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SEMICONDUCTORS AND SEMIMETALS VOLUME 14 Lasers, Junctions, Transport

This Page Intentionally Left Blank

SEMICONDUCTORS AND SEMIMETALS Edited by R . K . WILLARDSON ELECTRONIC MATERIALS DIVISION COMINCO AMERICAN INCORPORATED SPOKANE, WASHINGTON

ALBERT C . BEER BATTELLE MEMORIAL INSTITUTE COLUMBUS LABORATORIES COLUMBUS, OHIO

VOLUME 14 Lasers, Junctions, Transport

1979

ACADEMIC PRESS New York Sun Francisco A Subsidiary of Harcourt Brace Jovanotich, Publishers

London

COPYRIGHT @ 1979, BY ACADEMIC PRESS, h C . ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRlTING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. 111 F

i Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD.

24/28 Oval Road, London NW1 7DX

Library of Congress Cataloging in Publication Data ed . Willardson, Robert K Semiconductors and semimetals. Includes bibliographical references. CONTENTS: v. Physics of Ill-V compounds.--v. 3. Optical properties of 111-V compounds. [etc.] v. 14. Lasers, junctions, transport. 1 . Semiconductors--Collected works. 2. Semimetals-Collected works. I. Beer, Albert C., joint ed. 11. Title. QC612.S4W5 537.622 65-26048 ISBN 0-12-752114-3 (v. 14)

PRINTED IN THE UNITED STATES OF AMERICA

798081 82838485

987654321

Contents LIST OF CONTRIBUTORS . . PREFACE. . . . . CONTENTS OF PREVIOUSVOLUMES

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vii ix xi

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1 4

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Chapter 1 Photopumped HI-V Semiconductor Lasers N . Holonyak. Jr., and M . H . Lee I . Introduction . . . . I1. Laser Threshold Requirements I11. Photopumping Methods . . IV . Binary 111-V Semiconductors . V . Alloy 111-V Semiconductors . VI . Carrier Lifetime . . . VII . Conclusions . . . .

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6 10 26 55

63

Chapter 2 Heterojunction Laser Diodes Henry Kressel and Jerome K . Butler I . Introduction . . . . . . . . I1 . Laser Diode Structures . . . . . . I11. Wave Propagation . . . . . . . IV . Relation between Electrical and Optical Properties . V . Laser Diode Technology . . . . . VI . Heterojunction Lasers of Alloys Other than GaAs-A1As . . . . . VII . Laser Diode Reliability . VIII . Devices for Special Applications . . . . IX . Distributed-Feedback Lasers . . . . . X . Laser Modulation and Transient Effects . . . List of Symbols . . . . . . .

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104 121

139

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151

161 175 185 192

195 199 217 229

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I . Introduction . . . . . . . . . I1. Single-Injection Space-Charge-Limited Solid-state Diodes . 111. Double-Injection Space-Charge-Limited Solid-state Diodes IV . Noise in Space-Charge-Limited Solid-state Diodes . . V . Applications . . . . . . . . .

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66 68 75

Chapter 3 Space-Charge-Limited Solid-state Diodes A . Van der Ziel

V

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. . 245

vi

CONTENTS

Chapter 4 Monte Carlo Calculation of Electron Transport in Solids Peter J . Price

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I . Introduction . . . . . . . I1 . Hot Electrons . . . . . . 111. Hot Electron Properties . . . . . IV . Spatial Structures . . . . . . V . Ohmic Conduction . . . . . . VI . Collective Effects . . . . . . Appendix A . Generation of a Gaussian Distribution Appendix B . Some Vector Geometry .

AUTHORINDEX SUBJECT INDEX

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-249 . 254 . 272 . 283 . 294 . 297 . 306

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307

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309 317

List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

JEROME K. BUTLER,Southern Methodist University, Dallas, Texas 75275 (65)

N. HOLONYAK, JR., Department of Electrical Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 (1) HENRYKRESSEL,RCA Laboratories, Princeton, New Jersey 08540 (65) M. H. LEE,'Department of Electrical Engineering, Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 ( 1 ) PETERJ . PRICE,IBM Thomas J . Watson Research Center, Yorktown Heights, New York 10598 (249) A.

Electrical Engineering Department, University of Minnesota, Minneapolis, Minnesota 55455 ( I 95)

V A N DER ZIEL,

'Present address: IBM Research Laboratory, San Jose, California 95 193. vii


Preface Of the four chapters in the present volume, two deal directly with transport phenomena and junction effects, and the other two chapters are concerned with devices involving such phenomena. Applications of semiconductor junctions are of course the basis of most of the devices which have revolutionized electronics technology. Recent developments, which are now proceeding at a rapid pace, involve electrooptical interactions. A major factor contributing to the extensive activity, especially in the communications and data processing fields, is the semiconductor laser. The first two chapters of this volume are concerned with such devices. Advances in materials technology, which have led to the preparation of high quality ternary and quaternary semiconductor compounds (more specifically, alloys or solid solutions involving 111-V compounds-often referred to as mixed crystals), have permitted the development of heterojunction lasers. These lasers are discussed in detail in Chapter 2. Heterojunctions have produced major improvements in laser performance, as well as greater flexibility, and the availability of various emission wavelengths. In particular, the threshold current densities at room temperature have been reduced by orders of magnitude, permitting continuous operation of the laser. Besides electron-hole pair injection by means of current in a junction, excess electron-hole pairs can be introduced into a semiconductor laser by two other common methods, namely electron-beam bombardment and photopumping (photoexcitation). Laser excitation by the latter process (Le., photoluminescence) is the subject of Chapter 1 of this volume. Although such an excitation scheme may not be convenient for many applications, the technique is useful for producing laser emission in semiconductors where junctions cannot readily be fabricated. Thus it is possible to study laser effects in homogeneous samples that have arbitrary doping levels and are not complicated by the impurity gradients inherent in junction structures. The chapter dealing with junction phenomena and injection effects (Chapter 3) is devoted specifically to space-charge-limited diodes. Consideration is given to both single- and double-injection cases, and effects of diffusion, trapping, and noise phenomena are analyzed. The last chapix

X

PREFACE

ter in the book discusses the Monte Carlo method of calculation of electron transport in solids. This technique is finding increasing applicability in problems that require numerical results but that involve physical processes or device geometries not capable of direct representation by readily soluble mathematical relationships. Obvious examples of such situations are hot electron phenomena, various spatial structures, and the existence of certain collective effects. The increased availability of large computers is undoubtedly an important factor in the popularity of Monte Carlo computations. The editors are indebted to the many contributors and their employers who make this treatise possible. They wish to express their appreciation to Cominco American incorporated and Battelle Memorial Institute for providing the facilities and environment necessary for such an endeavor. Special thanks are also due the editors’ wives for their patience and understanding.

R. K. WILLARDSON ALBERTC. BEER

Semimetals and Semiconductors Volume

Physics of 111-V Compounds

C . Hilsum, Some Key Features of Ill-V Compounds Franco Bassani. Methods of Band Calculations Applicable to 111-V Compounds E. 0 . Kane, The k . p Method V. L. Bonch-Bruevich, Effect of Heavy Doping on the Semiconductor Band Structure Donald Long, Energy Band Structures of Mixed Crystals of 111-V Compounds Laura M . Rorh and Petros N . Argyres, Magnetic Quantum Effects S. M . Puri and T . H . Geballe, Thermomagnetic Effects in the Quantum Region W . M . Becker, Band Characteristics near Principal Minima from Magetoresistance E. H . Putley, Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in InSb H . Weiss. Magnetoresistance Betsy Ancker-Johnson, Plasmas in Semiconductors and Semimetals

Volume 2 Physics of 111-V Compounds M . G . Holland, Thermal Conductivity S . I. Novkova. Thermal Expansion U.Piesbergen. Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J . R. Drabble, Elastic Properties A . U . Mac Rae and G . W . Gobeli, Low Energy Electron Diffraction Studies Robert Lee Mieher, Nuclear Magnetic Resonance Bernard Goldsrein. Electron Paramagnetic Resonance T . S . Moss. Photoconduction in 111-V Compounds E. A n t o d i k and J . Tauc. Quantum Efficiency of the Internal Photoelectric Effect in InSb G. W. Gobeli and F. G. Allen, Photoelectric Threshold and Work Function P.S. Pershun, Nonlinear Optics in 111-V Compounds M. Gershenzon, Radiative Recombination in the 111-V Compounds Frank Stern, Stimulated Emission in Semiconductors

Volume 3 Optical Properties of 111-V Compounds Marvin Hass, Lattice Reflection William G. Spitzer, Multiphonon Lattice Absorption D. L. Stienvalt and R. F. Porter, Emittance Studies H . R. Philipp and H . Ehrenreich, Ultraviolet Optical Properties Manuel Cardona. Optical Absorption above the Fundamental Edge Eurnest J . Johnson, Absorption near the Fundamental Edge John 0.Dimmock, Introduction to the Theory of Exciton States in Semiconductors xi

CONTENTS OF PREVIOUS VOLUMES

XU

B. Lax and J . G. Mavroides, Interband Magnetooptical Effects H. Y. Fan, Effects of Free Carriers on Optical Properties Edward D.Palik and George B. Wright, Free-Carrier Magnetooptical Effects Richard H. Bube. Photoelectronic Analysis B. 0.Seraphin and H. E. Bennett, Optical Constants

Volume 4 Physics of I I E V Compounds N. A. Goryunova. A . S. Eorschevskii, and D. N. Tretiakov, Hardness N . N. Sirora, Heats of Formation and Temperatures and Heats of Fusion of Compounds AmBv Don L . Kendall, Diffusion

A. G. Chynoweth. Charge Multiplication Phenomena Robert W. Keyes, The Effects of Hydrostatic Pressure on the Properties of 111-V Semiconductors L. W . Aukerman, Radiation Effects N . A . Goryunova, F. P.Kesamanly, and D. N. Nasledov. Phenomena in Solid Solutions R. T. Bate, Electrical Properties o f Nonuniform Crystals

Volume 5 Infrared Detectors Henry Levinstein, Characterization of Infrared Detectors Paul W.Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. E . Prince, Narrowband Self-Filtering Detectors b a r s Melngailis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides Donald Long and Joseph L. Schmir, Mercury-Cadmium Telluride and Closlely Related Alloys E. H . Putley, The Pyroelectric Detector Norman E. Stevens, Radiation Thermopiles R. J. Keyes and T. M. Quist, Low Level Coherent and Incoherent Detection in the Infrared M. C. Teich, Coherent Detection in the Infrared F. R.Arams, E. W . Sard, B. J . Peyton, and F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers, Jr., Microwave-Based Photoconductive Detector Robert Sehr and Rainer Zuleeg. Imaging and Display

Volume 6 Injection Phenomena Murray A. Lampert and Ronald B. Schilling, Current Injection in Solids: The Regional Approximation Method Richard Williams, Injection by Internal Photoemission Alien M. Barnett, Current Filament Formation R. Baron and J . W. Mayer, Double Injection in Semiconductors W.Ruppel, The Photoconductor-Metal Contact

Volume 7 Application and Devices: Part A John A. Copefand and Stephen Knight. Applications Utilizing Bulk Negative Resistance F. A. Padovani. The Voltage-Current Characteristics of Metal-Semiconductor Contacts


xiii

P. L. Hower, W . W . Hooper, E . R. Cairns, R. D. Fairman, and D. A. Tremere, The GaAs Field-Effect Transistor Marvin H . White. MOS Transistors C. R. Antell. Gallium Arsenide Transistors T. L. Tansley. Heterojunction Properties

Volume 7 Application and Devices: Part B T.Misawa, IMPATT Diodes H . C. Okean. Tunnel Diodes Robert B. Campbell ond Hung-Chi Chong. Silicon Carbide Junction Devices R. E. Enstrom, H. Kressel, and L . Krassner, High-Temperature Power Rectifiers of GaAs,-,PZ

Volume 8 Transport and Optical Phenomena Richard J . Stirn, Band Structure and Galvanomagnetic Effects in Ill-V Compounds with Indirect Band Gaps Roland W . Ure, Jr.. Thermoelectric Effects in Ill-V Compounds Herbert filler, Faraday Rotation H. Barry Bebb and E. W . Williams, Photoluminescence I : Theory E. W. Williams and H . Barry Bebb, Photoluminescence 11: Gallium Arsenide

Volume 9 Modulation Techniques B. 0. SPmophin, Electroreflectance R. L. Agganual. Modulated Interband Magnetooptics Daniel F. Blossey and Paul Handler, Electroabsorption Bruno Batz. Thermal and Wavelength Modulation Spectroscopy Ivar Balslev. Piezooptical Effects D. E. Aspnes and N. Bottka, Electric-Field Effects on the Dielectric Function of Semiconductors and Insulators

Volume 10 Transport Phenomena R. L. Rode. Low-Field Electron Transport J . D. Wiley, Mobility o f Holes in 111-V Compounds C. M. WoFe and C . E. Stillman. Apparent Mobility Enchancement in Inhomogeneous Crystals Robert L. Peterson. The Magnetophonon Effect

Volume 1 1

Solar Cells

Harofd J . Hovel, Introduction; Carrier Collection, Spectral Response, and Photocurrent; Solar Cell Electrical Characteristics; Efficiency; Thickness; Other Solar Cell Devices; Radiation Effects; Temperature and Intensity; Solar Cell Technology

Volume 12 Infrared Detectors (II) W. L. Eiseman, J . D. Merriam, and R. F. Potter, Operational Characteristics of Infrared Photodetectors

XiV


Peter R . Brutt, Impurity Germanium and Silicon Infrared Detectors E. H . Putley, InSb Submillimeter Photoconductive Detectors G. E . StiNman, C. M. Wove, and J . 0.Dimmock. Far-Infrared Photoconductivity in High Purity GaAs G.E. Sti[lman and C. M. Wore, Avalanche Photodiodes P . L. Richards, The Josephson Junction as a Detector of Microwave and Far-Infrared Radiation E. H. Putley, The Pyroelectric Detector-An Update

Volume 13 Cadmium Telluride Kenneth Zanio, Materials Prepartion; Physics; Defects; Applications

SEMICONDUCTORS A N D SEMIMETALS. VOL. 14

CHAPTER 1

Photopumped 111-V Semiconductor Lasers N . Holonyak, Jr., and M . H . Lee I. 11. 111.

IV.

V.

VI. VII.

INTRODUCTION . . . . . . . . . . . . . . LASERTHRFSHOLD REQUIREMENTS. . . . . . . . PHOTOPUMPING METHODS . . . . . . . . . . BINARY III-V SEMICONDUCTORS . . . . . . . . . I . Gallium Arsenide . . . . . . . . . . . . 2. Indium Phosphide . . . . . . . . . . . . ALLOYIll-V SEMICONINCTORS . . . . . . . . . 3. Indium GaIlium Phosphide . . . . . . . . . 4. Indium Gallium Phosphide Arsenide . . . . . . . 5 . Gallium Arsenide Phosphide (GaAs,-,P, untl GaAs,-,P,:N) 6. Indium Gallium Arsenide and Indium Arsenide Phosphide . CARRIER LIFETIME. . . . . . . . . . . . . CONCLUSIONS . . . . . . . . . . . . . .

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. . .

. . .

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I 4 6 10

10 74 76 28 35 3X

52 55

63

1. lntroduction

Since the first demonstration of stimulated emission in semiconductors p-ii junction lasers have been the subject (GaAs and GaAs,-,P,, 1962).1.2.3 of extensive research and development. Of all lasers, the semiconductor laser is smallest and is unique in that the active population, excess electron-hole pairs, can be injected directly with current, thus making the junction laser capable of direct modulation as is desirable, for example, in communication applications. Besides electron-hole pair injection via current in a homojunction or heterojunction structure. assuming such structures can be built for the material in question, excess electron-hole pairs can be introducted into a semiconductor laser sample by two other major methods of excitation: (1) electron-beam bombardment (cathodoluminescence, CL) and (2) photopumping or photoexcitation (photoluminescence, PL). Electron-beam bombardment and optical pumping provide convenient methods of excitation for study of laser effects in semiconductor materials in

' R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J . Soltys, and R. 0.Carlson, Phys. Reu. Lett. 9, 366 (1962). M. 1. Nathan. W. P. Dumke, G. Burns, F. H . Dill, and G . Lasher, Appl. Phys. Lett. 1, 62 (1962). N. Holonyak. Jr.. and S. F. Bevacqua. Appl. P l i j ~Lett. . 1, 82 (1962).

1 Copyright @ 1Y79 by Academic Press. Inc. All right of reproduction in any form reserved. ISBN 0-12-752114-3

2

N. HOLONYAK, JR., A N D M. H. LEE

which direct carrier injection (i.e.,junction injection) is not readily realizable or may not be desired. These excitation methods afford a means to study homogeneous samples of variable but controlled impurity content. This is advantageous in determining the behavior of energy bands and impurity levels and their effects on laser transitions without the complications of the impurity gradients inherent in junction structures. For example, it is possible to prepare and excite homogeneous samples of arbitrary doping levels, compensated or uncompensated, that cannot be readily prepared and excited in other structures, not even in heterojunction structures-not to mention that heterojunctions can be built in only a limited number of crystal systems. In any case, the effect ofimpurities can be more easily studied in homogeneous samples, and for this reason, if no other, electron-beam bombardment and photopumping are valuable. Of these two methods of excitation, electron-beam bombardment possesses the advantage of considerable primary beam penetration into the laser sample and thus provides relatively large-volumeexcitation. This is advantageous in, for example,TV projection device^.^ Because of its high vacuum and high voltage requirements, however, electron-beam bombardment is cumbersome compared to photopumping. Accordingly, in this review we are concerned with the latter, not so much as a semiconductor laser source' but as an analytical tool. We mention that it is questionable if any form of semiconductor laser is as practical, convenient, and useful as is a junction structure, but not for all purposes-materials analysis being perhaps the foremost exception. Photopumping (band-to-band excitation) is a particularly convenient method of exciting a homogeneous sample and has been achieved by several methods: (1) with a junction laser, the diode serving as a lamp,6g7(2) with gas lasers,8-" (3) with solid-state lasers,I2 (4) with gas discharge lamps,'j and, 0. V. Bogdankevich, Kuunfouaya Elcktron. (Moscow) No. 6 (18). 5-22 (1973) [English transl.: Sou. J . Quanf. Electron. 3(6), 455 (May-June, 1974)l. J. A. Rossi, S. R. Chinn, J. J. Hsieh, and M. C. Finn, J . Appl. Phys. 45,5383 (1974). R. J . Phelan. Jr., and R. H. Rediker, Appl. Phys. Letf.6,70 (1965). ' N. Holonyak, Jr., M. D. Sirkis, G. E. Stillman, and M . R. Johnson, Proc. IEEE54, 1068 (1966). * M. R. Johnson, N. Holonyak, Jr., M. D. Sirkis, and E. D. Boose, Appl. Phys. Lett. 10, 281 (1967). D. L. Keune ef al., J . Appi. Phys. 42, 2048 (1971). l o D. R. Scifres et al., Solid-Stare Electron. 14,949 (1971). 1 1 R. D. Burnham, N. Holonyak, Jr., D. L. Keune, and D. R. Scifres, Appl. Phys. Lett. 18, 160 (1971). N. G . Basov, A. Z . Grasyuk, 1. G. Zubarev, and V. A. Katulin, Fiz. Tverd. Tela 7, 3639 (1965)CEnglishtransl.: Sou. Phys.-Solid Sfate7,2932 (19661. l 3 R. J . Phelan, Jr., Proc. IEEE54, 1119 (1966).

1. PHOTOPUMPED

111-V SEMICONDUCTOR LASERS

3

more recently, ( 5 ) with dye lasers14 and optical parametric oscillators.s.'s Laser sources are especially useful as optical pump sources because of their high emission intensity, the ease with which their outputs can be focused, and the potentially good photon energy match between pump and sample, which minimizes heating. Perhaps the main problem in photopumping is the limited depth of penetration associated with the high absorption coefficient of direct-gap laser samples (l/n 5 1 pm). Except in special cases,I5 this leads to excitation of the sample near the surface and laser operation from edge to edge along the sample surface. Although this appears to be a limitation, it is not a very serious problem because the sample can be made very thin (1-5 pm) and can be heat sunk between a diamond or sapphire window and a compressible indium heat sink16; in this configuration the sample easily can be pumped to high levels. A thin sample can be pumped sufficiently to overcome surface effects, or in many cases these effects can be reduced or even suppre~sed.~.'' In any case, it is no problem to photopump and lase a thin sample, provided the material is capable of operating as a laser. For a suitable material, it turns out that the edge-to-edge path length of a thin sample need not be as long as even 10 pm for photopumped laser operation. Below we discuss a number of results that have been obtained by means of photopumped laser operation of Ill-V semiconductors. Among these are the demonstration of: (1) laser operation of GaAs on transitions involving the Ge acceptor" and the shallower or deeper Si acceptor"; (2) GaAs laser operation at energies as high as 60meV above the energy gap (7880& nd 1019/cm3)20;(3) dynamic Moss-Burstein shift in optical absorption induced by excess carriersz1; carrier lifetime shortening owing to stimulated (5) laser operation on emission in GaAs," GaAsl-xP,,22 and Inl~,GaxP23;

-

I'

Is

M. H. Lee, R. J. Nelson. and N. Holonyak, Jr. (unpublished data). S. R . Chinn, J . A. Rossi. C. M. Wolfe. and A. Mooradian, IEEE J . Quanium Electron. QE-9. 294 (1973).

N . Holonyak, Jr., and D . R . Scifres. K w . Sci.Insrrum. 42, 1885 (1971). P.D. Ddpkus el a / . , J . Appl. Phys. 41. 4194 (1970). R. D. Burnham, P. D. Dapkus. N . Holonyak, Jr.. and J. A. Rossi, Appl. Phys. Lett. 14, 190 (1969). 19

*'

J . A. Rossi, N . Holonyak, Jr., P. D . Dapkus. R. D . Burnham, and F. V. Williams, J . Appl. Phys. 40, 3289 (1969). P. D. Dapkus, N. Holonyak, Jr., J. A. Rossi, F. V . Williams. and D. A. High, J. Appl. Phys.

"

40. 3300 (1969). P. D . Dapkus, N . Holonyak, Jr.. R. D. Burnham, and D. L. Keune, Appl. fhys. Lett. 16, 93 ( I 970).

'' M . H . Lee, N . Holonyak, Jr., J . C. Campbell. W. 0. Groves, M. G. Craford, and D. L. 23

Keune, Appl. Phys. Let/. 24, 310 (1974). J. C. Campbell. W. R. Hitchens, N. Holonyak, Jr., M . H. Lee, M . J . Ludowise, and J . J . Coleman. A p p f . Phvs. Lett. 24, 327 (1974).

4

N . HOLONYAK, J R . , A N D M. H . LEE

the N-trap recombination transition in GaAs,,P, ,24 includingat the directindirect transition (x x 0.46,77"K) where the crystal behavior is indirect25; (6) the quasi-indirect behavior of above-gap (or below-gap) N isoelectronic trap states (laser states) in direct GaAs,-,P,26; and (7) the laser operation of In,p,Ga,P at wavelengths as short as 5500 A (green).27Many of these results, all in 111-V materials, have not yet been realized experimentally in either homojunctions or heterojunctions. 11. Laser Threshold Requirements Before considering some of the various laser data that have been obtained on photopumped 111-V semiconductors, we consider first the conditions required to establish stimulated emission and photon gain in a laser sample, which ordinarily is a direct-gap material-the only exception to date being N-doped GaAs,,P, adjusted in composition to be at the direct-indirect transition (x w 0.46,77"K).2s-28That it might be possible to achieve laser operation in a semiconductor can be argued from very elementary considerations. Assuming a sufficient population of excess electron-hole pairs (here produced by photopumping), we might expect that a beam of photons in the sample of proper energy would stimulate recombination and be exponentially amplified instead of absorbed. If this process exceeds, or can be made to exceed, the photon absorption in the material and also the photon escape from the crystal, it is possible to operate a semiconductor sample as a laser. No matter how photons are lost, absorption or transmission out of the sample, the recombination process producing the photons of interest must be capable of exceeding the photon loss rate for it to be possible to establish appreciable stimulated emission. These simple ideas can be converted readily into convenient expressions describing the conditions for stimulated emission in a s e m i c o n d ~ c t o r . ~ ~ The photon loss rate in a sample with Fabry-Perot reflecting edges is given by29 l/tc = (c/q)[a + (l/X)Ml/R1R2)]

(c/q)aeff

(1)

N. Holonyak, Jr., D. R. Scifres, R. D. Burnham, M. G. Craford, W. 0. Groves, and A. H. Herzog, Appl. Phys. Lett. 19,254 (1971). N. Holonyak, Jr., R. D. Dupuis, H. M . Macksey, G. W. Zack, M. G. Craford, and D. Finn, IEEE J . Quantum Electron. QE-9, 379 (1973). 2 6 M. H. Lee, N. Holonyak, Jr., J. C. Campbell, W. 0. Groves, and M. G. Craford, J. Appl. Phys. 45, 1775 (1974). *' H. M . Macksey, M. H. Lee, N. Holonyak, Jr., W. R. Hitchens, R. D. Dupuis, and J. C. Campbell, J. Appl. Phys. 44, 5035 (1973). 'LI N. Holonyak. Jr.. etal.. J . Appl. P l ~ y s44, . 5517 (1973). 29 B. A. Lengyel, "Lasers," 2nd ed., pp. 61-65. Wiley, New York, 1971. 24

''

1. PHOTOPUMPED

5

Ill-V SEMICONDUCTOR LASERS

where c is the speed of light in vacuum, q is the crystal index of refraction ( 3 3 , a is its absorption coefficient, I, is the sample or cavity length (sample edge-to-edge width), and R , and R , are the sample reflection coefficients at the two edges. For q z 3.5 and an edge-to-edge sample configuration, R , = R , z 0.31. The absorption in the sample can be fairly small if the carrier population is inverted and the electron and hole quasi-Fermi levels are located well into each band edge, or well into appropriate impurity tail states if dense enough. For sufficient pumping (i.e., population inversion E,, - E F p> hv E,, and low absorption loss) and a typical sample width of I, 25 pm, the effective loss coefficient aeffis due almost totally to photon transmission loss at the edges of the sample, giving from (1)a,,, z 470 cmCorresponding to this loss, the photon lifetime in the sample is rc sz 2.5 x sec. In this range, or for shorter samples and still shorter times, the photon lifetime t, may start to compete with intraband scattering times and lead to inhomogeneous rather than homogeneous line broadening. A small semiconductor resonator obviously has a poor Q, which is even worse if the cavity is shorter in length or if much absorption exists in the sample, as is possible if only part of the edge-to-edge length of the sample is pumped. It is interesting to note that a semiconductor sample as short as 1 pm (the thickness dimension of a thin platelet) has been operated as a photopumped laser,30and ordinarily it is a routine matter to photopump and operate as a laser a 10-pm-long, thin sample of, for example, GaAs, GaAs, -,P,, or In,-,Ga,P. Corresponding to the unusually short length of 1 pm, aeff lo4 cm-', and this can be exceeded by the gain process in the sample. For laser operation, the loss a,,, expressed by (1) must be overcome by gain, a,. Regarding the recombination transition as a simple two-level system,29 we obtain for the distributed gain a, near the line center (vo) +

-

-

'.

-

-

a, = (r.'/8nq2v; A v T ) N ~ ~ ,

-

(2)

where Av is the half-width of the spontaneous emission, r is the usual spontaneous electron-hole lifetime, and N,, is the density of excited states or impurities. For the type of crystals of interest here, for example, GaAs emitting at 8400 8, (3.57 x l O I 4 sec-') in a line of width -200 8, (8.5 x 10l2 sec-I) and with electron-hole lifetime T sec, a,

--

- 2.7

-

x 10-15Ne,

-

cm-I.

(3)

If a, is to exceed aeff 470 cm- ( I , 25 pm), as required for laser operation, then from (3) N,, 1.7 x 10"/cm3. This is a reasonable magnitude for

'' G . E. Stillman, M. D. Sirkis, J. A. Rossi. M . R. Johnson, and N . Holonyak, Jr.. Appl. Phys. Len. 9, 268 (1966).

6

N . HOLONYAK, JR., AND M. H . LEE

the density of excited states or impurities needed for stimulated emission for the sample and parameters chosen here. Sufficiently thin samples can easily be photopumped to well beyond this level, in fact, depending upon the sample geometry and pumping source, to N,, 10T9/cm3.Accordingly, photopumping can be a very powerful and useful excitation technique, particularly for the evaluation of laser materials. As already mentioned, photopumping has led to certain semiconductor laser results which still have not been duplicated via other excitation schemes, including via junction injection.

-

111. Photopumping Methods

The experimental apparatus used for photoluminescence studies consists basically of a light source, the experimental sample, and a detection system for the recombination radiation. While the accuracy of the data depends on the properties of the detection system, the range and the types of phenomena that can be observed are largely functions of the excitation technique and the method of sample preparation. In earlier photoluminescence studies, light from a quartz-iodine lamp, a mercury lamp, or other incoherent source has been passed through a monochromator or other appropriate filters and used for excitation. Although an incoherent excitation source is adequate to produce low-level luminescence in light-emitting semiconductors, it is not ordinarily a convenient, if adequate, source to create a condition of population inversion (EFn- E F p> hv E,). A typical density of electron-hole pairs that can be produced using low-power lamp sources is 1012/~m3,31 orders of magnitude too small for stimulated emission to be appreciable. These light sources are thus generally used in experiments where only phenomena associated with low excitation conditions are to be observed. High-power flashlamps have been used to produce stimulated emission in semiconductors,' but these sources do not possess the convenience of those now commonly in use. One of the higher-intensity and more easily controlled light sources for photoexcitation of semiconductors ( E , < 2.00 eV) is, quite appropriately, a p-n junction l a ~ e r .Properly ~.~ constructed laser diodes can produce typically watts of monochromatic radiation. This is adequate power to excite small samples, which ideally should be thin and arranged to capture all of the diode output in a narrow path across the sample. This can be accomplished by attaching a thin (1-5 pm) polished and etched semiconductor sample directly to the Fabry-Perot face of the junction diode with either vacuum grease or The platelet is cleaved to a width comparable to that of the active region of the diode. This technique of sample attachment and pumping

-

3’

B. Tuck, J. Phys. Chem. Solids 28, 2161 (1967).

-

1.

7

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

permits nearly uniform volume-excitation of the sample along its width, and the simplification of no intervening optical components between the sample and the extremely narrow emitting region of the diode. The use of laser-diode excitation sources has produced a number of important results, some of which are discussed later. The utility of laser diodes in this application is limited, however, by the energy of the emitted photon. The highest-energy pn junction laser reported thus far radiates at 2.10 eV (5900 A),32 but in more developed crystal systems laser operation has not been achieved above -2.00 eV.33This is a serious limitation on the range of materials that can be pumped with a diode laser. Gas lasers have become perhaps the most widely used light source for photoluminescence and laser studies. The most common of these are He-Ne, Ar', and N, lasers. Commercial He-Ne and Ar' lasers are available which can be operated mode-locked with several watts or more of peak output power, which is sufficient excitation for the laser operation of most thin semiconductor samples. Nitrogen lasers and newly developed Ar lasers can produce even higher peak powers. The fact that properly designed gas lasers can be mode-locked is of advantage for photoexcitation experiments : pulse excitation reduces heating problems in addition to increasing the peak excitation level. Even the wide wavelength separation between strong lines of gas lasers is diminishing as a limitation on certain photoluminescence experiments as the availability increases of tunable dye lasers pumped with gas lasers. In order to study the laser properties of a semiconductor sample with a moderate power gas laser pump source, the samples should be quite thin and well heat sunk. A mounting technique which has proved to be successful is shown in Fig. 1.16 The experimental material is polished and etched to a thickness of 1-5 pm and is cleaved into samples of width 10-50 p n ; these are compressed as shown into In wetted into a Cu heat sink. A sapphire or S i c window may be compressed onto the In for additional heat removal from the samples (Fig. 1). The 1-5-pm sample thickness insures that the excess carrier distribution is fairly uniform throughout the sample depth; as already mentioned the effects of the surface can be overcome by wide-gap epitaxial window^^.'^ or by sufficient pumping. Thin samples are also intrinsically good optical waveguides and, when mounted as shown in Fig. 1, the waveguide losses are further reduced because of the In metal at the sample back surface and folded up near the edges. The minimum diameter to which the pump beam can be +

'*

W. R. Hitchens, N . Holonyak, Jr., M . H. Lee, J. C. Campbell, and J . J . Coleman, Appl. Phys. Lett. 25, 352 (1974). 3 3 J . J. Coleman. W. R. Hitchens. N. Holonyak. Jr.. M. J . Ludowise, W. 0. Groves, and D. L. Keune, Appl. Phvs. Lett. 25, 725 (1974).

8

N . HOLONYAK, JR., AND M. H. LEE

/

Window

FIG.1. Sandwich heat sink fixture for optical pumping of thin semiconductor samples through either a diamond or s a p phire window. The samples are compressed into indium with the window. Indium seizes and holds both the samples and the window. (After Holonyak and Scifres.’6,

focused depends on the type of lens used but is typically 10 pm, comparable to the sample width. The laser cavity of the sample, bordered by its cleaved edges (which act as Fabry-Perot mirrors) and its front and back surfaces, can thus be fully pumped at the maximum power density available. Other sample preparation techniques for laser studies of semiconductors by optical pumping have also been used. Some are basically variations of the above technique in that the thin optical waveguide geometry is maintained. The front and perhaps the back surfaces of the “waveguide” are bounded by h e t e r o l a y e r ~ . ~ ,For ~ * ’excitation ~ with the photon energy just larger than the band gap, thicker samples can be used because of the lower pump a b ~ o r p t i o n . ~ . ”If*a~high-power ~ laser is used for the pump, such as a Q-switched Nd:YAG laser, the requirement for the samples to be thin can be eliminated ~ompletely.~ Besides serving simply as an excitation source, mode-locked gas lasers are useful also for carrier lifetime measurements at both low and high levels (spontaneous and stimulated recombination). While real-time techniques are adequate to observe carrier lifetimes in the tens of nanosecond range, shorter lifetimes are difficult or expensive to determine directly. The well-established optical phase shift method of lifetime m e a s ~ r e r n e n twhich , ~ ~ makes use of the delay between the input excitation (mode-locked gas laser) and the output recombination radiation, is a convenient and now commonly used method to N

J. A. Rossi and S. R. Chinn, J . Appl. Phys. 43,4806 (1972). J . A. Rossi, S. R. Chinn, and A. Mooradian, Appl. Phys. L e f f .20, 84 (1972). 36 E. A. Bailey and G. K. Rollefson, J . Chem. Phys. 21, 1315 (1953); R. J. Carbone and P. R. Longaker, Appl. Phys. Left.4,32 (1964);H. Merkelo, S. R. Hartman, T. Mar, G. S. Singhal, and Govindgee, Scienre 164, 301 (1969); D. L. Keune, N . Holonyak, Jr., P. D. Dapkus, and R. D. Burnham, Appl. Phys. Leu. 17. 42 (1970); C. H. Henry and K. Nassau, Phys. Rev. 5 1, 1628 (1970).

34

35

1.

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

9

Dewar

PJ Silicon

detector

Vector voltmeter

I

signalBandpass filters

FIG. 2. Experimental apparatus used for measurement of photoluminescence and carrier lifetimes in semiconductors by the optical phase-shift method. The 5145-A output of the Ar+ laser excitation source is mode-locked at 140 x lo6 Hz. From this repetition rate (o= 2n x 140 x lo6 Hz)and the measured phase shift 0, the luminescence lifetime T is given as T = (I/w) tan0 = 1.14tanO nsec. (After Lee rt ~ 1 . ~ ’ )

determine short lifetimes. This method for determining luminescence and carrier lifetime is based on the measurement of the phase difference between the fundamental Fourier component of the recombination radiation and that of the repetitive excitation from a source such as a mode-locked laser. Under the simplest approximations the luminescence lifetime z is related to the measured phase angle 0 by36 t = (l/w) tan 0,

(4)

where o is the angular frequency of the repetitive excitation source. The measured decay time is affected by many factors, some of which will be discussed below. A typical and convenient optical excitation and phase shift measurement apparatus is shown in Fig. 2.9*37 The Ar’ laser excitation source is operated at 5145 A and delivers 0.2-nsec pulses (peak power -5 W) at a repetition rate of 140 x lo6 Hz. At this frequency (l/w) = 1.14 nsec, giving T = ”

1.14tanO nsec.

(5)

M . H. Lee. N. Holonyak, Jr., R.J. Nelson, D. L. Keune. and W. 0. Groves. J . Appl. Phys. 46,323 (1975).

10

N. HOLONYAK, JR., AND M. H . LEE

As shown in Fig. 2 a small fraction of the excitation signal is detected by a Si photodiode and applied to the reference channel of a vector voltmeter. Excitation light scattered from the sample and sample fluorescence are individually passed through a 0.25-m monochromator, are detected by a photomultiplier, and applied to the signal channel of the vector voltmeter which determines the phase difference 8 between the fundamental Fourier components of the excitation and sample luminescence signals. Phase angles approaching 8 7r/2 obviously lie in an inconvenient measurement range, while values from 0 to 2n/5 correspond to times T = 0 to 3.5 nsec, a convenient range for the 111-V semiconductors of interest here.

-

IV. Binary 111-V Semicooductors

Besides being the first 111-V semiconductors to be synthesized and employed in practical applications, binary materials such as Gap, GaAs, GaSb, InP, InAs, InSb, etc., continue to grow in interest and in use. Probably the most used binary 111-V semiconductors are GaAs and Gap, but of these two only the first is a direct-gap material and of use in lasers. For the case of binary 111-V materials, stimulated emission has been demonstrated also in GaSb, InP, InAs, and InSb, but here we shall be concerned only with GaAs and InP, since these have been of most concern in photopumping experiments. 1 . GALLIUM ARSENIDE In addition to being one of the first semiconductor materials to exhibit stimulated emission, GaAs remains the most used injection laser material because, for one reason, it is readily fabricated sandwiched between wide-gap Ga,,AlXAs layers into room-temperature cw heterojunction lasers3’ (Jth5 700 A / c ~ * ) Hence, . ~ ~ we consider first GaAs. While the excitation mechanism differs for current-driven devices and optically pumped samples, the recombination processes are nevertheless determined in both cases by the properties of the material. It is therefore useful to compare the laser performance of GaAs for these two methods of excitation. Recent experiments by Rossi and co-workers5 on single- and double-heterojunction samples make possible a direct comparison of GaAs laser operation via current injection or photoexcitation. For this comparison Zh. 1. Alferov. V. M. Andreev, V. I. Korol’kov, E. L. Portnoi, and D. N. Tret’yakov, Fiz. Tekh. Poluprouodn, 2. 1545 (1968) [English transl. : Sou. Phys.-Semicond. 2, 1289 (19691; 1. Hayashi, M. B. Panish, and R. K. Reinhart, J. Appl. Phys. 42, 1929 (1971). 39 G. H. B. Thompson and P. A. Kirkby, Electron. Lerr. 9, 295 (1973); H. C. Casey, Jr., M. B. Panish. W. 0. Schlosser. and T. L. Paoli, J . Appl. Phys. 45, 322 (1974). 38

1.

11


an optical parametric oscillator is employed which allows the pump photon energy E , to be tuned between the band-gap energies of GaAs and the G a l - x AI,As heterojunction window on the sample. The pump beam is focused onto the Ga,_,AI,As surface by crossed cylindrical lenses which make it possible to excite the GaAs sample with a line image extending across its length from edge-to-edge. The results of the study are summarized in Table I.5 Note first that the comparison of the data for optically versus electrically excited sample is of limited significance since, among other factors, the generated excess carrier density is quite different for the two excitation schemes. The results are influenced by several factors which, although not amenable to simple calculations, are known generally to degrade the performance of the optically pumped lasers relative to the corresponding injection devices. These include effects which occur when the pump photon energy E, is close to the band-gap energy of GaAs. The most important are: (1) the incomplete absorption of the optical excitation, especially in the double heterojunction structure of Table I, and (2) the decreasing absorption that occurs with increased excitation level for optically pumped samples. The latter effect,21which influences the differential efficiency qdiff (Table I), will be discussed in more detail below. For the study leading to Table I s the experimental conditions also favor to some extent electrical excitation. It is thus surprising that the performance of the optically pumped sample compares so favorably with the junction injection case, especially for the single heterostructure device. The TABLE I PERFORMANCE OF OPTICALLY PUMPED A N D INJECTION GaAs LASERS'

Optical

Injection

Single heterostructure Double heterostructure Single heterostructure Double heterostructure

60 60

26 5.6

20.7 5

19 4.75

20

35

24

5

6

28

21

3.2

After Rossi et aL5 For optical pumping qdifr = !P,,,'(P," x (hvP/hs): qdifr= ( P / h v , ) ( I - I,,), where hv, is the pump photon energy and is the sample lasing photon energy. 'Same as footnote h but evaluated for P,,, = I , , = 0. Defined as the ratio of PJP,,.

12

N. HOLONYAK, JR., A N D M. H. LEE

> 20% quantum efficiency for the single heterojunction structure is nearly a factor of two larger than the best value previously reported for optically pumped GaAs.” While it is clear from the above results that optical pumping of 111-V compounds does not rival electrical injection as a practical means for obtaining laser operation, photoluminescencedata nevertheless have clari6ed many issues concerning the basic recombination processes that occur in lightemitting semiconductors.One of the important concepts in the understanding of recombination processes in 111-V semiconductors is that of the existence of quasi-equilibrium within the conduction and valence bands4’ It is known, for example, that intraband scattering times in GaAs are -lo-’’ set:' which is orders of magnitude faster than radiative recombination times. It is not immediately obvious, however, that scattering times within the donor bandtail of heavily doped GaAs should also occur rapidly; i.e., that quasiequilibrium should exist in the bandtail. Photoluminescence data on heavily donor-compensated p-type GaAs: Zn :Sn crystals have resolved this question4’ The GaAs crystal used for experimental measurement is grown from a Sn solution doped with Zn. The net acceptor concentration at room temperature is estimated” to be p x n, 102’/cm3 and the donor concentration nd 1019/cm3.A thin platelet of this sample is attached to a diode pump source (GaAs or GaAs,-,P,) and excited at 77°K. It is found that the laser operation of the platelet occurs at 1.41 eV. Since the band gap of GaAs is 1.51 eV (77°K) and the Fermi level of the compensated p-type sample is located near the valence band edge, the absorption event clearly occurs between the valence band and the donor tail states for the case of the low-energy pump (a GaAs laser diode, E , z 1.45 eV).42The fact that the laser energy of the sample does not vary in spite of band-to-band or valence band-to-donor pumping shows that thermalization processes, which lead to redistribution of the electrons to lower energies in the tail states, are fast relative to the radiative lifetime. The fast thermalization of carriers in the donor tail states of GaAs doped with shallow donors differs considerably from the behavior found in closely compensated Si-doped GaAs.43-45 Silicon is an amphoteric dopant in

-

-

W . Shockley, “Electrons and Holes in Semiconductors, p. 308.” Van Nostrand-Reinhold, Princeton, New Jersey, 1950. 41 E. M. Conwell and M . 0. Vassell, Phys. Rev. 166,797 (1968). 4 2 D. R. Scifres, N . Holonyak, Jr., P. D . Dapkus, and R. D. Burnham, J. Appl. Phys. 42,896 (1971). 43 M . G. Craford, A. H. Herzog, N. Holonyak, Jr., and D . L. Keune, J . Appl. Phys. 41,2648 (1970). 44 D. Redfield, J . I. Pankove, and J. P. Wittke, Bull. Am. Phys. SOC.14, 357 (1969); D . Redfield and J. P. Wittke, ibid. 15, 525 (1970). 4 5 D. Redfield. J. P. Wittke, and J. I. Pankove, Phys. Reu. B 2, 1830 (1970). 40

1, PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

13

GaAs,46p48serving as a shallow donor on a Ga site and as an acceptor on an As site. The distribution of the Si impurity on the two types of lattice sites depends on crystal growth conditions. Photoluminescence experiments on heavily doped closely compensated GaAs :Si have shown that quasi-equilibrium does not appear to exist in the band tail At lower pumping levels the rise and decay time of the radiation from transitions between the band tail states is nearly 1 pec, indicating that the thermalization times are quite large. These rise and decay times shorten appreciably, however, for higher-level pumping and laser operation.49 The relatively long time constants observed in Si-doped GaAs are characteristic of the Si. In addition, the close compensation of the crystals grown via liquid phase epitaxy (LPE) plays an important r ~ l e ~ ’ Recombination .~ in closely compensated GaAs appears to be analogous to donor-acceptor pair recombination transitions. For example, the time-resolved recombination spectra of closely compensated G a A ~ : sbear i ~ ~some resemblance to those obtained on the donoracceptor pair transition52p54in lightly doped GaAs, suggesting that similar recombination transitions are involved. For well-resolved discrete donoracceptor pair transitions, it is known that the recombination probability decreases exponentially with the separation of the pairs and hence the energy of the tran~ition.~Similarly, in closely compensated GaAs :Si crystals, the transitions are generally far below the band gap in energy, and thus the carrier lifetime should be long. Apart from the complications introduced by impurity compensation in GaAs :Si, the Si acceptor in GaAs has particularly interesting properties. For example, Si introduces two acceptor levels in G ~ A s . ~ ’ , ’ ~Ph , ~oto’ luminescence experiments on diode-pumped GaAs :Si platelets have shown

’

J. M. Whelan, J. D. Struthers, and J . A. Ditzenberger, Proc. Int. Cot$ Phys. Semicond., 6th, Pruyur, p. 943. Academic Press. New York, 1960. 4 7 H. Rupprecht, J . M . Woodall, K. Konnerth. and G . D. Pettit, Appl. Phys. Lett. 9,221 (1966). 48 H. J . Queisser, J. Appl. Phys. 37. 2909 (1967). 49 P. D. Dapkus, Ph.D. Thesis, Univ. of Illinois, 1970 (unpublished). 5 0 Zh. I. AlErov, V. M . Andreev, D . Z. Garbuzov. and M. K. Trukan, Fiz. Tekh. Poluprouodn. 6. 2015 (1972)[Enylish trunsl.: Soi?.Pliys.-Semicond. 6 . 1718 (1973)l. ’’ A. P. Levanyuk and V. V. Osipov. R z . Tekh. Poluprotwin. 7. 1058 (1973)[English trtltlsl.: Sor. Pliys.-SeniiconJ. 7 . 721 (1973)l. 5 2 R. Dingle and K . F. Rodgers. Jr., Appl. Phys. Leu. 14, 183 (1969). 53 R. Dingle, Phys. Reu. 184,788 (1969). 54 For a discussion of donor-acceptor pair recombination transitions in lightly doped GaAs see E. W. Williams and H. B. Bebb. in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 8, pp. 349 351. Academic Press, New York. 1972. See also J. A. Rossi, C. M. Wolfe. and J. 0. Dimmock, Phys. Rev. Lett. 25, 1614 (1970). 5 5 D. G . Thomas, J . J. Hopfield, and W. M. Augustyniak, Phys. Rev. 140, A 202 (1965). 5 6 E. W. Williams and D. M. Blacknall. Trans. AIME239, 387 (1967). ” H. Kressel, J. V. Dunse. H. Nelson, and F. Z. Hawrylo, J . Appl. Phys. 39, 2006 (1968). 46

14

N . HOLONYAK, JR., AND M . H . LEE

that the lifetime of the transition between the conduction band edge and the deeper acceptor level is as long as 10-100 n ~ e cThis . ~ ~is more than an order of magnitude longer than the lifetime of band-to-band transitions in GaAs, and accounts for a sizable fraction of the long radiative rise and fall times of compensated GaAs: Si.43*49In fact, the differences in the measured carrier lifetimes in p-type and closely compensated GaAs: Si may be principally the result of differencesin samples and in experimental methods and not actually fundamental differences. Despite the long lifetime observed in closely compensated or p-type GaAs:Si, photopumped laser operation has been obtained between the conduction band edge and the shallower and the deeper Si acceptor levels.’ Since the first demonstration of photopumped laser operation of these transitions, the Si impurity has been found to be useful also in double-heterojunction Al,Ga,-,As/GaAs lasers. The GaAs:Si crystals used in photopumped laser experiments are grown on (111)-A oriented GaAs seeds to insure that Si is incorporated only into acceptor sites.’’ Figure 3 shows the emission spectrum of a p-type GaAs: Si platelet, grown from a 0.05% Si + Ga melt, lasing simultaneously on both acceptor transitions.’ At lower excitation levels the broad spontaneous peaks of the transitions can be distinguished. Laser operation can also be achieved simultaneouslyon the band-to-band and the higher-energy acceptor transitions. The predominant transition observed depends on the exact geometry of the platelet and its relative orientation with respect to the output of the pump diode. If the platelet is excited completely from edge-to-edge,

’

1.d

GaAs:Si

77

8.8

OK

8.6

eV

rev 8.4

Wavelength (& x ~ O - ~ )

FIG.3. Emission spectra (77°K) of p-type GaAs:Si platelet sample pumped along only a portion of the edge-to-edge cavity length. Note simultaneous lasing on the two (shallower 1, and deeper A,) acceptor transitions. (After Rossi et ~ 1 . ’ ~ )

1.


