 reconstructions [781], in agreement with Ref. [788]. A 4 · 4 reconstruction was also observed at low temperature in Ref. [797]. A 2 · 2 reconstruction is also observed on the (0 0 0 1) surface during MBE growth with NH3 gas source [777] and could be pffiffirelated ffi pffiffiffi to the adsorption of H atoms.[787]. A case related to InGaN growth, a stable ( 3  3) R30 reconstruction (1 · 3 RHEED pattern) on the GaN(0 0 0 1) surface has been induced by In atoms after sufficient nitridation [83]. The same reconstruction has also been observed on InGaN surface under N-rich conditions [800].

3.5 The Art and Technology of Growth of Nitrides

The affect of in situ annealing on the surface structure has also been investigated using such techniques as LEED. The LEED patterns of clean GaN{0 0 0 1} 1 · 1 surfaces, if heated to temperatures above 830  C, have exhibited splitting of the normal order spots into circular sextets (multiplets of six spots) as a function of the primary electron energy [801]. These have been attributed to faceting of the surface [84] and alternatively the splitting may be owing to oppositely oriented, regular step arrays in the ½2 1 1 0, ½1 1 2 0, and ½1 2 1 0 directions on the GaN{0 0 0 1} surfaces [801]. The multiplets of sharp spots evolve from weakly diffused rings as the annealing temperature was increased above 830  C, which was attributed to the formation of straight step edges oriented perpendicular to the ½2  1 1 0, ½1 1  2 0, and ½ 12 1 0 directions, meaning rotated by 60  each. A quantitative exercise led to terrace widths of 11.0  1.0 Å and step heights of 5.25  0.2 Å. This surface structure may be the result of the development of “growth” spirals around the screw dislocations by thermal etching of the surface in much the same way as the development of growth spirals observed under Ga-rich conditions [782,785]. 3.5.6.1 MBE Buffer Layers As in the case of OMVPE, growth on sapphire with MBE requires a short period of nitridation of the sapphire substrate before the buffer layer growth, generally between 10 and 30 min, or until the RHEED image indicates a transition from sapphire to AlN with the accompanying 30 rotation. During the nitridation process, a thin AlO1
xNx film may form on the substrate surface [802] transforming Al2O3, in a natural way, through AlO1
xNx into AlN, and the epitaxial films grow on AlN, which has nearly the same lattice constant as GaN (mismatch 2–4% in c-plane). This is why the best films until now have been obtained on sapphire with an AlN buffer layer. It is obvious that nitridation parameters can greatly influence the properties of the epitaxial layer [803,804]. A set of RHEED patterns showing the nitridation stage and subsequent growth of AlN is shown in Figure 3.151. The other buffer layer employed is GaN, either a low-temperature or a high-temperature variety. In general, AlN and GaN buffer layers grown at relatively high temperatures lead to Ga- and

Figure 3.151 (a) RHEED image taken along the ½1 1 2 0 direction of AlN, rotated about 30 with respect to that of sapphire, during nitridation of sapphire; (b) RHEED image taken along the ½1 1  2 0 direction of AlN during AlN growth on sapphire.

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Figure 3.152 (a) RHEED imaging of GaN deposited on a nitridated sapphire, taken along ½1 1 2 0 direction at 800  C and represents N polarity of the film; (b) RHEED image of AlN grown on GaN of Figure 3.151 taken at 800  C along the ½1 1 2 0 azimuth, and represents a N-polarity film. The bright spots represent the specular reflection associated with each diffraction.

N-polarity films, respectively [805]. Those grown on buffers prepared at lower temperatures can lead to either polarity if grown on AlN buffer layers, depending on the particulars, as discussed in Section 3.5.6. When grown on low-temperature GaN buffer layers, the film has N-polarity. Shown in Figure 3.152 are RHEED images of GaN buffer layers deposited directly on nitrided sapphire and AlN grown on that GaN buffer layer. These images are for an N-polarity film and upon cooling do not show the characteristic 2 · 2 pattern observed in Ga-polarity films. Instead, it shows a 1 · 1 pattern, although a 3 · 3 diffraction should have been observed. A 1 · 1 pattern can also be observed for a Ga-polarity film for a range of group III/V ratios, but under the conditions used, Ga-polarity films lead to strong 2 · 2 patterns. Unlike Ga-polarity films where the edge dislocations are dominant and present in concentrations above 109 cm
2, N-polarity films have much fewer extended defect concentrations, and the concentrations of screw, edge, and mixed dislocations are comparable. Moreover, the N-polarity films exhibit wider symmetric diffraction peaks and higher photoluminescence efficiency, as compared to the Ga-polarity ones, which indicates that the edge dislocations are not benign, the details of which are discussed in Volume 2, Chapter 5 in terms of optical properties and in Chapter 4 in terms of electrical nature of dislocations. 3.5.7 Growth on ZnO Substrates

ZnO is considered a promising substrate (yet to really be implemented, but quality of GaN on ZnO has been improving steadily. Because GaN can be separated from ZnO easily, this method is considered by some as an attractive freestanding GaN template preparation method) for the III-N semiconductors because it has a close match for GaN c- and a-planes and an identical stacking order with GaN [806,807], and GaN follows the in- and out-plane orientations of ZnO. In the context of GaN, ZnO has been used in two capacities. One is the buffer layer, particularly for HVPE growth of

3.5 The Art and Technology of Growth of Nitrides

GaN, and the other as a substrate. Several techniques have been employed and are being developed to grow nitride-based compounds on different kinds of substrates. Vispute et al. [808] prepared GaN layers using ZnO as a buffer layer by the pulsed KrF excimer laser deposition technique with a wavelength of 248 nm and pulse duration of 25 ns ablating a ZnO (99.99%) target. The focused beam energy and pulse repetition rate were 2–3 J cm
2 and 5–15 Hz (20 ns pulse duration), respectively. The substrate temperature for the deposition of ZnO was in the range of 300–800  C. The background oxygen pressure was in the range of 10
5 to 10
2 Torr. In a similar fashion, GaN films have also been deposited on ZnO/sapphire at 850  C under a background NH3 pressure of 10
6 to 10
5 Torr. Similarly, but employing a Nd : YAG pulsed laser, with a wave length of 266 nm, a fluency of 0.8 J cm
2 per pulse and a repetition rate of 5 Hz, Wang et al. [809] also prepared GaN layers on sapphire using ZnO buffer layers, all deposited at 800  C. Uedo et al. [810] grew thick GaN layers on pulsed laser deposited ZnO buffer layers on sapphire by the HVPE technique. The growth was performed at 1000  C at the V/III ratio of 500 and a rate of 10 mm h
1. Detchprohm et al. [375], and Molnar et al. [11] used sputtered deposited ZnO on sapphire as buffer layers for GaN epitaxy. In the former case, a 10 cm diameter ZnO ceramic disk was used as target, and Ar or O2 was employed as sputtering gas with a chamber pressure of 0.13 and 0.4 Torr for Ar and O2, respectively. The discharging input and reflecting powers were 200 and 20 W, respectively, for both gases. The ZnO deposition rates were in the range of 150 and 5 nm h
1 for Ar and O2, respectively, and the deposition took place at room temperature. In exploring GaN deposition by the HVPE on ZnO-coated sapphire, Gu et al. [811] observed that the thin ZnO prelayer, which is thermodynamically unstable at the temperatures employed and reducing gases present, disappeared. To quantify this obvious conclusion, two 20 nm thick ZnO samples were deposited on sapphire. One was annealed at 1050  C for 10 min in N2 ambience and the other was not investigated. The unannealed one was dipped into HCl to dissolve the ZnO film. The XPS analysis showed no residues of ZnO or any other compound formed with sapphire. However, the annealed one, after dipping in HCl, showed traces of ZnAl2O4 by XPS analysis, as compared to that of a standard compound such as ZnAl2O4. One can conclude that there might be formation of ZnAl2O4, which may act as nucleation centers for GaN growth, which is not yet corroborated. Hamdani et al. [812] deposited GaN films on ZnO substrates in an MBE environment by reacting Ga and NH3 at 760  C and employing either GaN or AlN as the buffer layer. During the growth, the chamber pressure was maintained at 2–5 · 10
5 Torr. Matsuoka et al. [813] used degreased and etched ZnO substrates (the etching rate of the O-face ZnO is faster than that of the Zn-face by about one order of magnitude) to grow thin GaN films with OMVPE. Until attaining the required growth temperature, a nitrogen flow was maintained in the chamber, then ammonia and group III sources were allowed into the chamber. A vertical cold walled OMVPE reactor was used for the growth of GaInN and InGaAlN epitaxial layers. As expected, the incorporation of In increased when the substrate temperature was reduced from 800 to 500  C, which is because of the much higher vapor pressure of In compared to that of gallium. A typical ammonia flow rate was 20 l min
1. The growth

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temperature was 500–700  C and the V/III flow rate ratio was maintained at 20 000. At 800  C, the flow rate of TEG was kept 17 mmol min
1, which resulted in a V/III ratio of 4000. An XRD analysis showed that the FWHM of the rocking curve of the (0 0 0 2) diffraction values are 0.21, 0.20, and 0.52 for GaN/ZnO/Si(1 1 1), GaN/ZnO/ sapphire (0 0 0 1), and ZnO/sapphire, respectively [814]. The FWHM of the (0 0 0 2) reflection for ZnO films was 0.17 for a substrate temperature of 750  C and O2 pressure of 10
5 to 10
4 Torr. When ZnO on sapphire was used for GaN growth, the sapphire substrate and the epitaxial layer (ZnO and GaN) are oriented with each other by a 30 rotation of the unit cells. The in-plane epitaxial layer and substrate are in the form of ZnO½1 0 1 0kAl2O3½1 1  2 0, similar to AlN and GaN on sapphire. Wang et al. [809] reported that the FWHM of rocking curve of the (0 0 0 2) diffraction for GaN/ZnO/sapphire decreased from 0.45 to 0.22 with increasing deposition chamber pressure from 10
5 to 1 Torr. Similarly, this value decreased with increasing substrate temperature. The GaN films deposited at >500, 700, and 800  C under 0.1 Torr in N2 ambience showed amorphous, polycrystalline, and epitaxial nature, respectively. Ueda et al. [810] observed that the RHEED showed ring pattern indicating that a polycrystalline film had formed by a reaction between ammonia and the single-crystal ZnO. The (0 0 0 2) reflection was predominant when the GaN films were grown with ZnO buffer layer. An additional ð1 1  2 1Þ reflection was observed along with (0 0 0 2). In this sample, sharp rocklike three-dimensional structures were observed. The GaN layers grown on the O-face of ZnO exhibited a lack of tilting with respect to the substrate, in contrast to that on sapphire where noticeable tilting is observed. By observing the positions of the (0 0 0 2) diffraction peaks of GaN and ZnO for two X-ray incident beam directions that are 180 apart, one can determine the relative tilt of GaN with respect to ZnO. The data shown in Figure 3.153 clearly show that the direction of the incident beam did not affect the position of the GaN and ZnO diffractions. Consequently, one can argue that the GaN film is not tilted with respect to the ZnO substrate on which it was grown. Optical processes in GaN on ZnO have attracted a good deal of attention. Because ZnO and GaN transitions are close in energy, and there may be overlap in the PL spectra, a short discussion of ZnO PL spectra is warranted. ZnO substrates have improved dramatically, to the point where very sharp X-ray and PL peaks are possible now, with PL spectra showing low FWHM of 0.55 meV for the exciton bound to neutral donor peak (D X) at 3.3597 eV, as shown in Figure 3.154. Other sharp peaks shown in the inset of this figure are related to excited states and excitons bound to different donors. Identification of the exciton structure of ZnO is quite controversial in the literature, but suffice it to say that the main peak at 3.36 eV is repeated three times on the low-energy tail of exciton emission at energies that are multiples of the LO phonon energy (about 71 meV). Another sharp peak was observed at 3.3206 eV with LO phonon replicas at 3.2505 and 3.180 eV, which could be attributed to an exciton bound to acceptor. From defect-related features, only the broad band with the maximum at about 2.4 eV (green band) resolved. The low-temperature PL spectrum

3.5 The Art and Technology of Growth of Nitrides

ZnO (0 0 0 2)

Δα (a)

GaN (0 0 0 2)

Δα (b)

15.6

15.7

15.8

15.9

16.0

Diffraction Angle ( 2θ ) Figure 3.153 X-ray rocking curves for a GaN film grown on the Oface of ZnO with ammonia MBE. The solid and dotted lines associated with the (0002) diffraction peaks of GaN and ZnO are for two X-ray incident beam directions that are 180 apart. Because GaN and ZnO peaks do not show any directional dependence, one can conclude that the relative tilt of GaN with respect to ZnO is negligible.

of the Cermet sample is nearly identical to that of the Hanscom sample. However, the FWHM of 3.36 eV peak is lower by approximately three times that of Hanscom sample. The broad defect related band is observed at about 2.1 eV instead of the green band. In the Eagle Picher sample, the PL spectra from two faces (Zn and O) are compared at room temperature, which demonstrated total identity. Residual strain owing to the mismatch of thermal expansion coefficients of substrates and GaN layers is of special importance as it impacts optical, vibrational, properties, and, to some extent, the electrical properties of GaN. To this end, Hamdani et al. [812] demonstrated the relative benefit of GaN directly grown on ZnO as opposed to that grown on sapphire and SiC. X-ray and optical measurements indicated that GaN layers grown on sapphire substrate undergo a compressive strain and GaN layers grown on SiC substrate are under tensile strain. An accurate calculation of the temperature variation of the thermal strain (eth) owing to the difference in the thermal expansion coefficients between GaN and the substrate used is given by: eth ðTÞ ¼ ½Dal ðTÞ
Das ðTÞ=as ;

ð3:66Þ

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10

PL Intensity (au)

10

10

10

5

T = 10 K Bulk ZnO

4

3

2

3.355 10

10

3.36

3.365

1

0

2

2.4

2.8

3.2

3.6

Photon Energy (eV) Figure 3.154 PL spectrum from the Hanscom ZnO sample at 10 K. The inset enlarges fine structure in the near-band edge region.

where Dal(T) and Das(T) are the integrals representing the variation of the lattice parameter in the range of temperature between the growth temperature and the room temperature for the GaN layers and the substrate, respectively. Figure 3.155 shows the variation of the calculated thermal strain eth with growth temperature for different heterostructures, using the temperature dependence of the thermal expansion coefficient presented in Chapter 1. Owing to the close thermal expansion between GaN and ZnO, it is important to note that the thermal strain in GaN/ZnO is about half that of the GaN/SiC and AlN/ZnO heterostructures, established from Figure 3.155. The data show that the cracking of thick GaN, particularly AlGaN, layers observed on SiC substrates could be avoided in the case where ZnO is used as a substrate. Secondly, AlN is not very suitable to be used as buffer layer in the growth on ZnO substrate. We shall mention that the thermal strain eth has negative values for GaN/ZnO and AlN/ZnO indicating a compressive strain and positive values for GaN/SiC and AlN/SiC heterostructures, indicating a tensile strain. This is congruent with the shift of the energy position of free excitons observed in both reflectivity and photoluminescence spectra obtained in some of these heterostructures. It is important to note that the thermal strain increases linearly with increasing growth temperature, indicating that the MBE-grown samples undergo less thermal strain compared to the OMVPE-grown samples, which are grown at higher temperature.

3.5 The Art and Technology of Growth of Nitrides

0.15 GaN/SiC

0.050 0.0 GaN/ZnO

—0.050 —0.10

AlN/ZnO

—0.15 —0.20 200

400

600

800

Compressive strain

Thermal strainε th(%)

0.10

Tensile strain

0.20

1000 1200 1400

Growth Temperature (°C) Figure 3.155 Variation of the thermal strain with the growth temperature calculated for GaN/ZnO, AlN/ZnO and GaN/SiC heterostructures.

3.5.8 Growth on LiGaO2 and LiAlO2 Substrates

As alluded to earlier, in Section 3.2.6.1, the structure of LiGaO2 is similar to the wurtzitic structure, but because Li and Ga have different ionic radii, the crystal has orthorhombic structure. The atomic arrangement in the (0 0 1) face is hexagonal, which promotes the epitaxial growth of GaN(0 0 0 1), so that the epitaxial relationship GaN(0 0 0 1)/LiGaO2(0 0 1) is expected; the in-plane relationship is ½1 1  2 0 GaN|| [0 1 0]LiGaO2. The distance between the nearest cations in LiGaO2 is in the range of 3.133–3.189 Å, whereas the distance between nearest anions is in the range of 3.021–3.251 Å. The lattice mismatch between GaN(0 0 0 1) and LiGaO2(0 0 1) is then only 1–2%. Even though there are difficulties with surface preparation and stability of the material in the presence of H and high temperatures, not to mention the poor thermal conductivity, efforts have been expended to take advantage of this material for substrates. Promising results from material growths on lithium gallate (LGO) including aluminum, gallium, and indium nitride alloys by MBE [815] as well as OMVPE [816] growth have been reported. Yun et al. [817] examined the in-plane atomic arrangement of GaN with respect to LGO. To do this, two-dimensional c–j scans with 2y fixed at the ð1 0  1 2Þ orientation of GaN; the resultant pole figure is shown in Figure 3.156. From the contour plot of c–j space, six intensity maxima corresponding to GaN asymmetric ð1 0  1 2Þ

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Figure 3.156 XRD pole figure of GaN epilayer grown on LiGaO2 substrate, with 2y fixed at GaN ð1 0 1 2Þ orientation. Six intensity maxima are from GaN (1 0 2) diffraction. No rotation relative to the substrate is observed. The random patterns are from the background.

orientation are observed at 60 equiangularly. No lattice rotation within the equatorial plane relative to the LGO substrate is observed. This is significant in that it shows close lattice matching (Da/|a| < 1%) and may pave the way for high-quality epitaxial c-plane GaN films. Structural qualities of the films were evaluated by performing o-scan (rocking curve) in both symmetric (0 0 0 2) and the asymmetric ð1 0  1 2Þ directions. The FWHMs of 4.5 and 21.2 arcmin have been obtained on GaN/LGO for (0 0 0 2) and ð1 0 1 2Þ, respectively, in approximately 1 mm thick films [817]. The photoluminescence measurements of GaN on LGO showed a sizeable shallow donor–acceptor pair band in the range 3.1–3.3 eV in addition to the band edge emission. In the case of MBE, second-order reconstructions of the GaN surface on LGO, indicating smooth and well-ordered Ga-polarity surfaces have been reported. From a structural point of view, the X-ray reciprocal space mapping indicated higher structural quality than GaN on SiC and sapphire available in that particular laboratory, whereas TEM data indicate 6 · 108 cm
2 threading dislocation density [815]. However, GaN on sapphire and SiC, in general, prepared in other laboratories are superior to this particular report of GaN on LGO. By improving the surface finish of LGO by

3.5 The Art and Technology of Growth of Nitrides

polishing, the authors noted linear, as opposed to spiral, step-flow growth of AlGaN/ GaN during MBE growth. In this mode of growth, monolayer terraces are observed through AFM with 30 nm of Al0.25Ga0.75N on a 2.4 mm thick GaN layer, in contrast to the spiral step-flow growth observed previously for MBE-grown GaN samples on OMVPE buffer layers [818]. Additionally, the dislocation density derived from the small pits in the AFM image result in a threading dislocation density of 4–5 · 108 cm
3, which is in close agreement with the TEM data presented above. The FWHM of o
2y X-ray diffraction rocking curves as low as 85 arcsec for a 1 mm thick film were measured with the h0 0 0 4i reflection. However, one must be careful of this reflection as it is sensitive mostly to screw dislocations, which are miniscule in density compared to edge dislocations. Asymmetric reflections, such as the [102] reflection, would be a better gauge of the quality of the film. With OMVPE growth, it was reported that the films grown at 1000  C peeled off as soon as they were in contact with water vapor in the atmosphere. Moreover, hydrogen attacks LiGaO2, thus nitrogen must be used as a carrier gas in OMVPE growth of GaN to achieve high structural, high-quality GaN on LiGaO2. With the controlled low growth temperature (850  C) and extremely low input partial pressures of hydrogen, Duan et al. [819] successfully deposited GaN on both domains of LiGaO2 by OMVPE without the problem of peeling off. Fischer and coworkers [820] used LiAlO2 substrates with 1.5% lattice mismatch and 21% mismatch in the thermal expansion coefficients, respectively. As mentioned previously, in conjunction with substrate issues, the epitaxy relationships between GaN and LiAlO2 are expected to be ð0 1 1 0Þ=ð1 0 0Þ LiAlO2 with ð2  1 1 0Þ==ð0 0 1Þ LiAlO2. Unlike Al2O3 and 6H-SiC substrates with very smooth surfaces, the LiAlO2 substrate exhibited a wavelike surface with equidistant grooves about 10 nm deep, which could have originated from the mechanical surface polishing. Because of the high surface roughness, the interface between GaN and LiAlO2 was highly defective. Three-dimensional growth by OMVPE of monocrystalline GaN thin films on bLiGaO2 substrates was realized by Kung et al. [816]. The GaN layers were grown by OMVPE at temperatures between 600 and 1000  C, at deposition rates of about 0.7 mm h
1. The samples had smooth surfaces and the X-ray rocking curve was as narrow as 300 arcsec for substrate temperatures of 900  C. All GaN layers were n-type (n about 1020 cm
3) as determined by room-temperature Hall measurements. The electron mobility was about 10 cm2 V
1 s
1. The high electron concentrations may be because of the incorporation of oxygen, resulting from the decomposition of LiGaO2 at elevated substrate temperatures. This effect would probably be reduced by the use of low-temperature buffer layers. 3.5.9 Growth on GaN Templates

In the absence of large area GaN substrates in the conventional sense, the basic research of the homoepitaxial GaN growth has been on small templates of GaN platelets grown from the liquid phase under high hydrostatic pressure and at high temperatures. Growth was performed both by OMVPE [821–823] and MBE [824,825]

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on the aforementioned GaN platelets. In addition, GaN epitaxial layers have been grown by MBE on OMVPE [317] and HVPE [826] prepared templates and by OMVPE on HVPE-prepared templates [827]. More on this topic will be given shortly. Such studies serve to establish benchmark values for the optoelectronic properties of thin GaN films on native substrates. Both p- and n-type dopings were achieved, and a p–n junction was realized by OMVPE on GaN crystals grown from Ga solution. The deposition temperature was 1000–1050  C. The width of rocking curves for epitaxial layers was practically the same as that of the substrate. The photoluminescence spectra were dominated by exciton emission at low temperatures, pointing to high quality of the epitaxial layers. There are, in general, two possible approaches for suitable substrates for homoepitaxial growth: The desirable one is to grow a large bulk crystal, cut it, and polish the slices. This method is widely used for conventional semiconductors. It cannot be applied easily to GaN bulk growth, however, because only small pieces of GaN can be grown only at high temperatures and very high pressures (tens of kbar), taking into account that GaN begins to decompose at 800  C. The second possibility that has been explored is the growth of thick GaN films on foreign substrates (sapphire, SiC, or Si). It is well known that the quality of the heteroepitaxial film improves as the thickness increases. Very popular epitaxial deposition methods, such as OMVPE and MBE, have very slow growth rates, a few microns per hour at best, and cannot be used for the growth of thick films. As discussed earlier, the inorganic CVD method has high growth rates (up to 100 mm h
1 or larger). Thus, it is argued that the best way to grow homoepitaxial GaN film is to use a two-stage growth, a thick GaN substrate grown by inorganic CVD in 1–2 h followed by the device layer grown by OMVPE or MBE. Early efforts of homoepitaxy relied on growth by MBE on buffer layers grown by OMVPE or HVPE. Owing to smooth surfaces obtained by these two vapor techniques and avoidance of complications brought about by heteroepitaxy, atomically smooth layers could be grown by MBE. This permitted delineation of regimes leading to smooth and rough surfaces, namely Ga-rich and N-rich growth conditions. We should mention that prior to GaN, growth by MBE was always conducted under slightly group V rich conditions. As mentioned previously, homoepitaxy also includes growth on GaN templates or on epitaxial layers grown by HVPE or OMVPE. In those cases, in addition to misfit dislocations, other threading defects originating at the GaN substrate interface, such as edge dislocations, screw dislocations, and mixed dislocations, propagate to the surface unless they run into one another and loop, and their Burgers vectors cancel leading to dislocation looping [828]. GaN and related heterostructures have been grown by MBE on GaN/sapphire templates prepared by OMVPE [318]. In this particular work, the samples grown under nitrogen-rich conditions are not conductive, whereas the others are and have room temperature mobilities approaching 1200 cm2 V
1 s
1. The mobility increases as the Ga flux increases and turns over above a certain group III/V ratio where the Ga droplets begin to form. The regions where the Ga droplets form are of lower quality and degrade the overall mobility measured. The optimum properties are reached

3.5 The Art and Technology of Growth of Nitrides

when this Ga-adlayer coverage is maximized without the formation of Ga droplets (i.e., at the highest Ga flux within the intermediate regime). Extending GaN growth to heterostructures, the AlGaN/GaN 2DEG system has been prepared on GaN templates. Samples over a broad range of electron densities, ranging from ns ¼ 6.9 · 1011 to 1.1 · 1013 cm
2 were grown with the best mobility of 53 300 cm2 V
1 s
1 at a density of 2.8 · 1012 cm
2, and temperature T ¼ 4.2 K [829]. Magnetotransport studies on these samples display exceptionally clean signatures of the quantum Hall effect [830–832]. The investigation of the dependence of the 2DEG mobility on the carrier concentration suggests that the low-temperature mobility in these AlGaN/GaN heterostructures is currently limited by the interplay between the charged dislocation scattering and the interface roughness. The typical MBE overgrowth consisted of a 0.5 mm undoped GaN buffer layer capped by approximately 30 nm of Al0.09Ga0.91N. Changing the thickness of the Al0.09Ga0.91N caused a variation of the sheet carrier concentration (formed by electrons released from defect centers on or near the surface of Al0.09Ga0.91N), which is the screening charge in response to polarization charges present at the interface. Recently, these mobility figures were extended to about 73 000 cm2 V
1 s
1 and beyond by MBE growth on HVPE GaN layers using a RF N source and also on GaN bulk platelets using ammonia as the reactive nitrogen source. A nominally undoped 1.5 mm layer was recently grown by MBE on the Ga-face of a freestanding GaN template [833]. The template, in turn, was grown by HVPE on a cplane sapphire substrate and separated from the substrate by laser liftoff. Before the overgrowth, the GaN template was mechanically polished, dry etched, and finally etched in molten KOH. A 30 nm thick undoped AlxGa1
xN cap layer with x ¼ 12% has been deposited on top of the 1.5 mm GaN layer. Part of the 10 mm · 10 mm surface of the sample was mechanically shuttered during the MBE process, so that both the GaN substrate and the MBE-overgrown layer could be studied under the same experimental conditions. Steady-state PL was excited with a He–Cd laser (325 nm), dispersed by a 1200 rules per millimeter grating in a 0.5 m monochromator and detected by a photomultiplier tube. The best resolution of the PL setup was about 0.2 meV, the photon energy was calibrated with a mercury lamp accounting for the refraction index of air (1.0003). The temperature was varied from 15 to 300 K in the closed cycle cryostat. The excitation density in the range of 10
4 to 100 W cm
2 was obtained by using unfocused (2 mm diameter) and focused (0.1 mm diameter) laser beam attenuated with neutral density filters. Structural properties of the GaN template and overgrown MBE epilayer were characterized by high-resolution X-ray diffraction and AFM. The FWHM of the oscan (rocking curve) for the [0 0 2], [1 0 2], and [1 0 4] directions are 53, 145, and 54 arcsec, respectively, for the MBE-grown area and 52, 137, and 42 arcsec, respectively, for the GaN substrate. The similar characteristics of the substrate alone and the substrate with MBE overgrowth indicate similar crystal quality of GaN both in the substrate and overgrown layer. A comprehensive characterization of the quality of the GaN templates has been reported elsewhere [834] and will not be repeated here. Suffice it to say that the template quality is unmatched in terms of transport and

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optical properties, and extended and point defects. The surfaces of both substrate and MBE-overgrown layer are very smooth, with the root mean square roughness of about 0.2 nm in the 2 mm · 2 mm AFM images. The homoepitaxial growth of GaN has recently been achieved by using GaN(0 0 0 1) bulk substrate [835,836]. No buffer layer was used for the growth. The GaN bulk substrate, which was used for the growth, was cleaned first by boiling in aqua regia for 10 min and then in organic liquid with an ultrasonic cleaner prior to the growth. A horizontal OMVPE reactor was employed at atmospheric pressure to grow the GaN layer. TMG and ammonia were the source gases and H2 was the carrier gas. For the growth of Mg-doped GaN, bis-cyclopentadienyl magnesium was utilized. To protect the GaN substrate from the escape of nitrogen at high temperatures (>800  C), ammonia was fed into the reactor before the beginning of the substrate heating. The as-grown n-GaN films thus obtained were found to be high-quality single-crystal films with good surface morphology. The as-grown GaN : Mg films showed strong blue cathodoluminescence and photoluminescence spectra peaking at 445 nm even without LEEBI treatment or thermal annealing. GaN homoepitaxial layers grown by OMVPE on the highly conductive GaN bulk crystals grown at high hydrostatic pressure exhibit smaller free-electron concentrations in the layers in contrast to the substrates, which had about 2.5 · 1019 cm
3 of free electrons [837]. In X-ray diffraction [0 0 0 2] peaks for the substrate and epitaxial layers had rocking curves with half widths of about 20 arcsec. The photoluminescence spectrum exhibited by the epitaxial layer was less than 1 meV wide. GaN layers grown by OMVPE on freestanding GaN templates grown by HVPE also showed excellent properties. X-ray reflections, with a FWHM of as low as 20 arcsec, were obtained. The dislocation density was determined to be 2 · 107 cm
2. The lattice mismatch between the GaN substrate and the homoepitaxial layer was below 3 · 10
5 and the PL linewidth was about 0.5 meV [838]. Much more refined experiments followed these early attempts, which resulted in GaN with excellent properties as determined by photoluminescence [839–841]. One could realize unstrained GaN layers with dislocation densities comparable with that in the template on which the layers are grown. In the case of platelets prepared by the high-pressure technique, this dislocation density is several orders of magnitude lower than the best conventional heteroepitaxy. Through the use of dry etching techniques for surface preparation, a pathway was blazed to achieve high crystal quality in the overgrown epitaxial film [839]. The layers so grown reveal an exceptional optical quality as determined by a reduction of the low-temperature PL linewidth from 5 to 0.1 meV and a reduced symmetric XRD diffraction rocking curve width from 400 to 20 arcsec. The latter is not surprising as the epitaxial layer replicates the structural properties of the template unless the surface preparation is not optimum. Narrow PL linewidths paved the way for observing fine structure of the donor-bound exciton line at 3.471 eV with five fine features inclusive of the excited states of free excitons. Additionally, all three free excitons as well as their excited states were visible in the photoluminescence spectrum at 2 K. A PL scan taken at 2 K is shown in Figure 3.157. Moreover, InGaN/GaN multiple quantum well (MQW) structures as well as GaN p–n and InGaN/GaN double heterostructure LEDs on GaN bulk

3.5 The Art and Technology of Growth of Nitrides

n=1

8. (Do,X An = 1) 9. (D o,X C

PL Intensity (au)

104

o

3. (Ao,X An = 1)

n=1

4. (D ,X A

n=1

o

n=1

5. (D ,X B

n=1

) 6. (XA

n=1

) 11. (X C

) 10. (X A 4

)

) 12. (XBn = 1) 13. (X Cn = 1)

o

(A ,X An = 1): 0.09 meV

PL and PR @2K 3

4

3 6

103

5 2

1 2

7

2

10

8

9

10 11 12

3.44

Reflectance (au)

o

n=1 ) 2e1. (D o,XAn = 1)2e- 2. (D ,X B

3.46

3.48

3.50

13

1 3.52

Energy (eV) Figure 3.157 Photoluminescence and reflectance scans of a homoepitaxial GaN layer grown on GaN platelet grown by the high-pressure technique. Courtesy of M. Kamp.

single-crystal substrates have also been prepared [839]. These particular LEDs were reported to be twice as bright as their counterparts grown directly on sapphire. In addition, they exhibited improved high power characteristics, which are attributed to enhanced crystal quality and increased p-type doping. Although very preliminary, high-quality GaN layers have also been grown on freestanding templates, which, in turn, are prepared by HVPE. The PL spectrum of an undoped GaN epilayer grown by OMVPE on such a template is shown in Figure 3.158 for a wide range of photon energies. Besides exciton-related peaks, which will be considered later, the spectrum contains two broad bands: a YL band with a maximum at about 2.3 eV and a blue luminescence (BL) band with a maximum at about 3 eV. The YL band is the omnipresent feature in PL spectra of n-type GaN grown by different techniques and it is most commonly attributed to a structural defect, namely a complex of the gallium vacancy with oxygen or silicon atom [842]. The BL band is most probably related to the surface states of GaN [843]. Unlike the BL often observed in HVPE- and OMVPE-grown GaN [844], the BL in the sample strongly bleached with the laser exposure time, very similar to the behavior observed earlier [843]. The bleaching is attributed to photo-assisted desorption of oxygen atoms from the GaN surface. Note that with increasing excitation intensity, the relative contribution of the YL and BL bands decreases because of the saturation of the corresponding defects with holes, and at highest excitation density, these bands can be barely detected in the spectrum (intensity is five orders of magnitude weaker than the peak intensity of the main exciton peak).

j609 n=1

) 7. (X B

)

j 3 Growth and Growth Methods for Nitride Semiconductors

610

10

9

T = 15 K

Sample

PL Intensity (au)

410 H01 10

8

10

7

10

6

3.44

3.45

3.46

3.47

3.48

3.49

3.5

Photon energy (eV) Figure 3.158 Excitonic spectrum of the undoped epilayer (sample H01) and freestanding substrate (sample 410). Excitation density is 100 W cm
2 (focused laser beam).

Excitonic spectrum of the undoped GaN epilayer is very similar to the spectrum of the freestanding GaN template, except some difference in relative intensities of some peaks. The peak at 3.4673 eV is attributed to the A-exciton bound to unidentified shallow acceptor (A0 ; X An¼1 ) [845]. The most intense peaks at 3.4720 and 3.4728 eV are attributed to the A-exciton bound to two neutral shallow donors, D1 and D2: (D01 ; X An¼1 ) and (D02 ; X An¼1 ). The FWHM of these peaks is 0.6 meV at 15 K and their positions are the same with an accuracy of 0.2 meV for the substrate and epilayer. The peaks at 3.4758 and 3.4766 eV, having the same intensity ratio and energy separation 0 n¼1 as the (D01 ; X n¼1 A ) and (D2 ; X A ) peaks, are attributed to the B-exciton bound to the D1 0 n¼1 and D2 donors: (D1 ; X B ) and (D02 ; X Bn¼1 ), in agreement with Refs [845,846]. Two other peaks, also related to donor-bound excitons (DBE), are observed at 3.4475 and 3.4512 eV and attributed to the so-called two-electron transitions: 0 n¼1 ðD01 ; X n¼1 A Þ2e and ðD2 ; X A Þ2e . This type of the DBE recombination, first observed in GaP [847], involves radiative recombination of one electron with a hole leaving the neutral donor with second electron in an excited state. In the effective-mass approximation, the donor excitation energy from the ground to the n ¼ 2 state is three fourth the donor-binding energy ED. Consequently, from the energy separation between the principal DBE line and the associated two-electron satellite, we can find the binding energies of two shallow donors: E D1 ¼ 4=3  24:5 ¼ 32:6 meV and E D2 ¼ 4=3  21:6 ¼ 28:8 meV. It appears reasonable that a weak peak at 3.443 eV is the two-electron transition related to the second excited state (n ¼ 3) of the D1 donor because its separation from the (D01 ; X An¼1 ) peak is 8/9  32.6 ¼ 29.0 meV. Free excitons (FE) related to the A and B valence bands (X An¼1 and X Bn¼1 ) are identified at about 3.479 and 3.484 eV, respectively (the X Bn¼1 peak is seen as a shoulder). From the separation between the FE and DBE peaks, we find the binding

3.5 The Art and Technology of Growth of Nitrides

energies of the DBEs related to the D1 and D2 donors as 7.0 and 6.2 meV, respectively. According to empirical Haynes rule, the binding energy of the DBE is proportional to the binding energy of the corresponding donor. The proportionality constant (a) for the D1 and D2 donors is found to be 0.215, which is close to the result of Meyer (a ¼ 0.2  0.01) [846]. A peak at 3.4983 eV is attributed to the n ¼ 2 excited state of the A exciton (X An¼2 ), in agreement with Refs [848,849]. From the energy positions of the X An¼1 and X An¼2 peaks (3.479 and 3.4983 eV, respectively), we find the A-exciton binding energy in the hydrogen model as 25.7 meV and the bandgap Eg ¼ 3.5047 eV. In short, optical studies of GaN grown on GaN templates, both by MBE and OMVPE, have been conducted. Submillielectron volt luminescence linewidths allowed the delineation of many excitonic transitions involving intrinsic transitions and their excited states and excitons bound to impurities such as donors with two electron transitions. Consequently, exciton-binding energies as well donor-binding energies could be determined accurately. The details can be found in Volume 2, Chapter 5. 3.5.10 Growth on Spinel (MgAl2O4)

Cubic MgA12O4 has a spinel-type structure (Fd3m) with the oxygen atoms forming a face centered cubic sublattice and Mg and Al atoms occupying the tetrahedral and octahedral sites, respectively. Lattice mismatch with GaN, Dd/d is 10%. The crystals are stable at the GaN growth temperature. The spinel substrate has an advantage over the sapphire substrate for obtaining mirror laser facets by cleaving [90,850,851]. GaN crystals were grown on (1 0 0) and (1 1 1) oriented MgA12O4 substrates by OMVPE. As in the case of sapphire substrates, low-temperature buffer layers are employed followed by a few micrometers thick GaN films grown at about 1000  C. As expected, GaN films grown on (1 1 1) substrates are wurtzitic single crystals. The crystallinity of the films on spinel is not comparable to that on sapphire. The only attractive feature of spinel is the cleaved facet prospect, which with the recent advances on sapphire and SiC is becoming a moot point. 3.5.11 Growth on Non c-Plane Substrates

The layers and heterostructures on polar c-plane orientation are associated with polarization-induced charge, owing to strain and in the case of heterostructures to compositional gradient, which, for example, causes electric field to be present in quantum wells and barriers and consequently leads to a quantum confined stark effect (QCSE) [852]. This also manifests itself as a red shift in emission energy such as in LEDs, the degree of which depends on the injection level with higher injection leading to a blue shift (attempting to negate the red shift) [853] (see Figure 3.159). Although this polarization has been used to induce 2DEG at the interfaces such as that between AlGaN and GaN for field-effect transistors without introducing intentional dopants, the same field causes a spatial separation of electrons and holes in quantum wells of LED, thereby increasing the radiative lifetime owing to spatial

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4.0

(a)

3.8

3.6 Energy (eV)

3.6 Energy (eV)

(b)

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3.35 eV 0.4

3.5

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0.2 0.1

0.2

0.0 0.0 0

5

10 15

20 25

Thickness (nm)

0

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10 15

20 25

Thickness (nm)

Figure 3.159 Calculated band profiles in (5 nm GaN)/(10 nm Al0.1Ga0.9N) quantum wells by self-consistent effective mass Schr€ odinger–Poisson calculations. (a) The very large electrostatic fields in the [0 0 0 1] orientation result in a quantum confined Stark effect and poor electron-hole overlap. (b) The ½1 1 0 0 orientation is free of electrostatic fields, thus true flat-band conditions are established. Patterned after Ref. [853].

separation of carriers [854] and reducing the quantum efficiency [855]. The increased carrier lifetime and reduced recombination efficiency are more noticeable at low injection levels. At high injection levels, screening reduces the field and thus increases the wave function overlap between the electron and hole states. Moreover, the existence of spontaneous and piezoelectric polarization effects can induce large densities of free electrons or in principle holes (has not been observed irrefutably) at nitrides interface, which can sometimes be much larger than that induced by intentional doping. This can therefore reduce our ability to control the free carrier concentration at the interface. Tocircumventthisproblem,onecangrowcubicratherthanhexagonalGaN,whichis nonpolar along the cubic [0 0 1] direction, and therefore can avoid strong polarizationinduced electric field in heterointerfaces. Unfortunately, cubic GaN is metastable and also very difficult to achieve with quality comparable with that obtained for the wurtzite phase. Seemingly, more a attractive approach is to explore the a- or m-plane hexagonal GaN structures rather than c-plane GaN. The GaN layers with nonpolar orientation are termed nonpolar GaN, because the c-axis is parallel to the substrate surface and there would then be no polarization-induced electric field at nitrides interfaces. Figure 3.160 shows schematically the m-, a-, and c-plane of GaN. These planes are perpendicular to each other. However, as always, the case 1 problem is traded with another. Although growth GaN on SiC follows the substrate orientation (e.g., a-plane GaN results on aplaneSiCandm-planeGaNresultsonm-planeSiC),thesameisnottrueforsapphire.As discussed early on in this chapter, GaN on a-plane sapphire by MBE and OMVPE is low quality c-plane. Toobtain a-plane GaNonsapphire r-plane sapphire isused. Inaddition, growthonm-planesapphiredoesnotleadtoc-directiontobealongthesubstratesurface

3.5 The Art and Technology of Growth of Nitrides

Figure 3.160 Schematic representation of m-, a-, and c-planes of GaN. (Please find a color version of this figure on the color tables.)

plane. Instead, it is off the plane, leading to semipolar as opposed to nonpolar surfaces. Thegenesisofthesepeculiaritiesonsapphireisnotwellknownandrequiresagooddeal of investigation. Degrading device performance, however, is that the stacking fault formation energy in a-plane GaN is very low leading to high concentration of stacking faults that are radiative recombination killers. In addition, In incorporation of InN into the GaN lattice is difficult. Further, the LEDs fabrication on this plane provides optical powers in the submilliwatt range as opposed to 20 mW range obtained for c-plane. InN incorporation in m-plane GaN is much easier and LED power levels close to 1 mW are attainedinthedevelopmentandproductionlaboratories[856]andtensofmicrowattsin journal publications [857]. However, the forward turn on voltage is high (which may havetodowiththelowp-typedoping)andsurfaceisrelativelyrough,2 nm,asopposedto atomically smooth on c-plane, although turn-on voltages of about 3.2 V have been reported [857]. To improve the surface morphology, tilted substrates are used. The researchisitsearlystagesandmoredevelopmentsareneededforadefinitivestatement. 3.5.11.1 The a-Plane GaN Growth Usually ð1 1 2 0Þa-plane GaN is grown on ð1 1 0 2Þr-plane sapphire, a-plane SiC or aplane g-LiAlO2. Before the c-plane GaN gained the dominant status in GaN research, some early research has been devoted to the growth of a-plane GaN using OMVPE [858–860] and MBE [861–864]. In fact, growth of GaN by MBE on c-, r-, a-, and m-plane sapphire substrates has been explored in a systematic study that included the investigation of A1 and E1 phonon modes by Raman scattering and excitonic transitions by PL and photoreflectance on all the aforementioned surfaces

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[865]. But the crystalline quality was not so good on non-c-plane sapphire so the potential for device fabrication could not be realized. Owing to the maturity and to a great extent the built-in polarization issues of c-plane GaN, the growth of nonpolar GaN, that is, a-plane or m-plane GaN, has gained considerable interest. The epitaxial relationship of ð1 1 2 0Þa-plane GaN on ð1  1 0 2Þr-plane sapphire is ½1 1 2 0 GaNjj½1 1 0 2 sapphire, ½0 0 0 1 GaNjj½ 1 1 0 1 sapphire, and ½ 1 1 0 0 GaNjj ½1 1 2 0 sapphire [866] (refer to Figure 3.161). Craven et al. [866] have grown a-plane GaN on r-plane sapphire using OMVPE. Their cross-sectional TEM (see Figure 3.162a) results show that there was a large density of threading dislocations (TDs) originating at the sapphire/GaN interface with line directions parallel to the growth direction ½1 1 2 0. The TD density determined by plan view TEM (see Figure 3.162b) was 2.6  1010 cm
2. Pure screw dislocations will have Burgers vectors aligned along the growth direction (b ¼ ½1 1  2 0), whereas pure edge dislocations will have b ¼ ½0 0 0 1. In plan view TEM, stacking faults have also been found, with a direction parallel to [0 0 0 1]. In SEM images (see Figure 3.163), after full coalescence, we can observe surface striations that are uniformly aligned parallel to the GaN [0 0 0 1] direction [867]. The growth in c-direction is larger than in the ð1 1 0 0Þ direction, accounting for the elongated features to be aligned in the [0 0 0 1] direction. Striations may also point to some stacking faults. Omega rocking curves were measured for GaN on-axis ð1 1  2 0Þ and off-axis ð1 0  1 1Þ planes showing anisotropy with FWHM values of 0.29 –0.52 and 0.29 –0.43 ,

0

1]

Sapphire [0 0 0 1]

N

[0

0

GaN [0 0 0 1]

Ga

Sapphire [0 1 1 1]

c/2

GaN [1 1 0 0] Sapphire [2 1 1 0]

Figure 3.161 Epitaxial relationship between a-plane GaN and rplane sapphire. The hexagon lying on the r-plane of sapphire depict the GaN crystal. Note the a-plane is formed on r-plane sapphire with huge lattice distortions.

3.5 The Art and Technology of Growth of Nitrides

Figure 3.162 Cross-sectional TEM image (a) and plan view (b) TEM image of a-plane GaN on r-plane sapphire. The TDs in image (a) have a common line direction parallel to ½1 1 2 0. The stacking faults in image (b) have a common faulting plane parallel to the (0 0 01). Courtesy of J.S. Speck. Ref. [866].

respectively. Compared to c-plane GaN, the crystalline quality for a-plane GaN is not very good and needs to be optimized further for devices purpose. Several research groups have grown a-plane AlGaN/GaN MQWs on r-plane sapphire and made comparative studies with that of c-plane structures, showing the absence of polarization-induced electric filed in nonpolar GaN [868]. Ng [869] has grown GaN/AlGaN MQWs on r-plane sapphire using MBE technique. Roomtemperature photoluminescence results show that the photoluminescence intensity was 20–30 times higher for the ð1 1 2 0Þ MQWs compared to the (0 0 0 1) MQWs. This particularly holds at low injection levels as high injection levels screen the polarization-induced charge, the details of which are discussed in Volume 2,

Figure 3.163 SEM surface morphological evolution during early stage of a-plane GaN growth: (a) LT GaN buffer, (b) 20 nm GaN, (c) 50 nm GaN, (d) 100 nm GaN, (e) 300 nm GaN, (f) 1.5 mm fully coalesced GaN. Note that c direction is the stripe direction shown in (f), and all samples here have their c directions parallel to each other.

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Chapter 5. The fact that the peak transition energy, as a function of well width for the ð1 1 2 0Þ MQWs, followed the trend for rectangular potential profiles indicated the absence of built-in electrostatic fields. Craven et al. [870] confirmed the advantage of nonpolar GaN. The PL emission energy from a-plane MQWs followed a square well trend modeled using self-consistent Poisson–Schr€ odinger calculations, whereas the c-plane MQW emission showed a significant red shift with increasing well width, which is attributed to the quantum-confined Stark effect. Despite a higher dislocation density, the a-plane MQWs exhibit enhanced recombination efficiency as compared to the c-plane wells because well emission is no longer observed for c-plane wells wider than 50 Å. Because p-type doping (Mg coefficient) may be sensitive to polarity, p-type doping of a-plane GaN has been carried out for the first time by Armitage et al. [871]. The electron mobility in a-plane GaN : Si was only 18 cm2 V
1 s
1, compared to 200 cm2 V
1 s
1 for c-plane epilayers of similar Si doping level. On the contrary, the maximum conductivity of p-type a-plane GaN : Mg was as high as was typically found for c-plane GaN:Mg. A maximum hole concentration of 6 · 1017 cm
3 was found, with mobility of 2 cm2 V
1 s
1. A comparison suggested that the Mg sticking coefficient may be higher in ð1 1 2 0Þ GaN than in ð0 0 0  1Þ GaN. So the control of Mg doping will be easier in a-plane than in c-plane epilayers. Their result basically proved that sufficiently high p- and n-type doping can be achieved for applications in light-emitting diodes. Chakraborty et al. [872] also reported their research about ptype doping of a-plane GaN using OMVPE technique. Their maximum hole concentration achieved was 6.8 · 1017 cm
3. They also found that higher Mg incorporation could be achieved with increasing growth rate, higher V/III ratio, and lower growth temperature. In-plane anisotropy is an important characteristic for a-plane GaN when compared to c-plane GaN. In the surface morphology of a-plane GaN, several groups [873,874] have observed [0 0 0 1] oriented stripes features on the surface of a-plane GaN and discussed the anisotropic properties of GaN. Wang et al. [873] attributed the stripe features to the unequal growth rates along ½1  1 0 0 and [0 0 0 1] directions. The overall growth rate in [0 0 0 1] direction was higher than that in ½1  1 0 0 direction, resulting in a growth front with stripe features. The FWHM values of XRD o-scan were different with different azimuth angle, which is defined with respect to the [0 0 0 1] direction (see Figure 3.164). Li et al. [874] observed in-plane anisotropy of electronic property of a-plane GaN grown by OMVPE, in addition to the crystallographic anisotropy. The electron mobilities in the [0 0 0 1] and ½1  1 0 0 directions were 10 and 6 cm2 V
1 s
1, 18 respectively, when doped to 1.0  10 cm
3. 3.5.11.2 Epitaxial Lateral Overgrowth of a-plane GaN Although a variety of techniques have been demonstrated to grow a-plane GaN, the crystalline quality should be improved further for the fabrication of optoelectronic and other devices. ELO is a good technique in this direction. The lateral epitaxy on aplane GaN is interesting in that windows in the mask material are made parallel to ½1 1 0 0 and the growth emanating in the window regions extends laterally along the relatively fast growth [0 0 0 1] direction, representing the Ga-polar growth front or

3.5 The Art and Technology of Growth of Nitrides

Template

Intensity (a.u.)

5

(a)

Azimuth 0o

4 30o 3

60o

2

90o

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-0.5

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Δω (o) 0.6

Template

(b)

FWHM (o)

0.5

0.4

0.3

0.2 0

300 100 200 o Azimuth angle ( )

400

Figure 3.164 XRD ð1 1 2 0Þ Omega scan of a-plane GaN with different azimuth angles (a), and the FWHM values of ð1 1 2 0Þ Omega scan as a function of azimuth angles (b). Courtesy of M.A. Khan and Ref. [873].

½0 0 0 1 direction, representing the N-polar growth front. These two fronts advance toward one another and meet somewhere on the masked region forming the coalescence front, as shown in Figure 3.165a–c. Also to be noted in the figure is that Ga-front advances a factor of 3 faster than the N-front leading to shift in the coalescence point from the center line of the masked region. A scan of the yellow optical emission band across the wafer by near field scanning optical microcopy (NSOM) in a 20 mm  20 mm area is also consistent with the topographic images obtained by AFM in that the N-face front grows at about one third the rate as compared to the Ga-face front (Figure 3.165c). Note the window region in the NSOM image is dark and the dark line running parallel to the stripes in the wing region corresponds to the coalescence front.

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Figure 3.165 Cartoon of epitaxial lateral overgrowth (ELO) on a-plane GaN with growth along the [0 0 0 1] direction, representing the Gapolar growth front and along the ½0 0 0 1 direction, representing the N-polar growth front: (a) top view and (b) side view. The growth along the Ga-polar front is about a factor of g ¼ 3 times faster. Courtesy of VCU students Vishal Kasliwal

and Xianfeng Ni. (c) Left 30 mm · 30 mm AFM image for sample B. Right 4 mm · 4 mm AFM image near the window and N-polar wing boundary of sample B, showing different surface pit densities for the window and the wing. Courtesy of VCU students Vishal Kasliwal and Xianfeng Ni. (Please find a color version of this figure on the color tables.)

3.5 The Art and Technology of Growth of Nitrides

As in the case of c-plane GaN ELO, wing tilt is observed that can be reduced by adopting a two-stage growth where lower substrate temperature is employed for the first stage with relatively low lateral-to-vertical growth rate ratio [875]. This is followed by increased temperature growth that enhances the lateral growth rate over the vertical one. Because the gap to the bridged laterally is smaller, the height difference between the advancing Ga- and N-fronts at the coalescence boundary is smaller as shown in Figure 3.166. The wing tilt is also influenced by any miscut of the r-plane sapphire and deviations of the template surface from the exact a-plane GaN template. XRD rocking curves have been obtained with three different f angles, where f is the angle of rotation about the sample surface normal and is defined as 0 when the projection of incident X-ray beam is parallel to the SiO2 mask stripes, top determine the wing tilts. As shown in Figure 3.167a and b, for f ¼ 0 , only one diffraction peak from the a-plane of GaN can be observed for both samples, with a FWHM of 0.40 and 0.19 for samples A (single-step growth) and B (two-step growth), respectively. For f ¼ 90 (the projection of incident X-ray beam is normal to the mask stripes), however, samples A and B exhibit two and three peaks, respectively, the order of which is reversed for f ¼ 270 . The strongest peak observed for sample A is that from the Ga-polar wings, since the area of the Ga-polar wing is much larger than that of the N-polar wing and the window. The observed tilt angle of 0.86 is much larger than 0.25 obtained from the large angle convergent beam electron diffraction (LACBED), which may be attributed to the local nature of LACBED. The peak with the strongest intensity, shown in

Figure 3.166 Schematics of the height difference between two neighboring wings on a-plane GaN (not in exact proportion), with Ga- to N-polar wing width ratios of (a) 5 : 1 and (b) 1.6 : 1. (c) Schematic showing the inclination of the growth planes due to the 1.05 miscut of the r-plane sapphire toward its [0 0 0 1] c-axis [875].

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Figure 3.167 XRD rocking curve data for (a) sample A and (b) sample B with different f angles. The dashed lines correspond to the multiple Gaussian fits to the rocking curve data [875].

Figure3.167b, is again fromthe Ga-polar wings (0.44 tilt), while the central oneis from the crystal plane in windows and third is from the N-polar wings (0.37 tilt). NSOM measurements have been carried out on sample B. Figure 3.168a and b shows the AFM and NSOM images, respectively, taken from a 40 mm · 40 mm area. The window and wing regions and meeting fronts are clearly distinguishable in the NSOM image of Figure 3.168b where the windows appear dark straddled by bright wings. The narrow and the wide bright wings are because of N- and Ga-polar wings, respectively. Part of the observed PL intensity degradation at the meeting fronts may be because of an artifact associated with a probe-size-related effect, as a weak drop in intensity has been observed at the meeting fronts in the near field reflection measurements. However, the PL intensity variations in the window and the wing regions are inherent to the samples. The increased PL intensity in the wing regions suggests improved l quality by defect reduction. After their demonstration of epitaxial growth of a-plane GaN on r-plane sapphire, Craven et al. [876] reported laterally overgrown a-plane GaN, as demonstrated in Figure 3.169 in the form of a cross-sectional TEM image. They found that the

Figure 3.168 (a) AFM and (b) NSOM scans from a 40 mm · 40 mm area of a-GaN ELO sample B [875]. (Please find a color version of this figure on the color tables.)

3.5 The Art and Technology of Growth of Nitrides

Figure 3.169 Cross-sectional bright field TEM images of laterally overgrown a-plane GaN near the edge of SiO2 pattern (encompassing both window and overgrown regions) for ½ 1 1 00  (a and b the latter being an expanded view) and [0 0 0 1] (c) stripes. The dislocation lines are seen to bend from the window region into the

overgrown region for stripes aligned along [0 0 0 1]; whereas no dislocation bending is observed for ½1 1 0 0 stripes. The diffraction conditions are g ¼ ½1 1 2 0 for (a) g ¼ 0 0 0 6 for (b) and g ¼ 0 1 1 0 for (c). Courtesy of J.S. Speck and Ref. [876].

optimum stripe orientation of SiO2 mask is that along the ½ 1 1 0 0 direction. Reminiscence of pendeo-epitaxy, the a-plane GaN can be fully removed and growth can be initiated from resultant walls ((0 0 0 1) on one side and ð0 0 0  1Þ plane on the other) to bypass seed growth on relatively low-quality a-plane GaN. Threading dislocating densities by this method in the range of 106–107 cm
2, which are large, have been reported [877]. The evolution of growth is different from that on c-plane [875]. Shown in Figure 3.170 is the evolution of an overgrown GaN sample as viewed by plan view and cross-sectional SEM images after 2 and 5 h of growth. As displayed in Figure 3.168, the PL intensity is also much stronger in the wing regions of ELO a-plane GaN samples than that of the a-plane GaN template. From AFM and NSOM data, it is clear that a significant reduction in dislocation density in the wing region occurs. Laterally overgrown a-plane GaN on r-plane sapphire has also been reported by Chen et al. [878] also achieved fully coalesced overgrown GaN with smooth surface with a root mean square roughness as low as 4.6 Å for a 5 mm · 5 mm sample. Using an improved selective area lateral epitaxy for the growth of a-plane GaN [879], the crystallinity was further improved, with the FWHM value of XRD rocking curve for ½1  1 0 0 decreased from 0.17 to 0.09 . Conductive AFM (CAFM) can be used to produce premature current conduction maps over the surface area (see Section 4.2.5 for a short discussion of the method). Again the point should be made that the device formed by the CAFM tip (Schottky barrier) and the ohmic contacts on the sample must be biased in such a manner as to

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Figure 3.170 SEM images of ELO a-plane GaN samples on rplane sapphire. (a) Plan view and (b) cross-sectional SEM image for an a-plane GaN ELO sample after 2 h of growth at 1000  C. (c) Plan view and (d) cross-sectional SEM images for this sample after additional 3 h growth at 1050  C.

have only the defective (point defects) region(s) to conduct. Under this premise, the region over the windows (presumed to be more defective) and the line of coalescence (over the mask) would conduct first. The complicating factors involve much faster growth along the [0 0 0 1] direction as compared to the ½1  1 0 0 direction during the growth. This is true of course during the buffer layer growth with its own set of complications for natural lateral growth for complete coalescence. Therefore, when ELO is performed on top of this buffer layer, the ELO imposed surface features are added on top of the buffer features with its sets of point defect arrangement making the analysis of already complicated CAFM. In this spirit, we show topographic and CAFM images of an a-plane ELO GaN grown on r-plane sapphire substrate in Figure 3.171 for a reverse bias voltage of 12 V. The meeting front as well as the window regions appear darker than the wing regions. Assuming that the results are reproducible, this would imply that the defective regions are very highly resistive, which is why the lack of current, whereas the wing region is conductive. This goes counter to the underlying premise that the premature conduction in lower quality films should appear. Improvements in the growth procedure led to films where the meeting fronts and the windows represent the premature current conduction paths. As in the case of the growth on c-plane, ELO improves the layer quality, particularly when attention paid to growth conditions that affect the growth rates of Ga- and N-polarity growth fronts. When a two-stage growth process is used for the ELO process, the first promoting vertical growth and the second promoting lateral growth, much reduced wing tilts and thus improved quality has been obtained [880]. TRPL measurements conducted on standard a-plane GaN indicated a radiative recombination lifetime of about or less than 50 ps (system response). With one-stage ELO (two-stage leads to much better films) the radiative recombination lifetime improved to about 200 ps, but still much lower than about 2 ns observed in very best layers grown with ELO using nanonetworks. Koida et al. [881] confirmed the advantage of laterally grown nonpolar GaN owing to the absence of polarization-induced electric fields in quantum wells and higher crystalline quality. Their results also showed that the AlGaN/GaN MQWs grown on

3.5 The Art and Technology of Growth of Nitrides

Figure 3.171 AFM topographic image (a) and conductive AFM current map (b) forward direction and (c reverse direction) of an ELO GaN layer having 4 mm windows separated by 10 mm SiO2 masked stripes perpendicular to the [0 0 0 1] direction. Due to the experimental setup and the

instrument, there is a large voltage drop extrinsic to the tip-sample device, and the dark and bright spots in forward and reverse bias directions, respectively, correspond to larger values of the current. Courtesy of VCU students Chris Moore and Xianfeng Ni.

GaN prepared by ELO can have higher quantum efficiency compared to conventional a-plane MQWs. The above-mentioned reports show that ELO technique is a very promising way to reasonably grow high-quality a-plane GaN for the purpose of highperformance optoelectronic devices. In addition to using r-plane sapphire substrate for the growth of a-plane GaN, it can also be grown on a-plane SiC substrate. Craven et al. [882] reported the successful growth of a-plane GaN growth on a-plane SiC. The crystalline relationship between the substrate and GaN was [0 0 0 1]SiC||[0 0 0 1]GaN, and ½1  1 0 0SiCjj½1  1 0 0GaN. 3.5.11.3 The ð1  1 0 0Þ m-Plane GaN Growth Growth of GaN m-plane is of interest as the lattice mismatch on the face with m-plane sapphire is relatively small (3%) and it is nonpolar. In this vein, the growth of GaN on on-axis m-plane ð1 1 0 0Þ sapphire, m-plane sapphire rotated around the c-axis by as much as 20 , g-LiAlO2, and m-plane SiC [883,884] has been pursued. Although the surface roughness on the order of 2 nm has been obtained on on-axis m-plane sapphire (albeit without any observation of atomic steps that are characteristic of cplane GaN), the tilt of the wafer surface from the m-plane around the c-axis has been reported to improve the quality and surface morphology of GaN [885]. In spite of the fact that Matsuoka and Hagiwara [885] reported that the tilt angles of 15 and 20 led to single-crystalline GaN growth, growth experiments since then indicated not only

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single crystalline GaN but also reasonable LED performance for growth on on-axis mplane [856]. However, growth on exact ð1  1 0 0Þm-plane sapphire does not lead to cdirection of GaN to lie on the m-plane and leads to not fully nonpolar but partially polar (semipolar) surfaces [886]. Baker et al. [886] reported planar GaN films of ð1 0  1 3Þ and ð1 1  2 2Þ grown on ð1 0 1 0Þm-plane sapphire. The in-plane epitaxial relationship for ð1 0  1 3Þ GaN was reported to be ½3 0 3 2GaNjj½1 2 1 0sapphire and ½1  2 1 0GaNjj½0 0 0 1sapphire, whereas that for ð1 1 2 2Þ GaN was ½1 1  2 1GaNjj½0 0 0 1sapphire and ½1 1 0 0GaNjj½1 2 1 0sapphire. The ð1 1  2 2Þ films, however, were reported to have N-face sense polarity and a threading dislocation density of 9  108 cm
2. On the contrary, the ð1 1 2 2Þ films were noted to have Ga-face sense polarity and a threading dislocation density of 2  1010 cm
2. The basal plane stacking fault density was recorded at 2  105 cm
2 for both orientations. The investigation by Ni et al. [887], however, indicated that only the semipolar ð1 1  2 2Þ GaN film orientation results in nominally on-axis ð1 0 1 0Þm-plane sapphire substrates, regardless of the III/V ratio used. HRXRD results indicate a preferred ð1 1  2 2ÞGaNjjð1 0  1 0Þsapphire orientation as shown in Figure 3.172a, which depicts the on-axis 2y–o scans supporting the aforementioned epitaxial alignment. The inset of the figure suggests that ð0 0 0 2ÞGaN is oriented 180 away from ð0 1  1 2Þsapphire. The GaNð1 1  2 2Þ plane forms an angle of 58.4 with ð0 0 0 2ÞGaN, whereas the sapphire ð0 1  1 2Þ plane forms an angle of 32.4 with the sapphire ð1 0 1 0Þm-plane. After combining these results, one determines the epitaxial relationships for GaN on m-plane sapphire, as shown in Figure 3.172b, namely, ð1 1 2 2ÞGaN jjð1 0  1 0Þsapphire, ½1 0  1 0GaNjj½1  2 1 0   sapphire, ½1 2 1 1GaNjj½0 0 0 1sapphire. While on the topic of ð1 1 2 2Þ plane GaN, InGaN/GaN MWQ LED structures have actually been grown and investigated on ð1 1  2 2Þ oriented bulk GaN substrates (produced by cutting thick HVPE GaN templates) [888] in which case the epitaxial layer follows the orientation of the substrate [889]. Free A excitons were reported to dominate the PL spectrum at 10 K and with a weaker but sharp doublet emission associated with neutral donor-bound excitons. The PL decay obtained at 428 nm was fitted with the double exponential decay form with characteristic lifetimes of 46 and 142 ps at 10 K. These values are two orders of magnitude shorter than those in coriented QWs and signify not only the reduced polarization induced internal field but also increased nonradiative processes. The E field of emission from GaN bulk and MQWs was found to be polarized parallel to the ½1  1 0 0 direction, with polarization degrees of 0.46 and 0.69 , respectively, owing to the low crystal symmetry and dispersion in crystalline orientation. Packaged LEDs in epoxy with 320 mm  320 mm area exhibited output power and external quantum efficiency (EQE) 1.76 mW and 3.0%, respectively, for the blue LED, 1.91 mW and 4.1% for the green LED, and 0.54 mW and 1.3% for the amber LED at 20 mA. At 200 mA, the output powers were 19.0 mW (blue), 13.4 mW (green), and 1.9 mW (amber) with associated EQEs of 4.0% at 140 mA (blue), 4.9% at 0.2 mA (green), and 1.6% at 1 mA (amber). The details of GaN-based LEDs fabricated on c-plane are discussed in Volume 3, Chapter 1. In the investigation of Matsuoka and Hagiwara [885], the GaN layers grown on the surface tilted 15 from the m-plane around the c-axis exhibited a very smooth surfaces

3.5 The Art and Technology of Growth of Nitrides

Figure 3.172 (a) XRD 2y–o scan for GaN grown on m-plane sapphire, with the inset showing the off-axis f scans with different c tilt angles (i.e., pole figure) for GaN (0 0 0 2) and sapphire ð0 1  1 2Þ of GaN ð1 1 2 2Þ on m-plane sapphire. (b) A schematic representation depicting the epitaxial relationship derived from XRD measurements for ð1 1 2 2Þ GaN on m-plane sapphire.

and pure near band edge emission with strong PL intensity. The dislocation density in this GaN was reported to be 50% higher than in GaN grown on c-plane, which is expected to improve with further developments. The variation of the surface morphology of GaN on on-axis and vicinal m-plane sapphire grown by OMVPE is shown in Figure 3.173. To reduce the defect density, ELO has been carried out using HVPE [890], resulting in basal plane stacking fault and threading dislocation densities of less than 3 · 103 cm
1 (from 105 cm
1) and5 · 106 cm
2 (from109 cm
2),respectively, in the Ga-face (0 0 0 1) wing regions of the ELO material. These values are better than those reported for the a-plane GaN, making the m-plane GaN very promising for device applications. It should be mentioned that m-plane ELO GaN films were grown on MBE m-plane GaN films, which, in turn, are grown on either g-LiAlO2(1 0 0) or 6H-SiCð1  1 0 0Þ substrates, as the c-direction of the GaN films grown on ð1 1 0 0Þ sapphire does not lie on the ð1 1 0 0Þ m-plane of sapphire. The growth experiments were carried out in a three-zone horizontal directed-flow HVPE reactor. Moreover, unlike the ELO process on c- and

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Figure 3.173 Variation of the GaN surface morphology grown by OMVPE on sapphire with the surface tilted 0 , 8 , 15 , and 20 from m-plane around the c-axis of sapphire as observed with a differential interference optical microscope. Courtesy of T. Matsuoka.

a-plane GaN, X-ray diffraction experiments did not detect any wing tilt. Vertical growth rates ranged from 40 to 300 mm h
1 at substrate temperatures of 860–1060  C. Figure 3.174a shows a plan view SEM image for the m-plane ELO sample with SiO2 stripes oriented along the a-axis of the sapphire substrate obtained by Ni et al. [887]. The cross-sectional SEM image of Figure 3.174b shows that the wings for this mask orientation are inclined by 32 with respect to the substrate plane with well-defined aand c-plane surfaces. The growth along the c-axis advances faster than others. Therefore, the observation of inclined wings is consistent with the epitaxial relationships shown in Figure 3.172, which suggest a 32.4 angle between the c-axis of GaN and c-axis of sapphire. The upwardly inclined wings are suggested to be because of the Ga-polar (0 0 0 1) wings, since their growth fronts (GaN c-plane) are smooth. Because of the large incline angle, only the Ga-polar wings extend, whereas the Npolar wing growth is stymied by the template. With further growth, the Ga-polar wings advanced along the c-axis of GaN with negligible growth along the a-axis. The m-plane GaN growth has also been reported on g-LiAlO2(1 0 0) substrate rather than sapphire substrate. The epitaxial relationship between the m-plane GaN and LiAlO2 substrate is ½1 1 0 0GaNjj½1 0 0LiAlO2 and ½1 1  2 0GaNjj½0 0 1LiAlO2 [891]. Waltereit et al. [853] reported on their results concerning the successful growth of m-plane GaN on g-LiAlO2(1 0 0), which was free of polarization-induced electric field. Sun et al. [891] from the same research institute reported that the stoichiometric ratio of Ga and N for the GaN nucleation layer would influence the phase of subsequently grown nonpolar GaN layers. Chen et al.[878,879] reported their result on growth of m-plane GaN on freestanding ð1 1 0 0Þ GaN template, which was prepared by HVPE on g-LiAlO2(1 0 0) substrate. They also showed that the emission of AlGaN/GaN MQWs grown on this m-plane GaN template did not exhibit any quantum-confined stark effect. Researchers also observed in-plane anisotropic optical properties in m-plane GaN on g-LiAlO2(1 0 0) substrate. Ghosh et al. [892] studied the polarization dependence of absorption, reflectance, and photoreflectance spectra of compressively strained

3.5 The Art and Technology of Growth of Nitrides

Figure 3.174 (a) Plan view, (b) cross-sectional SEM, and (c) crosssectional TEM images of GaN ELO sample with SiO2 stripes oriented along the a-axis of sapphire. (d) Plan view, (e) crosssectional SEM, and (c) cross-sectional TEM images of the GaN ELO sample with SiO2 stripes oriented along the c-axis of sapphire.

m-plane GaN. They observed different optical energy bandgaps when varying the measurement angle with respect to the c-axis [0 0 0 1], which was attributed to the influence of in-plane compressive strain. Moreover, in-plane anisotropy of PL intensity for m-plane AlGaN/GaN MQWs was found by Rau et al. [893] and Kuokstis [894]. Sun et al. [895] studied the polarization anisotropy of PL for m-plane InGaN/GaN MQWs and found similar anisotropic phenomenon. The details of optical processes in m-plane GaN are discussed in Volume 2, Chapter 5. 3.5.12 Growth of p-Type GaN

Until 1989, efforts to achieve p-type GaN led to compensated high-resistivity material. Activation of Mg doping for p-type conductivity has been responsible for placing GaN and related materials in a commanding position as light emitters [896,897]. An inordinate number of early investigations were directed toward the potential of Zn as

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a p-type dopant in GaN. These investigations demonstrated that Zn impurities effectively compensate the semiconductor and result in high-resistivity material. All Zn-doped samples exhibited the commonly observed 2.8 eV emission with heavily doped samples, which exhibit room-temperature peaks at 1.8–1.9, 2.2, and 2.5–2.6 eV. A number of researchers have investigated Mg doping that is known to effectively compensate GaN. As the Mg concentration is increased, the Mg-related emission peak broadens and shifts to lower energies. Attempts to dope GaN with Cd failed to produce p-type conductivity, even though a Cd-related peak at 2.85 eV was observed. As for Be, early attempts led to compensated material as well while yielding high-resistivity GaN. Later efforts led to possibly p-type material but the hole concentrations obtained were too low to be measured by the Hall effect, although p–n junctions have been made, leaving Mg the only practical p-type dopant. Hg doping was also investigated with no electrical measurements. Hg, too, leads to optical transitions at energies much lower than the band edge, about 0.4–0.8 eVabove the valence band edge. Carbon has also been tried and produced only deep-level transitions reaching even the yellow region with no electrical conductivity. Li atoms have gone the same way and produced only compensated material with optical transitions of about 750 meV below the band edge [369]. A common point is that all p-type dopants including Mg, particularly at high concentrations, can lead to donorlike levels that act to compensate the semiconductor. In a sense, unsuccessful attempts end up generating defects and only when the undoped layers have a low background donor concentration, which is also applicable to the case of MBE growth, can the available data point to active p-type doping. The deep transitions have, in fact, been exploited to produce blue emitters before reliable p-type doping was realized. First achieving and then controlling p-type doping represented a formidable challenge for semiconductor nitride researchers. The first breakthrough came when Amano et al. [898] converted compensated Mg-doped GaN into conductive p-type material by low energy electron beam irradiation (LEEBI). Nakamura et al. [897] improved upon these results using LEEBI to achieve GaN with p ¼ 3 · 1018 cm
3 and a resistivity of 0.2 O cm with the follow-up of thermal annealing at 700  C under an N2 ambience, which converted the material to p-type equally well. Owing to the high binding energy (150–200 meV) of Mg, acceptor activation ratios of only 10
2 to 10
3 are typically achieved and require Mg chemical concentrations in the 1020 cm
3 range. However, OMVPE samples require postgrowth activation, the MBE-grown samples do not. Hydrogen has long been suspected of passivating Mg atoms and is driven off in the subsequent annealing treatment. The general issue of p-type doping in general and Mg doping in particular is discussed in Section 4.9.3. Theoretical insights have provided a plausible explanation for the success of the Mg acceptor in GaN compared to the other group II materials, which continue to compensate GaN even after LEEBI or annealing. Owing to the strong binding of the nitrogen anion, group III nitrides are considerably more ionic than typical group III semiconductors. Their calculated band structures resemble II–VI semiconductors in many ways, including a large splitting between upper and lower valence bands (LVBs). The Ga(3d) core-level energies have been predicted and observed to

3.5 The Art and Technology of Growth of Nitrides

overlap in energy with the N(2s)-like LVB states as a result of the LVBs being deeper in GaN than in GaAs, which is a more typical group III semiconductor. The resulting energy resonance causes the Ga(3d) electrons to strongly hybridize with both the upper and lower valence band s- and p-levels. Such hybridization is predicted to have a profound influence on GaN properties including such quantities as the energy bandgap, lattice constant, acceptor levels, and valence heterojunction offsets. It is known, in the cases of ZnS and ZnSe, that potential acceptors such as Cu have the p–d acceptor level raised because of antibonding repulsion between Cu(3d) and Se(4p) [S (3p)] resulting in a deeper level, whereas impurities without d-electron resonances form shallow acceptors. Mg has no d-electrons and turns out to be sufficiently shallow for room-temperature p-type doping of GaN. On the contrary, Zn, Cd, and Hg, all of which have d-electrons, form deep levels in GaN. These observations are consistent with theory. Enhanced hole concentrations can be obtained by codoping, such as the Mg–O pair, the details of which can be found in Chapter 4. 3.5.13 Growth of InN

InN is pivotal in the triad of the group III nitride system and holds the key for full exploitation of the group III nitride system for optical emitters, as it is the binary needed to form alloys with transition energies in the visible. This being the case, the bandgap of InN, particularly for considerable InN molar fractions, must be known accurately for predicting the resultant transition energy on the basis of the knowledge of alloy composition. The issue of bandgap has been controversial with earlier data pointing to a bandgap of 1.9 eV and the later data suggesting this parameter to be near 0.7 eV. The details regarding the bandgap-related issues are discussed in Section 2.9. In this section, the focus is placed primarily on growth-related issues. In part because of the low dissociation temperature (beginning at temperatures as low as approximately 500  C at low pressures), growth of high-quality InN films has proven difficult. The nitrogen equilibrium vapor pressure over InN is many orders of magnitude higher than that over both AlN and GaN (Figure 1.6). To circumvent this difficulty, a number of options, including low-temperature (less than 600  C) deposition, various growth techniques, such as reactive evaporation, ion plating, reactive RF sputtering, reactive magnetron sputtering, vapor phase epitaxy, microwave-excited OMVPE, laser-assisted CVD, halogen transport (HVPE), pulsed laser deposition (PLD), and MBE, have been used [369,899]. Unlike the case of GaN wherein the better quality films result inclusive of the surface morphology with Ga-polarity samples, properties of N-polarity samples InN are better. Most of the InN growths by MBE have been performed at below 500  C, for example, 475  C for In-polarity samples and below 600  C, for example, 575  C for Npolarity samples [900,901]. Essentially, the highest growth temperature that can be tolerated would lead to a better film. In this vein, the best films are obtained with high V/III ratios, as well as high indium resulting in growth rates over 1 mm h
1 in MBE growth. The high V/III ratio is to avoid In droplet formation on the surface. However, from the point of view of In adatom surface migration during growth, In-rich

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conditions are preferable. Therefore, the optimum point is a compromise between the two requirements that culminate in a surface stoichiometry that is slightly N rich. With N-polarity, the growth temperature can be nearly 100  C higher than that for Inpolarity growth. An example for the growth schedule for both the In and N polar InN films is shown in Figure 3.176. In early developments and in particular for films produced by sputtering, short thermal annealings (30 min, 450–500  C) have been found useful for improving the crystallinity of the as-deposited InN films. Particularly true for early varieties, InN films were generally polycrystalline with agglomerates of small columnar grains having various degrees of texture and epitaxy. The improvement in the crystalline quality is because of a rearrangement of these crystallites in the film. The electrical properties of the as-deposited InN films are dominated by conduction between the grains. Buffer layers used for the growth also play a role. For example, application of an AlN buffer layer on a sapphire substrate significantly alters the granular polycrystalline growth mode of InN and results in substantial improvements in the growth morphology and in electrical properties. However, the electrical transport properties are still dominated by intergranular interactions. It is well known that annealing helps mainly when the starting sample quality is not really all that good. There is no substitute in compound semiconductors to employ the best method and growth conditions to produce the highest quality film by growth alone. This is what transpired with efforts focusing on growth by MBE and OMVPE. In what follows, the growth and properties of InN by first MBE followed by OMVPE and finally by other techniques are discussed, keeping in mind that MBE and OMVPE produced films are the best available. In one effort, reasonably high-quality epilayers of InN on (0 0 0 1) sapphire substrates have [902] been achieved by radio frequency plasma excited (RF-MBE) by using low-temperature grown intermediate layers. Nitridation was carried out at 550  C for 1 h with a nitrogen flow rate of 1 sccm and RF plasma power of 300 W. An InN buffer layer was deposited at 300  C with a nitrogen flow rate of 2 sccm and RF plasma power of 330 W. The InN epilayers were grown at 550  C for 1 h with a nitrogen flow rate of 2 sccm and RF plasma power of 240 W. This was followed by an intermediate layer at 300  C with the same conditions as for the buffer layer and the process was repeated three times. SEM observations of the final epilayer showed uniform surface morphology. Layers with a thickness of 600 nm and with a carrier density of 1.0 · 1019 cm
3 and a reasonably high electron mobility of 830 cm2 V
1 s
1 have been reported. More details of electron transport in much improved InN layers can be found in Section 1.4.4. Fairly high-quality InN films have also been grown on Si(1 1 1) substrates using a sequence of processes whereby Si (7 · 7) structure was obtained followed by nitridation of the Si surface to form a few monolayers of Si3N4, which was followed by the growth on an AlN buffer layer. On top of this buffer layer, InN layer was grown all by RF-MBE. The interesting features of this approach is that lattice constant of layers/templates that are juxtaposed are indexed well in terms of the lattice constant. For example, a 2 : 1 indexing is provided by the Si : Si3N4 pair, a 5 : 4 in-plane lattice constant indexing is provided by the AlN : Si pair, and a 8 : 9 lattice constant indexing

3.5 The Art and Technology of Growth of Nitrides

Figure 3.175 Reflection high energy electron diffraction patterns obtained on (a) clean (1 1 1) surface indicating the characteristic 7 · 7 reconstruction, (b) b-Si3N4 with 8 · 8 reconstruction, (c) AlN with 3 · 3 recontruction, and (d) InN with 1 · 1 reconstruction. Courtesy of S. Gwo.

is provided by the InN : AlN pair. By using reflection high energy electron diffraction, images of which are shown in Figure 3.175, and cross-sectional transmission electron microscopy, Wu et al. [903] have noted the pseudomorphic temperature to commensurate lattice transition within the first monolayer of growth in switching growth from template to the layer in the aforementioned pairs. This resulted in an abrupt heterojunction at the atomic scale. This avenue of indexing among the otherwise lattice mismatched pairs allows the formation of commensurate and two-dimensional layer with minimal strain and reduced propagating dislocations. Use of AlN buffer layers on sapphire substrates is a good alternative for growth of good-quality InN layers by all MBE growth of In polarity samples with improved structural and electrical properties. An improved surface morphology, a monotonic

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(a)

Time (au)

(b)

500 nm N-polar GaN

Nitridation

~600 °C Med./temp. N-polar GaN

MBE In-polar InN

Sapphire thermal treatment

~500 °C

800 °C

Temperature (au)

800 °C MBE 500 nm Ga-polar GaN

OMVPE Ga-polar GaN:Thermal

Temperature (au)

632

InN:N-polar

Time (au)

Figure 3.176 Growth schedule employed in the preparation of Inpolar (a) and N-polar (b) InN on Ga-polar OMVPE-grown GaN and N-polar MBE-grown GaN buffer layers, respectively. Courtesy of A. Yoshikawa.

increase in the Hall mobility, and a decrease in the electron density result with increasing thickness of the AlN buffer layer. A Hall mobility of 52 eV, an amorphous interfacial layer of AlN was formed on the SiC, which inhibited epitaxial growth. Employment of UHV conditions and very pure nitrogen sputtering gas yielded reduced impurity levels to the extent that O contamination was 3.5 · 1018 cm
3. The RMS surface roughness and surface feature size of AlN films as a function of ion-assisted energy (Ei) during the growth were measured with AFM with the largest RMS roughness value of 10 resulting for an ion energy of 20 eV. Large Al–N binding energy allowed Ohta et al. [933] to pursue pulsed-laser deposition of epitaxial AlN films at room temperature on nearly lattice-matched (Mn, Zn) Fe2O4(1 1 1) substrates. In situ RHEED observations have shown that the growth starts in the two-dimensional mode followed by a transition to the threedimensional mode when the film thickness reaches about 2 nm. The heterointerface between AlN and (Mn, Zn) Fe2O4 was determined to be abrupt and approximately 90% of the lattice mismatch was released at the interface owing to the introduction of crystalline defects such as misfit dislocations. Differences in thermal expansion coefficients of the epilayer material, in this case AlN, and the substrate lead to residual strain upon cooling down from growth temperature. Using Raman, Liu et al. [934] analyzed stress in the AlN/SiC (AlN was deposited by sublimation). The stress distribution in AlN crystals seeded on 6H-SiC was modeled using Equations 3.67 and 3.68 based on the sketch shown in Figure 3.184.

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R L AlN

t1 t2

W

6H-SiC y Figure 3.184 Sketch of AlN/6H-SiC structure after cooling down. Patterned after Ref. [934].

The thermal strain is expressed as ðl ei ¼ ½ðal
ah Þ=ah  ¼ ai dT:

ð3:67Þ

h

The stress along the y-axis is expressed as Fi E ti  þ y
: sy ¼ ti w R 2

ð3:68Þ

The theoretical frequency shifts of the E1 (transverse optical) mode calculated from model-predicted stress were in good agreement with experimental values taken along the edges of crystal samples. The stress was linearly distributed along the depth of the samples and changed from compressive at the growing surface to tensile at the interface between AlN and SiC for a thickness range of several hundred micrometers. Large tensile stresses, up to 0.6 GPa, were detected in the AlN at the interface. It is predicted that the AlN on 6H-SiC must be at least 2 mm thick to prevent it from cracking while cooling down the sample from a growth temperature of 2000  C. In addition to the thermal and mechanical, electrical, and optical properties discussed in Section 1.3, a good deal of investigations exists on the electronic manifestation of defects that are assumed to be similar to those observed for GaN to a first extent except the energy levels of course discussed in Sections 4.1.6 and 4.3. 3.5.14.1 Surface Reconstruction of AlN MBE of AlN, as in the case of other two main binaries, lends itself to the investigation of surface reconstruction examined by in situ RHEED. The investigation can also be aided by vacuum connected scanning tunneling microscopy (STM), LEED, and AES. The detailed atomic arrangements for most of the reconstructions on both (0 0 0 1) and ð0 0 0 1Þ surfaces have been determined by a combination of STM data and firstprinciples theoretical analysis, as discussed in Section 3.5.14.1. A unique feature of the GaN surfaces is that they can be terminated by excess cation species, that is, the Ga

3.5 The Art and Technology of Growth of Nitrides

atoms. Termination by N atoms is energetically unfavorable, because those N atoms prefer to form N2 followed by desorption from the surface. Against a large body of work on identifying surface arrangement of GaN, the AlN cousin has not received as much attention. Several studies have reported the symmetry of AlN surface reconstructions as seen by RHEED [935,936]. Symmetries of 1  1, 2  2, and 2  6 are commonly reported, and a sequence consisting of 1  3, 3  3, and 6  6 is reported by several groups. It should be noted that for RHEED a surface with hexagonal pffiffiffion p ffiffiffi symmetry, a 1  3 pattern usually is indicative of a 3  3 R30 symmetry. This comes about from the usual notation for RHEED from a surface with hexagonal symmetry, namely, listing the number of fractional-order streaks plus one for the integer-order streak observed with the electron beam along 2  1 1 0 and 0 1  10 directions, respectively. The latter is not a primitive lattice vector so that a pffiffiffi pffiffiffi 3  3 R30 reconstruction leads to a 1  3 RHEED pattern [937]. AlN films investigated by Lee et al. [937] were grown by PAMBE on approximately 2 mm thick OMVPE GaN templates. Depositing AlN on top of 0.5 mm thick GaN film by MBE on the OMVPE template led to rough surface morphology for AlN thicknesses greater than about 10 nm, presumably because of the lattice mismatch. To alleviate this problem, 5–10 periods of approximately 1 nm thick AlN layers followed by approximately 2 nm thick GaN layers, doped with Si, were deposited at 750–800  C, in an attempt to gradually grade the lattice constant to that of AlN. An undoped AlN layer with thickness of about 10 nm was then deposited on top, which yielded an overall flatter morphology. In a few instances AlN was directly deposited on SiC and a flat morphology was also achieved in that case (lattice mismatch of þ1.0% of AlN relative to SiC). Low Al coverage (as in N-rich growth conditions), moderate Al coverage (as in metal-rich growth conditions), and high Al coverage samples have been investigated. For obtaining low Al coverage or nitrogen-rich conditions, Lee et al. [937] heated an as-grown surface to 700–750  C and exposing it to the N plasma for a period of about 20 min. During this time, the 1 · 1 RHEED pattern of the surface stays streaky and sharp. Upon cooling down under the N plasma, a brightening of the RHEED pattern is seen when the surface passes through the melting point of Al (660  C). At that temperature, the plasma is turned off to avoid surface roughening. The resulting RHEED pattern is 1 · 1 with sharp streaks, which is referred to it as “1 · 1 nitrided” by Lee et al. [937]. The Al content of such a surface can vary considerably as confirmed by AES (not shown). In some cases, this procedure of forming a 1 · 1 nitrided surface with low Al coverage in a 1 · 3 pattern, as shown in Figure 3.185, which is pffiffiffi presults ffiffiffi indicative of a 3  3 R30 surface reconstruction. This surface reconstruction can also be obtained from a 1  1 nitrided surface, with saturated N content, by careful Al deposition and annealing. pffiffiffi pffiffiffi It should be kept in mind that the range of Al coverages over which the 3  3 R30 surface is formed is quite narrow. A 2  2 N rich surface reconstruction has been occasionally observed when the MBE growth is performed under Al-poor conditions, that is, with reduced Al flux compared to the cases discussed above. A sharp, streaky 1  1 RHEED pattern is still found during growth. If the growth is terminated by closing the Al shutter followed by cooldown under the N-plasma, the 2  2 pattern forms. The 1  1 nitrided, 2  2, or

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Figure 3.185 RHEED patterns for low Al coverage samples displaying 1 · 3 pattern, along the (a) h0 1 1 0i azimuth and (b) along the h2 1 1 0i azimuth. The tick marks in (a) indicate the 3 spacing. Courtesy of R.M. Feenstra and Ref. [937].

pffiffiffi pffiffiffi 3  3R30 surfaces were not investigated by STM in detail. However, a few attempts at studying the 1  1 nitrided surface indicated a disordered surface arrangement. Figure 3.186 shows a terraced surface, in the moderate Al coverage category, consisting mainly of the long-period structure consisting of a hexagonal array of corrugation maxima whose separation is 25  1 nm. Some deviations and defects in the stacking arrangement of are evident. Moreover, another reconstruction can be seen on the terrace appearing at the lower left-hand corner of the image. A hexagonal array of corrugation maxima, with separation between maxima of 10  1 Å is found. The orientation of the short-period and long-period corrugation are identical in that they are not rotated by 30 relative to each other. The RHEED pattern during growth was 1 · 1 under Al-rich growth conditions and persisted during cooldown when the growth was terminated by simultaneously turning off the Al and N sources. However, on occasion, a 2 · 6 RHEED pattern appearing at a temperature of 500  C could result from this growth termination. This pattern can also be obtained by a postgrowth annealing at 800  C for about 10 min. Longer annealing results in a 1 · 1 pattern. Surfaces displaying the 2 · 2 pattern had surface reconstructions as seen by STM of the type shown in Figure 3.186. Typical 2 · 6 RHEED patterns with sixfold streaks along the h0 1  1 0i azimuth and half order streaks along the h2 1 1 0i azimuth are shown in Figure 3.187a and b. Additionally, a relatively intense satellite fringe for both azimuths was always seen at wave vectors larger than that of the first-order streaks, as indicated by the white arrows in Figure 3.187a and b. LEED investigations were performed for additional insight into the surface structure, the results of which are shown in Figure 3.187c. pffiffiffi The six first-order spots are apparent, with spacing relative to the origin of b ¼ 2=ðpffiffiffiffi 3ffiaÞ along h0 1 1 0i directions. At smaller wave vectors, additional spots with b=ð2 3Þspacing along both h2 1 1p0i h0ffiffiffi1 1 0i directions are visible. This arrangement is a result of ffiffiffi andp an underlying 2 3  2 3 R30 symmetry and leads to the conclusion that the basic symmetry of the surface reconstructions is rotated by 30 relative to the underlying 1  1 lattice. This observation helps clarify the symmetries of the structures observed inpSTM images. The 10  1 Å terrace spacing is consistent with either 3  3 or ffiffiffi p ffiffiffi 2 3  2 3R30 . However, the latter is the correct one based on the LEED.

