SEMICONDUCTORS & SEMIMETALS

SEMICONDUCTORS AND SEMIMETALS VOLUME 2 Physics of 111-V Compounds

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SEMICONDUCTORS AND SEMIMETALS Edited by R . K . WILLARDSON BELL AND HOWELL RESEARCH CENTER PASADENA, CALIFORNIA

ALBERT C . BEER BATTELLE MEMORIAL INSTITUTE COLUMBUS, OHIO

VOLUME 2 Physics of 111-V Compounds

1966

ACADEMIC PRESS New York and London

COPYRIGHT 0 1966, BY ACADEMIC PRESSINC. ALL RIGHTS RESWVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS I N C . 1 f 1 Fifth Avenue, New York. New York 10003

United Kingdom Edition publhhed by ACADEMIC PRESS I N C . (LONDON) LTD. Berkeley Square House, London, W. 1

LIBRARY OF C O N G ~CATAJJX S CARDNUMBER: 65-26048 PRINTED IN THE UNITED STA'ZES OF AMERICA

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

F. G. ALLEN,Bell Telephone Laboratories, Inc., Murray Hill, New Jersey (263) E. ANToNCiK, Institute of Solid State Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia (245) J. R. DRABBLE, Department of Physics, University of Exeter, Exeter, England (75)

M . GERSHENZON, Bell Telephone Laboratories, Inc., Murray Hill, New Jersey (289) G. GIESECKE, Research Laboratory, Siemens Schuckertwerke AG, Erlangen, Germany (63) G. W . GOBELI, Bell Telephone Laboratories, Inc., Murray HiN, New Jersey (115, 263) BERNARD GOLDSTEIN, RCA, David Sarnofl Research Center, Princeton, New Jersey ( 1 89)

M . G. HOLLAND,Raytheon Company, Research Division, Waltham, Massachusetts ( 3 )

A. U. MAC RAE,Bell Telephone Laboratories, Inc., Murray Hill, New Jersey (1 15) ROBERTLEEMJEHER, Department of Physics, Purdue University, Lafayette, Indiana (141)

T. S . Moss, Royal Aircraft Establishment, Farnborough, Hants, England (205)

S . I . NOVIKOVA, A.A. Baikov Metallurgical Institute, Moscow, U.S.S.R. (33)

P. S . PERSHAN, Harvard University, Cambridge, Massachusetts (283) V

vi

LIST OF CONTRIBUTORS

U. PIESBERGEN,' Physikalisch-Chemisches Institut, Universitiit Zurich, Zurich, Switzerland (49)

FRANKSTERN,IBM Watson Research Center, Yorktown Heights, New York (371)

J. TAUC,Institute of Solid State Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia (245)

'Present address : Emhart Zurich S.A., Zurich, Switzerland.

Preface The extensive research devoted to the physics of compound semiconductors and semimetals during the past decade has led to a more complete understanding of the physics of solids in general. This progress was made possible by significant advances in material preparation techniques. The availability of a large number of compounds with a wide variety of different and often unique properties enabled the investigators not only to discover new phenomena but to select optimum materials for definitive experimental and theoretical work. In a field growing at such a rapid rate, a sequence of books which will provide an integrated treatment of the experimental techniques and theoretical developments is a necessity. An important requirement is that the books contain not only the essence of the published literature, but also a considerable amount of new material. The highly specialized nature of each topic makes it imperative that each chapter be written by an authority. For this reason the editors have obtained contributions from ten to fifteen scientists to provide each volume with the required detail and completeness. Much of the information presented relates to basic contributions in the solid state field which will be of permanent value. While this sequence of volumes is primarily a reference work covering related major topics, the volumes will also be useful in graduate courses. Because of the important contributions which have resulted from studies of the 111-V compounds, the first few volumes of this series are devoted to the physics of these materials. Volume 1 reviews key features of the 111-V compounds, with special emphasis on band structure, magnetic field phenomena, and plasma effects. In Volume 2, the emphasis is on physical properties, thermal phenomena, magnetic resonances, and photoelectric effects, as well as radiative recombination and stimulated emission. Volume 3 is concerned with optical properties, including lattice effects, intrinsic absorption, free carrier phenomena, and photoelectronic effects. The editors are indebted to the many contributors and their employers who made this series possible. They wish to express their appreciation to the Bell & Howell Company and the Battelle Memorial Institute for providing the facilities and the environment necessary for such an endeavor. vii

viii

PREFACE

Thanks are also due to the U S . Air Force Offices of Scientific Research and Aerospace Research, whose support has enabled the editors to study many features of compound semiconductors. The assistance of Rosalind Drum, Jo Ann Gibel, Eleanor Quinan, and Inez Wheldon in handling the numerous details concerning the manuscripts and proofs is gratefully acknowledged. Finally, the editors wish to thank their wives for their patience and understanding. December, 1965

R. K. WILLARDSON ALBERTC. BEER

Contents LISTOF CONTRIBUTORS. . PREFACE . . . . OF OTHER VOLUMES CONTENTS

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THERMAL PHENOMENA I Chapter 1 Thermal Conductivity M . G . Holland I. 11. 111. IV. V. VI. VII.

Introduction . . . . . . . Theory . . . . . . . . Measurement Techniques . . . . . Thermal Conduction in Various 111-V Compounds. Special Effects . . . . . . . Summary . . . . . . . . Conclusions . . . . . . .

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Chapter 2 Thermal Expansion S . I . Nouikova . .

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Chapter 3 Heat Capacity and Debye Temperatures U . Piesbergen I. Introduction . . . . . . . 11. Heat Capacity . . . . . . . 111. Debye Temperature OD . . . . .

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49 50

63 65 73

I. Introduction . . . . . 11. Experimental Results . . . . 111. Calculation of the Griineisen Parameter.

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PHYSICAL PROPERTIES I Chapter 4 Lattice Constants G . Giesecke I. Introduction . . . . 11. Measurement of Lattice Constants 111. Summary . . . . .

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CONTENTS

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Chapter 5 Elastic Properties J . R . Drabble I . Introduction . . . . . . . . I1 . Thermodynamic and Atomistic Aspects . . . 111. The Propagation of Elastic Waves . . . . IV . Effects of Carrier Concentration on the Elastic Constants V . Experimental Results . . . . . . . VI . Conclusion . . . . . . . .

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Chapter 6 Low Energy Electron Diffraction Studies A . U . Mac Rae and G . W. Gobeli I . Introduction . . I1 . Experimental Techniques 111. Results . . . IV . Conclusions . .

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124 136

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MAGNETIC RESONANCES Chapter 7 Nuclear Magnetic Resonance Robert Lee Mieher I . Introduction . . . . . . . . I1. NMR Absorption Line 111. Relaxation. Saturation. and Polarization Appendix . . . . . .

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145 165 183

Chapter 8 Electron Paramagnetic Resonance Bernard Goldstein I . Introduction . . . . . .

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I1 . The Paramagnetic Resonance Condition and the Spin Hamiltonian . 111. Gallium Arsenide . . . . . . . . . IV . Indium Antimonide . . . . . . . . . . . V . Gallium Phosphide . . . . . . . . . . . . . VI. R h m k and Concluding Remarks

PHOTOELECI'RIC EFFECTS Chapter 9 Photoconduction in III-V Compounds T. S. Moss I . Introduction . . . . . . . I1 . Theory . . . . . . . . 111. Experimental Results . . . . . . List of Symbols . . . . . .

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191 199 200

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205 206 225 243

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CONTENTS

Chapter 10 Quantum Efficiency of Effect in InSb E. AnronEik and J . Tauc I. Introduction . . . 11. Experimental . . . 111. Theory . . . .

the Internal Photoelectric . . .

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Chapter 11 Photoelectric Threshold and Work Function G . W . Gobeli and F. G . Allen I. Introduction and Discussion . . . . .

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IV. Discussion

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11. Measurement Techniques

111. Results and Discussion

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PHOTON EMISSION Chapter 12 Nonlinear Optics in 111-V Compounds P . S . Pershan I. Introduction . . . . . . . 11. General Discussion . . . . . . 111. Theory

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IV. Experiment V. Conclusion

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Chapter 13 Radiative Recombination in the In-V Compounds M . Gershenzon I. Introduction . . . . . . . . . . 11. GaP . . 111. GaAs . . IV. Other Compounds V. Notes Added in Proof

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Chapter 14 Stimulated Emission in Semiconductors Frank Stern I. Introduction . . . . . . .

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11. Relation between Stimulated and Spontaneous Emission 111. Laser Structures. . . . . . . . IV. Modes, Directionality, and Coherence . . . . V. Quantum Efficiency . . . . . . . VI. Radiation Confinement, Threshold, and Loss . . VII. Laser Materials . . . . . . . . VIII. Effects of Ambients and External Fields . . .

AUTHOR INDEX . S m c r INDEX

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371 374 376 380 389 396 403 407

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Semiconductors and Semimetals Volume 1 Physics of 111-V Compounds C . Hilsum, Some Key Features of 111-V Compounds Franco Bassani, Methods of Band Calculations Applicable to 111-V Compounds E. 0. k h e , The k ' p Method V. L. Bonch-Brueuich, 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 P. N . Argyres, Magnetic Quantum Effects S . M . Puri and T . H . Gebalk, Thermomagnetic Effects in the Quantum Region W. M . Becker, Band Characteristics Near Principal Minima from Magnetoresistance E. H. Putley, Freeze-Out Effects, Hot Electron Effects, and Sub-mm Photoconductivity in lnSb H. Weiss, Magnetoresistance of the 111-VCompounds Betsy Ancker-Johnson, Plasmas in Semiconductors and Semimetals

Volume 3 Physics of III-V Compounds Marvin Hass, Lattice Reflection William G. Spitzm, Multiphonon Lattice Absorption D. L. Stierwalt and R . F . Potter, Emittance Studies H . R. Philipp and H . Ehremeich, Ultraviolet Optical Properties Manuel Cmdona, Optical Absorption Above the Fundamental Edge E. Johnson, Absorption Near the Fundamental Edge John 0. Dimmock, Exciton States in Semiconductors Benjamin Lax and John G. Mauroides, Interband Magneto-Optical Effects H . Y. Fan, EXects of Free Carriers on the Optical Properties Edward D . Palik and George B. Wright, Free Carrier Magneto-Optical Effects Richard H. Bube, Photoelectric Analysis B. 0. Seraphin and H . E . Bennett, Optical Constants

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Thermal Phenomena I

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CHAPTER 1

Thermal Conductivity M . G. Holland I. I1.

111.

INTRODUCTION .

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THEORY. . . . . 1 . Phonons . . . 2 . Electrons and Holes 3 . Other Heal Carriers

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MEASUREMENT TECHNIQUES. .

I v . THERMAL CONDUCTION IN VARIOUS 4 . InSb . . . . . . 5 . InAs . . . . . . . . 6.InP . . . . . . . . 1.GaAs . . . . . . . 8 . Ga S b . . . . . . . 9 . Other III-V Compounds . .

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13

1II-V COMPOUNDS

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VI .

SPECIAL EFFECTS . . . . . 10. Ternary Alloys . . . . . 1 1 . Magnetic Field Effects . . 12. Electron Irradiation Effects .

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SUMMARY. . . . . . . 13 . Boundary Scattering . . . 14. Impurity Scattering . . . IS. Resonance Scattering . . . . I6. Electron-Phonon Scattering . . 11. Three-Phonon Processes . . . 18 . Electronic Thermal Conductivity

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20 20 20 21 22 24 25 26 26 27 21 30

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I Introduction

The thermal conductivity of materials has been studied and understood qualitatively for many years. The early theoretical work of Debye’ and Peierls’ on materials in which the heat is carried predominantly by phonons P. Debye. in “Vortrage uber die kinetische Theorie der Materie und Elektrizitat.” Teubner. Berlin. 1914. R. Peierls. Ann . Physik 3. 1055 (1929).

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M. G . HOLLAND

indicated that the following behavior is to be expected: The thermal conductivity K at the lowest temperatures depends on the size and shape of the crystal (or crystallites). It increases with temperature with approximately the same temperature dependence as the specific heat and reaches a maximum. At temperatures above this maximum, K is limited by the scattering of phonons by phonons and is characteristic of the material. Near the maximum, K is sensitive to the imperfections and impurities in the material. Electrons, when available in sufficient quantities, can also carry heat. This electronic contribution is usually significant only at very high temperatures in semiconductors. There has been a renewed interest in thermal conductivity in the last few years because of the availability of new and more accurate data3 on a large number of pure elements and compounds. Better data on a wider range of materials have led to an increased understanding of some of the phenomena involved in heat conduction in Many unsolved problems remain however ;most of these involve the scattering phenomena4 The 111-V compounds are attractive for thermal conductivity studies. These materials offer a wide range of lattice and electronic properties. They can, for the most part, be obtained in highly pure form,6 so that impurity effects are minimized and the intrinsic properties can be investigated and compared. These materials can also usually be doped with known amounts of electrically active impurities, and the electronic effects can be compared. For these compounds information exists on properties such as sound velocity, Debye temperature, energy gap, electron and hole mobilities, and impurity ionization energy. But thermal conductivity measurements also assist in understanding the 111-V compounds. For example, information on impurities, both electrically inactive and active, can be obtained from the low temperature thermal conductivity.’ A comparison of the strength of the phonon-phonon interactions can be obtained from the data near the Debye temperature.’,* Values for the energy gap and mobilities near the melting point can, in principle, be deduced from the electronic thermal conductivity. These are all properties of importance. Thermal conductivity is clearly also of technological importance. The thermal conductivity value is necessary in calculating the figure of merit See, for example, P. G. Klemens, Solid State Phys. 7 , 1 (1958). P. Carruthers, Rev. Mod. Phys. 33, 92 (1961). H. Bross, Phys. Star. Sol. 2, 481 (1961). See, for example, “Compound Semiconductors” (R. K. Willardson and H. L. Goering, eds.), Vol. 1 : “Preparation of 111-V Compounds,” Reinhold, New York, 1962. M. G. Holland, Phys. Rev. 134, A471 (1964). E. F. Steigmeier and J. Kudman, Phys. Rev. 132, 508 (1963).

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1.

THERMAL CONDUCTIVITY

5

for thermoelectric devices9 Of greater importance would be the ability to control the processes which limit conduction so that the thermal conductivity could be altered. For example, decreasing the thermal conductivity while the electrical conductivity remained the same (or increased) would increase the efficiency of thermoelectric devices. For power dissipating devices such as diodes, transistors, or lasers it is useful to know the value of the thermal conductivity to assist in device and circuit design.” For example, for a GaAs injection laser operating at 4.2”K it is clearly better to remove the heat through the n-type material which has at least two orders of magnitude higher thermal conductivity than the p-type layer. There are other areas in which the thermal conductivity can be of importance. For phonon amplifiers’ or problems of microwave phonon attenuation,I2 thermal conductivity can provide information on the relaxation times for the high frequency (thermal) phonons. This is necessary to understand the interactions affecting the low frequency ( 5lo1’ cps)phonons which are being propagated and studied. The high thermal conductivity coupled with the low electrical conductivity of some of the purer 111-V compounds would make these materials excellent heat shields at low temperatures. Some of the properties of GaAs in a magnetic fieldI3 might lead to a useful heat switch at low temperatures. The same might be true of InSbI4 at high temperatures. In this chapter we first review the theory of thermal conductivity (Part 11) and give a short resume of the methods of measurement (Part 111). Part IV presents the data available for the 111-V compounds, with emphasis on the most up-to-date material, while Part V contains information on ternary alloys, magnetic field effects, and radiation damage effects. Part VI is a summary, followed by some general conclusions in Part VII. 11. Theory

There are several excellent reviews of the general theory of thermal A. F. Joffe, “Semiconductor Thermoelements and Thermoelectric Cooling” Infosearch, London, 1957. l o W. W. Gartner, “Transistors, Principles, Design, and Applications.” Van Nostrand, Princeton, New Jersey, 1960; W. N. Carr and G. E. Pittman, Appl. Phys. Letters 3, 173 (1963). D. L. White, J. Appl. Phys. 33, 2547 (1962). ”See, for example, A. H. Nethercot, Jr., and H. H. Nickle, Proc. 8th Intern. ConJ Low Temp. Phys., London, 1962, p. 300. Butterworth, London and Washington, D.C., 1963. l 3 M. G. Holland, in “Physics of Semi-Conductors” (Proc. Intern. Con€ Phys Semicond., Paris, 1964), p. 713. Dunod, Paris and Academic Press, New York, 1964. l4 D. Kh. Amirkhanova and R. I. Bashirov, Fir. Tverd. Teia 2, 1597 (1960) [English Transt.. Soviet Phys.-Solid State 2, 1447 (1961)].

6

M. G . HOLLAND

cond~ction,~~ so~ only ~ ' ~ ~a ' short ~ summary is presented here. In an elementary way we can describe the heat conducted by any excitation (phonon, electron, photon) by the equation K =

fCul,

(1)

where C is the heat capacity, u the propagation velocity, and 1 the mean free path of the excitation. If there are several types of excitations, IC is the sum of a contribution from each. If the mean free path is limited by several scattering processes, the effective mean free path is given by

where lj is the free path determined by the jth scattering process alone. 1. PHONONS Equation (1) must be generalized in order to account for the heat carried by all the phonons. The specific heat per normal phonon mode is given by' where x = hw/kT and w is the phonon frequency. If we now add up the contributions due to each mode we obtain the total specific heat

where q is the wave vector and A the polarization of the phonons. Now, if we generalize the mean free path to 1 = (v,,a)z(q,

4,

(5)

where u is the phonon velocity and t(q,A) is the relaxation time for the phonon (9, A), then

The frequency and wave vector have been related in the Debye approximation (for low wave vectors) by o = uq, and we have assumed an isotropic phonon spectrum. Equation (6) can also be obtained from a Boltzmann equation a p p r ~ a c h .In ~ ' place ~ of Eq. (2) we write

l6

K. Mendelssohn and H. M. Rosenberg, Solid State Phys. 12, 223 (1961). J. M. Ziman, "Electrons and Phonons." Oxford Univ. Press (Clarendon), London and New York, 1960.

1. THERMAL

CONDUCTIVITY

7

where each ':z, is the probability that a phonon is scattered by the jth process. The z i s are normally obtained by perturbation techniques. TABLE I TIMES' RELAXATION Scattering process Crystalline boundariesb Impurities'

Inverse relaxation times T;'

= vJLF

T;’

=

Am4,

A = (Vr)/4m,'

r = Z,J(AM~/M)Z

(mass difference) Three-phonon N processesd U processes'

TN

Four-phonon'

T;’

BNw"Tm B,o'T3 exp( -O/orT) = B,'w2T-high temperature cz 02T2/M202u,3 =

T; I =

Resonan& Electron phonon Zimanh Keyes'

Key to terms used in Table I : Constants Equivalent sample diameter F Geometrical factor V Molecular volume A Atomic fraction of the ith impurity whose mass is Mi M Average mass of an atom of the host lattice AM = M - M i Average phonon velocity us 0 Debye temperature Density P R, G Constants that contain the number of scattering centers Resonance frequency WO R Describes the damping of the resonance Measures the electron-phonon interC action m* Electron effective mass B L

&

= fmusz

Function of T x , q, and the Fermi temperature A Splitting of the donor or acceptor ground state ro Mean radius of the bound states n = 1 for transverse phonons, = 2 for longitudinal phonons m = 1 for high T = 3 for longitudinal phonons at low T = 4 for transverse phonons at low T From Casimir" and Berman et d.21.22 From K l e m e n ~ . ~ . ~ ~ From Klemens3 and Herring.26 ' From K l e m e n ~ ~and . ~ ' Holland.'8 From Pomeranchuk." (I From Poh13' and Walker and Pohl." From Ziman.33 i From Griffin and car rut her^.'^ w

8

M. G . HOLLAND

9

’

’

“ 1

’

’

‘

I

’

’

‘ “1

FIG.I . Thermal conductivity vs temperature for a typical 111-V compound. The ranges over which the various scattering times, T, are operative are shown, as are the ranges in which the electronic, K , , and phonon, tiprcomponents are usually important.