15

it will tend to lase at shorter wavelengths; pumping over lesser lengths tends to favor longer wavelengths because of the higher losses at shorter wavelengths in unpumped regions." Germanium,just as Si, is an amphoteric dopant in G ~ A s . In ~ ' a Ga site, Ge introduces a shallow acceptor level 35-40meV above the valence band edge.59*60 The radiative recombination time for transitions involving the Ge acceptor is much shorter, however, than for transitions involving the deeper Si acceptor.49 Photopumped laser operation of the Ge acceptor should thus be easier to obtain, as indeed is the case." Figure 4 shows the spectral emission (77°K) of a diode-pumped p-type Ge-doped GaAs platelet (n, > 3 x 10' ’/an3)at various excitation levels. At threshold the platelet operates as a laser on the band-to-impurity transition labeled A2 . As the pump level is increased, the band-edge transition 1, becomes more prominent and can be made to lase; this is the principal laser transition if the acceptor doping is relatively low (< lO"/cm3). In ordinary p-n junctions, the band-edge transition cannot be made to lase because of the difficulty in supplying a sufficient density of excess carriers in lightly doped crystals. While it is clear that many kinds of acceptors in GaAs can be involved in laser operation, given sufficient pump intensity, not all acceptors show this property. The Mn acceptor in GaAs, for example, has not been shown capable of supporting laser operation. Nor for that matter is it an easy matter to operate the Ge acceptor in a laser transition in a p-n junction6' unless a heterojunction structure is employed.62This is a consequence ofthe relatively long carrier lifetime in GaAs doped with amphoteric impurities. The above studies of optically pumped p-type GaAs:Si and GaAs:Ge show that for a certain range of impurity concentrations, depending upon the impurity, laser radiation may result from band-to-band or conduction bandto-acceptor transitions, or both. These observations suggest that such behavior may be common to all relatively shallow acceptors in GaAs. Experimental data on photopumped homogeneous samples of Zn- and Cd-doped G ~ A show s ~ that ~ over a large range of impurity concentrations laser transitions may indeed terminate on either valence band states or acceptor states, depending upon the gain profile of the laser resonator. J. 0. McCaldin and R. Harada, J . Appl. Phys. 31, 2065 (1960). H. Kressel, F. Z . Hawrylo, and P. LeFur, J . Appl. Phys. 39,4059 (1968). 6o F. E. Rosztoczy, F. Ermanis, 1. Hayashi, and B. Schwartz, J . Appl. Phys. 41, 264 (1970). 6 1 Zh. I . AlfErov, D. Z. Garbuzov, E. P. Morozov, and D. N. Tret'yakov, Fiz. Tekh. Poluprovodn. 3, 706 (1960).[English rransl. : Sou. Phys.-Semicond. 3, 600 (19691. 6 2 R. D. Burnham, P. D. Dapkus, N. Holonyak, Jr., D. L. Keune, and H. R. Zwicker, Solid Slate Electron. 13, 199 (1970). "J. A. Rossi, N. Holonyak, Jr., P. D. Dapkus, J. B. McNeely, and F. V. Williams, Appl. Phys. Leu. 15, 109 (1969). 59

16

N . HOLONYAK, JR., AND M. H. LEE

c

8.6

*IL

8.4

Wavelength

8.2

(8 x 1O-j)

-

FIG.4. Spectral emission ofa p-type Ge-doped GaAs platelet (n, 3 x 10’’ m-’)at various pump levels. At 6.5 A applied to the pump diode, recombination transitions occur at I.* (impurity) and Il (band edge). At 7.6 A in the pump diode, sample laser operation occurs at energy El, (impurity), and at 10.4 A the recombination to the band edge becomes more prominent. (After Burnham et ~ 1 . ” )

1. PHOTOPUMPED 111-V

17

SEMICONDUCTOR LASERS

f c

C

b c

1.46

W

c

5 0 1.44 S

a

' ' ' ' "I 1017

,

I

I

I l l 1 1

1018

I

I

I

, I , I I I

I

1

I

I ,

1019

Holes /cm3 FIG. 5. Photon energy at threshold (77 K)of optically pumped p-type GaAs platelets as a function of the crystal impurity concentration. Circular data points (solid curve) indicate laser photon energy at threshold. Triangular data points (dashed curve) show the photon energy of secondary transitions. which may or may not lase (depending upon pumping intensity and geometry). (After Rossi er a/.")

Figure 5 shows the laser photon energy (circular data points) at threshold (77°K) of photopumped p-type GaAs as a function of the room-temperature free-carrier concentration, which is close to the acceptor concentration. The photon energy of the secondary transition is indicated by the triangular data points. These data are most easily interpreted with the aid of Fig. 6, which shows the subthreshold emission spectrum of samples with acceptor concentrations in the range 10'6-10'8/cm3. The spectra show two optical transitions, which are identified by their wavelength (Fig. 6) or energy (Fig. 5 ) as being conduction band to valence band, icv,or conduction band to acceptor, &A.

For the lightest-doped sample considered (4.5 x 1016/cm3),the two emission bands (Fig. 6 ) are well separated and distinct. As the pumping is increased, the sample lases only on the higher-energy transitions, &, as shown by the solid line in Fig. 5. For higher acceptor concentrations, the band-toband transition shifts to somewhat longer wavelengths but remains the dominant laser transition to dopings - 2 x 101'/cm3. In the intermediate doping region 2 x 10"/cm3 5 I L , 5 10"/cm3, the acceptor states tend to dominate the laser recombination transition, although it is possible to lase the crystal also on the band-to-band transition if the platelet geometry is suitably chosen. At high acceptor concentrations ( > 1018/cm3),laser transitions terminate only on the acceptor states. If the doping is increased still further, the acceptor states penetrate deeper into the forbidden gap, and the laser photon energy at threshold shifts to increasingly longer wavelengths (cf., Fig. 5).

18

N . HOLONYAK, JR., A N D M . H . LEE

W

c

c

H

0 t

._ Y)

W

I

8.7

8.5

8.3

8.1

Wovelength ( A x ~ O - ~ )

FIG.6. Relative spectral emission of p-type GaAs platelets for various impurity concentrations. Note recombination transitions to valence band edge and to the acceptor impurity. Simultaneous laser transitions on I.,, and ACA are possible for crystals doped in the range 2 x 1017-101*/~rn3 and pumped in a low-loss configuration. (After Rossi et

As with p-type GaAs, the laser photon energy of n-type GaAs at threshold also shifts with increasing impurity concentration. The shift, however, is not to lower energy but to higher as the Fermi level is “doped” higher into the conduction band, a low density of states band. Laser transition data (77°K) on GaAs extending in doping from quite pure material ( N 1014/cm3)to the highest donor, acceptor, and donor-acceptor (compensated) dopings have been obtained to determine the laser photon energy as a function of impurity concentration.” Depending upon the crystal doping and the method of excitation, GaAs can be operated as a laser (77°K) anywhere from wavelengths longer than 9100 A (1.363eV) to wavelengths shorter than 7880 A (1.575 eV), a range exceeding 1200 A (0.210 eV). The former is possible in highly compensated p-type material; the latter is possible in heavily doped (nd lOI9/cm3)n-type material. This wide range of laser operation has been demonstrated by means of optical excitation of thin samples,’034’ and is confirmed on the low-energy end by the behavior of heavily compensated p-n junctions. The main results for the behavior of laser photon energy as a function of doping are shown in Fig. 7. As the impurity doping (donor, acceptor, or both) in a crystal is decreased, the laser photon energy approaches 1.495 eV (77°K) as shown by the portion of the curve labeled i for “intrinsic.” The 15-meVreduction ofthe laser energy from that of the band gap (- 1.510 eV)

-

-

-

1.


1.6

I

I

I

I

I

I

I

I

I

I

19

2, ) .

P c W

-

0 E

$ 1.5 a L

m

0 yl

J yl

a 0

W

1.4

t 10 4

1016

Impurity

lOl*

1020

Atorns/cm3

FIG.7. Dependence of laser photon energy in GaAs for various impurity concentrations. Samples of n-type are Se. Sn. or Te doped: p-type samples are Cd or Zn doped. Compensated crystal is Zn-Sn. Zn-Te, or Zn-Se doped. Laser photon energy becomes asymptotic to 1.495 eV 2 E , - 0.015 at low impurity concentrations. Top diagrams illustrate recombination processes applying to the various curves. The dashed lines in diagram “i” signify the reduction in electron-hole recombination energy caused by particle-lattice interaction at high excitation levels. The small dashed lines in n, p, and p n indicate impurity tail states. (After Dapkus et

is attributed to electron-hole-lattice (EHL) interaction^,^^*^^ which increase with the excitation level. EHL interactions in GaAs have been discussed in 64

65

N. G. Basov, 0. V. Bogdankevich. V. A. Goncharov, B. M . Lavrushin, and V. Yu. Sudzilovskii. Dokl. Akod. Nouk SSSR 168. 1283 (1966)[English trunsl.: Sou. Phys.-Dokl. 11. 522 (1966fl. J . A. Rossi, N. Holonyak, Jr., P. D. Dapkus. F. V. Williams, and J . W . Burd. Appl. Phys. Lerr. 13, 117 (1968).

20


detail elsewhere.66 The exact mechanism for the reduction of the threshold laser photon energy from that of band gap, however, is still in Nevertheless, it is clear that this reduction does occur. Furthermore the laser photon energy at threshold can be increased if the density of excess carriers can be reduced. This is demonstrated in photopumped ni/n/ni GaAs wafers where laser energies as high as 1.502 eV (77'K) have been obtained.'l The high donor doping in the’n regions helps to confine better the excess carriers in the n region, reduce the surface losses, improve the sample waveguiding properties, and reduce the optical losses. These improvements reduce the pumping requirements, the carrier population, and the degree of EHL interaction. As the doping of n-type GaAs crystal is increased beyond roughly the conduction-band effective density of states N , , the laser photon energy increases correspondingly and becomes larger than the energy gap E, at a nd 1018/cm3(cf., Fig. 7). In the doping range 10'8-10'y donors/cm3, the laser photon energy increases rapidly, commensurate with the higher position of the Fermi level in the conduction band. As indicated in diagram n of Fig. 7, the recombination transition occurs to the hole quasi-Fermi level EFp located near the valence-band edge. It is interesting to note that the transition can occur from high in the conduction band, from a relatively high density of states near the electron quasi-Fermi level EFn,despite the higher reabsorption at large photon energies. The laser emission spectrum of a heavily donor-doped( 10'y/cm3)GaAs platelet is shown in Fig. 8. This is the highest energy laser emission that has been observed in bulk GaAs and extends over a range from EFnto E, or 70 meV. From about the same impurity concentration at which uniform n-type platelets begin to lase at photon energies higher than 1.495 eV, p-type crystals begin to lase at lower energies. This is shown by curve p of Fig. 7 and more clearly by Figs. 5 and 6. The relatively high density of acceptor states near the valence-band edge is sufficient to permit transitions to the acceptor to dominate the recombination process in heavily doped (n, > 1017/cm3)p-type GaAs. In contrast, the more smeared energy distribution of donor states in heavily doped n-type crystals is not sufficiently dense to preclude strong recombination from higher in the band near the Fermi level.

-

-

66

-

-

J. A. Rossi. D. L. Keune, N. Holonyak, Jr.. P. D . Dapkus, and R. D . Burnham, J . Appl. Phys. 41,312 (1970); D . L. Keune. J . A. Rossi, N . Holonyak, Jr., and P. D. Dapkus, ibid. 40,1934 (1969). '' K . L. Shaklee, R. F. Leheny, and R. E. Nahory, Appl. Phys. Letr. 19, 302 (1971 ). '* N . Holonyak, Jr., D . R. Scifres. H. M. Macksey, and R. D. Dupuis, J . Appl. Phq's. 43, 2302 ( 1 972). 69 W. D. Johnston, Jr., Phys. Rer. B 6 , 1455 (1972). 'O W . F. Brinkrnan and P. A. Lee, Phys. Reo. Lett. 31,237 (1973). " P. D . Dapkus, N . Holonyak, Jr., D. L. Keune, and R. D. Burnham, J . Appl. Phys. 41, 5215 (1970).

1.


G a A s , Te - doped (from G a s o l u t i o n ) ,

-

21

10i9/Cm3

Pump G o ( A s P ) , 7200 & TW77'K

1

8.3

8 2

81

Wavelength

80

,

(

&x

i 79

I

7.8

- -

FIG.8. Laser emission spectrum of a heavily doped (n, 1019/cm3)n-type GaAs platelet operated as illustrated by the inset. The mode spacing of 5 A corresponds to a resonator cavity length from platelet edge-to-edge of 125 Itm; Te-doped sample grown from Ga solution. (After Dapkus cf dZ")

-

Thus, given an adequate pumping method, homogeneous n-type samples lase at increasingly higher energy with heavier doping, and homogeneous p-type samples tend to lase at lower energies. For compensated crystals the variation in laser photon energy as a function of doping is given in Fig. 7 by curve p n . The data are actually obtained from compensated p-type crystals (n, > nd) from which the effect of the excess p dopant is eliminated by using curve p. The data thus represent the energies of photons that would be observed from perfectly compensated crystals.

22

N . HOLONYAK. JR.. AND M. H. LEE

While this extension may not be completely justified, it is nevertheless clear that the energy reduction owing to compensation takes place. Furthermore, the energy difference between curves p and p-n should closely approximate the photon energy reduction attributable to the depth that the donor tail states penetrate significantly into the forbidden gap. In heavily doped n-type GaAs the Fermi level can be shifted to as much as 70 meV above the conduction band edge as is evident from the previously discussed laser data. This shift of the Fermi level into the band causes a socalled “Moss-Burstein of the absorption edge to higher energy. The shift occurs because only a small fraction of the conduction band states below the Fermi level participates in the absorption process. A similar Moss-Burstein shift can also be observed if the quasi-Fermi level is moved into the conduction band during sample excitation. This effect can be inferred indirectly from bandfilling in p n junctions,74 but has been seen directly in photoexcited platelets.21 The dynamic Moss-Burstein shift is not only an interesting phenomenon in itself but can be the basis for other experiments such as measurement of carrier lifetime shortening due to stimulated emission. As mentioned in the previous section, most lifetime data on recombination processes shorter than several nanoseconds are usually obtained using the optical phase-shift rnethod.j6 This is a reasonably direct method and can be quite accurate. An accurate dependence of lifetimes on excitation levels is more difficult to obtain, however, owing to pecularities of this method of measurement. Fortunately, this dependence in both the high-level spontaneous and stimulated recombination regimes can be obtained by analyzing the dependence of the Moss-Burstein shift of the absorption, at a given energy, on the level of optical excitation.’ This is accomplished by means of a combined excitationtransmission measurement. While this indirect method of measurement is limited in the sense that the approximations required to analyze the data hold only for the conditions of the experiment, the trend in lifetime decrease with excitation that is observed can be applied to the behavior of the average carrier lifetime in any homogeneous GaAs sample, and in other light emitting semiconductors as well. The excitation-transmission lifetime measurement is made on an n + / n / n + epitaxial wafer that is polished and etched to a total thickness of 7 pm. The center active region is -4 pm thick with a donor concentration of nd z 2-3 x lO1’/cm3. The n + layers are equal in thickness and doped sufficiently (nd z 3 x lO’*/cm3) to produce a Moss-Burstein shift in absorption to well

-

’

-

l2

T.S . Moss, Proc. Phys. SOC.B67,175 (1954).

” E. 74

Burstein, Phys. Reo. 93, 632 (1954). D. F. Nelson, M. Gershenzon, A. Ashkin, L. A . D’Asaro. and J . C. Sarace, Appl. Phys. Leu. 2, 182 (1963).

1.

23


above E,. The pump diode energy ( 1S33 eV, 77°K) is chosen to lie below the absorption edge of the n+ layer so that essentially all of the absorption occurs in the lightly doped region of the wafer. The power transmitted through the platelet is measured with a large-area photodiode. The n + / n / n + sample structure minimizes carrier loss at the n + / n boundaries so that nonradiative recombination becomes only a minor factor. In addition, low “surface” losses promote a rather uniform excess carrier distribution in the active layer.” This uniformity is particularly important because it leads to approximations that simplify the calculation of the carrier lifetime from the absorption data. Data obtained from transmission measurements on the n + / n / n + sample are shown in Fig. 9. The percent transmission is seen to increase with the incident power. The absorption coefficient, which is calculated from the transmission data by assuming that the absorption takes place uniformly in the 4-pm-thick n region ofthe sample, is seen to decrease initially very rapidly with increasing pump power. Above the laser threshold of the wafer (as observed in the emission spectrum), the absorption decreases less rapidly and is approximately constant near the highest powers used in the experiment. This saturation indicates that the quasi-Fermi levels tend to lock at the onset of laser operation. If the carrier concentration is assumed to be relatively constant spatially in the active region, the density of carriers An can be determined as a function of the absorption coefficient. The average carrier lifetime s is given simply as T = I

30 t

EV(An/P) I

(6) I

E(purnp1 = 1.533eV E(GaAs recomb. rod.) =1.49?eV 77 “K

-

3

E

V

-6-

-0,-

-

-

U

C

0

;2 0 E -

-4

za,

P

-

.L c

2 10 a, ? $ -

-2

*

-

P

a,

F

-

0 0

0

I

0

0.5

I

1.0

I

1.5

n

oa

2 .o

Incident Power ( W )

FIG.9. The relative transmission and the absorption coefficient for an n * / i i ; i i + GaAs sample as a function of incident power obtained from a low-energy GaAs, -,P, pump diode. Note saturation at highest powers owing to stimulated recombination of carriers. (After Dapkus et a / . ’ ’ )

24

N. HOLONYAK, JR., AND M. H. LEE

77 O K

= -1.0

I

Absorbed Power (mW)

1(

FIG. 10. Carrier lifetime and excess carrier concentration versus absorbed power of an n i . n / n f GaAs sample exhibiting the characteristics of Fig. 9. Note change of slope in carrier lifetime T and the tendency to saturation of excess concentration An when stimulated emission begins. (After Dapkus et ~ 1 . ” )

where I/ is the sample active volume, E the energy of the incident photon, and P the absorbed power. The generation rate during sample excitation is assumed to be constant since the pulse width of the pump-diode laser output is much longer than the carrier lifetime. The results of these measurements are shown in Fig. 10.’ The curves for T and An are related by Eq. (6) and exhibit two regions, one dominated by spontaneous recombination and the other stimulated. Below 300 mW, corresponding to spontaneous recombination, t decreases approximately as the square root of the absorbed power, as expected from a steady-state analysis of direct recombination under high-level condition^.'^ In the region of stimulated emission, which occurs at an absorbed power P 2 300 mW (Fig. lo), z varies inversely with P . This is clear from Eq. (6) since An is nearly constant above laser threshold. These data show quite clearly the effect of stimulated emission in speeding up carrier recombination in GaAs. Optical phase-shift lifetime measurements on GaAs,p,P,22 and on In 1p,Ga,P23, as described later, show similar behavior.

’

-

2. INDIUMPHOSPHIDE

While InP p-n junctions have been operated as lasers as early as 1963,76 little attention has been focused on this 111-V material until recently. GaAs has been most developed, and the need to study InP has not appeared to be urgent. For various reasons, however, InP could be a useful light emitter, l5 l6

J. S. Blakemore, “Semiconductor Statistics,” p. 208. Pergamon, Oxford, 1962. K. Weiser and R. S. Levitt, Appl. Phys. Lett. 2, 178 (1963).

1. PHOTOPUMPED


25

including the fact that it matches the transmission characteristics of glass fibers better than does GaAs. Also InP infrared light emitting diodes can be used to pump several rare-earth phosphors which fluoresce in the visible ~ p e c t r u m . ~ For ~ - ' ~this purpose InP appears in principle to be a better pump than GaAs." In addition, InP can be used with In,_,Ga,P,-,As, to make double heterojunctions that emit at longer wavelengths than InP or GaAs,8'*82and hence in heterojunction form may be developed into quite useful devices. Initial photoluminescence studies on InP have been focused mainly on understanding the origin of the various emission lines that can be observed at low temperatures. The emission spectra of InP and GaAs are similar at low temperature^,'^ and early speculation indicates that these lines are related to exciton recombination p r o c e ~ s e s .The ~ ~ lines , ~ ~ subsequently have been identified as band-to-band processes and donor-acceptor pair recombination transition~.~~-" Laser operation of InP has been obtained by photoexcitation with GaAs laser diode^,^**^^ gas lasers,68 and optical parametric oscillator^.^^ The behavior of the laser spectrum of InP appears similar to that of GaAs and to In,-,Ga,P of fairly low G a content." For example, the laser emission shifts progressively to lower energy with increasing excitation.68-88This shift cannot be attributed to such effects as heating and appears related only to the high excitation levels employed.b8 To be useful in exciting rare-earth phosphors or for other applications, the efficiency of InP diodes must be high. Photoexcitation of InP samples has demonstrated that total power-conversion efficiency of up to 49; at 300°K can 77

S. V. Galginaitis and G. E. Fenner, Pruc. In/. Cuqf Gallium Arsenide, 2nd. Dallas, 1968. p. 131. The Institute of Physics and the Physical Society. London. 1969. 7 8 H. J . Guggenheim and L. F. Johnson. Appl. PIijs. Lett. 15, 51 (1969). 7 9 L. G. Van Uitert. S. Singh, H. J . Levinstein. L. F. Johnson, W. H. Grodkiewicz, and J . E. Geusic, Appl. Phys. Lett. 15. 53 (1969). G . M . Blom and J . M. Woodall, Appl. P ~ ~ LLett. Y . 17, 373 (1970). 81 A. P. Bogatov, L. M. Dolginov. L. V. Druzhinina, P. G. Eliseev, B. N . Sverdlov, and E. G . Shevchenko, Koantocayo Elektron. (Moscow~)I , 2294 (1974)[English transl.: SOP.J . Quant. Electron. 4, 1281 (19751. 8 2 J . J . Coleman, N . Holonyak, Jr., M . J . Ludowise, P. D. Wright, W. 0. Groves, and D . L. Keune, IEEE Semicond. Laser Con/:. 4th. Norembar, 1974, Atlanta: IEEE J . Quantum Electron. QE-11,471 (1975). 8 3 W. J . Turner and G . D. Pettit, Appl. Phys. Lett. 3, 102 (1963). 84 W. J. Turner, W. E. Reese, and G. D. Pettit. Phj~s.Rev. 136. A 1467 (1964). *' R. C. C. Leite, Phys. Rev. 157,'672 (1967). '' U. Heim, SolidStare Commun. 7. 445 (1969). 87 E. W. Williams et al.. J . Electrochem. Suc. 121, 835 (1974). P. E. Eliseev, 1. Ismailov, and L. 1. Mikhailina, Zh. Eksp. Teor. Fiz. Pis'ma 6 , 479 (1967) [English transl. : Sur. Phxs.-JETP Lrrt. 6 . I5 (1967)l. 8 9 U . Heim, 0. Roder. and M. H. Pilkuhn. Solid State C'ummun. 7. 1173 (1969).

26


k v)

W 2

I-

I z

FIG. 1 I . Laser emission spectra (T = 300 K and I,,,, = 9150 A) obtained on InP pumped with a parametric oscillator at energy E, E, + kT.(After Rossi and chin^^.^^)

0 v) 2 L

-

W W

z

a -1 W

a

9450

9400

9350

9300

9250

WAVELENGTH (A)

be obtained.34 This efficiency is the ratio of the sample power emitted to the pump power absorbed. The samples exhibiting this relatively high conversion efficiency are obtain from Czochralski-grown material ( n 3 0 0 = 2 x 1015/cm3, ~ 3 0 'v 0 3200 cmZ/Vsec). The InP sample is excited with the tuned pulsed output of a parametric oscillator which is focused onto the sample surface in the form of a line image, with the cleaved edges of the sample acting as the Fabry-Perot mirrors of a laser cavity.34 Stimulated emission from one edge is analyzed by a high-resolution spectrometer. Output from the other edge is detected by a calibrated Si photodiode to determine the power output. Laser emission from such a photopumped InP sample is shown in Fig. 1l.34 Clearly InP is a useful laser material in the region of A 0.9 pm. This is evident also from recent improvements in the growth of InP, which have led to diodes of quantum efficiencies as high as l.SS,; at 300"K.90

-

V. Alloy 111-V Semiconductors The 111-V binary compounds possess the property of being miscible in all proportions. Consequently, ternary alloys of the type IIItIII~-xVand I11 . VtV?, as well as quaternary alloys of the type I I I ~ I I I ~ - x V ~ Vcan ~ - ,be , E. W. William. P. Porteous, M. G. Astles, and P. J. Dean, J. Elerlrochem. Sor. 120, 1757 (1973).

1.

27


synthesized. The discussion here is concerned mainly with ternary alloys but not exclusively inasmuch as recently the quaternary alloys have begun to gain in interest. The properties of 111-V alloys can be varied continuously between the properties of the constituent binary compounds by changing the crystal composition parameter, x (or y or z ) in the range 0 I x I 1. For efficient light generation, the “tunability” of the band gap with crystal composition is probably the most important property of the alloy. This can be seen, for example, in the GaAs,-,P, system. GaAs, as discussed above, is a direct semiconductor emitting in the infrared. GaP has a band gap corresponding to the green portion of the spectrum, but is indirect and, if not doped with isoelectronic traps or complexes such as N or Zn-0, is an inefficient light emitter. If these two binary compounds are mixed in the proper proportions. the direct characteristics of GaAs can be retained with efficient emission being obtained from other parts of the spectrum, including from well into the red. In fact, a large majority of visible light emitting semiconductor displays are fabricated from GaAso,60Po,40. These red emitters form, for example, most of the displays in pocket-sized electronic calculators and a wide range of other electronic instruments. Despite the obvious advantage of tunability in the 111-V ternary alloys, the use of these alloys is not more widespread owing principally to the difficulty of synthesizing high-quality crystals. The lattice constants of binary compounds frequently differ by several percent. If the composition of the alloy is not uniform or carefully graded throughout the crystal, nonradiative recombination centers can result from the strains and defects associated with poor lattice matching. For efficient light generation, the density of nonradiative recombination centers must be small. Thus, much of the work with the 111-V alloys involves producing uniform crystal growth and learning to dope and make junctions or heterojunctions without introducing unwanted defects into the material. The physical properties of the alloys are often much less accurately determined than those of the binary compounds owing to the lack of accuracy with which the additional parameter, the crystal composition, can be measured. Typically, the composition cannot be found to closer than Ax 0.01, particularly away from x = 0 or .x = 1. As an example, the energy shift of the luminescence peak as a function of donor concentration can be evaluated with much more accuracy in binary compounds. Nonetheless, some other physical effects can be analyzed in the alloy much more easily than in the binary compound. Some of these are discussed in detail below. Of the several 111-V ternary alloys that have been successfully grown, In,-,Ga,P and GaAs,-,P, are two of the most frequently investigated because of their large direct energy gaps and visible emission. From the point

-

28

N. HOLONYAK, JR.,

AND

M. H .

LEE

of view of the physics of crystals, these alloys are interesting since the binary constituents GaAs and InP are direct-gap semiconductors while GaP is indirect. Thus, either alloy can be direct or indirect, depending upon crystal composition. G a l -,AI,As has t h s property also, but its emission characteristics have not been as frequently investigated as GaAs,-,P,, which is the most widely used LED material. Owing to the close lattice match between GaAs (ao = 5.654 A) and AlAs (5.661 A), Ga,_,Al,As has been used principally for the inactive layers in Ga,-,AI,As/GaAs heterojunctions and in some cases for visible spectrum G a l ~xA1,As/Gal~,~AI,.As (x # x’) heterojunctions. At longer wavelengths (1.2 1 pm), 1 n , G a , ~ , A ~and ~ ’ ~InAs, ~~ . P lasers have been reported at various crystal compositions. These crystals are direct throughout their composition ranges and, although still not thoroughly developed, are useful for emission in the near infrared. In addition to the ternary alloys, several quaternaries have recently become more interesting and useful. The two receiving most attention, particulady for use in heterojunctions, are Gal~,A1,Asl gPy95-97 and In, -,Gax . P,-,As, .33*82The latter is especially interesting because of the wide wavelength range (A > 1 pm to I. 5500 A) over which it will ’operate as a laser, and the fact that it can be fabricated into various kinds of heteroj l i l l ~ t i o n s . ~ ~Below , ~ ~ ,we ~ ~consider * ~ ~ , ~ the ~ photopumped laser operation of various of the 111-V alloys mentioned above, considering first those that lase at higher energy.

-

3. INDIUMGALLIUM FHOSPHIDE Because the lattice constants of InP (5.869 A) and G a P (5.451 A) are so different, the composition of In,_,Ga,P, which is so far the highest energy junction material,*’ must be quite uniform in order to minimize the density of nonradiative recombination centers arising from composition-induced lattice change or mismatch. Despite this problem, interest in In,,Ga,P has continued because the direct-indirect transition of this ternary system occurs at high energy, well into the yellow or even the edge of the green.99 As the quality of the In,-,Ga,P crystals that can be grown has improved, the capabilities of the material have become more clear. First of all, the energy ”

’*

1. Melngailis, A . J . Straws, and R. H. Rediker. f r o c . IEEE51. Il54(1963).

H. M. Macksey, J . C. Campbell, G. W. Zack, and N. Holonyak, Jr., J . Appl. Phys. 43, 3533 (1972); C. J. Nuese, R. E. Enstrom, and M. Ettenberg. Appl. Pkys. Lett. 24,83 (1974). y 3 F. B. Alexander et al.. Appl. Phys. Lett. 4. 13 (1964). y 4 R. D. Burnham, N. Holonyak, Jr., and D. R. Scifres. Appl. fliys. Lett. 17,455 (1970). R. D. Burnham et a/., Appl. f h y s . Lett. 19.25 (1971). 9 6 R. D. Burnham et a!., Fiz. Tekh. Poluproimdn. 6. 97 (1972)CEnglish transl.: Sot. Phys.Semicond. 6, 77 (19721. ” G. A. Rozgonyi and M. B. Panish, Appl. f/iys. Le f t . 23. 533 (1973). J. J. Coleman e t a / . , fliys. Reo. Lett. 33,1566 (1974).

’’

1. PHOTOPUMPED

111-V SEMICONDUCTOR LASERS

29

gap is direct to 2.25 eV (300 K).99-107Thus, the eventual construction of efficient yellow-green electroluminescent devices has appeared certain,99 consistent with the fact that recently junction laser operation has been demonstrated in the This prospect received support in the early stages of the development of In,-,Ga,P by the demonstration that photopumped laser operation (x 0.3, i. < 7000 A, 77'K) could be achieved on uniform platelets prepared from bulk crystals. O 8 Photopumped laser operation showed that In,-,Ga,P crystals could be grown of sufficient quality to allow high-level light generation. This result contradicts the notion that In could not be added to Gap, or Ga to InP, without severely disturbing and creating defects in the crystal lattice. It is true, however, that In,-,Ga,P is not simple to grow, dope properly, and make into high-quality junctions. While polycrystalline In, _,Ga,P of fairly good quality can be easily synthesized,lo5 seeded single-crystal In ,-xGa,P is more difficult to grow. Owing to the large difference in the lattice constants of InP and Gap, considerable difficulty exists in grading the crystal composition continuously over a large range and then holding it constant over the dimensions desired. Nevertheless, reasonable success has been achieved in the growth of Inl-,. Ga,P by vapor phase epitaxy (VPE)'03~'06~'073'09 and by liquid phase epitaxy (LPE).23,27,' In the case of the former, the crystal composition is graded from that of G a P or GaAs to that of the desired In,_,Ga,P. For the LPE growth of In,_,Ga,P the substrates employed are generally GaAs or GaAs,_,P,(x % 0.52 + 0.48~).with the composition of the melt and the

-

99

M. H. Lee, N. Holonyak. Jr.. W. R. Hitchens. J. C. Campbell, and M. Altarelli. Solid Stare Commun. 15,981 (1974). l o o A. Onton and M. R. Lorenz, Prnc,. I970 Symp. GaAs and Related Compounds, Aaclien, Germany, p. 222. The Institute of Physics, London, 1971. l o ' M . R. Lorenz and A. Onton. Proc. Int. Conf: PIiys. Semirond., 10th. p. 444. At. Energy Commission, U.S. Oak Ridge. Tennessee. 1970. A. Onton. M. R. Lorenz, and W . Reuter, J . Appl. Phys. 42. 3420 (1971). l o 3 C. J. Nuese, D. Richman, and R. B. Clough. Met. Trans. 2, 789 (1971). G. B. Stringfellow, P. F. Lindquist. and R. A. Burmeister. J . Electron. Marer. 1,437 (1972). lo' H. M. Macksey, N. Holonyak. Jr.. R. D. Dupuis. J. C . Campbell. and G. W. Zack. J . Appl. Phys. 44. I333 (1973). l o 6 A. G . Sigai, C . J. Nuese, R. E. Enstrorn. and T. Zamerowski. J. Electrochem. Soc. 120,947 (1973). 107 C. J. Nuese. A. G. Sigai, M. S. A b r a h a m , and J. J. Gannon, J . Electrochem. Soc. 120. 956 (1973). l o * R. D. Burnham. N. Holonyak, Jr.. D. L. Keune, D. R. Scifres, and P. D. Dapkus, Appl. Phys. Lett. 17, 430 (1970). C. J. Nuese. A. G. Sigai, and J. J. Cannon. Appl. Phys. L e f t . 20, 431 (1972). G. B. Stringfellow, J . Appl. PhJis. 43, 3455 (1972). ’ I 1 W. R. Hitchens. N. Holonyak, Jr., M. H. Lee, and J. C . Campbell, J . Cryst. Growth 27, 154 (1974).

30

N. HOLONYAK, JR., AND M . H . LEE Energy (eV) I F

1.7

2.;

2.;

2;0,

E

'I 'Z w

I

I

I

66

6.4

6.2

I 6.0

1

Wavelength ( lo' A)

FIG. 12. Photoluminescencespectra (77'K) obtained on: (a)LPE n-type x = 0.52 In,-,Ga,P: Te well lattice-matched on a GaAs substrate and (b) partially compensated p-type x = 0.56 Inl-xGa,P:Zn:Te poorly matched on a GaAs substrate. The total emission intensity of the low dislocation density n-type crystal (a) is more than 10 x that of the poorly lattice-matched x = 0.56 In,-,Ga,P sample (b). (After Hitchens er al."')

temperature being adjusted carefully to permit lattice-matched epitaxial growth, and little or no grading of the lattice. The successful epitaxial growth of In,-,Ga,P has led to several notable laser results: (1) p-n junctions grown via VPE have operated at wavelengths as short as 1 6105 A (80"K,x = O.57),lo9 as compared to 1 7600 A (77"K,x 0.27) achieved earlier on junctions fabricated by Zn diffusion into polycrystalline material;"' (2) LPE In,-,Ga,P junctions grown on GaAslp, P, substrates have operated at 5900 A (77"K, x z 0.63);3' and (3) photo(x 0.70) has operated as a laser at 1 5500 A pumped LPE In,,Ga,P (green)." It turns out that the last result is the easiest to obtain; i.e., it is much easier to operate In,-xGa,P as a photopumped rather than junction laser. As shown by the laser data below on photopumped LPE samples, In,-,Ga,P clearly has the potential to become an important injection laser material. A comparison of the luminescence capabilities of well lattice-matched and poorly lattice-matched In,,Ga,P is shown in Fig. 12. The crystals used to obtain these data are grown on { 100) GaAs substrates.' l 1 The dashed curve in Fig. 12a is obtained on a crystal (x = 0.52) that is well lattice-matched to

- -

-

-

112

-

H. M. Macksey, N. Holonyak, Jr.. D. R. Scifres, R. D. Dupuis, and G. W. Zack, Appl. Phys. Lerr. 19, 271 (1971).

1.

31


the substrate ( y = 0) and is relatively free of dislocations (etch pit density I 5 x 103/cm2).The photoluminescence spectrum shows that radiative recombination occurs principally through band-to-band or donor-to-valence band transitions. The solid curve (Fig. 12b) shows the emission spectrum of an .Y = 0.56 sample that is photopumped at the same excitation level. Note that the total emission intensity of this sample, which is badly mismatched relative to the GaAs substrate (etch pit density > 105/cm2),is lox lower than that of the well lattice-matched sample. The magnitude of the decrease in the lumininescence output is typical of poorly lattice-matched In,-,Ga,P crystals, regardless of doping concentrations. Also present in the emission spectrum is a long low-energy tail characteristic of crystals with high defect densities. If the In,-,Ga,P layer is well lattice-matched to the substrate, the photoluminescence performance of the resultant LPE layer is far superior to that of polycrystalline In I-xGa,P samples grown by the modified Bridgman method.lo5 Some better In I-SGa,P crystals have even ~ u t p e r f o r m e d ~ ~ platelets prepared from the best GaAs,-,P, grown by VPE.IL3Among other factors, this may be due to the lower surface recombination velocity of In -,Ga,P. As an example of the excellent In,-xGaxP that can be grown by LPE. consider the performance of the .Y = 0.52 In,-,Ga,P:Te (lid 10'*/cm3) crystal of Fig. 13. Thin samples ofthis crystal are mounted compressed into In with a sapphire window16 and are excited(throughthe sapphire window) with an Ar+ laser which is operated cw for low-level excitation or mode-locked at a frequency of 140 x lo6 sec-' (0.2 nsec pulses) for high-level excitation. Figure 13 shows the 300-'K photoluminescence spectra obtained on these samples and, for comparison, the spontaneous emission spectra (500 W/cm2) at 77 and 196' K. In accord with the behavior of the energy gap, the peak of the emission spectrum shifts to lower energy with increasing temperature. In addition, the emission spectrum broadens, and the slope on the highenergy side of the peak decreases. This thermal spreading reflects the behavior of the Fermi function, and the carrier population, as a function of temperature. As the excitation level is increased, the emission spectrum broadens considerably, and laser operation (300 K ) occurs at an excitation level of 5 x lo4 W/cm2. Photopumped room-temperature laser operation of III-V semiconductors with the apparatus described here, where the pump photon energy is not tuned to the sample absorption. is very unusual. For example, the best visible spectrum GaAs, ,P, has not been observed to lase at room temperature under the pumping conditions described here. We mention that the In I-xGa,P samples of Fig. 13 also have been operated cw as a laser (77°K). which has not been possible yet with GaAs,-,P, under similar heat sinking and excitation conditions.

-

-

32

N . HOLONYAK, JR., AND M . H . LEE Energy (eV) I.So

1

7.0

1.8s

1.90

I 9S

2.00

I

I

I

I

I

I

I

6.8

6.6

6.4

Wavelength

2-05 I

I

I 6.2

6.0

(lo3A )

-

FIG. 13. Photoluminescence spectra (300°K) obtained on high-quality LPE In,_,Ga,P:Te ( X = 0.52, nd 10’8/cm3).Note the dashed reference spectra obtained on the same crystal at 77‘K and 196°K. As expected, the peak of the low-level (500 W/cm2)spontaneous spectrum shifts to lower energy with increasing temperature. At an excitation level of 5 x lo4 W/cm2, the spectral width broadens to 140 meV, and laser operation occurs on the low-energy side of the emission spectrum. (After Campbell er

+

Higher percentage In,-,Ga,P grown by LPE on GaAslp,P, (x % 0.52 0.48~~) possesses also excellent photoluminescence characterist~cs~’ for compositions below the direct-indirect transition (x < x, % 0.74).” This is illustrated in Fig. 14which shows photoluminescence and laser spectra (77°K) obtained on lightly doped n-type In,-,Ga,P (x 0.70, nd 10l6 Te/cm3). The half-width of the emission spectrum at an excitation level of 500 W/cm2 is only 12 meV. This width is extremely narrow and indicates the obvious local homogeneity of the crystal. Note also the absence of emission at energies far below the emission peak. This behavior is in contrast to that observed on the poorly lattice-matched crystal of Fig. 12. For the crystal of Fig. 14, at an excitation level of 5 x lo4 W/cm2 laser operation occurs at wavelengths , I 5500 A (yellow-green), which is shorter than achieved in other 111-V systems, except In,-,Ga,P ,-,Asz. For efficient light generation in In ,_,Ga,P homojunction devices, p-type material must also possess good luminescence characteristics. Figure 15 shows photoluminescence spectra (77°K) obtained on LPE InIp,Ga,P:Zn (n, 7 x 1018/cm3).27 At low excitation levels (500 W/cm2)the predominant

-

-

-

-

1.


33

Energy (eV1 2.1

2.2

I

I

2

30 A (0.012 eV)+

500 I

n

I

I

I

I

I

sn

S.6

Wavelength ( 103A)

I

-

Frci. 14. Photoluminescence spectra (77 K ) ofn-type LPE In,~.Ga,P (u 0.7).At low excitation levels, the band-to-band transition is narrow (0.012 eV) because of the low doping and the uniformity of the crystal. As the excitation level is increased, the recombination shifts t o somewhat lower energy. with laser operation occurring at 5 x lo4 W/cmZ. (After Macksey et ul.”)

recombination transition is conduction band-to-acceptor. In some samples a higher-energy shoulder corresponding to the band-to-band transition is also seen. As the excitation level is increased to 5 x lo4 W/cm2, laser operation occurs. Note the different lasing behavior of the p-type and n-type crystals of Figs. 15 and 14. For the lightly doped n-type crystal of Fig. 14, laser operation

34

,


2.00

6.2

Energy (eV) 2.;

2.;0

6.0

2.u

58

2.;

2.;

5.6

Wavelength ( I O3A )

-

FIG. IS. Photoluminesce'nce spectra (77°K) of p-type LPE In,-,Ga,P ( x 0.7). At low excitation levels band-to-band recombination is evident as a high-energy shoulder on the higher-intensity band-to-acceptor transition. As the excitation level is increased, the band-toacceptor recombination increases rapidly. and at 5 x 10" W/cm2 laser operation occurs. (After Macksey et d 2 ' )

occurs on the low-energy side of the emission peak owing to the high reabsorption of photons at the band edge. O n the other hand, laser modes for the p-type sample of Fig. 15 are seen to extend from the low- to the highenergy side of the spontaneous emission peak because reabsorption of the conduction band-to-acceptor recombination radiation is small in p-type material. For p-type samples, recombination occurs to the ''top'' of the acceptor and not down to the Fermi level E F p(cf., Fig. 7). The difference in the reabsorption of the laser photons between n- and p-type samples is suggested also by the laser mode spacing. The index dispersion expression n' = n - i.(dn/di.),

(7)

which appears in the usual laser mode spacing formula

A2 = (A2/2L)(n- I ( d n / d l ) ) - ',

(8)

1.


-

35

-

is found to be n’ 7 for the n-type sample and n’ 6.2 for the p-type sample. These values are consistent with laser mode spacing data taken on other samples of these crystals and show that the absorption losses are lesser for p-type samples. While excellent quality homogeneous n-type and p-type LPE In,-,Ga,P can be synthesized, consecutive growth of n-type and p-type layers to form p-n junctions has not been as successful. To date, laser operation (77°K) of In,-,Ga,P junctions formed by this technique has been obtained at wavelengths as short as 5900 A (2.10 eV),32by far the shortest laser wavelength from p-n junctions. The laser thresholds, however, are quite high, near lo5 A/cm2. It is clear that much work remains before In,_,Ga,P junction lasers will become practical devices. 4. INDIUM GALLIUM PHOSPHIDE ARSENIDE

Related to In,-,Ga,P, which is fairly difficult to grow, is In,~,Ga,P,_,As,, which can be grown much more easily and, in addition, can be incorporated into quaternary-ternary In, -,.Ga,P, _=AsZ/GaAs1 - ,P, and quaternary-quaternary In ,Ga,P -,As,/In _,.Ga,.P I-Z.As,, heterojunct ions that Although these devices are in their cover a wide spectral range.33.81.82*98 infancy, they promise to have considerable future, particularly quaternary heterojunctions. Thus, the photopumped behavior of 1n,~,Ga,P1-,As, is of interest here.’05 As described e l s e ~ h e r e , the ~ ~ problem .~~ of lattice-matching In,-,Ga,P on a GaAs,-,P, substrate is eased if the ternary is rendered a quaternary by the incorporation of a small amount ( 2 0.01) of As in the LPE layer changing it to In,-,Ga,P,-,As,. Small deviations A x in the Ga composition x of In,,Ga,P,_,As, from the lattice-match condition on GaAs,-,P, then are capable of being balanced by compensating small deviations A2 in As composition z that hardly change the electrical and optical behavior of the quaternary LPElayer. The quaternary itself for z 0.01 is not much different in electrical and optical behavior from the ternary In ,-,Ga,P that lattice matches the same GaAs,-,P, substrate. If an increase Ax in G a percentage occurs in the In,~,GaxP,~,Asz LPE layer from the proper composition for a lattice match on GaAs,-,P,, the lattice constant decreases (cf., Fig. 16).This change in lattice constant is compensated by a corresponding increase Az of As in the crystal, which increases the lattice constant sufficiently to keep it constant. The net effect is a small overall decrease in the energy gap in the resulting In,-,Ga,P,~zAs,, as is evident from Fig. 16 and the fact that the isoenergy gap lines (solid) cross the isolattice-constant lines (dashed, 5.5725.869 A). Because the isolattice-constant lines of Fig. 16 cross the isoenergygap lines, it is possible to grow, along an isolattice-constant line, a wide range

,

-

-

36


1.41

1.00

InAs FII3. 16. Energy-gap surface (77'K) as a function of crystal composition for the quaternary alloy systeim In,,Ga,F 'l-zA~,. The edges are divided into mole fractions of the individual binary constitutents. Plotted also on the surface are isolattice-constant lines (dashed, 5.869-5.572 A) and isoenergy-gap lines (solid, 1.41 -2.20 eV). From the fact that the isolattice-constant and isoenergy-gap lines intersect. it i s apparent that a wide range of compositions of In,-xGa,P,_,As, of higher-energy gap can be grown along an isolattice-constant line on GaAs,-,P, of lower energy (.K # x', z # 2') gap, making possible also the growth of In,-,Ga,P,_,As,/In,-,.Ga,.P,,.As, homoheterojunctions. (After Coleman et a/.*')

of compositions of Inl,GaxP,,As, of higher energy gap on GaAs,-,P, of lower energy gap or even on In,~xrGax.P,,.As,. of lower energy gap. Compared to homojunctions, the quality of the quaternary-ternary heterojunctions that have been constructed on GaAs,,P, suggests that latching (one lattice locking on the other)"O occurs when In,,GaxP,-,As, is grown by LPE on GaAsl,P,.s2 It is evident that the quaternary In,,Ga,P,,As, (z 0.01 or even larger) affords greater freedom and leeway than the ternary (z = 0) in fulfilling the basic lattice-match requirement on GaAs,_,P,, per-

-

1.


37

mitting in turn the LPE growth of high-quality In,p,GaxPI-,As, layers for heterojunctions or for photopumping experiments, as below. The behavior ofphotopumped s 0 . 7 0 , ~ 0.01 LPE In,-,Ga,P,-,As, in spontaneous and stimulated emission is shown in Fig. 17. Notice that the spontaneous peak at 300°K is well below 5700 (>2.175 eV, yellow) and at 77°K lies at 5470 A (2.267 eV, green). At 300°K the half-width of the spontaneous emission is AA 120 A (46 meV) and at 77'-K is AA 50 A (21 meV), which is as narrow as materials such as GaAs (comparable doping) and is indicative of the quality of the alloy. At the fairly modest pumping level of 2 x lo4 W/cm2, the sample exhibits laser operation with the shortest mode at 5520 A (green). No other 111-V semiconductors have given indication of being better high-energy lasers. Nor do any other 111-V materials have a better prospect of being incorporated into heterojunctions in the wavelength and energy range of the sample of Fig. 17.82Quite likely yellow-green junction lasers, as present experience indicate^,^^^'^ will be easier to build in InlpxGa,P,~zAsZ than in In,_,Ga,P ( Z = 0).

