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SEMICONDUCTORS AND SEMIMETALS VOLUME 14 Lasers, Junctions, Transport

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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

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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

CONTENTS OF PREVIOUS VOLUMES

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

CONTENTS OF PREVIOUS VOLUMES

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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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

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

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.

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

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

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

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

Ill-V SEMICONDUCTOR LASERS

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

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

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

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

,

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

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

-

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

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

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

__

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

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

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

<

Wavelength (IO’A)

FIG.20. Photoluminescence spectra (77‘K) obtained on .Y = 0.34 GaAs,-,P,:N+, nd = 1.8 x 101’/cm3. At low excitation levels (500 W/cm2), distinct peaks corresponding to the r band minimum and t o the N, N-trap level can be seen. At higher excitation levels, the relative intensity of the band-to-band transition increases. At 7 x lo4 W/cmz laser operation occurs at the peak of the N , band. The shaded portion represents 10 distinct and well-separated modes which are clearly seen in the original data. (After Craford and Holonyak.1z8)

relative to that of N,. At an excitation level of 7 x lo4 W/cm2, laser operation occurs clearly on the N, trap level. We note that the proximity of the r band edge to the N, levels (EN)exerts a band-structure-enhancement effect on the trap recombination oscillator strength, as is discussed later. The interaction between the conduction band edge and the N, states can be seen more clearly in photoluminescence spectra obtained on the same x = 0.34 GaAs,-,P, crystal as that of Fig. 20 after it has been made p-type by Zn diffusion. One of the purposes of the Zn diffusion and dopant is to permit laser operation of the crystal at a lower total electron population (than n-type crystal) so that transitions involving Nx states will be more dominant

1.

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

43

Energy (eV) 1.75 I

1.80

1.8s

1

I

1.90 I

1.9s I

h

8X

7.2

\

lo4 W/Em2/

7.0

6.8

6.6

6.4

Wavelength (103A)

FIG. 21, Photoluminescence spectra (77 K ) obtained on p-type s = 0.34 GaAs,-,P,:N+ :Zn, = 1.8 x 1017~cm'.A t an excitation of 500 W cm' the emission spectrum is that of the N, hand. As the excitation level is increased to 2 x lo4 W;cm', partial saturation of the N, band allows the electrons to occupy r - N states as shown by the bump in the emission at 6530 A. At an excitation level of 8 x 10' W 'cni'. laser operation occurs on N, transitions. (After Campbell r1 ul.l"o) ti,,

relative to band-to-band transitions. Also the absorption losses are reduced just as for the p-type sample of Fig. 15 relative to the n-type one of Fig. 14. Figure 21 shows the photoluminescence spectra (77°K) of the .Y = 0.34 GaAsl~,P,:Nf :Zn crystal of interest. The low-level photoluminescence spectrum of the crystal before it is converted t o p-type is shown dashed for comparison. The emission from transitions involving N, levels is seen to shift to lower energy by 29 meV in the p-type sample. This shift occurs because in direct p-type GaAs,-,P, electrons trapped in N-impurity potentials tend to recombine with holes bound to acceptors rather than decay as exciton^.'^^"^' At low excitation level (500 W/cm2), no emission from the band 13'

J. C. Campbell et a/.. J . Appl. P/ij.s. 45. 795 (1974). J. C. Campbell, N. Holonyak. Jr., A. B. Kunz. and M. G. Craford. Phvs. Rev. B 9 , 4314 (1974).

44

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

edge is observed because of the high concentration of N in the crystal (and its strong trapping capability). As the excitation level is increased, emission from the band edge (band-to-acceptor transition) can be seen. At an excitation level of 8 x lo4 W/cm2, the gain on N, transitions is sufficient for laser Operation. Emission from the band edge is no longer observed because electrons scatter to the N, states emptied by stimulated emission. This indicates that the scattering process is at least as fast as <0.5 nsec, as is shown by the carrier lifetime measurements described later. Emission involving the N-trap (N, or N,; cf., Fig. 19)in p-type GaAs,-,P,: N shifts to lower energy by 30 meV in direct crystal^.'^' This shift decreases considerably, however, for .Y > .Y, and is negligible much beyond the directindirect transition. This means that for indirect GaAs,-,P,: N: Zn recombination involving N tends to occur via the decay of excitons. Calculations131.1 3 2 show that the ratio of recombination involving electrons at N sites and holes bound to acceptors to that involving the decay of excitons is given by

-

1’ = (a,,/c43 exp( - 2R,/a,),

(9)

where aexand a, denote the screened Bohr radii for the excitonic hole and for the Zn-bound hole, respectively, and R , is the separation between the Zn and the N impurities. The ratio y decreases exponentially in the indirect composition range due to the increased binding energy of the Zn acceptor for x 2 x,. Accordingly, the Zn-N recombination transition becomes weaker in the indirect region (x > 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

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

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

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.

,

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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.

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

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.

PHOTOPUMPED 111-V SEMICONDUCTOR LASERS

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

Ill-V SEMICONDUCTOR LASERS

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

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60

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

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,

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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 (<< 0.1 nsec). With the absorption edge sufficiently above the laser energies, edge-to-edge modes do not quench rapidly since the recombination radiation tends not to be absorbed, even in the unexcited regions of the sample between the Fabry-Perot edges. In addition, as long as excess carriers are present the states emptied by stimulated emission are rapidly resupplied owing to the

-

-

14’

M . H. Lee, N. Holonyak, Jr., R. J. Nelson, W. 0.Groves, and D. L. Keune, J . Appl. Phys. 46,1290 (1975).

1.

PHOTOPUMPED Ill-V SEMICONDUCTOR LASERS

63

short scattering time in heavily doped GaAsIp,P,. This behavior differs from that in lightly doped crystals where the gain decreases due to decreasing EHL interactions. Thus, while stimulated emission increases the rate of radiative recombination, this increase is not necessarily observed in lifetime measurements as a sharp decrease of the type evident in Fig. 33. For the Ndoped GaAs lpxPxcrystal of Fig. 33, the decrease in the lifetime corresponding to the cavity modes can be seen far below the absorption edge. This is consistent with the relatively long scattering time (7,5 0.5 nsec) for the resupply of carriers to the N, states relative to the characteristic times of the stimulated emission process. VII. Conclusions

The wide variety of data presented above shows clearly the value of photopumping as a method of exciting semiconductor lasers. In fact, many of the results discussed here have not been duplicated by any other means of pumping, e.g., the laser operation of h p , G a , P at A 5500 A (green). In some cases, observations first made by means of photoexcitation, such as the laser operation of the G e or Si acceptor in GaAs, have been duplicated only later in double heterojunctions.' 43 Possessing low vapor pressures and being capable of supporting laser operation, these amphoteric impurities are convenient dopants in LPE crystal growth processes and are now widely used in various heterojunction devices. As might be expected, in any circumstance where it turns out to be a difficult task to build a p-n junction or heterojunction, as in In,-,Ga,P, photopumping is particularly useful as a means of generating excess carriers. In contrast to, for example, electron beam pumping, it is a simple and convenient excitation scheme and obviously will continue to be widely used, at least as an analytical tool. This is evident, for instance, from the unique character of the optical phase-shift lifetime data described above, where emission and lifetime spectra can be compared-including in the stimulated emission regime. Newer uses for photopumping are being found, and this promises to continue. Photoexcitation, with a gas laser, has recently been used to generate and examine dark line defects in double h e t e r o j ~ n c t i o n s . These ' ~ ~ ~ are ~~~ the same class of structures that (current-driven)operate cw at room temperature and are subject to reliability problems. In another area ofcurrent interest, photopumping has been used to first demonstrate that a semiconductor

-

lQ3 144

IQ5

M . B. Panish. 1. Hayashi, and S. Sumski. Appl. Phys. Lerr. 16, 326 (1970). W. D. Johnston, Jr. and B. I. Miller, Appl. Phys. L e u . 23, 192 (1973). P. Petroff, W . D. Johnston, Jr.. and R . L. Hartman. Appl. Phys. Letf. 25, 226 (1974).

64

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

laser can be operated in a distributed feedback ~0nfiguration.l~~ It is reasonable to assume that more such applications will be found, particularly since photoexcitation possesses the obvious advantage of not requiring metallic contacts on experimental samples. Unquestionably photopumping is a powerful and convenient method to excite laser operation in semiconductors. It is only fair to add, however, that, other than for purposes of analysis, photopumping ofa semiconductor cannot be regarded as serious practical competition to excess carrier injection via a p-n junction or heterojunction. The most effective means of photopumping, i.e., with a laser (say, a gas laser), generally involves the use of a relatively large, expensive “machine” to excite a small piece of material that, at least in principle, can be fabricated into a small, cheap injection laser. Add to this the fact that semiconductor laser materials are direct, highly absorptive, and cannot be pumped to very large depths (or very large volumes), and it is obvious that p-n junctions or heterojunctions represent the best use of this class of materials for lasers. Nevertheless, photopumping will continue to be a viable method for assessing material quality and for studying basic recombination and laser processes in semiconductors, including various heterojunction configurations. ACKNOWLEDGMENTS We are grateful to a number of our present and past colleagues for collaboration and help with work on the photoexcitation of semiconductors. Specifically. we wish to mention the support we have received in materials and ideas from M. G. Craford, W. 0. Groves, and D. L. Keune (Monsanto). We thank J. A. Rossi (Lincoln Laboratory, now at Monsanto) for helpful discussions on the photopumping and laser operation of semiconductors, and for the use of Figs. 11 and 28. Finally, we thank Yuri S. Moroz for expert technical contributions, and the IBM Corporation for providing M.H.L. with a post-doctoral fellowship. Further thanks are due to the National Science Foundation and to the Advanced Research Projects Agency for their support of much of the work described here.

‘46

Zh. I. Alferov et al., Fiz. Tekh. Poluprouodn. 8, 832 (1974) [English transl.: Sou. Phys.Semicond. 8, 541 (1974)].

SEMICONDUCTORS A N D SEMIMETALS. VOL . I4

CHAPTER 2

Heterojunction Laser Diodes Henry Kressel and Jeronw K . Butler

I . INTRODUCTION . . . . . . . . . . . . . I1. LASERDIODE STRUCTURES. . . . . . . . . . 1. Laser Topologj . . . . . . . . . . . . 2 . Vertical Geometry . . . . . . . . . . . 3. Carrier Confinement and Injected Carrier Utilization . . 111. WAVEPROPAGATION . . . . . . . . . . . . 4. General Con.siderations . . . . . . . . . . 5 . Near- and Far-Field Radiation Patterns . . . . . 6 . Optical Anomalies in Asymmetrical Heterojunction Lasers IV . RELATIONBETWEEN ELECTRICAL AND OPTICAL PROPERTIES . 7. Basic Considerations . . . . . . . . . . . 8 . Threshold Current Density and Diferential Quantuni Eficiency . . . . . . . . . . . . . . 9 . Temperature Dependence qf jth . . . . . . . . V . LASERDIODETECHNOLOGY . . . . . . . . . . 10. Epitu\-iul Growrh . . . . . . . . . . . . . I I . Efect of Dopants on the Emission Worelenyrh . . . . . . 12. Pertinent Properties o/.AI,Ga, -.As 13. Stripe-Contact Geometry . . . . . . . . . 14. Thermal Dissipation of Laser Diodes . . . . . . VI . HETEROJUNCTION LASERSOF ALLOYS OTHERTHANGaAs - N A s . . . . . . . . 15. Major Defects in Heteroepit~~ial Structures . . . . I6 . E.uperimenta1 Considerations . . . . . . . . 17. Laser Results . . . . . . . . . . . . . VII . LASERDIODERELIABILITY. . . . . . . . . . I8 . Catastrophic Degradation . . . . . . . . . 19. Gradual Degradation . . . . . . . . . . . VIII . DEVICES FOR SPECIAL APPLICATIONS . . . . . . . 20 . High Peak Power Luser Diodes . . . . . . . . . . . . . . . . . . . 21 . C W Laser Diodes 22 . Visible Emission Laser Diodes . . . . . . . I X . DISTRIBUTED-FEEDBACK LASERS . . . . . . . . 23 . Coupled Mode Analysis . . . . . . . . . . 24 . Solution of Coupled Modes . . . . . . . . . X . LASERMODULATION AND TRANSIENT EFFECTS . . . . . 25 . Introduction . . . . . . . . . . . . . 26 . The Rate Equations . . . . . . . . . . . 27 . Continuous Oscillations . . . . . . . . . .

66 68 68 69 72 75 75 78 100 104 105

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107 118 121 121 125 129 135 137 139 139 141 145 151 152 155 161 161 165 172 175

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65 Cop>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.

HETEROJUNCTION LASER DIODES

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

(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

HETEROJUNCTION LASER DIODES

-

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

HENRY KRESSEL AND JEROME K. BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HETEROJUNCTION LASER DIODES

-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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

(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

HENRY KRESSEL AND JEROME K . BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

'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

HETEROJUNCTION LASER DIODES

-

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

HETEKOJUNCTION LASER DIODES

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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

HETEROJUNCTION LASER DIODES

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.

HETEROJUNCTION LASER DIODES

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

,

>

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

-;-----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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL A N D JEROME K. BUTLER

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.

HETEROJUNCTION LASER DIODES

-

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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

HETEROJUNCTION LASER DIODES

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

(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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL A N D JEROME K . BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESEL AND JEROME K. BUTLER

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

HENRY KRESSEL AND JEROME K. BUTLER

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESEL AND JEROME K. BUTLER

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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.

HETEROJUNCTION LASER DIODES

135

In such thin regions the material remains strained; and AuJq, <, i.e.. misfit dislocation formation is prevented. 13. STRIPE-CONTACT GEOMETRY

The maximum optical power emission from a laser diode depends on the thickness of the waveguide region (i,e., the optical flux density), the width of the emitting region, and the pulse width (Section 18).Originally, cw operation was reported with broad-area diodes,' 24 but narrow stripe-contact diodes are now used exclusively for these room-temperature lasers for the following reasons :

(1) The radiation is emitted from a small area. This simplifies coupling of the radiation into low numerical aperture fibers which have typical diameters under 100 pm. (2) The operating current is low, usually well under 300 mA. (3) The thermal dissipation of the devices is improved because the heatgenerating emission region is partly imbedded in a larger passive semiconductor body. (4) The small active area improves the possibility of obtaining a low defect area conducive to long-term reliability. (5) The active region of the device can be isolated from open surfaces on its two major dimensions, which is essential for reliable operation (see Part VII). ( 6 ) Fundamental lateral mode operation can be obtained for narrow stripe devices. Several basic techniques for making stripe contacts are shown schematically in Fig. 44. The current path is defined as in (a) by using oxide masking to limit the metallic contact areal2'; by using selective diffusion into an ntype surface layer as in (b)' 2 6 ; by etching a narrow mesa as in ( c ) ' ~by ~ ;using (A1Ga)As to fill in the space between the mesas as in (d) to form a "buriedheterojunction"'28; or by using proton bombardment (or oxygen implants)'28a as in (e) to form high resistivity regions in selective areas.lZ9 124

125

lZ6

lZ7 lZ8

lZ9

1. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, Appl. Plzn. Lett. 17. 109(1970). J . C. Dyment, Appl. Phys. Letr. 10. 84 (1967);J . C. Dyment and L. A. D'Asaro. ihicl.. 1 1 .

292 (1967). H . Yonezu, 1. Sakuma, K. Kobayashi, T. Kamejima, M. Ueno. and Y . Nannichi, Jpn. J . Appl. Phys. 12, 1585 (1973). T. Tsukada. H. Nakashima, J . Umeda, and D. Nakada, Appl. Phys. Letr. 20, 344 (1972). T. Tsukada, J. Appl. Phys. 45,4899 (1974). J . M . Blum, J. C. McGroddy, P. G . McMullin, K. K. Shih, A. W. Smith, and J. F. Ziegler, IEEE J . Quantum Elecrron. QE-11, 413 (1975). J . C. Dyment, L. A. D'Asaro, J . C. North. B. I. Miller. and J. E. Ripper, Proc. IEEE60. 726 (1972).

136

HENRY KRESSEL AND JEROhlE K. BUTLER E TA L

OiFFUSED P-REGION

@ ,

rYETAL

SiOe

...

......

GaAI I Y I

I IN1

GaA8 tNorP1

GoAs(N1

(el

FIG.44. Various techniques for forming stripe-contact laser diode structures: (a) oxideisolated stripe-contact; (b) selective Zn diffused contact area into n-type surface layer; (c) mesastripe structure; (d) buried- eterojunction stripe-contact structure; e) isolation of the stripecontact region by lateral high-resistivityregions produced by ion implantation.

(Instead of using an oxide for isolation, stripe diodes have also been reported using an n-type A1,.,Ga0.,As surface layer in which a contact area was etched to reach the p-type regi~n.)'~' Laser diode technology is evolving and a definitivechoice of the best stripe technique is premature, particularly since the laser reliability is affected by the fabrication technique, An important difference between the various stripe structures is in their lateral current and radiation confinement which determines the eflectiue diode area for a given stripe area. A planar stripe lacking lateral blocking regions at the recombination region edges cannot be made arbitrarily narrow because the current and radiation spread outside the contact The current spreading is somewhat limited with proton bombardment (or selective diffusion) because of the high resistance of the I3O 13'

'33

K. Itoh, M. Inoue, and 1. Teramoto, IEEE J . Quunrurn Electron. Q E l l , 421 (1975). M . Cross and M. J. Adams, Solid-Stare Electron. 15,919 (1972). 8. W. Hakki, J. Appl. Phys. 44, 5021 (1973). B. W. Hakki, J. Appl. Phys. 46,2723 (1975).

2.

HETEROJUNCTION LASER DIODES

137

material surrounding the active area defined by the stripe. Nevertheless, radiation loss remains and there is still carrier loss by lateral diffusion within the recombination region in devices lacking potential barriers to laterally confine the carriers.' 32 The maximum lateral confinement is obtained with the deep mesa or the buried-heterojunction structure. Because of the simpler construction, however, planar stripe lasers are most popular. A planar oxide-isolated, selective diffused, or proton bombarded, stripe diode with a width of 10-15 pm offers reasonable operating current and adequate cw optical power for many applications. With these stripe widths, however, the threshold current density (through the contact) is typically 2-3 times higher than that of broad-area diodes fabricated from the same material because of the above spreading effects. 14. THERMAL DISSIPATION OF LASERDIODES Neglecting facet damage, the parameters affecting the maximum power emission from a laser diode are the temperature dependence of J,h and the thermal and electrical resistance. For heterojunction lasers (and many homojunction ones), the temperature-dependence of the threshold current has commonly the empirical form134 Ilh(

T)

eTITo,

(26)

where T o typically varies from 50 to 170"K, depending on the dopant concentration in the recombination region and the bandgap energy steps at the heterojunctions. A large step reduces the temperature dependence of J,h (Part IV). The electrical resistance of the diode is minimized by using highly doped ptype layers ( P > 1019 ~ m - under ~ ) the ohmic contact and thinning the various layers in series with the active region. For a typical 13-pm planar stripe-contact laser 400 pm long, a resistance of 0.5-1 R is considered reasonable. The thermal resistance of the diode can be calculated from the horizontal and vertical ge0met1-y.'~~ A rough estimate of the thermal resistance of the various regions of the laser can be made for a planar stripe-contact doubleheterojunction diode on the basis of the schematic shown in Fig. 45. The resistance of the AI,,,Gao,,As and GaAs contact regions, and of the indium solder used to mount the diode p-side down on a copper heat sink are important. Table VIII shows the calculated thermal resistance of each region and the temperature rise for a power dissipation of 0.5 W. With the 1-pm 134

J . 1. Pankove, IEEE J . Quantum Electron. Q E 4 , I19 (1968). W.B.Joyce and R. W. Dixon. J . A p p / . fhw.46. 855 (1975).

138

HENRY KRESSEL AND JEROME K . BUTLER

.-s+

RECOMBINATION REG ION, ,TEMPERATURE T

-.

GaAs

INDIUM SOLDER

COPPER HEAT SINK

FIG.45. Schematic cross section of stripe-contact laser diode. with oxide isolation, used to approx:imate the thermal resistance. The stripe width is S,.

TABLE VIII THERMAL RESISTANCE OF LASERSTRUCTURE OF FIGURE 45 a

Layer or Region Type A1,.,Ga0.,As GaAs Indium Copper (heat sink)

Thermal Conductivity Ki ( W/cm' C)

Layer Thickness (wn)

Thermal Resistance Rth ("CjW)

AT ("C)

1 1 I

14.8' 3.9' 2.4' 9.6d 30.7

1.4 1.2 1.2 4.8 15.4

0.13 0.5 0.8 4 Total

X'

~

Fabry-Perot cavity length, L = 400pm; stripe width, S, = 13 pm. Power dissipation, P = 0.5 W. Rth = di/KiA, where A is the diode active source area, LS,. R,, = In(4L/S,)/(aKcUL). where L >> S,.

thick regions indicated, the thermal resistance of the Al,,,Ga,,,As layer is about half of the total value because of its low thermal conductivity. The dissipation within the cw diode is about equal to the input power because the power efficiency near threshold is only a few percent. Hence, for a threshold current = 0.3 A, a bandgap energy E, 2 1.4 eV, and a series resistance of R , = 1 Q, the power input Pi = I&, + ZLRs z 0.51 W. Assuming a power conversion efficiency of 2%, the dissipated power is 0.5 W. The temperature rise of the active region above the heat sink temperature is calculated to be 15.4"C.This is in excess of the measured value on actual devices' 36 because lateral thermal spreading was neglected in the calculation. The maximum output from a pulsed laser occurs at a duty cycle determined by the thermal and electrical resistance, the threshold current, and the 136

H. Kressel and I. Ladany, unpublished

2.

HETEROJUNCTION LASER DIODES

139

temperature dependence of Jlh. Because the input power increases the temperature of the active region, lasing will be quenched by too high a duty cycle or diode current. Keyesl3'.I3' obtained the following condition for cw operation when the power dissipation due to the diode series resistance is small compared to the power dissipation in the junction itself (as was the case in the illustration considered above), and Eq. (26) is appropriate; IlhE,Rlh/eT,< 0.37.

(27)

is the Here we assume that the junction built-in potential equals E,/e and threshold current at the junction temperature. To illustrate the above expression, we calculate the highest possible value for which cw operation is possible using our illustrative structure of Fig. 45. Assuming T o = 70'K,

This threshold estimate exceeds the practical limit because it neglects ohmic heating. VI. Heterojunction Lasers of Alloys Other Than GaAs-A1As

15. MAJORDEFECTS I N HETEROEPITAXIAL STRUCTURES Major defect-related problems associated with heterojunction laser structures arise from the interfacial lattice-parameter mismatch. We first briefly review the simple theory and then discuss experimental results. Consider the joining of two simple cubic structure materials with dissimilar lattice parameters a; and a: (Fig. 46). Assuming complete plastic deformation to relieve the misfit strain, the lattice misfit gives rise to a set of dislocations that lie in the interface shown in Fig. 46, whose density pl is pt = ( a ; - uh)/ubab:2 Aa,/a;

cm-'.

(28)

Here, a, is the mean lattice parameter for the two cubic structures, and Aa, = ub: - a;. In the case of the sphalerite lattice structure, the Burgers vector differs from the lattice parameter. For small differences between a; and a:, an adequate linear dislocation density estimate for (100) planes is obtained using the expression (assuming a (1 10) type Burgers vector) p1 13’

= $(Aa,/a;)

R. W. Keyes, IBM J . Res. Deo. 9, 303 (1965). R.W. Keyes. IBM J . Res. Der. 15,401 (1971)

cm-'.

(29)

140

HENRY KRESSEL AND JEROME K. BUTLER

L2

L,

a,2/Ao,

FIG.46. Schematics showing the formation of an edge misfit dislocation in joining simple cubic crystals A and B with different lattice constants: (a) Separate crystals; (b) formation of edge dislocation when crystals are joined; (c) formation of dislocations at edge of crystal L , x L , . The distance between dislocations is L,.

The density of recombination centers N, introduced by dislocations lying in the interfacial plane can be calculated by assuming that each atom terminating an edge dislocation constitutes a recombination center. The calculated value of N,,depends on the crystal structure and orientation. For a sphalerite structure on the (100) interfacial plane,'39

N,, x 8Aao/aG and, for (111) plane, N,, x ( 8 / f i ) AaolaG-

The recombination velocity S at the heterojunction interface can be estimated by s = UthNssot (32a) where 0th is the thermal electron velocity and CT is the capture cross section for the center. The value of CT is uncertain for dislocation-induced centers, but CT g 10- cm2 is the probable order of magnitude. Hence, the predicted dependence of S on misfit strain is for the (100) plane, S g (2.5 x 1O7)Aao/a0 cm/sec. lJ9

D. B. Holt, J . Phys. Chem. Solids 27, 1053 (1966).

(32b)

2.

141

HETEROJUNCTION LASER DIODES

TABLE 1X SOMEPARAMETERS OF INTEREST I N THESYNTHESIS OF TERNARY ALLOYS ~

~

~~

~~

Lattice ParameteP Alloy System"

a0

b

a

Melting Temperature ("K)

(4

E , (300"Kr

a

b

Difference

a

b

a

b

6.479 6.058 5.869 5.661 6.057 5.451 6.095 6.057 6.479 5.653 6.057 6.095 6.057 6.479 6.095 6.135 6.135 5.451

6.095 5.653 5.45 I 5.653 5.870 5.45 1 6.135 5.661 6.135 5.45 1 5.451 5.653 6.479 5.869 5.45 1 5.451 5.661 5.661

6.1 6.9 7.3 0.14 3.2 0.01 0.65 6.7 4.7 3.6 10.5 7.5 6.8 10 11.1 11.8 8 3.8

803 1210 1343 2013 1210 2823 985 1210 803 1511 1343 985 1210 803 985 1323 1323 2823

985 1511 1738 1511 1343 1738 1323 2013 1323 1738 2823 1511 803 1343 1738 2823 2013 2013

0.17d 0.35 1.34d 2.16 0.36' 2.4 0.73' 0.35' 0.17' 1.42' 1.34d 0.73' 0.35' 0.17' 0.73' 1.65 1.65 2.4

0.73' 1.42d 2.26 1.42" 1.34d 2.26 1.65 2.16 1.65 2.26 2.4 1.42' 0.17d 1.34d 2.26 2.4 2.16 2.16

~

lnSb InAs 1nP AlAs lnAs

AIP GaSb InAs lnSb GaAs InP GaSb InAs lnSb GaSb AlSb AlSb AIP

GaSb GaAs GaP GaAs InP GaP AlSb AlAs AlSb GaP AIP GaAs InSb InP GaP AIP AlAs AlAs

-

Note: Bulk grown single crystal substrates are available of InSb, InP. GaP.GaSb, and GaAs. The lattice parameters generally vary nearly linearly with composition in the ternary alloys. However, this is rarely the case for the bandgap energy variation. The bandgap energy values are within 10 meV. 'Denotes direct bandgap transition material.

As we will see in the next section, this estimate is in reasonable agreement with experiment.

16. EXPERIMENTAL CONSIDERATIONS Many semiconductors are candidates for heterojunction structures,140 their utility being limited by interfacial defects resulting from lattice mismatch and availability of compatible synthesistechniques. In recent years, the 111-V compounds have received the greatest attention for practical optoelectronic devices. Table IX lists the 18 possible combinations of 111-V compounds that can be alloyed to form ternary alloys which span the bandgap range of 0.17 eV (for InSb) to 2.4 eV (for Alp). The AlAs-GaAs and A review of optoelectronic heterojunction devices is given by H. Kressel, J. Electron. Mafer.4, 1081 (1975).

142

HENRY KRESSEL AND JEROME K . BUTLER

6.2

-

-

6.0 * s p 5.9 . 6.1

'C

I-

0 u

w

0

5.8

.

I-

5 5.1 . IL

8 5.6 . 0

5.5

Eg - BAND6AP ENERGY ( r V ) I

"

'

a

.

1

I

a

I

4.0 3.0 2.5 2 0 1.6 1.4 1.2 1.1 1.0 0.9 0.8 0.75 0.7 065 0.6 0.55 DIODE WAVELENGTH (PIN)

FIG.47a. Lattice constant versus bandgap energy (and diode emission wavelength) for some Ill-V compounds. indirect bandgap; shaded region, InGaAsP field. Lattice-matching combinations are determined by drawing horizontal lines. The InGaAsP quaternary alloys are particularly useful because of their wide bandgap span.

-.

Alp-GaP alloys have the smallest dependence of lattice parameter on alloy composition, followed by A1As-GaSb and Gap-GaAs alloys. The AlAsGaAs alloys are presently the most important for heterojunction devices. GaSb-A1Sb alloys have only a limited range of direct-bandgap compositions, and AlP-GaP alloys cover only a very small range of bandgaps which are always indirect. Heterojunction laser structures can be made using lattice-matched combinations of ternary and quaternary alloys. Figure 47a shows the lattice parameter versus bandgap energy of various 111-V alloys. For example, the lattice parameter of InGao.49Po,5,matches that of GaAs (at the usual growth temperatures) giving heterojunctions with E, z 1.42 eV on one side and 1.9 eV on the other. Figure 47b shows transmission electron micrographs of the interface of InxGal,P layers grown by VPE on GaAs with no lattice mismatch (x = 0.49)and a lattice mismatch of 0.3% when x = O.45.l4l There are no observable misfit dislocations if the lattice is matched at the growth temperature. The ability to produce lattice-matched heterojunctions by the use of quaternary alloys greatly extends the useful device range of 111-V alloys. As shown in Fig. 47a, the InGaAsP alloys are particularly interesting in this respect because a wide bandgap energy range can be encompassed. For

-

I4l

G. H. Olsen, unpublished

2.

.. . -

x=.49

(a 1

GaAs

.

.

,

143

HETEROJUNCTION LASER DIODES

I

k-lprn-4

x s.45 (b 1

FIG. 47b. Transmission electron micrograph - . of a lattice-matched In, .asGan5,P-GaAs interface (a),compared to (b). which shows a mismatched interface with Au,, t i o = - 3 x 10made by growing by vapor phase epitaxy In 45Gao.55P on GaAs. (G. H. Olsen. unpublished.) ~

example, InGaAsP alloys can be lattice matched to InP to produce devices in the 1-pm spectral emission range. On the other end of the spectrum, we see that InGaAsP alloys can be matched to Ga(AsP) to produce visible emission devices. It is a common observation that some of the lattice misfit is accommodated by dislocation formation and some by strain. The distribution between these two effects depends on the growth temperature, thickness of the epitaxial layer, growth rate, cooling rate, and plasticity of the material, i.e., the ability of dislocation sources to generate new dislocations to relieve the strain. In general, the remaining (i.e., unrelieved by dislocations) strain is well under 0.17; for epitaxial layers more than several micrometers thick. However, very thin layers (<< 1 pm) can theoretically remain highly strained, on the order of 19;, without the formation of misfit dislocation^'^^^'^^ If the layer is sufficiently thin, the stored energy is low compared to the energy needed to form strain-relieving misfit dislocations. and the layer, therefore, remains elastically strained. For layer thicknesses of about 1 pm or less, an approximate theoretical expression for the maximum layer thickness h, grown without misfit dislocations on a substrate with lattice parameter difference Auo is h, 2 a ~ / 2 J ? ( A a 0 ) . Even if the lattice parameter is matched at one temperature, dissimilar thermal expansion coefficients (see Table VII for some values of interest) can 14’ 143

F. C . Frank and J. H. van der Merwe. Proc. ROJ. Soc. London A198. 205. 216 (1949). J. W. Mathews in “Epitaxial Growth“ ( J . W . Mathews. ed.). p. 562. Academic Press. New York, 1975.

144

HENRY KRESSEL A N D JEROME K. BUTLER

be troublesome. If the lattice is matched at room temperature, the mismatch at the growth temperature can result in misfit dislocations. On the other hand, if the lattice is matched at the growth temperature, the crystal will remain strained if insufficient misfit dislocations form as the crystal cools to room temperature. Fracture can even occur in extreme cases. The strain can be conveniently determined by x rays on the basis of the wafer bowing.’44 We have so far discussed interfacial misfit dislocations which lie in the junction plane, but some dislocations also turn upward into the epitaxial layer.’45 Some of these “inclined dislocations bend over in the plane of the second However, because of the dislocation network formation at the interface, device properties which rely on the heterojunction interface itself may not be improved. An important experimental observation is that the dislocation density in an epitaxial layer resulting from abrupt lattice parameter changes depends on whether the layer being deposited has a smaller or larger lattice parameter than the substrate. In VPE, it is found14’ that dislocations extending from the substrate tend to bend over at the interface if the epitaxial layer has a higher lattice parameter; i.e., it is in compression. However, dislocations do propagate into the layer if the lattice parameter of the layer is lower than in the substrate; i.e., the layer is in tension. A reduction in the diffusion length of minority carriers caused by inclined dislocations in regions adjoining a heterojunction represents a serious limitation in some devices. A good correlation has been established in GaAs between the average dislocation spacing and the diffusion length of minority carriers;’49 and a region around each dislocation core was found to be nonradiative.’ 5 0 In the simplest approximation, the diffusion length is limited to half the average distance between dislocations. For example, with an inclined dislocation density of lo6 cm- threading through a layer, half the average distance between dislocations (if uniformly distributed) is z 5 pm. Such a value may barely affect the minority carrier lifetime, because the bulklimited diffusionlength in highly doped GaAs material is only a few micrometers. Unfortunately, the inclined dislocation density can easily exceed 10’ cn- , which limits the diffusion length to only 1 pm.