The use of this combined relaxation time, Eq. (7), in the integral of Eq. (6) gives a very good approximation to the phonon thermal conductivity. Numerical integration of Eq. (6) by many has given excellent fits to data on a large number of materials. The important scattering times are listed in Table I, and Fig. 1 is a curve of thermal conductivity vs temperature for a typical semiconductor. The temperature regions over which various scattering mechanisms are operative are indicated. The first three scattering processes shown in the table have been used extensively in calculations. However, several points can be made concerning these relaxation times. In the boundary scattering term,” the factor F represents a correction due to both the smoothness of the sample surface and the finite length-to-thickness ratio of the specimen.21*22 In q ,the scattering parameter r is for mass-difference scattering al0ne.3.’~ For other types of impurity scattering, such as that due to strain fields or I’

l9

J. Callaway, Phys. Rev. 113, 1046 (1959). For an extensive list of references see M. G. Holland, Phys. Rev. 132,2461(1963). The equation is written in the form % / I ’ f“’(ex - I)-’ dx K = CTf -k K z , T-I

where C = (k/2~~u,)(k/h)~ and K~ is a correction due to the conservative nature of the normal three-phonon processes to be discussed later. ti2 has usually been neglected in calculations of ti. zo H. B. G. Casimir, Physica 5, 595 (1930). ” R. Berman, F. E. Simon, and J. M. Ziman, Proc. Roy. SOC. (London) A220, 171 (1953). ” R. Berman. E. L. Foster, and J. M. Ziman, Proc. Roy. Soc. (London)A231, 130 (1955). 2 3 P.G.Klemens, Proc. Phys. Soc. (London) A68, 1113 (1955).

1.

THERMAL CONDUCTIVITY

9

changes in the elastic constants of the interatomic linkages due to the point imperfections, the form of r is changed, and in some cases the frequency dependence is altered as well.3*24 The scattering due to the naturally occurring isotopes of the material is usually the largest cause of massdifference ~cattering~~-unlessthe material is exceptionally heavily doped (- IOl9 crnp3), in which case the doping atom is important. There are two types of three-phonon p r o c e ~ s e s . ~Normal * ~ ~ * (N) ~ ~ processes are momentum-conserving processes which alone could not cause thermal resistance. Umklapp (U) processes are responsible for most of the phonon scattering at high temperatures and are processes in which the phonon momentum is changed by a reciprocal lattice vector. The N processes are important in that they change low-momentum phonons to highermomentum phonons. A more complete analysis of the problem shows that, to a first approximation, the relaxation time for these N processes can be simply included in the total r. Various values of the exponents n and m have been used in rN18 depending on the phonon branch and temperature range. Various forms of 2u18*27have also been used, and in some cases the exponential is taken to be unity. Details of the dispersion of the phonon spectrum should be known in order to derive accurate expressions for these relaxation times. Four-phonon processes” have seldom been used in actual calculations. At high temperature these processes would lead to a T-’ temperature dependence in K but no real proof of their existence has been pre~ented.~.’~ The h a 1 two relaxation mechanisms, resonance and electron-phonon ~ c a t t e r i n g , ~ are ~ -relatively ~~ new. Both have been used to explain anomalous temperature dependences in recent data. Resonance scattering has been suggested as the cause of dips found in some data at low temperature3’ (p = 0) and at temperatures just above the maximum (p = 2).31 Electron-phonon scattering has been used to explain slopes greater than T 3 and abrupt slope changes at low temperat~res.~ Both P. Carruthers, Phys. Rev. 114, 995 (1959). G. A. Slack, Phys. Rev. 105, 829 (1957). 2 6 C. Herring, Phys. Rev. 95,954 (1954). ” P. G. Klemens, Proc. Roy. SOC. (London) AZ08, 108 (1951). 1. Pomeranchuk, Phys. Rev. 60,820 (1941). 2 9 C. J. Glassbrenner and G. A. Slack, Phys. Rev. 134, A1058 (1964). 3 0 R. 0. Pohl, Phys. Reu. Letters 8, 481 (1962). 3 1 C. T. Walker and P. 0. Pohl, Phys. Rev. 131, 1433 (1963). 32 M. Wagner, Phys. Rev. 131, 1443 (1963). 3 3 J. M. Ziman, Phil. Mag. 1, 191 (1956). 34 I. C. Pyle, Phil. Mag. 6, 609 (1961). 3 5 R. W. Keyes, Phys. Rev. 122, 1171 (1961). 36 A. Griffin and P. Carruthers, Phys. Rev. 131, 1976 (1963). 24

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M. G. HOLLAND

electron-phonon relaxation times-the Ziman form for heavy dopings including degenerate samples and the Keyes term for lighter dopingsprobably have validity. The Keyes term can be used to explain the variation in scattering caused by different chemical impurities. Electron-phonon scattering is commonly found in metals” and at high temperatures in semiconductors.29 2 . ELECTRONS AND HOLES The thermal conductivity due to electrons (K,) can be related to the electrical conductivity D by the Wiedemann-Franz law,3*’ K, = L T D , (8) watt-ohm/deg2in the usual where Lis the Lorenz number (L = 2.45 x case of degenerate carriers). The conditions on the electron and phonon scattering, which insures the accuracy of this expression, have been discussed thoroughly by Ziman.I6 Ambipolar diffusion of electrons and holes along a temperature gradient can be important in K, for semiconductors. A general form of K, which includes both the ambipolar term and the Wiedemann-Franz term is given by Eq. (9)37-39:

where D, and D , are the electrical conductivities of the electrons and holes, and E, is the energy gap at temperature T The coefficients A and B depend on the detailed relaxation process and are each about 2.

3. OTHERHEATCARRIERS In principle, any excitation can carry heat energy in a crystal-for example, photons, spin waves, or polarons. Of these we will consider only photons, since the others have not been observed in semiconductors. If the material being studied is transparent to photons of some frequency, energy can be radiated through the sample from the hot to the cold end The thermal conductivity for this situation was obtained by Gemel*’ using 16nZa0T3/(3a), (10) where n is the index of refraction, oo the Stefan-Boltzmann constant, and a the absorption coefficient averaged over frequency. Cody et aL41 have K, =

P. J. Price, Phil. Mag. 46, 1252 (1955). B. I. Davydov and I. M. Shmushkevitch, Usp. Fiz. Nauk 24,21 (1940). 39 J. R. Drabble and H. J. Goldsmid, “Thermal Conduction in Semiconductors,” Chap. 4. Macmillan (Pergamon), New York, 1961. 40 L. Genzel, Z. Physik 135, 177 (1953). 41 C. D. Cody, B. Abeles, and D. S. Beers, Bull. Am. Phys. SOC.8, 296 (1963). 37 38

1. THERMAL

CONDUCTIVITY

11

modified this equation to include time-dependent heat flow as used in some measurement techniques. The value of K,, is usually quite small, so that it is seen only in materials with very low values of lattice and electronic thermal conductivities. It has, however, been found in G ~ A s . ~ ’ ~ 111. Measurement Techniques The problems involved in the measurement of thermal conductivity differ somewhat with the method of measurement and the temperature range, particularly when the measurement is made below or above room temperature. The methods are discussed in the so only a brief review will be presented. Between 2” and 300”K, the measurement technique most widely is that of lateral heat flow in which electrical power (Q = V x I) supplied by a heater to one end of a rectangular bar is extracted by a heat sink at the other end of the bar, and the temperature gradient V T = AT/AX is measured along the bar. The thermal conductivity is then given by

(Q/A)/VT, (11) where A is the cross section of the bar. This is an absolute measurement and depends on accurate measurement of the temperature gradient and on minimum heat loss from the specimen. The accuracy is limited by the measurement of VT. Thermocouples can be used down to 2°K with an accuracy of a few percent. This method has also been used to 0.3”K, but accuracy is difficult to obtain in this region. Gas thermometers and carbon resistance thermometers4’ have also been used to obtain A T at low temperatures. The chief disadvantage of the lateral heat flow method is that it cannot be used with very small samples. The difficult problem of heat losses by radiation makes this method less accurate above 300°K. At room temperature and above, a comparison method46 and the Angstrom are often used. In the comparison method, the material to be measured is sandwiched between two specimens with known thermal conductivity. By measuring the temperature gradients along both K =

A. Amith, I. Kudman, and E. F. Steigmeier, Phys. Rev. 138, A1270 (1965). R. Berman, in “Experimental Cryophysics” (F. E. Hoare, L. C. Jackson, and N. Kurti, eds.). Butterworth, London and Washington, D.C., 1961. 43 M. G. Holland and L. G. Rubin, Reu. Sci. Instr. 33, 923 (1962). 44 G. A. Slack, Phys. Rev. 105, 832 (1957). 4 5 G. A. Slack, in “Encyclopedic Dictionary of Physics” (J. Therolis, ed.), Vol. 3, p. 601. Macmillan (Pergamon), New York, 1962. 46 A. D. Stukes and R. P. Chasmar, Rept. Meeting Semicond., Rugby, 1956, p. 119. Phys. SOC., London, 1956. 47 A. J. Angstrom, Ann. Physik 114, 513 (1861). 48 P. H. Sidles and G. C. Danielson, J . Appl. Phys. 25, 58 (1954). 49 B. Abeles, G. D. Cody, and D. S. Beers, J . Appl. Phys. 31, 1585 (1960). 41s

42

12

M. G . HOLLAND

the known and unknown samples, a determination of the actual heat flow through the unknown samples can be obtained, and the heat loss from the surface is thus taken into account. The members of the sandwich must be maintained in good thermal contact and the unit surrounded by a radiation shield to keep the surface losses minimal. The Angstrom method is not a steady-state method. In this method a sinusoidal heat input is imposed at one end of a specimen and the heat equation V *(kgrad T) = aT/at, k = ic/cp (12) is solved under appropriate boundary conditions. In this equation k is the thermal dsusivity, c is the specific heat, and p the density. If the velocity of propagation of the temperature wave, or the amplitude decrement of the wave, or both are measured, the diffusivity can be obtained, and the result is independent of the radiation from the sample. In both methods the heat loss is not as severe a problem as in the lateral heat flow method, but the accuracy depends on the accuracy of subsidiary data: in the former, the thermal conductivity of the comparison material, and in the latter the specific heat of the specimen. At very high temperatures (about 1000 C)the sources of error are difficult to eliminate and the measurements become inaccurate. The radial heat flow method” combines the advantages of the lateral heat flow method with a geometry which minimizes the heat loss problem. In this method the sample is a cylinder of length L. A heater is placed on the axis, and the temperature gradient is measured at two radii, rI and r2. The conductivity is given by K = Q ln(r2/rl)/(2nL AT). (13) Since the gradient is measured before the heat reaches the outer surfaces, the heat loss problem is not severe. The two difficulties in this approach are: (1)the gradient cannot be measured accurately because the measuring thermocouples must be quite close together unless the samples are very large; (2) heat loss from the ends of the sample is still possible. However, using this method, Slack and G l a s ~ b r e n n e rhave ~ ~ *obtained ~~ what appear to be highly reliable, absolute values of the thermal conductivities of Ge and Si from below room temperature to very near their melting points. All of the methods discussed, except the radial heat flow method, have been used for obtaining the various data presented in Part IV. The continuity of the data taken above room temperature for InSb and InAs, for example, indicates that the methods give agreement of about 5 % near room temperatures.

’’G. A. Slack and C. Glassbrenner, Phys. Reu. 12.0, 782 (1960); C. J. Glassbrenner, Rev. Sci. Instr. 36,984 (1965).

1.

THERMAL CONDUCTIVITY

13

IV. Thermal Conduction in Various III-V Compounds

In the following sections experimental results and theoretical analysis are presented for many of the 111-V compounds. Only the most recent results are presented and discussed.

4. InSb The thermal conductivity of InSb7,51*52 is shown in Fig. 2. Older results are not shown but have been discussed by Busch and Steigmeier and by Holland, nor have the data of Amirkhanova and Bashirov14 between 50" and 700°K been included, despite the fact that they are 25 to 50% lower than the results shown. Their results will be discussed later.

I

2

4

6810

100 Twnpemtue 1 K)

FIG.2. Thermal conductivity of InSb, InAs, and InP. InSb [InSb ( 1 ) after Holland,' InSb (2) after Busch and Steigmeier," InSb-P and InSb-N after Challis et lnAs (1)after S h a l ~ t , ~ ~ InAs (2) after Steigmeier and Kudman,' InP after Kudman and Steigmeier6'J The solid curves A, B, C, and ice are described in the text.

Curves A and B in Fig. 2 are calculations7 for InSb, using numerical integration of Eq. (6), in which the relaxation times included t b , T,, T ~ and tU. The experimental sample geometry was used to determine t b . A phonon-phonon relaxation time, obtained from fitting the data near 50"K,was used in the calculation. Curve A uses zI calculated from the normal isotope distribution, while curve B, which clearly is a better fit to the data, uses three times as much impurity scattering as could be attributed to the isotopes. This magnitude of mass-difference impurity scattering implies that there are of the order of loi9 impurities/cm3 in the material

'' L. J. Challis, J. D. N. Cheeke, and J. B. Harness, Phil. Mug. 7, 1941 (1962). 52

G. Busch and E. Steigmeier, Helu. Phgs. A C ~ 34, Q 1 (1961).

,

14

M. G. HOLLAND

(assuming the extreme case ( A M / M )= 1 in 7 , of Table I). Since the sample contained of the order of 1014 impurities/cm3 the cause of this large effect is not clear. It is probably too large to be due to strain effects.24However, this effect is found in many materials, including the III-V compounds. The fit is good in the boundary scattering region for the pure sample. The data on samples InSb-N5’ which contained 1.4 x 10”Te/cm3 and InSb-P5’ which contained 2 x 10l6Ge/cm3 are lower than would be predicted from boundary scattering (by about 60% for InSb-N and a factor of 10 for InSb-P). This effect in the boundary scattering region appears to be due to the impurities. It is often seen even with somewhat smaller dopings, and is usually more pronounced for p-type material than for n-type.791 3.5 3 Several scattering mechanisms have been used to explain this effect. Challis et al.” used calculations by Ziman33based on scattering of phonons by electrons in an impurity band and, alternatively, calculations by based on scattering of phonons by localized electrons capable of hopping to an adjacent site to fit their data on the p-type material. The agreement with Pyle’s model was not good and that using Ziman’s model was better. The Keyes m e c h a n i ~ m ,which ~ ~ . ~depends ~ on the sensitivity of the impurity ionization energy on strain, has not been completely developed for the acceptor state but has successfully explained similar anomalies for donors in Ge and GaSb. Since the theory should be similar for acceptors, it would be interesting to apply it to these InSb data. There is a slight, but real, dip in the data near 15°K. This may indicate resonance scattering31354and might also account for the large amount of impurity scattering near the maximum in K. The high temperature data of Busch and SteigmeierS2show both the lattice conductivity (curve IC,,)and the total conductivity, which includes the electronic component. The electronic component was obtained by using Eq. (9) and the resistivity and mobilities measured at high temperature. They deduce values of A = B = 2 in Eq. (9) and show that the WiedemamFranz contribution is greater than the ambipolar contribution, especially at the highest temperatures ( 700°K) where the energy gap has decreased substantially. 5. InAs The low temperature data of Shalyt” [InAs (l)]and the high temperature lattice component [ I d s (2)] deduced by Steigmeier and Kudman’ are N

G. Holland and L. J. Neuringer, Proc. Intern. Con$ Phys. Semicond., Exeter, Eng., 1962 p. 415. Institute of Phys. and Phys. SOC.,London, 1962. 5 4 G. A. Slack and S. Galginaitis, Phys. Rev. 133, A253 (1964). 5 5 S. S. Shalyt, Fiz. Tuerd. Tela 4, 1915 (1962) [English Transl.: Soviet Phys.-Solid State 4. 1403 (1963)l. 53 M.

1. THERMAL CONDUCTIVITY

15

shown in Fig. 2. Data by Sirota and Bergers6 on polycrystalline material covers the range 90" to 300"K, but does not agree with the data shown in Fig. 2 and has not been included. The dashed line in Fig. 2 is an interpolation between the low and high temperature results. Shalyt did not analyze his data except to compare them to the work of Mielczarek and Frederikse on InSb." However, we have obtained curve C of Fig. 2 from a numerical integration of Eq. (6). Since many of the cons t a n t ~(Debye ~ ~ temperature, density, and elastic constants c1 are similar to those of GaSb, we have used a similar phonon velocity of 3 x los cm/sec. A sample size of 4 x 4 x 40 mm3 as indicated by Shalyt was used to calculate the boundary scattering relaxation time. A very good fit was obtained in the low temperature region. However, to obtain a fit near the maximum, the value of impurity scattering had to be increased by a factor of about 10 over that calculated from the isotope scattering. Since the sample contained only 3 x 10I6 free electrons/cm3, the source of this impurity scattering is not clear. Since the ratio of Debye temperature in InAs and InSb is -260/200, and the other properties are similar, the higher value of thermal conductivity for InAs, compared to InSb, above the conductivity maximum may be indicative of weaker phonon-phonon scattering (T;’ and zN1) in InAs or stronger optical mode scattering in InSb. This will be discussed in Part VI. The effect of impurities on the high temperature thermal conductivity can be seen in Fig. 3. The n-type impurity increases the thermal conductivity through the electronic component K , . The magnitude of this component was obtained by Steigmeier and Kudman by using Eq. (9) and measured values of resistivity vs temperature. The values of A and B were obtained from An,p= r

+$

Bn*p=

5 + T,

and (15)

where r is given by T = TOE',

(16)

z is the electron relaxation time. A temperature-dependent r as calculated

by Ehrenreichs9 was used. For the heavily doped samples, the electron 56

N. N. Sirota and L. J. Berger, Inzh. -Fiz. Zh., Akad. Nauk Belorussk. SSR 1, 117 (1958).

'' E. V. Mielczarek and H. P. R. Frederikse, Phys. Rev. 115, 888 (1959). '* E. F. Steigmeier, Appl. Phys. Letters 3, 6 (1963). 59

H. Ehrenreich, J . Appl. Phys. 32, 2155 (1961).

16

M. G. HOLLAND

FIG.3. The effects of impurities on the high temperature thermal conductivity of InAs (after Steigmeier and Kudman').

- 100

300

400

500600700800 Trrpaatura ( O K )

m

FLG.4. The lattice thermal conductivity of InAs at high temperatures (after Steigmeier and Kudman').