-

-

-

-

Energy (eV) 2.15

2 20

2.2s

2.30

23

I

I

I

I

I

I

5.8

I

1

5.6

Wavelength ( 10' A )

I

I

5.4

-

-

FIG. 17. Photoluminescence spectra (77 K ) obtained on s 0.71. I 0.01 LPE In ,,Ga, P,-,As,:Te. The spontaneous spectrum of the same crystal at 300 K is shown dashed for comparison. At a relatively moderate excitation level of2 x lo4 W/cm2 laser operation occurs o n the low-energy side of the emission spectrum at 5520 A (1.246 eV, green). (After Mackscy er d.’"’)

-

38

N. HOLONYAK, JR., AND M . H. LEE

5. GALLIUM ARSENIDE PHOSPHIDE (GaAs,-,P, AND GaAs,-,P,:N)

As mentioned earlier, GaAs,-,P, is widely used as a light-emitting diode (LED) material. In fact, it is the most used LED material. This has occurred because large quantities of high-quality VPE GaAs,,P, can easily be grown on GaAs (or Gap) substrates by means of a cheap open-tube ASH,-PH, vapor transport process,' l 3 and the junctions are formed by the same type of diffusion technology used in integrated circuit technology. Not only are Nfree and N-doped GaAs,-,P, crystals important LED materials, they both exhibit an interesting variety of laser effects, some of which have not been observed in other materials. We describe below much of what has been learned about GaAs,-,P, and GaAs,-,P,:N by means of photopumping. Some of the data are unique in that junction devices have not been capable of exhibiting the same effects. The behavior of both N-free and N-doped GaAs,-,P, is considered below. The behavior of the former is clear in reference to the behavior ofGaAs,,P,:N, which is considered in detail. Apart from the difference in the emission wavelength, the luminescence behavior of GaAs,,P, is similar to that of GaAs over most of the direct composition range.'I4 Near the direct-indirect transition x = x, (x, z 0.46, 77°K; 0.49, 300°K)'' the luminescence properties are dominated by the heavy-mass indirect X conduction band minima and the donor states associated with the X minima.'16 These donor states are known to affect, for example, the free carrier concentration and the threshold of laser junctions as x + x,.' l 8 These effects in GaAs,,P, can be studied conveniently by changing the crystal composition and moving the conduction band minima relative to each other. As in GaP,"9*'20 N doping in indirect GaAslp,P,(x > x,) significantly enhances the efficiency of the radiative recombination process.' ' 5*121,122 For example, x = 0.7 GaAs,,P, LED'S doped with N are 14x brighter than similar devices without N doping.'23 Clearly, the N impurity adds an''9'

''' M . G. Craford and W. 0. Groves, Proc. IEEE61, 862 (1973). M. G. Craford. Progr. Solid State Cham. 8. 127-165 (1973). M. G. Craford, R.W. Shaw. A. H. Herzog, and W. 0.Groves, J. Appl. Phys. 43,4075 (1972). ’I6 M. G. Craford, G. E. Stillman, J. A. Rossi, and N. Holonyak, Jr., Phys. Recr. 168,867 (1968). ' N. Holonyak, Jr., C. J . Nuese, M. D. Sirkis, and G. E. Stillman, Appl. Phys. Letr. 8,83 (1966). C. J. Nuese, G. E. Stillman, M. D. Sirkis, and N. Holonyak, Jr., Solid-State Electron. 9, 735 (1966). ’I9 D. G. Thomas, J. J. Hopfield, and C. J. Frosch, Phys. Rev. Lett. 15,857 (1965). I2O D. G. Thomas and J. J. Hopfield, Phys. Rev. 150,680 (1966). W. 0. Groves, A. H. Herzog, and M. G. Craford. Appl. Phys. Lett. 19, 184 (1971). 1 2 2 J. C. Campbell, N. Holonyak, Jr., A. B. Kunz, and M. G. Craford, Appl. Phys. Lett. 25, 44 (1974). 12’ M . G. Craford, D. L. Keune, W . 0. Groves, and A. H. Herzog. J . Electron. Marer. 2, 137 (1973). ’I4

"

''

1. PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

39

other dimension to the behavior of GaAslp,P, and invites comparison with the behavior of the usual shallow donor and acceptor impurities. From the time of the first work on GaAs and GaAs,-,P, junction lasers (1962), it has been evident that donor and acceptor states, states due to Coulomb centers, are involved in a major way with electron-hole recombination and laser operation of direct-gap material. One can ask whether the N isoelectronic trap, as distinct from a donor or an acceptor possesses sufficient oscillator strength to be involved in stimulated emission or laser operation. If stimulated emission can be obtained, the extent to which the band structure influences the lasing process is an important issue. The N impurity in GaAslp,P, is isoelectronic and much more electronegative than even P. Lattice relaxation around the N atom reduces the effect of the electronegativity difference,’24 but an electron can still be bound to the N atom. The N potential, unlike the screened Coulomb potential which extends several lattice constants, is effective only in the central cell. The electron is tightly bound to the N trap, and, as a consequence, its wavefunction 4 is widely spread in k-space as sketched in Fig. 18.28This spread in 4 assists in the recombination process in an indirect-gap crystal. The electron captured at the N site has a much larger k = 0 wavefunction component ( - 100 times) than if it were “trapped” at a donor,’2s and has a larger “overlap” with k = 0 (r)holes. A hole (at k = 0) is attracted to the Coulomb center formed by the electron captured at the N site, and the resultant exciton can decay, emitting a photon. Nitrogen atoms on different lattice sites can also act in pairs to bind electrons.’ 2 o The binding energy becomes progressively larger as the NN-pair separation decreases, NN, representing the limit of a pair of N atoms on adjacent Column V lattice sites. The role of N in radiative recombination in GaAs,-,P, is similar in some respects to that in Gap. As the GaAs,-,P, composition is shifted downward from x = 1 (i.e., downward from Gap), the sharp lines in the emission spectrum due to the decay of excitons bound to single N atoms (A-line) and to pairs of N atoms (NN), and phonon replicas of these lines, tend to become smeared owing to local statistical fluctuations in the As-P ratio and hence in the potential around the N atoms. At compositions x 5 0.90, the importance of NN pairs is considerably diminished relative to individual N atoms ( ~ - i i ~ ~1 5). 1.2 6l . 1 2 7

Figure 19 shows the behavior of the N-trap emission peaks, one assoas a function of crystal composition x. In ciated with X and one with GaAs,-,P, the conduction band density of states at X is more than an order

”*

J . C. Phillips, Phys. Rec’. Lelt. 22, 2x5 (1969). P. J . Dean. J . Lumin. 1. 2. 398 (1970). N . Holonyak, Jr., R. D. Dupuis, H . M . Macksey, M. G . Craford, and W. 0. Groves, J. Appl. Phys. 43, 4148 (1972). R . J. Nelson el a / . ,Phys. Rec. B 14, 685 (1976).

40


__

n

E-k. GaP : N

-- - E-k' GaAs

EN

- I0 meV

2

z$j wC

E

1

c

E,(X) = 2.'3 (77 "K)

I IOOOl

k = % [I001

Reduced Wave Vector. k

FIG. 18. Simplified band structure of G a p : N. The conduction band of GaAs is shown dashed for reference. The shaded region represents the magnitude of the wavefunction of an electron dN(k)bound to a N isoelectronic trap. The binding energy is approximately 10 meV. The shortrange nature of the potential associated with the neutral N isoelectronic impurity causes &(k) to have an increased amplitude at k = 0 thereby enhancing the probability of a direct radiative recombination. (After Holonyak et a / . 2 8 )

of magnitude larger than at r. Consequently the energy of a bound electron state associated with the short-range N potential is strongly influenced by the minima at X and to some extent tends to follow the X band edge. As x decreases, however, the N-trap level associated with the X band edge (N, in Fig. 19) becomes deeper relative to the X band minima as a result of the increasing As-P ratio and the difference in electronegativity of As and P atoms. The N, emission peak of Fig. 19 and the direct band edge E , of GaAs,,P, are degenerate at crystal composition .x zz x N % 0.28.'26-'28If laser operation is to be observed on transitions involving N,, the crystal composition must be adjusted away from x = 0.28 (x > xN)in order to resolve the band edge and the N, emission. In a small crystal-composition interval between 128

M. G. Craford and N. Holonyak, Jr., in "Optical Properties of Solids" (B. 0. Seraphin, ed.). Chap. 5. pp. 187-253. North Holland Publ.. Amsterdam, 1976.

1.


41

2.3

2.2

2.1

-

2.a

5

9

wz

I9

1.8

1.i

1 .c 0.0

0.5

1.0

GaP FIG. 19. Energies (77 K ) ofthe N Vband I":l-line") and the higher energy N, state in nitrogen. of the r and S conduction band minima are shown for reference. doped G ~ A S , - , P , ~Energies For s 2 0.90. N N , and NN, pairs are distinguishable. but not for s 5 0.90 where the N i n e of G a P becomes a broadened N, band. The state labeled Nr is a second N-trap leveled (.Y 5 0.55) associated with the r band minimum. The downward shift in energy caused by the Zn acceptor I 5 .xL) is plotted as a dashed curve hcloH (lit r band edge and he lo^ the N,, band. (After GaAs

Nelson

x

~f c//."')

= 0.32 and 0.37, photopumped laser operation has been demonstrated ~ ~ .example, '~~ Fig. 20 shows unambiguously on the N, t r a n s i t i ~ n . ~ " . ' For photoluminescence spectra (77'K) obtained on x = 0.34 GaAs,-,P,: N', nd = 1.8 x 1017/cm3.'28The N + denotes a large density of N atoms, At low excitation levels (500 W/cm2), the emisestimated to be 1019/~m3. sion from the band edge (r)and from the N x trap level is easily resolved. As the pumping level is increased, the intensity of the emission increases .Y

-

'29

D. J. Wolford, B. G. Streetman. R. J . Nelson, and N. Holonyak, Jr.. Appl. P h p . L e f t . 28, 711 (1976).

42


Energy (eV)

I 7.0

1.80

1.85

1.90

1.95

I

I

1

I

I 6.8

I 6.6

2,

I 6.4

XJ. This behavior is further accentuated by the fact that uexdecreases for x >, x, because of the decreased influence of the lightmass r-conduction band minimum. The shift in the N, photoluminescence peak with crystal composition is shown summarized and compared with calculated results in Fig. 22.’ 31,1 3 2 Laser operation has not been demonstrated on the N, transition in GaAs,-,P, for crystal compositions x > 0.37.12’ This difficulty may be related in part to the large difference in energy between the conduction band and the N trap. The probability density at k = 0 of an electron trapped at a N site is enhanced by the proximity of the r conduction band edge. This band structure enhancement (BSE) is accentuated in GaAs,-,P,: N as the crystal composition is varied to bring Er nearer to the N trap level The oscillator strength for recombination on the N, transition is sufficient to produce stimulated emission at crystal compositions up to x = 0.37 but weakens considerably as x increases due to the increase in the energy differ13* 133

J . C. Campbell, N . Holonyak, Jr., M. H. Lee, and A. 9.Kunz, Phys. Reu. B 10, 1755 (1974). J. C. Campbell, N. Holonyak. Jr., M. G. Craford, and D. L. Keune, J . Appl. Phys. 45,4543 (1974).

1.

45


CaAs

GaP

GaAs,- .P, Composition.

FK;.22. Normalized spectral shift

.Y

the N, band peak in GaAs,-,P,:N versus crystal composition Y (circles). The shift in energy E, due to the Zn acceptor has been obtained from experiment. The value of the Zn acceptor energy E,, has been scaled linearly from that of GaAs to that of GaP in the composition region 0.3 5 v 5 1.0. The solid curve has been obtained by normalizing y at x = 0.3 and by adjusting riel (see text) to fit the data. (After Campbell et ~ 1 . ~ ~ ~ 1 of

ence Er - EN and the resultant decrease in band structure enhancement. 1 2 2.1 3 3 Being associated with the band edge, the N r N-trap level differs fundamentally from NX,'2 7 exhibits narrower 2 8 and possesses considerable oscillator strength to crystal compositions as high as .Y x,. While the N, and I- band edge emission appear similar in spontaneous emission, important differences can be seen at higher excitation levels. Figure 23 shows the photoluminescence spectra (77°K) obtained on x = 0.43 GaAs,_,P,:N+ (n, = 3.6 x 1016/cm3)and on an otherwise identical N-free sample (cross-hatched spectra) that obviously exhibits no N-trap emission.12*For the N-doped sample, at low and moderate excitation levels the N, line A-line is seen to lie at lower energy than the band edge emission (cf., Fig. 19).At higher excitation levels, the emission from the N-free sample broadens and shifts to lower energy owing to EHL interactions,6666 and laser operation occurs on the low-energy side of the spontaneous portion of the spectrum. Differing from the band edge process, the Nr emission (Ndoped sample) shifts much less with increasing excitation. In fact, laser operation on N r tends to occur near the peak of the spontaneous spectrum and af higher energ!' than laser operation on the band edge transition even though Er > EN.This is an interesting result which adds further weight to the contention that the EHL energy shift is indeed characteristic of a large

-

46

N . HOLONYAK, J R . , AND M . H . LEE Energy (eV) 1.80 I

I .90

r

I 6.8

I

1 6.6

I

2.00 1

1

I

6.4

6.2

I

I 6.0

Wavelength ( I 0 3 A )

FIG.23. Laser operation (77‘K, nd = 3.6 x 1O’”;cm’) of x = 0.43 GaAs,-,P,:N on the Ntrap state Nr associated with the r band minimum. For comparison, laser operation on the band edge is shown for an otherwise identical N-free sample prepared from just below the Ndoped VPE layer. Notice that the laser operation on N r does not shift with increasing pumping, whereas that on shifts to lower energy due to EHL interactions. (After Craford and Holonyak.‘ ’*)

electron-hole population in an otherwise undisturbed lattice. In contrast the N isoelectronic trap tends to fix the transition at an energy more characteristic of the impurity. The results of Fig. 23 are important in elucidating and distinguishing between two different recombination and laser processes: (1) band-to-band (r)recombination and laser operation, and (2) recombination and laser operation involving the N trap. Beyond crystal composition x 0.43,laser operation of N-free GaAs,,P, has not been achieved. This is due to the fact that, near ?I 0.43, the highdensity-of-states indirect minima ( X ) , and the donor states associated with X , ’ I 6 begin to compete successfully with the direct (r)band for excess electrons. Without some further agency to assist in the process, not enough electrons are available for “direct” recombination with k = 0 (r)holes for stimulated emission to occur-thus the need for the N trap. Although influenced by the indirect ( X ) minima (x k x,, Fig. 19),12 the N, level (and its proximity to the r band edge) insures a strong wavefunction component for the electron at k = 0 and thus for “direct” electron recombination with k = 0 holes. For example, at crystal composition x x, x 0.46 (77”K), the

-

-

-

1.


41

Energy (eV) 1.8s

1.90

1.95

2.00

2.05

1

I

1

I

I

n

1 5 x 104

w&

;h h

1

6.8

I 6.6

I 6.4

Wavelength

1

I

1

6.2

6.0

( 1 O’A)

FIG.24. Photopumped luminescence and laser spectra (77 K ) of N-doped .Y = 0.46 GaAs,-, . P,:N. nd = 2.9 x 101‘,cm3 Note that the direct-indirect transition in GaAs,-,P, occurs at x = xC= 0.46 ( 7 7 ’ K ) :because N,(X) > N,(T), at Y = s, the crystal is in effect indirect. and is inefficient and will not lase if there is n o N doping. (After Holonyak er al.’’)

N trap to a large extent defeats the indirect character of the crystal, making possible enough “direct” recombination to lead to stimulated emission. This is demonstrated by the photopumped sample and data of Fig. 24.” Notice that the donor doping in the sample (nd = 2.9 x 1016/cm3)is kept low deliberately to prevent interference from indirect ( X ) donor states. Consistent with these results, N-free GaAs,_,P, of composition .Y x, emits much less light than otherwise identical N-doped crystals, and has not operated as a laser. Note that these measurements have been made on VPE crystals where the last 10-25 pm ofepitaxial growth is N-doped and can be removed to provide N-free comparison samples.’ 33a

-

133a

Recently, Aspnes [D. E. Aspnes. Phys. R w . B 14, 5331 (1976)] has assembled data and arguments indicating that the L band minima in GaAs are lower in energy than are the X minima. For GaAs,-,P,:N in the composition region of interest here, however. the r and X band minima are lower than L, thus making it unlikely that L is involved in the laser operation described above. For a more complete presentation of laser data on N-doped GaAsl-,P,(0.38 5 I 2 0.47 > xC),including the effect of the L band minima, see Holonyak et al., J . Appl. Phss. 48. 1963 (1977).

48


Although the upper limit of the crystal composition for which laser operation can be demonstrated in GaAs,~,P,:N has not been established, it probably will not extend much beyond the direct-indirect transition x, . The recombination probability at a N site is related to the modulus of the wavefunction l&0)l2 of the trapped electron at k = 0. The latter is given by 1 2 2 , 13 3

\+(‘)I2

= Q(271)-3(E, - EJ2[Sp(E)dE/(&

- EN)’]-’,

(10)

where EN is the energy of the N state, P ( E ) is the conduction band density of states, F,,is the conduction band energy, and R is the volume of a unit cell. From the form of (lo),it is clear that the recombination probability (at k = 0) varies with the energy separation of the N trap and the direct conduction band minimum. Figure 25 shows a plot of the calculated value l4(0)l2as a function of crystal composition. Since the modulus of the wavefunction decreases rapidly with increasing crystal composition, laser operation should accordingly become increasingly difficult to achieve [cf., Eq. (2)], particularly for .Y > X, 0.46 (77’K). The wavefunction of an electron trapped at a N impurity is widely spread in k-space. This property results in the enhancement of the radiative recombina-

-

10’

lo’

-

0,

9 -

10’

id Crystal Composition, x FIG.25. Modulus (&O)(’ (at k = 0) ofthe wavefunction of the trapped electron in GaAs,-,P,: N as a function of crystal composition .Y. The increase in Id(0)l’ for decreasing x results in an increase recombination rate as E , approaches EN.(After Campbell et

1.

,


49

Energy ( e V ) 1;s

1.B"

If'

I;XII

1.9s

I

1

I

I

I

J 2

7 11

hH

66

6.4

2

61

Wavelength ( l o ' # )

FIG.26. Photoluminescence spectra ( 7 7 K ) of .v = 0.24 GaAs,-,,P,:Te possessing a large (10'" cm3) doped-in electron density. The peak of the emission spectrum is -80 meV greater

than E,(r)= 1.80 eV. At higher excitation level ( 5 x 10' W cm') the edge-to-edge modes increase and laser operation occurs with /I\, > Ex(r). (After Lee t'i t r / . ' h )

tion efficiency of GaAs,-,P, in the indirect-gap composition range. On the other hand, it may also detract from the radiative recombination process under some conditions, namely, in the portion of the direct region where ordinarily the indirect ( X )minima have no effect. This fact is made clear from photoluminescence experiments on heavily Te-doped (lo' 9/cm3) and Ndoped ( - 1019/cm3).Y = 0.24 G ~ A S , ~ , P , : N ' .The ~ ~ crystal composition is chosen so that N, states, if still "attached" to the indirect ( X ) conduction band minima (at x = 0.24), should lie in the continuum above the direct (r) conduction band minimum.'34 If N, states are dominant, as is likely,'27 recombination will occur from below the band edge. Otherwise-identical N-free GaAsIP,P, samples are used for comparison. Photoluminescence spectra (77 K ) obtained on N-free x = 0.24 GaAslPx P,:Te samples are shown in Fig. 26. First to be noted on the bottom solid

50

N . HOLONYAK. JR., AND M. H . LEE

curve is the location of the band-gap energy E,(T). The peak of the emission spectrum obtained at low excitation levels is 80 meV greater than E,(T) = 1.80 eV, as would be expected from the behavior of curve n of Fig. 7. As the excitation level is increased, the edge-to-edgemodes on the lower energy side and the peak of the spectrum become more pronounced, with laser operation occurring on the lower-energy side of the spectrum (but at hv > Eg)at a pumping level of 5 x lo4 W/cm2. The presence of modes at low excitation levels indicates fairly low absorption of the emitted radiation due to the The laser spectrum of this Moss-Burstein shift of the absorption heavily donor-doped GaAs,,P, sample is in fact similar to that of comparably doped GaAs (cf.. Fig. 8). The comparison photoluminescencespectra (77°K)obtained on otherwiseidentical N-doped x = 0.24 GaAs,,P,:N+:Te are shown in Fig. 27. The purpose of the heavy Te doping (EF > EN E , ; cf. inset) is to insure a large supply of electrons in the N-trap states. This should favor laser operation, if possible at all, on the N-trap states. The dashed reference (taken from the bottom curve of Fig. 26) shows the spectrum of N-free GaAs,,P,:Te at an excitation level of 500 W/cm2. Compared to the reference, the spectrum of the GaAs l-xP,:N+:Te sample is shifted toward lower energy by -25 meV, an effect due mainly to the N doping in the crystal and the Nr state (Fig. 19).12’ At an excitation level of 8 x lo4 W/cm2, laser operation occurs on the lower-energy side of the emission spectrum just below E,(T), again suggesting the dominance of the Nr level. Relative to the N-free crystal (Fig. 26), the laser operation in Fig. 27 is very much reduced in energy because of the N doping. Also, in spite of the large doped-in supply of electrons, laser operation occurs in a region of lower density of states, below the band edge rather than above as in Fig. 26. To determine further the role of the N-trap states in radiative recombination, photoluminescence data have been obtained on samples of the same Te+N-doped GaAs,-,P, (x = 0.24) that have been converted to p-type by Zn-diffusion. Conversion of the sample to p-type drains electrons from the N-trap states, and fast intraband scattering in the band tail4’ insures that recombination occurs from donor tail states and not from the N levels. In terms of highest photoluminescence efficiency and lowest laser threshold, the best sample is the one doped with only Te, followed by the one doped with Te + N + Zn. Since Zn-diffusion, creating more defects, is known to detract from the luminescence properties of GaAslpxPx,this order suggests that the N impurity itself does not unduly damage the crystal. The laser threshold is in fact lower for the Zn-diffused sample than for the sample doped with just Te+N. The point is that in the Zn-doped sample the N does not play an active role; that is, the N levels lie above the donor tail states involved in the recombination process. In GaAs,-,P,: N + :Te, however, the N-trap states

-

-

1.

51

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS Energy (eV)

,

175

I85

1.80 I

I

I90

I95

I

!

I

L

I

I

12

70

68

66

I

I 64

Wavelength (IO'A)

FIG. 27. Photoluminescence spectra (77 K ) of Y = 0.24 GaAs,-,P,:N+ + Te with a large doped-in electron population (1019/cm3)and a large N-trap density ( N + 10i9/cm3).The recombination involves N-trap states ( N r and ? Nx), Note the reference spectrum (dashed 500 Wicm') of the same crystal without N doping ( N + layer removed), and also the laser operation(8 x 104 W/cmz)which occursat hv E,(T)orwell below theenergyofthesampleofFig. 26. (After Lee et ~ 1 . ' ~ )

-

-

are involved in recombination and play a detrimental role in the radiative recombination process (because of the nature of the trap). This agrees with heterojunction data on the behavior of the N, state (A-line) in x = 0.38 and 0.40 ~ r y s t a l s . The ~ ~ detrimental ~'~~ effect of the N occurs because of the large spread in the N-trap wavefunction. In other words, for x < x,, the N trap dilutes or robs from direct recombination. The N trap insures that there is both direct and indirect recombination for x 5 xc,which is an advantage for x 2 x, but not for x < x,. 135

J. J. Coleman, N. Holonyak, Jr., A. B. Kunz, W. 0. Groves, D. L. Keune, and M. G. Craford, Solid Slate Commun. 16, 319 (1975).

52

N. HOLONYAK, JR., AND

M. H .

LEE

6. INDIUM GALLIUM ARSENIDE AND INDIUM ARSENIDE PHOSPHIDE

Just as the binary GaAs can be modified to become the wider-gap material GaAs,-,P, by the substitution of the lighter atom P for As (as in Fig. 19), similarly it can be shifted to narrower gap with the substitution of heavier elements for lighter elements. For example, with the addition of In to the Column I11 sublattice, GaAs shifts to lower band gap, becoming In,Ga,,As. Likewise, G a substituted for In in InP yields the wider-gap crystal In,_,Ga,P (cf., Fig. 16)and As substituted for the lighter atom P yields the narrower-gap material InAs,P,-,. As shown when junction lasers were first demonstrated (1962), alloy semiconductors can be successfully fabricated into p-n junction lasers,3 including also In,Galp,As 91 and InAs,P1-x.93 These two materials with In I-xGa,P,_,As, are now perhaps of greatest interest in the neighborhood of 1.1 pm (1.13 eV), i.e., in the region of the spectrum of lowest transmission loss in optical fibers and waveguides. Also, this is a convenient spectral region for infrared sources and detectors for use in integrated optics based upon the use of GaAs and InP substrates.’ 36 For these applications In,,Ga,P,-,As, should be mentioned but this quaternary alloy is less well developed but is likely to receive much attention because it can be used in Al-free double heterojunctions. Besides being studied in p-n junction lasers,91q92137 In, G al-,As has been the subject of some photopumping experiments. The most important of these are the room-temperature experiments of Rossi and Chinn34 in which the pump source is a tunable parametric oscillators and the sample is -20 pm wide and 5-10 ,urn thick. Note that such small dimensions, or smaller as in most of the data presented above, are typical of photopumped semiconductor lasers, which are possible because of the high gain that is characteristic of such a laser (see Section 11). To decrease surface losses and invert the carrier population to as great a depth as possible, the pump photon energy is selected to be only -kT ( = 26 meV, 300°K) higher than the energy gap E, . This does not necessarily lead to the lowest pumping power needed to achieve stimulated emission in In Gal-,As,, but is advantageous for appreciable emission from the sample since its volume (still small) is utilized more effectively.Although the sample is quite small, it is possible to obtain a pulsed output of more than 1 W with a total power conversion efficiency of 3 -4%.34 Typical emission spectra obtained well above threshold on x = 0.06 In,Galp,As are shown in Fig. 28.34 136

13

G . E. Stillman, C. M. Wolfe, and I . Melngailis, Appl. P h y ~Lerf. . 25, 36 (1974). M. Ettenberg, C. J. Nuese, J. R. Appert, J. J. Cannon, and R. E. Enstrom. J. Eleclron. M u m . 4, 37 (1975).

1.

53


I

ll

FIG.28. Laser emission spectra (300 K ) of In,Ga,_,As (.w = 0.06)pumped with an optical parametric oscillator at energy E, E , + kT (ipUmp) = 9350 A). (After Rossi and C h i r ~ n . ~ ~ )

-

I

1

9750

1

I

I

9700

9653

9600

I 95M)

I

9500

WAVELENGTH (A)

A interesting property of th ternaries In,Ga,,As and InAs,P,-, is that their electron effective masses are low, leading to a low effective density of states N,. Since these materials can be doped to appreciable levels (nd > 1018/cm3),the Fermi level E , can be shifted well up into the conduction band, making possible photoluminescence over a very wide spectral range. For example, for an x = 0.13 InAs,P,_, crystal (me 0.071m0) of doping nd 3 x 1018/cm3,we estimate (77'-'K)that the Fermi level lies 100 meV above the band edge.'38 Photoluminescence from such a sample should occur in a range A2 700 A, which is fairly close to that observed in Fig. 29. The wide spread in the emission of the InAs,P,.-, sample of Fig. 29 makes it possible to couple another sample (CdS)to the active sample and observe the effect that a low-loss auxiliary cavity introduces in the laser operation. As shown, both platelet samples are imbedded into indium.16 The InAs,P,-, platelet is 1-2 pm thick and has a width I, 130 pm. The CdS platelet is 67 pm wide and has a thickness w, 20 pm, which is relatively large compared to that of the active sample. Notice that at an excitation level of 4 x lo3 W/cm2 (mode-locked Ar' laser, 5145 A) broad modes of 90 A spacing are observed. These modes correspond properly to the CdS thickness dimension w,. If w, is reduced, the modes are broadened still further and finally (w,= 0) are determined in spacing by only the active sample thickness. In

-

-

-

'"

-

-

-

-

R. D. Dupuis, N. Holonyak. Jr., H . M. Macksey. and G. W. Zack. J. Appl. Phys. 43.3801 (1972).

54

N . HOLONYAK, JR., AND M. H . LEE Energy ( e V )

1 S

Y I

10.0

I

9.8 9.6 Wavelength (lo3A )

I

< 2

9.4

FIG.29. Photoluminescence spectra (77 K ) showing spatially orthogonal mode-coupling ellectsinan InAs,P,-,(.u 0 . 1 3 , ~=~ 3 x 10'*/cm3)plateletrnountedonaCdSauxiliarycavity. At an excitation level of 4 x lo3 Wkm*. broad modes of -90 ,& spacing corresponding to the CdS thickness w, = 20 pm are observed. At an excitation level of 4 x lo4 W/cm2, edge-toedge (longitudinal)laser oscillations of the active InAs,P,.., platelet occur on one of the peaks of the thin-dimension broad modes of the CdS. (After Dupuis er u I . ' ~ ' )

-

other words, the thickness dimension of the sample is a source of cross-modes, which are not observed if the sample thickness is small enough or if the emission width is small. Depending upon the focus of the excitation source and its position on the sample of Fig. 29, small edge-to-edge modes exist across the entire spectrum of broader modes. At a sufficient excitation level (4 x lo4 W/cm2),edge-toedge (I,) laser oscillations occur on one (Fig. 29) or more of the peaks of the broader modes. For I, 150 pm and AA(1nAsP) 7 A, the mode spacing 5. For laser diodes diffused into the expression (8) yields (n - Ah/&) same InAs,P,-, crystal, at higher injection levels and more band filling (n - Ah/&) 4.6. This is reasonable agreement and taken with the fact that the CdS dimensions cannot account for AA 7 A [Fig. 29 and Eq. (S)] insures that the CdS affects only the broader mode spacing. It is evident from Fig. 29 that the CdS platelet enhances recombination selectively. That is, the close-spaced edge-to-edge InAs,P,-, modes are strongest at the peaks of the broader thin-dimension CdS modes. The effect of the CdS auxiliary cavity is to exert a certain wavelength selectivity on the laser operation of the active

-

-

-

-

-

1.


55

sample. We note finally that an auxiliary cavity can be coupled to the active sample in such a way as to affect the edge-to-edge (longitudinal) modes themselves (cf., Dupuis et ~ 1 . ' ~ ~ ) . V1. Carrier Lifetime

Accurate characterization of the radiative processes vital in light-emitting semiconductors and lasers requires a knowledge of the lifetime of excess carriers undergoing radiative and competing nonradiative recombination. For the study of direct semiconductors, in which laser operation is possible, the lifetime measurement technique must be sufficiently fast to detect recombination processes extending down into the subnanosecond range, or else an indirect measurement process, such as the transmission measurements of Section 1 (Figs. 9 and lo), must be employed. Preferably the lifetime data should be obtained on homogeneous samples. The behavior of radiative lifetime in p-n junctions, for example, is complicated frequently by nonuniform carrier distribution, nonuniformly doped regions, or often regions of uncertain doping. A direct method to determine short radiative lifetimes is the optical phase shift method of measurement3' discussed in Part 111. This method takes advantage ofthe short repetitive excitation pulses available from, for example, a mode-locked gas laser; only the phase difference between the fundamental Fourier component of the recombination radiation and the excitation source need be determined. This in turn permits the use of a relatively slow detection system. An optical excitation source generates also essentially an instantaneous excess electron-hole population, which is not the case in p-n junctions. This is an obvious advantage in many measurements. As discussed in Part I11 laser operation of photopumped samples can generally be obtained only on very thin (1-2 pm) platelets. Owing to the thinness of the samples, surface recombination may have a large effect on the measured lifetime T, and in some cases corrections must be made in order to obtain bulk carrier lifetimes. Other factors not associated with bulk carrier lifetime per se may also distort the measured lifetime. These include the effects of inhomogeneous pump absorption, carrier diffusion in some cases, and variable absorption of the recombination radiation. The effect of these factors on the measured phase angle (and lifetime) can be determined for uniformly excited thin platelet samples' 3 9 and semi-infinite samples. 140 Figure 30 shows the behavior (the influence) of the surface recombination velocity s as a function of the measured phase angle 0 for GaAs excited with a 139

H.

R.Zwicker, D. L. Keune, N. Holonyak, Jr., and R. D. Burnham, Solid-State Electron.

14. 1023 (1971). I4O

C. J. Hwang, J . Appl. Phys. 42,4408 (1971).

56

N . HOLONYAK, J R . , AND M . H . LEE 10’

103

0

50

0

Meosured

Phase

Difference

50

8 (deg)

FIG.30. Effect of surface recombination velocity s on optical phase-shift angle 0 of a 2-pm GaAs platelet. In panel (a) the ambipolar diffusion constant D* is varied and has small effect for s < lo5 cm/sec; in (b) the thickness of the platelet is varied and for the GaAs parameters chosen rapidly approaches semi-infinite behavior for I 10 pm; and in (c) the absorption constant a is varied and has a very weak effect. The other constants remain the same from (a)through (c)except that D* is reduced from 20 cm*/secto 5 cm3/sec in panel (c).(After Zwicker et

He-Ne laser (6328 A).’ 39 The parameters affecting the measurement include the ambipolar diffusion constant D*,the absorption constant 01, and the sample thickness 1. The excitation pulses are assumed to be &functions in time with a period of t, = 1.33 nsec; as shown a bulk carrier lifetime of z = 2.1 nsec corresponds to a phase angle of 6' = 45" for s = 0. The results of Fig. 30 indicate that for s 5 lo4 cm/sec little difference exists between the actual bulk carrier lifetime and the measured lifetime. For larger surface recombination velocities, the measured phase angle may decrease considerably, depending on the material parameters (but with the effect of c( small). For uniformly excited semi-infinite samples, the reabsorption of the emitted photon and the carrier diffusion length may have a large effect on the measured phase angle. 140 These effects can be important even for measurements on thin platelets when the photoexcitation pattern on the sample is not uniform and carrier diffusion occurs. While calculations of the effect of various limiting parameters on the measured phase angle involve simplifying assumptions, the results nevertheless indicate the trends that can be expected. In particular, the effect of surface recombination can easily be seen experimentally. Optical phase-shift mea-

1.


57

surements on lightly doped “surface-free” GaAs excited with the mode-locked output of a He-Ne laser (6328 A) show that the measured “lifetime” is -2.u as long as that obtained on similarly doped GaAs with unprotected (polished and etched) surface^.^ The “surface-free” condition can be attained, for example, by sandwiching the GaAs layer between heavily doped Ga,-,AI,As and GaAs layers in a Galp,A1,As (n+)/GaAs(n)/GaAs(n+)configuration.’ Similar results have been obtained also on GaAs,-,P, samples with protected surfaces.’ O Even without meticulous corrections for the various factors affecting the measured lifetime, much information can still be obtained from lifetime data. Optical phase-shift measurements on GaAs,_,P, show that a good estimate of the defect density in the bulk of the material can be obtained by observing the dependence of the measured phase angle on the excitation i n t e n ~ i t y . ’ ~ ~ The bulk carrier lifetime T is given by l/T =

l/rR

+

l/tNR,

(11)

where zR and TNR are the radiative lifetime and the nonradiative lifetime, respectively. The nonradiative lifetime T N R is assumed to be of the form ~ N d t= )

1/CN,(t).

(12)

where N,(t) is the density of traps available at time t and C is a constant. N, = NT(0)denotes the total trap density and T N R = t N R ( 0 ) the corresponding lifetime. If N T is small compared to the density of carriers An that can be created in the excitation process, the traps can be saturated. The measured lifetime at high excitation levels would then mainly reflect T R . At low excitation levels N T > An, and the measured lifetime is largely influenced by rNR. Figure 31 shows calculated phase-shift data as a function of the electron or hole density generated. If nonradiative recombination is negligible (TNR = m), the measured lifetime (dashed curve) is governed mainly by the bimolecular recombination rate for spontaneous emission r, where I’ =

BAn(An + no).

(13)

Here B is the band-to-band recombination constant and no the doped-in electron (or hole) concentration. For zNR = 1.5 nsec (solid curve) the phase shift decreases initially with increasing excitation, but increases as An approaches N,. As An increases further, the phase-shift curve approaches that obtained for TNR = m. Lifetime data (77 ‘ K )obtained on x = 0.28 GaAsIp,P, follow closely the behavior shown in Fig. 31 with N , z 3.5 x 10’6/~m3.’41 Above laser threshold the average carrier lifetime decreases. This is observed in many types of experiments, including optical phase-shift lifetime 14

H. R.Zwicker, D.

R. Scifrea. N . Holonyak. Jr.. R. D. Dupuis. R. D. Burnham. J . W. Burd. and Zh. I . Alfgrov. S o l i d . S / t r / ~( ‘ o f ~ i i ) r i u i 9. . 587 (1971 ).

58

N . HOLONYAK, J R . , A N D M . H. LEE

TNR.

m

q

30

60

Phase Difference , B (degl

FIG.31, Calculated optical phase-shift data for samples with both bimolecular and nonradia= ce) tive recombination processes. Data for only bimolecular recombination processes (lNR and for a constant nonradiative lifetime T~~ = 1.5 nsec are illustrated by the dashed curves. Data for bimolecular recombination (constant B = 5 x cm3sec-') along with timedependent nonradiative trapping (trap density NT)of one carrier are given by the solid curves (no = l O I 4 cm-'). (After Zwicker et ~ 1 . ' ~ ' )

m e a ~ u r e m e n t s .Exactly ~ - ~ ~ how the increased radiative transition rate during stimulated emission results in shorter measured lifetime is not straightforward. Fortunately, however, a clearer understanding of this process can be derived from lifetime data obtained as a function of wavelength (or energy),22.23.37 which is an inherent capability of phase shift measurements. For example, the role of the N isoelectronic trap in radiative recombination in GaAs,-,P, is clarified by comparison of data obtained on N-free and Ndoped crystals. In order to introduce and understand the information contained in lifetime data, particularly on N-doped GaAs,,P,, we consider first data obtained on the somewhat simpler case of N-free n-type GaAs,-,P,. Photoluminescence lifetime and the corresponding emission spectra (77°K) obtained on x = 0.34 GaAs,_,P, (nd = 4.5 x lOI6/cm3) are shown in Fig. 32. These data are typical of those obtained on GaAs,,P, in this crystal composition and doping range. As the excitation intensity is increased from lo3 W/cm2 (dotted curve) to 2 x lo4 W/cm2 (solid curve), the emission spectrum broadens and shifts to lower energy, the well-known EHL effect.6L66 The corresponding lifetime spectrum shifts to lower values with increased pumping owing to predominantly an increase in bimolecular recombination. The trapping effect discussed previ~usly'~'(Fig. 3 1) is not present, probably

1. PHOTOPUMPED


59

t-ncrgy ( r V ) I.ns

I UII

I .vs

FIG.32. Photoluminescence lifetime TI;.) and corresponding emission spectra (77°K)obtained on z = 0.34 GaAs, J r . )id= 4.5 x 10'" cm'. The lifetime T(j.) shifts to higher values with decreased sample excitation. The behavior of the lifetime as a function of wavelength can be accounted for by known processes, the local maximum occurring near 171' E , (see text). (After Lee era/.")

-

because of the higher crystalline quality of the samples used to obtain lifetime spectra. The average lifetime varies considerably among samples, but the shape of the spectrum is always approximately the same for comparably doped crystals of similar crystal composition. The most notable and consistent feature of the lifetime spectra obtained on GaAs,-,P, crystals in the doping range nd 5 10"/cm3 is the presence of a sharp relative maximum. For the sample of Fig. 32, this peak lies at 6430 A (1.93 eV). The lifetime peak moves slightly to lower energy ( 2.5 meV) when the sample excitation is increased from lo3 W/cm* to 2 x lo4 W/cm2, while the emission peak shifts more ( -c 6.5 meV). With decreasing excitation, the emission peak approaches the relative maximum of the lifetime curve. While the emission peak at low excitation levels agrees well with the band-gap energy, its position depends on excitation level and is generally somewhat lower than E , owing to EHL interactions.6L66 The location of the local

- -

60


maximum in the lifetime suggests rather strongly that this peak is associated with the band edge. More important than the agreement in the lifetime and emission peaks is the large lifetime decrease on either side of the lifetime maximum, which indicates a difference in carrier scattering and luminescence characteristic^^^ and perhaps also a change in the absorption coefficient in the two regions.14' In lightly doped n-type GaAs,,P, crystals, the Fermi level lies at or below the conduction band edge, and the band edge appears fairly sharp (i.e., not appreciably disturbed by impurity states). If large changes occur in the carrier scattering and luminescence characteristics of the crystal, these should occur at or near the band edge. On either side of the relative maximum, the lifetime decreases. On the high-energy side the decrease appears to be due principally to the movement of the quasi-Fermi levels as the excess carriers recombine.22 On the lowerenergy side the decrease is probably mainly the result of EHL interactions64-66 ( E < E g ) and the associated effect of some stimulated emission. In either case the lifetime on either side of the band-gap lifetime maximum is lower because these states lose electrons by both scattering -and recombinati~n.~~ Significant changes in the behavior of the lifetime spectrum of n-type GaAs,-,P, can be observed as functions of the donor concentration and the crystal c o m p ~ s i t i o n For . ~ ~ example, some of these changes occur because of the effect of the indirect ( X ) conduction band minima, and the donor states associated with X, on the competition for carriers and hence on the radiative recombination processes in GaAs,-,P,. Spontaneous lifetime spectra, together with the corresponding emission spectra, provide in fact information on recombination and on scattering processes that are either difficult or impossible to obtain from spectral data alone. This is especially true of GaAsl-,P,: N in which the various scattering and recombination processes are not very well understood. While the lifetime spectra z(1) of N-free GaAs,-,P, vary predictably with the donor concentration and crystal comp~sition,~'similar spectra for Ndoped direct GaAs,-,P, vary in a different manner.'42 Figure 33 shows the photoluminescence lifetime and corresponding emission spectra (77°K) obtained on x = 0.40 GaAs,,P, : N + with nd = 5.7 x lOl7/crn3.The spectra of the same crystal, but with the N-doped layer removed, are also included for reference (dotted curves). Note that the lifetime spectrum of the N-free sample of Fig. 33 exhibits shallower changes compared to those of the more lightly donor-doped GaAs,,P, of Fig. 32. This is a consequence of the location of the Fermi level, which lies above the conduction band edge for nd 2 1017/cm3.37The carrier lifetime of the N-doped sample (Fig. 33) is nearly flat in the region of the r transition, and its average value is lower (z 5

1.

61

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS Energy ( e V l 5

06

FIG.33. Photoluminescence lifetime and corresponding emission spectra (77-K ) obtained on 5.7 x 10”’cm3. Spectra obtained on the same crystal. but with N-doped layer removed, are also shown (dotted) for comparison. The average lifetime in the region of the N, band is only -0.4 nsec. which is much lower than that in lightly donor-doped crystals. The mode structure in the lifetime and emission spectra of the GaAs,_,P,:N+ crystals corresponds t o a condition of stimulated emission. (After Lee et u / . ’ ~ ’ )

Y

= 0.40 GaAs,~.P,:N+. nd =

0.4 nsec) than for the corresponding N-free samples. For direct GaAs,-,P, :N crystals of various compositions and doping concentrations (N and impurity doping) for which N, states lie below the r band edge (0.28 < .Y 5 xc). the average lifetime in the region of the T-N, transition is always 0.2 nsec at moderate excitation levels. This suggests that the lifetime in this wavelength region is controlled mainly by the N impurity. Below the energy region of T-N, transitions, the lifetime increases with wavelength. For more lightly doped n-type GaAs,,P,: N, the lifetime on N, transitions increases much more rapidly and is approximately constant ) several beyond the peak of the N, emission. In this region ~ (isiusually nanoseconds. If the crystal is converted to p-type by Zn diffusion, the lifetime spectrum shifts to lower energy by E,, 2 30 meV compared to n-type samples,

-

62

N. HOLONYAK. JR., AND M. H. LEE

but is otherwise similar in magnitude and behavior to lightly donor-doped GaAs,-,P,: N throughout the spectrum regardless of the donor doping ~oncentration.'~~ At an excitation level of 2 x lo4 W/cm2, the GaAs,_,P,:N sample of Fig. 33 exhibits laser operation. The lifetime spectrum in the region of the laser modes decreases while the remainder of the spectrum shows little change. Notice that cavity modes are seen in the emission spectrum from the peak of the T-N transition to below the peak of the Nx emission. These local emission maxima correspond to the local minima in the lifetime spectrum. The modes in the emission spectrum indicate that reabsorption in the spectral region EN < E is sufficiently low that photons can make multiple passes in the cavity. This situation is not uncommon, particularly in p-type and in more heavily doped n-type GaAs,-,P, crystals. We note that the presence of modes in the emission spectrum is not always accompanied by modes in the lifetime spectrum. Samples with mode structure in the emission but not the lifetime spectrum can be made to show modes in both spectra, however, by increasing the excitation level. This suggests that the presence of modes in lifetime spectra indicates stimulated emission. Extensive data on N-free2Z.37and N-doped GaAs,-,P, 22*142 and on In,,Ga,P23 indicate that the decrease in the lifetime at the laser mode energies is related principally to the rapidity with which stimulated emission turns off, which can be fast compared to the spontaneous lifetime.37The speed with which stimulated emission turns off depends in part on the sample absorption at the laser photon energies and on the resupply of carriers to the states undergoing stimulated emission (after the excitation pulse). For lightly doped N-free GaAs,,P,, as for GaAs, laser operation (77°K) usually occurs within 20 meV of the absorption edge (cf., Fig. 7). The effective gain below the band gap is large at laser threshold due to EHL interactions, but decreases rapidly with stimulated emission because of the decreasing excess carrier population. The measured lifetime-decrease at laser energies (hv E, 20 meV) just above threshold may be 20.2 nsec. For p - t ~ p e ' ~or ' heavily donor doped (na 2 3 x 101*/cm3)N-free GaAs,-,P,, laser operation usually occurs at energies 230 meV below the absorption edge, and the observed lifetime-decrease caused by stimulated emission may be negligible (righi @ 1'179 hy Academic Press. Inc . All nghts of reproduction in any form reserved. ISBN 0-12-752114-3

66

HENRY KRESSEL AND JEROME K. BUTLER

28. Oscillations Related to Nonuniform Population Inversion . . . . . . . . . . . . LISTOF SYMBOLS . . . . . . . . . . . . .

19I 192

I. Introduction This chapter is concerned with the design and operating characteristics of heterojunction semiconductor laser diodes. Stern, in an earlier chapter in this series,’ discussed the fundamental aspects of stimulated emission in semiconductors and, in particular, homojunction laser structures. Figure l a shows the general dependence of the optical output from a laser diode on current. The key parameters are the threshold current Ith, which depends on the diode area and the threshold current density Jth, and the differential quantum efficiency uext.Both Jrh and vex,depend on the internal device structure and vary with temperature. The addition of one or more heterojunctions to laser diodes, first demonstrated in 1968 in the AlAs-GaAs alloy system, has resulted in major improvements of their performance, flexibility, and emission wavelength range. In particular, the threshold current densities at room temperature have been reduced by orders of magnitude (see Fig. lb), permitting cw operation. The concurrent improvements in the materials technology have allowed the construction of useful devices for a variety of applications. An extensive literature deals with various aspects of optoelectronic devices, and several booksZ.2a-2c and review^^.^"^^ offer introductory treatments. It is not possible to cover all aspects of laser diodes because of the increasing sophistication of heterojunction structures and the wealth of literature published in recent years. The objective of this chapter is to concentrate on basic F. Stern, Stimulated emission in semiconductors, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 2. Academic Press, New York, 1966. A. Yariv, “Quantum Electronics.” Wiley, New York, 1975; J. 1. Pankove, “Optical Processes in Semiconductors.” Prentice-Hall, Englewood Cliffs, New Jersey, 1971. Za C. H. Gooch (ed.), “GaAs Lasers.” Wiley, New York, 1969. Zb C. H. Gooch, “Injection Electroluminescent Devices.” Wiley, New York, 1973. 2c T. S. Moss, G. J. Burrell, and B. Ellis, “Semiconductor Opto-Electronics.” Wiley, New York, 1973. M. H. Pilkuhn, Phys. Status Solidi 25, 9 (1967). H. Kressel and J. K. Butler, “Semiconductor Lasers and Heterojunction LEDs.” Academic Press. New York, 1977. 3b H. Kressel, Semiconductor lasers: Devices, in “Laser Handbook” (F. T. Arecchi and E. 0. Schulz-DuBois, eds.). North Holland Publ., Amsterdam, 1972. 3c P. G. Eliseev, Kuantooaya Elrktron. 2. 3 (1972) [English transl.: Sou. J . Quantum Electron. 2 , 505 (1973)l. 3d M. B. Panish and I. Hayashi, in “Applied Solid State Science” (R. Wolfe, ed.), Vol. 4. Academic Press, New York, 1974.