-

G. A. Rozgonyi and T. J. Ciesielka, Rev. Sci. fnstrum. 44, 1053 (1973). M. S. Abrahams, L. R. Weisberg, C. J. Buiocchi, and J. Blanc, J . Mater. Sci. 4,223 (1969). ‘46 R. H. Saul, J . Elecrrochem. SOC.118, 793 (1973). 14 M. Ettenberg, S. H . McFarlane, and S. L. Gilbert, in “Gallium Arsenide and Related Compounds” (Proc. f n t . Symp., 4th, Boulder, 1972). p. 23. Inst. Phys. and Phys. Soc., London, 1973. I 4 8 G. H. Olsen, M. S. Abrahams, C. J. Buiocchi, and T. J . Zamerowski J . Appl. Phys. 46, 144 145

1643 (1975). ‘49

Is*

M. Ettenberg, J . Appl. Phys. 45,901 (1974). W. Heinkl and H. J. Queisser, Phys. Reo. Leu. 33, 1082 (1974).

2.

HETEROJUNCTION LASER DIODES

145

The preparation of heterostructures by vapor phase or molecular beam epitaxy offers one important advantage over preparation by LPE-greater control of gradients and more flexibility for growing dissimilar materials. For example, vapor phase epitaxy has been used to grow (1nGa)As lasers on GaAs substrates. The alloy composition was gradually changed between the substrate and the p-n junction region in order to reduce the dislocation density in the active region of the diode. The next section discusses these and other structures which are more complicated than the (A1Ga)Asones. For dissimilar alloys, the problem in LPE is that, unless the solution is saturated with the constituents on the substrate, some dissolution of the substrate material will occur on first contact with the substrate. Even if the growth occurs under nonequilibrium conditions (for example, by rapid cooling rate), the first material to crystallize (i.e., the new epitaxial layersubstrate interface) may include more constituents than intended. There are few studies correlating the lattice mismatch at III-V compound heterojunctions to the surface recombination velocity. Studies' 5' of(1nGa)PGaAs interfaces show that Eq. (32b) is experimentally observed. Thus, when Aao/ao z l%, S 2 8 x lo5 cm/sec. With regard to AI,Ga,_,As-GaAs interfaces, S 6 5 x lo3 cm/sec for x x 0.2.'52 Here, misfit dislocations are not responsible for the interfacial recombination, but the origin of the nonradiative centers is uncertain. 17. LASERRFSULTS

Major spectral areas of interest include the 1-1.2-pm, far-infrared, and the visible emission ranges. (Visible emission lasers are discussed in greater detail in Section 22.) Because of the interest in 1- 1.2-pm spectral range emitters for fiber optical communications, there has been substantial work in producing heterojunctions for this purpose. Three III-V alloy systems are illustrative of possibilities in this range: ( 1 ) (ZnGa)P/(ZnGa)Ason GaAs substrates. Here lattice matching at the heterojunctions is possible, but the epitaxial layers are mismatched to the substrate by about 17". (2) InGaAsP on InP substrures. This is the only combination which produces a lattice-matched heterojunction with no substrate lattice mismatch problems, and it is therefore the most desirable one. (3) AlGaAsSb/Ga(AsSb) on GuAs substrates. The lattice mismatch to the substrate is l"/o.However, the Al/Ga substitution is convenient in LPE growth and essentially lattice-matched heterojunctions are produced.

-

M . Ettenberg and G . H. Olsen. J . Appl. fhys. 48,4275 (1977). M . Ettenberg and H. Kressel. .I. App/. Phys. 47, 1538 (1976).

TABLE X HETEROJUNCTION LASERDIODESUSINGALLOYS OTHERTHAN AIAs-GaAs

Growth Technique

J,,

Materials GaAsP-GaAs

Laser Type SH

(A/cm')

1.4 x 105

1 to -10

GaAsSb-AIGaAsSb InGao,,Po,,-GaAs InGaP-InGa As InGaAsP-GaAsP AIGaAsP-GaAsP AIGaAsP-GaAsP PbSnTe-PbTe PbSnTe-PbTe PbSnTe-PbTe InGao.sPo.sA1o..wGao,,zAs PbS0.72SeO.28Pbo.78Seo.22 1n0.66Ga0.12As0,~3P0.,,-lnP

Ga Asp-InCaP

DH LOC LOC SH SH DH SH DH DH SH SH DH LOC DH

2.5 x 103

--2.8200

x lo3

5.5 x 104 3.37 103

-

0.9

3w

VPE

0.98 - 0.89 1.075

5

0.63 -0.81 0.845 8.9 -8.35 8.4

300 300 300 77 300 300 77 77 77

LPE VPE VPE LPE + VPE LPE LPE + VPE Evaporation LPE LPE

1

0.638

77

LPE

Schul and Mischel'"k

-3 0.6 -4

4.78 1.1 0.675 0.703

12 300 273 300

Evaporation LPE VPE VPE

Sleger et a/.1s21 Hsieh' 5 2 m Ladany er ~ 1 1 . ' ~ ~ " Kressel et a1.1s2"

8

lo4 9 103 8 x lo3 1.49 104 6.2 x 104 6.6 x lo4 1.6 x lo4 780 4.2 x 103 1.3 x 103

References

0.8 2.5 1.6 1 1.24 1.25 -

9

0.2

-

~

Distance between the two heterojunctions: d, for DH structure, (1- for LOC structure.

Craford et a1.152n Sugiyama and Saito's2b Nuese et d . ' s 2 c Nuese and Olsen'52d Coleman er al.lsze Burnham et a1.1s2r Burnham er a1.1s2* Walpole et a/.152h Groves ei 01.' Tomasetta and Fonstad'"'

2.

HETEROJUNCTION LASER DIODES

147

A second major group of heterojunction structures consists of Pb-salt lasers emitting in thefar infrared. A third area of interest is in extending laser diode operation into the visible portion of the spectrum beyond the range of (A1Ga)Asdevices. The use of (InGa)P/Ga(AsP), InGaAsP/Ga(AsP), and AlGaAsP/Ga(AsP) structures offers the possibility of producing red-yellow light-emitting lasers. However, the lattice mismatch to the GaAs substrate on which the Ga(AsP) is grown is again quite severe. Although LPE or VPE are most widely used for the 111-V alloys, the above structures could also be produced by molecular beam epitaxy. The choice of the best technology can only be determined on the basis of the device quality and reliability. In view of the sensitivity of the laser reliability to defect density, extensive testing is needed to establish the comparative merits of various material synthesis technologies. a. 111-V Compound Lasers

The successful heterojunction lasers have a close lattice match (Au,/a, 5 0.3",). The degree of success is defined by the achieved reduction in Jthcompared to homojunction lasers operating at the same temperature and emission wavelength. Representative laser data, from a variety of sources, are shown in Table X. ______

~

M. G. Craford. W. 0. Groves, and M. J. Fox, J. Electrochem. SOC.118, 355 (1971). The abrupt heterojunction barrier was formed using GaAs,.,P,,,-GaAs (AE8 : 0.1 eV). Graded heterojunctions with 4 J oP/pm were also studied with similar device results. K. Sugiyama and H. Saito, Jpn. J . Appl. Phys. 11, 1057 (1972). C. J. Nuese, M. Ettenberg, and G. H. Olsen, Appl.. Phys. Lett. 25. 612 (1974). C. J . Nuese and G. Olsen, Appl. Phys. Lett. 26, 528 (1975). The heterojunction barrier consisted of the near-lattice matching compositions In,,,,Ga,,,,P-In,,,,Ga,,~~As. The highest Aa,ia,. value used was 0.2",,. ’’51 J. J. Coleman, W. R. Hitchens, N. Holonyak, Jr., M. J. Ludowise, W.0. Groves, and D. L. Keune, Appl. Phys. Lett. 25,725 (1974). 1 5 2 f R. D. Burnham, N . Holonyak, Jr., and D. R. Scifres, Appl. Phys. Lett. 17, 455 (1970). 1 5 2 y R. D. Burnham el a/., Appl. Phys. Lett. 19, 25 (1971). I S Z h J . N. Walpole, A. R . Calawa, R. W. Ralston. T. C. Harman, and J. P. McVittie, Appl. Phys. Lerr. 23.620( 1973).The structure used consisted of Pbo,88Sn,,,,Te-PbTe(Aa,/a,2 0.24";). The J,, values obtained were reported to be 113 of those measured for homojunction lasers. S. H. Groves, K. W. Nill, and A. J . Straws, Appl. Phys. Lett. 25, 331 (1974). 1 5 2 1 L. R. Tomasetta and C. Fonstad, Appl. Phys. Lett. 25, 440 (1974); see also 24,567 (1974). A high dislocation density (108-10' cm-') in substrate was noted. 15" G. Schul and P. Mischel, Appl. Phys. Lett. 26, 394 (1975). "" K. G. Sleger, G. F. McLane, U. Strom, S. G. Bishop, and D. Mitchell, J . Appl. Phys. 45, 5069 (1974);Aa,ia, 2 O.2"k at the n-n heterojunction. 1 5 2 m J. J. Hsieh, Appl. Phys. Lett. 28, 283 (1976). "’51 I . Ladany, H. Kressel and C. J. Nuese, unpublished. "" H. Kressel, G. H. Olsen. and C. J. Nuese, A p p / . Phys. Lett. 30,249 (1977). 152a

-

148

HENRY KRESSEL AND JEROME K. BUTLER

Some structures were constructed as alternative to (AlGa)As/GaAslasers. For example, the GaAs,.,P,. ,/GaAs single-heterojunction laser has been studied in detail' 5 3 using both an abrupt and a slightly graded heterojunction. The lattice mismatch here is about 0.36%.The heterojunction lasers are, at best, only comparable to homojunctions, indicating serious defect-related problems in the addition of the heterojunction. The In,.,Ga,, ,P/GaAs/In,, &a,, 5PLOC heterojunction diodes' 54 have a close lattice match and their characteristics (except for poorer reliability) are similar to those of an (AIGa)As/GaAs LOC with equally spaced heterojunctions. These diodes were prepared by VPE, as were DH lasers with J t h z 1000-2000 A / c ~ ' . ' ~ ~ " In the 1-1.2-pm spectral range, lasers have been fabricated using various combinations. The first devices in this range were LOC laser diodes with In,,68Ga,~,~P/In,,16Gao~84As on GaAs and showed a factor of about 5 reduction in Jthat room temperature compared to homojunctions emitting at 1 z 1 Figure 48 shows the cross section of this LOC laser (made by VPE) with the p-n junction between the two lattice-matched heterojunctions. Figure 49 shows the temperature dependence of J f h for the (InGa)P/(InGa)As lasers (with the lattice mismatch indicated). Note that improved performance compared to the homojunction at room temperature was observed with Aa,/a, z 0.2%. Fp-In.l,Ga,a4As - _

-

"cap"

J

P- 1n68Ga32P -*z$nlGGawAs LASER CAVITY

I_-= c

FIG.48. (InGa)As/(lnGa)P LOC laser structure designed for lasing at 1.075 pn. (After Nuese and Olsen.'55)

It is interesting to analyze the (InGa)As/(InGa)P LOC lasers with lattice parameter mismatch Aa,/a, = 0.2%, having the threshold characteristics shown in Fig. 49. It should be noted that J , h at room temperature is 2-3 times higher than that of comparable-dimension GaAs LOC lasers, and factors other than nonradiative interfacial recombination could well be responsible. In the following, we estimate the interfacial surface recombination velocity M . G . Craford, W. 0. Groves, and M. J. Fox, J . Electrochem. SOC.118, 355 (1971). C. J. Nuese, M. Ettenberg, and G. H. Olsen, Appl. Phys. Lett. 25,612 (1974). lS4' C. J. Nuese, M. Ettenberg, and G. H. Olsen, Appl. Phys. Lett. 29, 54 (1976). C. J . Nuese and G. H. Olsen, Appl. Phys. Lett. 26, 528 (1975).

ls3

54

2.

149

HETEROJUNCTION LASER DIODES

--

.215 .312 0.01 ,176 ,343 0.2 0 ,145 ,432 0.2

0

5 -

-

1.145 1.075 1.025

2

I

I

1

1

I

I

S needed to fully account for the high laser threshold, by assuming y* = 0.4 (see Part 11).The p-type recombination region is 0.7 pm thick, and we assume L, = 3 pm, D = 50 cm2/sec, and d , / L , = 0.23. From Fig. 5 we estimate SLJD = 0.5 from which we conclude that S I 8 x lo4 cm/sec. This value compares to S = 5 x lo4 cm/sec calculated from (32b), assuming that all of the atomic sites associated with misfit interfacial dislocations contribute to the formation of nonradiative recombination centers. By improved control of the growth of very thin active regions (which reduced the interfacial misfit defects),l 5 (InGa)As/In(GaP)double-heterojunction laser diodes have been produced with J , , z lo00 A/cm2 capable lSSa

The very thin active region can be dislocation-free, Section 16, even if some lattice misfit exists.

150

HENRY KRESSEL AND JEROME K . BUTLER

of room-temperature cw operation in the I-LI-pm range.'55b These had heterojunction spacings of a fraction of a micrometer (0.1-0.3 pm). Other devices for the 1-pm spectral range were produced by LPE. The first AlGaAsSb/Ga(AsSb) DH lasers emitted at 0.98 pm and had a Jth value about 4-5 times lower than a homojunction laser emitting at the same wavelength.'56 Subsequently,JIh values of 2000 A/cm2at & % 1 pm were achieved, making room-temperature cw operation possible.' 56a These lasers had active region widths d3 = 0.45 pm, and a composition A10.,Gao.6Aso.88Sbo.12/ GaAso.88Sbo,12. To minimize the dislocation density in the recombination region, three layers of Ga(AsSb),with increasing Sb content, were grown on the GaAs substrate prior to the deposition of the AlGaAsSb n-type layer. Probably the most successful devices capable of cw lasing at 1.1 pm at room temperature were DH structures of Ino,88Gao,12Aso.23Po,,7/InP prepared by LPE.'56b These lasers, with Jlh= 2800 A / m 2 (d3 = 0.6 pm), were grown on p-type InP substrates with (11l)B orientation. (The first such devices did not lase at room temperature.)' 56c Some of the early quaternary alloy heterojunction laser^'^^-'^^ listed in in Table X first demonstrated the concept of using these alloys for lattice matching, but their performance at room temperature did not match that achieved at the same wavelength with (A1Ga)As diodes. They probably contained many metallurgical defects, including nonplanar junctions, which further technological improvements would eliminate.

-

-

b. ZV-VI Compound Lasers

Injection lasers of Pbl -,Sn,Te and PbS, -,Sex can provide emission from 2.5 to about 33 pm (Fig. 50). In addition to varying the emission wavelength by choice of the appropriate alloy composition, a valuable feature of these alloys is that appreciable spectral tuning is possible by changing external factors such as temperature, magnetic field, or pressure. Practically, however, the diode current changes the emission wavelength by simply changing the junction temperature. The Pb-salt lasers operate only below 3WK, and the smaller the bandgap energy the lower the operating temperature. They have C. J. Nuese, G . H. Olsen, and M. Ettenberg, Appl. Phys. Lett. 29, 141 (1976). K. Sugiyama and H. Saito, Jpn. J. Appl. Phys. 11,1057 (1972). R. E. Nahory, M. A. Pollak, E. D. Beebe, J. C. DeWinter, and R.W. Dixon, Appl. Phys. Lett. 28, 19 (1976). 156b J. J. Hsieh, Appl. Phys. Lett. 28, 283 (1976); J . J. Hsieh, J. A. Rossi, and J. P. Donnelly, Appl. Phys. Lett. 28, 709 (1976). 1 5 6 c A. P. Bogatov et al., Sou. J . Quantum Electron. 4, 1281 (1975). R.D. Burnham et al., Appl. Phys. Lett. 19,25 (1971). 1 5 * J. J. Coleman, W. R.Hitchens, N. Holonyak, Jr., M. J. Ludowise, W. 0.Groves, and D. L. Keune, Appl. Phys. Lett. 25, 725 (1974). R. D. Burnham, N. Holonyak, Jr., and D. R.Scifres, Appl. Phys. Lett. 17,455 (1970). I"

2.

151

HETEROJUNCTION LASER DIODES

--A00

-0.003

Pbl_,Sn,Se a -+

00

-=a000 +- - I

Pb,_,,Sn ,Te

A00

a0

P b Sl-,Se,

ha0 --0.03 1-1 00

Pbl-,Ge,Te

*-+ Pbl-,Ge ,S

*---I

---

Pbl-xCdXS

I

I

I

2

3

4

I

I

I

I

6 8 1 0 20 WAVELENGTH ( p m )

I

1

1

30 4 0 50

FIG.50. Range of heterojunction device possibilities in IV-VI alloys. Shown is the maximum lattice parameter mismatch between the recombination region of the heterojunction structure and the higher bandgap bounding layers. (A. Groves, unpublished.)

been used in high-resolution spectroscopy, including the monitoring of air pollutants, where tunability is highly desirable. The addition of heterojunctions has greatly improved the laser performance compared to earlier homojunction lasers. However, the fabrication of heterojunction Pb-salt structures involves devices with substantial lattice parameter mismatch. Nevertheless, experimental Pb,,,,Sn,, ,Te/PbTe single- and double-heterojunction diodes, with Aa,/a, = 0.24q2, showed improvement in performance over their homojunction c0unterparts.l 60-1 6 2 a Although data in the literature indicate that J,, is still substantially above the theoretical limit, interfacial defects may have a lesser impact on laser performance in the IV-VI lasers than in the 111-V ones. Representative laser data are shown in Table X. VII. Laser Diode Reliability Two basic failure modes which may limit laser life are denoted catastrophic and gradual degradation. The first depends on the optical flux density (and the pulse width) and results in facet damagv; the second is mainly a function J. N. Walpole, A. R.Calawa, R. W. Ralston, T. C . Harman, and J. P. McVittie,,Appl. Phys. Lett 23, 620 (1973).

S. H.Groves, K. W. Nill, and A. J. Straws, Appl. Phys. Lett. 25, 331 (1974). L.R.TomasettaandC. Fonstad, Appl. Phys. Lett.25,440(1974);seealsoibid.24,567(1974). K. J. Sleger, F. G. McLane, U. Strom, S. G. Bishop, and D. L. Mitchell, J. Appl. Phys. 45, 5069 (1974).

152

HENRY KRESEL AND JEROME K. BUTLER

FIG. 51. Appearance of laser diode facet following catastrophicdegradation as seen by scanning electron microscopy.

of the current density, the duty cycle, and the details of the laser fabrication process. 18. CATASTROPHIC DEGRADATION

Complete or partial laser failure may occur as a result of mechanical The damage of the facet in the region of intense optical nature of the damage suggests local dissociation of the material’65 (see Fig. 51), and the damage sometimes extends a significant distance into the crystal. 6 6 It is well established that the damage is caused by the optical flux density at the laser facet and it is not a function of the operating current.’63 The effective width of the emitting region and the pulse width are important factors determining the failure level. A commonly used figure of merit is the H. Kressel and H. P. Mierop, J . Appl. Phys. 38, 5419 (1967). C. D. Dobson and F. S. Keeble, in “Gallium Arsenide” (Proc. Int. Symp., Reading, 1966). p. 68, Inst. Phys. and Phys. Soc.,London, 1967. 165 D. A. Shaw and P. R. Thornton, Solid-state Elecfron. 13,919 (1970). B. W. Hakki and F. R. Nash, J . Appl. Phys. 45, 3907 (1974).

‘64

2.

HETEROJUNCTION LASER DIODES

153

FIG.52. Dependence oflinear power density for facet damage, P,, on pulse width for uncoated facet lasers. The upper curve is for single-heterojunction lasers. The lower curve is for fourheterojunction lasers. (After Lockwood and Kressel.16’)

critical damage level in watts per cm of emitting facet, P,. For a given operating condition, P, is expected to decrease with the recombination region width of double-heterojunction lasers (when d , > 0.4pm). For example, with a pulse width of 100 nsec, P, = 200 W/cm for a broad-area DH (A1Ga)AsGaAs laser with d3 = 1 pm, while P, = 400 W/cm with d3 = 2 pm. (It is important to note that the damage threshold can be reduced if flaws exist at the laser facet; damage initiated at these faults will usually spread into the other regionsof the facet.)167 The determination of the optical power density is not easily made at the failure point because of the nonuniform optical energy distribution in the direction perpendicular to the junction plane. Furthermore, in stripe-contact lasers the optical intensity is not uniform in the plane of the junction. Therefore, the power density estimates are at best within a factor of two. The experimental results obtained for double-heterojunction, broad-area, devices operated with 100-nsec long pulses indicate that failure occurs at about 2-4 x lo6 W/cm2.168 Studies of stripe-contact lasers gave an estimate of 4-8 x lo6 W/cm2.’66 For a given device, P, decreases with increasing pulse width as as illustrated in Fig. 52 for various devices which include FH, and single-heterojunction devices. (Data reported by E l i ~ e e v ’for ~ ~DH lasers follow a similar behavior.) A comprehensive study of FH lasers, discussed in Thompson et ~ l . , ’ ~ ’relates facet damage to the structural parameters. H . F. Lockwood and H. Kressel, unpublished. V. T. Borodulin er 01. Kuuntouuyu Electron. (Moscow) 2, 108 (1972) [English trunsl. :Sou. J . Quuntum Electron. 2, 294 (1972)l. 169 Extensive DH laser data are given by P. G . Eliseev in “Semiconductor Light Emitters and Detectors.” (A. Frova, ed.), North-Holland Publ., Amsterdam, 1973. 170 G . H. B. Thompson, G. D. Henshall, J. E. A. Whiteaway, and P. A. Kirkby, J. Appl. Phys. 47, 1501 (1976).

Ic7

154

HENRY KRESSEL AND JEROME K. BUTLER

‘i

-$ .o I

+

.

+ +

SINGLE- HETEROJUNCTION

a

8

.2 IO-~

lo-‘

10-2

FACET REFLECTIVITY R

FIG.53. Ratio of the linear power density at catastrophiclaser failure, Pc to the power density for failure for GaAs-air interface, P , ( R = 0.32), with decreasing facet reflectivity R of singleheterojunction lasers. Solid line is best empirical fit. (After Ettenberg et dl )

The safe operating level is increased by antireflecting films on the laser facet (SiO, for example)which lower the ratio between the optical flux density inside and outside the crystal. It has been shown that a factor of three17’ in the allowable peak power is possible with very low facet reflectivity. This is illustrated in Fig. 53 for single-heterojunction lasers where the facet reflectivity R was changed. Despite the significant scatter in the data, the trend of improved power levels with reduced R is evident. It has been suggested’66 that facet failure occurs at a constant electric field intensity of 120 kV/cm (for a 100-nsecpulse width) and that the ratio Pb/Pc for a facet reflectivity R can be expressed as

l’b/Pc

= n[(l - R)/(1

+ R”2)2],

(33)

where n is the GaAs refractive index (- 3.6) and P , is the measured value for the GaAs-air interface. The “best-fit’’ curve to the data of Fig. 53 agrees quite well with (33). Lasers operating cw can also fail by facet damage if the optical flux density is excessive.’72For uncoated-facet lasers with stripe widths of 13 and 50 pm, the values found were 20-40 W/cm or 2-4 x lo5 W/cmZ.(The above laser diodes were thin DH lasers, d3 z 0.2-0.3 pm, with Alo,,Gao,,Asin the recom17’

M. Ettenberg, H. S. Sommers, Jr., H. Kressel, and H. F. Lockwood, Appl. Phys. Lett. 18, 571 (1971).

172

H. Kressel and I. Ladany, RCA Rev. 36, 230 (1975).

2.

HETEROJUNCTION LASER DIODES

155

bination region and A1,.,Ga0,,As in the adjoining n- and p-type regions.) The facet failure is generally initiated in the central portion of the stripecontact were the optical flux density is highest. The above discussion concerns a failure mode that occurs in a short period of time. Facet damage (“erosion”) can also appear gradually over a long operating time for diodes operated below P, ,an effect accelerated by moisture on the facet.’ 7 3 The use of half-wave thick dielectric facet coatings (such as A1,0,) minimizes the formation of such facet damage.

’”

19. GRADUAL DEGRADATION a . General

The reduction with time in the laser diode output at constant current (without evidence of facet damage) typically results from a reduction in the differential quantum efficiency and an increase in J,, . Erratic reliability was characteristic of early laser diodes, which suggested imperfect control of the device metallurgy. However, as the relevant material parameters have been brought into control, long-lived devices have become a reality. Major changes in the bulk of a laser diode operated at high current density were first observed in LPE GaAs homojunction lasers by Kressel and B ~ e r . ’ , ~ The lasers showed a drop in internal efficiency and an increase in internal absorption. Similar results were obtained on double-heterojunction (AlGa)As/GaAs diodes’ 7 6 which also showed that operation-induced changes within the recombination region are mainly responsible for degradation. A drop in internal quantum efficiency lowers the output of both the spontaneous and coherent emission, as wilt an increase in internal absorption. The effects may differ quantitatively, however, because the spectral patterns and internal dynamics change when lasing occurs. b. Experimental Observations

Results of various investigators often differ, but the following observations on the changes as a function of operating time appear to be generally valid in devices which degrade: (1) The threshold current density increases while the differential quantum efficiency frequently (but not always) decreases.

’

1. Ladany and H . Kressel, Appl. Phj3.s. Letr. 25, 708 (1974). 74

’’’

For a review of the literature until 1973 see. H. Kressel and H. F. Lockwood, J . Phys. (France)

SUPPI.4 35, C3-223 (1974). H. Kressel and N. E. Byer, Proc. IEEE 38, 25 (1969). D. H. Newman, S . Ritchie, and S. O’Hara, IEEE J . Quantum Electron. QE-8, 379 (1972).

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HENRY KRESSEL A N D JEROME K . BUTLER

(2) Heterojunction diodes operated at high current densities degrade similarly whether lasing or not (as long as facet damage does not occur), the degradation rate being a super-linear function of the operating current density. The degradation rate may be a quadratic function of the average diode current density175-’77; LED studies suggest a J3” dependence.177a (3) The minority carrier lifetime decreases with the spontaneous efficiency.178The change of T with qi suggests the formation of nonradiative recombination centers. Although the changes in T and .Ithare often found to be relate~i,”~ good correlation between the reduction in T and the increase in J,,, is not possible, perhaps because any large area defects also created will strongly increase the absorption (4) Degradation does not require the presence of a p-n junction. It has been observed both above and below lasing threshold’82 in n-type GaAs optically pumped. No evidence of damage was observed in the transparent (A1Ga)As window region of the heterojunction structure. Apparently, the recombination of electron-hole pairs is necessary for degradation to occur. ( 5 ) There are successive degradation stages. When the density of newly formed nonradiative recombination centers is low, the spontaneous efficiency decreases with only minor effects on the lasing properties. As degradation proceeds, “dark lines” (commonly (100)) may appear when the emission is viewed through the surface of the diode. The “dark lines” constitute regions of concentrated nonradiative centers and are believed responsible for the spotty near-field emission pattern of degraded lasers, and the rapid fall-off in the output of cw lasers are room temperature.’81’183 The concentrated nonradiative centers can have several origins, including impurity precipitation at existing dislocations. In the specific structures examined by transmission electron microscopy, the dark regions were dislocation networks formed by climb starting at existing dislocation sources.184,185 Since dislocation climb requires vacancy or interstitial diffusion to existing dislocations, dark line formation suggests that a point defect diffusion process is a basic element of gradual degradation. N. E. Byer, I E E E J . Quantum Electron.QE-5, 242 (1969). H. Kressel. M . Ettenberg, and H. F. Lockwood. J . Electron. Mutvr. 6. 467 (1977). 17’ E. S. Yang, J . Appl. Phys. 42, 5635 (1971). L79 N. Chinone, R. Ito, and 0. Nakada. I E E E J . Quantum Electron. QE-10, 81 (1974). B. W. Hakki and T. L. Paoli, J . Appl. Phys. 44,4113(1973). H. Yonezu, M . Urno, T. Kamejima, and 1. Sakuma, Jpn. J . Appl. Phys. 13.835 (1974). W. D. Johnston and B. I. Miller, Appl. Phys. Letr. 23. 192 (1973). B. C.DeLoach, B. W. Hakki, R. L. Hartman,and L. A. D’Asaro, Proc.IEEE61.1042(1973). P. Petroff and R . L. Hartman, Appl. Phys. Lett. 23,469 (1973). P. W. Hutchinson, P. S. Dobson. S. 0. O’Hara, and D . H . Newman, Appl. Phys. Letr. 26, 250 (1975). 177

177a

2.

HETEKOJUNCTION LASER DIODES

157

In one study the dark lines were found to originate in regions of the crystal of low initial radiative efficiency as measured by photoluminescence.' 86 By using intense external laser photoexcitation, it has been possible to promote the rapid growth of these dark regions. (6) The degradation rate increases with temperature, as established from studies of (A1Ga)AsDH lasers.'*' (7) High-frequency oscillations are sometimes seen in the output of degraded lasers.' c. Materials-Related Factors in Gradual Degradation

Experimental studies have uncovered major metallurgical factors which contribute to degradation. Most of these studies have been conducted on GaAs and (AlGa)As devices. The first direct evidence of strong metallurgical correlation with the degradation rate was obtained by a study of the effect of dislocations on GaAs homojunction lasers made by LPE.I9' It was shown that a high dislocation density (introduced by plastic deformation at elevated temperatures) led to erratic degradation rates as shown in Fig. 54. Such behavior is consistent with a nonuniform dislocation distribution introduced by the plastic deformation. In similar experiments, it was shown that diodes made using different acceptor dopants to form the p n junction degrade at different rates. For example, Be-doped diodes degrade much more rapidly than Zn-doped ones. The effect was attributed to the smaller ionic radius of Be with its consequent higher probability of displacement by the energy released in nonradiative electron-hole recombination. The vacancy so formed constitutes a nonradiative recombination center. Contaminants such as Cu in GaAs are definitely deterimental; experiments with Cu in GaAs: Zn diffused h o r n o j u n ~ t i o n syielded ' ~ ~ an activation energy for the spontaneous emission efficiency decrease of 0.45 & 0.1 eV, suggestive of Cu diffusion. However, since Cu can be effectively removed by various leaching techniques, it is unlikely that Cu is significant for the degradation of R. Ito, H. Nakashima, and 0. Nakada, Jpn. J . Appl. fhvs. 13, 1321 (1974). R. L. Hartman and R. W. Dixon, Appl. f h y s . Lett. 26, 239 (1975). I. Ladany and H. Kressel, in "Gallium Arsenide and Related Compounds" (Proc. Inr. Symp., Srh, Deaucille, 1974), p. 192. Institute of Physics, Conf. Series No. 24, London, 1975. 1 8 9 E. S. Yang, P. G. McMullin, A. W. Smith, 3. Blum, and K. K. Shih, Appl. f h y s . Lert. 24, 324 (1974). T. Paoli. IEEE J . Quantum Electron. QE-13. 351 (1977). 19’ H. Kressel et al., Metall. Trans. 1. 635 (1970). ’91 A. Bahraman and W. G. Oldham, J . Appl. f h y s . 43,2383 (1972).

158

HENRY KRESSEL AND JEROME K. BUTLER

0.1

0

10

20

30

40

50 60 70 80 90 OPERATING TIM E I H R S )

100

110

120

130

140

FIG.54. Power output as a function of operating time (100-nsecpulse width, 4-kHzrepetition rate) of LPE homojunction GaAs lasers made from as-grown and plastically deformed material to introduce a high dislocation density. Solid line, control; dashed line. deformed. Note the random degradation rate of the lasers containing a high dislocation density. (After Kressel et. a/.’”)

diodes made by liquid phase epitaxy, where the Ga melt is an excellent getter for Cu. Care is needed in the assembly of diodes to prevent stressing the diode. This in particular occurs when “ h a r d solders are used to bond the diodes to the heat sink.193The effect of stress on accelerating degradation has been well known in the commercial LED and laser diode field, but the more demanding operating conditions of cw laser diodes make it even more imperative to avoid diode stress. Therefore, a soft indium solder is commonly used for diode mounting. 19

R. L. Hartman and A. R. Hartman. Appl. Phys. L e f f .23, 147 (1973).

2.

HETEROJUNCTION LASER DIODES

159

There are important differences in the degradation rates of different alloys. It has been found that diodes with recombination regions of A1,Ga,p,As (to x 5 0.1) are less susceptible to gradual degradation than comparable diodes with GaAs in the recombination region.’94 On the low bandgap energy side of GaAs, (1nGa)As diodes showed a reduced degradation rate with increase in In.19s (These experiments were limited to a spectral range around 1.1 pm at room temperature.) Exposed edges where electron-hole recombination occurs can greatly contribute to the degradation.’ 7 7 a as demonstrated by the performance of comparable oxide stripe and broad-area cw laser diodes cut from the same wafer.’ 7 3 - 1 8 8The broad-area diodes which had sides formed by sawing (thus highly damaged) degraded in a few hours, whereas the stripe-contact diodes were quite stable for thousands of hours. These experiments also showed that a possible source of defects enhancing degradation is a highly doped Zndiffused region in close proximity to the recombination region. The growth rate used to prepare the GaAs:Ge recombination region by liquid phase epitaxy can strongly affect the degradation rate.’ 96 This effect on the degradation rate was attributed to the influence of the stoichiometry of the material grown at the various rates. The most reliable material was obtained at a very high growth rate where the vacancy concentration is believed to be reduced. Work has also been reported on diodes with GaAs in the recombination region but with phosphorous added to the outer (A1Ga)As regions in order to match their thermal coefficient of expansion and hence reduce the elastic strain. There is insufficient information at this time to determine whether this process improves the laser reliability.’ 8 7 d. Basic Causes of Gradual Degradation

It is clear that nonradiative recombination centers are introduced into the recombination region during forward bias and that the typical degradation is spatially nonuniform. That vacancies or interstitials are formed or diffuse into the recombination region is established by the observed growth of dislocation networks by climb. The origin of these point defects is still the subject of speculation. Gold and Weisberg, 97 in their tunnel diode degradation studies, suggested that nonradiative electron-hole recombination at an impurity center could result in its displacement into an interstitial position leaving a vacancy behind. This is the basic “phonon-kick” model in which multiphonon emission gives an intense vibration of the recombination center.