1.

17

THERMAL CONDUCTIVITY

relaxation time was modified to include scattering by optical phonons. The resulting lattice thermal conductivity is shown in Fig. 4 for the four samples. The source of the large increase in scattering for the impure n-type sample was not identified, but the estimated mass difference scattering due to the impurities was insufficient to cause the effect. This large increase in scattering could be due to electron-phonon scattering. 6. InP

The high temperature thermal conductivity of InP6’ is also shown in Fig. 2. This presents the results on two samples with carrier concentration between 1015 and l O I 7 ~ m - The ~ . calculated electronic contribution K, 60

,

,

,

u

300

403

600

800

I000

Temperature ( K )

FIG.5. (a) The thermal conductivity of GaAs (after H ~ l l a n d ’ . ’ ~ The ) . curves A, B, and C are described in the text. (b) The high temperature lattice thermal conductivity of GaAs (after Amith et 0 1 . ~ ~ 9 .

was found to be negligible for both samples. Kudman and Steigmeier indicate that the dominant scattering mechanism in this material in this temperature range involves only acoustical phonons-no optical phonons participate. This is discussed in Part VI. 7. GaAs

The thermal conductivity of several samples of GaAs7.l3 is shown in Fig. 5(a). The solid lines A, B, and C were calculated using Eq. (6) and serve to illustrate several points. Sample GaAs-2 is the pure specimen, GaAs-1 reduced in cross section. In the low temperature region, curve C is obtained by increasing the boundary scattering to take the smaller cross section into 1. Kudman and E. F. Steigmeier, Phys. Reo. 133, A1665 (1964).

18

M. G . HOLLAND

account. Thus, the theoretical value of boundary scattering is seen to lead to an excellent fit between the data and the calculation for these pure (c ~ m - samples. ~ ) As is again the case, isotope scattering, curve A, does not provide sufficient impurity scattering to account for the maximum in thermal conductivity. Curve B is obtained using about twice as much scattering as would be provided by the isotopes. This is much more scattering than would be expected from the = loi6 C M - ~impurities. There is a dip in the data near 20°K similar to that noted in the InSb data. This is possibly due to resonance scattering and may account for the magnitude of the conductivity near the maximum. The remaining data in Fig. 5(a) show the effects of impurities on the lowtemperature thermal conductivity. As was the case for InSb, the n-type impurities (in this case Te) do not cause as large a decrease in thermal conductivity as comparable amounts of the p-type impurities, Zn, Cd, and Mn. This was also the case in Si and Ge. The high temperature lattice thermal conductivity of GaAs is shown in Fig. 5(b).41aThe decrease in K with increasing free carrier concentration is attributed to scattering of phonons by electrons. The anomolous curve shape for samples 1 and 4 are the effect of heat transport by photons?' 8. GaSb

The thermal conductivity of four samples of GaSb' are shown in Fig. 6(a). The data of KopeC61 are in good agreement with this data above 100"K, but are somewhat lower near the maximum. Curve A in Fig. 6(a)is the calculated thermal conductivity obtained from Eq. (6) in the boundary scattering region. This calculated curve overestimates the data by almost a factor of 100. The n-type samples, although containing 10 times more impurity than the p-type samples, have, again, a higher value of IC over most of the lowtemperature region. An attempt was made to fit the data by using electron-phonon scattering. Curve I C ~in Fig. 6(a) was obtained using an analysis due to Ziman33 which treats scattering of phonons by electrons in a degenerate band. While the magnitude of the thermal conductivity is correctly predicted, this scattering does not account for the change in slope in the p-type material nor can it account for the dependence on impurity density. Figure 6(b) shows an attempt to fit the GaSb data using the electronphonon mechanism suggested by K e y e ~ . In ~ ~this . ~case, ~ the scattering is due to the strain sensitivity of the donor (or acceptor) ground-state energy. The change in slope near 5°K obtained in the analysis is highly encouraging. Since the phonons will be scattered only when they have a wavelength 61

2. KopeC, Actu Phys. Polon. 17, 265 (1958).

19

1. THERMAL CONDUCTIVITY

Ten~perolure (OK I

FIG. 6. (a) The thermal conductivity of GaSb (after Holland'). (b) Thermal conductivity of ) - ~ , GaSb (after Holland'). The curves are obtained using 7;: = yx4T4(l + ~ X ~ T ' and w < Alh.

-

0.1

0.08-

M

FIG. 7. The high temperature lattice thermal conductivity of GaSb, Gap, and AlSb (after Steigmeier and Kudman6'").

2Q

M. G . HOLLAND

comparable to, or larger than, the electron (or hole) orbit, the higher energy phonons are not scattered, and the mechanism dies out as the temperature is increased. The high temperature lattice thermal conductivity of GaSb is shown in Fig. 7.61aThe decrease in thermal conductivity with increasing free carrier concentration is attributed to scattering of phonons by electrons as in the case of GaAs. Some optical phonon scattering has also been identified. 9. OTHER 1II-V COMPOUNDS

The high temperature lattice thermal conductivity of GaP and AlSb is shown in Fig. 7.61a The electronic contributions are negligible. In both cases the temperature dependence is stronger than T - I , thus, three-phonon scattering processes will not be sufficient to explain the data. V. Special Effects

10. TERNARY ALLOYS The data for two groups of ternary alloys are shown in Fig. 8.62 Abeles has done calculations using an analysis which was reasonably successful Md hoam of In&

GOAS

0

InAs

0.1

0.2

0.3

0.4

I

I

I

1

0.5 1

06 -

1

InAs 0.7

0.8

0.9

I

I

I

M d fractionof InP

1 0 .

I

InP

FIG.8. Thermal resistivity of (Ga, In) As and In(As, P) alloys as a function of composition (after Abeles6*).

on Ge-Si alloys between 300" and 900°K. This analysis uses Eq. (6) with three relaxation times ,z = BN'02(three-phonon N processes),2; = BU 'a2 (three-phonon U processes), and z; = Arm4 (strain as well as mass point defects). r is the disorder parameter and is a function of the masses and the 61a

E. F. Steigmeier and I. Kudman, Phys. Rev. 141, 767 (1966). B. Abeles, Phys. Rev. 131, 1906 (1963).

1.

21

THERMAL CONDUCTIVITY

radii of the constituent atoms. Using a ratio of N-process to U-process scattering of 2.5 and several minor adjustable parameters, the solid curves shown in Fig. 8 were obtained. Abeles concluded that the large thermal resistance in these ternary alloys is predominantly due to strain scattering, and not mass-defect scattering. 1 1 . MAGNETIC FIELDEFFECTS There are two separate effects expected and found for a 111-V compound in a magnetic field. If there is an electronic component in the thermal conductivity, this component will change because of changes in the number of carriers and in the carrier mobility in the magnetic field [see Eq. (9)].

am

2-

- I-

%

-

E

\ o -

-r"I Y

-

-

104

@ & 2.7~10"Zn o -5xIO'Mn -5 xlOR Mn D

-

I

I

8 . [1

' I 2

I

l

l

1

I

1

4 6 810

1

1

1

i I

100 200

Ternpemtwe (OK)

FIG. 9. Thermal conductivity of GaAs (after H ~ l l a n d ' ~ )The . closed symbols represent measurement in a magnetic field of 12.7 kOe.

If, at low temperatures, electron-phonon scattering exists, the amount of scattering can be affected Magnetic fields have not been found to influence the phonon distribution or velocity directly. The thermal conductivity of many 111-V compounds has been investigated in a magnetic field In the low-temperature region InSb,'4*51InAs," GaSb,63 and pure GaAsI3 are not affected by magnetic field. The effect of a magnetic field on the low-temperature thermal conductivity of Mn- and M. G. Holland, unpublished measurements.

22

M. G. HOLLAND

Zn-doped GaAs is shown in Fig. 9.13These effects are assumed to be related to changes in the lattice thermal conductivity due to the effect of the magnetic field on the electron-phonon scattering. Analysis of the zero field data using the Keyes relaxation time35 as modified by Griffin and car rut her^^^ leads to the following conclusions: (1) The dips in the data for the Mn-doped samples imply that the splitting A of the Mn acceptor ground states is between 3 and 4 meV. (2) The mean radius of the hole bound to the Zn ion is greater than 30 A. Furthermore, the splitting of the Zn acceptor ground state is at least twice as large as that of the Mn acceptor. The relative change in lattice thermal conductivity in a magnetic field H can be written' AK 2gpH for - x f2gpH+1 and h o + A , (17) K A A where gPH describes the effective splitting of the ground state level in a magnetic field and A is the splitting for H = 0. Using this equation and the measured change in K with field, the ground-state splittings obtained were A(Mn) 3 meV, A(Zn) > 8 meV-consistent with the zero field analysis. The effect of a magnetic field on the high temperature thermal conductivity of InSbI4 has been investigated. The effect is not large, seldom exceeding lo%, and is due to the decrease in the electronic component of thermal conductivity. Figure 10 shows the temperature dependence of the relative change in thermal conductivity of InSb at 23.7 kOe. Sample No. 14 is a highly doped n-type sample. The remaining samples are reasonably pure. Amirkhanova and Bashirov show that their data follow an expected H Zlaw for low fields and tend to saturate at high fields. This effect is shown in Fig. 11. One important result of this work is that for the pure samples the relative change AK/K in a saturating magnetic field is in good agreement with the ratio of electronic to total thermal conductivity calculated from Eq. (9). That is, the electronic contribution was entirely removed with a high magnetic field, leaving only the lattice thermal conductivity. The authors also indicate that the electron-phonon scattering parameter r [in Eq. (16) for example] can be investigated through this type of study, but with the small effects at low fields the problem is difficult.

-

12. ELECTRON IRRADIATIONEFFECTS V O O examined ~ ~ ~ the thermal conductivity of InSb and GaAs after irradiation by 2-MeV electrons. The change in thermal resistance with irradiation is shown in Fig. 12. The irradiations were carried out at 50 K 64F.L. Vook, Phys. Rev. 135, A1742 and A1750 (1964).

23

1. THERMAL CONDUCTIVITY

Tenpaotun (OK1

FIG. 10. Relative change in thermal conductivity vs temperature at 23.7 kOe (after 1-2 and 1-3 were p-type; 2a and 14 were n-type; and Amirkhanova and B a s h i r ~ v ' ~Samples ). No. 14 was the impure specimen.

-

10

*S

d

No.14 No.1-2

D

No.20 No 1-3

8

~~.m-%e~

FIG. 11. Relative change in thermal conductivity as a function of the square of the magnetic field at various temperatures (after Amirkhanova and B a ~ h i r o v ' ~ ) .

0 Number d e!acllals/a2

FIG.12. Change in thermal resistivity with irradiation for GaAs and InSb (after V00k~~).

24

M. G . HOLLAND

for GaAs and at 50" and 18°K for InSb. The additive thermal resistivity of InSb increases as the three-fourth power of 4, the number of electrons per cm2 passed through the sample, and hence as the three-fourth power of the number of defects produced. At high defect concentration this dependence approaches the one-half power of 4. For GaAs, the dependence goes as the first power of 9. The results also show that the same number of electrons per cm2 causes a much larger change in the thermal resistivity of InSb than in GaAs. This effect is unexpected since, at 50"K, the thermal resistivity of GaAs should be more sensitive to impurities than that of InSb, because of the dominance of three-phonon processes at 50°K in InSb. Furthermore, both materials exhibit the same change in lattice strain per defect so that the cause of the increased scattering is not necessarily the lattice strains. 40,

I

I I I I II

I

I

I

I I ,,,I

I

I

I

I

t

0'1

/,

I

10

I

,

I I 1 I II

, 1 1 1 1

I

100

I

I

I

I

1

1

I I I I,

I , , ,L

KKX)

Tempmture ( O K )

FIG. 13. Thermal conductivity of GaAs, low temperature annealing temperature (after

V00k~~). ( K / L ) - ' is the boundary scattering relaxation time (see Table I).

Figure 13 shows some of the results of isochronal anneals on the thermal conductivity of GaAs. The minimum near 20°K appears similar to the minimum attributed to resonance scattering in other materials. Annealing at temperatures between 325" and 575°K caused decreases in the low temperature thermal conductivity and a slight increase in the room temperature value. The data are shown in Fig. 14. For InSb, isochronal anneals at 395°K caused almost complete recovery of the thermal conductivity and at least two competing annealing processes were indicated. VI. Summary In summarizing the work on the 111-V compounds it is convenient to examine the various scattering times separately.

1.

25

THERMAL CONDUCTIVITY

G-3unrrndnsd

E

-5

2.0-

1.0-

ae a6

-

a4 -

I01 8 10

I

20

4

I

1

1

1 1 I I L l l l l l l U l

40 6080103 Ternperabn

1

I

1

I I

xx)3CXJ4W

FIG. 14. Thermal conductivity of GaAs, high temperature annealing (after V 0 0 k ~ ~ ) .

We will first consider those relaxation mechanisms operative below room temperature. In Table I1 the ratios of theoretical to experimental relaxation times for boundary scattering (zb) and impurity scattering (z,) are listed. These ratios were obtained from the curves obtained using Eq. (6). Similar ratios for pure Ge and Si are also i n ~ l u d e d . ~ " ~ 13. BOUNDARYSCATTERING

It is significant that the boundary scattering relaxation time ratio (zbtheory/zbexp) = 1 for all the pure semiconductors, although this is rarely TABLE I1 RELAXATION TIMERATIOS' Sample Ge Si 0-Sic InSb InAsJ G a As- 1 GaSb

'b Lhcory/'b

erp

'1 I h e o r y / T l e r p

1 1

1 1

1

-5 -3 10

-1

1 I > 50

-- 2

-

B N + 8, (sec/deg3)

2.8 3.8 3.8 1.9

x

10-23

x x 10-z4

x lo-'' 5.5 x 1 0 - 2 3 3.6 x 1 0 - 2 3 -

n 1.2b 1 .2b -

l.ld 1.2' 1.25g

> lh

Unless otherwise indicated, data are from Holland.' The parameter n is from K oc T - " for high 7: * From Glassbrenner and Slack." Silicon containing about 10" oxygen atoms/cm3. From Busch and Steigmeier.52 From Steigmeier and Kudman.* Most of the results are from this work. From Amith, Kudman, and Steigmeier.41a From Steigmeier and Kudman.6'"

26

M. G. HOLLAND

the case for other materials. However, with relatively low concentrations of impurities the ratio increases considerably. This effect is more pronounced for p-type impurities than n-type impurities. An apparent increase in boundary scattering with addition of impurity was found in boron-doped Si53 and has been noted in other r n a t e r i a l ~ ,but ~ ~ the scattering mechanism responsible for the effect has not been identified. It is doubtful that clusters of impurities44can be the cause of this effect since it is evident in so large a variety of samples, many in which the doping levels are not particularly high. Electron-phonon scattering may help explain these effects. 14. IMPURITY

SCATTERING

Of all the semiconductors studied, only in Si and Ge could the maximum in the thermal conductivity be fitted by using the isotopic contribution to the impurity scattering relaxation time.7 The amount of impurity scattering calculated from the known impurity concentrations in the materials is more than two orders of magnitude less than the isotope contribution, since the isotope concentrations are normally measured in the percent range in these elements. Nevertheless, the impurity scattering must be increased by factors of 2 to 10 as indicated by the ratios ( T ~ ~ ~ ~ ~in / Table 11. The effects of lattice strains24 have not been included in q ,and these effects may resolve part of the discrepancy. It is possible that electrically inactive impurities, dissolved gases, for example, are partially responsible for these effects.

15. RESONANCESCATTERING In most of the data, dips are apparent in the region on the high temperature side of the thermal conductivity maximum. The results of Pohl and Walker indicate that these dips are possibly due to resonance scattering of phonons and are associated with impurities. If one examines older data on Si66 a similar dip can be noted. However, when oxygen-free S P 7 was measured there was no dip in the data and it was found that isotope scattering alone could be used to obtain a good fit. Thus the impurity effect ( T ~ ~ # ~1) may ~ ~sometimes ~ ~ /be associated T ~ ~ with ~ ~ dissolved gases. Wagner32 has shown that a relaxation time which can account for the high temperature dips is obtained by considering inelastic scattering of phonons by localized modes associated with the impurities. The effect appears above the maximum in the same temperature region in which the 6 5 R . 0. Pohl, Phys. Rev. 118, 1499 (1960). M. G. Holland, Proc. 7th Intern Con$ Low Temp. Phys., Toronto, Ont., 1960 p. 280. Univ. of Toronto Press, Toronto, 1961.

66

/ T , ~ ~ ~

1. THERMAL CONDUCTIVITY

27

Umklapp and normal three-phonon processes are dominant, so that an exact comparison of data with theory cannot be carried out. Dips and resonance scattering have also been seen at low temperatures in the boundary scattering region.30Some of the effects noted for the doped samples at low temperatures may be related to this type of scattering. 16. ELECTRON-PHONON SCATTERING Attempts to use electron-phonon scattering as suggested by Ziman and by Keyes have thus far been most successful on Ge.35*36The large effect of impurities on the thermal conductivity of InSb, GaAs, and GaSb has led to consideration of electron-phonon scattering for these materials as well. Attempts to use the Ziman effect in InSb and both the Ziman and Keyes term in GaSb were only moderately successful. The Keyes mechanism as modified by Griffin and Carruthers was used to explain the magnetic field dependence in Mn- and Zn-doped GaAs. One of the prime difficulties is that these effects are strongest in p-type materials for which the Keyes theory is not adequately developed. 17. THREE-PHONON PROCESSES

An over-all interpretation of the three-phonon relaxation times in the various 111-V compounds is very difficult to construct because of the complexity of the three-phonon processes. Initial work of Leibfried and Schloemann6' indicated that the lattice conductivity due to three-phonon processes at high temperatures is given by

where M is the mean atomic mass, I/ the atomic volume, and y is the Griineisen anharmonicity parameter. Equation (6) can be solved at high temperature ( T > 0) and gives an equation with the same temperature dependence (neglecting impurity scattering) as Eq. (18),68-69 K =

(q) 27c2US

0 h T(B"+ B"')

Leibfried and E. Schloemann, Nachr. Akad. Wiss. Gottingen, Math.-Physik. KI. l l a No. 4, 71 (1954). '* J. Callaway and H. C. van Baeyer, Phys. Reo. 120, 1149 (1960). 6 9 V. Ambegoakar, Phys. Rev. 114, 488 (1959).

" G.

28

M. G . HOLLAND

where the three-phonon relaxation time that must be used is

'FN

+ ti ' = (BN'+ BU')Td.

(20)

This is different from the low temperature form 'FN

+ F'; ' = (BN + B")T302.