2.

HETEROJUNCTION LASER DIODES

67

a" I3

n I-

3 a

W

Bn SPONTANEOUS EMISSION

DIODE CURRENT

FIG. la. Schematic curve of power output as a function of diode current. which defines the threshold current and the differential quantum efficiency.

HOMoJUNCTlONS

-

1AIGa)Ar-GaAs

HETEROJUNCTIONS

lo'

1965

1967

c

1969

1971

1973

1975

YEAR

FIG. 1 h. Historical decrease in the threshold current density of homojunction GaAs and

(AIGa)As-GaAs heterojunction laser diodes.

design and operating aspects of the more important structures. Part I1 reviews briefly the evolution of these structures. Part I11 concentrates on radiation guiding in heterojunction lasers, while Part IV considers the relationship of the electrical and optical properties to the threshold current density. Part V reviews some of the basic technological

68

HENRY KRESSEL AND JEROME K . BUTLER

and materials parameters in the construction and design of laser diodes. Part VI discusses the broader aspects of heterojunction laser fabrication in alloy systems in which lattice matching is more difficult to achieve than in the Al As-GaAs system. Factors influencing reliability and the achievement of long operating life are discussed in Part VII. State-of-the-art characteristics of lasers for various major applications are discussed in Part VIII. Part IX discusses distributed-feedback lasers, and Part X reviews transient effects. 11. Laser Diode Structures 1. LASERTOPOLOGY

Contemporary laser topologies are shown in Fig. 2. In general a FabryPerot resonator is formed by cleaving two parallel facets which produces a

FIG.2 . Laser diode geometries: (a) broad-area diode with sawed sides and cleaved facets; (b) stripe-contact diode; (c) model for dielectric waveguide.

2.


69

resonant condition for lasing along the cavity axis (direction perpendicular to the facets). A reflecting film is sometimes placed on one facet to increase the useful output at the opposite one. Waves propagating parallel to the cleaved facets must be suppressed by the introduction of high losses peculiar to them. For broad-area contacts, sawing the sidewalls achieves this. In the stripe-contact diode, the active junction area is restricted by the carrier flow from a stripe. This restriction produces a small change in the dielectric constant thus forming a dielectric waveguide confining the optical field to a region below the stripe. Various methods of forming stripe-contact structures are discussed in Part V. 2. VERTICAL GEOMETRY

The laser diode requires an active region in which electron-hole pair recombination generates the optical flux and a mode confinement region which overlaps the active region. The optical confinement is controlled by the refractive index profile. The extent of the recombination region is limited either by the minority carrier diffusion length or by a potential barrier to minority carrier outdiffusion. In heterojunction lasers the potential barrier at the interface is several kT high, whereas in homojunction lasers a small change in potential associated with p+-p or n - n impurity distributions provides limited carrier confinement. The realization that an internally formed dielectric waveguide was essential for laser operation came early in the history of the semiconductor laser.44b However, it was only with the development of lattice-matching AlAs-GaAs alloys that heterojunction structures could be constructed in which the magnitude of the dielectric discontinuity was greatly increased (without lattice defect formation), thus greatly increasing the optical confinement to the vicinity of the recombination region. With the flexibility now made possible with the multilayer epitaxial technology (Part V), it is possible to construct lasers in which the waveguide region either practically coincides with the recombination region or extends a controlled distance beyond it. Figure 3 shows a section through the five-layer structure which can be considered the generalized laser diode. The laser consists of a radiative recombination (active)region 3 (width d3)4cin which the inverted carrier population produces the required optical gain, and a waveguide region for optical confinement of thickness do = d 2 d 3 d4. The higher bandgap regions 1 and 5 provide optical barriers because of their reduced index of refraction at the lasing photon energy of the recombination region.

+ +

A . L. McWhorter, Solid-State Electron. 6, 417 (1963). A. L. McWhorter, H. J. Zeiger, and B. Lax, J . Appl. Phys. 34, 235 (1963). 4b A. Yariv and R . C. C. Leite. Appl. Ph.v.7. Leu. 2. 55 (1963). 4c The nomenclature of d, will be used throughout for the recombination region width 4a

70

HENRY KRESSEL AND JEROME K. BUTLER REFRACTIVE INDEX, n

As RECOMBINATION

I

I

n-TYPE

I

n-TYPE

I

I

I

I I

I

p-TYPE

I

p-TYPE I

I -do-

FIG.3. Cross section of generalized laser structure showing the refractive index at the lasing wavelength. The radiativerecombinationoccurs in region 3, whereas regions 2,3, and 4 constitute the nominal waveguide.

Figure 4 shows the idealized cross section of important classes of laser diodes using from 1 to 4 heterojunctions. For each structure, we show the energy diagram, the refractive index profile, distribution of the optical energy, and the position of the recombination region. These structures (listed in historical stage of evolution) are of increasing complexity : (a) In the homojunction laser' ' there are no abrupt index steps for optical confinement or high potential barriers for carrier confinement. The recombination region is determined by the minority carrier diffusion length. The limited radiation confinement results from small index gradients produced by dopant concentration gradients and carrier concentration differences. Typically a p+-p-n configuration is used, where the p + - p interface provides a small potential barrier. (b) In the single-heterojunction (close-~onfinement)~*~ diode, a pf-p heterojunction forms one boundary of the waveguide as well as a potential barrier for carrier confinement within the p-type recombination region. The index step at the p + - p heterojunction is much larger (typically a factor of 5 ) than that at the p-n homojunction. Thus, this is an asymmetrical waveguide. The threshold current densities are typically $ to f of the homojunction values at room temperature (- 10 kA/cm2 versus 50 kA/cm2). (c) In the double-heterojunction (DH) laser the recombination region is bounded by two higher bandgap regions to confine the carriers and the

-

R. N. Hall, G . E. Fenner, J. D. Kingsley, T. J. Soltys, and R. 0. Carlson, Phys. Rev. Left. 9, 366 (1962). ' M. 1. Nathan, W. P. Dumke, G. Burns, F. H. Dill, Jr., and G. Lasher, Appl. Phys. Lett. 1, 63 (1962). T. M. Quist e f at., Appl. Phys. Lett. 1, 91 (1962). H. Kressel and H. Nelson, RCA Rev. 30,106 (1969). I. Hayashi, M. B. Panish, and P. W. Foy, IEEE J . Quantum Elecfron. 5, 211 (1969).

'

2.

71


1

(8)

(0

FIG.4. Schematic cross section of various laser structures showing the electric field distribution E in the active region, variation of the bandgap energy E , and of the refractive index n at the lasing photon energy. (a) Homojunction laser. (b) single-heterojunction "close-confined" laser, (c) double-heterojunction laser, ( d )large optical cavity (LOC)laser, (e) very narrow doubleheterojunction laser, and (f) five-layer heterojunction laser with centered recombination region (four heteroj unct ions).

radiation. The device can be made either symmetrical or asymmetrical. In the symmetrical heterojunction barrier case, referring to Figs. 3 and 4, d4 = dz = 0, n, = n5, and d3 = do. The first reported DH laser had J,, = 4OOO A/cm*.'' The reduction to a threshold current density at room temperature of about 2000 A/cm2 was thereafter achieved with GaAs' ' and with (A1Ga)Asin the recombination region. lo

'I

l2

Zh. I. Alferov. V. M. Andreev, E. L. Portnoi, and M . K . Trukan. Fiz. Tekh. Poluprov. 3, 3. 1107 (197O)l. 1328 (1969). [Etiylish / r u m / . : S O I . .P/i~..\.-Scr,iic,oritl. M. B. Panish, 1. Hayashi, and S. Sumski, Appl. Phys. Lett. 16, 326 (1970). H. Kressel, and F. Z. Hawrylo, Appl. Phys. Lett. 17. 169 (1970).

72


(d) In the large-optical-cavity (LOC) laser, d2 = 0, n3 z n4, and do = d3 d4. The waveguide region is wider than the recombination region which occupies one side of the space between the two major heterojunctions.I3 The device was basically intended for efficient pulsed power operation. (e) The very narrowly spaced double-heterojunction laser is a subclass of the basic DH device, but it is designed with an extremely thin recombination region d3 < 0.3 pm, with n , and n5 adjusted to permit the wave to spread This is to minimize outside d3 but still provide full carrier ~0nfinement.l~ the beam divergence while keeping .Ith< 2000 A/cm2, a value desirable for room-temperature cw operation. (f) In the four-heterojunction (FH) laser, five regions are included. A submicron thick recombination region is bracketed by two heterojunctions that are in turn enclosed by two other heterojunctions.I5 By adjusting the refractive index steps between the recombination region and the adjoining regions 2 and 4,the fraction ofradiation within d, and the interaction with the two outer heterojunctions can be adjusted. This structure allows the maximum design flexibility.

+

The properties of the various heterojunction structures will be discussed in detail in the subsequent sections of this chapter. 3. CARRIER CONFINEMENT AND INJECTEDCARRIER UTILIZATION

The recombination region of the DH and FH lasers can be p-type or n-type. At room temperature, a bandgap energy difference of at least 0.2 eV is desirable for good carrier confinement. If the barrier is too low, carrier loss from the recombination region increases the threshold current density. Consider the simpiest case of low injected minority carrier density in the low bandgap energy E,, side, and equal dopant densities on the sides of a heterojunction. The illustration of Fig. 5a for a p - p heterojunction shows that the potential barrier confining electrons is Q1 2 E S 2 - Es3 z AE,.I6 The situation is more complex when the doping levels differ on the two sides of the heterojunction, and the quasi-Fermi level shift in the low bandgap side is substantial because of high injection. The illustration of Fig. 5b is appropriate for this case. The effective barrier height is now reduced, as indicated, to reflect the relative positions of the Fermi and quasi-Fermi levels with respect to the band edges. It is particularly important to note that a low hole concentration in the high bandgap side 2 shifts the Fermi level into the gap in that region which reduces 4. Furthermore, a high electron H. F. Lockwood, H. Kressel, H. S. Sommers, Jr.. and F. Z. Hawrylo, Appl. Phys. Lett. 17, 499 (1970). l 4 H. Kressel, J . K. Butler, F. 2. Hawrylo, H. F. Lockwood, and M. Ettenberg, RCA Rev. 32, 393 (1971). l 5 G. H. B. Thompson and P. A. Kirkby, IEEE J. Quantum Electron. QE-9, 31 1 (1973). H. Kressel, H. F. Lockwood, and J . K. Butler, J . Appl. Phys. 44,4095 (1973).

l3

2.

73


-

RECOMBINATIONREGION @

'

(b) h I--

w

-

W

REDUCED RECOMBINATION VELOCITY, SLelO ( C )

FIG.5. Electron confinement by an ideal p-IJ heterojunction between materials with bandgap energy E,, and E g 3 .respectively. (a)Equal doping on both sides and low electron injection. The barrier height 4 2 E , , - E g 3 .(b) Lower hole concentration in the high bandgap side combined with high injection into the low bandgap energy side. ( c ) Relative radiation recombination current ;'* as a function of the reduced recombination velocity; 4 is the potential barrier for minority carrier5 at the heterojunction: T = 300 K and 41 2 0.2 cV. (After Burnhani e / (//.I8)

concentration in the low bandgap active region can reduce 4 substantially. The upward shift of the quasi-Fermi level with injection will reduce the effective bandgap energy step by the approximate distance between the conduction band edge and the quasi-Fermi level given at T = O K by

6,

=

3.64 x 10-'s((m,/rn~)(N,)2'3 eV,

~ and N, is the where m,*is the effective electron mass ( 0 . 0 6 8 ~in~GaAs) injected electron density.16" For example, at T = 0°K. 6, = 61 meV for N , = 10l8cm-3 and, at 300 K, 4, = 40 meV. Ih.'

For 7 > 0 K. d, is obtained using I h c Frrmi integral.

74


The loss of carrier confinement and the resultant effect on the threshold current density has been discussed in detail.”.’ 7a In the case of (AlGa)As/GaAs heterojunctions an A1 concentration difference of as much as 50% may be needed to ensure full carrier confinement at temperatures 100°C. The minority carriers escaping over the confining potential barrier give rise to a diffusion current in the adjoining region which adds to the measured current at threshold, thus effectively increasing the threshold current density. A simple expression for this excess “leakage” current J,,, due to electrons escaping over a p-p heterojunction barrier can be derived assuming

-

6,, x 0;

E,, - E,, 2 4kT;

6,,

2

kTIn(P,/N,,),

(where P, is the hole concentration in region 2 and N,, is the effective density of valence band states in that region). Then J,,,

(eDe,/Le,)(N,,/P,)Nc,

exp[ - (AEg - d,,)/kT],

(1)

where e is the electron charge, D,, is the electron diffusion constant in the high bandgap p-type region 2, and L,, is the electron diffusion length in that region. If the width of region 2 is less than Le2,then that width replaces L,, in Eq. (1). A similar expression can be derived for J,,, due to hole loss across an n-n heterojunction barrier. In (A1Ga)As typical values are D,, = 50 cm2/sec,L,, = 2-10 pm, and N,, x 10’’ an-,. The above assumes that the heterojunction acts as an ideal barrier; i.e., a barrier without nonradiative interfacial recombination. In practice, interfacial recombination centers exist, and their importance depends on the width of the recombination region relative to the diffusion length. Burnham et a/.” and Eliseevlg have calculated the fraction of injected carriers that recombine radiatively for an abrupt heterojunction, and James” has studied the problem for a graded heterojunction. Assuming a uniform p-type recombination region with a high potential barrier (4 = 0.2 eV or >>kT)which repels electrons (placed a distance d , from the injecting junction), the fraction of the injected electrons which recombine radiatively, y*, is given by1

l7

l9

*’

D . L. Rode, J. Appl. Phys. 45,3887 (1974). A. R. Goodwin, J. R . Peters, M. Pion, G. H. B. Thompson, and J. E. A. Whiteaway, J. Appl. Phys. 46,3126 (1975). R. D. Burnham, P. D. Dapkus, N. Holonyak, Jr., D. L. Keune, and H. R. Zwicker, SolidState EIecrron. 13, 199 (1970). P. G. Eliseev, Sou. J . Quantum Electron. 2,505 (1973). L.W. James, J. Appl. Phys. 45, 1326 (1974).

2.


75

Here S is the interfacial surface recombination velocity (nonradiative centers are assumed); L, is electron diffusion length as limited by the bulk minority carrier lifetime T and diffusion constant D, [L, = (DT)”’]. (A similar expression holds for an n-type recombination region with appropriate L, and D.) Figure 5c shows y* as a function of SL,/D for various values of d3/L, at room temperature. Since L, is typically 2-5 pm in GaAs, it is evident that SLJD must be less than lo-’ for devices with the narrow recombination regions ( d 3 / L ,= 0.1) required for the lowest threshold laser diodes. For example, 90% utilization of the injected carriers in the case of d 3 / L , = 0.1 (assuming L, = 5 pm and D = 50 cm2/sec) requires S z lo3 cm/sec. This S value compares to S z lo6 cm/sec for a free GaAs surface. The materials requirements for obtaining low S values will be discussed in Part VI. We will not be concerned here with the nonideal current-voltage characteristics of the p-n heterojunction.” It is experimentally found that (A1Ga)As-GaAs forward-biased heterojunctions frequently exhibit tunneling current at relatively low bias-voltage values.22,22aFurthermore, junction surface leakage can be important. However, at high bias, thermal injection dominates and this is the operating regime of laser diodes. 111. Wave Propagation

4. GENERAL CONSIDERATIONS The cavity modes of the simple “box” laser cavity are separated into two independent sets of TE (transverse electric, E, = 0) and TM (transverse magnetic, H , = 0) modes. The modes of each set are characterized by three numbers, q, s, m, which define the number of antinodes of the field along the major axes of the cavity. By longitudinal modes we mean those associated with change in q at fixed m and s. In particular, q determines the principal structure in the frequency spectrum. Similarly, lateral modes are associated with changes in s which give the character of the lateral profile of the laser beam. Our major concern in this section is with the transverse modes associated with the index m, which involve the electromagnetic field and beam profile in the direction perpendicular to the junction plane. Throughout the discussion of the modal behavior of laser diodes, we will not concern ourselves with the internal dynamics of the lasing process. We neglect the effect of the optical field on the gain coefficient. Therefore, the 21

Extensive reviews are given by: A. G. Milnes and D. L. Feucht, “Heterojunctions and Metal-Semiconductor Junctions.” Academic Press, New York, 1972: B. L. Sharma and R. K. Purohit, “Semiconductor Heterojunctions.” Pergamon. Oxford, New York, 1974, L. J. van Ruyven, Ann. Rev. Marer. Sri. 2, 501 (1972). C. Constantinescu and A. Goldenblum, Phys. Status Solid ( a ) I , 551 (1970). J . F. Womac and R. H . Rediker. J . Appl. Phys. 43,4129 (1972).

76


parameters deduced are basically threshold values, although they provide useful insights into the lasing condition. a. Longitudinal Modes

Longitudinal modes are related to the length L of the cavity and to the index of refraction. Since the optical cavity is composed of regions with different indices of refraction, the propagating mode sees an averaged index n. The allowed longitudinal modes are determined from elementary considerations for a cavity of length L : qiL/2n= L ,

(3)

where iLis the free-space wavelength. Because the medium is dispersive, the Fabry-Perot (longitudinal) mode spacing is (AAL) =

- L t / [ 2 L ( n - Adn/dA)].

(4)

The mode spacing is typically a few angstroms. For a rigorous description of mode spacing and dispersive properties of dielectric waveguides see Kressel and Butler3"(p. 159).

b. Lateral Modes The lateral modes in the plane of the junction are dependent on the preparation of the side walls. These modes generally have small s numbers for typical narrow stripe-contact laser diodes. A detailed analysis of a loworder spectrum of stripe-contact laser diodes was made by Zachos and Ripper" and by Paoli et aLZ4who were able to associate with specific Hermite-Gaussian modes the fine structure of longitudinal modes. The wavelength separation due to the lateral modes is usually only a fraction of an angstrom unit compared to a few angstrom units for the wavelength spacing of the longitudinal modes. SommersZ5has similarly studied the modes of a broad-area laser and identified the various modes. In Section 21 we will discuss some practical implications of lateral mode change with current in cw laser diodes. c. Transverse Modes

Transverse modes (direction perpendicular to the junction plane) are confined by the dielectric variations perpendicular to the junction plane. The theoretical analysis of this problem is very important because the interaction between the propagating wave along the waveguide and the T. H. Zachos and J. E. Ripper, IEEE J . Quanfum Eleetron. QE-5.29 (1969). T. L. Paoli, J. E. Ripper, and T. H. Zachos, IEEE J . Quantum Electron. QE-5, 271 (1969). 2 5 H. S. Sommers, Jr., J . Appl. Phys. 44, 3601 (1973).

23 24

2.


77

optical gain region (recombination region) within the waveguide is a controlling factor in the laser characteristics, including the radiation pattern and the threshold current density. Several factors contribute to the dielectric constant profile at the lasing wavelength. Carrier density controls the plasma resonance frequency, whose contribution to the index can be predicted by considering the simple classical oscillator. Most of the major factors affecting the index can be understood from the Kramers-Kronig relation between the absorption coefficient and the refractive index which shows how a spatial variation of the absorption coefficient gives a corresponding variation of the index. Some of the specific factors affecting the index profile are: (1) The free carrier concentrtrtion. Free carriers depress the index. The dielectric discontinuity between a depleted region and one containing a density P of free holes is AE/E w ~ / ( where d wp is the plasma frequency (Pe2/mz&)”2and m; is the effective hole mass. A similar expression holds for electrons. ( 2 ) Negatitle absorption coeficient. A region of population inversion (i.e., of optical gain) has a higher imaginary component of the dielectric constant than a region without gain. (3) The shape of the absorption d y e . This depends on the doping level, and there is a related dependence of the refractive index in the vicinity of the bandgap energy.26 The dependence of the GaAs refractive index on enerm and doping level has been directly measured and also calculated from available absorption data. (4) The r‘ariation of the nomintrl buridgap by the herrrc+mctions. This introduces the largest differences in the refractive index. For example, it is easy to change the refractive index by many percent by changing the Al content of (A1Ga)As(see Part V). The very large changes in refractive index with bandgap are the basis of heterojunction laser design.

-

An early treatment of waveguiding in homojunction laser diodes was presented by Cooley and Stern.” Closed-form solutions of Maxwell’s equations have been made for a few specific structures with continuously variable index profiles in the direction perpendicular to the These solutions were possible because Maxwell’s equations reduced to differential equations with well-known solutions. However, numerical techniques make it feasible to solve the field equations for many index profiles.

*’ ’?

28 29 30

F. Stern. Pk,vs. Rer. 133. A1653 (1964). J . W. Cooley and F. Stern. IBM J . Rev. D r r . 9. 405 (1965). D. F. Nelson and J . McKenna. J . .4ppl. Phyx 38,4057 (1967). J . Hatz and E. Mohn, IEEE J . Qimiiuni Electron. QE-3. 656 (1967). R. G. Allakhverdyan. A. N. Oraevskii. and A. F. Suchkov. Fi:. Tekh. Poluprowdn. 4, 341 o r i277 [ / , (1971)]. (1970) [Erlgli.vh / r m x / .: S o r . P h ~ . . ~ - . S [ ~ r i i i c ~4.

78


The continuously variable index profile is appropriate for modeling diffused homojunction lasers where the p + - p interface is generally more extended than in diodes fabricated by epitaxial growth. A three-layer model was initially assumed for heterojunction lasers31-33 where the refractive index is constant in each of the layers, similar in concept to Anderson’s34 model for homojunction lasers. This was later extended to a five-layer which forms the theoretical basis for calculating the waveguidedependent properties of complex heterojunction laser structures.

5. NEAR-AND FAR-FIELD RADIATION PATTERNS Much effort has been devoted to the control of the far-field radiation patterns of laser diodes because of the need to collect the emitted radiation with low numerical aperture optical systems. In the simplest approximation, a uniformly illuminated aperture radiates a single lobe whose full width at half-intensity (denoted 0,) changes inversely with the thickness of the emitting region do,

8, z 1.2,1L/d0. Similarly, for a diode with strong lateral confinement of the radiation (such as a mesa diode W wide), the beam width in the plane of the junction is, on the assumption of uniform illumination,

o,,

%

1.2,lJW.

Higher order modes give rise to a more complex radiation pattern with two or more lobes and the above approximations are no longer useful as a measure of the radiation pattern. In particular, for the high-order lateral modes, the lateral beam divergence is in excess of (5b). The theoretical basis of calculating the far-field from the near-field is presented in Kressel and Butler3a(Chapter 7). Some important structures are discussed in this section where the major differences are related to the position of the recombination region within the waveguide, and the dielectric symmetry. a. Eflect of Facet Reflectivity on Transverse Mode Selectivity

The gain/loss distribution and the end-facet reflectivity play a major role in determining the relative powers of the cavity modes. The gain dis” 32

’’ ’’ 34

H. Kressel, H. Nelson, and F. Z. Hawrylo, J . Appl. Phys. 41. 2019 (1970). N. E. Byer and J . K. Butler, IEEE J . Quantum Electron. QE-6, 291 (1970). M. J. Adams and M. Cross, Phys. Lett. 32A, 207 (1970); Solid-State Electron. 14,865 (1971). W.W. Anderson, IEEE J . Quantum Electron. Q E l , 228 (1965). J. K. Butler, J . Appl. Phys. 42, 4447 (1971).

2.


79

tribution produces mode discrimination because of the way modes interact with the active region. For example, the mode which has the largest fraction of its field intensity in the gain region will have the largest optical gain. The facet reflectivity also affects the relative thresholds of the modes; however, its major effect is on the p ~ l a r i z a t i o n . ~ ~ . ~ ~ ” A mode in a slab waveguide can be decomposed into a set of plane waves propagating at symmetrical angles with respect to the facet normal. The reflectivity of the mode is termed Fresnel r e f l e ~ t i v i t y . ~ ’A. ~simple ~ understanding of the mode reflectivity and its effect on the polarization of the emitted radiation can be obtained by considering the Fresnel reflection coefficient of a plane wave. A TE wave has E perpendicular to the direction of propagation and parallel to the facet. For E in the facet plane, the reflectivity is a maximum. Hence, in any given decomposition of the wave, TE polarization has lower facet loss than other polarizations and is therefore favored. The orthogonal polarization has the magnetic field perpendicular to the propagation direction; E will not lie in the plane of the facet and its reflectivity will consequently be lower. Accordingly, it is easy to see that the reflectivity of TE modes is greater than that for TM modes. Consequently, the equivalent absorption coefficient associated with cavity-end losses satisfies C(,.dTE < ctendTM. The facet reflectivity for a given mode can be derived by a rigorous technique which includes the effect of the part of the modes which is outside the heterojunctions. Basically, when a given mode strikes the waveguide-air interface, the reflected energy is distributed among all modes. The “airmodes” which form a continuous spectrum will of course be different from the waveguide modes. The distribution of the incident energy across all of the reflected and transmitted modes is determined from the solution of a rather complicated boundary value problem obtained by matching the tangential fields on each side of the facet. The reflectivity R, of the mth trapped mode is then defined as the ratio of the reflected energy in the mode and the energy of the incident mode. The reflectivities which we show below were obtained by this technique. In Fig. 6 we show the facet reflectivity of a waveguide-air interface for the fundamental mode as a function of the waveguide parameter^.^^ The field solutions are determined for a simple symmetrical double-heterojunction structure as a function of the cavity width d3 and index step An. M . J. Adams, Electron. L e t / . 7. 569 (1971). D. 0. North, J . Quantum Electron. QE-12. 616 (1976). 3 7 F. K . Reinhart, I . Hayashi, and M . B. Panish, J . Appl. H i p . 42, 4466 (1971). E. I . Gordon, IEEE J . Quantum Electron. QE-9. 772 (1973). 39 T. Ikegami, IEEE J . Quantum Electron. QE-8,470 (1972). 3h

36a

80

HENRY KRESSEL AND JEROME K. BUTLER 43

4c

-s I-

G

35

0 IL IL W

0

MODE NUMBER m = I

0

z 3a 0

W I -I 1 .

w a 25

21

0.5

I

I

0.I

0.5

1.0

d3 (pm1

I ,

,,I

1.0

I

2 . 0 3.0

d3(pm)

FIG. 6 FIG.7 FIG. 6. Facet reflection coefficient for the fundamental transverse waveguide mode of a double-heterojunction laser as a function of the thickness of the active layer d , (TE wave, solid line; TM wave, dashed line). Parameter An is the index step between the waveguide and the surrounding material. (After Ikegami.j9) FIG. 7. Plots of ln(I/Rm) for the mth transverse mode as a function of d,, in a double-heterojunction laser for TE waves (solid lines) and TM waves (dashed lines). The index step An = 0.18. (After Ikega~ni.~’)

The lasing wavelength AL = 0.86 pm. Note that the reflectivity for the TE modes increases with the dielectric step, whereas the reflectivity decreases with An for the T M modes. To determine the role played by the facet reflectivity in modal discrimination, we show in Fig. 7 the quantity LcL,,~ = h(l/R,,,)for the trapped waveguide modes as a function of d3.39The index step An = 0.18. It is clear that the high-order TE modes have the smallest cavity-end losses. b. Symmetrical Structures-Double-Heterojunction

( D H )Lasers

The highest order transverse mode (in a cavity where losses are neglected) depends on the thickness of the waveguide region and on the index steps at its boundaries. Figure 8 shows, for example, the field intensity distribu-

2.

81


-5

-

-10-

-

m -15P

z -20a w

c -259

u

-45

d

-50 -5 5

-80 D I S T A N C E ACROSS OPTICAL CbVlTY l p m 1

-60 -40 -20

ANGULAR

0

20

40

60

80

D I S P L A C E M E N T F R O M FACET N O R M A L (DEGREES1

FIG.X. Laser near- and far-field patterns of the (a) fundamental transverse mode ( m = 1 ) and ( b ) second mode (iii = 1).These illustrations are applicable. for example. 10 DH and LOC lasers.

82

HENRY KRESEL AND JEROME K. BUTLER

tion for the fundamental mode (m = 1) with a single high intensity maximum in the field intensity, and mode 2 (two high intensity maxima) and the corresponding far-field distribution. Information concerning the dominant transverse mode can be deduced from either near- or far-field measurements, but experimental considerations make the far-field the more reliable source. The order of the dominating mode of the cavity can be deduced from the

1

-40

.

L

,

t-14 0 20

I

-20

1

,

40

€(Id e g r e e s )

Ifdb)

,TPjq. -4

\

\+I I :

(b)

I ;-lo I

- 4 0 -20

1-12 !-I4 0 20

40

a1.41

0 (degrees)

I

-40

-20 0 20 f? t degrees)

FIG.9. Comparisonofthe experimental and theoretical far-field patterns for double-heterojunctionlasers with varying heterojunction spacing (theory, dashed line; experiment, solid line): (a) d3 = 0.7 pm, fundamental transverse mode only; (b) d., = 1 . 1 pm, second mode only; (c) d3 = 2.8 pm, third mode with small admixture of the fourth mode. (After Butler

40

2.


83

number of lobes in the beam profile. The fundamental mode gives rise to a single major lobe while higher order modes give rise to other lobes; the mode number for m > 1 is given by the number of maxima, which includes two large lobes. Other useful data deduced from the radiation pattern are the angular separation between the two large lobes and the angular width of the lobe. The angular separarion between the lobes depends mainly on the dielectric step at the heterojunctions, increasing with in, whereas the lohe w,idths are related to the width of the waveguide region.”’ The simplest case to understand is the symmetrical DH laser. Figure 9 illustrates the change in the observed transverse mode number with increasing heterojunction spacing d3 of a DH laser, keeping the refractive index steps An z 0.08.41 With d 3 = 0.7 pm, only the fundamental mode ( m = 1) is excited. Increasing d 3 to 1.1 pm results in a dominating second ( m = 2) mode, while, with d 3 = 2.8 pm, the third ( m = 3) mode is dominant (with admixture of the fourth, m = 4,mode). Note in Fig. 9 that the angular width of the major lobes decreases with increasing active region width, consistent with the increase in the thickness of the waveguide region (i.e., source size). It is evident that simply increasing the width of the waveguiding region to improve the collimation is not always helpful. Arbitrarily increasing the width of the waveguiding region not only increases the threshold current density (see Part IV) but, as we have seen above, results in the propagation of high-order transverse modes and consequently “rabbit-ear” beams. Conversely, decreasing the heterojunction spacing (while keeping the radiation confinement constant) can decrease the threshold current density but at the expense of a broad beam. A practical compromise between low threshold and moderate beam width is found by using “thin” double-heterojunction structures where the recombination region (either n-type or p-type) is very narrow and the refractive index steps are moderate, producing optical tails spreading into the adjoining higher bandgap regions.I4 This thin DH structure yields a very practical device for efficient room-temperature cw operation. In the following, we present a series of theoretical plots that show the relationship between the internal device configuration and the near- and far-field distribution. Figure 10 shows the optical intensity distribution for various heterojunction spacings and An values. Since the optical power is the same in each, the increase in the peak reflects the increase of confinement 40

4’

J . K. Butler and H . Kressel. J . A ~ / J Phj.s. /. 43. 3403 (1971). W . P. Dumke. f E E E J . Quonrimi Elc~rrori.QE-11, 400 (1975). J . K. Butler, H. S. Sommers. Jr.. and ti. Kressel, Appl. P/IJ.s.Lcrr. 17. 403 (1970).

84


t-

z

a21 x

- DISTANCE

ACROSS JUNCTION REGION (pm)

ANGULAR DISPLACEMENT FROM NORMAL (DEGREES)

(a)

FIG.10. Near- and far-field patterns for symmetrical DH lasers with various heterojunction spacings d3 and refractive index steps An. The lasing wavelength is 0.9pm. (The near-field patterns are normalized in each figure to equal area under the curve.) (a) Constant An = 0.1 and d3 = 0.3, 0.2 and 0.1 pm; (b) Constant An = 0.1 and d , = 0.3 and 0.6 pm; (c) Constant width d , = 0.3 pm and An = 0.06,0.10, and 0.22.

as d3 is increased. Conversely, as d3 is decreased, an increasing fraction of the power propagates outside the region between the heterojunctions.This near-field distribution is reflected in the breadth of the transverse profile of the beam as shown in Fig. 10: The beam width narrows as d3 is decreased

2.


85

0.5

-

-

v)

k 0.4 z 3

n=3.60

-40 -20 0 20 40 60 ANGULAR DISPLACEMENT FROM NORMAL

-60

(DEGREES)

(b 1 FIG.10 (C'onrinued)

because the radiation spreading beyond the heterojunction boundaries yields a wider source. Figure 11 summarizes the change in the peak within the recombination region as a function of d3 and An. The above examples were chosen to illustrate the main features of the dependence of the radiation pattern on An and d 3 for small d 3 . Useful summary plots covering a broad range of values of practical interest for DH lasers

86


(C)

FIG. 10 (Conti17ued)

on d, are shown in Figs. 12 and 13. Figure 12 shows the dependence of (adjusted for the lasing wavelength) for An ranging from 0.04 to 0.62 [which encompasses the (A1Ga)As alloy system]. Dumke40a has derived an approximate expression valid for small heterojunction spacings of symmetric DH diodes: 1

AWJL + (A/l.2)(d3/AL)’

rad,

2.

87

HETEKOJUNCTION LASER DIODES

v)

k

z

I3

w 0.5 -

2

F

4w 0.4 -

K I-

i?~ 0.3z W

I-

z

;0.2 J

wLc y

a

0.1 -

W

n

0

I

I

I

I

I

0.2

0

0.4

I

I

I

0.6

0.6

FIG. 11 double-heterojunction spacing d , . The modal power is equal for each tl, value. A reduction in the peak intensity results from a change of the spatial distribution.

tL

80

-70 -60 -

/

/

/

/

/'

.---. '. \

0.22 An

88


t

I30

I-

4

I

i

I

\

\

\

'\1\\062=

An

\\

\'

0.42

FIIIIl

na3.6

where A = 4 3 4 - nf). Figure 12 shows that Eq. (5c) is not useful beyond d , 5 0.1 pm. The effect of changing the laser parameters on the optical confinement within the two heterojunction spacings is shown in Fig. 13. The confinement factor r, representing the fraction of the radiation power within the recombination region, is given as a function of An and d 3 . The relation of r to the threshold current density and differential quantum efficiency is discussed in Part IV.

2.


89

0.9 d 3 ( p m )

x

(a)

0.I

0.2 0.4 0.6 0.8 EFFECTIVE CAVITY WIDTH (rm)

3

[

+3]

(b) R c i . 13. The radiation confinement factor

r within the effective cavity width of a symmetrical

double-heterojunction laser diode as a function of the refractive index step An, (a) An ranges from 0.1 to 0.62; (b) An ranges from 0.04 to 0.22. (After Bulter P I 0 1 . ~ ~ ) 42

J. K. Butler, H. Kressel, and 1. Ladany, IEEE J . Quanrum Electron. QE-11,402 (1975).

90


c. Symmetrical Structures-Four-Heterojunction

( F H )Lasers

As discussed in the previous section, the double-heterojunction configuration can be adjusted to yield a desired beam pattern. Addition of two more heterojunctions makes the control of the device properties more precise, but at the expense of additional fabrication complexity. The wave

""23

-d4443-c - i +y -

_

An12

ACTIVE REGION

5

4

3

2

I

X

FIG. 14. Vertical geometry of symmetrical four-heterojunction laser diode,

1

0

1

2

3

I 4

OPT! & WITY WITH d' (pm) (b)

FIG. 15. Summary of FH device performance for fundamental transverse mode operation. ( a ) The radiation confinement factor within as a function of device geometry and (b) the radiation half-power beam width corresponding to the structural paranietcrs of (a). Refer to Fig. 14 for the nomenclature.

2.

91


properties of these structures can be calculated using the five-layer model described in Kressel and Butler3a(Chapter 5). In this section, we present the results of some important symmetrical dielectric configurations of the basic FH structure shown in schematic form in Fig. 14. In Figs. 15 and 16 we show plots of the radiation confinement r factor within the recombination region 3, as well as 0, for various values of the refractive index steps, recombination region width d , , and total outer heterojunction spacing d". Note that r increases with decreasing do only when do is not too large compared to d 3 . With large rl", the optical confinement is little affected by the outer heterojunctions. The field distribution due to inner heterojunctions depends, of course, on d 3 and on the size of the index steps enclosing it relative to that of the outer heterojunctions.

I 2 3 OPTICAL CAVITY WIDTH d'(pm)

a

I 0

1

2

3

1

OPTICAL CPWlTY WIDTH d ' l p l

(a) (b) Fic;. 16. Calculations for FH laser aresimilar to those illustrated in Fig. 15 but with different device parameters. ( a )Thu radiation confirienieni factor and ( b ) the radiation half-power beam width.

Figure 17 summarizes the beam width for fixed do = 2 pm as a function the recombination region width for selected values of the index steps. As the recombination region widens, the confinement factor increases and the beam broadens. In the above, the recombination region was centered within the outer heterojunctions. In the FH structure, off-center placement of the recombination region affects the preferred transverse mode, as illustrated in the

92


't I

2 20

9

20

-

0.1

a2

03

FIG. 17. Summary plot for FH laser operating in the fundamental transverse mode showing the far-field beam width for a fixed outer heterojunction spacing d" = 2 pm as a function of the recombination width d , for selected values of the refractive index steps. (a) An,, = 0.06 and (b) An,, = 0.04.Refer to Fig. 14 for the nomenclature.

0.4

d,"

following examples. Figure 18a shows the dimensions of a four-heterojunction diode with a GaAs:Si recombination region d, = 0.5 pm. The adjacent regions 2 and 4 consist of Alo~03Gao~,7As, while regions 1 and 5 consist of Alo.lsGao.ssAs. The separation between the outer heterojunctions d" = 3.7 pm. The radiation pattern of this laser, which operates in the fundamental mode, is shown on the left side of Fig. 18a.43 It consists of a single lobe 0, = 17" which corresponds to the approximate diffraction limit of a 3 pm aperture. Both the inner and outer sets of heterojunctions contribute to the radiation confinement, with the inner ones tending to peak the field in the recombination region and promote fundamental transverse mode operation. A theoretical analysis of the gain for the various tranverse modes of a FH laser of the type shown in Fig. 18a is presented in Fig. 19. It is evident that the fundamental mode is theoretically preferred since it reaches threshold before the higher-order modes. Furthermore, Fig. 19 shows that the position of the recombination region within the waveguide region of the specific FH structures studied is not critical, as experimentally confirmed. The reason is that when the index steps at the two inner heterojunctions are relatively large there is strong field confinement to region 3. For example, it was found4, that structures similar, except that the recombination region was placed either 1.2 or 2.5 pm from the p-n interface, had radiation patterns 43

H. F. Lockwood and H. Kressel, J . Crysr. Growth 27,97 (1974).

2.

93


-

P

,

'/'b:.

72.Opm

"'

GaAi n*- AI,Ga

1-

As

Jth = 13 kA/cm2

1

1

3 7pm

I

1 \ M n*- At, Gal-

Jth

, As

19 kA/cm2

FIG. 18. Structures, far-field patterns. and threshold current densities of lasers with different internal placement of heterojunctions. (After Lockwood and Kre~sel.~')

identical to those of the structure shown in Fig. 18a. A structure similar to the above, but with higher A1 content in regions 2 and 4 [x2 5 x4 = 0.061, has tll 35", indicative of much stronger peaking of the field near the recombination region.

-

94

HENRY KRESEL AND JEROME K . BUTLER

c

2 0 W 100 W

a W

L

c u

a

MODF

I

50L 30

t 10

0.5 1.0 I.5 ACTIVE REGION DISPLACEMENT d2 bm)

FIG.19. Threshold gain curves for four-heterojunction structure of Fig. 18a as a function of the displacement of the recombination region from the p-side edge of the waveguide region. Each curve is labeled with a specific transverse mode number; 1 is the fundamental mode.

When the index steps of the two inner heterojunctions are small, as opposed to the example above, the placement of the gain region within the cavity of width do does effect strongly transverse mode selection. The structures of Figs. 18b and 18c show two examples where the GaAs:Si recombination region is differently placed within the waveguide region (without inner heterojunctions). In Fig. 18b, the recombination region is near the p+-p interface; while in Fig. 18c, the recombination region is nearer the center of the waveguide region, thus similar to the FH structure (a) but without the inner heterojunctions. In contrast to the FH laser in (a), the far-field radiation patterns of (b) and (c) contain high-order transverse modes and displacing the position of the recombination region changes the far-field pattern. With the recombination region centered as in (c), a relatively pure m = 4 mode is seen, but with

2.

95


the recombination region near the edge of the waveguide as in (b), several modes share the power. This behavior is qualitatively consistent with the change of coupling of the radiation to the recombination region. It is important to keep in mind the requirement for carrier confinement within the recombination region of FH structures. If the heterojunction barrier is too low, carrier loss occurs which raises the threshold current density. This is illustrated by the devices of Fig. 18, which shows the threshold current density for each structure. In (a) and (b), J,, is under 4 kA/cm2 per micron of optical cavity thickness, but it is much higher for (c).The reason for the difference is the poor carrier confinement in (c),resulting from the absence of the inner heterojunctions. In fact, the spontaneous spectra reflect the emission from both the GaAs:Si and the GaAs:Ge region in (c) as can be seen in Fig. 20. Since the stimulated emission occurs in the GaAs: Si region, the carriers diffusing beyond that region are wasted, resulting in a relatively high threshold current density.

10000 9900

9800

9700

9600

9500

9400 9300 9200 WAVELENGTH (%,

9100

9000

8900

8800

8700

Fic. 20. Spontaneous spectra from lasers shown in Fig. 18a and c (amplitudes arbitrary). The short wavelength emission (B) discernible i n unit 507N is radiation from the GaAs:Ge passive region. Both spectra are distorted on the high-energy side due to selective internal absorption. (Edge emission at 300 K and I = 5 mA.) (After Lockwood and K r e ~ s e l . ~ ~ )

d. Asymmetrical Structures-Single-Heterojunction (Close-Confinement)Lasers In the single-heterojunction (close-confinement) laser, the dielectric step on the p + - p side of the waveguide is formed by a heterojunction, whereas the other is the result of differences in carrier concentration between the p-type recombination region, generally formed by Zn diffusion, and the GaAs n-type substrate (2-4 x 10l8 ~ r n - ~ ) . Guided wave propagation is impossible in an asymmetrical waveguide whose thickness is below a critical value set by the dielectric asymmetry and

96

HENRY KRESSEL AND JEROME K . BUTLER RECOMBWATDN REGION

+

7

' QI

'?31

X

0

’0

w

0 0

0.01

0.02 0.03 0.04 0.05 0.06 0.07 REFRACTIVE INDEX DIFFERENCE (An3B)

0.08

(b)

FIG.21. (a) The asymmetrical waveguide of the single-heterojunction type. (b) The minimum d , value for fundamental TE mode propagation. (After Kressel et a/.44)

index step values. Consider the asymmetrical waveguide of Fig. 21a in which the p-type recombination region is enclosed on one side with a large dielectric step A E ~ Iand a smaller step on the other side. The dielectric asymmetry q is defined q

= AE3i/AE35.

(6)

Mode guiding within d3 is only possible if the following conditions are ~ a t i s f i e d For . ~ ~ TE waves, d3

'2?t(n: A.ns) -

tan-'(q -

and, for TM waves,

44

H. Kressel, H. F. Lockwood, and F. Z. Hawrylo, J. Appl. Phys. 43, 561 (1972).

(74

2.

28 26


97

-

Figure 21b shows a plot of Eq. (7a) for TE waves assuming an asymmetry q = 5, which is appropriate for single-heterojunction laser diodes. The loss of mode propagation because of the reduction in the radiation confinement within the recombination region is reflected in an increase in the threshold current density. Figure 22 shows experimental values of Jthin single-heterojunction (SH)lasers as a function of the recombination region width; it shows that the lowest Jthis obtained with d3 z 2 pm, and that no lasing occurs below d3 z 1 pm. Since these experimental devices had an estimated Arts5 z 0.01 1. Hayashi, M. B. Panish, and F. K. Reinhart, J . Appl. Phys. 42, 1929 (1971). J . Camassel, D. Auvergne, and H . Mathieu, J . Appl. Phys. 46, 2683 (1975). " Zh. I. Alferov, et af.,Fiz. Tekh. Poluprouodn. 8, 1270 (1974) [English iransl.: Sou. Phys. Semirond. 8, 826 (1975)l. 45

46

98


at the p-n homojunction, the calculated cutoff value for d 3 should be 0.6 pm, which can be considered in close agreement in view of the approximations made. Because of this small An, these devices can be temperature sensitive as discussed below. Single-heterojunction lasers usually operate in the fundamental transverse mode, but higher-order modes can be excited if the recombination region is widened to about 2.5 ~ m . ~For ' the typical good-quality SH laser, d3 is between 2 and 2.5 pm, and 0, = 20"; reasonable agreement between the calculated and experimental far-field radiation pattern has been obtained.32 e. Large-Optical-Cavity (LOC)Lasers-Symmetrical and Asymmetrical Structures

In the large optical cavity (LOC) laser the recombination region is placed at one edge of the waveguide region defined by two outer heterojunctions spaced a distance d" apart. Figure 23 shows the symmetrical LOC with equal refractive indices in the outer regions, the three-heterojunction version of the device with the p-n inner homojunction replaced by a low barrier heterojunction, and the asymmetrical LOC configuration. The choice of LOC structure is based on the desired radiation pattern. Fundamental mode operation with the smallest possible beam width is favored by : (1) moderate (under 1 pm) heterojunction spacing, (2) the use of three heterojunctions, and ( 3 ) dielectric asymmetry. RECOMBINATION REG I O N b

L O C WITH INTERNAL HOMO J U N CTlON

LOC WITH INTERNAL H E T E R O J U N C T ION

ASYMMETRICAL LOc

FIG.23. Schematic cross sections of three variations of the large-optical-cavity structure: symmetrical LOC with p-n homojunction, LOC with p n heterojunction, and asymmetrical LOC. Shading indicates the recombination region.

2.


99

A detailed study of the preferred mode as a function of device parameters has been presented by Butler and KresseL4' We consider here some examples in Fig. 24 where the gain to reach threshold for various transverse modes is plotted as a function of the waveguide thickness do. Note that for the chosen index step An = 0.06, the second mode is predicted to be the first to reach threshold in a 2-pm cavity, whereas the third mode reaches threshold first in a 3-pm cavity, as indeed found e~perimentally.~' Also, for large cavity widths, the modal selection between the high-order modes becomes less pronounced and end losses are important as competing factors in modal selection as discussed in Section 5a above.

MODE NUMBER

1

do

2

3

CAVITY WIDTH

4

S

6

(pm)

FIG. 24. Threshold gain ( G t h )curves for the different cavity modes of a symmetrical LOC structure. The pertinent cavity parametera are: d , = 0. n , = n 5 = 3.54. t i j = n4 = 3.6, x l = 20 cm- I , a4 = a, = l o - ' . For d < 0.5 pm. t/ z d 3 and, for d > 0.5 pm. d3 = 0.5 pm. (Refer to Fig. 23 for the nomenclature.)

It is clear from Fig. 24 that the threshold gain for mode 1 becomes extremely large in wide optical cavities. The fact that the threshold for a particular mode is infinite below a certain cavity width simply indicates that the mode cannot propagate in the cavity because it is too lossy. Asymmetrical LOC structures can be designed43 to propagate only the fundamental, as illustrated in Fig. 25, which shows the profile of a device having H, = 14 . In the above device, there is a homojunction between the two heterojunctions. The use of a triple-heterojunction structure (Fig. 23),

100


t

----0 . 8 p n Go As:Te

I . O p m ALYGaImyAs :Te v)

z

0.1

x.0.2 y = 0.02

W

I-

P

I/

0.01

/

\ ANGLE

8

FIG. 25. Far-field pattern of asymmetrical LOC structure operating in fundamental transverse mode. The dimensions of the structure are shown. (After Lockwood and Kre~sel.4~)

in which the bandgap energy of the n-type region within the outer heterojunctions is about 40 meV higher than in the p-type recombination region, is another way of promoting fundamental mode operation.48Also, an undesired mode can be suppressed by a suitable coat designed for low reflection of that mode. Hakki and Hwang4' used a facet coating consisting of Al,O,/ZnS to obtain maximum transmission loss at 30" to extend the range of fundamental operation to higher power.