’

194

19’

19 19’

M . Ettenberg, H. Kressel. and H. F. Lockwood. Appl. Phys. Lett. 25, 82 (1974). M . Ettenberg and C. J . Nuese. J . Appl. Phy.s. 46,2137 (1975). M . Ettenberg and H. Kressel. Appl. Phys. Lerr. 26, 478 (1975). R. D. Gold and L. R. Weisberg, Solid-Stute Electron. 7, 81 1 (1964).

160

HENRY KRESSEL A N D JEROME K . BUTLER

Applying this mechanism to electroluminescence, the assumption is that the vacancy and interstitial formed would have a large cross section for nonradiative recombination. Repeated “kicks” would gradually move it to internal sinks. Another possible process contributing to the introduction of dislocations into the recombination region is glide under the influence of stress. It is possible that the glide process is eased by nonradiative electron-hole recombinations.’97a Experimental results are consistent with the hypothesis that the energy released in electron-hole recombination is a factor in the degradation process. The (1nGa)Asdiode experiments mentioned earlier showed’95 degradation rates to decrease with increasing In (i.e.,decreasing bandgap energy), although in (A1Ga)As the trend is opposite. The experiments of Lang and Kimerling’98 provided more direct support for the hypothesis of enhanced point defect migration by showing that lattice defects introduced into GaAs diodes by irradiation with 1 MeV electrons anneal much more readily under forward bias. The activation energy for motion of the unidentified defects was reduced to -0.34 eV by forward bias from 1.4 eV in the unbiased diode. In summary, the major degradation effects can be explained by the idea that the energy released in nonradiative carrier recombination enhances the diffusion of vacancies or interstitials. (Whether any point defects are actually formed within the recombination region remains unclear.) If nonradiative electron-hole recombination occurs, for example at the damaged surface of a diode (as in the case of the sawed-edge diode experiment described above), it accelerates the motion of point defects into the active region of the device. Another source of defects could be regions containing point defects close to the active region where nonradiative electron-hole recombination can occur. Differences in initial vacancy concentration could contribute to the improved radiation resistance of (A1Ga)Asdiodes (although a lower oxygen concentration could be a factor).’98a Finally, regions where nonradiative recombination occurs will tend to grow in size, leading the strongly nonuniform degradation process commonly observed. From the available results, control of gradual degradation seems to be basically a metallurgical problem. With increasing material and device perfection, operating lifetimes comparable to those obtained from conventional solid-state devices are being achieved. The feasibility of LED’Soperating at loo0 A/cmZ with lifetimes in excess of 20,000 h has been established with

-

I9”See, for example, T. Fujiward it al.. IEEE J . Quantum Electron. QE-13, 616 (1977). 19’ D. V. Lang and L. C. Kimerling, Phys. Reo. L R f f .33,489 (1974). 19** The Al in the solution used in LPE tends to combine with oxygen, thus reducing the concentration available for incorporation into the grown (A1Ga)As layer. Another factor could be the reduction in the active region strain with small Al additions.

2.

HETEKOJUNCTION LASER DIODES

161

laboratory and cw lasers operating at a few milliwatts have reached more than 40,000 h of operating life. Excellent stability has been obtained with fully A1,03 facet coated oxide-stripe lasers, with > 15,000 h with negligible degradation being consistently obtained.’ 98b Because the gradual degradation rate increases with temperature, it is possible to accelerate laser aging by high-temperature operation. Various studies have been conducted to determine the temperature dependence of the laser degradation rate (for gradual degradation). It is frequently found that the degradation rate increases with temperature following an approximate expression exp(E/kT), where E is an effective “activation energy” which varies from -0.7 to 1 eV. Such experiments suggest that the extrapolated room-temperature operating life of the best laser diodes exceeds 100,OOO h, but it is substantially shorter at 70°C.’ y8c

-

8 7 7 1

VIII. Devices for Special Applications

Various aspects of laser design have been reviewed in the preceding sections. Although specific characteristics can be obtained by suitable structural design, no single device can perform all of the functions required in various optoelectronic systems. In this section, we consider devices designed for high peak power operation, cw operation, and lasers emitting in the visible or near-visible spectral range. 20. HIGHPEAKPOWERLASERDIODES

Laser diodes emitting peak power in excess of 5 W are useful in ranging and infrared illumination. These usually operate at duty cycles of about 0.1% with pulse widths of 50-200nsec. Operating current densities can be 40,000 A/cm2, the maximum being limited by catastrophic damage (Part VII). The emitting width is adjusted for the required peak power. Commercial pulsed units have widths ranging from 75 to 600 pm. Very wide diodes are not feasible because of the possibility of cross-lasing, although widths of 1500 pm have produced 100 W peak power. Series-connected diode arrays are used when power beyond the capability of a single device is desired. Figure 55 shows such an array of diodes mounted on a metallized Be0 stubstrate. A strip of diodes is first soldered to a Be0 substrate, each diode isolated from its neighbor by sawing through the strip, and then the individual lasers are wired in series. Diodes can also be seriesconnected by stacking, but the thermal resistance of such a structure is relatively high and the duty cycle limited to less than a fraction of 1%.

- -

198b 198c

I. Ladany, M. Ettenberg, H. F. Lockwood. and H. Kressel, Appl. Phys. Lett. 30,87 (1977). H. Kressel, M. Ettenberg, and I. Ladany. Appl. Phys. Lett. 5. 305 (1978).

162

HENRY KRESSEL AND JEROME K . BUTLER

EMISSION

FIG.55. Series-connected diode array designed for high peak power operation: (a)Construction; (b) photograph of assembled array. (Courtesy RCA Solid State Division, Lancaster, Pennsylvania.)

2.

163

HETEROJUNCTION LASER DIODES

The internal diode structure varies with the required operating temperature. The single-heterojunction laser, which has been produced in commercial quantities for several years, is capable of reliable pulsed operation in the 300-400 W/cm range. The beam width B,z 20F (fundamental transverse mode) offers relative ease of beam collimation. It is limited to operating temperatures below about 80 C because of the loss of optical confinement (see Part 111). The LOC laser with an optical cavity width of 2-3 ,urn has a lower threshold current density than the single-heterojunction laser (Fig. 56) and superior high-temperature performance. The resistance to catastrophic

F

0

8,000

f 4,000 3

u

OO

2

6

4

8

do ( p )

(01

A

10

4L-Ud-L OO

2

4

6

8

I

do(p)

(b)

FIG.56. ( a ) Threshold current density and (b) differential quantum efficiency (total) as a function of the thickness of the mode guiding region tl" of LOC lasers at 300 K. (c) Power output and power conversion efficiency (two sides) of a low threshold LOC laser (2000 A/cm2), with L = 400 pm.The device is operated pulsed with a duty cycle of loo. (After Kressel et

164

HENRY KRESSEL AND JEROME K . BUTLER

CURRENT (A) (C

1

FIG.56 (Conrinued)

degradation is a function of the heterojunction spacing, and antireflective facet coatings are used to prevent facet failure. The four-heterojunction laser can fill a function similar to the LOC laser, but the fabrication is more complex. The beam width of either device can be designed to give 8, values under 30". The power conversion efficiency depends on the basic diode parameters and the operating current relative to the threshold current. It is given by ?p

=

Po

12R,

+ (E$e)l'

(34)

where R, is the diode resistance, I the diode current, and Pe is the output power. S ~ m m e r shas ' ~ ~calculated the relationship between the device properties and qP. The highest-reported power efficiency (two-sided emission) 199

H. S. Sommers, Jr., Solid-State Electron. 11, 909 (1968).

2.

HETEROJUNCTION LASER DIODES

165

at room temperature is -220/,, as shown in Fig. 56c for a LOC diode operating at a duty cycle of I%, and a current I = 41L,,. 21. CW LASERDIODES

The development of optical fibers with low (< 10 dB/km) absorption windows in the vicinity of 0.8-0.9 and 1-1.2 pm has given impetus to the development of optical communication systems using diode sources. The requirements depend on the bandwidth and transmission distance. In general, diodes should be capable of modulation at very high frequencies, the spectral linewidth should be small to reduce pulse dispersion, and the beamwidth narrow to improve the coupling into the fibers. Milliwatts of laser power are usually adequate. The cw laser diode with its potential modulation capability extending to above 1 GHz (when biased to threshold to remove the delay effect mentioned in Part II), narrow spectral linewidth, and power in the milliwatt range is an ideal source for optical communications.200,201 In this section, we consider state-of-the-art cw laser diodes of (A1Ca)As.These are the most developed. Other cw laser structures designed for emission at 1-1.2 pm (such as InGaAsP/InP lasers) will have comparable properties, and the design problems are similar. The laser diode designed for cw operation at room temperature requires sophisticated technologies from wafer fabrication (see Part V) through diode assembly. We have previously considered the device requirements for low threshold current operation in Parts 111-V,and in Part VII we have discussed the reliability of the laser. Because of the higher reliability (and fiber loss minimum at II = 0.82 to 0.85 pm),the most widely used cw laser diodes have A1 fractions of 5-10% in the recombination region. There is little or no penalty in diode threshold current density from addition of up to 1004 Al, and the diode technology is similar in other respects to that of diodes with GaAs in the recombination region. Planar stripe diodes (Part V) have proven the most popular because the ease of fabrication and relatively good life outweigh the imperfect lateral confinement. Stripe widths of 10-20 pm are widely used because they provide a good compromise between low operating current and useful power emission. Figure 57 shows a cross section of a practical cw laser diode using oxide isolation for stripe formation. These lasers have room-temperature threshold T. L. Paoli and J. E. Ripper, Proc. IEEE 58. 1457 (1970). M . K. Barnoski (ed.),“Fundamentals of Optical Fiber Communications.” Academic Press, New York. 1976.

166

HENRY KRESSEL AND JEROME K . BUTLER

P""""

"'"1

/METAL

G a A i SUBSTRATE

rlON

0.1 TO

OXIDE METAL SOLDER

I------I

STRIPE WIDTH 10- 5 O p m

FIG.57. Typical cross section (not to scale) of oxide-isolated stripe-contact (A1Ga)As cw laser diode designed for emission in the 0.8-0.83-pm spectral range.

currents between 75 and 300 mA, with emitted power of 10-20 mW. However, cw threshold currents as low as 18 mA have been obtained by using the "buried-heterojunction" structure (Fig. 44)when the emitting region width was reduced to only 1 pm.'28 The useful output from such a device will be lower than from a wider stripe because of facet damage, and long-term reliability data are needed to establish operating levels. The power emission from stripe lasers can be increased (as will the threshold current) by increasing the stripe width, but, with excessive stripe widening, the cw power output is limited by the nonuniform current distribution resulting from an increase in temperature in the central region. The reduction in the bandgap energy with local heating results in current crowding. Even with a 100-pm stripe width, the maximum cw power is thermally limited to about 100 rnV.''' However, this limit could well be raised with improved technology. The temperature dependence of the cw lasing threshold follows that of the pulsed threshold over a significant temperature range above room temperature, but for a fixed diode current the output power is rather strongly temperature dependent. To eliminate these power fluctuations, the heat sink temperature can be stabilized (by using a small thermoelectric unit, for example) or an optical feedback circuit can be used in which a reduction in the power output is automatically prevented by a compensating increase in the diode current. Figure 58a shows a typical power versus current curve of an oxide-defined stripe-contact laser (13-pm stripe width) operating cw at various heat sink temperatures. Figure 58b shows the increase in the cw threshold current with temperature. In practice, the maximum cw operating temperature is not limited by the threshold current increase, but rather by the increased degradation rate at elevated temperatures (Part VII).

-

*02

I. Ladany and H. Kressel, Final Report, NASA NASI-11421, December 1974.

2.

167

HETEROJUNCTION LASER DIODES

CURRENT ( m A )

FIG. 58a. Power output at various heat sink temperatures as a function of current from an (AIGa)As double-heterojunction laser diode with a stripe width of 13 pm and 1 = 8200 A. (After Kressel and Ladany, unpublished.)

5801,

0

I

10

,

, , , ~, 20

, ,

40 50 TEMPERATURE (OC)

30

, , , , 60

70

FIG. 58b. Cw threshold current as a function of temperature

The lasing spectral emission of cw lasers is typically multimoded. There are several longitudinal modes, each having its own “satellites” due to the lateral cavity modes. However, some diodes show a decrease in the number of dominant longitudinal modes with increasing current as shown in Fig. 59. The spectral purity has been related to the material uniformity in the

I1360mA 1.1 mW

,I

1=420mA lOmW

J_

8200

u 8280

8300

8180

8190

8290 8272 WAVELENGTH (A)

8160

8170

,

8140

8150

,

8130

Cil FIG.59. Emission spectrum of room-temperaturecw lasers as a function of current showing the diversityofspectral purity seen. Both lasers are oxide-defined, 13-pm stripe devices.(a)“Pure” spectrum at relatively high drive with one dominant longitudinal mode. (b) More typical multimode spectrum. The power values indicated are emission from one facet only. WAVELENGTH

168

2.

HETEROJUNCTION LASER DIODES

169

recombination region.203There is also a tendency for shorter cavity lasers to operate with fewer longitudinal modes. The lateral mode content ofa stripe laser diode depends on the stripe width, effective refractive index profile, and, frequently, on the operating current above threshold. Considering a laser with high and sharply defined dielectric sidewalls, such as the “buried-heterojunction” laser, the conditions for sustaining fundamental lateral mode operation are similar to those for maintaining fundamental transuerse mode operation. In effect, a diode width under 2 pm is typically needed to maintain fundamental lateral mode operation. Such lasers with wider active regions will operate in higher-order lateral modes, but these should not be strongly current dependent because the built-in dielectric step is large relative to dielectric changes introduced by changing the laser current. In planar stripe-contact lasers, however, where the lateral sides are not well defined except by the current distribution, the lateral dielectric profile is a complex function of the gain coefficient, and of the current distribution, and is expected to change with increasing current. The mode guiding in the junction plane is affected by two competing effects on the dielectric profile. The free carrier concentration under the stripe is higher than beyond its edges, an effect which depresses the dielectric constant under the stripe thus producing an “antiguiding” mechanism. Counteracting this effect is the higher gain coefficient under the stripe which increases the imaginary part of the dielectric constant, thus producing a mode guiding mechanism. The latter effect is generally believed to dominate. However, it is not surprising that with increasing current the dielectric profile as well as the current distribution (i.e., free carrier concentration) in the junction plane can change. Therefore, the order of the preferred lateral mode will change, with higher order modes reaching threshold with increasing current. In addition, the temperature profile in the junction plane can change with increasing current, thus further contributing to changes in dielectric profile. The experimental observation is that fundamental mode operation is frequently seen in planar stripe lasers with stripe widths of 5 13 pm but higher-order modes proliferate when the emitted power level exceeds a few milliwatts. The factors responsible for lateral mode proliferation with drive are difficult to resolve. Shifts in the current distribution with drive and spatial hole burning are possible factors influencing the change in dominant lateral mode with ~ u r r e n t . ~ ~ ” - * ~ ~

’04 ’OS

’06 ’O’

See. for example, P. G. Eliseev and V . P. Strakhov. W . Eksp. 7eor. Fiz. Pis’rna Red. 16. 606 (1972) [English rransl.: JETP Lerr. 16,42X (1972)J. T. L. Paoli. IEEE 1. Quanlum Electron. QE-13, 662 (1977). P. A. Kirkby, A. R . Goodwin. G. H . 8. Thompson, and P. R. Selway. IEEE J . Quunrurn Elwrron. QE-13, 705 (1977). 8. W. Hakki and T. L. Paoli, J . Appl. Phys. 4 6 , 1299 (1975). D. D. Cook and F. R. Nash. J . App’pl. P h ~ s46. . 1660 (1975).

170

HENRY KRESSEL AND JEROME K. BUTLER

A possible effect promoting change from the fundamental to the next-order mode with increasing current involves spatial hole burning wherein the gain profile is distorted with increasing optical power because the region under the center of the stripe is depleted of carriers, with a corresponding reduction of gain. It is to be noted that this distortion changes the gain profile and also tends to promote the effective gain coupling to the higher order modes. The key principle which governs the restriction of laser operation to a single mode is that the differences between the propagation losses of the fundamental and the higher order modes be as large as possible. As mentioned above, restricting the stripe width of buried-heterojunction lasers to very small values of 1 to 2 pm is one method of achieving this objective, although at the expense of the useful power from the device. Another approach consists of incorporating regions in the device that produce a greater internal absorption coefficient for the higher order modes than for the fundamental one. Structures involving buried channels207a and restricted active region thickness207bhave been described in order to achieve this objective. A basic phenomenon promoting mode change with current has been discussed by Sommers and North whereby the gain in the active region of a laser does not remain constant above threshold.208-210Barring any spatial changes in the gain profile, this phenomenon has been postulated to lead to the threshold of modes having increasing losses. Lateral mode changes with current are frequently reflected in kinks in the curve of power emitted versus current, as shown for laser A in Fig. 60a. The laser shown starts in the fundamental lateral mode, but the power emitted in this mode is reduced with increasing drive as the threshold for the second mode is reached, which then dominates. The slope change is associated with the transition from the fundamental to the dominant second mode. Kinks are not generally seen in very wide lasers where many lateral modes are present and modal shifts do not have a detectable effect on the power versus current curves. The above kink should not be confused with nonlinearities in the power versus current curve due to filaments (Fig. 60b). Lasers that operate in separate segments because they are nonuniform commonly exhibit sharp slope changes and slope reversals in the power versus current curves. An example is shown in laser B of Fig. 60a. The technological importance of the kinks results from noise generation in the optical output as the current K.Aiki et al., IEEE J . Quantum Electron. QE-14,89 (1978). D. Botez, Appl. Phys. Lett.. 33, 872 (1978). H. S . Sommers, Jr. and D. 0. North, Solid-State Electron. 8, 675 (1976). ’09 H. S. Sommers, Jr., J. Appl. Phys. 44, 1263 (1973). ”’H. S. Sommers, Jr., and D. 0. North, J . Appl. Phys. 45, 1787 (1974). 20 b

’08

2. 14

171

HETEROJUNCTION LASER DIODES I

-3 1 2 E

L

!loIn w

80 W

c

k 6I w

LL

w 4 -

3 a

0

20-

100

I

I

2 00

300

I

400

B

A

(b) FIG. 60. (a) Two types of “kinks” in the light versus current curves of cw lasers. Laser B exhibits filamentary emission due to defects as shown in the near-field photograph of part (b)of the figure. Laser A is representative of behavior related to change in the dominant lateral mode from the fundamental to the second mode with no evidence of “beady” emission due to defects.

changes through the nonlinear portions of the curve. Furthermore, amplitude modulation is rendered difficult because of the nonlinear power versus current curve. As discussed in Part 111, the far-field depends on the internal diode geometry, and practical DH structures represent a compromise in design. Figure 61 shows the far-field pattern of a typical cw laser in the planes perpendicular and parallel to the junction plane. In good lasers there is no significant change in these beam patterns with increasing temperature, at least in the useful

172

HENRY KRESSEL AND JEROME K . BUTLER

t Z

ij

‘1

t v)

Z

w m

(0)

{ b)

FIG.61. Typical radiation patterns (a) in the plane of the junction (11) and (b) perpendicular to the junction (I) for an oxide-isolated DH (A1Ga)Aslaser diode with a 13-pm stripe width.

range below 100°C. From f to $ the power emitted from one laser facet can be coupled into a step index fiber with a numerical aperture of 0.14”Oa The coupling efficiency into the fiber can be substantially improved compared to a flat termination by using a globular termination at the end of the multimode fiber formed by melting the fiber tip.21’*212 The type of laser diode package used for cw operation will depend on the application. Sealed packages have been reported2I3 where the diode is inside a case with glass walls and a small lens is internally placed which focuses the radiation through the walls of the package. Another package incorporates a short fiber length (“pigtail”)convenient for coupling to long fibers in optical communication systems.214.2 I

22. VISIBLEEMISSION LASERDIODES The development of laser diodes emitting closer to the visible spectral region began very early, with the first Ga(AsP) homojunction lasers operating at cryogenic temperatures.2 Subsequent work extended the lasing region both at 77°K and room temperature. The lowest threshold lasers of The numerical aperture NA = sin 0”. where 0, is the maximum acccptance angle measured from the fiber axis. 2 L 1 D. Kato, J. Appl. Phys. 44,2756 (1973). 2 1 2 C. A. Brackett, J. Appl. Phys. 45,2636 (1974). 2 L 3 T. Uchida, Optical telecommunication systems using fibers, in “Photonics” (M. Balkanski and P. Lallemand, eds.), p. 341. Gauthier-Villars, Paris, 1974. 214 J. P. Wittke, M. Ettenberg, and H. Kressel. R C A Rev. 37, 159 (1976). For a review of sources for optical communications, H. Kressel, I. Ladany, M. Ettenberg, and H. F. Lockwood, Phys. Today 29, 38 (1976). 216 N. Holonyak, Jr., and S. F. Bevacqua, Appl. Phys. LPII.1, 82 (1962).

2.

HETEROJUNCTION LASER DIODES

173

FIG. 62. Dependence of the lasing wavelength on the Al fraction x in the recombination region. Some variation due to doping in the recombination region may occur. (After Kressel era/.*’*)

this alloy were prepared by vapor phase epitaxy on GaAs substrates in which both the p - and n-type region were grown rather than diffu~ed.’~’ These have lased at 6750 (300“K, Jth = lo6 A/cm2) and at 6350A (77°K). In the (A1Ga)As alloy system, single-heterojunction,” double-heteroj u n ~ t i o n ’ ~ . 9,220, ~ ’ . ~ ~and LOC lasersZoZhave been made in the direct bandgap region. Figure 62 shows the dependence of the emission wavelength on the alloy composition; Jth against lasing wavelength at 300 and 77°K is shown in Fig. 63 for broad-area single- and double-heterojunction lasers. The room-temperature threshold current density is nearly constant between

”’ J. J. Tietjen, J. I. Pankove, I. J. Hegyi, and H. Nelson, Trans. A I M E 239,385 (1967).

’” H.Kressel, H. F. Lockwood, and H. Nelson, IEEE J . Quantum Electron. QE-6,278(1970).

2L9

B. I . Miller, J. E. Ripper, J. C. Dyment, E. Pinkas, and M. B. Panish, Appl. Phys. Lett.

”O

18,403 (1971). Zh. I. Alferov et a / . , Fiz. Tekh. Poluprooodn. 6. 568 (1972) [English transl.: SOL). Phys. Semicond. 6, 495 (1972)].

174

HENRY KRESSEL AND JEROME K. BUTLER

/ z 111

-

.

SINGLE HETEROJUNCTION

/

0

0 I v)

0

OSINGLE- HETEROJUNCTION ODOUBLE- HETEROJUNCTION

10.

LASING WAVELENGTH 1 0

I

I

1

01

02

03

1

X

(b) FIG.63. Threshold current density as a function of lasing wavelength of single and doubleheterojunction Al,Ga, -,As lasers. (a) Room-temperature (300°K) pulsed operation. (b) 77 K pulsed operation. The single-heterojunction data ( d , z 2 pm) are from Kressel.2'8 The doubleheterojunction (77'K) data (d3 E 1 pm) are from Kressel and Hawrylo,2z1 while the roomtemperature data are from Kressel and Hawrylo221"with d , > 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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K. BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL A N D JEROME K. BUTLER

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

(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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

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.

HETEROJUNCTION LASER DIODES

-

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

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL A N D JEROME K . BUTLER

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

HENRY KRESSEL AND JEROME K . BUTLER

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.

HETEROJUNCTION LASER DIODES

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

HENRY KRESSEL AND JEROME K . BUTLER

-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.

HETEROJUNCTION LASER DIODES

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.

HETEKOJUNCTION LASER DIODES

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)

HENRY KRESSEL A N D JEROME K. BUTLER

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 << Na,)the acceptor concentration of the x-region. Let a space-charge region in the left-hand p-region extend from -.yo to 0 and a corresponding region in the right p-region from d to d + .yo (Fig. 1). The equations of the system are in that case (1) current equation

J,

= ep,pF

-

ppkT L1ppld.u = 0,

where J , is the hole current density, p is the hole concentration, F = - d$/dx the field strength, $(x) the potential, p, the carrier mobility, e the electron charge, k Boltzmann's constant, and T the temperature. Consequently, if $ = 0 in the left-hand p-region (.Y < -.yo), and, since p = N,, in that region, p

=

N , , exp( - e$/kT).

( 2 ) Poisson's equation

where $ = O a t at .Y = 0.

' P. S. Stone, A

.Y

= - . ~ ~ , d $ / d= x O a t s = -xo

a n d a t .Y = i d , and $

=

Ic/o

Numerical Investigation of the Steady State Behavior of Silicon Diodes. Ph.D Thesis, Univ. of Minnesota, Minneapolis, Minnesota (1971).

198

A. VAN DER ZIEL

By multiplying the latter equations by 2d$. substituting (la) and integrating, we obtain

(d$ld.)2

= f($X

(4)

where f($)is a known function of $ for each of the regions - x o < x < 0 and 0 < x < *d. There are three unknowns, xo, $o,

and

$(id),

and there are three relationships,

t,b = llfo at x=O,

dllfldx = O

at x = i d d ,

llf = $($d)

at x = i d

so that the potential distribution can be completely evaluated. When a negative voltage is applied to the right-hand p-region, d$/dx < 0 for most of the i-region, and it is clear from Fig. 1 that a potential maximum, limiting the current flow, will be formed near x = 0. In the same way there will be a potential minimum near the cathode (x = 0) of an n-v-n diode when a positive voltage is applied to the anode (x = d ) . The metal-insulator-metal problem at zero bias was solved by Wright3 and by Skinner3”;we refer to their papers for details. 3. THEEFFECTOF TRAPS The behavior described so far prevails for devices without traps. If traps are present, much more complicated behavior can be expected (Parts I1 and 111).For most applications traps are a nuisance and steps should be taken to prevent their presence. From the point of view of device physics, however, the effects of traps are quite interesting. Lampert and Mark’s book’ goes into considerable length about trapping effects. The same is true for several papers in Vol. 6 of “Semiconductors and semi metal^".^ Lampert and Schilling discuss the regional approximation method,4a Baron and M a ~ e discuss r ~ ~ trapping effects in double injection, whereas Barnett4c discusses the phenomenon of filament formation associated with the presence of traps. G. T. Wright, Solid Srute Electron. 2, 165 (1961). M . Skinner, J . Appl. Phys. 26,498 (1955). R. K. Willardson and A. C. Beer (eds.). “Semiconductors and Semimetals.” Vol. 6. Injection Phenomena. Academic Press, New York. 1970. 4a M. A. Lampert and R. B. Schilling. Current injection in solids: The regional approximation method, in “Semiconductors and Semimetals“ (R. K. Willardson and A. C. Beer, eds.), Vol. 6, pp. 1-96. Academic Press, New York. 1970. 4b R. W. Baron and J. W. Mayer. Double injection in semiconductors. in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 6. pp. 201-313. Academic Press, New York, 1970. 4c A. M. Barnett, Current filament formation, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 6, pp. 141-200. Academic Press, New York, 1970. ” S.

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

199

11. Single-Injection Space-Charge-Limited Solid-state Diodes

4. THETRAP-FREE CASE’” a. Basic Equation for a One-Dimensional Geometry

and no traps

We start with the one-dimensional equations for an n-v-n diode; the equations for a p n - p diode are similar. The p i - n single-injection diode is discussed in Part 11, Section 6. We assume here that the material has no traps; later we shall investigate the effect of traps (Part 11, Section 5). We take the direction of positive current in the positive x-direction, i.e., from the cathode (at x = 0) to the anode (at x = d ) . Let the sample have a uniform donor concentration N , , an electron concentration n(x, t), and a field distribution F(x, t ) , then the electron current density J,(x, t ) satisfies the equation J,(x, t ) = ep,nF

+ eD, dn/dx,

(5)

where e is the electron charge, p, the electron mobility, and D, the electron diffusion constant. The field F(x, t ) is related to the space-charge density - e(n - Nd) by Gauss’s law

dF/dx =

- e(n - Nd)/EEO,

(6)

where c0 = 8.85 x 10- l 2 F/m and E is the relative dielectric constant of the material. The electron concentration n(x, t ) is related to the electron current density J,(x, t ) by the continuity equation e an/&

=

dJ,/dx.

(7)

Substituting for n with the help of Eq. ( 6 )yields d(J,

+

EEO

3F/dt)/i3x = 0,

(8)

which indicates that the total current density ~ ( t= ) J,(X, t )

+

CEO

aF/at

@a)

is independent of x, and hence a function of t only. Equations 5-8a are generally valid as long as the material has no traps. As mentioned before, there is a potential minimum near x = 0, say at ?I = d,; at that point V = - V,. We shall assume from here on that d , << d, the distance between cathode and anode, and that V,<< V,, the applied M. Lampert and A. Rose, Phys. Rev. 121,26 (1960).

200

A. VAN DER ZIEL

voltage at the anode. We may then put F = - dV/dx = 0 and V = 0 at x = 0 in good approximation. For most of the region 0 < x < d the drift term in Eq. ( 5 ) predominates over the diffusion term; the diffusion term is significant close to the potential minimum only. We make thus little error, except near the cathode, by putting J,

(54

= ep,nF.

Especially in materials with a high extrinsic resistivity, N , can be neglected in comparison with n. Consequently, Eq. (6) may then be written dF/dx

=

en/^^,.

(64

All other equations remain unchanged. b. Elementary Theory: N ,

= 0, p, = p, =

Constant

Here we start with Eqs. (5a), (6a), and (8a) and substitute for n. This yields

+

J ( t ) = - ~ E E O ~ O ? ( F ~ )CEO / I ~?F/dt. X

We now substitute F

= - S V / S x where

V ( x ,t ) is the voltage at x , and assume

V ( d )= V ,

V(0)= 0,

(9)

+ valexp(jwt),

(10)

where u,, exp(jut) is a small-signal ac term. To find the solution of (9) for the applied voltage (10) we write J ( t )= J o

+ J , exp(jwt),

F

= F,

+ F , exp(jwt),

V

=

V, + V, exp(jwt), (11)

where the zero subscript describes dc terms and the one subscript smallsignal ac terms. Substituting Eq. (11) into Eq. (9),neglecting second-order ac terms, and separating dc and ac terms yields: (a) The dc equation Jo

=

d(Fi)/dx

with the initial conditions F, V,

=0 =0

(potential minimum) at x at x = 0, Vo(d)= V,.

= 0,

(b) The small-signal ac equation J,

=

+

- p o ~ ~ O d ( F o F l ) / d ~WEE OF,

(13)

and the initial conditions F , = 0,

V, = 0 at .x

= 0;

V l ( d )= t i a I .

(134

3.

20 1

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

1.00

1.00

Fa

Y

t

t

”a

0.50

0.50

0

FIG.2. Normalized field distribution F;F, and normalized potential distribution

L’/K

in

a single-injection diode.

We first solve the dc equation (12) under the initial conditions (12a). Integrating (12) once yields or dVo/d.x

-Jox = +poeeoF:.

= (-

(note that dV0/d.u > 0 in this case, except at V,(.Y) =

g( - 3J,,:poCco)’ ‘z.Y3’2

Y .

2JO/p0~~0)1 2.~1(14) Z

= 0). Integrating

(14) yields

= l/,(.x/d)3’2

(15)

so that ~2 ( - 2 3 ~ / / 1 ~=~ Ve , ~/ L) ~~ ~’ ’~~or,

-Jo

=~

~ ~ ~ p ~ V , 2(16) / d ~ ,

indicating that positive current flows from anode to cathode. If phonon scattering predominates. po varies as T 3 I 2 , where T is the absolute temperature of the material, so that - J o increases with decreasing temperature. Equation (16) is the semiconductor equivalent of Child’s law. Figure 2 shows the normalized potential distribution and the normalized field distribution in the single-injection diode. Next we solve the small-signal ac equation under the initial conditions (13a).6To that end we substitute s = ( . ~ / d ) ” *as a new independent variable. Since Vo(s)= Ks3, according to (1 5), Fo

= - d V0ld.u =

-

( l/,/d )s,

F

= - ( d V, /d.~)/2sd.

S. T. Hsu and A . van der Ziel, Pro(,. IEEE 54, 1194 (1966)

(17)

202

A. VAN DER ZIEL

Substituting into (13) yields

+

dZVl/ds2 j w z d V , / d s

=

-(6Jl/go)s,

(18)

where g o is the low-frequency ac conductance per unit area and z the transit time of the carriers, go = -dJo/dl/, = $po~coT/,/d3,

z = $d2/pol/,.

( 184

The initial conditions are now V, = 0 and dVl/ds = 0 at s = 0 (x =O), VI = u a 1 at s = 1 (x = d ) .