(21)

Neither form is precisely correct for both longitudinal and transverse phonons18 (see Table I). There are two facts which indicate some inadequacies in the formulation indicated by Eqs. (18) and (19): (1) The T-' temperature dependence is seldom found, the exponent is more usually 1.1 or 1.2 (see Table 11). (2) In general the region where K a T-", n 1, persists well below 0 (the Debye temperature). Recent analysis" indicates that these facts can be accounted for if the transport of heat by longitudinal and transverse phonons is considered separately. The high temperature slope is then due to a combination of T from Umklapp scattered transverse phonons and T - 3 from normal (or Umklapp) scattered longitudinal phonons, and the temperature corresponding to the zone boundary frequency of the transverse acoustical phonon replaces the Debye temperature in significance. Thus both difficulties are remedied. A second type of analysis' accounts for the temperature dependence by allowing y, the Gruneisen parameter, to be temperature dependent. The temperature dependence is considered to be due to the occurrence of higher order processes. For example, four-phonon processes would give a T-' dependence for the conductivity. Neither this nor the above approach seems conclusive at this time. Steigmeier and Kudman have presented a discussion of the variation of y2 with each material at its Debye temperature.' The appropriate parameters are listed in Table 111, and the curve of y2 vs the mass ratio for the various 111-V compounds is shown in Fig. 15. The behavior shown is explained in terms of scattering of acoustical (A) phonons by the optical (0)phonons. Beginning with an analysis by Bla~kman,'~ the authors conclude that the number of Umklapp scattering processes of the form A + A = 0 increases as the mass ratio increases, goes through a maximum, and then goes to zero as the ratio increases further. Since y 2 a K-’, the results show that the thermal resistance varies as the amount of this type of Umklapp scattering. This implies that there is little scattering by the optical mode in high mass ratio materials, such as InP and AlSb. A later analysis6'" based on an equation derived by AbeleP2 implies the same result.

-

'O

M.Blackman, Phil. Mag. 19, 989 (1935).

29

1. THERMAL CONDUCTIVITY TABLE 111

THEINFLUENCE OF OPTICAL MODESCATTERING ON THE LATTICETHERMAL CONOUCTIVITY OF GROUP IV AND GROUP111-V SEMICONDUCTORS~ 300°K

M,

KI

Y

M

(OK)

(W/cm-deg)

T=Q

28.1 72.6 1 18.4 72.3 94.8 95.7 50.4 72.9 74.4 29.0 50.9

647.8 374.0 202.5 344.2 249.0 265.5 435.0 321.5 292.0 588.0 417.0

1.412 0.606 0.166 0.455 0.273 0.390 0.77 0.680 0.57 0.9b OXb

0.94 0.95 0.97 0.98 0.9 1 0.86

0

-

M2 Si Ge InSb GaAs InAs GaSb GaP InP AlSb AIP AIAs

1 1 1.06 1.07 1.53 1.75 2.25 3.70 4.51 1.15 2.78

-

0.76 0.79

After Steigmeier and Kudman.8-61a Estimated.

O.$g

I

I

Gar

,*

\

l

I

l

'\

\

\* 08 -

InAs

\

\

T=B

\

N

k

G&b

0.7-

\

-

\

\\

\

\

\

\ \\

Alab

\

0.6-

-

\ \-\InP

0.55,

I

I

I

1

1

15

2

2.5

3

4

5

FIG. 15. The anharmonicity parameter of group 1V and 111-V semiconductors at T = 0 as a function of the mass ratio of the constituent elements (after Steigmeier and Kudman*). Revised data appear in Ref. 61a.

30

M. G. HOLLAND

For completeness the parameters B, + B, of Eq. (21), used in the low temperature curve fitting, are listed in Table 11. Since these are connected with U and N processes at lower temperatures (i.e., lower phonon frequencies), optical mode scattering is not important. No correlation has been obtained between the parameters and any of the 111-V propertie~.’’-’~

18. ELECTRONIC THERMAL CONDUCTIVITY For the most part, in order to obtain the lattice component of thermal conductivity, workers have calculated the electronic thermal conductivity and subtracted it from the measured value. However, since K , contains such important properties as the energy gap, mobility, and other scattering coefficients, information about these parameters for materials in which they are not known can in principle be obtained from IC,.It is possible that measurements in high magnetic fields could provide K , directly, and one could thus check the calculated values. However, the experimental problems are severe.

VII. Conclusions Despite a good deal of effort which has gone into measurement and analysis of thermal conductivity, there are still some serious problems. Much of the data can be explained, but there remain in particular two unsatisfactory areas : (1) Defect effects, including electron-phonon scattering, strain effects, and resonance scattering, are not completely understood. (2) Three-phonon processes cannot be calculated from first principles, nor is the method of combining the several three-phonon processes clear. In particular, in using 111-V compounds for studying thermal conductivity, the most obvious difficulty is that these compounds have far more scattering to be accounted for than can be explained by the impurity densities determined by electrical methods. But, the 111-V compounds are supposed to be significantly purer and more defect-free than most other materials (such as ruby, garnets, alkali halides, metals, and the like). Consequently, one must conclude that one cannot really expect to get good agreement between theory and experiment in any of these materials. On the other hand, when one uses thermal conductivity results as a means of understanding the behavior of 111-V compounds, this very 71

A. M. Toxen, Phys. Rev. 122,450 (1961).

73

P. Carruthers, Phys. Rev. 126, 1448 (1962).

-I’ B. Abeles, D. S. Beers, G. D. Cody, and J. P. Dismukes, Phys. Rev. 125, 44 (19621

1.

THERMAL CONDUCTIVITY

31

difficulty indicates that more defects are present, and can be significant in determining the properties of the material, than are to be found by electrical measurement alone. For example, inert dissolved gas and dislocations apparently play a role in thermal conductivity and may also have unappreciated significance in other areas (phenomena). Further, it is quite clear that these compounds are somehow quite different from silicon and germanium. One other possible source of impurity scattering in the 111-V compounds is a lack of stoi~hiometry.’~ Misplaced atoms would, of course, act like scattering centers. However, the numbers of these defects must be several orders of magnitude greater than that deduced from the electrical data (donor and acceptor concentrations) in order to account for the effects noted. Even for InSb where (71theory/7,exp) 3, about 1019 defects/cm3 with (AMIM) 1 are needed to explain the large amount of scattering. The two newest scattering mechanisms, resonance scattering and scattering of phonons by electrons, have both been identified in the 111-V compounds, so that these compounds can be useful in studying the new mechanisms. However, a study of electron-phonon scattering can provide some of the properties of the impurity ionization energy states in the 111-V compounds. Thus we can conclude that, notwithstanding all the problems, the 111-V compounds are useful in a study of thermal conductivity in general, and thermal conductivity studies can be useful in a general study of the 111-V compounds.

-

-

ACKNOWLEDGMENTS I would like to thank E. F. Steigmeier for providing his data and analysis before publication, R. K. Willardson for pointing out the prevalence of lack of stoichiometry in several 111-V compounds, and D. M. Warschauer for help in clarifying the manuscript.

74

R. K. Willardson, Conf Purification oJ Materials, New York, 1965 to be published

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CHAPTER 2

Thermal Expansion S. I . Novikova I. INTRODUCTION . . . . . . . . . . . . . . . . I . General Considerations . . . . . . . . . . . . . 2. The Variation of the Griineisen Parameter with Temperature . . 3 . The Possibility of a Negative Expansion Coeficient . . . . . 11. EXPERIMENTAL RESULTS . 4. Low Temperatures . . 5 . High Temperatures . .

.

.

.

.

.

.

.

.

.

.

.

.

33 33

35 36

31

. . . . . . . . . . . . 31 . . . . . . . . . . . . 42

111. CALCULATION OF THE GRUNEISEN PARAMETER . . . . . . . 44 6. The Gruneisen Law . . . . . . . . . . . . . . 44 1. The Gilvarry Equation . . . . . . . . . . . . . 46

I. Introduction 1 . GENERAL CONSIDERATIONS

Harmonic approximations are frequently employed in solid state theory. The theoretically predicted results for several physical parameters agree both qualitatively and in their order of magnitude with experimental measurements. However, some anharmonic phenomena such as thermal conductivity, thermal expansion, the pressure and temperature dependence of the elastic constants, as well as the deviation of the specific heat from the Dulong and Petit law at high temperatures, cannot be described by the harmonic approximation. The mathematics involved in the computation of the anharmonic terms in the expansion of the crystal energy in terms of the displacement from the equilibrium position is very difficult. Therefore, experimental investigations of physical properties associated with anharmonicity are of considerable interest. The study of the thermal expansion of solids is one of the methods used in investigating the anharmonicity in forces acting in a crystal. The volume coefficient of thermal expansion is defined as

B=

;(g)p 9

where V is the volume, T is the temperature, and P is the pressure. 33

34

S. I. NOVIKOVA

The temperature dependence of the expansion coefficient is given by Eqs. (2) and (3) below’ :

(’)

j = A -T- 3V d P ii3 ’

Tee,

where 0 is the Debye temperature, ii is the velocity of sound, and A is a constant, C , dv B = - -Vij dP’ T B 0, (3) where C , is the specific heat at constant volume, and F is the average oscillation frequency of the atoms in the crystal. Thus, when T 4 0, /3 T3, and when T p 0, /Iapproaches a limiting value that is independent of the temperature. At absolute zero p becomes zero, in accordance with the Nernst theorem. It follows from Eqs. (2) and (3) that the sign of the expansion coefficient is determined by the sign of dv/dP. When the atoms in a solid are brought together by increasing the pressure, the amplitude of their vibrations decreases and the frequency increases so that dv/dP > 0. Consequently, the expansion coefficient /3 should be a positive quantity and solids should expand with increasing temperature. The relationship between B and other thermodynamic quantities has been established by G r u n e i ~ e n ,whose ~ * ~ equations show that the ratio of the thermal expansion to the specific heat of a solid is independent of temperature. Griineisen’s law is easy to derive by assuming an isotropic solid with a Debye frequency distribution :

-

CVXT

B =Y y

(4)

where

is the isothermal compressibility, and

d log 0 d log V

y = -~

is the Gruneisen constant. L. Landau and E. Lifshitq “Statisticheskaya Fizika, p. 216. Gos. Izd-vo Tekhniko-Teoret. Lit-ry, Moskva-Leningrad, 1951 [English Transl.: L. Landau and E. Lifshitz, “Statistical Physics,” p. 191. Pergamon Press, London and Addison-Wesley, Reading, Massachusetts, 19581. E. Gruneisen, Ann. Physik 39, 289 (1912). E. Gruneisen in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 10, Chap. I, p. 1. Springer, Berlin, 1926.

2.

35

THERMAL EXPANSION

All the quantities entering Eq. (4)are intrinsically positive so that the sign of fl is determined by the sign of y. It is seen from Eq. ( 5 ) that y is related to the frequency spectrum of the solid through the Debye temperature. The knowledge of y gives an idea of the dependence of the natural frequencies of the solid on the volume (or on the pressure). Consequently, by studying the thermal expansion we obtain information about those properties of a solid that cannot be obtained by studying the specific heat, the neutron diffraction spectra, or the Mossbauer effect. Usually y is computed from Eq. (4)where p, C,, xT, and V are experimentally measured quantities. At temperatures close to the Debye temperature, y is a constant whose value depends on the nature of the forces present in the crystal and ranges from 1 to 3. The Gruneisen parameter will be a constant if all the atoms in the solid vibrate independently of each other with a single frequency-the Einstein approximation-or if the dependence of the frequency v on volume is the same for all vibrations, which is true for the Debye model. Recent experimental and theoretical studies have shown conclusively that y decreases with a decrease in temperature. This indicates that the various frequency spectra depend differently on the volume (or on the pressure). The quantity y, describing the deviation of the spectrum from the harmonic approximation, can serve as a measure of the anharmonicity of the vibrations of the atoms in solids. 2 . THE VARIATIONOF

THE

GRUNEISEN PARAMETER WITH TEMPERATURE

The general theory of the temperature dependence of the Gruneisen constant was developed by Barron4 on the basis of Born’s dynamic theory of crystal lattices. His analysis applies to all crystal lattices. Calling y the weighted mean of yi, where d log vi y i = -~ d log V’ one can define yo and ym as the low-temperature and high-temperature limits of the Gruneisen constant, respectively. At sufficiently low temperatures, where only acoustic waves are important, y. =

- d log(V- ‘I3ui) dlog V ’

where ui is the propagation velocity of the ith wave (i temperature limit is 4T.H. Barron, Phil. Mag. 46, 720 (1955).

=

1,2,3). The low-

36

S. I. NOMKOVA

where the integration is performed in all directions in the crystal (w is the solid angle). The high-temperature limit is

where N is the number of particles in the crystal. A sharp change in y begins at temperatures T- 0.28. The amount of variation in the Griineisen constant (7, - yo) depends on the structure of the material and on the nature of the interaction between the particles. A specific calculation for close-packed cubic structure assuming a central-force interaction of only nearest neighbors gave a value of ym - yo = 0.3; for the structure of the solid inert gases, taking account of the interactions between all the atoms, ym - y o = 0.15.

Thus, Barron’s theory explained the variation of the Gruneisen constant at low temperatures. The qualitative picture of y = f(T)thereby obtained agrees quite well with experimental data. 3. THE POSSIBILITY

OF A

NEGATIVE EXPANSION COEFFICIENT

Using Barron’s theoretical treatment, Blackman showed, in a series of papers, the possibility of the existence of a negative expansion ~oefficient.~-’ Starting with Eq. (6),he wrote y i in the form

where ui is the propagation velocity of the ith wave, p is the density, r is one-half the lattice constant, and C,, = puiz is the elastic constant. The major difficulty consisted in writing an analytical expression for the elastic constants as a function of the distance r. Blackman considered two models: a NaCl and a ZnS-type lattice with the assumption of purely ionic interaction. The interaction potential was represented by a sum of two terms: a Coulombic attraction term and a -a/? repulsion term. Three cases were considered for the NaCl model with n = 7, n = 10, and n = 21. M. Blackman, Proc. Phys. SOC.(London)B70, 829 (1957). M.Blackman, Phil. Mag. 3, 831 (1958). M. Blackman, Proc. Phys. Soc. (London)74, 17 (1959).

’

2.

37

THERMAL EXPANSION

It was found that the y i corresponding to transverse vibrations whose ,, elastic constant assume negative propagation velocity depends on the C values. The absolute value of the negative quantity yi increases as the exponent n increases: y i = -0.23 for n = 7, y i = -0.74 for n = 10. This, however, is not enough to cause the average value of y to be negative. For n = 21 (a case which is not realized in reality) y o = -0.62. When the ZnS model was considered, n was assumed to be equal to six. Three values of y i were obtained for vibrations whose propagation velocities were determined by the C , , , C ,, and f-(CI1- CI2)elastic constants. The corresponding values of y i are 1.24, -0.766, and -5.09. The negative values of y i obtained here are associated with transverse vibrations, just as in the case of the NaCl model. Consequently, the existence of a negative value of the Gruneisen parameter y is closely related to the character of the frequency spectrum of the solid and is determined by transverse vibrations. At low temperatures, when only acoustic vibrations are present, the average value y can also become negative. In this case the thermal expansion coefficient becomes negative.

II.Experimental Results 4. Low TEMPERATURES

The study of the thermal expansion of A"'BV type compounds has been extremely inadequate. The results of thermal expansion measurements are shown in Fig. 1 for the antimonides of aluminum, gallium, and indium and for gallium arsenide in the temperature interval from 25" to 340°K; also 6

5

-

4

m

$ 3 0

2

0 1

0 -1 7

-L

20

60

100

140

180

220

260

300

340

T OK

FIG. 1. Temperature dependence of the linear expansion coefficient for the antimonides of gallium arsenide,' and the average value of a for indium aluminum,1ogallium,1° arsenide."

38

S. I. NOVIKOVA 7 6

5 4 Tm

3

0 a,

a

0 r a

2 1

0 -1

-2 -3 -4

0 T OK

FIG.2. Temperature dependence of linear expansion coefficient for Ge, GaAs, and ZnSe.

shown is the average value of the expansion coefficient of indium arsenide at 184 K. We studied the thermal expansion of AlSb, GaSb, InSb, and GaAs8-" in a low-temperature quartz dilatometer. The purity of the elements from which the compounds were formed was no less than 99.999%. The specimens studied were macrocrystalline. Gibbons studied two single-crystal specimens of indium antimonide" in the [lo01 direction using an interference method. The carrier concentration was 1014 and 10I6cmP3. His data (small circles in Fig. 1) and the results of our measurements agree.

-

T

-

OK

FIG.3. Temperature dependence of linear expansion coefficient for a-Sn, InSb, and CdTe.

2.

THERMAL EXPANSION

39

Sirota and co-workers" obtained a value for the expansion coefficient of InAs using a low-temperature X-ray camera. The sample purity is not specified. It is seen in Fig. 1 that the linear expansion coefficient a of all the compounds studied is negative for temperatures T I0.28 (For cubic structures B = 3 4 . This anomalous temperature dependence of the expansion coefficient is characteristic of elements in the fourth group of the periodic tablesilicon,11.13-15 germanium,I6 gray tin,* and A"BV' The nature of the effect of chemical bonds in crystals on the temperature dependence of the expansion coefficient can be established by considering a = f(T) for the isoelectronic sequences of germanium and gray tin (Figs. 2 and 3). Germanium and gray tin crystallize in the diamond structure; compounds of the isoelectronic sequences crystallize in the sphalerite structure which is similar to the diamond structure and differs from it in that it does not consist of identical atoms but of two different atoms in an alternating sequence. The chemical bond in crystals of germanium and gray tin is covalent. In compounds of the isoelectronic sequences an ionic component is added to the covalent bond. The degree of ionic binding is insignificant in GaAs and InSb and increases in ZnSe and CdTe. The lattice constants and Debye temperatures are similar for materials from each of these two sequences. It is seen from Figs. 2 and 3 and from Table I that the transition temperature T, at which a becomes negative increases, and the minimum in a becomes deeper as one goes from Ge to ZnSe and from a-Sn to CdTe. This corresponds to the increase in the ionic component in the binding forces. By considering the series of antimonide compounds of aluminum, gallium, and indium one can explain the effect on the anomalous behavior of a = f(T) produced by replacing one of the elements in the compound, in particular, an element of group 111 by a heavier element of the same group.

* S. I. Novikova, Fiz. Tverd. Tela 2 2341 (1960) [English Transl.: Soviet Phys.-Solid

State 2, 2087 (1961)]. ' S . I. Novikova, Fiz. Tuerd. Tela 3, 178 (1961) [English Transl.: Soviet Phys.-Solid State 3, 129 (1961)l. l o S. I. Novikova and N. Kh. Abrikosov, Fiz. Tuerd. Tela 5, 2138 (1963) [English Transl.: Soviet Phys.-Solid State 5, 1558 (1964)l. D. F. Gibbons, Phys. Rev. 112, 136 (1958). N. N. Sirota and Yu. I. Pashintsev, Dokl. Akad. Nauk SSSR 127, 609 (1959) [English Transl.: Proc. Acad. Sci. USSR, Phys. Chem. Sect. 127, 627 (1959)l. l 3 S. Valentiner and J. Wallof Ann. Physik 46, 837 (1915). l4 H. D. Erfling, Ann. Physik 41, 467 (1942). l 5 S. I. Novikova and P. G. Strelkov, Fiz. Tuerd. Tela 1, 1841 (1959) [English Transl.: Soviet Phys.-Solid State 1, 1687 (1960)l. l 6 S. I. Novikova, Fiz. Tverd. Tela 2, 43 (1960) [English Transl.: Soviet Phys.-Solid State 2, 37 (1960)l.