3.5 The Art and Technology of Growth of Nitrides

Figure 3.186 STM images of a surface with multiple reconstructions, obtained with sample voltages of (a) þ3 V and (b) þ2 V after a background subtraction routine with which the background is formed by averaging the image over a window of 3 nm · 3 nm. The surface morphology consists mainly of five terraces,

pffiffiffi pffiffiffi labeled A–E. A region of 2 3  2 3
R30 structure is seen in terrace of pffiffiEffi andpremainder ffiffiffi the terraces consists of 5 3  5 3
R30 structure. Distortions of the stacking sequence are marked by the dashed lines and arrows. Courtesy of R.M. Feenstra and Ref. [937].

pffiffiffi pffiffiffi  A pffiffi2ffi 3p ffiffiffi 2 3R30 surface periodicity produces a 2  6 pattern in RHEED. The 3  3 R30 periodicity produces a 1  3 RHEED pattern. For the long-period with a period of 25  3 Å, the surface structure is pffiffiffi pstructure ffiffiffi identified as 5 3  5 3 R30 , the genesis of which is probably because of a surface lattice that is contracted and slightly rotated with respect to the underlying AlN in such a way that the Al atom density in this surface lattice is close to the bulk Al. Additional splitting of the LEED spots is apparent in Figure 3.187c and are shown in an expanded view in Figure 3.187d, where the first-order LEED spot is labeled by A.

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Figure 3.187 Diffraction patterns from a moderately Al-rich AlN (0 0 0 1) surface. (a) Reverse-contrast RHEED pattern with electron beam along a h0 1 1 0i direction. Black tic marks indicate the location of 6 streaks. White arrows mark additional streaks seen at higher wavevectors. (b) Reverse-contrast RHEED pattern, with electron beam along a h2 1 1 0i

direction. White arrows mark the location of streaks occurring at high wavevector. (c) LEED pattern, acquired at 80 eV electron energy. The left-most first-order peak is slightly distorted due to a stray light reflection in the optical system used to acquire the image. (d) Expanded view of first-order reflection and satellite peaks. Courtesy of R.M. Feenstra and Ref. [937].

The satellite spots labeled B–F, oriented at approximately 30 intervals around a ring of radius (0.125  0.007)b surround the first-order spot. The radius of this ring agrees well with the observed spacing of the satellite fringes in the RHEED pattern of Figure 3.187b, which corresponds to a cut through the first-order spots of the LEED pattern and have a spacing (13  1)% larger than the first-order fringe spacing. Moreover, the inverse of this radius, 21.5  1.2 nm, is inpagreement with the spacing of the ffiffiffi corrugation lines seen in the STM images ð 3=2Þð25  3Þ nm ¼ 22  3 nm. The reconstructions were found to depend on the Al coverage, which can vary considerably because continued Al deposition at substrate temperatures below about 750  C produces thick and flat films on the surface. The Al content of our surfaces was estimated from the AES measurements. It should be pointed out that in the case of Ga on GaN where any excess surface Ga above about two ML condenses into droplets. For Al coverage of two to three monolayers (ML ¼ 1.19 · 1015 atoms cm
2) a characteristic 2 · 6 RHEED pattern was observed. Invariant line intensities for the 2 · 6 patterns from surface to surface and across a given surface points to the

3.5 The Art and Technology of Growth of Nitrides

pffiffiffi pffiffiffi pffiffiffi pffiffiffi presence of a mixture of 2 3  2 3 R30 and 5 3  5 3 R30 phases that was seen in STM images. Guided by pdensity pseudopotential ffiffiffi pfunctional ffiffiffi pffiffiffi pffiffiffi calculations, Lee et al. [937] argued that the 2 3  2 3 R30 and the 5 3  5 3 R30 is probably incorporating a laterally contracted monolayer or bilayer structure containing a 4/3 ML Al layer that is contracted and rotated by 30 so that it fits the underlying 1  1 lattice. For Al films with thickness greater than a few ML, a characteristic 1  1 RHEED pattern was observed but with the diffraction streak spacing of about 6% greater than 1  1 AlN spacing. This implies a contraction of the lattice such that the Al atom density is close to that of bulk Al. For lower Al coverages additional reconstructions with symmetry 2  2 and 1  3 was also observed. During metal-rich growth conditions, high Al coverage occurs readily that can be simulated by depositing pure Al at a substrate temperature of approximately 150  C onto the 2 · 6 surface discussed above or onto a 1 · 1 nitrided surface about to be discussed [937]. Figure 3.188a shows an STM image of a surface that is covered by this type of thick (5 ML) Al layer. Unlike the case of moderate Al coverage, highmagnification STM images generally do not reveal any atomic corrugation on the surface, except for an occasional and very weak corrugation with an approximate 1· spacing can be seen, as in Figure 3.188b. This type of surface displays sharp firstorder RHEED, streaks, located at distinctly larger wave vectors than for a 1 · 1 AlN surface as illustrated in Figure 3.188c, which is referred to as “1 · 1-Al,” with the observed streaks being located at wave vectors (6  1)% larger than that for the AlN 1 · 1 surface. Thermal annealing at temperatures above about 750  C for 5–10 min effectively removes the “thick” Al film from the surface but leaves micron-sized Al droplets [938]. The RHEED pattern of Figure 3.188d was acquired at the elevated temperature from such an annealed surface. With sufficient annealing time, the Al/N Auger ratio returns to a value close to that of the 2 · 6 structure and the RHEED pattern displays the 2 · 6 symmetry, as shown in Figure 3.188e. However, when the film was annealed for an insufficient time, a 2 · 6 structure resulted as shown in Figure 3.188f. The pattern displays streaks at the 1· positions of AlN and neighboring streaks located at wave vectors (10  1)% larger than the 1· spacing. The wave vector of these latter streaks is thus intermediate between those of the 1 · 1 Al pattern (Figure 3.188c) and those of the 2 · 6 pattern (Figure 3.188e or Figure 3.187b). LEED analyses were also attempted for high Al coverage sample in an attempt to determine the nature of the surface structure as was done for moderate coverage that did not prove to be straightforward in that LEED displayed a threefold splitting of the spots that may be attributed to nearly hexagonal (1 1 1) oriented Al layers. It should also be pointed out that LEED samples not only the surface layer but also a few subsurface layers. The occurrence of nearly hexagonal, (1 1 1) oriented Al layers in the film is consistent with the diffraction data in that a uniform (6  1)% contraction is observed in the first-order streak in RHEED. The observed LEED pattern probably arises from multiple domains of distorted Al(1 1 1) layers, and based on the similarity pffiffiffi pofffiffiffi their respective diffraction patterns, Lee et al. [937] propose that the 5 3  5 3 R30 structure may serve as a template for subsequent growth of the thin Al film. First-principles total energy calculations have been performed for a large number of possible AlN(0 0 0 1) surface reconstructions to gauge the likelihood of the surface

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Figure 3.188 (a) Large-scale STM image of AlN surface covered with an Al film obtained with a sample voltage ¼
4 V and a gray scale range ¼ 12 Å. (b) Higher magnification view of the same surface acquired at a sample voltage ¼
0.5 V and gray scale range ¼ 0.3 Å. (c–f) show the RHEED images along the h2  1 1 0i azimuth with (c) taken at 20  C (surface covered by 5 ML thick Al film) and (d) taken at

750  C (15 min anneal of the same surface as in (c)) and (e) taken at 20  C, following cooldown as the same surface as in (d); (f) taken at 20  C, following cooldown of a similar surface as in (d), which was annealed for only 10 min. White solid lines indicate the location of the first-order AlN streaks and dashed lines indicate the first-order streaks associated with the Al overlayer as in (c). Courtesy of R.M. Feenstra and Ref. [937].

reconstruction observed experimentally [937]. The calculations employ the local density functional theory and the electron–ion interaction is treated using firstprinciples pseudopotentials [939,940]. In previous studies of the AlN(0 0 0 1) surface, the focus was on structures having doubly occupied N dangling bonds and empty Al

3.5 The Art and Technology of Growth of Nitrides

dangling bonds and the guiding principle for the choice of structures was to satisfy the electron-counting rule (ECR) [941], which typically necessitates a 2 · 2 reconstruction on III–V (0 0 0 1) surfaces. In the light of the experimental observations discussed above, the theory pffiffiffi was pffiffiffi augmented and extended by performing calculations for structures having 3  3 symmetry and comparing the energies of these pffiffiffi with pffiffiffi the lowest energy 2  2 structures found previously. As discussed above, a 3  3 symmetry is observed experimentally in some cases. Moreover, this type of unit cell allows us to consider the laterally contracted metal adlayer structures that have been shown to be stable on GaN(0 0 0 1) surfaces under Ga-rich growth conditions [942]. As we are about to discuss below, metallic laterally contracted Al adlayer structures are found to be energetically favorable under Al-rich conditions in comparison to the standard 2  2 structures so that in this respect there is a similarity between AlN and GaN. The supercell employed in the calculations consists of four layers of Al and four layers of N. A layer of pseudo-hydrogen atoms having three fourth the charge was also employed to passivate the back side of the slab. The (0 0 0 1) surface on the opposite side of the H-terminated surface may be decorated by Al or N adatoms or even one or two pffiffiffi adlayers pffiffiffi of Al may be present. The plane wave cutoff was chosen at 50 Ry. For the 3  3 and 2  2 structures, a mesh of six special k-points is employed to sample the Brillouin zone. The relative energies are calculated as a function of the chemical potential of Al, which varies between a lower limit of mAl ¼ mAlðbulkÞ
DH and an upper limit of mAl ¼ mAlðbulkÞ as discussed in the literature [941,943]. For AlN the formation energy from bulk Al and molecular N2 (at zero temperature) is DH¼ 3.3 eV [941]. The calculated energies for a subset of the structures considered in this work are shown in Figure 3.189. For a good portion of chemical potentials on the N-rich side of the phase diagram, the most stable structure is the 2 · 2 N adatom model, in which the adatom occupies an H3 site [941]. As the Al chemical potential is increased, the 2 · 2 Al T4 adatom structure becomes stable in a small region of the chemical potential space [941]. The 2 · 2 structure observed experimentally could in principle to either of these pffiffiffi correspond pffiffiffi possibilities. As for the Adatom structures having 3  3 symmetry, both the N-H3 and Al-T4 structures are energetically unfavorable for all conditions, as seen in Figure 3.189. Because these structures do not contain rest atoms, the electron-counting pffiffiffi pffiffiffi rule is not satisfied (as doing so would require subsurface donors). Thus, the 3  3 adatom are energetically unstable on impurity-free surfaces, including pffiffiffi structures pffiffiffi pffiffiAlN ffi p ffiffiffi the 3  3 Al vacancy structure and a number of other 3  3 structures. With increasing Al chemical potential, metallic structures with more than one ML of Al eventually become more pffiffiffi p ffiffiffi stable than those in the 2 · 2 Al adatom model. One such model is of 3  3 symmetry and contains 4/3 ML of Al above the ideal surface that could be described as a laterally contracted monolayer (LCM). pffiffiffiIn this model, the hexagonal lattice of Al withplattice vectors is reduced by a factor 3=2 and ffiffiffi pffiffiffi rotated by 30 to be in registry with a 3  3 cell of the substrate. This structure is stable with respect to the Al T4 adatom structure forpmffiffiffiAl >m
0:29 eV as shown pAlðbulkÞ ffiffiffi in Figure 3.189. A schematic representation of the 3  3 LCM model is shown in Figure 3.190.

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Figure pffiffiffi3.189 pffiffiffi Relative energies for several 2 · 2 and 3  3 surface reconstructions of the AlN (0 0 0 1) surface plotted as a function of the chemical potential of Al, which is relative to that of the 1  1 relaxed ideal surface. The zero of the Al chemical potential corresponds to bulk fcc Al. In the N-rich limit, where mAl is lower than the maximum value by 3.3 eV, the 2 · 2 N adatom model has the lowest surface energy. As the chemical potential is increased the 2 · 2 Al adatom model becomes lower in energy, and

pffiffiffi pffiffiffi then the 3  3 LCM pffiffiffistructure pffiffiffi is preferred. In the Al-rich limit, the 3  3 LCB structure is preferred by a small amount. The numbers in parentheses refer to the excess (or deficit) in the number of Al atoms relative to the ideal 1  1 surface, per 1  1 unit cell. Many possible models (not shown here) can be excluded on the basis of calculations performed in Ref. [941]. Courtesy of J. Northrup and R.M. Feenstra and Ref. [937].

In the high Al-coverage case, where mAl ¼ mAlðbulkÞ , it is slightly energetically favorable to add another layer of Al to the surface and form a laterally contracted bilayer structure with 7/3 ML ofpexcess ffiffiffi pAl, ffiffiffi the model structure of which is shown in Figure 3.190b and c. In each 3  3 cell, there are four atoms in the laterally contracted top layer and three atoms in T1 sites in the layer below. One of the atoms in the top layer is directly above an Al atom in the T1 layer below, whereas the other three being in bridge sites. The top layer then exhibits substantial corrugation: The atoms above the T1 sites reside approximately 0.53 Å above the other three atoms in the top layer. The 1  1 Al adlayer structures having one or two MLs of Al are tensile strained because the Al–Al spacing in the adlayer is too large, which can be illustrated with two examples. Compared to a 1  1 adlayer structure with one ML of Al in T1 sites, the Al–Al spacing in the 4/3 ML laterally contracted monolayer is reduced by pffiffiffi pffiffiffi structure  13% and the energy also is lowered by 0:59 eV= 3  3 cell in the Al-rich limit. This holds when comparing structures with two adlayers also. The 7/3 ML laterally contracted bilayer structure is more pffiffiffi than pffiffiffi stable  a two ML structure, with adlayers in T1 and T4 registry, by 0:27 eV= 3  3 cell in the Al-rich limit. The calculations demonstrate that the 2 · 2 N-H3 adatom model is relatively stable over a substantial region of the chemical potential space and is, therefore, considered a plausible model for the 2 · 2 structure seen in N-rich conditions. The most likely

3.5 The Art and Technology of Growth of Nitrides

Figure 3.190 Schematic top views of the LCM and LCB models, respectively. Open and solid circles represent Al and N atoms, respectively with the diameter of the circles representative of the height of the atoms relative to the surface, and the layer numbers are depicted in parentheses. In each case, layer 1 is the outermost layer. (c) A 3D rendition of a side view of the LCB model where gray and black spheres

represent Al and N atoms, respectively. A projection slightly rotated from h0 1 1 0i is illustrates the vertical corrugation of the Al adlayers where the Al atoms in layer 2 are positioned directly above those in layer 3. The Al atoms in layer 1 that are positioned directly above those in layers 2 and 3 are higher than the other atoms in layer 1 by approximately 0.5 Å. Courtesy of J. Northrup and R.M. Feenstra and Ref. [937].

pffiffiffi pffiffiffi model to account for the observation of a 3  3 symmetry is the 4/3 ML laterally contracted monolayer. In very Al-rich conditions, the metallic adlayer structures having more than one ML of excess Al are energetically favorable and these adlayers are stabilized by a contraction offfiffiffi the Al–Al spacing to relieve tensile stress. pffiffiffi p Experimentally observed 2 3  2 3 R30 structure at this coverage may be formed from the 7/3 ML structure by an appropriate buckling of the surface atoms or

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 alternatively by a more complex rearrangement. In any case, of either pffiffiffia 30 protation ffiffiffi  the first or second layers (or both) seems likely.p As for the 5 3  5 3 R30 structure ffiffiffi pffiffiffi observed at Al coverage close to that of the 2 3  2 3 R30 , this structure arises from an approximately 4 rotation and approximately 9% contraction of the surface lattice of Al, which produces a surface Al–Al separation close to that of unstrained Al (1 1 1) planes. The unit cell of the resulting structure is too large for the first-principles theory to deal with. Finally, at higher Al coverages, the formation of Al layers with a contracted lattice compared pffiffiffi topAlN ffiffiffi is observed. The nature of this distortion appears to follow that of the 5 3  5 3 R30 structure, the net result being that the Al–Al separation in each plane of the film is close to bulk Al. To summarize, forpAl-poor 2 · 2 and 1 · 3 RHEED patterns, the latter may ffiffiffi pffiffisurfaces, ffi  be associated with a 3  3 R30 reconstruction, pffiffiffi pffiffiffi are observed. For moderately Alpffiffiffi pffiffiffi rich surfaces, 2 3  2 3 R30 and 5 3  5 3 R30 reconstructions are observed. The former produces a 2  6 RHEED pattern and the latter then adds some satellite features and complex intensity variations (as discerned from diffraction spot splitting) in that pattern. For even larger Al surface coverage, flat films of epitaxial Al are found to form on the surface, with a characteristic 1  1-Al RHEED pattern with fringe spacing slightly expanded relative to the 1  1 AlN spacing and a LEED pattern with threefold spot pattern.

3.5.15 Growth of Ternary and Quaternary Alloys

By alloying InN together with GaN and AlN, the bandgap of the resulting alloy(s) can be increased from 0.8 to 6.2 eV, which is critical for making high-efficiency visible light sources, ultraviolet detectors, and high-performance electronic devices. The quaternary alloys (InGaAlN) have the added advantage that their lattice constants can be tuned, at least to ZnO, whereas its bandgap can be adjusted to meet the need set forth by the device design. Detection of missiles requires solar-blind detectors, meaning those that do not get affected by the sun’s radiation. Nitride-based ultraviolet detectors employing high fractions of AlN would not be blinded by the large infrared, visible, and UV components of the sunlight, but would be able to detect missiles by detecting their plumes. Incorporation of indium in these alloys is not easy, even though the InN molar fraction in InGaN has been pushed almost to 70%. To prevent InN dissociation, the InGaN crystal was originally grown by vacuum deposition at low temperatures (about 500  C). However, the use of a high nitrogen flux rate allowed the high-temperature (800  C) growth of high-quality InGaN and InGaAlN films on (0 0 0 1) sapphire substrates. It was noted that the incorporation of indium strongly depends on the flow rate ratio of TMI, TMA, and TMG (or TEG), with larger flow rates of TMG producing larger InN mole fractions. The indium incorporation efficiency decreases quite rapidly with increasing growth temperatures to values above 500  C. The crystalline quality of InGaN was observed to be superior when grown on the well-matched ZnO substrate to that grown on a bare (0 0 0 1) sapphire substrate. Employment of a buffer layer provides further improvements in the crystalline quality of InGaN. As in the case of GaN, InGaN films grown on

3.5 The Art and Technology of Growth of Nitrides

sapphire substrates with low-temperature GaN buffer layers exhibit much better optical properties. This has to do with the nucleation issues on the substrate. The pathways to deposition of high-quality AlGaN layers are similar to those for GaN with the exception that higher deposition temperatures can be employed. One concern is the reactivity of AIN and oxygen. The higher the temperature, the lower the incorporation of oxygen, the source of which is attributed to ammonia. Despite the above-mentioned complications, high-quality InGaN/AlGaN, InGaN/GaN, and InxGa1
xN/InGal-yN superlattices have been grown. 3.5.15.1 Growth of AlGaN The AlxGa1
xN alloy forms the barrier for all the GaN-based devices and as such is of paramount interest. In all the devices, the AlxGa1
xN alloy constitutes a pivotal component and determines such properties as carrier and light confinement and sheet carrier density at heterointerfaces. With the advent of short wavelength sources and detectors, as discussed in Volume 3, Chapters 1–3, AlGaN is increasingly taking the role of active emission and absorption medium as well. The bandgap of this material, which is direct, lies in a typical range of 3.42 (x ¼ 0) to 6.0 (x ¼ 1) at T ¼ 300 K. Although a number of these devices have been reported, there are still serious issues to be addressed. Among them is the deteriorating radiative recombination efficiency with increasing Al content [944] and control of the mole fraction. The influence of growth conditions on the aluminum mole fraction and the quality of AlxGa1
xN film have been studied in some detail [945,946], as will be expanded upon below. The MBE growth of AlxGa1
xN on sapphire is less well understood, except for some regularities of the growth established for the MBE growth employing electron cyclotron resonance-microwave plasma source of nitrogen [947]. In this section, mainly the growth issues and structural properties of AlGaN are addressed, leaving the optical properties for Volume 2, Chapter 5. The OMVPE growth of AlGaN is often accompanied by parasitic gas-phase chemical reactions, which not only diminishes the group III deposition efficiency but also makes it difficult to control the alloy composition. Such chemical mechanisms proposed usually are initiated by the formation of adducts between the group III metalorganic and NH3. Mihopoulos [948] proposed a mechanism for AlGaN OMVPE growth that mainly focused on the eventual formation of dimer and trimer particles large enough to be kept from the surface by thermophoretic forces. This proposal sufficiently explained the reduction in AlN growth rate at typical OMVPE conditions and was confirmed by Creighton [949] who directly observed the formation of gasphase nanoparticles during AlN, GaN, and AlGaN OMVPE growth. Keeping the gas stream cool and/or allowing Al- and N-containing gases to be in contact only near the sample surface reduces this pregrowth interaction. Aluminum nitride films can be grown through the reaction of AlCl3 with NH3 or ammonolysis of AlCl3NH3 [950]. In this pyrolytic process, the reactant gases are supplied to the reaction zone either separately or in the form of the compound AlCl3NH3 in appropriate composition. In early varieties, vertical reactors and substrate temperatures in the range of 700–1300  C were used for pyrolytic decomposition of reactant species, which resulted in deposition of AlN films. Evaporation

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temperatures for AlCl3 and AlCl3 NH3 are 80–140 and 180–400  C, respectively, which are conveniently accessible. Deposition rates in the range of 0.6–6 mm h
1 have been attained. Following the binary growth, AlxGa1
xN (x ¼ 0.45) layers were grown on sapphire substrates at 1050  C by Baranov et al. [951]. A mixture of GaCl, AlCl, and ammonia was introduced to obtain the ternary material. The growth was initiated with GaN, which was then followed by the deposition of AlGaN. Simply raising the temperature of the Al source from 600 to 750  C, increased the Al content. All undoped AlGaN samples showed n-type conductivity with temperature-independent electron concentrations and mobilities between 77 and 300 K. With increasing Al content up to values of x ¼ 0 : 4 or greater, the electron concentration decreased from 5 · 1019 to 1 · 1019 cm
3 and the mobility from 100 to 10 cm2 V
1 s
1. The results obtained thus far indicate that high-quality GaN films and templates can be grown on sapphire and other substrates by HVPE. These layers and templates served the basis for further growth on them by MBE and OMVPE. As such, the technique is very useful. The extent to which the method can be useful can be expanded if high-resistivity films can also be obtained. Owing to the high temperatures employed, it has not yet been possible to grow InGaN ternary with this method. Similarly, p-type doping has so far been lacking, which means that the HVPE technique alone is not in a position to produce device structures requiring p–n junctions. Although Si and O incorporation from the walls has been an issue, the high quality of the recent layers indicates that this issue is not as serious as it used to be. Traditionally, AlGaN layers by MBE were grown under N abundance. In that case, relative fluxes of Ga and Al would determine the mole fraction. For GaN, however, Ga abundant, in which case the N flux determines the growth rate, and N abundant, in which case the Ga flux determines the growth rate, conditions have been applied. Attempts to control the Al mole fraction in AlGaN were undertaken by varying the Al cell temperature (thus the Ga flux), Ga cell temperature, both Ga and Al fluxes with the fixed Al/Ga ratio, and Al/Ga ratio with the fixed total flux. However, a new realm of metal-rich growth of the AlGaN ternary has been applied that resulted in AlGaN with high radiative efficiency over the entire composition range. He et al. [952] reported on growth and properties of AlxGa1
xN epilayers deposited on (0 0 0 1) sapphire substrates by MBE under Ga-rich conditions for a wide range of aluminum compositions (0.13–0.92). Some details are as follows: An overabundant Ga flux (two Ga sources) and moderate N pressure were kept to maintain N-limited (Ga-rich) growth conditions. The Al mole fraction in the alloy was controlled by fixing the Al and Ga source temperatures and varying the N flow. Variation of the nitrogen partial pressure in the MBE chamber in the range of 2.7 · 10
6 to 9 · 10
6 Torr during growth provided AlxGa1
xN with almost a complete range of x. Care, however, must be exercised because of the temperature-dependent Ga desorption rate from the surface. This means that at higher temperatures, the AlN component in AlGaN will increase even when the cell temperatures and N flux are kept the same, necessitating a good control over the substrate temperature. Shown in Figure 3.191, the dependence of Al mole fraction in AlxGa1
xN epilayers grown under Ga-rich conditions at constant substrate temperature versus the MBE

3.5 The Art and Technology of Growth of Nitrides

Aluminum mole fraction, x

1.0

0.8

0.6

0.4

0.2

–6

2.0 x 10

–6

4.0 x 10

–6

6.0 x 10

–6

8.0 x 10

–5

1.0 x 10

Pressure (Torr) Figure 3.191 Aluminum mole fraction x in AlxGa1
x N alloys as a function of nitrogen plasma pressure under metal-rich condition.

chamber pressure, which represents the N flux. The Al mole fraction is about inversely proportional to the N flow and the Al concentration increases rapidly when the pressure is lower than 4 · 10
6 Torr. This phenomenon can be explained by preferential formation of the AlN component in the AlxGa1
xN alloy under Ga-rich growth conditions [947]. Indeed, because the Al–N bond is stronger than the Ga–N bond, it is reasonable to surmise that the available N would preferentially be used to form AlN. The remaining N would then form the GaN bonds for the alloy. Consequently, the N arrival rate would determine the mole fraction in the presence of an abundant amount of Ga on the surface in some range of substrate temperatures. As expected, under these conditions, the Al mole fraction in AlxGa1
xN remained the same even with increasing temperature of the gallium sources. Keeping the pressure of nitrogen plasma fixed, the aluminum mole fraction in AlxGa1
xN is a function of aluminum flux, or the source temperature, and it could not be changed when the temperature of Ga sources was varied as shown Figure 3.192. Implicit in Figure 3.192 is that the higher the aluminum source temperature or flux, the higher the aluminum mole fraction x in AlxGa1
xN alloys. This can be inferred from observing that the Al mole fraction is increased by reducing the N flux, which leads to reduced contribution from Ga, and thus the growth rate diminishes. PL intensity for the samples grown under Ga-rich conditions generally quenched at higher sample temperatures, as compared to the N-rich samples. Quantum efficiency of PL at 15 K was markedly higher for the AlxGa1
xN layers grown under Ga-rich conditions (3–48%) compared to the layers grown under N-rich conditions (1–10%). These values in turn are much higher than radiative efficiencies obtained by us for a large set of GaN layers (Ga polarity) grown in similar conditions: 0.01–0.3% for Ga-rich and 0.1–2% for N-rich GaN. Such improvement of radiative efficiency in AlxGa1
xN, especially under the Ga-rich conditions points to reduced density of dislocations, although carrier confinement cannot be excluded.

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0.5

Growth rate (Πm h–1)

0.4

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Aluminum mole fraction, x Figure 3.192 Growth rate of AlxGa1
x N alloys as a function of mole fraction and indirectly (in conjunction with Figure 3.191) nitrogen-plasma pressure under Ga-rich conditions. The growth conditions are identical to those used for Figure 3.191.

TEM was used to characterize defects and the microstructure of AlGaN layers grown under N- and Ga-rich conditions [953]. The AlxGa1
xN layers investigated had nominal thicknesses in the range of 0.5–1 mm, Al mole fraction x in the range of 0.10–0.25, and were grown on (0 0 0 1) sapphire substrates by MBE. The dominant defects present in films grown under N-rich conditions were threading dislocations as determined from multibeam imaging conditions. The TEM bright-field images of Figure 3.193a–c associated with a sample of N-rich conditions show the density of dislocations in the vicinity of the substrate to decrease from 5 · 1010 cm
2 at the interface with the substrate to 2 · 1010 cm
2 at the surface. To estimate the relative distribution of different types of dislocations (edge, screw, and mixed), bright-filed images were recorded under two beam conditions for a g-vector parallel to the [0 0 0 2] and ½1 1 2 0 directions, respectively. A pair of such images is shown in Figure 3.193b and c. In the image recorded for a g-vector parallel to the [0 0 0 2] direction, the edge dislocations are out of contrast, whereas the screw dislocations are out of contrast in the image recorded for a g-vector parallel to the ½1 1 2 0 direction. However, mixed-type dislocations appear in both images. In the case of layers grown under N-rich conditions, the majority (95%) of all dislocations were out of contrast in images recorded for a g-vector parallel to the [0 0 0 2] direction meaning that they were pure edge type. The microstructure and defect distribution in layers grown under Ga-rich conditions differed significantly. First, these layers had a rough surface compared to layers grown under N-rich conditions (see Figure 3.193d). Moreover, despite similar densities of threading dislocations in layers grown under Ga-rich conditions

3.5 The Art and Technology of Growth of Nitrides

Figure 3.193 TEM Bright-field images of layer grown under Ga-rich (left panels)/N-rich (right panels) conditions recorded under (a)/(d) multibeam condition, (b)/(e) two-beam condition with g-vector parallel to [0 0 0 2] direction, and (c)/(f) two-beam condition with g-vector parallel to ð1 0 1 0Þ direction.

(7 · 1010–8 · 1010 cm
2 in the vicinity of the substrate and 5 · 109 cm
2 at the layer surface) to those measured for layers grown under N-rich conditions, there was a significant difference in their type distribution. In layers grown under Ga-rich conditions, edge dislocations are again the majority (70%), although other dislocations are also in significant densities (see Figure 3.193e and f). Iliopoulos et al. [954] reported the growth kinetics of AlGaN films by plasmaassisted MBE on (0 0 0 1) sapphire substrates at 750–800  C. The incorporation probabilities of Al and Ga were determined by measuring the growth rate and composition of the films. In both the N-rich and Ga-rich conditions, the incorporation probability of Al is unity for the entire investigated temperature range. The incorporation probability of Ga kept constant and equaled 0.75 at a substrate temperature of 750  C only under the N-rich growth condition. The temperature dependence of the incorporation probability in this regime has an activation energy of 2.88 eV, which is consistent with Ga desorption from the surface. In the group III rich growth regime, the incorporation probability of Ga decreases monotonically with group III fluxes owing to the competition with Al for the available active nitrogen atoms. In this regime, the GaN phase growth rate was determined by the capture probability of the available active nitrogen from the gallium surface adlayer.

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As is the case for all ternaries and quaternaries in all compound semiconductors, high-quality layers containing large fractions of Al or In are difficult to grow, which is particularly so in AlGaN. Although the crystalline quality of AlGaN on sapphire can be improved by using a LT insertion layer, the quality progressively worsens with increasing AlN molar fraction [955]. However, the crystalline quality can be significantly improved if the AlGaN layer is grown on a high-quality GaN layer. However, one has to be mindful of the high-density crack network, which originates from the tensile stress induced by the lattice mismatch between AlGaN and GaN when the thickness of AlGaN exceeds a critical value [956]. The growth can be interrupted after some thickness beyond the critical thickness, the sample be allowed to cool and crack by plastic relaxation, and growth is then resumed [413]. Cracking of AlGaN layers for higher compositions and for thicker layers is a serious problem. Recall, the origin of dislocations in AlGaN is the lattice misfit and the different thermal expansion coefficients between GaN and AlN, which result in stress in AlGaN epilayers. Einfeldt et al. [957] studied the relaxation of tensile strain in AlGaN layers of different compositions grown on GaN/sapphire. Extended crack channels along the h2 1 1 0i directions are formed, as shown in Figure 3.194, if the Al content exceeds a critical value that decreases with increasing layer thickness. The surface gap at the crack was found to increase linearly with stress. Annealing of the AlGaN sample above the growth temperature introduced additional tensile stress upon cooldown owing to the mismatch in thermal expansion coefficients. This tensile stress in AlGaN layers was relieved not only by the formation of additional cracks but also by the extension of these cracks. Other authors also studied the generation of dislocations and cracks in AlGaN epilayers [958,959]. In one such case, high-resolution X-ray diffractometry was employed to investigate the mechanism of stress reduction [960]. It was found that the tensile stress decreased with decreasing interlayer growth temperature, as shown in Figure 3.195. From reciprocal space maps it could be observed that the AlN interlayers grown at high temperatures were pseudomorphic, whereas interlayers grown at low temperatures were strain relaxed. Therefore, the AlGaN layers grown on a low-temperature AlN

δo

b t w

(a)

(b)

(c)

Figure 3.194 Artistic view of various cracking modes of tensilestressed AlGaN film. The crack type (a) propagates toward the interface; (b) channels across the interface; and (c) along the plane of the substrate as if to peel the layer. Patterned after Ref. [957].

3.5 The Art and Technology of Growth of Nitrides

1.4 1.2

Tensile stress (GPa)

1.0 0.8 0.6 0.4 0.2 0.0 500

600

700 800 900 1000 AlN growth temperature(ºC)

1100

Figure 3.195 Room temperature average tensile stress in 1–3 mm GaN layers grown on Si(1 1 1) substrates determined from the in-plane a-lattice constant by high resolution X-ray diffraction measurements. The only LT AlN interlayer was inserted at indicated growth temperatures following a 0.5 mm GaN [960].

grow under compressive interlayer-induced strain. The reduction of mismatchinduced tensile stress and suppression of crack formation during growth of AlxGa1
xN was also realized by inserting a low-temperature AlGaN interlayer between GaN and AlxGa1
xN [961]. The interlayer was found to mediate the elastic tensile mismatch between the adjacent layers and extend the critical thickness before the onset of cracking. The use of LT insertion layers (interlayers) and their effect on overlaying GaN is discussed in Section 3.5.5.5. In addition to the stress and associated cracking issue, another major problem with AlGaN is the presence of a large density of defects that play a detrimental role in the performance and reliability of AlGaN-based devices [962–964]. Thick, heavily doped and high-quality AlGaN layers are essential to developing GaN-based devices. For dislocation reduction, several approaches centered on buffer and interlayers are employed [965]. Amano et al. and others [966–969] reported the growth of low dislocation density AlGaN on GaN underlayers combining the low-temperature AlN interlayer and the trenched substrates technique. But for deep UV LEDs, GaN buffers greatly decrease the light extraction efficiency from the substrate side and does not allow light propagation in back illuminated detectors. Another approach relying on AlN/AlGaN superlattices, which is common in the GaAs-based systems, has been employed for reducing the threading dislocation density and growing thick AlGaN on sapphire [970]. Doing so allowed, crack free 3 mm thick AlxGa1
xN layers to be achieved. A thin AlN interlayer grown at low temperatures, “LT-interlayer,” is often incorporated to attain crack-free AlGaN layers on GaN on sapphire [971]. Inserting the LT interlayer reduces the tensile stress during growth and the also the number of threading dislocations with a screw component. An added benefit of LT interlayer, as in the case of GaN growth, is that the quality of AlGaN also improved with LT buffer

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insertion as demonstrated by Amano et al. [972]. Although the crystalline quality of AlGaN on sapphire improves by using an LT buffer layer, it progressively degrades with increasing AlN content. When the underlying GaN is improved, so did the AlGaN quality, but a network of high-density cracks caused by the residual thermal tensile stress is generated when the thickness of AlGaN exceeds a critical value. This problem is alleviated by LT AlN insertion. Amano et al. [972] inserted a LT AlN between the underlying GaN layer and the upper AlGaN layer. The LT interlayer reduces the tensile stress and threading dislocations that have screw components. Figure 3.196 shows the difference of the grown-in stress of the Al0.18Ga0.82N on GaN (a) with and (b) without the LT AlN interlayer. Nearly strain-free AlGaN could be grown on the LT interlayer, whereas relaxation occurs during a growth of Al0.18Ga0.82N, which is confirmed by the steep decrease of the stress and thickness product in Figure 3.196. It is important to emphasize that the crystalline quality of 500

Refraction intensity

Al0.18Ga0.82N

1

0.05(GPa)

300

0 200

100

–1 1.4 μm

1.7 μm

Stress×thickness(GPa×µm)

LT-AlN

GaN 400

0 3000

(a)

4000

5000

6000

7000

Growth time (s)

Al0.18Ga0.82N

GaN

LT-AlN insersion layer LT-AlN nucleation layer

Sapphire (b) Figure 3.196 (a) In situ measured stress and thickness product and reflectivity for AlGaN/ GaN heterostructure grown on sapphire with LT insertion layer. (b) Schematic diagram of the heterostructure with the insertion layer. (c) In situ measured stress and thickness product and

reflectivity for AlGaN/GaN heterostructure grown on sapphire without the insertion layer. (d) Schematic diagram of the heterostructure without the insertion layer. Courtesy of H. Amano and Ref. [972].

3.5 The Art and Technology of Growth of Nitrides

Al0.18Ga0.82N

GaN 1.6 (GPa)

1.15 (GPa)

Refraction intensity

400

1

300 0

0.62(GPa)

200

100

–1 1.3 µm

1.4 µm

0

3000

(c)

Stress×thickness (GPa× µm)

500

4000 Growth time (s)

5000

Al0.18Ga0.82N GaN

LT-AlN nucleation layer

Sapphire (d) Figure 3.196 (Continued)

this AlGaN is much superior to that grown on sapphire covered with only one LT buffer layer directly on sapphire, as confirmed by TEM. The reduction of the density of screw and mixed threading dislocations leads to a reduction in the leakage current in solar-blind UV photoconductors and pin photodiodes confirming that these defects act as a current leakage path. The fact that one LT interlayer is effective paved the way for the use of periodically spaced LT insertion layers [434,973]. The LT interlayer process was also used for a distributed Bragg reflector (DBR) mirror based on an AlGaN/GaN multilayered structure. For an expansive treatment of the topic, se Ref. [974]. The disadvantage of the LT interlayer technique is that it increases the density of edge dislocations, which also act as nonradiative recombination centers, as discussed in Volume 1, Chapter 4 and Volume 2, Chapter 5. Therefore, the fabrication of highly luminescent AlGaN is difficult. The lateral epitaxial overgrowth used to reduce the dislocation density, discussed in Section 3.5.5.2, cannot simply be adopted in the growth of AlGaN, particularly with a high AlN molar fraction, because of the deposition of polycrystalline islands on the dielectric mask. Another method of growing low dislocation density AlGaN is to use grooved GaN [972]. However, LT interlayers are still necessary to alleviate the cracking problem.

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The best overall results are obtained when the LT interlayer in used in conjunction with grooved templates underneath, as shown in Figure 3.197a and b. It should be emphasized that the LT interlayer is essential for alleviating fracture of the layer. Figure 3.197c shows the cross-sectional TEM image of the AlGaN layer grown on grooved GaN with LT AlN interlayer. Figure 3.198 shows the low-temperature CL image of an Al0.25Ga0.75N layer grown on grooved GaN covered with the LT interlayer. Several dark spots are visible in the grooved region, whereas it is entirely dark in the terraced regions, which implies that edge dislocations present in the terraced region act as nonradiative recombination centers. The upper limit for the density of the dark spots on the grooved region is around 107 cm
2. Therefore, a reduction in the threading dislocation density by as many as two orders of magnitude can be achieved. Amano [974] also reported that the efficiency of the AlGaN/GaN MQW with low dislocation density on grooved and LT interlayers is comparable to that of the GaN/GaInN MQW. Without the grooved LT interlayer employment, the AlGaN/GaN variety is much inferior, a point that was reflected in the performance of UV LEDs. The growth processes for grooved and LT interlayer concepts are rather complex because etching and regrowth are unavoidable. In addition, the underlying GaN layer is necessary for growth but acts as an absorption layer for UV emission, which is not desirable. To alleviate the absorption by the GaN template, a process that utilizes a grooved substrate has been proposed [975–977]. In this process, grooves are initially formed on the surface of sapphire, SiC, or Si substrates. AlGaN is then grown utilizing an LTnucleation buffer layer of AlN. The grooves should be sufficiently deep so that the laterally grown AlGaN initiating on the terraces will coalesce and bridge over the grooved region before the AlGaN initiated on the troughs reaches the laterally grown AlGaN. The practical advantages of this process is that it does not require etching or regrowth after the growth of GaN. As alluded to above, the absence of a GaN layer eliminates absorption of the UV emission. The thermal stress caused by the differences in the thermal expansion coefficients between AlGaN and the substrate used can also be reduced. To gain some insight into the behavior of threading dislocations and the mechanism of stress relief, cross-sectional TEM observations were conducted for GaN/ AlGaN/IL-AlN/GaN [978]. Figure 3.199a and b shows dark-field (DF) images of a specimen with GaN layer thickness of 580 nm, taken with the reflection of g||0002 and ½0 1 1 0, respectively. Using the well-known selection rule that indicates that screw dislocations are out of contrast for g  b ¼ 0 diffraction conditions, and in contrast for g ¼ (0 0 0 2) diffraction conditions, Burgers vector b can be determined. Here g is the diffraction vector. In the image recorded for g-vector parallel to the [0 0 0 2] direction, the edge dislocations are out of contrast, whereas the screw dislocations are out of contrast in the image recorded for a g-vector parallel to the ½0  1 1 0 direction, see Section 4.1.1.2 for a description. TDs appearing in Figure 3.199a are of screw and mixed type, and those in Figure 3.199b are of edge type or mixed type. Because Figure 3.199b happened to be taken with the ð0  1 1 0Þ reflection, there can be hidden TDs that disappear from Figure 3.199a and b. Because most of the TDs in Figure 3.199a appear also in Figure 3.199b, they are identified to be mixed type,

3.5 The Art and Technology of Growth of Nitrides

Figure 3.197 (a) SEM image and the schematic structure of Al0.19Ga0.81N on grooved GaN with low temperature deposited AlN interlayer. (b) SEM image and the schematic structure of Al0.19Ga0.81N on grooved GaN without low temperature deposited AlN interlayer. (c) Cross-sectional TEM image of the sample shown in (a). Courtesy of H. Amano.

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Figure 3.198 CL image, taken at cryogenic temperature, of Al0.25Ga0.75N grown on grooved GaN. Courtesy of H. Amano and Ref. [972].

whereas those that appear only in Figure 3.199b are of edge type. One can see that TDs that are formed at the interface of GaN/LT GaN are mostly of mixed type with a few of edge type. The density of TDs in the GaN layer was roughly estimated to be of the order of 108 cm
2. The TDs of mixed type penetrate the AlGaN layer and propagate up to the top surface. This indicates that the IL-AlN has no effect in the termination of mixed type TDs. Figure 3.199b clearly indicates that many edge type TDs are generated at the interface of IL-AlN and propagate into the AlGaN layer. The mechanism of the preferential formation of edge type TDs at the IL-AlN is not yet clear. The density of the TDs is estimated to be above 109 cm
2 in the AlGaN layer but decreases sharply in the top GaN layer. Al segregation was observed around the threading dislocations in Al0.1Ga0.9N and Al0.3Ga0.7N layers grown on 6H-SiC by OMVPE by TEM [979]. Up to 70% more Al concentration was found around dislocations than those regions free of dislocations. The Al-depleted regions were observed to be within a few nanometers of the dislocation lines. Pecz et al. [980] also reported the segregated Al in the middle of each V pit that is on the surface of AlGaN grown on GaN by OMVPE. The origin and morphological evolution of a self-terminating V-shaped defect in AlGaN/GaN superlattices was studied by TEM [981]. These V-shaped defects, which generally originated from pits on the top plane of the undoped GaN layers with a critical thickness below 500 nm and deposited with inclined superlattice structure during growth of superlattice, could terminate in the superlattice region after propagating to a distance of about 800–1000 nm. The structural variations of the defects could be attributed to the different growth rates between the surrounding matrix and the domains within the defects at various growth stages. Intentional impurities provided in high concentrations are known to help reduce the structural and point defects, as has been demonstrated in GaAs and GaN using, for example, Si and to some extent isoelectronic species such as In in low concentrations as to still maintain the binary nature of GaN. However, the case with Mg is

3.5 The Art and Technology of Growth of Nitrides

Figure 3.199 Dark-field TEM images of GaN (580 nm)/ Al028Ga0.72N/IL-AlN/GaN/LT GaN/sapphire for (a) g ¼ 0 0 0 2 and (b) g==0  1 1 0. Courtesy of H. Amano and Ref. [978].

complicated in that low concentrations follow the aforementioned trends, whereas the high concentrations introduce structural defects dubbed the pyramidal or V defects as discussed in Section 4.2.3. In this vein, while leaving the details to the aforementioned section, Cho et al. [982] reported the influence of Mg doping on structural defects in Al0.13Ga0.87N layers grown on sapphire substrates by OMVPE. The reduction of dislocation density occurred up to the Mg precursor flow rate of 0.103 mmol min
1. However, the reduction from 4 · 109 cm
2 (undoped) to 1.3 · 109 cm
2 (moderately Mg doped) is not very significant and may therefore not represent a true trend. The inversion domain boundaries present in the films were observed to have horizontally multifaceted shapes. The vertical type inversion domain boundaries were also observed in Al0.13Ga0.87N grown with low Mg precursor flow rate. In addition, isoelectronic In doping was also used to improve the crystallinity of AlGaN. Nakamura et al. [983] reported the structural analysis of

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Si-doped AlGaN/GaN multiquantum wells. The effect of Si doping on the structural properties of AlGaN/GaN multiquantum well layers grown on GaN was investigated. V-shaped defects were formed by Si doping. PL intensity gradually increased with Si doping. But when the Si concentration exceeded 4.2 · 1019 cm
3, PL intensity rapidly decreased with the formation of V-shaped defect. See Section 4.2.3 for description of V defects. For UV emitters and detectors, highly conductive n-type AlGaN alloys with high Al contents are indispensable. But, AlxGa1
xN alloys with high x are very difficult to grow. For silicon-doped AlxGa1
xN, the conductivity decreases by six orders of magnitude from x ¼ 0.2 to x ¼ 1. Undoped AlxGa1
xN alloys with high x (x > 0.4) generally have high resistivity, possibly because of the sharp increase of the carrier localization energy around x ¼ 0.4. Although n-type AlxGa1
xN (x ¼ 0.58, conductivity ¼ 0.08 O
1 cm
1) with Si doping and n-type AlxGa1
xN (conductivity 5 O
1 cm
1) with indium–silicon codoping were realized [984,985], highly conductive n-type AlxGa1
xN alloys with Al contents as high as 0.6–0.7 are still needed. Nam et al. [986] reported the OMVPE growth of highly conductive n-type Al0.65Ga0.35N with a conductivity of 6.7 O
1 cm
1 (free electron concentration 2.1 · 1018 cm
3 and mobility of 20 cm2 V
1 s
1 at room temperature). Their experimental results also revealed that the conductivity of AlxGa1
xN alloys continuously increases with an increase in Si doping level for a fixed Al content value and that a critical Si doping level of about 1 · 1018 cm
3 is needed to render insulating AlxGa1
xN (x > 0.4) n-type. Hall-effect measurements show that the activation energy of the Si donor increases linearly from 0.02 eV in GaN to 0.32 eV in AlN [987]. Yoshida et al. [988] noted a decrease in the free carrier concentration from 1020 cm
3 in GaN to 1017 cm
3 for AlGaN with x ¼ 0.3 in nominally undoped films grown by MBE. Steude et al. [989] obtained the localization energy Eloc ¼ 0.08 eV and the donorbinding energy ED ¼ 0.4 eV from Eloc ¼ aED, which showed that in high-quality undoped AlGaN films with room temperature free carrier concentrations below 1017 cm
3, Si is residual donor. 3.5.15.1.1 Growth of p-Type AlGaN In p–n junction devices such as LEDs and lasers, p-type layers are imperative. Although the device design in HBTs and detectors could be manipulated to rely only on p-type GaN, this additional restriction is not welcome. This being the cases, efforts have been undertaken to produce p-type AlGaN. Although applicable to all ternaries in general, to reduce the parasitic reactions between metalorganic sources, particularly with TMA and NH3, a dualflow channel reactor was employed [990]. For activating Mg in AlGaN, flash-lamp annealing under atmospheric pressure nitrogen flow at 1140  C for 60 s was used. One problem with p-type nitrides, progressively getting worse as the AlN mole fraction increases, is the poor ohmic contact, which makes measurements difficult and less accurate. Measured temperature dependencies of the hole concentration indicate that the hole concentration drops off with decreasing temperature from about 1018 cm
3 at 500 K to about 1013 cm
3 at 125 K. Follow-up studies of Mg-doped AlxGa1
xN have resulted in p-type conductivity for Al mole fractions up to 27% [991–995]. Essentially, the binding energy of Mg progressively gets larger leading to lower and lower hole

3.5 The Art and Technology of Growth of Nitrides

0.38

10 19

0.36 10

Mg activation energy, EA (eV)

0.32

P(cm –3)

0.34

x=0.27 EA=0.31eV

17

10 15

0.30 0.28

10 13

2

0.26

3

4 5 1000/T(K –1)

6

0.24 0.22 0.20 0.18 0.16

GaN

0.14 0.0

0.1 0.2 Al content, x

Figure 3.200 Activation energies of Mg acceptors in Mg-doped p-type AlxGa1
xN as a function of Al content x. Solid squares, solid and hollow circles, and triangles are data from Refs [993–995], respectively, all obtained by Hall measurements. Open circles indicate data

0.3 obtained by PL measurements from Ref [995]. The inset shows measured temperature dependence of Hall concentration p in the Mgdoped p-type Al0.27Ga0.73N sample from which EA ¼ 0.310 eV was obtained [995].

concentrations as the Al mole fraction is increased, making it difficult for device applications requiring p-doping and large mole fractions. Among them are the UV detectors and lasers. Figure 3.200 shows activation energies of Mg acceptors in Mgdoped p-type AlxGa1
xN as a function of Al content x. For x ¼ 0.27, activation energy increases from 150 meV in GaN to 310 meV in Al0.27Ga0.73N as determined by variable temperature Hall measurements that include screening. From the measured EA versus x in Mg-doped p-type AlxGa1
xN, the resistivity versus x can be estimated as follows [994]: rðAlx Ga1
x N Þ

¼ r0 exp½E A ðAlx Ga1
x N Þ=kT ; ¼ r0 expf½ðE A ðGaNÞ þ DE A =kT Þg; ¼ rðGaNÞÞexpðDE A =kT Þ;

ð3:69Þ

where DE A ¼ E A ðAlx Ga1
x N Þ
E A ðGaNÞ. A typical p-type GaN has a resistivity of about 1.0 W-cm, which, using Equation 3.69, leads to an activation energy and a resistivity of 0.4 eV and 2.2  104 W-cm, respectively, for the Al0.45Ga0.55N alloy, which

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is needed for solar-blind detectors. Obviously, the problem of high resistivity is of paramount concern for solar-blind detectors and deeper UV emitters. 3.5.15.1.2 Ordering in AlGaN Because the misfit between AlN and GaN is only 2.5%, one would expect a stable growth of AlGaN with good control of composition. However, the atomic long-range ordering along the c-axis had also been observed in the form of AlN/GaN (1 : 1), Al0.25Ga0.75N (3 : 1) and Al0.16Ga0.84N (10 : 2) [996]. These three chemical orderings were found to coexist in the same layers and lead to inhomogeneous composition in AlGaN films. Korakakis et al. [997] first reported the observation of long-range order in AlGaN films grown by MBE on sapphire and 6H-SiC. In XRD studies of AlGaN films grown under nitrogen-rich conditions, other than the (0 0 0 2) and (0 0 0 4) diffraction peaks expected in hexagonal AlGaN, they observed the normally forbidden diffraction peaks (0 0 0 1), (0 0 0 3), and (0 0 0 5), as shown in Figure 3.201. The presence of such normally forbidden diffraction peaks indicates the long-term atomic ordering existing in AlGaN. AlGaN films with different Al contents were investigated, the relative intensity of these peaks was found to be largest for Al content in the 30–50% range, in qualitative agreement with expectations for an ordered structure of ideal Al0.5Ga0.5N stoichiometry. The maximum ordering was found to occur in Al0.5Ga0.5N, just as expected. The average size of the ordered domains in the films was found to be within a factor of 4 of the films’ thickness. Iliopoulos et al. [947] found that the ratio of group III and group V fluxes influences the relative incorporation of gallium and aluminum in AlGaN. Different types of spontaneously formed superlattice structures with periodicities of 2, 7, and 12 ML were found. Figure 3.202 is a cross-sectional TEM image with selected area diffrac-

Al0.25Ga0.75N

Al0.25Ga 0.75N ×1 (0 0 0 2)

Intensity (au)

×30 (0 0 0 1)

GaN x1 (0 0 0 2)

17.0

17.5

34.5

35.0

2θ (degrees) Figure 3.201 y-2y XRD scan of a bilayer GaN–Al0.25Ga0.75N grown on c-plane sapphire. Courtesy of T.D. Moustakas and Ref. [997].

3.5 The Art and Technology of Growth of Nitrides

Figure 3.202 Cross-sectional TEM and SAD pattern of an Al0.89Ga0.11N film grown under group III-rich condition. Courtesy of T.D. Moustakas and Ref. [947].

tion (SAD) pattern of an Al0.89Ga0.11N film grown under group III rich condition. The SAD indicated a more complex superlattice than that reported earlier by the same group [997]. Figure 3.203 is an on-axis y
2y XRD spectra of three AlGaN films with different III/V ratios and Al mole fraction, but other growth conditions were similar. AlGaN (AlN mole fraction is 55  2% by RBS and 45.5% by XRD) grown under N-rich condition (III/V ¼ 0.6) showed a long range atomic order consisting of a superlattice

Figure 3.203 On-axis XRD spectra of the three AlGaN films grown with different III/V ratios and Al mole fractions, 1, 0.9, and 0.6 for samples A (89% Al mole fraction), B (74% Al mole fraction) and C (55% Al mole fraction), respectively. Courtesy of T.D. Moustakas and Ref. [947].

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with 2 ML periodicity reported previously [997]. AlGaN films grown under group III rich conditions showed a number of additional peaks, suggesting superlattice structures. By fitting the data to Voight functions, it was concluded that AlGaN grown under group III rich conditions has two intermixed superstructures with different periodicities of 7 and 12 ML. The origin and the exact structure of these spontaneously formed superlattices have not yet been established. In Iliopoulos’s work [947], the three kinds of superlattice structures were found in different AlGaN layers grown under different conditions. However, Ruterana et al. [996] reported the coexistence of different types of ordering in the same AlGaN film. The sample they investigated was grown by OMVPE with good optoelectronic properties. A close examination by diffraction analysis as shown in Figure 3.204 showed the AlGaN sample had the usual 1 : 1 (Ga : Al) ordered domains [996] and the 3 : 1 ordering, which was reported in InGaN [1027]. Moreover, another type of ordering is shown to take place in which a series of five GaN cells and one AlN form the new supercell (i.e., 10 GaN monolayers and 2 AlN monolayers). The average composition of such ordered phase corresponds to Al0.66Ga83.4N and the electron diffraction analysis was made inside a 14% Al composition layer. The occurrence of such ordered phases gives rise to compounds with lower symmetry than the parent wurtzite GaN with the same space group (P3m1). However, a more accurate spatial distribution and the structural relationships between the ordered phases and the matrix is needed, requiring further investigation in an effort to understand formation mechanisms involved.

Figure 3.204 Diffraction patterns with superlattice spots. (a) 1 : 1 ordered phase along the ½1 0 1 0 zone axis, g1//[0 0 0 1], g2 ==½1 1 2 0; (b) 3 : 1 with three spots between 0 0 0 0 and 0 0 0 2, along ½1 0 1 0 zone axis; and (c) superlattice spots, d is approximately 3.1 nm, diffraction pattern along the ½1 1 2 0 direction, g1: [0 0 0 1], g 2 ==½1 0 1 0. Courtesy of P. Ruterana and Ref. [996].

3.5 The Art and Technology of Growth of Nitrides

Figure 3.204 (Continued )

3.5.15.2 Growth of InGaN The growth of GaInN alloys has proven to be relatively more difficult. The large difference in interatomic spacing between GaN and InN and high nitrogen pressure over InN due to N volatility are the causes that give rise to a solid phase miscibility gap. Growth at high temperature (800  C) results in higher crystalline quality, but the amount of InN in the alloy is low and very high V/III ratios, meaning primarily high NH3 flow rates, are required. It should be mentioned that the flow rate for a given

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V/III ration is system dependent. In author’s laboratory the ammonia flow rate is about 10 l min
1. Conversely, growth at lower temperatures (500  C) increases the InN concentration but at the expense of low crystalline quality [998]. Increasing In pressure in the vapor results in the formation of In droplets on the surface [999]. The early attempts for realizing single crystalline InGaN by OMVPE were made by Nagatomo et al. in 1989 [1000] and Matsuoka et al. [1001] again in 1989, and followed by Yoshimoto et al. in 1991 [1002]. Since then, considerable work has been expended worldwide. Matsuoka et al. [1003] discovered that lowering the growth temperature to 500  C from nominal temperatures, such as 800  C, increased the In content in the layers, but at the expense of reduced quality. Efforts to increase the In concentration by raising the indium precursor temperature or the carrier gas flow rate resulted in degradation of the structural and surface morphology so much so that In droplets were formed on the surface, which is common to all the other growth methods, including sputtering [1004]. Because InGaN layers are typically used as thin active layers and are generally straddled by GaN and/or AlGaN layers, and gas source consumption is considerable, only thin layers are grown in the form of quantum wells. Consequently, the bulk of the reports in this material system deal with quantum wells. It is therefore instructive to discuss the aspects of InGaN quantum wells in the context of growth related issues. Defects and in particular V defects garnered a good deal of attention as reflected by number of publications. In an effort to map the quality of InGaN layers Reed et al. [1005] prepared 5, 10, and 15% InGaN double heterostructures by OMVPE and studied the PL, X-ray, and Hall mobilities to determine the best conditions for InGaN layers. The structures consisted of a LT GaN buffer(550  C) followed by 1 mm GaN then InGaN growth at 800  C (5%), 775  C (10%), and 750  C (16%), and finally a 100 nm GaN cap at 1000  C (5–10%) and 880  C (16%). The premise of this particular exercise is that there is a critical range of parameters, namely the thickness and intertwined with it the In molar fraction, within which GaN/InGaN structures display the best electrical and optical properties. The electron mobility increases with increasing InGaN thickness up to 100 nm for 5% and 50 nm for 10% In molar fraction, beyond which that parameter degrades. The authors suggest that all of these results can be related to the strain/relaxation state of the InGaN material as well as continued relaxation during any thermal loads encountered by the samples even after the MQW has been capped. Moving from the dimensional parameters associated with the structures to the growth-related parameters, Piscopiello et al. [1006] investigated the effects of H2/N2 ratio as carrier gas during InGaN growth. Their OMVPE growth on sapphire commenced with a 50 nm nucleation layer formed at 560  C followed by 1 mm GaN grown at 1150  C and finally 100 nm InGaN grown at 790  C under H2 and N2/H2 (4 : 1). While the surface roughness between the two is about the same, in the area of a few nanometers, a large difference is in the indium incorporation, which increases as the N2/H2 ratio increases, was observed. However, the defect density and structure are reported to be very similar between the two types of samples. Focusing on the effects of growth-related parameters, Kim et al. [1007] set out to determine the effects of temperature and carrier gas on the quantum barriers in the MQW. They did so by using thermally precracked NH3 and growing a 30 nm LT

3.5 The Art and Technology of Growth of Nitrides

buffer (500  C) followed by 2 mm GaN (1080  C at 300 Torr) then a five period In0.28Ga0.72N (3 nm)/GaN (7 nm) MQW with InGaN at 700–740  C and GaN at 840–880  C. Those authors explored constant growth temperature under N2 ambience for both the wells and barriers, ramping up the temperature without capping for the barriers under N2 ambience, ramping up the temperature with capping for the barriers under N2 ambience, and ramping up the temperature with capping for the barriers under H2 ambience. The results show that the sample with H2 ambience for the capping layer had the best PL results but a higher XRD value. The authors also suggest optimization of capping layers as they saw degradation of PL if the cap was over 1 nm. This degradation would more likely be due to temperature loads of the cap layer instead of layer thickness according picture presented by Reed et al. [1005] regarding the critical layer thickness. Along similar lines, Lefebvre et al. [1008] attempted to determine the effects on PL of barriers versus QWs and AlGaN caps versus GaN caps. Their MBE growth on sapphire consisted of a 1–5 mm GaN buffer (800  C) followed by 15–20% InGaN (550  C) with a GaN or AlGaN cap. The samples investigated had 2 nm thick quantum wells with either GaN or Al0.14Ga0.86N barriers bulk and/or surface barriers. The buried and surface GaN barrier layer thicknesses of 800 and 40 nm, respectively, were employed. In another set, buried GaN barrier layers of 500 nm and Al0.14Ga0.86N surface barrier of 30 nm were employed. The authors found increased PL intensity at higher temperatures for the layers with the AlGaN cap versus the GaN cap. They attribute this to interdiffusion of aluminum atoms introducing potential barriers, reducing the in-plane mobility of carriers between different recombinations centers and toward nonradiative centers. The optical emission seen was attributed primarily to localized electron–hole pairs even at room temperature. Furthermore, the authors did not note any benefit because of quantum boxes over quantum boxes because the localization was smaller than the quantum box dimensions. Ting et al. [1009] studied the evolution of defects and related morphology in InGaN/ GaN MQWs as well as subsequent device characteristics. The growth was conducted by OMVPE on sapphire with a 300 Å GaN buffer layer (grown at 520  C) followed by 3 mm n-GaN (4 · 1018 cm
3) then 1 mm unintentionally doped GaN (1050  C). The growth continued with a 2500 Å n-GaN regrowth burying layer followed by a 20 Å GaN barrier layer and finally a five period 30 Å In0.3Ga0.7N/20 Å GaN MQWs. Single quantum well (QW) structures were also grown and studied and included a 120 Å GaN barrier on top. Much of the analysis relied on AFM imaging and LED device fabrication and testing. Applying the best growth conditions based on the materials characterization, the authors created a thermally robust 525 nm LED. It was concluded that inclusion formation being evident only in InGaN layers could not be the result of LT GaN growth, but that the inclusions continued to grow during GaN periods of the multiple quantum well structure. The authors also observed that adding H2 or ramping the temperature during the GaN barrier layer growth would planarize the InGaN surface and improve barrier layer properties. The authors believe the V defect is a result of transitioning from InGaN 3D growth to GaN 2D growth and it is this interface that is the most important because InGaN layers seem to have 3D growth no matter how good the GaN

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layer underneath looks. The PL on the MQWs shows that the structures grown with H2 used in the barrier show three times less thermal degradation when subjected to subsequent thermal processes than the ones with N2 used in the barrier layer. Named after their shapre V defects are fairly common in InGaN and InGaNcontaining quantum wells. Many studies have been devoted to the investigation of these defects including their genesis. Among them, Sharma et al. [1010] also discussed V defects found in InGaN/GaN MQWs including their formation and chemical composition. These V-shaped voids, which have also been discussed in the same context by Cho et al. [1011], should not be confused with pyramidal defect observed in Mg-doped GaN and AlGaN layers, a topic discussed in Section 4.2.3. The samples used for this investigation were grown by OMVPE using low-temperature GaN initiation layers. The structures investigated in the effort of Sharma et al. were grown using a typical two-step procedure as follows: a 50 nm nucleation layer was deposited at a susceptor temperature 560  C followed by 150 nm of undoped GaN. Then a 2.8 mm layer of n-type Si-doped GaN was deposited at 1065  C. The MQW was composed of five periods of InGaN/GaN grown at 790 and 950  C, respectively. Finally, the entire structure was capped by a 150 nm p-type Mg-doped GaN layer grown at 1045  C. Energy filtered cross-sectional TEM chemical mapping of the quantum well structures was used to concluded that the V defects nucleate at dislocations in the first layer and propagate through the entire active layers that can be filled with p-type GaN. Cho et al. [1011] reported, however, that the V pits can be nucleated at a stacking mismatch boundary, which, in turn, is caused by a stacking fault. A TEM image of the threading dislocation induced V defect is shown in Figure 3.205, whereas models for threading dislocation nucleated and stacking mismatch boundary nucleated models for Vdefects are shown in Figure 3.206. Further, TEM and annular dark field imaging (ADF) studies of the samples show these V defects to be approximately 100–200 nm apart along the MQW. Moreover, it has been stated that each V defect incorporates a pure edge ðb ¼ 1=3 h1 1 2 0iÞ dislocation, which runs through the apex of the V defect up to the free surface. The lack of growth where these dislocations are is most likely because of preferential evaporation of reactive species at the dislocation site at the particular temperature dictated by InGaN growth. However, growth of the p-type layer at a higher temperature, 1045  C, enhances reactive specie diffusion at the surface ensuing lateral growth that together leads to filling of the V groove defect as shown in Figure 3.206. Along the same lines, Kim et al. [1012] prepared MQWs and also thicker (bulk) InGaN to study V defects, indium mole ratio, and thickness. The authors prepared bulk (considered bulk in terms of exciton radius and de Broglie wavelength) GaN by OMVPE and MBE on OMVPE templates followed by a MQW structure grown by OMVPE on sapphire. It consisted of a 25 nm buffer layer (grown at 550  C) followed by 1.5 mm GaN grown at 700  C and then a 2 : 3 thickness ratio, 5 period InGaN/GaN MQW grown at 800  C. Relying on TEM images, Kim et al. [1012] have argued that only mixed and screw dislocations terminate in V defects on the surface, with pure edge having no surface characteristic. Therefore, Kim et al. [1012] disagree with Sharma et al. [1010] in regard to the type of dislocation that initiates a V-defect site. Kim et al. [1012] observed a clear degradation in surface quality as the In incorporation increased. This poor morphology has been also linked to increased strain in the InGaN layer due to increasing indium

3.5 The Art and Technology of Growth of Nitrides

Indium concentration (%)

15

10

5

0 (c)

0

20

40

60

80

Profile width (nm) Figure 3.205 Energy filtered TEM (EFTEM) indium ratio map (a) and gallium elemental map (b), which show that OMVPE-grown InGaN quantum wells end abruptly at the V-defect boundary. Quantitative EFTEM from a line profile across the MQW (b) shows a small variation in the indium concentration between the wells (c). The V-defects can be filled with the p-type GaN cap layer. Courtesy of C. Humphreys and Ref. [1010].

mole ratio and increasing thickness. Poor XRD results for the thicker layers with higher indium incorporation have been attributed to phase segregation in the InGaN. Other substrates were also used for InGaN growth. It was reported that the crystalline quality of InGaN was superior when grown with the composition that lattice matches ZnO substrate to that grown on bare (0 0 0 1) sapphire sub-

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Filled V-pit V-pit

V-pit p-GaN

SMB SF n-GaN

Threading dislocation

InGaN GaN SMB: stacking mismatch boundary SF: stacking fault Figure 3.206 A model for the formation of the V-defect nucleated at a threading dislocation defect site or due to a combination of a stacking fault (SF) and a SMB. Growth of p-type GaN has been reported to fill the V-shaped pit. GaN. Patterned after Refs [1010,1011].

strate [1002,1003]. In the same investigations, it was observed that InGaN films grown on sapphire substrates using GaN as buffer layers exhibited much better optical properties than InGaN films grown directly on sapphire substrates [1013]. For a given set of growth conditions, an increase of InN in InGaN can be achieved by reducing the hydrogen flow [1014]. The overarching issue in In containing ternary and quaternary nitride semiconductors is the phase separation and compositional instability caused by great disparities between In and other group III metals used, such as between atomic radii, equilibrium N vapor pressure over the metal, surface diffusion barrier, and other aspects. Although the wrenching issues in this vein are left to be discussed in Section 3.5.15.2.2, a glossary is given here. In this vein, Ho and Stringfellow [1015] investigated the temperature dependence of the binodal and spinodal boundaries in the InGaN system with a modified valence force field model. The calculation of the extent of the miscibility gap yielded an equilibrium InN mole fraction in GaN of less than 6% at 800  C [1015]. In the annealing experiments in argon ambience, the phase separation of an InxGa1
x N alloy with x  0.1 was observed at 600 and 700  C [1016], pointing to the large region of solid immiscibility of these alloys. However, under nonequilibrium growth conditions, Ga1
x InxN layers were grown in the entire range of compositions. However, the decomposition into two phases upon annealing of the InxGa1
xN alloys (x ¼ 0.11 and x ¼ 0.29) at 600 and 700  C was observed pointing to the existence of the miscibility gap. For some alloys with x ¼ 0.6, the phase separation could not be observed at 600  C. Above 800  C, the alloy samples with x ¼ 0.1 actively evaporated from the substrate. These results suggest that the solid solutions are grown in metastable conditions and decompose under annealing conditions.

3.5 The Art and Technology of Growth of Nitrides

Koukitu et al. [1017] performed a thermodynamical analysis of InGaN alloys grown by OMVPE. They found that in contrast to other III–III–V alloy systems where the solid composition is a linear function of the molar ratio of the group III metalorganic precursors at constant partial pressure of group V gas, the solid composition of InGaN deviates significantly from a linear function at high substrate temperatures. Kawaguchi et al. [1018] reported on the so-called InGaN composition pulling effect in which the indium fraction is smaller during the initial stages of growth but increases with increasing growth thickness. This observation was to a first extent independent of the underlying layer whether it was GaN or AlGaN. The authors suggested that this effect is caused by strain due to the lattice mismatch at the interface. It was found that a larger lattice mismatch between InGaN and the bottom epitaxial layers was accompanied by a larger change in the In content. What one can glean from this is that the indium distribution mechanism in InGaN alloy is caused by the lattice deformation due to the lattice mismatch. With increasing thickness, the lattice strain is relaxed owing to the formation of structural defects, which weakens the compositional pulling. As always, progress is made this time on the front of better understanding processes involved in the growth of In-containing ternaries. Reaction pathways in effect during growth of In-based alloys have been investigated with good progress [1019]. Such insight led to the attainment of InGaN films with high InN molar fraction and improved structural and optical properties. It is to be noted that the phase separation is a real concern in this alloy, particularly for high InN molar fraction. Using a unique susceptor, Bedair et al. [1019] reported In0.1Ga0.9N layers with symmetric X-ray FWHM values of as little as 6 arcmin. The PL was characterized as being intense band edge type with occasional deep level transitions. In this particular approach, dubbed molecular stream epitaxy (MSE), the substrate is first exposed to a mixed gas composition inclusive of group III and V species. The sample is then rotated away from the impinging gas along the axis of the susceptor. After a complete rotation, the substrate is again exposed to reactant gases. The state of affairs is such that presence of In brings about compositional nonuniformities and localization, which makes the picture highly complex. Although higher temperatures and ensuing inhomogeneities have been mastered to extract efficient light from InGaN for LEDs and lasers, the same is not helpful for electron transport. Consequently, the expected relatively higher mobilities visceral to InGaN have not materialized. The electrical properties of InGaN quantum wells, in the form similar to single-period modulation doped or undoped single heterostructures have also been investigated in structures grown by both MBE and OMVPE. Because the implications are more pertinent to FETs, the some details are discussed in Volume 2, Chapter 5. In an effort to enhance the quality of InGaN in MQW structures, Pozina et al. [1020] employed a mass transport overgrowth technique to grow high-quality InGaN/GaN MQWs with good optical properties. The OMVPE growth on sapphire started with an AlN buffer followed by 7 mm unintentionally GaN. Micrometer size trenches (3, 5, 10 mm) were then patterned using RIE and the sample was annealed at 1100  C in ammonia and N2 ambience resulting in mass transport overgrowth. Finally, 35 Å

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InGaN/105 Å GaN (Si doped to 1018) three-period MQWs were grown. The CL images taken by Pozina et al. [1020] suggest improved optical and physical properties of the mass transport regions. These improvements include a dislocation density reduction from >108 to 0.1 following annealing in an argon ambience at 600 and 700  C. The large region of solid immiscibility may in part be responsible for the difficulties involved in the epitaxial growth of these alloys. Rigorous first principles thermodynamics calculations with a focus on phase separation in the InGaN have been undertaken by Teles et al. [1040]. The calculations are based on the generalized quasichemical approach to disorder and composition effects and a pseudopotential plane wave approximation for the total energy. Those authors generalized the cluster treatment to study the influence of biaxial strain, which pointed to a remarkable suppression of phase separation in InxGa1
xN. These are consistent with experiments. The calculations of Teles et al. [1040] led to the conclusion that a broad miscibility gap exits for growth temperatures around 1000 K for unstrained InxGa1
xN alloys. Those authors also calculated a critical temperature of 1295 K for phase separation, which is some 250 K lower than that calculated by Ho and Stringfellow [1015]. However, the resulting structural properties, such as the dependence of the lattice constant on composition, the bond lengths and the second nearest neighbor distances are in agreement with other calculations or measurements, and that the structural parameters for intermediate In molar fractions would exhibit notable fluctuations. Fortuitously, the biaxial strain has far reaching consequences in terms of the miscibility behavior of the alloys in that miscibility gap is reduced [1040]. This is particularly the case for cubic InxGa1
xN, which is conjectured to represent wurtzitic alloy in this respect, as compared to AlxIn1
xN. Recall that AlxGa1
xN does not exhibit miscibility gap and as such the effect of strain is a mute point. However, for the InxGa1
xN case, the higher the InN molar fraction, the more noticeable the effect. Invariably, the region of spontaneous decomposition is reduced by strain. By considering an inhomogeneous strain distribution over the clusters contributing to the ternary alloy, Teles et al. [1040] sought to show that a regular behavior of the structural and elastic properties are obtained. In this limit, the miscibility gap and the critical temperature are remarkably reduced. For coherent layers, that is, the in-plane lattice constants of the underlying binary compounds, the phase separation is even completely suppressed. As the in-plane strain increased from the relaxed case to e// ¼
0.01 to
0.05, the excess free energy is decreased by the elastic energy due to the buildup strain. Consequently, the miscibility gap decreased with increased strain as shown in the T–x phase diagram of Figure 3.208. For a biaxial strain of 21%, the strain-induced change in the critical temperature is not notable, except that the phase diagram becomes only slightly more asymmetric owing to small elastic energy. However, because the strain energy is proportional to the square of the in-plane strain e// further increases as the critical temperature decreases and reaches a low of 120 K for 25% strain. Strain also reduces the miscibility gap as well as the region of spontaneous decomposition, particularly for larger In molar fractions. In short, the in-plane compressive strain suppresses the phase separation.

3.5 The Art and Technology of Growth of Nitrides

1600

Binodal Spinodal

ε = – 0.001

1400

Temperature (K)

1200

1000 ε = – 0.05

800

600

400

200

0 0.0

0.2

0.4

0.6

0.8

1.0

Composition, x Figure 3.208 T–x phase diagram for homogeneously strained InxGa1
xN. Heavy and light lines depict the binodal decomposition curves while the corresponding dashed ones depict the spinodal decomposition curves [1040].

There are indications that the assumption of an equal strain in all microclusters of the alloy is not suitable. Teles et al. [1040] observed that the c12/c11 elastic stiffness constant ratio is nonlinear versus composition for both tensile and a compressive strain of 1%. It should be mentioned that the arithmetic average of the two curves gives an almost linear variation of the ratio c12/c11 with composition between the values known for GaN and InN. Therefore, an accurate representation requires the inhomogeneous strain distribution in the alloy on a microscopic length scale be considered. Congruent with one of the basic assumptions that each cluster takes its own volume, Teles et al. [1040] assumed a coherently strained alloy, which means that each cluster j is strained according to the change in the in-plane lattice constant a//. The in-plane strain in each cluster is therefore given by ej// ¼ (a//
aj)/aj. Consider two inhomogeneous strain cases: one representing the coherent growth of the ternary InxGa1
xN alloy on a GaN substrate with a// ¼ aGaN and the other on InN substrate with a// ¼ aInN. The thermodynamic potential taking inhomogeneous strain into consideration is shown in Figure 3.209. The inhomogeneous strain distribution, albeit assumed in this model, substantially affects the mixing free energy of the system by changing primarily the mixing enthalpy.

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Mixing free energy, ΔF (eV/pair)

0.0

on GaN on InN

–0.01

T (K)

–0.02

700

–0.03

950

–0.04 700

–0.05 –0.06 –0.07 0.0

950

0.2

0.4

0.6

0.8

1.0

Composition, x Figure 3.209 Compositional dependence of the mixing free energy of inhomogeneously strained InxGa1
xN alloys for T ¼ 700 and 950 K. The biaxial strain is defined by a// ¼ aGaN and depicted with solid lines or in the case of growth on InN a// ¼ aInN and depicted with dashed lines [1040].

Ho and Stringfellow [1015] demonstrated, using a valence-force-field model, that at growth temperature of 800  C, the solubility of In in GaN is less than 6%, as shown in Figure 3.207 assuming of course that the thermodynamic equilibrium holds. Equilibrium conditions do not really apply to a large extent to MBE and apply more readily to OMVPE grown layers. Regardless of the extent, phase separation is synonymous with InGaN. Ho and Stringfellow found a critical temperature of 1250  C for phase separation of InGaN, which is higher than the melting point of InN. It should be noted that at nominal growth temperatures, the alloy is unstable over the entire composition. With strain, however, as shown comparatively in Figure 3.210a and b, the solid solubility of the alloy can be increased remarkably to a value of about 60% at 800  C from the 6% value effective for the relaxed case [1032]. Behbehani et al. [1048] observed phase separation in InxGa1
xN samples grown by metal organic chemical vapor deposition (OMVPE). For x ¼ 0.49, phase separation and c-plane ordering were observed by both XRD and SAD analyses. A typical transmission electron micrograph image for a nominally In0.49Ga0.51N and its SAD image are shown in Figure 3.211 where the dark regions contain high InN composition and lower percentage of InN regions showing ordering. Extra spots in the SAD image are due to ordering and some corresponding to In0.25Ga0.75N. Samples with less than x ¼ 0.25, did not show phase separation. Surprisingly, in thin samples, for x ¼ 0.21 extra spots were observed by SAD analysis; however, thick layers did not show any sign of phase separation. The SAD pattern revealed that both In0.5Ga0.5N and In0.21Ga0.79N samples showed ordering pattern. In general, InGaN samples containing a high percentage of InN show ordering [1015]. The extra spots might be due to defects. The nonideal diffraction patterns were observed, which are

3.5 The Art and Technology of Growth of Nitrides

(a)

Binodal

Spinodal

1600

Temperature (K)

1200

Relaxed InGaN

800

400

0 0.0

0.2

GaN

0.4

0.6

0.8

1.0

InN

Indium fraction, x

1200 (b) 1000

Wurtzite to c-axis

interface

Temperature (K)

800 600

Binodal

Strained InGaN

400 200 0 0.0

Spinodal

0.2

0.4

0.6

0.8

1.0

InN

GaN Indium fraction, x

Figure 3.210 Phase diagram of InN–GaN system (Binodal–solid line, spinodal–dashed line); (a) for relaxed InGaN and (b) for strained InGaN where an asymmetric shift to of the equilibrium InN fraction in the alloy to higher values occurs [1032].

due to the interference of the 1 : 1 and 1 : 3 ordered phases for x ¼ 0.25. The ordered phase has stacking Ga and In atoms with ratio of 3 : 1 along the c-axis (Ga–Ga– Ga–In–Ga–Ga–Ga–In–Ga–). The authors constructed theoretical diffraction patterns for 1 : 3 and 1 : 1 orderings, as shown in Figure 3.212. Simulated lattice images by Fourier transforms indicated that the interference of the 1 : 1 (left) and 1 : 3 (right) ordered phases showed the nonideal diffraction patterns A 2–111 twin plane with 35 rotation around ½0 1 1 0 [1048]. Sing et al. [1049] studied the segregation of secondary phases in InGaN thin layers and heterostructures grown by MBE. The InGaN layers (x > 0.30) showed phase separation by both XRD and optical absorption analysis in that InN, In0.37Ga0.63N,

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Figure 3.211 (a) TEM image of a nominally In0.49Ga0.51N layer where the dark regions contain high InN composition and lower percentage of InN regions showed ordering; (b) electron diffraction pattern of a nominally In0.49Ga0.51N layer, which shows extra spots due to ordering and corresponding to In0.25Ga0.75N. Courtesy of N. El-Masry and Ref. [1048].

Ga In Ga In

Ga In Ga Ga Ga In Ga Ga

50% In

Ga

Ga: In

Ga: In

1:1

3:1

One unit cells 50% InGaN

Two unit cells 25% InGaN

Figure 3.212 Theoretical diffraction pattern of 1 : 3 (right) and 1 : 1 (left) ordering where the smaller and extra spots are due to ordering along with schematically layered structure for each case. Courtesy of N. El-Masry and Ref. [1048].

25% In

3.5 The Art and Technology of Growth of Nitrides

GaN

104

In0.37Ga0.83N

103

Intensity (au)

InN

102

101

100

–8000 –6000 –4000

–2000

0

2000 4000

6000

Δθ (arcsec) Figure 3.213 X-ray rocking curve of In0.37Ga0.63N epilayers in which InN is phase separated [1049].

and GaN phases were observed in the X-ray data, as shown in Figure 3.213 . However, in an InGaN/GaN heterostructure, only the In0.53Ga0.47N and GaN phases were observed by X-ray rocking curve, as shown in Figure 3.214. Note that the heterostructure suppressed the segregation of InN in the InGaN layers. The optical absorption studies showed a bandgap of 2.0 eV for In0.37Ga0.63N. However, no phase separation was observed by TEM analysis owing to high density of defects. In the double heterostructures, no InN phase separation was detected because the strainassociated thin InGaN layers could prevent phase separation by stabilizing the alloy, as shown in Figure 3.210. El-Masry et al. [1050] extensively studied phase separation in two different types of samples grown at 710  C, which contained 20 and 49% of InN, respectively. The layers grown at 710  C showed In0.2Ga0.8N and GaN phases, whereas those that contained more than 40% of InN in InGaN, showed the intended In0.49Ga0.51N and extra 95, 36, and 14% InN multiphase structures by XRD analysis. Samples grown with 40% InN showed single crystalline nature. SAD pattern also indicated that the layers exhibited single-crystal nature with InN compositions of 10, 28, and 49. InN with 10% phase did not show any sign of extra phases, however, the remaining 28 and 49% InN phase was mixed with some other phases indicating extra phases. In the 49% InN sample, “tweed” nature was observed because of spinodal decomposition showing pseudobinary system of GaN–InN. The solubility of In in GaN is 6% at 800  C, whereas for spinodal composition, it is 22% at the same growth temperature. Moon et al. [1051] studied the suppression of In segregation by use of MQW structures. The segregated In on the surface of the well disturbs the abruptness of the

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GaN

105

Intensity (au)

104

103

In0.53Ga0.47N 102

101

100

–8000 –6000 –4000

–2000

0

2000

4000

6000

Δθ (arcsec) Figure 3.214 X-ray rocking curve of In0.53Ga0.47N/GaN double heterostructure, it shows no sign of InN phase [1049].

interface between the well and barrier and diffuses into the barrier. As-grown films exhibited abrupt and flat interfaces, as revealed by a TEM analysis, the sharpness of which degraded as the interruption time was increased. From XRD analysis, the In composition for InGaN layers in InGaN/GaN MQW was measured 27, 25, 22, and 16% for the growth interruption times of 0, 0.5, 1, and 2 min, respectively. H2 purging eliminates impurities such as C, O, and H at the interface. Again using MQW structures, Moon et al. [1052] reported that randomly distributed indium-segregated regions, which were formed near the upper interface of the InGaN well layers during the subsequent high-temperature growth process were found to act as nonradiative recombination centers and these could be effectively removed by introducing H2 gas during the growth interruption period. McCluskey et al. [1053] observed phase separation in InGaN/GaN quantum well structures grown by OMVPE, which consisted of In0.27Ga0.73N (20 Å)/GaN between a 4 mm GaN : Si layer on sapphire and a 0.2 mm thick GaN : Mg layer. The phase separation was found in the QW structure after annealing at a temperature of 950  C for 40 h, which is related to In-rich InGaN phases in the active region. Optical absorption studies indicated the absorption threshold at 2.65 eV corresponding to InGaN phase with x ¼ 0.35 in the annealed sample. However, XRD analysis showed slightly higher In composition in the sample. In TEM analysis, voids and precipitates were seen. Moire fringes were observed in the In-rich InGaN precipitates, which are due to the difference between the lattice constants of the precipitates and neighboring regions. Energy dispersive X-ray (EDX) analysis revealed that in the precipitates, the In content is 40%, which is much higher than the surrounding areas.

3.5 The Art and Technology of Growth of Nitrides

In another investigation of MQW structures, Narukawa et al. [1054] observed a high density of dark spots in the well region by TEM analysis with a diameter in the range of 2–5 nm, with most being 3 nm. The origin of dark spots was from selfassembled isotropic dotlike structures with an areal density of dots of 5 · 1011 cm
2. The observed In composition was higher than that in the neighboring well region. These observations are very similar to those by Chichibu et al. [1055] in In0.2Ga0.8N for which the experimental artifacts were ruled out. The arisen quantum dots in the well are due to compositional fluctuations because of the intrinsic nature of the InGaN alloy unlike interface fluctuations in the GaAs/AlAs and CdSe/ZnSe quantum wells. Cho et al. [1056] also studied In clusters in the QW structures. TEM analysis showed that as-grown samples exhibited a strong lateral variation in contrast to growth interrupted samples, owing to the presence of indium clusters in the QWs. Similarly, energy filtered TEM (EFTEM) analysis showed clear clusters of Indium in the as-grown samples, as compared to that of growth-interrupted samples. Line scan In composition profiles showed the In compositional fluctuation to be between 5 and 40% in the as-grown samples, which provided direct evidence of the strong indium clustering. In the growth interruption case, samples with 31% composition (with 5 s interruption) and 24% composition (with 30 s interruption), the In compositional fluctuation could not be detected. This does not, however, mean that fluctuations do not occur with growth interruption. It simply means that fluctuations are not as extensive and interruption suppresses them. Again employing TEM, Ruterana et al. [1057] investigated the different strain inhomogeneities introduced by compositional inhomogeneities in InGaN QWs. The OMVPE growth on sapphire included thick GaN buffers and nominally 2.4 nm InGaN QWs. The In content and thus the compositional inhomogeneities were therefore determine, as shown in Figure 3.215. Those authors have found differing thickness in QWs with each subsequent QW being thinner from 3.4–1.7 nm. They also reported nominal In incorporation of 17% but strong segregation with areas between 10 and 45% based on TEM analysis. The overriding conclusion was that the indium has segregated into quantum dots (or quantum boxes) with 2 nm height and 3 nm width. 3.5.15.2.3 Surface Reconstruction of InGaN Unlike its GaN counterpart, InGaN surface has not been heavily reported on. However, sufficient surface studies exist to draw conclusions on the surface reconstruction of In rich InGaN surface and its surface morphology when grown by RF MBE [1058]. In addition to the standard questions dealing with morphology of N- and Ga-polarity films, the question of In from the point of view of surfactant must also be considered. As in the case of GaN, InGaN grown on the metal-polarity surfaces tend to be smoother and In acts as a surfactant. Again, as in the case of GaN, the N-polarity films are rough in nature. Metal-rich metal-polarity (0 0 0 1) (cation-polarity) surface tends to rough surface as one deviates from the metal rich toward the N-rich p conditions. ffiffiffi pffiffiffi Near the smooth/ rough transition of InGaN(0 0 0 1) growth, a bright ð 3  3Þ has been observed. The same surface structure is also observed when the metal flux is turned off while

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Figure 3.215 (a) TEM image of a sample with three InGaN quantum wells with a nominal In molar fraction of 17% and thickness of 2.4 nm. The arrow tips show the local regions with contrast change corresponding to variations in strain. (b) Typical HREM image of the QWs in area close to edge encompassing the maximum thickness of 7 nm. Courtesy of P. Ruterana and Ref. [1057].

pffiffiffi pffiffiffi the N flux is maintained at the growth temperature. The ð 3  3Þ reconstruction in fact persists during cooldown at temperatures as low as 300  C. Chen et al. [1058] noted that the In atoms occupy the top two surface layers when Ga-polarity is used, see Figure 3.216. Based on theoretical results for the surface structures, they argue that strain arising from the presence of In in the second (S3) layer leads to the formation of small pits on the surface. Inside, and at the border of (a)

A1 S3

(0 0 0 1)

(b)

(0 0 0 1)

In N Ga

Figure 3.216 Basic structure of In-rich InGaN surfaces: (a) (0 0 0 1), assuming full coverage of the top hexagonal layer by In, and (b) ð0 0 0 1Þ. The A1 and S3 layers are indicated in (a) [1058].