(18b)

It is easily seen by substitution, and it can be verified by the method of variation of parameters, that the expression Vl = - [l - j o z s

+ +(jwrs)’-

exp( - j ~ z s ) ] 6 J ~ / g ~ ( j o z ) (19) ~

is a solution of(18).Also, it satisfies the boundary conditions at x = 0. Putting V, = ual at s = 1 gives a linear relationship between J 1 and ual, so that the device admittance Y is J Y=-’=g ua 1

0

1 - joz

;(jw~)~ = g +joC. - exp( -joz)

+ +(jwz)’

(20)

If now the conductance g and the capacitance C are evaluated as functions of oz,one obtains the following: (a) The conductance g is equal to go for uz << 1, as expected, goes through a minimum of about 0.58g0 at oz ‘v 7, and oscillates around the highfrequency value * g o for large 017. Figure 3 shows g / g o plotted versus or. (b) C changes from the low-frequency value $goz for oz << 1 to the highfrequency value i g o z for oz >> 1. We shall see that the latter corresponds to the electrode capacitance cd = a O / d per unit area. Figure 3 shows c/cd plotted versus wr. The single-injection space-charge-limited diode thus has a very good frequency response if z is small (equals small electrode spacing d). Next we derive the expression for the transit time z. Since polFol is the drift velocity of the carriers at x, ~-

-J O P O

Finally

4 dZ

3 pol/,’

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

203

FIG. 3. y/yo and C/C, for a single-injection diode as a function of COT

One can thus evaluate the electrode distance d from capacitance measurements. When the theory is compared with experiment one usually finds satisfactory agreement, both for the dc characteristic and for the ac admittance. Even if the material has traps, one finds that Eq. (16) is valid at sufficiently large anode voltages, where all the traps are filled. Sometimes one finds that a small correction must be made for the dc characteristic as follows: -J

0

+ v,)2/d3,

(21)

- 2se~;opo(Ij -

where V, is a small built-in potential difference between cathode and anode (see Part 11, Section 6 ) . c. Efect of N , ; p,, = p o

=

Constant, N o Traps’

Here we start with Eqs. ( 5 ) and ( 6 )and substitute for n. Since en =

8F/?x

-E E ~

+ eN,

we have J ( t )=

(7(F2)/8+ . ~ ep0NdF

- + E E O ~ ~

+

EEO

SF/?t.

(22)

204

A . VAN DER ZlEL

Substituting F

= - 8 V / d x and

J ( t )= J o

putting

+ J , exp(jot), V = Vo

+

F = Fo + F , exp(jwt), V, exp(jor),

(224

where the subscripts zero and one again denote dc terms and small-signal ac terms, respectively, neglecting second-order ac terms, and separating dc and ac terms yields the following: (a) The dc equation J0

-

-12

where F , = 0, Vo = 0 at x (b) The ac equation

W W x + epoNdFo,

= 0, and

V, = V , at x

J 1 = - C L ~ E E d(F,F,)/d.x ~

where F , = 0, V, = 0 at x

(23)

~ 0 ~ ~ 0

= 0,

= d.

+ e,uoNdF, + jwseoFl,

(24)

and V, = D,,at x = d.

We first find the characteristic for two limiting cases:

'

(a) Verq' small V,. Here injection is negligible and hence the first term in Eq. (23) is negligible. In that case Fo = - V,/dSaor -J, = -epoNdFo = epoNdV,/d = J , ,

(25)

where J , , with the change of sign, is the current density for small V,. (b) Very large V,. Here injection predominates, and hence the second term in Eq. (23) is negligible. This yields the solution of Section 4, b, or - J O = g&&OC(O V:/d3 = J b .

(26)

where J b , with change of sign, is the current density for large

v,.

We define parameters Vooand J o o by the condition = Jb = J o o

J,

i.e., Voois the value of V , for which J ,

at = J,.

V , = Voo;

(27)

It follows that

In this notation Eqs. (25) and (26) may be written

6a

-=-10

Va

Joo

Voo

for

~

v, << 1,

voo

2

Joo

This holds for practically the whole i-region. except near x with decreasing x so that F , = 0 at x = 0.

Va >> 1. (28) for -

vo0

= 0. Near

x = 0, F , decreases

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

205

To find the transition from the linear to the quadratic regime we start with the equations' ( a O / e ) d F 0 / d x= -(n

J o = eponFo,

Nd).

-

(29)

and introduce dimensionless parameters

measuring, respectively, the dc field F,. the distance .Y, and the dc voltage In that case Eq. (29) may be written rtwlliu = u / ( 1 - u).

&.

(29b)

Solving this equation for the appropriate boundary conditions yields the parametric solution M’ =

r

-u -

In( 1 - u),

ut/w

=

= -+id2

( 30) -

u

-

In(1 - u).

(31)

Applying this solution to the anode, where M’ = w, and L' = ra, gives the characteristic in parametric form. Introducing J o , and V,, yields the universal characteristic - J o / J o o versus V,/V,, in parametric form: (32)

where t’, and w, can be expressed in terms of the parameter u, with the help of Eqs. (30) and (31). We now have the following limiting cases: (a) V , very small: u,

= 1, r, 2 wa 2

-Jo/Joo

1

(b) V , very large: 11, << 1, M’, pansions in u, for w, and ",,

-J,/Joo

2

-In(l - u,), or (linear regime).

V,/V,,

2

iu,’,

( I$'Voo)2

11,

2

(324

fu,", as follows from Taylor ex-

(quadratic regime).

(3%)

In between, the full equations (32),(30),(3I ) must be used.

Figure 4 shows the normalized characteristic - J 0 / J , , plotted versus

V,/voo. We solve the ac equation for the limiting cases of very small and very large

V,. For very large V, the second term in the right hand of (24) is negligible and we again obtain the small signal ac admittance (20). For very small V, the first term on the right hand of (24) is negligible and F , = - ral/d.Hence

206

A. VAN DER ZIEL

I000

-

80.0

4oD --Je 601)

Joa

-

20.0

10.080

-

40

-

2.0

-

6.0 -

1.0 0.8

0.6

-

04 -

-

0.2

/

oh

1.

62

0:s

;o

OS&b

iiJ

Waok

+v,/v,

FIG.

4. Normalized - J 0 , V , characteristicof a single-injection diode without traps.

the small-signal ac admittance is Y = -J,/v,i

= epL,Nd/d

+ jdEEL,/d)= gd + j m c d ,

(33)

where g,, = epoN,/d is the intrinsic conductance per unit area and Cd = &cO/d the electrode capacitance per unit area. In between there is a transition from (33) to (20).

d. Hot Electron Effects in n-v-n Devices If the field in the device becomes very large, the mobility pn becomes field dependent and decreases with increasing IF[. We shall see that we can solve this problem in closed form6bif the drift velocity ud, defined as ~ d ( l F o l )= -pn(lFol)FO = ~ n ( ~ ~ o ~ ) \ ~ o ~

is a known function of lFol. 6b

We owe this suggestion to Dr. M-A. Nicolet and his co-workers.’

(34)

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

207

We solve this problem first for the case of arbitrary intrinsic conductivity (Nd# 0) and then let Nd go to zero. We then have, instead of Eq. (23),

Solving for dx, introducing IF,( as a new variable, and bearing in mind that - J , is a positive quantity, yields

or, integrating over the device length,

where 1F.l is the absolute value ofthe field strength at the anode. Furthermore,

The calculation of the characteristic now proceeds as follows: We assume a value of - J o and integrate the right-hand side of Eq. (36) up to the limit for which we obtain the value d ; this determines the value of \Fa\.We then evaluate the integral (37) and obtain V,. The calculation becomes inaccurate if the term E E , u ~ ~ ( F ,is( /small ~ x in comparison with the term f ? N d U d . But in that case it is obvious that -JO

= eNdud((FO().

(38)

This is the case for relatively small field strengths. If V , is so small that pn = p o , we have IFo\ = K / d and hence -J,

= ep&"V,/d.

(384

At low values of V , the characteristic is thus linear. There is another limiting case to be considered. At sufficiently high field strength, the term eN,ud is small in comparison with azOuddIFOl/dx.In that case we may neglect the term e N d U d in comparison to - J o . Equations (36)(37) then become

208

A. VAN DER ZIEL

so that the characteristic is obtained in parametric form. Assuming a value of (Fa(and knowing ud( IFol) as a function of IFo[,the integrals can be evaluated. - J o and V, are now functions of Fa and hence of each other.' We thus have the two limiting characteristics, given by (39)-(40) and (38)(38a), respectively. In between there is a transition from the one limiting characteristic to the other that is governed by Eqs. (36)-(37). This transition can go in various ways:

(a) N, quite small. The characteristic goes from linear to quadratic, as described in Section 5,c, and then terminates in the high-field form described by Eqs. (39)-(40). (b) Nd quite large. The characteristic goes from linear [Eq. (38a)l to sublinear [Eq. (38)] to the high-field form described by Eqs. (39)-(40). (c) In between there is a narrow range of N , for which the characteristic goes directly from the linear to the high-field form. We thus see that it is not necessary to have explicit expressions for pn(IFo(); the characteristics can always be evaluated numerically by means of Eqs. (36)-(37) or (39)-(40). Previously the characteristic was calculated for the following cases8:

(4

p,, = p o ( ~ c / ~ F o [ ) ' ~Nd z , negligible.

(41)

Here F , is a critical field strength determined by the material. In that case 10 5

- J o = T(3)

V,3'2 E & o P o F Y d3'2'

where u, is the limiting velocity of the carriers at high fields. In that case - J o = ?"EoU,V,/d2. (44) Since these field dependences are often not met in practical situations, they must be used with some caution. Equations (36)-(40) do not suffer from this defect.

e. Pulse Response in n-v-n Diodes' We shall now investigate the pulse response in an n-v-n diode with negligible extrinsic conductivity (", 2 0 ) . Let at the instant t = 0 a voltage V , be applied to the anode. What then is the current transient for t > O?

' J. L. Tandon. H. R. Bilger, and M-A. Nicolet, So/ir/ Srure Elcwrou. 18, 113 (1975). Compare. e g . A. van der Ziel, "Solid State Physical Electronics." 2nd ed.. Chapter 18. Problems 6 and 7. Prentice Hall, Englewood Cliffs, New Jersey, 1968.

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

209

Before solving this problem let us first draw a physical picture. When the voltage is applied, a space-charge front starts to move from cathode to anode. Between the space-charge front and the cathode the current flow is mostly by drift, but between the space-charge front and the anode there is no charge, and hence the current is displacement current. This picture is sufficient to solve the initial phase of the problem. The current equation is given by (9).

?F2/r'x + C E O c'F/'c?t.

j ( t ) = -$t:EO/lO

(45)

Integrating over the device length tl and bearing in mind that

since V, is constant and F ( 0 ) = 0, we obtain, j(t)=

- ( ~ ~ 0 / ~ 0 / 2F;f(t), (/)

(46)

where Fa is the field strength at the anode. But the currentj(r) is also equal to the displacement current at the anode. or j ( t ) = cco

tlFa/di.

(47)

Equating the two expressions and integrating, we obtain

where t o = LI2//J0v,.

Substituting into (46) yields

.i(r) = - . i o / [ I

-

%r/b)]23

where

,in= c t : , , ~ i ~V:,Qt13. )

(494

We now evaluate the time t , it takes the space-charge front to travel from cathode to anode. Let u d = -/lot.', be the drift velocity of the front, then

=

-2dIn(l

-

tt,/r,)

or

r , = 2to( 1

-

e - ' '')

= 0.786t0.

210

A. VAN DER ZIEL

1.4 1.2

1.0 0.8

I01 0.6

a4

0.2 0 0

a4

0.8

12

,

1.6

2.0

2.4

FIG. 5. Pulse response I ( t ) of a diode subjected to a step voltage for the cases of no trapping and of trapping. Here z is the trapping time in units t o . (From A. Many and G . Rakavy, P h p . Rer. 126, 1980 (1962); reproduced in Fig. 1.6 of Lampert and Mark.')

The current j, at that point is

After t > f 1 the space-chargeregion relaxes and the current changes gradually to [see Eq. (16)] j ( co) = - g

~ vf/d3 ~ = ~- 2j , p.

~

(52)

The latter part of the transient was evaluated by Many and Rakavy.' Figure 5 shows -j / j oplotted versus [ / t o . We see that there is some overshoot given by (51)followed by a slight undershoot and that the transient is completed in a time interval of the order of to. If there are trapping effects in the device, the transient starts out at the same current -j / j o at t = 0, obviously stays lower than for the case without traps for t > 0, peaks at the same time as without traps, and drops to much lower values after the overshoot (see Fig. 5 and the paper by Many and Rakavy ). Up to here we have discussed the problem of the current transient due to an applied step voltage. Zee and Lampert" have also evaluated the voltage transient due to a step current. Practical applications are discussed in Part V. lo

A. Many and G. Rakavy, Phys. Rev. 126, 1980 (1962). B. Zee and M. A. Lampert, J . Appl. Phys. 45,4416 (1974).

3.

211

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

5. SPACE-CHARGE-LIMITED FLOWWITH TRAPS u. Deep and Shallow Traps in n-Tj,pe Murerial

Let no and n,, be the equilibrium densities of free and trapped electrons, respectively, and let n and n, be the densities of free and trapped electrons under injection conditions. Then tio + IZ,, = N,, where N, is the donor density. Let N, be the trap density and N, the effective density of states in the conduction band. Let E, be the energy of the bottom edge of the conduction band, El the energy of the trap level, E,, the equilibrium Fermi level and E , the Fermi level under injection conditions. Then in equilibrium

no = N,exp[(E,,

-

E,)/h-T],

where g is the degeneracy factor (or statistical weight) of the trap level. In nonequilibrium n

=

N , exp(E, - E , ) / k T ] ,

N,

n1 =

1

N,

-

(54)

+ (l/g)exp[(E, - E,)/kT] - 1 + N/gn'

where in both cases N

=

N,exp(E, - E , ) / k T ] .

(544

If we have deep traps, that is, if -exp

~

Y

<< 1

or

then

If we have shallow traps, that is, if

then

where fl

= N/gN,

is a constant << 1.

N -

9n

<< 1

and

N -

9"o

<< 1,

(55)

212

A. VAN DER ZIEL

We now apply these results to the evaluation of the device characteristic with traps.

6. The Case of Deep Traps in n-Type Material The equations of the system are now -(EEO/e)dFo/dx= n - no

+ n, - ntO,

J o = enpoFo.

(59)

Substituting Eq. (56) into the first half of Eq. (59) yields - ( ~ ~ ~ / e ) d F=~ n/ d-xN,N/gn

+ N,N/gno - no.

(60)

We now introduce dimensionless variables u = - =no -

n

w = - - e*n&ox,

enopoF0 ,

3 3 2v

u = e nopo

0

E E ~ J. ~

EEoJo

JO

(61)

Carrying out the appropriate mathematical manipulations, Eq. (60) may be rewritten 1

)du

1

= dw,

where A = N,N/gng

is the so-called trap parameter. If there are many deep traps, A >> 1. Since u = 0 at w = 0, the solution of Eq. (62) is w

= [ -h(l

and the potential is u = J : u d w = - [ - ( l1 l+A

- u) - (l/A)ln(l + Au)]/(l + A )

(63)

+ ~ ) ~ - I n ( l - u ) + 1~ l n ( l + A u ) A

1

so that the potential distribution has been obtained in parametric form. Note that Eqs. (30) and (31) are obtained in the limit N , -,0 (i.e. A -+ 0). Let u, u, and w have the values u,, u,, and w, at the anode, respectively; then Eqs. (63)-(64) give a parametric solution of w, and 0, in terms of u,. Putting

the device characteristic may be written as [cf. Eq. (32)]

- J o / J b o = Z(~/W,),

VJVbo = PVJW,'.

(66)

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

- -

213

As before, we now consider two limiting cases:

(a) V, relatively small : u,

-Jo/Jbo

-

1, L’, w, z In( 1 - u,)/( 1

V,/Vbo (linear regime).

1

(b) V, quite large: Au, << 1, w,

-Jo/Jbo

+ A). Then

1

(664

iu,’. D,z iu,".Then

( V,/Vbo)2 (quadratic regime).

(66b)

In between, there is a sharp transition from the linear to the quadratic regime. If A is quite large, this occurs for large values of V,/Vbo,as demonstrated in Fig. 6. For deep trap distributions, see Lampert and Mark's book.' Solutions can also be obtained by the regional approximation method. C.

The Case of Shallow Traps in n-Type Material'

Here Eqs. (59)are still valid, but different substitutions must be made for n1 and n,, . Making these substitutions, we obtain

and the solution is just as in the trap-free case, except that E must be replaced by &/(1 0). The characteristic in the quadratic regime is therefore [cf. Eq. (2611 -J 0 - -;[8/( 1 + fl)]E&OcLO V . W . (68)

+

FIG.6. Normalized - J , . V, characterIstic of a single-injection diode with traps. (From M. A. Lampert and P. Mark. "Current Injection in Solids." Academic Press, New York, 1970.)

214

A . VAN DER ZIEL

If there are several sets of shallow traps, well separated in energy, then the smallest corresponding 0, appears in Eqs. (67) and (68). 6. EFFECTS OF DIFFUSION IN INSULATORS (N,

= 0)

The effects of diffusion on the characteristic of insulator devices have been discussed by Adirovich," Wright,3 Lindmayer et al." and others. We follow here Wright's discussion. The current density equation is Jo = e,u,nF

+ e,uo(kT/e)dn/dx,

(69)

where use has been made of the Einstein relation. Poisson's equation is

dF1d.x

= - en/&&,,

(70)

Eliminating n yields

Wright now introduces dimensionless variables c = eVlkT, .f = edF/kT, j = -e2d3Jo/~~,pok2T2, b = e2d2N,/EEokT, s = x/d,

(714

where N , is the effective density of conduction band states. The potential difference V is measured with respect to the cathode metal as zero. He further introduces V, = (4, - x)/e, where 4o is the work function of the cathode metal, x the electron affinity of the diode, and V, = (dl - x)/e, where cjlis the work function of the anode metal. Integrating once yields

dflds

+ .f 2/2 - ( j s + a') = 0,

(72)

where a2 is a constant of integration. Next new variables are introduced by writing

2112

21P

( - j s - a2)3/2 (74) 3.i Real z is selected if j s + u2 > 0 and imaginary z is selected

z=-(js+a)

3J

where i "

=

2 312,

or

2

=i

~

J-1.

E. I. Adirovich, Fiz. Tuerd. Tela 2, 1410 (1960) [English transl.: Sou. Phys. Solid State 2, 1282 (1961)l. J . Lindmayer. J. Reynolds, and C. Wrigley, J . Appl. Phys. 34,809 (1963).

3. if (js

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

+ a’) < 0. Making this substitution Wright obtains d’Y/dz’ + (1132) d y l h - = 0

215

(75)

for real z. This has a solution in terms of Bessel I-functions, .y = C Z ” ~ [ / ~1/3(z) I-

+ 11,3(z)],

(76)

where 1,,3(z)and I-1,3(z) are modified Bessel functions of the order 1/3. For imaginary 2 the solution is in terms of Bessel J-functions. The electric potential difference - P~ for real z at any point in the diode is now given by 11

where zo is the value of z at the cathode (s = 0).The electric field intensity for real 2 is

and the electron density for real 2 follows from bn/No = - df/ds, or

Here zo is the value of 2 at s = 0. To proceed further, the integration constants a’ and p must be evaluated. Wright has done this carefully. Two cases must be considered: (a) The low roltage range. In that case j << a’, a’ > 0 and js + a’ > 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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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 << 1. Experimentally Tsai found this relationship satisfied in coaxially mounted samples up to lo00 MHz." 8. EFFECTS OF DIFFUSION a. The FIetcher Solution.21AC Admittance

Fletcher has given a simplified discussion of the p-i-n diode that incorporates diffusion. He considers the device as a p-i junction, an i-region, and an i-n junction in series. Assuming complete symmetry and D, = D, and z, = z p at each point in the device he finds for small applied voltages J,

=

C , exp(eV+dcT),

(127)

where is the junction voltage and m lies between 1 and 2,21awhereas for larger voltages

J o = C2(V - 2yy.

(127a)

This means that for relatively short diodes we obtain a diode characteristic at small applied voltages and a space-charge-limited characteristic due to the i-region at larger voltages. This solution does not seem to agree with the one given in Section 7,b. Nevertheless, the idea of three devices in series is a good one. Bearing this in mind, the most general solution of the ac impedance per unit area of the A. van der Ziel, Solid State Electron. 13, 195 (1970). Van der Ziel's paper contains two misprints. In Eq. (8) j$E,t/E,,'E, should be j?'E,Eo&. In Eq. (12a) ( - j w ~ J + ' should be ( -jwTds)"+’/(n + 1). His equation (lo),which corresponds to Eq. (1261, is correct, however. T. N. Tsai Impedance and Noise Measurements of Double Injection Space-Charge-Limited Diodes. Ph. D. Thesis, Univ. of Minnesota, Minneapolis, Minnesota (1972); T. N. Tsai and A. van der Ziel, Physicu 79B,76 (1975). N. Fletcher, Proc. IRE45, 862 (1957). * l a For relatively thin i-regions, rn = 1 : for thlcker i-regions. m = 2 for small 6 and m 1 1 for larger V,.

l9 19’

3.

227

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

device in the semiconductor regime should thus be written"

where the first and the third terms are diode impedance terms. Usually z 1 = T~ = z3 = zo and g1 = g3 = go. I f Eqs. (127) and (127a) are valid, then go = eJ,/(mkT) and g2 = J o / ( E - 2y). Tsai2' has verified Eq. (128) for a retatively short diode; Fig. 10 shows excellent agreement between the theory and his measurements. For a longer diode (e.g., d Y 4-5 mm), the first and third term in (128) are negligible and V , >> 2 y , so that Eq. (121) should be satisfied. This too agrees with experiment.

b. The Regional Approximation

If diffusion is taken into account, Eqs. (82) and (83) become J , = 4~

J,

= e(n

+ PT)P,F

- eD,?p/?.u,

(129)

+ nT)pnF+ eD,Sn/Sx.

(130)

Going through the same manipulations as in Section 7,a and making use of the Einstein relation, the dc equation (90) now becomes EEO

d

lo3 - -CALCUATED

ULLUE FROM THE FORMULA

___ 5 dVa

3mA rI (PSI

18

ImA 20.8

0.3mA 35.5

0.I3mA 35.5 1

228

A. VAN DER ZIEL

In the regional approximation method one splits the region 0 < x < d into three parts:

Region 1,0 < x < xl: Here the term with d2no/dxz (diffusion term) predominates and all other terms on the left are negligible. Region2, x 1 < x < x 2 : Here the term with d2no/dx2 is negligible and the other terms predominate. For this region the space-charge-limited solution is valid. Region 3. x 2 < x < d : Here the term with d2n,ldx2 again predominates and the other terms on the left are negligible. The equations are solved for the three regions separately, and the boundaries are taken such that the full equation (131) is valid at x = x 1 and x = x 2 ; then the fields are matched at x 1 and x 2 . This must be done for the semiconductor regime and for the insulator regime separately. Lampert and Mark’ have discussed this approximation method. There is good agreement with Baron’s computer solution” for relatively large ratios d / L , , where La is the ambipolar diffusion length of the carriers. It is doubtful whether the approximation method is valid in the ac case, since x1 and x 2 now depend on frequency. We must thus rely on Eq. (128) for the time being. 9. NEGATIVE RESISTANCE EFFECTS CAUSED BY TRAPS

We shall see that in the presence of traps, negative resistance effects can occur. We shall review the study’ of such effects for a sample with NR recombination centers per cubic centimeter with capture cross sections (T, and ( T for ~ electrons and holes, respectively such that on<< ( T ~ . At low injection levels these centers act mainly as hole traps and the hole lifetime zp,low

(132)

= l/NR(L1(Tp)*

where u is the hole velocity and the averaging is carried out over the hole velocity distribution. At high injection levels the centers act as true recombination centers, n z p , and the lifetime of the electrons and holes are equal, Tp,high 2: Tn.high

=

l/Nk(t’an>

>’

Tp.low

(133)

since the capture cross section for electrons now determines the rate of recombination. In that case the previous theory, developed for the semiconductor regime, is valid and - J o varies as V,Z.

*’

R. Baron, J . Appl. Phys. 39, 1435 (1968).

3.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

229

What is interesting, however, is the characteristic in the intermediate current range. In that current range the hole lifetime T~ increases from z~,,,,~ to T ~ . That ~ ~is, ~the~more . holes are injected the easier it is for them to traverse the crystal. Hence if we control the dc current density - J , and deterVo , actually , decreases with mine the anode voltage V , as a function of -.I increasing - J , . In other words the characteristic - J,, V,exhibits a negative slope. This continues until V , reaches its minimum value V,,, and the current - J,, is associated with it. At that point rP reaches its maximum value Tp,high ; from then on the characteristic is quadratic and is given by the theory of Section 7.h. Waxman and Lampert analyzed the negative resistance problem by the regional approximation m e t h ~ d . We ~ ~ refer . ~ ~ to these papers for details. According to R i d l e ~uniform ~ ~ current flow in a regime of current-controlled, bulk-distributed negative resistance is unstable against filament formation. This problem was investigated by Barnett.4b34c.25 The current density distribution in the filament can be studied with the help of recombination radiation. 10. PULSERESPONSE

IN DOUBLE

INJECTION

Pulse response in double-injection diodes was discussed by various authors. We refer to Baron and Mayer’s review paper for details.4b IV. Noise in Space-Charge-Limited Solid-state Diodes26 1 1. DISCUSSION OF NOISESOURCES”

Let X(r) be a noise signal, and let it be developed into a Fourier series for 0 5 t 5 T . If X(r) is properly defined at discontinuities and redefined at the boundaries such that X ( 0 ) = X ( T ) , then

X ( r )=

a,exp(jw,r). n

1

a, = -JOT

T

X(r)exp(-.jw,t)dt,

(134)

where w, = 2xf, = 2xn/T. The spectral intensity of the noise signal X ( t ) is now defined as

A. Waxman and M. A. Lampert. Phys. Rev. El. 2735 (1970). B. K. Ridley, Prof. Phys. Soc. London 82. 954 ( I 963). ” A. M. Barnett and A. G. Milnes, J . Appl. P/iy.\. 37,4215 (1966). 26 M-A. Nicolet, H. R. Bilger, and R. J . J . Zijlstra. Noise in single and double injection currents in solids (review paper), Phys. Stutus Solid ( h )70, 9 (1975). A. van der Ziel, “Noise : Sources, Characterization, Measurement.” Prentice Hall, Englewood Cliffs, New Jersey, 1970. 23

24

’’

230

A. VAN DER ZIEL

where the average is taken over an ensemble of identical systems subjected to independent fluctuations. Define a number M such that

/p

I 1I-r

X ( t ) X ( t + s)cosw,sds <<

+

X ( t ) X ( t S)COSW,SdS

and take T >> M . Then, according to (134) 2 2a,a,* = - JOT

JOT

T2

=

$

2 =T2

du

s'

X ( u ) X ( u exp[jo,( )

-u

+ u)] du du

jyi"X ( u ) X ( u+ s)exp(jo,s)ds

du J!M

0

X ( u ) X ( u+ s) exp(jco,s) ds

where s = o - u. The two domains of integration in the second and third step are nearly equal for T >> 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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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 << 1. where z is the time constant of Au,(t). Here

D, is the diffusion constant for electrons.

232

A. VAN DER ZIEL

To prove the expression for D,, we observe that Au,(t) is also the cause of diffusion. That is, the fluctuation A x in distance traveled during the time t is Ax =

li Au,(u) du,

Ji Ji A u , ( u ) A u , ( w ) ~ ~ ~liw du J:," AL~,(u)AL~,(u + = Ji du AuX(u)At1,(u+ 2t Au,(u)Ao,(u + s ) d s

L\xz =

=

s)ds =

m: J

so"'

s)ds

= 2D,t,

(138c) according to Einstein's well-known relation. The step from the second to the third expression is valid when t >> 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

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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,

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

= 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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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)<< V, at all times 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.

SPACE-CHARGE-LIMITED SOLID-STATE DIODES

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.

This Page Intentionally Left Blank

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

250

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).

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

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).

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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.

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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)

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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)

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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.

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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).

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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).

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MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

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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).

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MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

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.

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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.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

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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)

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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,,,
1 P,. m= 1

This formula is easily implemented, given the P,. We return to the implementation of (19). A final state of a scattering is specified by more than one variable, and therefore must be determined by more than one random number (with the exception-see below-of specification by a discrete variable and one continuous variable).Normally in practice there are distinct modes of scattering, each having a continuum of final-state p vectors, pa. These alternative scattering “channels” are conveniently distinguished by values of a discrete variable, which we denote by i. The usual computational procedure is to first choose an i value and hence a channel, in accordance with their competing probabilities P i ;then choose coordinates of pa. For elastic scattering (such as from impurities in a solid’*) the possible pa vectors lie on a surface of constant energy, E(p,) equal to E,. For scattering by absorption or emission of acoustic-mode phonons, if the phonon energy be neglected (as is usually, though not always, appropriate) then it is the same surface, though a distinct channel. Emission and absorption of an optical-mode phonon provides two channels per branch of the phonon spectrum. If variation of phonon energy within the branch may be neglected, for the states reached by scattering in practice, then the final states of the channel lie on a surface of constant energy

E,

= E,

f Aw,.

The same applies to intervalley, and also interband, scattering, where the final states in practice lie within a relatively small part of the Brillouin zone. If the foregoing conditions are combined with (24), or its tensor-mass generalization,” and with a constant scattering (matrix element(’ in each channel, then the scattering is isotropic and the scattering rate is given by (39). l* 9

See, however, N . Friedman, Phys. Reo. 8 7 , 1463 (1973). The tensor-mass case reduces to that of (24), by a linear transformation of p space. See. for example, Ref. 10.

4.

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267

In this case the selection of the final state is simple. Because of the spherical symmetry of the system, in the absence of a magnetic field we need only E the electron energy and X the cosine of the angle 6 between p and 8.(The azimuthal angle 4 is of no interest, unless we are calculating quantities such as the transverse component of diffusivity.) The final state distributions of i and x are independent. The former is obtained by Method lD, with p i = C,(E + E p / r

(53)

(with r greater than the values (39) in the allowed energy range as indicated in Section 2). The latter is obtained by a simple instance of Method 1 : X

=

1 - 2R.

(54)

As indicated above, a single pseudorandom number per scattering is sufficient in such cases. A hypothetical piece of FORTRAN source code will illustrate this (where R is a pseudorandom number, generated by a preceding statement):

sl=... IF(R. LE . S1) GO TO 150 R1 = R - Sl s2= ... IF(R1 . LE . S2) GO TO 250 R2 = R1 - S2

s3= . . . ......................... 250

CONTINUE EA = EB - EOPT RR = R1/S2 PA= ...

......................... For the channel branching to Statement 250, values of R R will be uniformly distributed in (0,l), and can be used to determine the final state for that channel; and similarly for other channels. A moregeneral situation applies to many cases ofinterest: The band system is still spherically symmetrical, in the sense that E(p) depends only on the vector modulus p and the scattering probability in a given channel depends on the angle between Pa and pa but not on their separate orientations. Then (still in the absence of a magnetic field) the hot electron distribution has rotational symmetry about the direction of 8,and we still have f = f’(E, X ) . In selecting a scattering, however, the relevant direction variable is not X but the angle between pa and pb. After this angle is selected, a further random number is needed to determine x,. The following particular procedure has been used for

268

PETER J . PRICE

semiconductors of the GaAsZo and CdTe’ types where a predominant scattering process is absorption or emission of an optical-mode phonon coupled to the electron by the “polar” electric field induced by the lattice displacements.

(a) Select one of the scattering channels, by Method lD, using for the Pi the channel total scattering frequencies divided by r. Compute the new energy as in (52). If the selected channel has isotropic scattering, determine X, by (54).Otherwise: (b) From the scattering distribution function P i ( Y ) given that the scattering occurs in this channel, determine Y the cosine of the angle ’+’between pa and p,. This could be by one of Methods 2-4. (c) With yet another random number, determine the angle R between the (8, pb) and (pa Pb) planes R = 271R. (55) 7

Then compute the final-state X value by

x, = X,Y - [(l - X,Z)(l - YZ)]’/2COSR.

(56)

The new azimuthal angle for the (8,p) plane about 8, +a , could then be obtained, if required, from (4, - 4,) given by Eqs. (B5) and (B6) of Appendix B. This is not necessarily the only available procedure. For example Y could be determined first, as in (54); then a scattering channel would be selected by using the resulting values of the S function for each channel as scattering probabilities. One might, similarly,first choose a value for A 4 = 4 b - 4a,the azimuthal angle between the two (8,p) planes, by use of a random number as in (55), and then determine X, from the dependence of the S function on

Y = x,x, + [(1

-

X,Z)(l - X,Z)]’/Zcos(4, - 4,).

(57)

In particular, for polar optical-mode scattering with the “parabolic” energy function (24) the square of the matrix element is proportion to )pa- Pbl-’, which is given by Eq. (B9) of Appendix B. In this case, however, use of a random number to obtain a A 4 value is not necessary, so long as 4, values are not needed in themselves. The average of the S function over azimuthal angles, with fixed p and 8 values, is given by

- 4p;p,2(1 - XZ)(1 - X : ) } 2o

(58)

Fawcett et a1.’ The formula for acoustic-mode phonon scattering given in this paper is incorrect in the “nonparabolic” case because the coupling matrix element should be that of a plane-wave factor times a deformation-potential operator.