40

S. 1. NOVIKOVA

TABLE I DATAON COMPOUNDS

Substance

IN SEVERAL ISOELECTRONIC SEQUENCES"

Ge GaAs ZnSe

5.65 5.64 5.65

a-Sn InSb CdTe

6.49 6.46 6.41

T ("K)

Ax

8 395 -370 400

0.4 0.8

-230 214 200

0.2 0.4

-

amin

(lo6deg- * )

48 f 1.5 55 k 1 64 1.5

-0.4 - 0.5 -3.1

45 f 1.5 58 +_ 1 72 f 2

-0.9

~

-

- 1.6 - 3.3

"The quantity a denotes the lattice constant and Ax denotes the difference in electronegativities, that is, the value which characterizes the magnitude of the ionic binding component. The other parameters are defined in the text.

Curves of a =f(T)are plotted in Fig. 4 for the antimonides of Al, Ga, and In. In this sequence of compounds the metallic component of the binding force between the atoms increases when aluminum is replaced by gallium and indium, while at the same time the degree of ionic binding does not change (Table 11). It is known, for example, that an increase in metallic binding in the Si, Ge, a-Sn series lowers the transition temperature at which M becomes negative. Hence, it could be assumed that the temperature at which a changes its sign should decrease from AlSb to InSb. Actually, as is seen from Fig. 4 and Table 11, the transition temperature is maximum for AlSb in this sequence of compounds. But for InSb, a becomes negative at a higher temperature than for GaSb. This does not agree with the hypothesis. The nature of the temperature dependence of the expansion coefficient is determined by the frequency spectrum of the solid. The frequencies vi, in turn, depend on the interaction potential U between particles and on the mass ratio m1/m2 of the atoms that enter into the TABLE I1 DATAON SEVERAL ANTIMONIDE COMPOUNDS Substance

ml/m2

Ax

T (OK)

AlSb GaSb InSb

4.54 1.75 1.06

0.3

0.2 0.2

85 f 2 52 1 67 f 1

2.

41

THERMAL EXPANSION

compound. In the case of compounds of the isoelectronic sequences of germanium and gray tin, the values of m,/rn, differ slightly from unity. Values of this ratio for several antimonide compounds are listed in Table 11. It can be assumed that an increase in the value of ml/m2 changes the force field present in the solid in such a manner that as a result the temperature at which changes sign, T,, decreases. It is evident that the effect of the opposing factors, the increase in metallic binding and the decrease in the value of m1/m2 for InSb, leads to the result that T(1nSb) > K(GaSb).

6

5 -

4

ID

o " 3 (D

0

2

l y l

0 -1

- 2 1 " I ' " ' 20 60 100

140

1

1

180

1

1

220

1

'

260

I 300 " '

T "K

FIG.4. Temperature dependence of linear expansion coefficient for AISb, GaSb, and InSb.

Consequently, the expansion coefficients of ALLLBV compounds assume negative values at low temperatures. The temperature at which a changes sign and the minimum value of a depend on the nature of the forces within the crystal and on the ratio of the masses of the atoms comprising the compound. With regard to the question of which factors can cause a negative expansion coefficient, one can attempt to arrive at an answer by considering the nature of the frequency spectrum distribution of the solid. A positive expansion coefficient results from the assumption that dv/dP > 0 [see Eqs. ( 2 ) and (3)]. This condition is fulfilled for a Debye frequency distribution, g(v) = Cv2. With the application of pressure, the cutoff frequency v,,, shifts toward higher values so that some additional high-frequency vibrations are excited. Since the total number of vibrations excited remains constant, there is a reduction in the number of low-frequency vibrations so that dv/dP > 0. It is known that the frequency spectrum distribution of a solid differs from the Debye distribution, particularly at the middle frequencies which exist at low temperatures. Therefore, it is actually impossible to apply automatically the result obtained for the high-frequency end

42

S. I. NOVIKOVA

of the spectrum to the low-frequency portion of the spectrum. A distribution g(v) is possible such that an increase in pressure leads, in some frequency interval, to an increase in the number of low-frequency vibrations, with the result that dv/dP < 0. Apparently, materials whose expansion coefficients become negative have just such a frequency distribution." In such a frequency distribution we are interested in the low-frequency region, up to some frequency v', which satisfies the condition dv/dP < 0. Consequently, in the temperature region at which all frequencies up to v' are excited, and frequencies v > v' are not excited, the expansion coefficient u will be negative. From the thermodynamic equations for the volume coefficient of thermal expansion it follows that

where S is the entropy. In the present case (aS/aP), > 0 in the region of negative p so that, as was pointed out above, the number of low-frequency vibrations increases with an increase in pressure. Evidently, the nature of the frequency distribution in materials with diamond or sphalerite structures is of the type considered above. The region of anomalous pressure dependence of the number of low-frequency vibrations depends on the nature of the particle interactions. With an increase in ionic binding forces and a decrease in metallic binding forces this region is shifted toward higher frequencies, corresponding to the change in the temperature range in which the expansion coefficient is negative. The assumption made concerning the characteristics of the frequency spectrum distribution in the materials under consideration agrees with the results obtained in studies of the specific heats of elements in group IV of the periodic table'* and of A"'BV compound^.'^ All of these, with the exception of diamond, have vibration spectra that differ greatly from the Debye frequency distribution in the long wavelength region. This is indicated by the deep minimum in the Debye temperature, appearing at T < 0.28.

5. HIGHTEMPERATURES At high temperatures, thermal expansion of the antimonides, arsenides, and phosphides of indium and gallium was investigated by Bernstein and The measurements were performed by an interference method from room temperature to 500-700°C. The antimonides and arsenides were I'

M. J. Klein and R. D. Mountain, J . Phys. Chem. Solids 23,425 (1962).

* J. C. Phillips, Phys. Rev. 113, 147 (1959). l9

U. Piesbergen, 2. Naturforsch. 1% 141 (1963). L. Bernstein and R. J. Beals, J . Appl. Phys. 32, 122 (1961).

2.

THERMAL EXPANSION

T

43

"C

FIG. The relative elongation of the antimonides, arsenides, and phosphi 's of gallium and of indium. (Curves from Ref. 20; points A and A from Ref. 12.)

-

single crystals and were measured in the [ 1111 direction ; impurities amounted to 10'' cm-j for the antimonides, 3 4 x 1 O I 6 cm-3 for InAs, and - 2 4 x l O " ~ m - ~for GaAs. The phosphide samples were polycrystalline. The temperature dependence of the relative expansion is shown in Fig. 5. It is seen from the figure that a sharp deviation from linearity is observed for GaSb in the interval 30@400"C,and it was impossible to make measurements beyond 436°C. Deviation from linearity is observed in the

T "C

FIG. 6. The temperature dependence of the linear expansion coefficient for the arsenides of aluminum, gallium, and indium. (After Sirota and Pashintsev'*.)

44

S. I. NOVIKOVA

other compounds starting at 400°C and continuing up to the maximum temperatures investigated. The curve of Al/l = f ( T ) for InSb has a kink at T 500 C.This is apparently explained by the effect of creep as the melting point is approached (the melting temperature of InSb is 525°C). The thermal expansion coefficient for the arsenides of In, Ga, and A1 in the temperature interval 100-350 C is shown in Fig. 6. These results were obtained by Sirota and co-workers" with a high-temperature X-ray camera. The sample purity is not specified. The relative expansions of GaAs and InSb at 200" and 300"C, computed from these data (triangles in Fig. 5), are in good agreement with the curves of Al/l =f(T)obtained by Bernstein and Beak."

-

111. Calculation of the Griineisen Parameter

6. THE GRUNEISEN LAW The Griineisen parameter can be computed from Eq. (4)if the values of the specific heat, the thermal expansion coefficient, and the compressibility are known. This equation has been derived on the assumption that y does not vary with temperature. Consequently, if y depends on the temperature and if we compute y from Eq. (4),we obtain the value of some parameter which, generally speaking, is not the usual Gruneisen parameter = - d log 8/d log K Let us use the symbol y' to represent the Griineisen parameter y computed from the Griineisen equation in its normal form when y =f(T). We shall now ascertain its physical meaning. Let us use the thermodynamic relations

From (7) we obtain

or

Hence,

2.

THERMAL EXPANSION

45

Thus, y characterizes the change in temperature during an adiabatic expansion of the solid. If B is positive, then the temperature decreases during an adiabatic expansion ; if B is negative, the temperature increases. At low temperatures, when the fl T 3 law is satisfied, and at high temperatures, when all frequencies are excited, y is equal to the Gruneisen parameter y. Consequently, y and y have the same low-temperature and high-temperature limits y o and ym.

-

T -

9 FIG.7. The temperature dependence of y’ for the antimonides of aluminum, gallium, and indium and for gallium arsenide.

For intermediate temperatures, y’ can be computed from Eq. (8). The temperature dependence of y is shown in Fig. 7 for the antimonides of aluminum, gallium, and indium and for gallium arsenide. From the graphs it is seen that, at temperatures near the Debye temperature, y approaches a constant limiting value. Consequently, at T 0, y is independent of the temperature and is the same as the Griineisen parameter y. At low temperatures y has a region of negative values which coincides with the region of negative values for the expansion coefficient. With a further decrease in temperature, y’ again crosses the abscissa axis, becomes positive, and approaches a constant, positive, limiting value yo, in the same way as for germanium.” Consequently, for materials whose expansion coefficient becomes negative, y changes sign twice. The temperatures at which y changes sign coincide with the temperatures at which c1 changes sign. The relationship (ym - yi) = f ( T / 0 ) is plotted in Fig. 8 for the antimonides of aluminum, gallium, and indium and for gallium arsenide in

-

’’ R. D. McCammon and G. K. Weit, Phys. Rev. Letters 10, 234 (1963).

46

S. I. NOVIKOVA

----

SI Ge a-Sn

o

GoAs

0.8

-- AlSb

x* 1.2 1.6

‘

@

GoSb

- lnSb I

I

I

I

I

I

1

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

I

T -

e

FIG.8. The temperature dependence of the difference between Y < and the high-temperature limit ym.

comparison with (ym - yT’) = f ( T / B )for silicon, germanium, and gray tin. (The value of y’ at T 0 is taken as ym). As seen from the figure, within the limits of the experimental errors the measured values fall into two categories : the first category includes the elements silicon, germanium, and gray tin ; the second category includes the compounds comprising the antimonides of aluminum, gallium, and indium. The values obtained for GaAs cannot be placed in either of these categories. Apparently the quantity (y, - yT ) characterizes the nature of the frequency spectrum. The atomic interaction in silicon, germanium, and gray tin is covalent in nature with the degree of metallic binding increasing from Si to cr-Sn. Evidently the appearance of metallic-type binding does not qualitatively alter the frequency spectrum so that the values of (y, - );y are similar for Si, Ge, and a-Sn. This hypothesis also applies to the AlSb, GaSb, and lnSb sequence of compounds, with the degree of metallic binding increasing from AlSb to InSb. The appearance of the ionic binding probably alters the nature of the frequency distribution. The difference between y; and ym for GaAs in comparison with Ge and particularly for InSb in comparison with a-Sn is greater in magnitude. The decrease of y‘ with the decrease of the temperature for GaAs and InSb begins at much higher temperatures (0.68 and 0.88 correspondingly), than for germanium and gray tin, for which y‘ begins to decrease with T - 0.58. This indicates the appearance of an ionic component in the binding forces of GaAs and InSb.

-

7. THE GILVARRY EQUATION

The relationship between the thermal expansion coefficient and the specific heat, when y is temperature dependent, was derived by Gilvarry” from the equation of state for a solid. 22

J. J. Gilvarry, J . Appl. Phys. 28, 1255 (1957).

2.

THERMAL EXPANSION

47

In the Debye approximation the equation of state for a solid is of the form

p =

dEO -__

dV

+ y--Evib

V ’

(9)

where E , is the lattice energy in the zero vibration state, Evibis the vibrational component of the energy, and y = - d log B/d log r/: the Griineisen parameter. Let us now differentiate Eq. (9) with respect to 7: Taking into account that (BPIBT), = f ? / ~we , obtain the Gilvarry equation :

Thus, we have obtained an expression for p which reduces to the Griineisen relationship, Eq. (4), for the special case when (ay/dT), = 0. To compute y it is necessary to solve the equation

In order to obtain a general solution of Eq. (10) it is necessary to have another equation for (ayldv),. This restricts us to a consideration of only a few special cases. a. Region in Which the Cv

-

T 3 Law Applies

Let us substitute in Eq. (10) expressions for Cv, Evib,and fl that are valid in the T 3 region. By means of simple mathematical manipulations one finds that in this approximation (ay/dT), = 0; i.e., y is a constant quantity equal to yo. Consequently, in the T 3 region one can use the Gruneisen relationship, Eq. (4), to compute y. b. The Temperature Region in Which fi Becomes Equal to Zero

Since B = 0 and, consequently, the volume remains constant, Eq. (10) can be written as

Replacing Cv by dE/dT, we obtain

48

S. I. NOVIKOVA

The solution of this equation is y=-

A

(11)

Evib’

where A is a constant. From Eq. (11) it follows that when u = 0 the Gruneisen parameter y, unlike y’, does not become zero. c. y = o In this case Eq. (10) simplifies to

(2) V

3uv

=-.

XEvib

Let us determine the temperature at which y becomes zero. To do this we integrate Eq. (12) and find the value T, at which y becomes zero, i.e., when the integral is equal to - y o :

-

where TI is the limiting temperature of the region where the C, T 3 law is valid and where y = yo, and where T, is the temperature at which y = 0. A numerical integration performed for germanium showed that the temperature at which y changes its sign is shifted toward higher temperature values than that at which u changes its sign. This shift depends on the value of the low-temperature limit yo. The higher yo, the greater the shift. At certain values of y and for certain ranges of negative values for the expansion coefficient, the parameter y can remain positive at all temperatures while y’ necessarily has a region of negative values coinciding with the region of negative values for a. Consequently, when considering the equation of state (9), it is impossible to use the value of y computed from Eq. (4)if y is temperature dependent.

CHAPTER 3

Heat Capacity and Debye Temperatures U.Piesbergen I.

INTRODUCTION. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

49

11.

HEATCAPACITY.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

50

111. DEBYETEMPERATURE OD . . . . 1. OD Values from Heat Capaciry Data

2 . O,, Values from Elastic Constants

. . . . . . . . . 53 . . . . . . . . . 53 . . . . . . . . . . 53

I. Introduction

It was a major step forward in our knowledge regarding the energy content of solids when Einstein’ applied the quantum theory of Planck to the motion of particles in a solid. This led to the expression for the heat capacity of one gram atom at constant volume: C , = 3R(hv/kT)’ exp(hv/kT)/[exp(hvfkT)- 11’ = 3R(@E/T)’ exp(@,/T)/[exp(@,/T)

- 112

(1)

Here v stands for the frequency of the oscillator; h, k, and R have their usual meaning, whereas 0,= hv/k defines the characteristic temperature of the Einstein oscillator. In 1912 Debye’ introduced instead of the single Einstein frequency the frequency distribution of the isotropic elastic continuum :

Z(V)= 4nV(1/uL3+ 2/ut3)v2= uv2,

(2) where Vis the volume of the solid and v, and u, are the “sound” velocities of longitudinal and transverse waves, respectively. Taking into account that the total number of normal vibrations of the N particles in the solid is 3N, Debye defined a cutoff frequency vD by Z(v)dv = 3N, A. Einstein, Ann. Physik. 22, 180 (1907). P. Debye, Ann. Physik., 39, 789 (1912).

49

(3)

50

U. PIESBERGEN

which means that the frequency distribution Z(v) only holds up to this Debye frequency, v,. The heat capacity per mole at constant volume can then be written as a sum over all Einstein functions, i.e.,

where 0, = hv/k is the characteristic Debye temperature. The function D(@$T) is called the Debye function for which detailed tables exist3 The main features of the Debye theory are: (1) It gives a universal function for all solids depending on only one parameter, OD/?: (2) It connects the characteristic temperature 0, with the elastic properties of the solid because the velocities of transverse and longitudinal waves can be expressed in terms of the elastic constants cij.

II. Heat Capacity The heat capacity at constant pressure

where AQ is the heat input and A T the corresponding change in temperature, can be determined with very high accuracy in a wide temperature range. In order to obtain the heat capacity at constant volume, C,, which is the quantity usually resulting from theoretical calculations, one uses the thermodynamic formula

c, - c, = (3.)’

Vo2T VP

( 3 ~ )V ’ T x-,

B

(5)

where a is the coefficient of linear expansion and the isothermal compressibility. If the temperature dependence of a and 8 is not known, the formula

c, - c, = AC,’T

(6) may be used, where A is a constant which can be determined by comparing Eqs. ( 5 ) and (6). “Landolt-Bornstein Physikalisch-Chemische Tabellen” (W. Roth and K. Scheel, eds.), 5th ed., 1. Erg.-Band (1st Suppl.), p. 705. Springer, Berlin, 1927.; F. Simon in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 10, p. 367. Springer, Berlin, 1926.

3.

HEAT CAPACITY A N D DEBYE TEMPERATURE

51

For the 111-V compounds, very few measurements of heat capacity are available. Gul'tyaev and Petrov4 measured InSb in a Nernst calorimeter between 80" and 300°K and tabulated their values of C , , C , , and 0,for every 10 degrees. The authors also give C , values for AlSb and GaSb at 80°K. These measurements are in agreement with those of Piesbergens who presents tables of C,, C , - C,, C , , and 0 for GaAs, GaSb, InP, InAs, and InSb over the temperature range 12" to 273°K and for AlSb from 20" to 273°K. From these data also the entropies at 298.2"K are calculated. Here a compilation of Renner6 must be mentioned in which enthalpy of formation, free energy of formation, and entropy for the same six compounds are calculated from data collected from various sources. For higher temperatures, the only measurements available are those of Kochetkova and Rezukhina,' who find the following equation for the specific heat of GaSb at constant pressure: cp = 0.04351 + 4.635 x lo-' T

cal/g-deg,

valid between 20" and 700°C. In the same temperature range Steigmeier et aL8v9 used extrapolated values for the specific heat of InP and InAs, which the authors expressed as CP3000K

+ 6.5 x

1 0 - 4 ( ~ -300)

Cp = CP3000K

+ 7.7 x

10-4(T - 300) J/cm3 deg ,

cp =

and

respectively. In Table I the specific heat of certain 111-V compounds at 298°K are shown. The value for AlAs was calculated from a Debye temperature given by Pashintsev and Sirota" for room temperature, but it might be too high by several percent. 4P.V. Gul'tyaev and k V. Petrov, Fiz. Tuerd. Tela 1, 368 (1959) [English Transl.: Souiet Phys.-Solid State 1, 330 (1959)J U. Piesbergen, Z . Naturforsch. 18a, 141 (1963). Th. Renner, Solid-State Electron. 1, 39 (1960). ' N. M. Kochetkova and T. N. Rezukhina. Vopr. Met. i Fit. Poluprou., Akad. Nauk SSSR, Tr. 4-go [Chetuertogn] Soveshch., Moscow, 1961, p. 34 [English abstract given in Chem. Abstracts 56, 4167e (1962)J * E. F. Steigmeier and I. Kudman, Phys. Reo. 132, 508 (1963). I. Kudman and E. F. Steigmeier, Phys. Rev. 133, A1665 (1964). l o Yu 1. Pashintsev and N. N. Sirota, Dokl. Akad. Nauk BSSR 3, 38 (19593. Quoted also in N. N. Sirota and Yu. I. Pashintsev, Dokl. Akad. Nauk SSSR 127,609 (1959) [English Transl.: Proc. Acad. Sci. U S S R , Phys. Chem. Sect. 127, 627 (1959)l.