6. OPTICAL ANOMALIES IN ASYMMETRICAL HETEROJUNCTION LASERS Asymmetrical structures can exhibit dramatic temperature-dependent changes in radiation confinement from reduction of the index step (at the lasing wavelength)at the n-n barrier with increasingtemperature. An example of the decrease of optical confinement with temperature is illustrated in Fig. 26, which shows at 22°C a 6 , = 20"beam, whereas, at 75"C,O1 is reduced to This beam reduction results from an effective aperture doubling due to field spreading mainly into the (A1Ga)As n-type region. Another perspective on the effect of this radiation confinement loss is obtained from the data shown in Fig. 27. The threshold current density of such a structure

49

B. W. Hakki, IEEE J . Quuntum Elecfron. QE-11, 149 (1975). B. W. Hakki and C. J. Hwang, J . Appl. Phys. 45,2168 (1974).

2.

t

101


TE

1

ANGLE

e

D

(b) FIG. 26. Far-field pattern of an asymmetrical LOC laser diode at (a)22°C (TE/TM = 5.5 dB) and (b) 75'C (TEflM = 2 dB) showing the effect of a narrowing ofthe far-field due to a reduction in the refractive index step with increasing temperature. (After Lockwood and K r e ~ s e l . ~ ~ )

increases sharply with temperature, Fig. 27a; as the Ear-field beam narrows radiation confinement is decreased, Fig. 27b. For comparison we also show the Jthdependence on temperatures of a LOC device with high heterojunction barriers. In such a strongly radiation confined unit (where 0, is constant) Jth only doubles between 22 and 70°C, whereas in the weakly confined asymmetrical unit it increases by a factor of 10. The index discontinuity at the n-n barrier in the above device was created by the incorporation of small (typically < 3%) amounts of A1 in the n-type

-

102

HENRY KRESSEL A N D JEROME K. BUTLER

0’

26

o:

a0 o: $0 T E M P E R A T U R E I'C)

7b

810

TEMPERATURE ('C)

(b)

FIG. 27. (a) Change of the threshold current density with temperature of asymmetrical LOC laser with temperature-dependent radiation pattern change (curve A), compared with LOC laser having temperature-independent radiation pattern (curve B). (b) Angular divergence 8, corresponding to laser of curve A . (After Lockwood and Kre~sel.~')

dielectric wall of the cavity. The single-heterojunction laser achieves similar index discontinuities by compensation in the p-type recombination region and heavy doping outside (n-type). Since the index discontinuity at the lasing wavelength created by doping is quite temperature sensitive, the performance of single-heterojunction lasers is generally more temperature dependent than that of the multiple heterojunction devices. In contrast to the asymmetrical LOC laser where the reduction of the refractive index step with temperature at the n-n boundary explains the radiation confinement loss, in the single-heterojunction laser there is a further contribution to the An3, reduction due to the injected carrier density in the recombination region (see Part V). As the threshold increases with

2.


FIG.28. Current and optical output pulse of a strongly asymmetrical LOC laser with properties shown in Fig. 27 as a function of temperature. The peak current is indicated with each figure. (a) T = 22 C. I=lOA.(b)T=60C,I=22A.(~)jr= 77.5 C. I = 37 A. The deterioration o f the optical pulse shape is evident at high temperature.

103

104

HENRY KRESSEL A N D JEROME K . BUTLER

temperature, the injected carrier density increases, contributing to a decreasing refractive index step at the p-n j u n c t i ~ n . ~The ~ * ~initial ’ (roomtemperature) refractive index step, at a pn junction diffused into heavily n-type GaAs (2-4 x lo’* CII-,), is estimated (from analysis of singleheterojunction lasers) to be only about 0.01. A doubling of the threshold current density, with a consequent injection of an additional pair concentration of 2 x 10 ~ m - reduces ~ , An,, by -0.003 or 30%. The strong reduction in optical confinement with increasing temperature may affect the shape of the radiated optical pulse. One finds that the optical pulse shape deteriorates markedly with evidence of instability in the output. In Fig. 28, we show the current and optical pulse shape at 22,60, and 77.5”C of the device of Fig. 26.” Note that the pulse deterioration becomes particularly noticeable at 77.5 C,where there is also a marked increase in the threshold current. Similar pulse anomalies have been seen in single-heterojunction lasers; the effect is clearly connected with a partial loss of optical confinement. There have been attempts to explain the optical anomalies in terms of the interaction of saturable absorbers in the n-type region of the single-heterojunction laser with the radiation “leaking” from the recombination reg i ~ n . ~ At ~ .this ’ ~ time, these models are purely qualitative. There is no independent evidence for saturable absorbing centers in the n-type GaAs used for laser fabrication. The related problem of optical anomalies in other laser structures has been reviewed.”

IV. Relation between Electrical and Optical Properties In this part we analyze the relationships between the vertical device geometry, threshold current density, and differential quantum efficiency using the results of the wave propagation analysis of Part 111. In Section 7 we consider the relationship between the current density, the injected carrier density, and the gain coefficient. In Section 8 we discuss the threshold current density and differential quantum efficiency of various laser structures and show that G. H. B. Thompson, P. R. Selway, G. D. Henshall, and J. E. A. Whiteaway, Electron. Lett. 10,456 (1974). 51

P. R. Selway, G. H. B. Thompson, G. D. Henshall, and J. E. A. Whiteaway, Electron. Lett. 10,453 (1974).

H. Kressel and H. F. Lockwood, unpublished. M. J. Adams, S. Griindorfer. B. Thomas, C. F. L. Davies, and D. Mistry. IEEEJ. Quantum Electron. QE-9, 325 (1973). 5 4 S. Griindorfer, M. J. Adams, and B. Thomas, Electron. Let[. 10, 354 (1974). 5 5 J. E. Ripper and J. A. Rossi, IEEE J. Quanfum Electron. QE-10, 435 (1974).

52

53

2.


105

the major device performance parameters can be predicted once the internal geometry is accurately known. 7. BASIC CONSIDERATIONS

The gain of optical power from the recombination region at lasing threshold must equal the total loss of power occurring within and outside that region (if the mode is not fully confined) as well as from the radiation. The recombination region gain coefficient is a function of the injected carrier pair density N , , which is related to the width of the recombination region d , and the minority carrier lifetime T : N,

J~/ed,,

where e is the electronic charge and full carrier confinement is assumed. In lightly doped material, the minority carrier lifetime should decrease with increasing injected carrier density" because of increasing bimolecular recombination. An estimate of the "effective T" for spontaneous recombination can be obtained immediately at threshold by measuring the time delay t, between the current pulse to the laser diode and the light e m i s ~ i o n ~ ~ . ~ ~ : where I and Ithare the amplitude of the current pulse and of the threshold current, respectively. The time delay t , is important because it affects the rate at which a laser diode can be turned on at a given current density [see Part XI. Also, (9) shows that the threshold current is higher for pulses of shorter duration. In typical DH lasers, T = 2-3 nsec at thresholu. Once a laser diode is operating above threshold, the carrier lifetime for stimulated emission into a given mode is shortened inversely with the photon density in that mode. In practice, it is found that the lifetime is shortened to values in the lO-"-sec range when the operating current is much greater than the threshold Just above threshold, stimulated carrier lift time is already below sec which allows very fast modulation of the light output as long as the laser remains biased to threshold. Therefore, to obtain modulation rates above a few tens of megahertz it is essential to avoid current modulation through the threshold region. It is evident from Eq. (8) that a reduction in the minority carrier lifetime at threshold results in an undesirable reduction in N , for a given diode current density. Hwang and DymentS9 have studied the effect of increasing (by H. Narnizaki, H. Kan, M. Ishi, and A. Ito, Appl. Phys. Lerr. 24. 486 (1974). K. Konnerth and C. Lanza, Appl. Phys. Lett. 4, 120 (1964). 5 8 J. E. Ripper, J . Appl. Phys. 43, 1762 (1972). "' N. G . Basov era[., Sov. Phys.-Solid Sate 8,2254 (1967). 59 C. J. Hwang and J. C. Dyment, J . Appl. Phys. 44,3240 (1973).

56

57

106


,

>

102

m L

1 NORMALIZED THRESHOLD CURRENT DENSITY l o Ir SPONTANEMJS LIFETIME AT THRESHOLD

w z W

0 0 I-

z

-

$2

-

W n

ONi A

510

$2

W Y

I n-

w 0

3

a

L

,

r r l l l l

I I

,-lo7 107

f

- -m

0 v)

-

1

w I n

t

1%

-;1% --

- L L I1

N _I

I

1 1 v 1 1 1

- 2 W

F

-

- z5

d I

I

, , , , , , I

1 I I

,I

d9

I

, , ,,

$

1020

ACCEPTOR CONCENTRATION lcm-3)

FIG.29. Dependence of the normalized threshold current density and spontaneous recombination lifetime of electrons on Ge acceptor concentration in the recombination region of a double-heterojunction laser. (After Hwang and Dyrnen~.~’)

doping) the hole concentration of the p-type GaAs: Ge recombination region of double-heterojunction lasers. A correlation was established between the reduction in the minority carrier lifetime (with increasing hole concentration) and the increase in threshold current density, as shown in Fig. 29. Hence, it is desirable to minimize the initial free carrier concentration in the recombination region; this is also desirable because of the reduced free carrier absorption, as discussed below. The relation between N , (or, more usefully, the current density) and the gain coefficient can be calculated on the basis of various assumptions concerning the material. The theoretical plots relating the nominal current density and the active gain region coefficient are of the form

where J , is a constant, and J,,, is the current density for a laser with a micrometer-thick recombination region with unity quantum efficiency. The current density for a device J is therefore related to J,,, by J = J,,,d3/qi. The exponent b (between 1 and 3) and the numerical g versus nominal current density relationship depend on the temperature, the assumed density of states distribution, the assumption concerning momentum conservation for conduction-to-valence-band transitions, and the doping level. Stern,60-61

-

6o

61

F. Stern, IEEE J . Quantum Electron. 9, 290 (1973). F. Stern in “Laser Handbook” (F. T. Arecchi and E. 0. Schulz-DuBois, eds.), NorthHolland Publ., Amsterdam, 1972.

2.


107

Hwang,62and Landsberg and have presented calculated results and reviews to which we refer. For the present purpose it suffices to consider some examples of these calculations for gain values of interest for typical heterojunction laser diodes (30-100 cm- ’). Figure 30 shows Stern’s calculations. In (a) the gain coefficient is calculated as a function of the nominal current density for undoped GaAs.60 In (b) the same data are plotted in a linear form to show that the gain coefficient can be assumed to be a linear function of the current density (above a minimum value) in the narrow gain value range of typical device interest. In (c) is shown the nominal current density needed to reach an active region gain coefficient of 50 cm-’ in compensated n- and p-type samples between 10 and 300‘ K. The calculated gain values are useful in predicting experimental J , h Values, as will be discussed in the next section. For the moment we note that Landsberg and ad am^^^ have compared their calculations to experimental data for DH lasers with good agreement. For example, for a D H laser with d 3 = 1 pm and r 2 1 having a lightly doped (5 x 10l6 cmP3)ra-type GaAs recombination r e g i ~ n ,the ~~ threshold ? ~ ~ current density forg,, = 50 em- * at room temperature is predicted to be 5000 A/cm2.

-

8. THRESHOLD C U R R E N T DENSITY A N D DIFFERENTIAL QUANTUM EFFICIENCY

To make the design curves as general as possible, we will introduce certain parameters relating only to the internal geometry of the diode perpendicular to the junction plane. In the multilayer structure used to model the lasers, one layer is designated the “gain region” (or “active” region) in which stimulated emission occurs. We assume that this region is fully inverted with only free carrier absorption occurring within it. The other regions surrounding the gain region are “passive” and only absorb the stimulated radiation. The term absorption coejicient is used here in two ways. In the first instance, it refers to a bulk parameter of the material in a passive region of the laser. (This value is determined from conventionally measured absorption coefficient data using bulk samples.) In the second instance, the absorption coefficient is the value seen by a guided mode propagating in the plane of the junction. Here, to determine the absorption coqficient qfthe mode we must 62 63

C. J. Hwang, Phys. Reo. B2,4117, 4126(1971). P. T. Landsberg and M. J. Adams, in “The Physics and Technology of Semiconductor Light Emitters and Detectors.” (A. Frova, ed.). p. 3. North-Holland Publi.. Amsterdam,

1973. E. 0. Kane. Phys. Rec. 131, 79 (1962). 6 5 H. Kressel and H. F. Lockwood. Appl. Phys. Lett. 20, 175 (1972). 6 6 H. Kressel, H. F. Lockwood, F. H. Nicoll, and M. Ettenberg, IEEE J . Quantum Ekcrron. QE-9. 383 (1973).

64

108


-;-----I I2

300-

! ! !

p

100-

(a 1

LL

8 u I

30 -

4 (1

10

Id

5

3 10' 3 10' 3 10 NOMINAL CURRENT DENSITY ( A /cmcpm' I

TEMPERATURE

(OK)

FIG.30. (a) Gain coefficient versus nominal current density for undoped GaAs calculated using the bandstructure model of (The nominal current density J,,, = qiJ/d, .) (b) Data of (a)using straight line fits to the calculated curves for 30 5 g 5 100 cm- I . (c) Nominal current density needed to reach threshold versus temperature for three GaAs ptype samples and two n-type samples. The integers labeling each curve give the donor and acceptor concentration, respectively, in units of lo1*C I I - ~ .(After Stem6')

2.


109

know its intensity distribution in the direction transverse to the junction plane. Each region of the laser, including the gain region, now makes a prorated contribution to the modal absorption coefficient dependent on the fraction of the radiation in that region. The material in various regions of the laser is characterized by its complex relative dielectric constant K . The imaginary part of K is related to the absorption coefficient whereas the real part is related to the index of refraction. The contributions to the effective absorption coefficient include free carrier absorption within the recombination region, af, , a weighted absorption coefficient Bo from all passive regions which takes into account the fraction of the radiation in each region and the cavity-end loss a,,,, . We consider now the waveguide modal field as the medium for transferring energy from the gain region to the passive regions, ignoring for the moment the radiation losses at the end facets. The fraction of the wave power confined to the active region is defined as r. We now define a quantity G such that TC is proportional to the power from the active region going into the waveguide mode, while Bo is proportional to the power drained from the waveguide mode by the passive regions. For a mode to propagate without magnitude change, the gain and loss must be balanced. From this condition, G,h is defined as rG,h = Z O .

(11)

The above expression can be modified if each of the passive regions has the same bulk absorption coefficient ai, Gfh = Ui(1

-

r)/r

(12)

Therefore, Gth represents the recombination region gain coefficient at the threshold for a laser with no free carrier absorprion within the recombination region (afF= 0) and no cavity-end loss (aend= 0). The calculation of Gth is very convenient since it allows us to isolate some key device properties, including inherent modal preference, without introducing confusing nonvarying factors which enter into the device performance. Once G,h is calculated, the active region gain coefficient 9th at threshold for a laser ofjnite length and known a,, value is easily calculated as follows. Equating the net gain coefficient of the recombination region to the losses outside that region, we obtain

110


The external differential quantum efficiency qextis then given by

or

where qi is the the internal quantum efficiency.66aA cavity of length L and facet reflectivities R, and Rb has @-end

= (1/2L)1n(1/RaRb).

(16)

The value of afcdepends on the equilibrium (initial carrier concentration in the recombination region No) and on the injected carrier density 2N,. Thus, the total free carrier concentration is N = No + 2N,.

(17)

The value of Ne for a given current density and active region width depends

on the minority carrier lifetime, Eq. (8). Figure 31 shows the experimentally determined afcat room temperature for increasing values of No in a DH laser.67 A reasonable approximation at room temperature is afc 2 0.5 x 10-17N.

(18)

For low initial concentrations (below 10" ~ m - ~the ) , injected carrier concentration determines aft, and a value of -10 cm-' is obtained. In single-heterojunction lasers, higher values of afc are obtained because the recombination region is Zn-doped to an average level in the mid-10'8-cm-3 range. A detailed analysis3' of such structures shows that afcE 30 cm-’ at room temperature but decreasing with temperature. Once gth is known, the threshold current density is determined by the thickness d3 of the recombination region, the internal quantum efficiency,

66a

67

The internal quantum efficiency is not a welldefined quantity in laser diodes. Available measurements show that qi = 0.6--0.7 at room temperature and that qi approaches unity at very low temperatures. The difficulty in defining a unique value of qi results from the impact of stimulated recombination above threshold, which lowers the radiative lifetime. Therefore, qi is not generally the same as the internal quantum efficiency for spontaneous recombination as determined in LEDs, for example. E. Pinkas, B. I. Miller, I. Hayashi. and P. W. Foy, IEEE J . Quantum Electron. QE-9. 281 (1973).

2.

111


a (FREE CARRIER) FOR

BULK

GoAs

0 0

loo

L ,016

0

I

I

I

I

10”

10’8

1019

1020

N , P ,( c ~ n - ~ )

FIG.31. Comparison of bulk free carrier absorption coefficient ‘x and laser free carrier absorption coefficient zlCas a function ofcarrier concentration. 0 , ti-type active layer; 0,p-type active layer. The solid line is the bulk free carrier absorption coefficient in GaAs just below the band edge. The circles represent the laser free carrier absorption coefficient a t various concentrations. (After Pinkas et 0 1 . ~ ’ )

and the relationship between the injected carrier density and the gain coefficient. A useful relationship between J t h and 9 t h is obtained from Eq. (10): 9 t h = fls(qiJth/d3

- Jl)b,

(19)

where p, and J , are constants. A simple expression for Jthis obtained if the GaAs recombination region is undoped. Then the exponent h is unity for gain coefficient values between 30 and about 100 cm-’ (see Fig. 30b). j t h = (d3/qi)(gth/Ps

+ 51).

(204

At 300‘K, 6, = 0.044 pm-cm/A and J 1 = 4100 A/cm2-pm. Assuming a symmetrical DH structure with equal absorption coefficient mi in the two confining layers [see Eq. (12)], Jlhat 300‘K can be expressed as follows:

It is convenient to express the total internal cavity absorption coefficient by the quantity d = rmfc+ (1 - r)mi.The differential quantum efficiency

112


can then be written from (15),

For illustration consider typical AI,Ga, -,As-GaAs double-heterojunction lasers with x = 0.2-0.3 (where radiation confinement is near unity over a wide d3 range), and the initial carrier concentration in the recombination region is typically about 10’’ ~ m - Assume ~ . a typical Fabry-Perot cavity length L = 400 pm, with uncoated facets (R = 0.32); hence, aend= 27 cm- l. Assume r = 1, and afc = 10 cm-’. Thus, from (14), gth= 37 cm-’. For devices of this type, it is experimentally found (see Fig. 22) that Jth/dJ

= 4.0 k 0.5 x lo3 A/cm2-,um,

(22)

where d3 is between -0.3 and -2.Opm. Equation (20b) predicts values within the range of the above experimental values. When d3 5 0.3 pm, J t h no longer decreases linearly with d3 for typical DH structures. This effect is mainly due to the decreased radiation confinement discussed in Part 111 (I- < 1) which increases g t h . We now turn our attention to such devices. A careful comparison between theory and experiment with regard to J,, is possible in devices with accurately known recombination region width, heterojunction barrier height, and absorption coefficient values. Figure 32 shows the data of Kressel and Ettenberg67afor AI,Ga,-,As DH lasers with lightly doped n-type recombination regions, compared to J t h values computed from Eq. (20) using r values from Fig. 13. The index step was related to the A1 concentration difference Ax at the heterojunctions by An = 0.62 Ax (Section 12b). Within the uncertainty of the internal quantum efficiency value (a value of unity was used in Fig. 32), the agreement with theory is satisfactory. Changes in doping level of the recombination region will change the gain versus current density relations, and thus affect the calculated J t h expression (20b). It is frequently interesting to determine the diode internal parameters from the experimental Jthand qextvalues. For a given device structure, one can determine qi and E by measuring the differential quantum efficiency as a function of the cavity-end loss. One may study diodes of different length selected from the same wafer, but the best technique is to vary the facet reflectivity of a single device by changing the thickness of the dielectric facet coating.68 From the change in J , with cavity-end loss one estimates E + raJl.Note that internally circulating modes (i.e., trapped modes be67a

68

H. Kressel and M. Ettenberg, J. Appl. Phys. 47, 3533 (1976).

M.Ettenberg and H. Kressel, J . Appl. Phys. 43, 1204 (1972).

2.


-

113

/ An.Q.4

Y

cause of their angle of incidence at the facets) can perturb the results obtained with very short cavity lasers when their width becomes comparable to their length. In fact, one may find that the differential quantum efficiency of short lasers is lower than that of long ones in contradiction to Eq. (21). We now turn our attention to specific device structures of practical importance where the radiation spread beyond the recombination region may lead to troublesome complications. G,, can be calculated for structures of varying complexity by appropriate solution of the wave equations as discussed in Kressel and B ~ t l e r . Particularly ~" important are D H structures with d3 5 0.3 pm used for room-temperature cw operation. As shown schematically in Fig. 33 (for an asymmetrical structure) the field may extend through the thin p-type (A1Ga)As region 1 into the pf GaAs contact layer

1

"3 n2 "4

REGION I

Y 0

c

a

n

4 0

-Fi

c)

3.59 1o4crn-I 7

I

Y

+a

n 2

n 5 . 3 538 1

1

,

1

1

1

n-TYPE As

( AiGo)

n,= 3.425 o,=~~cm"

'n+-GoAs O23pm--

P-(AlGa)Ar ACTIVE REGION

dq= I 5 p m

-2 0

10

-I 0

DISTANCE

A C R O S S OPTICAL CAVITY

FIG.33. Near-field calculated for an experimental asymmetrical DH laser [schematic in (a)] in which the radiation is preferentially spreading toward the surface of the structure. The index values are indicated as well as the thickness of each relevant region. The index values were first estimated from the (AIGa)As compositions in each region and then "fine-tuned to match the experimental far-field pattern. (b) The thicknesses were measured on the actual laser. (After Butler tv 0 1 . ~ ' )

114

2.

115


FIG.34. The active region gain coefficient (iL= 0.8 pm) at threshold G,,. calculated for various device parameters with quantities noted in the insert sketch held constant. Both the refractive index step An and the width d, ofthe p-type (AIGaIAs region 2 are varied. (After Butler ('I ul.")

500

200 -

100

(C) 50 -

20 10 -

5’

'

015 '

P-(AlGalAs

Ib

'

1'5

' 210

'

'

REGION WIDTH d p (MICRONS)

at the device surface, which is highly absorbing at the lasing wavelength. The results of a series of theoretical calculation^^^ for symmetrical structures are shown in Fig. 34 where Gth is calculated for various An values and distance d 2 between the Ale,, Gao.,As recombination region (AL 2 0.8 pm) and the GaAs surface "cap" region. T o illustrate the use of these calculated

116


TABLE I

CALCULATED DEPENDENCE OF J,, ON DISTANCE d2 BETWEEN A10.1Ga,.,As RECOMBINATION REGION AND GaAs:Ge CAPLAYER( D H LASER)" d , (pm)

G, (cm-'lh

rc

0.5 0.75 1 .o

200 40 15

0.5 0.5 0.5

Jth

(A/cmi)d

vene

2886 1847 1684

0.14 0.36 0.48

Assumed parameters (refer to Fig. 33 for basic structure):d,=0.2pm;d4=2pm;al = a 5 = 1.5 x 104cm-'; An = 0.1; ,IL= 0.8 pm;acrid = (l/L)ln(l/R) = 27 cm-'; arc = 10 cm-'; a2 = a4 = 10 cm-l; '1; = 0.7. From Fig. 34. ' From Fig. 13. From Eq. (20). ' From Eq. (15).

values to determine device properties, we show in Table I values of Jth and vex' calculated using Fig. 34 to determine Gth, Fig. 13 to determine r, Eq. (20) for J l h , and Eq. (15) for qexI.Table I clearly shows the calculated effect of having the absorbing GaAs "cap" region too close to the recombination region. For example, with a spacing d2 = 0.5 pm, J t h is over loo0 A/cm2 greater than with d 3 = 1 pm. Even if the recombination region contains GaAs (A, = 0.9 pm) instead of Alo.lGao.9As,the effect of the absorption in the surface GaAs layer is also important because the high free carrier concentration in the "cap" ( >loi9 results in a high absorption coefficient 2 100 cm- '. Experimental data indeed show that a spacing of 2 0.8 pm between contact and surface layer is needed to eliminate the deleterious effect of the surface GaAs layer69on DH laser performance. The effect of changing the width of the recombination region on Gth can be seen from Fig. 35. As d3 is decreased, more of the wave propagates outside the recombination region, and the device loss becomes more sensitive to the distance between the recombination region and the GaAs "cap" region. To illustrate the use of the theoretical curves in predicting the major laser parameters, we present in Table I1 data for two well-characterized Calculated and measured values of Al,Ga,_, As-AI,Ga,-,As DH 1a~er.s.~' Jth, vex,. and 8,, also shown in Table 11, are all in reasonable agreement, indicating how closely it is possible to predict the device performance from 69

’O

H. C. Casey, Jr. and M. B. Panish, J. Appl. Phys. 46,1393 (1975). I. Ladany and H. Kressel, unpublished.

2.

117


5 00

200

-5

7

100

f (3

50

D J

0

I u)

w

a

I

c

20

k

a 3 10

z 0 W

W

a

?

w

2

IV Q

2

0.I

0.2

.3

ACTIVE REGION WIDTH d3(MICRONS)

FIG. 35. The active region gain coefficient (A, = 0.8 p n ) at threshold as a function of the active region width d , for various An values. (After Bulter et ~ 1 . ~ ’ )

basic parameters. This is predicated, of course, on good junction quality as discussed in Parts V and VI. There is limited information available concerning four-heterojunction lasers compared to double-heterojunction lasers. Among the lowest reported J t h have been with FH lasers in which d 3 5 0.1 pm, as summarized in Table III.71-73 The lowest Jthof -600 A/cm2 is to be compared to J t h z loo0 A/ cm2 routinely achieved with DH structures. G . H. B. Thompson and P. A. Kirkby, Electron. Lett. 9,295 (1973). M. B. Panish, H. C. Casey, Jr., S. Sumski, and P. W. Foy, Appl. fhys. Lett. 11,590(1973). 7 3 H. C. Casey, Jr., M. B. Panish, W. 0.Schlosser,andT. L. Paoli, J . Appl. fhys. 45.322(1974). 72

118

HENRY KRESSEL AND JEROME K. BUTLER TABLE I1

COMPARISON OF EXPERIMENTAL AND CALCULATED Jlh, O,

AND

q.%, FOR THINDH LASER

A1 Concentration

Diodeno. 31 78

d3

x z L ? x3'

Ax

xqC

a

gih (calc.)

88 50

31 78

Gth(cm-') a,,(cm-') 7 7

0.18 0.29 0.08 0.30 0.21 0.13 0.45 0.18 0.31 0.08 0.31 0.23 0.14 0.48 Jlh

Diode no.

r

An

IAicm')

10 10

L ( p m ) aend(cm-l) 340 710

32 16

81 (deg)

Vex1

calc."

observed

calc.

observed

calc.'

observed

1569 1346

1950 1140

0.57' 0.47'

0.49 0.44

37 37

45 41

(A1Ga)Asp-type region, a2 = 10 cm-'.

' Recombination region, a,, = 10 cm-' (A1Ga)Asn-type region, a., = 10 cmFrom Eq. (20). Assuming qi = 0.7. From Fig. 12.

(IL2 0.8 pn).

'.

TABLE I11

Low THRESHOLD FOUR-HETEROJUNCTION LASERDATA Al Concentrations"

1030 670 527 625

650 900 575 720

51 41

55 52

33 20 26 30

Panish et a!.’’ Thompson and Kirkby7' Thompson and Kirkby" Thompson and Kirkby7'

Refer to Fig. 14 for definition of layer numbers. Some small Al concentration (x3 < 0.03) may exist in these layers judging from the emission wavelengths reported. a

9. TEMPERATURE DEPENDENCE OF Jlh The threshold current density increases with temperature in all types of semiconductor lasers, but no single expression is rigorously valid for all devices or temperature ranges. It is convenient to use an approximation

2.

119


x exp(T/T,), but it has no theoretical basis because many factors can affect the J[h temperature dependence:

J[h

(1) The carrier confinement can change if the potential barrier at the heterojunction is relatively low (Part 11). ( 2 ) The radiation confinement can change for small An values (Part 111).

(3) The internal quantum efficiency can decrease with increasing temperature, although this effect is usually small in good-quality heterojunction lasers.

/

d t -

I

40 ~ 60 80 100 l

1

200

I

400

1

LL

0

600O

f

3

2

TEMPERATURE ( O K ) (a) FIG.36. (a) Temperature dependence of the threshold current density Jlh (0) of a DH laser diode with a lightly doped GaAs ( N o = 2 x 10" cm-3) recombination region d , = 1.3 pn. Also shown are the theoretical dependence60 of the nominal current density J,,, ( x ) needed to maintain a gain coefficient of 50 cm ' in undoped GaAs and the corresponding injected carrier pair density (A).(b) Ratio of the threshold current density at 70 and at 22'C. (After Kressel and Ettenberg.67")(c) Ratio of the differential quantum efficiency at 7 0 C to its value at 22-C for AI,Ga, _,As double-heterojunction laser diodes for varying values of x (and corres(0) ponding heterojunction barrier height). Data of Goodwin et a/.' 7a for 5,,(65 )/J,,(10)(0). undoped active region; (A) Ge-doped active region. ~

120


(4) The effective absorption coefficient may be strongly temperature dependent if r is changing. (5) If the above factors are unimportant, then J t h increases because more injected carriers are needed with increasing temperature to maintain a given gain coefficient.

The temperature dependence of the nominal current density J,, (and of the corresponding injected carrier pair density) for a constant gain coefficient g = 50 crr-' has been calculated by Stern6' between 80 and 400 K for an undoped GaAs active region. Figure 36a shows that J,,, a T'.33. The same figure also shows the measured f t h K to 300°K for a DH laser with a lightly doped ( N o = 2 x 10l6~ m - recombination ~ ) region (d3 = 1.3 pm), in good theoretical agreement. Above room temperature, if the internal losses remain constant (i.e., gth is constant with temperature), the theoretical prediction is that J t h should increase by a factor of about 1.23 between 300 and 350"K.60 A convenient

2.


121

measure to gauge the temperature dependence of J t h is the ratio of the two values at 70 and 22°C. Figure 36c shows that one finds experimentally’7a967a that a barrier height of at least 0.3-0.4 eV is needed to obtain a threshold current density increase of 1.5 in that temperature interval, For lower barrier heights, the steeper temperature dependence of the threshold current density is mainly attributed to decreasing carrier confinement with increasing temperature. Note in Fig. 36b that the differential quantum efficiency is nearly temperature independent in the strongly confined lasers, whereas the loss of carriers decreases the internal quantum efficiency in the other structures. V. Laser Diode Technology 10. EPITAXIAL GROWTH

Liquid phase epitaxy (LPE)74 is the preferred technique for fabricating (A1Ga)As heterojunction lasers, and is widely used for other materials as well. A detailed review of LPE theory and technology has been p r e ~ e n t e d , ~ ’ . ~ ~ including specific aspects bearing on heterojunction structures. The present discussion is intended as an introductory treatment only. Although epitaxial growth of AlAs-GaAs alloys on GaAs substrates is aided by the almost perfect match in lattice parameter between these two compounds, multiple layer growth of heterojunction structures is complex because it requires layers with different compositions and dopings. In addition, the interfaces between these layers must be flat; there must be no contamination from one growth solution to the next; the layer thickness must be precisely controlled to submicron tolerance; and the final surface of the processed wafer must be free of any solution, to permit the application of ohmic contacts. The technique used for GaAs or (A1Ga)Asgrowth consists of sequentially sliding a GaAs substrate into bins containing various solutions. There are many possible designs of the growth apparatus, but variations of the linear multiple-bin graphite boat77-80are the most popular. In a model shown in Fig. 37% a GaAs source wafer, usually polycrystalline, precedes the 74

H. Nelson, RCA RPU.24, 603 (1963).

’’ A comprehensive review is presented by H . Kressel and H. Nelson, Properties and application of 111-V compound films deposited by liquid phase epitaxy, in “Physics of Thin Films” (G. Hass. M . Francombe. and R. W. Hoffman, eds.). Vol. 7. Academic Press. New York. 1973. ’6 A special issue of the J . Cryst. Growth is devoted to liquid phase epitaxy (Vol. 27, 1974) ” H. Nelson. U. S. Patent No. 3,565,702 (1971). M . B. Panish, S. Sumski, and I. Hayashi, Trans. A I M E 2 , 795 (1971). 79 J. M. Blum and K. K. Shih, J . Appi. Phys. 43. 1394 (1972). H. F. Lockwood and M . Ettenberg, J . Cryst. Growth 15, 81 (1972).

122


I

GRAPHITE

PHlTE SPACER

FIG.37. Schematic illustration of liquid phase epitaxy growth boat. In (a) the saturation of the solutions is completed by the source wafer preceding the substrate wafer while in (b) each solution is saturated by its own source wafer. (After Lockwood and Ettenberg8")

substrate wafer to assure saturation of the solution in each bin before growth on the substrate is initiated." Figure 37b is an improvement over this design in that each solution has its own source wafer, leading to improved layer control. In a well-fabricated wafer, there are gradual steps on the surface with a height of 0.1 pm or less; thus, the wafer can be processed for ohmic contact without final polishing. This is especially important in the cw laser where the active layers of the structure are within a few micrometers of the surface. The performance of laser diodes is sensitive to the metallurgical perfection of the active region. It is important to have a very high internal quantum efficiency (i.e.,freedom from nonradiative centers),planar junctions for radiation guiding, and freedom from clustered defects which produce absorption in the vicinity of the p-n junction. Table IV lists the effects of various metallurgical defects on laser properties. The simplest heterojunction structure to prepare is the single-heterojunction laser, which requires the growth of only a single Zn-doped (A1Ga)As epitaxial layer on an n-type (2-4 x 10" ~ m - substrate. ~ ) The Zn diffuses from the epitaxial layer into the substrate a distance of about 2 pm.31 The diffusion step can accompany the epitaxial growth or be separate. Because the recombination region is actually within the substrate, its crystalline perfection as well as the carrier concentration are imporant. Silicon-doped GaAs grown by the Bridgman technique generally yields the best devices because of relative freedom from precipitates, a significant difference from GaAs:Te.'l Note that GaAs:Si grown from the melt is always n-type, in H. Kressel, H. Nelson, S. H. McFarlane. M. S . Abrahams, P. LeFur, and C. J. Buiocchi, J . Appl. Phys. 40,3587 (1969).

2.


123

TABLE IV

IMPERFECTIONS AkHx-TINC; PARTICULAR LASERPARAMETERS Laser Parameter Low internal quantum etliciency"

High absorption coefficient"

Nonuniform emissionh

Imperfection Small precipitates Nonradiative centers either in bulk or at heterojunction interfaces Precipitates High free carrier density Nonplanar junctions Spatial variations in Al (or other alloy) concentration Nonuniform dopant density Precipitates Nonradiative centers Nonplanar junctions Damaged reflecting facets

* These affect the threshold current density and differential quantum efficiency.

May produce lasers with anomalous "kinks" in the power versus current curve.

contrast to LPE-grown material where the growth temperature determines the majority carrier type.75 The refractive index step between the p-type Zn-diffused recombination region and the n-type region strongly affects the device characteristics, particularly at high temperature (Part 111). The smaller the step, the lower the temperature at which mode guiding is lost, and at which the threshold current density increases steeply. Part of the index difference at the p-n junction is due to the high electron concentration in the rz-type region. As indicated in Section 6, the refractive index decreases with increasing free carrier concentration. Carrier concentrations of -4 x 10l8 cm-3 are therefore desirable for obtaining the best high-temperature diode operation; however, the defect density in such crystals is sometimes too high. Instead of forming the recombination region by diffusion. it is also possible to separately grow the various regions of the single-heterojunction laser using a multiple-bin boat.82 However, the added complexity in multiplelayer fabrication is generally justified only for the LOC laser because of its improved performance over the single-heterojunction device. Of the acceptors used to fabricate multiple-layer lasers, the most widely used is Ge rather than Zn, because of its lower vapor pressure which reduces the cross-contamination of the epitaxial layers during growth. The ionization energy of Ge increases with A l concentration in (A1Ga)As to a much 82

H. T. Minden and R. Premo. J . A p p / . Phrs. 45.4520 (1974).

124


greater extent than Zn, which limits the utility of Ge in Al-rich alloys.83 Although Ge is amphoteric in G ~ A Swhen , ~ ~prepared by liquid phase epitaxy in the usual growth temperature range below 900°C it introduces predominantly a shallow acceptor (Ei 2 0.038 eV). Hole concentrations up to -3 x 1019 anF3can be ~btained.'~ The recombination region is sometimes doped with Si to produce a p-type, closely compensated region, or lightly doped with Ge (order 10'' crnp3), or left undoped to produce a region that is generally n-type (- 10l6~ m - ~ ) . The first LOC lasers' had recombination regions formed by Zn out-diffusion from the p+-(AlGa)As layer, but the growth of a Si-doped region is generally preferred and has yielded the best devices.44 By adjusting the growth conditions, both the p- and the n-type regions can be produced using only Si.86*87 With regard to the donors, both Sn and Te are used to provide electron concentrations up to - 4 x 10" c n P 3 .The lower segregation coefficient of Sn compared to Te75eases the precise control of the donor concentration. Both Sn and Te are shallow donors (Ei < 0.01 eV) in direct-gap (AlGa)As, but reach ionization values of 0.05-0.06 eV in the indirect-bandgap region of the alloy.88 The minority carrier diffusion length of GaAs doped with various impurities is shown in Table V. It is a guide to the maximum width of the recombination region. Vapor phase epitaxy (VPE)89is not commonly used for the preparation of (A1Ga)As alloys, partly because of the reactivity of the gases transporting the Al. Furthermore, the choice of dopants is more restricted than in LPE. In particular, the use of Ge as an acceptor, desirable because of its low diffusion coefficient, is not possible because it enters GaAs as a donor.g0However, H. Kressel, J . Elecrron. Muter. 3, 747 (1974). H. Kressel, F. Z. Hawrylo, and P. LeFur, J. Appl. Phys. 39,4059 (1968). 8 5 F. E. Rosztoczy, F. Ermanis, 1. Hayashi, and 9. Schwartz, J . Appl. Phys. 41, 264 (1970). 8 6 B. H. Ahn, C. W. Trussel, and R. R. Shurtz, Appl. Phys. Lett. 19,408 (1971). F. H. Doerbeck, D. M. Blacknall, and R. L. Carroll, J. Appl. Phys. 44, 529 (1973). H. Kressel, F. H. Nicoll, F. Z. Hawrylo, and H. F. Lockwood, J . Appl. Phys. 41,4692 (1970). R 8 a H. Schade, H. Nelson, and H. Kressel, Appl. Phys. Len. 18, 121 (1971). K. L. Ashley and F. H. Doerbeck, J . Appl. Phys. 42, 4493 (1971). 8 8 c L. W. James, G. A. Antypas, J. Edgecumbe, R. L. Moon, and R. L. Bell, J . Appl. Phys. 42, 2976 (1971). S. Garbe and G. Frank, in "Gallium Arsenide and Related Compounds" (Proc.Int. Symp., 3rd. Aachen, 1970), p. 208. Inst. Phys, and Phys, SOC.,Conf. Ser. No. 9, London 1971. Zh. 1. Alferov, V. M. Andreev, V. 1. Murygin, and V. I. Stremin, Fiz. Tekh. Poluprouodn. 3, 1470 (1969) [English trunsl.: Sou. Pbys.-Semicond. 3, 1234 (1970)l. 88r M. Ettenberg, H. Kressel, and S. L. Gilbert, J. Appl. Phys. 44,827 (1973). ""LJ H. C. Casey, Jr., 8.1.Miller, and E. Pinkas, J . Appl. Phys. 44, 1281 (1973). G. A. Acket, W. Nijman, and H. 't Lam, J. Appl. Phys. 45. 3033 (1974). J . J. Tietjen, Ann. Rev. Muter. Sci. 3, 317 (1973). E. W. Williams, Solid Stute Commun. 4, 585 (1966). 83

84

2.

125

HETEROJUNCTION LASER DIODES TABLE V

MINORITY CARRIER DIFFUSION LENGTHIN LPE GaAs

Type

Dopant

P

P P P P P P

Ge Ge Ge Ge Ge Si Zn Si Si Zn Cd

n n n

Sn Uodoped Sn

P

P P

P

Carrier Concentration (cm-') 5 6 2.0 8 1.1 5 7 5 5 5

x 1OlJ x 10" x 10l8

x 10" x 1019

Not stated 1018 x 10l8 x lorJ x 1OI6 x 10l6 x

7 x 1OlJ-5 x 10" 5 x 1045 5 x 1O16-10"

Diffusion Length (pm)

Ref.

6-7 20 10.5 2.5 5.5 6 -7 6 -3 -2-3.6 5 3

Schade ef ~ 1 . ~ Ettenberg et aLssr Ettenberg ef dssr Ackert ef a/.8ah Ettenberg ef Ashley and Doerbeckssb Jones ef ~ 1 . ~ ~ ' Garbe and Frankssd Alferov et Alferov et Alferov et

2.8-2.5 5-11 4-0.3

Alferov et Alferov et Casey er d S 8 g

VPE is superior to LPE for graded composition heteroepitaxial layers, because grading reduces the dislocation density in the active region of the device. Work is also proceeding on molecular beam epitaxy," which provides good control of layer thickness, and lasers have been made.92 Acceptor concentrations in GaAs or (AIGa)As appear relatively low compared to the lOl9 from other synthesis techniques. Combinations of molecular beam and liquid phase epitaxy offer interesting possibilities. 1 1. EFFECTOF DOPANTS ON THE EMISSION WAVELENGTH

For a given material, the lasing wavelength shifts somewhat with the dopant type and concentration. At room temperature, the GaAs lasing wavelength can be as short as 0.85 pm for highly Te-doped GaAs (3-4 x 10'' ~ m - ~or ) ,as long as 0.95 pm for heavy Si doping (order 1019cm-3).93 In n-type material, the lasing energy follows the shift of the Fermi level into the conduction band while, in p-type material, the lasing transitions involve conduction-band tail states. However, excessively high impurity concentrations in the recombination region are undesirable because of formation of nonradiative centers and increased free carrier absorption (Part IV). A. Y. Cho. J. Vac. Sci. Techno/. 8,531 (1971); J . Appl. fhys. 46. 1733 (1975). A. Y. Cho and H. C. Casey. Jr.. Appl. f h y s . Leu. 25, 288 (1974); H. Casey. Jr., A. Y. Cho and P. A. Barnes, IEEE J . Quantum Elecfron. QE-11, 467 (1975). 9 3 J. A. Rossi and J. J. Hsieh, Appl. fhys. Lerr. 21. 287 (1972). 91

92

~

~

126


g= 1 4 4 8 n

101’

1018

10 l9

Holes/cm3

FIG.38. Photon energy at threshold (77°K) of optically pumped p-type GaAs platelets as a function of the hole concentration. Circular data points (solid curve) indicate the laser photon energy at threshold. Triangular data points (dashed curve) show the photon energy of secondary transitions, which may or may not lase depending upon pumping intensity and geometry. (After Rossi er

The shift of lasing energy with acceptor concentration is best studied at low temperatures. Figure 38 shows the lasing peak energies observed in p-type (Ge and Cd-doped) GaAs optically pumped at 77"K.94For low concentrations of acceptors relative to the injected carrier density, the higher peak (near 1.5 eV) is a near-bandgap transition seen in undoped GaAs (as discussed below),whereas the peak at 1.48 eV involves the acceptors. Both lasing lines can be seen simultaneously at certain levels of doping and carrier injection, but at high doping levels (> 10l8 only the low-energy lasing transitions are observed. At this point, bandtailing moves states further into the forbidden gap with increasing acceptor concentration. Figure 39 illustrates this effect at 77°K for n-type, p-type, and compensated G ~ A s . ~ ~ ' The lasing transitions in relatively pure GaAs have been extensively studied since discovery that the lasing photon energy is always less than the bandgap ene~-gy.~'-~' It appears that, at low temperatures, the k selection rule is not obeyed for lasing transition in very pure G ~ A s . ~ ~ .A' "detailed study of

-

J. A. Rossi, N. Holonyak, Jr., P. D. Dapkus, J. B. McNeely, and F. V. Williams, Appl. Phys. Lett. 15, 109 (1969). 94a P. D. Dapkus, N. Hotonyak, Jr., J. A. Rossi, F. V. Williams, and D. A. High, J . Appl. Phys. 40,3300 (1969). 9 5 N. G. Basov, 0. V. Bogdankevich, V. A. Goncharov, B. M. Lavrushin, and V. Yu. Sudzilovskii, Dokl, Akad. Nauk SSSR 168, 1283 (1966) [English transl.: Sou. Phys-Dokl. 11, 522 (1966)l. 96 J. A. Rossi, N. Holonyak, Jr., P. D. Dapkus, F. V. Williams, and J. W. Burd, Appl. Phys. Lett. 13, 119 (1968). 97 K. L. Shaklee, R. F. Leheny, and R. E. Nahory, Appl. Phys. Left. 19, 302, (1971). 9 8 H. Kressel and H. F.Lockwood, Appl. Phys. Lett. 20, 175 (1972). 9q E. Gobel, Appl. Phys. Lett. 24,492 (1974). loo E. Gobel and M. Pilkuhn, J. Phys. (France)Suppl. 435, C3-191 (1974). 94

2.

127


i

P

n

(a)

1Ol6 DOPANT CONCENTRATION (cm-')

10"'

(b) FIG.39. Dependence of the laser photon energy on impurity concentration at 77 K in an optically pumped GaAs laser. The rr-type samples are Se. Sn, or Te doped; the p-type samples are Cd or Zn doped. The compensated material is Zn-Sn, Zn-Te, or Zn-Se doped. The laser photon energy becomes asymptotic to hv = E, - 0.015 eV at low impurity concentrations. (a)The diagrams illustrate the recombination processes appropriate to the various curves in (b). The dashed curves in (i)denote the reduced effective gap at high injection. (After Dapkus et a/.94a)

lasing has been made as a function of temperature in double-heterojunction laser structures where both the stimulated and spontaneous emission could be observed through a surface window of high bandgap (AlGa)As, thus eliminating the spectral distortions in the spontaneous emission viewed through the edge of conventional laser diode^.^^,^^ Figure 4046*'0'compares the bandgap energy and the lasing peak energy; their difference decreases I01

D. D. Sell, Proc. In!. Conf. Phys. .Scmic,orrtl. 11th Warsaw, 1972

128


1.400

0

I00

2 00

300

T( K)

FIG.40. Temperature dependence of the “one-electron” (i.e., low injection level) bandgap energy of GaAs as determined by Camassel Ct ( x ); Selllo’ ( ); Sturge”” (0).The experimentally determined variation of the lasing photon energy in “pure” GaAs is from the data of Kressel and Lockwood,6’ (m);Chinn, er u1.’02 (0).The theoretical curve is from Camassel er

with temperature from 20 to 30 meV at room temperature. Also shown in Fig. 40 is the temperature dependence of the lasing peak energy in pure GaAs as determined by optical excitation of the material.102The values are similar to those obtained in the laser diode.65 It is believed that the reduced lasing energy results from “bandgap shrinkage” and bandtailing due to the high injected carrier density, effects theoretically treated in Camassel et al.46and Brinkman and Lee.lo3The bandtailing effects involve the extension of states into the forbidden gap, whose density increases with the injected carrier concentrations. For further details concerning doping effects on (AlGa)As/GaAs doubleheterojunction lasers, we refer to the studies of Pinkas et d l o 4 M. D. Sturge, Phys. Rev. 127,768 (1962). S . R.Chinn, J. A. Rossi, and C. M. Wolfe, Appl. Phys. Lett. 23,699 (1973). l o 3 W. F. Brinkman and P.A. Lee. Phys. Rev. Lett. 31, 237 (1973). ‘04 E. Pinkas, B. 1. Miller, I. Hayashi, and P. W. Foy, J. Appl. Phys. 43, 2827 (1972).

lo*

2.