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

269

The result of substituting (58) is an effective S function from which values of X, could be obtained by the foregoing methods. When the total physical scattering rate 1/t varies widely over the electron states of the Monte Carlo history, for a substantial fraction of these states tT will be large. In the latter case, in most executions of the procedure selecting a scattering mode the physical scattering rates (of the modes) will be computed but then the outcome will after all be a self scattering. This wasteful situation may be ameliorated by using a conveniently varying r and corresponding generalization of (21).9 Alternatively, with the usual constant r: Let 1 be an easily computable function of the electron state, predetermined in the initialization phase of the Monte Carlo “run”, which is positive and such that the self scattering rate r - l/t exceeds it by only a relatively small amount (positive semidefinite function) A2 . (In particular, the range of the electron energy E could be divided into equal intervals and il could be a step function of E, constant in each interval, with predetermined step values.) Then, in the scattering mode selection, the condition 1 > 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.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

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.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

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.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

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
(103)

where x is the scattering operator defined in (10).A similar equation isz7

sfb - BH,,= (68/d)(g,

- .fa),

(104)

where B is here the “accelerate, then scatter” operator that carries f, into itself.27 Such an equation suggests a procedure in which “source states” provided by a Monte Carlo history give contributions to the right-hand side, which are each the beginning of an “after-effect” Monte Carlo sequence giving a contribution to 6 u ; there is a formal similarity to the “autocorrelation” procedure for calculation of diffusivity, discussed in Section 8. A calculation of differential mobility on these lines has been reported.28 Time dependences which are not periodic, but are of limited duration, can also be calculated by Monte Carlo simulation. This is done by computing a set of “histories” for a single time interval. They form an ensemble, with their starting states belonging to a prescribed initial distribution (for example, the thermal equilibrium distribution just before an electric field is “switched on”), all describing the same situation with the same physical parameters, but of course generated with different pseudorandom number^.'^ The scheme of Section 15 is a version of this. 8. DIFFUSION The calculation of diffusion by Monte Carlo simulation introduces the position coordinates of the particle. An integral of (2) in analytical form is available for particular energy functions like (24), but not in general. For the field direction. however, the component of r is given by an adaptation of (28); it is equal to (Z AE)/ed. The diffusivity involves a correlation between particle variables at different times, and hence it is not given by an estimator is the operator combination “C .4” in P. J. Price. IEM J. 14, 12 (1970); denoted by “T” in Ref. 28. The distribution 9. is delined in Ref. 27. C. Hammar. Phys. Re[?.B 4, 2560 (I97 I ) . 29 One may also calculate thermalization in terms of “distance travelled” instead of time. See: J. G. Ruch. IEEE Trans ED 19. 652 (1972); T. J . Maloney and J. Frey, J . Appl. Phys. 48. 781 (1977).

” It

’’

280

PETER J. PRICE

of the simple type (30);a special algorithm is required for the computation. In what follows, we deal with the components x, u, and u of position vector, drift velocity, and particle velocity in some specific but arbitrary direction. Diffusion refers to the spreading out of a system of many particles as they tend to uniformity in space. According to Fick’s Law, it is given by &/at

= D d2n/dx2-

u dn@x

(105)

in the present case, where u(&) is the drift velocity, n(x,t ) is the space distribution function, and D the diffusivity in this coordinate direction. On defining moments M(’)=

s

xndx

and

we have from (105) dM“’/dt

= U,

dM‘”/dt = 2 0 ,

and Thus. ( d / d t ) ( i / ( x- ( - ~ ) ) 2 > =

(111)

where the ( ) means an average over the ensemble of particles. The validity of this result depends on the validity of (105). The latter is certainly not a complete description. As the particles spread out, from some origin in space, their p distribution, which is suppressed in n(x,t), will become not entirely independent of x. One can, however, see from microscopic considerations that the left-hand side of (1 11) should reach a limiting value as t increases; the argument is developed in the following paragraph. Then D may be defined as equal to this limitingvalue. For a Monte Carlo calculation, the ensemble of distances x may be obtained from effectively independent portions, each of duration t, of a single electron history30or from a set of independent histories ~ ’ it is found in practice that M‘2), the second each providing an ~ ( t ) . Then A. Alberigi Quaranta, V. Borsari, C. Jacoboni, and G. Zanarini, Appl. Phys. Lett. 22, 103 (1973). Frequency dependence of D is calculated, using a generalization of (1 14). by G. Hill, P. N. Robson and W. Fawcett, J. Appl. Phys. 50,356 (1979). 31 P. A. Lebwohl and P. J. Price, Solid State Commun. 9, 1221 (1971).

30

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

281

moment of x, eventually for large enough t increases linearly with t ; half the limiting rate of increase is the resulting value of D. By taking x as x(0, t ) ,the time integral of I‘ from 0 t o t. and differentiating the averaged quantity inside the ( ) on the left of (1 1 l), we get

D

=

(lim)((r(r + s)[.u(t,t + s ) - (x)]))

where the “(lim)” means that we expect the two sides to be equal in the limit of large enough time interval s, and the change of notation for the time variable emphasizes the independence of the starting time. On alternatively taking for x the integral of 2’ from - r to 0, and differentiating, we get from (111)

+

D = (lim)((~~(t)[x(t,rs) - (.Y)]))

and therefore

The (( )) represents a double averaging: over all initial states and over all final states resulting from each initial state. With the latter average taken first, the expectation of the second factor ( r - u ) in (114), for a given state at the initial time t, will tend to zero with increasing time t’. Correspondingly we expect the final-state average of (.u - (x)) in (113) to tend to a limiting value, which will be a function of the particular initial state, with increasing time interval s. Consequently the right-hand sides of (1 13) and (114) will tend to a limiting value with increasing s; then D is equal to this limit. Written as D = J x nr’(((a(r) - u ) ( u ( t ’ )- u ) ) )

(115)

the result has the familiar autocorrelation form. The form (113) can be evaluated directly in a Monte Carlo simulation, in the same way as M ( ” ; one then calculates a limiting value rather than a limiting trend, for long enough time intervals. The result ( 1 13) agrees with the expression (v 1’) for the diffusivity tensor ofhot electrons where I’ is the differential mean free path, obtained by treating a concentration gradient as a perturbation in the Boltzmann equation. l 2 Although this agreement verifies the use being made here of the second moment of position, it would still be desirable to have an unambiguous “operational” basis for defining and computing diffusivity. and further to be able to examine departures from Fick’s Law for diffusion on a small spatial scale. Accordingly, one may specify the phenomenon in a way analogous to the general definition of the dielectric function. Suppose that, for the hot electron steady state of Section 2, there is an additional scattering process

282

PETER J . PRICE

with scattering function S’(B, A) = d3(p, - p b ) y [ 1

+ A COS(Y.~,)].

(116)

The additional scattering rate is a constant y independent of initial (B) state. In the final (A) state, the p value is unchanged and the particles are redistributed in position with a sinusoidal distribution, in accordance with the final factor of (1 16). (Of course 1, 5 I .) The effect is like an absorption and reemission of particles which does not change the p distribution due to the constant electric field. The resulting steady-state distribution in space will be a constant plus a linear response (sinusoidal with, in general, a phase shift) to the I term. Accordingly. it is convenient to make the customary substitution cos(qs,)

+ exp(iqx,).

(117)

Then the space distribution is n ( x ) = no[ 1

+ p exp(iqx)],

(118)

with p a complex number proportional t o 1~1. By adding the “scattering” effect given by (116) and (117) to the right-hand side of (105), and setting the total rate of change equal to zero to represent the steady state, we get for the coefficient in ( 1 18) p / i = y/(y

+ uiq + Dq’).

(119)

The idea is to take (119)as the definition of D ;it will give in general a complex function D(q,y). The limit D(q,O) is the generalization of D to nonlocal diffusion. We expect D(0,O) to be equal to the Fick’s Law D. From the microscopic point of view, since the p distribution of the A states in the process given by (116) is the same as for the homogeneous steady state, we have = Je-’qxtl.y

i

Jox e - ” ’ y t l t n ( x , t ) .

(120)

The right-hand side is calculated3* by adding a scattering channel, with rate y, to the ordinary Monte Carlo particle history; taking the distances x traveled between successive pairs of these scatterings as an ensemble; and computing the average of exp(iyx) over this ensemble. The value of this estimator for (120) is equated to the right-hand side of (1 19), and the solution for D is then the value of D(q,y). The latter is in general a complex number. The imaginary part represents the particle flux in quadrature with iinldx, in excess of the contribution from the second term on the right of (105).The limit D(0,O) will be real, being the ordinary diffusivity, and it should agree with the D value obtained as described earlier. The more general function D(q,y), and the limit

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

-501 103

I

I

I

104

105

106

283

cm-‘

FIG. 3. Nonlocal diffusivity versus wavevector. for n-GaAs at 300 K in a field of 10,000V/cm Curves 1 and 2, respectively, are the “real” and “imaginary” components. (After Jacoboni.”)

Dfq, 0) especially, are of interest in themselves. The Fourier transform with Applicarespect to q expresses directly the nonlocal aspect of diff~sion.~’ tion of the foregoing procedure to diffusion in the electric field direction in H - G ~ Agives s ~ ~curves of the real and imaginary parts of D ( q , y ) versus q, for fixed field and values, that tend to limiting D(q,O) curves. Figure 3 shows a pair of curves close to the limiting + 0 ones, for a field of 10 kV/cm and temperature 300 K. (The hump in the curve of the real part is reminiscent of the maximum in the ordinary diffusivity versus field, which latter is evidently due to the contribution from intervalley transitions.) ; !

;I

IV. Spatial Structures 9. INTRODUCTION

The discussion of diffusion, in Section 8, illustrates that position coordinates may be included in the set of particle variables, and may be used to calculate a spatial phenomenon. The latter was, in that case, an effect independent of position in the substrate solid; the calculation thus depended on translation-invariant correlations rather than on absolute positions of the particles. The original use of Monte Carlo simulation (in modern times) was to calculate localized effects depending on position in a spatial structure of arbitrary geometry-the passage of neutrons, accompanied by fission and scattering events, through a nuclear reactor. It is relatively simple for Monte Carlo programming, as compared with the numerical solution of the corresponding Boltzmann equation, to take account of complicated boundaries between regions with differing properties. We expect to be able to accomplish analogous calculations for electron transport effects in solid structures by Monte Carlo simulation. One may distinguish three types of these: escape and penetration phenomena. size effects, and junction properties. 32

C . Jacoboni. Phrs. Stafus S o l i d ( h )65, 61 (1974)

284

PETER J . PRICE

10. ESCAPEAND PENETRATION PHENOMENA

In these, electrons are “generated”-excited to states at energies above thermal-in a confined region, and make their way to another place where they are detected and their distribution over a spectrum of states may be determined; or a beam of electrons (ideally, monoenergetic and well collimated) enters a solid where the electrons are scattered and partially thermalized. In the former case (the escape phenomena) the purpose may be to investigate the initial excitation itself, in which case calculation is needed to allow for the effect of the intervening medium on the final distribution of states at the detector; or the purpose may be to investigate this transformation of the generated initial electron distribution, as a property of the solid medium through which the electrons pass. Usually “the geometry is planeparallel,” as when the region through which the electrons escape is a thin film deposited on a plane surface of a substrate and the electrons are generated, in the film or the substrate, uniformly over a parallel subregion. In the latter case (penetration) the incident electrons may be the scanning beam of an electron microscope; or they may be used for selectively etching a mask covering a semiconductor, to delineate a device structure in the latter. In photoemission experiments a beam of incident light penetrates to a short distance below the surface of a solid, and excites electrons which emerge from the solid into vacuum where their spectrum can be determined33 or which pass in the other direction into a semiconductor substrate where they produce a measured current.34 In secondary emission experiments an incident electron beam excites electrons of the solid, and these make their way to a detecting region.35 In either case the emerging electrons have to surmount an energy barrier at the solid surface or interface, and to do so must retain some part of their excitation energy. Otherwise they are lost to the detector. The critical aspect of the intervening scattering processes is therefore the net energy loss. An important process is electron-electron (e-e) scattering, which is very inelastic for the individual electrons. The usual “phonon scattering” will provide appreciable electron energy changes, as well as randomizing the direction of motion. Scattering off defects and grain boundaries may also be important, contributing to the randomization. In addition the electron’s state will be modified when it either crosses or is turned back at a solid boundary; the reflection, or the transmission, could be “specular” or “diffuse” or some combination of these. At least if generation of secondary electrons is negligible, the needed Monte Carlo calculations are technically straightforward. For each successive path F. Wooten, J . P. Hernandez, and W. E. Spicer, J. A p p / . PIiys. 44, 1 I12 (1973). R. Stuart. F. Wooten, and W. E. Spicer, PIiw Rev. 135. A495 (1964). 3 5 F. Wooten,J. Appl. Phys.44, 1118(1973).

33

34

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

285

between scatterings, the position of the electron is computed. If the path would cross a boundary, it is instead terminated at the boundary and is followed by the path generated by a boundary transmission or reflection process. The latter is, therefore, not included in the schedule of scattering processes that determines the path durations. The possible difficulties in such calculations are not in the programming principles, but rather in the complexity of the scattering process (and possibly the excitation process) that may need to be incorporated in the model, and especially the lack of detailed information on these. Experiments of the type discussed here may be an effective means of augmenting this information. The complexity may make Monte Carlo calculation, in relating experimental results to the possible scattering details, a practical necessity. In e-e scattering there are two initial and two final particles. Since in practice one of the initial electron states belongs to the Fermi distribution of the solid in the absence of excitation, this state is given by the random number scheme used to generate the scattering event. We then have in effect one initial and two final particles. If both of the final electrons have enough energy to lead to contributions to the detected “escaping” flux, or other calculated outcome, then both must be accounted for in the further computation. This could be done by arranging to store the state variables of many electrons, and following all the products of successive particlemultiplying scatterings simultaneously in the simulation. The occasions when the number of daughter electrons exceeded that provided for, in the allocated storage, could be counted on to be rare. Alternatively, a weighting could be included with the electron state variables, to represent the number of daughter electrons resulting from the initial one. When an eee scattering producing two electrons (each with sufficient energy to be subsequently accounted for) occurred, one of the two final states would be adopted by a random choice as the initial state for the further simulation, and the weight would be doubled. (The initial and subsequent weights would therefore be 1,2,4,8, 16 . . . .) The final weight of the emerging electron would be used in the calculation of the yield from a given source excitation. One could, for the sake of an improved estimator, use a hybrid procedure in which, when a doubling causes the number of daughter electrons to be one too many for the allocated storage array, one electron chosen at random is eliminated and a weighting represents the true total number thereafter. More complicated system “geometries” than plane-parallel need not entail serious difficulty. The distribution of the electron trajectories in space, and in particular the lateral spread of the electrons dispersed from a point or pencil-beam source, is of interest for electron m i c r o s ~ o p yand ~ ~ for mask 36

R . Shimizu, T. Ikuta, and K. Murata. J . Appl. Phys. 43.4233 (1972);T. Koshikawa and R. Shimizu, J . P h w D 7. I303 (19741.

286

PETER J. PRICE

etching in device fabri~ation.~’ The thin films grown in practice are commonly polycrystalline. Therefore one may want to incorporate grain boundary scattering into the calculation. A plausible way of doing this would be to generate a specific crystallite structure, designed to be representative of the actual ones, by the use of random numbers. The electron simulations are then for one or more of these sample films. 11. SIZEEFFECTS

The size effects are the influence of a boundary of a solid on transport phenomena occurring within it. For example the electrical conductance of a metal wire, normally proportional to its cross section, falls below this proportionality when the radius of the wire is not large compared to the electron mean free path. The current is decreased by the additional scattering of electrons at the surface. A similar phenomenon in semiconductors is an aspect of field effect conduction. When a semiconductor is made one electrode of a capacitor, an appropriate applied voltage gives an accumulation of carriers on the semiconductor side of the interface with the dielectric. The semiconductor current due to an additional field parallel to the interface is given by a conductance that is increased by this accumulation. The increase may be less than proportional to the number of excess current carriers, however, because the latter are scattered at the interface and consequently have a mobility lower than the bulk value. It is this mobility field effect that is discussed below. In an idealized view of the field effect, the interface is macroscopically a plane surface and the equipotentials of the retaining field are planes parallel to the interface. The carrier paths, between scatterings, are given by the combined retaining and driving fields (which are mutually perpendicular) in accordance with Eqs. (2) and (3), so long as the retaining field is not so strong that the motion normal to the interface must be taken as quantized. The retaining field is not constant; it decreases to zero from the interface to the interior of the semiconductor. The situation is simplest if this variation (deviation from the value at the interface) may be neglected; this will be so if the field times the depth at which it first varies appreciably is large compared to prevailing energies (divided by electron charge) for the excess carriers. We are then assured of a simple relation between the initial and the final path states. The state of the excess-carrier system then depends on the two fields, driving and retaining; on the “bulk” scattering scheme and band structure; and on the details of the “boundary process,” reflection and scattering from the interface. In the illustrative calculation to be described here, to examine the dependence of the behavior on these boundary processes, a simple bulk 3’

R . J . Hawryluk, A. M. Hawryluk, and H . I. Smith, J. Appl. f h y s . 45, 2551 (1974).

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

287

system which would still give results of practical interest was adopted. For this model solid, it is assumed that there is a single band with spherical symmetry and energy given by (24).The scattering is assumed to be as described in Section 4 in relation to (52) and (53): it occurs with emission or absorption of a lattice phonon, either acoustic-mode with energy quantum to be neglected or optical-mode with a constant energy quantum htu, = E , ; and (as is appropriate to a nonpolar semiconductor) it is isotropic in each of these three “channels.” The dependence of the scattering rate on electron energy and lattice temperature is then of the form T(El =S1

(i;)(:)”’ -

-

{ (“a“.)’;’

+S2 N ,

~

where S, and S, are constants, and N,

=

[exp(E,/kT) - I]-’

(122)

is the excitation factor for the optical-mode phonons. The parameter values used in these calculations were S , = 1.28 x 1012 sec-’.

S, = 2.35 x lo’, E,jk = 430 K, m*i(free electron mass) = 0.3.

sec-’,

( 123)

These values are intended, so far as possible within this model, to represent holes in germanium. A lattice temperature T of 300 K was used. The frequency of boundary encounters depends on the fields and on the bulk scattering. It is given directly by the distribution function close to the boundary. The boundary processes, in turn, affect the distribution function and hence the transport phenomena. These processes may be specular reflection or diffuse scattering, and elastic or inelastic. They are assumed here to be purely elastic. There exist two standard models for the distribution of “outgoing” wavevectors, following the boundary impact, over the hemisphere of possible directions. In the model introduced by Fuchs (in his classic theory of the size effect for metals), there is a fixed probability p s , independent of incident direction, for specular reflection ; else with probability (l-ps)there is diffuse scattering. In the former case, the wavevector component normal to the surface changes sign while the parallel component is unchanged; in the latter, the outgoing wavevector has a uniform (isotropic) distribution, over the hemisphere, of possible values. The size effects then depend on a single

288

PETER J . PRICE

parameter characterizing the boundary, p s . The realism of this model has been questioned, on the grounds that specular reflection is likely to be more probable for incident-wavevector directions more nearly parallel to the surface.38 In the model proposed by Parrott, this dependence is taken into account in a simple way: The process is entirely specular if the direction cosine for the incident wavevector is less than a critical value:

case, < ps,

( 124)

where O1 is the angle between wavevector and normal. Otherwise, the process is diffuse. The detailed-balance principle requires that in the latter case the values of the outgoing direction cosine, cos 0 2 ,have a uniform distribution over ( p s , 1). (In the computations, COSO rather than O is the variable that is stored and used.) The parameter on the right of (124) is the probability of specular reflection, when the incident distribution is isotropic. Accordingly, it is appropriate to compare the consequences of the two models for the same value of p, . The provision of path and bulk scattering algorithms of appropriate dimensionality is straightforward. Only the programming of the boundary encounters needs comment here. It was convenient in these computations to treat the motion preceding and following a boundary process as part of a single path, beginning and ending in bulk scattering events. The path duration, following a bulk scattering, was obtained from (21), and the position of the carrier (the coordinate in the direction of the boundary normal) at the final end of this path was calculated. If the path would leave the semiconductor, it was instead terminated at the boundary. Then the particle position was updated to the boundary, the path duration was updated by subtracting the time elapsed in reaching the boundary, and these new values were made the initial values following the boundary process. Further boundary encounters, with position and remaining path duration again updated, could occur as part of the same complete path. The estimator scheme of Eqs. (30) and (31) was used, with a B state population consisting as usual of the initial states of the bulk scatterings (physical and “self-”). Where the path following a boundary encounter can enter another medium, with different scattering processes and a different suitable value of l-, it may be appropriate instead to program it as a new path with a fresh value of path duration from (21). For both Fuchs and Parrott surface models, with the bulk model described above, there was no great difficulty in programming and no undue increase in required CPU time, relative to the basic static-homogeneous bulk case. Evidently this would be true also for a more general dependence of specular 38

J . E. Parrott, Proc. Phys. SOC.85. 1143 (1965); R. F. Greene and R. W. O’Donnell, Phys. Reu. 147,599 (1966).

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

0S 0

289

i 0.2

O

06 0.8 10 FIG. 4. Semiconductor field effect. Effect of varying p,, from 0 to 1, for fixed driving field, 5000 V/cm, and retaining field, 100.000 V/cm. The curves rising from bottom left are drift velocity (left-hand scale). The curves descending from top left are total rate of boundary encounters (right-hand scale). 04

reflection probability on incident direction. Some results of these illustrative computations are given in Figs. 4 and 5. Figure 4 shows the effect of varying p s for fixed values of the retaining field and driving field, respectively 100,000 and 5000 V/cm. At p s = 0 and p , = 1, the two models are equivalent and necessarily give the same results. In between, for a given p s value the Parrott model has the higher drift velocity u, as one might expect, associated with the lower total boundary event rate and hence the lower rate of diffuse scatterings. Figure 5 shows the dependence of u on retaining field, for a driving field of 5000 V/cm and three p , values. For p s = 1, of course, the retaining field has no effect and u has its bulk value. For ps = 0, Fuchs and Parrott

JO !

" 20

'

40 " 60

I

'

80

I

100

xl000 V/cm

FIG. 5. Semiconductor field effect. Dependence of drift velocity on retaining field for fixed driving field. 5000 V/cm. and the p , values shown.

290

PETER J. PRICE

necessarily give the same results, represented by the curve. For ps = 0.5, the two drift velocities do not differ much except at the highest retaining-field values; so only one is shown. The two curves for ps = 0.75 show about the largest difference between the two surface models. Since the Parrott model is an extreme of anisotropy-the opposite extreme would have the inequality (124) reversed-while the Fuchs model is central, one may conclude that surface scattering anisotropy has appreciable but modest effects. 12. JUNCTIONS In the systems considered here, charge carriers move through a structure with parallel-plane symmetry, from source region to sink region, experiencing electric fields normal to the plane. We are concerned with the current due to to an applied potential between source and sink regions, and attendant phenomena. The illustrative calculation here is for an n-n or p p semiconductor homojunction. Semiconductor heterojunctions and p-n junctions, and metal-semiconductor structures, etc., would be similarly investigated. The peculiar combination of strong electric fields with space inhomogeneity in a semiconductor junction, such that there can be a small or zero current in a strong field because the field is balanced by the carrier concentration gradient, is hard to treat correctly by analytical methods. With Monte Carlo simulation there is no particular technical difficulty ; but a special formulation is required. A brief account is given here. The basic idea is as follows. Let two parallel planes in the “lead” regions outside the actual junction, one on each side of it, be A and B. Let the charge carrier flux across A in the direction of B be NA, and let PA, be the probability that a carrier which crosses A in this direction will cross B before it returns across A. Similary N, and PBA for the reverse direction. The net current density across the junction is then (We are still, for convenience and definiteness, taking the carrier charge to be +e.) Given the distribution represented by NA, the transit probability PA, is obtained by simulating many carrier histories, with beginning states belonging to this “injection population,” each terminating at the first crossing of either A or B. It is sufficient for our purpose to assume the currents in the lead regions to be at ohmic levels, with PA, and PBA small. We may then take the fields in the lead regions to be zero, since diffusion-caused and field-caused currents will be equivalenf there. Accordingly NA

= nAuo

where nA is the carrier concentration at A and 2u0 is the average of the normal velocity (taken always positive) for the thermal distribution. For spherical

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

291

band symmetry we have the kinetic-theory-of-gases result 11,

(127)

=+(P).

With the energy function (24) in particular, (”) = (2kT/7rrn*)”2.

Similarly for N , , n,: we assume-though it is not essential to do so-that the temperature and substance are the same at A and B. It can be shown that

~/’PAB = ( u o / D ) t l , + (1/1)An)

+ (uO/D)(P,A/PA,)d,,

(129)

where D is the carrier diffusion constant and d,, d , are the lengths of the field-free regions adjoining the A, B planes. This formula gives the dependence of PA, on d , and d,, and defines an effective transit probability pABfor the actual junction. (The latter is arbitrary to the extent that we may define inner parts of the two field-free regions to be included in the junction.) Since D / U , has the order of magnitude of the carrier free paths. Eq. (129) shows that PA, must be small when d , exceeds many mean free paths (and similarly P B A ) ; but pABneed not therefore be small. Because of the detailed-balance relation, only one of PA, and P,, need be computed. and naturally one chooses the larger of them; for definiteness we take this to be PA,. We have PBA!PAB = PBA/PAu

= 01A/%)

exp( - r k J k T ) .

( 130)

where the applied potential V,,, is positive when the net current density J,, from A to B is positive. In the resulting formula for the current.

the second and third terms in the denominator represent resistance in series with the junction itself. They are not conceptually relevant, therefore, and as the junction current may be written (dropping subscripts of )iAand I),,) J

= euorlp[ 1

-

exp( - e V / k T ) ] .

(132)

In practice, p depends only on the upper part (the A end, in the foregoing) of the junction field region. Once a carrier that was initially at the A end has “descended farther than this, it has little chance of getting back all the wayalmost certainly it will terminate at the B end, and so is predestined to contribute to p . The scheme of computation is as follows. The space between the outer A and B planes is divided, by parallel planes, into regions in each of which the field is constant (or varies linearly) so that the particle dynamics is given by a

292

PETER J. PRICE

simple algorithm. At the beginning of a path, a path duration is selected as usual by (21). The state (including position) at the end of this interval is computed as though the trajectory remained in the same region-ie., the field value, or dependence of field on position, remains that of the initial region. If the entire trajectory, so computed, is in this region, then the path is complete and the computation passes to the subsequent scattering, followed by the next path. If, instead, the trajectory enters a neighboring region, on either side of the initial region, then (a) the state (including position) is set equal to that at the trajectory point where the first crossing occurs; (b) the path duration is reduced by the elasped time to reach that point; (c) the program branches back to immediately after the initial selection of a path duration at the beginning of a path. If, instead, the trajectory crosses an outer (A or B) plane, the trial is terminated, and the count of AB transits is “updated.” A trial begins with a path, for which the initial position is the A plane and the initial p value is selected from a Boltzmann distribution with the normal component of velocity taken in the inward direction. In addition to computing P A , from the fraction of trials that are transits (A to B), space histograms and other measures of the ensemble of carrier particles in the junction may be obtained. For these, the initial states of all scatterings may be used as the representative population, as described earlier. In the illustrative computations reported here, the model junction consisted of the two field-free regions of length d, and d,, and between them a single region of width d with constant field 8.The model semiconductor was the same as in Section 11 and Figs. 4 and 5. Representative values ofjunction width and of field are of order cm and of order lo4 V/cm. Carriers diffusing into the central, field, region from the “upstream” end (the A end) are found to turn back and leave at the A end, if at all, within a few times cm into the field region. Carriers that go any farther downstream almost certainly reach the B end (and so contribute to PA,). Accordingly, while p ( d ) is in general a decreasing function of d, when d is increased beyond a value oforder cm (depending on 8 ) p reaches a limitingvalue unchanged for larger d. This limiting value, p,(b), is plotted in Fig. 6. When it is substituted for p in (132),together with V = db,it gives the effective current-voltage characteristic for this model junction. The current depends here on a boundary phenomenon at the edge of the field region, not on a “bulk” formula for drift velocity in a constant field combined with a concentration gradient. A histogram for the space distribution of the diffusing and drifting carriers is obtained by dividing the space with equally spaced parallel planes into a large number of cells (usually 100 of them in practice) and counting the total number of scatterings occuring in each cell, as an estimator for the time spent in each cell. The counts may be weighted with a particle variable, such as

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

0 0

1 0

293

20

x ~ 0 3V/cm

FIG.6. Transit probability p o versus field I in a model semiconductor junction.

velocity or energy or velocity times energy, to give a density histogram for that variable. The count weighted by E, divided by the unweighted count, gives the average energy for particles when in that cell. Figure 7 shows the average-energy distribution in space (obtained by drawing a curve through the 100 histogram points) for the same case as Fig. 6, with d = 20,000 V/cm. cm; the energy is in units The width of the field region, as shown, is 2 x of the optical-mode quantum ( E , of (123)).Beyond the neighborhood of the A edge of the field region, the population is all carriers in transit. Figure 7 shows that they “thermalize” in the field over a distance of about 1 x cm, and hence over an interval of about 10- sec. The thermalization on leaving the field region at the B end, however, is faster. Other estimators may be freely incorporated into the Monte Carlo program, to provide results on various interesting properties of the system. The average transit time (for those carriers that accomplish the AB passage) may be obtained; its linear dependence on d gives a value for the drift velocity in the field 8.

0 FIG.7. Average particle energy, in units E , ) . as a function of position in the semiconductor junction. The field E is 20,000 V/cm, and the width of the field region, d, is 2 pm.The direction A to B is left to right.

294

PETER J . PRICE

V. Ohmic Conduction 13. OHMICCONDUCTION

‘hematerial concerned with conduction has been, up to here, specifical Y for “hot electrons.” The ohmic electron transport properties-electrical conductivity especially, but also thermal conductivity and thermoelectricity, in the situation where the driving field causes only a small deviation from thermal equilibrium, with a flux proportional to the field-are of great importance in solid-state physics. Their theory especially invites the application of Monte Carlo computation, because there is an emphasis on precisely accounting for numerous details of electron band structure and scatterings, perhaps with the complication of an applied magnetic field or of size effects. A special Monte Carlo technique is, however, needed for the ohmic regime. It might seem that, at least for the calculation of the mobility, we need only compute the drift velocity for decreasing values of field 8 until the ohmic proportionality appears. Methods for analysis of hot electron systems have a way of going bad for small electric fields, however, and Monte Carlo is not an exception. The procedures that have been described are successful, at high fields, because the “PoincarC period”-the time it takes for the particle variables to sufficiently sample the appropriate domain-is short enough in practice to require only a tolerable number of computer C P U operations for its generation. This fortunate situation is liable to break down as the electric field, and hence its contribution to the particle’s meandering through phase space, becomes small. With the scattering scheme of( 121), it is found that as 8 decreases toward zero the distribution of states obtained in the simulation does not finally approach the thermal distribution (13).The correct behavior can sometimes be restored by including the small phonon energies in the acoustic-mode scattering, provided that any approximations in the scattering algorithm are such that the detailed-balance relation, between the rates for a transition and for its reverse, remains precisely satisfied. While it may be possible in some cases to make valid calculations of ohmic mobilities using hot electron programming, there is always the difficulty that the amount of computation required, to obtain the drift velocity with a given relative accuracy, increases as the value of the drift velocity decreases. Furthermore, much of the interest in ohmic transport constants is for metals, where the electron system that is slightly perturbed by the electric field has a Fermi distribution function, whereas hot electron Monte Carlo simulation has been for Boltzmann statistics. It would be better to have a procedure in which the appropriate thermal-equilibrium distribution is generated directly and is used as a source population for the computation of a transport property based on it.

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295

Because of the Einstein relation for Boltzmann statistics p = Dr/kT,

(1 33)

calculation of ohmic mobility p and of diffusion constant D are equivalent. Four methods for obtaining D have appeared here. In each of them, as applied to the ohmic case, a set of trials is generated with initial states drawn from a Maxwell-Boltzmann distribution, and Brownian-motion displacements x ( r ) are computed for these trials. They differ in the way the trials are terminated and in the way the s values are used in the estimator for D . They are : (a) Use of (129) in the form ( l / P )- 1

= (u,/D)d

(134)

for a single field-free region. (b) Use of the procedure indicated by (1 19) and (120),and the discussion following, in the thermal equilibrium case. (The trials are terminated by imposing a distribution with probabilities proportional to exp( - 7s) on their durations s.) (c) Use of (114),or an equivalent, in the thermal equilibrium case (where, as for (119) in (b), u = 0). (d) Similarly, use of (1 11) in the thermal equilibrium case ((r) = 0). Both ( c ) and ~ ~ (d)" have been used in practical calculations of ohmic conduction. For the ohmic mobility, ( 1 14) gives

' . the ~ ~case of a degenerate electron where 1 is the vector mean free ~ a t h . ~For distribution, as in a metal, the average ( ) in (135)must be taken not over the Maxwell-Boltzmann distribution ( 1 3) but over .fF( 1 - &), where &E) is the Fermi distribution function, with a normalization factor appropriate to the definition of mobility or conductivity; and with inelastic scattering the scattering function S(1,2) must, in the calculation of I, be multiplied by (1'- ,f&) )/(I - j F ( i )), where the transition is (1 4 2).

'' B. Liithi and P. Wyder. Heh. Phjs. Acta 33. 667 (1960). 4' 42

C. Canali, C. Jacoboni. F. Nava. G . Oltaviani, and A. Alberigi Quaranta. P/ij.s. Rer. B 12. 2265 (1975); L. Gherardi, A. Pellacani, and C. Jacoboni, L e f t . Nuoro Cimenfo 14, 225 (1975). P. J . Price. IBM J . 1, 239 (1957). P. J . Price. IBM J . 2, 200 (1958).