3.5 The Art and Technology of Growth of Nitrides

the pits, the surface concentration of indium is found to be larger than that far from the pits, which may lead to inhomogeneous indium incorporation in the bulk film. Typically, the InGaN films are grown on GaN buffer layers, which in turn are grown on generally SiC or sapphire substrates. The growth procedures employed for GaN are detailed in Section 3.5. Due to the high vapor pressure of N on InN, relatively large, in relation to Ga, desorption rate of In, and relatively small binding energy of In with N, as compared to Ga and N, the InGaN films are grown at relatively lower temperatures, typically between 550 and 650  C. These substrate temperatures represent a reduction of about 100  C, in some cases higher, as compared to GaN. In the author’s laboratory, the molar fraction of InGaN is controlled with control of In and Ga with abundance of N. In this growth mode, the affinity of Ga and N is such that the Ga flux on the surface is satisfied with N first and the remaining N is available for In for InN formation. The substrate temperature is such that accumulation of In on the surface, beyond the surface layers, is avoided. The molar fraction is dependent of the substrate temperature and higher the substrate temperature, the larger the In flux must be to maintain the same In molar fraction. The general issue of surface In coverage has been addressed well by Chen et al. [1058] by employing a combination of Auger and scanning tunneling microscope (STM) probes. As discussed in Section 3.4.2.9.4, the Ga-rich growth conditions on (0 0 0 1) surface leads to smoother surface in MBE growth of GaN than N-rich. In addition, the ð0 0 0 1Þ surfaces are always relatively rough regardless of whether Ga- or N-rich conditions are used. In the case of InGaN, the role of In on the surface morphology must also be considered. For device structures requiring smooth heterointerfaces, smooth surfaces may be required in which case, as in the case of GaN, it is imperative that conditions leading to smooth InGaN surfaces are found and the smooth to rough transition point determined. Widmann et al. [1059] reported that indium atoms on the surface act as a surfactant, keeping the growth in the smooth regime when the gallium flux is slightly reduced below the transition flux. Feenstra et al. [1060] investigating N-polarity InGaN on GaN ð0 0 0 1Þ did not observe such a surfactant effect on this nominally rough surface. To gain insight on this issue on both the cation and anion surfaces, Chen et al. [1058] undertook a detailed study on the smooth/ rough transition of InGaN growth on both the ð0 0 0  1Þ and (0 0 0 1) faces, the results of which are shown in Figure 3.217. In experiments which dealt with both metal-polar and N-polar surfaces, the nitrogen flux was kept constant, followed by exposing the surface to a certain indium flux while adjusting the gallium flux to determine the smooth to rough transition point. The dashed lines in Figure 3.217 show where the total metal flux (indium gallium) is constant. Figure 3.217a shows the case for the (0 0 0 1) face where it was found that when the indium flux is applied the gallium flux can be greatly reduced (by an amount considerably greater than that of the added indium flux) before the growth transition to a rough. This is interpreted as evidence that indium on the (0 0 0 1) surface serves as a surfactant, consistent with the observations of Widmann et al. [1059]. In contrast, Figure 3.217b shows that for the ð0 0 0  1Þ face, even when a large indium flux is applied, the gallium flux can only be reduced slightly before the growth becomes rough.

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4 (0 0 0 1)

2 Smooth Smooth

1

In flux (1014 cm–2s–1)

In flux (1014 cm–2s–1)

(a)

0

(0 0 0 1)

(b)

0 3

Smooth Rough 2

1

Rough 0

1

2

Rough

3

Ga flux (1014 cm–2s–1)

0

1

2

3

Ga flux (1014 cm–2s–1) Figure 3.217 Smooth/rough transition on (a) (0 0 0 1) cation face (metal-polar) and (b) ð0 0 0  1Þ anion face (N polar) for a fixed nitrogen flux and a substrate temperature of 600  C. Experimental data are shown with dots representing the experimental data, each with an

error bar. A dashed line is drawn in each figure (-1 slope) for comparison denoting the line with constant total metal flux. To the right of the transition lines, demarked by solid lines, the growth is smooth [1058].

As mentioned above, the competition between Ga and In is really lopsided in favor of Ga so much so that under typical Ga rich conditions In incorporation is not possible. This disparity is made worse because the indium atoms tend to segregate to the surface so that when there is abundance of gallium, the indium atoms do not incorporate. To incorporate In, the gallium flux must be lower than the transition flux of gallium in the absence of indium [1061,1062]. Consequently, to grow InGaN with significant indium content, requires that growth on the (0 0 0 1) face must occur in the smooth regime, whereas it can take place in the rough growth regime for the ð0 0 0  1Þ face. In concert with GaN, the (0 0 0 1) surface is better suited for better quality InGaN growth. The ensuing smooth surface on this polarity bodes well with faster lateral surface diffusion (up to a limit, as excessive diffusion is not desirable either) during growth. Additional insight can be gleaned from the surfactant effect on the (0 0 0 1) surface in that more nitrogen would have to desorb from the surface when the gallium flux is reduced to maintain stoichiometric growth. Moreover, now the total metal flux is lower than that during GaN growth without indium. This increased desorption might result from either higher nitrogen surface diffusivity or higher nitrogen surface concentration. In either case, the nitrogen atoms have more opportunity to meet and form molecules and then leave the surface. For GaN growth, it was pointed out that nitrogen accumulation leads to paffiffiffi higher pffiffiffi diffusion barrier and hence a rough growth [256]. With a nitrogen-rich ð 3  3Þ reconstruction, however, smooth surfaces can be obtained, indicating that indium lowers the diffusion barrier even when the surface has high nitrogen concentration. In PA MBE of GaN surface reconstruction is generally not observed during growth. Although a 2  2 reconstruction occurring during growth [1063–1065] has been reported, this has later been attributed to arsenic present in the growth

3.5 The Art and Technology of Growth of Nitrides

Figure 3.218 RHEED pattern along the ½1 1 0 0 azimuth on InGaN at the growth temperature of 540  C; (a) with In and Ga fluxes as well as the N flux on and (b) with metal fluxes off and N flux on. The InGaN layer was grown under N-rich conditions.

environment [1066]. It should be mentioned that for growth ammonia as the nitrogen source, a different, intrinsic 2  2 structure occurs [1067]. At least in the case of Lin et al. [1063], the system used for GaN was previously used for GaAs and it is conceivable the As p atoms have been present. Unlike GaN, when grown on ffiffiffi pcould ffiffiffi (0 0 0 1) surface, a ð 3  3Þ surface reconstruction is observed in author’s laboratory as well as that reported in by Chen et al. [1058] with no As present in the environment. In fact, in the case of the author’s laboratory, the system used for this experiment had never been exposed to As. While only a 1 · 1 structure is observed when metal and N fluxes are on, when the metal fluxes are turned off, a 3x reconstruction occurs as shown in Figure 3.218. It should be stressed that the Nrich growth conditions were employed during the InGaN growth on GaN templates, which were grown under Ga-richpconditions. ffiffiffi pffiffiffi Further, the GaN barrier layers grown on InGaN quantum well with ð 3  3Þ surface reconstruction maintained the pffiffiffi pffiffiffi ð 3  3Þ surface reconstruction even at the growth temperature with metal fluxes off but N flux on. Figure 3.218 shows the corresponding RHEED pattern, which is a 3x structure when viewed along the ½1 1 0 0 azimuth and 1x structure along the ½1 1 2 0 azimuth, the latter not shown. According to Chen et al. [1058], this reconstruction is observed when the Ga flux is near or below the transition point between rough and smooth growth in the presence of indium. In addition, Chen et al. [1058] have varied the growth conditions and surface stoichiometry in an effort to determine the geometrical structure of the surface reconstruction. In a series of experiments, indium atoms were deposited onto a (0 0 0 1) smooth GaN surface up topaffiffiffifewp monolayers (1 ML ¼ 1.14 · 1015 atoms cm
2) ffiffiffi
1 at a rate of 0.5–2 ML min . No ð 3  3Þ reconstruction was observed during the deposition nor was it observed when the same process was repeated with indium deposition that pffiffiffi pffiffiffi carried out at the growth temperature. These observations indicate pffiffiffi p ffiffiffi ð 3  3Þ reconstruction is not due to an adlayer of indium with ð 3  3Þ structure on top of a gallium-terminated GaN surface. However,pif RF ffiffiffi thepnitrogen ffiffiffi plasma source was turned on when indium was deposited the ð 3  3Þ appeared very quickly. The same result occurred in the nitrogen-rich growthpregime ffiffiffi pffiffiffiwhen indium was deposited during the GaN growth. For the latter case, the ð 3  3Þ was clearly seen when about 1/10 ML of indium was deposited, the intensity reaching a

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maximum when 1/3 ML of indium was deposited. If the gallium flux is well above the smooth to rough transition point, the intensity of the reconstruction gradually decreases and finally disappears when more indium is deposited. Moreover, when GaN is grown in the regime (without indium) followed by deposition of pffiffiffigallium-rich pffiffiffi indium atoms, a ð 3  3 Þ reconstruction does not result. Chen et al. [1058] found pffiffiffi pffiffiffi the ð 3  3Þ reconstruction to persist when the nitrogen plasma is onpwhile ffiffiffi pthe ffiffiffi metal fluxes are off. Also, deposition of In on GaN surface did not lead to a ð 3  3Þ pffiffiffi pffiffiffi reconstruction with N source off, but a ð 3  3Þ reconstruction was observed when the N source is turned on. It can, pffiffitherefore, ffi pffiffiffi be stated that N-rich conditions are required for the formation of the ð 3  3Þ structure. In short, it can be stated that pffiffiffi pffiffiffi the ð 3  3Þ reconstruction forms under N-rich conditions and contains about 1/3 ML of indium p atoms. ffiffiffi pffiffiItffi is worth noting that growth can still be smooth with the nitrogen-rich ð 3  3Þ reconstruction, which implies relatively low surface diffusion barriers for this structure. Chen et al. [1058] also employed scanning tunneling microscopy to study the structure of the InGaN(0 0 0 1) surface, as shown in Figure 3.219. The film shown in Figure 3.219 was grown at 610  C, with Ga and In flux rates of 1.7 · 1014 and 4.8 · 1013 cm
2 s
1, respectively, well above the smooth to rough transition line. From prior theoretical work by the same group one expects the possible presence of surface structures containing In atoms either in the top surface layer or in the top two layers [1068], as illustrated in Figure 3.216a. In Figure 3.219 in the lower part of the image a region of uniform 1 · 1 corrugation is visible. The structure of this region is

Figure 3.219 STM image of InGaN (0 0 0 1) surface containing 0.9  0.2 ML of indium. Image was acquired at a sample voltage of 11.25 V, and with tunnel current of 0.075 nA. Gray scale range is 0.5 Å. Different surface regions are labeled A1 and A1 þ S3 [1058].

3.5 The Art and Technology of Growth of Nitrides

assigned to having In only in the top layer, which is referred to as the “A1 phase.” The bright corrugation maxima observed elsewhere in the image are attributed to In atoms in the S3 layer. The height of the observed bright maxima, typically 0.2 Å above the nominal height of the 1 · 1 region, is consistent with theoretical result of 0.30 Å for the change in surface height caused by a second layer In atom. The small black pits appearing on the surface appear dark (lower surface height) for both positive and negative sample bias voltage, indicating that they are some type of surface vacancy island. Another salient feature in Figure 3.219 is that the indium concentration in the second layer is higher around pits. This inhomogeneous surface In concentration may contribute to the formation of the widely observed indium compositional fluctuation in the bulk InGaN discussed in detail in Volume 2, Chapter 5. The In compositional fluctuation could be grown in from the inhomogeneous surface, caused by the processes of In surface segregation [1068], and formation of a strained surface layer owing to lattice mismatch between InN and GaN. This strain is even larger than typical from bulk properties of these binaries because the In surface concentration is much higher than its bulk concentration. 3.5.15.3 Growth of AlInN Interest in AlInN stems from its tunable bandgap by compositional change during deposition, as in the case of InGaN with the added benefit that the Al0.82In0.18N composition the alloy lattice matches GaN. The alloy for this composition has a bandgap of 4.7 eV, which is larger than that of GaN, and forms a barrier layer for modulation-doped structures. Lattice matching has added consequences in polar semiconductors such as nitrides in that piezoelectric-induced polarization due to mismatch vanishes leaving only the spontaneous polarization owing to compositional grading across the interface of an Al0.82In0.18N/GaN heterojunction. Similar to the case of InGaN, the bandgap and particularly the bandgap bowing parameter of this ternary remained elusive and confusing for quite some time owing in part to the incorrect assignment of the InN fundamental gap. Details of the bowing parameter and related issues are discussed in detail in Section 1.5.3. Moreover, the particulars of the polarization charge involving this ternary together with GaN are treated in Section 2.12. The discussion here will be limited to growth issue. The potential applications of Al1
xInxN include, but not limited to cladding layers that are lattice matched to GaN or as an active layer for LEDs and LDs emitting in the ultraviolet (UV) to infrared (IR) region. This alloy as in the case of other nitride semiconductors has been grown primarily by either OMVPE or MBE. The disparity between InN and AlN in terms of atomic radii and binding energy of N does not set the stage for an easy task. The relatively high equilibrium pressure of N on InN necessitates the relatively low temperatures be used for growth. However, the early growth efforts relied on sputtering such as magnetron RF sputtering [1069]. On the sputtering side, thin films of III–V nitride semiconductors (AlN, GaN, InN), mixed-crystalline films (AlxIn1
xN), as well as multilayered films (GaN/InN)n were grown by RF magnetron sputtering at low substrate temperatures below 500  C [1070]. These films were characterized by X-ray diffraction, Raman scattering,

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optical absorption, and electrical measurements. Dependence of the bandgap energy of AlxIn1
xN on composition x was determined. On the OMVPE side, ternary AlInN thin films were grown using, in one case [1071], a horizontal atmospheric pressure OMVPE reactor. In contrast to extensive efforts invested in Ga1–yInyN by OMVPE and Al1-zGazN for optical device applications such as light-emitting diodes (LEDs) and laser diodes (LDs), efforts, particularly early on, expended for AlxIn1
xN has been sparse. This is to a great extent owing to the difficulties experienced in the growth of high-quality AlxIn1
xN. This ternary is also characterized by its thermodynamic instability. The unstable region in mixing has been calculated from the free energy of mixing using the strictly regular solution model [1072]. From this calculation, one can conclude that the ternary InAlN always shows, InGaN sometimes shows, and GaAlN never shows unstable mixing region at temperatures below 3000  C. As is typical, the precursors for Al and In metals were TMA and TMI, respectively, and ammonia (NH3) for N source. To reduce any parasitic chemical reactions, the TMA, TMI, and NH3 were mixed at the entrance of the reactor chamber. The substrate temperature range, as determined by measuring the temperature of the susceptor on which the substrate rests during growth, was 650–900  C. Sapphire (Al2O3) substrates with different orientations have been used. In another growth experiment with OMVPE, Yamaguchi et al. investigated growth-related processes as well as the determination of the optical properties of this alloy [1073]. In this investigation, Yamaguchi et al. prepared Al1
xInxN films covering a composition range of 0.14 · 0.58 on GaN buffer layers by OMVPE on c-plane sapphire substrates. The growth temperature was 720  C. The FWHM in 2y obtained from an o–2y scan of the (0 0 0 2) XRD peak (denoted by D2yc) increased from about 10 arcmin for AlN to over 60 arcmin for the largest In fraction (x ¼ 0.58). The dispersion in 2y is indicative of the degree of distribution of the lattice constant c, namely, the fluctuation of the alloy composition x in AlxIn1
xN. The FWHM in o obtained from an o-scan of the symmetric (0 0 0 2) peak (Y-axis, denoted by Doc) ranged from about 10 arcmin for x ¼ 0 and 40 arcmin for x ¼ 0.58. The dispersion in the o-scan of the ð1 0 1 0Þ peak (R-axis, denoted by Doa) range from 25 arcmin for AlN (x ¼ 0) to about 40 arcmin for x ¼ 0.58. The spreads depicted by Doc and Doa indicate the degree of tilting and twisting components of the crystalline mosaicity, respectively. Lukitsch et al. [1074] prepared epitaxial AlxIn1
xN thin films with 0 x 1 by plasma source molecular beam epitaxy on sapphire (0 0 0 1) substrates at a low temperature of 375  C. In-plane orientation of AlxIn1
xN on c-plane sapphire was investigated by both reflection high-energy electron diffraction and X-ray diffraction measurements with the result being nitride [0 0 0 1]||sapphire[0 0 0 1] and nitrideh0 1 1 0i||sapphire h2 1 1 0i. This means that, as in the case of GaN, the c-plane of nitride lattice is rotated by 30 with respect to the c-plane of sapphire lattice to reduce the lattice mismatch. But, the broad and diffused spots in RHEED images indicate that the nitride films have some degree of crystalline mosaicity, both along the normal and in the plane of the film. The degree of crystalline mosaicity and the compositional fluctuation was reported to increase with decreasing x (increasing InN fraction). Similarly, XRD FWHM,

3.5 The Art and Technology of Growth of Nitrides

D(2y) in y
2y scan and D(o) in rocking curve scans, both increased with increasing InN composition. The D(2y) widening is indicative of the degree of distribution of lattice constant c, which has been construed as varied alloy composition. The D(o) widening indicates the dispersion in the degree of tilt of the c-axis throughout the mosaic structure of the film. These observations are similar to the high-temperature (720  C) grown AlxIn1
xN films on GaN by organometallic vapor phase epitaxy [1075]. The direct energy bandgap, determined using optical transmission and reflection measurements, of the alloy was used to comment on the bowing parameter discussed in Section 1.5.3. Resistivity and Hall-effect measurements indicated n-type conduction with carrier concentrations n  1019 cm
3 for In-rich alloys and n 1010 cm
3 for Al-rich alloys. 3.5.15.3.1 Miscibility Gap in InAlN Phase separation in InGaN has been covered with more rigor than in AlInN and AlGaInN alloys, in part because of the critical role InGaN plays in light emitters. There are a few theoretical studies of the thermodynamic stability of the AlInN alloy and all of them use a very simplified model. Matsuoka [1076] expanded the material system of interest in terms of decomposition to include Al in addition to Ga-containing alloys formed with InN, namely the wurtzite quaternary alloy system. The unstable region in mixing was calculated from the AlxGa1-xyInyN free energy of mixing using the strictly regular solution model. The interaction parameter used in the calculation of Matsuoka was obtained using the delta-latticeparameter method with the result that AlInN always, InGaN sometimes and AlGaN hardly ever, has an unstable mixing region at the temperature below 3000  C. Unlike GaN, relatively less is known about the other binaries and much less about the ternaries and quaternary. The least is known about AlInN and the quaternary, particularly on the theoretical side concerning electronic properties [1077,1078]. To make matters more challenging, the experimental results, progressively less so, are often somewhat contradictory [1039,1079–1081]. For structural properties of AlxIn1
xN, there is not much theoretical work which discusses, in an equal footing, both a reasonably sized model supercell and the statistics of the alloy. Rigorous calculations are also available in regard to thermodynamic properties of the AlxIn1
xN alloy. Teles et al. [1082] present a rigorous theoretical study of thermodynamic, structural and electronic properties of ternary c-AlxIn1
xN bulk nitride alloys. The calculations performed for AlxIn1
xN are based on an ab initio pseudopotential plane wave method, within the framework of the density-functional theory (DFT) and the local-density approximation (LDA), the so-called Vienna ab initio simulation package (VASP), and a generalization of the quasi-chemical approach combined with a cluster expansion of the thermodynamic potentials, as was performed for AlGaN and InGaN [1082]. It has been demonstrated that this model is able to successfully describe the physical properties of group III nitride alloys, even including those involving boron, such as BGaN and BAlN [1082–1085]. Focusing on the topicathand, weare goingto focusinparticular onthephasediagram of thec-InxAl1
xN alloy, its chemical bonds, through first- and second nearest neighbor distances and bond angles and the bandgap behavior as functions of the alloy composition x. The reader is referred to the paper by Teles et al. [1086] for a more in-depth treatment, but suffice it to state that the mixing free-energy DF is calculated as a

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function of x and T at a fixed pressure, which allows us to access the temperature–composition (T–x) phase diagram and obtain the critical temperature for the miscibility. As in the case of the calculations of Ho and Stringfellow [1015], the zinc blende system is modeled as it represents well the thermodynamic properties of the wurtzitic system. The calculations of the mixing free energies were carried out by combining the cluster expansion method within the framework of the generalized quasichemical approximation (GQCA) and the self-consistent total-energy pseudopotential (the VASP) calculations. In the calculations of Teles et al. [1086], eight-atom supercells have been used as the basic clusters to describe the fully relaxed alloys. The structure of each cluster has been optimized in terms of its lattice constant by minimization of the total energy. The calculations indicated a propensity for substantial deviation from randomness toward phase separation. The T–x phase diagram for the InxAl1
xN ternary alloys calculated within the GQCA and the ab initio total-energy method is shown in Figure 3.220, where the binodal and spinodal curves are depicted with solid and broken lines, respectively. Note that the spinodal curve marks the equilibrium solubility limit, that is, the miscibility gap. The behavior displayed in Figure 3.220 is similar to that observed in InGaN due to a rigorous thermodynamics calculations by Teles et al. [1087]. One salient exception is that the critical temperature for AlInN, Tcrit ¼ 1485 K, is higher than that for InGaN. The results depicted in Figure 3.220 indicate that for typical growth temperatures,

1600 Inx Al1–xN Binodal

1400

Temperature (K)

1200 Spinodal

1000 800 600 400 200 0 0.0

0.2

0.6 0.4 Composition, x

0.8

Figure 3.220 Calculated T–x phase diagram for unstrained c–InxAl1
xN. Binodal curve: solid line; spinodal curve: dashed line [1086].

1.0

3.5 The Art and Technology of Growth of Nitrides

phase separation for a wide range of compositions should be observed if grown under thermodynamical equilibrium. Considering a growth temperature of, for example, T ¼ 1000 K, a large decomposition tendency is clearly seen for In molar fraction between 15 and 79%. This result is in good agreement with experimental findings that show a tendency of phase separation for x > 0.17 [1039]. 3.5.15.4 InGaAlN Quaternary Alloy This quaternary material is explored for the simple reason that its bandgap can be changed while maintaining lattice match to GaN, which would pave the way to reduced residual stress and thus associated piezoelectric effects as well as alleviating cracking problems in high mole fraction and/or thick AlGaN layers used in MODFETs and lasers. Use of this quaternary material allows almost independent control of the band offset in AlInGaN-based heterostructures. With the advent of UV emitters and detectors, near or in the solar-blind regions, details of which are discussed in Volume 3, Chapter 4, interest in the AlxGa1
x
yInyN quaternary increased. In this vein, LEDs emitting at 280 nm have been demonstrated using AlGaInN/AlN superlattices [1088]. Moreover, laser diodes emitting at 366.4 nm have also been demonstrated using AlGaInN quantum wells as active media [1089]. AlGaInN quaternary alloys potential might reduce the differential dilatation coefficient in heterostructures, which could be an important advantage in epitaxial growth. The carrier localization induced by In can also lead to higher quantum efficiencies as opposed to AlGaN for wavelengths shorter than that associated with the fundamental gap of GaN, details of which are discussed in Volume 2, Chapter 5. However, the growth of quaternary AlGaInN is extremely challenging due to the different bond lengths and decomposition temperatures of the binary compounds making up the quaternary, namely, AlN, GaN, and InN. Further, the vastly different surface mobility and desorption temperature of the impinging species make matters even more complex. The optimal growth temperature is paradoxical, as aluminumbased compounds generally require higher growth temperatures and In-based ones require lower temperatures. Higher temperatures are also desirable for reducing the O incorporation in the growing film as oxides of Ga and In desorb from the surface. Higher temperatures are also desirable for reducing the O incorporation in the growing film as oxides of Ga and In desorb from the surface. The growth temperature will therefore govern the limits of In and Al incorporation into the AlGaInN quaternary alloy [1090]. Furthermore, there are fundamental issues such as immiscibility, which have been predicted for the InN–AlGaN system [1072], and must be dealt with. The feasibility of AlGaInN has been demonstrated using OMVPE [1091,1092]. This quaternary has also been grown by MBE, which brings into the arena additional strengths, particularly in fundamental investigations of growth and surface structure [1093–1095]. A detailed analysis of MBE growth kinetics and a systematic study of material properties as a function of composition has been reported using plasma-assisted MBE for the controlled growth of quaternary AlxGa1
x
yInyN [1096]. Monroy et al. [1096] prepared AlxGa1
x
yInyN (0 < x < 0.5, 0 < y < 0.2) epilayers with thicknesses in the range of 0.2–0.7 mm by MBE on GaN templates, 2–4 mm thick

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that were grown on c-sapphire by OMVPE. The growth rate was fixed to 0.20 ML s
1 for all the alloy compositions. Prior to AlGaInN growth, a thin (10 nm) GaN buffer layer was deposited at 730  C. Quaternary compounds were then grown at substrate temperatures in the range of 590 and 650  C. The growth kinetics was analyzed in situ by RHEED. To remind the reader, the GaN and AlN layers are typically grown in the 700–800 and 750–900  C ranges, respectively, but In desorbs at temperatures above 550  C. The upper limit for the growth temperature of AlxGa1
x
yInyN is thus determined by In incorporation in AlGaN while maintaining quality that sets the window to 650–610  C. Because the incorporation of N is more favorable adjacent to Ga- and Al-occupied sites, to get In incorporation the sum of Ga and Al fluxes, FGa þ FAl, must be less than the nitrogen flux, FN, while the excess metal is provided by the In flux. In other words, the compositional control is achieved by controlling the fluxes of Al, Ga, and N. The N flux control is not as well developed as the other when using RF nitrogen sources, unless special precautions are taken into consideration. Monroy et al. [1096] determined the adequate amount of In flux by wetting of the AlGaN surface in such a way to produce stable one ML thick In layer [1094,1097]. Evolution of RHEED specular beam intensity for AlGaN and also Al0.36Ga0.61In0.03N (Al, Ga, and N fluxes are the same for growth of both compounds) was monitored, shown in Figure 3.221, which shows some transient oscillations assigned to the formation of the In film at the surface. The growth is said to proceed with N diffusing through the In film. As the intensity oscillations indicate, the growth proceeds layer by layer. The difference in the growth rates measured from the oscillation period for the quaternary and the ternary compounds, 0.245 versus 0.235 ML s
1, respectively, can be used to determine In incorporation. After the initial transient, the RHEED intensity remains stable throughout the growth, indicating the lack of excess In accumulation beyond the one ML stable In layer. This adsorbed In film can be removed from the surface by a postgrowth thermal cycle of 1 min at a substrate temperature of 650  C. The final RHEED image, shown in the inset of Figure 3.221, consists of a streaky pattern, indicative of 2D growth. If the substrate temperature is decreased abruptly at the end of the growth, the In film segregating at the surface does not have sufficient time to desorb. Monroy et al. [1096] also investigated the composition of the quaternary alloy versus Al molar fraction (recall that Al and then Ga bonds with N must be satisfied first before In bonds are served and increasing Al and Ga fluxes would lead to reduced In incorporation everything else being the same) by Rutherford backscattering spectrometry (RBS). To determine the maximum amount of In content that can be incorporated, AlGaInN layers have been grown under In-rich conditions at a fixed substrate temperature and Al flux. When the Ga flux is reduced, the RHEED intensity oscillation indicates no change in the growth rate indicating that In replaces Ga. Naturally, if pushed further, the growth rate decreases because of In incorporation dynamics at that particular substrate temperature. However, the system is still in the In-rich regime and only a limited amount of In is incorporated with the rest segregating to the surface. Figure 3.222 presents the maximum In content measured by RBS as a function of substrate temperature and Al molar fraction. The decrease in

3.5 The Art and Technology of Growth of Nitrides

Figure 3.221 Variation of the RHEED intensity with time when starting the growth of ternary and quaternary alloys at a substrate temperature Ts ¼ 640  C, using the same Al, Ga, and N fluxes. In the inset, photograph of the final RHEED pattern of AlGaInN after 2 h of growth (360 nm). Courtesy of B. Daudin and Ref. [1096].

In incorporation with increasing temperature is common to In-containing nitrides inclusive of InGaN and AlGaInN across the growth methods, both MBE [1098,1099] and OMVPE [1100]. In addition to the temperature, In incorporation also decreases with increasing Al molar fractions at all substrate temperatures, which contradicts a

In molar fraction (%)

50

T s = 610 ºC T s = 630 ºC T s = 650 ºC

40 30 20 10 0 0

20

40 60 Al molar fraction (%)

80

100

Figure 3.222 Maximum In incorporation in the quaternary compound as a function of the Al mole fraction and the substrate temperature TS. Courtesy of B. Daudin and Ref. [1096].

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report on the same material deposited by OMVPE [1101]. The impetus force for In segregation and limitation of its incorporation for both InGaN of AlGaInN requires further research, although this has been attributed to re-evaporation of adsorbed In atoms, this does not apply to the case as there is an equilibrium monolayer of In. It has also been argued that the immiscibility of InN and AlGaN is responsible for limit for In incorporation. A standing model, which accounts for the nature of surface segregation, is preferential segregation that occurs during growth of ternary and quaternary III–V semiconductor alloys and drives one of the column III elements involved in the surface [1102]. This process reflects the competition between strain and/or binding energies (which promotes the driving of certain atoms to the surface) and entropic factors that favor the incorporation of a fraction of these atoms in growing layers that are successively buried below the surface. Body of evidence in arsenides, phosphides, and nitrides suggests that the direction and extent of the segregation process follows the In > Ga > Al order, whereas in antimonides Al tends to segregate (Al > Ga). Specific to the case of AlGaInN quaternary alloys, exchange of In with Ga or Al is energetically favorable due to the different binding energies of, in ascending order, InN (8 eV), GaN (9 eV), and AlN (12 eV) [1103]. It is therefore argued, as alluded to above, the larger binding energy of AlN as compared to that of GaN is most likely the genesis for In segregation with increased Al. Moreover, elastic strain also would favor In segregation because In–N bond is longer than both Ga–N and Al–N. Strain may also be involved. The structural analysis of the quaternary alloy was carried out using HRXRD. The (0 0 0 1) oriented III nitride epilayers present a columnar structure characterized by the average tilt and twist of the columnar domains typical of nitride semiconductors. The broadening of symmetric (000) o-scans is sensitive to the grain size and the column tilt such as the rotation of columns out of the growth axis and has been attributed to screw dislocations with Burgers vector b ¼ h0 0 0 1i. The FWHM of the (0 0 0 2) o-scans of AlxGa1
x
yInyN lied in the 300–500 arcsec range, which is comparable with 370  50 arcsec obtained in OMVPE GaN templates. This implies that the mosaicity of the quaternary layer is mostly determined by the substrate quality. In contrast, FWHM of the (0 0 0 2) y
2y scans of the alloy increases with Al content (independent of the In content) from 150 to 300 arcsec and compares with 35  4 arcsec for the GaN template. It should be noted that broadening, caused by alloy disorder with increasing Al content, is similar to that obtained in AlxGa1
xN alloys grown with the help of In surfactant [258]. It appears therefore that the broadening is mainly caused by Al. The X-ray diffraction broadening for the quaternary layers obtained by MBE are similar to those reported in quaternary layers grown by OMVPE, with compositional range of 0 < x < 0.2 [1104] and 0 < y < 0.06 [1105]. Additionally, the quaternary layers exhibit low-sheet-resistance that could be of importance in reducing the current crowding in deep-ultraviolet lightemitting diodes over sapphire substrates, but it has not been based on fact as yet [1106]. Aumer et al. [1107] reported the OMVPE growth of the quaternary alloy AlInGaN and argued that it is desirable to grow quaternary films at temperatures greater than

3.5 The Art and Technology of Growth of Nitrides

855  C to suppress deep level emissions in the room-temperature photoluminescence using two sets of layers. One set was a sequence of relaxed thick films (430 nm) grown with 11% InN and different aluminum compositions. The second set of films comprises strained thin films (60 nm) with 6–10% InN and up to 26% AlN molar fractions. First, a 35 nm AlN buffer layer is grown by atomic layer epitaxy (ALE), followed by standard OMVPE growth of 2 mm of GaN at which point the growth of the quaternary alloy is initiated. The relationships between composition and bandgap (or lattice constant) can be predicted following a routine that was originally developed for the InGaAsP system [1108], as discussed in Section 1.5.4 and Figures 3.140–3.142. Here, the experimental efforts dealing with bandgap determination of the quaternary material is discussed. The bandgaps of the strained and relaxed quaternary films are plotted versus the aluminum content in Figure 3.223a and b.The bandgap variation with aluminum for

3.30 (a)

3.28

Bandgap (eV)

3.25 3.23 3.20 3.18 3.15 3.13 3.10

0

0.02

0.04 0.06 x (%AlN)

0.08

0.1

3.50 (b)

Bandgap (eV)

3.40 3.30 3.20 3.10 3.00 2.90 2.80

0

0.05

0.1

0.15

0.2

0.25

0.3

x (%AlN) Figure 3.223 (a) Bandgap versus x for thin films of AlxInyGa1
x
yN (triangles, y ¼ 0.06; squares, y ¼ 0.08). (b) Bandgap for AlxIn0.08Ga0.92
xN (triangles, thick films; squares, thin films; solid line, theoretical Eg using bAlGaN ¼
1, bInGaN ¼
4.5, and bAlInN ¼
5). Courtesy of S.M. Bedair and Ref. [1107].

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thin films with less than 9% indium is nearly linear. Thin films with that composition experience biaxial strain from the underlying GaN. As more aluminum is added to the films, two competing effects alter the bandgap. Increasing aluminum content results in films with a larger bandgap; however, the smaller lattice constant reduces the biaxial strain in the thin film, thereby reducing the strain-related blue shift of the bandgap. These effects are observed by differences in the bandgap of thin and thick films. Chen et al. [1109] reported on the growth of quaternary AlxInyGa1-x-yN layers using a slight variant of OMVPE-based pulsed ALE. TMA, TMI, and TEG were pulsed during the growth at a temperature of 750  C while keeping the ammonia flow rate constant throughout the growth. It was found that the existence of ammonia increases three-dimensional growth, which results in increasing the density of localized tail states because of stronger alloy compositional fluctuations. The layers exhibited intense room temperature photoluminescence spanning from 320 to 350 nm. Lima et al. [1110] reported on the AlInGaN growth on a GaN buffer layer by plasmainduced MBE. They obtained different alloy compositions by varying the growth temperature with constant Al, In, Ga, and N fluxes because the incorporation rate of In and, to a lesser extent, Ga depends on the substrate temperature, decreasing with increasing substrate temperature. The In content in the alloy, measured by Rutherford backscattering spectroscopy, increased from 0.4 to 14.5% when the substrate temperature was decreased from 775 to 665  C. X-ray reciprocal space maps of asymmetric AlInGaN reflections were used to measure the lattice constants and to verify the lattice match between the quaternary alloy and the underlying GaN buffer layer. They found that the alloys with lower In concentrations, that is, grown at a higher temperature, have a smaller lattice constant. The samples were also characterized for their optical emission characteristics [1111] and as expected, the samples grown at higher temperatures, with lower InN component, showed stronger photoluminescence. The following expression can be used to find the lattice constant of the quaternary from the composition and binary lattice constants. aAlx Iny Ga1
x
y ¼ xaAlN þ yaInN þ ð1
x
yÞaGaN :

ð3:70Þ

This formula is based on the assumption that a solid solution of the binary constituents is present in the quaternary alloy. Having discussed the two ternaries, InGaN and AlInN, which are prone to phase separation, attention can be directed to the lone quaternary system in the nitride semiconductors. Matsuoka [1076] regarded the In1-x
yGaxAlyN alloy as a pseudoternary system because mixing of In, Ga, and Al atoms occurs at one of the sublattices. That author employed the strictly regular solution approximation to calculate the immiscibility gap, a method originally applied to a ternary mixture by Meijering [1112] with the assumption of pairwise interactions among nearestneighbor atoms and random mixing of the constituent atoms in a single phase. In this approach, the calculation for the miscibility gap can be focused on the calculation of binodal isotherms.

3.5 The Art and Technology of Growth of Nitrides

AlN

0.8

0.6 3 00 0 oC

0.4

oC 0 1 00 0 0 0 o C 2

Y

0.2

InN

0.2

0.6

0.4

0.8

GaN

X Figure 3.224 Calculated spinodal isotherms for In1
x
y GaxAlyN system at 0–3000  C for the constant K of 1.15 · 107 cal mol
1 Å
2.5 [1076].

The calculated spinodal isotherms of In1-x-yGaxAlyN for 0–3000  C are shown in Figure 3.224 for a reasonable value of the parameter K, which enter into the interaction parameter. The regular interaction parameter that enters in the calculation of the molar free energy of mixing is dependent in a complex form on the lattice parameters of the binaries forming the ternary and a constant T, which has the unit of cal/mol Å2.5, for details see [1015,1076]. The region surrounded by the InN–AlN line and isotherms shows the unstable composition of In1-x-yGaxAlyN at the growth method under the equilibrium conditions. As clearly seen, the ternary InAlN alloy always has an unstable region. The AlGaN, however, could be grown at all compositions without instability because the isotherms are parallel to the GaN–AlN line. For InGaN, the unstable composition appears to depends on growth temperature. The tendency toward instability for InAlN and InGaN is determined by the value of each interaction parameter, namely, the difference in lattice constants between the binary compounds. In the calculation mentioned above, only the constant K that enters into the calculation interaction parameter, which, in turn, enters into the determination of the free energy of mixing is uncertain. Therefore, in discussing the unstable mixing region of In1-x-yGaxAlyN, it is imperative that one is clear in regard to the dependence of spinodal isotherms on the K value. Figure 3.225 shows the calculated spinodal isotherms of In1-x
yGaxAlyN at 1000  C for the K value of 0.6 · 107–1.0 · 107 cal mol
1 Å
2.5. As can be seen in the figure, the unstable mixing region becomes narrower as the K value decreases. In particular, this value affects the instability of the InGaN ternary alloy.

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AlN 1000 ºC

0.8

0.6

Y 0.4

0.6 0.8

0.2

1.0

InN

0.2

0.4

0.6

0.8

GaN

X Figure 3.225 Dependence of calculated spinodal isotherms for In1
x
yGaxAlyN system at 1000  C on the constant K (cal mol
1 Å
2.5) [1076].

3.5.16 Growth of Quantum Dots

Semiconductor quantum dots (QDs) have been investigated very extensively [1113,1114]. As compared to bulk (3D) materials and quantum well (QW) (2D) structures, QD is the prototype of a zero-dimensional system, which, in the limit, spatially localizes the electrons, and the energy is fully quantized, similar to the atomic system, as shown in Figure 2.17. In addition to quantization, the quantum dots have other benefits such as localization that leads to reduced internal quantum yield degradation. Gerard et al. [1115] pointed out that once the carriers are captured by QDs, they become strongly localized and their migration toward nonradiative recombination centers is made difficult. Furthermore, the increased localization gives rise to increased radiative recombination rates, which brings one to the expected low threshold for lasers that have already been experimentally observed in the InGaAs system. A flurry of interest in low-dimensional GaN and other III nitrides is in part due to a desire to develop new optoelectronic devices with improved quality and wider applications. Development of the light emitters with QDs, for example, is expected to have a lower threshold current in LDs and a higher thermal stability [1116]. The quantization causes the electronic density of states near the bandgap to be higher than in 3D and 2D systems, leading to a higher probability for optical transitions. Furthermore, the electron localization may dramatically reduce the scattering of electrons by bulk defects and does reduce the rate of nonradiative recombination. In the GaN system, additional impetus is due to dots tending to nucleate because of reduced strain and increased nucleation probability, where

3.5 The Art and Technology of Growth of Nitrides

the extended defects are then driven to the surface of dots for more effective annihilation. The size of QDs and of its distribution as well as density over the wafer are important parameters. The typical value of size is on the order of a few nanometer, which necessitates a large assembly of QDs rather than a single one, although probes are continually developed to interrogate individual dots. The fluctuation in dot size produces an inhomogeneous broadening in quantized energy levels and may destroy the very fundamental properties expected from a single QD. The random rather than well-ordered distribution may destroy the coherence of the optical and electronic waves propagating through the structure. Similar to other semiconductor heterostructures, the surface or interface of the QDs must also be free of defects (see recent chapters for details) [1117,1118]. Otherwise, the surface/interface may become the effective scattering center for electrons. In metal dots, minimization of the chemical potential leads to merging of small dots with large ones, which leads to convergence to more or less uniform dots on the surface. Initial variation of the dots size obviously is caused by disorder in the system. In semiconductor dots, however, the surface mobility and mass is not available, and other methods, such as strain and periodic topological features such atomic steps on vicinal substrates, must be used to the extent possible to accomplish this. The holy grail of quantum dots is the nature of the confined states and the resultant density of states. If we consider a semiconductor whose constant energy surface for conduction band in k-space is a sphere, such as the case in GaN, the volume of that sphere in k-space is proportional to k3 in terms of momentum and E3/2 in terms of energy as shown in Figure 2.19a. The density of states associated with this system is proportional to E1/2, again as shown in Figure 2.19a. The area in k-space in an ideal system confined in one direction only (representing quantum wells), which is often the z or the growth direction, is proportional to k2 or E, as shown in Figure 2.19b. The density of states in this case is given by m =ph2 and forms a staircase as shown in Figure 2.19b. If we continue and place a confinement in the x-direction in addition to the z-direction, which represents quantum wires, the line length in k-space is proportional to k in terms of momentum and E1/2 in terms of energy as shown in Figure 2.19c. The corresponding density of states depends on E
1/2, again as shown in Figure 2.19c. If confinement is imposed in all three directions, which represent the pseudoatomic or quantum dot state, the energy is discretized in all direction and the resultant density of states takes on a deltalike function in energy, as shown in Figure 2.19d. The quantum confinement effect shifts the bandgap to a higher energy. This shift, called the confinement energy, depends on the size and shape as well as the material properties of both QDs and surrounding matrix. Here, we estimate the confinement energies for two simplified cases: a plate or disk and a sphere. Assuming an infinite barrier, the confinement energy of the ground state for an electron in a rectangular box is given by ! h2 1 1 1 ; ð3:71Þ þ þ E¼ 8 mx d2x my d2y mz d2z

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where dj (j ¼ x, y, and z) are the dimensions of the box and mj are the electron mass in j-directions. For a plate- or disklike dot, in which the in-plane sizes dx and dy are much larger than the height dz ¼ d, the confinement energy is simplified to h2/(8mzd2). The shift in the bandgap is calculated using 1/mz ¼ 1/me,z þ 1/mh,z, where me,z and mh,z are, respectively, the effective masses of the electrons and holes in III nitride along the z-direction. For a cubic box of size d, the confinement energy is given by the same expression, h2/(8mzd2), but with 1/mz replaced by (2/me,t þ 2/mh,t þ 1/me,z þ 1/mh,z), where me,t and mh,t are the transverse effective masses of the electrons and holes. In Figure 9.173, the confinement energies as a function of d for a plate and a sphere have been plotted [1119]. In both cases, the effective masses of 0.22m0 for electrons and 1.1m0 for holes were assumed. For small dots of a few nanometer in size, the confinement energy is very sensitive to the dot size in that when the size decreases from 10 to 2 nm, the confinement energy changes from 20 to more than 1 eV depending on the shape of the dots. The curves shown in Figure 9.176 represent good lower and upper bounds for the effective confinement energy in GaN QDs, both free and embedded in an AlN matrix, if the size is not too small. For electrons, the barrier height in GaN/AlN interface (0.75 DEg or 2.1 eV) [1120] is high and the mass anisotropy is small. For holes, the barrier is lower (0.7 eV) and the mass anisotropy is larger and the effect on confinement energy is much reduced due to the large mass. For a specific GaN QD, the actual energy shift from the bulk value is expected to be within those two bounds mentioned earlier. In the case of self-assembled QDs, the platelike results may be more suitable because the aspect ration is very large. In other cases, such as GaN nanocrystallines, spherelike shape may be more appropriate. To attain the advantages of the QDs, many requirements must be met in material preparation. The most important one is the size and distribution of QDs. Depending on the material and the dot shape, the maximum size should be near or less than some characteristic length of the electrons in the bulk III nitride, such as exciton radius. The typical value is on the order of a few nanometer. With such a small size, the practical applications are thus often associated with a large assembly of QDs rather than a single one. This implies that the size uniformity of the dot assembly is critical. The fluctuation in dot size produces an inhomogeneous broadening in quantized energy levels and may destroy the very properties expected from a single QD. In addition to the size uniformity, the spatial position of each QD is also important in many applications. The random rather than well-ordered distribution may damage the coherence of the optical and electronic waves propagating through the system. Similar to other semiconductor heterostructures, the surface or interface of the QDs must also be free of defects. Otherwise, the surface/interface may become the effective scattering centers to electrons. Fabrication of QD assembly with small and uniform size, attainment of high density, well-ordered placement, and production of defect-free material remain as great challenges today in any semiconductor system, especially in III nitride materials. The majority of III nitride QDs are grown by MBE and metalorganic chemical vapor deposition (OMVPE). Due to the lack of a suitable material both lattice and thermally matched to GaN, III nitride heterostructures are commonly grown on

3.5 The Art and Technology of Growth of Nitrides

foreign substrates, as mentioned above [1121]. Sapphire (a-Al2O3), 6H-SiC(0 0 0 1), or 4H-SiC, Si(1 1 1) and cubic 3C-SiC(0 0 1) (for cubic GaN QDs) have been used for growth and GaN by HVPE is used as a template for further MBE growth. All of these mean that suitable buffer layers must first be grown before quantum dot growth can be attempted. In the case of GaN dots driven by strain, either AlGaN or AlN buffer layers must be used. The composition of the ternary and the thickness of these layers depend on the level of strain desired, as thin barrier layers tend not to relax fully. In the case of InGaN dots, GaN underlayer becomes an option also. In addition to strain driven dots, antisurfactants such as Si (the exact mechanism of which has evolved quite substantially in the case of GaN and is now believed to be due to SiNx formation [1122]) and lithographic processes have been explored. In the latter, lithography in conjunction with anisotropic nature of growth associated with OMVPE has been employed to further reduce the dimensions to what is closer to Bohr radius. There are many other methods among which are as colloidal forms and inorganic matrices. In this paper, however, only GaN dots driven by strain caused by underlying AlN layers will be discussed. Growth of GaN self-assembled QDs on AlGaN, with the aid of a sub- to monolayer Si layer, which are then covered with AlGaN has been reported by Tanaka et al. [1123] and Shen et al. [1124]. Other approaches have been reported by Widmann et al. [1125,1126] and Damilano et al. [1127] who used AlN layers that provide a larger lattice mismatch to GaN than AlGaN, and in turn provide the impetus for a 3D growth. In addition, the surface topology of AlN is smoother, which removes the surface features from being the nucleation sites for dots. Dots have been demonstrated on 6H-SiC [1123,1124] and sapphire (0 0 0 1) [1125,1127]. Blue-light emission has been reported from such QD structures [1127]. By changing the size of the quantum dots, one can in fact tune the color of the emission owing to large polarization-induced band bending [1127]. However, the wavelength would depend on the injection level, as injected carriers tend to screen the polarization charge. The epitaxial growth by MBE and OMVPE is essentially a nonequilibrium process. However, it is very useful to categorize it into three different modes as in the equilibrium theory. The typical growth modes in any epitaxial deposition are Frank–van der Merwe (FV or FM), which results in 2D growth, desired for epitaxial growth with lateral uniformity [1128,1129]. The other mode is VW mode, which results in 3D growth from the get go and typically occurs in metals. The third mode is the Stranski–Krastanov (SK) mode for which strain is the driving force. After an initial 2D growth, called the wetting layer, the built-in strain drives the system to a 2D growth for strain minimization. Obviously, the wetting layer must be grown on a buffer whose lattice constant is smaller than that of the dot material. The SK mode is the exploited for the growth of semiconductor QDs. The investigation of InAs QDs grown on GaAs shows that the 2D to 3D transition in SK mode is in fact a first-order phase transition [1130]. Schematic representation of the aforementioned processes as shown in Figure 3.226. On vicinal substrates that often consist of a staircase of equally spaced steps in tilting direction, one may expect the approximate 2D analogues of all the 3D equilibrium growth modes, for example, FV, SK, and VW shown in Figure 3.227.

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(a)

(b)

(c)

Frank–van der Merwe (FV)

Volmer–Weber (VW)(VW) Volmer-Weber

Stranski–Krastanov (SK) Stranski-Krastanov (SK)

Figure 3.226 Various growth modes occurring in general on exactly oriented templates. (a) FV mode, layer-by-layer 2D growth mode; (b) Volmer–Weber mode applicable mostly to metals and leads to 3D growth, (c) Stranski–Krastanov mode, which is driven by strain and is typical of semiconductors.

Which mode would be in effect is determined by the balance between the interstep energy (laterally between the film and the substrate steps, equivalent to interfacial energy in 3D) and energies of film steps and substrate steps (equivalent to surface energies in 3D) [1131]. For homooepitaxy on a vicinal substrate (Figure 3.227a), one would expect “ideal step-flow growth.” In the case of heteroepitaxy, there is misfit strain, and depending on the extent of the strain, different modes occur. For example, with wetting layer (stripe in this case), Figure 3.227b the VM mode occurs, and without the wetting stripe, the SK mode occurs (Figure 3.227c). The wetting stripe corresponds to the wetting layer in the 3D case. Experimentally, the growth mode depends not only on the materials of both the epilayer and substrate but also on the growth conditions such as substrate temperature and flow rates of various sources. Essentially, it is the result of a competition between the kinetic energy of adatoms and the free energies of bulk, surfaces, and interface. For a lattice-matched system, in the limit of equilibrium growth, the layer-by-layer growth is favored if the energy of the substrate surface is higher than the sum of the epilayer surface energy and the interface energy. Island or 3D growth can be realized by changing the surface and interface energies. In a lattice-mismatched system, the bulk elastic energy in the epilayer induced by strain plays an important role. Because it increases with layer thickness, a strain

3.5 The Art and Technology of Growth of Nitrides

(a)

(b)

(c)

Frank–van der Merwe (FV)

Volmer–Weber (VW)

Stranski–Krastanov (SK)

Figure 3.227 2D analogues of 3D equilibrium growth mode, which occurs during step flow growth on vicinal substrates. (a) FV mode, rowby-row growth, which is termed the step flow growth. In the case of GaAs, there will be no RHEED oscillations observed in this mode.

Continuous stripes formed under “ideal” step flow. (b) VM mode, island growth. (c) SK mode, row-by-row followed by island growth. The island being square in footprint is just an assumption for simplicity.

relaxation is expected when the layer thickness is increased beyond a critical value. In fact, the SK mode is only observed when the epilayer is subject to compressive strain. In this case, the stress field tends to force the adatoms to coalesce. The strain energy can be partially released by the formation of islands through elastic relaxation, without any dislocations of the islands. When the epilayer is subject to tensile strain, the growth will continue to be 2D and the strain energy is released through plastic relaxation with the creation of dislocations. The spontaneous growth of QDs by either 3D or SK mode is known as self-organized or selfassembled growth. The methods used to produce GaN and InxGa1
xN QDs quantum dots are discussed below. 3.5.16.1 Quantum Dots by MBE Both RF nitrogen plasma [1125,1126,1132–1135] and ammonia (NH3) [1127,1136] have been used as nitrogen sources during the growth of quantum dots. A thin (10–100 nm) AlN buffer layer is grown on substrates followed by the active layer, which can be either a single layer of GaN QDs or repeated layers of GaN QDs separated by AlN or ternary AlGaN spacer layers. The latter is necessary for many practical applications requiring higher QD density. In this case, the AlN is used to not only isolate the adjacent QD layer but also provide a flat surface for the growth of the following QD layer so that topological features do not impede dot formation driven by

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strain. The top layer of QDs may or may not be capped by AlN depending on the measurement to be performed. Extensive investigation of GaN QDs grown by MBE has been carried out by Daudin and coworkers [1125,1126,1132–1134,1137,1138]. Typically, after substrate nitridation, a thin (10–30 nm) AlN buffer layer was grown at a temperature between 500 and 550  C, followed by a thick (0.2–1.5 mm) AlN layer grown at a higher temperature of 650–730  C [1138]. Sometimes the growth of the thick AlN layer was preceded by a thick (2 mm) GaN buffer layer [1132]. The GaN QDs were grown on AlN by depositing two to four MLs of GaN at temperatures ranging from 680 to 730  C. Due to the 2.5% lattice mismatch between GaN and AlN, under the growth conditions that were used, the growth follows a SK mode. After the 2D growth of a GaN wetting layer (about 2 MLs), 3D growth follows, and the GaN QDs are formed [1132]. It was found that the growth mode is sensitive to the substrate temperature. At growth temperatures below 620  C, the growth was purely 2D. Only at the elevated temperatures (680–730  C) did the growth transitions from 2D to 3D, that is, the SK mode, take place [1132]. RHEED is a powerful method for real-time observation of dot formation. The intensity, shape, and rod spacing can be used to monitor the transition from wetting layer to 3D growth where the dots form and evolution of the lattice constant as shown in Figure 3.228. The self-assembled GaN QDs have a disklike shape, or more accurately, a truncated pyramid with a hexagonal base, with a base diameter a few times larger than the height. The dot size and density depend on growth condition, deposition time, as well as postgrowth treatment. AFM has been widely used to image the general morphology of the QDs that are not covered by any capping layer. Figure 3.229 shows the typical AFM images of the GaN QDs grown on AlN at three different temperatures near 700  C [1125]. The dot density is higher than 1011 cm
2 and decreases with growth temperature. The density can be effectively reduced through a postgrowth reorganization, called the ripening effect [1132], after GaN growth is finished. During this period, the sample was exposed to N plasma and kept at high temperature for approximately 50 s. Figure 3.230 clearly demonstrates the ripening effect. In this particular case, as compared to the sample without 50 s ripening process, the dot density reduced from 5 · 1011 to 5 · 1010 cm
2, whereas the average size (height/ diameter) increased from 2/20 to 5/25 nm. A detailed investigation, with the help of RHEED, shows a similar ripening effect when the samples are either exposed to nitrogen or held at high temperature, typically comparable to the growth temperature, in vacuum [1125]. At low temperature, however, the dot size and density remain unchanged under the nitrogen plasma. The reason was suggested that the Ga diffusion on the surface might be inhibited in the presence of nitrogen [1125]. The detailed structure of GaN QDs has been investigated by high-resolution transmission electron microscopy (HRTEM) [1135,1137]. Figure 3.231a gives an example of the HRTEM image of a GaN dot embedded in AlN, taken along the ½0 1 1 0 axis. A 3D schematic view derived from the HRTEM analysis is shown in Figure 3.231c. The sample is composed of repeated layers of about 2 GaN MLs and thick AlN layers. The analysis of the HRTEM image reveals the following results: (1)

3.5 The Art and Technology of Growth of Nitrides

Figure 3.228 Evolution of lattice parameter and the RHEED images during quantum dot formation. Constructed using data by B. Daudin and coworkers.

The QD has the shape of a truncated pyramid of hexagonal base with a ¼ 30 . The dimensions in this particular case are measured to be 3.3 nm in height and 15.3 nm in diameter. (2) The QD is fully strained and is dislocation free. (3) A wetting layer of

Figure 3.229 200 nm · 200 nm AFM images from GaN quantum dots grown on AlN surface at (a) 725, (b) 705, and (c) 685  C. The growth follows SK mode. The dot density decreases with growth temperature [1125].

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Figure 3.230 (a) AFM image of smooth AlN surface. (b) GaN quantum dots formed by depositing the equivalent of four GaN monolayers on the smooth AlN surface immediately followed by cooling under vacuum. (c) GaN quantum dots formed by depositing the

equivalent of two GaN monolayers on the smooth AlN surface immediately followed by exposure to N plasma for 50 s. The structure reorganization or ripening effect is observed in (c) [1132].

two ML GaN is demonstrated. Only a small amount of Ga and Al atoms (15% in this case) is exchanged across the GaN/AlN interface. It has been established earlier that the strained islands such as InAs grown in successive layers separated by a spacer layer such as GaAs could lead to vertical correlation if the thickness of the spacer layer is appropriate [1139,1140]. The driving force for such vertical self-organization is schematically shown in Figure 3.232 [1139]. Islands in the first layer produce a tensile strain in the spacer above the islands, whereas little stress exists in the spacer away from the islands. Indium adatoms impinging on the surface would be driven by the strained field to accumulate on the top of the islands where they can achieve an energetically lower state due to lower lattice mismatch between the new islands and the spacer. The vertical correlation of self-assembled multilayer QDs was also demonstrated in GaN/AlN systems [1125,1126]. A HRTEM image is shown in Figure 3.233 [1132]. It reveals such a correlation for an AlN spacer of 8 nm. For a thicker AlN layer of 20 nm, no vertical correlation is observed [1132]. In addition to the strain-induced vertical correlation of GaN QD arrays, a correlation between the QD growth and the threading edge dislocations propagating in AlN has been noted [1134]. The conventional TEM and HRTEM images shown in Figure 3.234 demonstrate that the GaN QDs may be more likely to form adjacent to the edge dislocations. In this experiment, the dislocation density in the thick AlN layer is 1.8 · 1011 cm
2, comparable with the density of GaN QDs (1.1 · 1011 cm
2). The strain field near the edge dislocation favors the nucleation of QDs where the AlN lattice is stretched and the mismatch to GaN is smaller. If the dislocation density is high, as in this case, the vertical correlation of the QDs may be disturbed by the presence of a dislocation line that is slightly inclined. Instead of following the vertical positions of the QDs in the previous layer, the QDs seem rather likely to follow the dislocation line. This effect may be unimportant if the dislocation density is lower than the QD density.

3.5 The Art and Technology of Growth of Nitrides

The growth of GaN QDs on Si(1 1 1) [1127,1141] and SiC(0 0 1) [1142] substrates by MBE were also reported. The purpose of growing GaN QDs on Si substrates is mainly for the integration of light-emitting devices with Si technology. The growth processes

Figure 3.231 (a) HRTEM image of a GaN dot, taken along the ½0  1 1 0 axis. The region of the truncated pyramidal dot is outlined. (b) Fourier filtered image of (a) obtained by using all the (0 0 0 1) frequencies except those belonging to AlN. (c) Schematic view of the dot within the

cross-sectional sample as deduced from fits. Note that only half of the pyramidal dot is within the cross-sectional thickness. (d) Experimental and simulated interplanar distance profiles of the dot [1137].

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Impinging In flux

d Surface stress field GaAs spacer

(2)

(1)

Stressed region I

II Low or no stress region

ls InAs island GaAs (1 0 0)

l

z o

x

Figure 3.232 A schematic representation showing the two major processes for the In adatom migration on the stressed surface using the InAs/GaAs system as an example: (1) directional diffusion under mechanochemical

InAs wetting layer potential gradient contributing toward vertical self-organization and (2) largely symmetric thermal migration in regions from the islands contributing to initiation of new islands not vertically aligned with islands below [1139].

in the former case are essentially the same with those on sapphire substrates. By controlling the size of the GaN QDs in AlN matrix, intense RT PL with different colors from blue to orange as well as white were demonstrated [1127]. When GaN is grown on 3C-SiC(0 0 1) surface, the QDs with cubic rather than hexagonal structure can be obtained. The zinc blende GaN islands were formed on AlN buffer by RF MBE with

Figure 3.233 HRTEM image, taken along the ½0 1 1 0 direction of a superlattice of GaN dots capped by AlN. Because of the low magnification of the printed image, the atomic columns are not seen although they are present. The vertical correlation of the GaN dots is evident. The two-dimensional GaN wetting layer is also clearly visible. Note the dislocation line running through the column of dots at the left-hand side [1125].

3.5 The Art and Technology of Growth of Nitrides

an average height of 1.6 nm and a diameter of 13 nm. The island density is 1.3 · 1011 cm
2. In addition to SK mode, the 3D growth of GaN QDs on AlxGa1
xN was possible by using a so-called antisurfactant Si [1136]. In this experiment, a smooth AlxGa1
xN layer was prepared on 6H-SiC(0 0 0 1) by OMVPE and used as the substrate for MBE regrowth. The GaN QDs were grown by MBE in which NH3 was used as the N source and CH3SiH3 was used as the Si source. The AlxGa1
xN surface was exposed to Si flux before the GaN growth and the NH3 flow was stopped for this step. The subsequent GaN growth was carried out with and without introducing Si on two different samples. In the sample without Si flux during the GaN growth, the growth was twodimensional and the streaky RHEED patterns were observed. With Si flow, a change of GaN growth mode from 2D to 3D was observed and the RHEED patterns turned out to be spotty. Formation of GaN QDs was confirmed by AFM. The dot density could be changed by the variation of the Si flux and the growth temperature. The dot density decreased by a factor of 103 and the dot sizes increased from 4/50 to 10/200 nm by raising temperature from 660 to 740  C [1136]. More investigations on the antisurfactant growth scheme by OMVPE will be presented in the next subsection. For applications to light-emitting devices, the InxGa1
xN alloy is more frequently used as the active layer. The bandgap and the emitting wavelength are easily modified by alloy composition. The quantum efficiency of light emission from InxGa1
xN QWs is usually higher than that in GaN. As compared to GaN QDs, however, fewer investigations have been published on the growth of InxGa1
xN QDs. The fluctuation in alloy composition or phase segregation during the growth may complicate the growth and the origin of light emission. Using a conventional MBE with RF plasma source, the growth of InxGa1
xN QDs on GaN in SK mode was demonstrated [1143]. In this experiment, the substrate was a 2 mm thick GaN layer grown by OMVPE on sapphire. The growth parameters were monitored by RHEED and the fluctuation in In mole fraction was estimated to be 3%. At the substrate temperature of 580  C, a layer-by-layer growth of In0.35Ga0.65N on GaN was observed during the first 1.7 ML deposition. Beyond 1.7 ML, the growth mode was changed from 2D to 3D and the InxGa1
xN islands were formed. The AFM image of the InxGa1
xN surface with five ML deposition shows a high island density of 1011 cm
3. The average diameter and height of the islands are 27 and 2.9 nm, respectively. This investigation shows that the transition of 2D–3D growth can be realized for In content from 18 to 100%. For an In content below 18%, the growth mode remains 2D. The multiple layers of InxGa1
xN QDs can also be formed by the overgrowth of a GaN layer of 4–5 nm, which smoothes the surface. The InxGa1
xN QDs grown by SK mode can also be realized by MBE using NH3 as the nitrogen source [1144]. Before the growth of InxGa1
xN, a GaN buffer of a few micrometers was grown at 820  C. InxGa1
xN was then grown at temperatures from 530 to 570  C and a growth rate of 0.1–0.2 mm h
1. The In composition was kept at 0.15, which is larger than the critical value of 2D–3D transition (0.12) determined in the same experiments [1145]. The 2D-to-3D transition was observed after the deposition of four to five ML (11 Å) of In0.15Ga0.85N. The average island size is about 35 nm in diameter and 4 nm in height. The island density was approximately

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3.5 The Art and Technology of Growth of Nitrides

5 · 1010 cm
3 and greater than the dislocation density in the GaN buffer layer, which was approximately 5 · 109 cm
3. A typical surface morphology of InxGa1
xN QDs on GaN imaged by AFM is shown in Figure 3.235. As compared to GaN, the density and the sizes of InxGa1
xN QDs are more difficult to control in growth. Although AFM and RHEED investigations have been performed, no detailed lattice structures have been imaged by HRTEM for InxGa1
xN QDs. 3.5.16.2 Quantum Dots by OMVPE The III sources used in the OMVPE growth of III nitrides are TMG, TMA, and TMI, carried by nitrogen (N2) or hydrogen (H2) gas. The nitrogen source is ammonia (NH3). The self-organized growth of GaN QDs by OMVPE was first reported by Dmitriev et al. [1146] who grew GaN QDs directly on 6H-SiC substrates. In this case, the lattice mismatch between the GaN and SiC is large enough to lead to the island growth. The main contributions of the growth of GaN QDs on AlxGa1
xN (x < 0.2) by OMVPE are by Tanaka and coworkers [1147–1150]. They developed a method called the antisurfactant, which can change the growth mode from 2D to 3D. The selfassembly of GaN QDs is realized in this small lattice-mismatched system by exposing the AlxGa1
xN surface to Si during growth. The Si is from tetraethyl-silane [Si (C2H5)4: TESi: 0.041 mmol] (TESi) and carried by H2. Samples were grown on Si-face of 6H-SiC(0 0 0 1) substrates [1147–1150]. Typically, after depositing a thin (1.5 nm) AlN buffer layer, a thick (0.6 mm) AlxGa1
xN cladding layer was grown. The Al content x varied from 0.07 to 0.2. GaN was then grown on the top of this AlxGa1
xN with a short supply (5 s) of TMG and NH3 during which TESi may or may not have been used. The QDs may be covered by a 60 nm AlxGa1
xN layer for optical studies or left without a capping layer for AFM studies. If TESi is not supplied, a step-flow growth of GaN with a smooth surface is observed. Only when the AlxGa1
xN surface is exposed to TESi are the GaN QDs effectively grown. A transformation of the surface morphology with and without Si exposure is shown in Figure 3.236. The step-flow growth observed without TESi flux (Figure 3.236c) was explained by a fairly small lattice mismatch between GaN and AlxGa1
xN (0.37% for x ¼ 0.15). Under the exposure of small Si dose, large GaN islands were formed. These islands transformed into isolated small dots under a higher Si flux (Figure 3.236a and b). The dot density in this case could be controlled from 107 to 1011cm
2 by changing the TESi flow rate, growth temperature, growth time, and alloy composition. The density was found to be very sensitive to the growth temperature, varying by a factor of 3 Figure 3.234 (a) Weak-beam image of a cross section of the GaN QDs in the AlN matrix ðg ¼ f2;  1;  1; 0gÞ. Only the dislocations with a Burgers vector along the c ¼ [0 0 0 1] direction are visible and most of are of screw type. (b) A weakbeam image with g ¼ (0 0 0 1) of the same area. Only the dislocations with Burgers vector of the form 1=3 < 2;  1;  1; 0> (in-plane) are visible. (c) A

 0> HRTEM image. The slightly off-axis < 0; 1; 1; first eight QD layers of the samples can be seen. Note that the shape of the QDs in the first three QDs layers above the thick AlN layers are less well defined than the shape of the QDs of other layers. Traces of edge dislocations are outlined by dark arrows for convenience. As clearly seen, the QDs are vertically aligned [1134].

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Figure 3.235 A 1 mm · 1 mm AFM image of nonburied selfassembled InxGa1
x N islands grown on GaN by MBE [1144].

Figure 3.236 AFM image of GaN quantum dots assembled on an AlxGa1
x N surface using TESi as an antisurfactant, (a) plane view; (b) bird’s eye view. (c) An AFM image of GaN grown on AlxGa1
xN surface without TESi doping, showing a step-flow growth [1147].

103 between 1060 and 1100  C. Figure 3.237 illustrates the AFM images from a typical set of GaN QDs grown on AlxGa1
xN surface (x ¼ 0.2) [1150]. The hexagonal-shaped GaN dots have an average height of  6 nm and diameter of  40 nm. The dot thickness/diameter ratio could be changed from 1/6 to 1/2 by varying the growth temperature and Si dose [1147,1148]. The dot density is 3 · 109 cm
2, more than one

3.5 The Art and Technology of Growth of Nitrides

Figure 3.237 Cross-sectional HRTEM image of uncapped GaN quantum dots grown on an AlxGa1
xN surface. The upper bright part is glue used in the HRTEM sample preparation [1150].

order of magnitude lower than the GaN QDs grown by MBE. For a fixed growth temperature (Ts ¼ 1080  C) the densities of 5 · 109 and 5 · 108 cm
2 were obtained with a TESi doping rate of 44 and 176 nmol min
1, respectively. By increasing the GaN growth time from 5 to 50 s, the dot size was changed from 6/40 to 100/120 nm. The PL and the stimulated emission [1151] from the similar QDs will be discussed later. Using a similar method, Hirayama et al. [1152] fabricated InxGa1
xN (x from 0.22 to 0.52) QDs on an AlxGa1
xN (x ¼ 0.12) surface. A two-layer buffer structure was used. First a 300 nm Al0.24Ga0.76N layer and then a 100 nm Al0.12Ga0.88N layer were grown on a SiC substrate, both at 1100  C. Prior to InxGa1
xN growth, a small amount of Si antisurfactant was deposited at 1120  C. Then the temperature was cooled to 800  C for QD growth. The dot density was as high as 1011 cm
2 and decreased with increasing Si dose. The average dot height and diameter were 5 and 10 nm, respectively, as determined from AFM images. The microscopic mechanism of antisurfactant in the growth of GaN QDs is not well understood. Incorporation of Si in the growth process is assumed to change the surface energy of the AlxGa1
xN layer so that the growth mode is modified. An effective surfactant usually raises the surface energy so that a 2D layer-by-layer growth is favored. For GaN growth, the opposite is assumed. Incorporation of Si is assumed to reduce the surface energy of AlxGa1
xN and increase the diffusion length of the adatoms. As a result, the adatoms are more likely to coalesce in order to reduce the total energy. InxGa1
xN QDs can also be grown on GaN by OMVPE without using antisurfactant [1153]. The equipment is an atmospheric pressure two-flow system with a horizontal quartz reactor. A GaN buffer layer was first grown on (0 0 0 1) sapphire at a temperature of 1075  Cand a V/III ratio of 2000. The InxGa1
xN portion was grown at a reduced temperature of 700  C with a growth time 10 s. The growth rate is estimated to be 0.17 nm s
1. The thickness of InxGa1
xN is about 10 MLs. The average In composition is low (x < 0.1). AFM images show the density of the QDs increasing with growth time. If the growth time is short (6.4 ML), two kinds of QDs, bigger and smaller sizes, were observed. The bigger QDs have a diameter of 15.5 nm and height of 5.4 nm. The smaller QDs have a diameter of 9.3 nm and height of 4.2 nm. When the growth temperature is increased, the dot density decreases monotonically. The important difference in this case from the typical SK mode is the formation of the InxGa1
xN QDs even at a long growth time (19 ML). Thus, the formation of the QDs may be mainly attributed to phase segregation rather than strain-induced coalescence.

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Formation of QD-like structures in semiconductor alloys and QWs induced by alloy fluctuation or phase segregation has a great effect on the material properties [1154]. The In-rich clusters in InxGa1
xN QWs were suggested to be the origin of high luminescence efficiency in InxGa1
xN/GaN LEDs [1155]. Existence of alloy fluctuation/phase separation in InxGa1
xN grown on GaN by OMVPE was confirmed by HRTEM [1154]. The spherical QDs were observed in HRTEM images of a 280 nm thick In0.22Ga0.78N layer. A typical dot consists of a core and a surrounding strain zone. The lattice parameters inside the core are slightly larger than that in the surrounding matrix and approach those for InN. The size estimated from an HRTEM image is in the range of 1.5–3 nm. A method very different from self-assembly is selective growth of QDs. InxGa1
xN QDs have been grown on Si-patterned GaN/sapphire substrates by OMVPE [1156,1157]. As shown in Figure 3.238, a Si film of 50 nm was first deposited onto the surface of GaN/sapphire substrate by electron beam evaporation at room temperature. Nanoscale circular windows were then opened in the Si mask by focused ion-beam irradiation followed by photoassisted wet chemical etching. GaN/ InxGa1
xN multilayers were finally epitaxially grown on the GaN plinths with shapes

Figure 3.238 Upper panel: Schematic diagram of the focused-ion-beam/OMVPE process used for the fabrication of InxGa1
xN/GaN nanostructures. Sixty percent of the Si layer was sputtered by focused-ion-beam whereas the rest was removed by photo-assisted-wet etching. Five periods of InxGa1
xN/GaN QDs were then selectively grown on GaN plinths exhibiting a

small density of dislocations. Lower left panel: A top SEM view of GaN plinths laterally overgrown on circular windows with diameters of 600 nm (upper row) and 300 nm (lower row). The regrown GaN has a hexagonal pyramid shape with six f1 1 0 1g side facets. Lower right panel: Schematic drawing of InxGa1
xN quantum dot structures [1156,1157].

3.5 The Art and Technology of Growth of Nitrides

of hexagonal pyramids. Both InxGa1
xN QWs and QDs were formed in the structure but the QDs appeared only on the top of the pyramids. As compared to self-organized growth, the selective growth on pattered substrates could in principle provide a better way of controlling the position, size, and density of the QDs. Ripening after the initial formation of quantum dots has also been employed to shape the dots. In addition, Ga spraying techniques followed by nitridation and conversion to GaN have been used in the author’s laboratory to achieve a high density of dots. Spraying at temperatures where Ga migration occurs leads to ball formation. 3.5.16.3 Quantum Dots by Other Techniques In addition to MBE and OMVPE, growth of GaN QDs by other techniques was also reported. Goodwin et al. [1158] have fabricated nanocrystalline GaN by reactive laser ablation of pure Ga metal in a high-purity N2 atmosphere. The samples were collected from the surface of a membrane filter and then thermally annealed at 800  C in a high-purity ammonia atmosphere. TEM dark field images show a log-normal size distribution with a mean diameter of 12 nm and a standard deviation of 8 nm. Selected-area electron diffraction pattern confirms the hexagonal phase. The quantum confinement effect was observed from the blue shift of the size-selective PL and PLE spectra. Nanocrystalline GaN thin films were also fabricated recently on quartz substrates by RF sputtering using GaAs as a target material at a nitrogen pressure of 3.5 · 10
5 bar [1159]. The average particle size of the nanocrystalline GaN increased from 3 to 16 nm when the substrate temperature was raised from 400 to 550  C. Crystalline GaN particles can be synthesized by simple inorganic reactions at various temperatures. Well et al. [1160,1161] reported a method of nanosized GaN synthesis by pyrolysis of gallium imide {Ga(NH)3/2}n at high temperatures. Dimeric amidogallium [Ga2N(CH3)2]6 was first synthesized by mixing anhydrous GaCl3 with LiN(CH3)2 in hexane. This dimer was then used to prepare polymetric {Ga(NH)3/2}n by reacting [Ga2N(CH3)2]6 with gaseous NH3 at room temperature for 24 h. The GaN QDs were prepared underammoniaflow by slowlyheating{Ga(NH)3/2}n in trioctylamine to 360  C, cooling it to 220  C to add and stir a mixture of trioctylamine and hexadecylamine, and finally cooling it to room temperature. Formation of isolated spherical QDs in colloidal GaN solution was confirmed by TEM images, as shown in Figure 3.239 [1162]. The image reveals that GaN has a zinc blende structure with dots formed having diameters ranging from 2.3 to 4.5 nm. The absorption and PL peaks were observed to shift to a higher energy as compared to bulk GaN. In addition to GaN, an AlGaN nanoparticle/polymer composite was also synthesized using a similar method and the microstructure of zinc blende QDs was confirmed by HRTEM [1163]. Optical properties of dots are discussed in Volume 2, Chapter 5. Because colloidal formation of quantum dots has been very successful in other materials such as conventional III–Vs and particularly II–VIs with aspect ratios approaching a real sphere, a compelling case can at least be made to make mention ofGaNdots producedbythis wetchemicalmethod. Micic etal. [1164] reported on the use of this colloidal chemistry to synthesize GaN quantum dots. The process begins by heating a GaN precursor, polymeric gallium imide, {Ga(NH)3/2}n, which was prepared by the reaction of dimeric amidogallium with ammonia at room temperature, in

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Figure 3.239 TEM image of GaN QDs taken in bright field. The particles are well dispersed and not agglomerated. Top panel shows low magnification of QDs and some linear alignment. Bottom two right panels show high magnification and lattice fringes of QD oriented with the h1 1 1i axis in the plane of the micrograph. Bottom left panel shows electron diffraction pattern of GaN QDs indicating zinc blende structure [1162].

trioctylamine at 360  C for 1 day to produce GaN nanocrystals. The resultant GaN particles were separated, purified and partially dispersed in a nonpolar solvent that culminated in transparent colloidal solutions that consisted of individual GaN particles. The GaN nanocrystals had a spherical shape and mean diameter of about 30  2 Å, unlike very unfavorable aspect ratios that result in epitaxial methods. The spectroscopic behavior of colloidal transparent dispersion has been investigated and showed slightly

3.5 The Art and Technology of Growth of Nitrides

blue shifted near band edge emission, indicating quantum confinement. The PL spectrum recorded at 10 K (excited at 310 nm) showed band edge emission with several emission peaks in the range between 3.2 and 3.8 eV, whereas the PL excitation spectrum showed two excited-state transitions at higher energies. With the advent on “nano” the scope of the game changed and nano implicitly meant all kinds of nanowires and tubes, and so on, as if dots were not nano in dimension. Below, we will follow the great masses, mainly the chemists, and discuss nanowires and the methods used to produce them under the auspices of “nanostructures” that do not include quantum dots. 3.5.16.4 Preparation and Properties of Nanostructures The usual justification used for exploring nanostructures is that they exhibit electronic and optical properties that are novel, primarily owing to confinement owing to dimensions that are comparable to intrinsic process lengths in semiconductors, although many nanostructures reported in nitrides are not sufficiently small to exhibit confinement in optical experiments. In some cases, the goal is to produce less defective material by limiting the contact area of the growing nanostructure with the mismatched substrate and/or cause the nanostructure to form freely. Just to develop a sense of relative dimensions, the length scales of the well-known processes in semiconductors are illustrated in Figure 3.240. In optical processes, the exciton Bohr radius is one of the most if not the most critical dimension. Other processes such as phonon and electron mean free paths, the Debye length (depends on doping level as well as the material) and the exciton diffusion length for certain polymers are in the range of 1–500 nm. It is hoped that Critical magnetic single domain size Exciton diffusion length in polymers Phonon mean free length

Exciton Bohr radius Debye length Fermi wavelength in metals

10–1

100

101

102

103

Characteristic length (nm) Figure 3.240 Characteristic lengths of a few of the well-known processes occurring in semiconductors at 300 K. The length scales of over a few tens of nanometers are accessible by advanced lithography. Below that length scale chemical processes are used, however, the production is not yet conducive to semiconductor fabrication processes that revolutionized our world through high performance electronics with reduced cost per function.

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synthesized nanowires with 5–100 nm in diameter might allow access to the realm of mesoscopic phenomena. These structures are often beyond the realm of lithography, albeit the limit for lithography has steadily reduced to the point where dimensions near 100 nm are easily producedandarationalforsynthesizingthemmustbedevelopedfortheirself-formation such as the quantum dots discussed in Section 3.5.16. The blue shift of the bandgap expected from an ideal confined system in the simplest sense is given by Equation 3.71, which predictsthefigurefor a cubic quantum box model to beh2/(8md2), with d being the dimension of the cube’s edge. In practice, the blue shift is of the form 1/dn (1 n 2). More detailed calculations for the blue shift in a 2D plate, a sphere with a diameter of d, with d varied, are shown in Figure 9.173. The cubic box falls in between the 2D plate and the sphere. An InP-based study [1165] indicated the blue shift to be DEg  1/d1.35 for dots and DEg  1/d1.45 for wires. The calculations mentioned assume no band bending due to surface states and the accurate extraction of the confined system size from the blue shift would require that the surface band bending be taken into account. The bandgap tunability with a controlled size variation would have applications in emitters, detectors, andsolarcells.Interestingstructures such asperiodic quantumwells along thenanorods or wiresashasbeen implemented inZnO wellsseparated byZn0.8Mg0.2Obarriers[1166] and coresheathnanowires[1167] implemented inGaN/Al0.75Ga0.25N in thesame veinas quantum dots coated with a larger bandgap barriers for enhanced quantum efficiency by removing the high surface recombination velocity out of the picture. 3.5.16.4.1 Approaches for Synthesis The reader would be well served with a discussion of the merits and limitations of often used processes for producing nanowires. A very good review of the field in the context of group III nitrides can be found in Ref. [1168]. In addition to desired ability to produce high-quality material, pivotal issues to look for are the understanding of the one-dimensional growth process based on kinetic and thermodynamic rationale and predictability of the process as well as being applicable to a wide variety of materials systems [1169]. Nanowire growth, in general, comes about by instilling and taking advantage of anisotropic growth that often times can be promoted by templates having at least one-dimensional morphologies. Introduction of liquid/solid interface in the process to reduce the symmetry of the seed and use of an appropriate capping reagent to control the growth rates of various facets of the seed are among the methods garnered. Many of the early methods did not have the requisite attributes such as reproducibility, uniformity, scalability, cost effectiveness, and the knowledge of the basis for growth. It is of course instructive to know a priori whether core/sheath (coaxial) or longitudinal heterostructures are desired as end product. Let us now give a succinct review of methods used to produce nanostructures, among which are growth from the vapor phase and vapor–liquid–solid (VLS), inclusive of self-catalytic and vapor phase varieties, vapor–solid (VS) growth, and confined chemical reaction such as that utilizing carbon nanotubes for preparation. 3.5.16.4.2 Vapor Phase Growth This is by far the most successful and most commonly employed method for producing low-dimensional structures, the term that was popular before the nano craze, such as whiskers (was considered a failure by some of the

3.5 The Art and Technology of Growth of Nitrides

pioneers who stumbled on them), nanorods, and nanowires. Numerous methods have been developed for precursors to make this process possible, including laser ablation, chemical vapor deposition, chemical vapor transport methods, molecular beam epitaxy, and sputtering. As in any growth mechanism, measures must be taken to avoid or suppress parasitic secondary nucleation events, leaving only the nanowire synthesis as the dominant process. Vapor–Liquid–Solid Process As the name suggests, this is a process wherein constituents in the vapor phase diffuse through a metal catalyst and deposit underneath as a solid. During theentireprocess,thecatalystrides asliquidontop ofthegrowthfront.The catalyst is involved during growth but does not incorporate inthe final product. This is by far the most widely used method among the vapor-based processes in generating large quantities of single crystalline nanowires. Typically, the process takes place in a two- or three-zone furnace, allowing independent temperature control of the source and substrate regions. The source isnaturally kept at a higher temperature than the substrate thatis downstream, asshown inFigure 3.241.The group IIIelements, mainly Ga or In in the case of GaN or InGaN, in the form of foil or powder is placed in an alumina boat that providesvaporsourceuponheating.Thesubstrateisplaceddownstreaminacolderzone of the furnace after precoating it with a catalysts. The substrate could be preloaded with catalyticmetalstrip/powders orprecoatedwithathinlayerofcatalyticmetalfilmbyvapor transport of by evaporation in a separate vacuum system. Introducing catalysts via vapor transport on a patterned substrate by either shadow masking or lithography offers the opportunity for selective area growth. A variant of this process is one wherein the catalytic growth may also be attained with source materials in a solution form. This process is thus termed as solution–liquid–solid (SLS) growth. The VLS process dates back to 1960s, albeit with dimen-

Three-zone furnace

Quartz tube

Flow controller

NH3

Powder

O-ring joint

Substrate

Bubbler Figure 3.241 Schematic representation of a VLS reactor for the growth of GaN-based nanowires and nanorods. The setup can be used in either the “single-boat” configuration in which the sample is collected on a substrate placed on top of the crucible containing reactants, or the “two-boat” configuration in which the sample is collected on a separate substrate which is placed downstream of the reactants (shown).

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sions in micron and submicron regimes, when it was applied to produce whiskers [1170,1171], which, following the development of the thermodynamic and kinetic basis [1172], paved the way for generating nanowires and nanorods from a rich variety of inorganic materials [1173–1175]. Even with improved techniques that culminated in the reduction of dimensions and production of a plethora of shapes, the control of the orientation, size, nucleation sites (unless aided by lithography) with sufficient precision remains challenging. In addition, the catalyst used for nucleating and feeding the growth front remains with the nanorods (nanowires) and must somehow be removed. To circumvent this problem altogether, catalyst-free methods have been developed, mainly MBE and OMVPE like approaches, which will be discussed shortly. Let us take the example of Ge nanorod growth, as it provides a very good example of the process, because of the manner in which the rods are produced, to shed some light on the evolution of the growth processes in VLS epitaxy. Using an environmental transmission electron microscope in the realm of Ge nanorod growth with Au as the liquid catalyst, Wu and Yang [1176] monitored the VLS growth in real time. They found that the process commences with the dissolution of gaseous reactants into nanosized liquid droplets of a catalyst metal, which pave the way to nucleation and growth of single-crystalline rods and then to wires. The liquid droplets, whose sizes remain essentially unchanged during the entire process of wire growth, are pivotal in initiating and maintaining growth. The evolution of the VLS process is illustrated in Figure 3.242, with the growth of a Ge nanowire observed by in situ TEM. Based on the Ge–Au binary phase diagram, Ge and Au form liquid alloys when the temperature is raised above the eutectic point (361  C). Once the liquid droplet is supersaturated with Ge, nanowire growth will start to occur at the solid–liquid interface. The

Figure 3.242 In situ TEM images recorded in an environmental transmission electron microscope during the VLS process of Ge nanowire growth using Au dots as catalyst. (a) Au nanoclusters in solid state form at 500  C; (b) alloying is initiated at 800  C with Au still existing

in mainly in a solid state; (c) liquid Au/Ge alloy formation; (d) commencing of the nucleation of Ge nanocrystal on the alloy surface; (e) Ge nanocrystal elongates with further Ge condensation, and (f) eventual forming of a wire. Courtesy of P. Yang and Ref. [1169].

3.5 The Art and Technology of Growth of Nitrides

establishment of the symmetry-breaking solid–liquid interface is the key step for the pseudo-one-dimensional nanocrystal growth in this process, whereas the stoichiometry and lattice symmetry of the semiconductor material systems are less relevant. One of the main challenges for catalyst assisted growth has to do with the catalyst itself that is to produce a uniform dispersion and yet maintain the small and uniform size of the catalyst. In general, the correlation in the size of the catalytic nanoparticles and resulting diameters of the nanowires or nanorods is reasonably good. In other words, in the VLS process the diameter of each nanowire is to a large extent determined by the size of the catalyst particle. It then follows that smaller catalyst islands yield thinner nanowires and dimensional control is strictly based on the control on the initial size of the catalyst used. Having said that it should be noted that the average diameter of the nanowires (nanorods) is typically larger than that of the catalyst itself. Poorly dispersed catalytic nanoparticles and thin film coating often lead to significant agglomeration of catalysts during heating. In addition to the agglomeration problem of the catalyst, the nonuniform distribution of the constituent group III elements and/or N in the catalytic reaction can also cause a broad diameter distribution of the nanowires. Because the process temperature for nanowires is relatively high, considerable self- and surface-diffusion could occur. To circumvent the diffusion issues, diffusion barrier material and the catalyst are mixed, a process that is effective in preventing agglomeration of the catalyst. In this vein, reaction of Ga and SiO2 mixtures with NH3 in the presence of the Fe2O3 catalyst supported by Al2O3 have been used to produce GaN nanowires [1177]. The addition of SiO2 is thought to reduce the melting temperature of the catalyst by forming FeSi2 as well as enhancing the production of high-pressure Ga2O gas through formation of a eutectic Ga–SiO2 or the following reaction: 4Gaðs; lÞ þ SiO2 ðsÞ ! 2Ga2 OðgÞ þ SiðsÞ:

ð3:72Þ

In this case, the catalysts continuously dissolve the gaseous Ga2O and NH3, leading to the formation of GaN as discussed in Section 3.5.16.4.2. In this process, although Al2O3 does not participate directly in the catalytic reaction, together with Fe2O3, it seems to prevent the agglomeration of the Fe-containing catalytic droplets, paving the way for GaN nanowire formation with a much smaller diameter (10–50 nm) than those produced with Fe2O3 only (80–200 nm) [1168]. In addition to the agglomeration problem of the catalyst mentioned above, obviously the nonuniform distribution of the constituent group III elements and/ or N in the catalytic reaction can also cause a dispersion in the diameter of the nanowires. Group III elements are typically placed in a boat at the hot zone in the furnace close to the gas inlet and the substrate on which the rods are to form is placed in the colder section downstream. The distance between the source and substrate typically has an impact on the dispersion in the diameter, with diameter decreasing with increasing distance. This leads to larger diameter forms at the leading edge of the substrate than the trailing edge. In this respect, a long furnace with a large source to substrate distance would reduce the diameter variation across the substrate.