0

129

PHOTOLUMINESCENCE AND MICROPROBE DATA

ALUMINUM FRACTI0N.r

FIG.41. The E,, line is the bandgap energy as a function of x for AI,Ga, -,As in the Ga-rich portion of the alloy system as determined from electroreflectance data.lo5 The experimental data are from Ladany and Kressel (unpublished).

12.

PERTINENT

PROPERTIFS

OF

Al,Ga, -,As

a. Bandgap Energy The bandgap energy Egr as a function of alloy composition (determined from electroreflectance measurements) can be analytically expressed by (see Fig. 41)'" as Egr = 1.424 1 . 2 6 6 ~+ 0.266~'. (23)

+

The dependence of the X minima on composition is not well known, except for the endpoint values of 1.86 eV in GaAs'06 and 2.16 eV in AlAs."' As shown in Fig. 41, the T-X crossover energy Eg z 1.92 eV, a value consistent with other data.'08-"0 The corresponding x value is still somewhat uncertain, but is in the vicinity of 0.37-0.42. Recent experimental data indicate that the L conduction band minima (at 1.72 eV), rather than the X minima, are closest in energy to the r minimum, contrary to long established belief.' lo' As a result, the indirect minima are closer to the direct conduction band valley than previously believed over

-

0. Berolo and J. C. Wooley. C’un. J. Pliys. 49, 1335 (1971). 1. Balslev, Phys. Rev. 173, 762 (1968). lo' M. R.Lorenz, R.Chicotka, G. D. Pettit, and P. J. Dean, Solid State Commun. 8,693 (1970). ' 0 8 H. C. Casey. Jr. and M.B. Panish. J. Appl. Phys. 40,4910(1969). log H. Nelson and H. Kressel, Appl. Phys. Let/. 15, 7 (1969). H. Kressel, H. F. Lockwood, and H. Nelson, IEEE J. Quunrum Electron. QE4.278 (1970). D. E. Aspnes, C. G. Olson, and D. W.Lynch, Phys. Reo. Left. 37, 766 (1976).

130


a substantial portion of the direct bandgap alloy. However, because in AlAs the L minima are believed to be substantially above the X minima, the X-T valley crossover controls the direct-to-indirect bandgap transition. With respect to semiconductor lasers of alloys incorporating GaAs, the position of L conduction band minima will have some effect on the internal quantum efficiency dependence on alloy composition. Barring the availability of sufficient bandstructure data, however, we will assume in Section 12d that only the T-X separation is relevant to the internal quantum efficiency. b. Refractive Index

The effective refractive index step An in GaAs-(A1Ga)As heterojunction lasers has been determinedI6 as a function of the bandgap energy step from the radiation patterns. These data are shown in the form of An versus x at A z 0.9 pm in Fig. 42 (where x is the A1 concentration in the high bandgap side of the junction). Figure 42 also shows An versus x (at A z 0.9 pm)deduced from epitaxial layer measurements."' Up to x z 0.38, An z 0.75x, with bowing at high x values. (The estimated accuracy in determining x is generally k 0.02). For comparison we note that the measured refractive index at I = 0.9 pm is 3.59 in GaAs and 2.971 in A1As.'l2 Assuming a linear dependence of n on the A1 concentration, at 1. = 0.9 pm we estimate an average value of An = 0.62~. The carrier and dopant concentrations affect the refractive index. The refractive index has been calculated for GaAs at 300 and 77°K from absorption coefficient data,lI3 and measured for photon energies between 1.2 and 1.8 eV at room temperature as a function of d ~ p i n g . " ~ ~ " ~ In addition to changes in the density of states distribution with doping (which changes the shape of the absorption curve), the index is depressed by a contribution to the refractive index due to intraband absorption of free carriers. The approximate expressions given by Stern26 are useful in estimating the difference in refractive index at p-n junctions, p + - p or nf-n interfaces. For n-type material, the only significant contribution is from intraband absorption ;

Anincraz -9.6 x 10-"NlnE2,

(24)

where N , n, and E are the electron concentration, index of refraction, and photon energy, respectively. H . C. Casey, Jr., D. D. Sell, and M. B. Panish, Appl. Phys. Left.24,63 (1974). A. Onton, M. R. Lorenz, and J . M. Woodall, Bull. A m . Phys. Soc. 16. 371 (1971). J . Zoroofchi and J . K . Butler, J. Appl. Phys. 44,3697 (1973). D. T. F. Marple, J . Appl. Phys. 35, 1241 (1964). D. D. Sell, H. C. Casey, Jr., and K. W. Wecht, J . Appl. Phys. 45,2650 (1974).

'I1

I"

’I3 ’I4 ’15

2.


8.

131

/ /

0 A\

FRACTION x

FIG. 42. Refractive index (at -0.9 pm) difference between the GaAs recombination region and the outer AI,Ga, -,As region as a function of z.n-(GaAs) = 3.59, w(AIAs) = 2.971. Curve A is estimated from laser radiation pattern measurements," curve B is from refractive index measurements.' I '

For p-type material, interbund transitions are significant also, and the total free carrier contribution is Aninter+ Anintra2 - 1.8 x 10-21P/nE2- 6.3 x 10-22P/E2, (25) where P is the hole concentration. In GaAs, as an example, for 10l8cm-3 electrons, An = -0.0014; with 1 O I 8 emp3 holes, An = -0.00026 0.00032 = - 5.8 x In the case of injection into a recombination region, the refractive index will be reduced as a result of both the injected electrons and holes. This reduction can significantly lower the index at a p-n junction, an effect important in single-heterojunction lasers as discussed in Part 111. c. Thermal Conducticity

The thermal resistivity of AI,Ga, -,As affects the thermal resistance of lasing structures. Figure 43a shows the data of Afromowitz'I6 compared to the theoretical curve of Abeles.'" It is evident that the (A1Ga)As layers 'I'

M . A . Afromowitz. J . Appl. Phrs. 44,1292 (1973). B. Abeles, Phys. Rev. 131, 1906 (1963).

132


FIG.43a. The thermal resisitivity of AI,Ga, -,As alloys as a function of x. The solid line is a theoretical fit to the data*l6 using the model developed by Abele~."~ TABLE V1 AND CROSSOVER COMPOSITION VALUESx, BANDGAP ENERGY

Compound

Bandgap Energy Range (eV) E,,

GaAs,P,-, AI,Ga, _,As In, -,Ga,P AI,In, -,P

1.42-2.26 1.42-2.16 1.34-2.16 1.34-2.45

x,

Lattice Constant a. Range (A)

0.452 0.37' 0.616 0.394

5.6533-5.4506 5.6533-5.6607 5.8694-5.4506 5.8694-5.4625

FOR

FOURTERNARY ALLOYS'

Best Substrate near E,,

Maximum Practicalb Photon Energy, i, for Direct Bandgap Emission (A), Color

GaAs' 1.99 1.92d 2.17 2.23

(ao = 5.6533 A)

GaAs GaAs' GaAs'

1.89 6560 1.82 6800 2.07 6O00 2.13 5820

Red

Red Yellow Green

Data collected by Archer,117aexcept as noted in (d). Assuming hv = E,, - 0.1 eV. Substantial lattice mismatch between epitaxial layer and the substrate. Based on electroreflectance data of Berolo and Wooley.' 17b

between the active region and the heat sink should be thin, requiring a compromise with the need for proper radiation confinement as discussed in Part 111.

d. Internal Quantum Eficiency As the energy of the r and L or X minima approach each other with increasing A1 concentration (see Fig. 41), an increasing fraction of the carriers 'I7'

R. J. Archer, J. Electron. Muter.

1, 128 (1972). 0. Berolo and J. C. Wooley, Can. J. Phys. 49, 1335 (1971).

2.

133


injected into the recombination region will be transferred by thermal activation from the r to the L and X minima. It is generally assumed that the carriers in the indirect conduction band minima do not contribute to radiative recombination or stimulated emission.Table VI lists the bandgap parameters for four ternary alloy systems of present or potential interest in laser diode fabrication. Neglecting the shift of the quasi-Fenni levels with carrier density, and neglecting the L minima, the fraction of carriers in the X minima is determined by the ratio of the density of states and the energy difference BE between the r and X minima. The relative internal quantum efficiency as a function of alloy composition has been estimated experimentally in (AlGa)As,(InGa)P, and Ga(AsP). The data (obtained from spontaneous luminescence measurement^)"^^ are shown in Fig. 43b as a function of emission photon energy.' The calculated curves,119assuming a density of states ratio of 50 for the states in the indirect and the direct conduction band minima for each of the alIoy systems, is

'*

FIG. 43b. Calculated relative external quantum efficiency119 for AI,Ga, -.As, GaAs, -,P,, In, -,Ga,P, and In, -,AI,P lightemitting diodes as a function of photon emission energy. The data points for AI,Ga, _xA~119'and for GaAs, -xPx119b are relative electroluminescence efficiencies. The In, -,Ga,P data"" are relative cathodoluminescence efficiency values. The internal quantum efficiency may be assumed to follow the externally measured efficiency.

PHOTON ENERGY ( t V ]

The internal quantum efficiency is assumed to scale with the measured external efficiency. Note that the values in Fig. 43b are relative ones, being normalized to the binary value (x = 0). C . J. Nuese, H. Kressel, and 1. Ladany, IEEE Specirwn 9.28 (1972). R. J. Archer, J. Elecrron. Muter. 1, 128 (1972). ’I9’ H.Kressel,F. Z. Hawrylo, and N. Almeleh, J. Appl. Phys. 40,2248 (1969). A. H. Herzog, W.0.Groves, and M.G. Craford, J . Appl. Phys. 40, 1830 (1969). 119c A. Onton, M. R. Lorenz, and W. Reuter, J. Appl. Phys. 42. 3420 (1971).

117c

134


TABLE VII

THERMAL EXPANSION COEFFICIENT OF SELECTED MATERIALS a(

InP GaP GaAs Ge AlAs InAs

C l ) *

(4.75 f 0.1) x (5.91 f 0.1) x (6.63 0.1) x 5.75 x (5.20 f 0.05) x (5.16 0.1) x lo-'

an (27'C)

a. ( - 6 W C )

Ref.

4.8697 5.4510 5.6525 5.6570 5.6605 6.057

5.8870 5.4742 5.680 5.6603 -5.6790 6.080

Kudman and Paff'20a Pierron er al.lzob Stranmanis and Krumme120' and PafflZod Gibbons12oe Ettenberg and Paff'20f PafflZod

' The thermal coefficient of expansion may be assumed to vary linearly with composition in ternary alloys. The a,, values shown may differ slightly from those reported by other investigators.

superimposed on the experimental data. There is approximate agreement, although the precision of the data is not sufficient to warrant a more detailed analysis, particularly in view of the uncertainties in the bandstructure within the direct-bandgap region of the alloys. Note that there are no experimental data for (1nAI)Palloys. e. Lattice Parameter

The lattice parameters of AlAs and GaAs are equal at about 900"C,'z0 but differ at room temperature where a, = 5.661 (AlAs)and 5.653 A (GaAs), because of different thermal expansion coefficients (Table VII); a, can be assumed linear in A1 composition in this alloy system. The elastic strains at heterojunction structures have been measured.'*' The use of quaternary alloys allows an additional degree of freedom to help match both the lattice parameter and the thermal coefficient of expansion for any bandgap. By adding small amounts of phosphorous to the outer (A1Ga)As layers of a double-heterojunction laser with GaAs in the recombination region, it is possible to obtain a matching thermal coefficient of expansion of the outer and inner layers of the d e v i ~ e . ' ~However, ~-'~~ there is then a lattice mismatch at the growth temperature and misfit dislocations can easily be formed-unless the width of the active region is < 1 pm M . Ettenberg and R. J. Paff, J.Appl. Phys. 41, 3926 (1970). I. Kudman and R. J. Paff, J. Appl. Phys. 43,3760 (1972). l Z n b E. D. Pierron, D. L. Parker, and J. B. McNeely, J. Appl. Phys. 38,4669 (1967). l Z o eM. E. Stranmanis and J. P. Krumme, J. Elecfrochem. Suc. 114, 640 (1967). I2Od R. J. Paff, private communication. 1 2 0 e D. F. Gibbons, Phys. Reu. 112, 136 (1958). "Of M. Ettenberg and R. .I. Paff, J . Appl. Phys. 41,3926 (1970). ''I F. K. Reinhart and R. A. Logan, J. Appl. Phys. 44, 3171 (1973). l Z 2 G. A. Rozgonyi, P. N. Petroff, and M . B. Pdnish, J. Cryst. Growth 27, 106 (1974). R. L. Brown and R. G. Sobers, J. Appl. P h p . 45. 4735 (1974).

2.


135

In such thin regions the material remains strained; and AuJq, 0.1 -0.3 pm.

-

9000 and 8000 A. Below 7800 A, however, Jthincreases mainly because of the reduction of the internal quantum efficiency discussed in Part V. In addition, the material becomes more difficult to prepare. 221 221a

H. Kressel and F. Z. Hawrylo, J . Appl. Phys. 44,4222 (1973). H. Kressel and F. Z. Hawrylo, Appl. Phys. Lett. 28, 598 (1976).

2.


175

Double-heterojunction stripe-contact lasers have been made that lase pulsed at 300°K to 6880 A (J,,, = lo5 A/cm2); cw operation at 7400 A was also At 77”K, lasing is easily obtained to about 6300&221 with the lowest reported (A1Ga)Aslaser emitting at 6190 A ( J t h = 300 A/cm2).222Continuous operation at 6500-6600 A has been obtained with emitted power in excess of 50 mW.221 The shortest-wavelength low-temperature (77°K) laser diode emission is (Jthz lo4 A/cm2) from a double-heterojunction device consisting of InGaAsP layers grown by LPE on Ga(AsP) substrates.222aThe shortestwavelength cw laser operation near room temperature reported is at 7030 A (heat sink temperature of 10°C). These devices were In,,34Gao,6,P/ GaAs,.,P,,, double-heterojunction structures with an active region width of 0.2 pm.222h IX. Distributed-Feedback Lasers The desire for integrated optical c i r c ~ i t s , in~ ~ which ~ , ~modulators, ~~ switches, waveguides, and radiation sources are formed monolithically, has led to interest in laser diodes not requiring cleaved facets for optical feedback. In addition to the elimination of facets, distributed-feedback structures can give a “pure” spectral emission by limiting the longitudinal modes. Whereas conventional Fabry-Perot (FP) lasers employ discrete end reflectors, distributed-feedback (DFB) lasers make use of periodic dielectric variations along the direction of propagation. Longitudinal mode selection in FP lasers is quite different from that in DFB lasers. Typically one finds a proliferation of modes in FP lasers while D F B s offer longitudinal mode control. An illustrative double-heterojunction structure shown in Fig. 64 has periodic variations of the dielectric constant along the z direction (propagation direction). Each length defining a corrugation period produces a scattering of the propagating wave into the opposite direction. The field reflected from a traveling wave has a phase term. Thus, to obtain positive feedback there must be a phase connection between the reflected field of each corrugation and the phase of the traveling wave. This fact gives rise to a strict relationship between the waveguide mode wavelength and the grating period, which can be approximated by

lLg =2A, 222

1 = 1,2,3,. . . ,

K. Itoh, Appl. Phys. Lett. 24, 127 (1974). W. R. Hitchens, N. Holonyak, Jr., P. D. Wright, and J. J. Coleman, Appl. Phys. Lett.

27, 245 (1975). H , essel, G. H. Olsen, and C. J. Nuese, Appl. Phys. Lett. 30, 249 (1977). 2 2 3 S. E. Miller, Bell Syst. Tech. J . 48, 2059 (1969). 2 2 4 P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 14, 291 (1969). Z22b

176


II .

L

iI

p - T Y P E (AIGolAS REGION I

L

LASER OUTPUT

4

2 -o

LASER OUTPUT

n - T Y P E GOAS( ACTIVE REGION 3)

n - T Y P E (A1Ga)As REGION 5

F;7//////////////////llllllm----FIG.64. Distributed feedback. double-heterojunction diode.

where I is the Bragg reflection order, 1, is the guide wavelength, and A is the grating period. The waves traveling in the positive and negative z directions (contradirectional waves) are coupled via the gratings. In addition to the energy scattered into the forward and backward waves, there is scattering of power into a radiation field. The DFB structure can thus be viewed as a phased antenna array where each corrugation is an element of the array. The phase between each radiating element, depending upon the grating spacing and Bragg order I, determine the radiation direction of the main lobe. Kogelnik and Shank225*226 have investigated the characteristics of DBF dye lasers. Nakamura and c o - ~ o r k e r s ~ and ~ ~ -Shank ~ * ~ and Schmidt230 have observed lasing in corrugated structures of GaAsqA1Ga)As crystals by optical pumping at low temperatures (about 77°K). Injection DFB lasers H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18, 152 (1971). H. Kogelnik and C. V. Shank, J . Appl. Phys. 43, 2327 (1972). 2 2 7 M . Nakamura, A. Yariv, H . W. Yen, S. Somekh. and H . L. Garvin, Appl. Phys. Lett. 22, 315 (1973). 2 2 8 M . Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, and H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973). z29 M. Nakamura, K. Aiki, J. Umeda, A. Yariv, H. W. Yen, and T. Morikawa, Appl. Phys. Len. 25,487 (1974). 230 C . V. Shank and R. V . Schmidt, Appl. Phys. Left. 25, 200 (1974).

225

226

2.


177

first operated at low temperature^,'^ but (A1Ga)As devices with separate recombination and mode guiding regions (using the concept of the LOC and FH structures) have operated at room t e m p e r a t ~ r e ; ’ ~cw ~ .operation ~~~ has also been achieved.233a

23. COUPLED MODEANALYSIS Various technique can be used for the analysis of DFB lasers, but the application of the concepts of coupled wave theory to grated structures appears to be most fruitf~l.’~”2 3 8 The use of this perturbation theory allows one to determine the threshold gain and frequency characteristics in terms of the grating and waveguide geometry. For simplicity. we will consider only the TE modes in a three-layer waveguide. Assume regions 1 and 5 bracket a center slab region 3 with n3 > n n 5 . The trapped waveguide modes are of the form

E,

= Il/,(x)exp[i(wt

f B,,,:)],

(35)

which are solutions to Maxwell’s equations for an unperturbed reactive structure. The transverse behavior of the mth mode is $,(x) and 8, is the longitudinal propagation constant. We now introduce corrugations in the structure as indicated in Fig. 64. The set of trapped modes and the leaky modes236 form a complete set of eigenfunctions so that the fields in the perturbed structure can be expended in terms of the unperturbed modes. Limiting our discussion to only the coupling between a forward and a backward trapped mode of identical order, the waveguide field is E,

=

+

[Am+(z)L>-iB*,,=A,(:) e’!’m-‘]$,(.u),

(36)

where Am+ and A; satisfy the following differential equations:

+ rifbeiAZA;,

dA;/dz

= -aA:

dA;/dz

= ribFKiAZAc

+ aA;.

(374 (37W

D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. PhFs. Letr. 25, 203 (1974). K . Aiki, M . Nakamura. J . Umeda, A. Yariv, A . Katzir. and H. W. Yen. ,4ppl. Pliys. Lert. 27, 145 (1975). 2 3 3 H. C. Casey. Jr., S. Somekh. and M . Illegems, Appl. Phys. Lett. 27, 142 (1975). M . Nakamura, K. Aiki. J . Umeda. and A. Yariv, Appl. Phys. Lett. 27, 403 (1975). 234 D. Marcuse, “Light Transmission Optics.” Van Nostrand-Reinhold. Princeton. New Jersey. 1972. 2 3 5 A . Yariv, IEEE J . Quantum Electron. QE-9,919 (1973). 236 H. F. Taylor and A. Yariv, Proc. IEEE 62, 1044 (1974). 2 3 7 H. Kogelnick, Bell Sysr. Tech. J . 48, 2909 (1969). S. Wang, J. Appl. Phys. 44, 767 (1973).

’’I

”*

178


Here Kfb and Kbf are the coupling coefficients, A is a phase factor depending on and the grating geometry, and a represents the wave attenuation due to losses in all regions which we treat here as part of the perturbed dielectric constant. From Fig. 64, the real part of the perturbed dielectric constant can be written as

am

AK' = (nz - n:)[u(x - d3/2 + a) - u(x - d 3 / 2 ) ] f ( z ) ,

(38)

where u(x - xo) is the unit step turning on at xo and f ( z ) is a periodic function whose Fourier expansion is a

f(z) =

1

Ctei('"/')z,

(39)

I=-m

with

Equation (38) assumes the form AK'(x, z)=

2 AK;(x) ei(2tniA)z.

(41)

I

The imaginary part of the perturbed dielectric constant is AK"(x) = (l/k,)n(x)a(x),

(42)

where n ( x ) is the unperturbed index distribution and a(x) is the absorption constant. Note that region 3, -d,/2 < x < d3/2, has a(x) = a, = (afc- 9). The total dielectric perturbation AK = AK' - i k " . Using the standard analytical techniques, we substitute (36)into Maxwell's equations which give

where we have normalized the wavefunctions according to

Note that the term AK in the integrand of (43)contains spatial harmonics along z. The product of these spatial harmonics with the term in parenthesis gives phase terms varying with z. The terms of primary importance are those which give small phase terms, or produce the so-called matched-phase condition, This condition can be understood if we assume that A: and A; are

2.

179


slowly varying functions of 2. If a phase term is large, then the product of A: or A; and a rapidly oscillating function produces a highly oscillating function. On the other hand, some terms will be slowly varying with z. If we multiply both sides of (43) by dz and integrate, those terms with small phase form the major part of the integration while the terms with rapid oscillation contribute very little. The perturbed dielectric constant has both a real and an imaginary part. Only the real part contains spatial harmonics. The “dc component”

of the expansion in (43a) has a zero phase term which we will not consider here. Physically, this term relates to the fact that the effective width of the waveguide is modified by the corrugations which in turn modifies the propagation constant of both forward and backward waves. Substituting the perturbed dielectric constant into (43) and comparing with (37) one finds

c1=

A

+

( k o / 2 B m ) ( n , ~ , a 1~

= 2bm-

2h/A

26

Y

3

+

~ 4 5

4

,

(Mb) (MC)

0,

where a, is the portion of the unperturbed mode intensity in the ith region. The quantity 6 represents the deviation of the wave number of the unperturbed mode and ln/A;I is the spatial harmonic responsible for the scattering. Consider now the case where m = 1, i.e., fundamental transverse mode operation. For a symmetric structure, n , = n5,the fundamental mode wave function $, = IC/ is

“:1

-cos(h3d3/2)exp[h,(d3/2

- x)],

d3/2 < x

-cos(h3d3/2)exp[h,(d3/2

+ x)],

-d3/2 > x

where the normalization factor N, is

[

N i = -1 -+d3 sin h3d3 2 2 2h3

+ cosz(h3d3/2)] hl

(45)

180


(b-Akb = 0.25

-I

n

n, = 3.4

3

0

u

3.6 "5’3.4

-

104

d3= I p m A@ 0 5 0 0 8

I

L

I

I

L

I

I

I

The coupling coefficient is

The backward-forward coupling coefficient

satisfies

tiif = ( X i ' ) * .

(47)

It is important to note here that in (44) the integral is equal to the product of the dielectric step and the fraction of the unperturbed field in the corrugation region which we define as R. The distributed feedback coefficient K' = JKG'/ is tif = (

k~/2fl)CI( nnt)R ~ z koC,(n3- nl)R.

(48)

The feedback coefficient K’ has been calculated for various grating con65~we f i g u r a t i o n ~ In . ~ Fig. ~~~ ~ ~show the coupling coefficient Id) for a double-heterojunction laser with rectangular teeth. The laser is operating in W. Streifer, D . R. Scifres, and R. D. Burnham, IEEE J. Quantum Electron. QE-11, 867 ( I 975). 240 S. Wang, IEEE J. Quantum Eluctron. QE-13. 176 (1977). 239

2.

181


the fundamental (m= 1) transverse mode. Note that the coupling coefficient for I = 1 (first-order Bragg scattering) is larger than for high orders. As an pm-', and pm-', lx21 4 x example, at a, = 0.1 pm, Id)2 6 x 2 x 10-~ pm-'.

-

-

OF COUPLED MODES 24. SOLUTION

Consider now a structure with corrugations extending from

2 =

0 to

z = L. The lasing modes are determined by solving (37) with appropriate

boundary conditions. At 2 = 0. the forward wave is launched with zero amplitude, while the backward one starts at z = L with no energy. The waves in the DFB region upon reaching the boundaries at 2 = 0, L, transfer all of their energies into the uncorrugated waveguide regions. Consequently, the appropriate boundary conditions are A:=,

= A Z = L= 0,

(49a)

The wave solutions, found by elementary techniques, are A+(:)

=

A , sinh(;Iz)d",

A - ( z ) = & A,sinh;l(L

(50a) -

~)e-~".

(5W

where 7

+ + i6)2]"2.

= [ ( K ' ) ~ (a

(51)

The secular equation defining the different longitudinal modes is ),cothpL =

-(M

+ id).

(52)

Equation (52) contains complex quantities, and thus the roots ;I, will be complex. There are several regimes where (52)can be simplified; for example, the high-gain region where - M >> K corresponds to weak feedback compared to the active region gain. Equation (52) can thus be separated into its real and imaginary parts leading to 1%

2 tan- - + 26,L a9

d , L(K ' y a;

+ h,Z = (24

+ 6;)

ezzqL/(Myz

q

=

-

l)n,

(53b)

4/(.')2,

= 0, f 1,

f2.

Near the Bragg frequency, the modes are given by 6,L

=

( q - +)EL,

(54)

182


which corresponds to the modes in a Fabry-Perot cavity. The condition defining the optical cavity gain gth at threshold for a specific longitudinal mode number q is

where r is the optical wave confinement factor and G l h is the active region gain required to offset losses in the regions exterior to the active region as discussed in the previous sections. An important result deduced from (55) is that there is strong discrimination between the various longitudinal modes, which helps to produce spectral purity of the output. The threshold gain gth derived above has not taken into consideration the coupling between different transverse modes which includes the leaky modes. Equation ( 5 5 ) applies to the case of weak feedback. However, the gain coefficient glh can be machine calculated directly from (52) as illustrated by Kogelnik and Shank.z26In Fig. 66 we show the normalized gain coefficient of various longitudinal modes as a function of the normalized feedback coefficient IKIL,where L is the grating length. (The coefficient 1 has been dropped because the feedback applies to any Bragg order. The order I is selected from the grating period and the operating wavelength.) We now illustrate how the threshold current density of a DFB laser diode is calculated. Consider the structure of Fig. 64 with the following values: I . = 0.85 pm, a, = 0.1 pm, d3 = 1 pm, n, = n5 = 3.4, n3 = 3.6, ( b - A)/A = 0.25, A = 0.35 pm, and L = 200 pm. Assume fundamental transverse mode operation. In the waveguide I , 0.85/3.6 = 0.236 pm so that I = 3. From

-

I

I

I

I

2

3

NORMALIZED FEEDBACK COEFFICIENT

I 4 IKl

L

FIG. 66. Normalized gain coefficient as a function of the normalized feedback Coefficient. (Data from Kogelnik and Shank.’”) Integer parameters are q values.

2.


-

183

--

Fig. 65 we find 2 x l o p 3pm-’. Therefore, I K ~ ~ 0.4. L Now turning to Fig. 66 we find for q = 0 that r ( & h - Glh - cc,,)L 3. From Fig. 13a we 1. As a consequence there are no losses external to the mode guiding see region; i.e., = 0. Assuming a reasonable value of arc= 10 cm-’ we conclude gth = 10 + 3/0.02 = 160 cmThe computation of Jlhis now similar to that of a Fabry-Perot laser. Assuming an undoped active region,240aEq. (20a) predicts Jth = (160/0.044 4100)/qi = 7700/qi A / m 2 . Of course, this assumes an ideal structure. Experimental values can be higher if the periodicity of the grating is imperfect. Note that J , h can be reduced by increasing L and by using a LOC or FH structure where d3 is reduced compared to the DH example chosen above. The best experimental DFB J , , values, at 3 W K , are about 2000 A / c ~ ~ . ~ ~ The reduction to practice of the injection DFB structures has centered mostly on the use of (A1Ga)As-GaAs. The corrugations are commonly introduced by ion milling (although etching can also be used), and regrowth into the grooves so formed. The periodicity used is -0.3-0.4 pm. The most successful devices to date (operating at 300°K with J t h < 5000 A/cm2) have used the large optical cavity concept in which the recombination region is smaller than the waveguide region, thus allowing part of the optical energy ~ ~introduction ~,~~~ to couple to the periodic structures in the w a v e g ~ i d e .The of corrugations within the double-heterojunction has not been a successful approach perhaps because of excessive nonradiative recombination due to defects introduced in the process of ion milling. However, such structures have exhibited the basic DFB behavior. Figure 67a shows a schematic cross section of a doubie-heterojunction DFB laser in which corrugations separate the p-type GaAs from the p-type ( A I G ~ ) A S .Figure ’ ~ ~ 67b shows the spectra from a device operating in a single longitudinal mode. A completed DFB laser is shown in Fig. 67c in a stripecontact format suitable for cw operation. Note that operation in a single longitudinal mode does not ensure “single-mode” operation, even with control of the transverse modes, because of lateral mode proliferation in typical stripe-contact devices. However, a single mode could be produced in a sufficiently narrow stripe device. The maximum power level emitted into a single mode remains to be established. In addition to the structure we discussed above, which has a feedback mechanism built into the body of the laser, a second type (Bragg-reflector laser) has a grating appended to the laser body. These structures are discussed in detail by Wang.240

-

’.

+

240a

We also assume that the gain coefficient versus J,,, relationship at the lasing wavelength follows the curves of Fig. 30a.

184

HENRY KRESSEL AND JEROME K. BUTLER Au-Cr contact

p-GaAs

output

pGa0.7Alo.,As p-GaAs n-Ga, ,AI,,As

+

n-GaAs substrate

Au-Ge-Ni contact (a)

ASO-2-4P

4k

I

4t-

2 SA

8100 Wavelength (

8200

A 1

(b)

Excited region

contact

-lo2 film

b/’/’

rate

F A u - G e - N i contact

(C

f

FIG.67. (a) Schematic cross section of the double-heterojunction distributed-feedback laser. and (b)emission spectra from a double-heterojunction distributed-feedback laser (82°K). The active region t13 = 1.3 pm. L = 630 pn, ti, = 0.09 pm. and A = 0.3416 pm.'2' (c) Construction of DH DFB laser diode in stripe-contact configuration. (Courtesy of M. Nakamura.)

2.

185


X. Laser Modulation and Transient Effects

25. INTRODUCTION The laser diode output can be modulated at rates in excess of 1 GHz because the stimulated carrier recombination lifetime is very short, but various phenomena affect the usefulness of the devices.241 When a laser is turned on, high-frequency damped oscillations occur (following an initial time delay) related to the interdependence of the electron and photon populations in the cavity. Because of the same basic effect, quantum shot noise produces oscillations in the output of a continuously operating laser, but the noise level is generally very low and therefore negligible under practical conditions. Whereas the above effects are inherent in the physical behavior of lasers, other harmful effects are due to uneven population inversion in the cavity and filamentary laser behavior. Being the result of "nonideal" laser properties and defects, modeling of these phenomena is difficult. 26. THE RATEEQUATIONS When the laser is switched on, Fig. 68, a damped oscillatory opticat output is observed following an initial delay. The delay t, between the current and stimulated emission pulse follows Eq. (9); hence, the higher the overdrive above I,,, the shorter the delay. In order to avoid this delay, it is essential to bias the laser to threshold and restrict the modulating current range to the stimulated emission region.

FIG.68. (a) Simple circuit with laser diode in series with a resistor. At I = 0 the switch is opened so that current from the current generator is supplied to the diode. (b) The current density applied to the diode is a step function beginning at f = 0. (c) The photon density transient solution. At r = 0 the photon density grows according to a solution of the nonlinear rate equations. For larger values oft. the solution is represented by a decaying exponential modulating a sinusoidal as t + r.(The laser function; N , , 4 turn-on delay is not shown.)

m,,

1479

E

&I-

DIODE

(a)

~

i

(b)

f

i/V"-, I

f, I

Rph

2;w

0

(C)

241

For a review of diode modulation see G. Arnold and P. Russer. Appl. PI7j.s.. 14. 255 (1977).

186


The damped oscillations depend on the diode current relative to the threshold current, and on the spontaneous carrier lifetime and photon lifetime. These relaxation oscillation effects are also important when it is desired to modulate the laser output at frequencies near the oscillation frequency. Basically, the functional dependence of the high output behavior is analogous to the current and voltages in a tuned circuit. Consequently, there is a stability condition which depends on the average light output, the current density, and the driving frequency. Depending upon the system application some modulation schemes may be optimized, but, for straight AM modulation, distortions can occur at frequencies near resonance. We analyze the laser kinetics using the coupled rate equation^^^^.'^^ with the following simplifying assumptions: (i) The laser is operating in a single mode above threshold; (ii) we consider an ideal cavity with homogeneous population inversion.The spontaneous carrier lifetime T~and photon lifetime Tph are constant, and the quantum efficiency is unity; (iii) the gain coefficient is a linear function of the injected carrier density Ne ; (iv) noise sources are excluded. We describe the rate of change of the injected electron density Ne (in ptype material) and of the photon density Nph(in the single mode):

dNe/dt =

J/ed3

-

AN,Nph

-carnerdffreasedue dNpddt

=

-

N e b s 1_

’ ,

lo stimulated emmion

decrease due to spontaneousrecombination

(KJ

(R,)

ANeNph

-

+

N ~ h / ~ p h

z&z?stimulated photon

(56)

y~(Ne/~s)

into the mode

(57)

emission

Here A (in cubic centimeters per second) is a proportionality constant, d3 is the width of the recombination region, and ys is the probability that an emitted photon is in the mode. The photon lifetime is given by

+ (l/L)ln(l/R)],

l l T p h = (c/n)[E

(58)

where c is the velocity of light in vacuum, n is the refractive index at LL, Z is the effective absorption coefficient, L is the Fabry-Perot cavity length, and R is the facet reflectivity. 242

243

H. Statz and G. DeMars, in “Quantum Electronics.”(C. H. Townes, ed.). Columbia Univ. Press, New York, 1960. A comprehensiveanalysis of the maser rate equations is given by D. A. Kleinman, Bell Syst. Tech. J . 43, 1505 (1964).

2. HETEROJUNCTION

187

LASER DIODES

We assume a current step function which gives the steady-state values and mph.243-247 However, during the transient, the electron and photon populations deviate from their equilibrium values by ANe and AN,,, respectively. This assumption will allow us to obtain a solution to the equations by linearization. Thus,

me

AN, = Ne -

me

and

AN,h

= N,h - m p h .

(59)

It is assumed that the deviations of the electron and photon populations are sufficiently small to permit the following Taylor expansion around the median values for the stimulated and spontaneous recombination rates, R,, and Rsprrespectively.

R,,

Z

R,, + (aR,,/c?N,)ANe + (aR,,/aN,h) ANph,

(604

+

(60b) R,, ? R,, (dRsP/?Ne)ANe. The second term of (60a) does not appear in (60b) because the spontaneous recombination rate is independent of the photon density. Using (60) in the rate equations and neglecting the spontaneous emission into a single mode, we obtain identical differential equations for AN, and AN,h : = 0,

(61a)

A solution to (61)is of the form

ANe = (AN,),exp[-(a

- io,)t]

ANph = (ANp&eXp[ -(a - i0,)f].

(62)

The values of a and o,are

+ 1)

a 2 (1/22,)(5/5,h 0,=

(63)

2njc [(l/SSSph)(J/Jth

- 1)]”2.

(64)

The photon lifetime is of order 10- l 2 sec in a typical laser, and T , is about sec. From (63)we see that the oscillations are damped in a period of the 244

245 246

24’

W. Kaiser, C. G. B. Garrett, and D. L. Wood, Phys. Rev. 123, 766 (1961). P. P. Sorokin, M. J. Stevenson, J . R. Lankard, and G. D. Pettit, Phys. Reu. 127,503 (1962). An introductory discussion of transient behavior in pulsed laser is given by W. V. Smith and P. P. Sorokin, “The Laser,” p. 86. McGraw-Hill, New York, 1966. T. Ikegami and Y. Suematsu, IEEE J . Quantum Electron. QE4. 148 (1968).

188


order of the spontaneous carrier lifetime. For example, with J = 2J,,, sec and rph= t, = 2 x sec, f: z 4 GHz. The oscillatory effect described above also impacts the modulation efficiency of the laser above threshold, and not just the turn-on properties. Suppose we bias the laser well above threshold and superimpose a small amplitude sinusoidal current I = I , exp(iwt). We find that the modulation efficiency will peak at w = w,, with the value of w, changing with the constant bias current. Above the resonant frequency w,, the modulation efficiency will be found to decrease rather steeply with frequency. The resonant effect in modulation has been seen by Ikegami and S ~ e m a t s u ~in~ homojunction ’ GaAs lasers, and moderate agreement with theory has been obtained using values of 7, = 2 nsec and rphz sec. Moderate agreement with regard to the predicted ,f,also was found in (A1Ga)AsDH laser diodes operating at room temperature.248

OSCILLATIONS 27. CONTINUOUS We discussed above the damped oscillations theoretically predicted following laser turn-on by a step-function current. These oscillations are predicted to last only for a period of time about equal to the spontaneous carrier lifetime. Self-sustained oscillations of cw lasers also exist where both the light output and the current can exhibit high-frequency oscillations. These effects are attributed to quantum shot noise, i.e., intrinsic fluctuations in the photon generation rate and hence of the carrier c o n ~ e n t r a t i o n . ’ ~ In ~ - general ~~~ the noise level generated by this process is very low and therefore not limiting in practical applications. The starting point for the theoretical analysis again involves the coupled rate equations (56) and (57). with an added shot noise term calculated from the photon density. rhe high-frequency oscillations peak at a frequency f , (J/J,h Detailed calculations of the noise spectrum require assumptions concerning the relationship between the gain coefficient and the carrier concentration, as well as a determination of the spontaneous emission rates at various temperatures and for various material parameters. HaugZs1 calculated the GaAs laser noise spectrum assuming parabolic bands with k selection for the electron-hole recombination process. The following observations are relevant concerning the theoretical calculations: (i) The frequency where the noise peaks can vary from the megahertz

’“ H . Yanai. M . Yano. and T. Kimiya. IEEE J . QUUIIIUM E/wtroti. QE-11, 519 (1975). 249

25’

252

D. E. McCumber, Phys. Rev. 141, 306 (1966). H. Haug and H . Haken, 2. Phys. 204,262 (1967). H. Haug. Phys. Rev. 184, 338 (1969). D. J . Morgan and M. J. Adams, Phys. Sratus Solidi ( a ) 11, 243 (1972).

2.


189

region well into the gigahertz region, depending on the pump rate and the temperature; (ii) the relative magnitude of the noise spectrum decreases with increasing pump rate; i.e., maximum noise is expected near threshold and it rapidly decreases above threshold. Hence, by biasing the laser diode above threshold, the laser should be much quieter than with bias very near threshold. A major difficulty in quantitative comparison of theory to experiment is the single-mode restriction, since practical laser diodes are generally rnultimode devices. In addition, uniform inversion of the active region is difficult to achieve and filamentary behavior is not uncommon. In practice, sections of the laser often operate nearly independently of each other, with the effect of a different noise spectrum from each section. Furthermore, as we noted earlier the modal content of the laser diodes is a function of the diode current, with changes in both the longitudinal and lateral mode. Therefore, any description of the diode in terms of a simple model based on a one mode, or even several modes which remain invariant with current, must be treated with caution. Among the earliest observations of microwave oscillations which could be qualitatively related to theory were those reported by D’Asaro et who studied cw GaAs homojunction lasers at low temperatures. Oscillations in the 0.5-3-GHz range were observed in the optical output and the diode current, depending on the temperature and diode current. In fact, the peak resonant frequency of the noise spectrum was found to increase with an increase in J above J t h , qualitatively consistent with theory. Room-temperature studies of DH (A1Ga)Aslaser diodeszs4have produced a complex picture not easily related to theory beyond the fact that highfrequency (measured at 4 GHz) oscillations are seen, but these are not necessarily reduced in intensity as the current is increased above threshold as theoretically expected. However, as shown in Fig. 69, the low-frequency noise (measured at 50 MHz) does exhibit a first maximum in the vicinity of J , h . It is probable that the complicated behavior of various cw lasers is related to their structural perfection, uniformity, and mode content stability. For example, Fig. 69 shows that the low-frequency noise power peaks near threshold, then increases again beyond the kink at 200 mA, suggesting the onset of another small lasing region going through its threshold behavior, or perhaps a change in the number of lateral modes excited. It is evident from the above that the noise characterization of cw lasers is very complex and that detailed comparisons between theory and experiment require carefully selected devices. From the practical point of view, it appears 253

’y

L. A. D’Asaro, J. M. Cherlow, and T. L. Paoli. IEEE J . Quanrum Electron. QE-4. 164 (1968). T. L. Paoli, IEEE J . Quantum Electron. QE-l1,276 (1975).

190


-p z 3 *a

700

10

a

E-

t m

>

500

z

a

3

a a c a

500

cm

B

U

a

400

W

; cVI

II

8

%

U

W

z

1.0

W I-

t

300

+ a

f

-I W

a

200

4

,_

0

__

0

8850

8900

I00

INTENSPd

--

--4-

140

I20 I

I

6

7

l

I

I

160 180 LASER CURRENT ( m A ) 1

I

1

1

1

200 I

8 9 10 1.1 NORMALIZED CURRENT ( 1 / l t h )

I

1

~

5

220 1

I 2

I I3

FIG.69. Variations of the relative noise power at 50 MHz and the total intensity of the laser emission of a device exhibiting a kink in its characteristics. The longitudinal mode spectra are obtained for selected currents as shown. Note the peak in the noise at threshold and the additional increase just beyond the kink. (After P a ~ l i . * ’ ~ )

that the noise is affected by structural parameters. For example, the modulation of diodes in a current range which traverses a kink could well result in the emission of random pulses in a frequency range corresponding to the modulation frequency of the device and hence bothersome from the systems point of view in optical communications. However, since devices free of kinks

2.


191

(over a useful power emission level) are fabricated, this is not an inherent limitation. 28. OSCILLATIONS RELATED TO NONUNIFORM POPULATION INVERSION Lasher2” proposed that self-sustaining pulse generation from a laser can be produced by placing an emitting region in tandem with an absorbing region. Both regions are encompassed within the same optical cavity because the two facets of the Fabry-Perot cavity enclose both regions. The instability occurs because of saturable absorption; i.e., the absorption coefficient is a function of the photon density. A detailed analysis of such structures can be found in the original paper by Lasher,2ss and in Lee and Roldan2s6 and Basov et The basic concept of oscillations due to inhomogeneous population inversion is a general one. It is possible that optical anomalies in devices containing defects within the active region are related to this process, although characterization of such devices via reasonable models is not possible unless the detailed internal configuration is known. In fact, self-sustaining oscillations increasing in frequency with current in some room-temperature DH (A1Ga)Aslaser diodes has been attributed to this effect.2se It is noteworthy that saturable absorption involving traps has been suggested as responsible for the Q-switching seen in certain diffused GaAs diodes at cryogenic temperatures. It was that lasing will not occur during the application of the current pulse because the optical losses due to absorption by the traps is too large. At the end of the current pulse, however, the dynamics of recombination transfer the traps from the absorbing to the nonabsorbing state. Thus, the internal loss decreases rapidly as the charge on the traps changes, and recombination of the remaining carriers is sufficient to give transient lasing. While the mathematical model postulated does produce the required effect, no evidence exists of such traps in GaAs. ACKNOWLEDGMENTS We are indebted to H. S. Sommers, Jr. for his most helpful comments on the manuscript and to H. F. Lockwood, M. Ettenberg, I. Ladany, C. J. Nuese, G. Olsen, J. Wittke, and J. I. Pankove for discussions. G. J . Lasher, Solid-State Electron. 7, 707 (1964). T. Lee and R. H. R. Roldan, IEEE. J . Quantum Electron. QE-6, 338 (1970). 2 5 7 N. G. Basov, V. N. Morozov, V. V. Nikitin, and A. S. Semenov, Fiz. Tekh. Poluprovodn. 1570 (1967) [English transl.: Soo. Phys.-Semirond. 1, 1305 (1968)l. 2s8 T. Ohmi, T. Suzuki, and M. Nishimaki, Oyo Bufuri Suppl. 41. 102 (1971). 2 5 9 J . E. Ripper and J. C. Dyment, IEEE J . Quantum Electron. QE-5, 396 (1969).

255

256

192

HENRY KRESSEL AND JEROME K. BUTLER List qfSymbols

h h,

damping constant for transient oscillations lattice parameter corrugation amplitude in distributed feedback laser fraction of modal power in ith layer forward wave amplitude constant backward wave amplitude constant exponent to relate gain coefficient versus current density (in Part IX, distance between steps in DFB laser) continuous wave distributed feedback double-heterojunction laser active region width total width of optical waveguide perpendicular to the junction minority carrier diffusion constant slab width with d , the recombination region width electron charge Fermi level bandgap energy; AEg bandgap energy step at heterojunction electric field components four-heterojunction laser resonant frequency gain coefficient of the recombination region gain coefficient of the recombination region at threshold gain coefficient of the recombination region at threshold with no free carrier absorption in the recombination region and no cavity-end losses Planck's constant critical epitaxial layer thickness for dislocation-free growth in lattice mismatched structure transverse propagation constant in ith layer magnetic field components diode current diode threshold current current density, threshold current density. and increase in Jlhdue to carrier loss from recombination region, respectively nominal current density Boltzmann constant free space wave number (27~/&J thermal conductivity of ith layer Fabry-Perot cavity length Bragg order large optical cavity laser liquid phase epitaxy minority carrier diffusion length for electrons and holes, respectively transverse mode number effective electron mass (electron mass in vacuum) effective hole mass molecular beam epitaxy refractive index, with ni refractive index in region i

2.


193

effective refractive index for a particular mode electron concentration initial electron (hole)concentration in recombination region. i.e., prior to injection photon density in cavity (single mode) injected electron concentration in p-type recombination region surface state density hole concentration linear power density ( i n W cm) for catastrophic damage at semiconductor-air in terface longitudinal mode number facet reflectivity of facets u and h facet reflectivity of mth transverse mode diode series resistance diode thermal resistance lateral mode number surface recombination velocity single-heterojunction (close-confinement) laser stripe width used t o define active diode area time delay between current and laser emission pulse pulse width temperature constant appearing in equation for temperature dependence of threshold current thermal velocity of carriers vapor phase epitaxy diode width A1 fraction in region (AI,,Ga, -.,As) difference in Al fraction between two layers of (A1Ga)As equivalent absorption coetticient associated with cavity-end radiation loss free carrier absorption of the recombination region weighted modal absorption coeflicient due to passive regions average modal absorption coefficient due to all factors bulk absorption coefficient of the ith layer longitudinal mode propagation constant constant in gain versus current density relationship minority carrier utilization (radiative recombination) complex propagation constant fraction of wave energy confined to the recombination region deviation of the modal propagation constant from the Bragg condition energy separation of Fermi or quasi-Fermi level from the conduction band edge (ialence band edge) index step between region i and j : for symmetrical double- eterojunction structures An denotes the index step between the recombination region and the outer regions dielectric constant dielectric asymmetry factor internal quantum efficiency differential quantum efficiency power conversion efficiency free space wave impedance full angular beamwidth at half power (direction perpendicular to junction plane)


relative complex dielectric constant ( E / E ~ ) real part of dielectric constant imaginary part of dielectric constant forward and backward distributed reflection constant distributed feedback coefficient of Ith Fourier component free space wavelength lasing wavelength grating spacing in DFB laser carrier mobility linear dislocation density capture cross section of surface states material conductivity seen by the optical field minority carrier lifetime photon lifetime in cavity minority carrier lifetime for spontaneous radiative recombination minority carrier lifetime for nonradiative recombination minority carrier potential barrier at isotype heterojunction modal wave function plasma frequency angular frequency

SEMICONDUCTORS AND SEMIMETALS. VOL. 14

CHAPTER 3

Space-Charge-Limited Solid-state Diodes A , Van der Ziel I. INTRODLJC~ON . . . . . . . . . . . . 1. Space-Charge-Limited Current . . . . . . 2. Potential Distribution in the Derices at Zero Bias . 3 . The Efect of Traps . . . . . . . . . 11, SINGLE-INJECTION SPACE-CHARGE-LIMITED SOLID-STATE DIODES . . . . . . . . . . 4. The Trap-Free Case . . . . . . . . . 5 . Space-Charge-Limited Flow with Traps . . . . 6 . Effects of Diffusion in Insulators ( N , = 0) . . . 111.