296

PETER J. PRICE

Not just a diagonal component, pxxor oXx,may be required. In particular, the Hall effect, and magnetoconductivity in general, may be calculated on the same lines. Equation (135) remains valid in the presence of a magnetic field provided that v(t), in (135) or in calculating I, represents the path motion with the magnetic field acting (as in Section 6 without the electric field). The linear Hall effect might be obtained from the limit of the effect of a magnetic field as it becomes small. It is also given directly by a formula in terms of 1 in the absence of a magnetic field4’ which in suitable cases may be evaluated directly as a thermal-distribution average of an expression that is quadratic in 1 instead of linear. The size effects are also subsumed in (135); their calculat i ~ merely n ~ ~entails including the particle position in the path computation and thereby including the surface scattering in v(t) and 1, and then averaging appropriately over “initial” position in ( ). The increment of x from a single path is in the absence of any fields equal to u times the duration of the path; this may be replaced by its expectation TO. When there is a magnetic field the increment of x, being equal to the time integral of o(t), may still be equated to the “B value” of tu that terminates the path.42 Thus43

(Vl)

1

i( V

(TV)6”’),

(136)

n

where ( T V ) ~ )is the value at the end of the nth path (just before the next scattering) following the initial state with velocity v. (Of course if we have a constant scattering rate r then z is replaced by a factor ljr outside the averaging over the simulated histories.) In practice the time interval in (139, and equivalently the sum in (136), must be finite. The convergence of the latter (as the sum from n = 1 to n = N , say) is to be assessed from the sum values obtained for an increasing series of values of N . Time dependence, in the form of frequency-dependent mobilities pfw), is obtained by including a phase factor exp(iot) in the integrand of (135), and hence in the B values of (136); the exponent is, of course, cumulative over successive paths. Calculation of time-dependent magnetoconductivity, and thus analysis of cyclotron resonance, should accordingly be possible by elaboration of the same procedures. The thermoelectric constants would be obtained by replacing either v factor in the autocorrelation (135) by (E- ()v, where is the Fermi energy. The same substitution in both v factors gives the quantity required in the calculation of the electron contribution to thermal conductivity. 43

Corresponding to the sum in Eq. (136) is the series in the scattering operator, for vector mean free path. given in Section 5 of Ref. 41. It is equivalent to the direct scattering-series solution for the Boltzmann equation, which was utilized by D. L. Rode, Phys. Reu. B 2, 1012 (1970).

4.

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297

VI. Collective Effects

14. INTRODUCTION In the calculations described thus far, only a single particle is represented in the simulation. This is appropriate for the many phenomena in which the mobile carriers of the solid do not interact appreciably, and the “substrate” (the lattice variables, etc., of the carriers’ environment) is not significantly displaced from thermal equilibrium. But there are electron transport phenomena that one also wishes to calculate, in which such collective effects are appreciable and may need to be included in a simulation. In particular: (a) long-range coulomb interaction between carriers, describable by a space charge ; (b) short-range interaction describable as carrier-carrier scattering; (c) varying occupation and excitation of localized, nonmobile states; (d) avalanche and Auger processes; (e) the “Fermi statistics” reduction in electron scattering rates due to the exclusion principle; (f) hot phonon effects. where the lattice mode excitations are changed by interaction with a nonthermal electron system; (g) phonon drag effect (the ohmic equivalent of the hot phonon phenomenon). Monte Carlo procedures have been extended to deal with a few of these, but it is not evident that pure Monte Carlo procedures by themselves can deal with all of them. In Part VI a proven extension of Monte Carlo is described first; then extensions of pure Monte Carlo for the foregoing calculations are discussed; and, finally, possible adaptations of Monte Carlo by combination with other procedures are pointed out. 15. MANY-PARTICLE MONTECARLO

One may attempt to represent an actual system of electrons by computing a substantial number N of particle Monte Carlo histories “in parallel.” That is, the N sets of particle variables are stored simultaneously in the machine memory, and an increment of each particle history is generated in one cycle of the program, corresponding to a time step. The evolution of this simulation with successive time steps may represent an actual time dependence, or it may be a means of generating a steady state. At each step, many-particle effects are included. The number N will (except perhaps in some device simulations) be much smaller than the actual number of electrons in the system being analyzed. It need only be large enough for the system quantities of importance to be represented sufficiently accurately and without undue

298

PETER J . PRICE

fluctuations (due to the finiteness of the number of particles contributing to the estimator); the required N value can be expected to depend on the particular calculation. A technical problem concerns the nature of the program cycle that is to have the role of a time step, and thereby the programmed interaction between particles that is to simulate the simultaneous, continuous interaction in the actual system. The simplest procedure is to take the intervals between successive scatterings as effective time steps. The N particle states immediately preceding the nth scattering of each particle are treated as representing the actual system after a time n/T, where r is the constant scattering rate (including self scattering) of a particle, and the interactions are calculated for each of these successive B-state ensembles. The elapsed time t, to the nth scattering event of a particle is actually a statistical variable, of course; the N values belong to a population given by a distribution function Q,,(t),such that the integral [Qndt over a given time range is the probability for the nth scattering of a given particle to fall in this range. Then Q A t ) = rpn- l(tL

(137)

where P,(t) is the probability for an interval (0,t )to contain n scattering events. 0bviously Po = exp( - rt)

(138)

and

or, substituting for Q , from (137) and (138), P,+l(t)= r e +

S:, ~ , ( s ) e ‘ ” d s .

(140)

The standard result pn(t)= [(rt)./n!]

(141)

e-‘I

(the Poisson distribution) follows from (138) and (140). By (137) and (141), similarly Q,(t) = [(rt)”-’/(n- l)!]re-rr.

(142)

From (142) we obtain the moments oft,, the elapsed time for n scatterings:

(rnP> = JOT Q,(r) t P d t =

(p

+n

l)! r-p. ( n - l)! -

(143)

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

299

From (143) we have in particular r(t,) =

r2((rn2) (r,)Z) -

= n.

(144)

Thus f, in the sample ensemble will be distributed about its expectation value n/T with a standard deviation h / r . (Similarly, the variance of the number of scatterings in a fixed time interval, from (141). is that of a normal distribution.) As n and ( t , ) increase, we may accordingly expect an increasing degradation of the time resolution obtained with use of the ensemble of states preceding the nth scattering as surrogate for the simultaneous states at time n/T. even though in some cases the agreement between the two has been found t o be quite ~atisfactory.~’ The second disadvantage of this procedure is that it might be desirable to compute a particle interaction, or to obtain an output of results for the state of the system, more frequently than the scattering rate. The alternative scheme is to compute the particle histories between successive “observation times” t‘”, t ( 2 ’ .. . . that are predetermined and are the same for each particle. With all N particle states known at time f@), for each particle in turn a Monte Carlo .procedure is used to compute the further history up to time tCP+I). In this procedure, the time variable of the particle is incremented by a “free path time” (?I), the resulting new value of the time ‘ I , and the path connecting scatterings is then variable is compared with computed up to the lesser of these two times; the process is repeated until P+ is)reached. The scatterings and paths are calculated in the usual way, other than that each segment of a particle’s history is made to terminate at an observation time, rather than at the next impending scattering time, and the next following path then begins at this observation time. It can be shown that the interval between the last scattering before a given observation time and the first scattering after it has the same distribution over its possible values as ordinarily (that is, it is given by Ql),just as if it did not span an observation time; and likewise for the correlation of this interval with preceding and following path durations. The sequence of electron states will not be the same, with this special observation-time procedure, as it would otherwise be (because the number of pseudorandom numbers required, for a given time interval, will differ. and hence the train of pseudorandom numbers themselves will be different); but the sequences of states in the two cases should belong to the same statistical population. It is unnecessary to continue the same path from before to after an observation time, and to do so requires additional computer memory since the remainder of the path duration at the observation time needs to be stored for each particle. Normally the intervals between observation times, At = Pf ’ ) - t(P),will be all equal, but obviously this is not essential; one might, for example, have them spaced more closely for one part of the time range in a calculation. The

300

PETER J. PRICE

spacing At can be greater than l/r,or less than l/r,as a p p r ~ p r i a t eThe .~~ CPU time required will normally depend, substantially, on the number of scatterings rather than the number of observation times. Interactions between particles (for example, the space charge, and the resulting potential, given by their positions) are calculated at observation times-at some or at all of these, as appropriate. The observation times may be made as frequent as necessary for this purpose, with an addition to CPU time that does not include additional scattering computations. Practical experience suggests3’ that there is not good reason for not using this “true time” scheme, rather than the “scattering clock‘’ alternative described earlier. The scheme can be used without interaction between particles, to calculate thermalization

effect^.^' This procedure has been applied to the simulation of Gunn domains45 and of a field effect transistor (FET).46For both of these the collective effect was space charge. In the former, the macroscopic system was assumed to have a one-dimensional “geometry.” The space between the electrodes at the two ends of the crystal was divided into parallel-plane “cells” of equal thickness; and the field averaged over each plane was computed from the space charge on one side of it (i.e., the number of mobile electrons minus the average value given by the doping concentration), and the external emf. The Monte Carlo paths were then calculated for a stepwise-constant field equal in each cell to the average of the foregoing values for the two boundaries of the cell. (For the path procedure appropriate to a field of this form, see the paragraph after that containing Eq. (132), in Section 12.) A simulation model was needed for the passage of electrons into and out of the electrodes. The. model for the semiconductor represented n-GaAs. In the simul a t i o n ~ ~the ~ .initial ~ ’ state of the system was thermal equilibrium (as though the applied emf has been “switched on at t = 0 ) . Instability behavior4* in agreement with observation was obtained. Evidently a correct analysis Since no path duration actually is going to exceed At, we could replace the argument of the logarithm in Eq. (21) by l/(l - a R), where a is a fixed number chosen so that none or few of the pseudorandom numbers R result in s values greater than At and hence are wasted. If r. At is small we might. with small and known maximum error. replace the logarithm by a simple polynomial: a R, or a R - f (a R) . or ..... 4 5 P. A. Lebwohl and P. J. Price. Appl. Phys. Lett. 19,530 (1972). 46 R. W. Hockney. R. A. Warriner, and M. Reiser, Electron. Lett. 10, 484 (1974); R. A. Warriner, Solid-Stare and Electron Devices 1, 105 (1977). 47 M. Abe, S. Yanagisawa, 0. Wada, and H. Takanashi, Jpn. J. Appl. Phys. 14, 70 (1975), using the same procedures, have examined the behavior with a “notch” (a step in the doping concentration) near the cathode. 48 In small bars of n-Ga As, the negative-differential-mobility feature of the variation of electron drift velocity with field, together with time lag in the dynamical response to a varying field, results for suitable macroscopic conditions in high-field domains travelling from cathode to anode: Gunn domains. See, for example, J. E. Caroll, “Hot Electron Microwave Generators.” American Elsevier, New York, 1970. 44

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

301

of the domain phenomenon, with its complex macroscopic and microscopic detail, is possible by this means.49 In the FET simulation the one-dimensional array of space-charge cells was replaced by a two-dimensional lattice of cells; and the field assumed in computing the paths was stepwise constant in each cell (but now two-dimensional, of course). The calculation of this field from the particle distribution, for the particular device geometry, is a much more substantial and elaborate c o m p u t a t i ~ nThe , ~ ~ Monte Carlo procedure was otherwise substantially the same as described above. The model semiconductor was GaAs (with n-channel conduction). Convincing simulations of FET phenomena were, evidently, obtained.46 Obviously, more elaborate junction and other structures could be attempted. Because these simulations are based on representation of the actual microscopic state of the carriers, they are of potential interest for the shrinking time and space scales of present semiconductor device technology. 16. CARRIER-CARRIER SCATTERING

The effect of carrier-carrier scattering (the short-range coulomb interaction between individual particles) in hot electron systems, at concentrations where it is significant, has been a matter of supposition rather than knowledge. Numerical analysis of the phenomenon should be possible with the many-particle Monte Carlo scheme of the preceding section, with the collective effect now in the scattering rather than in the paths.” A pair scattering function S“(pl, pz ;p; ,pi) must be assumed in any such calculation, such as to incorporate many-particle correlations in some approximate way. The final states (pi, pi) of these mutual scatterings are those allowed by energy and crystal momentum conservation. In particular the allowed values of the momentum transfer vector

AP = P;

- Pt = P 2 -

P;

(145)

lie on a surface (or surfaces). In the procedure to be discussed below, for generating these scatterings in the many-particle Monte Carlo scheme, the successive steps are : (a) For a given initial particle state p,,, choose an initial companion state pu randomly from the other N - 1 single-particle states. R. A. Warriner, Solid-Stale and Electron Derices 1, 97 (1977) reports simulating a device with width equal to one quarter the length between the electrodes, using the same “Poisson solver” as for Ref. 46; but he does not describe any lateral dependences in the results. The paper does, however, give the fullest account of one-dimensional Gunn domain phenomena calculated by this Monte Carlo scheme. See, for example, J. W. Eastwood and R. W . Hockney, J . Comput. Phys. 16, 342 (1974). ” The many-particle ensemble is obtained with a single-particle Monte Carlo procedure, by saving the N preceding states in the history (where in practice N was 1500). III A. Matulionis. J. Pozela. and A. Reklaitis, Solid Stafr. Comnrun. 16, 1133 (1975).

49

302

PETER J. PRICE

(b) Use the total mutual scattering rate for this initial pair to decide if the scattering of the vth particle, in this scattering/path cycle of the manyparticle history, is a mutual scattering or is one of the single-particle scatterings. (c) If step (b) results in a mutual scattering, then select values of the coordinates of Ap (and hence of p:,~;) by a process such that they will be distributed in accordance with the form of S". A suitable assumption for the scattering function is the Brooks-Herring formula

(146)

where ii is the mobile carrier density, p , is a shielding constant, and K is a dielectric constant. The normalization here is such that the result of integrating (146) over p space for the p;,p; variables, ;rI1(pl,p2),is the mutual scattering probability per unit time for the initial pair (1,2) multiplied by the total number of particles contributing to this process.' (Averaging p2 in 1,2) over this particle population gives the mutual scattering rate for a single particle in state p1.) If the single-particle energy function ;Ii1(

E applies-which

= p2/2m*

we assume from here on-then

(24) (146) gives

where

rrl= (4ne4/K2)(m*n/p:) and PI2

= P2 - P 1 .

(149)

The maximum value, l-:ax, of y" is in this case YI', or a smaller value if the range of particle energy E in the computation does not extend to (&~~)~/2rn*. With step (a) above carried out for every particle scattering in the scattering/ path sequence, r;,,,,should be included in the minimum value of r in (21);and y"(v, p)/l-, given by (147),is the probability of mutual scattering that competes with the probabilities of other scattering modes in step (b). If r:Jr is small, however, one might save some computation by executing steps (a)-(c) for only a fraction a of the particle scatterings, while replacing all the 7'' in step (b) by ylI/a and similarly for the contribution to r.

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT IN SOLIDS

303

The locus of the p‘ vectors, for allowed mutual scatterings, is a sphere with p1 and p2 the poles of a diameter. The distribution of scatterings on the sphere depends on the latitude, relative to this diameter, but not on the longitude (azimuthal angle). If ( 150)

-y EE p i 2 ‘ AP/P122

with the definitions (145)and (149),then the distribution 0f.x values is given by P(.u)= P h ( l h r 2

+

2 , J , 2 ) / ( P 1 2 .y

+ Pt)’?

(151)

(where J ’ P d sover a given range is the probability of .Y lying in that range). Here Method 1 of Section 4 applies, for choosing an .v value: .Y =

Rp,Z/[p,Z

+ (1 -

R ) ~ 1 2 ~ ] .

(152)

To execute step (c), a fresh R value is substituted in (152). The algebra of going from coordinates of (p,, p2) to ( p Iz , Ap) and thence to coordinates of (pi,pi) will be passed over here. 17. LOCALIZED STATES ; AVALANCHE PHENOMENA

Semiconductors contain electrons in localized states, and other microscopic bodies such as crystal lattice defects, that can be excited above their energy ground states. Since these active “flaws” interact with the mobile electrons, they may have to be considered as part of the electron transport system. With the members of one species (such as, a particular donor atom) each described by a finite number 1’ of quantum states, they may be represented collectively by the occupation probabilities P , , P , , . . . , P,, . I t is evident that the Monte Carlo schemes discussed previously may be suitably extended by supplementing the mobile-carrier variables with the additional variables [ P i ; .Transitions due to interaction with a mobile carrier will have probabilities proportional to Pib r the initial state and to (1 - P j ) for the final state (with nondegenerate levels). Spontaneous local-state transitions ( i +j), not involving a mobile carrier, obviously could be included among the transition modes. In addition to “collisions” in which a mobile-carrier state changes and a localized state also changes, there exist processes in which a mobile carrier is captured into or emitted from a localized bound state. Since the bound state may be regarded as another band or valley, obviously this complication can be readily included in the Monte Carlo scheme. ( I t may be expedient to multiply all bound-to-mobile transition rates by a constant factor, with a corresponding weighting factor in the estimators, so as to have a convenient fraction of simulation time spent in bound states.) There are similar processes involving more than one electron: impact ionization, in which the initial state has one mobile and one bound carrier and the final state has two mobile

304

PETER J. PRICE

carriers, and the Auger process in which these initial and final states are interchanged. Impact ionization could be included in either single-particleor many-particle Monte Carlo, as discussed towards the end of Section 10; the Auger process requires a many-particle version, however, because there are two initial states to be drawn from the carrier population as in Section 16. There are other such ionization and recombination processes that may need to be represented; the foregoing discussion should be sufficient to indicate what is evidently involved.52 Avalanche ionization rates may be calculated by the Monte Carlo procedures already d e ~ c r i b e d . ~ ’ .The ~ bipolar space-dependent and timedependent avalanche phenomena that are of practical interest54 could evidently be included in the many-particle Monte Carlo scheme-generalized, when appropriate, to simultaneous electron and hole ensembles. With the electron-hole radiative recombination process and Auger inverse-avalanche process, however, there is a difficulty in including these processes in full detail (rather than perhaps by an appropriately simplified model). For radiative recombination, we would need to select an initial electron state and an initial hole state from the two ensembles of particle variables, and then choose between the recombination process for this pair and a competing scattering process, on the same lines as in Section 16 for carrier-carrier scattering. In the present case, however, because of the momentum selection rule associated with the translational invariance, an arbitrary pair of electron and hole states does not in general constitute a possible initial state of the process; similarly for the three electrons and one hole (or vice versa) of the initial state for the Auger avalanche process. It appears necessary in such cases to retreat from “pure Monte Carlo” by introducing an auxiliary distribution function f A , as discussed in the following paragraph. For definiteness, consider the case of radiative recombination, where a single photon is the annihilation product of an electron and hole, accounting for their energy and crystal momentum. (Similar considerations will apply to the case of Auger avalanche recombination.) A possible procedure would be to evaluate the recombination probability (tobe used in the choice between recombination and “competing” scattering processes) for an initial electron state pe chosen from the many-electron Monte Carlo ensemble and an initial hole state p,, selected, in the manner discussed in Section 4,out of the surfacelike domain of possible hole states that could recombine with pe. Substitution For a general reference on such processes, see (for example)Chapter 6 of J. L. Moll, “Physics of Semiconductors.” McGraw-Hill, New York, 1964. Where electron-hole recombination through traps is significant it may be necessary to include both mobile electrons and mobile holes in the Monte Carlo scheme. 5 3 R . C. Curby and D. K. Ferry, Phys. Status Solidi (a)15, 319 (1973). ’* See J. E. Carroll. Ref. 48. Chapters 8 and 9. 52

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

305

of this %in the auxiliary hole distribution function ft(p)gives the appropriate ensemble weighting for the probability of the recombination process. The f t is to be obtained from the Monte Carlo hole ensemble; it represents the smoothed density of ensemble members in the neighborhood of the given p value. One can think of a number of procedures for this purpose: (a) An analytical form, such as “displaced Maxwellian” or an expansion over spherical harmonics, could be assumed for the whole of the p space; the coefficients in it would be determined by comparison with the current values of certain moments of the Monte Carlo ensemble of particle states.55 (b) Only a limited region of p space, containing the value to be substituted into f A , would be considered. A suitable analytical form, such as a polynomial, could be similarly equated to the Monte Carlo ensemble of p values in that region (the simplest, and primary, “moment” in this case being the total number of particles currently in the region). (c) A version of (b) in which the local region is the “shell” between two surfaces of constant energy, the bounding energies being two successive values in a set defining an energy histogram. The fitted function within this region (which could be a spherical harmonic expansion) would be multiplied by the normalized number of particles in the shell-the histogram measure of the energy distribution function. The latter could be replaced by the smoothed value obtained by an interpolation through neighboring histogram values. The best procedure in a particular case may be expected to be shown by experience. The use of an approximate or idealized value for one of the factors in a transition rate must to some extent degrade the accuracy of the complete calculation, compared to the potential accuracy of a pure Monte Carlo calculation where all quantities representing the system can be exact. A measure of the error might be computed, from the auxiliary function and the Monte Carlo ensemble, at the same time as the calculation itself. 1 8. AUXILIARY FUNCTION APPLICATIONS Hot electron phenomena occur predominantly at modest electron densities so that the effect of the final-state factor (1 - f )in transition rates is not appreciable. It can be of interest, however, to analyze situations where this factor should be taken into account. If transitions could in general occur between any given pair of states, a possible procedure might be to use complementary electron and hole Monte Carlo ensembles to represent the same electron system with particle densities corresponding to f and ( 1 - f ) 55

Such a displaced-Maxwellian function, fitted to moments obtained from the preceding part of a single-electron Monte Carlo history, has been used to generate electron-electron scattering in the computation in Ref. 53.

306

PETER J . PRICE

respectively; in a transition, an electron state from one particle ensemble and a hole state from the other would interchange roles-ach transferring to the other ensemble. Since in practice arbitrary pairs of states that happen to be members of a Monte Carlo ensemble are not the initial and final states of a possible scattering transition, it appears necessary to make use of an auxiliary function, as discussed in the preceding section, to provide the final-state ( 1 -3) factor in transition probabilitie~.’~ An analogous procedure appears necessary for the calculation of “hot phonon” phenomena, in situations where the electron and phonon distributions mutually control each other.57 The phonon distribution function gives the absolute quantum excitations of the lattice modes, and the electron-phonon transition rates depend on these. It does not appear possible to use an ensemble of phonons already generated, to determine the scatterings in a Monte Carlo electron ensemble, because in general an arbitrary member of the latter will not scatter with any of the lattice wavevectors that happen to be included in the phonon ensemble. Useful calculations might be made, however, by use of an analytically specified “model” phonon distribution function, analogous to f A above, together with a single Monte Carlo electron variable or a many-electron ensemble. One would expect to determine (the parameters of) this phonon distribution function by equating the change of suitable moments of it to the corresponding quantitites for the phonon emissions and absorptions. Appendix A. Generation of a Gaussian Distribution

An illustration of Method 3 of Section 4 is provided by the generation of values of x with the distribution function P(x) = A exp( - s 2 ) ,

(Al)

where the normalization constant A is equal to 2/& when the domain of x is zero to infinity. (Method 1 is liable to give a too high proportion of rejections, from the upper part of the x range for typical applications.) The thermal distribution of one component p of momentum, for particles with energy function (24), is given by ( A l ) with .x = (2rn*kT)-’i2p.

(‘42)

The procedure described here was used to provide the initial particle states in the computations, described in Section 12, of semiconductor junction characteristics. 56 57

This has been done by S. Bosi and C. Jacoboni. J. Phjs. C 9, 315 (1976). For example, E. M . Conwell, P/~.F:Y. Rec. 135, A 814 (1964).

4.

MONTE CARLO CALCULATION OF ELECTRON TRANSPORT I N SOLIDS

307

A suitable form for (45) in the present case is -

x = RI/(l

+ E - Rl).

(A3)

The positive constant E determines the maximum x value, b = 1/E; the minimum value is a = 0. The density of the values generated by (A3) is proportional to (1 + X)-,, so the fraction accepted, in obtaining the distribution (Al), does not tend to zero as b increases. Then (46) takes the form

B(1 + .u),exp( - X2) 2 R,,

(A4)

where B I [2/(3

+ f i ) ]exp[(3 - 8 ) / 2 ] = 0.559642 . . . .

('45)

The fraction of (R,,R,) pairs giving an accepted .x value is B/A. When the upper limit of x values, b, is large ( E is small), so that A is nearly equal to 2/&, then this acceptance fraction is nearly equal to 0.496; that is, almost half of the executions of this algorithm give an x value used in the generated Gaussian distribution. Appendix B. Some Vector Geometry

Trigonometric formulas pertaining to material in Sections 4 and 6 are presented here. They concern the relationship of two common-origin vectors OP,, OP, and a fixed direction OR.The azimuthal angles of the P,OR and P,OR planes about O R are 4, and 4, ;the radial (OP,, OR) and (OP,, OR) angles are 8, and 0,; the azimuthal angle of the P,OP, plane about OP, is R; and the (OP,, OP,) angle is (1. The perpendicular projection of P, on OP, isQ. The radial coordinates of the vectors transform to Cartesian coordinates (suppressing the subscript 1 or 2): .x = p cos 0,

(B1)

=p

sin 9 cos 4,

(B2)

z =p

sin 8 sin 4,

(B3)

where p is the length of CP. By comparing these expressions, for OP,, with those obtained by adding the projections of its components OQ and QP, we obtain the basic equations connecting angles for the two vectors: cos 8 2 cos(4,

-

bl)sin8,

sin(4, - +,)sinO,

= cos t,h

cos 81 - sin $ sin 81 cos 51,

(B4)

+ sint,hcos8,cos51,

(B5)

= cost,hsin9, = sin+

sin0

(B6)

308

PETER J. PRICE

(where the first terms on the right of (B4) and (B5) are contributed by OQ, the remaining three terms on the right by QP,). By eliminating cosR between (B4) and (B5) we obtain cos t j= cos 0, cos 0,

+ sin O1 sin 0, cos(4, -

This result may be substituted for cos Jplpz12= P:

037)

+ in the standard formula

+ P:

- 2P,P,COS+

(B8)

+

for the length of the difference vector PIP,.Then

IP1P21’= p f

+ p:

- 2p,pzcosd1 cosd,

-

2p1p,sind, sine,cos(&,

+

-

41). (B9)

In Section 4, where cos 0 is denoted by X and cos by Y, Eq. (56) comes from (B4)above and Eq. (57) from (B7). The result (58) comes from (B9) together with the standard formula

=:f.

-

a

28

+ b c o s 4 - (a2 - b2)’’2‘

Author Index Numbers in parentheses are footnote numbers and are inserted t o enable the reader t o locate those cross references where the author’s name does not appear at the point of reference in the text.

A Aas, E. J.. 277 Abe, M., 300 Abeles, B., 131, 132 Abrahams, M.S., 29,122, I 4 4 Acket, G. A., 125 Adams, M. J., 78, 79, 104, 107, 136, 188 Adirovich, E. I., 214 Afromowitz, M. A., 131 Ahn, B. H.,124 Aiki, K., 170, 176, 177, 183(229), 184(229) Alexander, F. B., 28, 52(93) Alferov, Zh. I., 10, 13, 15, 64,71, 97, 125, 173(47) Akemade, C. Th. J., 239 Allakhverdyan, R. G., 77 Almeleh, N., 133 Altarelli, M.,29, 32(99) Andersen, D. L., 277 Anderson, W. W., 78 Andreev, V. M., 10, 13, 71, 125(88c) Antypas, G. A., 125(88c) Appert, J. R.. 52 Archer, R. J., 132, 133 Arnold, G., 185 Ashley. K. L., 125 Ashkin, A., 22 Aspnes, D. E., 47 Astles, M. G., 26 Augustyniak, W. M., 13 Auvergne, D., 97, 127(46), 128

B Bahraman, A., 157 Bailey, E. A., 8, 9(36), 22(36), 55(36)

Barnes, P. A., 125 Barnett, A. M., 198, 229(4c) Barnoski, M. K., 165 Baron, R. W., 198, 223, 228, 229(4b) Bartelink, D. J., 237 Basov, N. G., 2, 19, 45(64), 58(64), 59(64), 60(64),105, 126, 191 Bebb, H. B., 13 Beebe, E. D., 150 Beer, A. C., 198 Bell, R. L., 125(88c) Berolo, O., 132 Bevacqua, S. F., 1, 6(3), 52(3), 172 Bilger, H. R., 208, 229, 238, 239, 245 Bishop, S. G., l47(152), 151 Blacknall, D. M.,13, 124 Blakemore, J. S., 24 Blanc, J., 144 Blom, G. M., 25 Blum, J. M.,121,135, 157 Boardman, A. D., 257,260,268(8), 272(11) Bogatov, A. P., 25,28(81), 35(81), 52(81), I50 Bogdankevich, 0. V., 2, 19,45(64),58(64), 59(64), 6O(W, 126 Boose, E. D., 2 Borodulin, V. T., 153 Borsari, V., 258,268(9), 269(9), 280 Bosi, S.,306 Botez, D., 170 Bougalis, D. N., 237,238 Brackett, C. A., 172 Braslau, N., 279 Brinkman, W. F., 20, 128 Brown, R. L., 134 Buiocchi, C. J., 122, 144 Burd, J. W., 21,45(65), 58(65),60(65), 126 Burmeister, R. A,, 29 Burnham, R. D., 2,3,4,8,9(36), 11(21), 12,

309

310

AUTHOR INDEX

14(19), 15(18,19), 16,18(42), 20,22(21, 36). 28,29(11), 31(11), 41(24), 45(66), 50(42), 55(36), 56(139), 57,58(66, 1411, 59(66), 73(18), 74, 147,150,177, 180 Bums, G., 1,70 Burrell, G. J., 66 Burstein, E., 22,50(73) Butler, J. K.,72,76,78,82,83(14), 89.91, 98(32), 99, 113,114, 115(42), 117, 130(116), 131 Byer, N . E.,78,98(32), 155, 156(175), 159

C Calawa, A. R., 147(152h),151 Camassel, J.,%', 127(46), 128 Campbell, J. C., 3,4,7,28(27,99), 29(27, 32), 30(27,32), 31(23, 105), 32(23,27,99), 33(27), 34(27), 35(32,105), 37(32,105), 38, 43,44(122, 130, 131),45(122), 48(122, 133), 49(26), 51(26), 52(92), 58(22,23), 62(22,23) Canali, C., 237,295 Carbone, R. J., 8,9(36), 22(36), 55(36) Carlson, R. 0..I , 70 Caroll, J. E., 300,304 Carroll, R. L., 124 Casey, H. C., Jr., 10,I16,117,125, 130(111), 131, 177 Cherlow, J. M.,I89 Chinn, S. R., 2,3(5), 8(5), 10(5), 11(5), 12(15),25(34), 26(34), 52(34), 53, 128 Chinone, N., 156 Cho, A. Y.,125 Ciesielka, T. J., 144 Clough,R. B.,29 Coleman, J. J., 3,7,25,28(33,82), 2x23, 32), 30(32), 31(23), 32(23),35(32,33,82, 98),36(82), 37(32,82), 51(98, 135). 52(82), 58(23), 62(23). 147, 150,175 Constantinescu, C., 75 Conwell, E. M.,12,306 Cook, D. D., 169 Cooley, J. W.J7 Craford, M. G., 3,4, 12, 13(43). 14(43), 31(113), 38,39(115), 40(126), 41(24, 128). 42,43,44(122, 131),45(122, 126, 128, 131). 46(116), 47(25), 48(122, 133). 49(26), 51(26,135), 58(22), 62(22), 133,147, 148

Cross, M.,78, 136 Curby, R. C., 304 D Dapkus, P.D., 3,8(17), 9(36), 11(21), 12(20), 13, 14(19,49), 1518, 19,49), 16(18), 17(63), 18(20,42,63), 19,20,21,22(17,21,36), 23,24,29,45(65,66), 50(42), 55(36), 58(65, 66),59(66), 60(65,66), 73(18), 74,126, 127 D'Asaro, L. A.,22,135,156, 189 Davies, C. F. L.,104 Davis, E. A.,269,270 Dean, P.J., 26.39 DeLoach, B. C., 156 DeMars, G.. 186 DeWinter, J. C., 150

Dill,F.H.,Jr.,1.70 Dingle, R., 13 Ditzenberger, J. A., 13 Dixon, R.W., 137, 150, 157,159,161(187) Dobson, C. D., 152 Dobson, P.S.,156 Doerbeck, F.H., 124, 125(88b) Dolginov, L. M.,25,28(81), 35(81), 52(81) DonneUy, J. P.,150 Driedonks, F.,223,238,239,244 Druzhinina, L. V.,25,28(81), 35(81), 52(81) Dumke, W.P.,1,70,83,86 Dunse, J. V., 13 Dupuis, R. D., 4,20,25(68), 28(27), 29(27), 30(27), 31(105), 32(27), 33(27), 34(27), 35(105), 37(105),39,40(126),45(126), 47(25), 53,54,57,58(141) Dyment, J. C., 105,106,135,173,191

E Eastwood, J. W.,301 Edgecumbe, J., 125(88c) Eliseev, P.G., 25,28(81), 35(81), 52(81), 66, 74,169 Ellis, B., 66 Enstrom, R. E.,28,52 Ermanis, F., 15,124 Ettenberg, M.,28,52,72,83( l4), 107, 112, 113, 119, 121(67a), 122(80), 125, 127(66), 134,144,145,147(152), 148,150,154,156, 159, 160(195), 161(177a),172

311

AUTHOR INDEX

F

Fawcett, W., 257,259,260,268(8), 272( 11) Fenner, G. E., 1,25,70 Ferry, D. K.,304 Feucht, D. L., 75 Finn, D.. 4,47(25) Finn, M. C., 2,3(5), 8(5), 10(5), 1 I(5) Fletcher, N., 226,227(21) Fonstad, C., 147, 151 Fox, M.J., 147(152a), 148 Foy, P. W.,70, 110, 111(67), 117, 128. 135 Frank, F. C., I43 Frank, G., 125(88d) Friedman, A,, 235 Friedman, N., 266 Frosh, C. J., 38 Fujiwara, T., 160