TABLE I SPECIFICHEATAT CONSTANT PRFSSUREOF 111-V COMPOUNDS'

AlSb

GaAs

GaSb

InP

In&

InSb

0.108'

0.07443

0.07640

0.06058

0.07449

0.06019

0.04996

Values for specific heat are given in units of calories per gram degree. Calculated from a 0 value given in Ref. 10. The other values were calculated from the molar heat capacity given in Ref. 5. a

C

AlAs

3.

HEAT CAPACITY A N D DEBYE TEMPERATURE

53

In. Debye Temperature 0, The Debye temperature 0, defined as 0, = hv/k, where v is either a characteristic or some average frequency, is a very useful parameter in solid state problems because of its inherent relationship to lattice vibration. There are several ways to evaluate the characteristic temperatures, e.g., from melting point, expansion coefficient, compressibility, thermal and electrical resistance, infrared, X-ray, and Mossbauer effect data. For a more extensive compilation the reader should refer to the chapter on Lattice Dynamics by D. Kleinman, or to the article of Blackman.’Oa It should be noted, however, that 0 values derived from different physical properties need not necessarily be equal. It is obvious that the cutoff frequency which is determined by Eq. ( 4 ) cannot be the same as the single frequencies calculated from infrared absorption bands.”,” 1. 0,VALUES FROM HEATCAPACITY DATA

It has become a practice to represent heat capacities by a plot of characteristic temperature @ versus temperature. In the case of the III-V compounds the 0 values run through a minimum at low temperatures, as can easily be seen from Table 11. This minimum is in agreement with the results from the lattice absorption bands in the infrared,”,” which give a relatively low energy for the transverse acoustic phonons in these compounds. The decrease in 0 at higher temperatures is believed to be due to anharmonic effects in the lattice vibrations. In order to give an estimate of this contribution, the last column of Table I1 gives 0, values calculated by Thirring expansion.

2. O,, VALUESFROM ELASTIC CONSTANTS At very low temperatures, where only the low frequencies contribute to the atomic heat, the characteristic temperature @,(O) is given by

where N is the number of atoms in volume I/ of the solid, and v, is the mean velocity of sound propagation averaged over all directions. This mean velocity determines, exactly as in the Debye theory, the magnitude of the heat capacity of a crystal. loaM. Blackman, in “Handbuch der Physik” (S. Fliigge, ed.),Vol. VII/I, p. 325, Springer, Berlin, 1955. ‘ I S . J. Fray, F. A. Johnson, and R H. Jones, Proc. Phys. SOC.(London) 76,939 (1960). W. Cochran, S. J. Fray, J. E. Quarrington, and N. Williams, J . Appl. Phys. 32, 2102 (1961).

54

U. PIESBERGEN

For an evaluation of u,, several methods may be chosen, if the elastic constants of the solid are known: (a) Numerical integration method used by DeLaunay13 for certain cubic crystals. (b) Series expansion method established by Hopf and Lechner,I4 modified by Quimby and Sutton,” or Houston’s’6 method developed by Betts et a1.” Numerical integration as well as series expansion after Betts were used by Joshi and Mitra’* to calculate characteristic temperatures of GaAs and GaSb at O’K, while Verma et aL2’ used the latter and the VRHG approximation for AISb, GaAs, GaSb, InAs, InSb, and 12 other semiconducting elements and compounds. Table I1 gives characteristic temperatures 0,,(0) calculated after Sutton’s formula together with the elastic constants used. These values correspond to the circles with a cross in the figures. There is strong evidence that the characteristic temperatures calculated from elastic and from heat capacity data coincide, if both are taken at sufficiently low temperature. Therefore it is permissible to replace 0,,(0) by O,,(O),as it is done here. The uncertainty brought about by using the elastic constants at room temperature does not exceed 2% in 0,as the elastic constants vary only very slightly with temperature. In the fourth column 0 values, taken from a paper of Steigmeier,” are shown. He made use of an observation of Keyes” that the elastic constants of IV, 111-V, and 11-VI elements and compounds depend only on functions of their corresponding lattice parameters. Thus Steigmeier was able to calculate, using the Marcus-Kennedy2’ formula and a slight interpolation, the 0 values for all the 111-V compounds for which elastic data are not available. DeLaunay, Solid State Phys. 2, 286 (1956); J. DeLaunay, J . Chem. Phys. 22, 1676 (1954); J. DeLaunay, J . Chem. Phys. 21, 1975 (1953). l4 L. Hopf and G. Lechner, Verhandl. Deut. Phys. Ges. 16, 643 (1914). 15S. L Quimby and P. M. Sutton, P h y s Rev. 91, 1122 (1953); P. M. Sutton, Phys. Rev. 99, l 3 J.

1826 (1955).

W. V. Houston, Rev. Mod. Phys. 20, 161 (1948). D. D. Betts, A. B. Bhatia, and G. K. Horton, Phys. Rev. 104, 43 (1956); D. D. Betts, A. B. Bhatia, and M. Wyman, Phys. Rev. 104, 37 (1956). S. K. Joshi and S. S. Mitra, Proc. Phys. SOC.(London)76, 295 (1960). E. F. Steigmeier, Appl. Phys. Letters 3, 6 (1963). 20R.W. Keyes, J . Appl. Phys. 33, 3371 (1962). P. M. Marcus and A. J. Kennedy, Phys. Rev. 114, 459 (1959). ’* J. K. D. Verma, B. D. Nag, and P. S. Nair, Z . Naturforsch. 19a, 1561 (1964). l6

17

3. HEAT CAPACITY A N D DEBYE TEMPERATURE

rr)

Q

G

55

TABLE

11-cotit.

CHARACTERISTIC TEMPERATURES OF I1I-V COMPOUNDS-COnt . Compound

Elastic constants O,,(O)" (10" dyne/cm2) (OK)

GaP

@,,(0)b (OK)

@ =

f (C")

@m

("K)

446

e C , , = 11.88'

C,,

=

5.38

GaAs

344

344

T

=

314

GaAs

C,, = 5.98

298°K 0

T'K

-

)

3.

E

0

3 L

1

B

8

0

P

HEAT CAPACITY AND DEBYE TEMPERATURE

-*+

57

TABLE 11-conr. CHARACTERISTIC

Compound

Elastic constants (10" dyne/cm2)

0,,(0)"

("K)

TEMPERATURES OF 1II-V

COMPOUNDS-Con?.

@=f(CJ

Oe,(0)b

0, ("K)

("K)

45*L

400

e

w

a

% a

Jn P InP

321

460

2

TABLE II-cont. CHARACTERISTIC TEMPERATURES OF 1II-V COMPOUNDS-COFZr. Compound

Elastic constants (10" dyne/cm2)

O,,(Op ("K)

O,,(0)b ( O K )

C , , = 8.329' C,, = 4.526

InAs

c,,

247

262

= 3.959

T = 300°K

100

fK-

2

s

TABLE II-cont. CHARACTERISTIC TEMPERATURES OF 1II-V Compound

Elastic constants O,,(O)” (10” dyne/cm2) (“K)

@,,(O)b (“K)

COMPOUNDS-COnf

@=

.

f(CJ

0,

(“K)

C , , = 6.75#

c,,

= 3.47

InSb

208 C ,,

278

203

= 3.16

T=0 K

loo

* After Sutton.

’After Steigmeier.

‘D. I. Bolef and M. Menes, J . Appl. Phys. 31, 1426 (1960). dT. B. Bateman, H. J. McSkimin, and J. M. Whelan, J . Appl. Phys. 30, 544 (1959). HJ. McSkimin, W. L Bond, G. L Pearson, and H. J. Hrostowskk Bull. A m Phys. SOC. 1, 111 (1956); H. B. Huntington, Solid State Phys. 7 , 213 (1958).

I D . Gerlich, Bull. Am. Phys. SOC.8, 472 (1963). @RoyF. Potter, BulL Am Phys. SOC.1, 53 (1956); L, J. Slutsky and C. W. Garland, Phys. Rev. 113, 168 (1959); see also Ref. e.

c

Physical Properties I

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CHAPTER 4

Lattice Constants G.Giesecke I . INTRODUCTION . . . . . . . . MEASUREMENT OF LATTICE CONSTANTS. 1 . BN Structure . . . . . . . . 2. Wurtzite Structure . . . . . . 3 . Zinc BIende Structure . . . . . 4. Rhombohedra1 Structure . . . . . 111. SUMMARY . . . . . . . . . . 11.

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

63 65 65 66 61 72 13

I. Introduction

The more important semiconductor compounds of elements of groups 111 and V of the Periodic Table are the nine produced by compounding Al, Ga, and In with P, As, and Sb. These compounds crystallize in the cubic zinc blende structure (Fig. 1). Further compounds are those of B with P and As, which likewise crystallize in the zinc blende lattice; with these two, however, there also exist rhombohedra1 modifications, which form at high temperatures. The compounds of the elements Al, Ga, and In with nitrogen likewise have semiconductor characteristics, but crystallize in the wurtzite lattice (Fig. 2). In general BN crystallizes in a graphite-like structure (Fig. 3); however, a zinc blende modification is also known. The compounds of the heavy elements such as T1 and Bi are no longer semiconductive, but have metallic character. Thus, altogether, fifteen semiconducting 111-V compounds are known. This chapter deals with their lattice constants. The structures given above for the group 111-V compounds apply under normal conditions. Although it is possible to change these structures while using high pressures and high temperatures,' only modifications which are stable under normal pressure and temperature conditions will be considered here. For example, for InSb at 25,000 kg/crnz Rooyrnans" indicates a NaCl structure with a = 5.84 A. Is C. J. M. Rooyrnans, Phys. Letters 4, 186 (1963).

63

64

G . GIESECKE

FIG.1. Zinc blende lattice (B3 type). Two interlaced identically oriented fcc lattices displaced by (a/4,a/4, a/4), where a is the edge length of the elementary cube. Every atom has four atoms of the other type placed at the comers of a tetrahedron Each two nearest neighbors are atoms of different elements whose spacing is (44)

A.

FIG.2. Wurtzite lattice (B4type). Two interlaced cph lattices. With c/a = 1.633 and u = 3/8, each atom of the one type is surrounded exactly tetrahedrically by atoms of the other kind.

FIG.3. Boron nitride lattice? Similar to the graphite lattice, stacked sheets of six-membered rings with alternating B and N atoms.

4. LATTICE

CONSTANTS

65

For the lattice constants of mixed crystals of 111-V compounds reference should be made to the article by Woolley entitled “Solid Solution of 111-V Compounds.” ’a On the basis of the atomic radii given by Pauling and hug gin^,^ the distances of the nearest neighbors can be approximately calculated by summing the atomic radii. However exact values can be obtained only by measurements. 11. Measurement of Lattice Constants

1. BN STRUCTURE BN

The hexagonal form of boron nitride, BN, can be obtained in a fairly pure state. Its crystal structure (Fig. 3) is quite similar to that of graphite, with stacked sheets of six-membered rings. In each ring B and N atoms alternate. This structure was described by Pease,’ who used commercially prepared boron nitride after recrystallization by heating the material at 2050°C for 2 h in a stream of nitrogen. Chemical analysis detected only traces of Mg, C, Si, and some metallic elements, amounting in all to less than 0.6% by weight. For the X-ray examination of powder specimens (about 3 mm diameter) Cu K and Mn K radiations were employed. The camera temperature was stabilized at 35” k 0.5”C. The principal high-angle reflections used for determining a were the Cu Ka 120 and 122 and C u K P 300 and 302, the last being reflected at 8 = 80.5”. For the determination of c, the 006 reflection of M n K a at 0 = 71” was the best available. Using an extrapolation technique the dimensions obtained at 35”C, assuming C U K E , = 1.54051 A are a

=

2.50399 5 0.00005,

c = 6.6612 f 0.0005 A.

The absence of large systematic errors was confirmed using the same equipment and technique to determine the a dimension of graphite. The value obtained agreed well with the value of other authors. The unit cell dimensions were further determined in the range 0-800”C ; the thermal expansion coefficients are 40.5 x deg-‘ for c-this dimension expands linearly over the range investigated-and - 2.9 x degat 20°C falling to zero at 770°C for a.

’

R. S. Pease, Acra Cryst. 5, 356 (1952). J. C. Woolley in “Compound Semiconductors” (R. K. Willardson and H. L. Goering, eds.), Vol. 1 : “Preparation of 111-V Compounds,” p. 3. Reinhold, New York, 1962. L. Pauling and M. L. Huggins, Z. Krisr. 87, 205 (1934).

66

G. GIESECKE

2. WURTZITESTRUCTURE a. AlN

As early as 1924 Ott4 examined the structure of AlN and determined a wurtzite structure with a = 3.113,c = 4.9g1A,c/a = 1.601,while Stackelberg and Spies5 indicate a wurtzite structure with a = 3.104 0.005 c = 4.965 i-0.005 c/a = 1.600. For this, AlN was prepared from pure A1 in a stream of both N, and NH3. In some cases the preparations contained A1 which had not yet been converted. Powder photographs were made with Cu radiation, using NaCl as the reference substance. The deviation of the structure of A1N with c/a = 1.600 from the ideal wurtzite structure with the axial ratio of 1.633 prompted new investigations by Jeffrey and Parry6 and by Jeffrey et af.’ The lattice parameters measured from the .higher-order reflections (Cu Ka radiation) were a = 3.111, c = 4.978 A, C/U = 1.600.

A,

b. GUN Gallium nitride was examined by Juza and Hahn’ and the suspected wurtzite structure determined. The authors took Debye-Scherrer photographs of GaN powder preparations in 0.7-mm diameter Mark tubes with Cu Ka radiation. The distances of the interferences were corrected to rod thickness and the angles obtained were corrected to a reference substance (KCl). Taking as a basis a value for A Cu Ka = 1.539 A, the dimensions for the unit cell obtained from two photographs were a = 3.18,, k 0.004 A, c = 5.16, f 0.005 A,C/U = 1.625. Shdanow and Lirmanng give different values: a = 3.160 k 0.008 A, c = 5.125 0.010A,C/U = 1.622. c. ZnN

Juza and Hahn’ confirmed the suspected wurtzite structure for InN. Debye-Scherrer patterns of preparations in Mark tubes, to which KCl was added as a reference substance, were photographed with Co Ka radiation. The measured distances of the interferences were corrected to preparation thickness and the angles obtained corrected to the KCI interferences. Two photographs were measured, which gave lattice constants agreeing within the indicated error limits. Taking as a basis Co Ka = 1.787A the H. Ott, 2. Physik. 22, 201 (1924). M. v. Stackelberg and K. F. Spiess, 2. Physik. Chem. (Leipzig)A175, 140 (1935). G. A. Jeffrey and G. S. Parry, J. Chem. Phys. 23,406 (1955). ’G. A. Jeffrey, G. S. Parry, and R. L. Mozzi, f . Chem. Phys. 25, 1024 (1956). * R. Juza and H. Hahn, Z . Anorg. Allgem Chem. 239, 282 (1938). G. S. Shdanow and G. W. Lirmann, Zh. Eksperim. i. Teor. Fiz. 6, 1201 (1936).

4.

67

LATTICE CONSTANTS

measurements of the unit cell are a = 3.53, f 0.004k c c/a = 1.611.

=

5.693

0.004A,

3. ZINC BLENDESTRUCTURE a. B Compounds

(1) BN. The cubic form of boron nitride was described by Wentorf'o*ll and prepared as a very hard material at temperatures and pressures of about 1800°C and 85,000atm. A careful measurement indicated that the cubic material has a zinc blende structure with a unit cell edge length of 3.615 f 0.001 A at 25°C. A spectrographic analysis detected the presence of an impurity in the sample from the growing medium, but no data are given about the character of these impurities. At 50,000 atm and about 2500°C the cubic material reverts to a substance, identified as hexagonal BN by its Debye-Scherrer patterns, thus confirming the cubic material to be really BN. (2) BP. Boron phosphide of zinc blende structure was prepared in 1957 by Popper and InglesI2 by the reaction of the elements in an evacuated tube at 1100°C. Analysis by X-ray diffraction showed a zinc blende structure with a lattice parameter 4.55 A, in good agreement with the covalent tetrahedral radii by Pauling and hug gin^.^ The material was not very pure, traces of tungsten carbide being present. Williams and R ~ e h r w e i n 'give ~ some general methods for preparation of cubic boron phosphide. The lattice parameter of this cubic BP was determined to be 4.537 A. In 1958 the lattice constant of BP having the zinc blende structure was determined by Perri et to be 4.538 A. The last reflection available( 4 4 0 t i s reflected at 6' = 73.75" using Cu radiation. (3) BAS.The same authors who gave the preceding information about the production and investigation of BP carried out the corresponding investigation for BAS. The structure of BAS prepared from the elements by Williams and R~ehrwein'~ depends upon the temperature and the arsenic pressure. At 700-800°C and arsenic pressures greater than 1 atm a cubic material with a boron to arsenic atom ratio approaching unity was obtained. The lattice constant of this material with zinc blende structure is 4.777 A. Perri et ~ 1 . prepared ' ~ BAS also by direct union of the elements at temperatures of 800°C. The compound crystallized with the zinc blende structure, R. H. Wentorf, J . Chem. Phys. 26, 956 (1957). R. H. Wentorf, J. Chem. Phys. 34, 809 (1961). P. Popper and T. A. Ingles, Nature 179, 1075 (1957). l 3 F. V. Williams and R. A. Ruehrwein, J . Am. Chem. SOC. 82, 1330 (1960). l4 J. A. Perri, S. LaPlaca, and B. Post, Acta Cryst. 11, 310 (1958). lo

68

G. GIESECKE

and powder diffraction data gave the same value for the lattice constant as reported above: a = 4.777 A ; the last reflection used (600) is reflected (Cu radiation) at 0 = 75.3'.

b. A1 Compounds (1) AZP. G o l d ~ c h m i d t ' ~ reported in 1926 that aluminum phosphide possesses the zinc blende structure with a unit cell edge length a = 5.451 kX (5.462 A) and an AI-P distance of 2.360kX (2.365 A). The powder diagram of the sample, produced by passing vaporized P in a stream of Hz over an A1 powder at 500'C, showed still a considerable amount of metallic Al. Measurements of White and BusheyI6 on AlP prepared in their laboratory gave values for the lattice constant of 5.445 to 5.451 A. These authors conclude that only one composition AlP exists and that different results of other authors are caused by mixtures of AlP with free aluminum and aluminum oxide. A d d a m i a n ~ 'used ~ Zn,P, and A1 for producing Alp. The zinc phosphide was purified by subliming the compound in a stream of argon at llOO'C, the A1 powder was assumed to be pure. For most experiments a stoichiometric mixture of aluminum and zinc phosphide was used, and by heating for several hours at 800"-900'C AlP was produced. X-ray powder photographs of different samples were consistent with the existence of a zinc blende structure. The measured value of the cell edge is a = 5.451 A (Cu Kcr radiation, I,, = 1.5405 The intensities of the lines, however, were found to be quite different from the data reported by other authors; and a recalculation was given' 7,1* for the correct intensities.