. . . .

. . . .

. . . .

195 195 197 198

. . . . . .

199 199 21 1 214

DOUBLE-INJECTION SPACE-CHARGE-LIMITED

SOLID-STATE DIODES . . . . . . . . . . 7. The Trap-Free Case . . . . . . . . . 8 . Effects of Diffusion . . . . . . . . . . 9. Negative Resistance Effects Caused by Traps . . . 10. Pulse Response in Double Injection . . . . . . IV. NOISE I N SPACE-CHARGE-LIMITED SOLID-STATE DIODES . 1I . Discussion of Noise Sources . . . . . . . 12. Single-Injection Diodes . . , . . . . . 13. Double-Injection Diodes . . . . . . . . . V . APPLICATIONS . . . . . . . . . . . . . . 14. Applications of Single-Injection Diodes and Triodes 15. Applications of Double-Injection Diodes . . . .

. . . .

. . . . . .

217 217 226 228 229 229 229 233 239 245 245 247

I. Introduction 1.

SPACE-CHARGE-LIMITED

CURRENT'

If an insulator or nearly intrinsic semiconductor is provided with a carrierinjecting metal contact on the one side and a carrier-collecting metal contact on the other side, and a voltage is applied between these contacts, then spacecharge neutrality cannot be maintained in the material, and hence the current through the device becomes space-charge limited. For example, let the one contact (cathode) inject electrons in the material and let the other contact (anode) be a noninjecting or blocking contact. One thus has electron flow from cathode to anode when a positive voltage is applied to the anode, and this current is limited by the space charge. This is called single-injection spacecharge-limited flow.

' M . A . Lampert and P. Mark. "Current Injection in Solids." Academic Press. New York. 1970.

195 Copyright @ 1979 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-7521 14-3

196

A . VAN DER ZIEL

Such a situation also occurs in n+-v-n+ and p+-7c-p+ devices. Here n+ means strongly n-type, v means weakly n-type, p + means strongly p-type, and 7c means weakly p-type. These devices act as resistors at low applied voltages, where the effect of the equilibrium carrier concentration predominates. At higher voltages carrier injection takes over at the one contact (cathode), the other contact becomes a carrier-collecting contact (anode), and the current becomes space-charge limited. For example, in an n+-v-nf device the negative electrode acts as an electron-injecting contact (cathode) and the positive electrode as an electron-collecting contact (anode). The current at higher voltages is thus single-injection space-charge-limited flow. There are several possibilities for p+-i-n+ structures, where i means intrinsic or near-intrinsic semiconducing material. If the p-region provides a hole-injecting contact and the n-region a hole-collecting contact, but the n-region is not electron injecting, then the device is a single (hole) injection space-charge-limited diode. If the n-region provides an electron-injecting contact and the p-region an electron-collecting contact, but the p-region is not hole injecting, then the device is a single (electron) injection space-chargelimited diode. If both regions are carrier injecting and carrier collecting, such that the p-region collects electrons and injects holes and the n-region collects holes and injects electrons, then the device is a double-injection spacecharge-limited diode. There is a very significant difference between single and double injection. In the single-injection diode there is only one type of carrier in the material, so that an appreciable space charge is developed, which strongly limits the current. In the double-injection diode there are two types of carriers of opposite charge in the material, so that their charges mostly neutralize each other; the net space charge is then much smaller than in the previous case. As a consequence, the current density for the same dimensions and the same applied voltage is much larger than in the single-injection diode. There is a considerable difference between the metal-semiconductormetal diode on the one hand and the p+-n-p+ and the n+-v-n+ diodes on the other hand. In the first case the characteristic is of the form Z,[exp(eVJ k T ) - 13 at low applied voltages and of the form C(V,+ V,)' at higher applied voltages, characteristic for true single-injection space-charge-limited current; here V, is a built-in potential. In the second case the characteristic is linear for low applied voltages and of the form CV: at higher applied voltages, as expected for single-injection space-charge-limited current flow. Here C is a constant in either case, depending upon the geometry and the semiconducting material used in the device. In double-injection space-charge-limited diodes the characteristic is linear at low applied voltages (ohmic region), quadratic at intermediate applied voltages (semiconductor regime),and cubic at higher applied voltages (insulator regime). These three regimes are distinguished by the predomi-

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

+(XI

197

Jl(+d)

i

nating terms in the differential equation describing the field distribution in the device (Part 111).

2.

POTENTIAL

DISTRIBUTION IN THE DEVICES AT

ZERO

BIAS

We first discuss the potential distribution in a p--71-p diode.2 Let the junctions be abrupt junctions and let the width of the x-region be denoted b y (1. Let N,, be the acceptor concentration of the p-region and N,,(Na2 0. The current flow is mainly by diffusion and the current density is given by - Jo = 2’/’

(an,p,kT/d)[exp(eV,/kT)

-

11,

(80)

where V , is the externally applied voltage, and n , = N,exp(eV,/kT). This is the diode part of the characteristic. (b) The high tdrage range. Here E is negative and la’] is large but .j >> la2[ numerically. However, ,is may or may not be smaller than 1a21.12a In this case there is a potential minimum close to the cathode at x = x,. The situation must now be described by Bessel J-functions for s < ,s = x,/d and by Bessel I-functions for s > s,. In that case Wright obtains the limiting characteristic -J

+

- ’&xOpo(I/a V, -

v0)’/d3

so that V, of Eq. (21) is equal to (V, - Vo). 122

(js

+ r2)

0 for 0 5 Y

5 .Y, and ( is

+ r 2 ) 2 0 for s 2 s,,

(81)

216

A. VAN DER ZIEL

FIG.7. Characteristic of a CdS diode showing the diode characteristicat low currents going over into a quadratic characteristicat high currents. (S. T. Hsu.13)

Between the cases (a)and (b) there is a transition region, where the characteristic changes from exponential to quadratic. We have discussed this situation so extensively, because it seems to occur in CdS, and because the exponential regime of the characteristic is missing in the analysis of Lindmayer et af.” Figure 7 shows Hsu’s results for CdS.13 The sudden rise in the current I, around V, > 25 mV is not due to trapping effects (such rises in I, usually occur at much larger voltages) but represents the exponential part of the characteristic predicted by Wright’s analy~is.~ l3

S. T. Hsu, Noise in Space-Charge-LimitedCurrent Flow in Solid State Devices. Ph. D. Thesis, Univ. of Minnesota, Minneapolis, Minnesota (1966).

3.


217

111. Double-Injection Space-Charge-Limited Solid-state Diodes'. ' 7. THETRAP-FREE CASE (1.

Basic Equations for a One-Dimensional Geometry

In this discussion we neglect traps but take recombination centers into account. As in the single-injection case we neglect diff~si0n.l~" The basic equations are then

Here J , and J , are hole and electron current densities, p T and nT are equilibrium hole and electron carrier densities, p and n the injected hole and electron carrier densities, p, and p, the hole and electron mobilities, R the recombination rate per unit volume, and F the field strength. We assume that x = 0 at the boundary with the n-region and x = d at the boundary with the p-region. Equation (86)neglects the presence of electrons and holes in recombination centers. This is allowed if the density of these centers is relatively small. If the recombination is via Shockley-Read-Hall (SRH) centers, R = 4 7 , where T is the carrier lifetime. Usually there is approximate space-charge neutrality, or n u p . For direct recombination R = p n p , where p is a constant. The problem of direct recombination is discussed by Lampert and Mark' and by others. We now substitute R = n/s, multiply Eq. (84) by b = p n / p p ,add it to Eq. (85), replace n-p by Eq. (86)and substitute p Y n in all other terms (approximate space-charge neutrality). This yields

l4 14’

M . Lampert and A. Rose, Phys. R m . 121. 26 (1960). This is not allowed near the p - and n-regions of a p-i-n diode, since there is a potential maximum near the p-region and a potential minimum near the n-region; near those points the current flows by diffusion (compare the single-injection case). We neglect these effects.

218

A. VAN DER ZIEL

with the initial condition that F (84)and substituting (86)yields

Jff)

=J,

+J, +

= 0 at

r'F EEO - 2

at

the boundaries. Subtracting (85) from

en(pp

+ p,JF + e(nTpn+ pTpp)F,

(88)

which is independent of x. Since the displacement current density E E 8F/& ~ is only important at microwave frequencies, we have neglected it in the second half of Eq. (88). We now put J =Jo F

= Fo

+ J, exp(j w r ) ;

+ F , exp(jwr),

+ n , exp(j w t ) , R = n/.r = no/z + (n,/z)exp(jwt), n = no

(89)

where the subscript zero indicates dc terms and the subscript one small signal ac terms. We thus have for dc

and for ac

Jl

= 4 P P + PnNnoFl

+ n1Fo) + 4 P p P T + P L n M - 1 .

(93)

b. Dc Characteristics We can here distinguish between three regions of the dc characteristic. (1) Ohmic regime. Here no is very small, and the first terms in (90) and (91)are negligible. Hence dFo/dx = 0 or F , = - V,/d, where V , is the device voltage and d the device length. Consequently -JO

= e(PpPT

+ PztnT)%/d.

(94)

In this case we thus obtain a linear characteristic. (2) Semiconductor regime. Here no is quite large, so that the last term in (91)is negligible. Moreover, the first term in (90)is still negligible. The equations thus are

3.


219

Substituting for no yields the approximate equation

F , dLF = - 1- d(Fi)Jole du 2 d.u ( n , - pT)pPpn~'

(97)

which must be solved under the appropriate initial conditions. We choose Vo = 0 at .Y = 0 and V, = V , at s = ri, but the condition F , = 0 can now only occur at one boundary. Which boundary that is depends on whether nT > pT (weakly n-type material) or p, > nT (weakly p-type material). It should be borne in mind that J , and F , are both negative. Iff?, > p T . the right side of Eq. (97)is negative, or d(Fg)/dx < Oeverywhere. Since F , 5 0, this is only possible if F , < 0 at x = 0 and F , = 0 at .Y = d. The condition F , = 0 at .y = ti expresses the fact that at that contact the current flows by diffusion. Consequently, integrating once, and bearing in mind that F,(r/) = 0, yields, if F , = -dV,/A?c,

=

K[

--j.

,P2- ( d -_ _.Y)3'2 ,/3'2

(99)

Hence

or

If p , > n T , the right-hand side of (97) is positive, since J , 'is negative. Consequently d ( F g ) / d x > 0; this is only possible if F , = 0 at .Y = 0 and F , < 0 for x > 0. Consequently, integrating once, and bearing in mind that F,(O) = 0, yields

220

A. VAN DER ZIEL

Hence

or

In both cases we thus obtain a quadratic characteristic. The approximate field distributions in this case are shown by the broken lines in Fig. 8a and b for the case pT < nT and pT > nT, respectively. The accurate curves, obtained by integrating Eq. (90),are shown schematically by the full-drawn curves. The position of minimum field strength lies close to x = 0 for pT < nT and close to x = d for nT c p T . If the anode voltage is 0

d

--X

(a)

0

-X

d

FIG. 8. (a) Field distribution in the semiconductor regime for p7 .E nT. (b) Field distribution in the semiconductor regime for pT > nT. (c) Field distribution in the dielectric regime according to Eq. (107). The full-drawn curves in (a) and (b) represent the accurate values of the field that would be obtained by solving the full equation (90). The broken curves represent Eqs. (99) and (99a), respectively.

3.

221


increased, the position at which the field strength is a minimum shifts towards the center and arrives at the center when the insulator regime of the characteristic is reached (Fig. 8c). As long as the positions of minimum field strength lie close to the endpoints x = 0 or .Y = d, little error is made by extrapolating the field curve to x = 0 (for p T < n T ) or to x = d (for P T > nr). We then obtain the potential distributions (99) or (99a), respectively. The particular shape of the field distributions comes about because the condition

is not satisfied near .Y = 0 for p T < nr (because of an excess electron concentration near x = 0) or near s = d for p r > nr (because of an excess hole concentration near x = d ) , respectively. (3) Insulator regime. Here

In that case Eq. (90) may be written

Substituting for no into Eq. (103)yields F0 X(F0

2)

-J

,

=

=

c,

where C is a positive constant, independent of x. This equation must be solved under the initial conditions F,

=0

at x = 0 (cathode),

F,

=0

at x

=d

(anode). (105a)

To solve the equation, we introduce the new variable y by the definition F o d / d x = -d/dy, where J' = 0 at .Y

or

= 0 and J' = jidat .Y = d.

F o = -dx/dy,

( 106)

The equation then becomes

d2Fo/dJj2= C

(106a)

222

with F ,

A. VAN DER ZIEL

= 0 at J’ = 0 and J’ = Y d .

Hence

F0 ‘c ,(,> - J ’ d ) -2 J .

so that dx -=

d.Y

icy(J‘d

-

(107a)

y).

The behavior of F , according to the above relations is shown in Fig. 8c. From the above equations, we obtain

where 2

= y/y6.

Furthermore

Consequently, eliminating y d , we obtain 125 V 3 c=--” 18 d 5 ’

or

-J,

125 18

v,

=E E ~ ~ = ~ ~ , EsE C , ~ ~ ~ ,, T ~

ti5

.

(110)

We thus obtain a cubic characteristic. The ( I , , V,)characteristic thus goes from linear (case 1) through quadratic (case 2) to cubic (case 3). These kinds of characteristics are indeed found in germanium’ and silicon double-injection diodes (Compare Fig. 9). Lampert and Mark’ have discussed transition points between the different regimes. They also have discussed the possible transition from single to double injection. The above theory holds for relatively long diodes with d ir 4-5 mm. For much shorter diodes, deviations in the characteristic occur that will be discussed in Sections 8,a and 8,b. The reason is obvious. In the discussion we neglected diffusion. But near the electrodes, diffusion predominates. As a matter of fact, diffusion predominates over a distance of the order of the ambipolar diffusion length La = from each electrode, where D, is the ambipolar diffusion constant, D, = 2D,Dp/(D, + Dp),and D, and D, are the diffusion constants for electrons and holes, respectively. The condition is therefore d >> ~ ( D , T ) ” ~ . For silicon D, = 19.4 cm2/sec, and La = 0.10 mm for z = 5 x lops sec.

’

J . H. Liao, Characteristic, Admittanceand Noise in Double-Injection Space-Charge-Limited Solid State Diodes. Ph. D. Thesis, Univ. of Minnesota, Minneapolis, Minnesota (1971).

3.


223

V (VOLTS)+

FIG.9. Characteristic of a double-injection Ge diode. showing the linear. the quadratic, and the cubic regimes. (J. H . Liao.I5)

We have in our discussion neglected the effect of the occupancy of recombination centers. This is allowed if the number of centers is relatively small. The case where this is not allowed was discussed by Baron and M a ~ e r . ~ ~ c. Admittance of Double-Injection diode^^^*'^* We shall now show that in each regime the ac admittance per unit area may be written as

A. van der Ziel, Electron. Lett. 5. 298 (1969). "F. Driedonks, Physica 46, 291 (1970).

l6

224

A. VAN

DER ZIEL

where n = 1 for the linear (ohmic) regime, n = 2 for the quadratic (semiconductor) regime, and n = 3 for the cubic (insulator) regime. The linear case is obvious, since the device acts as a resistor of conductance lJ,,l/K per unit area at all frequencies. For the other two regimes we need proof of the above equation. (1) The semiconductor regime. Here the term - ( ~ ~ ~ / e ) d ~ ( F , F ~in) / d x ~ Eq. (92) is negligible and hence (92) may be written

and the dc equation is

Furthermore, if we divide (93) by (91)

+ %/no + &nT)Fo.

J,/J, =Fl/F,

since we can neglect the term e(P+ Dividing Eq. (1 12) by Eq. (1 13) and substituting for n /no with the help of Eq. (1 14) yields

,

dFl/dFo with F ,

= 0 at

+ (1 + j o z ) F , / F ,

= (1

+j o t ) J 1 / J o ,

(115)

F , = 0 (x = 0). We now substitute F , = uF0,

where u may be a function of F,. Then F , du/dF,

+ (2 + j w ) u = ( 1 + jwz)J,/J,.

(117)

It is easily seen that u=--

1 2

+j o z J , + jwz J o

is a solution of (117) and that = AF;(~+~oT)

( 1 18a)

is the solution of the homogeneous part of (117). Hence the full solution of (1 17) is

3.


225

and

This solution blows up at F , = 0 unless A = 0. Integrating over the device length and taking the ac voltage at x = d as V,, yieIds

as had to be proved.

(2) The insulator regime. Here the term (nT - p , ) d F , / d x is negligible and n , / n , follows from ( 1 14). Hence

As in the previous case (Fl/Fo)= u is a constant so that u = $ ( I + . j ~ ~ ~- )u),( ~ ~ / ~ ~

or

or, by integrating over the length of the device and taking the ac voltage at V,, yields

x = d as

as had to be proved. For a more rigorous proof see Driedonks.' ' Van der Ziel' * has calculated the admittance in the transition region between the ohmic and the semiconductor regimes and has expressed the results in terms of an infinite series. He has also calculated the microwave A. van der Ziel, Solid Sfote Electron. 13. 191 (1970).

226

A. VAN DER ZIEL

admittance of a double-injection diode in the semiconductor regime by taking into account the displacement current in the diode.lg He obtains the following infinite series expansion that converges at all frequencies :

+ 3 ( - j w ~ , , ) ~ [ (+.jwt)'/(3 l +jwr)(4 + jwz)] + .

a

.

where zda = &&,F,/JOis the dielectric relaxation time at the anode (x = d ) and Fa is the dc field strength at the anode.' 9s This indicates that Y = - J o / V , for wz >> 1 and wda > M, and in that case the integrals differ by a negligible amount. The last step involves extending M to infinity, which is allowed because the integral converges absolutely. Consequently, since the imaginary part of the integrand gives no contribution, because X ( t ) X ( r + s) is symmetric in s, S,(f) = 2

JTE X ( t ) X ( r + s)coso,sds.

(135)

This is called the Wiener-K hintchine theorem. By inversion X(t)X(t

+ s) =so" S,(j,)coso,sdf,

so that

-

x 2 = Jo= Sx(fn)dfn.

(135a)

The most important noise sources are: (1) Shot noise. Current is carried by carriers of charge e at the average rate it. Then the average current T= eii, but the instantaneous current shows fluctuations of spectral intensity S,(f) = 2ezii = 2eT.

(136)

This is known as Schottky's rheorem. This occurs, e.g., in saturated thermionic diodes, solid-state diodes, etc. Any current fluctuations of spectral intensity S , ( f ) can thus be represented by an equivalent saturated diode current I,, by

3.


231

the equation

S,(.f) = 2el,,.

(136a)

( 2 ) Thermal noise of a resistance R at the temperature T . It is caused by the random motion of carriers in the sample. The spectral intensities S , , ( f ) of the open-circuit voltage and S,(f ) of the short-circuit current are then given by Nyquist's theorems S , ( f ) = S , . ( f ) / R 2= 4kT/R = 4kTy,

S , ( f ) = 4kTR,

(137)

where y = 1/R and k the Boltzmann constant. Any noise phenomenon in a device with internal resistance R, represented by voltage fluctuations of spectral intensity Sv.(f )or current fluctuations of spectral intensity S , ( f ) , can be represented by an equivalent noise resistance R , or a noise conductance g, by the definitions S,.(f) = 4kTR,.

S,(,f')= 4kTR,g2 = 4kTg,,

( 137a)

where g = 1/R. (3) Dzflusion noise or velocity fluctuation noise due to the random collisions ofcarriers with the lattice. In electron conductors the noise can be represented by a current source H ( t )due to electrons moving from left to right and from right to left. Assuming full shot noise of these currents yields a cross-spectral noise intensity (representing shot noise of the two currents),

where A is the cross-sectional area of the device under discussion, D, the electron diffusion constant, and the &function indicates that fluctuations at x and x’ are uncorrelated. In thermal equilibrium situations eD, = k T p , , where p, is the electron mobility, and hence

corresponding to thermal noise of the conductance for unit length at .x'. To prove Eq. (138)we make a Fourier analysis of the spontaneous velocity fluctuation Ac,(t) = z>Jt)- of a single electron. Applying the WienerKhintchine theorem gives

=4

Jox

Az.,(r)Ar,(t

+ s)ds = 4 0 ,

(138b)

for frequencies such that wz > t. If we now consider a conductor of length Ax and an electron in it moves with a velocity u,(t), then the current in the external due to that electron is i ( t ) = eu,(t)/Ax and hence its spectrum is

But if n is the carrier density of the conductor, there are AN = n Ax A electrons in the sample. Since the velocity of each electron fluctuates independently,

S , ( f ) = n Ax A S A i ( f )= 4e2D,nA/Ax, and hence the current density fluctuation is S,(f)

= (1/A2)S,(f= ) 4e2D,n/AAx.

To switch over to cross-correlation spectra we must replace l / A x by 6(x' - x). Bearing in mind the possible dependence of n(x) upon position yields Eq. (138). In nonthermal equilibrium situations (hot carriers) we can always define an equivalent temperature T , by the definition eD, = kT,p,

( 139)

in which case (139a)

corresponding to thermal noise at the temperature T , . T h ~ r n b e r ~has '~ derived the expression d (139b) eD, = k T , dF -( p , F ) 27b

A. van der Ziel, "Noise." Prentice Hall, Englewood Cliffs, New Jersey, 1954. K. K. Thornber, Bell Sysr. Tech. J . 53, 1041 (1974).

3.

233


from the fluctuation-dissipation theorem. This is rigorously valid for equilibrium situations ( T , = T) only, but can be used as a definition of the electron temperature T , in the hot electron regime. (4) Trapping noise. Let carriers be generated at the average rate g(n) and be recombined at the average rate r(n), where n is the carrier density. The fluctuations in these rates consists of a series of independent random events and hence can be represented by shot noise terms. As a consequence the cross-spectral intensity S, - Jx, d , f ) can be represented as

where A is the cross-sectional area of the sample. In thermal equilibrium g(n) = r(n) and also for nonthermal equilibrium involving traps. In injection diodes generation is absent and (140) becomes

12. SINGLE-INJECTION DIODES

a. Thermal Noise and Shot Noise In Part I1 we investigated the case where the characteristic was linear for low voltages and quadratic for larger voltages. In the linear regime the device acts as a resistance R = K / I , , where V , is the anode voltage and I , the device current. One would now expect thermal noise to be associated with this resistance R, so that the open-circuit voltage has a spectral intensity S,(f) = 4kTR = 4kTI/,/I,

= 4kTR,,

or

R,

=

K/Ia.

(141)

It will be shown that the second half ofEq. (141)remains valid in the quadratic regime. Since g = 21a/K in the quadratic regime, the current noise in (141a)

S A f ) = S,(f’)g2 = 8k7-g.

This is twice the thermal noise of the ac conductance g. Besides this “thermal” noise there might also be generation-recombination noise; this noise predominates at lower frequencies. In Part I1 we also investigated the case where the characteristic was a diode characteristic for lower voltages, going into a quadratic characteristic at higher voltages. In the diode regime I,

= l,,exp(eT/,/kT),

g = dI,/dV,

= el,/kT,

(142)

where g is the differential conductance. This current should show full shot noise or S,(.f) = 2eI, = 2kTg (143)

234

A. VAN DER ZIEL

so that

This should occur in well-prepared CdS diodes such as represented in Fig. 7. Van der ZieI2* assumed that in the quadratic regime the noise was caused by fluctuations in the depth V, of the potential minimum in front of the cathode, which, in turn, were driven by the shot noise in the current passing the potential minimum. He was then able to show that the noise resistance R , was in this case given by Eq. (143a). SergiescuZ9took into account the carrier collisions between the cathode and the potential minimum; this smooths the fluctuations in the potential minimum to a certain extent. He then obtained R,

= +[(kT/e)/1,]5,

(143b)

where ( is the collision smoothing factor of the noise. He evaluated 5 to be about $ so that the noise was even smaller than evaluated by van der Ziel. While these effects will certainly exist, they are usually completely masked by the thermal noise, since under most circumstances K>>$kT/e in the quadratic regime. b. The Salami Method3' Van der Ziel proposed the following method of calculating the thermal noise. The device is sliced up into sections Ax, with the faces of the slices parallel to the electrodes, hence the name. In view of Eq. (138a) one would now expect the noise in each section A x to be represented by an emf of spectral intensity s A , , ( , f ) = 4kT AR = 4kT AV/la,

(144)

where A V is the dc voltage developed across the section Ax. Assuming the noise in individual sections to be independent, he found by adding the noise of all sections S r R ( , f= ) CSA,.(.f)

= 4kTC

A V v, - = 4kT - = 4kTR,, la

'a

or

R

v,

= -. 'a

(1444 A. van der Ziel, Solid State Electron. 9, 123 (1966). V. Sergiescu, Brir. J . Appl. Phys. 16. 1435 (1965). 30 A. van der Ziel. Solid State Electron. 9, 899 (1966). 28 29

3.


235

That this is correct is accidental, for a more detailed theory shows that the noise in different sections A.Y is While it is possible to modify the approach and make it fully correct, other methods are simpler. c. The Latigerin

I n the Langevin method a random source function H , ( s . t ) is added to the current density. For an 11-typesample we then have the equations J,, = r p o / ~ F + H,(.Y,t ) , (7 Fli7.y =

e?n/?t

=

(145)

-~II/'EEO.

(146)

-c'J,/?r.

(147)

Manipulating these equations yields that the total current density

+ H,(.u,t ) is independent of .Y [cf. Eqs. (81 and (9)]. Putting F = F , + ,f,(.u, J ( t )=

- - ~ E E , / I , (1F2/c'.y

+ EE,

?F,'?t

(148)

t ) and J ( t ) = J, + j l ( t ) , where F , and J , are dc terms and . j ; and j , are small-signal ac terms, putting ,jl = 0 (i.e.,open-circuiting the device for hf )- and neglecting second-order terms yields -EE,/L,?(FO.~;)/?S

Introducing u

= (.Y/[/)"~

+ E E , ] t f ; / ? t + H , ( . Y . =~ )0.

(149)

as an independent variable, putting 1'1 =

-

so" ' .r;

( 149a)

tix

as a dependent variable, putting h , ( s ,t ) = A H , ( . Y ,t ) , and using the relations V, = l / , ( . ~ / d ) F~, ' ~=, - dV,/d.x = - + ( i ; / d ) u yields

where yo = $ / L , E E , I / , A / L /and ~ , T = $ d 2 / p , t is the dc transit time. Making a Fourier analysis of t., for 0 5 t 5 T and introducing the Fourier coefficients V , , and A,,, we obtain

d 2 V,"/nu2

+ ,j(urd V,,,/ d U = ( 6/g,)h

with the initial conditions V,, V,, 31

32

=

1

=0

and

1 (, U)U,

and dVln/du = 0 at u

= 0.

V , , = exp( -jam)

(151)

Since (152)

K. M . van Vliet. A. Friedman, R . J . J . Zijlstra, A. Gisolf. and A. van der Ziel, J . Appl. Phq's. 46,1804. 1813 (1975).

A . van der Ziel, Solid

Stritc' Electiou. 9 , 1139 (1966).

236

A. VAN DER ZIEL

are solutions of the homogeneous equation, the full solution of Eq. (151) can be obtained by the method of variation of parameters. This yields

x (1 - exp[ -jwz(l - w , ) ] }

(155)

x Shi(Xi,Xz,f)Wldw1 w z d w , .

Here x 1 = w:d and x , = wid. Switching to x , as a variable, expressing no(xl) in terms of w l , and carrying out the integrations yields Svs(f) =

8kT

6

~

(wz - sin wz) = 8kTRe(Z),

90 ( 0 9 )

where Re stands for “real part of,” Z = 1/Y, and Y is given by Eq. (20).Hence Sr,(f) = S,,(f)l Yl2

(156a)

= 8kTg.

Other methods give the same result. Since g = 21a/K at low frequencies, the low-frequency value of S,,(f) may be written Sv,(f) = 4kTE/Z,,

or

R,

=

E/Za.

(156b)

For an alternate approach, see Rigaud et d. Verijication of the Thermal Noise Theory

Liu34 verified the expression for R, by plotting R, versus K/Za for a CdS diode (Fig. 11). Nicolet and his c o - ~ o r k e r verified s ~ ~ the thermal noise preA. Rigaud, M-A. Nicolet, and M. Savelli, Phys. Status Sdidi (a) 18, 531 (1972). S. T. Liu, Solid State Electron. 10,253 (1967); S. T. Liu, Noise in Solid State Devices. Ph. D. Thesis, Univ. of Minnesota, Minneapolis, Minnesota (1967). 35 M-A. Nicolet and J . Golder, Phys. Srurur Solidi (a)17, K 49 (1973).

33

34

3.

0

231


10

20

30

60 %Ksz)--r

I0

FIG. I I . R , versus C; ‘I, Tor a single-injection diode. (S. T. L I u . ~ * )

diction for carefully prepared trap-free samples and also verified the predicted temperature dependence of SVa(,f). B o u g a l i ~measured ~~ noise of hot holes in short Si diodes and found that if Tois the lattice temperature and T , the noise temperature, then ( T J , - 1) varied as I/:. Mrs. Abdel Rahman and van der Zie13’ showed that this could only be the case because the equivalent electron temperature varied with the electric field F in such a manner that ( T , / T o - 1) varied as F 2 . Unfortunately this calculation was done with the help of the salami method which gives noticeably incorrect results in this case. It may be shown, however, that this would not upset the conclusion about (T,/To - 1). What is much more serious is that Bougalis’s resuts are incompatible with recent m e a s ~ r e m e n t s about ~ ~ , ~the ~ ~field dependence of the hole diffusion D. N . Bougalis. Noise in Space-Charge-Limited Solid State Devices. Ph. D. Thesis. Univ. of Minnesota, Minneapolis, Minnesota (1970); D. N . Bougalis and A. van der Ziel, Solid Stare Electron. 14, 265 (1971). 3 7 M. Abdel Rahman and A . van der Ziel. Solid Sfutr Elecrron. 15,665 (1972). 3 8 G. Persky and D. J. Bartelink, J . Appl. Phys. 42, 4414 (1971). 38a C. Canali, G . Ottaviani, and A. Alberigi Quaranta, J . Phys. Chem. Solids 32, 1707 (1971). 36

238

A. VAN DER ZIEL

constant Dpl,and the hole mobility pplI(both measured parallel to the field). Defining T , by the relation eDp11= kTePpII

(157)

yields that T , is only a very slow function of the applied field F , in contradiction to Bougalis’s data which seem to indicate that T , / T o could be as large as 2-7. There seems to be reasonable agreement between measurements in p-type silicon by Tandon ef a/.’ and the field dependence of DPlland pLpll. on hot hole noise in Ge. The theory There are also data by Nicolet et for this case was developed by Gisolf and Zijl~tra.~’ Assuming a Druyvesteyn distribution for the velocities of the hot holes they could obtain reasonable agreement with experiment at high fields. There is some ambiguity in the definition of the noise temperature. B ~ u g a l i used s ~ ~ the definition S,(O) = 4kT,1/,/Za

(157a)

so that T , = T in the linear and the quadratic regime. It is uncertain whether this definition is appropriate in the hot electron regime. Gisolf4’” introduces a parameter a by the definition = St,(0)/4kTR(0),

(157b)

where R(0)= dK/dZa is the differential resistance of the device. This gives cx = 1 in the linear regime, c1 = 2 in the quadratic regime, and > 2 in the hot electron regime. If one uses definition (139b) for T e and then arbitrarily puts T , = T , one obtains at high injection St,(0 ) = 8kTR0,

(157c)

so that a = 2 in this case. However, in general, T , # T and T , will depend on position so that (157c) is not valid in the hot electron regime.

e. Trapping Noise At lower frequencies the noise is much larger than thermal noise. This is due to carrier trapping. The theory for this effect was developed by Zijlstra and Driedonks4’ and by others. M-A. Nicolet, H. Bilger, and A. Shumka, Solid Stare Electron. 14, 667 (1971). A . Gisolf and R. J. J. Zijlstra, Solid Stare Electron. 16, 571 (1973). 40a A. Gisolf, Int. Conf. Phjw. Aspects ojNoise in Solid Slate Devices, 4th, Noordwijkerhout, The Netherlands, September 9-1 I, Conf. Rep. p. I 1 (1975). 4 1 R. J . J. Zijlstra and F. Driedonks, Physicu 50, 331 (1970). 39

40

3.


239

The basic rate equation for trapping is c%,/?t = P n ( N , -

n , ) - an,

+ H(t),

(158)

where p and a are constants, n is the free carrier density, n, the trapped carrier density, N , the trap density, and H ( t ) a random source function describing the effects of trapping and detrapping. For the linear regime they obtain in the case of shallow traps (see Part 11)

+

where 115, = B ( N , - n,,), 1/r2 = ct /hio, l / = ~ I/T, + I / T ~t,, = d 2 / ( p , , E ) the carrier transit time. d the device length, I , the device current, V , the device voltage, and the subscript zero denotes equilibrium values. For the spacecharge-limited regime the results are much more complicated and it is beyond the scope of this review paper to discuss the details. ~ Figure 12 gives a typical trapping noise spectrum observed by H S U . ’This is probably a case of relatively deep traps so that the time constant depends on position, resulting in “smeared-out’’ spectra, and on voltage, resulting in an increase of turnover frequency (frequency at which the noise is one half the value observed at low frequencies) with increasing current I , . 13.

DOUBLE-INJECTION

DIODES

It was found by several investigators in Ge and Si device^^^-^^ for COT >> 1, where T is the carrier lifetime, that Si( f ’ ) = a.4k7-g,

( 160)

with a = 1.0; here y 1 la/E is the hf conductance. At that time the theory was not developed. Van der Ziel conject~red,~’ since in double-injection diodes p I n, that the fundamental noise source could be written as an ambipolar noise source S,(,f’) = 4e2D,n(x’)S(x’- x),

(161)

+

where D, = 2DpDn/(D, D,)is the ambipolar diffusion constant. He then argued that this would lead to

+

= 4 ~ p ~ - n / ( ~pn)2. p

(161a)

M-A. Nicolet, H. R. Bilger, and E. R . McCarter. Appl. Phys. L e f t . 9, 434 (1966). S . T. Liu, S. Yamamoto, and A. van der Ziel, Appl. Phys. Lerr. 10, 308 (1967). 44 F. Driedonks, R. J . J. Zijlstra, and C. Th. J . Alkemade, Appl. Phys. Letr. 11, 318 (1967). 4 5 H. R . Bilger, D. H. Lee, M-A. Nicolet. and E. R . McCarter, J . Appl. Phys. 39,5913 (1968). 46 J . H. Liao, Electron. Lerr. 4,402 (1968). 47 A. van der Ziel, IEEE Trans. MTT-16.308 (1968). 42

43

240

A. VAN DER ZIEL

FREO. IN MHz +

FIG. 12. Equivalent saturated diode current versus frequency for a single-injection diode with relatively deep-lying traps. (S. T. H s u . ' ~ )

To test this conjecture conclusively, Liao and van der Zie14* measured a for InSb, which has a very high mobility ratio p n / p p;they found a = 1 in this case, which refutes the conjecture (161). In addition, the theory of Section 13,a does not lead to ambipolar diffusion noise. a. The Basic Equations and Their Asymptotic Solution

The basic equations for the double-injection diodes are J, 48

= ep,(p

+ p T ) F - eD,dp/dx + eh,,

J . H . Liao and A. van der Ziel, Physica 67, 113 (1973).

3. J,


= ep,(n

=

+ n,)F + eD, anlax - eh,, a

-R-p,-[(p+p,)F]+D ?X

241 (163)

a2p ah -----r, ax

pax2

(164)

_an_- - ~ + - 1dJ "-r e iix

at =

-R

+ p, a

-

?X

[(n

d2n ah, + n,)F] + D, - r, axz ax -

d F / d x = ( e / m o ) ( p- n),

(165) ( 166)

where the symbols have the same meaning as before, hp and h, are diffusion noise sources and r the generation-recombination noise source. It is obvious that the ohmic or linear regime must give thermal noise, since the device behaves there as an ordinary resistor. We thus only must investigate the solutions for the semiconductor and the dielectric regimes. To do so, we multiply Eq. (164) by pn/ppradd the result to (165), substitute for p n , putting p N n everywhere else because of space-charge neutrality, and set R = n / T . This yields

-?Z(F:) e ax

+(n,

-

p T ) d- F +-2kTd2n ax e axz

-= p +p pppn

But we also have that J(t) = J,

+ J, +

E E dF/dt ~

(168)

is independent of x . Now the displacement term E E iiF/dt ~ is only significant at microwave frequencies, so that it may be neglected. We furthermore neglect the intrinsic conductivity term e(p,p, + p,nT) since we are not operating in the linear regime. We then have J(t) = e ( p p

+ p,)nF + e ( D , - D p ) d n / d x+ e(h, - h,).

(169)

We now put J = Jo + J , , n = no + n l , F = Fo + F , , V = Vo + V,, where V is the potential; the zero subscripts denote dc and the subscripts one denote

242

A. VAN

DER ZIEL

small-signal ac quantities. We further make a Fourier analysis for the ac quantities for 0 < f < T and introduce Fourier coefficients J , , , n l n , F , , , V,,, h,,, h,,, and r,. Open-circuiting the device for ac, i.e., putting J , , = 0, yields

or, solving for Fin, F,,

=

FO

--n,,(x)

+ kT

0

-

e

n0

p,

p p -~ 1 dn,,(x) - e(h,, - h,,,) + P p no dx 0,+ P h o ' -

___

cc.

(171a)

Here

S,( s.s’,j ) =

2n0(x')6(x'- N) AT

(173)

) r(no)r [compare Eq. (140a)l. It is easily shown that h,, - h,, where M ~ ( - Y ' = and h,,/p,, + h,,/pp are independent noise sources. To find the asymptotic ~ o l u t i o nof~ Eqs. ~ * ~(170)-(171a) ~ we observe that in Eq. (1 70) the term with n , ,contains the coefficient ( 1 + joz). Since nothing spectacular happens if at 4 Y,. it follows that n,,(s) must go to zero for OT + Thus F , , approaches KI.

Fl,(x) = -4h,,

- ~,,)/4cc, + PAn0

=

-4h,,

- h,,)Fo/Jo

(174)

so that

49

C. H. Huang, Study of Thermal Noise in Double-Injection Space-Charge-Limited Solid State Diodes. Ph. D. Thesis, Univ. of Minnesota, Minneapolis, Minnesota (1973). C. H. Huang and A. van der Ziel, Physicu 78.220 (1974).

3.


Now substitute J ,

243

+ pfl)noF,;this yields

= e(pp

where V, is the potential at x. This equation was verified by Huang for full diodes (x = d ) and partial diodes (x < d ) s l . It is thus obvious that for w >> 1 the device shows thermal noise all along the device. Any theory that does not give this must be treated with caution [see comment of Eq. (179)l.

b. Generation-Recombination Noise For generation-recombination noise, diffusion may be neglected. One must solve the equation

=

C(PP + Pfl)/(PpPfl)lrfl.

(176)

~ ~both the semiconductor and the insulaThis was done by D r i e d ~ n k s ” ,for tor regimes and by H ~ a n for g ~the ~ semiconductor regime. The results are

for the semiconductor regime; for the insulator regime

5‘ 52

C. H. Huang and A. van der Ziel, Phvsicu 76, 172 (1974). F. Driedonks, Electrical Conduction and Noise in Solid State Injection Diodes. Ph. D. Thesis, Univ. of Utrecht, Utrecht, The Netherlands (1970).

244

A. VAN DER ZIEL

These formulas were experimentally verified by Driedonks.' This noise is important for relatively short diodes. c. Thermal Noise Theory Reconsidered

Zijlstra and GisolP3 solved the noise problem for the insulator regime under neglection of the diffusion term (2kT/e)d2n,,JdxZ.They obtained

V , (1 + 0 2 t 2 ) 16kTd z +I , (9+ 02z2) E E ~ A 9 + ozzz'

Sv,(f) = 4 k T -

(179)

Here the first expression must be attributed to the noise term (h, - hnn)and the second expression to the noise term d(h,/p,, + h,,,,/pn)/dx.It should be noted that for oz >> 1 this reduces to (175) for the case x = d, V, = V,. For 0 < x d their method of solution will not work and it is doubtful that Eq. (175) can be proved for x d and 07 >> 1. This casts some doubt upon the derivation. Various investigators have tried to solve the diffusion noise problem for the semiconductor regime. If one neglects again the diffusion term, one either obtains divergent integrals, or, if one manages to avoid these divergent integrals, one is left with an undetermined integration constant. The divergent integrals stem from the term d(h,,/p, + h,,,,/p,,)/dx.If one ignores this term and solves the problem one does not obtain Eq. (175) for x = d and oz >> 1.49*50 According to H ~ a n g ? ~ this , ~is~due to the fact that the diffusion term (2kT/e)dZn,,,/dx2cannot be ignored in the semiconductor regime. If this term is taken into account, and one neglects instead the term with dF,,,/dx in Eq. (170) one obtains Eq. (175) for 07 >> 1. In addition, one obtains a very small term due to the noise source d(h,,,/pt, + h,/p,,)/dx for short diodes that disappears for o7 >> 1. The conclusion is therefore that there should always be thermal noise as given by Eq. (175). In addition there is a noise term due to the source d(h,,,/ I(,, + h,,,/p,,)/dx that disappears for WT >> 1. For oz >> 1 we thus have

-=

Sv,(f)

-=

= 4kT

v,

-, I,

Sr,(f) = 4 k T

V , IYI2 = 4kTg

-

1,

= 4kT

la

-

V,

(180)

since Y = I J V , for oz >> 1. This agrees quite well with experiments, except for very short diodes. Liao" and Tsai2' found that the equation S,,(f) = 4kTg was always valid for oz >> 1, whereas for short devices the expression 4 k T I J K must be replaced by 4kTI,/v, where 6 is the voltage across the 53

R.J. J. Zijlstra and A. Gisolf, Solid Stare Electron. 15, 877 (1972).

3.

245


i-region. For a recent review of the noise properties of single- and doubleinjection diodes see a paper by Nicolet, Bilger, and Zijlstra.26 V. Applications

14. APPLICATIONS OF SINGLE-INJECTION DIODESAND TRIODES a. Applications of Single-Injection Diode Circuits

Single-injection diodes could be used in detectors and in logic circuits. In detectors one must bear in mind that it is required to have OT < 7.6 This limits the frequency range for which the detector is operable. One must make to = poI/,/d2 as small as possible. Here t o is 75% of the full transit time 7.This is in principle not difficult. If one wants to use the device in pulse circuits, then the solution discussed in Section 4,e is not applicable. For we are not interested in the current transient due to a constant voltage pulse nor in the voltage transient due to a constant current pulse, but we want to transmit pulses of duration T. For that one wants the circuit of Fig. 13 with V ( t )= V’ for 0 < t < T, where V, is the pulse height. In order to have a fast response, to = poV,/dz should be as small as possible, which is basically not difficult. In order to transmit pulses we must require that the initial output voltage Vooand the final output voltage V,, are comparable to V,, or, in other words, we require I/,(t) to be relatively small at all times. Though K ( t ) is not constant in this case, we can use the solution of Section 4,e to clarify what is going on. Let KO and V,, be the initial and final diode voltage and io and i , the initial and final current in the circuit. Then if A is the device area and R the resistance across which Vo(t)is developed, V, = V,,

+ vo,,

V, = V,,

+ Vo,,

io = ( ~ ~ ~ p ~ A / 2 d=~Voo/R, )l/$

(181)

i ,=8 ( E E ~ ~ ~ A /= ~~ Vom/R, )VI,

(182)

from which Vooand Vo, can be determined. As said before, for a good operation one wants Vo(t)= V, and E(t) 0. In that case the overshoot disappears to a great extent and the

FIG. 13. The single-injection diode as a transmitter of pulses. The pulse supplied by the generator has a height V,,.

I

W0tt)

I

246

A. VAN n-TYPE

0 8 8 0 8

DER ZIEL

ANOM

0(=lD CATHODE

(0)

CATHODE

I

SAPPHIRE SUBSTRATE

FIG. 14. Zuleeg's space-charge-limited triodes.

I

response becomes almost instantaneous. It is then not necessary to find the exact transient response of the circuit.

b. Single-Injection Triodes Just as in vacuum tubes one can make triodes by adding a grid between the cathode and the anode of a vacuum diode, so in the case of single-injection space-charge-limited devices one can make triodes from diodes by adding another electrode. Figure 14a shows a version that closely resembles a vacuum triode. A weakly n-type semiconductor is provided with two electrodes acting as cathode and anode, respectively, and a p-type grid structure is imbedded in the n-region. The anode is biased positively with respect to the cathode, whereas the grid is at zero, or slightly negative, bias with respect to the cathode. The device was proposed by S h ~ c k l e yand ~ ~built by Z ~ l e e g . ' ~ Zuleeg's units operated up to about 100 MHz, but this is probably not the limit. The device also resembles a junction FET, but the characteristic is triode-like rather than FET-like. Figure 14b shows another version of the device that closely resembles a MOSFET; the difference is again that the characteristic is triode-like rather than FET-like. Here a single-crystal layer of silicon is grown on a sapphire substrate and it is provided with ohmic contacts. On the cathode side an oxide layer is evaporated and a gate electrode is deposited upon it. The anode is biased positively with respect to the cathode and the gate is biased at zero voltage or slightly negatively. The gate voltage controls the potential distribution in front of the cathode; this gives the devices its triode characteristic. The device was built by Z ~ l e e gHis . ~ ~units operated up to 500 MHz, but, by making the distance between anode and cathode shorter, the limit of " W. 55 56

Shockley, Proc. ZRE40, 1289 (1952). R. Zuleeg, Solid Stare Electron. 10,449 (1967). R. Zuleeg and P. Knoll, Proc. ZEEE 54, 1197 (1966).

3.


247

operation might be pushed into the lower microwave region. The device can easily be built in integrated circuit form. It is doubtful, however, whether these structures have much future, since they must compete with junction and MOS field effect transistors whose development has progressed much farther. Moreover, those devices have a pentode-type characteristic, which is much more favorable than a triode characteristic. Nevertheless the development shows that these devices can be made. 15.

h P L l C A T l O N S OF DOUBLE-INJECTION

DIODES

Double-injection diodes have found even fewer applications than singleinjection diodes. This is not surprising, for they operate best for oz < 1, where z is the lifetime of the carriers. In many cases is of the order of 10100 p e c , so that only low-frequency operation is possible. One might try to reduce the carrier lifetime, but it is doubtful that one can go very far in this direction. There is one high-frequency application that could be feasible. We saw that the device has an ac conductance g, equal to I J K , up into the microwave region for wz >> 1.” The device might therefore be used as a voltage-controlled termination of transmission lines or waveguides. Since the noise of the devices corresponds to thermal noise of the conductance g, this would be quite acceptable. Another application is to use devices with traps in switching circuits. We saw that these devices could show a characteristic with a negative resistance regime and therefore they could be used in switching circuits. It is doubtful whether they can be made very fast, since one is limited by the carrier lifetime. But even if that hurdle were passed, one would still only have a two-terminal switch. Bipolar transistors and field effect transistors are used as threeterminal switches with separate input and output and are therefore much more versatile than two terminal ones.


SEMICONDUCTORS A N D SEMIMETALS. VOL . 14

CHAPTER 4

Monte Carlo Calculation of Electron Transport in Solids Peter J . Price I . INTRODUCTION 1. Introduction . . . I1 . HOT ELECTRONS . . . 2 . General . . . . . 3 . Averages and Estimators 4. Scattering . . . . 5 . DisorderedSolids . . 111. HOTELECTRON PROPERTIES 6. Magnetic Field Effects . 1. Time Dependence . . 8. Diyusion . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

.

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iv . SPATIAL STRUCTURES . . . . . . . . . . . 9. Introduction . . . . . . . . . . . . . 10. Escape and Penetration Phenomena . . . . . . 11 . Size Effects . . . . . . . . . . . . . 12. Junctions . . . . . . . . . . . . . V . OHMIC CONDUCTION . . . . . . . . . . . 13. Ohmic Conduction . . . . . . . . . . . VI . COLLECTIVE EFFECTS . . . . . . . . . . . 14. Introduction . . . . . . . . . . . . 15 . Many-Particle Monte Carlo . . . . . . . . 16. Carrier-Carrier Scattering . . . . . . . . 11. Localized States; Avalanche Phenomena . . . . 18. Auxiliary Function Applications . . . . . . . Appendix A. Generation of a Gaussian Distribution . Appendix B . Some Vector Geometry . . . . .

. . . . . .

. . . .

. . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. .