G

Galginaitis, S.V.,25 Gannon, J. J., 29,30(109), 52 Garbe, S., 125 Garbuzov, D. Z., 13, 15 Garmire, E., 176 Garrett, C. G. B., 187 Garvin, H. L., 176 Gershenson, M.,22 Geusic, J. E., 25 Gherardi, L., 295 Gibbons, D. F., 134 Gilbert, S. L., 125(88f), 144 Gisolf, A., 235,238,244 Gobel, E., 126 Gold, R. D., 159 Goldberger, M. L., 25 1 Goldenblum, A., 75 Golder, J., 236 Goncharov, V. A., 19,45(64), 58(64), 59(64), 6 O ( W 126 Gooch, C. H., 66 Goodman, A. S.. 252 Goodwin,A. R.,74, 119, 121(17a), 169 Gordon, E. I., 79 Govindjee, R.,8,9(36), 22(36), 55(36) Grasyuk, A. Z., 2 Greene, R. F., 288 Grodkiewicz, W. H., 25

Groves, S.H., I51 Groves, W. O., 3,4,7,9,25,28(33,82), 31(113), 35(33,82),36(82),38,40(126), 41(24), 45( 126), 49(26), 51(26, 135). 5482). 58(22,37), 59(37), 6O(37, 142), 62(22,37), 133,147(152a, 152e). 148, 150 Griindorfer, S., 104 Guggenheim, H. J., 25

H Haken, H., 188 Hakki, B. W.,100,136,137(133), 152, 153(166), 154166). 156, 169 Hall, R. N., 1,70 Hammar, C., 263,279 Hammersley, J. M., 251,261 Handscomb, D. C., 251,263 Harada, R., 15 Harman,T. C., 147(152h). 151 Hartman, A. R., 158 Hartman, R. L., 63, 156, 157, 158, 159, 161(187) Hartman, S. R.,8,9(36), 22(36), 55(36) Hatz, J., 77 Haug, H., 188 Hauge, P. S., 279 Hawrylo, F. Z., 13, 15,71,72,78,83( l4), %, 97(44),110(31), 122(31), 124(13,44), 133, 163(44), 173(12), 174, 175(221a,221) Hawryluk, A. M.,286 Hawryluk, R. J., 286 Hayashi, I., 10, 15,63,66,70,71,79,97, 110,lI 1(67), 121,124,128, 135 Hegyi, 1. J., 173 Heim, U.,25 Heinkl, W., 144 Henry, C. H., 8,9(36), 22(36), 55(36) Henshall, G. D., 104, 153 Hernandez, J. P., 284 Herzog, A. H., 4, 12, 13(43), 14(43),38, 39(115),41(24), 133 High, D. A., 3, 12(20), 18(20), 19(20), 21(20), 126, 127(94a) Hillbrand, H. A., 275 Hitchens, W. R., 3,4,7,28(27,33,99), 29(27,32), 30(27,32), 31(23), 32(23,27, 99), 33(27), 34(27), 35(32,33), 37(32). 58(23), 62(23), 147(152e), 150, 175

312

AUTHOR INDEX

Hockney, R. W.,300,301(46) Holonyak, N., Jr., 1,2,3,4,5,6(3,7,30), 7(16), 8,9(36), 11(21), 12(20), 13(43), 14(19,43), 15(18, 19), 16(18), 17(63), 18(20,42,63), 19(20), 20,21(20), 22(21, 36), 25(68), 28(27,33,82,99), 29(11,23, 27,32), 30(27,32), 31( 11, 16.23, 105), 32(23,27,99), 33(27), 34(27), 35(32,33,82, 105), 36(82), 37(32,82, 105), 39(115), 40, 41(24,128),42,43,44(122, 129, 131), 45(65,66, 122 126, 128, 131),46(116),47, 48(122, 133).49(26), 50(42), 51(26, 135). 52(3,82,92), 53(16), 54(138), 55(36), 56(139),57,58(22,23,37,65,66, 141), 59(37,66), 60(37,65,66,142), 16(142), 62(22,23,37), 73(18), 74,126, 127(94a), 147(152e, 152f),ISO, 172, 175 Holt, D. B., 140 Hopfield, J. J., 13,38,39(120) Hsieh, J. 1. 2,3(5), 8(5), 10(5), 125, 147, 150 Hsu, S.T., 201,216,239 Huang, C. H.,242,243,244 Hutchinson, D. W.,252 Hutchinson. P.W.,156 Hwang,C. J.,55,56(140), 100, 105, 106, 107 I

Ikegami, T., 79.80, 187 Ikuta,T., 285 IUegems, M.,177 Inoue, M.,136 Ishi, M.,105 Ismailov, I., 25 Ito, R., 156 Ito, A., 105 Itoh, K., 136, 175

J Jacoboni, C., 258,261,268(9), 269(9), 280, 282(32), 283,295,306 James, L. W., 74, 125 Johnson, L. F.,25 Johnson, M.R., 2,5. 6(7,30) Johnston, W.D. Jr.,20,63,156 Joyce, W.B., 137 K Kaiser, W., 187 Kamejima,T., 135, 156

Kan, H., I05 Kane, E. O., 107,108 Kato, D.,172 Katulin, V. A., 2 Katzir, A., 177 Keeble, F. S., 152 Kelvin, Lord, 251 Keune, D. L., 2,3(9), 7(9), 8(9), 9(36), I1(21), 12, 13(43), 14(43), 15, 20,22(21, 36), 25,28(33,82),29(11,23), 31(11), 35(33,82), 36(82), 38,44,45(66), 48(133), 51(135),52(82), 55(36), 56(139),57(9), 58(9,22,37,66), 59(37,66), W37.66, 142), 61(142), 62(22,37), 73(18), 74, 147(152e), 150 Keyes, R. W.,139 Kimerling, L. C., 160 Kirniya, T., 188 Kingsley, J. D., I , 70 Kirkby, P.A., 10,72, 117, 118, 153, 169 Kleinman, D. A., 186 Kobayshi, K., 135 Kogelnik, H.176, 177, 182 Konnerth, K., 13, 105 Korol’Kov, V. I., 10 Koshikawa, T., 285 Kressel, H., 13, 15, 66,70,71,72,76,78, 82(41),83(14),89,91,92,93,95, %, 97, 99(43), 100(43), 101, 102,104,107, 110(31), 112, 113, 11442). 115(42),116, 117(42), 119, 121(67a), 122(31), 123(75), 124(13,44), 125(88a, 88f). 126, 127(65,66), 128(65), 130(16), 133. 138,141, 145, 147(152n), 152, 153, 154, 155(172), 156(175), 157,158, 159(173, I S ) , 161(177a),163,166, 172, 173(12,202), 174, 175(221a, 221) Krumme, J. P., 134 Kudman, I., 134 Kunz,A. B., 38,43,44(122,131). 45(122, 131). 48(122),51( 135) Kurosawa, T., 25 I , 261 1

Ladany, I,. 89, 114(42), 115(42),116, 117, 133, 138, 147, 154, 155(172), 157, 159(173, 188), 161,166,172,173(202) Lam, H.7.. 125(88h) Lampert,M. A., 195,198,199(1), 199, 203(5),205(5), 208(1),210,211(1,4a),

313

AUTHOR INDEX

213(1,4a), 217(1), 222,223(4a), 227(1,4a), 228(1), 229(4a) Landsberg, P. T., 107 Lang, D. V.,160 Lankard, J. R., 187 Lanza, C., 105 Lasher, G., I , 70,191 Lavrushin, B. M., 19,45(64),58(64),59(64), 6o(w

Lax, B., 69 Lebwohl, P. A., 275.277(25), 280,299(31), 300(31), 304(31) Lee, D. H., 239 Lee, M. H., 3,4,7,9,28(99), 29(23,27,32), 30(27,32), 31(23), 32(23,27,99), 33(27), 58(22.23,37), 59, W37, 142),61,62(22, 23.37) Lee, P. A., 20,28(27), 128 Lee, T.. 191 LeFur, P.,15,122,124 Leheny, R. F., 20,126 Leite, R. C. C., 25, 169 Lengyel, B. A,, 4.5(29) Levanyk, A. P., 13 Levinstein, H.J., 25 Levitt, R.S., 24 Lewis, P. A. W., 252 Liao, J. H., 222,223,239,240,244 Lindmayer, J., 214,216(12) Lindquist, P. F., 29 Liu, S. T.. 236,237,239 Lockwood, H.F., 72,83( l4), 92,93,95, %, 97(44), 99(43), lOO(43), 101, 102, 104, 107, 121, 122(80), 124(13,44), 126, 127(65,66), 128(65), 130(16), 153, 154, 155, 156, 161(177a), 163(44), 172, 374(218) Logan, R. A., 134 Longaker, P. R., 8,9(36), 22(36), 55(36) Lorenz, M. R., 29, 131, 133 Ludowise, M. J., 3,7,25,28(33,82), 29(23), 32(23), 35(33,82), 36(82), 37(82), 52(82), 58(23),62(23), 147(152e), 150 Luthi, B., 251,295,296(39)

M McCaldin. J. O., I5 McCarter, E. R., 239 McCumber, D. E., 188 McFarlane, S. H., 122, 144 McGroddy, J. C., 135

McKenna, J., 77 Macksey, H. M., 4,20,25(68), 28(27), 29, 30(27), 31(105),32(27), 33,34,35( 105),37, 39,40(126), 47(25), 52(92), 53,54(128) McLane, G. F.,147(152),151 McMullin,P. G., 135, 157 McNeely, J. B., 15, 17(63), 18(63), 126, 134 McVittie, J. P., 147(152h), IS1 McWhorter, A. L., 69 Maeda, H., 263 Many, A., 210 Mar, T., 8,9(36), 22(36), 55(36) Marcuse, D., 177 Mark,P., 195, 198, 199(1),208(1),211(1), 213(1), 217(1),222,227(1), 228(1) Marple, D. T. F., 131 Martin, R.J., 175 Mathews, J. W., 143 Mathieu, H., 97, 128(46) Matulionis, A,, 301 Mayer, J. W., 198,223,229(4b) Melngailis, I., 28,52(91) Merkelo, H., 8,9(36), 22(36), 55(36) Mierop, H. P. 152 Mikhailina, L. I., 25 Miller, B. I., 63, 110, I 1 1(67), 125(88g). 128, 135, 156, 173 Miller, J. M., 252 Miller, S. E., 175 Milnes, A. G., 75,229 Minden, H. T., 123 Mischel, P., 147 Mistry, D., 104 Mitchell, D. L.,147, 151 Mohn, E. 77 Moll, J. L., 304 Moon, R. L., 125(88c) Mooradian, A., 3,8, 12(15) Morgan, D. J., 188 Morikawa, T., 176, 183(229), 184(229) Morozov, E. P., 15, 191 Moss, T. S., 22,50(72), 66 Mott, N. F., 269,270 Murata, K. 285 Murygin, V. I., 125(88e)

N Nahory, R. E., 20, 126, 150 Nakada, O., 135, 156, 157 Nakamura, M., 176,177,183(229), 1841229)

314

AUTHOR INDEX

Nakashima, H., 135, 157 Namizaki, H.,105 Nash, F. R., 152, 153(166),154(166), 169 Nassau, K.,8,9(36), 22(36), 55(36) Nathan, M. I., 1,70 Nava, F., 295 Nelson, D. F., 22, 150(127),77 Nelson, H.,13,70,78,110(31), 121, 122(31), 123(75), 125(88a), 173, 174(218) Nelson, R. J., 3,9,39,40(127), 44(129), 45(127), 46(127), 49(127), 58(37), 59(37), 60(37, 142),61(142), 62(37) Newman, D. H., 155.156(176) Nicolet, M.-A.,206,208,226,236,238,239, 245 NicoU, F. H., 107, 124, 127(66) Nijman, W., 325(88h) Nikitin, V. V.,191 N d , K. W.,147(152i), 151 Nishimaki, M.,191 North, D. O., 79,170 North, J. C., 135 Nuese, C. J., 28,29,30(109), 38,52, 133, 147(152n, 1520). 148, 149,150, 160(195), 175 0

O’DonneU, R. W.,288 O’Hara, S., 155, 156(176) Ohmi, T., 191 Oldham, W. G., 157 Olsen, G. H.,142, 144(141), 145, 147(152c), 148, 149, 150, 175 Onton, A., 29, 131,133 Oraevskii, A. N., 77 Osipov, V. V.,13 Ottaviani, G., 237,261,295

P P a , R. J., 134 Paige, E. G. S.,259 Panish, M. B., 10,28,63,66,70,71,79,97, 116,117, 121, 130,134,135, 173 Pankove, J. I., 12, 13(44,45), 137, 173 Pa0li.T. L., 10,76,77(24), 117, 156,157, 165, 169,189,190 Parker, D. L., 134

Parrott, J. E., 288 Pellacani, A., 295 Persky, G., 237 Peters, J. R., 74, 119(17a), 121(17a) Petroff, P., 63,134,156 Pettit, G. D., 13,25, 187 Phelan, R. J., Jr., 2,6(6, 13) Philips, A., Jr., 264 Phillips, J. C., 39 Pierron, E. D., 134 Pilkuhn, M.H.,25,66,126 Pinkas, E., 110, 11 1, 125(88g), 128, 173 Pion, M.,74, 119(17a), 121(17a) Pollak, M.A., 150 Porteous ,P., 26 Portnoi, E. L., 10.71 Pozela, J., 301 Premo, R., 123 Price, P. J., 254,262,264,272(14), 277, 278(26),279(14), 280,295,296(41,42), 299(31), 300(31),304(31) Purohit, R. K., 75

Q Quaranta, A. A., 237,261,280,295 Queisser, H. J., 13,144 Quist, T. M., 70

R Rahman, M. A., 237 Rakavy, G., 210 Ralston, R. W., 147(152h),151 Redfield, D., 12, 13(44,45) Rediker, R. H., 2,6(6), 28,52(91), 75 Rees, H. D., 250 Reese, W.E., 25 Reinhart, F. K., 10,79,97,127(46), 134 Reiser, M.,300,301(46) Reklailis, A., 301 Reuter, W.,29, 133 Reynolds, J., 214,216(12) Richman, D., 29 Ridley, B. K., 229 Rigaud, A., 236 Ripper, J. E., 76,77(23,24), 104, 105, 135, 165, 173, 191

315

AUTHOR INDEX

Ritchie, S.. 155, 156(176) Rode, D. L., 74,2% Riider, O . , 25 Rodgers, K. F., Jr., 13 Roldan, R. H. R., 191 RoUefson, G. K., 8,9(36), 22(36), 55(36) Rose, A., 199,203(5),205(5), 217 Rossi, J . A., 2,3(5), 5,6(30), 8(5), 10, I1(5), 12(15,20), 14(19), 15(18, 19), 16(18), 17, 18(20,42), 19(20), 20,21(20), 25(34), 26(34), 38,45(65,66), 46(116), 52(34), 53, 58(65,66), 59(66), m65.66). 104, 125, 126, 127(94a),128,150 Rosztoczy, F. E., IS, 124 Rozgonyi, G. A.. 28, 134, 144 Ruch, J. G., 260,272(1 I ) Rupprecht, H., 13 Russer, P., 185

S Saito, H., 147, 150

Sakuma, I., 135, 156 Sarace, J. C., 22 Saul, R. H.,144 Savelli, M.,236 Schade, H., 125 Schilling, R. B., 198,211(4a), 213(4a), 223(4a), 227(4a), 229(4a) Schlosser, W.O., 10, 117 Schmidt, R. V., 176 Schwartz, B., 15, 124 Scifres, D. R.,2,3(10),4,7(10, 16),8(10), 12, 18(42),20,25(68), 28,29(1 I), 30,31( 11, 16). 41(24), 49,50(42), 53(16), 57,58(10, 141), 147(152f),150, 177, 180 Sell, D. D., 127, 128, 131 Selway, P. R., 104,169 Semenov, A. S., 191 Sergiescu, V., 234 Shaklee, K. L., 20, 126 Shank, C. V., 176, 182 Sharma, B. L., 75 Shaw, D. A., 152 38,39(115) Shaw. R. W., Shevchenko, E. G., 25,28(81), 35(81), 52(81) Shih, K. K., 121,135,157 Shimizu, R.,285 Shockley, W.,12,60(40), 246

Shul, G., 147 Shurnka, A., 238 Shurtz, R. R., 124 Sigai, A. G., 29,30(109) Singh, S., 25 Singhal, G.S., 8,9(36), 22(36), 55(36) Sirkis, M.D., 2,5,6(7,30), 38 Skinner, S. M.,198 Sleger, K. J., 147,151 Smith, A. W.,135, 157 Smith, H. I., 286 Smith, M. V., 187 Sobers, R.G., 134 Soltys, T. J., 1.70 Somekh, S., 176, 177 Sommers, H. S., Jr., 72,76,82(41), 83, 124(13), 154, 164, 170 Sorokin, P. P., 187 Spicer, W.E., 284 Statz, H., 186 Stem, F.,66,77, 106, 107(6@),108, 120, 131 Stevenson, M.J., 187 Stillman, G. E., 2,5,6(7,30), 38,46(116), 52 Stone, P. S., 197,205(2) Strakhov, V. P., 169 Stranmanis, M.E., I34 Strauss, A. J., 28,52(91), 147(152i),151 Streetman, B. G., 41,44(129) Streifer, W.,177, 180 Stremin, V. I., 125(88e) Stringfellow, G. B., 29 Strom, U., 147(1521), 151 Struthers, J. D., 13 Stuart, R., 284 Sturge, M.D., 128 Suchkov, A. F., 77 Sudzilovskii, V. Yu., 19,45(64), 58(64), 59(64), 60(64),126 Suematsu, Y., 187 Sugiyama, K., 147, 150 Sumski, S.,63,71, 117, 121, 135 Suzuki, T., 191 Sverdlov, B. N., 25,28(81), 35(81), 52(81) Swain, S., 257,268(8)

T Takanashi, H., 300 Tandon, J. L., 208,238

316

AUTHOR INDEX

Taylor, N.F., 177 Teramoto, I., 136 Thomas, B., 104 Thomas, D. G., 13,38,39(120) Thompson, G. H. B., 10,72,74,104,117, 118, 119(17a), 121(17a), 153, 169 Thornber, K.K.,232 Thornton, P. R., 152 Tien, P. K., 175 Tietien, J. J., 124, 173 Tomasetta, L. R., 147,151 Tret’yakov, D. N., 10, I5 Trukan, M. K.,13,71 Trussel, C. W., 124 Tsai, T. N.,226,227,244,247(20) Tsukada,T., 135, 166(128) Tuck, B., 6 Turner, W. J., 25

U Uchida, T., 172 Ulam, S. M.,251 Ulrich, R., 175 Umeda,J., 135, 176, 177. 183(229), 184(229) Umo, M.,156

V vander Menve, J. H., 143 van der Ziel, A., 201,208,223,225,226,229, 232,234,235,237,238(27a), 239,240,242, 243,244,247(20) van Ruyven, L. J., 75 Van Uitert, L. G., 25 van Wet, K.M.,235 VasseU, M.O., 12 von Neumann, J., 251,252,264 W

Wada, 0.. 300 Walpole, J. N., 147,151 Wang, S., 177, 180, 183 Waniner, R. A., 300,301(46) Waxman, A., 229 Wecht, K.W., 131 Weisberg, L. R., 144,159 Weiser, K., 24 Whelan, J. M.,13

Whiteaway, J. E. A., 74, I04,119(17a), 121(17a), 153 Willardson, R.K.,198 Williams, E. W., 13,25,26, 124 Williams, F. V.,3,12(20), 14(19), 15(19), 17(63), 18(20,63), 19(20), 21(20), 45(65), 58(65), 60(65), 126, 127(94a) Wittke, J. P.,12,13(44,45), 172 Wolfe, C. M.,3, 12(15), 52,128 Wolf0rd.D. J.,41,44(129) Womac, J. F.,75 Wood, D. L., 187 Woodall. J. M.,13,25, 131 Wooley, J. C., 132 Wooten, F.,284 Wright, G. T., 198,214 Wright, P. D., 25,28(82), 35(82), 36(82), 37(82), 52(82), 175 Wrigley, C., 214,216(12) Wyder, P., 251,295,296(39)

Y Yamamoto, S.,239 Yanagisawa, S.,300 Yang,E. S., 156,157 Yanai, H.,188 Yano, M.,188 Yariv, A., 66.69, 176,177,183(229), 184(229) Yen, H. W., 176, 177, 183(229), 184(229) Yonezu, H., 135, 156

Z

Zachos, T. H.,76,77(23,24) Zack,G. W.,4,28,29,30,31(105), 35(105), 37(105), 47(25), 52(92), 53,54 Zamerowski,T. J., 144 Zanarini, G., 280 Zee, B., 210 Zeiger, H. J., 69 Ziegler, J. F., 135 Zijlstra, R. J. J., 229,235,238,239,244,245 Zoroofchi, J., 131 Zubarev, 1. G., 2 Zuleeg, R.,246 Zwicker, H. R., IS, 55,56(139), 57.58(141), 73( 18), 74

Subject Index A

Absorption, 56,62,seealso Absorption coefficient free carrier, 109. I 1 1 GaAs, 23 interband, 131 intraband, 131 quenching of edge-to-edge modes, 62 trap involvement, 191 Absorption coefficient, 107, 109, I 1 I , see also Absorption AlAs lattice constant, 28, 134 thermal expansion coefficient, 134 (AlGafAs, 10,11, 14.28 band-gap energy, 129,132,141 photonenergy, 132 CW lasers, 165-172 DFB structures, 177,183,184, see also DFB laser DH laser threshold current density, 71,

(AIGa)(AsP),28 laser properties, 146 (AIGa)(AsSb),laser properties, 146,150 (AIGa)P bandgap, 141 lattice parameter, 141 melting temperature, 141 (A1Ga)Sb bandgap, 141, 142 lattice parameter, 141,142 melting temperature, 141 (AI1n)As bandgap, 141 lattice parameter, 141 melting temperature, 141 (AI1n)P bandgap energy, 132, 141 external quantum efficiency, 133 lattice constant, 132,141 maximum emission energy, composition,

132

melting temperature, 141 116 (AIInfSb double heterojunction lasers, 10,1 I , 14, bandgap, 141 112,I13 lattice parameter, 141 external quantum efficiency, 133 melting temperature, 141 FH laser threshold current density, 93,95, AI(PAs) 118 bandgap, I41 laser properties, 146,see also Optical vs. lattice parameter, 141 electrical property relationships melting temperature, 141 lattice constant, 132,141 AI(SbAs) lattice match, 28 bandgap, 141,142 maximum emission energy, composition, lattice parameter, 141,142 132,173 melting temperature, 141 melting temperature, 141 AI(SbP) refractive index, 130, 131 bandgap, 141 carrier concentration dependence, 13I lattice parameter, 141 doping dependence, 13 I melting temperature, 141 SH laser properties, 146 Ambipolar diffusion constant, 56,222 thermal conductivity, 131, 132,138 Amorphous semiconductor, 269,see also Disordered solid thermal resistance, laser structure, 138 visible emission lasers, 173-175 Arrays, laser, 161,162 composition, 173 Asymmetrical-structure laser shortest wavelength. 175 DH structure, 1 14,see also DH laser

317

318

SUBJECT INDEX

field intensity, 114 geometry, 114 LOC structure, 98-100, see also LOC laser beam width, 102 far-field pattern, 100, 101 fundamental mode, 100 output, 103 temperature-dependent changes, 100104

threshold current density, 102 optical anomalies, 100-104 SH structure, 96-98, see also SH laser Avalanche phenomena, 303-305 Auger process, 303 impact ionization, 303 rates, 304 radiative recombination, 304,305

B Band gap, see also Energy gap (AlGa)As, 129.132 composition dependence, (AIGa)As, 129, 132

lasing-energy dependence, 126-129 numerous ternary alloys, 141,142 temperature dependence, GaAs, 128 variation with alloy composition, 27, 35-37

Band tail states, 13,128 reduced lasing energy, 128 Boltzmann equation, 255,270 inhomogeneous, 279 steady state, 279 Boundary effects, 284-290 diffuse scattering, 287,288 semiconductor field effect, 289,290 specular scattering, 287,288 Fuchs model, 287 Parrott model, 288

C Carrier concentration, 255 effect on laser wavelength, 125-128 effect on refractive index, 131 relation to current density, 199.255, see also Continuity equation

Carrier confinement, 20,72-75, see also Confinement FH laser, 95 loss, effect on threshold, 74,95 potential barrier, 72-74 Carrier lifetime, 55-63, 105, see also Lifetime bimolecular recombination, 57.58,105 bulk lifetime, 57 determination, 55 optical phase shift method, 55 nonradiative. 55,57,58 radiative, 55,57 Spectra, 59-61 cavity modes, 62 N-doped Ga(AsP), 60,61 N - f r e Ga(AsP), 60,61 spectrum shift with pumping, 58 surface recombination, 55,56, see also Surface recombination velocity Carriers, injected, 105, 110,see also Double-injectiondiode; Single-injection diode diffusion length, 125 effect of dislocations 144 rate equations, 186 related to gain coefficient, I10 thermal. 75 tunneling, 75 utilization, 72-74 Cathodoluminescence 1 Cavity-end loss, 109, 112 Cavity modes, 62,75-77 CW lasers, 167-170 DH laser, 80 FH laser, 94 lateral, 75,76, 169,170 lifetime spectra, 62 LOC laser, 99,100 longitudinal, 7576,167 control (DFB laser), 175,182 longitudinal mode selection, 175,182 relationship near Bragg frequency, DFB laser, 181 SH laser, 98 single-mode operation, 170 buried channels, 170 TE, TM,75,80 transverse, 75-77,169 CdS, auxiliary cavity with In(AsP) 53.54 Child's law, semiconductor equivkent, 201

319

SUBJECT INDEX

Collective effects avalanche phenomena, 303-305, see also Avalanche phenomena carrier-carrier scattering. 301-303 many-particle Monte Carlo, 297-301 Monte Carlo simulation, 297-306 Conductivity, 270 nonohmic, hot electrons, 254-283, see also Hot electrons; Monte Carlo calculations ohmic, 270.294-296 Einstein relation, 295 mobility, 295 Monte Carlo calculations, 270,2942%. see also Monte Carlo calculations Confinement carrier, 20.72, see also Carrier confinement optical, 69,83,88 confinement factor, 88,89, 109. 120, see also Confinement factor (optical) stripe-contact diodes, 136,137 temperature dependent, 104 Confinement factor (optical), 88,89, 109, 120, see also Confinement; Radiation confinement factor Continuity equation, 199,217,241 Coordinate transformations, 307,308 Coupled-mode analysis (DFB laser), 177184

Coupling coefficient, DFB laser, 180 Current linear regime, 205,218 quadratic regime, 205,220 space-charge-limited, 195-197 double injection, 1% single injection, 195, 1% Current equation, 197.199-206,217-223, 240-242

diode characteristic, 215,216 n-v-n or p-7r-p diode, 197,198,199-201, 2 17-228

dc equation, 200,204 diffusion term, 200,214 double injection, 217-228 drift term, 200 single injection, 197-201 small-signalac equation, 200,201-206 quadratic characteristic, 215,216 Current-voltage characteristics

insulator regime, 215,216,221-223 cubic characteristic, 222 diode characteristic, 215,216 quadratic characteristic, 215,216,220, 223

ohmic regime, 218 semiconductor regime, 218-221 quadratic region, 220 CW laser diode, 150,165-172, see also Laser devices for special applications; Lasers; Visible emission laser diode (AlGa)As, 165-172 coupling to fiber, 165, 172 DFB structures, 177 far-field pattern, 171, 172 (InGa)(AsP), 165 noise power, 190 noise spectrum, 188-190 power output, 167 heat sink temperatures, 167 single mode operation, 170 buried channels, 170 spectral emission, 167-170 stripe contact structure, 165-172 threshold currents, 166,167 temperature dependence, 167

D Dark line defects, 63, 156 impurity precipation, 156 nonradiative centers, 156 Degradation, see also Laser diode reliability catastrophic, 151-155,163,164 LOC laser, 163,164 facet damage, 152-155, see also Facet damage gradual, 151, 155-161 avoidance of strain, 157-159 basic causes, 159-161 contaminants, 157 dark lines, 156, 157, see also Dark line defects dislocations, 157, 158 materials-related factors, 157- 159 observations, 155-157 temperature dependence, 161 temperature effect, 157 DFB (distributed-feedback) laser, 175-184 coupled mode analysis, 177-184

320

SUBJECT INDEX

coupling coefficient, 180 feedback coefficient, 180 matched phase condition, 178 DH DFB structure, 176,184 emission spectra, 184 fabrication, 183 frequency characteristics, 177 gain coefficient, 182 longitudinalmode control, 175,182 mode relationship, near Bragg frequency, 181

optical cavity gain, 182 threshold current density, 182, 183 threshold gain, 177,182 DH (double heterojunction) laser, 10, 11, 14, 25,7672, see also Heterojunction laser; Laser diode structures asymmetrical, 113, see also Asymmetrical-structure laser beamwidth, 87,88 cavity modes, 80 confinement factor, 88,89,120, CW, 150, 165-172 DFB structure, 176, 183,184,seealso DFB laser differential quantum efficiency, 119 temperature dependence, 119 doping effects, 128 heterojunction spacing, 82,83,87 peak field intensities, 87 properties, various materials, 146 radiation pattern, 80-89,171, 172 room-temperature CW, 150 shortest wavelength, 175 stripe-contact structure, 138,165-172 thermal resistance, 138 thin structures, 83 efficient room-temperature operation, 83 threshold current density, 71, 112, 113, 116-120, 146-151, 174

temperature dependence, 119, 120 visible wavelengths, 173-175 Dielectric constant asymmetry, 95 periodic, 175, see also DFB laser fabrication, 183 profile, 77, 169, 170, see also Refractive index profile stripe contact laser, 169

shallow maximum, dielectric waveguide, 69,169

step FH laser, index steps, 92 lobe separation, 83 SH laser, 95 Dielectric waveguide, 69 Differential quantum efficiency, 66,67,107, 110-113, 116,118, 163

LOC laser, 163 tabulated values, 116, 118 temperature dependence, 119-121 Diffusion Eienstein relation, 295 Fick’s law, 280 frequency dependence, 280 Monte Carlo simulation, 270,279-283 random walk process, 270 nonlocal, 282,283 intervalley transitions, 283 space distribution function, 280 moments,280 Diffusion term, see Current equation Difisivity, 261,267,279-281,295,see also Diffusion autocorrelation form. 281 nonlocal, 283 intervalley transitions, 283 tensor, 281 Dislocations, due to lattice misfit, 144, 145 effect on minority carrier diffusion length, 144

Disordered solid, see also Amorphous semiconductor density of states, 269 diffusion, 270 in energy, 269 random walk process, 270 mobility edge, 270,271 mobility function, 269 Monte Carlo calculations, 269-271, see also Monte Carlo calculations ohmic conductivity, 270 Distributed feedback lasers, 175-184, see also DFB laser Distributed gain, 5 Distribution function, see also Diffusion charge density, 255 current density, 255 degenerate statistics, 295,305,306

321

SUBJECT INDEX

generation, 264-266 Maxwell-Boltzmann, 256,270,295,305 "displaced", 305 Double-injection diodes, I%, 217-228, see also Space-charge-limiteddiode ac impedance, 223-226 insulator regime, 225 semiconductor regime, 224 admittance, 223-226 cubic (insulator) regime, 223,225,226 linear (ohmic) regime, 223,224 quadratic (semiconductor) regime, 223-225 applications, 247 basic equations, 217-223.240-243 dc characteristics, I%, 218-222 cubic behavior, 222-224 insulator regime, I%, 221-223,224 ohmic regime, I%, 218,223,224 quadratic behavior, 220,223,224,226 semiconductor regime. 1%, 218-221, 224 diffusion effects, 222,226-228 Fletcher solution, 226,227 regional approximation method, 227, 228, see also Regional approximation method filament formation, 198 generation-recombination noise, 243, see also Noise, in space-charge-limited diodes noise, 239-245 ambipolar solution, 239,240 asymptotic solution, 240-243 recombination, 217 SRH centers, 217 space charge small, 1% thermal noise, 241,244, see also Noise, in space-charge-limiteddiodes trapping effects, 198 Drift term, see Current equation Drift velocity, 259.260-262.274 estimator, 259,260-262 Fourier components, 276 harmonic waveform, 274-276 harmonics, 274,276 injunction, 293 linear response, 274 nonlinear response, 274,276 square waveform, 275,276

E Effective mass tensor, 259,262 EHL interaction, see Electron-hole-lattice interactions Einstein relation, 295 Electric field double injection diode, 220 n-u-n or p-r-p diode, 199 distribution. 201 Electron and phonon distributions, coupled, 306 Electron-beam bombardment, 1 Electron energy function, 259 spherical harmonics, 260 Electron-hole-lattice (EHL) interactions, 19,20,45,58,63 GaAs, 19,20 Ga(AsP), 45.63 Electron-photon interactions, 258, see also Transition Electron trajectories, Monte Carlo calculation, 251 Electron transport, see also Hall effect; Mobility current, 255 cyclotron resonance behavior, 277 differential mobility, 274 diffusion, 279-283, see also Diffusion drift velocity, 274, see also Drift velocity harmonics, 274 nonlinear, 274 Monte Carlo calculations, 24%. see also Monte Carlo calculations time dependent effects, 273-283, see also Time dependent transport superlattice materials, 277, see also Superlattice materials Electronegativity differences, N-center, 39, 40 Emission spectra CW lasers, 167-170 mode content, 167-170 DFB lasers, 177, 184, see also DFB laser GaAs laser, 14, 16, 18.21 (InGa)P, 30-34 InP laser, 25,26 longitudinal mode control, 175 DFB laser, 175, see also DFB laser Emission wavelength, see also Photon emis-