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Obviously, the deposition (growth) time is also a factor in determining the nanowire diameter and length. In addition to solid sources such as powders used for group III elements, gas sources as in the standard vapor phase deposition techniques have also been employed. Although increasing the complexity of the experimental setup, this feature provides a better control over the group III element supply and also paves the way for heterostructure growth. An example is the combined use of NH3 gas for N, and ferrocene (C10H10Fe) and gallium dimethylamide (Ga2[N(CH3)2]6) for catalyst and group III element, respectively [1178]. By this method, GaN nanorods with diameters of 15–70 nm and lengths of 3–30 mm have been produced. However, due to the presence of carbon in the organometallic sources used, carbon nanotubes were also formed at process temperatures of 900 and 1000  C. Other organometallic sources [1179] for Ga such as (CH3COCHCOCH3)3Ga have also been explored for GaN nanowire growth at temperatures as low as 620  C. The correlation between the catalyst size and resulting rod size has been demonstrated in Si and GaP nanowires in that any specific size could be obtained by controlling the diameter of monodispersed gold colloids serving as the catalyst [1180,1181]. Analogous to the thickness in films grown by MBE and OMVPE, the length of the wire is proportional to the growth time. Because VLS lives or dies with the catalyst, the selection of catalyst is of paramount importance. In the selection criterion of the catalyst, the main requirement, in addition to availability in sizes desired, is that the catalyst must be capable of forming an alloy with the target material, ideally an eutectic compound, which brings about the pseudobinary phase diagram between the metal catalyst and the solid material of interest. It should be mentioned that VLS cannot be applied to the production of metal nanocrystals because of alloy formation between the metal and catalyst. In terms of the semiconductor nanocrystal growth by VLS, one of the major drawbacks, at least potentially, is the contamination of semiconductor nanocrystal by the metal catalyst. There has been considerable activity in GaN and to a lesser extent AlN based nanorods and nanowires in this respect. InN, however, did not received as much attention due to the high vapor pressure of N over In and thus low thermal stability that requires the growth temperature to be low ( 2ND and NA  2ND. For the NA > 2ND, all donors have sufficient number of acceptors with which to combine. Therefore, the concentration of ADA complex will be ND. The remaining acceptor and ADA complex will contribute together acting as ionized centers. The neutrality equation in this case can be expressed as  ¼ p; N A þ N ADA

ð4:59Þ

where  ¼ N ADA

ND  : 1 þ g exp kTE F

ð4:60Þ

For the NA  2ND, case because there are abundance of donors, all acceptors are captured by donors to form the complex and some donors are left as isolated ions, N Dþ . The charge neutrality equation becomes  N ADA ¼ p þ N Dþ

ð4:61Þ

where  ¼ N ADA

N A =2  ; 1 þ g exp kTE F

ð4:62Þ

and N Dþ ¼ N D  N A =2

ð4:63Þ

The simulated hole concentration versus donor concentration within the context of the ADA model are indexed in Figure 4.118 as (i) for NA ¼ 1 · 1019 cm3 and (ii) for 1 · 1020 cm3, which is similar to that reported in Ref. [492] with the exception of a more abrupt change near the inflection point. Shown with diamond symbols is the hole concentration under the condition that NA ¼ 2ND, which represents the upper limit of this method. As argued, an increase in the hole concentration with an

j1023

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j 4 Extended and Point Defects, Doping, and Magnetism

Figure 4.118 Hole concentration versus ND where the acceptor–donor–acceptor complex model of Ref. [495] is shown with thin and bold lines for NA ¼ 1 · 1019 cm3 (i) and 1 · 1020 cm3 (ii), respectively. The optimum hole concentration where NA ¼ 2ND, as expected from the complex formation, is shown with diamonds. For comparative purpose, the simple compensation model which assumes a

single donor and (unpaired) acceptor is depicted with thin and thick lines for NA ¼ 1 · 1019 cm3 (iii) and 1 · 1020 (iv), respectively. The random pair model is also plotted with thin and thick lines for NA ¼ 1 · 1019 cm3 (v) and 1 · 1020 cm3 (vi), respectively. Discussions with Dr R. Korotkov are acknowledged. (Please find a color version of this figure on the color tables.)

increase in the donor concentration, ND, is seen up to a level of NA ¼ 2ND. Because the maximum ADA complex concentration is determined by the acceptor concentration a decrease in the hole concentration for donor concentrations ND > 0.5NA follows. The curve deduced from the simple compensation model [495] is also displayed, as curves iii and iv again for acceptor concentrations of NA ¼ 1 · 1019 and 1 · 1020 cm3.

4.9 Intentional Doping Table 4.9 Parameters used in codoping calculations.

Effective mass for electron Effective mass for hole Activation energy for donor Activation energy for acceptor GaN dielectric constant Temperature Acceptor degeneracy factor

0.22m0 2.0m0 30 meV 200 meV 8.9 300 K 3.6

Let us now turn our attention to the paired ADA model in which the formation of ADA complex is based on the random pairing mechanism. The acceptors can be viewed as being composed of two parts, namely, ionized acceptors and neutral acceptors. The donors (all ionized as the Fermi level is close to the valence band) will first interact with ionized donors to form close pairs. A pair of this kind would capture one isolated neutral acceptor to form the complex. Also activation energy is not zero any more but has the relationship with total impurity density described as E 0A ¼ E A 

q2 ; 4per

ð4:64Þ

where the donor–acceptor distance is given by  1=3 3 : r¼ 4pðN A þ N D Þ

ð4:65Þ

To simplify the algorithm used for calculation, as done previously, different regimes can be considered, similar to that performed in the context of the ADA ðnÞ model, but with different terms. First, let us denote N A as the neutral acceptors within the semiconductor given by ðnÞ

N A ¼ N A  N A : If

ðnÞ N A >N D ,

N A

ð4:66Þ

all donors will form ADA complexes, resulting in

  N D þ N ADA ¼p

 where N ADA ¼ ND:

ð4:67Þ

ðnÞ

If, on the contrary N A  N D, only a portion of donors can form complexes, resulting in  ¼p N A  N D þ N ADA ðnÞ

where

ðnÞ

 N ADA ¼ NA :

ð4:68Þ

Because N A is not predetermined, iterations are needed to converge on a hole concentration. The hole concentrations for the random distribution model as a function of donor concentration are shown in Figure 4.118 as (v) for Nt ¼ 1 · 1019 cm3 and (vi) for 1 · 1020 cm3. The hole concentration is not seen to decrease as rapidly relative to the simple compensation model (curves iii and iv). However, only a small increase in p for ND  5 · 1019 cm3 is noted.

j1025

j 4 Extended and Point Defects, Doping, and Magnetism To evaluate the effect of donor concentration through the ADA complex formation on the hole concentration, the hole density can be calculated as a function of acceptor concentration using Equation 4.56 with the donor concentration being a parameter (complementary figure to Figure 4.118). These hole concentration versus the acceptor concentration for two donor concentrations, namely, for ND ¼ 1018 and 1019 cm3, are shown with solid lines in Figure 4.119. Hole concentrations obtained using the simple compensation model with one donor, one acceptor and E 0A ¼ 200 meV are also displayed with dashed lines for comparison. An increase describable by a square root dependence of hole concentration is expected from the simple compensation model for high acceptor concentrations [496]. Instead, a very weak dependence of the hole density is observed for the ADA model at high acceptor concentrations. Katayama-Yoshida and coworkers [497–499] investigated the role of n-type dopants, Si and O, in codoping of p-type GaN doped with Be or Mg using ab initio electronic band structure calculations. They found (1) the total energy calculations showed that the formation of Be (Mg)–O–Be (Mg) structures or Be (Mg)–N–Si–N–Be (Mg) ones were energetically favorable; and (2) the variation of the impurity levels caused by the strong interactions between Si (O) and Be (Mg) would enhance the hole concentrations in p-type codoped GaN. They predicted that the p-type codoped GaN using Be or Mg as acceptors and Si or O as donors would exhibit an increased

1019 -3 19 ADA N D =10 cm

ADA ND =1018 cm-3

1018

Holeconcentration(cm-3)

1026

18

1017

= 10

-3

cm

ND

1016

18

= 10

-3

cm

ND 1015 1018

1019

Acceptor concentration (cm-3) Figure 4.119 Calculated hole concentration as a function of the acceptor concentration NA for two donor concentrations of ND ¼ 1 · 1018 and 1 · 1019 cm3 as a parameter for the acceptor–donor–acceptor and simple compensation models that are shown with bold and thin lines, respectively. Courtesy of R. Korotkov and B.W. Wessels.

1020

4.9 Intentional Doping

incorporation of Be or Mg acceptors compared with p-type GaN doped with the acceptors alone. They recently also calculated the codoping method in p-type AlN: [O þ 2C] and n-type diamond [500]. It is demonstrated the codoping method is the efficient and universal doping method by which one can avoid carrier compensation with an increase in the solubility of the dopant, increase in the activation rate by decreasing the ionization energy of acceptors and donors, and an increase in the carrier mobility. Moving onto the technology, predictions regarding codoping are complemented well with experiments in that low-resistivity p-type GaN has been obtained. For example, Korotkov et al. [501] successfully obtained highly conductive p-type GaN by using Mg and O codoping during OMVPE growth. The resistivity of codoped layers decreased from 8 to 0.2 O cm upon oxygen codoping. The activation energy of Mg decreased from 170 to 135 meV (not as much as predicted by the ADA complex theory, a point which is elaborated on below), and the hole concentration was as high as 2 · 1018 cm3. To amplify, to determining the effect of oxygen codoping on carrier concentration for p-type GaN by Korotkov et al. [492] undertook two sets of experiments. The epitaxial films for the first set of experiments were grown at a constant Mg dopant concentration in the gas phase, while the concentration of oxygen was varied from 0 to 80 ppm [501]. The room-temperature carrier concentrations as a function of oxygen doping for this set are displayed in Figure 4.120a. As seen, the hole concentration of the codoped epilayers increased super linearly with oxygen doping. A hole concentration as high as 2 · 1018 cm3 was attained. However, for an oxygen concentration of 30 ppm, the conductivity type changed from hole to electron conduction as indicated by a change in the sign of the Hall effect. For the second set, Korotkov et al. [492] varied the Mg doping while keeping the oxygen partial pressure during growth constant at 4 ppm. The resultant dependence of hole density on Mg flow rate for codoped samples is shown in Figure 4.120b. The hole concentration is seen to increase by more than an order of magnitude up to 1 · 1018 cm3, followed by saturation. For comparison, Mg doping without oxygen grown under similar conditions produced p-type samples with a concentration of 2 · 1017 cm3 and a resistivity of 3.5 O cm. A qualitative agreement between the observed hole dependence on dopant partial pressure and that predicted using the ADA complex model calculations is present. For example, an increase in hole density was observed experimentally with increasing donor dopant as seen in Figure 4.120a, which is predicted by theory. In addition, the data presented in Figure 4.120b indicate that the hole density is nearly fixed at a specific concentration by the donor concentration, ND, which again is in agreement with theory. The solid line in Figure 4.120a (NA is taken as a fitting parameter) and b (ND is taken as a fitting parameter) represents results of calculation using Equation 4.56. Consistent with the work of Katayama-Yoshida and coworkers [495], for these calculations an acceptor binding energy of 0 was assumed for the ADA complex and 200 meV for the isolated acceptor. The value of ND ¼ 1.3 · 1018 cm3 in Figure 4.120b is used to fit the data, and there is again a good agreement between theory and experiment. However, in Figure 4.120a, the increase in the experimental hole concentration is steeper than

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j 4 Extended and Point Defects, Doping, and Magnetism Donor concentration (cm-3) 1018

1019

1020

Carrierconcentration(cm-3)

T = 300 K

1019

1018

p-type

1017 1

n-type

10 Oxygen concentration (ppm)

100

Acceptor concentration, fit (cm-3) 1018

1019

T = 300 K 1018

Hole concentration (cm-3)

1028

1017

1016

0.05

0.1

0.5

Mg flow rate (μmol/min) (ppm) Figure 4.120 (a) Experimental carrier concentration as a function of oxygen concentration during growth. The solid line is obtained by plotting Equation 4.58 with NA ¼ 1 · 1019 cm3. The solid circle symbols are used for p-type while solid square are used to depict n-type conductivity. (b) The hole

concentration is plotted as a function of the Cp2Mg flow rate during growth. The oxygen flow rate was held constant at 4 ppm. The solid line is a fit of Equation 4.58 with ND ¼ 1.3 · 1018 cm3. Courtesy of R. Korotkov and B.W. Wessels.

4.9 Intentional Doping

that predicted by theory. For the data in Figure 4.120a, the fitting parameter NA was taken as 1 · 1019 cm3. The sharp increase observed in the measured hole concentration might to some extent be due to other compensation mechanisms for low hole concentrations. Interestingly, the usual bell-shaped dependence of hole concentration on acceptor concentration typical of GaN:Mg was not observed. In some studies [496] this was attributed to the formation of compensating donor defects. It should be pointed out that the ADA model does not include compensation by donor complexes such as [MgGa–VN]2þ, which might in part account for the discrepancy. Experimental data, presented in Figure 4.120a, are for the heavy doping case (p ¼ 8 · 1016 cm3 and was obtained without oxygen present) with NA on the right-hand side of the bell-shaped maximum [501]. It is likely that oxygen substitutes on the V3Nþ sites, allowing more MgGa acceptors to become active upon codoping with oxygen because the [MgGa–VN]2 þ complex will not form, providing an additional increase of the hole density. While there is a good qualitative agreement between the experiments and the ADA complex model results, several unanswered questions still remain. Among them is the fact that a degenerate acceptor level EA  0 meV was utilized in the calculations, but nonzero activation energies were measured for the samples shown in Figure 4.120 [501] and also in Ref. [504]. In spite of this, degenerate p-type GaN was observed in other investigations, see, for example, Ref. [502], along with the enhancement of hole densities by codoping. The different behavior of Si codoping as compared to O in GaN can be tentatively explained within the framework of the ADA model. The tendency for ADA complex or pair formation depends on the bonding strength, and because the Mg–O bonding is stronger than the Mg–Si bonds, it is natural to expect O codoping to be more effective. Therefore, NA not being dependent on Si donor density is attributed to a donor concentration that is inadequate to produce significant enhancement of acceptors incorporation. There is also an inherent problem with Si in that both Mg and Si compete for substituting for the same (Ga) lattice site. Competitive adsorption also affects the mutual incorporation of Mg and Si. Similarly, the highest hole concentrations were attained in GaN grown with a decreased gallium flow rate with respect to the optimum Ga-flux, as the formation energy of MgGa increases with gallium chemical potential. To reiterate, the ADA model explains some of the observed codoping behavior but not the lack of low ionization energy for acceptors in some codoping studies. This might indicate that there are other important mechanisms in play. One plausible explanation that was forwarded has to do with screening by impurities [493,494]. It is most likely that a combination of several factors, including complex formation, decrease of compensation, increase in acceptor incorporation, and screening effects that lead to reduced ionization energy could all be responsible for the observed increases in hole concentration to varying extents. Interestingly, a potential fluctuation model was also used to describe the observed increase in the hole concentration by codoping, which also predicts very weak temperature dependence of the hole concentration [503].

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j 4 Extended and Point Defects, Doping, and Magnetism Other experiments have been also been conducted. For example, Kipshidze et al. [504] also reported on their successful Mg and O codoping in p-type GaN and AlxGa1xN (x < 0.08) using GSMBE. In MBE growth, it is known that incorporation of Mg takes place on Ga sites of a Ga polarity surface [505]. The stoichiometry of the growing surface is thus important, which can be controlled by adjusting the NH3/Ga and NH3/(Ga,Al) flux ratios. For a fixed Ga and Al flux, the flux of NH3 is increased until the growth rate saturates. The point at which the growth rate becomes independent of the flux of NH3 corresponds to the stoichiometric surface [456]. Because the presence of Al results in more efficient decomposition of NH3, the condition of stoichiometry in AlGaN is obtained at lower fluxes of NH3. In these experiments, the optimum growth rates were in the range 0.5–0.6 mm h1 for both GaN and Al0.08Ga0.92N. Doping with Mg was carried out close to the optimum point, in the region where the growth rate is independent of the NH3 flux (f NH3 ). At lower fluxes, where the growth rate depends on the NH3 flux, p-type doping was not possible and all the samples were n-type or heavily compensated. The decrease in hole concentration for the smallest values of f NH3 is related to the loss of stoichiometric growth condition, that is, when the growing surface becomes Ga rich and the Ga vacancy concentration is greatly reduced. Kim et al. [507] used Mg–Zn codoping in GaN by OMVPE. A low resistivity of 0.72 and hole concentration of 8.5 · 1017 was achieved. Bis(cyclopentadienyl) magnesium (Cp2Mg) is becoming a popular Mg source. Although Cp2Mg is a white crystalline solid with very low vapor pressure, which will lead to transport problems, recently 1 · 1018 hole concentration was obtained in OMVPE growth of GaN [508]. Coimplantation was also used to get high hole concentration. Kent et al. [509] used coimplantation of Be þ O and Mg þ O into GaN to achieve high activation efficiency. Yu et al. [510] used Be implantation into Mg-doped GaN samples. The hole concentration after annealing showed an increase from 5.5 · 1016 to 8.1 · 1019 cm3 determined by Hall measurements. Figure 4.121a shows the change in the hole concentration for GaN (squares) and Al0.08Ga0.92N (circles) as a function of the ammonia flux. These samples were grown consecutively, with the same background pressures of O, as judged by in situ mass spectrometry data and confirmed later by SIMS analysis. The substrate temperature and the growth rate were the same for each sample. Figure 4.121b plots hole concentrations in samples of GaN (squares) and AlGaN (circles) as a function of Mg/O ratio in the layer, as determined by SIMS. The hole concentration strongly depends on the Mg/O ratio. The Mg/O ratio of 40 results in the highest hole concentration of 2 · 1018 cm3 in GaN and in AlGaN. For these samples, SIMS data give Mg and O concentrations of 1.6 · 1020 and 4 · 1018 cm3, respectively. For the best samples the acceptor activation efficiency is thus about 1.3%. The excellent electrical quality of these samples, grown on sapphire, was confirmed by room temperature Hall mobilities of 40  10 cm2 V1 s1 for GaN and 8  2 cm2 V1 s1 for AlGaN. Although it might at first sound counterintuitive, and in fact counter to common perception, codoping with donor impurities along with acceptor impurities has been theoretically and experimentally shown to lead to increased hole concentrations.

Hole concentration(cm-3)

4.9 Intentional Doping

18

10

17

10

15

10

NH 3 flow rate (sccm)

(a)

Hole concentration(cm-3)

25

20

1018

1017

1016

1015 20

(b)

40

60

80

Mg/O ratio

Figure 4.121 (a) Hole concentration versus ammonia flux, solid square: GaN, and solid circle: Al0.08Ga0.92N. (b) Hole concentration versus Mg/O ratio. Solid square: GaN, and solid circle: Al0.08Ga0.92N. Mg concentration in the layer is (1–2) · 1020 cm3. Courtesy of Nikishin and coworkers [504].

However, chemical nature of donor impurities in GaN lattice and their relative tendency for chemical attraction appear to determine which donor acceptor combinations would be best. Because Mg is the most efficient acceptor impurity, this leaves the donor side open for experimentation. Among the candidates, codoping with O appears to be more effective than with Si. Oxygen and silicon donors behave differently as codopants in GaN:Mg, which indicates that the observed codoping effects are not simply due to Fermi level effects. There is experimental evidence that shallow acceptor complexes form upon codoping, as deduced from the measured

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j 4 Extended and Point Defects, Doping, and Magnetism dependence of hole concentration on Mg dopant concentration for oxygen codoped samples. One thing is certain that codoping is a complex phenomenon and more detailed experiments in well-controlled templates are needed. 4.9.3.1.2 Use of Superlattices for Improving p-Type Conductivity Another promising method to get higher acceptor activation and lower resistivity is using AlxGa1xN/ GaN doped superlattices [511–513]. The idea of using AlGaN/GaN heterojunction for enhancement of p-type doping [514–516] has been proposed as a technique to increase the average hole concentration. Increased hole concentration at room temperature through the use of AlGaN/GaN superlattices have been reported [517,518]. The mechanism for hole enhancement is the periodic oscillation of the valence band edge. Acceptors are ionized where the band edge is far below the Fermi energy and the resulting holes accumulate where the band edge is close to the Fermi level, forming a confined sheet of carriers. Although the free carriers are separated into parallel sheets, their spatially averaged density will be much higher than in a simple bulk film. High electric fields due to both spontaneous and piezoelectric polarization within the strained AlGaN layers are expected to strongly impact the band bending within the superlattice. Hole concentration expected from an AlGaN/GaN superlattice both with and without the polarization fields taken into account has been calculated [516]. The fields create a periodic sawtooth variation in the band diagram as shown in Figure 4.122. When the polarization fields are present,

0.2

0.1

A

B

A

B

EMg EF

0.0

Energy (eV)

1032

-0.1

-0.2

EV

-0.3

-0.4

AlGaN

GaN

Figure 4.122 Calculated valence band diagram for the Mg-doped Al0.2Ga0.8N/GaN superlattice with spontaneous and piezoelectric polarization fields taken into account. The thickness of each layer is L ¼ 8 nm. The dashed line indicates the Fermi energy, and the circles represent the energy of the Mg acceptor with

AlGaN solid circles indicating the ionized form. Regions A and B defining each interface are also indicated. The sapphire substrate is on the left and the free surface of the film is on the right side of the figure. Courtesy of DenBaars and coworkers [516].

Spatially average hole concentration (cm -3)

4.9 Intentional Doping

18

10

GaN

AlGaN 17

Holemobility (cm2/Vs)

10

GaN

16

AlGaN 12

8

2

4

6

8

10

12

14

Well (barrier) width (nm) Figure 4.123 Room temperature Hall effect measurements on uniformly Mg-doped OMVPE-grown Al0.2Ga0.8N/GaN superlattices. The superlattice dimension L is varied. The arrows indicate values obtained on bulk samples of Mg-doped GaN and Al0.1Ga0.9N. The dashed lines are eye guides. Courtesy of DenBaars and coworkers [516].

a strong dependence on the superlattice dimensions is obtained: thicker layers yield larger potential changes from the polarization fields, and therefore, higher cracking. The measured hole concentration, spatially averaged, and mobility for the Mg-doped Al0.2Ga0.8N/GaN superlattices are shown in Figure 4.123 as a function of L, the thickness of the GaN and AlGaN layers (the layer thicknesses were kept equal so that the superlattice period is 2L). Further improvements were obtained by minimizing the ionized and neutral impurity scattering mechanisms through modulation doping that maximize the separation of the dopants (ionized scattering centers) from the multiple two-dimensional hole gases (2DHG). In this respect, the effect of modulation doping in Al0.20Ga0.80N/GaN using Hall effect and C–V profiling techniques has been investigated [518]. Mobility, resistivity, and carrier concentration were measured as a function of temperature. The modulation-doped (MD) and shifted-modulationdoped (SMD) samples were shown to have superior electrical properties compared to uniformly doped (UD) samples, especially at low temperatures. Furthermore, C–V profiles were presented that showed the 2DHG of the superlattice structure.

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j 4 Extended and Point Defects, Doping, and Magnetism Figure 4.124 shows self-consistent valence band diagrams of an MD SL (represents superlattice), a SMD SL, and a UD SL. The three ground state hole energies are E0  EF ¼ 5.9 , 21.7, and 21.7 meV, respectively. The term P is the self-consistently solved free hole concentration. The ground state is the only occupied subband at 90 K. The epilayer surface is on the left-hand side. The band diagrams shown in Figure 4.124 are calculated self-consistently using a one-dimensional (1D) Schr€ odinger–Poisson solver. An AlxGa1xN hole mass of (1.76 þ 1.77x) me,z, valence band discontinuity of 0.3 DEg, and energy gap Eg(x) ¼ (3.425 þ 2.71x) eV are used, x being the aluminum concentration. The calculated free hole concentration is shown in Figure 4.124 for each type of SL. At 90 K, only the ground states are occupied. The ionization energy of Mg in AlxGa1xN is not known precisely. However, the available data indicate that it increases from about 170 meV for x ¼ 0 (depending on the hole concentration which affects the screening) to 360 meV for x ¼ 0.27. A simple Vegard-like relationship can be used and put the acceptor level, EA, at (170 þ 704x) meV above the valence band. The calculated average free hole concentrations at 90 K are 2.3 · 1018, 3.3 · 1018, and 3.3 · 1018 cm3 for the MD, SMD, and UD SL, respectively. This agrees favorably with the measured values. A large improvement in mobility for the MD and SMD SLs versus the UD SL was achieved, especially, at low temperatures. At 90 K, the mobility (cm2 V1 s1) was 36, 18, and 2.0 for SMD, MD, and UD SLs, respectively. The free hole concentration versus temperature is presented in Figure 4.125. The effective acceptor activation energies are 16, 30, and 13 meV for the MD, UD, and SMD SLs, respectively, as determined in the temperature range of 250–390 K. These values are much smaller than activation energies of about 200 meV, depending on the doping level found in bulk GaN. Using Mg–Zn codoping in GaN by OMVPE, a low resistivity of 0.72 O cm and hole concentration of 8.5 · 1017 have been achieved [519]. It should be noted that Bis (cyclopentadienyl) magnesium (Cp2Mg) is a very popular Mg source. Although Cp2Mg is a white crystalline solid with very low vapor pressure, which leads to gas transport problems, hole concentrations as high as 1 · 1018 have been reported in OMVPE-grown GaN [520]. Coimplantation was also used to get high hole concentrations. Similarly, coimplantation of Be þ O and Mg þ O into GaN to achieve a high activation efficiency [521]. Again, in terms of codoping Be has been implanted into Mg-doped GaN [522]. The hole concentration after annealing showed an increase from 5.5 · 1016 to 8.1 · 1019 cm3 determined by Hall measurement, a figure that is certainly much higher than other reports, which may necessitate some degree of scrutiny. Despite gallant efforts to obtain p-type doping in GaN and AlGaN with high hole concentration, the problem is still a limitation for optoelectronic devices, particularly for shorter wavelength devices such as real solar-blind (l < 290 nm) detectors, or UV LEDs or LDs, which need p-type AlGaN with high Al composition. For example, a UV LED using AlGaN p–n junction with emission wavelength shorter than 290 nm, can only give submilliwatt or milliwatt power [523]. 4.9.3.1.3 Role of Hydrogen and Defects in Mg-Doped GaN Unlike undoped and n-type GaN, hydrogen has been reported to be very beneficial for p-type growth. It is

4.9 Intentional Doping

Al 0.20Ga0.80 N/GaN superlattices,T = 90 K shaded regions are Mg doped at 1019 cm-3 200

Modulation doped

EF - ---- EA

0 -200 -400

ΨGS

2

-600

EV

Energy E (meV)

GaN

AlGaN

Shifted modulation doped

200 0

- - -200 -

- - -

ΨGS

-400 -600

2

EF - EA

EV

GaN

AlGaN

Uniformly doped

200 0

- - -200 - -

-400

ΨGS

2

-600 600 620 640 660 680 700720 740 Position x (Å)

- - -

EF - EA

EV 760

780

800

Figure 4.124 Self-consistent valence band diagrams of an MD SL, an SMD SL, and a UD SL. The three ground state hole energies are E0 – EF ¼ 5.9 meV, 21.7 meV, and 21.7 meV, respectively. P is the self-consistently solved free hole concentration. The ground state is the only occupied subband at 90 K. The epilayer surface is on the left-hand side. Courtesy of Schubert et al. [518].

well established that growth of Mg-doped GaN in hydrogen ambient improves p-type conductivity after consequent annealing in N2 at temperatures above 700  C [524]. The need for postgrowth annealing to activate Mg in OMVPE-grown samples and reverting them to high-resistivity compensated material after ammonia annealing have both received a good deal of attention. It has been suggested that hydrogen passivates Mg [525], which is also supported by the calculations [526]. Moreover, they demonstrated that hydrogen is beneficial to p-type doping by Mg when compared to the hydrogen-free case, because the hydrogen passivates Mg during growth and thus inhibits the formation of native donors that self-compensate the acceptors. For OMVPE-grown films, it has been argued that H is in its positive charge state, the Hþ proton, and passivates the Mg acceptors that are in their negative charge state during growth; this would prevent the compensating donor-like defects from forming. During post-growth annealing, Hþ is driven out, which results in p-type

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j 4 Extended and Point Defects, Doping, and Magnetism 1.0 0.9

Al 0.20 Ga0.80 N/GaN superlattices 100 Å /100 Å, Mg doped

0.8

Resistivity, ρ (Ω cm)

1036

Uniformly doped

0.7 0.6 0.5

Shifted modulation doped

0.4 0.3 0.2 0.1 0.0

Modulation doped

100

150

200

250

300

350

400

Temperature, T (K) Figure 4.125 Variable temperature three-dimensional carrier concentration data of an MD, an SMD, and a UD Al0.20Ga0.80N/ GaN SL. Courtesy of Schubert et al. [518].

GaN due to the negatively charged Mg acceptors. Although it has been mentioned frequently that Mg and H form a complex in Mg-doped GaN films, the exact mechanism of formation and the release of H upon a postgrowth treatment have not been sufficiently elucidated. In addition, predictions of the position of the H atom from first principles calculation are not consistent. For example, in the Neugebauer et al. [215] calculations, the antibonding site (i.e., a hydrogen sitting next to a N atom, about 1 Å away from the N site, at the antibonding position of the Mg-N bond) is lower in energy than the bond center site. This follows from the argument that the BC position requires an outward motion of the Mg and N atoms, which is very energy consuming in GaN, as GaN is a very hard material. On the contrary, an argument in favor of the BC location for H between substitutional Mg and N nearest neighbors has been provided [527]. In general, H takes the BC position in all other semiconductors. Figure 4.126 displays the passivating H at both the BC and the antibonding sites. Also useful is the determination of the local vibrational modes (LVMs) of the Mg–H complex in GaN, which provides not only the confirmation of hydrogen in GaN but also gives significant information on the structure of the complex. At the present time the stretch frequency is, in fact, the only reliably established physical parameter available from experiment for the Mg–H complex in GaN. To support their spectroscopic identification, Fourier-transform infrared-absorption spectroscopy has been performed on three Mg-doped GaN layers grown by OMVPE [424]. The first sample was as-grown and electrically semi-insulating; the second one was subjected to a thermal anneal and displayed p-type conductivity; and the third sample was exposed to monatomic deuterium at 600  C for 2 h, which increased the resistivity of the material. The as-grown sample displayed an LVM at 3125 cml, which is in very good

4.9 Intentional Doping

Figure 4.126 Proposed bond center and antibonding site incorporation of H in GaN and its passivation of Mg during growth (the Mg atom is directly below the H atom). In part courtesy of C. Van de Walle. (Please find a color version of this figure on the color tables.)

agreement with 3360 cm1 predicted by calculations [215]. After thermal activation of Mg, the intensity of this absorption line is reduced. After deuteration, a new absorption line appeared at 2321 cm1, which disappeared after a thermal activation treatment. The isotopic shift clearly establishes the presence of hydrogen in the complex. The controversy regarding the location of H at Mg–Ga in GaN can be resolved by a Rutherford backscattering experiment with a good-quality crystal and moderately heavy doping [528]. In this method a well-collimated beam of ions is directed down one of the channel axes and observe those that bounce back from a host atom that is not in its perfect lattice site. If H were bond centered then Mg would protrude out into the channel, which was calculated to be energetically unfavorable [215]. If H were in the interstitial space (antibonding site), then the incident ions would strike neither Mg, Ga, or N, but rather H; if there were any detectable recoil ions, which would be doubtful, they would easily be distinguished from those that struck other atoms. Additional support for the interstitial, antibonding site can be discerned from the following phenomenological argument [528]. The Hþ proton should act very much like an idealized positive test charge as it does not have to have its wave function

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j 4 Extended and Point Defects, Doping, and Magnetism orthogonalized to any of the electronic states. The very massive proton is also effectively a point charge, because the energy to confine its wave function to a small volume is less than that to confine the electron to the same volume. Another positive test charge is the positron eþ. A positron’s wave function does not have to be orthogonalized to those of the electrons, but the energy to localize it in a small volume is the same as for an electron. In Si, Ge, GaAs, GaP, InSb, and most likely in GaN, and other semiconductors, eþ travels between interstitial spaces in a perfect crystal, whereas e has the least wave function density. This is a natural consequence of maintaining the mass and switching the sign of the charge. However, eþ also needs sufficient volume for its wave function so that the localization energy is not too large, and it finds this volume in the interstitial space. The analogy between eþ and Hþ suggests that Hþ will be in the interstitial site surrounded by N ions. However, because Hþ does not need all of the interstitial space, it might find a lower energy site with a much smaller volume. One might think that the bond site would be a logical candidate. There is a local peak in the electron density at the bond site and it does not result from the Coulomb potential of the ions. If it did result from the Coulomb potential, then obviously Hþ would be repelled rather than attracted. The extra electron density in the bond site results from the constructive interference between the atomic wave function centered on the two atoms participating in the bond; this would be a quantum mechanical effect. Thus, Hþ might reside in a low-energy state. However, if it does, it will then certainly perturb all the bonds in the area rather strongly, which is not energetically favorable over the interstitial antibonding site [215]. Unlike the strongly covalent semiconductors where Hþ is attracted to the bond center, the ionic nature of GaN would tend to expel Hþ into the interstitial site. Thus, it is the interstitial site that will offer the lowest energy [528]. The compensating nature of H is not unique to GaN in that it is known for other group III–V semiconductors [529]. Owing to the incorporation of a relatively large concentration of shallow-acceptor impurities as compared to the concentration of native donor-like defects in GaN, one finds that the Mg acceptor concentration is accomplished primarily through the presence of Mg-H complexes. This compensation is lifted (i) when samples are irradiated with an electron beam of 5–15 keV incident energy for several hours, or (ii) when samples are thermally annealed at constant temperature for half an hour. Also, first principles calculations demonstrate that the same amount of both Mg and H incorporated into the GaN films when they are grown under an H-ambient growth condition, such as that of OMVPE [526]. Furthermore, the calculations predict that more Mg can be incorporated into GaN film when there are more H atoms present. A large body of work exists on optical properties of Mg-doped GaN, which is discussed in Volume 2, Chapter 5. 4.9.3.2 Beryllium Doping Earlier work on Be doping by means of CVD was unsuccessful and led to the incorporation of Be as deep centers [530,531]. Predictions on the basis of ab initio calculations [532] point out that Be on Ga sites would form a shallow acceptor with a thermal ionization energy of 60 meV, and a double donor, if incorporated, as

4.9 Intentional Doping

interstitial. The error in this method of calculation could be 100 meV. Potentially, Be represents the shallowest acceptor level in wurtzite GaN. Until the experimental investigation of Salvador et al. [533], it was believed that Be doping leads to compensated material and yields high-resistivity GaN. Optical measurements in early samples, dating back to the 1970s, yielded a broad emission at energy of about 2.2 eV. It should be noted that this is at about the same energy as the notorious yellow peak found in GaN with native defects (Volume 2, Chapter 5). In an experiment with RMBE-grown films [533], the 380 nm emission peak is the dominant one for low Be doping levels. In samples with higher doping, 420–430 nm emission is observed. Dewsnip et al. [534], Sanchez et al. [535], and Ptak et al. [536] indeed observed the PL band that can be attributed to the shallow acceptor level introduced by BeGa acceptor in Be-doped GaN grown by MBE (Figure 4.127). The band represents a set of peaks separated by the LO phonon energy in GaN (about 91 meV). The peak at 3.38 eV has been attributed to the zero-phonon transition from the shallow donor to the BeGa acceptor [534–536], while that at 3.397 eV to the zero-phonon transition from the conduction band to the BeGa acceptor [536]. The ratio between the intensities of the zero-phonon peak and its phonon replicas is typical of the shallow acceptors with small Huang–Rhys factor (compare with the UVL band caused by the unidentified shallow acceptor in GaN). The assignment of the 3.38 and 3.397 eV peaks to the DAP and e–A transitions has been confirmed through their temperatures and excitation intensity dependencies [534–536]. The 3.38 eV peak decayed nonexponentially and shifted to lower energies with some time delay after pulse excitation, as it is expected

10

7

(e,A):3.397eV A,X:3.472eV 10

DAPand(A,X)-LO 3.38eV

6

PL intensity (au)

DAP(?)and3.27eV 10

3.30eV

AX 3.482eV

5

I 4(?) 3.14eV 10

4

BX LO

10

10

LO

LO

3

2

3.0

3.1

3.2

3.3

3.4

Energy (eV) Figure 4.127 Low-temperature PL from a Be-doped GaN grown by MBE. Courtesy of Myers and coworkers [536].

3.5

3.6

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j 4 Extended and Point Defects, Doping, and Magnetism for the shallow DAP transitions [535]. From the energy position of the DAP and e–A peaks, the ionization energy of the BeGa acceptor can be estimated as 100  10 meV, much lower than that of MgGa [534–536]. In another report [564] the 3.38 eV peak was not observed in Be-doped GaN, and the usual UVL band with the main peak at about 3.27 eV, observed in these works, is apparently related to residual shallow acceptor in GaN (SiN, CN, or MgGa). A small peak observed at 3.35 eV [537,677], which was attributed to BeGa, is questionable. Enhancement of the YL band with the maximum at about 2.2 eV has been noted in GaN samples implanted with Be [377,564,565]. Possibly it is not related directly to Be, but caused by implantation damage [538,564,565]. In the samples doped with Be during growth, an enhancement of emission in the visible part of the spectra has also been reported [534,535]. The emission represented very broad band peaking at 2.4–2.5 eVand related to some deep level or levels apparently formed by Be. In the early work of Salvador et al. [533], the Hall measurements in Be-doped GaN did not indicate p-type conductivity. However, hot-probe measurements indicated p-type conductivity with p–n junctions attained. Later investigations by Yu et al. [539,540] noted an increase in the room temperature hole concentration up to 2.6 · 1018 and even to 8 · 1019 cm3 [540] in Mg-doped GaN after Be implantation. Astonishingly though, Be–O codoping in cubic GaN grown on GaN has been reported to result in a high Be activation in excess of 1020 cm3, with hole mobilities in the 100 cm2 V1 s1 range [541]. One can conclude that extreme care must be exercised in determining the acceptor binding energies by photoluminescence measurements. 4.9.3.3 Mercury Doping Several groups have investigated Hg doping of GaN, but none reported on electrical measurements [406]. Pankove and Hutchby [542] performed optical measurements and observed 2.43 eV emission in Hg-doped samples. This is lower than the 2.9 eV emission reported by Ejder and Grimmeiss [543]. On the basis of measured emission, Ejder and Grimmeiss estimated that the Hg-acceptor level lies about 410 meV above the valence band. We should note that dopant incorporation in nitrides is a rather complex process. Many metals promote defect creation within the gap. Such defects are efficient optical centers and can easily be mistaken for acceptors due to metal incorporation. To this end, available data appear to suggest that unsuccessful attempts end up generating defects. Successful p-type doping, as confirmed by Hall measurements, can be achieved when the unintentional doping level is low, for example, at or below 1017 cm3. 4.9.3.4 Carbon Doping Carbon in both GaN and AlN is amphoteric in nature [210]. Its salient features in both materials are similar: carbon on a Ga or Al site, Ccation, is predicted to be an effective mass donor while carbon on a N site, Canion, is an effective mass acceptor. The ionization energies of the acceptor have been predicted to be equal to 0.2 in GaN and 0.4 eV in AlN. Incorporation of C on a nitrogen site is preferable because the CNformation energy is lower under both Ga- and N-rich conditions of growth. Carbon on a cation site, Ccation, can also assume a metastable DX-like configuration C GaorAl that

4.9 Intentional Doping

requires a broken bond between C and one neighbor. In this configuration, both the host N and carbon atoms are significantly displaced. C GaorAl introduces a singlet at approximately 0.4 and 0.3 eV above the valence bands of GaN and AlN, respectively, and a singlet occupied by one electron at about 0.3 and 1.0 eV below the conduction bands of GaN and AlN, respectively. Investigation of the formation of the þ  Ccanion  Ccanion nearest ion pairs brought to light that the binding energy is substanþ  and C canion . tial and lower than the sum of the formation energies for C canion Therefore, the tendency toward self-compensation is expected to be large. However, for the preferred coordination, CN acceptors compensate residual shallow donors, resulting in semi-insulating GaN [544,545]. As the C concentration is increased the resistivity of GaN decreases, apparently due to formation of deep-level donors [544]. Several deep-level defects have been detected in C-doped GaN by employing methods such as thermally stimulated current spectroscopy [544], photoionization spectroscopy [545], and deep-level optical spectroscopy [546,547]. Most of these defects, however, are apparently not effects directly related to carbon [544]. In one report, hole concentrations up to levels of about 3 · 1017 cm3 with mobilities of about 102 cm2 V1 s1 in CCl4-doped GaN grown by MOMBE on GaAs substrates have been obtained. Unless the partial cubic nature of the resultant films is responsible for this seemingly high mobility, the results are otherwise considered controversial as the mobility reported is characteristic of n-type mobilities [548]. Annealing at 800  C did not increase the hole concentration that construed that the hydrogen passivation of acceptors is not significant. On the contrary, highly resistive CHx-doped GaN films resulted from plasma-assisted OMVPE. SIMS analyses indicated large amounts of C and H ( 1019–1020 cm3) in the samples. After annealing under a nitrogen atmosphere, the films remained highly resistive and suggest that the C–H complex is thermally stable [549]. As et al. [550–552] conducted a series of experiments dealing with C doping in c-GaN. Hole concentrations as high as 6.1 · 1018 cm3 with a mobility of 23.5 cm2 V1 s1 have been obtained. In samples with the hole concentration below 1017 cm3 the hole mobility exceeded 200 cm2 V1 s1. The e–A optical transition in a PL experiment was used to deduce an activation energy for C on the N site of about 215 meV in cubic GaN. Because carbon is present as a contaminant in most growth reactors (although graphite parts have been suggested as the likely source [554–561], the most likely source is hydrocarbons in MBE environment because properly cleaned high purity pyrolytic graphite has been used for crucibles with high quality layers resulting) and in metalorganic sources and possibly graphite parts exposed to high temperatures and hydrogen in OMVPE processes, its unintentional role in the compensation and doping of GaN layers might nevertheless be significant. It should be pointed out that any carbon contamination is most likely contributed by sources other than solid sources such as graphite parts, as the vapor pressure of graphite at growth temperatures is very low, although the effect on H should be considered. Of course, this picture is only applicable when the reaction pathways with those solid sources are rendered inconsequential. It is likely that C doping may be an attractive means of obtaining high-resistivity GaN, as opposed to, for example, Fe-doped GaN with its hysteresis effects.

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j 4 Extended and Point Defects, Doping, and Magnetism In terms of optical transitions, a correlation between the YL intensity and C doping has been attempted [554–561], although the properties of the YL band in undoped and C-doped GaN are apparently different [562,563] from those in undoped GaN. Enhancement of the YL band has also been observed in GaN after implantation of carbon [542,564,565], which may also create VGa resulting in the YL. The intensity of the UVL band apparently also increased with C doping [561]. However, in another report [566], a broad blue band peaking at 3.0 eV, instead of the UVL band, has been observed in highly resistive GaN:C. Seager et al. [566] attributed this blue band to transitions from CGa donor to CN acceptor, even though CN is more favorable. Polyakov et al. [556] also noted an enhancement of the blue band peaking at 3.05 eV in GaN samples heavily contaminated with carbon. Reuter et al. [559] observed a broad red luminescence band with the maximum at 1.64 eV in C-doped GaN. This band could be excited resonantly, and its excitation spectrum involved one or two broad bands in the range from 2.1 to 2.9 eV. The zero-phonon transition for this red band has been estimated at 2.0  0.2 eV [559]. When excited above the bandgap, the red band was buried under the stronger YL peaking at 2.2 eV. 4.9.3.5 Zinc Doping Attempts to achieve p-type conductivity by Zn doping of OMVPE-grown films were unsuccessful. Highly resistive films with resistivities up to 109 O cm were grown [567,568]. It has well been established [567] that among others, Zn causes a dominant level that is about 0.5 eV deep both in GaN and InGaN with a low InN mole fraction. In fact, this deep level was exploited in the earlier versions of InGaN LEDs marketed by Nichia Chemical Ltd in an effort to keep the InN mole fraction in the emission layer small while achieving 450 nm emission that is required by the display society. The nature of Zn centers has received some theoretical attention [569]. As many as four different Zn centers with energies in the forbidden gap, corresponding to emission between 1.8 and 2.9 eV, have been observed experimentally [570] and predicted theoretically [569]. While the 2.9 eV peak is attributed to substitutional Zn on Ga sites, the other three peaks have been assigned by experimentalists to various charged states of Zn on N sites, namely, ZnN  ;ZnN2  ;ZnN3  , with Zn atoms binding to as many as three electrons having binding energies of 0.65, 1.02, and 1.43 eV. Because Ga and Zn are both somewhat comparable in size, it is not too difficult to imagine Zn in Ga sites. However, Zn in N sites requires large formation energies and are unlikely to form. The thermal activation energy of Zn in Ga sites has been estimated to be 0.33 eV [570]. Optical properties of GaN doped with Zn are discussed in Volume 2, Chapter 5. 4.9.3.6 Calcium Doping Suggestions have been made that Ca may form a shallow acceptor level in the GaN bandgap [571]. Ca p-type doping of GaN was achieved by ion implantation of Caþ ions or by a coimplantation of Caþ and Pþ followed by a rapid thermal annealing at temperatures 1100  C [572]. The ionization energy of 169 meV was found by temperature-dependent measurements of the hole concentration in a sheet, which should be considered with a good deal of caution. Ca acceptors, like those of Mg, can

4.9 Intentional Doping

be passivated by atomic hydrogen at low temperatures (250  C), and they can be reactivated by thermal annealing at 500  C for 1 h. Pankove and Hutchby [542], Monteiro et al. [573], and Chen and Skromme [574] studied PL from GaN implanted with Ca. A strong green luminescence band with a maximum at about 2.5 eV ensued Ca implantation and subsequent annealing [542,573]. Chen and Skromme [574] also observed another green luminescence band (with a maximum at 2.35 eV) in GaN:Ca; however, they noted that the 2.35 eValso appeared in their semi-insulating GaN samples doped with Mg and Zn and therefore may not be related to Ca. With increasing temperature from 14 to 300 K, the green band shifted from about 2.52 to 2.59 eV [573]. Monteiro et al. [573], proposed that the 2.5 eV emission is due to transitions from a deep compensating donor (0.6 eV below the conduction band) to the Ca-related acceptor (0.3 eV above the valence band). This identification is questioned as the deep DAP transitions are characterized with enormous shift of the band with excitation intensity [366] or with time decay after pulse excitation [479]. Note that no shifts of the green band with excitation intensity nor time delay were detected in Ca-doped GaN [573]. A more likely scenario is that the 2.5 eV band is related to transitions from the conduction band (or shallow donors at low temperature) to the CaGa acceptor. In this case the ionization energy of CaGa can be estimated at about 0.6–0.7 eV. Indeed, the zero-phonon transition for the 2.5 eV band can be expected at 2.8–2.9 eV from the shape of this band at low temperature. 4.9.3.7 Cadmium Doping Cadmium on Ga site (CdGa) is an acceptor. PL features in GaN doped with Cd have been the topic of a few investigations [377,574–577]. Cd-doped GaN exhibits a broad blue band peaking at 2.7–2.9 eV. Lagerstedt and Monemar [576] observed fine structure on the high-energy side of this band with the zero-phonon line at 2.937 eV and a set of LO and what appears to be local phonon replicas. The fine structure, caused by electron–phonon coupling, is similar to that in the Zn-related BL band, although the position of the PL band in Cd-doped GaN is red shifted by about 0.1 eV [228,439,574,576]. Lower photon energy of the Cd-related PL, as compared to Zn-related PL, is consistent with theoretical predictions on ionization energies of these two acceptors. Assuming that the assigning of the 2.937 eV peak in Ref. [576] is reliable, the ionization energy of Cd should be about 0.56–0.57 eV. At 77 K the decay of the Cd-related blue band is nonexponential, in the microseconds range [577]. No obvious variation in the shape and position of this band was observed at different time delays after pulse excitation. Transitions from the conduction band to the CdGa acceptor overlap with transitions from the shallow donors to the same acceptor may be responsible for the nonexponential decay of PL [577]. 4.9.3.8 Other Acceptors in GaN A plethora of group II elements have been used in an attempt to attain acceptors in GaN. In addition to the abovementioned acceptors, Pankove and Hutchby [377] examined the effect of Hg implantation on PL in GaN. Implantation of Hg resulted in a broad green band with a maximum at 2.43 eV. Assumption that this band is due to transition from the conduction band or shallow donor to the HgGa acceptor, the

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j 4 Extended and Point Defects, Doping, and Magnetism activation energy of the HgGa acceptor can be estimated as 0.8  0.2 eV, based on position and shape of this broad band. Pankove and Hutchby [377] also implanted several other elements in GaN that may act as acceptors such as Cu, Ag, Au, Sr, Ba, Li, Na, K, Sc, Zr, Fe, Co, Ni, Dy, and Er. All these elements gave broad PL bands with maxima in the range from 1.5 to 2.2 eV. In all cases deep levels are responsible for the observed PL. However, it is difficult to distinguish emission possibly related to these elements and arising from native defects formed by implantation damage. In case of transitional metals or rare earth elements, the emission in the red part of the spectrum may be also caused by internal transitions in d or f shells. The element Fe has been used for compensating unintentional donor toward achieving semi-insulating (SI) GaN. In this vein Vaudo et al. [578] obtained SI freestanding substrates by hydride vapor phase epitaxy. Variable temperature resistivity measurements were used to determine the resistivity of an iron-doped GaN sample to be approximately 3 · 105 O cm at 250  C that extrapolates to 2 · 109 O cm at room temperature by linear fitting. The near-infrared photoluminescence at 1.6 K exhibited sharp emission at 1.3 eV, which is most likely associated with the 4 T1 ðGÞ ! 6 A1 ð6Þ internal transition of the Fe3 þ charge state. Refer to Section 4.9.6.2 for discussion of optical transitions associated with Fe-doped GaN. 4.9.4 Doping with Isoelectronic Impurities

Isoelectronic impurities, such as AsN and PN, are considered as promising candidates for the mixed anion nitride alloys such as GaAsxN1x and GaPxN1x. The dilute nitride semiconductors in terms of their electronic/optical properties are discussed in Section 1.5.5. In this section, the focus is on GaN doped with small concentrations of isoelectronic impurities. Of particular interest here is the formation of deep levels in these alloys, attributed to AsN and PN, which would induce discontinuous changes in the size of the bandgap [385]. 4.9.4.1 Arsenic Doping Arsenic has been used as a surfactant for the growth on GaN. In addition, As incorporated in GaN introduces a broad blue band peaking at about 2.6 eV [377,574,579–582]. This blue band is usually structureless, although a welldefined phonon-related fine structure associated with it has also been reported [574]. The intensity of the 2.6 eV band increases monotonically with increasing As doping [435,437,580,582], while the concentration of free electrons remains nearly unchanged [583,584]. Several reports noted that intensity of this band is nearly insensitive to temperature in the range from 10 to 300 K [579,580,583]. An activation energy of 50 meV has been estimated associated with quenching the 2.6 eV band between 100 and 300 K [579]. Note that the total decrease in intensity in the work of Li et al. [579] work was only about a factor of two. It can be argued the small decrease might be related to the variation in the capture cross section [142], and the real quenching of this band might take place above room temperature with an unknown activation energy.

4.9 Intentional Doping

The time-resolved PL study revealed that the decay of the 2.6 eV band is nearly exponential even at low temperatures, with a characteristic time of about 0.1 ms [579,581]. The lifetime decreased from 92 to 77 ns with increasing temperature from 8 to 100 K, and subsequently increased up to 148 ns with further increase of temperature up to 300 K [581].Such behavior is typical for internal transitions, rather than those involving the states associated with the conduction or valence bands. Chen and Skromme [574] observed the fine structure on the high-energy side of the blue band related to As implantation. The ZPL was observed at 2.952 eV, followed by two LO phonon replicas. Moreover, the spectrum contained peaks separated from the ZPL by 38 and 75 meV that were attributed to other lattice phonon modes [574]. If the 2.6 eV band were due to recombination of exciton bound to the As-related defect, as it was suggested in Refs [574,579,581], its localization energy would be estimated as 535 meV, the energy difference between the zero-phonon line and free-exciton line [574]. However, if transitions from the conduction band to the As-related defect were responsible for the 2.6 eV emission, the energy level of the As-related defect would be about 560 meV above the valence band. A more detailed analysis, including comparison of the PL lifetime with concentration of free electrons at different temperatures, is warranted for definitive identification of the 2.6 eV band in Asdoped GaN. 4.9.4.2 Phosphorus Doping Similar to the case of As, implantation of P in GaN results in a broad blue–violet luminescence band peaking at about 2.9 eV [377,574,585]. This band has been reported to quench above 100–150 K with an activation energy of about 168–180 meV [150]. Chen and Skromme [574] observed the fine structure of the blue–violet band in GaN:P. The zero-phonon line at 3.200 eV was followed by two LO phonon replicas (91 meV) and a few other phonon replicas (separated by 39, 59, and 77 meV from the zero-phonon line) attributed to various lattice phonon modes [574]. The energy difference between the zero-phonon line and the free-exciton line (287 meV) gives the exciton localization energy if the blue–violet band is due to the isoelectronic PN bound exciton [574]. 4.9.5 Doping with Rare Earths

Strong emission sources in the range of 1.3–1.54 mm are in some demand for optical communications on the basis of silica fibers. For these applications, wide bandgap semiconductors present an advantage in terms of room-temperature stability compared to systems based on narrow-bandgap semiconductors. In addition, wide bandgap nature of GaN and related compounds allow transitions due to rare earths in visiblewavelengths.Initialworkfocusedonrareearth ErinGaNandAlNlayers. Among them is one dealing with AlN doped with Er that produced a strong PL at 1.54 mm. Incorporation of levels in the range of 3 · 1017–2 · 1021 were achieved in MOMBE systems with a solid Er source [586]. A plethora of activity intended particularly for visible applications ensued, which is discussed in Volume 2, Chapter 5.

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j 4 Extended and Point Defects, Doping, and Magnetism 4.9.6 Doping with Transition Metals and Rare Earths

Transition metal impurities are interesting from two points of view. One deals with acceptor-like doping in the context of electronic properties. The other deals with magnetic properties when the transition element concentration is relatively high but still within the dilute limits as to not change the main structural nature of the GaN matrix. On the magnetic side, extending the carrier-mediated magnetic interaction from diluted GaAs:Mn to zinc blende GaN:Mn, Dietl et al. [587] predicted hightemperature ferromagnetism in diluted MnxGa1xN with x 0.05 and potential application of this material to spin transport electronics (spintronics). This started a flurry of activity in GaN doped with Mn and also other transition metals with impetus being provided by the lure of electronics using charge and spin, primarily the latter. 4.9.6.1 Manganese Doping for Electronic Properties In contrast to GaAs where Mn is a relatively shallow acceptor (EA 0.1 eV) providing high concentration of free holes at room temperature, the Mn level in GaN is almost in the middle of the bandgap [588–591], and one cannot expect detectable transport of holes bound to Mn acceptor even at very high concentrations of Mn. When the host material is doped with Mn, emission energies associated with the internal transitions within the Mn atom are modified by the crystal field. Moreover, transitions involving levels induced by Mn in the host material appear, the specifics of which depend on the interaction of the Mn atom with the host material. The nature of the internal transitions in Mn as an isolated atom as well as that affected by the tetrahedral crystal field such as that in GaN is shown in Figure 4.129 for the Mn2þ state. The Mn2þ (d5) levels depicted in Figure 4.129 are expected to hybridize with s–p bands of the host and broaden into d bands, still narrow. An insight can thus be gained by discussing the nature of intratransitions in free Mn atom followed by the same in isolated Mn2þ occupying a cation site in the host material. Transition metal elements have valence electrons corresponding to the 4s orbital, and have partially filled 3d shells, thus the name 3d transition metals (i.e., Mn with the shell structure of 1s2 2s2 2p6 3s2 3p6 3d5 4s2). The partially filled shells of transition metal ions warrant a discussion of protocol involved in labeling the ground and excited states. Consider a free or isolated ion all the electronic shells of which are filled except one which is the 3d shell in transition metal elements and 4f shell in rare earth elements. Suppose the electron levels in the partially empty shell are characterized by an orbital angular momentum l that assumes the value of 2 for 3d states (here n ¼ 3, and l ¼ n  1 ¼ 2). In the particular shell there are 2l þ 1 states [l, (l  1), . . . , 0, . . . , (l  1), l] (assign a letter designation for each of lz) each of which can have two electrons, one with spin up and one with spin down, which would result in 10 states for the 3d shell. If the electrons were not to interact with each other, the ionic ground state would be degenerate. However, this degeneracy, albeit not completely, is lifted by electron–electron Coulomb interaction and electron spin–orbit interaction. The lowest levels after the degeneracy is lifted are governed by a simple set of rules, Russel–Saunders coupling (or LS coupling) and Hund’s rules, which come

4.9 Intentional Doping

about as a result of complex calculations [592]. The former rule states that the Hamiltonian is commutative with the total electronic spin angular momentum (S), orbital angular momentum (L), and the total electronic angular momentum (J ¼ L þ S). Because the filled states have zero orbital spin (L ¼ 0), the eigenvalues determine the quantum numbers that in turn describe the configuration of the partially filled shell and the ion. The latter rule has three components, one of which states that the electrons that lie lowest in energy have the largest total spin while adhering to the exclusion principle. This means, for example, that all the spin-up electrons must occupy the partially empty shell while adhering to the exclusion principle. The second Hund’s rule prescribes that the total angular momentum L of the lowest lying states has the largest value without violating the first Hund’s rule and the exclusion principle. The value is equal to the largest magnitude that lz can have, which means that the first electron will go into the state with the largest |lz|. Because the first Hund’s rule indicates that the second electron must have a spin to maximize the spin, it must go into the second state with the same spin as the first. Continuing on with P this rule in mind, the value of L can be calculated using L ¼ j lz j. With the first and second Hund’s rules, one can determine the values of S and L, leaving (2L þ 1)(2S þ 1) states that can be further configured according their total angular momentum J. The third rule then helps determine the J values as J ¼ |L  S| for n  (2l þ 1) and J ¼ |L þ S| for n (2l þ 1) or J ¼ (L  S#), and (L þ S"). The 3d band of the Mn2þ ion is exactly half-filled with five electrons among the 10 available states, with a gap between the up-spin (") occupied states and empty down-spin (#) states. For other transition metals, such as Fe, Co, Ni, one of the bands is usually partially filled (up or down), as shown in Figure 4.128. Table 4.10 shows the oxidation and charge states for some of the transition metals in ZnO and GaN [593]. The TM–d bands of the transition metal hybridize with the host valence bands (N–p bands in GaN) to form the tetrahedral bonding. This hybridization gives rise to the exchange interaction between the localized 3d spins and the carriers in the host valence band. In the simplest of pictures, the s-band of the conduction band does not mix with the TM–d bands, but it is still influenced by the magnetic ion. For an element with five 3d electrons (Mn), it means that all five have electrons that would have spin up. The total spin S is calculated at S ¼ (1/2)(n#  n") for which Mn (3d) would be (1/2)(5) or 5/2 because all d shell electrons have the same spin. In Fe, however, there are six 3d electrons and one of them would have to have a spin-down configuration. In this case the total spin would be 1/2(5  1) ¼ 2. The methodology for arranging the electrons in the d shell for transition metals including orbital angular momentum, spin angular momentum, total momentum, and the designation for each of the available 10 d shell states is shown in Table 4.11. Because the electronic shell structure of rare earths is not as readily in the mind of the casual reader, it is provided in Table 4.12. The same is tabulated in Table 4.13 for rare earth elements in terms of the 4f shell configuration using the Hund’s rules discussed above. As mentioned in the preceding paragraph, the element Mn is a unique case in that it has only five electrons in its 3d shell, half of all the available states, which are all in

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j 4 Extended and Point Defects, Doping, and Magnetism V

Cr

Mn

Fe

Co

Ni

4s23d3

4s13d5

4s23d5

4s23d6

4s23d7

4s23d8

Cu 4s13d10

4s 3d 3 p6 3 s2 2 p6 2 s2 1 s2

Figure 4.128 Electronic configuration of the 3d-states and 4ssates of transition metal elements (from V to Cu).

their ground state, following the Hund’s rule, which calls for all electrons to maximize the spin angular momentum ("""""). As tabulated in Table 4.11, the electrons in their ground d shell (in Mn 3d5 state) have their orbital angular momentum L ¼ 0 and spin angular momentum S ¼ (1/2)(5) ¼ 5/2 (the orbital angular momentum L is quenched). The excited states of the d shell electrons would have L ¼ 1, 2, 3, and 4 values and are designated by the letter, in the same order P, D, F, G. For electron to make the transition to one of these excited states, one electron must flip its spin (""""#) that changes the spin quantum number to S ¼ (1/2)(3) ¼ 3/2. In the notation of spectroscopy, the states are labeled with left superscript 2S þ 1 XJ . Because the ground state has S ¼ 5/2 and L ¼ 0, the nomenclature used is 6 S5=25=2 . Similarly, L ¼ 1, 2, 3, and 4 states (with S ¼ 3/2) are labeled as 4 G, 4 P, 4 D, and 4 F. Among these excited states, the 4 G (S ¼ 3/2, L ¼ 4) level has the lowest energy as shown in Figure 4.129. The transition from the 6 S ! 4 G dominates the optical spectra involving a free Mn atom. Because of this reason, the S ¼ 1/2 states ("""##), where an additional electron flips its spin, which lie above the 4 G level, are not discussed. Let us now turn our attention to optical properties of Mn-doped GaN. Some background information is necessary to facilitate the discussion as well as getting

Table 4.10 Expected oxidation and charge state of some candidate transition metals in ZnO and GaN [593].

ZnO Acceptor (negative charge) Neutral Donor (positive charge) Double donor (2þ charge)

3d3

3d4

Cr3þ Mn4þ

Cr2þ Mn3þ Fe4þ

3d5

3d6

Crþ Mn2þ Fe3þ

Mnþ Fe2þ

GaN

Acceptor Neutral Donor

1 2 3 4 5 4 6 7 8 9 10

lz ¼ 1 " " " " " " "# "# "# "#

lz ¼ 2

" " " " " " "# "# "# "# "# " " " " " " "# "# "#

lz ¼ 0

The up and down arrows represent the spin up and spin down. P L ¼ j lz j ¼ jðl  2 Þ þ ðl  1 Þ þ ðl0 Þ þ ðl þ 1 Þ þ ðl þ 2 Þj The label name X is determined as follows: L ¼ 0; 1; 2; 3; 4; 5; 6 X ¼ S; P; D; F; G; H; I

Cu,Zn

Cr,Mn (3d5) Mn (3d4) Fe Co Ni

Sc Ti V

Element

d shell electrons (l ¼ 2) n

" " " " " " "# "#

lz ¼ þ 1

" " " " "#

"

lz ¼ þ 2 1/2 1 3/2 2 5/2 2 2 3/2 1 1/2 0

S ¼ (1/2)(n"  n#)

Table 4.11 The electronic configuration for the d shell in transition metals, and the relevant spin, orbital, and total angular momentum along with the nomenclature used.

2 3 3 2 0 2 2 3 3 2 0

L¼j

P lz j 3/2 2 3/2 0 5/2 0 4 9/2 4 5/2 0

J ¼ (L  S"), and (L þ S#)

D3=23=2 F2 4 F3=23=2 5 D0 6 S5=25=2 5 D0 5 D4 4 F9=29=2 3 F4 2 D5=25=2 1 S0 3

2

Symbol x in the form 2Sþ1XJ

4.9 Intentional Doping

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j 4 Extended and Point Defects, Doping, and Magnetism Table 4.12 The electronic shell structure of some of the rare earth elements that are pertinent to the content of this text.

Number

57 58 59 60 61 62 63 64 65 66 67 68 69 70

Element

K

L

M

N

O

P

Q

2 sp 26 26 26 26 26 26 26 26 26 26 26 26 26 26

3 spd 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10

4 spdf 2 6 10 2 6 10 2a 2 6 10 3 2 6 10 4 2 6 10 5 2 6 10 6 2 6 10 7 2 6 10 7 2 6 10 9a 2 6 10 10 2 6 10 11 2 6 10 12 2 6 10 13 2 6 10 14

5 spdf 26126-26-26-26-26-26-26126-26-26-26-26-26--

6 spdf 2 2 2 2 2 2 2 2 2 2 2 2 2 2

7 s

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

1 s 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Note that Eu and Gd both have seven electrons in their 4f shells. a Note irregularity.

acquainted with the terminology used. When Mn is introduced into GaN, the neutral configuration of Mn is Mn3þ (3d4) or (A0) when viewed as replacing Ga3þ in the lattice. However, in a II–VI material only the 4s2 electrons would be needed for bonding and as such the Mnþ2(3d5) configuration would be the neutral state. Owing to the metal oxide compound terminology, this state is also called the oxidation state. The configuration of Mn as an ionized acceptor is Mn2þ(3d5), with five tightly bound electrons in the Mn d shell. The neutral configuration of Mn3þ(A0) may be realized in two ways, as Mn3þ(3d4) with four electrons localized in the Mn d shell [594] or as an Mn2þ(3d5) þ hole with five electrons in the Mn d shell and a band hole weakly bound in a delocalized orbit [595]. When an Mn atom is substitutionally placed into a tetrahedral crystal of the II–VI type or said to be in its oxidation state (to be more descriptive Mn2þ or Mn2þ(3d5), which also indicates that only two 4s2 electrons participate in bonding), the four neighboring anion atoms exert a crystal field. This field causes the ninefold degenerate 4 G state to split into 4 A1 (non degenerate), 4 E (twofold degenerate), 4 T2 (threefold degenerate), and 4 T2 (threefold degenerate) states. Of these, 4 A1 and 4 E states nearly coincide and are hardly affected by the crystal field as shown in Figure 4.129. In the notation used to describe the intra-Mn2 þ transitions, the ground state label 6 S gives way to 6 A1 that is spherically symmetric and nondegenerate. The calculations show that the crystal field lowers the energies of both the 4 T2 and 4 T1 states. Transitions between the 6 S (S ¼ 5/2, L ¼ 0) and any of the excited states (S ¼ 3/2, L ¼ 1, 2, 3, 4) in free Mn atoms are forbidden because DS ¼ 0 spin parity requirement is

Tb Dy Ho Er Tm Yb

lz ¼ 2 " " " " " " " "# "# "# "# "# #"

lz ¼ 3

" " " " " " " "# "# "# "# "# "# "#

f shell electrons (l ¼ 3) n

1 2 3 4 5 6 7 8 9 10 11 12 13 14 " " " " " " " "# "# "# "# "#

lz ¼ 1

" " " " " " " "# "# "# "#

lz ¼ 0

" " " " " " " "# "# "#

lz ¼ þ 1

" " " " " " " "# "#

lz ¼ þ 2

" " " " " " " "#

lz ¼ þ 3 1/2 1 3/2 2 5/2 3 7/2 3 5/2 2 3/2 1 1/2 0

S ¼ (1/2) (n"  n#) 3 5 6 6 5 3 0 3 5 6 6 5 3 0

P L ¼ j lz j

The up and down arrows represent the spin up and spin down. Note that both Eu and Gd have seven 4f shell electrons. P L ¼ j lz j ¼ jðl  Þ þ ðl  2 Þ þ ðl  1 Þ þ ðl0 Þ þ ðl þ 1 Þ þ ðl þ 2 Þ þ ðl þ Þj The label name x is determined as follows: L ¼ 0; 1; 2; 3; 4; 5; 6 X ¼ S; P; D; F; G; H; I

Ce Pr Nd Pm Sm Eu,Gd

element

Table 4.13 The electronic configuration for the f shell in transition metals, and the relevant spin, orbital, and total angular momentum along with the nomenclature used.

5/2 4 9/2 4 5/2 0 7/2 6 15/2 8 15/2 6 7/2 0

J ¼ (L  S"), and (L þ S#)

F5=25=2 H4 4 I9=29=2 5 I4 6 H5=25=2 7 F0 8 S7=27=2 7 F6 6 H15=215=2 5 I8 4 I15=215=2 3 H6 2 F7=27=2 1 S0 3

2

Symbol x in the form

2S þ 1

XJ

4.9 Intentional Doping

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j 4 Extended and Point Defects, Doping, and Magnetism 4F 4D 4P

CB (S = 3/2,L = 2,3,4)

1, 4s 2

4G

T2 3d 4

1.5 eV E 1.8 eV

6S

VB

(S=5/2,L=0)

Isolated Mn atom

Isolated Mn 3d 4 atom

Figure 4.129 A schematic diagram showing the splitting of the lowest excited states of the 3d5 level ð4 GÞ relative to the ground state ð6 SÞ for an Mn þ þ (or Mn2 þ (d5)) ion in isolated case (left) and in the presence of a tetrahedral crystal field. The arrows indicate possible intra-Mn transitions. The picture is similar for the Mn3 þ

Mn+++ (Mn3+) in a tetrahedral crystal field

shell. The transition with energy of around 1.8 eV in GaN is assigned to the direct emission of holes from Mn3 þ acceptors to the valence band, and the transition observed at around 1.5 eV is assigned to the internal spin-allowed 5 E ! 5 T transition of the deep neutral Mn3 þ state.

not satisfied. For Mn2þ isolated ions in the crystal, the selection rules are relaxed by the lack of inversion symmetry and crystal field. Therefore, during transitions from 6 A1 ground state to the excited states, derivatives of the 4 G states become possible. This is one reasonwhyadifferentnomenclature,fromthatusedforintratransitions infree Mn, is used for isolated Mn2 þ ion. Of the possible transitions, 6 A1 ! 4 T1 has the lowest energy and therefore constitutes the most important transition. The energetics of this transition are discussed in GaN lattice environment later in this section. As mentioned above, the Mn2þ ion is ideally suited in conjunction with discussions dealing with II–VI materials wherein Mn donates two of its electrons (4s2) to bonding which statistically in turn spend more of their time in the orbitals of the O atom making it O2. However, in III–V compounds such as GaAs and GaN, three electrons are needed for forming bonds with anions. This means that either one of the 3d5 electrons would have to be given up to satisfy bonding or an electron would be taken from a donor if the semiconductor contains some, which all compound semiconductors do to varying degree. The former gives rise to an Mn3þ(d4) ion. This further means that both Mn2þ and Mn3þ configuration would be present simultaneously and both must be considered. The Mn3þ configuration has intratransitions that are similarly split due to the crystal field exerted on the ion. In Td symmetry that is applicable to GaN and GaAs, electron spin resonance (ESR) measurements have been undertaken to investigate the splitting of the M3þ levels.

4.9 Intentional Doping

Graf et al. [590,591] have demonstrated that holes bound to Mn in GaN:Mn are located in the d-shell, in contrast to GaAs [596]. The transition associated with Mn3þ and Mn2þ levels, also referred to as the Mn3þ/2þ acceptor level, is located at 1.8 eV above the valence band to an accuracy of about 0.2 eV. The majority of Mn is present in the neutral Mn3þ state in (Ga,Mn)N with 0.2–0.6% of Mn grown by plasma-enhanced MBE. The deep level of Mn states would hinder the presence of free electrons or holes in (Ga,Mn)N. Graf et al. also observed the characteristic absorption spectrum with the ZPL at 1.42 eV, same as that observed by Korotkov et al. [588,589], who inferred their results from internal transition 5 E ! 5 T in Mn3 þ (d4). Transition of holes from Mn3 þ to the valence band can also be seen in the absorption spectrum at photon energies above 1.8 eV [590,591]. Note that when the GaN:Mn sample is codoped with Si, electron is captured by Mn acceptor, converting it from the Mn3 þ (d4) state into Mg2 þ (d5) state. Attempts have been made to determine the nature of intraion levels in the GaN environment. One such investigation is the X-ray absorption spectroscopy (XAS). In Mn L3,2 X-ray absorption, 2p core electrons are excited to unfilled 3d states; therefore, this technique is a direct probe of the magnetically active Mn 2d band. Mn L3,2 X-ray absorption is also highly sensitive to the oxidation state of 3d transition metal ions, and thus can yield valuable information when applied to transition metal doped semiconductors. X-ray absorption in (Ga,Mn)As and (Ga,Mn)N showed that the hybridization between Mn states and charge carriers is reduced in (Ga,Mn)N, which is consistent with the absence of ferromagnetism order across the sample [597]. Similar conclusion was drawn recently from electron spin resonance studies of (Ga, Mn)N. The multiplet structures in XAS is characteristic of a mostly d5 ground state, indicating predominantly Mn2þ impurities [597]. EPR is a technique that can be brought to bear to determine the energy levels associated with magnetic ions in the GaN environment. Wolos et al. [598] carried out EPR magnetization and optical absorption studies in bulk GaN crystals doped with Mn and some samples codoped with Mg acceptors. They concluded that the charge state of the Mn ion in GaN depends on the Fermi level. In n-type samples, Mn is an ionized acceptor A center of Mn2þ(d5). However, in the highly resistive samples, with lowered Fermi level, Mn is most probably in neutral configuration, A0, in the form of either a localized Mn3þ(d4) or a delocalized Mn2þ(d5) þ hole center. Optical absorption spectra of Mn-doped and Mn/Mg codoped samples show typical absorption bands related to Mn. Those authors interpreted these as arising from photoionization of Mn2þ(d5) to GaN conduction band in n-type samples and photoionization of neutral Mn A0 to GaN valence band in highly resistive samples. The location of the Mn acceptor level was derived as 1.8  0.1 eV below the bottom of the GaN conduction band. Note that the accuracy of these measurements is at best about 0.2 eV. Given the fact that GaN has a low temperature bandgap of 3.5 eV, this result agrees with the previous discussion [590]. The observations were interpreted with the help of a coordination diagram shown in Figure 4.130 describing the valence band to Mn2þ level and Mn2þ level to the conduction band. Moreover, Graf et al. [590,591] reported the Mn3þ/2þ level to be 1.8 eV above the valence band maximum. Relevant to the dilute magnetic semiconductor properties, the magnetization data reveal

j1053

j 4 Extended and Point Defects, Doping, and Magnetism 5

A 0+e CB

4

Energy (eV)

1054

Mn 2+(d5)+h ν 3

A -(Mn 2+)

GaN 2

A 0+hν 1

A 0+eCB

E CB opt

Mn 2+(d5)+holeVB

A 0+e VB

E VB opt

0

Q1

Q2

Configuration coordinate Figure 4.130 Configuration coordinate diagram showing Mn-related optical transitions in GaN where parabolas represent the energy levels. The optical transitions occur without changing the configuration coordinate. For example, at point Q1 the direct E VB opt optical transition represents A0 þ hn ! Mn2 þ

ðd5 Þ þ holeVB and at point Q2 the direct E CB opt transition represents Mn2 þ ðd5 Þ þ hn ! 0 A þ eCB . The thermal energies are determined by the minima in the configuration parabola, which are indirect. The relaxation energy is then the difference between the optical transition energy and the thermal energy [598].

Brillouin-type magnetization with Mn2þ(d5) (Mn spin S ¼ 5/2), whereas the highly resistive samples show magnetic anisotropy characteristics in n-type crystals. The highly resistive samples show magnetic anisotropy characteristic for nonspherical transition metal configurations. As mentioned in the previous paragraph, the d5 ground state of Mn was also observed by Edmonds et al. [597] via their Mn L3,2 X-ray absorption study. Wolos et al. [599] also investigated the magneto-optical properties of intracenter absorption band related to neutral Mn acceptor in GaN bulk crystal. The band is built of a ZPL at 1.4166 eV followed by GaN phonon spectrum. No splitting of the ZPL was observed. The only characteristic feature recorded for high magnetic fields was a steplike behavior of the spectral position of ZPL measured in Faraday configuration. The step appeared at a magnetic field of about 7 T, with a shift in energy of about 1.3 meV. The behavior of the band in magnetic field may be reasonably explained in terms of a model of Mn3þ (d4) ion in trigonal crystal field, undergoing Jahn–Teller distortion and spin–orbit coupling, strongly supporting the localized character of Mn neutral acceptor in GaN. Turning our attention to standard photoluminescence measurements, Korotkov et al. [444,600] observed PL transition due to Mn acceptor. It represents a broad structureless band with a maximum at 1.27 eV and FWHM of about 0.26 eV. With increasing temperature from 13 to 300 K, the Mn peak position remained nearly independent of temperature. Transient PL study of this band revealed a very slow

4.9 Intentional Doping

exponential decay of the PL intensity with the characteristic lifetime of about 8 ms [588]. A slow exponential decay of PL is typical of internal transitions. Consequently, it is most likely that the 1.27 eV band is caused by transitions of holes from the excited states of Mg3þ, located close to the valence band, to its ground 5 T level. The Mn-related optical properties in GaN have also been studied as a function of Mn concentration [601] as determined by the so-called elastic recoil detection method. In this experiment, the Mn doping level was varied between 5 · 1019 and 2.3 · 1020 cm3. The optical absorption spectra recorded at 2 K of nominally undoped GaN, (Ga,Mn)N, and (Ga, Mn)N:Si are replotted in Figure 4.131. The absorption peak at 1.414 eV can be attributed to an internal 5 T2 ! 5 E transition of the neutral Mn3 þ state. The intensity of this peak was found to scale with the Mn3 þ concentration in transmission spectroscopy measured at 2 K. In addition, the CL measurements showed that Mn doping concentrations of around 1020 cm  3 reduced the near band

GaN Mn (2.3 x 1020 cm-3)

Absorbance (au)

GaN Mn (1.1 x 10 20 cm-3 )

GaN Mn (5 x 1019 cm-3)

GaN Mn Si

GAN(u) 1.40

1.42

1.44

1.46

1.48

1.50

1.52

Energy(eV) Figure 4.131 Optical absorption spectra at 2 K for MBE-grown undoped, Mn-doped, and Mn þ Si codoped GaN. The Mn concentration varied from 5.4 to 23 · 1019 cm3. The ZPL is at 1.414  0.002 eV. Courtesy of Gelhausen et al. [601].

1.54

j1055

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j 4 Extended and Point Defects, Doping, and Magnetism edge emission intensity by about 1 order of magnitude. A complete quenching of the donor–acceptor-pair band at 3.27 eV and strong decrease of the yellow luminescence centered at 2.2 eV were attributed to a reduced concentration of VGa. In the infrared spectral range of 0.8–1.4 eV, three broad Mn doping related CL emission bands centered at 1.01, 1.09, and 1.25 eV were observed, the genesis of which may be from deep donor complexes generated as a result of the heavy Mn doping. The electronic states of (Ga,Mn)N codoped with Mg acceptors have also been investigated utilizing both PL [602] and PLE [603] spectroscopies. It was found that by adding Mg acceptors in (Ga,Mn)N, the weak Mn-related PL band at 1.3 eV was quenched. The change in PL spectra indicates that the Mg addition stabilizes the Mn4þ charge state by decreasing the Fermi level. A series of sharp PL peaks were observed at 1 eV in codoped epilayers and were attributed to the intra-d-shell transition 4 T2 ðFÞ  4 T1 ðFÞ of Mn4 þ ions and their phonon replicas are believed to be involved as well. The relative intensities of the sharp peaks were found to be strongly dependent on the excitation wavelength, indicating that the optically active Mn4 þ centers involved in the separate peaks are different. The temperature dependence of the PL spectrum suggests the presence of at least three distinct Mn4 þ complex centers. On the basis of the optical studies of Han et al. [603], the energy levels of Mn4þ ions in the GaN band diagram can be given as drawn in Figure 4.132. The energy levels in Figure 4.132 can be used to understand the PLE spectrum (shown in Figure 4.133) of the Mn4 þ ½4 T2 ðFÞ  4 T1 ðFÞ
transition, which reveals intracenter excitation processes via the excited states of Mn4 þ ions. The contribution involving the contribution by the 1.1 eV transition using the Luckovsky fit [604] is shown in dashed lines in Figure 4.133a. PLE peaks at 1.79 and 2.33 eV shown in Figure 4.133b, after removing the Luckovsky fit, are attributed to the intra-d-shell transitions from the ground 4 T1 (F) state to the 4 T1 (P) and 4 A2 (F) excited states,

Conduction band Mn4+(4A 1(F))

2.3 eV

Mn4+(4T1(P))

1.8 eV Mn4+(4T

2(F))

3.3 eV

1.0 eV Mn4+/3+

Mn4+(4T1(F))

1.11 eV (Lucovsky fit) Mg3+/2+

Valence band Figure 4.132 Energy levels of Mn4þ ions in wurtzite GaN. The energies are given for a crystal temperature of 20 K. Courtesy of Han et al. [603].

4.9 Intentional Doping

Lamp background

PL Intensity (arb. units)

Calibrated PLE response 10

Lucovsky fit

E =1.11 eV

T=20K

th

1 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Energy (eV)

Figure 4.133 Calibrated PLE spectrum of (Ga,Mn)N codoped with Mg at 20 K; (b) the 20 K PLE spectrum after subtracting the Lucovsky fit. Courtesy of Han et al. [603].

respectively. In addition to the intrashell excitation processes, a broad PLE band involving charge-transfer transition of the Mn4 þ /3 þ (or 4 þ /3 þ ) was observed. As determined from the onset of this PLE band, the position of the Mn4 þ /3 þ deep level is 1.11 eV above the valence band maximum, which is consistent with ab initio calculations by Gertsman [605], where the Mn4 þ /3 þ transition level is calculated to be 1.1 eV above the VBM (see Figure 4.135a for GaN transition metal transition levels). PLE results further indicate that 4 þ is the predominant oxidation state of Mn ions in p-type GaN:Mn when the Fermi energy is below 1.11 eV, as measured from the top of the valence band maximum. The optical and electrical properties of GaN films implanted with Mn have been jointly studied [606] by means of optical transmission spectra, microcathodolumines-

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j 4 Extended and Point Defects, Doping, and Magnetism 1A

1S

1

1T 1G

1E

Cr 4+ 3P

1D

2

A1

Cr 4+

V 3+

T2

4.303

3.789

E

4.264

3.743

T1

3.316

3.050

T2

3.174

2.773

3.145

2.743 2.645 2.537

1T

1

3T

1

E

1A

1

T1 A1 A1 T2

V 3+

1T 1A

1

1E 3T

1

1E 3F 3T

2

3A

2

2

2.827

A1 E

2.740

A1 T1

1.958

1.580

T2 E

1.917 1.670

1.537

T1 E

1.208

0.941

T2 A2

1.188

0.925

T2

0 eV

0 eV

1.658

Figure 4.134 Energy level before and after crystal field splitting for the d2-type V3þ and Cr4þ ions on the Ga site in GaN calculated by a numerical diagonalization of the static crystal field energy matrices, neglecting the effects due to the wurtzite trigonal field [608].

cence spectra, capacitance–voltage and capacitance–frequency curves, temperature dependence of resistivity, and DLTS with both electrical and optical injection. Optical transmission on n-type GaN samples implanted with high doses of Mn (3 · 1016 cm2) shows that Mn forms a deep acceptor near Ev þ 1.8 eV. The Mn complexes formed with native defects are deep electron traps with a level near Ec  0.5 eV, which are most likely responsible for a strong blue luminescence band with energy near 2.9 eV.

4.9 Intentional Doping

3+/2+ 3

2+/+

3+/2+ 4+/3+ 3+/2+

2

4+/3+

GaN Eg

Energy (eV)

3+/2+ 3+/2+

4+/3+

3+/2+ 2+

3+/2+

4+/3+ 1 4+/3+

5+/4+ 4+/3+ 0 Ti

V

Cr

Mn

Fe

Co

Ni

Cu

(a)

Figure 4.135 (a) Charge transfer levels of transition metal impurities in GaN. Data in this figure include the results from Gerstman et al. [605] (n), Mahadevan and Zunger [618,619] (?), Heitz et al. [620] (?), Graf et al. [590] (*), Baur et al. [611] (n), Van Schilfgaarde and Myrasov [621] (s), and Han et al. [603] ( ). (b) Calculated charge transfer levels of isolated 3d TM defects in group III

nitrides, AlN, GaN, and InN. The valence and conduction band edges determined via the Langer-Heinrich rule using Cr4þ/3þ as the reference level are also shown to get a glimpse of transition element levels in the nitride host. Note that the 1.9 and 6.2 eV bandgaps for InN and AlN, respectively, have not been updated to 0.8 and 6 eV. Courtesy of A. Blumenau and Ref. [605].

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j 4 Extended and Point Defects, Doping, and Magnetism It is worth noting that, apart from Mn forming deep-acceptor levels in (Ga,Mn)N as reported in the literature, an optical study by Yoon et al. [607] pointed out the presence of an Mn acceptor level at about E ¼ 3.08 eV for a low Mn content (Ga,Mn)N. This suggests that the Mn-bound holes in III nitride exhibit the shallow impurity states. Arrhenius plots of the intensity of the Mn acceptor give an activation energy of E ¼ 310 meV, indicating that the observed thermal quenching of the Mn-related PL peak is due to the dissociation of an acceptor-bound hole. The observation by Yoon et al. [607] is consistent with predictions based on the effective mass theory of the shallow hydrogenic impurity state model. 4.9.6.2 Other TM Doping for Electronic Properties While Mn, for traditional reason, has been investigated in detail in many semiconductors with GaN not providing the exception, other transition and rare earth elements may turn out to be more, if not just as, important. Among the transition elements, V, Cr, Fe have also been investigated and employed for possible ferromagnetism. Baur et al. [608] discussed the optically active transitions associated with these three transition metals in unintentionally doped GaN (Fe3þ and Cr4þ in OMVPE films and V3þin HVPE layers). In terms of the interaction with the host material, hybridization through the second nearest neighbor interaction between the Cr 3d levels and Ga 4s levels leads to gap states, the details of which can be found in Ref. [609]. The focus here is not to entertain the question as to what the source of unintentional impurities is, but rather to what the configuration and energy levels associated with d–d transitions are. Serendipitously, many GaN samples studied by Baur et al. [610] have shown an infrared PL band with a ZPL at 1.30 eV. This line has been previously assigned to the 4 T1 ðGÞ ! 6 A1 ðSÞ crystal field transition of Fe3 þ impurities (internal transition). Moreover, a similar luminescence line has been reported in AlN also and used in conjunction with GaN to determine the band discontinuities [611], as discussed in Volume 2, Chapter 4. The luminescent lifetime of the Fe3 þ PL in GaN is considerably long at 4.5 ms [612], because the radiative transition involved is spin forbidden. Optical properties of p-type GaN epilayers prepared by metalorganic chemical vapor deposition and subsequently implanted with Fe þ ions have also been investigated [613]. The implant activation was accomplished at 850  C for 30 s. Optical transitions observed during photoluminescence measurements at 2.5 eV and around 3.1 eV have been attributed to the presence of Fe. The photoluminescence peak at 2.5 eV has been identified as a donor–Fe-acceptor transition and that at 3.1 eV as a conduction band–Fe acceptor transition. In investigating V in GaN, Baur et al. [608] observed a new PL band with a ZPL at 0.931 eV with a lifetime less than 0.4 ms. This particular band was observed in most GaN HVPE samples but not in the OMVPE samples investigated. At elevated temperatures a “hot” ZPL appeared 13 cm1 above the main ZPL. Some phonon replicas of the 0.931 eV ZPL were in proximity in energy to intrinsic lattice phonons of GaN [614]. This particular band has been assigned to neutral vanadium impurities with a configuration of V3þ (3d2). Due to the relative weakness of the 0.931 eV PL line,

4.9 Intentional Doping

Baur et al. were not able to measure the excitation spectra. In addition, inherently thick HVPE layers showing the V3þ PL line also exhibited absorption bands at 2.54 and 2.74 eV. Baur et al. [608] also investigated transitions associated with Cr in GaN with particular characteristics of the PL band, with a ZPL at 1.19 eV appearing in all OMVPE and most HVPE GaN layers. Well-resolved TO phonon replicas appeared at about 550 cm1 below the ZPL. At elevated temperatures a hot ZPL appeared at 25 cm1 above the main ZPL. The PL band in question could be very efficiently excited in the red spectral range as well as with various Krþ laser lines ranging between 2.4 and 3.5 eV albeit with the efficiency being at least a factor of 20 less than in the peak of the band. In comparing OMVPE samples with different positions of the Fermi level EF (induced by changing the doping level such as n-type undoped, lightly and heavily Mg acceptor doped), one observes a dramatic increase in excitation efficiency with a drop of EF from near the conduction band to a position deep in the forbidden gap. The genesis of transitions associated with V and Cr warrant further discussion. As the case made for Mn, transition element ground and excited states are split further by the crystal field present. Typically, the transitions to the ground state in crystal field configuration are active. The PL bands associated with V and Cr discussed above can be accounted for by intraelement transitions that are termed as d–d transitions. The PL band observed at 0.931 eV at 2 K can be ascribed to V3þ(3d2) based primarily on the associated ZPL doublet structure and a comparison with the well-understood PL spectra of V3þ in other semiconductors such as GaAs, GaP, and InP [615,616]. The bands observed in those semiconductors result from the 3 T2 ðFÞ ! 3 A2 ðFÞ crystal field transition of V3 þ . Within 3 T2 ðFÞ the first-order spin–orbit splitting is quenched by a moderately strong dynamic Jahn–Teller effect [615]. In this case the four 3 T2 ðFÞ spin–orbit levels, namely, A2, T2, E, and T1 levels, shown in Figure 4.134, merge into a lower lying triply and a higher lying sixfold quasi-degenerate levels separated by a second-order spin–orbit splitting. The cold and the hot ZPL observed correlate well with transitions from these two levels to the T32 A2 ðFÞ ground state (1.88 and 0.925 eV for Cr4 þ and V3 þ , respectively). The splitting between the two lines amounts to 10, 15, and 12 cm  1 for GaAs, GaP, and InP, respectively, appears to be D ¼ 13 cm1 in GaN, and falls into the range observed for V3þ in the conventional III–V semiconductors. Baur et al. [608] also noted that the trigonal field splitting due to the axial field in wurtzite GaN is not observable in PL, indicating that either the trigonal field is small or that such splitting is also quenched within 3 T2 ðFÞ by the Jahn–Teller interaction. The energetic position of the V3 þ 3 T2 ðFÞ ! 3 A2 ðFÞ PL band is to a first extent determined by the cubic crystal field parameter Dq that in GaN is 18% larger than in GaP [617]. For comparison, the 3 T2 ðFÞ ! 3 A2 ðFÞ main ZPL transition in ZnO occurs near 0.85 eV. However, the cation replaced by the transition elements is divalent in ZnO as opposed to trivalent in III–V semiconductors such as GaP:V3 þ and GaN:V3 þ in that the charge state of the host ion replaced (2 þ in ZnO) differs from the impurity charge state (3 þ ).