A).

(2) AIAs. Gold~chmidt'~ obtained AlAs in the same way as the phosphide, the AlAs showing the zinc blende structure too. Measurements from X-ray photographs gave a value of a = 5.628kX (5.639 A) and an A1-As distance of 2.437kX (2.442A).In accordance with these results is the value a = 5.62A, determined by Natta and P a ~ s e r i n i 'from ~ powder photographs. Gorjunova" gives in her book, with reference to a report of Yu. I. Pashintsev (concerning his dissertation, Minsk, 1959), a lattice constant a = 5.6622 A for AlAs. V. M. Goldschmidt, Skrqter Norske Videnskaps Akad. Oslo, Z: Mat. Naturv. K1. 1926, VIII. W. E. White and A. H. Bushey, J . Am. Chem. SOC.66, 1671 (1944). l 7 A. Addamiano, J . Am. Chem. SOC. 82, 1537 (1960). '* A. Addamiano, Acta Cryst. 13, 505 (1960). l9 G. Natta and L. Passerini, Gazz. Chim. Ztal. 58, 458 (1928). *' N. A. Gorjunova, "Khimiya almazopodobnykh poluprovodnikov" ("Chemistry of Diamond-Like Semiconductors"-in Russian) Izd-vo Leningradskogo Universiteta, 1963. l5

l6

4.

69

LATTICE CONSTANTS

(3) AlSb. As far back as 1924 the lattice constant of AlSb, which likewise has a zinc blende structure, was found by Owen and Preston21 to be a = 6.126kX (6.138Al whereas Goldt~chrnidt'~ gives a value of a = 6.091kX = 6.103 A. An accurate determination of the lattice constant was carried out by Giesecke and Pfister.22 In the latter work, crystals of very high purity were obtained by melting together the components. The basic materials were purified beforehand and the preparations further purified by zone refining until spectroscopically pure. For the powder photographs, preparations with a diameter of 0.1 to 0.15 mm were used. The temperature was kept constant at 18" _+ 0.1"C. In the determination of the lattice constant by the asymmetric method, after Straumanis, the 622-reflection with C o K a radiation (at 8 = 75") and the 553-reflection with Cu Ka radiation (at 0 = 75") were used. By taking into account an absorption correction due to Taylor and Sinclairz3 and Nelson and Rileyz4 a value of a = 6.1355 _+ O.OOO1 A at 18°C was obtained.

c. Ga Compounds (1) CUP. Gallium phosphide was prepared and examined by Goldschmidt," who indicated a zinc blende structure with a = 5.436kX (5.447A). An accurate determination was carried out by Giesecke and Pfister." Here, as with AISb, spectroscopically pure material was used, and the same measurement and evaluation techniques were employed. From the three photographs taken with Co K a radiation the evaluation of the 531-reflection at 8 = 76" gave a lattice constant of a = 5.4505 0.0001 A at 18°C. Addamiano"*'* indicates a value of a = 5.4504 A. The photographs with Cu Ka radiation were taken on pure material (prepared from 99.99 % pure Ga and Zn,Pz purified by sublimation), which showed only GaP reflections. For the calculation of the lattice constant from the measured lattice spacings, the wavelength Cu K a , = 1.5405 was used. No information was given on the temperature at which this lattice constant was measured or on the absorption corrections. However, the main purpose was to show the measure of agreement between the calculated and the measured intensities. (2) GaAs. Gallium arsenide, too, has been prepared and examined by Golds~hmidt.'~ The compound is of the zinc blende type. The lattice constant is given by Goldschmidt as a = 5.635kX (5.646 A). Barrie et aLZ5 reported X-ray diffraction measurements of material

A

E. A. Owen and G. D. Preston, Nature 113, 914 (1924). G. Giesecke and H. Pfister, Acta Cryst. 11, 369 (1958). 23 A. Taylor and H. Sinclair, Proc. Phys. Soc. (London)57, 126 (1945). 24 J. B. Nelson and D. P. Riley, Proc. Phys. Soc. (London)57, 160 (1945). 2 5 R. Barrie, F. A. Cunnell, J. T. Edmond, and I. M. Ross, Physica 20, 1087 (1954). 21

22

70

G . GIESECKE

prepared by heating together the elements of high purity, in stoichiometric ratio, confirmed the zinc blende structure, and gave the unit cell edge as 5.64 dL An accurate determination by Giesecke and fisterz2was carried out by the asymmetric method after Straumanis from powder photographs of spectroscopically pure material at constant temperature (18" & 0. 1°C). From three photographs taken with Cu K a radiation (see under AISb), the evaluation of the 551-reflection at t3 = 77", taking into acFount the absorption correction, indicated a value of a = 5.6534 f 0.0002 A. A further accurate determination was carried out by Ozolin'sh et a1.26 likewise by means of the asymmetric method. Samples with stoichiometric proportions and others with deviations on both sides of stoichiometry were prepared from pure components (Ga, 99.99% and As, 99.98%). Several photographs were taken at a constant temperature of 25" k 0.2"C of each of the preparations, which consisted of thin Lindemann glass fibers dusted with the sample powder. The powder size was controlled by passing it through a sieve with 10,OOO to 15,000 meshes/cm2. Measurements on the photographs were made to an accuracy of 0.005 mm. The position of the last reflections were in the range t3 > 55". In the calculation the- following values were used for the Cr K wavelengths (Cr K a , :2.28962 A, Cr K a : 2.2909 A, Cr K/3 :2.08480 A). The calculation lattice constants were corrected by the method of Nelson and Riley.24 For stoichiometric GaAs a value of a = 5.65317 f 0.00010 A was obtained; for GaAs + As (0.5% and 1.25%) a value of a = 5.65324 f 0.00005 A; and for GaAs Ga (0.5 % and 1.25 %) a value of a = 5.65320 f 0.00010 A-in all cases from two preparations. The most probable value for the lattice constant of GaAs as obtained by extrapolation of the mean values is given as a = 5.65315 f 0.000lOA. The lattice constant is therefore independent of the stoichiometry of the mixture of the components used in the preparation of GaAs, i.e., stoichiometric GaAs can be prepared from either arsenic- or gallium-rich mixtures.

+

(3) GaSb. Gallium antimonide was prepared and examined by Goldschmidt.15 Its structure was found to be zinc blende and the lattice constant determined to be a = 6.093kX (6.105 A). The accurate determination carried out by Giesecke and Pfisterz2 on spectroscopically pure material, in the manner described under AlSb, indicated a value of a = 6.0954 ? 0.0001 A. Use was made here of the 731reflection with Cu Ka radiation at 8 = 76". Three photographs were taken and an absorption correction was made. 26

G. V. Ozolin'sh, G. K. Averkieva, A. F. Ievin'sh, and N. A. Gorjunova, Kristal/ografiya 7,850 (1962) [English Transl.: Soviet Phys.-Cryst. 7 , 691 (1963)].

4.

LATTICE CONSTANTS

71

d. I n Compounds (1) ZnP. For the lattice constant of InP, Iandelli27 gives a value of a = 5.86,kX (5.873 A) obtained from powder photographs using Cu radiation (AKa = 1539, LKB= 1389 xu) and NaCl and KCl as reference substances. The compound was obtained by synthesis from the elements. However, the In used contained 2 % Ge impurity. Shafer and Weiser2" produced InP from the elements of high purity at different temperatures. The InP gave no traces of excess indium or phosphorus. The authors employed filtered Cu K a radiation and give data for InP spacings and intensities from diffractometer measurements. An accurate determination of the lattice constant of InP by Giesecke and Pfister22 was carried out by the same method given under AISb. Two samples were examined with Cu Kcl radiation. By evaluation of the 642reflection at 6 = 80", taking into account the absorption correction, a mean value from a = 5.86875 f 0.00OlOA for the lattice constant of InP was obtained. Gorjunova ef ~ 1 report . ~ accurate ~ lattice constant determinations of stoichiometric InP and (InP + P) and (InP + In) preparations. The lattice constant is considered to be independent of the exact stpichiometry of the starting materials and is given as a = 5.8693 f 0.0003 A at a temperature of 25" f 0.5"C and using Cu K a radiation with LKal = 1.54050A. (2) InAs. For InAs, Iandelli2' reports a lattice constant of a = 6.036 kX, which was determined under the same conditions as for InP. Here again, the basic material is In with 2 % Ge impurity. From the accurate determination by Giesecke and Pfister22 carried out on spectroscopically pure material, as-for AlSb, GaAs, etc., a value of the lattice constant a = 6.0584 f 0.0001 A was measured. For this, the 553reflection of CuKcl radiation at 6 = 78" was used. The value given was obtained from four photographs and an absorption correction was included. Ozolin'sh et ~ 1 likewise . ~ carried ~ out an investigation on the dependence of the lattice constant of InAs on stoichiometry. Pure materials were used (In : 99.99 %, As : 99.98 %). The methods of determination, preparation, measurement, and extrapolation were the same as those used by the same authors for GaAs. Here, however, additional use was made of photographs with Ni radiation (Ni K a , = 1.65784A). The values obtained from preparations containing an excess of As (1.0% and 1.5 %) and In (0.1%, 1 %, and 2%) are in agreement with the value of the lattice constant for stoichiometrically crystallized InAs, within the accuracy of measurement. A mean

''A. Iandelli, Gazz. Chim. frat. 71, 58 (1941). '* M. Shafer and K. Weiser, J . Phys. Chem. 61, 1424 (1957). ''N. A. Gorjunova,N. N. Fedorova, and W. J. Sokolova, Zh. Tekhn. Fiz. 28,1672 (1958 [English Trans{.; Sooiet Phys.-Tech.

Phys. 3, 1542 (1959)l.

72

G. GIESECKE

value for the lattice constant of all the InAs samples examined is therefore given as a = 6.05838 k 0.00005 A (at 25°C). (3) InSb. Indium antimonide likewise belongs to the group III-V compounds examined by Gold~chmidt.'~ This author indicates a zinc blende structure and a lattice constant value of a = 6.452kX (6.465 A). Iandelli2' reports that the lattice constant of InSb, containing 2 % Ge impurity in the In, is a = 6.46,kX (6.474A). Breckenridge et aL3' report that the purity used in their stoichiometric preparations of InSb was 99.95% for Sb and 99.954% for In. The InSb pieces obtained were further purified by repeated zone refining. X-ray analysis showed a value of a = 6.4782 A at 25°C. In the accurate determination by Giesecke and Pfister," carried out on spectroscopically pure material as indicated under AlSb, InAs, etc., two InSb samples were analyzed with Co Kcc radiation and the reflection 551 at 8 = 80" evaluated. Two photographs of one sample as well as one of a second including the absorption correction indicate a = 6.47877 k O.ooOo5 A.

4. RHOMBOHEDRAL STRUCTURE a. B-P Compound

A lower boron phosphide is described by M a t k ~ v i c h , ~which ' was prepared by the decomposition of BP in graphite crucibles in an inert gas atmosphere at temperatures of 1400"-1700"C. The unit cell data were obtained from single-crystal rotation photographs and powder photographs. The unit cell was found to be rhombohedra1 with the hexagonal dimensions a = 5.984, c = 11.850 A. All data obtained are in good agreement with the composition Bl,P2. A chemical analysis shows a total impurity content of about 3 %. b. B-As Compound

At temperatures of 10oo"-llOO°C and arsenic pressure less than 1 atm, Williams and R ~ e h r w e i n 'obtained ~ compositions of boron to arsenic of 5-7 to 1, which had an orthorhombic structure. The same orthorhombic structure was prepared by Perri et a1.,14 who found that the zinc blende structure of BAS is stable only to 920°C; above that temperature it undergoes an irreversible transformation to an orthorhombic modification, which is remarkably stable. The dimensions of the unit cell are u = 9.710, b = 4.343, c = 3.066A. 3 o R . G. Breckenridge, R F. Blunt, W. R. Hosler, H. P. R. Frederikse, J. H Becker, and W. Oshinsky, Phys. Reu. 96, 571 (1954). 3 1 V. I. Matkovich, A m Crysr. 14, 93, (1961).

4.

73

LATTICE CONSTANTS

111. Summary

In Table I, the most reliable data are summarized. TABLE I LATTICE CONSTANTS OF 111-V COMPOUNDS Compound

Lattice constant

Temperature

(A) BN (Ref. 3)

a = 2.50399 0.00005 c = 6.6612 f 0.0005

Wurtzite structure AIN (Ref. 6) a c GaN (Ref. 8) a c InN (Ref. 8) a c

= 3.111 = 4.978

Radiation

(4

(“C)

I C u Ka,

35 f 0.5

=

1.54051

Cu K a

cja = 1.600

= 3.18, f 0.004 c,a = 1,625 = 5.16, f 0.005

A Cu K a

=

1.539

= 3.53, +_ 0.004 c,a = 1,611 = 5.69, k 0.004

d Co K a

=

1.787

Zinc blende structure BN (Ref. 10) a = 3.615 5 0.001 BP (Ref. 14) a = 4.538 BAS (Ref. 14) a = 4.777 AIP (Ref. 17) a = 5.451 AlAs (Ref. 20) a = 5.6622

25 Cu Ka Cu K a 1Cu Ka,

AlSb (Ref. 22)

a = 6.1355 5 0.W1

18 f 0.1

GaP(Refs.22,17) GaAs (Ref. 26) GaSb (Ref. 22) InP (Ref. 22)

a = 5.4505 f 0.0001 a = 5.65315 f 0.00010 a = 6.0954 f 0.0001 a = 5.86875 f 0.00010

18 k 0.1 25 0.2 18 5 0.1 18 f 0.1

InAs (Ref. 26)

a = 6.05838 f 0.00005

25 f 0.2

lnSb (Ref. 22)

a = 6.47877 f 0.00005

18 5 0.1

Rhombohedra1 structure E P (Ref. 31) a = 5.984 (hexagonal) c = 11.850 E A s (Ref. 14) a = 9.710 b = 4.343 (orthorhombic) c = 3.066

I

=

1.5405

1 Cu K a , = 1.540500, A C o K a , = 1.78889, A C o K a , = 1.78889, iCr K a , = 2.2896, , iCu K a , = 1.540500, i C u K a , = 1.540500, A Cr K a , = 2.2896, I N i K a , = 1.6578, I Co K a , = 1.78889,

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CHAPTER 5

Elastic Properties J . R.Drabble I . INTRODUCTION . . . 1 . General Introduction .

. . . . . . . . . . . . . . 11. THERMODYNAMIC AND ATOMISTIC ASPECTS . . . . . . 2. Definitions of the Elastic Constants . . . . . . . . 3. Thermodynamics of the Elastic Constants . . . . . . .

.

,

.

. .

.

.

4. Atomistic Theories of the Elastic Properties

.

. . . . . . . . . . . . . . . . . . , .

. . . . . .

111. THEPROPAGATION OF ELASTIC WAVES . . . . 5 . Introduction . . . . , . . . , . . 6. Small Strain Theory of Wave Propagation-the Measurement of Second-Order Elastic Constants . . . . . . . . . . . 7. Propagation of Waves in a Stressed Medium--the Measurement of Third-Order Elastic Constants . . . . . . . . . , .

Iv.

EFFECTS OF CARRIER

CONCENTRATION ON THE

8. General . . . . . 9. Effects Associated with the Energy. . . . . . 10. Electron-Phonon Interaction

.

.

.

ELASTIC CONSTANTS

. . . .

.

. .

.

. .

75 75

77 71 84 87 88 88 89 94 97 97

Contributions of Carriers to the Free

. .

.

. . . . . . . . . . . . . . . . . V. EXPERIMENTAL RESULTS , . . . . . . . . . . . . 1 1. General Suroey . . . . , . . . . . . . . . . 12. Room Temperature Values of the Second-Order Constants . . . 13. Temperature Dependence oJ the Elastic Constants. . . , . . 14. Relations between the Elastic Constants . . . . . . . . VI. CONCLUSION . . . . . . . . . . . . . . . . . .

Effects.

98 107 109 109 109 1 10 111 114

I. Introduction 1 . GENERAL INTRODUCTION

The basic approach which I propose to adopt in this chapter is that the 111-V compounds form a group of materials in which a number of effects of fundamental interest associated with the elastic properties are to be expected but which so far have been almost completely unexplored. Before proceeding specifically with the 111-V compounds it will be useful to review briefly some of the developments in the study of the elastic properties of semiconductors over the last few years. These have been in 75

76

J. R . DRABBLE

two main categories. First, considerable progress has been made in what has always been one of the central problems of elasticity, viz., the explanation of the elastic properties in terms of atomistic theories based on the known atomic structure. Second, it is now well established that the classically defined elastic “constants” are not in fact intrinsic properties of a particular material but may depend on the detailed composition and on the physical environment. Studies in the first category up to about 1960 were almost exclusively based on the approach laid down by Born and co-workers which is described in the classic book by Born and Huang.’ The basis of this was that the atomic cores were regarded as point masses and that the forces between them could be specified by empirical parameters (coupling parameters or force constants). Procedures could then be established for setting up the equations for the normal modes of vibration of the lattice in terms of these empirical parameters. Complications occurred when Coulomb forces between the nuclei had to be taken into account. These are of a long range nature or, in alternative terms, it is not possible to restrict their influence to only close neighbors of a particular atom. This is because the Coulomb force is of an inverse square type and the number of atoms at a particular distance r from a given atom is proportional to r2. Procedures due to Ewald can be used to sum these effects and, together with the coupling parameters for close neighbors, this allowed the setting up of the secular equation governing the frequencies of the normal modes. Such equations can be solved to a high degree of complexity by computers. Calculations could be, and often were, made to the stage where the number of empirical parameters used in the theory exceeded the number of experimental observations with which the calculations could be compared. The elastic constants, which could be deduced from the normal mode distribution, played an important part in such comparisons. In general, it was found that the above procedure did not lead to a satisfactory agreement between theory and experiment, even for the simple semiconductors like germanium.2 It was realized that an important physical process, viz., the polarization of the atoms, was being neglected, and considerable progress toward agreement with experiment was achieved when a way of including this effect in the so-called shell model was found. This agreement, however, is still confined to a comparatively small number of materials and, as discussed in Section 4, has not yet been achieved for the 111-V compounds. In the second category mentioned above, there are a number of effects M. Born and K. Huang, “Dynamical Theory of Crystal Lattices.” Oxford Univ. Press (Clarendon), London and New York, 1954. F. Herman, J . Phys. Chem. Solids 8, 405 (1959).

5.