322

SUBJECT INDEX

sion energy; Transition; Visible emission laser diode (AlGa)As, 132,167-170,173-175 shortest wavelength, 175 (AUn)P. 132 CW lasers, 167-170 effect ofdopants, 125-128 Ga(AsP), 132,173 (InGa)P, 132 numerous ternary alloys, 142 shortest wavelength, 175 various lasers, 146 Energy gap, see also Band gap isoenergy-gap lines, 35,36 Energy relaxation, 261 Estimators, Monte Carlo calculations, 258, 259,276, see also Monte Carlo calculations B ensemble method, 259,260 drift velocity, 259,260-262 fluctuations, 261,262 fluctuations, 260,261 Expectation values, 251,256

F Fabry-Perot (FP) lasers, 175 Fabry-Perot reflecting edges, 4,8,26, see also Laser topology Facet damage, 152-155,164,seealsoDegradation acceleration by moisture, 155 effect of facet reflectivity, 154 antireflectingfilm. 154,164 optical power dependence, 153 related to structural parameters, 153 Facet elimination, 175-184, see DFB laser Facet reflectivity, 78-80,154 role in degradation, 154, see also Facet damage antireflecting film, 154 Feedback coefficient, DFB laser, 180, 182 FH (four-heterojunction) laser, 72,90, 164 beam width, 90-92,118 canier confinement, 95 cavity modes, 94 DFB structures, 177,183 differential quantum efficiency radiation confinement factor, 90-92.95 radiation pattern, 90-95

threshold current density, 93,95,1 I8 threshold gain, 92-94.118 vertical geometry, 90,93 Field effect, 289,290, see also Boundary effects Field-effect transitor, Monte Carlo simulation, 300 Filament formation, 198 FP (Fabry-Perot) lasers, 175 contrasted to D F3 design, 175 Fresnel reflectivity, 79

G GaAs. 10-24, 121-129 absorption, 23

bandgap energy, temperature dependence, 128 bandtail states, 13,128 band-to-band transitions, 14, 17 dopants, 3,12-15,123-128 amphoteric, 12,15 Cd, 15, 125 Ge, 3, 15,95, 123, 124 Mn, 15 Si, 3.12-14,95, 124,125 Sn, 12,124 Zn, 12,15,123-125 EHL interactions, 19,20 gain coefficient, 108 heterojunctions, 10-14, see also Heterojunctions laser, 10-24, see also Laser emission spectra, 14, 16,18,21 frequency range, 18 Ge acceptor, 3,95 photon emmision energy, 17- 19,125128

photopumped, 3 quantum efficiency, I I , 12 Si acceptor, 3,94,95 threshold current, 11, 12 lasing transitions, 126-128 lattice constant, 28, 134 minority carrier diffusion length, 124, 125 Moss-Burstein shift, 3.22 quasi Fermi level locking, 22,23 recombination, 13,24 surface, 57

323

SUBJECT INDEX

refractive index, carrier concentration effects, 131 surface recombination velocity, 57 thermal conductivity, 138 thermal expansion coefficient, 134 transmission, 23 Ga(AsP), 4,27,38-51 bandgap energy, 132,141 band minima, 38 effect of N centers, 38-51 defect density, 57 dopants N. 38-51 Te, 49-51 Zn, 41,43 edge-toedge modes, 62 external quantum efficiency, 133 junction lasers, 39,172,173,see also Laser emission tkequency, 173 threshold current, 173 laser properties, 146 lattice constant, 132.141 LED (light emitting diodes), 27,38 lifetime data, 57-61 N-doped, 60,61 N - t k ~60,61 , maximum emission energy, composition, 132 melting temperature, 141 N-dowd, 4,38-51 Nx emission peak, 40-51 optical phase shift measurement, 57 shortest wavelength, room temperature CW laser, 175 Ga(AsSb) bandgap, 141 laser properties, 146 lattice parameter, 141 melting temperature, 141 Gain, 5,105-107,seealso Gaincoefficient; Optical gain; Threshold gain gaidoss distribution, 78 laser operation, 5 Gaincoefficient, 108, 110,111,115-118, 169,182 DFB laser, 182 related to carrier injection, 110 stripe contact laser, 169 undoped GaAs, 108

Gap, see also Band gap; Transition direct, 27-29 indirect, 27,28,3&51 GaP lattice constant, 28,134 nitrogen doped, band structure, 40 thermal expansion coefficient, 134 Ga(SbP) bandgap, 141,142 lattice parameter, 141,142 melting temperature, 141 Gaussian distribution, generation, 306,307 Gauss’s law, 199,241 Ge lattice constant, 134 thermal expansion coefficient, 134 Geometrical relationships, 307,308 Gunn domains, Monte Carlo simulation, 300

H Hall effect, 272 Monte Carlo calculation, 272,273,2% Heat sinks, 7,see also Thermal dissipation, laser diodes Heterojunction laser, 658,see also Asymmetrical-structure laser; DH laser: FH laser; Heterojunctions; LOC laser; SH laser; specific materials listings double heterojunction, 10,11,14,25, 70-72 asymmetrical, 71 dark line defects, 63,156 symmetrical, 71 threshold current, 71 quaternary alloys, 28,35-37,142,146, 150 successful designs, 147-150 Heterojunctions, 10-14,68-75,see also DH laser; FH laser; Heterojunction laser; Laser diode structure; LOC laser; SH laser alloys other than (AlGa)As, 139-151 dark line defects, 63,156 major defects, 139-141 lattice mismatch, 139-141 quaternary alloys, 142,146,150 quaternary-ternary materials, 36,142, 146

324

SUBJECT INDEX

ternary alloys, 141,142 bandgap. 141 lattice parameter, 141 melting temperature, 141 thermal expansion, 143 Histogram, Monte Carlo calculations, 251, 259,260,277,282

energy, 260,305 Hot electrons amorphous semiconductors, 269-271 carrier-carrier scattering 284,285,301303

coupling of electron and phonon distributions, 306 diffusivity tensor, 281 magnetic field effects, 272,273 Monte Carlo calculations, 254-271, see also Monte Carlo calculations distribution function, 255 equations of motion, 254,272,273 n-v-n diode, 206-208 photon interactions, 258 photon transitions, 258 superlattice material, 277, see olso Superlattice material I

Imperfections, see also Degradation effect on laser parameters, 123 various 111-V compound lasers, 147-150 Impurity center, see also specific materials listings GaAs, 3,12-15, 123 effect on diffusion length, 125 effect on lasting wavelength, 125-128 effect on refractive index, 131 Ids lattice constant, 134 thermal expansion coefficient, 134 In(AsP), 28,52-54 bandgap, 141 coupled to CdS cavity, 53,54 edge-to-edge modes, 54 lasers, 28,52-54 lattice parameter, 141 low effective mass, 53 melting temperature, 141 mode spacing, 54 wavelength for optical fiber, 52

In(AsSb) bandgap, 141 lattice parameter, 141 melting temperature, 141 Index dispersion, 34.35 (InGa)As, 28,52,53 bandgap, 141,142 laser properties, 146 Lasers, 28,52,53 lattice parameter, 141,142 low effective mass, 53 melting temperature, 141 photopumped, 52.53 wavelength for optical fiber, 52 (InGa)(AsP), 28.35-37.52 bandgap vs. composition, 35-37 laser properties, 146,150 lattice matching, 35,36 room temperature CW laser, 150,165 shortest wavelength laser, 175 Te doped, 37 wavelength for optical fiber, 52 wide wavelength range, 28,35-37 (InGa)P, 4,25,27-35, 132 bandgap, 29,132,141 composition grading, 29 CW laser, 31 dopants Te, 30-32 Zn, 30,32 Zn:Te, 30 double heterojunction lasers, 25 external quantum efficiency, 133 growth, 29 LPE, 29 VPE, 29 laser properties, 146 lasers, 4,25,29-31 different lattice matchings, 30,31 mode spacing, 34.35 lattice constant, 132, 141 maximum emission energy, composition, 132

melting temperature, 141 photoluminescence spectra, 30-34 radiative recombination, 31 shortest wavelength room-temperature CW laser, 175 (1nGa)Sb bandgap, 141

SUBJECT INDEX

lattice parameter, 141 melting temperature, 141 InP, 24-26,28, 134 DH laser, 25 laser operation, 24-26 conversion efficiency, 25 emission spectra, 26 match to glass fiber, 25 quantum efficiency, 26 lattice constant, 28,134 LED, pump for phosphors, 25 thermal expansion coefficient, 134 In(SbP) bandgap, 141 lattice parameter, 141 melting temperature, 141 Internal quantum efficiency (AIGa)As, 133 (AIIn)P, 133 Ga(AsP), 133 (InGa)P, 133 Isoelectronic traps, 27.38-51, see also N-center

J Junctions, Monte Carlo simulation, 290293,300,301 current, 291 detailed balance relation, 291 drift velocity, 293 histogram, space distribution, 292 thermalization, 293 transition probability, 290,291,293 transit time, 293

L Laser, see also Asymmetrical-structure laser; DH laser; FH laser; Heterojunction laser; Laser diode structure: LOC laser; SH laser; specific materials listings (AIGa)As, 10, 11, 14, see also (A1Ga)As alloys other than (AIGa)As, 139-151 CW, 165-172. see also CW laser diode defect-related problems, 139 degradation, see Degradation devices for special applications, I6 I , see Laser devices for special applications

325

discussion, 111-Vcompound heterojunctions, 147-150 distributed feedback, 175-184, see also DFB laser contrasted to Fabry-Perot design, 175 edge-to-edge modes, 54,62 effect of materials imperfections, 123 emission wavelength, see also specific materials listings alloy composition for maximum frequency, 132 dependence on band gap energy, 127129 effect of dopants, 125-128 mode content, 167-170 various materials, 146 visible light, 172-175 Fabry-Perot reflecting edges, 4 , 8 Fabry-Perot vs. DFB, 175 facet reflectivity, see Facet reflectivity GaAs,10-24, see also GaAs emission spectra, 14. 16, 18,21 performance data, I I , I2 photon emission energy, 17-20 Ga(AsP), 38-51 transitions involving Nx, 40-5 1 gain, 5,105-107, see also Gain heat removal, 7 , 8 heterojunction diodes, 65ff. see also Heterojunctions (InGa)P, 25.30-35 (InGa)(PAs), 35-37 InP, 24-26 loss coefficient, 5 mode spacing, 34, 167-170 modulation, 185 noise spectrum, 188-190 nonuniform population inversion, 191 oscillations, 185- 191 Pb salt materials, 146,147.I50, 151 photopumped, I J ~ GaAs, 3 Ga(AsP), 3 . 4 (InGa)P, 3 , 4 photon lifetime, 5, 187 photon loss rate, 4 , 5 population inversion, 5 power conversion efficiency, 164 LOC laser, 164, 165 power output, see also Power output

326

SUBJECT INDEX

CW laser, 167 pulsed laser, 161, 163.164 properties, various materials, 146 Qswitching, 191 radiation pattern, see Radiation pattern reliability, see Laser diode reliability single mode operation, 170 spontaneouscarrier lifetime, 188,see also Carrier lifetime; Lifetime stripe contact diodes, 135-138,165-172, see also Stripe contact diode thermal dissipation, 137-139, see also Thermal dissipation,laser diodes threshold current density, 146-151,see also Threshold current density threshold requirements,4-6 visible light, see Visible emission laser diode Laser devices for special applications CW, 165-172,see also CW laser diode distributed-feedback lasers, 175-184, see also DFB laser highpeakpower, 161-165 arrays, 161, 162 LOC laser, 163-165 visible light, 172-175 shortest wavelength, 175 Laser diode fabrication,seeLaser diode technology Laser diode reliability, 151-161,see also Degradation; Reliability possible operating lifetimes, 160, 161 Laser diode structure, 68-75. see also Laser diode technology; Laser topology arrays, 161,162 buried channels, 170 single mode operation, 170 distributedfeedback, 175-184, see also DFB laser double heterojunction,70-72, see also, DH laser electric field distribution,71 four-heterojunction,72, see also FH laser generalized laser diode, 69-72 homojunction, 70,71,149 large optical cavity, 70-72, see also LOC laser single heterojunction,70,71, see also SH laser stripe contact diode, 135-138,165-172, see also Stripe contact diode

thermal dissipation, 137-139, see also Thermal dissipation,laser diodes thermal resistance, 138 vertical geometry, 69-72 Laser diode technology, 121-139,165 avoidance of stress, dislocations, 157-159 DFB structure, 183 LPE, 121-124, see also Liquid phase epitaxy molecular beam epitaxy, 125,145 stripe contact diode, 135-137,165-172, see also Stripe contact diode types of acceptors, 123,124 VPE, 124,125, see also Vapor phase epitaxy Laser modulation and transient effects, 185- I91 modulation, 185 efficiency, 188 rate equations, 185-187 noise spectrum, 188-190 photon lifetime, 187 self-sustained oscillations, 188 spontaneouscarrier lifetime, 188 Laser topology, 68, see also Laser diode structure broad-area contacts, 69 dielectricwaveguide, 69 Fabry-Perot resonator, 68, see also Fabry-Perot reflecting edges reflecting film, 69 stripe contact diode, 69 wave suppression, 69 Latching, 36 Lattice constant,see also specific materials listings AIAs, 134 (AIGa)As, 132 (AIIn)P, 132 GaAs, 28,134 (GaAs)P, 132 GaP, 134 Ge, 134 I d s , 134 (InGa)P, 132 InP, 134 isolattice-constantlines, 35,36 numerous ternary alloys, 141,142 Lattice matching, 28-31,35,139-142,145, 147-151 dislocations, 144

327

SUBJECT INDEX

effect on photoluminescence, 30 effect on threshold current density, 1481so latching, 36 quaternary alloys, 142,150 specific examples, 145 Lattice relaxation, around N-center, 39 LED (Lightemitting diode), 27.38 operating lifetimes, 160 Lifetime device operation, see Laser diode reliabdity; LED, Reliability luminescence, 9 photon, 5,187 radiative, 12,see also Radiative recombination shortening, stimulated emission, 3,24 spontaneous, carrier, 188, see also Carrier lifetime Lifetime measurement, 8-10,22 excitation-transmission, 22,23 optical phase shift method, 8-10,22,see also Optical phase shift method Liquid phase epitaxy (LPE), 13,29,35, 121-124 difhsion step, 122 lattice matching, 30,35 multiple-bingraphiteboat, 121,122 types ofimperfections, 123 LOC (Large-optical-cavity) laser, 70-72, 98-100,163-165, see also Heterojunction laser; Laser diode structure asymmetric structure, 98, see also Asymmetrical-structure laser beam width, 102 cavity modes, 100 current, 103 far-field pattern, 100,101 geometry, 98 optical confinement, 104 optical output, 103 temperaturedependent changes, 100104

threshold current density, 102 DFB structures, 177,183.see also DFB laser differentid quantum efficiency, 163 emitted power, 164 power conversion efficiency, 164, 165 properties, various materials, 146, 149

resistance to catastrophic degradation, 163,164 antireflective coating, 163 superior high temperature performance, 163 symmetrical structure, 98 cavity modes. 99 geometry, 98 threshold gain, 99 temperaturedependent effects, 1W104, 149,163 threshold current density, 163 temperature dependence, 149 visible light, 173 Loss coefficient, 5 Luminescence lifetime, 9

M Magnetic field effects Monte Carlo calculations, 272,273,2% equations of motion, 272,273 Hall effect, 272.2% magnetoconductivity, 2% Magnetoconductivity, Monte Carlo calculations, 2% Mean free path, vector, 262,295 Melting temperature, see also specific materials listings numerous ternary alloys, 141 Metal-insulator-metal diode, I%, 198,214 Wright’s solution, 214 Minority carrier diffusion length, 125 effect of misfit dislocations, 144 GaAs,124,125 Mobility amorphous materials, 269,see also Disordered solid chordal, 262 differential, 274,276 frequency dependence, 276 superlattice materials, 279 electric field dependent, 206 frequency dependent, 2% mobility edge, 270,271 ohmic, 295 time dependent, 275,276,2% Mode discrimination, 78-80, 170,175,182, see also Cavity modes Modes, see Cavity modes

328

SUBJECT INDEX

Mode spacing, 34,54 In(AsP), 54 (InGa)P, 34,35 Modulation, 188-191, see also Laser modulation and transient effects Molecular beam epitaxy, 125,145 Monte Car10 calculations, 2 4 w , see also Electron transport advantages, 250,251,253 after-effect, 279 amorphous semiconductors, 269, see also Disordered solid avalanche phenomena, 303-305, see also Avalanche phenomena B states, 259 charge density, 255 chordal mobility, 262 collective effects, 297-306, see also Collective effects coupling of electron and phonon distributions, 306 current, 255 cyclotron resonance behavior, 277 detailed balance relation, 256 diffusion, 270,279-283, see also Diffusion disadvantages, 252,253 disordered solid, 269, see also Disordered solid distribution function, 251,258, see also Distribution function drift velocity, 274, see also Drift velocity harmonics, 274 nonlinear, 274 electron trajectories, 251 equations of motion, 254,272,273 estimators, 258,259,276 B ensemble method, 259,260 drift velocity, 259,260-262 fluctuations, 260-262 expectation values, 251,256 histograms, 251,259 history, 256 hot electrons, 254-271, see also Hot electrons magnetic field effects, 272,273,2%, see also Magnetic field effects many particles, 297-301 nonrejection criterion, 265 ohmic conduction, 270.294-2% ohmic conductivity, 270 path, 256

path duration, 257 path term, 271 pseudorandom numbers, 252,257,264, 267 random variables, 251,252 rejection method, 257,264-266 scattering, 271, see also Scattering isotropic, 262 scattering time, 256 source states, 279.2W spatial structures, 283-293, see also Spatial structures junctions, 290-293 superlattice material, 277-279, see also Superlattice material time-dependent effects, 273-279, see also Time-dependent transport thermal conductivity, 2% thermoelectric constants, 2% trial, 257 variance reduction, 263,264 Moss-Burstein shift GaAs, 3.22 Ga(AsP), 50

N N-center, 4,38-51 direct and indirect recombination, 51 effect on carrier lifetime, 60,61 Ga(AsP), Nx band emission shift with composition, 44,45 G e N , band structure, 40 recombination probability, 48 Negative resistance, 228,229 due to trapping, 228,229 Noise, in space-charge-limiteddiodes, 217228 double injection diode, 239-245, see also Double-injection diode basic equations, 240-243 generation-recombination noise, 243 thermal noise, 241,244 Nyquist’s theorem, 231 Schottky’s theorem, 230 single-injectiondiode, 233-239, see also Single-injection diode Langevin method, 235,236 salami method, 234 thermal and shot noise, 233,234 trapping, 238,239

329

SUBJECT INDEX

sources, 229-233 diffusion (velocity fluctuations), 23 I-

threshold current density, 106,107,110113,116-118,146

dependence on acceptor concentration,

233 shot, 230

106

spectral intensity, 229 thermal, 231 trapping, 233 spectral intensity, 229,230 Wiener-Khintchine theorem, 230 Noise spectrum, see also Noise, in spacecharge-limited diodes CW laser, 188-190 spectral intensity, 229,230 Noise theorems Nyquist, 23 1 Schottky, 230 Wiener-Khintchine, 230 n-v-n diode, 1%-210 hot electron effects, 206-208 pulse response, 208-21 I trapping effects, 210 0

Optical fiber, 25,52, 165,172 coupling efficiency, 172 low absorption window, 165 use with CW laser diodes, 165 Optical gain, 69,105-107, see also Gain; Gain coefficient; Threshold gain DFB laser, 182 Optical phase shift method bimolecular recombination, 58 defect density, 57 Ga(AsP), 57 lifetime measurement, 8-10,22,55-58 nonradiative recombination, 58 Optical vs. electrical property relationships, 104-121,146,seealso Laser; specific materials listings absorption coefficient, 107,109 carrier lifetime, I05 cavity end loss, 109 differential quantum efficiency, 107, 110. 111-113,116-118

gain, 105-107 coefficient, 107,108, 110, I 1 I , 116-1 18 spontaneous recombination, 105 dependence on carrier concentration, 106

stimulated emission, carrier lifetime, 105

temperature dependence, 118-121 Optoelectronic devices, 66

P (PbCd)S, lasing wavelength, 151 (PbGe)S, lasing wavelength, 151 (PbGe)Te, lasing wavelength, 151 PqSSe), laser properties, 1% PbSe, laser properties, 146 (PbSn)Te, laser properties, 146, I51 Photoexcitation, 1, see also Photopumping Photoluminescence, I , see also Photopump ing spectra Ga(AsP), 42-47,49-51 (InGafP, 30-34 studies, 6 GaAs:Zn:Sn crystals, 12 InP, 25 Photon emission energy, see also Emission wavelength composition for maximum (AlGa)As, 132 (AlIn)P, 132 Ga(AsP), 132 (InGa)P, 132 GaAs, 17 dependence on doping, 18-21 Photon lifetime, 5, 187, see also Lifetime Photon loss rate, 4,5 Photopumped semiconductor laser, Iff, see also Photopumping emission spectra, 14, 16.30 GaAs, 10-24 Ga(AsP), Nxtransition, 40-51 In(AsP), 52-54 coupled to CdS cavity, 53.54 (InGa)As, 52.53 (InGa)P, 28-35 (InGa)(PAs), 35-37 InP, 24-26 photon emission energy, 17-21 Photopumping, see also Photopumped semiconductor laser distributed feedback configuration, 64 methods, 2,3,6-10

330

SUBJECT INDEX

penetration, 3 study of dark line defects, 63 threshold requirements, 4.20 P-i-n diode, 195, I%, 226 ac solution, 226 Fletcher's solution, 226 Poisson distribution, 298 Poisson's equation, 197 Population inversion, 5 6 , see also Laser; Photopumping Potential, built-in, I%, 203 Potential distribution, 197,220 metal-insulator-metal structure, 198 p v - p diode, 197,198,201 potential maximum, 198 potential minimum, 198,199 Power conversion efficiency, 164 high values, 164 LOC laser, 164, I65 Poweroutput, 161,163, 164,167, 171 vs. current, 167, 171 "kinks", 171 CW, 167 "kinks", 170, 171 pulsed, 161, 163,164 P-m-p diode, 1% current equation, 197 potential distribution, 197, 198 Pseudorandom numbers, Monte Carlo calculations, 252,257,264,267 Pulse response, 208,229,245 double-injectiondiode, 229 single-injection diode, 208-21 1,245 trappingeffects, 210

Q Quantum efficiency, see also Differential quantum efficiency; Internal quantum efficiency differential, 66,67, 110, see also Differential quantum efficiency external, 133 various alloys, 133 GaAsSHlasers, 11,12 InP photopumped diodes, 26 internal, 112-133 various alloys, 133 Quasi Fermi level, 5,22,23 locking, 22,23 shift, 72,73

Quaternary alloys, 26,28,35-37, 142,150, see also specific materials listings

R Radiation confinement, see also, Confinement stripe contact diodes, 136,137 Radiation confinement factor, 88-90, see also Confinement factor (optical) DH laser, 88.89, 120 FH laser, 90-92,95 Radiation pattern, 78-101 beam width, 87,88, 102, 118 DH lasers, 80-89, 171, 172 far-field, 78,81,82,84-86, 100. 101,171, 172 FH lasers, 90-95 lobe separation, 83 LOC lasers, 100-102 near-field, 78,81,84-86 temperature dependent, 101,102 Radiative recombination, 55,74, see also Recombination process; Transition GaAs, 13, 15-24 Ga(AsP) 38-51 N-centers, 4,38-51 (InGa)P, 31-34 (InGa)(PAs), 37 InP, 25 Monte Carlo simulation, 304.305 Random variables, Monte Carlo calculation, 251,252 Random walk process, 270 diffusion, 270 Recombination process, 12,13,19,20,55, 304,305, see also Carrier lifetime; Radiative recombination; Transition band-to-band, 57 bimolecular, 57,58, 105 GaAs, 13,15, 19,20,22,24 Ga(AsP), 38-51 (InGa)P, 3 1-34 nonradiative centers, 27.28, 156, 159.160 degradation, 159 lattice mismatch, 28,31 N-trap, 38-51 probability, 48 spontaneous, 105 SRH (Shockley-Read-Hall) centers, 217

SUBJECT INDEX

stimulated emission, 105 surface, 55.56 Recombination region confinement, 69 DH and FH lasers, 72,73 Refractive index, see also specific materials listings (AJGa)As. 130, 131 carrier concentration dependence, 131 Refractive index profile, 69.77,78, see also Refractive index step continuously variable, 78 major considerations absorption, 77 bandgap variation, 77 carrier density, 77 SH laser, % Refractive index step, see also Refractive index profile FH laser, 92 LOC laser, 100-104 SH laser, 102,104 temperature dependent changes, 100- 104 Regional approximation method, 227,228 negative resistance problem, 229 Rejection method, Monte Car10 calculation, 257 von Neumann’s, 251 Reliability laser diodes, 151-161, see also Laser diode reliability LED’S, 160

S Scattering, 255.262-269.271, see also Transition acoustic mode, 262,266 azimuth angle, 267,268 final state, 268 Brooks-Herring relation, 302 channels, 266-268,282 electron-electron, 284,285,301-303 final states, 257 grain boundary, 286 interband, 256,266 interface, 286-290 intervalley, 263,266 intraband, GaAs,12 isotropic, 262,263,266 nonisotropic, 263

331

occupation probability, 255 optical mode, 262,266 polar optical mode, 268 rate function, 255,258 self-scattering, 257,260,276 thermalization, 12,293 time, 256 SH (single heterojunction) laser cavity modes, 98 confinement region, 98 dielectric asymmetry, 95, see also Asymmetrical-structure laser dielectric step, 95 geometry, % operating temperature limitation, 163 properties, various materials, 146 threshold current density, 97,174 visible light, 173, 174 Single-injectiondiodes, 195, 1%. 199-216, 233, see also Space-charge-limiteddiodes applications, 245-247 diodes, 245,246 triodes, 246,.247 characteristics, 200-206 ac, 200-206 dc, 200-206 Child’s law, 201 diffusion, in insulator, 214-216 diffusion effects, 214-216 frequency response, 202 hot electron effects, 206-208 injection negligible, 204 injection predominant, 204 large space charge, 1% linear regime, 205 noise, 233-239 Langevin method, 235 salami method, 234 shot, 233 thermal, 233 trapping, 238 pulse response, 208-21 I trappingeffects, 210 quadratic regime, 205 thermal noise and shot noise, 233-238, see also Noise, in space-chargelimited diodes Langevin method, 235,236 salami method, 234 verification of theory, 236-238

332

SUBJECT INDEX

transit time, 202 trapping effects, 210-213,see also Trapping effects trapping noise, 238,239, see also Noise. in space-charge-limiteddiodes spectrum, 239,240 Space charge in diodes, 1 9 5 i limitation of current, 195-197 double-injection diode, 1%, 217 single-injection diode, 195, 1% Space-charge-limiteddiode, 195fl. see also Double-injection diode; Single-injection diode ac conductance, 202,203,223,226,227 frequency dependence, 227 admittance, 202,206 capacitance, 202,203 diffusion effects, 222,226-228 Fletcher solution, 226,227 regional approximation method, 227, 228, see also Regional approximation method double injection, 1%, 217-228, see also Double-injectiondiodes frequency response, 202 insulator regime, 1%, 214-216.221-224 cubic characteristic, 224 diffusion, 214 diode characteristic, 215,216 electric field, 215 electron density, 215 high voltage range, 215,216 low voltage range, 215,216 potential difference, 215 quadratic characteristic, 215,216 ohmic regime, 1%, 218,223,224 noise, 229-245, see also Noise, in spacecharge-limiteddiodes semiconductor regime, I%, 218-221,224 single injection, 195-216, see also Singleinjection diode transition between single and double injection, 222 transit time, 202 Spatial structures, 283 boundaries, 284-286 reflection, 285 transmission, 285 escape and penetration phenomena, 284-286

electron-electron scattering, 284,285 photoemission, 284 secondary emission, 284 junctions, 290-293, see also Junctions current, 291 transit probability, 290,291 Monte Carlo simulation, 283-293 size effects, 286-290 field effect, 286 interface scattering, 286-290, see also Boundary effects related to mean free path, 286 Spontaneous recombination, 105, see also Lifetime; Recombination process lifetime, 105, 188 SRH (Shockley-Read-Hall) recombination centers, 217 Stimulated emission, 5.6, see also Laser; Transition carrier lifetime shortening, 3,22,24, 57-63 lifetime, 105 Stripe contact diode, 69,135-138, 165-172 C W lasers, 165-172 mode content, 167-180 radiation confinement, 136,137 thermal resistance, 138 visible emission laser, 175 Superlattice material, 277-279 after effect, 279 differential mobility, 279 response, 278 source state, 279 velocity time dependence, 278 Surface recombination velocity, 55.56, 148, I 49 high-threshold current densities, 149 measurement, 56 absorption, 56 ambipolar diffusion constant, 56

/

T Temperature dependent changes bandgap energy, G A S ,128 differential quantum efficiency, 119, 120 large effects in asymmetrical LOC lasers, 100-104 beam width, 102 current, 103

SUBJECT INDEX

far-field pattern, 101 output, 103 threshold current density, 102 lasing threshold, 166 operating lifetime, 161 superior high-temperature performance. LOC laser, 163 thresholdcurrent density, 118-121, 149, 167 Ternary alloys, 26-29, 141, see also specific materials Listings bandgap, 141 lattice parameter, 141 melting temperature, 141 Thermal conductivity (AIGa)As, 131, 132 Monte Carlo calculation, 2% Thermal dissipation, laser diodes, 137-139 effect on threshold current, 137 heat sink temperature dependence, 167 laser output, 167 heat sinks, 7 thermal resistance, stripe contact DH diode, 137, 138 Thermal expansion coefficient NAs, 134 GaAs, 134 Gap, 134 Ge, 134 InAs, 134 InP, 134 unequal, at interface, 143, 144 Thermoelectric constants, Monte Carlo calculations, 2% 111-V (Three-five compound) semiconductor alloys, 26-55, see also specific materials listings composition parameter, 26 quaternary, 26,28 successful lasers, 147-150 ternary, 26,27 Threshold current density, 66,67,75, 106, 107, 110-113, 116-118, 146-151, 163, 166, 167,173-175, 182,183,seealso listings of specific materials and laser designs canier confinement loss. 74 CW lasers, 166, 167 dependence on acceptor concentration, 106

DFB laser, 182, 183

333

DHlaser,71,97, 106, 107, 112,113, 116, 146, 167 effects of lattice mismatch, 148-150 FH laser, 93.95 (GaA1)As laser, 7 I GaAs laser, 10- 12 homojunction laser, 149 (1nGa)P laser, 35 LOC laser, 102, 146, 149, 163 reduction with heterojunction, 147 SH laser, 97,146 temperature dependence, 102, 118-121, 149, 167 various lasers, 146 visible emission lasers, 173-175 Threshold gain, see also Gain DFB laser, 177. 182, see also DFB laser FH laser, 92-94 LOC laser, 99 Time-dependent transport, 273-279 cyclotron resonance behavior, 277 harmonic waveform, 274-276 linear response, 274 mobility, 2% nonlinear response, 274,276 nonperiodic. 279-283 periodic. 273-279 square waveform, 275,276 Transition, 3,4,258, see also Emission wavelength: Scattering dependence on bandgap, 127- 129 direct, 27.40, 132 GaAs, 14-17, 126-128 band-to-band, 14, 17 Ge acceptor, 15 Si acceptor, 15 Ga(AsP) system, 27,38-51 Nxband, 41-51 hot electrons, 258 indirect, 4.27 (1nGa)P band-to-acceptor, 34 band-to-band, 31,34 direct-indirect, 32 donor-to-valence band, 31 InP, band-to-band, 25 intervalley, 263,283 isoelectronic trap, 27 lasing energy shift with dopant, 125128 maximum energy, composition, 132

334

SUBJECT INDEX

low injection level, hole trap, 228 N-trap, 4,27,38-51 shallow, 211,213 quasi indirect, 4 trap parameter, 212 Transmission, GaAs, 23 Tunneling current, 75 Trap-free case, I%, 199,217 double-injectiondiode, 217-228 single-injection diode, 197-210 Trappingeffects, 198,211-214,228,229,see V also Traps deep trap, 211-213 Vapor phase epitaxy (VPE). 29,124,125 linear regime, 213 composition grading, 29,125 quadratic regime, 213 Visible emission laser diode, 172- 175 trap parameter, 212 (AIGa)As, 173-175 double-injection diode, 228 composition, 173 filament formation, 198,229 DHlaser, 173-175 negative resistance effects, 229 LOC laser, 173 filament formation, 198,229 SH laser, 173,174 negative resistance, 228,229 threshold current density, 173-175 shallow traps, 211,213,214 (GaAs)P, 172,173 single-injectiondiode, 21 1 shortest wavelength, 175 deep trap, 212 shallow trap, 213 Traps, 198, 203.21 I , 228, see also Trapping W effects Wave propagation, junction laser, 75-104 deep, 211,212 Wiener-Khintchine theorem, 230, see also high injection level, recombination cenNoise, in space-charge-limiteddiodes ters, 228

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