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j 4 Extended and Point Defects, Doping, and Magnetism While it is natural to ascribe the Cr4þ PL excitation band in GaN to the higher lying 1E(D) and 3Tþ (F) crystal field states of the 3d2 TM element configuration, the origin of the 2.54 and 2.75 eV absorption bands associated with V3þ is less evident. To gain some insight, Baur et al. [608] numerically diagonalized the static crystal field energy matrices, including Coulomb interaction (parameters B and C), the cubic crystal field (parameter Dq), and the spin–orbit coupling (parameter l). The results for V3þ and Cr4þ using the parameter sets listed in Table 4.14 are displayed in Figure 4.134. The results support the assignment of the Cr4þ PL excitation band to the 1 E(D) and 3 T1 (F) states. Further, they suggest that the 2.54 and 2.74 eV absorption bands are related to the 1 T þ (D), and the 3 T þ (P) states of V3 þ , respectively. Once the spin–orbit splitting of 3 T2 (F) predicted by static crystal field theory is known, the dynamic Jahn–Teller model developed earlier for this state can be applied. The dynamic Jahn–Teller model predicts a splitting, D, that matches the observed splitting for V3þ and Cr4þ in GaN. This further supports the Cr4þ assignments discussed above. Before closing the discussion of (Ga,Mn)N, the charge transfer levels of transition metal impurities in GaN reported in the literature reported by Gerstman et al. [605], Mahadevan and Zunger [618,619], Heitz et al. [620], Graf et al. [590], Baur et al. [611], Van Schilfgaarde and Myrasov [621], and Han et al. [603] are summarized in Figure 4.135a. These are of fundamental interest in interpreting and predicting the magnetic properties of GaN-based diluted magnetic semiconductors (DMSs. They are also naturally critical in interpreting the optical transitions in TM-doped GaN. To be complete, the calculated charge transfer levels of transition metal elements Ti, V, Cr, Mn, Fe, Co, and Ni in all the three nitride binaries (cubic AlN, GaN, and InN) obtained by Gerstman et al. [605] are shown in Figure 4.135b. Assuming that the Langer- Heinrich rule is valid, one can also estimate the valence band discontinuities, which leads to 0.91, 1.05, and 2.05 eV for cubic AlN/GaN, GaN/InN, and AlN/InN, respectively. In producing the figure, the Cr4þ/3þ donor level has been taken as the reference level for the alignment of the band edges of all host nitride materials. Figure 4.135b shows that the rule represents a reasonably good approximation, especially for the acceptor levels Co3þ/2þ and Ni3þ/2þ, and also for the donor level V4þ/3þ. The data of Figure 4.135b

Table 4.14 Crystal field parameters in units of cm1 for V3þ and Cr4 þ in GaN.

Cr4þ

k B C Dq

V3þ

Free ion

GaN

Free ion

GaN

11 77 1 040 4 264

115 800 3720 986

105 860 4128

89 810 3660 754

4.9 Intentional Doping

also include the experimentally determined charge transfer levels of transition elements in GaP reported in Refs [622,623] to make the point that the internal reference rule for, for example, determining band discontinuities, to be less reliable when the group III nitrides and other III–V semiconductors are combined, which is not all that atypical to do. The genesis for this inapplicability may have to do with the difference in electronegativity between nitrogen (3.04) and phosphorus (2.19) [624]. In summary, it is believed that a combination of optical, magneto-optical, EPR, ESR, electrical and magnetic measurements, not necessarily in that order, may be required for a full characterization of the state of these impurities in GaN, also applicable to ZnO, and their participation in any magnetization. 4.9.6.3 General Remarks About Dilute Magnetic Semiconductors Transition metal doped semiconductors, dating back to II–VIs such as (Zn–Mn)S, (Zn–Mn)Se, and (Cd–Mn)Te, [625] and later to arsenide-based III–Vs, such as In1xMnxAs [626] and Ga1xMnxAs [627] followed by GaN and ZnO, have been considered for magentoelectronics. If the magnetic moment of transition metal elements can be made aligned, the degree to which that alignment is controlled could pave the way for many device concepts. The II–VI system suffers from a low critical temperature ( 0 are called paramagnetic and electronic paramagnetism arises from, for example, atoms, molecules, and lattice defects, and unpaired electrons, causing a nonzero total spin. Free atoms and ions with a partially filled inner shell such as transition elements, ions isoelectronic with transition elements, rare earth, and actinide elements have unpaired electrons in those shells that lead to nonvanishing spin. Examples include V2þ, Cr2þ, Mn2þ, Fe2þ, Co2þ, and Ni2þ among the transition elements and Gd3þ among the rare earths, the latter of which is used very readily in semiconductors and attracted the most attention. The magnetization versus magnetic field H curves in these materials follow a linear relationship, as shown in Figure 4.136. When the magnetic moments of magnetic ions and electron spins are ordered, the material becomes ferromagnetic with nonzero magnetic moment (the susceptibility

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j 4 Extended and Point Defects, Doping, and Magnetism Ferromagnetic

Paramagnetic

Superparamagnetic

G

N

T

Figure 4.136 A cartoon showing ferromagnetic, paramagnetic, and superparamagnetic cases.

is positive and large), even for zero applied field, called the spontaneous magnetic moment. This alignment is broken due to thermal agitations above a certain critical temperature called the Curie temperature. The magnetization versus H curve for a ferromagnet is shown in Figure 4.136 also for temperatures below the Curie temperature. Above the Curie temperature an otherwise ferromagnetic material would behave as a paramagnetic material with linear dependence of magnetization on the H field. For completeness, the magnetic susceptibility in antiferromagnetic (AFM) materials is positive but small, and in ferromagnetic materials it is positive and very large. If the alignment is caused by exchange interaction with carriers, as in the case of GaAs:Mn, the control over the hole concentration would lead to control of the degree to which the material is ferromagnetic. The spin ordering can be simple or helical. If spins are antialigned in an alternating fashion, either simple ordering or canted ordering, the material is called antiferromagnetic. In transition metals having negative exchange energy such as Cr and Mn, electronic spins of adjacent atoms are held in opposite orientations that locally cancel each other. In Cr, which is body centered cubic, the body center atomic spins are directed opposite to those, but in equal number at the cube corners. In this particular case, the saturation magnetization would reduce as the temperature is increased up to a critical temperature called the Neel temperature beyond that the thermal agitations would render the material paramagnetic (in other words the thermal agitations break up the ordering) and thus magnetization would be a linear function of H field. Moreover, if portions of the material are ferromagnetic and others paramagnetic, the material is called superparamagnetic. Artistic views showing paramagnetic, ferromagnetic, and superparamagnetic cases are shown in Figure 4.136. In the paramagnetic case, the magnetic moments

4.9 Intentional Doping

associated with magnetic ions is randomly distributed. In the ferromagnetic case, the magnetic moments are aligned, the mechanisms for which are the topic of this section. In the superparamagnetic case, portions of the material is ferromagnetic and rest is paramagnetic. A schematic representation or M–H curves for diamagnetic, paramagnetic, and ferromagnetic cases is shown in Figure 4.137. In the paramagnetic case, the magnetization would eventually saturates as all the magnetic ions would have their magnetic moments aligned, to the extent possible, by the magnetic field. In the superparamagnetic case, the portions of the material are already ferromagnetic and for temperatures below the Curie temperature and TB (blocking temperature represents the temperature at which the metastable hysteretic response is lost), the M–H curves have the typical hysteretic shape as in the case of ferromagnetic materials. For temperatures above the Curie temperature but below TB, the M–H curves still have their hysteretic behavior but the curves do not saturate. Above both the Curie and TB temperatures, the material behaves as paramagnetic because the thermal agitations destroy any magnetic ordering.

Ferromagnetic M MS Mrem HC

•

•

Diamagnetic M

T TC the system is a disordered paramagnet, and for T < TC the system is a ferromagnet. We should mention that the main idea behind MFT is to replace all interactions to any one body with an average or effective interaction. This reduces any multibody problem, which is generally very difficult to solve exactly, into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a relatively low cost even at the expense of loosing some accuracy. Let us find l in terms of TC. If wparamag is the susceptibility, then the magnetization is given by M ¼ cparamag ðHa þ HE Þ in cgs units m0 M ¼ cparamag ðBa þ BE Þ

in SI units

ð4:89Þ

where Ha and Ba are the applied fields. The susceptibility of a paramagnet is given by the Curie law cparamag ¼

C : T

ð4:90Þ

Using Equations 4.88 and 4.89, we obtain MT ¼ CðH a þ lMÞ and c¼

M C ¼ Ha T  Cl

in cgs system:

ð4:91Þ

For TC ¼ Cl a singularity occurs in that w ! 1. For C < T < l, we have spontaneous magnetization and c¼

C T  TC

or

c¼

C T q

in cgs units;

ð4:92Þ

which represents the Curie–Weiss law. In fact, the Curie law is a special case of the more general Curie–Weiss law. It should be mentioned that y is often used for the Curie temperature as well. We should also mention that in an antiferromagnetic material, the Curie law takes the form c ¼ C=ðT þ T C Þ

or c ¼ C=ðT þ qÞ in cgs units:

ð4:93Þ

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j 4 Extended and Point Defects, Doping, and Magnetism In the above equation y can either be positive, negative, or zero. The case when y ¼ 0 corresponds to the case when the Curie–Weiss law equates to the Curie law depicted in Equations 4.76 and 4.90. A nonzero y implies that there is an interaction between neighboring magnetic moments and the material is only paramagnetic above a certain transition temperature. If, on the contrary, y is positive, the material is ferromagnetic below the transition temperature and the value of y corresponds to the transition temperature (Curie temperature, TC). If y is negative, the material is antiferromagnetic below the transition temperature (Neel temperature, TN); however, the value of y does not relate to TN. We should note that this equation is only valid when the material is in a paramagnetic state. Similarly, it is not valid for many metals as the electrons contributing to the magnetic moment are not localized. However, the law does apply to some metals, such as the rare earths, where the 4f electrons that create the magnetic moment are closely bound. Detailed calculations show that [628] c¼

C ðT  T C Þ1:33

;

ð4:94Þ

and

l¼

T 3kT ¼ : T C Ng 2 SðS þ 1Þm2B

ð4:95Þ

Let us apply what we learned so far to some known magnetic materials so as to gain an appreciation for the exchange interaction field BE. For Fe, TC ¼ 1000 K, g is about 2, and S is about 1 and l ! 5000, Ms 1700; thus BE ¼ lM ¼ 1700 · 5000 ¼ 8.5 · 106 Gauss. This exchange field in Fe is huge and much larger than the magnetic field due to ions in the crystal. A magnetic ion produces a field of BE mB/a3 103 G at the neighboring lattice point. From an experimental point of view, measured magnetization data are conveniently plotted following the method reported by Arrott [629] to determine the Curie temperature and susceptibility. The necessary analytical treatment used by Arrott is given here to facilitate the analysis of magnetization data. In the Weiss–Brillouin formalism, the magnetization is given by M ¼ Ms tan h

  mðH þ NMÞ ; kT

ð4:96Þ

where Ms is the spontaneous magnetization at absolute zero, m is the magnetic moment per atom, and N is the molecular field constant. This expression can be rewritten as mH mM þN ¼ tanh  1 ðM=Ms Þ: kT kT

ð4:97Þ

4.9 Intentional Doping

And for M Ms, the above expression can be expanded in a power series as

 mH mM M 1 M 3 1 M 5 þ þ þ ...: þN ¼ kT kT Ms 3 M2 5 M2

ð4:98Þ

Taking the first two of the series on the right-hand side, we can write m H e 1 1 þ M2 ; ¼ kT M Ms 3 M 3s

ð4:99Þ

where e ¼ 1(Tc/T ). At the Curie point (w1 ! 0) m0 H 1 1 M2 : ¼ kT c M 3 M3s

ð4:100Þ

Using M ¼ wH and in the limit of M ! 0, we can deduce from Equation 4.99 that 1 kT  N; ¼ c mM s

ð4:101Þ

and at the Curie point the left-hand side tends to zero, yielding Tc ¼

mN Ms : k

ð4:102Þ

Equation 4.99 can be rewritten as M2 ¼  3eM 2s þ

3mM 3s H : kT M

By plotting M2 versus H/M, one can determine w1 from the intercept with the horizontal (H/M) axis. Extrapolating w1 versus temperature and noting that w1 ! 0 at the Curie point leads to the Curie temperature with the aid of Equation 4.102. At Curie temperature, e ¼ 0, and thus M2 versus H/M curve goes through the origin, as shown in Figure 4.142. The Arrott plots are utilized in the discussion in Section 4.9.6.3. 4.9.6.4 General Remarks About Spintronics Having discussed the optical signature and likely energy levels caused by Mn in the gap of GaN and given a short introduction to the types of pertinent magnetism, let us now turn our attention to spintronics on the assumption that GaN can somehow be made ferromagnetic at high temperature. The use of spin or spin and electrons in a device has attracted much attention, spanning over several decades. While switching and amplification are performed by devices utilizing the charge nature of electrons, magnetic storage media relies on spin. A device combining the charge and spin nature of electrons or just the spin nature would represent a new realm. This realm can be implemented, depending on the application, using magnetic materials suitably integrated with semiconductors and diluted magnetic semiconductors that are alloys between a magnetic elements, such as Mn, Fe, Co, Cr to name a few, and a semiconductor that can inject coherent spin. In other words, DMS depicts a class of

j1075

j 4 Extended and Point Defects, Doping, and Magnetism M 2 (magnetization, abu)

1076

ε< 0 ε= 0 ε>0

H/M

0 FE

AFE

Figure 4.142 The Arrott plot, magnetization squared versus the magnetic field divided by magnetization below the Curie temperature ((e < 0), at the Curie temperature (e ¼ 0) and above the Curie temperature (e > 0). The inverse of magnetic susceptibility, w1, can be determined from the intercept of M2 versus H/ M curve with the horizontal (H/M) axis. Repeating the procedure versus temperature and noting that w1 ! 0 at the Curie temperature, one can obtain the Curie temperature. Practically, this is when the M2

versus H/M curve, which would form a straight line at that point, goes through the origin. If the extrapolations intercept the horizontal axis on the negative side, the sample is ferromagnetic (FE) or the measurement temperature is below the Curie point. If the intercept occurs on the positive side of the horizontal axis, the sample is antiferromagnetic or the measurement temperature is above the Curie point. If the extrapolation goes through the origin, the measurement temperature is identical to the Curie temperature.

semiconductors in which magnetic ions of the same chemical nature but different charge states coexist [630]. Using spin-up and spin-down state in a two-state (binary) system, information can be stored with electron spin. Spin information can be transported, as spin is associated with charge carriers, electrons in this case. At least in metal the spin relaxation length is much longer than the momentum relaxation length that relaxes the eventual device dimensions in a spin-based structure. Spin can be detected optically, through charge–spin coupling, and by polarizers and analyzers made of ferromagnetic materials. Use of spin and charge together could lead to storage and processing capability or use of spin in ferromagnetic semiconductor/ semiconductor heterostructures might pave the way to such applications as quantum computing at the so-called qubit level (quantum bit) [626]. Purported potential advantages are nonvolatility, increased data processing speed, and decreased power consumption, to cite a few [631]. The ultimate hope or fiction for applications is that new spin-based multifunctional devices such as spin-FETs, spin-LEDs, spin resonant tunneling device (RTD), terahertz frequency optical switches, modulators, encoders, decoders, and quantum bits for quantum computation and communication will someday be possible, perhaps even replace the all mighty MOSFET. It should be mentioned that magnetic metals are used for recording and storage. Ferromagnetic semiconductors are researched for expanding the application areas of spin.

4.9 Intentional Doping

Spin transport differs from charge transport in that spin is a nonconserved quantity in solids due to spin–orbit and hyperfine coupling. Spin and charge dynamics can be controlled by external electric and magnetic fields and illumination by light, which can be used to create new functionalities not feasible or ineffective with conventional electronics. The details of these approaches can be found in review articles [632,633]. To take advantage of the spin degree of freedom in semiconductors, spin-polarized carriers must be created, sustained, controlled, and of course detected. The most plausible means for generating spin polarization electrically is by injection of spin-polarized electrons. Doing so with ferromagnetic metal–semiconductor junctions has not been easy, most likely due to scattering at the magnetic metal (Schottky barrier) semiconductor interface. Owing to the fact that efforts in GaN are very sketchy, examples utilizing other semiconductors are provided for the reader to glean an understanding of the physics and applications involved. Tunneling from a ferromagnetic metal through vacuum into a semiconductor has been shown to provide a high degree of polarization [634]. Various combinations of magnetic metal/ semiconductor interfaces have considered for spin-polarized injection with various degrees of success, such as hybrid Au/Co/Cu/NiFe/n-GaAs spin valve Schottky barrier [635], and semiconductor light-emitting diode structure with ferromagnetic metal contact having an injection efficiency of 30% that persists to room temperature [636]. Light emission results from injection of spin-polarized electrons under reverse bias with the aid of Schottky barrier formed at the Fe/AlGaAs interface that provides a natural tunnel barrier. The injected carriers radiatively recombine and emit circularly polarized light. Along similar vein, ballistic spin-polarized transport through a diluted magnetic semiconductor heterostructure with the inclusion of a nonmagnetic barrier has also been investigated wherein, for suitable magnetic fields, the output current exhibited a nearly 100% spin polarization for large forward bias [637]. Spin-polarized tunneling of electrons in a GaInAs/GaAs quantum well lightemitting diode structure from the valence band of GaMnAs into the conduction band of n-type GaAs with Si delta doping at the interface has also been explored [638]. Injected spin-polarized electrons have been detected in the form of circularly polarized light emission from the quantum well that corresponded to magnetooptical Kerr effect (MOKE) loops. Because the angular momentum selection rules are simplified by the strain-induced heavy-hole (HH)/light-hole (LH) splitting, a direct relation between circular polarization and spin polarization can be obtained. On the transistor realm, Si-based spin valve transistors operative at room temperature have been demonstrated using spin-dependent transport over Schottky barriers [639]. That a very good interface between ferromagnet and semiconductor is imperative cannot be overstated. How long the injected spin can exist depends on the spin relaxation time, which can be quite long in lightly doped nonmagnetic semiconductors [640]. For control of spin in a ferromagnetic semiconductor such as (Ga,Mn)As where the effect is carrier spin interaction induced, ferromagnetism might be used. By using gate-induced depletion of the carrier density, ferromagnetism may be turned on and off. For illustrative purposes, photo-generated carriers have been used to induce ferromagnetism in (In,Mn)As [641].

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j 4 Extended and Point Defects, Doping, and Magnetism A succinct glossary of spin polarization is given here for completeness. Spinpolarized transport occurs naturally in any material in which there is an imbalance of the spin populations at the Fermi level, similar to that shown in Figure 4.139b depicting the case without a and with b magnetic field. The disparity between spin-up and spin-down electrons in the ferromagnet or a diluted magnetic semiconductor below Curie temperature versus the normal metal or nonmagnetic materials does not require magnetic field for it to take place. As depicted in the figure, the necessary imbalance occurs in ferromagnetic metals owing to the density of states available for spin-up and spin-down electrons, while often nearly identical; they are shifted in energy with respect to each other [642]. Unequal filling of the bands ensues, providing the source of the net magnetic moment for the ferromagnetic materials. Moreover, this can also cause the spin-up and spin-down carriers at the Fermi level to be unequal in concentration (and mobility), which can lead to a net spin polarization in a transport measurement. However, the sign and magnitude of the resulting polarization is dependent on the specifics. For example, a ferromagnetic metal may be used as a source of spin-polarized carriers for injection into a semiconductor (most likely in conjunction with an insulating barrier to tunnel as in the case of spin FETs discussed below) or a normal metal (as in the case magnetic sensor heads). Obviously, there is a need for 100% spin-polarized conducting materials. These are materials that have only one occupied spin band at the Fermi level. Materials such as Fe, Co, Ni, and their alloys are only partially polarized and have a polarization P of 40–50% [643] but are adequate to develop useful devices. P is defined in terms of the number of carriers n that have spin up (n") or spin down (n#), as Pspin ¼ ðn"  n#Þ=ðn" þ n#Þ:

ð4:103Þ

Clearly, the impetus is in place for 100% polarization in the case of which the only states that are available to the carriers are those for which the spin is parallel to the spin direction of those states at the Fermi level; for reference, spin polarization of over 95% has been achieved, albeit at 4 K, in La2/3Sr1/3MnO3/SrTiO3/La2/3Sr1/3MnO3magnetic tunnel junctions (MTJs) [644]. In this particular device, spin polarization can be measured through the tunneling magnetoresistance (TMR) ratio observed when switching the magnetic configuration of one of the FM electrodes from parallel to antiparallel while leaving the other in the parallel configuration. Spin-polarized tunneling (SPT) was reported by Meservey et al. [645], and Tedrow and Meservey [646]. In the associated experimental investigations, determining the spin-polarized tunneling was done by tunneling from a ferromagnetic film at the Fermi level into a Zeeman spin split superconducting Al film that reflects the spin polarization of the tunneling electrons coming from the ferromagnet. These experiments showed that conduction of electrons in ferromagnetic metals are spin polarized and that the spin is conserved in the tunneling process. Julliere [647] formula for ferromagnetic/insulator/ferromagnetic materials tunneling describes the tunnel junction magnetic resistance (JMR). For informational value, the JMR is defined in two manners, “pessimistic” and “optimistic.” In the former, JMR ¼ [R(0)  R(HS)]/R(0) where R(0) and R(HS) depict the tunnel junction resistance at H ¼ 0 and at saturation magnetic field H ¼ HS, respectively, the latter being much smaller, and represent the case when the magnetic

4.9 Intentional Doping

moments of the two ferromagnetic electrodes are aligned. The former, without the magnetic field, represents the case when the magnetic moments of the two electrodes are antialigned. In the “optimistic” case, JMR ¼ ½Rð0Þ  RðHS Þ
=RðH S Þ:

ð4:104Þ

In the so-called “pessimistic” case, the tunnel junction resistance is then given by JMR ¼

DR 2P1 P2 : ¼ ðR"#  R"" Þ=R"# ¼ 1 þ P1 P2 R

ð4:105Þ

Here, R"# and R"" represent the resistances for antiparallel and parallel spin alignment states of the two ferromagnetic layers in the tunnel junction composed of two ferromagnetic materials (magnetic metal or ferromagnetic semiconductor separated by an insulator). It should be mentioned that R"# R"". In addition, P1 and P2 represent the spin polarization values for injector and analyzer ferromagnetic electrodes with the additional assumption that the magnetization in both materials is normal to the interface. They are defined as P¼

DðE F Þ"  DðE F Þ# DðE F Þ" þ DðE F Þ#

;

ð4:106Þ

where D" ðE F Þ and D# ðE F Þ represent the density of states of spin-up and spin-down electrons, respectively, at the Fermi level in a given ferromagnetic material (consistent with P ¼ (n"  n#)/(n" þ n#) given in Equation 4.103). For equal values of the polarization associated with the two ferromagnetic materials, JMR ¼ 2P2/(1 þ P2). For example, if the ferromagnet has only the spin-up electrons, the spin polarization would be 100%. In nonmagnetic materials there are no spin-unpaired electrons, and therefore D"(EF) and D#(EF) are equal to each other making P ¼ 0. For 100% polarization of ferromagnetic materials on both side of the tunnel junction, the JMR value would be 1. The implicit assumption here is that spin-polarized electrons can be injected into the insulator very efficiently and wherein they do not decohere. In a sense the spin is assumed conserved in tunneling, that is, the tunneling current flows in up- and down-spin channels as if in two parallel but separate wires. If we consider the effect of spin decoherence as well, the JMR expression can be modified as DR 2P1 P2 e  d=ls ; ¼ R 1 þ 2P 1 P 2 e  d=ls

ð4:107Þ

where d and ls represent the thickness of the medium where the spin-polarized electron transport is considered and the spin coherence length, respectively. In the so-called “optimistic” approach, the tunnel magnetoresistance (chosen arbitrarily instead of junction magnetic resistance) is defined as [648] TMR ¼ ðR""  R"# Þ=R"" ¼

2P 1 P 2 : 1  P1 P2

ð4:108Þ

For equal polarization of the two electrodes, ferromagnetic materials, TMR ¼ 2P2/(1  P2). The optimistic and pessimistic definitions of TMR differ from each

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j 4 Extended and Point Defects, Doping, and Magnetism other with the sign of the summation in the denominator. The TMR value in the optimistic case explodes for P1P2 ¼ 1. It should be mentioned that the Julliere formula characterizes the magnetoconductance solely in terms of the tunneling spin polarization. This model does not accurately describe the magnetoconductance of free electrons tunneling through a barrier. Another method forwarded by Slonczewski [649] (which also takes into account the orientation of the magnetization vector in relation to the interface) provides a good approximation to the exact expression for free electrons in the limit of thick barriers. In spite of the fact that Slonczewski’s model provides a good approximation to the exact expression for free electrons in the limit of thick barriers, MacLaren et al. [650] found that the tunneling of band electrons shows features that are not described well by any free electron picture. This suggests that the details of the band structure of ferromagnetic material at the Fermi energy must be taken into consideration. Relative value of the two models has been debated by others as well [651]. Rigorous theories of tunneling in magnetic junctions have been developed [652], some predicting temperature dependence by taking the thermal smearing into consideration [653]. On the technological side, the seminal work of Meservey and Tedrow [645,646] provided the experimental evidence that the current in a ferromagnetic material and the current that tunneled across a barrier into another material has a net spin polarization. This confirmed the much earlier work of Sir N. Mott, predicting that current in a ferromagnet is spin polarized [654]. Julliere recognized the possibility of a converse effect, of that reported by Meservey and Tedrow [645,646], that is, if the metal on the other side of the tunnel junction (the counterelectrode) is also a ferromagnet, then the density of states available to receive tunnel current would also be different for up- and down-spin electrons. At that time this effect was very small and did not receive much attention which later changed as magnetic tunnel junctions became very popular with room-temperature magnetic tunnel junctions [655], giant magnetoresistance (GMR) taking advantage of the additive nature of the effect in the multiple layers used with GMR values potentially approaching 100% (the group of Fert) [656], the basis for which might go to the work of Gr€ unberg and his colleagues [657]. Getting back to magnetic tunnel junctions, if the magnetization of the materials is reversed, the spin direction of those states also reverses. Consequently, depending on the direction of magnetization of a material relative to the spin polarization of the current, a material can function as either a conductor or an insulator for electrons of a specific spin polarization. An analogy can be made with polarized light passing through an analyzer, which has been made already in conjunction with the proposal for a spin FET, as will be elaborated further in the text. The main difference being that in the optical case crossing the polarization axis at 90 blocks fully the transmission of the light, whereas for spin-polarized electrons, the magnetization must be rotated 180 for the same to occur. Magnetic-recording industry has been rapidly developing devices that rely on spin transport. When the two ferromagnetic metals straddle a normal metal, there can be transport if the polarizations (magnetization or the magnetic moments) of the injecting (polarizer) and collecting (analyzer) ferromagnetic materials are

4.9 Intentional Doping

aligned [633]. When they are antialigned, no transport is possible for the 100% spin polarization case. Technologically, both ferromagnetic and normal materials are deposited in thin films, and unlike the aforementioned scenario, the current is passed in the plane of the films, at the expense of making the spin-polarized transport more complicated. In this case, the effect of the spin exclusion in antialigned films is still observed which relies on interface scattering and channeling of the current into narrowed pathways. When the polarizations of the ferromagnetic films are aligned, both of these effects are removed, and the device would be in the low-resistance state. This two-layer system goes with the nomenclature of spin valve. It is constructed in such as way that the magnetic moment of one of the ferromagnetic layers is very difficult to reverse in an applied magnetic field, while the moment of the other layer is very easy to reverse. This easily reversed or soft layer then acts as the valve control by an external field. The device can be used to measure or monitor those external fields and can have numerous applications among which is the magnetic read head. These concepts can be applied to magnetic recording and nonvolatile memories, which are called magnetic random access memory (MRAM), which gained considerable interest [658,659]. It should be pointed out that because of the announcement of giant magnetoresistance, the magnetic hard disk read technology revolutionized. Ultimately, a new philosophy in computer memory in which the line between storage memory and active memory is blurred, may be spawned. With respect to issues dealing with magnetism, commonly used semiconductors are not magnetic. However, when they are doped appropriately with elements with spin-unpaired shell structures, they exhibit ferromagnetic properties. However, even then the ferromagnetism observed is not retained at room temperature. Remarkable progress has been made in GaAs [626], with transition temperatures in the vicinity of 170 K beyond which the magnetic ordering is destroyed. While the effort in GaAs is predictable and orderly, the same does not necessarily apply to GaN and other nitrides such as AlN. Although the dependability and physical basis are open to debate [660,661], theoretical [587,662,663] and a number of experimental investigations on GaN doped with Mn indicate Curie transition temperatures in excess at room temperature. Although Mn-doped GaN has been explored, as group II elements are potentially dopants that could produce p-type conductivity in GaN. The optical and other manifestations of Mn-doped GaN in this particular context are discussed in Section 4.9.6.1. Here, the focus is on the magnetic properties. The nascent state of GaN doped with transition metals and rare earth elements is such that the definitive observations of standard properties expected of dilute magnetic semiconductors are not yet available with the required clarity. The first group of semiconductor families that exhibited the clearest of all magnetic properties is the II–VI group, which unfortunately suffers from very low Curie temperatures. The semiconductor doped with Mn transition metal GaAs has a relatively higher Curie temperature, near 170 K, and is predictable by carrier exchange interaction induced ferromagnetism. In what follows, a succinct discussion of the theoretical basis for ferromagnetism in dilute magnetic semiconductors is given. When and if II–VI and GaAs examples are able to describe the mechanisms in a better way and/or provide reliable and well-behaved experimental results, they are included in the

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j 4 Extended and Point Defects, Doping, and Magnetism discussion. The results and conclusions of various theoretical approaches applied to GaN are naturally discussed. The discussion also extends to cover possible devices such as spin transistors, generically referred to as spintronics, and polarized lightemitting devices. 4.9.6.5 Theoretical Aspects of Dilute Magnetic Semiconductor The important characteristic of a ferromagnetic material is the spontaneous magnetization below the Curie (TC) temperature, also referred to as the critical temperature. As shown in Figure 4.139 in ferromagnetic materials [642], the d band is divided into spin-up and spin-down subbands, and the up and down states are displaced with respect to one another. The latter is displaced in energy so that the spin-up band is filled first, and the spin-down states contain the remaining, if any, electrons. The difference in the number of spin-up and spin-down electrons gives rise to the observed spontaneous magnetic moment. Above TC, the ferromagnetic material loses its permanent magnetism due to thermal agitations. To have practical applications in functional devices, it would be desirable, to put it mildly, to have a Curie temperature well above room temperature. Further for some device applications, it is also desirable to have the ferromagnetism be of carrier-induced origin, so that the magnetic properties of the DMS can be manipulated by external means such as through manipulation of the hole concentration. A better understanding of the underlying mechanisms will certainly provide the much-needed guidance for material design. To gain insight into the processes involved, a brief tutorial of the recently proposed mechanisms for ferromagnetism in DMS materials is presented in this section. A more detailed treatment of the theoretical results from recent literature about the mechanism of ferromagnetism in TM-doped ZnO and GaN then follows. Essentially, ferromagnetism results when the magnetic moments associated with magnetic ion impurities are polarized to be primarily in the same direction. As will be shown, the nature of the interaction of electrons of magnetic ions with those of conduction and valence bands of the host determine whether the resultant material is ferromagnetic or antiferromagnetic. Intuitively, everything else being equal, the closer the distance between the magnetic ions that occupy the cation sites and anions of the host the better the coupling expected to be. The strength of magnetic ordering in relation to thermal agitation is characterized by the Curie temperature below which ordering is not destroyed by agitations. Empirically, the Curie temperature is then expected to be high for semiconductors with smaller lattice parameters. Normalizing the lattice constant to the cubic form, the measured or the predicted Curie temperatures for various DMS semiconductors are shown in Figure 4.143. Getting back to a brief discussion of the nature of ferromagnetism in DMS, the mechanisms pertinent to magnetism are super direct exchange (antiferromagnetic), free carrier polarization, band polarization, indirect superexchange (could be ferromagnetic), double exchange (ferromagnetic and stronger when mediated by holes), and magnetic polarons, to cite a few. Let us now discuss the various theories developed to address ferromagnetism in magnetic ion doped semiconductors. First, a short glossary for each of the

4.9 Intentional Doping

500

GaN

Curie temperature (K)

400 ZnO

InN

300

AlP

1/a3

200

AlAs

Si 100

CdS

ZnS

GaP

CdSe Ge

CdTe InP

ZnTe

0 4.5

5.0

5.5

6.0

6.5

Lattice constant, a (Å) Figure 4.143 Measured or predicted critical temperature versus the lattice parameter for a group of semiconductors of general interest in terms of DMS. The solid line represents a fit with a3 dependence, where a is the lattice parameter. Courtesy of T. Dietl and J. Furdyna.

mechanisms is presented. What distinguishes DMS materials from more common semiconductors is that in DMS there are two systems, one comprising the semiconductor host and the other comprising the magnetic ion (either transition metal with partially filled d shells or rare earths with partially filled f shells) with their own somewhat preserved properties and their limited interactions. The semiconductor system can be characterized by delocalized band electrons that can be described by extended states. The magnetic ion, however, is characterized by localized 3d or 4f shell. Effective mass theory does well in describing the mobile carriers, that is, electrons in the conduction band and holes in the valence band. In this treatment the crystal Hamiltonian and pertinent perturbations can be described by the Luttinger basis. In addition, tight-binding calculations for wide bandgap semiconductors of interest here indicate that the G6 conduction band Luttinger functions mainly consist of cation s-orbitals whereas the G8 and G7 valence band associated functions to a first extent consist of anion p-orbitals. Electrical and optical properties of the semiconductor are determined primarily by the effective mass carriers. However, the localized magnetic moments associated with the magnetic ions and their interaction with the host semiconductor determine the magnetic properties. The interaction that is responsible for the desired magnetic behavior is sp–d in the case of transition metal magnetic ions and sp–f in the case of rare earth magnetic ions. Spin interactions are

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j 4 Extended and Point Defects, Doping, and Magnetism typically accounted for by the spin-dependent part of the Hamiltonian, which in metals give rise to the Kondo effect. It is for this reason, that the Hamiltonian, describing the spin-dependent coupling between the localized magnetic moments, associated with magnetic ions and band carriers is termed as the Kondo Hamiltonian. Such a Hamiltonian contains two independent mechanisms, namely, the kinetic exchange terms mediated by hybridization and direct Coulomb exchange. A succinct review of this topic is treated by Kacman [630]. 4.9.6.5.1 Carrier – Single Magnetic Ion Interaction Let us discuss the interaction between the host and one ionic magnetic impurity, involving conduction band electrons and the valence band electrons, with conduction band first. Direct Coulomb exchange is a first-order perturbation effect, and for s-like conduction bands, which is the case for the semiconductors under consideration here, and transition metals with their open d-shell, the direct Coulomb exchange interaction leads to ferromagnetic Kondo Hamiltonian. Coupling between the localized magnetic moment and the electron under the influence of strong spin–orbit interactions in the p-like valence band can be described by replacing the spin of the magnetic impurity in the Kondo Hamiltonian with the total eigenvalues of the angular momentum. For rare earths the spin–orbit coupling in the ion should be considered with the outcome that the ionic spin is replaced by the total angular momentum. To account for many magnetic ions in the crystal, a summation over magnetic ions is performed. The magnetic impurities can be described by mole fraction of the magnetic impurity, and in molecular field approximation, the spin operators are replaced with their averages. In the end the carrier-ion direct-exchange Hamiltonian for G6 conduction band electrons (describing s–d interaction in the case of transition metal ions) takes the form

 Hex ¼  xN 0 a S== s== ; ð4:109Þ

where xN0 is the concentration of the magnetic ions with x representing the mole

 fraction, a denotes the exchange constant for s-like electrons, S== represents the component of the thermodynamical average of magnetic ion spin along the magnetic field, and s//is the component of band electron spin along the magnetic field. Often times the magnetic field is applied along the growth or the z-direction, in the case of which the term hSz i is used to describe the average of magnetic ion spin. The exchange Hamiltonian is a measure of the extent of the exchange interaction and thus the extent of Zeeman splitting of the energy levels in thehost material. The extent of that splitting is generally taken proportional to xN 0 a S== , in the case of which N0a is treated as spin exchange integral for the conduction band. In II–VI DMS materials the parameter a is on the order of about 0.2 eV. Experiments performed mainly in conjunction with II–VI DMS indicated that this parameter for the valence band, going by the nomenclature b, has the opposite sign and is significantly larger than a by a factor of some 5–10 or even more, depending on the host material. As for the valence band, the picture is relatively complicated and there are several approaches that have been employed. For example, within the k-space perturbational approach, the p–d exchange interaction can be visualized as virtual transitions between p-electrons and ionic d-shell. A singly occupied orbital as in the case of

4.9 Intentional Doping

Mn with five electrons in 10 possible states and all electrons having the same spin can participate in virtual transitions that result in both creation and annihilation of p-like band electrons. The spin exchange interaction is governed in part by the Pauli principle that allows only the virtual transitions decreasing the total spin of the ion by removing the electron from d-orbital or by adding one with opposite spin. On the contrary, for an empty or doubly occupied orbital, the Pauli principle does not favor one spin polarity over the other, but the interaction remains spin dependent as the Hund’s rule requires that the ion to increase its spin. To take this into consideration, the theory must account for the many electron intershell correlations. In considering a single hole near the top of the G8 valence band and the magnetic ion with five d orbitals occupied by N electrons, the Hamiltonian that is applicable would have ionic, crystal, and hybridization components. The exchange interaction Hamiltonian for p-electrons with a degree of complication in that how the crystal field splits the internal levels associated and which internal levels is strongly coupled to the host p-orbitals must be known. Consider the case of coupling between a G8 valence band p-like electron with three one electron t2g states of the magnetic ion. The hybridization-mediated kinetic exchange depends on the filling of only the t2g orbitals, not all the one-electron d-orbitals of the magnetic ion. The spin-dependent part of the exchange Hamiltonian for interaction between the G8 valence band p-like electrons and all t2g d-orbitals occupied by one electron can be described as Hex ¼ 

 1 xN 0 b S== J == ; 3

ð4:110Þ

where b is the exchange constant, J// is the component of the total angular momentum of the G8 valence band p-like electron parallel to the magnetic field. As in the case of conduction band, the exchange spin Hamiltonian for the valence band electron determines the Zeeman splitting associated with the valence band states of the host. The term b is proportional to what is called the hybridization constant and inversely proportional to the sum of two terms that describe energies needed for transferring a d-shell electron from the magnetic ion into the band and transferring an electron from the band to the d-shell of the magnetic ion. For Mn2þ and Fe2þ, that is, for singly occupied d-orbitals, virtual transition of electrons from the ion to the valence band and also from the valence band to the ion is possible. In the case of Sc2þ or Ti2þ, only the virtual transitions from the valence band to the d-shell of the ion are possible. Band electrons with both spin up and spin down can be transferred. According to the Hund’s rule that applies to both types of ions mentioned, the transition that increases the total spin of the ion leads to lower energy. In the case of Fe2þ and Co2þ, the sixth electron of Fe and Co occupy the eg orbitals lying at a lower energy than the t2g orbitals. Consequently, bMn < bFe < bCo. For magnetic ions with empty t2g orbitals, only the transfer of a band electron regardless of its spin into the d-shell of the magnetic ion is possible, and thus the Hamiltonian again has two components in the denominator, but with energies needed for transferring an electron from the band to the d-shell of the magnetic ion with one spin and another spin being subtracted from each other. Hund’s rule, however, causes one of these energies to be smaller than the other, making the spin

j1085

1086

j 4 Extended and Point Defects, Doping, and Magnetism exchange interaction, which in this case goes by the nomenclature of g, positive, which means that in this treatment one would expect ferromagnetic interaction for Sc2þ and Ti2þ. When some of the t2g orbitals are singly occupied and some empty or doubly occupied, such as in Cr2þ that has two of its t2g orbitals singly occupied and one empty, strong exchange interactions (due to strong static tetragonal Jahn–Teller effect) result and rather complex splitting occurs. Naturally, exchange interaction has both b and g types of exchange interaction constants. In the opposite limit, when Jahn–Teller effect in none of the three crystallographic directions and the cubic symmetry is conserved, the mean exchange Hamiltonian has the simple Heisenberglike form again composed of b and g types of exchange interaction constants as  multipliers with different weighing factors to S== . This represents the case of Cr2 þ in zinc chalcogenides, which has a ferromagnetic character. 4.9.6.5.2 Interaction Between Magnetic Ions Magnetic ion to magnetic ion interaction, or in the case of transition metal ions, d–d interactions could also take place, which is dependent on the density of the magnetic impurity in the host. The exchange coupling tensor in effect can be described with three distinct interactions such as Heisenberg, Dzyaloshinsky–Moriya (DM), and pseudodipolar (PD). Even this partitioning does not lead to any microscopic model of the interaction but only the form of the interaction can be discerned. When spin–orbit interaction is not taken into account, the DM and PD interactions vanish. Further, the DM component of the Hamiltonian vanishes in crystal with the center of inversion. As an example, the DM and PD terms for Mn in II–VI DMS are at least an order of magnitude smaller than the Heisenberg term. The nature of the dominant mechanism determines the sign and strength of the nearest neighbor and the next nearest neighbor (NNN) interaction constants. Let us now consider four main mechanisms leading to spin–spin interaction between magnetic ions. In the Ruderman–Kittel–Kayusa–Yoshida (RKKY) mechanism, the sp–d exchange leads to the polarization of free electrons. In the Blombergen–Rowland (BR) interactions, the sp–d exchange leads to band polarization. In both cases, the spin–spin interaction results from the induced polarization. In the other two remaining mechanisms, namely, the superexchange and double exchange mechanisms, the interaction, within the atomic picture, can be thought of in terms of virtual transitions between the ions (situated at the cation sites) and neighboring anions. The RKKY, BR, and superexchange interaction can be described by an effective Kondo-like Hamiltonian [630]. Superexchange Mechanism The superexchange mechanism is a process wherein the spins of two ions are correlated owing to the spin-dependent kinetic exchange interaction between each of the two ions and the p-like valence band. For superexchange, it is more proper to employ the band picture to describe the spin–spin interaction in semiconductors as opposed to the atomic picture. In II–VI DMS, the superexchange resulting from the sp–d hybridization is by far the dominant spin–spin interaction mechanism for the observed isotropic (Heisenberg) and anisotropic (DM) exchange constants.

4.9 Intentional Doping

Conduction band 2

1 D

C

A

k

B

Valence band

k'

Figure 4.144 A cartoon of the four virtual transitions depicting the superexchange ion–ion interactions. The arrows show the direction of electron transfer for the path ABCD. For the path CADB, all the directions should be reversed. Similar to that in Kacman [630].

Visualizing the virtual transition picture, one can construe the superexchange mechanism as being due to four virtual transitions, namely, from the p-like valence band state to the ions and back to the valence band characterized by arrows A, B, C, and D in Figure 4.144. This can take place in six sequences, namely, ABCD, ABDC, BACD, BADC, ACBD, and CADB. The directions shown are for the transitions ABCD. What is common to all of the sequences is that they all start with the transfer of a band electron k to one of the ions that produces a hole in the band but leads to different intermediate states in the perturbation Hamiltonian matrix depending on the sequence. The unperturbed valence band states, which must be summed up over the entire Brillouin zone, are typically described within the empirical tight-binding model [630]. Blombergen–Rowland Mechanism The Blombergen–Rowland mechanism is a process wherein the spins of two ions are correlated owing to the spin-dependent kinetic exchange interaction between each of the two ions and the p-like valence band and slike conduction band. The BR process differs from the superexchange mechanism only by the specifics of the intermediate states and that it also allows for virtual transitions to the empty conduction band, as illustrated in Figure 4.145. Naturally, this mechanism is less likely than the superexchange mechanism, particularly in large bandgap semiconductors, as the path is more complex. In this method, the intermediate states, jni and jn0 i, in the fourth-order perturbation matrix may correspond to one ion with N electrons and another with N þ 1 electrons and one hole in the p-like valence band. The hole may be considered to be created by a valence band electron being transferred to the ionic d-shell, as in the case of superexchange. In addition, a correspondence to one ion with N electrons and another with N  1 electrons, and one electron in the conduction band is also present. The electron in question in the conduction band is one that is transferred from the d-shell of the ion to the conduction band. In the intermediate jli states, there are N electrons of the two ions involved and a hole in the valence band, and an electron in the conduction band, or there are N þ 1 and N  1 electrons on the ions with neither holes in the valence band nor electrons in the conduction band. This can pave the way to a ferromagnetic d–d interaction even for the Mn2þ ions. However, this interaction

j1087

1088

j 4 Extended and Point Defects, Doping, and Magnetism k'

Conduction band

B

C

1

2 D

A k

Valence band

Figure 4.145 A cartoon of the virtual transitions involved in the Blombergen–Rowland interactions. Unlike the six sequences available in the superexchange mechanism, there are twelve different orders for the four A, B, C and D transitions. These processes commence with

the transfer of an electron either from the valence band to one of the ions (ABCD shown by the arrows, DCBA, ADBC, etc.) or from an ion to the conduction band (e.g., BACD, CBDA, etc.). Patterned after Ref. [630].

was shown to be an order of magnitude less effective than the superexchange [630]. One might expect that Blombergen–Rowland interactions are much enhanced in narrow-gap alloys. Double Exchange Interaction The double exchange interaction, proposed by Zener [664], is based on coupling magnetic ions in different charge states by virtual hopping of the extra electron from one ion to the other, specifically, an Mn2þ–Mn3þ (or d4–d5) pair of ions with one d electron hopping virtually from one ion to the other via the p-orbitals of neighboring anions. Naturally, then the Zener double exchange interaction cannot be the mechanism leading to ferromagnetic correlation between the distant Mn spins, because the magnetic electrons remain localized at the magnetic ion and do not contribute to the charge transport. In this vein Dietl in Matsukura et al. [665] suggested that the holes in the extended or weakly localized states mediate the longrange interactions between the localized spins. The same processes apply to Cr1þ–Cr2þ magnetic ion pairs as well, and also the superexchange interaction may also be in effect. The DMS fitting into this category is not all that uncommon, and the best studied examples are the zero-gap II–VI compounds (HgSe and HgS) with Fe ions, for which the 2þ/3þ donor level is degenerate with the conduction band [630]. The different charge states of a given magnetic ion have also been observed in III–V DMS. The case of GaN doped with Mn where evidence for both charge states is available is discussed in Section 4.9.6.1. The zero-gap II–VI compounds (HgSe and HgS) with Fe ions for which the 2þ/3þ donor level is degenerate with the conduction band have also been investigated [666]. Although it is believed that the Mn ions in GaAs and InAs are in the high spin 2þ charge state [667], the precise nature of the centers associated with Mn ions has not yet been definitively determined. The coexistence of Mn2þ and Mn3þ ions leading to double exchange has been suggested for Mn-doped ZnO [668] and the chalcopyrite semiconductor CdGeP2 [669]. The double exchange mechanism has been successfully used to explain the ferromagnetism observed in (In,Mn)As [664,670,671]. In a DMS material, if

4.9 Intentional Doping

neighboring TM magnetic moments are in the same direction, the TM–d band is widened by the hybridization between the up-spin states. Therefore, in the ferromagnetic configuration the band energy can be lowered by introducing carriers in the d band. In these cases, the 3d electron in the partially occupied 3d-orbitals of the TM is allowed to hop to the 3d-orbitals of the neighboring TM, if neighboring TM ions have parallel magnetic moments. As a result, the d-electron lowers its kinetic energy by hopping in the ferromagnetic state. This is the so-called double exchange mechanism. Theoretical investigations could not progress much because of the fact that both disorder and interactions are strong and must be taken into account nonperturbatively. This model is shown schematically in Figure 4.146. This kind of model can explain the physics of the dilute Mn limit, and can also be easily adopted to include the holes that are localized on ionized antisite defects, such as As in (Ga,Mn) As, rather than Mn acceptors. The double exchange mechanism proposed by Zener couples magnetic ions in different charge states by virtual hopping of the extra electron from one ion to the other. This mechanism was employed by Anderson and Hasegawa [672] to explain the magnetic properties of manganites of perovskite structure. They treated an Mn2þ–Mn3þ pair of ions with one d electron hopping virtually from one ion to the other via the p orbitals of neighboring anions. The relevance of this mechanism in mixed-valence DMS has been discussed in the literature [630]. The DMS materials with the aforementioned property are not that unusual. In addition to the well celebrated II–VI and III–V GaAs [595], GaN will soon be discussed. 4.9.6.5.3 Zener, Mean Field, RKKY, and Ab Initio Treatments Let us now consider these various mechanisms in more detail with pertinence to GaN as needed. In the Zener model, the direct interaction between d shells of the adjacent Mn atoms (superexchange) leads to an antiferromagnetic configuration of the d shell spins because the Mn–d shell is half-filled. On the contrary, the indirect coupling of spins through the conduction electrons tends to align the spins of the incomplete d shells in a ferromagnetic manner. It is only when this dominates over the direct superexchange coupling between adjacent d shells that ferromagnetism is possible. Accordingly, the mean field approach assumes that the ferromagnetism occurs

Mn

Mn As

As

N

N

Mn

Mn

Figure 4.146 Local magnetic moments (Mn2þ(d5)) with spin S ¼ 5/2 are antiferromagnetically coupled to itinerant carriers with spin S ¼ 1/2 in the model semiconductor GaAs:Mn (upper) and GaN:Mn (lower).

j1089

1090

j 4 Extended and Point Defects, Doping, and Magnetism through interactions between the local moments of the Mn atoms mediated by free holes in the material. The spin–spin coupling is also assumed to be a long-range interaction, allowing the use of a mean field approximation. The mean field model calculates the effective spin-density due to the Mn ion distribution. The direct Mn–Mn interactions are antiferromagnetic so that the Curie temperature TC, for a given material with a specific Mn concentration and hole density (derived from Mn acceptors and/or intentional shallow-level acceptor doping), is determined by a competition between the ferromagnetic and antiferromagnetic interactions. It should be mentioned that the Zener theory does not take into consideration the itinerant character of the magnetic electrons and the quantum (Friedel) oscillations of the electron spin polarization around the localized spins. Both of these were later established to be critical concepts for the theory of magnetic metals [587]. Because the mean distance between the carriers is greater than that between the spins, the effect of the Friedel oscillations averages to be zero in semiconductors. Early attempts to understand the magnetic behavior of DMS systems are based on models in which the local magnetic moments are assumed to interact with each other via RKKY-type interactions. The RKKY mechanism was originally introduced to explain the interactions between nuclear spins in metals via the conduction electrons and as such it is efficient only in cases where a high concentration of free carriers is present. Therefore, it is not really suitable to describe ion–ion interactions in DMS. Nevertheless, it formed the basis for the carrier-mediated interionic spin interactions in metals and highly degenerate semiconductors. The basic idea behind the RKKY interaction is based on the exchange coupling between the magnetic ion and the band electrons described by the s–d Kondo Hamiltonian, which is a first-order perturbation effect. The Zener model would become equivalent to RKKY if the presence of the quantum (Friedel) oscillations of the electron spin polarization around the localized spins are taken explicitly into account. It should be mentioned that s and d wave functions are orthogonal and would not lead to any interaction in perfect one electron system in cubic semiconductors. The conduction electron gas is magnetized in the vicinity of the magnetic ion, with the polarization decaying with distance from the magnetic atom in an oscillatory fashion. This oscillation causes an indirect exchange interaction (RKKY) between two magnetic ions on the nearest or next nearest magnetic neighbors. This coupling may result in a parallel (ferromagnetic) or an antiparallel (antiferromagnetic) setting of the moments dependent on the separation of the interacting atoms. The Zener model and RKKY interaction cannot explain well the free carrier based ferromagnetism in such systems. The RKKY interaction between Mn spins via delocalized carriers has been used to explain the ferromagnetism observed in PbSnMnTe [673]. However, if the carriers come from Mn–d states and are localized, which are far from being free electron like, the RKKY interaction may not be realistic. The mean field theory is based on double exchange hole interaction, meaning electronic in nature in p-type GaN and also ZnO. The theory dealing with ferromagnetism driven by the exchange interaction between carriers and localized magnetic ions (localized spin) was first proposed by Zener [664,674,675]. The mean field Zener model proposed by Dietl et al. [587] has been successful in explaining the transition

4.9 Intentional Doping

temperatures observed for p-(Ga,Mn)As, which is roughly 2000· percentile fraction of Mn ions in the matrix (K) and (Zn,Mn)Te. The mean field Zener theory is based on the original model of Zener [675] and the RKKY interaction. The mean field theory basically assumes that the ferromagnetism is a result of interactions between the local moments of the Mn atoms mediated by free holes in the material (double exchange interaction). The spin–spin coupling is also assumed to be a long-range process that allows the use of a mean field approximation. On the experimental side, magnetism even in n-type GaN has been reported. As compared to the RKKY interaction, the mean field Zener model takes into account the anisotropy of the carrier-mediated exchange interaction associated with the spin–orbit coupling in the host material. This process reveals the important effect of the spin–orbit coupling in the valence band in determining the magnitude of the TC and the direction of the easy axis in p-type ferromagnetic semiconductors. On the basis of this model, it was predicted that TM-doped p-type GaN and also ZnO are the most promising candidates for ferromagnetic DMS with high Curie temperature. However, these predictions are predicated on the incorporation of some 5% transition metal element and hole concentrations of above 1020 cm3. Notwithstanding these seemingly yet to be demonstrated high hole concentration (which may in fact never be attainable), this prediction stimulated a plethora of activity to achieve high Curie temperature ferromagnetism by using ZnO- and GaN-based DMSs. Another simple model [676] has been put forth to explain the possible mechanism with specific attention paid to (Ga,Mn)As. In this model, holes are assumed to hop only between Mn acceptor sites, where they interact with the Mn moments via phenomenological exchange interactions. In some other models [587] the ferromagnetic correlation mediated by holes originating from shallow acceptors in the ensemble of localized spins and a concentration of free holes ( 3.5 · 1023 cm3) have been assumed for (Ga,Mn)As. With due reverence and deference to the models mentioned above, it is increasingly becoming clear that a true picture can only be obtained by performing first principles calculations. For example, it is stated that the mean field theory overestimates the critical temperature substantially when the magnetic ion density is small. It has also become clear that in Mn-doped GaN magnetic ion concentration has a profound effect on the way the levels split and whether the ferromagnetic or antiferromagnetic state is stable. Having made the case for ab initio calculations, in the first principles approach the total energy and electronic structures are calculated by using the DFT. At temperature T ¼ 0, the ground state structure of the system corresponds to a minimum of the total energy. DFT [677,678] is a successful approach for the description of ground state properties of metals, semiconductors, and insulators. Implementation of DFT is based on approximations for the exchange correlation potential, which arises from the overlap of the electron wave functions due to chemical bond formation. One effective and common approximation is the local (spin) density approximation that locally allows substitution of the exchange correlation energy density of an inhomogeneous system by that of an electron gas evaluated at the local density, and generalized gradient approximation (GGA) that locally substitutes the exchange correlation energy density by that of an electron gas

j1091

1092

j 4 Extended and Point Defects, Doping, and Magnetism evaluated at the local density and its gradient and higher terms. The magnetic state of the DMSs can be investigated by calculating the electronic structure of a ferromagnetic DMS (all the magnetic moments of TMs are parallel to each other) and that of a spin glass like (magnetic moments of TM point randomly with respect to each other) DMS. The TE is calculated for both states as a function of transition metal density. Then, DE ¼ TE (spin glass state)  TE (ferromagnetic state) is calculated to determine the stability of the ferromagnetic state, that is, when DE is positive, the ferromagnetic state is more stable than the spin glass state. In addition to the models mentioned above, an alternative model considers whether ferromagnetic ordering of the Mn moments could originate from carriers (holes) that are present in the material, but localized at the transition metal impurity [679,680]. Furthermore, ferromagnetism in DMS has been accounted for by percolation of bound magnetic polarons (BMPs) [679–686]. This in a sense relies on localized carrier, creating a spin polarization of the magnetic moments within the span of its wave function. The basic idea is schematically illustrated in Figure 4.147. The localized holes of the polarons act on the transition metal impurities

Figure 4.147 Representation of magnetic polarons. A donor electron in its hydrogenic orbit couples with its spin antiparallel to impurities with a 3d shell that is half-full or more than half-full. The figure is drawn for magnetic cation concentration x ¼ 0.1 and when the

orbital radius of the magnetic cation is sufficiently large. Cation sites are represented by small circles. Oxygen is not shown; the unoccupied oxygen sites are represented by squares (after Ref. [684]).

4.9 Intentional Doping

Figure 4.147 (Continued)

surrounding them, thus producing an effective magnetic field and aligning all spins. Transition to the insulating state takes place due to localization of the charge carriers (basically holes) at temperatures higher than the Curie temperature. As temperature decreases the interaction distance (boundary) grows. Below the Curie temperature, the neighboring magnetic polarons overlap and interact via magnetic impurities forming correlated clusters of polarons. A ferromagnetic transition is seen when the size of such clusters is equal to the size of sample. This model is inherently attractive for low carrier density systems such as many of the electronic oxides. The polaron model is applicable to both p- and n-type host materials [681]. Even though the direct exchange interaction of the localized holes is antiferromagnetic, the interaction

j1093

1094

j 4 Extended and Point Defects, Doping, and Magnetism between bound magnetic polarons may be ferromagnetic for sufficiently large concentrations of magnetic impurities. This enables ferromagnetic ordering of the Mn ions in an otherwise insulating or semi-insulating material. Even though the various theoretical approaches discussed above shed light on our understanding, albeit limited, of ferromagnetism in GaN, rigorous calculations must be used to get the true picture. In this vein, the first principles approach has also been used to elucidate magnetism in GaN-based DMS materials. As compared to the shallow acceptor Mn level in (Ga,Mn)As, the addition of Mn to GaN produces a deep impurity band within the GaN bandgap. Litvinov and Dugaev [687] questioned the RKKY interaction employed in the mean field theory and proposed that ferromagnetism in DMS systems is due to localized spins in the magnetic impurity acceptor level of the semiconductor crystal, and these localized spin excite band electrons due to p–d exchange interaction. Their model provided a detailed and quantitative predication of the dependence of the Curie temperature on the Mn concentration for various wurtzitic III–N alloys. Katayama-Yoshida et al. [688] studied GaN with 5% of various transition metals [689] and Ga1xMnxN with x ¼ 0.25 [690]. For a half-filled or less than half-filled d shell such as that in Mn, Cr, and V, the ferromagnetic state in GaN is stable. For a low concentration of Mn, ferromagnetism is favored, whereas for the high concentration the spin glass phase is stable. On the basis of local spin-density approximation functional calculations, Van Schilfgaarde and Myrasov [691] reported that for zinc blende GaN doped with 1–5% concentrations of Mn, Cr, and Fe, the exchange interactions are anomalous and behave quite differently from the picture assumed in simple models such as RKKY [587,692]. Those authors went on to argue that strong attraction between the magnetic elements tends to group them together in small nanoclusters of a few atoms. The magnetic coupling between doped Mn atoms in clusters and crystals of GaN has been predicted to be ferromagnetic by Das et al. [693] who used first principles calculations within GGA. Das et al. suggested that Mn atoms tend to cluster and bind more strongly to N atoms than to Ga atoms, which points out that the Mn concentration in GaN may be increased by using a porous substrate to offer substitutional surface sites. Their calculation also showed that the Fermi level passes right through the fattened impurity band (majority spin), thereby confirming that the impurity level acts as an effective mass acceptor. In a follow-up investigation, Wang et al. [694] examined two different cases where Mn atoms are bonded in bulk GaN as well as thin film forms with ð1 1
2 0Þ surface by allowing full structural relaxation within GGA. The study shows that in the (Ga,Mn)N system, the Mn–Mn separation plays a critical role in magnetic coupling of Mn ions. If the Mn atoms are incorporated into the GaN bulk, they couple ferromagnetically with or without structural relaxation. On the contrary, the coupling in unrelaxed thin film is ferromagnetic, which then converts into antiferromagnetic after relaxation. This may explain some controversial experimental observations for thick and thin film (Ga,Mn)N. Sato et al. [695] calculated the magnetic properties of (Ga,Mn)N from first principles using the Korringa–Kohn–Rostoker coherent potential approximation (KKR–CPR) method. It was found that the range of the exchange interaction in (Ga, Mn)N, being dominated by the double exchange mechanism, is very short ranged

4.9 Intentional Doping

due to the exponential decay of the impurity wave function in the gap. (Ga,Mn)N shows no high-temperature ferromagnetism for low Mn concentrations, so that the experimentally observed very high TC values in GaN should be attributed to small ferromagnetic MnN clusters and segregated MnN phases. The calculated TC for GaMnN is very low as compared to that obtained from the mean field theory that overestimates TC. Kronik et al. [696] reported the ab initio calculations for the electronic structure of (Ga,Mn)N with x ¼ 0.063. The introduction of Mn results in the formation of a 100% spin-polarized 1.5 eV wide impurity band, due to the hybridization of Mn 3d and N 2p orbitals (Figure 4.148). This result is qualitatively different from the case of GaAs due to the different Mn level position in GaAs the shallow acceptor, is 0.1 eV above the VBM in the isolated impurity limit, whereas in GaN deep-acceptor level, it is approximately 1.4 eV above VBM. So, in GaAs Mn hybridizes primarily with the valence band. However, in GaN its interaction with the valence band is therefore much smaller and the introduction of Mn barely polarizes the valence band. For an Mn composition of 6.3%, the impurity band does not hybridize to an extent sufficient for merging with the valence band. The impurity band renders the material half metallic and supports the effective-mass transport within it, which implies that (Ga, Mn)N is a highly suitable material for spin injectors. Kulatov et al. [697] studied the electronic structure and the properties of zinc blende (Ga,Mn)N with Mn concentration ranging from 1.56 to 12.5%. The calculations showed that the ferromagnetic state is lower in energy than those for the paramagnetic and antiferromagnetic states. The magnetic interaction of Mn atoms is short ranged. Their results also showed the important difference between GaAs and GaN in the energy position and localization of the Mn spin-majority states, as in Ref. [696]. 300

Majority spin 200

Density of states (au)

Mn 3d 100

0

−100

−200

Minority spin

N 2p EF

−300 −8

−6

−4

−2

0

2

Energy eV Figure 4.148 Partial density-of-states curves for wurtzite Mn0.063Ga0.937N. Thicker lines: Mn 3d. Thinner lines: N 2p [696]).

4

j1095

j 4 Extended and Point Defects, Doping, and Magnetism Codoping of O for N and Zn for Ga in (Ga,Mn)N shows that the atoms of O and Zn change the occupation of Mn bands and strongly affect both the magnetic moments and the conductivity. In addition, O codoping drastically enhances the ferromagnetic state and also causes a significant increase in the Curie temperature of (Ga,Mn)N. Further, Zn codoping causes strong hybridization of the eg states of Mn with VB and decreases the FM. Sanyal et al. [698] investigated the effect of varying Mn concentration on the electronic and magnetic properties in wurtzite GaMnN by using a first principles plan-wave method. They showed that the d states of Mn form an impurity band completely separated from the valence band states of the host GaN for dilute Mn concentration. Up to x ¼ 0.25, the Fermi level lies only in the spin-up density of states, as there is no state at the Fermi level for the spin-down channel, so that the system is half metallic with a high magnetic moment. For x ¼ 0.5, the system behaves like a ferromagnetic metal with a reduced magnetic moment due to the partially filled spin-down channel. The authors also showed that the DOS for a zinc blende structure is similar to that of a wurtzite structure, as shown in Figure 4.149.

8

(Wurtzite)

4

0

−4 DOS (states/eV)

1096

Mn−d N−p Total

–8

(Zinc blende)

4

0

−4

−8 −10

−5

0 E−E F (eV)

5

10

Figure 4.149 Spin resolved density of states of (Ga,Mn)N for (a) wurtzite and (b) zinc blende structures. Here, the Mn concentration is 6.25%. Courtesy of Sanyal et al. [698].

4.9 Intentional Doping

Mnd projected PDOS (/eV cell)

GaN: Mn t+

e-

t+

e+ t-

GaP: Mn

e-

t+ e+

t+

ee+

t-

t+

tt+

GaSb: Mn

e-

e+ t–3

t-

t+

t-

GaAs: Mn

–4

t-

–2

t-

t+ –1

–F Energy (eV)

1

2

3

Figure 4.150 In Mn d projected partial density of states for a single Mn in GaN, GaP, GaAs, and GaSb, where the symmetry (t2 and e) as well as the spin (þ and ) have been indicated. The shaded region represents the t2þ states (after Ref. [699]). (Please find a color version of this figure on the color tables.)

Mahadevan and Zunger [699] used first principles total-energy calculations to study the trends of Mn in GaN, GaP, GaAs, and GaSb. Figure 4.150 shows the calculated Mn d projected local density of states for neutral substitutional Mn (Mn3þ) in four GaX (X ¼ N, P, As, Sb) compounds. As can be seen, the antibonding t2 level, which is in the neutral state of the impurity, is occupied with two electrons (and therefore one hole) and strongly Mn localized. Moreover, the degree of Mn localization of the hole level decreases along the series GaN ! GaP ! GaAs ! GaSb. The acceptor level of Mn2þ is shown to be very deep in the (Ga,Mn)N, 1.4 eV, and it becomes progressively shallower as the anion x becomes heavier. They predicted a strong ferromagnetic stabilization in (Ga,Mn)N due to the p–d interaction that couples the tþ level of Mn ions to the p-like dangling bond states of the Ga vacancy, despite the fact that the hole orbital is a highly localized deep acceptor. This is in contrast to the model provided by Dietl [662] that assumes a host-like delocalized hole for all materials. In another study, Mahadevan and Zunger [700] explained the electronic structures of Cr and Mn in GaN as follows: introduction of a transition metal impurity in III–V semiconductor introduces a pair of levels with t2 symmetry – one localized primarily on the transition metal atom, referred to as crystal field resonance (CFR), and the other localized primarily on the host-anion atoms next to the transition metal impurity, referred to as the dangling bond hybrid (DBH). In addition, a set of nonbonding states with e symmetry, localized on the transition metal atom, are also introduced. Each of the levels is also spin split. Considering Mn in GaN shown in Figure 4.151, the 3d levels are well above the host cation dangling bonds. The dangling bond states are shown on the right-hand side, and the crystal

j1097

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j 4 Extended and Point Defects, Doping, and Magnetism 3d ion d n-1 t-(d )

e-(d )

t+(d)

Mn on Ga site t-CFR

Anion dangling bonds V Ga3-

CFR

e-

t+CFR t+(p)

e+(d) VBM

CFR

e+

DBH

t-

t-(p)

DBH

t+

Figure 4.151 A schematic energy-level diagram for the levels (central panel) generated from the interaction between the crystal field and exchange-split levels on the 3d transition metal ion (left panel) with the anion dangling bond levels (right panel), when the TM d levels are energetically shallower than the dangling bond levels (after Ref. [700]). (Please find a color version of this figure on the color tables.)

field and exchange split Mn d levels are shown on the left-hand side. The t2 (p) levels of the anion dangling bond hybridize with the t2 (d) levels of the transition metal. The levels generated after hybridization are shown in the central panel. The hybridization CFR in the t2-channel creates bonding, transition metal localized CFRs, tCFR þ and t  , as DBH DHB well as the host-anion localized antibonding DBHs t þ and t  , whereas the e CFR DHB channel creates the nonbonding eCFR þ and e  states. The hole resides in the t þ level deep in the bandgap. The symmetry (e versus t2) and the character (DBH versus CFR), as well as the occupancy of the gap level, determine the magnetic ground state favored by the transition metal impurity. The ab initio band structure and total energy calculations [696–699] seem to agree that Mn 3d levels are located in the gap, and that the interaction between substitutional Mn ions is ferromagnetic, at least in the not so high Mn concentration range. On the basis of these understandings and by using a band structure approach and level repulsion model, Dalpian et al. [701,702] proposed a unified picture to account for the magnetic ordering in Mn-doped III–V and II–VI semiconductors. The model of the host p states (VBM) and Mn d levels and level repulsion caused by p–d exchange coupling and d–d coupling between them in ferromagnetic and antiferromagnetic configurations is shown in the schematic diagram of Figure 4.152. The Mn d levels

4.9 Intentional Doping

Antiferromagnetic

Ferromagnetic

(a)

(b )

Spin up (c)

(b ’)

(a’)

(a)

(b )

Spin down (c)

(b ’)

(a’)

t 2d

t 2d

t 2p

t 2d

t 2d

t 2p

t 2p

t 2p

t 2d

t 2d t 2d

t 2d t 2p

t 2p

t 2p

t 2p

or showing the spin up and spin down cases separately

A nti f erromagneti c Ferromagneti c

(a)

(b)

Spin down (c)

(a’ )

t 2d

t 2d

t 2p

t 2p

t 2d t 2d t 2p

t 2p

and Spin-up

Antiferromagnetic

Ferromagnetic

(a)

(b)

(c)

(b’ )

(a’ )

t 2d

t 2d

t 2p

t 2p

t 2d t 2d t 2p

t 2p

Figure 4.152 A schematic model showing the position of the p and d levels and level repulsion between them in FM and AFM configurations for spin-up and spin-down cases. Note that the

Mn d levels are above the VBM. Also note that in (b), (b0 ), and (c), the states have mixed pd characters. Courtesy of Dalpian and coworkers [701].

j1099

1100

j 4 Extended and Point Defects, Doping, and Magnetism are above the VBM, which is the case for (Ga,Mn)N. In the ferromagnetic (FM) configuration, the majority spin state of neighboring Mn atoms couple to each other. Specifically, the spin-up channel of one Mn atom couples to the spin-up channel of neighboring Mn atom. Considering the spin-up channel, step (a) to (b) and step (a0 ) to (b0 ) describe the p–d exchange coupling, which results in raising the t2d state upward by 2D1pd and lowering the t2p state by the same amount. Steps (b) or (b0 ) to (c) illustrate the d–d coupling between the two t2d states of neighboring Mn atoms. Consequently, the holes are placed at a high energy level and electrons are placed at an energy level that is lowered by 2D1dd. The net energy gain in the spin-up channel is therefore 2nhD1pd  nhD1dd , where nh is the number of holes in the VBM. In the spin-down channel, the p–d coupling lowers the energy of occupied t2p spin-down state by 2D2pd . The net energy gain in the spin-down channel is  12D2pd for the six electrons in the t2p state. The total net energy gain in the FM configuration is  2nhD1pd  nhD1dd –12D2pd . In the antiferromagnetic configuration, the majority spin state of one Mn atom couples only to the minority spin states of the other Mn atom with opposite moment. The situation can be analyzed similarly as in the FM configuration. The total net energy gain in the AFM 1;2 1 2 configuration is  (6  nh)D1;2 dd  2nhDpd  12Dpd , where Ddd term represents the level repulsion caused by the coupling between the majority spin d state and the minority spin d state. The energy difference between the FM and AFM state is 1 therefore (6  mh)D1;2 dd  nhDdd . This indicates that when the system has holes at the t2d level instead of VBM, the stabilization of the FM or AFM state is not directly related to the p–d exchange splitting, but to the d–d coupling term D1dd (double exchange) and D1;2 dd . In the case of (Zn,Mn)O, there is no hole, so the AFM state is more stable. The FM state is stable for (Ga,Mn)N for moderate Mn concentration (e.g., 6.25% Mn), but it gives way to the AMF state when Mn concentration increases due to the increase of the AFM stabilization energy D1;2 dd . In addition to Mn magnetic ion, theoretical investigations have also been reported for other transition metals such as Cr [690,703,704], Co [705], and vanadium (V) [690,706] in GaN. Das et al. [703] studied the electronic structure, energy bands, and magnetic properties of Cr-doped GaN from first principles with GGA approximation. The coupling between Cr atoms was found to be ferromagnetic in both crystal and small cluster forms of GaN. Kim et al. [704] reported on the electronic structure of Cr-doped GaN observed by hard X-ray photoemission spectroscopy and first principle calculations. They proposed that the ferromagnetic interaction between distinct Cr atoms may be mediated by the Cr 3d–N 2p–Ga 4s hybridization, and more data from magnetic and optical characterization would lend more credence. Switching gears to another transition element, 6.25% Co-doped GaN was calculated for its magnetic properties by Hong and Wu [705] using the fullpotential linearized augmented plan wave (LAPW) method. The authors found that Co atoms prefer to remain close to each other in GaN and form ferromagnetic ordering, with approximately 10 meV lower energy than the ferromagnetic phase. In a similar fashion, vanadium (V) was predicted [690] to be ferromagnetic when doped in GaN according to ab initio calculations within the local spin density approximation, which gives the electronic structure of the 3d metal doped GaN by

4.9 Intentional Doping

the KKR–CPR method. However, at least in one experiment, V-implanted GaN showed paramagnetic behavior up to 320 K [706]. In terms of the theory in regard to Ga1xGdxN, albeit in the cubic form, Dalpian and Wei [707] undertook ab initio band structure calculations with symmetry arguments showing that the magnetic properties of Ga1xGdxN are notably different from that of TM-doped GaN. The coupling between the Gd atoms in the alloy is antiferromagnetic but the 4f orbitals in the rare earth elements are shielded from the host material and more localized as compared to the transition metal atoms; thus, the direct coupling between the 4f ions is expected to be weak. As compared to the 3d transition metal elements, the 4f rare earth elements can have larger magnetic moments and can couple strongly with the host s electrons. Therefore, the ferromagnetic state can be stabilized by introducing shallow donors that are present in unintentionally doped GaN. The large magnetic moments observed can be explained by polarization of donor electrons. Therefore, the electron-mediated ferromagnetism is in effect here. This discussion has a good deal of relevance to the experimental observations discussed in Section 4.9.7. Because many exchange interactions have been invoked, Table 4.16 summarizes salient features of various interactions. It appears that DFT calculations augmented by relevant approximations for exchange correlation energies have the best chance of determining a reasonably true picture, and it is compelling to tabulate the attributes of various approaches within DFT, as done in Table 4.17. Experimental observations of ferromagnetism in both p-type and n-type GaN DMS materials have been amply reported in spite of the preponderance of the theoretical results, mainly centering around non-ab initio calculations, which require p-type GaN for a strong p–d interaction involving holes as compared to s–d interaction involving electrons. The discrepancy between these theories and experiments could perhaps be better understood by invoking the possibility that some observations might be clouded by magnetic contamination, imperfections of material quality specific to growth techniques, such as dislocations in the host materials [708], clusters [709,710] Table 4.16 A comparison of magnetic interactions.

Interaction

Definition

RKKY

Indirect exchange coupling of magnetic moments over relatively large distance via band electrons due to the Coulomb exchange. It becomes efficient when a high concentration of free carriers is present such as in metals for which it was developed. Direct coupling of magnetic ions through overlap of magnetic orbitals Spins of two magnetic ions are correlated due to the exchange interaction between each of the two ions and the valence p-band. Couples magnetic ions in different charge state by virtual hopping of the “extra” electron from one ion to the other through interaction with p-orbitals.

Direct superexchange Indirect superexchange Double exchange

j1101

Definition Density functional theory

Local density approximation

Generalized gradient approximation

Becke three-parameter Lee–Yang–Parr

LDA

GGA

B3LYP

Method

DFT

Approximation for the exchange correlation energy

Uses hybrid exchange energy functionals and gradientcorrected correlation functionals

Locally substitutes the exchange correlation energy density by that of an electron gas evaluated at the local density and its gradient and higher terms

For regions of a material where the charge density is slowly varying, the local charge density can be considered to be the density of an equivalent uniform homogeneous electron gas

Uses the charge density as the fundamental system variable, and describes the ground state properties by using certain functionals of the charge density

Assumptions

The most popular DFT method. Produces more accurate results

Yields improvement over LDA in the description of finite systems

Simple and produces moderately accurate results in most cases.

Simplifies the many body problem by using the electron charge density as fundamental variable rather than the wave function

Advantages

Table 4.17 A comparison of different methods in DFT calculations to approximate the exchange correlation energy.

Occasional large errors

Overcorrects the lattice constant for semiconductor systems compared to LDA

Underpredict atomic ground state energies and ionization energies, and over predicts binding energies

Applicable only for the ground state, needs approximation for exchange correlation energy functional

Limitations

1102

j 4 Extended and Point Defects, Doping, and Magnetism

4.9 Intentional Doping

precipitates [711,712], antisite defects [713], and nonsubstitutional impurity sites [714]. On the theory side, ab initio calculations, which are comprehensive by their nature in terms of being inclusive of all the mechanisms that may take place, are a must for an accurate determination of whether the material is FM and if so what the Curie temperature is. Already, the mean field theory appears to overestimate the Curie temperature substantially for lower magnetic ion densities. It should be mentioned, however, that the mean field theory is pretty accurate for predicting the Curie temperature in Mn-doped GaAs, at least for concentrations above 5%. Additionally, we must not lose sight of the fact that the results from different theoretical approaches do not agree well to say the least. Presumably, no single model is capable of explaining the properties of a wide class of dilute magnetic semiconductors including ZnO and GaN with their many variants. This being the case, multisourced possible explanations may have to be taken into consideration [715]. In particular, if the solubility limit of the magnetic dopant is exceeded, nonuniform ferromagnetic behavior or precipitates exhibiting ferromagnetism may form. Naturally, the magnetic properties of any such precipitates will depend on the growth conditions. In general, the contribution by precipitates to the overall magnetic properties of the bulk DMS cannot be excluded. A clear need exists for further research in this field if clarity is to be obtained. In this vein, extended X-ray absorption fine structure studies of DMS can shed light on the detailed microscopic structure of the lattice, which is more complicated than assumed in at least some of the theoretical approaches taken. Such careful structural investigations have been undertaken in standard bearer and traditional II–VI compounds [716], and to a lesser extent in GaN, which are discussed in Section 4.9.7. It is believed that with more progress in the synthesis and characterization techniques for DMS materials, improvements in various theories to understand the underlying mechanism will ensue. What is certain is that in due time, the science will correct its course. Before we delve into discussing the properties of DMS materials, particularly those discerned by magnetoelectrical and magnetotransport measurements, it is imperative that a working familiarity with these methods is developed. It is with this motivation that we venture into the world of the aforementioned measurements. 4.9.6.6 A Primer to Magnetotransport Measurements Due to the presence of the anomalous Hall effect, which is also known as the extraordinary or spin Hall effect, the Hall resistance RHall is empirically known to be a sum of the ordinary and anomalous Hall terms. The Hall resistivity in a magnetic semiconductor is given by

rxy ¼ r0xy þ rsxy ¼ Ro B þ Rs ðrxx ÞM ? or RHall

ð4:111Þ R0 RS M? ; ¼ Bþ t t

where the normal contribution r0xy represents the nonmagnetic (normal) component of the off-diagonal resistivity matrix element and is proportional to the external

j1103

1104

j 4 Extended and Point Defects, Doping, and Magnetism magnetic field B, and rsxy represents the magnetic (anomalous) component of the offdiagonal resistivity matrix element and is proportional to the macroscopic magnetization M?, which is normal to the sample surface. The terms R0 ¼ 1/pe and Rs (which is a function of rxx) represent the normal and anomalous Hall coefficients, and t stands for the thickness of the film. Further, Rs arises from the spin–orbit interaction, which induces anisotropy between scattering of spin-up and spin-down electrons. The Rs term is related to the sheet resistance through Rgsheet , where g is a constant and can assume values between 1 and 2, the end points corresponding to skew scattering and side jump mechanism, respectively. The constant g obviously determines the magnitude of the anomalous Hall effect term and scales with the extent of the spin–orbit coupling for the carriers at the Fermi level and the exchange energy that describes the ratio of carrier spin polarization to the magnetization normal to the surface. For a given normal component of magnetization, the anomalous Hall effect is much stronger for holes than for electrons in the tetrahedrally coordinated semiconductors. Moreover, anomalous Hall effect depends on the extent to which the electrons are spin polarized; the effect ceases to be proportional to the magnetization when carrier spin splitting becomes comparable to the Fermi energy. Because magnetotransport measurements are very sensitive to magnetization, they have become a cornerstone in characterizing dilute magnetic semiconductors, and further this method is well applicable to materials where the magnetic ordering or moment is very small. In ferromagnetic samples the anomalous component dominates and the normal component can be neglected. If additionally, the skew scattering is the dominant process, the anomalous Hall coefficient would be proportional to the sheet resistance of the sample [626]. Thus, M can be calculated from the above expression as being proportional to the ratio of the Hall coefficient to the sheet resistance. The Hall measurements should be carried out in the applied magnetic field limit where the magnetization is fully saturated (i.e., at low temperature and high magnetic field). Eventually, if and when GaN-based DMS materials advance to the point where reliable Hall measurements can be made, the anomalous Hall effect would be a reliable means for determining whether the material is ferromagnetic and further what the Curie temperature is. Representative data for the well-established system of GaMnAs are presented in Section 4.9.6.7.2. 4.9.6.6.1 Faraday Rotation, Kerr Effect, and Magnetic Circular Dichroism (MCD) While magnetization measurements with SQUID are imperative in the early stages of development, eventually the extent of magnetic ordering must be such that electrical and optical measurements are feasible and used for further research and transitioning to development. The SQUID measurements are sensitive to any magnetization, inclusive of magnetic impurities, precipitates, clusters, and mixed magnetic phases. The magnetoresistance (MR) measurements and optical measurements, particularly the combination of the two, are therefore the most reliable methods for unequivocally determining whether the material in question has ordered magnetization. The details of the former in nonmagnetic materials are amply discussed in Volume 2, Chapter 3, and a succinct discussion of it is given in Section 4.9.6.6 for magnetic semiconductors. One of the optical measurements relies

4.9 Intentional Doping

on Faraday rotation (FR) that can be simply viewed as the rotation in polarization plane of a linearly polarized light as it propagates through a magnetic medium [717]. Let us take a moment to review linearly and circularly polarized light. Let the in! ! plane coordinates of the material be in x and y and the normal to the surface be in ! ! ! the z direction. The functional dependence of linearly polarized light in x , y , and ! z directions on time can be expressed as C^ x e  iwt ; C^y e  iwt ;

and

!