249 249 254 254 258 264 269 212 212 213 219 283 283 284 286 290 294 294 297 291 297 301 303 305 306 307

I . Introduction 1. INTRODUCTION

This article reflects the growing emphasis on the use of numerical solution by computer. in the theory of electron transport phenomena in solids. In most cases of interest. the electron system may be given a quasi-classical description in which an individual mobile electron is assigned a Bloch band and wavevector (and hence a “momentum” equal to h times the wavevector) and is simultaneously assigned a position. and the state of the system is specified by a distribution function in these variables. The distribution 249

.

Copyright @ 1979 by Academic Press Inc. All rights of reproduction in any form reserved. ISBN 0-12-752114-3

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PETER J. PRICE

function changes in response to external driving fields, the perturbed distribution function is governed by the appropriate Boltzmann equation, and hence solution of this Boltzmann equation provides a theoretical description of the behaviour of the system. Ideally, the Boltzmann equation can be solved by analytical means; and then we have an explicit mathematical formula for properties of the electron system in terms of the governing parameters and variables: for example, mobility versus magnetic field as a function of band masses and electron-lattice coupling constants. This is possible, however, only for some particular cases of high symmetry and simplicity, which occur less frequently in practice than in textbooks and journals. There are also cases, in particular the calculation of ohmic mobility when the thermal region of the band and the carrier scattering function in this region have spherical symmetry, where the applicable Boltzmann equation reduces to an equation in a single variable (in particular, the energy), and numerical results may accordingly be obtained with a modest amount of computation by straightforward procedures. If there is not this high symmetry, then we have a Boltzmann equation in several variables. (And space dependence, when it occurs, adds to the dimensionality.) Numerical solution of such an equation entails representing the distribution function by its values on a grid of points in the space of the carrier variables. If we still have spherical symmetry for carrier band and scattering, then for the hot electron situation with a large electric field, but with space homogeneity and no magnetic field, the Boltzmann equation is a homogeneous integrodifferentialequation in two dimensions (wavevector parallel to the field and radially perpendicular to it). In this case, numerical solution by computer has been accomplished, and extensive and valuable results obtained. even with time dependence and higher electron bands included.' Some results have been obtained, by special means, with a space variable also included (so that Gunn domains could be studied).' Such procedures are, however, subject to technical limitations that are narrower than the requirements of the problems of present interest. To represent the distribution function in sufficient detail with the needed dimensionality can entail an impractical number of grid points. The grid representation of the details of scattering and motion of the electrons, as a problem in numerical analysis and programming, can become a disproportionate burden. The situation is that we now have, for many solids, an essentially simple model of the mobile-electron system with a lot of complex detail, and it is desired to obtain at least numerical values for transport effects which reflect the model, so that by comparison with experiment the model can be tested and parameter values in it can be determined,

' See for example H. D. Rees, IBM J . 13, 537 (1969). H.D. Rees, J. Phys. C 6,262 (1973).

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251

or so that new phenomena or new domains of action for known phenomena can be investigated. For this purpose we need an easy and adaptable method of broad scope, for computing the phenomena with the full physical detail of the model included, and readily allowing revisions in the model. Although this is a probably unattainable ideal, Monte Carlo calculation goes quite far in its d i r e ~ t i o n . ~ Monte Carlo computation is used over a great range of problems, from direct simulation, of which the subject matter of the present work is typical, through calculations such as evaluating a partition sum by a sampling path through the domain of the summand, to computational procedures which have seemingly no relation to what is being calculated. In the Monte Carlo method used in transport theory, calculation of the distribution function is replaced by computer simulation of individual particle motions, which are treated as sample members of an ensemble. Expectation values of physical quantities (such as drift velocity) are given by the appropriate cumulative averages; distribution functions are obtained by counting the number of appearances of a variable in each “histogram” interval. The motion of a mobile electron in a solid, in the quasi-classical representation, is a sequence of trajectories governed by the external fields (and by any internal macroscopic structure as in a junction) alternating with scatterings determined by interaction with lattice motion and other microscopic deviations from crystal perfection. Both the duration of a trajectory (“path”) before termination by a scattering, and the electron state resulting from the scattering, are to be treated as random variables with given distributions. These random variables are produced in a determinate way as functions of one or more values of a standard random variable. For a good survey of Monte Carlo see J. M. Hammersley and D. C. Handscomb, “Monte Carlo Methods,” Wiley, New York. 1965. A more detailed work is Yu. A. Shrieder (ed.), ”The Monte Carlo Method.” Pergamon. Oxford. 1966. The Monte Carlo method was introduced into modern physics by Ulam (1946) and von Neumann. See: correspondence between von Neumann and Richtmyer (1974) reproduced In J. von Neumann, “Collected Works”(A. H. Taub.ed.). Vol. 5. pp. 751-764. Macmillan, New York, 1963.and S. M. Ulam and J . von Neumann, Bull. Am. Math. SOC.53, 1120 (1947). A personal account will be found in Chapter 10 of Ulam’s autobiography “Adventures of a Mathematician” (1976). Their immediate concern was with nuclear reactor design, but there soon was work on a wide range of applications. The scope and flavor of this activity is indicated by the papers at a 1949 conference - - “Monte Carlo Method.” National Bureau of Standards, Applied Mathematics Series, no. 12 (1951). and by the bibliography with abstracts in “Symposium on Monte Carlo Methods” (H. A . Meyer, ed.). Wiley, New York, 1956. An interesting early paper with content related to that of the present work is M. L. Goldberger. Phys. Rev. 74. 1269 (1948). Pioneer papers on electron transport in solids were Liithi and W ~ d e r ~ ~ . and T. Kurosawa, J . Phys. Soc. Jpn. Supple. 21, 424 (1966). An earlier use of Monte Carlo in physics, in which results of the kinetic theory of gases were tested by simulating the molecular motions, was Lord Kelvin, Phil, May. (6th ser.) 2, 1 (1901).

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PETER J. PRICE

Use of a random variable in a computer calculation seems a paradoxical idea. In reality these computations are fully determinate ;a repetition using the same initial numbers will give precisely the same particle “histories” and results. (Indeed, for purposes of program development and debugging it is ) calculation is of course a determinate important that this be ~ 0 .A~computer sequence of machine stored “words” which themselves can have only a finite and discrete set of values, and therefore can represent the real numbers only approximately; and so a calculation involving real numbers has a kind of indeterminacy, in that there is in general a cumulative roundoff error which depends on the actual sequence of machine instructions used to implement the algorithm. The pseudorandom numbers used in Monte Carlo calculations5 are similarly artifacts, but in this case deliberately so, of a truncation process. The successive values of the standard random variable are generated by a program which is designed to produce a determinate sequence of numbers with, for practical purposes, no correlations between them. At the same time, they tend in the aggregate to a predetermined distribution; the practice is to have them uniformly distributed in the interval (0,l). The effectivenessof the Monte Carlo method is related to the fact that the particle variables, instead of forming the basis of a grid of representative points between which a distribution function must be interpolated, are explicitly computed, as the termini of successive paths and final states of successive scatterings. Complicated paths, including physical boundaries in position space, and complicated scatterings, including particle absorption and generation, normally require only a manageable elaboration of the algorithms for the simulated particle “history.”The electron states generated in the history have a cumulative distribution which favors those ranges of the particle variables that are important in the situation being simulated. Monte Carlo calculation does, however, have a serious general disadvantage. The computed results are “estimator” values. The error difference from In an early discussion of the subject, von Neumann” remarks: “.......we could build a physical instrument to feed random digits directly into a high-speed computing machine ....... The real objection to this procedure is the practical need for checking computations. If we suspect that a calculation is wrong, almost any reasonable check involves repeating something done before. At that point the introduction of new random numbers would be intolerable.” The sequence of values of a physical variable produced in a particular computer “run” depends on the initial value of the seed integer in-the pseudorandom number generator’; but the estimator values given by the computation are useful results to the extent that they are independent of the initial seed integer. The pseudorandom number generator that was used in the unpublished calculations described here, and in Refs. 25, 26, 31, and 45, is an implementation of the Lehmer method for IBM 360 machines. See: D. W. Hutchinson, Commun. ACM 9, 432 (1966); P. A. W. Lewis, A. S. Goodman, and J. M. Miller, IBM Sysr. J . 8, 136 (1969).

4.

MONTE CARLO CALCULATIONOF ELECTRON TRANSPORT IN SOLIDS

253

a required true value does not decrease steadily with further execution cycles of the unit computation; it is a random variable. The associated variance, for a convenient quantity of computation, may be untowardly large. Furthermore the convergence process can be a wayward one, hard to monitor and estimate. Appropriate care should be exercised in controlling and assessing these result errors; the process is perhaps more like experimental than conventional theoretical physics.6 Nevertheless, the estimator variance for a given calculation is not immutable. It can be reduced by alternative algorithms and computational devices, such as will be described. One might argue that the special advantages of Monte Carlo are not fundamentally connected with the randomness aspect; and there should be a way to dissociate these so as to retain the “direct construction” aspect of the programming but get rid of the variance. The present article, however, deals with what is now known and practiced. This survey article reflects my belief that Monte Carlo has a place in physics as an accessible technique, available for use by researchers whose principal interest is in the content of the particular application, rather than a self-contained branch of applied mathematics. Since details of the model and the quantities that one wishes to calculate are liable to change as a physics research proceeds, the computer program should be “transparent” and simple in structure, and thereby flexible. Both these considerations limit the degree of mathematical sophistication, in the Monte Carlo procedures used, that it is useful to cultivate. A great deal can in fact be accomplished, in the field that is dealt with here, by means of simple computational ideas. This article attempts to provide a “state ofthe art” presentation of methods, and at the same time survey the accomplished-and some potential-applications of Monte Carlo to topics in the physics of electron transport in solids. These applications will be indicated by the table of contents. It may be helpful to itemize here predominant techniques and their principal locations in the text: (a) rejection methods: Section 4, Methods 2-4; (b) self-scattering: Eq. (20) and containing paragraph; (c) B state estimators: Eqs. (29),(941, and containing paragraphs. The literature belonging to this subject is becoming large; no attempt has been made to reference it comprehensively. Publications are cited where they usefully amplify the text; others that traverse the same ground are From another point of view, the fluctuations themselves can be what is calculated in a Monte Carlo simulation. An instance of this, use of a “fluctuation-dissipation theorem” to calculate a linear response coefficient, appears in Section 8. A fluctuation is, of course, given by an estimator that has its own variance.

254

PETER J. PRICE

omitted. A fuller bibliography will be found in my 1977 review paper “Calculation of Hot Electron Phenomena.”6a 11. Hot Electrons

2. GENERAL The principal use of Monte Carlo computation for electrons in solids has been for the hot electron phenomenon, especially where all or most of the following apply: (a) the electron states of the mobile carriers are Bloch states, and the quasi-classical description of their motion applies; (b) the “driving force” displacing the electron system from thermal equilibrium to the hot electron state is an external electric field; (c) only the mobile electrons are redistributed among their possible states (while the phonon system in particular may be considered to be undisturbed); (d) the hot electron system is homogeneous in space; (e) the driving force is constant in time, and the solid may be taken to be in a steady state; (f) the mobile electron density is low, so that collective electron effectsspace charge and mutual scattering-may be neglected, and Fermi statistics reduces to Boltzmann statistics. These conditions define what will be taken as the basic problem; Monte Carlo calculation will be introduced and described in terms of it, in the present section, after a summary of the corresponding analytical formulation. For each Bloch band and spin state, the electron’s state is specified by wavevector k and hence a momentum p=hk

(1)

and energy E(k) or E(p). The equations of motion are then cirldt

= V,

dpldt = F,

(2)

(3)

where r is the position vector. The electron’s quasi-classical velocity is v(p) = dE/dp

and the force F may be taken as given by

P.J. Price, Solid Stare Elecrron., 21, 9 (1978).

(4)

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

255

The first term of ( 5 ) is the force due to external electric and magnetic fields & and 9.The second term could be due to a strain gradient in the solid (such as that due to a sound wave) or a space inhomogeneity of composition and hence of the band system of the solid. According to assumption (d) we drop the second term and take &’ and B as constant in space; and according to (e) we take them as constant in time. The macroscopic-average state may be specified by a distribution function ,f(n,p, r, I)where , n stands for band and spin, and where the last two variables are to be dropped according to (d) and (e). The charge density and current density, for example, are p =

f r Jd3Pf

and J

=

+e

J d3pvf

(7)

As in (6)and (7), band and spin indices and summations over them are discarded throughout, except where the context requires them; the reader may mentally insert them elsewhere. For convenience the carrier charge will be taken as +e, even though we speak of “electrons.” According to (2) and (3),f should satisfy the Boltzmann equation

where the final term x { f }is the effect of scattering processes which are outside the quasi-classical motions:

xm

= Jd3P”f(P‘)S(P’,P)(l

- . f ( P ) ) -.T(P)S(P3P‘)(1 -.7(P’))17

(9)

where S is the scattering-rate function, and? is the absolute probability of occupation of the particular one-electron state specified by the argument variables. The notation f is used for other normalizations, and for no particular normalization. In Eqs. (6) and (7), .f is in general normalized by an integration over both momentum space and position space; but in Eq. (94) it is normalized by an integration over momentum space alone. (Spin, and band o r valley, where applicable, are suppressed for convenience.) Equations such as (8) with (10). being linear-homogeneous in ,f, are independent of normalization. T h e 3 of Eq. (9), etc., is not subject to normalization. In (9), electron-electron scattering has been disregarded. Also in accordance with assumption (f) one may discard the (1 - 3)factors:

x{.f} = Jd3Pt [.f(P”P’,P)

- f(P)S(P,P’)l.

(10)

256

PETER J. PRICE

Consequently (8) is linear homogeneous in f ;for theoretical analysis at least, this simplification is of critical importance.’ Where interband scattering is significant, we actually need a set of coupled equations like (8), with interband terms like (9) or (10) providing the coupling. When, in accordance with (c), the rest of the solid (lattice modes, and other “moving parts” such as excitable impurities) is in thermal equilibrium, the detailed-balance relation

2)(1 -fd2))= fF(2)S(29

(11) holds, wheref, is the Fermi function. With assumption (f) this reduces to fF(l)s(l?

fMB(1)S(L2) = fkLW(2, I),

- fF(l))

(12)

where fMB is the Maxwell-Boltzmann function: fMB(P)

= const exp[ -E(p)/kT].

(13)

If, as we are now assuming, the fields giving the Lorentz force in ( 5 ) are independent of time, the most important solution of (8) is constant in time. Then the ensemble average represented by integrals over f,such as (6) and (7), is equal to the corresponding average over time for a single electron. The expectation of an electron variable Q(p) is

The electron “history” represented by the argument of Q on the right of (14) consists of alternating “paths,” given by the equation of motion (3), and scatterings. The duration of a path between scatterings depends, according to (lo), on the scattering time t(p) where 1 -

T

=

[ S(p,p’)d3p’.

Then the probability that, starting from the state p, the elapsed time to the next scattering will exceed s is

where the argument (PI t ) means the state (p value) reached along a path after a time interval t , starting from p. The initial and final p values at the nth scattering will be denoted by pr) and pr), respectively, so that the nth path

’The linearizarion

of (8) and (9) to describe Ohmic conduction gives an inhomogeneous equation, for the deviation off from the thermal equilibriumfunction, in which the scattering term is not algebraically the same as (10).See Part V.

4.


257

between scatterings goes from pf) to p t + l ) . The possible values of pf) will, according to (lo), have the distribution function W(pp, pb“’)= t(pP))S(pp, pb”)).

(17) The prescription for a Monte Carlo calculation in this case is, then, to use the successivepseudorandom numbers R, belonging to a uniform distribution in (0, l), to select path durations s by the solution of R = QP,, s)

(18) and to select initial states of paths (final states of scatterings) pa by some algorithm satisfying dR

=

W(pb,pa)d3p~

(19)

Expectations of electron variables of interest are obtained as cumulative averages, over the sequence of states constituting the “history,” equivalent to (14). Although (16) and (17) were originally implemented in the literal form presented above, subsequent work has used the “self-scattering” artifice to make the computation simpler or practicable.8 The scattering function S is modified by adding a null term: S(P,P‘)

S(P,P )+ 0-- 1/7)S3(P - P),

(20) where r is not less than the maximum value of the physical scattering rate l/r given by (15) before the substitution, and d3 means the three-dimensional Dirac function. Then the scattering rate becomes r. It is usual to make r a constant, and then (16) and (18) become --+

s = ( l / r ) In(l/R).

(21)

Physical results are unchanged, so long as the appropriate fraction 1 - (l/Tt) of the scatterings generated as path terminations according to (21) are taken as null processes in which pa = P b . The scheme is analogous to von Neumann’s rejection method of generating a set of numbers with a prescribed distribution function. The advantage in using it is in the simplicity of generating the path durations, instead of what (16) otherwise entails. Probably some calculations would be impossible without it. The simplification of the path-duration algorithm to (21) means a saving in computer time per “trial” (path/scattering cycle). If the physical scattering rate 1 / t varies by a large factor, over the range of electron states included in the history, however, then the requirement Ts 5 1 can result in there being more of these trials, in a

* W. Fawcett, A. D. Boardman, and S. Swain, J . Phys. Chem. Solids 31, 1963 (1970).Other references are in S. L. Lin and J. N . Bardsley. Computer Phys. Cornrnun. 15, 161 (1978).

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PETER J . PRICE

history of given duration, by a large factor. It may then be advantageous to use a varying, though still piece-wise constant, r.9An alternative procedure is suggested in Section 4 (page 269). When there is no natural upper bound for l/z, in the semiconductor model, one may have to artificially limit the electron states to a domain such that within it r is finite and not inconveniently large compared to l,k, while the excluded remainder does not have an appreciable probability of being occupied if not thus excluded. For example, if the scattering rate increases without limit as the electron energy E increases, one may impose an upper limit on E artificially, with the self-scattering rate falling to zero at the upper limit. (It may be possible to avoid this, however, by so arranging the part of the computer program corresponding to (19) that in practice an electron reaching a state for which Tr < 1 is quickly scattered to a lower energy out of the domain of these states. The fact that states having a negative self-scattering rate, as analytically defined, can be reached is then of no consequence.) An aspect of hot electron physics which is not explicitly considered here is the interaction of the carriers with light. Transitions due to incident radiation can act as driving force, either instead of or in addition to an electric field. These transitions would have the same formal role in the Monte Carlo procedures as the ordinary scattering processes. Absorption, emission, and scattering of light, phenomena giving information on the distribution function, may be calculated as expectations of electron variables, averaged over the distribution; no discussion of technique seems necessary. Interaction with coherent radiation of large amplitude at optical frequencies, and fields at frequencies intermediate between optical and quasi-classical values, need special consideration as physics. 3. AVERAGES AND

ESTIMATORS

Purposes of the Monte Carlo simulation of sample electron "histories" are to calculate both expectation quantities such as the drift velocity u

=

(v)

and interesting projections of the distribution function f, such as the distribution over energy P(E') = ( 6 ( E - E ' ) )

(23)

(where E' in (23)is the independent variable, and E(p) is a function of electron state), and occupation probabilities of individual bands or band valleys. Quantities like (22) are given by (14). and hence may be obtained by a time V. Borsari and C. Jacoboni, Phys. Srarus Solidi (h)54,649 (1972).

4.

MONTE CARLO CALCULATlON OF ELECTRON TRANSPORT IN SOLIDS

259

average over each path and then a cumulative average over paths. A direct evaluation of (23), and similar quantities, would entail counting passages though prescribed “histogram” ranges of E, weighted by the time spent in each passage. This literal implementation of the time averaging is useful only in some special situations. For example, if the band energy function is “spherical parabolic,” E

= p2/2m*,

(24)

then for an electric field alone in (5) we have

ji v( p I t )df = $ s(v, + vb), where v, = v(? = 0) and vb = v(t = s) are initial and final values on the path. Thus in this case only a sum over the terminal states of all paths is required. There are, however, alternative estimators for the quantities of interest, a situation typical of Monte Carlo calculation. A drift velocity estimator that applies generally is provided by the formulas

t l E / d t = e808

(26)

tlp,Jdt

(27)

and = eb,

where the force F is given by the electric and magnetic field terms of ( 5 ) , and the subscript means the vector component in the electric field direction. On applying (26) and (27) to (14) and (22) we have U8 =

1 A E l 1 AP87 paths

(28)

paths

where A means the final (b) value minus the initial (a) value on a single path. A similar formula for the expectation of E applies only in the case of (24) and its generalization from scalar to tensor effective mass.’ O An estimator that applies to all quantities like (14) and all cases of interest is provided by what we will call the B-Ensemble method. The conventional distribution function f(p) corresponds to an ensemble of p values selected at arbitrary times t from a history p(t). The states which terminate the paths, the “B states,” may equally be considered an ensemble and characterized by a distribution function ,fb(p).Then it can be shown that

f ( P ) = const Z(P).f,(P) lo

W. Fawcett and E. G . S. Paige. J . P l i y ~C4. . 1801 (1971).

(29)

260

PETER J. PRICE

(where the constant provides for normalization). Therefore, expectations (14) may be replaced by averages over B states, weighted by T :

Distributions can be similarly obtained by counting the number of paths terminating in each histogram interval, with the weighting T. For spherical surfaces of constant energy E in p space, it is of interest to compute energy histograms with various of the spherical harmonics of the (p,&) angle as a weighting. This gives the spherical-harmonic projections off: With self-scattering, if all trials including those in which the path is terminated by a self-scattering are included in the sum (30a), T is replaced by l/r; and if r is a constant then the weighting is superflous, and we have

and in particular

where N is the total number of trials (scatterings)in the sum. For B-Ensemble estimators we require from the path algorithm only the terminal state pb resulting from the state pa after a prescribed interval s, not the trajectory between them. With a constant electric field only in (5), this is obtained just by adding se& to pa. With a magnetic field also, however, only for a simple band energy function like (24) is there still a straightforward formula for P b that can be directly implemented.' (Otherwise, presumably, some kind of numerical integration of the trajectory is required for this purpose.) The equality of (14) and (30) or of (28) and (31) is an ensemble theorem valid in the limit N -,00. The two estimator values of the drift velocity, in particular, approach each other with increasing N, and to some extent this mutual convergence can serve as an indicator of their common convergence to the true value. Inspection of many plots like Fig. 1 discloses, however, that the fluctuating values of (28) and (31), versus increasing N, are rather closely correlated; and their variances are evidently of the same order of magnitude. Their common deviation from the true value is therefore liable to be greater than the difference between their individual values. We are especially concerned with the fluctuations of these estimators. The variance A. D. Boardman, W. Fawcett, and J. G. Ruch, Phys. Starus Solidi (a) 4, 133 (1971); D. Chattopadhyay,J. Appl. Phys. 45,4931 (1974).

4.


261

x lo6 TRIALS FIG. 1. A representative plot ofestimator values, for drift velocity, versus number oftrials. The estimators given by Eqs. (28), (31), and (36) in the text are represented by points zero, plus, and asterisk, respectively.

of the time integral x(s) =

so”

u,,(t)dt

of the velocity component in the direction of u is related not just to the fluctuation of the drift velocity but to the autocorrelation of i)ll(t).In fact, the variance of the quantity (32)is equal to 2sDll,where Dll is the longitudinal diffusivity.” The correlation time 511

= ql/

(33)

is liable to be that characteristic of energy relaxation, typically much larger than z. Taking s = N/T, we have for the drift velocity as a time average (variance)’” (2rD11)”2 2 q I ‘ I 2 ( ( q(quantity) - W 2 u =(7) u

(34)

One might suppose that for (31) as estimator (34) should be replaced by ((q - u ) ~ ) ’ ~ ~ / u Nsmaller ’ ’ ~ , by better than a factor - ( T / Z ~ , ) ’ / ~ . This is incorrect because successive vb are not uncorrelated. As remarked above, in practice the fluctuations of the two estimators for the drift velocity are about the same in magnitude; indeed, they are correlated. An estimator of the variance of the calculated quantity, such as u, may be provided by breaking up the simulated history into many shorter ones and computing the variance of the set of results obtained.l3

’’ See Chapter 8 in “Fluctuation Phenomena in Solids” (R. E. Burgess, ed.). Academic Press, New York, 1965. Alberigi Quaranta, C. Jacoboni, and G . Ottaviani, Rittsta dei Nuoro Cimenfo 1, 445 (1 971).

” A.

262

PETER J . PRICE

The final factor in (34) is large, and for this reason especially the fluctuation of the drift velocity estimator is large compared to u / a . I n physical terms: the drift velocity is the average of a quantity u that is odd in p, and is normally small compared to the prevailing “thermal” values of I>. Clearly, it would be desirable to substitute an estimator that is the expectation of a quantity even in p. Such an estimator exists for the drift velocity. It can be shown that

where I is the vector mean free path.14 The coefficient in (35), the “chordal mobility,” is to be averaged over the hot electron distribution as in (30).In particular for the longitudinal component

Although (35)is of quite general validity, this application is restricted to cases where a suitable formula for the quantity averaged is available. If the scattering is isotropic, for example, then (37)

1 = TV

and if also z is a function of energy only, T

= z(E(p)), then

? 6 dz ---l=z,v+-vv. (P cp dE

For (24) or its tensor-mass generalization (the latter being normally valid for n-Ge and n-Si) the coefficient of T is just l/m* times the unit tensor, or the inverse mass tensor. In the model used for the test computations represented here by Fig. 1, these assumptions were made; and the z ( E ) function was of form 1 - = C Ci(E+ Ei)li2 (39) T

i

with one of the Ei zero (for acoustic-mode-phonon scattering) and two other terms with equal and opposite Ei values (for optical-mode-phonon scattering). Thus it might be a suitable model for a single conduction-band valley of a nonpolar semiconductor like Ge or Si. It was used for test purposes because, while simple, it is in many ways representative of the actual systems of interest. Figure 1 shows values of the three estimators, (28),(31), and (36),of drift velocity, plotted against number of trials N . It is clear that fluctuations of all three are correlated, and that (36)gives a much smaller variance. l4

For example, P. J. Price, J. Phys. Chem. Solids 8, 136 (1959).

4.


263

A quite different means for variance reduction” is significant both because of its general applicability, and hence importance for calculating the drift velocity, and because it represents an approach of broader interest in Monte Carlo calculation-to substitute precise known values for quantities subject to the random fluctuations, wherever possible, within the calculation. (Another example of this approach, in the direct calculation of differential mobilities, will be found in Ref. 25.) Although this procedure is a modification of the scattering algorithm, it is presented here in advance of the general discussion of scattering. In isotropic scattering the pairs of states kp’, with velocities v’, are equally probable A states. Their relative frequency fluctuates around equality, however, as a result of the random choice between them in Monte Carlo calculations. This fluctuation is an important source of fluctuation in the sum of the vp’. In the method being described, both states of the pair are selected, in a scattering, and used in the calculation. The two following paths are generated with the same random number, and hence (with a constant r) have the same duration. The average of the two path contributions to the estimator of u is used. For the next trial, one of the two B states is chosen at random and used as the initial state ofthe next scattering. If scattering is not isotropic, then the pair of A states is selected, in a scattering, with a probability given by the average of s@, p‘) and s@,-p‘); and a relative weight equal to the ratio of these two scattering-function values is applied to the pair of paths in their contribution to the estimator and in the selection of a B state for the next scattering. Evidently this procedure could usefully be generalized so that a complete symmetry star of Bloch states is used to represent the result of the scattering. Another source of variance, the fluctuation in the relative occupation of band valleys as N increases, may be suppressed by a method in the same spirit. The elements of the matrix of intervalley transition rates are obtained by evaluation of

from the distribution over p in the mth valley. (Again, the time average may be replaced by an average over B states.) The occupation probabilities of the valleys, P,, are then obtained from the rate equations

and substituted for the Pnvalues given by the Monte Carlo simulation itself. l5

C. Hammar, Phys. Ree. B 4, 417 (1971); T. Kurosawa and H. Maeda, J. Phys. SOC.Jpn. 31, 668 (1971). Compare Section 5.6 in J . M. Hammersley and D. C. Handscomb, “Monte Carlo Methods.” Wiley, New York, 1965.

264

PETER J. PRICE

Since (40)is not affected by the fluctuations in the latter, the variance can be reduced in this way. It should be noted that the validity of the method described does not require that the intervalley scattering be weak. The procedure remains valid when the distribution functions which determine the elements (40) are substantially affected by the intervalley scattering. There is a special difficulty with variance if one is investigating a part ofthe domain of the electron variables which has a small probability of occupation-the “tail of the distribution.” In this case a computer-generated “history” of feasible length could have very few or no trials ending in that part of the domain; would be comparable to the expectation, for the corresponding estimators. This difficulty has been overcome by a special Monte Carlo procedure in which appropriate parts of the simulation are iterated many times.16

4. SCATTERING As was indicated in Part I, complexity in the model of the solid being simulated, and resulting complexity of the scattering function, is not a bar to the use of the Monte Carlo method. There is, evidently, always a way to implement (19),selecting A states with a correct distribution, in the limit, to represent the scattering function. It can be, however, desirable to choose the algorithm used with some care to avoid needlessly large computer time per trial. It is expedient to first explain the available methods more abstractly, in terms of the general problem of generating a prescribed distribution, before discussing the implementations of (19) in practice. We first consider a single continuous variable x in (a,b), with distribution function P ( x ) normalized in this interval.

Method 1 Equations (16) and (18) are an instance of this. If the relation R=

s.” P W d Y

(42)

can be conveniently inverted to give x(R), then we use the inverted formula to generate the x values from R values. Otherwise, one may use

Method 2 The rejection method,” in which two random numbers R,, R , (of course, independent of each other) are used to get one x value. First, a trial value R l6

A. Philips. Jr. and P. J. Price, Appl. Phys. Leu. 30, 528 (1977).

’’ J. von Neumann. in “Monte Carlo Method” (Proc. 1949 Conf.), National Bureau of Standards, Applied Mathematics Series. no. 12, pp. 36-38 (1951). This is reprinted in J. von Neumann, “Collected Works”(A. H. Taub, ed.). Vol. 5. pp. 768-770. Macmillan. New York, 1963.

4.


265

is computed from

x = a + (b - a)R,. Then, if the condition P(K) 2 CR,

is satisfied the K value is accepted and used as a member of the x population being generated; otherwise, the process is repeated, with two new random numbers; and so on. The constant c must be not less than the maximum value of P ( x in (a,b)). This method works no matter how complicated P ( x ) may be; but it can entail many cycles, on average, before an x value satisfying (44)is obtained (if P is too far from constant), and hence be wasteful of computer time. In that case, more efficiency may be obtainable by Method 3 With a suitable function F(R), set

K = F(R,)

(45)

P(K)F'(R,) 2 cR,,

(46)

and accept K as an x value if where F is the derivative and c is not less than the maximum value of FP. The idea is to choose a function F such that both F(R) and F'(R) are economically computed, and at the same time there are not too high a proportion of rejections. (If F’P were actually constant, we would be back to Method 1.) It can be shown that-with P ( x ) normalized in the domain of K given by (45)--the fraction of R,, R , pairs resulting in an accepted x value is l/c. An example of the use of (45) and (46) is presented in Appendix A. A variant of this procedure, which has been used to compute scattering angles,g is Method 4 Suppose C(x, y) is a function such that (a) the analog of (42)

R=

s' G ( z , y ) d z

(47)

can be inverted to compute x ( R ,y), and (b) the distribution function is P ( x ) = G(x, x).

(48)

The procedure is to predetermine a suitable value of y and compute R(R,,y) by (47),then treat this K as a trial value and apply Method 3. The nonrejection criterion (46) becomes G(K, K)/G(K, y ) 2 cR,.

If C(x, y) depends only weakly on y, then the rejection rate will be low.

(49)

266

PETER J . PRICE

If there is a single discrete, rather than continuous, variable, then the equivalent of Method 1 is always applicable: Method ID

The variable is n ; the probabilities are P,, with normalization

c Pn=l. M

n= 1

Then the n value is a step function of R, given by n- 1

n

P,,, RT is tested first. If it is satisfied, this procedure terminates immediately with a self-scattering as outcome. Otherwise, the physical scattering rates are successively compared with the residuals of RT (with or without preselection of an angle) as usual. The probability of computing them all and then obtaining a self scattering is only Az/T. It is evident that more complicated band structures and scattering functions can be handled, by an elaboration of the same kind of procedure. One might, for instance, introduce as an additional particle variable a weight w to be changed at each scattering by multiplying it by a function factored out of S( , ).

5 . DISORDERED SOLIDS This section presents an illustration of a somewhat different use of Monte Carlo simulation, to calculate a distribution resulting from a diffusion process. The illustration is provided by hot electrons in amorphous semiconductors, the diffusion “space” is the electron energy. In these material?’ mobile electron states are not describable by Bloch wavefunctions with wavevector as a good quantum number. The electron states have localized wavefunctions, which nevertheless can provide a current because they overlap. They may be characterized by energy E alone (and where appropriate, of course, position, and spin, etc.). Electronic properties of the system are then given by quantities depending on E : the density of states g ( E ) per unit energy and per unit volume of the substance; a mobility function p ( E ) ; a scattering function, for phonon absorption and emission, S ( E , , E 2 ) ; and a

’’ For the physics referred to here see Section 2.9 in Mott and Davis, “Electronic Processes in Non-Crystalline Materials.” Oxford Univ. Press, London and New York. 1971.

270

PETER J. PRICE

similar “oscillator strength” function giving the rate of transitions induced by radiation. The ohmic conductivity is then cr = J ~ gE( E )(-

d j / / d ~ekTp(E), )

(59)

where J(E) is the thermal-equilibrium function. Characteristically, in place of the forbidden gap of crystalline semiconductors, g ( E ) has a broad, shallow minimum; and the mobility function is zero over an energy range that includes this low-g region. If now the Fermi energy lies many times kT below the upper mobility edge (upper boundary of the immobility range) then f(E) in (59) becomes the Boltmann function (13), and we have a kind of nondegenerate semiconductor. A strong electric field, in these materials, has a direct effect on the electronic states themselves.22In addition it can be expected to cause a redistribution among the states, just as in crystalline semiconductors. The applicable “Boltzmann equation” for this redistribution is

where

9 = e k T 8 * p ( E ) 8. The quantity 9 (E) has the dimensions (energy)’/(time) and has the role in (60) of a diffusivity. The inelastic scattering processes that appear in the scattering term of (60) will contribute also to p and hence to the diffusion term. Since diffusion may be represented by a random-walk process in a Monte Carlo simulation, the diffusion term of (60)may be replaced by an expression just like the second term of (60) but with a scattering function S,(E1,E2) generating the random walk. The ensemble of Monte Carlo histories has a particle density per unit energy E equal to gf (appropriately normalized); and the particle scattering rate per unit final energy E2 is S ( E , , E2)g(E2). Whereas S in (60) satisfies (12), the surrogate scattering function must be symmetric: S,(E,,

E 2 ) = S,(E2,

El).

(62)

The maximum energy change ( E , - El( must of course be small compared to the energy range in which any of the quantities in (60)change appreciably. 22

See Section 7.8 in Mott and Davis, “Electronic Processes in Non-Crystalhe Materials.” Oxford Univ. Press, London and New York, 1971.

4. MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS 271 Then for the first term of (60) to be simulated we must have

If &(El, E d

= h(El)h(EZ)r(pl - E Z J ) ,

(64)

so” r(e)cZdc.

(65)

then we require

9/h2g

=

The steady state is obtained by the procedures of the preceding sections, with the scattering term only--no path term-in a trial. Since the scattering is one-dimensional a relatively simple scattering algorithm will apply. The distribution function below the mobility edge will tend to (13), and unless the minimum in g ( E ) is zero a lower bound on E must be imposed. Exploratory computations have been performed for the simplest case of interest, in which g ( E ) was taken as a constant go over the range of interest and an isotropic p ( E ) was assumed to be a step function equal to zero if E I E, and otherwise a constant p o . The function (64) was taken as

for E , > E,, with A& chosen appropriately small. When the true scattering is taken to be either elastic or given by (52),with scattering rates independent of E, the resulting distribution function has discontinuities at intervals of hwo. The discontinuities below the mobility edge E , may be eliminated by replacing the elastic scattering with the following model of inelastic acousticmode scattering in which the final state is given by an additional pseudorandom number: E , - Eb = kT In( (1

-

R)e-Aa‘kT+ ReAaIkT 3 )

(67)

where Aa is a small energy, representative of the effective acoustic-mode frequencies. With a fixed number of carriers almost all well below E,, the nonohmic conductivity and mobility are proportional to the value of f ( E ) just above E , divided by the limit of f(E)exp[(E - E , ) / k T ] as E decreases, below E,, to the Maxwell-Boltzmann regionz3 23

With this model (having constant density of states, step function mobility, etc.) the overall mobility decreases with increasing field, because of an inversion effect at the lower energies above E,. The interest in these unpublished computation results is limited by the evident divergence between the model and physical systems of present interest.

272

PETER J . PRICE

111. HOT ELECTRON PROPERTIES 6 . MAGNETIC FIELDEFFECTS

Addition of a magnetic field causes the current due to a given electric field to change direction (Hall effect) and change in magnitude (magnetoresistance), and it also disturbs the distribution function. For a not too large magnetic field strength (in particular, the cyclotron orbit frequency must remain small compared to kTlh or its hot electron equivalent) the quasiclassical description of the electron states still applies. Then the general formulation and results of Sections 2-4 apply unchanged. The paths are given by Eqs. (2) and (3), with both d and a terms in (5). The scattering function is unchanged. The estimator (28) still applies; so do (29) and the estimator (30).Equation (35)also still holds, provided that the vector mean free path is correctly defined.14 In general, the path motion in combined electric and magnetic fields is complicated, however, so that some form of numerical integration of it may be needed in a Monte Carlo simulation. For the energy function (24), the equation of motion (3) has a simple analytical solution, and for this case Monte Carlo calculation has been used successfully.' The path motion is given by dp'ldr

= w,

x p'

+ eb',

where

is the e.xtric field component in the magnetic fie... direction ant

P’ = P - Po, po = 8 x Am*cb/g w,

= ae/m*c.

(The camer charge is taken here as + e.)In the time interval s, the p' vector is displaced along the &?? direction by

bp, = e&s

(73)

and rotated about this direction through an angle d#' = w,s.

(74)

Thus a natural system would be cylindrical coordinates, with the magnetic field direction as axis and with azimuthal angle 4 referred to the (&??,po)

4.


273

plane, related to the system used in Part I1 by pr = psin 0,

(75)

p cos 0,

(76)

p4) =

and to the primed equivalent defined in (70) by p, sin 4 = p: sin @, prcos+ = p:coscb'

(77)

+ Po.

(78)

The alternating path and scattering steps, of successive trials, evidently require these or equivalent transformations for their computation-the scattering being computed as discussed in Section 4,with the azimuthal angle of the final state included. Obviously, a significant increase in the amount of computation per trial is entailed. Components of the drift velocity and other quantities of interest can be obtained as indicated above by estimators like those of Section 3. Since the procedures of this section and the following one may be combined, cyclotron resonance phenomena can be calculated. 7. TIMEDEPENDENCE

The application of Monte Carlo computation to electron transport in solids is not limited to steady states. Time-dependent states may be calculated also, when the frequencies involved times Planck's constant are small compared to appropriate energies characterizing the system so that the quasiclassical description remains valid. For the steady state, the time averages (14) are equivalent to ensemble averages, over the electrons of the system being simulated. If we were averaging over the simultaneous states of these electrons, in a calculation, then any time dependence in the system could be represented; the Monte Carlo realization of such a treatment is described in Part VI. The simulated history of a single electron does not directly represent an ensemble of electrons. Time-dependent phenomena in the system can still be represented and computed, however, so long as the time dependence is periodic. That is, &(t)= &(t

+ At) = &(t + 2 At) = . .

*

,

(79)

and similarly for any other driving force, and consequently

+ At) = ,f(t + 2 A t ) = . . . . (80) The states of the single-electron history at times t , t + Ar, r + 2At, . . . can f ( t ) = f(t

then be averaged over, as the equivalent of an ensemble of electrons all at time t. In practice, just as for the steady-state calculations, the quantities of

274

PETER J. PRICE

interest are obtained as sums over the A and B states of successive trials of the history. It is convenient to write the electric field strength satisfying (79)as b(t)= 8 0

+ 68,

(81)

where g o is constant and equal to the time average of 8,so that the time average of Sb is zero. (It is not necessary, in what follows, for these two components to be parallel, or even for the direction of the latter to be constant.) A basic form for the time-dependent part is 6 8 = 8, sin at,

(82)

w At = 2 ~ .

(83)

where

If the amplitude 8, is small enough, the deviation of the distribution function (80) from the steady state will be proportional to the amplitude, and also sinusoidal. The drift velocity in particular will be given by u = uo = u,,

+ csu

+ ul sin wr + u2 cos ux,

(84)

with UI = P

l b ) * b,,,

u2 = p*(o)* gc0,

(85)

defining the two components of the differential mobility, and uo = U(b0).

(86)

where u(8) is the steady-state dependence of drift velocity on field. For larger amplitudes 8,. harmonics of the drift velocity will become appreciable, and the fundamental will no longer be proportional to 8, ; the constant component uo will no longer be given by (86). For both linear and nonlinear regimes, however, the amplitudes of the fundamental will be given by u1 = 2u(t) sin wt = 2( l/At)

joA' u(t) sin wt dt,

u2 = 2u(t) cos tor = 2(l/Ar)

joA' u(t)coswtdr.

An alternative to (82) is

68 = 8,sq(t/At),

(87)

4.

MONTE CARLO CALCULATIONOF ELECTRON TRANSPORT I N SOLIDS

275

where sq( ) is the “square sine” function sq(.u)=

+ 1,

{-

1,

+,

0 < s(mod 1) I < .u(mod 1) SO,

-4

and similarly the “square cosine” is cq(u) = sq(.Y

+ a).

(90)

For At large enough, one would expect the response to be a square waveform (except for some distortion at the corners. which could be ignored), the analog of (84) with sq(r/At) and cq(t/Ar) replacing the sine and cosine. If in addition to At being large the amplitude R is small, (86)will be satisfied. In any case response amplitudes

u, = u(r)sq(r/At), u2 = u(r)cq(r/At), may be defined in a way analogous to (87).Then, when 6, is small enough for the linear regime to hold at all frequencies, we will have

Thus the mobilities defined by (92)give a measure of the sinusoidal mobilities at frequency l/Ar mixed with those of the odd harmonics. Both these schemes, for an electric field (79), (81) with square waveformz4 and with harmonic waveform,2s.26can be and have been implemented, by moderate elaborations of the procedures of Part 11. In the absence of a magnetic field, the elaboration of the path algorithm giving pf’” = pr’ + A(”)pby integration of (3) is trivial, so long as self-scattering is used with a constant I-. For the square form, the part of A(”)pdue to SF is given by the portion of the path duration s for which sq( ) in (88) is positive minus the portion for which it is negative. For the harmonic form, it is given by A(“’cos(wr). The scattering algorithm is unchanged, except insofar as one 24

l5

H. A. Hillbrand. J . P/?J~Y. C 5. 3491 (1972). P. A. Lebwohl. J. App/. P h j ~44, . 1744 (1973).(Comparable results for fi-Si are obtained by J. Zimmermann. Y . Leroy. and E. Constant. J. ,4pp/. P / ~ J .49.3278 c. (1978). usinga somewhat different procedure.)

276

PETER J . PRICE

may now wish to generate more details of the A state; for example, the component of v perpendicular to domay be newly needed because a transverse component of the differential mobility tensor is being calculated and it is consequently necessary to generate the v, values. Quantities of interest may be obtained, from the simulated history, by the appropriate elaborations of estimators described in Section 3. In particular, the generalization of (29),

holds25for periodic time dependence (80), and consequently (30a) becomes

when Q has the same periodicity: Q(P,

+ A t ) = Q(p, t).

(96)

Here tb means the value of t at the scattering event. Since in practice selfscattering with constant is used in these calculations, (95) is replaced by the analog of (30b):

where again N is the total number of trials in the history. For calculation of the differential mobilities, in particular, the left-hand side of (95) or (97) would be the time average of

as in (87), or for (91) similarly the time average of

The procedure indicated above may be used equally to calculate the nonlinear response to large amplitudes of Sb‘. For sufficiently large 6, in (82), one may calculate the Fourier components of 6u at the same frequency, the amplitudes of the harmonics of 6u, and the change in the static component uo, each as a function of gc0. The emphasis has been on the frequency dependence of the linear-response differential mobilities, which is of considerable theoretical and practical interest. By combining the procedures

4.


277

of this section with those of the previous one, it should be possible to calculate the cyclotron resonance behavior, at least for the electron energy function (24).One may, of course, obtain the time-dependent system average of any other electron variable as well as v, for example the energy E. Actual time dependences, (Q(p)) versus t, may be obtained on the basis of (94). They result from replacing the Q function in (95) or (97) by

(to give the value at time rl). They are computed in practice as histograms, by averaging separately in (95) or (97) the terms for which t,(modAt) falls in each of the time intervals of the histogram.25 Corresponding to a function Q(p) which is equal to one when the electron is in a particular band valley (or band), and zero otherwise, would be the calculation of the time dependence of the fraction of electrons in that valley. The estimators based on (94) are valid for any band structure and scattering function. A variety of responses of the system to a periodic driving force may be computed, as indicated, without any great additional difficulty in programming, though the computer time required for a reasonably accurate answer may be considerably more than for just the steady state in the same model solid. Generalization to several frequencies (fundamental and harmonics) in the driving field and in the fourier-analyzed response, and hence the study of nonlinear harmonic and conversion effects, are inherent in this scheme. (As indicated, any effect periodic in time can, at least in principle, be studied.) With a single drivingfield frequency, effects beyond linearity in the sinusoidal field amplitude, including nonlinear dependence of the resulting drift velocity component at that frequency, can be calculated. As illustration, such results are shown here for the semiconductor superlattice. In superlattice materials, a periodicity of chemical composition, on a larger scale than that of the crystal lattice, results in electron subbands (with a reciprocal-space periodicity on a correspondingly smaller scale than that of the zone scheme). Their hot electron properties, including the effect of a superposed sinusoidal electric field as in the foregoing discussion, have been calculated by Monte Carlo computation for a model intended to approximate the experimental materials.26 Figure 2 shows, for the same model, the response to a sinusoidal field in the absence of a static field. The fourier coefficients u 1 and u2 defined in (87) are plotted as curves “1” and “2,” respectively, against the coefficient 6 , in (82);all are in the reduced units of the P. J. Price, IBM J . 17, 39 (1973). The results shown in the present work are for Case I11 of the model. For a similar calculation of the steady state only, see: D. L. Andersen and E. J. Aas. J . Appl. Phys. 44, 3721 (1973).

278

PETER J . PRICE 0.21

-0.3 0

I

10

20

30

40

2.0

50

FIG.2. Response of a semiconductor superlattice to a large sinusoidal electric field. Fourier components of drift velocity (full curves, left scale; dashed curve, right scale) versus sinusoidal field amplitude, all in the reduced units. Details in the text.

and the reduced frequency is 1.0. The out-of-phase coefficient u2 shows an oscillatory dependence with declining amplitude, which can be interpreted in terms of a free (unscattered) particle. For the latter the wavevector k will oscillate sinusoidally, with amplitude (eb,/hw) and phase lag 7r/2. In this model the particle velocity is proportional to sin (ak),where LI is the superlattice constant. Then the time dependence of the velocity may be expanded by the Jacobi series sin(x sin 0) = 2

1

n = 1.3. 5 . . _

J,(x) sin(n0),

(101)

where the J,(x) are Bessel functions; and in particular the fundamental (n = 1) term for the velocity is (in reduced units) - 2[Jl(aeF,,/hw)]coswt.

(102)

In Fig. 2 the dashed line is the coefficient of cos(wt) in (102),plotted at reduced scale as noted. The fivefold reduction in amplitude of curve “2,” compared to (102),must contain the effect of the dispersion in phase among the particles of the ensemble represented by the simulation. An additional effect of the scattering is shown by the phase displacement of the oscillations. An additional static field, as in (81), results in a static term of the drift velocity (as well as modifying the sinusoidal terms, an effect that will not be displayed here). The curve 0 shows the static term due to a simultaneous static field of 1.0

4.


279

in reduced units. It too has an oscillatory dependence on the sinusoidal field amplitude, including small negative swings. The differential mobility corresponds to a perturbation of the steady-state Boltzmann equation by a small change of the field, and it can be discussed analytically in terms of the resulting inhomogeneous Boltzmann equation.’ For the case of a change in field Sb parallel to the initial field, it can be shown that the resulting first-order change bf satisfies j((4f)

- (ed - i / ? p ) 6 f = (sa/a)(,f, - fJ

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