C z e  iwt ;

ð4:112Þ

where C is a constant (amplitude) and no phase shift is introduced. A right-hand (þ) and left-hand () polarized elliptical polarization would take the form 1 ^  ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða^ s x  ib^yÞe  iwt : a2 þ b2

ð4:113Þ

A special case of which is circular polarization when a ¼ b, in the case of which we can write 1 ^  ¼ pffiffiffi ð^ ð4:114Þ x  i^yÞe  iwt : s 2 Right-hand and left-hand circular polarizations arise from material exhibiting two different refractive indices, nþ and n, for positive and negative helicity. In semiconductors, application of a magnetic field in the z-direction, parallel to surface normal, causes Zeeman splitting of the electronic energy levels. The transitions associated with some are right-hand and others left-hand circularly polarized in addition to transitions with linear polarization. Faraday Rotation

qF ¼

Faraday rotation is given by the well-known expression as [718]

El  ðn  n þ Þ; 2
hc

ð4:115Þ

where nþ and n are the refractive indices of the medium for right-hand and left-hand circularly polarized light, respectively. Different refractive indices for right-hand and left-hand circularly polarized light forms the basis for circular bifringence. E is the photon energy, l is the length traversed by photons,
h is the reduced Planck’s constant, and c is the speed of light in vacuum. In the case of H ¼ 0, the dispersion in the refractive index has the form n2  1 /

X 2 i;j E ij

f ij  E2

;

ð4:116Þ

where Eij is the transition energy between levels i and j, and fij is the oscillator strength. Interband, intraband, internal d–d transitions associated with the magnetic ion in the host material, and those associated with impurities and defects in principle contribute to the circular bifringence and, through this, to Faraday effect, provided that there is Zeeman splitting. It should be noted that in a DMS, as soon will be discussed, the spin exchange interaction/magnetic field causes large Zeeman splitting of the electronic energy levels.

j1105

1106

j 4 Extended and Point Defects, Doping, and Magnetism For frequencies near Eij, the refractive index given by Equation 4.116 is dominated by the term corresponding to that particular transition. The difference between nþ and n ^þ can then be related to the energy difference DEij between transitions observed in s  ^ polarizations. By Taylor expansion, it can be shown that the refractive index and s difference for left-hand and right-hand circularly polarized light can be expressed as ðn   n þ Þ ¼

qn DE ij : qE

ð4:117Þ

Then, the Faraday rotation becomes qF

El qn DE ij : 2
hc qE

ð4:118Þ

If we focus on a single oscillator of energy E0 in Equation 4.116, representing, for example, the transition at the G point in the Brillouin zone, the refractive index can be written as n2 ¼ n20 þ F 0 ðE 20  E 2 Þ  1 ;

ð4:119Þ

where F0 is a constant associated with the oscillator strength of the excitonic transition, and n represents all contributions to n in Equation 4.117. One can then obtain for Faraday rotation 1 F0l E2 DE 0 ; ð4:120Þ n 2
hc ðE 20  E 2 Þ2 where n represents the refractive index in the absence of magnetic field. For E close to E0 we can assume that n2 ffi F 0 ðE 20  E 2 Þ  1 , which leads to pffiffiffiffiffi  El  1 y2 F0 l þ qF ¼ ; ð4:121Þ ðn  n Þ ¼ DE 0 2
hc E 0 ð1  y2 Þ3=2 2
hc qF ¼

where y ¼ E/E0. Shallow impurity and intraband transitions are not taken into consideration in Equation 4.121. By defining pffiffiffiffiffi D y2 F 0 qDE 0 VðEÞ ¼ ; and D ¼ E 0 ð1  y2 Þ3=2 2
hc qH the Faraday rotation expression, expressing the derivative of the rotation angle with respect to the magnetic field, can be simplified to
 qqF El qn e qn lH ¼ VðEÞl or qF DE or qF or qF ¼ VðEÞHl; qH 2
hc qE 2mc 2 ql ð4:122Þ where l is the layer thickness in this case, l is the wavelength, and V(E) is a constant named Verdet constant that depends on the material and the photon energy, E. In nonmagnetic materials, the Verdet constant is very small, for example, 3.25 · 104 ( cm1 Oe1) which prohibits the use of these kinds of materials for magneto-optical effects. However, this constant is large in ferromagnetic materials such as rare earth transition metals. In addition to magnetic properties, the medium should

4.9 Intentional Doping

also be transparent to the wavelength of the light source in use. How the Faraday rotation is additionally related to spin exchange interaction and magnetic circular dichroism is discussed following the discussion of Zeeman splitting. The experimental implementation of the above mentioned magneto-optical effect has quite a few applications, in addition to materials characterization, among which is an optical isolator. A linearly polarized light through a medium is rotated by an angle y. If and when the transmitted light is reflected by a mirror and fed back into the same medium, an additional polarization rotation in the amount of y occurs, leading to a total rotation of the light polarization by 2y, as shown schematically in Figure 4.153. In ferromagnetic materials, the rotation angle is typically plotted versus the magnetic field at temperatures below and above Curie temperature. Ideally any rotation would be minimal above Curie temperature and sizable below Curie temperature. Because the magnetization saturates with magnetic field, the rotation angle would also saturate. Essentially, the rotation angle would be proportional to magnetization. If the magnetization and length of the medium are such that the rotation angle y is equal to 45 , a total rotation of 90 is obtained the second pass through which forms the basis for an optical isolator, as depicted in Figure 4.153. For example, a semiconductor laser could be separated from the medium, such as an optical fiber, which is situated to receive the laser radiation by an optical isolator; there would not be any reflected light coming back into the semiconductor laser to interfere with its operation [719]. Ferromagnetic GaN or AlN for that matter would

l θ

H

I I 2θ

θ

H

I Figure 4.153 A schematic showing the Faraday rotation suffered by light in propagating from left to right (upper figure) and then again from right to left after reflection (lower figure), leading to a total rotation of 2y.

j1107

1108

j 4 Extended and Point Defects, Doping, and Magnetism pave the way for Faraday rotation based isolators operative in a wide range of wavelengths due to the large bandgaps of these semiconductors. Magneto-Optical Kerr Effect The other related phenomenon, called the magnetooptical Kerr effect or Kerr rotation (KR), represents rotation of the polarization plane of the reflected light from the surface of a magnetic material where the magnetization is perpendicular to the surface (polar Kerr rotation), as shown in Figure 4.154. Moreover, the direction of the aforementioned rotation is dependent on the direction of the magnetization M. This is useful in reading the information stored in the form of direction of magnetization and for studying the domain structure of magnetization. From a practical point of view, the value of DE can be inferred from Equation 4.122, using experimentally determined @n/@E. This value is about 35 meV, and is fairly independent of E between 1.25 and 1.6 eV for Ga1xMnxAs. Using DE ¼ N0x(b  a)

l

θ

H

I

k

k

M

Figure 4.154 Faraday rotation in reflection is termed the magneto-optical Kerr effect or Kerr rotation wherein the polarization angle of the reflected light changes upon reflection from a magnetized medium such as ferromagnets.

4.9 Intentional Doping

hSzi (a more detailed discussion is given in conjunction with the discussion of magnetic circular dichroism and Equation 4.132), which is valid only at the band edge, N0x(b  a)hSzi 1 ev is obtained. The terms N0a and N0b are the spin exchange integrals for the conduction band and valence band, respectively, and hSzi is the thermal average of the transition metal such as Mn spins in the direction of B, typically determined by an independent magnetization measurement. The positive value of N0b  N0a reflects the positive sense of the Faraday rotation, which is opposite to that of II–VI DMSs such as undoped (Cd,Mn)Te discussed in conjunction with MCD below. This surprising result has been attributed to a large Burstein–Moss shift due to the high hole concentration specific to Ga1xMnxAs DMS [663,720]. In the context of these pages, it is not the applications that we seek, but rather it is the use of this phenomenon for material characterization that we desire. In experimental conditions concerning ferromagnetic materials, the degree of rotation versus the magnetic field in the ferromagnetic regime, that is, below the Curie temperature, above the critical temperature, that is, the paramagnetic regime, and preferably as a function of temperature can be conducted. From the collected data, one can discern information about the magnetic nature of the material. Magnetic Circular Dichroism (MCD) The other optical effect is the magnetic circular dichroism that is caused by the difference in absorption or transmission of right-hand (sþ) and left-hand (s) circularly polarized light. If the associated transmitted intensities and absorption for right-hand (positive superscript) and left-hand (negative superscript) polarized light are defined as Iþ and I, and aþand a, respectively, the magnitude of the MCD signal in degrees is defined as

Pcirc ¼ ða þ  a  Þ=ða  þ a þ Þ ðT þ  T  Þ=ðT þ þ T  Þ

ð4:123Þ

To gain an understanding of this effect in the particular semiconductor of interest, polarization of each transition involving heavy, light, and spin–orbit split-off bands must be known. In addition, the transition probabilities (oscillator strengths), at least the relative values, for each band are also needed. Continuing on, the knowledge of the selection rules as to the polarization for spin-up and spin-down electron in each of the bands are also required. These conditions reduce what needs to be known to a relationship between Pspin (given by Equation 4.103) and Pcirc (given by Equation 4.121). In this vein, a zinc blende semiconductor such as GaAs, and a generic wurtzitic semiconductor have been chosen for discussion. While the latter choice is obvious, the former is made because of its well-understood properties in the context of optical processes, the nature, and the degree to which those transitions are affected by strain and Zeeman splitting. These processes are intricately tied in with optical properties, the details of which are discussed in Volume 2, Chapter 5. Because the selection rules for GaAs are well established and MCD in ferromagnetic GaAs is, without comparison, more advanced than that in GaN, the MCD data for GaAs are discussed here for model. In many cases, depending on the excitation energy of the incident optical beam in relation to transitions involving the valence band states and conduction band, the right-hand and left-hand circularly polarized light cancel each

j1109

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j 4 Extended and Point Defects, Doping, and Magnetism other in the absence of magnetic field. However, with magnetic field (internal, external, or both) one or the other would be enhanced. To reiterate, MCD effect essentially relies on relative strengths of circular polarization and spin polarization. The former depends on the normalized relative intensities of right-hand and left-hand circularly polarized light defined by Equation 4.121. The latter depends on the spin polarization defined by Equation 4.103. Zeeman splitting of valence and conduction band states is caused by external magnetic field or the inherent magnetic ordering. When circularly polarized light is absorbed by a semiconductor, the selection rules are such that the electrons created would be spin polarized. Likewise, if spin selected transitions are dominant in a semiconductor (relative oscillator strengths of the transitions involved), the emitted light would be circularly polarized. By studying the light polarization, one can deduce information about the manner in which and degree to which the band states, particularly the valence band states, are split and can discern Zeeman splitting and therefore magnetization. At the very least the presence of the former can be used to decide if the material under investigation is ferromagnetic or not. In the case of Zeeman splitting by magnetic ordering, the degree to which that splitting occurs depends on the exchange interaction causing the ordered magnetic state. By performing the measurements versus temperature, one can determine the transition temperature. It should be mentioned that spin polarization is lost over time due to several processes such as interaction with magnetic impurities, spin–orbit interaction, to some extent electric field, and so on, which are critical for device applications of spin-polarized materials. In an effort to gain a degree of familiarity, let use first discuss GaAs, as a representative of many III–V zinc blende semiconductors, with no magnetic field and no strain. As will soon be described due to strong spin–orbit interaction in GaAs (DSO ¼ 340 meV), selected circularly polarized light (e.g., by injection of spin-polarized electron) is easily obtained even for bulk samples with degenerate heavy-hole and light-hole states, as the optical transitions involving the HH states are three times stronger than those involving the LH states (therefore, Pspin ¼ 0.5 Pcirc as detailed below). Let us now discuss the details of the process. Various energy levels associated with conduction and valence band states are subject to selection rules as to their participation in optical excitation and recombination. Assuming that spin-polarized carriers (electrons and holes) are generated, they would exist for a time before they recombine. If a fraction of the carriers’ initial orientation survives longer than the recombination time, if t < ts, where ts is the spin relaxation time, the luminescence (recombination radiation) will be partially polarized [632]. By measuring the circular polarization of the luminescence, it would then be possible to investigate the spin dynamics of the nonequilibrium carriers in semiconductors. Of device relevance, it would be possible to extract pertinent parameters such as the spin orientation, the recombination time, or the spin relaxation time of the carriers. The band structure of GaAs is depicted in Figure 4.155a. The 0 K bandgap of GaAs is Eg ¼ 1.52 eV, the light- and heavy-hole bands are degenerate at the zone center (which can be split by strain and magnetic field) while the spin split-off band is separated from the degenerate heavy- and light-hole bands by the spin–orbit split-off

j

4.9 Intentional Doping 1111 " Figure 4.155 Interband transitions in GaAs selected because of its well-known band structure and also its well-established and wellcharacterized properties in terms of magnetic ion doped diluted magnetic semiconductors: (arrows indicate emission but the concept is just as applicable to transitions from the valence band subband to the conduction band as in absorption). (a) Schematic band structure of GaAs near the G point, the center of the Brillouin zone. As for the terms, Eg is the bandgap and DSO the spin–orbit splitting; CB, conduction band; HH, valence heavy hole; LH, light hole; SO, spin–orbit split-off subbands; G6,7,8 are the corresponding symmetries at the k ¼ 0 point representing conduction, HH, LH, spin–orbit (SO) bands, or, more precisely, the irreducible representations of the tetrahedron group Td (see, e.g., Ref. [721]). The terms s1/2 and p3/2 and p1/2 represent the conduction band (s-like) and valence band (p-like) type of orbitals. (b) Selection rules for interband transitions between the Jz projection of the angular momentum along z-direction, sublevels for circularly polarized light sþ(righthand circular polarization or positive helicity that results from transitions between the Jz ¼ 1/2 conduction band states and Jz ¼ 3/ 2 heavy-hole states, and Jz ¼ þ1/2 conduction band and Jz ¼ 1/2 light-hole states), and s (left-hand circular polarization or negative helicity, which results from transitions between the Jz ¼ þ1/2 conduction band states and Jz ¼ þ3/2 heavy-hole states, and Jz ¼ 1/2 conduction band and Jz ¼ þ1/2 light-hole states). The numbers by each transition indicate the relative transition intensities, with respect to the light-hole subband to the conduction band (absorption or excitation of carriers to higher bands), or the conduction band to the light-hole subband transition (emission), which apply to both excitation and radiative recombination (depicted by the arrows). The circular polarization (s polarization) for light energies that would not excite the spin split-off band is ideally 50%, which becomes 0 if the spin–orbit split-off band is also excited. For completeness, the transitions between the Jz ¼ 1/2 conduction band states and Jz ¼ 1/2 light-hole states, and Jz ¼ þ1/2 conduction band states and Jz ¼ þ1/ 2 light-hole states, which are linearly polarized (p polarization), are also shown as depicted by

two-way arrows in the figure. The transition probability or the emission intensity normalized to the Jz ¼ 1/2 conduction state to the Jz ¼1/ 2 state transitions (indicated with 1) are also indicated in numbers for GaAs. The circular polarization resulting from the conduction band to the heavy-hole states are three times more intense than the circular polarization resulting from the conduction band states to the light-hole valence band states. The linearly polarized transitions are twice as intense as the circular polarization involving light-hole states. (c) Removal of the valence band heavy- and light-hole degeneracy by, for example, strain inducing either by lattice mismatch or by confinement in a quantum well, which increases the electron polarization to nearly 100%. Note that heavy- and light-hole states are no longer degenerate. Both the tensile (left) and compressive (right) in-plane biaxial strain cases are shown. The respective ratios of various transitions (oscillator strengths) have been assumed to be the same as in the relaxed case. Note that spin is indifferent to strain, which means that spin-up and spin-down states are moved in the same direction by strain, but not to magnetic field, as spin-up and spin-down states in a given band are split and moved in opposite directions as shown in (d). In part courtesy of W. Chen, Linko1ping University. (d) Removal of the valence band heavy- and lighthole degeneracy as well as splitting the spin-up and spin-down states by application of magnetic field. The total splitting is enhanced due to sp-d interaction in DMS materials in the form of xN0a<Sz> for the conduction band states, xN0b<Sz> for the HH and LH valence band states, and (1/3)xN0a<Sz> for the spin–orbit split-off band. Here, N0, x, a, b, <Sz> represent the number of cations per unit volume, mole fraction of magnetic ions, the product of Bohr magneton and the g factor for the respective bands, and average spin for each magnetic ion site, respectively. Note that magnetic field/magnetization causes Zeeman splitting, and direction of splitting either up or down in energy is spin dependent. If the semiconductor is ferromagnetic as is the case of GaMnAs, one can either couple polarized light to the symmetry/splitting allowed bands or cause polarized light emission by tuning the wavelength. (Please find a color version of this figure on the color tables.)

1112

j 4 Extended and Point Defects, Doping, and Magnetism

Figure 4.155 (Continued )

4.9 Intentional Doping

Figure 4.155 (Continued )

j1113

1114

j 4 Extended and Point Defects, Doping, and Magnetism Table 4.18 Angular and spin part of the wave function at the G point in a zinc blende symmetry (GaAs) [632].

 J; J z

Symmetry G6

j1=2; 1=2i j1=2;  1=2i

G7

j1=2; 1=2i j1=2;  1=2i j3=2; 3=2i

G8

j3=2; 1=2i j3=2;  1=2i j3=2;  3=2i

Wave function jS"i jS#i E 1=2  ð1=3Þ ½ðX þ iYÞ#  Z"
E 1=2 ð1=3Þ ½ðX  iYÞ" þ Z#
E 1=2 ð1=2Þ ðX þ iYÞ" E 1=2 ð1=6Þ ½ðX þ iYÞ# þ 2Z"
E 1=2  ð1=6Þ ½ðX  iYÞ"  2Z#
E 1=2 ð1=2Þ ðX  iYÞ#

parameter, DSO ¼ 0.34 eV. Following Ichenko and Pikus [721], Bloch states are denoted according to the total angular momentum J and its projection onto the can be calculated as in atomic positive z-axis Jz as J; J z . The allowed transitions  J ¼ 3=2; J ¼ 3=2 (heavy hole) or physics between the ground states with z  J ¼ 3=2; J ¼ 1=2 (light hole) and the excited levels with J ¼ 1=2; J ¼ z z  1=2i. Further, expressing the wave functions with the symmetry of s, px, py, and pz orbitals as jSi, jY i, and jZi, respectively, the band wave functions can be written, following Ref. [632] that relied on Ref. [722] with some typos removed, as listed in Table 4.18. To obtain the excitation (or recombination) probability, which is necessary for understanding dichroism, let us consider the photon propagation vector to be along the z-direction. Further, let us assume that sþ and s represent the righthand and left-hand circular polarization (helicity of the exciting light). The dipole operator corresponding to the sþand s optical transitions can be represented as m being proportional to ðX  iYÞ / Y 1 1 , where Y l is the spherical harmonic, using Table 4.18. Doing so with the help of h1=2;  1=2jY 1 j3=2;  3=2i 2 1 ¼ 3; h1=2; 1=2jY 1 j3=2;  1=2i 2

ð4:124Þ

1

would allow determination of the relative intensity of the sþ transition between the heavy-hole (Jz ¼ 3/2) subband and conduction band (between the conduction band and the heavy-hole subband in the case of emission) and the light-hole (Jz ¼ 1/2) subband and the conduction band (between the conduction band and light-hole subband in the case of emission) [632]. The relative intensities are indicated in Figure 4.155b with number as 3 for the heavy hole related transition and 1 for the light hole related transition, and 2 for the conduction band to the spin–orbit split-off band transitions. The relative transition probabilities of the other transitions can be found in a similar fashion and are indicated with numbers in Figure 4.155b. If the outgoing

4.9 Intentional Doping

light in the case of emission (incoming light in the case of absorption) is in the zdirection, the helicities are reversed. It should be mentioned that the polarization vector also describes the electron dipole motion in that electric dipole radiates normal to its own motion and does not radiate in the parallel direction. This discussion segues into the selection rules for light absorption and emission. To begin with, only those dipoles in the plane can absorb or radiate. The probability of transitions involving electrons and heavy holes and light holes, even when they are degenerate as in relaxed GaN, are not the same because of different oscillator strengths, the heavyhole transition being three times larger than that that for light holes for GaAs. The transitions involving the spin–orbit split-off bands, when they are excited, are twice as strong as those involving light-hole states. Removal of the valence band heavy and light-hole degeneracy by, for example, strain inducing either by lattice mismatch or by confinement in a quantum well increases the electron polarization to nearly 100%. Note that strain breaks down the heavy- and light-hole state degeneracy regardless of the sign of the strain, that is, tensile of compressive. The sign of the strain naturally has an effect on which valence band states move up in energy and which move down as well as dispersion of the band away from the G point. The relative positions of CB, HH, LH, and SO bands are shown in Figure 4.155c. The respective ratios of various transitions (oscillator strengths) assumed to be the same as in the relaxed case and are indicated below the symbols depicting the nature of polarization. Note that spin is indifferent to strain, which means that spin-up and spin-down states are moved in the same direction by strain. The knowledge of how the bands split with magnetic field in a DMS material, which can be large, can be used in an MCD experiment, which is about to be discussed. In the case of absorption, the spin polarization of the excited electrons is dependent on the photon energy
hw. Note that the holes are initially spin polarized also, but they lose spin polarization fast, on the order of momentum relaxation time, and are therefore omitted from this particular discussion. For
hw between Eg and Eg þ DSO, only the light- and heavy-hole subbands contribute, leaving the spin–orbit split-off band out of the picture. Denoting the density of electrons polarized parallel (Jz ¼ 1/2) and antiparallel (Jz ¼ 1/2) to the direction of light propagation by nþ and n, one can define the spin polarization (given in Equation 4.103 but repeated here for convenience) as [632] Pn ¼ ðn þ  n  Þ=ðn þ þ n  Þ:

ð4:125Þ

Applying the above equation to the zinc blende symmetry and without magnetic field (nonferromagnetic) and denoting the transitions involving light- and heavyhole transitions as |1/2, 1/2| representing nþ whose normalized intensity is 1 parallel to the direction of light propagation, and |1/2, 3/2| representing n whose normalized intensity is 3 antiparallel to the direction of light propagation, one obtains Pn ¼ ð1  3Þ=ð1 þ 3Þ ¼  1=2

or  50%;

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j 4 Extended and Point Defects, Doping, and Magnetism for the spin polarization at the moment of photoexcitation (absorption measurements). The spin is oriented against the direction of light propagation, because there are more transitions from the heavy-hole than from the light-hole subbands. The circular polarization of the luminescence in a zinc blende structure is defined as [632] Pcirc ¼ ðI þ  I  Þ=ðI þ þ I  Þ; þ

ð4:126Þ



þ



where I and I represent the radiation intensity for helicity s and s or, righthand and left-hand circular polarizations, respectively. In this particular case, the |Jz ¼ 1/2, 1/2| transition, 1/2 conduction band state to 1/2 valence band light-hole transition, represents right-hand circular polarization parallel to the direction of light with an electron polarization of nþ; the |Jz ¼ 1/2, 3/2| transition, 1/2 conduction band state to 3/2 valence band heavy-hole transition, represents left-hand circular polarization parallel to the direction of light with an electron polarization of 3nþ; the |Jz ¼ 1/2, 3/2| transition, 1/2 conduction band state to the 3/2 heavy-hole state, represents right-hand circular polarization and antiparallel to the light with an electron polarization of 3n; and |Jz ¼ 1/2, 1/2| transition, 1/2 conduction band state to 1/2 light-hole state, represents the left-hand circular polarization antiparallel to the light with an electron polarization of n. Utilizing these delineations and the fact that transitions involving the conduction band states to the heavyhole states are three times more intense that those from the conduction band to the light-hole states, one can calculate the electron polarization for right-hand circularly polarized light as Pcirc ¼

ðn þ þ 3n  Þ  ð3n þ þ n  Þ Pn 1 ¼  ¼ : 2 ðn þ þ 3n  Þ þ ð3n þ þ n  Þ 4

ð4:127Þ

If the excitation involves transitions from the spin split-off band, that is, if
w E g þ DSO , the electrons will not be spin polarized (Pn ¼ Pcirc ¼ 0), underlining h the vital role of spin–orbit coupling for spin orientation. On the contrary, removal of the valence band heavy- and light-hole degeneracy either by strain, confinement or by magnetic field, as shown in Figure 4.155, can substantially increase Pn, perhaps even up to the limit of complete spin polarization [723]. An increase in Pn and Pcirc in strained GaAs owing to a lattice mismatch with the buffer layer/substrate, or due to confinement such as in quantum well heterostructures, has been reported by Vasilev et al. [724] and Oskotskij et al. [725], with Pn values greater than 0.9. A magnetic field, however, causes Zeeman splitting with considerable impact on the band structure, as shown in Figure 4.155d, at the G point in a zinc blende symmetry. In DMS materials, the splitting can be quite large due to the strong sp–d interaction. In a sense, the effect of the magnetic field on the s and p band electrons is amplified by the magnetic moment of the magnetic ion through the sp–d exchange interaction. Let us now discuss the extent and nature of splitting for each of the four bands. For the spin-up and spin-down states in the conduction band: E "c ¼ E 0g  ð1=2ÞxN 0 ahSz i and

E #c ¼ E 0g þ ð1=2ÞxN 0 ahSz i;

ð4:128Þ

4.9 Intentional Doping

where E 0g is the bandgap with zero magnetic field, and N0, x, a, hSzi represent the number of cations per unit volume, mole fraction of magnetic ions, a constant that represents the s–d exchange interaction integral, and thermal average spin for each magnetic ion site along the direction of the external magnetic field, respectively. The term a is positive and the term hSzi is normally negative for zinc blende symmetry that means that the spin-up band moves up in energy and spin-down moves down in the presence of magnetic field. The total splitting of conduction band spin-up and spin-down states is xN0ahSzi. Whether a given band moves up or down in energy is also dependent on the sign of a. The G8 degenerate heavy- and light-hole bands split into four bands, 2 HH (Jz ¼ 3/2) and 2 LH (Jz ¼ 1/2) bands with HH energies: DðE HH ; J z ¼ þ 3=2Þ ¼  ð1=2ÞxN 0 bhSz i and DðE HH ; J z ¼  3=2Þ ¼ ð1=2ÞxN 0 bhSz i;

ð4:129Þ

where b is the p–d exchange interaction integral for the valence band. Recognizing that the spin exchange interaction term iSzh is normally negative and b is negative for zinc blende symmetry, the Jz ¼ þ3/2 would move down in energy and the Jz ¼ 3/2 state would move up in energy. Similarly, for the G8 LH bands split according to DðE LH ; J z ¼ þ 1=2Þ ¼  ð1=4ÞxN 0 bhSz i and DðE LH ; J z ¼  1=2Þ ¼ ð1=4ÞxN 0 bhSz i:

ð4:130Þ

As in the case of the G8 HH states, recognizing that the spin exchange interaction term hSzi and b are both negative for the zinc blende symmetry, the Jz ¼ þ1/2 move down in energy and the Jz ¼ 1/2 state would move up in energy with magnetic field. Moreover, because the HH states split by an amount twice as that for the LH states, both the Jz ¼ 1/2 and Jz ¼ þ1/2 states are straddled by the Jz ¼ 3/2 and Jz ¼ þ3/2 HH states, as shown in Figure 4.155d. A similar treatment for the G7 SO band leads to DðE SO ; J z ¼ þ1=2Þ ¼ þð1=6ÞxN 0 bhSz i and DðE LH ; J z ¼ 1=2Þ ¼ ð1=6ÞxN 0 bhSz i:

ð4:131Þ

Again, noting that the spin exchange interaction term hSzi and b are both negative for the zinc blende symmetry, the Jz ¼ þ1/2 SO state moves up in energy and the Jz ¼ 1/2 SO state moves down in energy with magnetic field. Note that the direction of splitting is opposite to that of the G8 HH and LH states. In addition, the total splitting is only one third and two third of the HH and LH states, respectively. This too is shown in Figure 4.155d. In the Faraday geometry where both the applied magnetic field and the light propagation are along the crystal growth direction, both the right-hand (sþ) and the left-hand (s) circular polarizations are allowed because of symmetry considerations. As Figure 4.155d indicates, the optical transition energies for right and left circular polarization are different. For example, the energy for transition (s)is larger than that for (sþ) by xN0(b  a)hSzi, which represents the Zeeman splitting due to

j1117

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j 4 Extended and Point Defects, Doping, and Magnetism sp–d spin exchange interaction. When the effect of the external magnetic field is also included, the energy difference between the abovementioned transitions becomes DE ¼  hSz iN 0 xða  bÞ þ g mB H:

ð4:132Þ

The measured energy separation, for example, associated with the two transitions cited above can be used to determine the Zeeman splitting whose magnitude contains information about the magnetic nature of the DMS material under investigation. In addition to the transitions associated with the Brillouin zone center, the G point, the transitions at the L point, between G and L points but near the L point, and also at the L point, both along the [1 1 1] direction, contain quite a bit of information regarding band structure in general and band splitting in particular. The E 1 ¼ Lc6 ! Lv4;5 transition is a case in point. At the L point, the G8 band at the zone center splits giving rise to Lv6 and Lv4;5 levels, the separation of which is typically termed D1. Therefore, analogous to the E0 þ DSO (or E0 þ D0) transition at the zone center, we have E1 þ D1 transition ðE 1 þ D1 ¼ Lc6 ! Lv6 Þ at the L point that can be harnessed in transmission measurements in conjunction with magnetic circular dichroism investigations. The Zeeman splitting due only to sp–d spin exchange interaction for the E1 þ D1 transition is similar to that for the E0 þ DSO and takes into account splitting of the E1 and D1 values. The total splitting is then expressed as DðE 1 þ D1 Þ ¼  r hSz ixN 0 ða  bÞ;

ð4:133Þ

where r is a numerical factor and takes values between 0.4 and 0.5 for zinc blende DMS materials depending on the orientation of the magnetic field. To reiterate, Zeeman splitting, inclusive of that induced by external magnetic field and also by the sp–d exchange interaction, can be used to determine the magnitude of the spin exchange interaction. Further, the p–d exchange interaction is stronger than the s–d interaction and |N0b| (associated with the former) is larger than N0a (associated with the latter). Cumulatively, the polarity of N0b can be determined from the polarity of Zeeman splitting of the most intense E0 transition. If Zeeman splitting of two different optical transitions are known along the value of the magnetic moment, both N0b and N0a can be determined independently. The magnitude of the Zeeman splitting can be tens of millielectron volts in typical but real DMS materials, allowing easy measurement. However, if the magnetization is low, and the optical transition spectrum is broadened by doping or by lack of high crystalline quality, the measurements become more difficult to interpret. Regardless, the measurements are made easier if one uses polarization modulation technique in conjunction with MCD. Now that we have gained sufficient knowledge as to the processes in the DMS material, let us now turn our attention to the optical aspects of MCD. As the above discussion laid the ground work, Zeeman splitting causes polarization-dependent optical anisotropy which is the source of MCD. The anisotropy of the refractive index causes the Faraday effect, which simply is the rotation of the polarization plane of the linearly polarized light as it traverses through a magnetic material. Both the Faraday rotation and MCD are related in that they are based on identical electronic structural parameters. Having discussed the Zeeman splitting of the electronic states and polarization associated with each of the transitions between various electronic states, let us relate

4.9 Intentional Doping

the Faraday rotation to magnetization. The transition between the G6 (1/2) and G8 (3/2), with energy Ea, is right circularly polarized whereas that between G6 (þ1/2) and G8 (þ3/2), with energy Eb, is the left circularly polarized (see Figure 4.155d). The energy difference between these two transitions in a DMS is given by ^ þ Þ  E b ðs ^Þ ¼ DE 0 ¼ E a ðs

ba M; g eff mB

ð4:134Þ

where M is the magnetization. When Equation 4.134 is substituted into Equation 4.121, the Faraday rotation in a DMS material is expressed as pffiffiffiffiffi  1 y2 F0 b  a M l: ð4:135Þ qF ¼ E 0 ð1  y2 Þ3=2 2
hc gmB Let us use the Faraday rotation as a segue to extend our discussion to MCD. In the absence of external magnetic field, the transmitted light intensity through a sample is given by I ¼ I 0 e  kðEÞl ;

ð4:136Þ

where k(E) represents the energy-dependent absorption coefficient of the material (aspired by the notation used for the imaginary part of the refractive index, which represents the loss term). l is the length of the medium which is the thickness of the sample in this case, and I0 is the intensity of the incident optical signal. The absorption coefficients for right-hand (kþ) and left-hand (k) circularly polarized light can expressed as     DE DE k þ ðEÞ ¼ k E þ and k  ðEÞ ¼ k E  ; ð4:137Þ 2 2 where DE represents the total Zeeman splitting for spin-up and spin-down electrons at energy E given by Equation 4.132. The magnetic circular dichroism in terms of degrees is expressed as (defined in Equation 4.123 but repeated here for convenience) [726] qMCD ¼

180  ðk  k þ Þ; also given in terms of the transmitted intensities as 4p

qMCD ¼

90 I þ  I  ; p Iþ þI ð4:138Þ

þ



where I and I represent the intensities of the right-hand and left-hand circularly polarized light. Utilizing Equation 4.137, yMCD can be rewritten as qMCD ¼ 

qMCD ¼

45 dkðEÞ DE ; or with spin exchange interaction elaborated p dE

45 dkðEÞ ; hSz iN 0 ða  bÞ p dE

ð4:139Þ

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j 4 Extended and Point Defects, Doping, and Magnetism where dk(E)/dE describes the rate of change of the absorption coefficient with respect to energy or the energy derivative of the absorption coefficient. In other words, the MCD signal is proportional to the product of the Zeeman splitting and the energy derivative of the light absorption coefficient, dk/dE. Recall that |N0b| > N0a > 0 because the p–d spin exchange interaction is stronger than the s–d spin exchange interaction. It is deemed instructive to succinctly discuss the experimental aspects of measuring the MCD signal. A typical experimental setup used by Ando et al. [726] for measuring MCD consists of a monochromatic and linearly polarized light impinging on a photoelastic modulator (PEM). PEM then converts the linearly polarized light into alternating right-hand sþ and left-hand s light circularly polarized light that is made to pass through the sample under investigation. The transmitted signal in phase lock with PEM is then detected, as shown in Figure 4.156a. The measurement is repeated for varying light wavelengths, and the absorbance of light for both sþ and s polarization is plotted as the function of energy for a set of magnetic fields and at a set of sample temperatures. Shown in Figure 4.156b is the energy dependence of absorbance for zero (top) and nonzero (bottom) magnetic field. In the case without the magnetic field, the right-hand and left-hand circularly polarized light spectra are identical giving rise to zero difference between the two. However, with Zeeman splitting induced by a nonzero magnetic field and/or spin exchange interaction, the energy levels generating the right- and left-hand circularly polarized light shift, which give rise to a nonzero difference between the transmitted intensities of two circularly polarized lights. This forms the basis of the use of MCD for materials characterization. The photo elastic modulator (PEM) in turn consists of a rectangular bar of a suitable transparent (to the wavelength of light used) material such as fused silica attached to a piezoelectric transducer. The principle of operation of PEM is based on the photoelastic effect, in which mechanically stressed samples exhibit birefringence proportional to the strain generated. By applying an electric field to the PEZ material, which is in mechanical contact with PEM, the strain can be induced. Strain changes the refractive index of the material. The strength of the electric field controls the amount of strain induced, and the polarity of the field determines whether sþ or s polarization is attained, as depicted in Figure 4.157. The PEMs is usually operated in the ultrasound frequencies in the range of 20–100 KHz. Variation in the strain causes a variation in the change of refractive index and phase of light at the exit of PEM, as shown in Figure 4.158a. If the refractive index is made larger and smaller by changing the sign of the strain periodically, the phase shift of the emerging light can be controlled. For l/4 phase shift, righthand and left-hand circularly polarized light is obtained. If the phase shift is smaller, elliptically right-hand and left-hand polarized light would be produced, as depicted in Figure 4.158b. Because GaAs:Mn and transition metal doped II–VI systems are relatively well advanced and the electrical and optical measurements do indeed indicate what the Curie temperature is, a conscious decision was made to discuss the GaAs and II–VI systems to some degree through which it is hoped that the reader would be

4.9 Intentional Doping

Linear

σˆ − σˆ +

Detector

Monochromator Light

Sample Lock-In Amp.

Photoelastic modulator

~

Absorbance of light

Absorbance of light

(a)

LCP No difference between LCP and RCP

RCP

LCP

Difference between LCP and RCP

RCP

(b) Figure 4.156 (a) Experimental setup for measuring magnetic circular dichroism is composed of a monochromatic light being incident on a photoelastic modulator that is capable of generating alternating right-hand sþ and left-hand s light that is made to pass through the sample. The transmitted signal, in phase lock with PEM, is then detected. (b) Energy dependence of absorbance at zero (top) and nonzero (bottom) magnetic field. In the

case without the magnetic field, the right-hand and left-hand circularly polarized light spectra are identical giving rise to no difference. However, with Zeeman splitting induced by nonzero magnetic field and/or spin exchange interaction, the energy levels generating righthand and left-hand circularly polarized light shift, giving rise to a nonzero difference between the transmitted intensities of two circularly polarized lights.

appraised of the type of magnetoelectrical and magneto-optical data that could be expected of GaN when FM state is obtained. In addition, device results associated with GaAs:Mn are also provided for the same reasons. The particulars of DMS and its measurement in GaN are discussed in Section 4.9.7.3, and those of ZnO are discussed in Zinc Oxide: Fundamentals, and Materials and Devices by Morkoc, H. and Özg€ ur, Ü. Wiley 2008.

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j 4 Extended and Point Defects, Doping, and Magnetism Fused silica

PZE

Figure 4.157 Schematic representation of the principle of operation of photoelastic modulator. A piezoelectric crystal that is in mechanical contact with a birefringent transparent material such as fused silica is subjected to strain controlled by a piezoelectric transducer. The direction of the

(a)

field applied to the transducer determines the polarity of light passed through PEM, sþ or s polarization. The figure in the middle indicates compressive strain applied to fused silica while the bottom is for tensile strain. The latter produces right-hand circular polarization. Courtesy of Umit Ozgur.

Ex

Ey

E = xE ˆ y ˆ x+yE

λ /4 Ex = Eocos(ωt–kz) Ex = Eocos(ωt–kz– θ)

t or

z

with θ = π /2,3π/4 (b)

λ /4

λ /4

Figure 4.158 (a) Waveforms of two sinusoidal signals that are phase shifted by l/4, which results in right-hand circular polarization (lefthand circular polarization for l/4 (or 90 ) or left-hand l/4. (b) The polarization of the resultant light with respect to the phase shift

induced. Note that for l/4 peak phase shift, polarization oscillates between right-hand and left-hand circular and linear and elliptical polarization states are observed in between. Courtesy of Umit Ozgur.

4.9 Intentional Doping

4.9.6.7 II–VI and GaAs-Based Dilute Magnetic Semiconductors To provide a clear picture of dilute magnetic semiconductors, because the picture with GaN is not yet clear, magnetic properties of well-established semiconductors doped with transition metals will briefly be discussed. The II–VI- and III–V-based diluted magnetic semiconductors such as Cd1xMnxTe and Ga1xMnx As have attracted considerable attention for the same reason that the spin-dependent magnetic phenomena can be manipulated in these low-dimensional tailored magnetic thin films for various spin-based devices to unprecedented capabilities [727]. Generally, 3d transition metal ions (some species of magnetic ions, i.e., ions bearing a net magnetic moment) are substituted for the cations of the host semiconductors. As a consequence, the electronic structure of the substituted 3d transition metal impurities in semiconductors is influenced by two competing factors: strong 3d-host hybridization and strong Coulomb interactions between 3d–3d electrons. The latter is responsible for the multiplet structures observed in d–d optical absorption spectra. On the contrary, the hybridization between the transition metal 3d and the host valence band gives rise to the magnetic interaction between the localized 3d spins and the carrier in the host valence band [625]. Principally, the majority of DMS studied in an extensive way involved Mn2þ ion as a legitimate member of the transition metal family to be embedded in various AIIBVI as well as III–V hosts. Some of the valid reasons for this choice are as follows: (a) Mn2 þ has a relatively large magnetic moment (spin S ¼ 5/2 and angular momentum L ¼ 0) with a characteristic of a half-filled d-shell; (b) Mn2 þ can be incorporated in sizable amounts into the AIIBVI (up to 80% of Mn) host without affecting much the crystallographic quality of the DMS, whereas about 5 and 35% of Mn are tolerable for III–V-based and ZnO hosts; (c) Mn2 þ is electrically neutral in AIIBVI hosts, acting neither as an acceptor nor as a donor, while it acts as an acceptor in AIIBVI-based DMS. With the significant advance in the materials engineering of thin film and nanoscale heterostructures, the quantized energy levels in semiconductor nanostructures can be coupled either with local magnetic fields created by integrated submicron ferromagnetic structures or with the exchange fields of magnetic ions into the semiconductor lattice itself, hence providing a foundation for quantum spin devices [642] including single-electron spin transistors that rely on spin-dependent tunneling into a magnetic quantum dot (QD) and magnetic field effect transistors (M-FETs) that employ carrier injection into polarized spin transport channels [728]. In this section, an overview of the physics and some generic features of AIIBVIand III–V-based DMS heterostructures are provided. Because several interesting quantum structures derived from the AIIBVI-based DMS (particularly, substituting A-site by Mn) provide a valuable framework for understanding spin transport and dynamics in magnetically active quantum semiconductor structures, some generic features of II–VI-based DMS are described. This is followed by a brief summary of the main experimental properties of (III,Mn)V DMS, and some aspects of semiphenomenological theory have been put forth for the explanation of the ferromagnetism in these compound semiconductors.

j1123

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j 4 Extended and Point Defects, Doping, and Magnetism 4.9.6.7.1 II–VI-Based Dilute Magnetic Semiconductors Most of the research works performed so far on II–VI-based DMS materials have been devoted to Mn, which represents rather a simple magnetic system. The family of AII1xMnxBVI alloys [729], along with their crystal structure is presented schematically in Figure 4.159. The phase diagram shows that Cd1xMnxTe forms a ternary alloy of a stable zinc blende structure for x  0.77, whereas Zn1xMnxSe shows wurtzite structure for 0.3 < x < 0.55. The range of the AII1xMnxBVI solid solutions is considerably large. The easy substitution of Mn for the group II elements in both zinc blende and wurtzite structures occurs mainly because the 3d orbitals of Mn2þ are exactly half-filled. As a consequence, all five spins are parallel in this orbital by the Hund’s rule, as it will cost CdTe

Cubic

HgTe Cu bi c

HgSe

Cu bi c

0.7 5

CdSe

bic Cu

0.77 6 0.8 MnTe

Telluride group

HgS 7 0.3

0.3 8

MnSe

x He

ZnTe

Selenide group

Sulfide group

bi c Cu

MnS 0.9 4 0.5

0.5

0.6 0.5

Hex

Hex

Cubic

Cubic 0.1

0.35

ZnS

ZnSe A schematic diagram of the AII 1x

VI

Figure 4.159 Mnx B telluride-, selenide-, and sulfide-based alloys and their crystal structures. The bold lines represent ranges of the molar fraction x for which homogeneous crystal phases form. Hex (hexagonal) and cubic structure represent wurtzite and zinc blende structure, respectively.

He x

CdS

4.9 Intentional Doping (a)

(Zn,Mn)Se

(Zn,Mn)Se

ZnSe ZnSe

ZnSe

(b)

(Zn,Cd)Se

ZnSe

(Zn,Cd)Se

ZnSe

MnSe monolayers

MnSe monolayers n–ZnSe

Figure 4.160 Some examples of various spinengineered DMS heterostructures: (a) left, a magnetic barrier QW structure in which the magnetic ions are located in the barrier; right, a magnetically coupled double QW; (b) left, a digital magnetic heterostructure in which the

magnetic ions are incorporated into the QW region in discrete, quasi-2D layers; right, a magnetic 2DEG in which modulation doping is employed to create a 2D Fermi sea that is in contact with magnetic ions.

a considerable amount of energy to add an electron with opposite spin to the atom, resulting in a complete 3d5 orbit. In view of this, Mn atom resembles a group II element, hence attracted many researchers to fabricate these materials using both thin film and crystal growth techniques. On the basis of the above phase diagram, several quantum structures, mainly strained layer superlattices and quantum wells (QWs) of (Cd,Mn)Te/CdTe and (Zn,Mn)Se/ZnSe on GaAs substrates, were grown by MBE [730]. With the advancement of enhanced MBE growth techniques, heterostructures containing a variety of band alignments, strain configuration and DMS alloys were fabricated. Some of the examples are illustrated in Figure 4.160. One may employ bandgap engineering techniques to create new DMS materials by manipulating the various physical parameters involved. Apart from this, the presence of local moments allows the spin engineering of new phenomena through the exploitation of two classes of exchange interactions. The first one is the d–d superexchange between d-electrons of the magnetic ions, and the second one is the sp–d exchange between the d-electrons and the band electrons or holes. This interaction is ferromagnetic for conduction band states and mainly antiferromagnetic for valence bands. This interaction determines the spin splitting of the band states in an external magnetic field. This gives rise to interesting magneto-optical and magneto-transport response in DMS samples. In dilute magnetic semiconductors (when the concentration of M2þ is small, i.e., x < 0.01), the M2þ spins can be regarded as isolated from one another, in the case

j1125

1126

j 4 Extended and Point Defects, Doping, and Magnetism of which the magnetization can be described by Brillouin function and we can write M ¼ xN 0 g Mn mB hSz i;

ð4:140Þ

where x is the Mn ion mole fraction, N0 is the number of cations per unit volume (xN0 therefore representing the number density of Mn ions), and hSzi is the thermal mean of the Mn spin along the z-direction (applied magnetic field direction). In the low-field and high-temperature limit, where gMnmBSH/(kBT) 1 (S ¼ 5/2 for Mn2þ), M is linear in H and static (DC) magnetic susceptibility w, defined by M ¼ wH, is of the Curie form c ¼ xC 0 =T with C0 ¼ N 0 ðg Mn mB Þ2 SðS þ 1Þ=3kB :

ð4:141Þ

For DMS with arbitrary magnetic ion concentration, x, the magnetization M cannot be expressed by the standard Brillouin function because of the M2þ–Mn2þ interactions. At low magnetic fields, M has been found to be linear in H as in the dilute case, which allows for the magnetic susceptibility, w, to be defined as provided below. At high temperatures and low fields, w follows a Curie–Weiss behavior described as c ¼ cd þ CðxÞ=½T  qðxÞ
;

ð4:142Þ

where y(x) is the Curie–Weiss temperature (the nomenclature Tc is also commonly used as is the case in this text), C(x) is the Curie constant, and wd is the diamagnetic susceptibility of the host. The inverse susceptibility w1 as a function of temperature, with the Mn concentration to be a parameter, for Cd1xMnxSe has been obtained for Mn mole fractions in the range of 5–45% [731], and is shown in Figure 4.161. The inverse susceptibility w1 shows a characteristic departure from the Curie–Weiss law at low temperatures (below about 40 K and shown in solid circles), deemed to be a characteristic of a cluster glass transition [732]. For an external magnetic field B applied along the z-direction, the magnetization Mz of a DMS alloy containing Mn2þ ions is empirically written as Mz ¼ xN 0 hSz i þ xN 0 Ssat B5=2 ð5mB B=kT eff Þ;

ð4:143Þ

where xN0 is the number density of Mn2þ ions and B5/2 (x) is the Brillouin function for S ¼ 5/2, Ssat is the saturation value for the spin of an individual Mn2þ ion (i.e., smaller than 5/2), and Teff ¼ T þ T0 is the rescaled temperature. Along with the distribution of magnetic ions on a DMS lattice, isolated spins, pairs of spins, and triplets are also distributed. Hence, magnetization is dominated by the paramagnetic response of isolated single spins, which are antiferromagnetically coupled. However, if we consider DMS heterostructure, the ferromagnetic s, p–d exchange interaction between conduction electrons and local moments results in an enhanced electronic spin splitting as described below (similar to Equation 4.132 discussed in conjunction with the GaAs case): DE ¼ gmB B  xN 0 ½ f ðYÞa  gðYÞb
hSz i;

ð4:144Þ

4.9 Intentional Doping

Cd1–xMnxSe

120

χ–1 (mol Mn)

100

Mn: x = 0.45 0.35

80

0.25 60 0.05

40

20

0 0

100

200

300

Temperature(K) Figure 4.161 Inverse susceptibility, w1, of Cd1xMnxSe for Mn percentile contents of 5, 25, 35, and 45% as a function of temperature for zero-field-cooled (ZFC) sample at a field 30 G in the low temperature (below about 25 K shown with solid circles) region and 8.5 kg in the high

temperature (above 25 K shown with open circles) region. Note the linear Curie–Weiss behavior above about 40 K. The characteristic downturn of w1 below 40 K and cups observed in all but the data for 5% Mn is indicative of the spin glass transition [731].

where a and b are the s–d and p–d exchange integrals, respectively, and f(C) and g(C) are corresponding factors representing the wave-function overlaps of the conduction and valence band states with the local moments, and a is the s–d exchange integral. The first term represents the simple Zeeman splitting due to the application of a magnetic field. The second term is the splitting caused by the sp–d spin exchange interaction that is dominant in Equation 4.144, in part because the intrinsic g-factor for electrons in II–VI semiconductors is small and in part because of strong sp–d interaction. The exchange integral for the heavy-hole states is typically approximately five times larger than that for the conduction band and light-hole states. Because of this reason, in most optical experiments, which probe heavy-hole excitations, the spin splitting is dominated by that of the valence band states. The effect of magnetic field on the confined electronic states in DMS heterostructures is generally probed using magneto-optical spectroscopy, such as magneto-photoluminescence, magnetic circular dichroism, magneto-absorption, and Faraday/Kerr effect; the latter methods are discussed in Section 4.9.6.6.1. In (II,Mn) VI heterostructures, the spin injection from the DMS layer into the semiconductors happens at very low temperatures of about 5 K and at relatively high magnetic fields of about 1 T [733–735]. In (II,Mn) VI semiconductors, nearly 100% polarization of the electron spins in the conduction band of the semiconductor can be achieved via the giant Zeeman

j1127

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j 4 Extended and Point Defects, Doping, and Magnetism splitting of the conduction band states due to the s–d exchange interaction between the spins of the extended band states and the S ¼ 5/2 spins of the localized Mn2þ ions. The (II,Mn) VI layer in the DMS/semiconductor acts as a spin aligner into the adjacent layer. However, the main constraint for applications of this class of paramagnetic semiconductors is that the Zeeman splitting decreases rapidly with increasing temperature and, at room temperature, is orders of magnitude smaller to spin polarize. In addition to Mn, Cr magnetic ions have been exploited particularly in the ZnTe host. This system has been found to be one of the most promising material systems from ab initio calculations. Using gradient-corrected density functional theory and a supercell slab technique, Wang et al. [736] showed that the ferromagnetic phase of Zn1xCrxTe thin film is energetically the most preferable state irrespective of Cr concentration and/or configuration. Therefore, the ZnCrTe system can be unequivocally classified as a DMS system. Specifically, the hybridization between Cr 3d and Te 4p orbitals leads to a ferromagnetic coupling between the Cr spins in Zn1xCrxTe thin films. Furthermore, Wang et al. pointed out that in clusters of CrxN, the coupling between Cr spins has been found to be ferromagnetic, which results from the hybridization between Cr 3d and N 2p electrons. It should be pointed out that Cr2 dimer as well as bulk Cr are antiferromagnetic. Because the Zn1xCrxTe material system has been studied very comprehensively both with magnetization measurements and magneto-optical measurements with excellent agreement, MCD data for this material [737] are discussed first with a follow-up discussion of magneto-optics studies conducted in the MnTe system. Further motivation for the choice of material is provided by the fact that Zn1xCrxTe (with x  0.20) has been reported to show ferromagnetism at room temperature using MCD and magnetization measurements [738,739]. The investigated films by Saito et al. [737] were grown by molecular beam epitaxy on GaAs substrates. The MCD spectra were measured in reflection geometry with the magnetic field being normal to the film surface. Alternating circularly polarized light (right hand and left hand) at a frequency of 50 kHz produced by silica quartz stresser was applied, see Figures 4.156–4.158. The direction of the impinging and reflected light deviated from the film normal by 10 to allow for the measurements to be performed. Magnetization measurements were also performed using a SQUID magnetometer with the H field being normal to the film surface. The diamagnetic contribution from the substrate was accounted for as far as the magnetization data are concerned. Magnetotransport measurements were also performed for a comprehensive evaluation of the films [739]. TEM images and TEM diffraction images were also taken to ascertain that the films did not contain any phases other than the ZB ZnCrTe structure, which could be responsible for the observed magnetization. Displayed in Figure 4.162 is standard SQUID magnetization data for the Zn1xCrxTe (x ¼ 0.20) film investigated. As a standard procedure the diamagnetic contribution from the substrate has been accounted for. From the data obtained at 20 K, which show a large hysteresis, the magnetic moment at m0H ¼ 1 T is about 2.6 mB per Cr ion (requires the knowledge of total number of Cr atom incorporated in the lattice) which is consistent with films containing much lower Cr concentration

4.9 Intentional Doping

Zn1–xCrxTe

T = 20 K

x = 0.20

200 K

2

300 K

0

8

2

20 K 100 K

M

Magnetization M (µB Cr)–1

4

–2

300 K 0 0

–4

200 K

–1.0

–0.5

0.0

μ oH/M

0.5

1

1.0

Magnetic field μ0H ( T ) Figure 4.162 Magnetization curves of Zn1xCrxTe (x ¼ 0 : 20) film at various T. The inset shows the Arrott plots of the magnetization data. Courtesy of Saito et al. [738].

(0.035). Note that the ferromagnetic feature persists at temperatures all the way up to 300 K. For an accurate determination of the spontaneous magnetization and the Curie temperature, Arrott plots, plotting M2 versus m0H/M, were constructed, and intercept of the linear extrapolation from high magnetic fields to m0H/M ¼ 0 gave the quantity M2s, see the inset in Figure 4.162. The Arrott plot analysis is deemed the most reliable method for an accurate determination of Ms and Tc because this method relies on data taken at higher magnetic fields where the effect of magnetic anisotropy and the formation of a magnetic domain are negligible. An intercept at the positive values of M2s is indicative of spontaneous magnetization and at the Curie temperature linear extrapolation goes through the origin and a close inspection of the inset leads to the conclusion that the Curie temperature is Tc ¼ 300  10 K at which the spontaneous magnetization vanishes. To develop further confidence in the conclusions based on the magnetization data only, MCD measurements at m0H ¼ 1 Tand at various temperatures up to 300 K were carried out and analyzed. A few comments on the spectral features of the DMS signal are warranted. For example, a 100 nm thick ZnTe film showed weak MCD intensities at about the G (2.4 eV) and L (3.7 and 4.2 eV) critical points (CP) due to the diamagnetic Zeeman effect. On the contrary, an 80 nm thick Zn1xCrxTe (x ¼ 0.20) film on a thin ZnTe buffer exhibited pronounced MCD spectra at photon energies corresponding to the L-CPs of ZnTe. Moreover, the polarities of the MCD peaks of ZnTe were positive at both L-CPs. However, the Zn1xCrxTe film showed a positive MCD peak at 3.7 eV and a negative signal at 4.2 eV. These particular MCD spectra are indicative of the opposing polarities of the Zeeman splitting for the two L-CPs of Zn1xCrxTe, which is consistent with the general features of the Zeeman splitting caused by the sp–d exchange interaction. Therefore, the Zn1xCrxTe film under discussion was deemed to be a DMS. However, to be certain, the magnetic field

j1129

j 4 Extended and Point Defects, Doping, and Magnetism – MCD ( kdeg cm–1)

20

Zn1-xCrxTe

T = 20 K

x = 0.20 E = 2.2 eV

10

293 K

0 –10

–20 –1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

Magnetic field μ0H ( T ) Figure 4.163 Magnetic field dependence of MCD intensity of Zn1xCrxTe (x ¼ 0.20) film at E ¼ 2 : 2 eV and T ¼ 20 and 293 K. Courtesy of Saito et al. [738].

20 K

300 120 K

200

MCD andM S (a.u.)

dependence of MCD intensity for Zn1xCrxTe (x ¼ 0.20) film on a thick ZnTe buffer near G-CP at T ¼ 20 and 293 K was analyzed to seek consistency with the magnetization data, as shown in Figure 4.163. Clearly, the MCD data agree with the magnetization data of Figure 4.162, confirming that the ferromagnetism persists at room temperature. To establish unequivocal confidence in the conclusion that the sample in question is ferromagnetic, the Arrott plots of the MCD data were plotted, as displayed in Figure 4.164. The inset shows the temperature dependence of MCD

MCD (kdeg2/cm2)

1130

0

MCD MS 0

180 K

100 200 T (K )

300

Zn1–xCrxTe 100

x = 0.20 E = 2.2eV

250 K

293 K

0

0

0.1

0.2

0.3

-μoH/MCD (T/kdeg. cm) Figure 4.164 Arrott plots of MCD intensity obtained in a Zn1xCrxTe (x ¼ 0.20) film at a photon energy of E ¼ 2.2 eV at temperatures up to room temperature. The inset shows the temperature dependence of MCD intensity (open circles) extrapolated to m0H ¼ 0 T, attained Arrott plots, together with the spontaneous polarization, MS (solid circles). Courtesy of Saito et al. [738].

4.9 Intentional Doping

intensity at m0H ¼ 0 T obtained by the Arrott plots, together with that of spontaneous magnetization, Ms. The MCD data plotted in the form of Arrott plots confirm the critical temperature TC to Tc ¼ 300  10 K, which is consistent with that obtained from Arrott plots of the magnetization data. The magnetization and magneto-optical data, particularly their conformity to Arrott plots, and the ensuing analysis clearly indicate that Zn1xCrxTe is ferromagnetic at room temperature. Saito et al. [739] also investigated electrical transport in their Zn1xCrxTe (x ¼ 0.20) samples that were grown on MBE-grown 200 nm thick ZnSe buffer layers that were in turn grown on high-resistivity (0 0 1) GaAs substrates. The film was of insulating nature, and showed a nonlinear behavior in log r versus T1 (here r is the resistivity measured), which means that the conduction is not due to thermally generated free carriers associated with either the conduction band or the valence band. The magnetic field dependence of the Hall resistivity rHall was too small to be measurable below 250 K, which sets an upper limit for the carrier mobility at 0.2 cm2 V1 s1 and points to hopping conductivity. Above room temperature positive sloped rHall versus H curves were noted with a p-type conductivity. With increasing temperature rHall increased, leading to deduction of 37 cm2 V1 s1 for the mobility at 350 K, indicative of band conduction. The magnetoresistance measurements taken at 20 and 150 K under magnetic fields, normal to the sample surface, ranging from 1 T to þ1 T indicated large negative slope with respect to the magnetic field regardless of its polarity. Similar results were also obtained for magnetic fields in the plane of the sample, ruling out any anisotropic magnetoresistance, which occurs in ordinary ferromagnetic metals, being responsible for the observed negative slope. Importantly, the magnetoresistance versus magnetic field curves showed hysteretic behavior at 20 K from which 0.6 T was deduced as the coercive field (HC) which is comparable to that obtained from magnetization measurements performed on the same sample. Defining a percentile change in MR (%) as Dr ¼ [r(H)  rmax]/rmax, where rmax is the maximum resistivity in the r(H) versus H curve, MR (%) versus temperature was measured and plotted as a function of temperature up to 350 K under a 1 T magnetic field, as shown in Figure 4.165. At a temperature of 20 K, an MR ratio of 26% (Dr ¼ 4100 O cm) was deduced. With increasing temperature, the MR (%) ratio decreased and its sign changed from being negative to positive between 300 and 350 K. Note that negative MR occurs in the hopping regime of conductance that emanates most likely from localized carriers as well as magnetization. However, the positive MR measured at 350 K is proportional to H2 and also mobility squared, as can be seen in the inset, which is indicative of ordinary MR effect caused by Lorentz force with the associated increase in mobility. The abovementioned results are consistent with other material systems, such as In1xMnxAs [626] and Ga1xMnxAs [740], which also exhibit high resistivity at low temperatures. It has been suggested that the negative MR in these materials systems is due to wave function expansion associated with localized carrier as a result of the application of magnetic field. This picture may also be applicable to the Zn1xCrxTe system. It has been suggested that dissociation of bound magnetic polarons could be the genesis of carrier delocalization upon application of a magnetic

j1131

j 4 Extended and Point Defects, Doping, and Magnetism 5 0 Zn0.8Cr0.2Te H = 10 kOe

–5

TC Magnetoresistancechange(%)

Magnetoresistanceratio(%)

1132

–10 –15 –20 –25

2 350 K

1 0 –1

300 K

–2 –40 –20

0

20

40

H(kOe)

–30

0

100

200

300

400

Temperature, T (K) Figure 4.165 Temperature dependence of magnetoresistance ratio for the Zn0.8Cr0.2Te film investigated at H ¼ 1 T. The Curie temperature TC of the film is indicated by an arrow to be slightly above 300 K, which is consistent with the magnetization measurements and magneto-optical measurements discussed above. The inset shows the magnetoresistance of the film at T ¼ 300 and 350 K. Courtesy of Saito et al. [739].

field [626,740,741]. It is also feasible that giant spin splitting of the Fermi energy induced by strong sp–d interaction might be responsible for wave function expansion [742,743]. Saito et al. [739], however, reasoned that the bound magnetic polaron binding energy, based on the saturation magnetic field being very small, 0.6 kOe, is only 0.05 meV and they would dissociate at room temperature anyhow. Therefore, this model may not be the one responsible for negative MR. More work is warranted to determine the basis for the observations. MCD measurements have been undertaken in MnTe. Addition of nonmagnetic ion into this material and possibility of multiple layers for possible magnetic ordering are of interest. It should be stated that Mn ions in the ZB MnTe lattice interact with their nearest neighbors antiferromagnetically. If one considers the aforementioned antiferromagnetic together with the next nearest neighbor interaction, which is also antiferromagnetic, a variety of magnetic ordering might be achieved. To this end, the CdMnTe system, in the context of CdTe/CdMnTe quantum wells [726,744], has been investigated extensively [745]. MCD signal has been observed and analyzed in MnTe successfully, which results from the Zeeman split G6 conduction band and G8 valence band states. Using thin films to reduce absorption for allowing transmission measurements, MCD measurements [726] have been accomplished with near band edge light and used to deduce the MCD signal, as schematically shown in Figure 4.156. The MCD signal was observed at 15 and 50 K indicating negative and

4.9 Intentional Doping

positive going peaks at 3.38 and 3.6 eV associated with G8  G6 transitions for lefthand and right-hand circularly polarized light. Another positive going peak at 3.9 eV has been attributed to the Zeeman split G7  G6 transition at the L point of the Brillouin zone [726]. The temperature dependence of the negative G8  G6 transition has been used to determine the N_eel temperature to be about 60 K. A word about the N_eel temperature is warranted. In antiferromagnetic solids the total energy of the system at low temperatures is lowest for zero external magnetic field when the dipoles of opposing magnetic moments alternate. This arrangement is stable at low temperatures with susceptibility being small. However, as the temperature is increased, the dipole–dipole interaction begins to falter and the susceptibility increases until the spins become free. The temperature at which this occurs is called the N_eel temperature. Above the N_eel temperature, the spins are free to respond to the magnetic field and the susceptibility follows the modified Curie law of Equation 4.93. In the form of general remarks, the magneto-optical measurements seek to exploit the Zeeman splitting, which is to enhance due to s,p–d interaction over simply that caused by the magnetic field applied. While MCD measurements are sensitive and seek to look at the difference of right-hand and left-hand circularly polarized light, PL measurements can also be made in response to, for example, right-hand and lefthand circularly polarized light to determine the associated transition energies from which one can deduce the Zeeman splitting, the extent of which can be used to discern whether the material is ferromagnetic. In addition, combination of magnetooptical measurements along with carefully designed spin injections can give rise to effects of importance either in terms of basic physics or potential device applications such as optically controlled spintronics. This has been reported in the context of a (CdMn)Te/CdTe (CdMnTe as a spin aligner) DMS [746], and CdTe/(Cd,Mn)Te with magnetically coupled double QW structure wherein the coupling between quantum wells is Zeeman splitting tuned using spin-dependent barrier [747]. In the former case the right-hand and left-hand circularly polarized light emissions clearly show large Zeeman splitting. Moreover, several types of ZnSe/ZnCdSe double QWs coupled by ZnMnSe barrier have been fabricated and studied using different optical spectroscopic techniques [748–750] to understand the magnetically induced changes in the band structure, exciton spin scattering, and coherence in these band structures. Furthermore, CdSe QDs were fabricated by strained layer epitaxy [751,752] to study the coupling of zero-dimensional states with magnetic ions. This of course predicated on the assumption that Mn ions are introduced only in the QDs. The additional motivation may be that the spin coherence time may be lengthened by reduced dimensional structures. 4.9.6.7.2 III–V-Based DMS: (GaMn)As The observation of ferromagnetic transition in (III,Mn) V semiconductors, such as GaMnAs [753–755], at increasingly high temperatures (having a Curie temperature of about 110 K which with modulation doping and annealing has been increased to 170 K) [756,757] has attracted an inordinate amount of attention. Because GaMnAs is a well-developed DMS and is also a III–V semiconductor such as GaN, albeit with different crystalline symmetry when it comes to hexagonal phase of GaN, the GaAs system is covered for the reader to again

j1133

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j 4 Extended and Point Defects, Doping, and Magnetism gain an acquaintance with appropriate magnetism, magneto-transport, and magnetooptical data as we have done with II–VI-based DMS materials. The demonstration of reasonably long spin coherence times in GaAs-based semiconductor structures (100 ns for T2 time at 4.2 K in n-GaAs, which represents the spin coherence time [758], 50 ms for the T1 time at 20 mK in GaAs quantum dots, which represents the spin flip time [759], with the predicted value for the T1 time being 1 ms in GaAs quantum dots [760]) and rapid development of information storage technology using ferromagnetic metals have heightened the research activities in this field. The spin coherence time reported for GaN is a few nanoseconds at low temperatures and 35 ps at room temperature [761]. For reference electron spin relaxation effects in semiconductor Si were studied as early as in 1954 by Honig [762] and Honig and Kip [763], which was extended by Feher and Gere [764]. As semiconductors have many potential advantages over metals because of their easy manipulation to form an appropriate heterostructure with impurities, gates, and optical excitation, the actual synthesis of room-temperature ferromagnetic semiconductors is very attractive for device applications. The discovery of ferromagnetism at temperatures much higher than the room temperature in (Ga,Mn)N [587] has fueled hopes that these materials will indeed have profound technological impact. In this section, some important established properties and physics of (Ga Mn)As and (Ga,Mn)N are described. TM-doped GaN is discussed in the following section. Optical and electron paramagnetic resonance experiments [765,766] confirmed that dilute concentration of Mn exhibits S ¼ 5/2 local moments in GaAs (Mn2þ(d5) configuration). Hence, the Mn-induced states near the Fermi energy play a key role in the origin of ferromagnetism and in the magnetotransport properties of (III,Mn)V DMS. Photoemission studies show that Mn states can be associated with the host semiconductor valence bands. Ferromagnetism is not observed [767] for Mn concentrations smaller than about 0.01%, which predicts that ferromagnetism does not occur when all valence band holes are trapped on individual Mn ions or on other defects, such as commonly observed antisite defects due to As in GaAs semiconductors grown at low-temperature by MBE. Although it is generally accepted that Mn acts as an acceptor when substituted for a cation in II–V semiconductor lattice, most holes are trapped not at the Mn acceptors but at other defects for very dilute Mn concentrations. The largest ferromagnetic transition temperature in (Ga,Mn)As occurs for x  5%, having a record Tc  110 K [753]. With a combination of modulation doping and bulk doping and sometimes incorporating InGaAs, this temperature has been raised to about 170 K for annealed layers [756,757]. The ferromagnetic transition temperature drops for higher x values, the exact cause of which is not yet fully understood. However, competition between interstitial and substitutional incorporation of Mn, and possibly clustering in GaAs, which might take place for higher concentrations of Mn might turn out to be the cause, the interstitial one not contributing to ferromagnetism [714,768]. Ideally, the mean field theory predicts the transition temperature to be about 2000 K times the fractional substitutional concentration of Mn. These DMS materials display large anomalous Hall resistivities [767], demonstrating that itinerant valence band carriers are fully participating in

4.9 Intentional Doping

the magnetism due to strong spin–orbit coupling present at the top of the valence band. The valence band in GaAs is degenerate, which can be lifted by Zeeman splitting through the application of an external magnetic field and spontaneous polarization through sp–d exchange interaction if the semiconductor is made ferromagnetic by doping with transition elements such as Mn. If the p layer in a p–n light-emitting diode is made ferromagnetic, the spin-polarized holes and spin-unpolarized electrons that are injected under DC forward bias recombine with electrons in the n-layer only if spins align, giving rise to increased circular polarization. An exchange interaction between localized spins of the Mn atoms and the holes leads to a large g factor and a large Zeeman splitting [769]. All the holes scatter to the lower Zeeman levels, yielding a high percentage of spin-polarized holes. These spin-polarized carriers recombine in the quantum well region to produce circularly polarized photons due to the selection rules. An implementation of this has been done using an InGaAs quantum well sandwiched between a p-type Mn-doped ferromagnetic GaAs and n-GaAs [770]. In this case spin-polarized holes and spin-unpolarized electrons are injected into the InGaAs recombination region. If the spin polarization injected from the p-layer were 100%, the light emission would be as well. The details are discussed in the Section 4.9.12 dealing with spin-based devices. While the GaN-based DMS is in the process of being developed, it is deemed instructive to provide examples of Faraday rotation, along with the follow-up MCD, obtained in established DMS materials such as GaAs [595,771,772]. Faraday rotation below and above the Curie point in a Ga1xMnxAs (x ¼ 0.043) sample obtained at 6 T as a function of photon energy by Kuriowa et al. [773] is shown in Figure 4.166a. The Faraday rotation measured in the same sample as a function of magnetic field at 300 K and a photon energy of 1.49 eV and at about 10 K for a photon energy of 1.55 eV photon energy is displayed in Figure 4.166b. Note that the Faraday rotation versus magnetic field has the general features of the magnetization curves. As discussed in Section 4.9.6.6.1, MCD signal may be obtained by measuring the difference in the reflectance spectra from the sample for right and left circularly polarized light. Typically, alternating right-hand and left-hand circularly polarized light of varying wavelength (chosen to probe the critical points in the band structure, such as E0 and E0 þ DE0 associated with the G minimum and E1 and E1 þ DE1 associated near the L point in the Brillouin zone) is made impingent on the sample at 10 off the sample normal in the experiments of Ando et al. [772] and the spectra of the reflected light is collected whose spectra are recorded. Because semiconductor GaAs is under discussion here, the band structure of GaAs is shown in Figure 4.167 for us to have a clear picture of the critical points. In these experiments 1 degree of MCD rotation corresponds to 7% difference in reflectivity. Shown in Figure 4.168 are the MCD spectra of a reference GaAs sample and two GaxMn1xAs samples with x ¼ 0.05 and 0.074. For the reference sample with x ¼ 0, features in the reflectance spectra appearing at E0 (involving the G6 conduction band states and G8 valence band states at the Brillouin zone center, see Figure 4.167), E0 þ DE0 (involving the G6 conduction band states and G7 valence band states at the Brillouin zone center E1

j1135

j 4 Extended and Point Defects, Doping, and Magnetism 15

(a) (Ga,Mn)As x = 0.043 6T

θ (º)

10

~10 K

5 300 K 0 1.4

1.6

1.8

Energy (eV) 15 10

(b) (Ga,Mn)As x = 0.043

300 K 1.49 eV

5

θ (º)

1136

0 –5

–10 –6

Magnetization (from transport)

~10 K 1.55 eV –3

0 B (T )

3

Figure 4.166 (a) Faraday rotation for a 2 m thick film of Ga1xMnxAs with x ¼ 0.043 measured in a magnetic field of 6 T at 10 and 300 K as a function of the photon energy. These two temperatures represent below and above the Curie points. (b) Faraday rotation as a function of the magnetic field at 10 K (below the Curie point), 1.55 eV, and at 300 K (above the Curie

6 point) 1.49 eV. Note that below the Curie point, the rotation resembles the typical magnetization curves minus the hysteresis. Solid lines show the magnetization determined from magnetotransport measurements at the given temperatures. They have been scaled to match the Faraday rotation data (open symbols) for convenience. Courtesy of H. Ohno.

(involving the L6 conduction band states and L4,5 valence band states at the L point in the Brillouin zone), and E1 þ DE1 critical points (involving the L6 conduction band states and L6 valence band states at the L point in the Brillouin zone) with very small MCD features caused by the external magnetic field induced Zeeman splitting. However, when GaxMn1xAs epitaxial layers doped with Mn (with x ¼ 0.05, middle panel, and x ¼ 0.074, lower panel), the associated MCD signal is significantly increased due to enhanced Zeeman splitting because of strong sp–d spin exchange interaction. Note that the signal from the GaAs reference sample is magnified by a factor of 10. To be certain, MCD data delineate ferromagnetism; those data as a function magnetic field should be attained and analyzed, as shown in Figure 4.169. The notable MCD signal is deemed to originate from the E1 critical point of Ga1xMnxAs and as such it cannot be attributed to other phases such as the MCD spectrum of MnAs that is distinctly different [774]. The MCD spectrum from a sample with x ¼ 0.005 showed optical interference related oscillations at the E0

4.9 Intentional Doping

L 4,5

6

Γ8

Γ8

L6

Γ7

4

Λ6 2

L6 0

L 4,5

Energy (eV)

X6

Γ6

Γ6 Γ8

Γ8

v Λ 4,5

Γ7

L6

–2

Γ7

X7

c

Γ7 X7 X6

–4 –6

L6

X6

L6

X6

–8

–10

Γ6

–12 L

Λ

Γ

Γ6 Δ

X

U,K

Σ

Γ

Reduced wave vector Figure 4.167 Band structure of semiconductor GaAs. A special attention is called for the critical points at the G and L minima with the associated critical points E0 (involving the G6 conduction band states and G8 valence band states at the Brillouin zone center), E0 þ DE0 (involving the G6 conduction band states and

G7 valence band states at the Brillouin zone center), E1 (involving the L6 conduction band states and L4,5 valence band states at the L point in the Brillouin zone), and E1 þ DE1 (involving the L6 conduction band states and L6 valence band states at the L point in the Brillouin zone).

critical point (consistent with the sample thickness) and a sharp negative MCD signal. The negative character of the MCD signal near E0 can be used to infer the character of the p–d exchange involving the upper valence bands, particularly N0b. For reference, in Cd1xMnxTe the p–d spin exchange interaction N0b < 0 and thus the MCD peak is positive, which points to antiferromagnetism. On the contrary, in Zn1xCrxSe the p–d spin exchange interaction N0b > 0 and thus the MCD peak is negative, which points to ferromagnetism. It should be recalled that spin exchange interaction involving the conduction band N0a is much smaller than the spin exchange interaction involving the valence band N0b, and therefore, the sign of N0b would determine the polarity of the MCD signal. A point to note is that the MCD

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j 4 Extended and Point Defects, Doping, and Magnetism

Figure 4.168 Magnetic circular dichroism spectra of a reference semi-insulating GaAs substrate with no magnetic ions (top), and two epitaxial Ga1xMnxAs films with x ¼ 0.05 (middle) and x ¼ 0.074 (bottom) at T ¼ 5 K and H ¼ 1 T. The spectrum associated with the GaAs reference sample is magnified by a factor of 10 because the signal is weaker than those of Mn-doped Ga1xMnxAs [772].

Figure 4.169 Magnetic field dependence of E1 MCD signal at a phonon energy of 2.83 eV for a Ga1xMnxAs (x ¼ 0.074) sample measured at T ¼ 5 K [744].

4.9 Intentional Doping

data at E1 and E1 þ DE1, associated with the L point, are merged into a broad band structure that is consistent with Mn incorporation into the GaAs lattice and can be attributed to strong sp–d hybridization. The effective g factor of 1.63 reported for GaAs is expected to also hold for Ga1xMnxAs. Additional support for negative N0b at the L point can be gained from the recognition that the sign of the effective g factor is determined from the sign of (N0a  N0b/4). Because N0a is positive, the conclusion that N0b is negative is consistent with negative effective g. As in the case of magneto-optical data and previous discussion of the II–VI DMS system, the GaAs:Mn system, because of its well behaved magnetotransport measurements, that is, in terms of anomalous Hall effect data and their predictability, is used here as a model system for displaying transport data expected from DMS. The behavior of rxx and rxy in magnetic fields has been documented in quite a few GaAs: Mn samples, one of which is that reported by Edwards et al. [775] for a sample with Mn content of 0.05. In these measurements the term rxx shows an initial positive magnetoresistance as is typically observed in (Ga,Mn)As with in-plane anisotropy [776], followed by a negative magnetoresistance beyond B 0.5 T. The term rxy shows a rapid rise at low B on the way to magnetic saturation of the ferromagnetic film, followed by a more gradual rise that has contribution from both R0 and Ra. Fitting the measured rxy versus B curves using Equation 4.111 yields the value for p. The hole density corresponds to 90% of the Mn concentration at low Mn concentration and has a maximum value of 1.0 · 1027 m3 when Tc ¼ 125 K. The data configured along the lines of the second form of Equation 4.111 describing the normal and anomalous components of the Hall data have been obtained [777] as well and are shown in Figure 4.170a in the form of the Hall resistance RHall at various temperatures, plotted as a function of magnetic field for 200 nm thick Ga0.947Mn0.053As. The inset shows the temperature dependence of the sheet resistance Rsheet. A close resemblance of the data to the magnetization data (not shown) is indicative of the fact that the contribution of the ordinary Hall term is negligible in the field and temperature range investigated. Consequently, assuming skew scattering, the second term in Equation 4.111, ðRgsheet  M ? Þ=t can be reformulated in the form of RHall cRsheetM, where c is a temperature-independent constant. Recognizing that RHall/Rsheet M, Arrott’s plots [629] can be employed to determine the temperature dependence of the spontaneous magnetization M. The results of transport measurements are summarized in Figure 4.170b and c. As noted in the prologue to this section, magnetotransport measurements with some caveat that are noted at the end of this paragraph can reliably determine the critical temperature, and in this particular case the critical temperature determined from the magnetotransport measurements is in good agreement with that determined from magnetization measurements. As shown in Figure 4.170c, the value of the Curie temperature follows the empirical relation TC 2000x  10 K, where x represents the mole fraction of Mn in GaAs, which is good up to about 5%, beyond which the critical temperature falls below the value predicted by the empirical expression. Although not known with certainty, this decrease may be due to increased incorporation of Mn on interstitial sites [714]. Unlike the case of GaN, the mean field Brillouin function, see Ref. [628] for an elementary treatment of paramagnetism and the associated Brillouin

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Figure 4.170 (a, top) Temperature dependence of the Hall resistance RHall for a 200 nm thick Ga0.947Mn0.053As sample for which direct magnetization measurements have been performed but not shown. The inset shows the temperature dependence of the sheet resistance Rsheet. (b, center) Temperature dependence of the saturation magnetization [RHall/Rsheet]S obtained using Arrott plots (solid circles) and inverse susceptibility 1/wHall (open circles), both deduced from the transport data

shown in (a). Solid lines depict [RHall/Rsheet]S and (c, bottom) 1/wHall (bottom, c) calculated using the mean field Brillouin theory with S ¼ 5/ 2 for the Mn spin and the Curie–Weiss law, respectively. The dependence of magnetic transition temperature TC on Mn composition as determined from the transport data. Courtesy of Ohno and Matsukura [777]. (Please find a color version of this figure on the color tables.)

4.9 Intentional Doping

function treatment, is reasonably close in predicting the saturation magnetization [RHall/Rsheet]S and 1/wHall, as shown with a solid lines in Figure 4.170b and c. The temperature dependence of spontaneous magnetization determined from magnetotransport and magnetization measurements may deviate somewhat. This may have its genesis in the fact that the anomalous Hall effect scales with the spin polarization of the carrier liquid. As such this polarization is proportional to the magnetization only if the spin splitting is much smaller than the Fermi energy and the contribution of the carriers to the total magnetization is negligible. Moreover, the anomalous Hall effect senses only the regions where the carriers venture out. Therefore, it is just in these regions where the carrier-mediated ferromagnetic interaction is strong. Owing to the fact that the carrier distribution is highly nonuniform near the metal–insulator transition, magnetotransport, and direct magnetic measurements may diverge some [587]. 4.9.7 Experimental Results of TM-Doped GaN

Having established the baseline for magnetization data, magneto-optical data and also magnetotransport data for the well-established II–VI and GaAs systems doped with magnetic ions, let us now change gear and bring into focus the GaN system. The demonstration of ferromagnetism in (Ga,Mn)As and the predication that high TC ferromagnetic DMS could be realized by TM-doped GaN have spawned a great deal of experimental interest, in addition to theory, in examining and understanding the magnetic behavior of GaN-based DMSs, albeit still sketchy. The magnetic properties of reported TM-doped GaN are listed in Table 4.19. In this section, the reader is treated to a detailed overview of the progress in experimental studies in Mn, Cr, and other transition metal doped GaN, inclusive of rare earth (Gd) doped variety. 4.9.7.1 Magnetotransport Properties TM-Doped GaN Electrical (magnetotransport) and optical (magneto-optics) measurements contain signature that can shed light into the state of magnetic semiconductors. However, (Ga,Mn)As-based materials being ferromagnetic exhibit anomalous Hall effect, which means that accurate determination of the hole concentration p requires an analysis which takes this fact into consideration [779]. As for GaN, the electrical data are scarce as many of the transition metal doped samples are high resistivity, which precludes reliable Hall measurements from being made. Of the few results that have been reported, the data were of preliminary nature and may not be as reliable. Against this background, attempts have been made to investigate the magnetic behavior of Mn-doped GaN by both vibrating sample magnetometer (VSM) and magnetoresistance measurements, with specific attention paid to the extraordinary Hall effect (EHE) component [780,784]. In this particular investigation, the magnetization of the Mn-doped GaN films was measured by VSM both parallel and perpendicular to the plane of the film. It should be noted that the EHE measures only the perpendicular magnetization component and that the magnetization has an easy axis that is normal

j1141

0.07–0.14

0.03 0.1

0.07 0.03–0.05

0.03–0.05 0.01–0.03

(Ga,Mn)N

(Ga,Cr)N (Ga,Fe)N (Ga, Cr)N (Ga,Mn)N (Ga, Co,V)N

(Ga,Cr)N (Ga,Mn)N

p-GaN p-GaN

4H-SiC (0 0 0 1) Bulk p-GaN c-Sapphire 4H-SiC (0 0 0 1) p-GaN

p-GaN c-Sapphire c-Sapphire p-GaN Bulk

Ion implantation Ion implantation

Sodium flux growtha Implantation ECR MBE MBE Ion implantation

Solid state diffusion MBE MBE Ion implantation Ammonothermal and resublimation MBE

Fabrication method

350 350

700 710 350

750

710

580–720 865 350 1200–1250

Growth temperature ( C)

5 min at 700  C 5 min at 950  C in N2

5 min at 700  C

700–900  C

>300 >300

280 >350 >400

750

220–370 940 10–25 250 >425

250–800  C

700–1000  C

TC (K)

Postannealing

Ferromagnetic, from Mn cluster Ferromagnetic Ferromagnetic Ferromagnetic Spin glass V: paramagnetic, Co: spin glass Ferromagnetic Ferromagnetic

Ferromagnetic Ferromagnetic Ferromagnetic Ferromagnetic Para- and ferromagnetic

Notes

Sodium flux method growth is essentially the vapor liquid solid (VLS) growth scheme for GaN growth in which “sodium flux” represents the N flux for the N source.

0.01–0.02 0.03 0.07 0.01–0.03 0.02

(Ga,Mn)N (Ga,Mn)N (Ga,Mn)N (Ga,Mn)N (Ga,Mn)N

a

TM content

Material

Substrate (for thin film) or bulk

Table 4.19 List of recently reported magnetic properties of TM-doped GaN, as complied in Ref. [778].

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j 4 Extended and Point Defects, Doping, and Magnetism

4.9 Intentional Doping

to the film plane. Briefly, the measured films exhibited saturation magnetization, and the coercive fields varied between 0.01 and 0.05 T (100 and 500 Oe or Gauss). Hysteresis loops could be observed for both parallel and normal field configuration in the VSM measurements with resolution compromised to some extent due to 30 nm thickness, and the resolution of the magnetization curves was limited. The temperature behavior of the extraordinary Hall effect curves was found to undergo a transition from a linear behavior to a typical S shape one with increasing applied magnetic field, the transition point of which was used to deduce the Curie temperature. The ferromagnetic response was observed beyond room temperature with a maximum Curie temperature believed to be 363 K. The ferromagnetic transition temperature was also confirmed from the measurements of the sheet resistance in zero-field at varying temperatures. Additional reports of these kinds of reports preferably in films exhibiting strong magnetization and conductive behavior would go a long way for building confidence that GaN DMS is really at a point where Hall measurements can be employed reliably to determine the critical parameters. 4.9.7.2 Magnetic Properties of Mn-Doped GaN Many of the Mn-doped GaN layers have been prepared by molecular beam epitaxy at relatively low temperatures to increase Mn incorporation. High concentration of magnetic ions, up to a point, is imperative for ferromagnetic behavior. Some researchers [781] have reportedly attained single-phase (Ga,Mn)N layers by gas source MBE, which contained 7.0% Mn, showing anomalous Hall effect, negative magnetoresistance, and magnetic hysteresis at 10 K. The authors [781] believe that Mn-doped GaN is ferromagnetic with a TC between 10 and 25 K because the anomalous Hall term vanished at 25 K. Later on, they prepared a series of Mn-doped GaN samples [782] with Mn composition varying from 3 to 12%, for which the room temperature M–H magnetization loops were observed for the samples with Mn concentrations in the range 3–9%. The M–H measurements indicate a weak but discernable remanent magnetization and hysteresis was reported, indicative of ferromagnetic ordering for Mn concentration up to 9%. No evidence of secondphase formation was observed by either powder X-ray diffraction or high-resolution cross-sectional TEM in the n-type GaN samples with 9% or less Mn. It should be mentioned that any lack of observation by the aforementioned methods does not automatically rule out such phases or small clusters. Of those samples, the material with 3% Mn composition showed the highest degree of magnetic ordering per Mn atom and a TC of 320 K. However, an Mn concentration higher than 9% was found to enhance the antiferromagnetic (AFM) coupling, resulting in a lower magnetic moment per Mn. An extremely high TC of as high as 940 K for Mn-doped GaN has been reported by Sasaki et al. [783] in samples on (0 0 0 1) sapphire prepared by reactive MBE using ammonia for N source. This indeed is astonishingly high as the thermal agitations would be substantial at temperature approaching that value. The authors already acknowledge the coexistence of ferromagnetic and paramagnetic phases in their samples. Reed et al. [784] verified ferromagnetism in (Ga,Mn)N fabricated by solid-

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j 4 Extended and Point Defects, Doping, and Magnetism state diffusion of Mn into OMVPE-grown p-type GaN and subsequent annealing. They used vibrating sample magnetometer and extraordinary Hall effect measurements, in the temperature range of 310–400 K, which is discussed in the preceding section dealing with electrical properties. Dhar et al. [710,712] presented investigations of the magnetic properties of (Ga,Mn)N layers grown on 4H-SiC substrates by reactive MBE. Employing a variety of techniques such as X-ray diffraction and TEM, they confirmed that homogeneous (Ga,Mn)N alloys of high crystallographic quality could be synthesized for Mn concentrations up to 10–12%. Relying on the measurements of temperaturedependent DC magnetization, isothermal remanent magnetization, and frequencyand field-dependent AC susceptibility, the authors concluded that the insulating (Ga, Mn)N alloys behaved as a Heisenberg spin glass with a spin-freezing temperature around 4.5 K. This was attributed to the deep acceptor nature of Mn in GaN, which in turn resulted in the insulating character of the compound. In other reports by the same group, Dhar et al. [712] and Ploog et al. [785] also showed evidence for Mn-rich clusters being embedded in the (Ga,Mn)N alloy matrix to which the ferromagnetic behavior was ascribed. Cubic GaMnN in its metallic phase has been predicted [786] by Monte Carlo simulations to be ferromagnetic with TC in excess of room temperature as well. In this vein, Chitta et al. [787] reported that a p-type cubic GaMnN exhibited roomtemperature ferromagnetism, determined by the M–H hysteresis. The cubic GaN material was grown by MBE on GaAs (0 0 1) substrates, followed by Mn ion implantation corresponding to Mn composition of 0.7–2.8%. After annealing at 950  C, all samples showed sustained M–H hysteresis up to 300 K. Temperaturedependent magnetization measurements for both FC and ZFC conditions confirmed the similarity to hexagonal GaN in terms of magnetic properties. The origin of the observed ferromagnetism was not clearly specified. In most of the reported experimental results, M–H hysteresis data have been used as a measure of ferromagnetism and Curie temperature. However, it is well known that the magnetization curves alone are not sufficient for a conclusive statement on the matter and could in fact be misleading at times. Pearton et al. [660] analyzed three samples, that is, (Ga,Mn)N with 5% Mn (single phase), 50% Mn (contains multiphase of GaxMny), or 5% Mn (contains small amount of multiphase GaxMny), all of which showed magnetization hysteresis loops at room temperature. But in the temperature behavior under FC and ZFC magnetization, which is a more instructive but not so airtight examination of ferromagnetism, three samples showed different behaviors as indicated in Figure 4.171. Only the single-phase (Ga,Mn)N sample with 5% Mn, shown in Figure 4.171 (top), shows ferromagnetism persisting above 300 K as " Figure 4.171 Temperature dependence of field cooled (FC) (top curve in each case) and zero field cooled (ZFC) (bottom curve in each case) magnetic moment for (Ga,Mn)N with 5 at.% Mn (optimized growth) at the top (a), 50 at.% Mn at the center (b), or 5 at.% Mn (termed as unoptimized growth) at bottom (c). Courtesy of Pearton et al. [660].

4.9 Intentional Doping

Magnetic moment (emu)

-4.0 x 10 -6

5% Mn Single Phase

(a)

-8.0 x 10 -6

-1.2 x 10 -5

FC

-1.6 x 10 -5

ZFC 0

100

200

300

Temperature (K) 6.0 x 10 -5

Magnetic moment (emu)

50% Mn multi phase

(b)

FC

4.0 x 10 -5

ZFC 2.0 x 10 -5

0.0

0

100

200

300

Temperature (K) 5.0 x 10 -5

Magnetic moment (emu)

5% Mn multi phase

(c)

4.5 x 10 -5

FC

ZFC 4.0 x 10 -5

3.5 x 10

-5

0

100

200

Temperature (K)

300

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j 4 Extended and Point Defects, Doping, and Magnetism evidenced by the separation between FC and ZFC curves. In contrast, the x ¼ 0.5 multiphase (Ga,Mn)N sample (Figure 4.171 center) showed behavior consistent with spin glass behavior below 100 K. The multiphase x ¼ 0.05 (Ga,Mn)N sample shows behavior consistent with the presence of at least two ferromagnetic phases as shown in Figure 4.171 bottom. These experiments demonstrated that the conclusion of the magnetic properties of DMS materials should not rely only on the magnetization hysteresis. Detailed characterization, especially about the magnetization state of the substitutional transition metal ions, are necessary for a better understanding of the genesis of the magnetism in these materials. Despite many reports of evidence for ferromagnetism in Mn-doped GaN, there are also other reports that are in contrast to these reports [788,789]. In an effort to show that precipitates could be the source of reported ferromagnetism, Zajac et al. [788] synthesized MnxNy precipitates in (Ga,Mn)N crystals by the ammonothermal and chemical transport methods. Both (Ga,Mn)N and the reference MnxNy samples were grown using similar growth parameters such as temperature and pressure. Then the magnetic properties of the samples were investigated in an attempt to determine whether the ferromagnetism observed in Mn-doped GaN could be due to magnetic material inclusions. On the basis of the ferromagnetic behavior observed from MnxNy, it was suggested that the ferromagnetic behavior in Mn-doped GaN could be from MnxNy precipitates that could form under the same growth conditions although they may be undetectable by X-ray diffraction. 4.9.7.3 Magneto-Optical Measurements in TM-Doped GaN As discussed in conjunction with II–VI and III–V GaAs doped with magnetic ions, magneto-optical spectroscopy is a very direct method for evaluating the s,p–d exchange interactions of DMS as it probes the Zeeman splitting of the transitions at critical points associated with electronic band structure of the host material [790]. As such it is not sensitive to other phases that might be present in the magnetic ion doped host material. In addition, the technique can also be used to determine if the DMS is in a paramagnetic or ferromagnetic state. Before delving into the topic of magneto-optics in GaN, it is essential that the band structure of GaN is revisited. While the detailed band structure of GaN without magnetic field is treated in Sections 2.3–2.6, it is imperative that we establish how the conduction and valence band states split in the presence of magnetic field. Specifically, the Zeeman splitting (enhanced by s, p–d, pinteraction) and also the direction in which various bands split must be known (i.e., the sign of the g-factor in addition to its value). The reader should be cautioned that while there is a reasonable degree of work available in freestanding GaN, the magneto-optics data in TM-doped GaN are nowhere near as complete for II–VI and more advanced III–V GaAs. The following discussion would segue into magnetotransport measurements in GaN. The band structure of GaN at the G point is shown in Figure 4.172. The conduction band and valence band HH, LH (spin–orbit split off), and CR (crystal field split off) VB;HH bands are depicted by GCB , G7VB;LH , and G7VB;CR , which in order represent the 7 , G9 conduction band (1/2), valence band heavy-hole G9VB;HH ð3=2Þ, valence band lighthole GVB;LH ð1=2Þ, and valence band spin–orbit split-off G7VB;CR ð1=2Þ states. Unlike 7

4.9 Intentional Doping

+1/2 CB

Jz = −1/2

σ+

σ− π

σ+ π

σ+ σ−

−3/2 −1/2

σ−

+3/2 HH +1/2 LH +1/2 CR

−1/2 VB;HH Figure 4.172 The GCB , G7VB;LH 7 conduction band and G9 VB;CR (spin–orbit split-off band), and G7 (crystal field split off band) valance bands in wurtzitic GaN at the G point along with polarization (sþ right-hand and s left-hand circular polarizations) of various transitions between the conduction and valence band states in the presence of a magnetic field. (Please find a color version of this figure on the color tables.)

the GaAs and II–VI zinc blende cases, due to the lower symmetry, the valence band HH and LH bands are not degenerate. The transitions involving GCB 7 ð  1=2Þ states and VB;LH G9VB;HH ð  3=2Þ states, GCB ð þ 1=2Þ states and valence band G ð  1=2Þ states, and 7 7 VB;CR ð þ 1=2Þ states and valence band G ð  1=2Þ states are right-hand circularly GCB 7 7 polarized, depicted as sþ. On the contrary, the transitions involving GCB ð þ 1=2Þ states 7 VB;LH and G9VB;HH ð þ 3=2Þ, GCB ð  1=2Þ states and valence band G ð þ 1=2Þ states, and 7 7 VB;CR GCB ð  1=2Þ states and valence band G ð þ 1=2Þ states are left-hand circularly 7 7 polarized, depicted as s. The numbers in parenthesis, as in the case of zinc blende symmetry represent Jz. Because (in the absence of Zeeman splitting) the spin-up and spin-down states are not separated in energy, in case of unpolarized light excitation or electron injection, the right-hand and left-hand circularly polarized emissions would cancel each other. This is true irrespective of the excitation wavelength as each allowed VB;HH pair of the GCB ð3=2Þ, G7VB;LH ð1=2Þ(the spin–orbit split-off 7 ð þ 1=2Þ to either G9 VB;CR band) G7 ð1=2Þ (the crystal field split-off band) states is at the same energy and unpolarized light would ensue. Additionally, one must consider the selection rules to ascertain whether a given transition is allowed in the optical geometry used, the details of which are provided in Volume 2, Chapter 5 as well as in Volume 2, Figure 5.42 dealing with free excitons. When a magnetic field is applied that could be parallel or perpendicular to the c-axis, the spin-up and spin-down states split and also change in energy with magnetic field due to Zeeman splitting effect (see Volume 2, Figure 5.41). It should be mentioned that Zeeman splitting could be large if there is a strong spin exchange interaction as in the case of the established diluted magnetic semiconductors. Otherwise, to see a sizable Zeeman splitting a very large magnetic field must be applied in addition to having very sharp optical transitions. Ste2pniewski et al. [791] undertook magnetoreflectivity measurements in a GaN epitaxial layer grown on bulk

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j 4 Extended and Point Defects, Doping, and Magnetism GaN substrate for! a detailed investigation of Zeeman splitting of states in GaN for ! both B ==c and B ? c configurations. They identified the energy positions of the excitonic resonances associated with transitions between the GCB 7 ð1=2Þ conduction band states and G9VB;HH ð3=2Þ, G7VB;LH ð1=2Þ, and GVB;CR ð1=2Þ valence band states 7 by employing the derivatives method, dR/dE, of the reflectivity spectra and assigning the maxima of (dR/dE) to the optically active transverse excitons. The energy positions of the Zeeman split excitons measured for different magnetic fields and for ! ! two experimental configurations ( B ==c and B ? c) are shown in Volume 2, Figure ! 5.41. A schematic diagram for clarity, but for only the B ==c orientation is shown in Volume 2, Figure 5.42. Furthermore, Ste2pniewski et al. [791] used the transitions observed in the magnetoreflectivity data to resolve the symmetry of three excitonic A, B, C resonances, emanating from the band splitting caused by the crystal field and spin–orbit effects on the valence band. The measurements allowed Ste2pniewski et al. [791] to determine the symmetry-breaking diagram of the twofold degenerate conduction band and the sixfold degenerate valence band states. Note that the magnetic field causes Zeeman splitting of each of the bands, the nature of which was also determined and is shown in a simplified form of band-to-band transitions in Volume 2, Figure 5.42. To underscore the point, magneto-optical spectroscopy is a direct method of evaluating s,p-d interaction in diluted magnetic semiconductors as the method seeks to exploit Zeeman splitting in critical point energies in the material, provided that the transition probabilities (oscillator strengths to be discussed briefly shortly) of the relevant transitions are favorable. Any contribution from other phases, if present, do not contribute to the critical point transitions. This marks the difference between magnetization-based measurements made possible by SQUID magnetometers and optical measurements such as magnetic circular dichroism based on magnetooptical transmission/absorption/reflection. For the sake of the argument, let us assume that the Zeeman splitting is very small and indiscernible. If an MCD experiment were conducted in this case, the signals associated with right-hand circularly polarized light (G9VB;HH ð  3=2Þ  GCB 7 ð  1=2Þ CB ð þ 3=2Þ  G and that for the left-hand circularly polarized light (GVB;HH 9 7 ð þ 1=2Þ occur at the same energy and would cancel each other leading to no MCD signal (the expression for the MCD signal is equations beginning from Equation 4.134). The same is true for the transitions involving LH bands. Because of the fact that the HH and LH states are only separated by 4–6 meV, which is very small, MCD signal would have a mixture of HH and LH hole contributions due to spectral overlap. It should be noted that comparable transition probabilities for these bands have also been mentioned [792–795]. In the context of eventual use of these structures for utilizing the spin properties in devices, lack of spin polarization in electroluminescence (EL) experiment in one particular attempt has been ascribed to fast spin relaxation [796]. For the case where spin-polarized electrons are injected, transitions allowed for that particular spin should be considered. Assuming that injected electrons are spin VB;HH up, then left-hand circularly polarized GCB ð þ 3=2Þ 7 ð1=2Þ conduction band to G9 CB valence band (HH) states, and right-hand circularly polarized G7 ð1=2Þ conduction

4.9 Intentional Doping

band to G7VB; LH ð  1=2Þvalence band (LH) states (see Volume 2, Figure 5.42) would take place. This means that both and right-hand and left-hand circularly polarized light would be emitted, albeit at slightly different wavelengths. However, if these transitions are close in energy and there is spectral overlap due to broadening, the emitted light would contain both polarizations unless cancelled. In GaAs the HHrelated transitions (A excitons) is a factor of 3 stronger than the LH transitions (B excitons), giving rise to a circular light polarization of 50% (see Equations 4.123 and 4.126). In GaN, however, the transition probabilities from the conduction band to the HH and LH states might be comparable in addition to these two valence band being close in energy, in the case of which the light emitted may not be polarized or if polarized the polarization would be small. It is clear that the disparity in transition probabilities is needed for light polarization. Using band-to-band calculations at the G point in wurtzite semiconductors, the transition strengths (or oscillator strengths) for the B (aB) and C (aC) bands relative to that for the A band (aA) can be obtained for light propagation perpendicular to the c-axis as aB/aA ¼ a2 and aC/aA ¼ (1  a2) using [794] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðD  D Þ þ ðD1  D2 Þ2 þ 8D23 1 2 1 pffiffiffi ; ð4:145Þ a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; x ¼ 2 2D3 x 1=x2 þ 1 whereD1 isthe crystalfield energy, and D2 and D3 are the spin–orbitcoupling parameters parallel and perpendicular, respectively, to the c-axis, which leads to a2 ¼ 0.97 and Pcirc ¼ 4% polarization for GaN if we used the parameters reported in Ref. [802] for GaN when both HHand LH bands are involved. If, on the contrary, we use the values given in Volume 2, Chapter 5 by Gil et al. [797], that is, D1 ¼ 10  0.1, D2 ¼ 6.2  0.1, and D3 ¼ 5.5  0.1 meV, we get a2 ¼ 0.62, which gives Pcirc ¼ 23% instead of 1.5% suggested byChoietal.[795]WeshouldpointoutthatChuangandChangarguethattheparameters used by Gil et al. are obtained by linearization while theirs are more precise parameters, that is, D1 ¼ 22 meV and D2 ¼ D3 ¼ 11/3 meV, by using the full expressions to fit the same experimental data. In the band-to-band model, the excitonic effects have been neglected in that the short-range electron-hole spin exchange interaction strongly mixes the excitons built from the G7 and G9 valence bands, but again this effect is small in GaN with a moderate exchange interaction (0.69 meV [798]) compared to the spin–orbit coupling and crystal field splitting parameters, and has almost no effect on the oscillator strengths of the optical transitions [799]. For information, the exchange interaction in ZnO is much larger (4.7 meV) [800,801]. The oscillator strengths and therefore the transition intensities can be modified by strain, induced by either thermal mismatch with the substrate and/or slightly lattice mismatched heterojunctions as in quantum wells. When integrated into the simple band-to-band model, the strain changes the relative oscillator strengths of the A, B, and C excitons drastically [802], as also verified by experiments in GaN [803]. It should be mentioned that the MCD experiments are different in that as long as band coupling to right-hand and left-hand circularly polarized light is sufficiently separate in energy due to Zeeman splitting, an MCD signal should be obtainable.

j1149

j 4 Extended and Point Defects, Doping, and Magnetism GaN:Mn H=1T 0.01 Magnetic circular dichroism (degree)

1150

300K

0 –0.01

0.01

6K

0 –0.01

1

2

3

4

Photon energy (eV) Figure 4.173 MCD spectra of GaN:Mn (6.8%) taken at 300 and 6 K with a magnetic field strength of 1 T. The inset associated with the 6 K data magnifies the MCD structure to show the contribution from diamagnetic GaN. Courtesy of Ando [804].

On the magneto-optics experimental side, Ando [804] employed MCD spectra to investigate possible ferromagnetic behavior in a 300 nm thick (Ga,Mn)N film grown on a 200 nm thick GaN which in was grown on (0 0 0 1) sapphire. The Mn concentration was 6.8% and no other phase, other than GaN, was detected in X-ray data. Further details related to the sample can be found in Ref. [805]. SQUID magnetometer measurement showed the typical hysteresis with an apparent Curie temperature above 300 K and a coercive field of 8.2 mTat the same temperature. The MCD spectra showed structures around the fundamental bandgap, 3.47 eV, as shown in Figure 4.173 wherein a broad MCD peak is visible at around a photon energy slightly below the fundamental band edge. It should be stated that there is no apparent contribution from the sapphire substrate to the MCD signal. Further, the magnetic field dependence of the MCD signal obtained at a photon energy of 3.54 eV is representative of a straight line that is not shown. Because the MCD signal is proportional to magnetization (Equation 4.139), the magnetic field dependence of the MCD signal should be consistent with ferromagnetism, which in this particular sample is not. Figure 4.174 shows the temperature dependencies of the MCD signal corresponding to the positive peak at a photon energy of 3.54 eVat 6 K and a negative peak at 3.20 eV. The dashed line shows the MCD intensity at room temperature, which arises from the Zeeman splitting of the diamagnetic host GaN given by g mBH. We should mention that all semiconductors are diamagnetic at room temperature unless made paramagnetic or

4.9 Intentional Doping

GaN: Mn H = 1T

Magnetic circular dichroism (degree)

0.05

3.54 eV (Γ)

0

3.20 eV

– 0.02 0

100

200

300

Temperature (K) Figure 4.174 Temperature dependences of the MCD intensities of the 3.54 eV peak and the negative signal at 3.20 eV. The dashed line shows the MCD intensity value at room temperature. Courtesy of Ando [804].

ferromagnetic by transition metal or rare earth dopants. Both MCD signal intensities at 3.54 and 3.2 eV show similar Curie–Weiss-type temperature dependencies, which indicate that the sample is paramagnetic and is responsible for the observed MCD spectrum at 6 K. Moreover, a clear MCD peak around the GaN fundamental gap energy (Figure 4.173) clearly suggests that this paramagnetic material has a common band structure with GaN, and its s,p-bands at the G critical point are influenced by the magnetization (d-electrons). Therefore, Ando [804] concluded that the sample is simply a paramagnetic, diluted magnetic semiconductor Ga1xMnxN. Furthermore, the small positive going shoulder on the strong 3.50 eV MCD peak, shown in the circular inset in Figure 4.173, is in reality the MCD signal of the diamagnetic GaN buffer layer, the peak position of which shifts to a higher energy due to the blue shift in the bandgap energy of GaN with decreasing temperature. Let us now analyze and interpret the observations, as was done by Ando [804]. In contrast to the negative MCD peak observed in Cd1xMnxTe [745], note that the MCD peak at 3.54 eV is positive. This particular MCD peak could have its genesis either in a superposition of the A (G9 ! G7) and B (G7 ! G7) transitions (see Figure 4.172 for a descriptive view of the transitions and note that the separation between the A- and B-transitions in GaN is very small [806]) or in the C-transition (G7 ! G7). Ando [804] argued that because GaN comprises light elements as its constituents, its spin–orbit interaction is small [806], in the case of which the Zeeman splitting of the A-, B-, and Ctransitions is expected to have roughly the same magnitude [807,808]. The sign of the

j1151

1152

j 4 Extended and Point Defects, Doping, and Magnetism Zeeman-splitting associated with the B-transition is therefore expected to be different from the ones for the A- and C transitions [807,808]. Consequently, the magneto-optical effect associated with the A- and B-transitions could possibly cancel each other if their transition energies completely fall on each other. This leaves only the magneto-optical effect associated with the C-transition. Ando [804] suggested that while such complete cancellation is generally rare, this possibility must be considered. However, as shown by Ste2pniewskiet al. [791] and inVolume2, Figure 5.41,following Zeeman splitting none of the spin-up and spin-down energies associated with A (G9 ! G7) and B(G7 ! G7) transitions are identical however close they may be. If the sample quality is not sufficiently high and the transitions associated with A and B excitons are broad and/ or both peaks are lumped together, it may not be possible to detect a clear MCD signal. In Mn-doped GaN, broad MCD structures below Eg are also present [804]. The investigations of the optical absorption spectra of lightly Mn-doped GaN have shown that two absorption bands below the bandgap are related to the Mn-acceptor level [588–591,603]. Therefore, the MCD structures observed by Ando [804] below 3.5 eV correspond to these two absorption bands. It should be noted that the same broad MCD structures below the bandgap were also observed in a paramagnetic GaN: Mn [809]. The sample investigated by Ando [804] exhibited possible ferromagnetism even at room temperature by the magnetization curves. However, no MCD signal was detected from that ferromagnetic material by Ando [804]. It is then fair to conclude that any ferromagnetism observed in the M–H curves must most likely have been associated with some unidentified ferromagnetic material that is different from that of Ga1xMnxN. As iterated, this simply is a demonstration that crystallographic techniques such as X-ray diffraction, and TEM would not always be able to detect any different phase responsible for magnetism. Moreover, the SQUID method is so sensitive that even the contribution by minute amounts of ferromagnetic impurities to magnetism could lead to false positive results. In the sample investigated by Ando [804], the volume fraction of the impurities was estimated approximately at about 1% and the measured value of the saturation magnetization of the sample, 7 emu cm3, is about 2 orders of magnitude smaller than the saturation magnetizations of typical ferromagnetic materials. The divergence of the SQUID data and magneto-optical measurements is not unique to GaN in that ZnO:Co [810] and ZnO: Ni [811], deemed to be ferromagnetic by SQUID measurements, did not agree with the paramagnetic or diamagnetic MCD data [812]. To make the point again, other III–V and also II–VI materials doped with transition elements showed ferromagnetism, as determined by SQUID measurements, and also showed magneto-optical response as MCD signal. Experimentally, the conventional II–VI DMS [745] and a paramagnetic III–V DMS Ga1xCrxAs [813] showed strong MCD enhancements at their CPs, as expected. The strong enhancements of the MCD signals at the CPs were also observed in Ga1xMnxAs [772], In1xMnxAs [814], and Zn1xCrxTe [737]. The field dependencies of their MCD intensities clearly proved that these three materials are the intrinsic ferromagnetic DMSs [737,772,814]. On an equally serious note, which does not bode well for Mn-doped GaN in terms of its prospects for ferromagnetism, the holes have been reported to be at the d shell of Mn (Mn2þ (d5)) by several

4.9 Intentional Doping

1.0 0.9

Magnetization (emu/cm3)

Saturationmagnetization(emu cm–3)

researchers [590,591,815], implicating that magnetization induced by hole exchange interaction may not be plausible in (Ga,Mn)N. The scrutiny and concern raised in the previous paragraph are equally applicable to another transition metal doped semiconductor, ZnO:Mn. In fact, the same may even be said with reasonable certainty about Co- and also Cr-doped ZnO. In the case of ZnO:Mn, the argument has been made that observation of hysteresis in magnetization curves alone is not sufficient for determining the Curie temperature. Even within the confines of magnetization measurements alone, the real test actually is more involved in that the diamagnetic component due to the substrate material and paramagnetic component provided by the transition metal alone must all be subtracted from the magnetization curves first. This involves the use of applicable theory in conjunction with parameters determined from experiments. A careful analysis has been undertaken for RF sputtering deposited Mn-doped ZnO, which concluded that the material is paramagnetic not ferromagnetic [816]. A more detailed discussion on ZnO can be found in Chapter 14. In contrast to the picture painted in the foregoing paragraphs, reports of ferromagnetism based on magnetization measurements continued. For example, Reed et al. [817] proposed that in (Ga,Mn)N when the Fermi position is such that there is a partial occupancy of electrons and holes in the Mn impurity band, ferromagnetic behavior occurs. This suggestion needs to be considered in terms of Mn-induced levels/bands within the bandgap to devise a scientific explanation of these observations. The dependence of the ferromagnetic properties on the Fermi level was demonstrated by doping (Ga,Mn)N with n-type or p-type dopants. Shown in Figure 4.175 is the effect of silane flow on the saturation magnetization of the (Ga,Mn)

0.8 0.7 0.6 0.5

1.0 0.5 0.0 –0.5 –1.0 –2000

0.4

Si = 2 nmol min-1 –1000

0

1000

2000

Applied field (Oe)

0.3 0.2 0.1 0.0 0

20

40

60

Silane flow (nmol min–1)

Figure 4.175 The influence of silane flow on the saturation magnetization in (Ga,Mn)N:Si films. Note the reduction in the magnetic strength with increased Si doping. Inset shows the magnetization curve for sample (b) at low magnetic fields, which is typical of the samples

80

used in this study. This sample has a residual magnetization of 0.154 emu cm3, a coercivity of 100 Oe, and saturation magnetization of 0.72 emu cm3 at 2.2 kOe, at room temperature. Courtesy of El-Masry and coworkers [817].

j1153

j 4 Extended and Point Defects, Doping, and Magnetism N:Si films. The data indicate that high silicon concentrations in (Ga,Mn)N:Si films eliminate the FM behavior determined by the magnetization curves, whereas a much stronger FM response was observed for low to moderately Si-doped samples. In the case of high Si doping, it is expected that EF is close to the conduction band; therefore, the argument goes that the deep Mn band is completely filled with electrons, leaving no available holes to mediate the magnetic exchange interaction. When doped with Mg, electrons in the Mn impurity band are absorbed by the normally empty Mg acceptor states that is 1.2 eV below the Mn band, so that these electrons have nearly zero probability of returning to the Mn energy band where they would facilitate the Mn–Mn exchange interaction. Therefore, the addition of Mg to an otherwise FM (Ga,Mn)N film is expected to reduce the FM response, and the introduction of Mg acceptor states at a concentration that is the same as that of Mn nearly eliminates the film’s FM response, as it has been observed. Arkun et al. [818] reported on the dependence of ferromagnetic properties of MOCVD-grown (Ga,Mn)N films on carrier transfer across adjacent layers. They found that the magnetic properties of (Ga,Mn)N, as part of (Ga,Mn)N/ GaN:Mg heterostructures, depend on the thickness of the GaN:Mg layer. These results were explained based on the occupancy of the Mn energy band and how the occupancy can be altered due to carrier transfer at the (Ga,Mn)N/GaN:Mg interfaces. The authors also found that the magnetic properties of the GaMnN/AlGaN/GaN:Mg depends on the thickness of the AlGaN barrier layer, as shown in Figure 4.176. The presence of this AlGaN barrier layer thus affects the carrier transfer from the GaMnN film to the GaN: Mg layers. These results seem to be consistent with the model proposed by Dalpian et al. [701] in that ferromagnetic state can only be stabilized when Mn levels have both holes and electrons. Mn-doped GaN samples have also been produced by MBE technique [781] and these samples have been reported to exhibit ferromagnetic properties at room 4

Magnetization (emu cm–3)

1154

3 2

AlGaN barrier thickness tb = 50 nm tb = 25 nm

1 0 –1 –2 –3 –4 –5000

–2500

0

2500

5000

Applied filed (Oe) Figure 4.176 Magnetization versus applied field for GaN:Mg/ AlGaN/(Ga,Mn)N/AlGaN/GaN:Mg DHS containing 25 and 50 nm thick AlGaN barriers The thickness of the (Ga,Mn)N and GaN:Mg layers are fixed at 0.38 and 0.75 mm, respectively. Courtesy of El-Masry and coworkers [818].

4.9 Intentional Doping

0.15

Cr: 1%

5K

0.2μ μ B /Cr

Cr: 5%

0.1 Magnetization (emu g–1)

Cr : 3% 0.05

Cr : 0.5% 0

–0.05

–0.1

–0.15 –10 000

–5000

0

5000

10 000

Magnetic field (Oe)

–1 Magnetization, (emu g )

0.05

Cr: 5%

Cr: 1% 0

–0.05 –1000

Hc = 100 Oe

0 Magnetic Field (Oe)

Figure 4.176 (Continued )

1000

j1155

1156

j 4 Extended and Point Defects, Doping, and Magnetism temperature for low concentration of Mn (3% of Mn). Optical properties of (Ga,Mn) N thin films are not available yet. However, optical injection on n-GaN samples [819] implanted with high doses of Mn (3 · 1016 cm2) reveals that Mn forms a deep acceptor near En þ 1.8 eV. The expected wonder of the reader as to whether Mn-doped GaN is ferromagnetic or not is well taken and reflects the murky state of the literature on the topic. In those instances where ferromagnetism has been reported, the Curie temperatures span from 20 to 940 K. Although the growth mechanism seems to play a vital role, the reason for such a wide variation of TC is not understood and it is reasonable to speculate about the likelihood of contributions from materials other than the (Ga, Mn)N proper, if any. There appears to be some hints from theory that ferromagnetism is very much dependent on the doping level and a narrow range of doping levels may lead to ferromagnetic state being stable. From an experimental point of view, it would be fair to conclude that perhaps Mn is therefore not the best choice for obtaining ferromagnetism in GaN. 4.9.8 Magnetic, Structural, Optical, and Electrical Properties of Cr-Doped GaN

Before delving into Cr-doped GaN, it is instructive to revisit briefly the section dealing with the theoretical simulations. Cr-doped GaN was found to have the most stable ferromagnetic state in transition metal doped GaN by Sato et al. [690] Das et al. [703] performed a first principles calculation within the framework of linearized muffin-tin orbital (LMTO) tight-binding method and gradient-corrected (GCA with approximation added toward the end) DFT. They predicted the coupling between Cr atoms to be ferromagnetic, with a magnetic moment of 2.69 mB for the case of bulk GaN and 4 mB for the case of GaN clusters. Prior to the abovementioned theoretical work, Cr-doped GaN single crystals grown by the flux method [820] and thin films grown by ECR MBE [821] had been grown and studied for their magnetic properties. The single-crystal (Ga,Cr)N samples fabricated by adding Cr to GaN single crystal by flux method were reported to show ferromagnetic TC at 280 K, whereas the thin films grown by ECR MBE were reported to display ferromagnetism with a Curie temperature above 400 K. The ferro- to paramagnetic transition at approximately 280 K was also observed in the temperature-dependent resistance measurement at zero-magnetic field. The M–H hysteresis loops yielded coercive fields of 54 and 92 Oe at 250 and 5 K, respectively. The carrier density in this crystal was about 9 · 1018 cm3 (n-type) with a mobility of 150 cm2 V1 s1. It should be pointed out that the electron mobility appears to be inconsistently high for the electron donor concentration not to mention any adverse effect of Cr doping. In another report [821], the (Ga,Cr)N thin films grown by ECR MBE were deemed very encouraging, because they showed ferromagnetism with TC above 400 K and remarkable M–H hysteresis loops with a saturation field of about 2000 Oe and a coercive field of about 55 Oe at 300 K. Again, caution must be exercised here as to the characterization of the observed magnetism.

4.9 Intentional Doping

0.15

Cr: 1%

5K

0.2 μ B /Cr

Cr: 5%

0.05

Cr: 3%

0.05 Cr: 0.5%

0

–0.05

Magnetization (emu g–1)

Magnetization (emu g–1)

0.1

j1157

–0.1 –0.15 –10 000

Cr : 5% Cr :1%

0

H c =100 Oe

–0.05 –1000

0

Magnetic Field (Oe)

–5000

0

5000

10 000

Magnetic Field (Oe) Figure 4.177 Magnetization curves for Crdoped GaN in atomic concentrations of 0.5, 1, 3, and 5% up to a magnetic field normal to the surface of 10 000 Oe (1 T). Note that the film containing 5% Cr does not show any saturation magnetization in the range measured and

appears to be paramagnetic. The blow-up version near the origin indicates hysteresis for 1% Cr sample and a coercive field of 100 Oe. Courtesy of F. Hasegawa. (Please find a color version of this figure on the color tables.)

The GaN films doped with Cr and grown with MBE using NH3 for nitrogen source (either on (1 1 1) Si or (1 1 1) GaAs substrates, which is inconsequential as far as the implications here are concerned) have been thoroughly investigated in terms of direct magnetization measurements and also by magneto-optical measurements such as MCD [822]. Films containing 0.5, 1, 3, and 5 at.% Cr were measured with a SQUID magnetometer with the film with 1% Cr showing the largest hysteresis and a saturation field of 1 T, as show in Figure 4.177. The saturation magnetization of 0.14 emu g1 at 5 K for the 1% sample corresponds to 0.2 mB per Cr atom. The diamagnetic component due to the Si substrate was subtracted, and that with 5% Cr behaved similar to a paramagnetic material and did not show any saturation. The antiferromagnetic behavior of the 5% sample may be due to antiferromagnetic CrN that may form in samples with high Cr concentration. In terms of the electrical conduction, unintentionally doped samples exhibited electron concentrations in the range of 1019 cm3 that decreased to 1015 cm3 by Cr doping initially but climbed back up to 1019 cm3 again for a Cr concentration of 10%. The temperature dependence of magnetization measured at H ¼ 200 Oe for 1, 3, and 5% Cr-containing GaN is shown in Figure 4.178. The temperature dependence in the low temperature region (circled) is consistent with ferromagnetic, combination of ferromagnetic and paramagnetic, and paramagnetic behavior for 1, 3, and 5% Cr (atomic percentile) containing GaN, respectively. Specifically, the dependence for 3 and 5% sample can be described with paramagnetic behavior. The 1% sample shows little temperature dependence, which is consistent with ferromagnetic behavior.

1000

j 4 Extended and Point Defects, Doping, and Magnetism 0.04 H = 200(Oe)

Magnetization (emu g–1)

1158

Ferromagnetic

Cr: 1%

0.02

Paramagnetic + ferromagnetic

Cr: 3 %

Paramagnetic

Cr : 5%

0 0

50

100

150 200 250 Temperature (K)

300

350

Figure 4.178 Temperature dependence of magnetization for 1, 3, and 5% Cr-containing GaN. As indicated, the film with 1% Cr is consistent with ferromagnetic behavior. The films with 3 and 5% Cr exhibit a combination of ferromagnetic and paramagnetic behavior, and paramagnetic behavior, respectively. Courtesy of F. Hasegawa. (Please find a color version of this figure on the color tables.)

Magnetic circular dichroism measurements in Cr-doped GaN (1.5% atomic percentile) samples have also been carried out at a magnetic field of 1 T that was normal to the surface. The MCD spectra displaying 1.65 and 3.49 eV peaks at 6 K are shown in Figure 4.179. The peak appearing at 1.65 eV is most likely associated with Cr4þ1 EðFÞ ! 3 A2 ðFÞ internal transition (d–d ) that occurs at 1.7 eV, see Figure 4.134. In samples grown using di-methyl-hydrazine (DMHy) for N source as opposed to NH3, another peak at 1.9 eV appeared that has been attributed to the 1 T1 ðFÞ ! 3 A2 ðFÞ internal d–d transition of Cr4 þ charge state, see Figure 4.134. The peaks appearing at 3.43 eV in the 295 K spectrum and 3.49 eV in the 6 K spectrum are associated with the band edge of GaN. However, the most striking data are that the MCD intensity of both the 1.9 and near 3.4 eV signals obtained at 6 K varied linearly with magnetic field, with no hysteresis loop [823]. This means that MCD data do not support the presence of any ferromagnetism in spite of the fact that magnetization measurements indicated otherwise. Moreover, another sample containing 1% Cr (which showed substantial hysteresis in SQUID measurements) was also measured in terms of the MCD band edge signal as a function of magnetic field (normal to the sample surface) at 6 K. The MCD and SQUID data measured versus magnetic field are shown together in Figure 4.180. Although the SQUID data show saturation and some hysteresis, the MCD signal varies linearly with magnetic field with no hysteresis [824]. The MCD data are in general in agreement with the Soft X-ray magnetic circular dichroism (XMCD) measurements reported by Makino et al. [825] and performed in Ga0.97Cr0.03N films grown by NH3 MBE in Cr-doped GaN. The temperature

4.9 Intentional Doping

Cr: 1.5% H = 10 kOe

(H ⊥ Sample)

3.43 eV

295 K 0 MCD

3.49 eV 6K 1.65 eV

0

1

2

3 4 5 Photon energy (eV)

6

Figure 4.179 MCD spectra obtained in Cr-doped GaN (1.5%) at 6 and 295 K for a magnetic field of 10 kOe, which was normal to the surface. The peak at 1.65 eV is most likely associated with Cr4þ d–d transitions. The near band edge transition (3.43 eV at 295 K and 3.49 eV at 6 K) is associated with band edge with no clear MCD signal. Courtesy of K. Yamaguchi and F. Hasegawa.

Figure 4.180 The magnetic field dependence of the band edge 3.4 eV MCD peak intensity measured at 6 K for 1 at.% Cr-doped GaN. The SQUID data of the same sample showing saturation and hysteresis are also displayed. The lack of saturation and hysteresis in the MCD data is in contrast to that of the magnetization data and implies that the film is not ferromagnetic. Courtesy of K. Yamaguchi and F. Hasegawa.

j1159

j 4 Extended and Point Defects, Doping, and Magnetism dependence of the XMCD intensity could very well be characterized by the Curie–Weiss law. This sample too showed ferromagnetism up to room temperature while the XMCD data did not show any ferromagnetism. The paramagnetic/diamagnetic nature of the MCD data suggests that homogenously Cr-doped GaN is not responsible for the magnetization data. This simply indicates that the magnetization measurement with a Curie temperature of greater than 350 K is most likely caused by another phase in the material not detected by X-ray measurements. One plausible explanation might be that films contain CrO2 precipitates. The Mn analogue might be MnN4 in Mn-doped GaN, which shows magnetism in magnetic measurements but not in optical measurements. For the sake of completeness, a discussion of optical transitions in Cr-doped GaN is warranted. In this vein, Hashimoto et al. [826] studied the structural and optical properties of (Ga,Cr)N layers grown by ECR MBE, which purportedly exhibited ferromagnetic behavior at room temperature. From XRD and EXAFS measurements, the authors could not detect any second-phase material. Some common compounds of Cr with Ga or N include CrGa4, CrN, Cr2N. Among them, CrGa4 and

3.47 eV

GaCr (Cr : 1.5%) 3.29 eV FWHM: 75 meV

GaN buffer layer

10 K

PL Intensity(au)

1160

30 K 50 K 80 K 120 K 160 K 200 K 300KK 300 2.4

2.8

3.2

Photon energy (eV) Figure 4.181 Temperature dependence of PL spectrum for the (Ga,Cr)N layer with the Cr concentration of 1.5% in the range of 10–300 K. Courtesy of Asahi and coworkers [826].

3.6

4.9 Intentional Doping

CrN have been shown to be nonferromagnetic [827–829], and the magnetic properties of Cr2N are not yet known. From the optical emission point of view, PL emission at 3.29 eV (10 K) was observed in Ga0.985Cr0.015N film, as shown in Figure 4.181, which was attributed to band-to-band transitions in GaCrN. The assignment was deduced from temperature- and excitation-power-density dependent PL measurements. It was also found that the PL emission peak energy in GaCrN decreases with increasing temperature in accordance with the Varshni formula similar to the GaN excitonic transition peak. If truly confirmed as being ferromagnetic, Cr-doped GaN may provide new hope for spintronics applications at room temperature and above. However, for Cr and particularly other transition metal doped GaN, further research is imperative to be certain of the observed results as being due to carrier-mediated ferromagnetism. At the very least, the basis for the observations need to be determined. In another investigation but still within the realm of Cr-doped GaN, Singh et al. [830] studied the effect of substitutional Cr on the magnetic properties of (Ga,Cr)N. For this particular investigation, several (Ga,Cr)N films with identical Cr concentration (3%) were grown at different temperatures by MBE. The fraction of Cr at the substitutional sites (CrGa) was determined from the RBS-channeling angular distribution of Cr and Ga in the h0 0 0 1i axial direction, as shown in Figure 4.182. The results indicate that 78–90% of Cr did indeed occupy the substitutional sites for GaN grown in the temperature range of 700–775  C, and HRTEM showed that Cr distributed uniformly in the lattice. However, only a small fraction of Cr (400 K) of Teraguchi et al. Therefore, the higher Curie temperature of

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j 4 Extended and Point Defects, Doping, and Magnetism Ga0.94Gd0.06N was deemed to be due to neither the Gd element nor the rocksalt structure of GdN. Rather, it was argued to be plausibly associated with the formation of (Ga,Gd)N ternary alloy. A later investigation performed by Dhar et al. [839] on a series of Gd-doped GaN samples with Gd concentration ranging from 7 · 1015 to 2 · 1019 cm3 demonstrated very high magnetization. Dhar et al. [839] also reported above room temperature ferromagnetism for all (Ga,Gd)N samples even with Gd concentrations less than 1016 cm3, for both MBE-grown (Ga,Gd)N and ion-implanted (Ga, Gd)N films. From the M–H curves as well as the M–T curves measured under both FC and ZFC conditions, the inferred Curie temperature is above 360 K. Further, the Curie temperature continually increased with increasing Gd concentration to well above room temperature. Most strikingly, magnetization measurements yielded an unprecedented magnetic moment of up to 4000 mB per Gd atom observed in the most lightly doped sample, 7 · 1015 cm3, as determined by SIMS analysis. Owing to the large Gd-Gd distance in the very lightly Gd-doped GaN, the microscopic origin of the magnetic coupling leading to ferromagnetism could neither be explained simply in terms of the direct exchange, double exchange, or superexchange interactions amongst the Gd atoms, nor be accounted for by the free carrier mediated RKKY-type long-range coupling, given the electrically highly resistive nature of the alloy after doping with Gd. Dhar et al. [839] believe that the high TC ferromagnetism is closely related to their observation of colossal magnetic moments of Gd in GaN, the latter of which may be explained by the long-range spin polarization of the GaN matrix induced by Gd atoms. This long-range spin polarization of the GaN matrix was reported to be consistent with the circular polarization in magneto-photoluminescence measurements in lightly Gd-doped GaN. The TC dependence on Gd concentration can also be qualitatively explained within the framework of percolation theory [841]. Additional support for ferromagnetism comes from the calculations of Dalpian and Wei [707] who argued that while Gd-doped zinc blende GaN doped with donors is stabilized in the antiferromagnetic state in that intruding electrons and strong exchange interaction with s electrons of the host would stabilize the ferromagnetic state. Magnetization data obtained in a Gd-implanted sample at concentration of 1018 cm3 are shown in Figure 4.184 in the form of magnetization versus magnetic field, magnetization at 1000 Oe versus temperature for FC and ZFC conditions, and Arrott plots determined from magnetization versus magnetic field data. It appears that a magnetization of 1000 mB per Gd atom is achieved, which is indicative of interaction with host material, as this figure is much larger than 8 mB expected from a Gd atom. These findings are very promising, because (Ga,Gd)N may be easily doped with donors (acceptors) with a concentration exceeding that of Gd, to generate spinpolarized electrons (holes) in the conduction band (valence band). Gd-doped GaN with its above room temperature TC might thus be a very attractive candidate for future semiconductor-based spintronics. Because cgs and SI units are both used in the literature, unfortunately sometimes interchangeably, a conversion table between the two as well as definition of pertinent magnetism parameters along with their units is tabulated in Table 4.15.

4.9 Intentional Doping

T=5K T = 300 K nGd = 1018 cm –3

500

0

600

Magnetization (µB /atom)

Magnetization( μ B/ atom–1)

1000

–500

400 200 0

–200 –400

–1000 –3000 –2000 –1000

–600 –500

0

1000

740

2000

3000

ZFC (H = 1000 Oe) FC (H = 1000 Oe)

720

–1 Magnetization(μ Batom )

500

Magnetic field (Oe)

(a)

700 680 660 640 620 Gd–3B

600

n

μ

0

Magnetic field (Oe)

580 0 (b)

18

Gd

= 10 cm

50

–3

100

150

200

250

300

350

Temperature (K)

Figure 4.184 (a) M–H curves showing hysteresis loops at 5 K (circles) and 300 K (squares), for a Gd-implanted sample with Gd concentration of 1018 cm3. The inset is that of (a) but for magnetic fields of 500 Oe. M-T curves at FC and ZFC conditions at H ¼ 1000 Oe are plotted in (b) for a sample with Gd concentration of 1018 cm3, which appear to indicate a Curie temperature above 300 K. The Arrott plot of the same sample measured at 5 and 300 K are shown in (c). If one were to

extrapolate from large magnetic fields on a straight line, those lines would intercept the horizontal axis on the negative side for both 5 and 300 K data indicating that the Curie temperature is above the measurement temperature. However, the Arrott plots do not follow a straight line away from zero magnetic field as they should, see Figure 4.142, and therefore it cannot be stated with requisite certainty what the Curie temperature is.

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j 4 Extended and Point Defects, Doping, and Magnetism 1x10

6

8x10

5

6x10

5

4x10

5

2x10

5

2

5K

2

M (μΒ Gd)–

1166

0 0

300 K

1

2

3

4

5

μ 0H/M Figure 4.184 (Continued)

4.9.10 TM-Doped Nanostructures

Incorporation of Mn into GaN nanowires has also been reported. Deepak et al. [842], Han et al. [843], Choi et al. [844], and Liu et al. [845] produced Mn-doped GaN nanowires with the impetus that small dimensions are needed to take advantage of the spin. These materials could be free of defects, perfectly single crystalline, and have been reported to have a homogeneous distribution of Mn. In addition, these Mndoped GaN nanowires have been reported to be ferromagnetic with a Curie temperature of up to 300 K. It should be pointed out that critical optical and electrical measurements are lacking to corroborate the magnetization measurements. The nanowires are expected to have interesting magnetoelectronic properties because of the confinement of carriers in the radial direction and large magnetic anisotropy energy. In this vein, Wang et al. [846] performed first principles calculations for Cr-doped GaN nanowires with diameters of 0.45 and 1 nm and predicted ferromagnetism regardless of the site occupancy of Cr is a each Cr atom carries a magnetic moment of about 2.5 mB (Bohr magneton). In contrast, in Mn-doped GaN nanowires the magnetic coupling between the Mn atoms is sensitive to the Mn-Mn and Mn-N distances [847]. The magnetic moment at the N site, however, was found to be small and aligned antiferromagnetically to the moments at the Cr atom. The magnetization axis was predicted to be perpendicular to the axis of the wire, but with a small anisotropy in energy. Wang et al. [846,847] also considered Mn-doped nanowires. Using first principles theory, the authors have also shown that Mn-doped GaN nanowires with diameters of 0.45 and 1 nm are ferromagnetic. However, this ferromagnetic coupling between the Mn spins, driven by a double exchange

4.9 Intentional Doping

mechanism, was found to be sensitive to the Mn-Mn and Mn-N distances. Furthermore, calculations of the anisotropic energy showed that the magnetic moment orients preferably along the ½1 0
1 0
direction while the wire axis points along the [0 0 0 1] direction. Deepak et al. [842] reported the synthesis of (Ga,Mn)N nanowires of two different groups with average diameters of 25 and 75 nm, respectively, and with Mn concentration of 1, 3, and 5%. All the samples showed magnetic hysteresis at 300 K. The Mn2þ-related peak was observed in PL measurements, with a significant blue shift of this Mn2þ-related emission for the 25 nm diameter nanowires as compared to the 75 nm ones. Han et al. [843] prepared their Mn-doped GaN nanowires by CVD growth utilizing the reaction of Ga/GaN/MnCl2 with NH3. The diameter of the resultant single-crystalline nanorods was about 50 nm with wurtzite structure and grown along the [0 0 0 1] c-direction. No other phase or clusters was detected by highresolution TEM analysis. The electron energy loss spectroscopy analysis as well as energy dispersive spectroscopy revealed homogenous doping with Mn over the entire nanowire with about 5 at.% of Mn. As repeated on many occasions, these techniques cannot unequivocally rule out the formation of magnetic phases that might be embedded in the diluted magnetic semiconductors as they either do not have the spatial resolution needed or they cannot probe the entire sample. The SQUID magnetometers are very sensitive and a minute amount of magnetic material inclusive of contamination from even tools used to handle the sample could give rise to hysteresis. Continuing on, Han et al. [843] observed hysteresis in magnetization curves at 5 and 300 K with the implication of room-temperature magnetization. In addition, the authors plotted the DM ¼ MFC  MZFC curve (using the temperature-dependent FC and ZFC measurements), which represents the difference between magnetization measured while cooling in zero field or cooling under H ¼ 100 Oe and that measured while the temperature is increased; DM dropped to zero at about 300 K. This was used to proclaim ferromagnetism at room temperature, which must yet be scrutinized. Han et al. also were able to perform magnetotransport measurements on individual nanowires in the temperature range 5–300 K, showing the negative MR at temperatures below 150 K.

4.9.11 Applications of Ferromagnetism and Representative Devices

The premise of spin-based electronic devices is predicated on transferring magnetic information from one part of the device to another by using extremely small magnetic elements to encode it onto the itinerant electron spin channels and subsequently reading it off. This coding may be changed by remagnetizing the medium comprising the small magnetic elements and thus enabling the creation of electronic components, the characteristics of which could be modified to respond to applied magnetic fields. The focus of these pages of course is centered

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j 4 Extended and Point Defects, Doping, and Magnetism on semiconductor media made magnetic by doping with magnetic ions such as elements with spin-unpaired electrons found in transition metals and rare earth elements. The use of spin and relevant properties mainly for information storage in media composed of ferromagnetic metals has been the topic of a tutorial review article [642]. The topic has been reviewed amply [631–633], and following a brief mentioning of some of the important device parameters, a few representative device-like concepts are discussed. Briefly, to utilize spin for electronic and optoelectronic applications, one must be able to inject or generate spin-polarized electrons. Second, the spin coherence times must be sufficiently long in relation to device dimensions so that spin coherence would only be perturbed mainly by external controls provided, such as a voltage applied to a gate. Spin-polarized electrons can in principle be injected from a magnetic medium, such as a Schottky barrier made of ferromagnetic metal or diluted magnetic semiconductor or oxide with proper precautions. Spin-polarized electrons can also be produced in a medium by optical excitation with carefully chosen polarized light, as discussed in Section 4.9.6.6.1 dealing with magnetic circular dichroism. If the aforementioned approaches turn out successfully, devices based on spin tunnel junctions, FET-like devices utilizing spin, optical emitters and detectors sensitive to polarization, and elements for quantum computing may be possible. The realization of high TC ferromagnetic DMS might promote the development of spintronic devices, the premise of which is exploitation of electron spin in addition to its charge, for creating a new class of devices with unprecedented functionality. Devices utilizing spin transport rather than charge transport are likely to have the advantages of faster data processing and operation with much less power consumption [631]. To realize spintronic devices for practical applications, spin-polarized carriers need to be first efficiently injected into semiconductors. Spin injection across a semiconductor-ferromagnetic material interface can be adversely affected by the presence of defects at the interface, resulting in low injection efficiencies [848] Diluted magnetic semiconductors with ferromagnetism above room temperature could pave the way for achieving spin polarization and highly efficient spin injection necessary for driving spintronics technologies. The prospects are exhilarating for device applications utilizing the ferromagnetic properties of transition metal doped wide bandgap GaN and ZnO. As discussed earlier in the theoretical studies of TM-doped GaN, Kronik et al. [696] found that the introduction of Mn, in its ordered ferromagnetic phase, results in the formation of a nearly 100% spin-polarized 1.5 eV wide impurity band, primarily due to hybridization of Mn 3d and N 2p orbitals, which renders the material half metallic and supports the effective-mass transport within it. This unique feature, in conjunction with possibly the above room temperature TC and the inherent compatibility with GaN technology, could make GaMnN a potentially significant material for spininjection applications for modern spintronics devices. The caveat is the unequivocal determination of ferromagnetism in this material, which is lacking. ZnO is also promising in achieving high spin-polarized injection efficiencies and carriers due in part to the fact that 3d transition metals have high solubility (up to 30% for Mn and Co). The addition of ferromagnetism to the already unique optical and electric

4.9 Intentional Doping

properties of GaN and ZnO could result in many novel multifunctional spintronics devices. Spin light-emitting diodes (spin LEDs) and spin tunneling junctions based on the current achievements in research of DMSs have already been reported. For the FETlike devices, although proposals have been around for many decades, experimental demonstration has proved to be difficult. Nonetheless, we discuss the principles of operation of a proposed spin FET. Figure 4.185a shows a schematic diagram of the spin FET proposed by Datta and Das [849] and reviewed in Ref. [632]. A metal analogue was also reported by Johnson [850] who performed a spin injection experiment with thin films using a device that became known as the spin transistor. In this device, carriers are injected into the channel from a spin-polarizing electrode (called the polarizer or source), which can be either a ferromagnetic metal or a dilute magnetic semiconductor, through a channel to be collected by another magnetically polarized electrode (called the analyzer or drain). The current through the transistor depends on the relative orientation of the electron spins and the source and drain magnetic moments, the current being maximum when they all align. In a spin FET, application of a relatively low gate voltage causes an interaction between the electric field and the spin precession of the carriers [851], via the so-called Rashba spin–orbit coupling effect [852,853]. It is equivalent to the action of an effective magnetic field Beff that lies within the plane of the 2D electron gas and is perpendicular to the instantaneous wave vector. This results in spatial modulation of the net spin polarization of the current, which can be controlled by an applied gate voltage. If this is sufficient to render the spin orientation of the carriers out of alignment in relation to that of the drain contact (analyzer), then the current is effectively shut off, and this can occur at much lower biases than is needed to shut off the current in a charge-controlled FET. Figure 4.185c represents the schematic of a proposed spin FET based on the GaN material system. Challenges to fabricate a spin FET in the GaN-based system include efficient spin-polarized injection, transport, control and manipulation, and detection of spin polarization as well as spin-polarized current. Devices are required to generate coherent spin injection, with dimensions comparable if not less than the spin coherence length. This, in the case of GaN, means that the channel length must be shorter than some 200 nm. New approaches such as the use of Mn-doped GaN nanotube arrays [854], which have been favorably predicted to enhance the spin polarization or to increase the dephasing length, might have to be brought to bear to realize a spin FET in GaN. Actually, electric field control of spin properties has been demonstrated in GaMnAs. In this system hole interaction is clearly the driving mechanism for ferromagnetism, and therefore if a reverse-biased metal insulator FM semiconductor is employed and made to cause hole depletion, the ferromagnetism can be made to vanish. In contrast, if the metal–insulator–semiconductor junction is forward biased, the depletion depth is reduced and therefore ferromagnetism remains. This is, however, limited to p-type material, which in compound semiconductors is not known for its transport properties and therefore not an attractive material. Ohno et al. [855] reported the demonstration of this concept in an insulated gate (In,Mn)As

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j 4 Extended and Point Defects, Doping, and Magnetism (a)

VG

n Ω

VG,off IDS = 0

k

Drain (spin detector)

Source (spin injector)

(b)

VG

n Ω

VG( = 0),IDS =I ON

k

drain Drain (spin (spin detector) detector)

source Source (spin (spin injector) injector)

(c) GaMnN

Gate

GaMnN

AlN GaN

Figure 4.185 (a) Schematic of the Datta–Daslike spin field effect transistor (SFET). In the case where the channel itself is not ferromagnetic and would not align the electron spin, spin-polarized electrons must be injected and transported all the way to the drain and collected for the drain current to occur. In this case, the source (spin injector) and the drain (spin detector) are ferromagnetic metals or semiconductors, with parallel magnetic moments, and it is assumed that spin-polarized

electrons can be injected with high efficiency. The injected spin-polarized electrons with wave ! vector k move ballistically along a quasi-onedimensional channel formed by, for example, an InGaAs/InAlAs heterojunction in a plane normal to n. Electron spins precess about the ! precession vector W, which arises from spin–orbit coupling and which is defined by the structure and the material properties of the channel. The magnitude of W is tunable by the gate voltage VG at the top of the channel. The

4.9 Intentional Doping

channel structure. Shown in Figure 4.186 are the gated Hall resistance (RHall) data measured at 22.5 K versus the magnetic field with the 125 V (reverse bias for p-type channel), 125 V (forward bias causing hole accumulation), and 0 V bias conditions. The 5-nm-thick (In,Mn)As channel of the insulating-gate field effect transistor (FET) (x ¼ 0.03) was grown on a 10 nm InAs/500 nm (Al,Ga)Sb buffer and a GaAs substrate. A 0.8 mm gate insulator and a metal electrode served to complete the structure. The hole concentration in the channel was estimated to be 5–8 · 1013 cm2 from the change in resistance with the gate voltage and Hall effect at room temperature. In Figure 4.186, the bias of VG ¼ 125 V reduces the sheet hole concentration by 3 · 1012 cm2. The application of a positive gate voltage depletes the channel and reduces the hole-mediated ferromagnetic interaction, which results in a paramagnetic behavior of the magnetization without any hysteresis. A negative gate voltage (125 V), however, causes hole accumulation, which leads to a clear hysteresis as shown in the figure. The magnetization curve resumes its original values as the gate voltage returns to 0 V. At zero gate bias, the channel is weakly ferromagnetic as can be seen from the presence of a small hysteresis. The 125 V swing gives rise to 6% change in the hole concentration and results in a TC change of 4% (1 K). The spin coherence length (dephasing length) is a critical parameter in that the initially spin-polarized electrons are subject to depolarization. The longer the distance between the injecting and collecting electrodes, the more severe is the depolarization. Depolarization strongly depends on material characteristics such as spin–orbit interaction. One benefit to device performance derives from the use of III-Nitride materials because of their small spin–orbit interaction compared to other III–V materials. Selection of the composition of III nitride alloys and of the maximum size of the channel must take into account the appropriate dephasing length, among other factors. To inject (collect) spin-polarized electrons into (from) the transistor channel, ferromagnetic electrodes, either semiconductor or metal varieties whichever could inject spin-polarized electrons efficiently, must be used. GaN-based materials offer certain advantages due to the piezoelectric and pyroelectric phenomena, as do the spin-splitting strained heterojunctions in conventional III–Vs. 3 current is large if the electron spin at the drain points in the initial direction (a); the current is small if the precession period is much larger than the time of flight, leading to a situation where the injected electron spin and detector spin polarization are not aligned or are opposite to one another (b) (patterned after Ref. [632]). In another scenario, the channel is made ferromagnetic by doping the semiconductor with magnetic ions. Under the assumption that ferromagnetism is achieved and device is operating below the Curie temperature,

electrons injected with any common source (spin unpolarized) would be spin polarized due to the ferromagnetic channel and be collected by the spin-polarized drain analyzer. When ferromagnetism is destroyed by some external means, then the spin polarization is not retained and the drain current would reduce. Schematic diagram of a proposed spin FET based on the GaN semiconductor system. InGaN can be substituted for GaN if the Rashba coupling is too small in GaN (c).

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j 4 Extended and Point Defects, Doping, and Magnetism VG < 0 R (kΩ) Hall

0.04

1

1.5 K 5K 10 K

0

20 K

0.02

RHall (kΩ)

–1 –0.5

0.0 B(T)

VG > 0

0.5

0.00

22.5 K

VG = 0V

–0.02

+125 V –125 V 0V –0.04 –1.0

–0.5

0.0

0.5

1.0

B (mT) Figure 4.186 Hall resistance RHall of an insulated gate (In,Mn)As field effect transistor at 22.5 K as a function of the magnetic field for three different gate voltages. RHall is proportional to the magnetization of the (In, Mn)As channel. Upper right inset shows the

temperature dependence of RHall. Left inset shows schematically the gate voltage control of the hole concentration and the corresponding change of the magnetic phase. Courtesy of Ohno et al. [855]. (Please find a color version of this figure on the color tables.)

Because the only attractive GaN FET is that based on two-dimensional electron gas (2DEG), a spin FET based on GaN could also take advantage of transport along the heterointerface. The proximity effects of the exchange splitting in a ferromagnetic metal on a 2DEG in a semiconductor with implications to FET-like devices have been reported [856]. The spin-dependent energy and lifetime in the 2DEG lead to a marked spin splitting in the in-plane current. The planar transport gives way to current leakage into the ferromagnetic layer through the interface, which leads to a competition between drift and diffusion, providing a basis for modulation. The transport theory of a proposed spin valve consisting of a FET with two ferromagnetic gates has been used to demonstrate the effect. The polarization charge at the GaN/AlxGa1xN interface provides an attractive avenue toward optimizing the spin splitting of electrons and holes and thus the performance of the spin-FET. Not much if any is known about the Rashba coupling strength in GaN or InGaN. The zero-field spin splitting in the nitride semiconductor

4.9 Intentional Doping

2DEG can be measured using the beating patterns in Shubnikov–de Haas oscillations [857]. The spin splitting is the combined effect of Rashba [852] and Dresselhaus [858] interactions. The former is due to structural inversion asymmetry and can be varied by an external gate voltage [859–862] while the latter is due to bulk crystalline inversion asymmetry and is insensitive to gate voltage. A spin valve structure can be used to measure the coherence length. At the time of writing this chapter, there was no device demonstration in GaN. However, electric-field control of ferromagnetism has been demonstrated in a (In,Mn)As-based FET at reduced temperatures [855], and optically controlled ferromagnetism has been reported in GaAs with Fe particles. In addition to switching devices, optical devices such as LEDs featuring, that is, ferromagnetic anode (p-type layer which is natural for transition element doping), are of interest because of polarized light emission [863] with applications to optical switching and optical communication with increased bandwidth. What is more is the modulation of the polarization of emitted light by application of an external magnetic field [864,865]. Spin LED allows modulating the polarization of the light emitted by the spin LED by application of an external magnetic field, which is of interest to optical switching and optical communication with increased bandwidth. Spinpolarized light emission has been demonstrated in II–VI semiconductors [866], III–V GaAs [770], and GaN recombination region but with ZnSe(Be:MN) spin aligner [866]. To give the reader a synopsis of the operation of spin LED, the GaAs device is discussed here, particularly because the GaN varieties are not yet available and there are some fundamental issues whether one circular polarization over the other is promoted. The sample structure of a GaAs-based spin LED is depicted in Figure 4.187. The structure contains a p-type ferromagnetic semiconductor (Ga,Mn) As and n-type nonmagnetic semiconductor GaAs, which were epitaxially grown by MBE. The use of ferromagnetic (Ga,Mn)As allows one to inject spins in the absence of magnetic field. Spontaneous magnetization develops below the Curie temperature in the ferromagnetic p-type (Ga,Mn)As semiconductor, indicated by the arrows in the (Ga,Mn)As layer. Under forward bias, spin-polarized holes are injected from the (Ga, Mn)As side into the nonmagnetic region and recombine with spin-unpolarized electrons injected from the n-type GaAs substrate in a nonmagnetic quantum well made of (In,Ga)As (hatched region), as shown in Figure 4.187a, through a spacer layer with thickness d, producing polarized electroluminescence [633]. Injection of spin-polarized carriers through a spacer layer enhances the injection efficiency of spin-polarized carriers. For perfectly spin-polarized carriers (i.e., 100%), the extent of optical polarization is 50%, as Popt ¼ (sþ  s)/(sþ þ s) where sþ and s represent the polarization of the HH and LH transitions, respectively. Due to the crystal symmetry of GaAs, injected spin-polarized electrons have been detected in the form of circularly polarized light emission from the quantum well, which corresponds to the magneto-optical Kerr effect loops and can be directly related to spin-polarization. The presence of spin polarization has been confirmed by measuring the polarization of the emitted light. Figure 4.187b shows the observed hysteresis in the degree of polarization of the emitted light at temperatures T ¼ 6–52 K.

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j 4 Extended and Point Defects, Doping, and Magnetism

Figure 4.187 Injection of spin-polarized holes into a light-emitting p–n diode using a ferromagnetic semiconductor (Ga,Mn)As. (a) Sample structure. Spin-polarized holes hþ travel through the nonmagnetic GaAs and recombine with spin-unpolarized electrons in the (In,Ga)As quantum well. I represents the current, and sþ represents circularly polarized light emitted from the edge of the quantum well. (b) Dependence of the polarization DP of the emitted light on the magnetic field B at temperatures of 6, 31, 52 K, the latter above the Curie value. The solid and hollow symbols represent the degree of polarization when the magnetic field is swept in the positive and

negative directions, respectively. The magnetic field was applied parallel to the surface along the easy axis of magnetization of the (Ga,Mn) As. The temperature dependence of the residual magnetization M in (Ga,Mn)As, where the degree of polarization of the zero magnetic field seen in the emitted light exhibits the same temperature dependence as the magnetization (not shown). (c) Dependence on temperature for B ¼ 0 of the change in the relative remanent polarization, DP, (hollow circles) and magnetic moment measured by a SQUID magnetometer (solid circles). Courtesy of Ohno and coworkers [770]. (Please find a color version of this figure on the color tables.)

4.9 Intentional Doping

∆P 1.00

T=6K

Magnetization

0.75

4 0.50

31 K

2

0.25

52 K 0.00 0

20

40

60

Temperature (K) (c) Figure 4.187 (Continued )

80

100

Magnetization (10-5emu)

Relative polarization,∆P (%)

6 16 K

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j 4 Extended and Point Defects, Doping, and Magnetism The polarized light emission concept has been extended to QD LED heterostructures [867], albeit in the relatively well-established GaAs system, grown by MBE on Sidoped nþ (0 0 1) GaAs substrate. The obvious choice for the QD over QW is due to expected longer spin relaxation times and smaller interband ground-state transition energy in (In,Ga)As/GaAs QD than the bandgap of ferromagnetic (Ga,Mn)As. In this (In,Ga)As/GaAs/(Ga,Mn)As heterostructure, (Ga,Mn)As DMS layer acts as a spin injector. Experimental method employed for producing the structure, however, is such that it is not clear if and why the quantum dot regions are only ones that would be doped with magnetic ions. Assuming this was the case, let us continue. A GaAs spacer layer was grown on an undoped active region, consisting of four coupled (In, Ga)As self-organized QD layers. The collective polarization in (Ga,Mn)As ferromagnetic DMS layer provides a large magnetic moment. Hence, the DMS provides localized magnetic moment and holes. The ferromagnetic behavior of (Ga,Mn)As in the heterostructure is shown in Figure 4.188. The magnetization exhibits the characteristic hysteresis behavior with a saturation field of 150 Oe at 8 K. However, the temperature-dependent magnetization shows a Curie temperature, Tc  44 K. All the holes scatter to the lower Zeeman levels, yielding a high percentage of spinpolarized holes. These spin-polarized carriers recombine in the QD recombination region to produce circularly polarized photons. Among many other measurements, electroluminescence turns out to be one of the popular experiments to reveal the properties of recombination of spin-polarized holes from DMS aligner with electrons in QD or QW. For (In,Ga)As/GaAs/ (Ga,Mn)As heterostructure [867], EL measurements were carried out at 5.1 K in an applied magnetic field of 1000 Oe. The unpolarized EL shows a peak at 1.27 eV with FWHM 67 meV. The light output of the LED was focused onto a zero-order quarterwave plate. The zero-order quarter-wave plate basically converts the circularly polarized output light into horizontal and vertical linearly polarized light, which is detected using an appropriate photomultiplier tube through scanning spectrometer. One such spectra of the circularly polarized light (sþ and s) output is shown in Figure 4.189 from the above DMS heterostructure. The spin degenerate conduction and valence band states in the QDs and the corresponding radiative transitions are shown in the inset according to the selection rule for this structure. The heavy-hole (sþ) and light-hole (s) transitions are polarized in opposite directions. It may also be noted that the transition probabilities of the sþ transitions are three times larger than that of the s transitions. The spin injection efficiency was calculated to be 36% at 5.1 K. Several other groups [746,868] have measured the EL and magnetic properties of (Ga,Mn)As-based heterostructures. Intuitively, GaN has a smaller lattice constant than conventional III–Vs such as GaAs; thus, the magnetic coupling strength is expected to be higher, which also would result in a higher Curie temperature, see Figure 4.143. However, optical experiments performed on GaN/InGaN quantum wells [792] show that the electron spin relaxation time in these structures is shorter than in bulk GaN as well as in GaAs/InGaAs quantum wells. The approach requiring the least semiconductor technology for achieving spin LED in GaN would be to implant Mn into the top contact p-GaN layer of the standard GaN/InGaN LED. Polyakov et al. [869] reported such spin LED

4.9 Intentional Doping

j1177

10 0.4 µm Ga0.974M0.026As Ts = 210 o C TCurie = 44K B = 0.1 T

8

6 M (emu cm—3)

Magnetization (emu cm–3)

15 10 4 5

–300

–200

–100

0

100

200 300 H (Gauss)

–5 2 —10 –15 0 0

100

200

300

Temperature (K) Figure 4.188 Magnetization versus temperature data for the (Ga,Mn)As spin aligner layer. Inset shows the characteristic hysteresis behavior for the (Ga,Mn)As layer at 8 K [867].

structure without polarized light emission. Although, the device produced electroluminescence of low intensity, much higher series resistance and turn-on voltage than those of ordinary LED, due to the difficulty in annealing all radiation damage, has been cited as the bottleneck. The same group studied the inverted spin LED structure grown by MBE with n-type GaMnN on the top also [870]. However, this approach was found not to be successful possibly due to the higher series resistance caused by the lower lateral conductivity of p-GaN inherent to these inverted structures. It should be mentioned that difficulties in obtaining high quality layers on top of a very heavily doped p-GaN are not trivial. The parasitic diode formed with the lower p-GaN layer has been reported as being detrimental to device performance [871]. Furthermore, the reported ferromagnetism in GaMnN was found to be unstable against the type of high temperature anneals used to minimize the contact resistance.

400

j 4 Extended and Point Defects, Doping, and Magnetism Similar spin LED structures [872] have also been reported, albeit without spinpolarized light emission, wherein the nonmagnetic parts of the device was grown by OMVPE and the magnetic component that is limited to the top layer was grown by MBE techniques. It contains four main regions, namely, (1) an intended spin injector of a 120 nm thick n-Ga0.97Mn0.03N layer, (2) a spin detector made up of five periods of nonmagnetic In0.4Ga0.6N quantum wells (3 nm) separated by Si-doped GaN barriers (10 nm each), (3) a nonmagnetic Si-doped GaN spacer (20 nm) (inserted between the aforementioned two regions to avoid direct overlap of electron and hole wave functions from the two regions and thus to ensure a predominant role of the spin injection as the source of spin polarization as monitored in the InGaN MQW spin detector), and (4) a 2-mm-thick layer of Mg-doped p-type GaN for electrical injection of holes into the spin detector. The structure was grown on sapphire substrates starting with a 2 mm thick undoped semi-insulating GaN and also has a top layer of n-type Si-doped GaN (100 nm). The structure shows none or very low optical (spin) polarization at zero field or 5 T, respectively. The weak polarization observed is attributed to the intrinsic optical polarization of the InGaN MQW associated with population distribution between spin sublevels at low temperatures. The device shows loss of efficiency of

B = 0.1 T T = 5.1 K d = 100 nm

σ+

2.0

1.5 EL intensity (au)

1178

σ1.0

0.5

0.0 1.1

1.2

1.3 Energy (eV)

Figure 4.189 Electroluminescence spectra for the right (sþ) and left (s) circularly polarized light measured at 5.1 K with an applied magnetic field of 1000 Oe. The inset shows the spin degenerate conduction and valence band states and the corresponding radiative transitions. See Figure 4.155 for the origin of the right sþ and left s circularly polarized light in GaAs.

1.4

1.5

4.9 Intentional Doping

spin polarization generated by optical spin orientation or electrical spin injection. The I–V characteristics also suffer from a parasitic junction between the (Ga,Mn)N and the n-GaN in the top contact layer due to low conductivity of the p-type layer. Transient and circularly polarized PL analyses of the spin injection dynamics [873] of a GaMnN/ InGaN MQW LED structure suggest that the spin loss is most likely due to fast spin relaxation within the InGaN MQW which destroys any spin polarization generated by either optical or electrical spin injection. In the wurtzite III–nitride system, biaxial strain at the interfaces of heterostructures gives rise to large piezoelectric fields directed along the growth axis that ultimately breaks the reflection symmetry of confining potential, leading to the presence of large Rashba term in the conduction band Hamiltonian. It is this effect that is assumed to adversely contribute to the short spin relaxation times. The adoption of additional stressor layers or even cubic phase of nitrides has been suggested as a cure for larger spin splitting. While the spin-FETs are sought and spin LEDs have been demonstrated, spintronics-based ultrasensitive magnetic sensors would have great impact in magnetic recording, the development of nonvolatile memory, and would greatly increase the magnetic recording density of hard disk drives. Among the devices that have been well developed and that utilize spin is the “spin valve,” shown in Figure 4.190a in the vertical geometry [642]. The structure utilizes a normal metal straddled by two ferromagnetic metals on either end. When the spins in ferromagnetic metals on either end are aligned parallel to each other, the system is in the low resistance state (top figure). When, on the contrary, the spin of the FM metal on the right is flipped by a magnetic field making the spins of the ferromagnetic metals antiparallel, a highresistance state is attained. A parallel transport configuration of the same concept is shown in Figure 4.190b that requires again two ferromagnetic materials separated from one another by a nonmagnetic conductor whose conductivity is not dominant in the parallel stack. When the magnetic moment of two ferromagnetic layers are antialigned, spin-polarized electrons experience high resistance due to the high interfacial scattering and channeling of the current into narrow pathways. When the spins in the upper and lower films become aligned, the device resistance decreases. To be able to control the spin of one layer only, one material is made “hard” and the other “soft.” Because the magnetic moment of one of the ferromagnetic layers is very difficult to be reversed in an applied magnetic field (the hard one), and the moment of the other layer is very easy to be reversed (soft layer), the soft layer then acts as the control valve and is sensitive to manipulation by an external field. The significant change in resistance in response to relatively small magnetic field in layered magnetic thin-film structures is called giant magnetoresistance effect, and is realized by using metallic layers or tunnel junctions in practical applications. In the vertical geometry, using a semiconductor resonant-tunneling diode in conjunction with a ferromagnetic emitter, large tunnel magnetoresistance has been observed in (Ga,Mn)As-AlAs-(Ga,Mn)As tri-layer structures (see Figure 4.191 for the layered structure, and the simplified in-plane band diagram, and the expected I–V characteristics) [874–876]. The I–V characteristics, particularly the derivative of the current with respect to voltage, consistent with resonance states associated with spin split (due to the sp–d exchange interaction) state have been observed. The structure

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j 4 Extended and Point Defects, Doping, and Magnetism

Figure 4.190 (a) Schematic representation of a spin valve, a normal metal straddled by two ferromagnetic metals. When the spins in ferromagnetic metals on either end are aligned parallel to each other, the system is in the lowresistance state top. When, for example, the spin of the FM metal on the right is flipped by a magnetic field, making the spins of the

ferromagnetic metals antiparallel, a high resistance state is attained. (b) Schematic representation of transport that is parallel to the plane of a layered magnetic metal sandwich structure for antialigned (upper figure – high resistance) and aligned (lower figure – low resistance) orientations. (Please find a color version of this figure on the color tables.)

studied by Ohno et al. [874] (see Figure 4.192a) consists of (from the surface that on the right side, down) 150 nm thick (Ga0.965Mn0.035)As, 15 nm undoped GaAs spacer, 5 nm undoped AlAs barrier, 5 nm undoped GaAs quantum well, 5 nm undoped AlAs barrier, 5 nm undoped GaAs spacer, 150 nm Be-doped GaAs (p ¼ 5 · 1017 cm3), 150 nm Be-doped GaAs (p ¼ 5 · 1018 cm3), and p1 GaAs substrate. All of the layers

GaAs: Be

AlAs: (i)

GaAs: (i)

GaAs: (i)

AlAs: (i)

4.9 Intentional Doping

GaAs: (i)

HH1 LH1 HH2

(Ga,Mn)As

HH3 LH2

HH4 (a)

E

E I

k//

V k//

0 (b)

GaAs quantum well

0 (Ga,Mn)As emitter

Figure 4.191 Schematic valence band diagram of a resonanttunneling diode structure (a), simplified diagram of energy versus wave vector parallel to the interface for the GaN quantum well and (Ga,Mn)As emitter, and resulting I–V curve by spin splitting of the valence band of (Ga,Mn)As emitter (b). Courtesy of Ohno and coworkers [874].

were grown at 650  C with the exception of the last (Ga,Mn)As layer, which was grown at 250  C. The derivative I/V characteristics obtained at 6 K indicated peaks associated with HH1, LH1, HH2, HH3, LH2, and HH4 resonant states. Spontaneous magnetization in ferromagnetic semiconductors gives rise to spin splitting of the conduction and valence bands due to the presence of exchange interaction. The magnitude of the splitting between the resonant states in the investigated structure was shown to be proportional to Ms (saturation magnetization) calculated from the Brillouin function, lending credence to the supposition that the origin of peak splitting is indeed the spin

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j 4 Extended and Point Defects, Doping, and Magnetism splitting in the valence band of ferromagnetic (Ga,Mn)As. One can therefore conclude that the splitting observed in the I–V curves is due to the spin splitting of the valence band associated with the development of spontaneous magnetization in (Ga,Mn)As. In the stepped part where the magnetization is antiparallel, the tunneling resistance increases. When the AlAs film that constitutes the tunneling barrier was made thin (1.6 nm), a TMR ratio over 70% was obtained. In this vein, a spin valve transistor with an epitaxial ferromagnetic Fe/Au/Fe(0 0 1) base has been integrated into n-GaAs [877,878]. Parallel transport in trilayer structures, such as the one discussed above, is characterized by the MR ratio (RH  R0)/R0, where R0 is the layer resistance in the absence of the external magnetic field, RH (H ¼ 0). For parallel magnetizations M, the structure consists of two (Ga,Mn)As layers and an AlN barrier separating the two Ga: MnAs layer with some thin GaAs spacer layers on either side (Figure 4.192a). The structure was grown with molecular beam epitaxy with the Be-doped buffer layer grown at a typical MBE temperature of 580  C while the Mn-doped layers utilized a low temperature of 250  C. With a moving shutter during the AlAs barrier growth, a wedge shape was achieved allowing the characterization of TMR as a function of barrier thickness [879]. Undoped GaAs spacer layers that were 1 nm thick were inserted on either side of the AlAs tunnel barrier to avoid Mn diffusion/incorporation into the AlAs barrier. Circular 200 mm device structures in which the AlAs thickness ranged between 1.3 and 2.8 nm, owing to the wedge-shaped growth of AlAs with the aid of a moving shutter in front of the substrate. Tunnel magnetoresistance effect in the aforementioned (Ga,Mn)As–AlAs–(Ga,Mn)As trilayer structure with an AlAs barrier thickness of 1.6 nm, measured at a temperature of 8 K, is shown in Figure 4.192b [879], when the magnetic field is parallel to the [0 0 1] axis in the plane. The measurements were performed at a bias of 1 mV, which obviates any possibility of hot carrier effects. The resistance for H ¼ 0 is slightly below 0.015 O cm2. The TMR values are 72% for H ¼ 110–120 Oe (solid curve) and 120–130 Oe when the magnetization of the Ga1xMnxAs becomes antiparallel. The minor loop represented by the thin line is indicative of the fact that the antiparallel as well as the parallel configuration is stable. The tunnel junction resistance measured at 8 K exhibited an exponential dependence on the barrier thickness ranging from the low 103 O cm2 for a barrier thickness of 1.4 nm to about 101 O cm2 for a barrier thickness of 2.8 nm. The barrier thickness dependence of TMR measured at 8 K for magnetic field parallel to [1 0 0] and ½1
1 0
axes is shown in Figure 4.193. A maximum value of 75% was attained for an AlAs barrier width of 1.46 nm when the field is along the [1 0 0] direction. However, the maximum was only about 30% when the magnetic field is parallel to the ½1
10
axis – again for the same barrier thickness of 1.46 nm. This dispersion is due to the cubic magnetocrystalline anisotropy induced by the zinc blende-type Ga1  xMnxAs crystalline structure, where its easy magnetization axis is h1 0 0i [880]. Above the barrier thickness of 1.46 nm, TMR drops rapidly for both orientations of the magnetic field. The drop in TMR below barrier thickness of 1.46 nm is not clearly understood but could be associated with the ferromagnetic exchange coupling between the two Ga1xMnxAs layers on either side of the barrier because of the thinness of the barrier. Moreover, the TMR decreased with increasing measurement

4.9 Intentional Doping

(Ga0. M 96 n0. )A 04 s(50nm GaAs(1n

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)

m)

AlAs GaAs (1 nm) (Ga0.967Mn0.033)As(50 nm) Be (p) GaAs (100 nm) p+-GaAs substrate (0 0 1)

(a)

60 40

0.020

TMR (%)

Resistance (Ω cm–2)

80

Minor loop

0.025

20 0.015

0 –200

–100

0

100

200

Magneticfield, H//[0 0 1] (Oe) (b) Figure 4.192 (a) Schematic illustration of a wedge-type ferromagnetic semiconductor trilayer heterostructure sample grown by LTMBE. (b) Tunnel magnetoresistance effect curves obtained at 8 K for a Ga1xMnxAs (x ¼ 4.0%, 50 nm)/AlAs (1.6 nm) Ga1xMnxAs (x ¼ 3.3%, 50 nm) trilayer tunnel junction of 200 mm in diameter. The bold solid and dotted curves were obtained while sweeping the magnetic field from positive to negative and from negative to positive directions, respectively (major loop). A minor loop is shown by the relatively thin line. The (Ga,Mn)As

layers are 50 nm thick, and the Mn composition in these layers are 0.04 and 0.033. The AlAs layer is 1.6 nm thick. Because the easy axis of magnetization lies within the plane of the sample, a magnetic field is applied parallel to the sample surface. When a magnetic field is applied along the [1 0 0] direction, a tunneling magnetoresistance effect of over 70% is observed, and when a magnetic field is applied along the [1 1 0] direction, this effect is approximately 30%, as detailed in Figure 4.193 [879].

j 4 Extended and Point Defects, Doping, and Magnetism 100 8K H//[1 0 0]

80

TMR (%)

1184

60

40 H//[1 1 0] 20

0 1.4

1.6

1.8

2.0

2.2

Barrier thickness (nm) Figure 4.193 The TMR values in Ga1xMnxAs (x ¼ 4.0%, 50 nm/ AlAs (dAlAs/Ga1xMnxAs (x ¼ 3.3%, 50 nm) tunnel junctions measured at 8 K versus the AlAs barrier thickness for magnetic field parallel to the [1 0 0] and ½1
1 0
axes [879].

temperature vanishing completely at 50 K, which corresponds to the Curie temperature. It should be pointed out that spin injector layer and medium into which the spinpolarized electrons are injected must match in terms of the ratio of the spin diffusion length and the conductivity to obtain sufficiently high spin-polarized electron injection and thus large TMR ratios. Semiconductor-to-semiconductor, oxide-to-oxide, and metal-to-metal varieties satisfy this requirement. However, this is not so in ferromagnetic metal to semiconductor varieties. To attain a reasonably matched case, a tunneling insulating barrier is typically inserted between the metal and the semiconductor. The spin precession length is a critical parameter for such a spin FET to function as intended. The initially spin-polarized electrons are subject to depolarization with distance, and the depolarization strongly depends on the material characteristics such as the spin–orbit interaction. On the contrary, the use of III nitrides may be beneficial because of possible longer spin coherence times, owing to their weak spin–orbit interaction compared to the more conventional III–V materials. Besides, due to the pronounced piezoelectric and pyroelectric effects in GaN-based materials, the spin splitting of electrons and holes in GaN/AlxGa1xN heterojunctions may be optimized. On the contrary, weak spin–orbit interaction leads to relatively small Rashba coupling. Spin–orbit interaction and the associated spin splitting in zinc blende III–V semiconductor heterostructures have been studied for more than a decade and are relatively well understood. Approaches used for the InGaAs system [859] wherein beating of two close frequencies due to spin splitting Shubnikov–de Haas oscilla-

4.9 Intentional Doping

tions [881], although may not be unambiguous [882,883], can potentially be used for GaN also in determining the spin–orbit interaction parameter. The intrinsic lack of inversion symmetry gives rise to the Dresselhaus [884] interaction. However, the structurally introduced inversion asymmetry such as heterostructures appears [851] to make the Rashba interaction [852,853] more plausible. As mentioned, a spin valve structure, for example, in the form of a spin valve transistor, can also be used to measure the coherence length (A spin valve – a term coined by IBM – is in general a structure that consists of a dedicated GMR trilayer in which the electrical resistance is high or low, depending on the direction rather than the strength of the magnetic field. The name may be somewhat misleading in that there is one fully open or closed valve. Instead, the change in resistance is typical in the range of 5–10%. A spin valve is typically made of only two ferromagnetic layers spaced by a layer of nonmagnetic metal. Contrary to a GMR multiplayer, the two ferromagnetic layers are magnetically decoupled. This is achieved by increasing the thickness of the spacer layer. As a further difference the magnetization of one of the ferromagnetic layers is spatially fixed (pinned) by an antiferromagnetic bottom layer. Thus, it is called the “pinned layer”, which is magnetically hard, and the other is called the “free layer” because it should easily follow the external magnetic field and is made of magnetically soft material, meaning it is very sensitive to small fields. The other, on the contrary, is made magnetically “hard” by various schemes – meaning it is insensitive to fields of moderate size. As the soft layer moves due to applied fields, the resistance of the entire structure changes. The analysis of weak antilocalization (WAL) [885] is much simpler to implement and determine the spin–orbit interaction parameter, as has been done for AlGaAs/GaAs/AlGaAs [886], AlSb(ZnTe)/InAs/AlSb [887], and InGaAs quantum wells with two-dimensional electron gas. As for the wurtzite AlGaN/GaN heterostructures, experiments based on Shubnikov–de Haas (SdH) [888,889], weak antilocalization [890,891], and circular photogalvanic [892] measurements have given conflicting results for the spin splitting. In particular, spin-splitting energies extracted from the beat pattern of SdH measurements are found to be as large as 9 meV, which is about an order of magnitude larger than the theoretical estimates based on the Rashba coupling mechanism for this material system [851]. To account for the discrepancy, Lo et al. [893] have proposed an additional spin-splitting mechanism for wurtzite structures and Tang et al. [894] proposed an alternative interpretation of such data based on magneto-intersubband scattering. In the context of weak antilocalization, Thillosen et al. [890] utilized the weak antilocalization effect to determine the dephasing time tf and the spin–orbit scattering time tSO (1.25 ps) in GaN with the aid of magnetoresistance measurement at very small magnetic fields (in the range of mT) in a two-dimensional electron gas system. The extent of the Rashba spin splitting is typically represented by the spin–orbit coupling (interaction) parameter a. Relying on the notion that the spin–orbit interaction is to a first extent due the asymmetry in the confining potential introduced by the heterostructure [851] and thus by the Rasha effect and with the help pffiffiffiffiffiffiffiffiffiffiffi of [895] aSO ¼
h=kF tSO tq (where tq is the quantum scattering time associated with cyclotron motion), Thillosen et al. [890] obtained a value of aSO  8.54 · 1013 eV m

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j 4 Extended and Point Defects, Doping, and Magnetism that compares with 6 · 1013 eVm obtained by Schumult et al. [891] The corresponding spin precession length lSO ¼
h2 =2am is approximately 200 nm, which is relatively short due in part to the large effective mass. The short value of lSO would indeed provide a tremendous challenge for device fabrication in terms of device lengths that can be used. The spin–orbit interaction parameter a is proportional to the average vertical field in the heterostructure a ¼ b<E>, where the b coefficient is inversely proportional to the effective mass and bandgap of the material [896]. Kurdak et al. [897] have also used weak antilocalization to determine the electron spin-splitting energies in the range of 0.3–0.7 meV (depending on the sheet electron concentration). This compares with DSO  0.3 meV at ne ¼ 1 · 1012 cm2 obtained by Schumult et al. [891] Consistent with such small spin-splitting energies, no beat feature in the SdH oscillations were observed. More importantly, as predicted by the Rashba mechanism, the measured spin-splitting energies were found to scale linearly with the Fermi wave vector. The spin–orbit times varied from 0.74 to 8.24 ps for sheet carrier densities of 5 · 1012 cm2 and slightly under 2 · 1012 cm2, respectively. The spin dephasing time varied between about 40 ps (at 1.8 K) and about 10 ps (at 4 K). Other potential spintronic applications include integration of magneto-optical effect in semiconductor optical isolators by using the Faraday effect to control the spin relaxation time and coherence for optical switches and quantum information processing. It has been shown that electron spin coherence is maintained for periods of at least a few nanoseconds in GaN [898], which raises the possibility of applications of confined electronic states to spin memory and manipulation. 4.9.12 Summarizing Comments on Ferromagnetism

A summary of the state of the possible DMS in GaN doped with magnetic ions is now warranted. With certainty the potential of room-temperature ferromagnetism has rejuvenated the field of dilute magnetic semiconductors. While the devices are the eventual driving force, the importance of understanding and producing ferromagneticdiluted semiconductor materials with an above room temperature Curie temperature, not to mention the junction temperature that is typically well above room temperature, cannot be overstated. The efforts on this topic could also potentially provide an opportunity for mainstream semiconductor device researchers to gain some insight into the world of magnetism, seen as a first and critical step toward realization of devices. Due to small spin–orbit coupling and small interatomic spacing, in addition to more intricate reasons, GaN and ZnO could potentially exhibit ferromagnetism when doped with ions having unpaired partially empty 3d (transition metal) and 4f (rare earth) shells. Intuitively, the smaller lattice constant of GaN and ZnO, compared to the conventional II–VI and III–V semiconductors, would serve to increase the magnetic coupling strength among the magnetic ions through indirect exchange interaction and lead to high Curie temperatures, as shown in Figure 4.143. However, the caveat is that, at least in the modified mean free field theory of Dietl, TM concentrations near 5% or higher and acceptor concentrations above 1020 cm3 are needed to observe above room temperature Curie temperatures. We must hasten to

4.9 Intentional Doping

mention that the mean field theory might overestimate the Curie temperature, particularly for low magnetic ion concentrations. While the above room temperature Curie temperature aspect of that theory is often cited, the conditions at which that temperature hinges are often left out of the critical discussion. As elaborated repeatedly, on the experimental side it is still not sufficiently clear if the uniformly doped host or some other phase is responsible for the observed magnetic hysteresis persisting in some cases well above room temperature. Moreover, the transition energies induced by transition elements, particularly in GaN, are not yet well understood. It is reasonable to expect that only certain states of the TM element (after they are split by crystal field splitting and Zeeman splitting which is to be enhanced by large spin exchange interaction) contribute to interactions supporting the ferromagnetic state. Data in favor and against ferromagnetism in both GaN and ZnO have been reported. Arguments and counterarguments for or against have been made. At least at the time of this writing, the field still appears too controversial and confusing, as the local environment and energy levels of magnetic ions are too sketchy and mechanism(s) leading to ferromagnetism are too many, not to mention the ever-varying sample quality. In the confines of this chapter, what could be described as a gallant attempt was made to discuss those reports with appropriate comments regarding any inconsistencies and difficulties as far as the mechanisms and experimental methods are concerned. In short, not only are the various reported experiments not consistent, the same also holds for the effort on the theoretical side. On the theoretical side, it is not clear which of the many mechanisms put forth is applicable, to what extent, and in what kind of samples. For example, super direct exchange between magnetic ions is antiferromagnetic in the realm of the Zener theory, but indirect exchange interaction between the magnetic ions can lead to ferromagnetism in the realm of Zener and RKKY theories. In the one-electron system, the d wave functions of the transition metals and s wave functions of the host (conduction electrons) are orthogonal and could not hybridize in a cubic system. However, the hybridization of d wave functions of the transition elements and p-states of the host (valence band) is strong and can lead to ferromagnetism. The picture in wide bandgap semiconductors is rather complex in that the level splitting and associated hybridization are magnetic ion and doping level dependent. Therefore, simple arguments, while instructive, may not strictly apply. The mean field theory, expanded by Dietl to include the anisotropy of the valence band spin–orbit interaction, indicates the hole-mediated mechanism to be dominant. In the realm of this theory, the material under investigation must be convincingly p-type and yet ferromagnetism in n-type samples has been reported. In fact, donor-doped samples along with TM doping have been reported to be ferromagnetic and the constructive role of conduction electrons has been touted. In contrast, doping with elements, which would tend to reduce the background electron concentration, has also been touted to be the mechanism for the experimentally observed ferromagnetism. Moreover, magnetization in electron codoped samples has been attributed to bound polarons, particularly in ZnO. A bound polaron is formed when the magnetic moment of the ion is aligned with the spin of an exciton and the magnetic ion

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j 4 Extended and Point Defects, Doping, and Magnetism within its radius. The exciton in turn renders the spin of nearby excitons to align, which in turn is parallel to the spin of a nearby magnetic ion, and so on. Clearly, the field is in its infancy and the picture is rather complex. It is quite possible that strict theories mentioned above are not necessarily applicable to all the samples. It is very likely that ab initio calculations are required. These calculations would also provide a guiding light for experimentalist to focus on exploring those transition or rare earth elements that are more likely to be ferromagnetic in the particular hosts under investigation. A case in point is Mn in these two hosts. The theory indicates that in lightly and heavily doped samples the ferromagnetic phase is not stabilized. The doping concentrations that lead to a particular class of levels splitting are required for GaN:Mn to be stabilized in the ferromagnetic state. This simply means that more detailed and probing investigations are necessary to sort out the various experimental data that have been reported. As has been the characteristic of scientific matters, more clarity will prevail in time as the recent discussions appear to be going in this direction. In closing the discussion of the potential of ferromagnetism and applications to devices, it is not an overstatement that the expected advantages of GaN- and ZnO-based spintronics are truly exciting although the efforts in materials science and devices are still in their embryonic stage. However, there are many challenges to consider including whether high ferromagnetic transition temperature and carrier-mediated ferromagnetism can be realized. In addition to a ferromagnetic semiconductor, successful operation of spintronic devices also requires the support of spin-polarized transport in the device and the amplification and detection of spin polarization (or spin current). With accelerated research efforts toward room-temperature ferromagnetism in GaN and ZnO, and perhaps others such as oxides-based DMS materials, it might be possible to realize semiconductor magnetoelectronics. Perhaps, the “real” utilization of the spin degree of freedom in spin-based all-semiconductor multifunctional devices such as spin FETs, spin LEDs, spin RTD, sensitive magnetic sensor, high-density nonvolatile memory, and quantum bits for quantum processing will be a reality even at a fraction of the present of expectation. If so, this would represent a truly new chapter in the annals of semiconductor-based devices.

4.10 Ion Implantation and Diffusion for Doping

Ion implantation and diffusion are industrial doping methods for Si. The reports on III-N doping by ion implantation are somewhat contradictory so far. As compared to classical semiconductors, group III-N semiconductors have larger binding energies of the constituent atoms and smaller interatomic distances. Consequently, substitutional impurity incorporation by ion implantation is intrinsically more difficult. On the contrary, the temperatures needed for annealing the implanted lattice defects are limited to the decomposition temperatures of group III-N compounds, which sometimes are not high enough to anneal out the damage created.

4.10 Ion Implantation and Diffusion for Doping

Similarly, it should come as no surprise that experimental reports on diffusion in group III-N semiconductors are limited. Diffusion of Mg at 800  C for 80 h into an unintentionally doped n-type GaN layer in a sealed nitrogen ampoule resulted in p-type material [899]. A hole concentration of 2 · 1016 cm3 and a mobility of 12 cm2 V1 s1 at room temperature were measured. Lower diffusion temperatures were not successful in converting the sample into p-type. However, higher diffusion temperatures caused the film to decompose and evaporate. High-energy implants (80–100 keV) increased strain and defects in GaN [900]. The damage caused by such high-energy implants could not be annealed out after 30 min at 800  C; no conversion into p-type was detected. However, films implanted with Mg ions at lower energies (40–60 keV) recovered after annealing at 800  C and retained their original lattice parameters, as determined by X-ray diffraction. Hot-probe tests showed conversion into p-type. n-Type and p-type conductivities were accomplished by ion implantation of Siþ (200 keV), Mgþ (180 keV) and Mgþ (180 keV) þ Pþ (250 keV) [900]. For these experiments, undoped layers with background electron concentrations of 1–4 · 1016 cm3 as well as a postimplant thermal anneal procedure in the range 700–1100  C for 10 s were employed. To prevent or otherwise limit nitrogen loss, another GaN film was firmly pressed against the surface of the implanted sample during annealing. Mgþ implantation alone did not produce any doping effect. Mgþ/Pþ coimplantation led to a conversion from n- to p-type conductivity after annealing at 1050–1100  C with an activation percentage of about 62%. The effect of coimplantation here is to increase the vacancy concentration. Siþ implantation resulted in a sharp increase in the n-type conductivity after annealing at 1050–1100  C with an activation percentage of 93%. Results somewhat contradictory to the above reports on GaN were obtained by Wilson et al. [901] who observed that no diffusion of implanted Mg and other impurities (Be, C, Zn, Si, Se, and Ge) was discernible even after an anneal at 800  C for 10 min. Only S showed a marked diffusion at temperatures higher than 600  C. Impinging energies of the ions were as follows: H, 40 keV; Li, 100 keV; Be, 100 keV; C, 260 keV; F, 100 keV; Na, 100 keV; Mg, 100 keV; Si, 150 keV; S, 200 keV; Zn, 300 keV; Ge, 500 keV; Se, 500 keV [902]. The lack of diffusion in the implanted samples could be attributed to the fact that the energetic ions are capable of producing defects that, in turn, can trap implanted impurities, causing them to cluster. The annealing temperatures may not be high enough to anneal out such defects. Spurred by the predictions that Ca might be a shallow acceptor [571], implantation of Ca has been attempted. Implantation of Caþ at 180 keV and coimplantation of Caþ with Pþ at 130 keV followed by rapid thermal annealing at T 1100  C produced p-type doping with an ionization energy of 169 meV for a Ca acceptor level, which is similar to that of Mg [903]. In some device fabrication schemes, selective-area conversion of the sample to high resistivity is very desirable. This serves to eliminate the unwanted current paths in devices such as field effect transistors. To this end, proton implantation has been undertaken and high-resistance samples have been obtained by implanting Hþ or Heþ [904], and Nþ or Fþ [905,906]. As endemic to any aspect of nitride semiconductor research and development, many more investigations are needed for improved results and reproducibility. We should, however, be prepared for disappointment if

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j 4 Extended and Point Defects, Doping, and Magnetism ion implantation does not turn out to be universally relevant in nitride development due to a large bond strength and the high N-vapor pressure in this material. On the contrary, one would like to have strong bond strength for stability, and on the contrary, one would like to have a semiconductor with small bond strength to allow the device engineer to modify its characteristics. Ion implantation for the purpose of doping the material with magnetic ions, when applicable, are discussed in conjunction with the applicable topic.

4.11 Summary

A simple conclusion that can be drawn is that defects in GaN-based semiconductors as in any other, are important. Furthermore, they are made more relevant because of relatively strong coupling between polarization, mechanical properties, and thermal properties in that any strain inhomogeneity caused by defects leads to polarization inhomogeneity that in turn could mitigate additional defect generation particularly under high field and/or temperature operation. Unfortunate for GaN, despite the tremendous advances made, the lack of native substrate in the desired form is synonymous with extended and point defects. Although epitaxial lateral overgrowth can be used to reduce the extended defect concentration and naturally the point defect concentration, the window regions as well as the coalescence boundaries still contain inordinate amount of defects. It is possible in some special device structure to use only the wing regions with very low defect concentration, but this is not necessarily applicable to large-area devices. In addition, point defects and p-type doping are forever intertwined, and the nature of active p-type impurity incorporation is not sufficiently understood. Site selection and defect creation to minimize the free energy will require more effort than that expended so far. The question of whether acceptors in GaN are effective-mass-like impurities still lingers despite the predictions that the thermal activation energy for Be is only 60 meV and that for Zn is a large 330 meV [907]. Experiments are lacking in that Hall measurements have not been possible in Be-doped GaN and that Zn-doped GaN often has high resistivity. The predicted thermal activation energy for Mg is 230 meV [907], which is in the range of experimentally reported figures. Acceptor-like impurity incorporation including codoping in nitride semiconductors represents one of the most pivotal issues. Doping with magnetic ions in the hope of achieving above room-temperature ferromagnetism is met with certain degree of stalemate and controversy. Unlike the case of GaN, magnetic ion doped GaN is not p-type conductive. Therefore, even if ferromagnetism is achieved, its control with hole concentration is precluded. In closing, analysis of defects (extended, point, and complexes) in GaN has seen substantial improvement in a relatively short period and helped the practitioners of GaN devices to enhance their craft to the point of being a strong contender for the lucrative and colossal general lighting.

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891 Schmult, S., Manfra, M.J., Punnoose, A., Sergent, A.M., Baldwin, K.W. and Molnar, R.J. (2006) Large Bychkov–Rashba spin–orbit coupling in high-mobility GaN/AlxGa1xN heterostructures. Physical Review B: Condensed Matter, 74, 033302. 892 Weber, W., Ganichev, S.D., Danilov, S.N., Weiss, D., Prettl, W., Kvon, Z.D., Belkov, V.V., Golub, L.E., Cho, H.-I. and Lee, J.-H. (2005) Applied Physics Letters, 87, 262101. 893 Lo, I., Wang, W.T., Gau, M.H., Tsay, S.F. and Chiang, J.C. (2005) Physical Review B: Condensed Matter, 72, 245329. 894 Tang, N., Shen, B., Wang, M.J., Yang, Z.J., Xu, K., Zhang, G.Y., Chen, D.J., Xia, Y., Shi, Y., Zhang, R. and Zheng, Y.D. (2006) Physical Review B: Condensed Matter, 73, 037301. 895 Schierholz, C., K€ ursten, R., Meier, G., Matsuyama, T. and Merkt, U. (2002) Physica Status Solidi b: Basic Research, 233, 436. 896 Lommer, G., Malcher, F. and R€ossler, U. (1988) Physical Review Letters, 60, 728. 897 Kurdak, Ç., Biyikli, N., Äzgur, à., Morkoc, H. and Litvinov, V.I. (2006) Physical Review B: Condensed Matter, 74, 113308. 898 Beschoten, B., Johnson-Halperin, E., Young, D.K., Poggio, M., Grimaldi, J.E., Keller, S., DenBaars, S.P., Mishra, U.K., Hu, E.L. and Awschalom, D.D. (2001) Physical Review B: Condensed Matter, 63, 121202. 899 Rubin, M., Newman, N., Chan, J.C., Fu, T.C. and Ross, J.T. (1994) Applied Physics Letters, 64, 64. 900 Pearton, S.J., Vartuli, C.B., Zolper, J.C., Yuan, C. and Stall, R.A. (1995) Applied Physics Letters, 67, 1435. 901 Wilson, R.G., Pearson, S.J., Abernathy, C.R. and Zavada, J.M. (1995) Applied Physics Letters, 66, 2238. 902 Wilson, R.G., Vartuli, C.B., Abernathy, C.R., Pearton, S.J. and Zavada, J.M. (1995) Solid State Electronics, 38, 11329.

References 903 Zolper, J.C., Wilson, R.G., Pearton, S.J. and Stall, R.A. (1996) Applied Physics Letters, 68, 1945. 904 Binari, S.C., Rowland, L.B., Kruppa, W., Kelner, G., Doverspike, K. and Gaskill, D.K. (1994) Electronics Letters, 30, 1248. 905 Pearton, S.J., Abernathy, C.R., Wisk, P.W., Hobson, W.S. and Ren, F. (1993) Applied Physics Letters, 63, 2238.

906 Zolper, J.C., Pearton, S.J., Abernathy, C.R. and Vartuli, C.B. (1995) Applied Physics Letters, 66, 3042. 907 Fiorentini, V., Bernardini, F., Bosin, A. and Vanderbilt, D. (1996) Ab initio shallow acceptor levels in gallium nitride. 23rd International Conference on the Physics of Semiconductors, vol. 4, part 4, World Scientific, Singapore, pp. 2877–28780.

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Index a ab initio band structure 1101 ab initio calculations 265, 281, 363, 405, 931, 992, 1038, 1091, 1100, 1103 ab initio tight-binding methods 141, 214, 366 ab initio treatments 1089 absorption, see photoreflectance spectra absorption, see reflectance 626 accelerating energy 420 acceptor 1000 – substitutional 1007 acceptor activation ratios 628 acceptor concentration 1026 acceptor-doped GaN 1018 acceptor-like isotropic resonances 1002 acceptor–nitrogen vacancy 936 acoustic–acoustic interactions, see acoustic– optic interactions acoustic–optic interactions 56 activation energy 581, 583, 657, 1025, 1044 ADA complex formation 1026 ADA model 1023, 1025, 1029 adsorbate–surface interaction 411 adsorption isotherm 390 adsorption process 389, 391, 400, 405 adsorption rate 412 AFM stabilization energy 1100 Al deposition 643 AlInN – growth 695 AlGaN – cracking 658 – growth 653 – low temperature interlayer 659 – ordering 668 – p-type growth 666 AlGaN alloy 90 AlGaN films 657

AlGaN layers 90, 91, 245, 265, 483, 485, 546, 602, 653, 656, 658, 659, 664, 670, 672, 699, 1032, 1033 AlInGaN-based heterostructures 99 AlInGaN multiple quantum wells 100 all-electron approach 136 all-optical method 288 Al mole fraction 654 Al-rich conditions 651 AlN – epitaxial relationship to Si (001) 383 – growth 638 – surface reconstruction 642 AlN buffer layer(s) 381, 437, 446, 505, 631, 733 AlN bulk crystals 176 AlN electron effective mass 180 AlN films 643 – growth 409 – molar fraction 276 aluminum-based compounds 99 aluminum mole fraction 655 aluminum nitride 62 aluminum nitride films 653 ammonia incident kinetic energy 440 ammonia-limited conditions 439 amorphous GaN films 394 amorphous material 1004 anion–cation bond length 7 anionic model potential parameters 141 anisotropic energy 1167 – splitting 159 anisotropic growth 732 anisotropic phenomenon 627 anisotropy 1139, 1188 annealing 643, 1018, 1035, 1178, 1189 annealing, see LEEBI treatment annealing temperatures 1190

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoç Copyright Ó 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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annihilation radiation 991 annular dark field imaging (ADF) 674 anodic alumina membranes 737 anodization parameters 504 antibonding site 1036, 1037 antiferromagnetic behavior 1157 antiferromagnetic configuration 1089 antiferromagnetic coupling 1143 antiferromagnetic solids 1133 antiferromagnetic state 1091 antiferromagnetism 1137 antisite defects 928, 934 antisites–Ga 935 antisites–N 935 antistandard 498 antisurfactant 721 – microscopic mechanism 721 a-plane GAN 373, 376, 484, 604, 612, 616 a-plane sapphire 293, 351, 371, 375 Archimedean displacement measurements 78 Arrhenius form 403 Arrhenius plots 452, 581, 964 Arrott plots 1074, 1129, 1131, 1164 arsenide semiconductors 371, 596 asymmetric reflections 605 atmospheric pressure chemical vapor deposition (APCVD) reactor 541 atmospheric pressure two-flow system 721 atom-centered basis functions 137 atomic arrangement 545, 603 atomic force microscopy (AFM) 550, 740, 832 – image 340, 570 atomic layer epitaxy (ALE) 79, 394, 528, 703 atomic sphere approximation (ASA) 136 Auger electron spectroscopy (AES) 69, 639, 341 augmented plane wave (APW) method 135 Austerman–Gehman model 846 Avogadro’s number 440 axial lattice parameter 7 axial ratio 7 azimuthal quantum number 1069

b backscattered electrons (BSE) 191 ball-and-stick model 837 band anticrossing (BAC) model 203 band-edge emission 193 band diagrams 1034 band edge emission 604 bandgap 100 – bowing parameters 214, 205 – issues 629

– low-temperature(s) 172, 179 – optical energy 627 – semiconductors 55, 92, 956 – variation 703 band structure 132 band tail model 170 band-to-band model 1149 band-to-band transitions 1161 band wave functions 1114 barrier–buffer interface 283 barrier height 271 basal GaN 373 basal plane lattice parameter 7 basal plane stacking faults 853 basal sapphire plane 373 basal-stacking fault 852 Bastard’s formalism 213 Bastard’s method 213 BC position 1036 beam equivalent pressure (BEP) 444, 993 biaxial relaxation coefficients 158 biaxial strain 226 biaxial stress 484 biexponential decay function 580 biexponential fits 578 biexponential PL decay curve 547 binary alloys 1010 binding energies 411 binodal 698 Bloch–Floquent theorem 133 Bloch function 135 Bloch wave 134 Blombergen–Rowland interactions 1088 – mechanism 1087 blue band 1042, 1044 blue luminescence (BL) 609 blue-shifted emission 561 blue–violet luminescence band 1045 Bohr magneton 1167 Bohr radius 213 Boltzmann constant 54, 941, 403 Boltzmann factor 956 Boltzmann transport equation 968 bond center (BC) 1016 bonding energies 412 bond rotation angle 368 boron nitride (BN) 361 Bose–Einstein expression 177 boundary-limited transport 326 bound magnetic polarons (BMPs) 1092 bound sheet density 267 bowing parameter 253, 269, 280 Bragg diffraction 419 Bragg mirrors 752

Index Bragg’s law 10, 463 Bragg spots 456, 457 Bragg–Williams approximation 390 bright field (BF) TEM images 472, 473, 657 Brillouin function 1126, 1139 Brillouin scattering 37 – measurements 63 Brillouin-type magnetization 1054 Brillouin zone 140, 1133, 1136 – boundaries 135 buffer layer(s) 94, 264 built-in electric field 157 built-in electrostatic fields 616 bulk crystal growth 384 bulk elastic coefficients 42 bulk material properties 214 bulk modulus 37, 38, 63 bulk rods 426 bulk SiC crystals 335 burgers circuit 831 burgers vector 289, 336, 473, 539, 571, 578, 614, 824, 823, 826, 832, 836, 837, 839, 843 Burstein–Moss shift 194, 195

c cantilever epitaxy (CE) 558 capacitance transient amplitude 946 capture process 194 carbon contamination 370, 396, 1041 carbon nanotubes (CNT) 735 carrier-ion direct-exchange 1084 carrier-mediated ferromagnetic interaction 1141 carrier–single magnetic ion interaction 1084 cathodoluminescence (CL) 537, 558 – data 177 – images 471 – spectra 1163 cation–anion bond 844 cation–anion distances 105 cationic model potential parameters 141 C–C bond strength 398 Cd-doped GaN 1043 cell internal parameter 105 central processing unit (CPU) 140 chalcopyrite-like (CH) structure 246, 254 charge-coupled device (CCD) 425 charge–spin coupling 1076 chemical etching behavior 845 chemical inertness 371 chemical mechanisms 653 chemical vapor deposition (CVD) furnace 751 chemical vapor phase epitaxy technique 1015

circular polarization 1110, 1116, 1117 cladding layer thickness 584 c-lattice parameter 6, 7, 36, 494 cleaving process 371 CL emission bands 1056 closed cycle cryostat 607 closed-shell interaction 139 coalescence boundaries 548 coalescence interface 551 coaxial heterostructures 755 codoping method 1018 coefficients of thermal expansion (CTE) 292, 483 collision-free growth mode 431 commercial LEC GaAs 331 common-cation rule 192 common-cation semiconductors 192 compensation model 1025 complexes 928 – Ga-vacancy 936 – H 937 – N-vacancy 936 compliant substrate 327, 328 composition-dependent bowing parameter 95 composition-induced reduction 211 composition pulling effect 94 composition, see bandgap 100 conduction band 216, 1114, 1128 conduction band Hamiltonian 1179 conduction band minimum (CBM) 143, 208, 219, 871 conduction band wave functions 147 conductive AFM (CAFM) 621, 912 confinement energy 220 confocal micro-Raman scattering 637 constant field approximation 288, 289 constant pressure (Cp) 60 continuum elastic theory 560 conventional Czochralsky melt pulling method 359 conventional semiconductor technology 465, 606 convergent beam electron diffraction (CBED) 363, 472, 587, 819 cooldown-induced thermal mismatch strain 260 – effect 260 corrugation maxima 644, 695 – hexagonal array 644 Coulomb exchange interaction 1084 Coulomb gap 1162 Coulombic defect 974 Coulombic potential profiles 945

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Coulombic repulsion 591 Coulomb interaction(s) 987, 1123 Coulomb repulsion 982 Coulomb tail 931 covalent semiconductors 1038 coverage-dependent function 412 c-plane sapphire 374 c-plane strain 484 crack-assisted pendeo-epitaxy (CAPE) 547 cracked-ammonia flux 444 crack-free AlGaN layers 659 crack-free samples 470 crack-free thick GaN 511 cross-sectional mapping 561 cross-sectional SEM image 530, 542, 626 – GaN layers 556 – SEM micrograph 546 cross-sectional TEM image 519, 551, 586, 620, 535, 668 cross-sectional transmission electron microscopy (XTEM) 486, 489, 631, 744 – micrograph 543 – observation 551, 662 crystal diffraction 418 crystal field 148, 1050 – parameters 1149 – splitting 148, 152, 1058, 1187 crystal-growth behavior 845 crystal-growth direction 1117 crystal-growth experiments 362 crystal-growth technique 323 crystalline GaN particles 723 crystalline grains 522 crystalline SiC 334 crystalline structure 514 crystallographic anisotropy 616 crystallographic directions 517 crystallographic orientation 531 crystallographic planes 529, 531 crystal-plane spacing 519 crystal symmetry 372 cubic crystal field 1061 cubic magnetocrystalline anisotropy 1184 cubic system 161 CuPt (CP) structure 246, 254 Curie temperature 1066, 1107, 1120, 1126, 1129, 1133, 1139, 1143, 1144, 1156, 1160, 1163, 1164, 1167, 1177, 1184, 1187 Curie–Weiss law 1126, 1160, 1073 Curie–Weiss-type temperature dependencies 1126, 1151 current–voltage characteristics 746 current–voltage measurements 575, 749 CVD-based catalytic 733

cyclotron measurements 169 Czochralski method 294, 332, 355, 358

d DAP transitions 1043 dashed curve 56 deBroglie wavelength 212 Debye expression 60 Debye length 264 Debye temperature 54, 55 Debye theory 79 decomposition 416 – processes 400 deep-level carrier emission 939 deep-level defects 194, 1041 deep-level emission 193 deep-level impurities 205 deep-level optical spectroscopy (DLOS) 956, 1041 deep-level transient Fourier spectroscopy (DLTFS) 963 deep-level transient spectroscopic measurements 818 deep-level transient spectroscopy 938, 939, 581 defect(s) – affected by hydrogen 924 – analysis by chemical etching 905 – analysis by deep-level transient spectroscopy 938 – analysis by surface probes 910 – role of hydrogen 1004 – V-type in InGaN quantum well 905 – V-type induced by Mg doping 894, 895, 897 deformation potentials 165 deionized water 329 d-electrons 139 delocalized metallic Ga–Ga bonds 449 density functional calculation 206 density functional theory (DFT) 137, 140, 819, 820 depolarization 1172, 1185 desorption 405, 412 desorption fluxes 406 desorption mass spectroscopy (DMS) 439, 443 desorption processes 389, 391 desorption spectra 51 d-hybridized upper valence band 139 diamagnetic component 1153, 1157 diamagnetic materials 1064 diamagnetism 1064 diamond-lattice structure 332 diatomic 399

Index dichloromethane 470 dielectric constant 18, 20, 21, 22, 27, 29, 32, 34, 74, 89, 90 dielectric function 20,22 dielectric mask 529 dielectric screening function 141 diethylchlorine gallium 539 differential polarization 285 differential temperature 292 diffusion-controlled processs 50 diffusion length 403 dilute GaAsN films 105, 330 dilute magnetic semiconductor(s) 1082, 1063,1123 – II-VI and GaAs based 1123 – II-VI based 1124 – ab initio theories 1088 – applications 1168 – comparison of magnetic interactions 1101 – GaAs based” 1133 – RKKY theory 1088 – theoretical aspects 1082, 1141 – nanowires 1167 – Zener theory 1088 dilute nitrides 105, 202, 210 dimethylhydrazine 512 direct magnetization measurements 1157 dislocations – 5/7 chain 828 – charge profile 874 – edge 821, 822, 823 – edge and screw 821, 823 – electronic structure 882 – mixed 823 – open core-5/7 atom ring 869 – open core-eight atom ring 869 dislocation density 394, 526 dislocation density, see electron mobility dislocation–dislocation interaction 820 dislocation-mediated surface morphology 460 dislocation reduction 557, 578 displacement vector D 264 distributed Bragg reflector (DBR) 97, 661 divacancy 934 DLTS – applied to GaN 948 – applied to ternaries of GaN 977 – basics 939 – dispersion in GaN data 970 DLTS method 938, 948 DLTS peaks 949 DLTS signal 944, 945 DLTS spectra 965

DMS heterostructures 1126, 1127, 1177 DMS materials 1116 DMS systems 1090 domain boundaries 495, 844 – annihilation 641 dominant compensating donor 937 donors – shallow 1007 donor–acceptor pair band 604, 1056 donor-binding energies 611 donor-bound excitons (DBE) 514, 610, 624 donor electrons 1101 donor–gallium vacancy complexes 936 doping 1006 – by point defects 917 – co-doping for p-type 1018 – magnesium 1014 – p-type 1013, 1014 doping with As 1044 doping with Be 1038 doping with C 1040 doping with Ca 1042 doping with Cd 1043 doping with Ge 1011 doping with Hg 1040 doping with P 1045 doping with rare earths 1045, 1046 doping with Se 1012 doping with Si 1010 doping with Si, Ge, O 1010 doping with transition metals 1046, 1060 doping with Zn 1042 Doppler broadening experiments 991 Doppler broadening spectroscopy 988 double-pulse DLTS 946, 974 double exchange interaction 1088 double exchange mechanism 1167 double positioning boundaries (DPB) 496, 818, 844 down-spin channels 1079 down-spin electrons 1080 dry processing techniques 329 d shell electrons 1048 dual-flow channel reactor 666 dual-interface heterostructures 132 dynamical theory 423

e Eagle Picher sample 601 e-beam evaporation 577, 722 EBIC measurements 978 EBIC method 981 ECR microwave plasma-assisted MBE (ECR-MBE) 450

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1236

edge dislocations 656, 824 – electronic structure 866 effective acceptor activation energies 1034 effective mass approximation 270, 272, 281 effective mass parameters 143 eight-atom core model 825, 837 eight-atom ring character 825 Einstein equation 403, 413 elastic constants 37, 132 elastic deformation 328 elastic moduli 23,78 elastic recoil detection analysis (ERDA) 1005 elastic recoil detection method 1055 elastic stiffness coefficients 63, 156, 159 elastic stiffness constants 224 electric field effect 972 electric field strength 954 electric force microscopy 587, 910 electrical spin injection, see spin orientation electro-optic 383 electroluminescence 1176, 1178 electroluminescence (EL) experiment 1148 electroluminescence (EL) images 744 electrolytic etching technique 35 electromagnetic theory 20 electron acoustic phonon scattering 191 electron beam incident 417 electron beam induced current (EBIC) 977 electron beam lithography 541 electron-counting rule (ECR) 649 electron concentration 525 electron cyclotron resonance microwave plasma assisted molecular beam epitaxy (ECR-MBE) 410 electron density 1038 electron diffraction intensities 423 electron dipole motion 1115 electron effective mass 197, 205, 210 electronegativity 66 electron–electron interaction 133 electron energy loss spectroscopy (EELS) 48 electron–hole pairs 288, 673 electron–hole recombination 279 electron-initiated excitation 559 electron–ion interaction 648 electron-irradiated undoped GaN 1000 electron mobility 526, 632, 966 electron mobility, see dislocation density 966 electron momentum 989 electron momentum distributions 992 electron–nuclear hyperne interaction 1001 electron paramagnetic resonance 998 1134 electron–phonon coupling 84, 955, 1043

electron–phonon interaction 39, 173, 193, 194 electron polarization 1115, 1116 electron spin coherence 1186 electron spin relaxation time 1178 electron spin resonance (ESR) 172, 1052 electron transport 84 electronic band structure 131 electronic degeneracy entropy 973 electronic devices 977 electronic energy levels 1105 electronic paramagnetism 1065 electronic Raman measurements 39 electronic states 1038 electronic structures 1091 electronic thermal conductivity 53 electronic wave functions 133 electrooptic modulators 743 ELO 559 – stripes along the direction in HVPE ELO 561 – three-step 557 – two-step 547 ELO process 622, 625 ELO technique 623 empirical pseudopotential approach 141 empirical pseudopotential method (EPM) 151, 199 empirical tight binding method (ETBM) 151 energy dispersive X-ray (EDX) 688 – analysis 191 – spectroscopy 744 energy-filtered TEM 1162 energy-loss spectroscopy (EELS) measurements 837 energy momentum 218 – dispersion diagram 218 energy transition site 414 enthalpy 50 envelope function approximation 219 environmental transmission electron microscope 728 epitaxial AlN films 639 epitaxial casting strategy 756 epitaxial core–sheath method 756 epitaxial deposition techniques 323 epitaxial film(s) 329, 448, 527, 597 epitaxial GaN 447, 495 epitaxial GaN film(s) 509, 522 epitaxial GaN layer(s) 487, 498 epitaxial growth temperature 525, 526, 603, 641 – effect 525 epitaxial lateral overgrowth (ELO) 6, 59, 324, 483, 511, 528, 1190

Index – HVPE 537 – maskless 557 – Nano-ELO 564 – point defects 558 – selective 537 – SiN and nanonets 569 – stripes along the direction in HVPE – W masks 583 epitaxial lateral over-growth on Si 539 – a-plane GaN 616 – pendeo-epitaxy 540 – pendeo-epitaxy on Si 544 – pendeo-epitaxy on SiC 542 epitaxial layer(s) 72, 259, 291, 327, 328, 382, 469, 474, 478, 482, 493, 841 epitaxial nitride layers 395 epitaxial process 527 epitaxial relationship 293, 372, 377 epitaxy 353, 369 EPR 998 EPR measurements 1003 equation of state (EOS) 37 equilibrium bond lengths 681 equilibrium growth 406 etch pitch density (EPD) 331, 528 etch pit delineation process 576 etching methods 552 evaporation coefficients 449 Ewald construction 417 Ewald sphere 419 excimer laser deposition technique 469, 599 excitation-induced carriers 283 excitation-power-density 1161 excitation probability 1114 exciton-binding energies 227, 611 excitonic emission 1012 excitonic spectrum 610 excitonic transitions 555 exciton spin scattering 1133 exothermic nature 413 expanded equilibrium vapor pressure data 51 ex-situ preparation 370 ex-situ processes 369 Extended defects – electronic structure 863 extended X-ray absorption fine structure (EXAFS) 933 external quantum efficiency (EQE) 624 extraordinary Hall effect (EHE) 1141

f face-centered cubic sublattices 144, 611 Faraday effect 1105, 1118, 1127, 1186

Faraday rotation 1104, 1105, 1106, 1107, 1108, 1118, 1119 Fermi–Dirac distribution 941 Fermi–Dirac function 196, 197 Fermi energy 1080, 1104, 1132, 1134, 1141 Fermi level position 189, 929, 1153 Fermi wave vector 1186 ferromagnetic behavior 1153 ferromagnetic coupling 1128 ferromagnetic-diluted semiconductor materials 1187 ferromagnetic electrodes 1079, 1172 ferromagnetic exchange coupling 1184 ferromagnetic film 1078, 1139 ferromagnetic (FM) configuration 1100 ferromagnetic GaN 1107, 1109 ferromagnetic interaction 1100, 1171 ferromagnetic layers 1180 ferromagnetic materials 1067, 1080 ferromagnetic metals 1078, 1131, 1168, 1179 ferromagnetic regime 1109 ferromagnetic samples 1071 ferromagnetic semiconductors 1076 ferromagnetic state 1070, 1072, 1101 ferromagnetic transition 1093, 1133 – temperature 1134, 1143 ferromagnetism 1123, 1130, 1134, 1136, 1144, 1150, 1152, 1156, 1158, 1160, 1163, 1164, 1167, 1169, 1171, 1178, 1187, 1188 field-effect transistors 326, 508, 1010 field emission (FE) SEM image 749 filled-core screw dislocations 832 finite difference time domain (FDTD) 742 finite-element method 538 first-order approximation 197 first-order desorption kinetics 402 first-order kinetic model 450 first-order LEED spot 645 first-order phonon Raman scattering 40 first-order Raman modes 738 first-principles calculations 166, 224 first-principles techniques 140 Float Zone (FZ) method 332 FM state 1121 forbidden energy regions 132 four hydrogen atoms 926 Fourier components 424 Fourier filtering 822 Fourier series 424 Fourier transform analysis 962 Fourier-transform infrared-absorption spectroscopy 1036 Fourier transform infrared (FTIR) 998 – measurements 998

j1237

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1238

four probe 54 F parameter 172 fractional ionic character (FIC) 12 Franks screw dislocation 732 – mechanism 732 Frank-van der Merve (FM) 415 free-carrier charge 272 free-carrier density 279 free carrier induced field 279 free-carrier plasma 89 free-carrier screening 270, 278 free-electron approach 132 free-electron approximation 133 free-electron concentration 87 free electron energy 135 free electron model 134 free electron picture 1080 free-electron recombination band (FERB) model 194 free electrons 929 free energy 536 free enthalpy 550 free excitons (FE) 610 free positron wave 988 freestanding bulk layer 580 FTIR 998 full-core screw dislocation 828, 831 full potential linearized augmented plane wave (FP-LAPW) method 138 full potential LMTO (FP-LMTO) calculations 137 full width half maximum (FWHM) 64, 178, 364

g GaAs 329, 1120 – surface preparation 331 GaAs analogy 388 GaAs-based superlattices 264 GaAs model 390 GaAs spacer layers 1182 GaAs substrate 332 GaAs system 174 GaAs technology 477 Ga-bilayer model 414, 596 Ga-containing alloys 697 Ga desorption mechanism 401 Ga-droplet growth regime 455 Ga-droplet regime 458 Ga-face profile 477 Ga-face sense polarity 624 Ga-filled-core model 831 Ga-filled-core screw dislocation 831 GaInAsN 210

GaInPN 212 gallium antisite 935 gallium arsenide (GaAs) 477 gallium interstitial 934 gallium nitride 30, 49 gallium nitride epitaxy 500 Ga melt 363 GaN 35, 47, 56, 61, 361, 362, 365, 372, 392, 398, 416, 438, 466, 506, 528, 608, 719, 1177 – alloy/multiple layer nucleation layers on SiC 491 – doped with Cr 1156 – doped with Gd 1163 – doped with rare earths 1163 – doped with V 1163 – epitaxial relationship to a-plane sapphire 375 – epitaxial relationship to c-plane sapphire 374 – epitaxial relationship to LiGaO2 and LiAlO2, and pervoskites 382 – epitaxial relationship to r-plane sapphire 376 – epitaxial relationship to sapphire 373 – epitaxial relationship to Si 381 – epitaxial relationship to SiC 381 – heteroepitaxial deposition 528 – high temperature nucleation layers on SiC 486, 489 – low temperature interlayers 584 – low temperature nucleation layers on SiC 488 – nucleation layers on SiC 482 – pertinent surfaces, c, a, m planes 365 – seeded growth 363 – seedless growth 362 GaN-AlN – nucleation layers on SiC by MBE 492 GaN and AlN – epitaxial relationship to sapphire 372 GaN-based electronic devices 62 GaN-based optical devices 62 GaN-based semiconductors 1190 GaN buffer layer(s) 446, 598, 636, 638 GaN-carbon composite nanotubes 758 GaN crystalline quality 404 GaN crystals 361, 362, 364 GaN decomposition 527 GaN deposition 382, 386 GaN desorption 399 GaN epitaxial layers 372, 492, 606, 817 GaN epitaxial relationship 372 GaN equivalent flux 440 GaN film orientation 624

Index GaN films 372, 374, 399, 407, 409, 491, 545, 569 GaN formation 417 GaN growth a-plane 613 GaN growth m-plane 623 GaN growth on GaAs 477 GaN growth on porous SiC (PSiC) 503 GaN growth on sapphire 512 – effect of III/V ratio on nucleation layer 523 – effect of pressure on nucleation layer 525 – low temperature nucleation layers 513 GaN growth on GaN templates 605 GaN growth on LiGaO2 and LiAlO2 603 GaN growth on non-c-plane substrates 611 GaN growth on Si 507 GaN growth on SiC 479, 499 GaN growth on spinel (MgAl2O4) 611 GaN growth on ZnO 598 GaN growth p-type 627 GaN growth technology 462 GaN kinetic model 398 GaN lattice parameter 47 GaN layers 384, 435, 456, 586 – structure 586 GaN low-temperature buffer layer 508 GaN on SiC – double positioning boundaries (SMB) 496 – inversion domain boundaries (IDB) 496 – stacking mismatch boundaries (SMB) 496 – substrate misorientation and domain boundaries 495 – polarity 498 GaN optical phonon energies 42 GaN phase diagram 61 GaN phase growth rate 657 GaN platelets 363 GaN-related materials 818 GaN stacking faults and epitaxial relationship to SiC 480 GaN stripes 531 GaN-substrate interface 287 GaN surface 370 – T and H sites 366, 368 GaN surface preparation 369 GaN synthesis reaction 399 GaN synthesis temperature 52 GaN templates 361, 363, 605 Ga-polar films 587, 591 Ga polarity 234, 381 – on sapphire 586, 597 Ga-polarity films 439, 598 Ga-polarity GaN 588 Ga polarity sample(s) 390, 500, 509 Ga-polarity surfaces 273, 365

Ga-polar matrix 587 Ga-polar sample 222 Ga-polar surface 363, 365, 402 Ga-polar wings 620 GaN QD arrays 714 Ga-rich conditions 367, 407, 457, 831, 935 Ga-rich dislocation core model 834 Ga-rich growth regime 455 Ga-rich regime 414, 460 Ga-rich structure 594 Ga solution 384 Ga sticking coefficient 440 Ga-substitutional acceptors 1007 Ga vacancy concentration 997 Ga-vacancy configuration energy 932 Ga-vacancy model 366 Ga vacancy-oxygen complexes 990 GaPN 210 gaseous species 388 gas manifolds 400 gas-phase diffusion rate 400 gas-phase etching 329 gas-phase product 396 gas-phase reactions 389, 488 gas-phase species 390 gas pressure technique 49, 52 gas-source MBE (GSMBE) 442 gas-source molecular beam epitaxy 210 Gaussian-like distribution 170 Gd-doped GaN samples 1164 Ge doping 1011 generalized gradient approximation (GGA) 9, 1091 generalized quasi chemical approximation (GQCA) 698 geometric theory 418 giant magnetoresistance (GMR) 1080 Gibbs free energy 49, 50, 431, 947 grain boundary (GB) 862 green luminescence band 1043 Green’s function theory 140 Green’s method 421 ground-state energy 283 growth chamber 369 growth kinetics 700, 993 growth of nitrides 462 growth phase 391 growth process 392 growth rate anisotropy 362 growth-related parameters 672

h half-order diffraction 594 halide precursor techniques 385

j1239

j Index

1240

Hall data 1139 Hall effect 87, 197, 1001, 1103, 1104, 1141, 1143, 1144 – measurements 59, 81, 91, 198, 475, 489, 666, 697, 1040, 1191 Hall electron concentration 197 Hall mobility 632 Hall resistance 1171 Hamiltonian matrix 213 Hamiltonian operator 135 Hanscom sample 601 Hartree-Fock method 140 Hartree-Fock (H-F) theory 137 HCl etching techniques 479 heat of evaporation 49 heat of sublimation 17, 49 heavy-hole bands 207, 1110, 1117 heavy hole (hh) effective masses 173 heavy-hole excitations 1127 heavy-hole subbands 1115 heavy-hole transition 1115 He-Cd laser 607 Heisenberg spin glass 1144 Hermann-Mauguin notation 2, 62 H-etched samples 340 heteroepitaxial film 327, 328, 606 heteroepitaxy 822 heterointerface 565 heterojunction bipolar transistors 110 heterojunction field effect transistor (HFET) 35 heterojunctions 384 heterostructure deposition systems 393 heterostructures 264, 425, 1125, 1179, 118, 1186 hexagonal cells 484 hexagonal close packed (hcp) 2, 143, 426 – sublattices 2, 143 – wurtzitic structures 839 hexagonal compact packed (hcp) structure 819 hexagonal GaN 331, 612 hexagonal lattice structure 374 hexagonal morphology 523, 526 hexagonal pits 458 hexagonal SiC substrates 381 hexagonal stacking 498 hexagonal structure 481 hexagonal symmetry 293, 643 hexagonal wurtzite structure 641 Hg-acceptor level 1040 Hg implantation 1043 high-density crack network 658 high electron momentum 990

high energy electron diffraction 631 high-energy O peak 370 high hydrostatic pressures 929 high-mist systems 327 high-mobility GaN 446 high nitrogen flux rate 100 high nitrogen pressure solution growth (HNPSG) 362 high-performance electronic devices 652 high-performance optical emitters 286 high piezoelectric constants 326 high power field-effect transistors 334 high-pressure anvil method 52 high-pressure apparatus 358 high-pressure growth 527 high-pressure method 365 high-pressure techniques 36 high-quality compound semiconductor growth 416 high-quality epilayers 630 high-quality epitaxy 632 high-quality freestanding GaN template 54 high-quality GaN films 468, 654 high-quality OMVPE 635 high-quality semiconductor films 387 high-quality single crystalline material 77 high-quality thin films 366 High-quality ZnO substrates 353 high-resistivity GaN 1039, 1041 high-resistivity material 627, 628 high-resistivity SiC substrates 327 high-resistivity substrates 334 high-resolution electron microscopy 850 high-resolution image 822 high-resolution TEM analysis 1167 high-resolution top-view lattice image 841 high-resolution transmission electron microscopy (HREM) 819 high-resolution X-ray diffraction (HRXRD) 474 – measurements 42 high-resolution X-ray diffractometry 658 high-resolution Z-contrast imaging 825 high-temperature AlN nucleation layers 486 high-temperature annealing technique 346 high-temperature ferromagnetism 1162 high-temperature GaN buffer layers 488, 489, 590 high thermal conductivity 62 hillocklike feature 58 HNPSG Method 363 hole concentration 1015, 1018, 1025, 1026, 1041, 1042

Index hole effective masses 132, 150, 159, 164, 168, 171, 173, 174, 175, 182, 184, 197, 202, 216, 225, 288, 708, 1025 hole exchange interaction 1153 hole wave functions 1179 hollow core screw dislocations 336 hollow pipes 840 Holt inversion domains 845 Holt-type IDB 846, 850 homoepitaxial AlN films 178 homoepitaxial growth 369, 456 homogeneous biaxial stress 158 homogeneous elastic strain 327 homogeneous emission 561 homogeneously mixed 387 homogeneous strain 638 honeycomb silicate adlayer 341 Hooke’s law 154, 156, 158 hopping conductivity 1131 horizontal dislocations 559 Hund’s rule 1047, 1046, 1048, 1124 HVPE-grown films 393 HVPE-grown samples 834 hybridization 1047, 1060, 1128, 1188 – constant 1085 hydrazine 436 hydride vapor phase epitaxy (HVPE) 36, 323, 326, 385, 387, 537, 818 – buffer layers 468, 836, 1061 – freestanding templates 468 – growth mechanism 388 – initial nucleation layer 388 – reactor 625, 997 – technique 599, 654 – template 472, 834 hydrocarbons 1005 – efficient desorption 1005 hydrogen 393, 397, 1004, 1005 – experimental results 1005 – role 1004 hydrogenation process 334 hydrostatic pressure coefficients 42, 64 hyperfine interaction tensor 1000 hypothetical epitaxy 846 hysteresis 1167, 1171, 1172 – loops 1143, 1158, 1163 hysteretic behavior 1131

i image dislocation 550 image-processing system 425 imperial expression 169 impinging species 516 implantation energy 986

InAlGaN – spinoidal decomposition 705 – quaternary 99 InAlN – alloy 97 – miscibility gap 697 – spinoidal decomposition 697, 698 InAsN 208 indirect gap semiconductor 210 indium-silicon codoping 666 indium doping 527 indium nitride 75, 77 inductively coupled plasma (ICP) 428 inferior carrier transport 453 infrared (IR) 432 – measurements 39 – region 695 – spectroscopic ellipsometry 470 InGaAlN – quaternary, growth 699 InGaAs quantum well 211, 1135 InGaN – alloys 92 – composition pulling effect 93 – doping 678 – growth 671 – layers 672 – phase separation 479 – surface reconstruction 689 – V-defects 674 InGaN/GaN quantum wells 286 inhomogeneous strain distribution 519, 682 injection laser experiments 371 InN – growth 629 in-plane components 218 in-plane current 1173 in-plane deformation potentials 160 in-plane heavy-hole mass 225 in-plane homogeneity 470 in-plane lattice mismatch 505 in-plane light-hole mass 225 in-plane rotation 334 in-plane strain 158, 278 in-plane strain anisotropy 155 in-plane stress-strain relationship 158 InPN 209 input fluxes 406 InSbN 209 in situ annealing 577, 597 in situ buffer morphology 482 in situ methods 402, 477 in situ preparation 370 in situ RHEED images 590

j1241

j Index

1242

intentional doping 1006 interaction between magnetic ions 1086 interfacial relationship 480 interlayer-induced strain 659 interlocking spiral ramps 456 internal cell parameter 103 internal magnetic induction 1064 internal parameter 6, 7, 103, 105, 107, 140, 156, 233, 240, 241, 251, 285 interstitial antibonding site 1038 interstitials 928, 934 – Ga 934 – N 935 intracenter absorption band 1054 intramagnetic ion levels 1071 inverse susceptibility 1126 inversion domains 818 inversion domain boundaries (IDBs) 324, 819, 844, 845, 901 – Holt-type 844, 847 – induced by Mg doping 900 – V-type, IBD
845, 847 inversion-type planar defects 844 inverted pyramids 457 ion-assisted reactive 641 ion beam assisted deposition (IBAD) 24 ionic bond 416 ionic core-core repulsion 820 ionic model potentials 141 ion implantation 1189, 1190 ion-ion interactions 1090 ionization energy 1034, 1040 ion scattering spectroscopy 587 IR ellipsometry measurements 190 island growth mode 415 isoelectronic impurities 1009, 1044 isothermal capacitance transient analysis 954 isothermal capacitance transient spectroscopy (ICTS) 945 isothermal remanent magnetization 1144 isotropic parabolic conduction band 160

j Jahn–Teller effect 999, 1061 Jahn–Teller model 1062

k Kane model 197 Kelvin microscopic investigation 563 Kerr effect 1104 Kerr rotation (KR) 1108 kinematical scattering theory 419 kinematic theory 423 kinetic considerations 392

kinetic energy 136 kinetic model 390, 405 kinetic parameters 436 kinetic theory 53 Knoop diamond indenter 63 Knoop’s hardness 15, 16, 23, 30, 63 Knudsen effusion cells 410 Koster-Slater model 947 k.p calculations 199 k.p model 143 k.p theory 160 Kronig-Penney approach 133

l Langer-Heinrich rule 1062 Langmuir evaporation data 449 large angle convergent beam electron diffraction (LACBED) 619 large bandgap bowing parameters 106 large stress fields 926 large thermal conductivity 334 laser-assisted CVD 394, 638 laser diode (LD) 696, 1014 laser-induced liftoff 472 laser liftoff (LLO) 462, 468, 471, 607 lasers employing films 371 lateral diffusion barrier 414 lateral dislocations 550 lateral epitaxial overgrowth (LEO) 324, 539, 546, 583 lateral growth 394 laterally contracted bilayer (LCB) 596 lateral propagation 543 lattice constants 102, 202, 351 lattice-matched AlInGaN alloy 265 lattice-matched barrier 97 lattice-matched conditions 62 lattice-matched system 710 lattice matching composition 262 lattice misfit 295 – strain 382 lattice-mismatched film 328 lattice-mismatched substrates 324, 326, 377, 752 lattice mismatch (lm) 185, 375 – component 290 – problem 328 lattice nuclei 1003 – hyperfine interaction 1003 lattice parameter 382, 494 lattice parameter mismatch 294 lattice specific heat 53 Lau condition 419, 424 Lau rings 420

Index Lau treatment 419 layer-plus-island growth mode 415 LEEBI treatment 1016 LEED patterns 597 Leibfried-Schloman scaling parameter 79 Lenz’s law 1065 Levenberg-Marquardt algorithm 580 LiAlO2 358 LiGaO2 355 light-hole subbands 1115 light-emitting devices 717 light-emitting diodes (LEDs) 616, 696, 817, 1014, 1077 light-hole bands 207, 1117 light-hole dispersion 209 linear combination of atomic orbitals (LCAO) 37, 133, 819 linear interpolation 282 linear muffin-tin orbitals method (LMTO) 78, 136, 139 linear polarization 284 linear system 161 linear thermal expansion coefficients 79 linearized augmented plan wave (LAPW) method 136, 1100, 1156 line-of-sight quadrupole mass spectrometry 401, 402 liquid encapsulated Czochralski (LEC) method 330 lithium gallate (LGO) 355, 603 lithographical step 547 lithographic/etching techniques 736 lithography 469 local density approximation (LDA) 9, 137 local density formalism (LDF) 138 local density functional theory 648 local probe analysis 563 local strain modification 551 local vibrational modes (LVMs) 1036 long haul fiber-based communications systems 211 longitudinal heterostructures 737 longitudinal optical (LO) phonon frequencies 20 long-period corrugation 644 long-period structure 644 long-range Coulomb interactions 820 long-wavelength lasers 110 LO phonon-plasmon coupled modes (LPP) 561, 563 Lorentz force 1065, 1131 Lorentzian peaks 64 low-energy crystal 411

low-energy electron beam irradiation (LEEBI) 628, 1014 low-energy electron diffraction (LEED) 341, 639 low-energy ion-assisted growth 641 low-energy state 1069 lower thermal conductivity materials 55 lower valence bands (LVBs) 628, 1017 lowest intensity atomic columns 836 low free-electron concentration 47 low-lying conduction electron state 219 low pressure (LP) 584 low-pressure Hg lamp 178 low substrate temperature buffer growth 330, 590 low-temperature buffer growth 590 low-temperature CL image 662 low-temperature data 177 low-temperature deposition processes 467 low-temperature free-exciton transition 177 low-temperature GaN buffer layer 478 low-temperature interlayer 584 low-temperature nitridation 590 low-temperature nucleation buffer layers 386, 467, 514 low-temperature photoluminescence mapping 470 low-temperature process 734 LT AlN insertion buffer layers 585 LT insertion layer 658 Luckovsky fit 1056 luminescence 1110, 1116 – band 1163 – efficiency 558 – images 471 – line 1060 luminescent lifetime 1060 Luttinger-Kohn model 215 Luttinger parameters 152, 161, 173, 174, 175, 183, 201 luzonite-like (LZ) structure 246, 254 Lyddane-Sach-Teller relationship 22

m magnesium doping 1014 magnetic anisotropy 1129 magnetic circular dichroism (MCD) 1104, 1109, 1119 – measurements 1158 magnetic coupling strength 1177 magnetic dopant 1103 magnetic electrons 1088, 1090 magnetic field 998, 1064, 1071, 1084, 1116, 1135, 1139, 1147, 1158, 1180, 1182, 1184

j1243

j Index

1244

magnetic field effect transistors (M-FETs) 1123 magnetic ions 1067, 1074, 1076, 1082, 1083, 1084, 1085, 1086 magnetic ion spin 1084 magnetic moment(s) 1063, 1066, 1068, 1069, 1070, 1079, 1080, 1082, 1133 magnetic polarization 1064 magnetic polaron binding energy 1132 magnetic properties – Mn-doped GaN 1143 magnetic quantum dot (MQD) 1123 magnetic random access memory (MRAM) 1081 magnetic-recording industry 1080 magnetic resonance 999 – methods 1000 – optical detection 999 magnetic semiconductor 1075, 1078 magnetic sensor heads 1078 magnetic susceptibility 1066, 1126 magnetic tunnel junctions (MTJs) 1078 magnetization 1108, 1118, 1119, 1126, 1139, 1164, 1167, 1168, 1171, 1173, 1176, 1182, 1188 – curves 1154 – data 1160 – hysteresis 1146 – measurements 1128, 1158, 1163, 1167 magneto-optical data 1139 magneto-optical effect 1107 magneto optical Kerr effect (MOKE) 1104, 1108 magneto-optical measurements 1133, 1157 magneto optical properties – TM doped GaN 1146 magneto-optical spectroscopy 1146, 1148 magneto-optics studies 1128 magneto-photoluminescence 1127 – measurements 1164 magneto electrical measurements 1103 magneto reflectivity data 1148 magneto reflectivity measurements 1147 magnetoresistance 1139, 1131, 1162 – measurements 1104, 1186 – effect 1180 magneto transport 607 – measurements 1103, 1168 magnetron sputter deposition 641 magnitude 420 maskless epitaxial lateral overgrowth 557 mass flow controllers (MFC) 408 mass spectrometer 49, 438, 447, 462 mass spectroscopic techniques 398

mass spectroscopy experiments 432 mass transport 391 material-specific parameters 212 maximum electron mobility 635 maximum growth rate (GR) 330 Maxwell-Garnett approximation 191 MBE 409, 460, 733 – adsorption 411 – buffer layers 597 – decomposition 416 – effect of III/V ratio on growth 455 – experiments 993, 1017 – GAN films 460, 832 – incorporation 415 – MBE growth – N species for growth 451 – PAMBE growth 435, 446 – reactive ion 435 – RF N source 430 – RF N species 431 – RMBE growth 435, 437 – surface diffusion 413 – surface reconstruction 426, 427, 428, 429 – UHV system 369 Magnetic Circular Dichroism (MCD) 1109 – data 1131, 1160 – effect 1110 – intensity 1130 – measurements 1129, 1132, 1133 – signal 1132 – spectra 1128, 1158 mean atomic column peak intensity 836 mean free path 56 mechanical chemical polish (MCP) 338 melting temperature 49, 68 membrane-based method 736 metallic adlayer structures 651 metallic gallium 387 metallic layers 1180 metalorganic chemical vapor deposition 385 metal-organic MBE (MOMBE) 637 metalorganic precursor 397 metalorganic pyrolysis 396 metalorganic vapor phase epitaxy 385 metal-rich growth 647 metal-semiconductor contacts 365 metastable growth process 416, 437 metastable zinc blende GaN epitaxial films 330 Mg-augmented ELO 536 Mg-doped GaN 1000, 1015, 1034, 1035, 1036 Mg-doped p-type GaN layer 753 Mg-doped samples 985, 991 M-H hysteresis loops 1143, 1144, 1156, 1163

Index microcathodoluminescence studies 76, 190 micro-Raman scattering 198 microwave-excited nitrogen 636 Mie resonances 76, 187, 191 Mie resonant absorption 86 Mie theory 191 migration-enhanced epitaxy (MEE) 404, 632 Miller-Bravais indices 343, 372 Miller index planes 531 minority carrier lifetime 977 minority carrier transient spectroscopy (MCTS) 963 miscibility gap – InAlN 697 misfit accommodation (MA) 327 misfit dislocations 822 misfit strain 293, 295 mist-induced piezoelectric polarization 260 mixed anion host materials 212 mixed cation ternary host materials 212 mixed dislocations 836, 841 – electronic structure 866 Mn complexes 1060 Mn-doped ferromagnetic GaAs 1103, 1135 Mn-doped GaN 1048 model dielectric matrix 141 moderate-temperature nucleation layers 514 MODFET structures 268 modulation doped field effect transistor (MODFET) 35 modulation-doped structures 169, 264 molar fraction 259 – function 259 molecular beam epitaxy (MBE) 323, 409, 450, 824 molecular cohesive energy 410 molecular stream epitaxy (MSE) 677 molecular weight 35 MOMBE systems 1045 monatomic interstitial hydrogen 924 monatomic nitrogen 438 monochromatic CL image 559 monocrystalline 605 monolayer units 412 monomethylsilane 1012 monovacancy defects 992 Monte Carlo calculations 83 Monte Carlo simulations 1144 Moss-Burstein effect 86, 186, 188, 189 Moss-Burstein shift 75, 76, 186, 187, 189 Mott critical density 189 Motts transition 197 – concentration 197

multibeam optical stress sensor (MOSS) system 491 multiheterolayer buffer structures 299 multiple interface heterostructures 278 multiple layer nucleation layers 491 multiple quantum disk (MQD) nanocolumn 753 multiple quantum well (MQW) 608 Murnaghan’s EOS 37

n

nanocrystalline GaN thin films 723 nanoheteroepitaxy (NHE) 503, 564 nanoimprint lithography 569 nanopipes 818, 840, 841 nanostructures 725 – coaxial heterostructures 755 – core multishell structures 744, 746 – FETs 746 – multiple quantum disk LEDs 753 – nanowires and heterostructures 737 – preparation 725 – properties 725 – synthesis 726 – synthesis by confined chemical reactions 735 – synthesis by template based-methods 735 – synthesis by self catalytic vapor-liquid-solid (VLS) process 730 – synthesis by vapor-liquid-solid process 727 – synthesis by vapor phase 726 – vapor-solid process 732 nanotubes 756 nanowire(s) 737 – growth 734 – optical emission 743 – stimulated emission 744 – synthesis in solution 734 nanowire FETs 749 native defects 929, 1000 N-beam case 424 N-containing gases 653 N-deficient layers 409 N-doped GaP 210 near field scanning optical microcopy (NSOM) 617 near-infrared photoluminescence 1044 negative-U system 1004 negative ion defects 995 neighbor bond lengths 105 N-face sense polarity 624 nitridation process 349, 513, 597 nitride-based optoelectronic devices 1015

j1245

j Index

1246

nitride-based semiconductors 75 nitride-based ultraviolet detectors 652 nitride crystal growth 361 nitride epitaxy 324 nitride family 75 nitride film 396 nitride growth techniques 384 nitride heterostructures 270 nitride semiconductor alloys 102, 269 nitride semiconductor family 102 nitride semiconductors 47, 131, 140, 323, 359, 383, 395 nitrides 1, 39, 851 nitrogen antisite 935 nitrogen concentrations 109 nitrogen dissociation pressure 49 nitrogen flux 454 nitrogen incorporation efficiency 440 nitrogen-limited growth 454 nitrogen precursor 585 nitrogen pressure 50 nitrogen-rich conditions 407 nitrogen-rich growth regime 693 nitrogen species 452 nitrogen vacancies 930, 931, 932 N-limited growth conditions 448 nonisomorphic substrates 496 nonlinear core corrections 139 nonlinearity of polarization 245 nonlinear optical properties 641 nonlinear optics properties 383 nonlinear polarization 276 nonmagnetic ion 1132 nonmagnetic materials 1078 nonmagnetic semiconductors 1064 nonplasma-based growth techniques 435 nonpolar surface 223 novel method 638 N-plasma conditions 634 N-polar films 591, 635 N-polarity 234 – films 598 – samples 266, 453 N-polar layer 994 N-polar surface 364, 365 N-polar wings 620 N-rich conditions 367, 450, 594 N-rich growth regime 455 N-rich regime 460 NSOM measurements 620 N-type GaN platelets 364 nucleation buffer layer 479, 480, 482, 492, 523 nuclei formation 527

o octahedral sites 611 ODMR method 998, 1000 Omega rocking curves 614 OMVPE – Ga and N precursor adsorption and desorption 400 – Ga and N precursor adsorption and desorption, activation energy 401 – Ga and N surface diffusion 403 – GaN desorption as it relates to growth 398 – growth mechanism 398 – kinetic model – balance between adsorption and desorption 405 – low-temperature nucleation buffer layers 513 – reactors 393 – samples 581, 628, 948, 952, 984, 1061 – systems 395, 408, 583 – technique 393, 636 OMVPE-grown films 1035 OMVPE-grown layers 1017 OMVPE-sources 467 one-dimensional growth mechanism 738 one-dimensional stress 292 one-dimensional system 269 one-electron model 132 one-electron picture 133 one-electron wave functions 137 one-monolayer adsorption model 390 open-core edge 841 open-core screw dislocation 828, 831, 840 optical absorption coefficient 71 optical absorption measurements 76, 95 optical absorption technique 191 optical beam 1109 optical detection of magnetic resonance 998 optical devices 977 optical emission 704 – spectra 432 optical emitters 629 optical excitation 1110 optically detected electron paramagnetic resonance (ODEPR) 935 optically pumped ultraviolet-blue stimulated emission 741 optical microscopy 459 optical processes 600 optical spectrum 492 optical surface reflectometry 521 optical transitions 1060, 1110, 1163 optical transmission measurements 95 optical transmission spectra 1057 optimal growth temperature 699

Index optimum stripe orientation 621 optoelectronic devices 225, 384, 623, 1034 orbital angular momentum 1072 order–disorder phase transitions 588, 594 organic chemical vapor deposition 684, 733 organometallic precursor gases 395 organometallic source 468, 730 organometallic vapor phase epitaxy (OMVPE) 323, 385, 393, 825 470, 565 orthogonalized plane wave (OPW) method 134 orthorhombic cell 382, 383 orthorhombic structure 382 out-of-plane component 218 out-of-plane deformation potentials 160 out-of-plane lattice constant 463 out-of-plane rocking curves 465 out-of-plane sound velocities 57 out-of-plane strain 276 out-of-plane stress 155 oxidation state 1050 oxygen-bonded silicon 341 oxygen doping 1013 oxygen-terminated surface 356

p parabolic band structure 216 parabolic conduction band 216 paramagnetic materials 1157, 1064 paramagnetic regime 1109 paramagnetism 1004 parametric variations 387 parasitic gas-phase chemical reactions 653 Pauling’s electronegativity 138 p-doping 667 peak electric field 946 peak electron drift velocity 83 peak transition energy 616 pendeo-epitaxial film 543 pendeo-epitaxial growth 540, 542 pendeo-epitaxial phenomenon 542 pendeo-epitaxy 507, 540, 541, 621 periodic stacking faults 852 perovskite cell 383 perovskite oxide(s) 296, 382 – substrates 295 phase separation 93 phase shift 1120 phonon dispersion curves 41 phonon energies 20 phonon frequency 20 phonon-phonon scattering 57, 79 phonon-phonon Umklapp scattering 53 photoelastic effect 1120

photoelastic modulator (PEM) 1120 photoelectron spectroscopy 208 photoemission experiments 139 photoemission methods 271 Photoemission studies 1134 photoemission transient capacitance methods 961, 976 photoexcitation 1116 photo-generated carriers 1077 photoionization spectroscopy 1041 photolithography step 557 photolithography techniques 530 photoluminescence 288, 407, 462, 635, 1018 – data 578 – efficiency 598 – experiments 185, 930 – intensity 615 – measurements 483, 604, 637, 1040 – spectra 435, 489, 587, 602, 606, 608 photomodulated transmission 95 photon energy resonates 961 photoreflectance (PR) spectra 87, 626 piezo component 276 piezoelectric (PE) coefficients 132, 180, 641 piezoelectric charge 262 piezoelectric constants 240, 259 piezoelectric effects 95, 182, 215, 588, 1185 piezoelectric fields 1179 – scattering carriers 679 piezoelectric material 948 piezoelectric polarization 157, 236, 259, 263, 289 – calculation 259 – effects 132 – non linearities 256 – properties 62 – transducer 1120 piezoelectric tensor 237 piezo-induced polarization 278 pillarlike interfacial layer 327 pin photodiodes 661 planar defects 844 plane wave expansion method 134 plane wave pseudopotential (PWPP) 78 Plank’s constant 942 plasma-assisted MBE (PAMBE) 428, 657 plasma-assisted OMVPE 636 plasma-enhanced CVD 394 plasma excitation 513 plastic deformation 327 plastic relaxation 483 PL bands 1061 PL emission 1161 PLE spectrum 1056

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PL intensity 1055 PL spectra 1056 PL transition 1054 point defects theory 919 Poisson effect 155 Poisson equation 270 Poisson ratio 484, 565 Poisson’s equation 266, 269, 271 Poisson’s ratio 37, 38, 158, 292, 483, 826 polarization charge 221, 267, 276, 1173 – Ga-polarity single AlGaN/GaN interface 272 – Ga-polarity single AlxIn1-xN/GaN Interface 276 polarization charge density 266 polarization difference 284 polarization effects 131, 230 polarization gradient 269 polarization-induced charge 221, 286, 287, 288, 615 polarization-induced electric field 282, 622, 626 polarization-induced field 283, 753 polarization-induced interface charge 266 polarization-induced interface sheet density 286 polarization-induced red shift 288 polarization-induced surface 266 polarization in heterostructures 264 polarization in quantum wells 278 – nonlinear 280 polarization vector 1115 polarized light 1109, 1110, 1119, 1176, 1177 polarized light emission 1173, 1178, 1179 polycrystalline film 600 polycrystalline growth mode 630 polymeric GaN 731 polymeric product 399 Poole-Frenkel barrier 954 Poole-Frenkel effect 939, 945, 947, 948, 950, 955, 972, 973 porous SiC (PSiC) substrates 503 porous silicon (PS) 509 positive polarization charge 273 positron annihilation experiments 982, 983, 984, 992 positron annihilation spectroscopy (PAS) 933 positron-electron momentum distribution 982 positron lifetime experiments 983, 989 positron lifetime spectroscopy 995 positron’s wave function 1038 postgrowth annealing 109, 110, 993 postgrowth evaluation 503

postimplant annealing treatment 1163 powder technique 77 Powell’s model 441 precursor-mediated model 449 precursor-mediated pathway 438 precursor flux 400 predigital thermometers 296 pre-epitaxy surface preparation 334 preexponential factor 399 pressure coefficient 14 prismatic growth 516 prismatic planes 533, 568 probe-size-related effect 620 process pressure 525 proton implantation 1190 proton irradiation 963 pseudo-hydrogen atoms 649 pseudomorphic temperature 631 pseudopotential calculations 138, 139 pseudopotential method 132, 135 pseudopotential plane wave calculations 42 pseudopotential plane wave method 38 p-type doping 1006, 1007, 1013, 1035 – role of hydrogen 1034 p-type doping with superllatices 1032 p-type GaN 627, 1163 p-type semiconductor 272 p-type impurity atoms 92 pulsed laser ablation/chemical vapor deposition (PLA-CVD) process 739 pulsed positron beam 991 pyramidal planar defects 818 pyramidal structures 559 pyroelectric effects 1185 pyrolysis temperatures 512 pyrolytic decomposition 653 pyrolytic process 653

q QD-like structures 722 quadratic equation 103 quadratic nonlinearity 280 quadrupole mass spectrometer 51 quantitative in situ method 401 quantum-confined Stark effect(QCSE) 264, 286, 611616 quantum dots 706, 707, 711, 723 – colloidal formation 723 – growth 706 – holy grail 707 quantum dots by MBE 712 quantum dots by OMVPE 719 quantum efficiency 612 quantum mechanical treatments 1065

Index quantum mechanics 218 quantum size effect 186 quantum well (QW) structures 90, 132, 213, 221, 215, 279, 280, 673, 706, 1125, 1149, 1179 – thickness 222 quasi-cubic approximation 166, 167, 174 quasi-cubic model 160, 168, 171, 179, 182 quasi-Fermi level 197 quasi-linear temperature gradient 361 quasi-particle approach 141 quasi-particle band structure energies 141 quasi-particle excitations 139 quaternary – InGaAlN 699 quaternary alloys 89, 100, 699, 652 – growth of 652

r radiation-induced defects 190 radiative recombination lifetime 573 radiative transitions 196, 1177 radio frequency plasma excited (RF-MBE) 98, 405, 630 Raman-active optical phonon modes 64 Raman frequencies 559 Raman measurements 483 Raman mode 39, 563 Raman scattering 40, 613, 695 – data 559 – experiments 193, 561 – measurements 61, 64 Raman spectroscopy 42, 79, 563 – measurements 563 Ramsdel notation 2-4 Rare earths 1046 Rashba coupling 1173 – mechanism 1186 Rashba–Sheka–Pikus (RSP) 214 Rashba spin–orbit coupling effect 1169 Rashba spin splitting 1186 RBS-channeling 1161 reactive ion etching (RIE) methods 511, 751, 966 reactive ion molecular beam epitaxy (RIMBE) 72, 435, 441 reactive molecular beam epitaxy (RMBE) 326, 963 rectangular cross section facet 533 reduced recombination efficiency 612 reduced screw dislocations 586 reflectance spectra 177, 1135 reflectance spectroscopy 84

reflection high-energy electron diffraction (RHEED) 6, 341, 417 – intensity 700 – patterns 633 – secular beam 700 refractive index 27, 32, 34, 73, 74, 89, 1120 relative transition probabilities 1114 RF-activated nitrogen 405 RF magnetron sputtering (RF MS) 98, 637, 638 RF plasma-assisted MBE 993 RF plasma source 452 RF-sputtered film 187 RKKY theories 1187 RKKY-type interactions 1090 reactive molecular beam epitaxy (RMBE) 436, 437 RMBE-grown films 1039 rock salt 1, 2, 14, 28, 36, 62 room-temperature absorption coefficient 73 room-temperature electron mobility 87 room-temperature ferromagnetic semiconductors 1134 room-temperature Hall measurements 605 room-temperature thermal conductivity 59 root mean square (rms) 196, 346 r-plane sapphire 293, 326, 373, 377 Russel–Saunders coupling 1046 Rutherford backscattering spectrometry (RBS) 91, 187, 188, 700, 704, 1162 – experiment 1037

s sacrificial buffer layers 469 Samsung Advanced Institute of Technology 468 Samsung freestanding template 1003 Samsung template 477, 1002, 1003 sandwich method 392 Sapphire 342, 343, 344, 346, 388, 452, 512, 709 – stacking order 351 – surface preparation 346 – unit cell 344 sapphire planes 372 sapphire substrates 72, 87, 293, 327, 382, 665, 752 satellite spots 646 saturation magnetization 1066 scanning capacitance microscopy (SCM) 910 scanning electron microscope (SEM) 978 – images 445 scanning thermal microscopy (SThM) 57

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scanning transmission electron microscope (STEM) 836 scanning tunneling microscopy (STM) 412, 588, 642, 691 – probes 691 scanning tunneling spectroscopy (STS) 367 scattering amplitude 422 Schmidt orthogonalization condition 135 Schoenflies notation 2 Schottky barrier 271, 621 – ultraviolet detectors 508 Schottky characteristics 575 Schrodinger–Poisson equations 265, 282 Schrodinger–Poisson solver 272, 281 Schrodinger’s equation 136, 266, 215, 219, 270, 420 screw burgers vectors 836 screw dislocation(s) 336, 550, 571, 823, 826, 828 – component 488 – full core 831, 865 – open core 831, 865 screening field 244 s–d exchange interaction 1117, 1126 SdH measurements 1186 secondary ion mass spectroscopy (SIMS) 91, 475, 985 second lithographic process 551 second-order elastic moduli 78 second-order perturbation theory 160, 172 Se doping electron concentrations 1012 segmented silicon lattice planes 567 selected area diffraction (SAD) 668 selective area epitaxy (SAE) 535 selective area growth (SAG) 583 selective epitaxial growth 529, 537 self-catalytic process 731 self catalytic vapor–liquid–solid (VLS) process 730 self-compensation process 1005 self-consistent field method 137 semiconductor-ferromagnetic material interface 1169 semiconductor lasers 216 semiconductor quantum dots (QDs) 706 semiconductors 369 semiconductor substrates 330 semiconductor technology 1006 semiempirical pseudopotential calculations 199 semi-insulating SiC 336 semi-insulating (SI) GaN 1044 semimetallic overlap 205 seven-atom ring core 837

shallow acceptors 1010, 1017 shallow donor acceptor 507 shallow donors 1007 sharper superimposed peak 522 sheet carrier concentration 268, 287 sheet carrier density 278, 653 short-range electron-hole spin exchange interaction 1149 Shubnikov-de Haas data 169 Shubnikov-de Haas measurements 169 Si 332 – surface preparation 333 Si-based electronics 507 SiC 334, 335 – epitaxial layers 503 – hydrogen surface etching 340 – stacking sequence 335 – surface preparation 338 SiC mechanical polishing techniques 479 Si-doped GaN layer 585 Si doping 1010 silica-based fiber dispersion 110 silicon-on-insulator (SOI) 509, 565 silicon substrates 544 SIMS 1005 SIMS analyses 1041 SIMS technique 475 single-crystal AlN films 396 single-crystal diffraction 462 single-crystal epitaxial thin films 73 single-crystalline AlN 638 single crystalline GaN 624 single-crystalline structure 741 single-electron spin transistors 1123 single-interface heterostructures 276 single magnetic ion interaction 1084 single-molecule precursors (SMPs) 740 single quantum well (SQW) 286, 559 – structure 286 single-step ELO technology 535 single-step substrate 496 single-walled carbon nanotubes (SWCNs) 749 Slater-Koster parameters 141 SMB threading defect 498 smooth coalesced films 558 smooth surface morphologies 634 solar-blind detectors 638, 652 solar-blind region 492 solar-blind UV photoconductors 661 solar cells 211 solid-liquid interface 729 space group 143 space grouping 2

Index spatial modulation 1169 sp-d exchange 1125 sp-d exchange interaction 1127, 1129 sp-d spin exchange interaction 1118, 1127 specific heat 17, 25, 29, 31, 53, 60, 61, 67, 68, 79, 80, 81 spectral position 998 spectroscopic ellipsometry 170, 177, 288 specular epitaxial films 523 specular reflection spot 417 sphalerite 2,4 spin angular momentum 1072 spin coherence length 1172 spin coherence times 1134, 1168 spin-dependent kinetic exchange interaction 1087 spin-dependent transport 1077 spin diffusion length 1184 spin-down channel 1100 spin-down electrons 1005, 1070, 1078, 1104, 1109 spin-down states 1115, 1116 spin exchange interaction 1085, 1117, 1137, 1147 spin FETs 1171, 1188 spin-freezing temperature 1144 spin interaction induced 1077 spin LED 1174 spin LED structures 1179 spin light-emitting diodes (spin LEDs) 1169 spinoidal decomposition – InAlGaN 705 spinoidal isotherms – InAlGaN 705 – spin–orbit interactions 148 spin–orbit splitting 152, 171 spin ordering 1066 spin orientation 1169, 1179 spin polarization 1110, 1169, 1176, 1179, 1188 spin-polarized carriers 1078, 1135 spin-polarized current 1171 spin-polarized electrons 1168, 1172, 1173, 1176, 1179 spin-polarized holes 1135, 1176 spin-polarized injection 1171 spin-polarized tunneling (SPT) 1078 spin-polarizing electrode 1169 spin–orbit band 225 spin–orbit coupling 1054, 1116, 1135, 1149, 1187 spin–orbit interaction(s) 148, 151, 174, 1084, 1104, 1110, 1172, 1185 spin–orbit interaction, see Jahn–Teller effect 999

spin–orbit levels 1061 spin-orbit split-off band coupling 215 spin-orbit split-off bands 1110, 1115 spin-orbit split-off band transitions 1114 spin–orbit split-off-hole effective mass 175 spin-orbit split-off mass 183, 201 spin-orbit splitting 148, 151, 152, 160, 161, 172, 183 spin resonant tunneling device 1076 spin splitting 1126 spin-splitting energies 1186 spin transport electronics 1046, 1077 spin tunnel junctions 1168, 1169 spin-unpaired electrons 818, 1168 spin-up channel 1100 spin-up electron 1047, 1070, 1079 spin valve 1180 spin valve structure 1173 spinel-type structure 611 spinodal curves 698 spinodal isotherms 705 spinodal phase separation phenomenon 97 spintronic devices 1168 spintronics 1063 – general remarks 1075 spiral hillock features 459 spiral hillocks 445, 460 split-off band 1110 split-off hole mass 152, 174 spontaneous polarization 231, 241, 265, 277 – charge density 266 – charges 588 – coefficients 132 – nonlinearities 253 sputtering technique 84, 637, 638 squared absorption coefficient 96 SQUID magnetometers 1167 SQUID measurements 1158 stacking-order mismatch 351 stacking fault (SF) 818, 851 – basal 853 – electronic structure 884 – induced by Mg doping 887 – prismatic 860 – type I and I1 852 – type II and I2 852 – type III 852 stacking mismatch boundaries (SMB) 329, 818, 844 stacking sequence 2,4,146 stamping method 569 standard framework 498 standard liter per minute (slm) 467, 486 stark effect 288

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stark shift 264, 288 steady-state capacitance 944 steady-state four-probe method 54, 59 step-flow growth 403 stick-and-ball representations 4, 846 sticking coefficients 400 STM data 642 stoichiometric boundary 633 stoichiometric growth 632 stoichiometric nitrides 131 stokes shift 170 strain-free films 328 strain-induced piezoelectric 277 strain-induced polarization 264 strain minimization 377 strain-relieving defects 260 strain–stress relationship 154 strained-layer superlattices (SLS) 213, 450 Stranski–Krastanov (SK) mode415 stress – strain-GaN on SiC 483 stress-related phenomena 502 stress–strain relationship 155 stress–temperature coefficient 484 strong exchange interactions 1086 sublimation sandwich method (SSM) 392 submicron seed posts 541 substitutional acceptors 1007 substrate surface preparation 494 substrate temperature 299 superexchange interaction 1086 superexchange mechanism 1086 supersaturation 363 surface acoustic wave measurements 63 surface diffusion 413 – length 404 surface energy 368 surface morphology 406, 455, 460, 845 surface-segregation processes 450 symmetry-conserving stress 42

t tandem shift 204 telecommunication purposes 110 TEM analyses 491 TEM bright-field images 656 TEM data 604 TEM diffraction images 1128 TEM images 1128 temperature controllers 296 temperature-dependent Boltzmann term 412 temperature-dependent CL spectra 178 temperature-dependent emission energy 170 temperature-dependent measurements 1042

temperature-dependent thermal conductivity 55 temperature-independent constant 1073 temperature-independent electron 654 temperature-programmed desorption (TPD) 443 temperature resistivity measurements 1044 template-based methods 735 template-based techniques 737 tensile biaxial strain 262 tensile strain 658 tensile uniaxial strain 160 tentative growth mechanism 505 terahertz frequency optical switches 1076 ternary – alloys 89, 90, 210, 652, 1010, 1124 – growth 652 terrace width 445 tetragonal unit cell 383 tetrahedral bonds 366 tetrahedral notation 852 theoretical frequency shifts 642 thermal activation 406 – energy 1042 thermal annealing 521, 608, 1015 thermal conductivity 53, 59, 79, 499, 638 thermal cracking 436 thermal decomposition rate 453 thermal desorption spectroscopy (TDS) technique 443 thermal emission 942 – techniques 956 thermal energy 412 thermal etching 597 thermal expansion coefficients (TEC) 47, 67, 479, 537, 601, 658, 662 thermal imaging 57 thermally detected optical absorption (TDOA) measurements 190 thermally stimulated current spectroscopy 1041 thermal mismatch 290, 326 – induced strain 290 thermal stability 396 thermal strain 510, 546 thermal stress 291, 297, 660 thermodynamic 436 thermodynamical data 388 thermodynamic equilibrium models 398, 451, 934 thermodynamical (low-stability) barriers 362, 392 thermogravimetric technique 398 thick single-crystalline GaN layers 386

Index thick wurtzite GaN films 329 threading dislocation (TD) 460, 473, 553, 586, 656, 660, 662, 818, 822 – densities 528, 585, 604 three-dimensional (3D) – diagrams 100 – growth 341 – mechanisms 565 – nucleation 544 – strain-minimizing shape 568 – stress relief 565 threefold spot pattern 652 three-step ELO process 557 three-step growth 527 tight binding (TB) approximation 133 tight binding calculation 208 tight-binding model 134, 499 tight binding realm 272 time-resolved luminescence 571 time-resolved PL (TRPL) 279, 573 time-resolved Raman measurements 39 TiN nanoporous blocking method 577 TM-doped GaN 1101, 1141, 1146 total energy (TE) 820 traditional compound semiconductors 205 transient PL study 1054 transient spectroscopy 462 transition elements 1046, 1060, 1063, 1188 transition energies 1187 transition growth regime 455 transition metal doped semiconductor 1153 transition metal impurities 1046 transition point 492 translation domain boundaries (TDBs) 818, 844 transmission cross-sectional TEM 472, 529, 565 transmission electron microscopy (TEM) 565, 817 – images 340, 684, 1128 transmission measurements 635 transverse acoustic (TA) mode 39 transverse optical (TO) phonon frequencies 20 trapezoidal crystals 516, 517 trapezoidal stripes 556 trenched substrates technique 659 trialkyl compounds 409, 468, 513 trialkyl precursors 467 triangular stripes 531, 538, 563 trichloroethane (TCE) 329 triethylgallium (TEG) 393, 395, 530, 1005 trigonal field splitting 1061 trimetal Ga (TMG) 1015

trimethylaluminum (TMA) 395 trimethylgallium (TMG) 395, 467 trimethylindium (TMI) 395 truncated hexagonal features 523 tunneling barrier 1182 tunneling magnetoresistance (TMR) ratio 1078, 1182 tunnel junction magnetic resistance (JMR) 1078 two-boat system 738 two-coupled conduction bands 203 two-dimensional buffer material 328 two-dimensional electron gas (2DEG) 586 two-dimensional growth mode 415, 641 two-dimensional imaging 839 two-dimensional polarization 266 two-dimensional stress 292 two-dimensional version 419 two-electron transitions 610, 1010 two-state (binary) system 1076 two-step ELO 551 two-step epitaxial lateral overgrowth 547

u UHV electron cyclotron resonance (ECR) 638 ultrahigh vacuum (UHV) 330 ultraviolet detectors 652, 667 ultraviolet LEDs 508 ultraviolet photoemission spectroscopy (UPS) 69 uniaxial stress 158, 373 universal Kane’s relation 194 unpolarized light 1147 UV-assisted atomic layer epitaxy 89 UV emitters 666 UV wavelengths 432

v vacancies – doping 983 – growth kinetic and thermal behavior 993 vacancy 928, 929 – Ga 929 – N 930 vacancy-impurity complexes 931, 988 vacancy-mediated self-diffusion 931 vacancy-related lifetime components 992 vacuum-deposition techniques 329 valence band 224, 1164 – confinement energies 226 – density 1017 – edge 173 – heterojunction offsets 139 – mass parameters 199

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1254

valence band maximum (VBM) 207, 828 valence band offset 211, 215 valence band structures 143 valence electrons 983 valence force field (VFF) 680 van der Waals bonding 328 van der Waals epitaxy 328 van der Waals forces 411 van der Waals substrates 328 vapor–liquid–solid (VLS) process 726, 727 vapor–solid (VS) growth 726 vapor–solid method 732 vapor phase composition 387 vapor phase epitaxy (VPE) 385 vapor phase growth 726 vapor phase reactions 520 variable-range hopping regime 1163 Varshni formula 203, 1161 Varshni parameters 170, 177, 192 V-defects – induced by Mg doping 894, 895, 897 – InGaN 674 – InGaN quantum well 905 Vegard’s law 85, 90, 94, 95 Verdet constant 1106 vertical cavity surface emitting laser (VCSEL) 110 vertical gradient freeze (VGF) methods 330 vibrating sample magnetometer (VSM) 1141 vibrational modes 39 Vinter notation 219 VLS process 729 void-assisted separation (VAS) process 576 Volmer–Weber (VW) mode 415 volume-conserving strain 284 V-shaped defect(s) 664, 666 V-type inversion domains 845

w Wafer bonding 471 wave pseudopotential methods 366 wave vector 420, 647 weak antilocalization (WAL) 1185 weak compressive strain 277 weak photoluminescence peaks 84 Weisbuch notation 219 well-characterized semiconductors 75 well-converged plane wave calculations 138 well-defined angular momentum 137 wet etching techniques 35 Wigner–Seitz cell 135 wurtzite (Wz) 1, 131, 1124 Wurtzite AlN, see bandgap semiconductor wurtzite crystal structure 28, 39

Wurtzite GaN bandgap 169 Wurtzite nitride growth 498 wurtzite phase 508, 612 wurtzite semiconductors 213, 1109, 1149 wurtzite structure(s) 2, 7, 143, 144, 148, 844 wurtzitic crystals 162 wurtzitic InN 185 wurtzitic phase 381, 477 wurtzitic systems 410

x X-ray absorption spectroscopy (XAS) 38, 1053 X-ray analysis methods 521 X-ray beam 474, 619 X-ray data 33, 34, 78, 393, 474, 487, 489, 521, 687, 1150 X-ray diffraction (XRD) 10, 11, 37, 38, 64, 187, 188, 276, 364, 372, 462, 465, 487, 514, 528, 537, 588, 590, 605, 607, 608, 637, 695, 702, 1144, 1146, 1189 – experiments 626 – in and out of plane diffractions 466 – peak 72, 91, 372, 463, 488, 519, 525, 570, 573, 581 – spectra 188 – spectral FWHM 636 – system 475 X-ray linewidth 407, 446 X-ray photoelectron diffraction 587 X-ray photoelectron spectroscopy 208, 301, 349, 639 X-ray photoemission spectroscopy (XPS) 48, 1100 X-ray reciprocal-space mapping 95 X-ray reflections 608 X-ray rocking curve 98, 364, 365, 465, 466, 474, 510, 513, 514, 576, 601, 605, 619, 687, 688 – analysis 636 – peaks 522 X-rays 462, 463 – techniques 67 – wavelength 463 – wave method 587 XRD analysis 77, 364, 489, 600, 687, 688 XRD rocking curves 474, 619, 620, 621

y yellow line (YL) emission 555 yellow luminescence 567, 886, 924, 926, 928, 982, 1000, 1056, 1198 YL band 563, 609, 1011, 1013, 1040, 1042

Index Young’s modulus 15, 16, 23, 24, 30, 33, 38, 63, 156, 157, 158, 291, 331, 333, 337, 345, 358, 483, 565, 566

z Zeeman effect 1129 Zeeman splitting 1110, 1116, 1117, 1118, 1119, 1120, 1127, 1129, 1133, 1135, 1136, 1146,1147,1148,1149,1150,1151,1152,1187 Zener theor(ies) 1090, 1091, 1187 zero-dimensional electronic systems 738 zero-order quarter-wave plate 1176 zero-phonon line (ZPL) 1000 zero-phonon peak 1011, 1039 zero-phonon transition 1039, 1042, 1043 zinc blende 1, 2, 6, 9, 14, 16, 17, 21, 22, 24, 28, 29, 40, 41, 43, 45, 57, 63, 77, 78, 90, 98, 109,

131, 132, 136, 138, 140, 141, 171, 176, 183, 184, 201, 202, 232, 294, 466, 681, 723, 851, 854, 858, 930, 932, 1118 – AlN 24, 28, 29, 183 – growth 507 – phase 9, 14, 16, 17, 22, 23, 24, 36, 477 – polytypes 36, 41, 62, 82, 90 – structure 2, 4, 39, 141, 143, 144, 1096, 1124 – symmetry 1117 zinc blende GaN 172 zinc blende InN 200 Zn-doped samples 628 ZnO 350, 351 – surface preparation 353 ZnO substrates 598 ZPL 1045

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Appendix

j1257

Appendix: Periodic Table of Pertinent Elements

atomic weight based 69.72 on carbon 12 Density 5.91 Ga Symbol solid&liquid:g/cm-3 Ar3d104s2p1 Electronic configuration Solid r r: Radioactive gas:g/liter at 273K&1 atm Name Gallium Atomic number 31

II A

IB

IIIB

IIB

9.0122

IVB

5

10.81 6 12.01 2.34 1.85 B 2.62 Be C 1s22s2p2 1s22s2p1 1s22s2 Solid Solid Solid Carbon Beryllium Boron 13 26.98 14 12 24.305 28.09 1.74 2.70 Al 2.33 Mg Si Solid Ne3s2p1 Ne3s2 Solid Solid Ne3s2p2 Magnesium Aluminum Silicon 20 72.59 40.08 29 65.38 31 69.72 32 63.55 30 Ca 8.96 Cu 7.14 Zn 5.91 Ga 5.32 Ge 1.55 Ar3d104s2p1 Ar3d104s2p2 Ar3d104s2 Ar3d104s1 Ar4s2 Solid Solid Solid Solid Solid Copper Zinc Calcium Germanium Gallium 47 112.41 49 118.69 107.87 48 114.82 50 10.5 Ag 8.65 Cd 7.31 In 7.30 Sn 4

VB 14.007 7 1.251 N 1s22s2p3 Gas Nitrogen 15 30.97

VIIIB 2

1.82 P Ne3s2p3 Solid Phosphorus 33 74.92 5.72 As 4.80 Se Ar3d104s2p3 Ar3d104s2p4 Solid Solid Selenium Arsenic 51 127.60 121.75 52 6.88 Sb 6.24 Te Kr4d105s2 Kr4d105s2p1 Kr4d105s2p2 Kr4d105s2p3 Kr4d105s2p4 Kr4d105s1 Solid Solid Solid Solid Solid Solid Tellerium Cadmium Silver Indium Tin Antimony 79 196.97 80 200.59 81 204.37 82 207.2 83 208.98 19.3 Au 13.53 Hg 11.85 Pb 9.8 Bi Tl 11.4 Xe4f144d106s1

Solid Gold

Xe4f145d106s2 Xe4f145d106s2p1 Xe4f145d106s2p2 Xe4f145d106s2p3

Solid Mercury

Solid Tallium

Solid Lead

Solid Bismuth

63.55

0.1787 He 1s2 Gas VIB Helium 16 10 20.18 8 1.429 O 0.901 Ne 1s22s2p4 1s22s2p6 Gas Gas Oxygen Neon 16 32.06 18 39.95 2.07 S 1.784 Ar Ne3s2p4 Ne3s2p6 Solid Gas Argon Sulfur 34 83.80 78.96 36 3.74 Kr Ar3d104s2p6 Gas Krypton 131.3 54 5.89 Xe Kr4d105s2p6 Gas Xenon