ELASTIC PROPERTIES

77

on the elastic properties which have been recently discovered or predicted in semiconductors. These may be classified here under three main headings : (a) Higher-order effects describing the variation of the elastic properties with strain or stress including, as a special case, high pressures. (b) Piezoelectric coupling effects on the frequency dependence of the elastic properties. (c) Carrier concentration effects. Studies under (a) and (c) have been largely confined to germanium and silicon and under (b) to the 11-VI compounds. The measurements of the elastic properties of the 111-V compounds which have so far been made have been confined to the determination of the three second-order constants, no attention having been paid to the above effects. It seems very likely that the elastic properties of the 111-V compounds will assume increasing importance as a proving ground for the development of atomistic theories and of the effects discussed above. These latter have already established themselves in their own right as being capable of yielding fundamental information. I t is this belief which has motivated the form and scope of this chapter. The existing measurements of elastic properties show some very interesting features insofar as they refer to a group of materials possessing a very similar structure, and it is possible to establish some well-defined trends and relations between them (see Section 14). The real interest of the elastic constants of 111-V compounds, however, lies in their potentiality for experimental and theoretical investigation, and this chapter will concentrate on these aspects.

11. Thermodynamic and Atomistic Aspects 2. DEFINITIONS OF THE ELASTIC CONSTANTS a. Introductory Remarks

The elastic constants of solids were historically introduced on an experimental basis as describing the linear relation between stress and strain for small strains. More fundamental and precise definitions may however be based on a thermodynamic approach and, in view of the objectives of this chapter, this will be adopted here. The starting point of any thermodynamic theory is the correct formulation of the extensive parameters which define the state of the system. For the thermodynamics of elastic media, this aspect itself has some difficulties which are not completely resolved. An excellent review of the problems ~ main difficulty involved may be found in the article by H ~ n t i n g t o n .The H.B. Huntington, Solid State Phys. 7, 213 (1958)

78

J. R . DRABBLE

is that the usual theories describe elastic deformation in terms of the components of a strain tensor which is defined so as to be independent of pure rotations. Consequently, if these components are used as state variables in a thermodynamic theory, the results cannot strictly be applied to materials in which local body torques are important, that is, when spontaneous polarization effects play an important role. This particular difficulty is largely one of principle only for the 111-V compounds and can be ignored. The main complication for these materials over the simplest type of thermodynamic theory is that they can, in principle, exhibit piezoelectric behavior although such measurements as exist indicate that the effects are small compared, for example, to those in the 11-VI compound^.^ Thus, in addition to the components of the strain tensor, other thermodynamic state variables are required. These are usually taken as the components of the electric displacement vector. The fundamental equation then expresses the internal energy of the solid as a function of the strain components, the electric displacement components, and the entropy. Starting from this equation, all physical properties of interest, including the elastic properties, can be defined and the conditions under which they are to be measured can be specified precisely. In view of the interests of this article, it is desirable to formulate the thermodynamics in such a way that both finite strain effects and piezoelectric effects are included. The treatments of piezoelectric effects given in the l i t e r a t ~ r e have ~ . ~ been confined to infinitesimal strains. Finite strain t h e o r i e ~ ~have . ~ not so far included piezoelectric effects. It is a simple process to combine these two, at least in the early stages of the theory, and to arrive at precise definitions of the elastic properties of the 111-V compounds. This will be done in the next three subsections.

b. T h e Elements of Finite Strain Theory It has been assumed that many readers will be unfamiliar with the description of nonlinear effects in the elastic range, and a brief review of finite strain theory will therefore be given here. The treatment follows closely that given by M ~ r n a g h a n . ~ The starting point of any description of strain is two sets of coordinates ai and x i (i = 1,2,3) giving, respectively, the positions of identifiable points of the medium before and after a displacement. In the customary description H. Kaplan and J. J. Sullivan, Plzys. Reo. 130, 120 (1963). J. F. Nye, “Physical Properties of Crystals.” Oxford Univ. Press (Clarendon), London and New York, 1957. W. G. Cady, “Piezoelectricity.” McGraw-Hill, New York, 1946. F. D. Murnaghan, “Finite Deformation of an Elastic Solid.” Wiley, New York, 1951. * K. Brugger, Phys. Rev. 133, A1611 (1964).

’

5.

ELASTIC PROPERTIES

79

(the Lagrangian description), the xi are regarded as dependent variables which are given as functions of the independent variables ai to specify a particular displacement. This description however includes rigid translations and rotations whose effects are not required for describing internal deformations (see, however, the remarks in the introduction to this section). These effects are eliminated as follows. Given the relations between the xi and the ai, it is possible to write down the expression for the transformation of any infinitesimal line element Aa in the original state to the corresponding infinitesimal line element Ax in the displaced state. This transformation is given by the equation Ax

=

JAa,

(1)

where J is the 3 x 3 Jacobian matrix of the transformation, i.e., J,, = ax,/aa,. The line elements Ax and Aa are to be treated as column vectors. The transposed row vectors will be written as Ax* and Aa*, and clearly Ax* = Aa*J*.

(2)

The square of the length of a displaced line element is

Ax** Ax

3

A a * J * * JAa.

For any rigid body motion, the lengths of all line elements remain unaltered and hence the product J*J must be equal to EJ, the unit matrix of order 3. Thus, for an arbitrary displacement, the difference between J*J and E3 can be used to define the strain. The finite strain matrix q is defined in practice as 9 = +{J*J- E3).

(3)

This is a symmetric matrix which in general would be a function of position. However, from now on, only homogeneous strain, in which by definition q is independent of position, will be considered. It is worth pointing out here the comparison with the usual theory of infinitesimal strain. In this latter, the same reference system is used for the coordinates before and after displacement, and xi is written in the form xi = ai

+ ui,

(4)

where ui is a function of all the aj. The quantities aui/daj are regarded as infinitesimals, so that any products of these of higher order than the first can be neglected. This leads to the result that the components o f J for this situation, as defined by Eq. (l), differ from the components of E, by infinitesimals. Neglecting higher products than those of first order, the strain

80

J . R. DRABBLE

matrix then reduces to the usual definition Unless otherwise stated, however, the full definition of q will be used in the following sections.

c. Thermodynamic Dejinitions of the Elastic Constants of order n Brugger* has recently discussed the finite strain formulation of the thermodynamics of elastic media without however including piezoelectric effects. The subsequent discussion follows his treatment closely but includes, among the extensive state variables, the components of the electric displacement vector in addition to the components of the strain tensor and the entropy. The internal energy is then a function of these variables and the following expression holds, using the tensor summation notation : U is the internal energy per unit mass and po is the density of the solid in the reference state of zero strain and electric displacement. The temperature T and the entropy per unit mass S form one pair of conjugate variables. The electric field components Ei and the components of the electric displacement Di (divided by po) form three other pairs of conjugate variables. Other pairs are provided by the thermodynamic stress comp o n e n t ~t,,~ and the components )I,,,, of the strain tensor (divided by po). The reasons for including po in the given manner are discussed, for example, by Callen.’” One point should be noted here. Since the strain components have been defined in such a way that qmn= q,,,,,, then there are only six independent components of the strain tensor entering into the expression for the internal energy. It turns out, however, that many of the derived formulas can be written in a much more compact form by assuming that the internal energy is written as a function of all nine strain components regarded as being independent of each other. This procedure has the consequence that any specific expressions for the internal energy as a function of the strain components, such as, for example, the Taylor expansions to be considered in the next section, must be written in a symmetrical form which is unaltered when )I,,,,, is replaced by )I,, for all m and n.7 Starting with Eq. (6),other thermodynamic potentials can be obtained in the usual way by Legendre transformation^.^" There are, of course, a very ’These are not the same as the actual stress components, except in the infinitesimal strain theory (see Section 3). 9a H. B. Callen “Thermodynamics.” Wiley, New York, 1960.

5.

81

ELASTIC PROPERTIES

large number of such possible potentials, but it is only necessary to consider three of them. First, the free energy per unit mass F defined as U - T S , is a function of T, q m n / P o , and Di/Po and dF = - S d T + tmn(dqmn/Po)+ Ei(dDi/Pol

(7)

Another potential, denoted by @ is defined as U - EiDi/po and is a function of S, qmnand Ei such that d@ = T d S

+ t m n ( d V m n l P 0 ) - Di(dEi/Po).

(8)

Finally, the function x = U - T S - EiD,/po is a function of T, qmn, and Ei and d~

=

-SdT

+ ~mn(dqmn/~o) - Di(dEi/Po)*

(9)

Physical properties are defined in terms of the second or higher partial derivatives of these functions. In particular, following the procedure of Brugger,' the elastic stiffness coefficients of order n are defined in terms of the nth partial derivatives of these potentials with respect to the strain variables, evaluated for the state of zero strain. Depending on the particular potential used, these elastic stiffness constants will correspond to different conditions of measurement. Thus, for example, nth-order elastic coefficients corresponding to constant entropy and cohstant electric displacement may be defined from the internal energy as S,D

mn,pq,.

.. =

..

~o(a"U/a~mna~pq .)S,D'

(10)

evaluated for a state of zero strain. These particular coefficients are then functions of the entropy and the electric displacement. The first-order coefficients are zero since the energy will be a minimum with respect to the strain in the equilibrium state. The potentials of Eqs. (7H9) allow definitions to be made for the nth-order elastic stiffness coefficients under other experimental conditions. Thus, for example, the isothermal moduli taken at constant electric displacement are T,D

c mn,pq,. .. =

PO(dnF/dqmndVpq

.

'

')T,D.

(11)

Similar expressions in terms of the derivatives of CD and x define the adiabatic and isothermal moduli, respectively, for conditions in which the electric field remains constant. These definitions may perhaps seem somewhat elaborate. It will be seen later, however, that the second-order constants defined in this way are

82

J . R . DRABBLE

consistent with the usual definition and usage. The definitions of the thirdorder constants which have been used in the literature have not however been consistent with the above definition and, further, have not been consistent as between different authors. Higher-order terms have not been considered so far. The author wishes to support here the suggested thermodynamic definitions based on the approach by Brugger. In addition to consistency, these possess the advantage that the nth-order coefficients form a contravariant tensor of order 2n and hence, when transforming from one set of axes to another, the transformation equations can be written down directly. The definitions of Eqs. (lOHll),together with the symmetry of the strain tensor, clearly lead to relations which reduce the number of independent constants of order n from the value of 3’”. This reduction is most conveniently described in terms of the matrix notation which is well established for the second-order constants and may with advantage be extended to the higher-order constants. Following the accepted convention, the quantities ylA are defined for A = 1 to 6 by 11 = 1 1 1 9

‘14 =

123

+ q329

‘12

= ‘122,

‘15

= ‘131

‘13 = ‘133,

+ q139

‘16

= ‘112

+ ‘121.

(12)

The inverse relations are

(13) ( l + 6ob)qA/2, where the double symbols transform into single symbols according to the Voigt notation 11 + 1, 22 -,2, 33 -,3, 23, 32 4 4, 31, 13 + 5, 12, 21 + 6. A contracted notation for the elastic stiffness coefficients of order n is introduced according to the direct transformation cab,ed... . (14) In the contracted notation Eq. (6), for example, becomes ‘1ab =

+

dU = T dS

+ t A ( d q A / p O ) + Ei(dDi/po)

9

(15)

with the convention of summation over repeated indices being, of course, implied, and Eq. (10) becomes (16) CS.D M , N , . , , = pO(a“u/a‘lMaqN ’ ’.)S,D’ In Eqs. (15) and (16), U is expressed as a function of the six independent components of the strain tensor and is to be obtained from the U of Eqs. (6) and (10) by substitution of Eq. (13). For a fuller discussion of these points and the corresponding definitions and transformations of the elastic compliance coefficients, reference should be made to the paper by Brugger.*

5.

ELASTIC PROPERTIES

83

It follows from Eq. (16) that the order of the subscripts in a particular elastic constant is unimportant, and it is found, when this is taken into account, that, for a general type of solid, there are 21 second-order and 56 third-order constants. In the case of Eq. (16), these are functions of the entropy and of the electric displacement. Similar forms may, of course, be written in terms of the potentials F, a, and x.

d. The Expansions of the Thermodynamic Potentials in Powers of the Strain Components The components of the strain tensor are always small in the elastic range, even for finite strains. The dependence of the thermodynamic potentials on the strain can therefore be expressed usefully in the form of a Taylor series expanded about the state of zero strain. Any desired degree of approximation can be achieved by terminating this series at an appropriate point. The argument is illustrated here in terms of the internal energy. Regarding U as a function of all nine strain components, the definitions of Eq. (10) clearly lead to the expansion POW

=

PO',

I + TCmn.pqqmnqpq + &Cmn,pq,rsqmnqpqqrsr

(17)

since the terms of first order are zero. The function U , and the various coefficients are, in this case, functions of the entropy and of the electric displacement. The appropriate superscripts have however been omitted in the interests of simplicity. This particular expansion will therefore be useful for adiabatic processes at constant electric displacement. The predominant dependence of U on the strain is given by the quadratic terms, and this is the point at which the series is terminated in the usual theory of infinitesimal strain. In this case, the form of q given by Eq. ( 5 ) may be used. For higher-order effects it is unnecessary, in practice, to proceed beyond the cubic terms in the expansion. In the contracted notation, when U is regarded as a function of the six independent strain components, the expansion analogous to Eq. (17) is p o u = pow0

+ +cpQrlPrlQ + &C~QRWIQVRI

(18)

where summations are taken over all values of the indices from 1 to 6 ignoring the equality of the elastic constants which follow from an interchange bet ween subscripts. The general expansion expressed by Eq. (18) is considerably simplified when the solid possesses symmetry operations. For solids possessing the symmetry of the 111-V compounds, as for germanium and silicon, the expression may be written specifically, when referred to the cube axes, as

84 POu

J . R . DRABBLE

+ 1 3 ’ ) + c44(142 + 1 5 ’ + 16’) + 2c12(1112 + 1 2 1 3 + ~ 3 4 1 ) + ) Hc111(113 + vZ3 + 1 S 3 ) + 3c112[112(12 f 1 3 ) + q Z Z ( 1 3 + 1 1 ) + r/3’(11 + V Z ) ] + 3c14d11142+ 1 2 1 5 ’ + 1 3 1 6 2 ) + 3c166[142(12 + 1 3 )

= POUO

f

~ ( c l l ( ~+ 1 212’

+ 1 5 2 ( 1 3 + V l ) + 162(11 + 1211 + 6c123ql)i1213f

6c4561415y/6}(19)

The “symmetrical” form of U is obtained by substituting the relations of Eq. (12) into this expression. There are three independent second-order constants and six independent third-order constants. It may be noted here that Eq. (19) differs from other expressions which have been given in the literature. In particular, the expression given by Birch” has been used as the basis for obtaining the third-order constants of germanium”*12 and silicon.” The third-order constants of Eq. (19) differ from those used by Birch as a result of the different definition, and the following relations exist between the two sets : clll

=

6cyll,

cllZ

=

2cy12,

c123

=

Cy23,

3. THERMODYNAMICS OF THE ELASTIC CONSTANTS In the preceding section, the elastic constants of a solid have been defined in a very general way for various conditions. We now commence the discussion of how these are measured. The current trend, almost to the exclusion of all other techniques, is to make use of the precision afforded by ultrasonic techniques by measuring the velocity of sound waves in the material. An effective elastic constant for a particular experimental situation is then defined by c = pu2 where u is the velocity and p is the density. Thus, the theoretical emphasis in this chapter will be on those aspects which govern the propagation of elastic waves under various conditions. It would, in principle, be possible to retain the generality of the preceding section and to derive a general theory of finite strain coupled with piezoelectric effects. This, however, would be a complicated and lengthy procedure, and it seems desirable at this stage to introduce some simplifications. These are to concentrate from now on on the primary effects. In particular, the piezoelectric effects referred to in Section 2,a, are expected to manifest lo

F. Birch Phys. Reo. 71, 809 (1947). T. Bateman. W. P. Mason, and H. J. McSkimin, J . Appl. Phys. 32, 928 (1961). J. R. Drabble and M. Gluyas, in “Lattice Dynamics” (Proc. Intern. Cod. Lattice Dynamics, Copenhagen, 1963), p. 607. Pergamon Press, London, 1965.

5 . ELASTIC PROPERTIES

85

themselves in the 111-V compounds principally through the dependence of the effective elastic properties on frequency, Such effects have been investigated for the propagation of waves in initially unstressed media. l 3 For this situation, the only strain components are those associated with the acoustic wave, and these may be regarded as infinitesimal. Thus, only the second-order elastic constants enter into the theory. On the other hand, the third-order constants associated with finite strains are measured by propagating elastic waves through a medium which is subject to a static This situation is one in which an infinitesimal alternating strain is superimposed on a finite static strain. So far, no measurements of these constants have been reported for piezoelectric materials. While it is certainly the case that a number of secondary effects are to be expected on the effective elastic constants of a piezoelectric medium subject to a finite strain, the two primary effects which will be considered here are (a) piezoelectric effects ignoring the complications of finite strain and (b) finite strain effects, i.e., measurement of third-order constants ignoring the complication of piezoelectric effects. These two aspects will be further considered in Sections 6 and 7, respectively. It is appropriate here, however, to consider some further thermodynamic aspects which form the basis of these sections. The theoretical basis of any method of measurement of the elastic properties of a solid, including the wave propagation method, is the relation between stress and strain. Quite generally, the components of the stress tensor z k j are given zkj

=

(P/PO)Jk/jqfpq’

(21)

where p is the density of the medium of initial density po, J k , is defined following Eq. (l), and the t p , are the components of the thermodynamic stress tensor defined for the appropriate conditions by one of Eqs. (6H9). Equation (21) is the starting point for the discussion of finite strain effects, i.e., for case (b) above. For infinitesimal strains [case (a)], however, a considerable simplification is introduced into the theory by the fact that the elements of J differ from those of the unit matrix of order 3 by infinitesimals. For this situation, Eq. (21) reduces to Tkj

=

tkj.

(22)

It is essential, for consistency in this limiting case, that in deriving t k j from the expansions of Section 2,d, only the quadratic terms are retained and, further, that the form of the strain components given by Eq. ( 5 ) should be used. R. Hutson and D. L. White, J . Appl. Phys. 33.40 (1962). R. N. Thurston and K.Brugger, Phys. Rea. 133, A1604 (1964).

l 3 A. l4

86

I. R . DRABBLE

The essential simplification arising from the application of Eq. (22)is that the stress components are derived extensive thermodynamic parameters conjugate to the (infinitesimal) strain components. This is not true in the general case. For the rest of this subsection, the consequences of Eq.(22) will be discussed, and, in particular, the equations which provide the basis for the discussion of wave motion under these conditions will be obtained. For this purpose the thermodynamic potential 0, defined by Eq. (8) is particularly suitable since it is expressed in terms of the entropy, the strain components, and the electric field components. The entropy remains constant in wave propagation, and the electric field is easily controllable. However, similar analyses may easily be carried out for different conditions. From Eq. (8), the following relations clearly hold tmn

= PO(~@/~V,)S.E ;

Di

= -P d d @ / d E i ) s . q -

(23)

These derived extensive parameters are clearly functions of S, qmn,and

Ei and, for isentropic processes we can write dtmn = dDi

=

(atmn/atlpq)s,E d q p q (aDi/aqpg)S,E

dqpq

4-

(atmn/aEk)S,q d E k ,

+ (dDi/aEk)S,q

dEk*

(244 (24b)

For the particular situation of interest, where small deviations from a state of zero strain and electric field are being considered, the quantities dt,,,,, dqp4,dDi, dEk may be replaced by their actual values, and the derivatives are then evaluated at the reference state. Thus (atmn/aqpq)S,E

=

pO(a20/a?mn~~pq)S,E

--f

S,E crnn,pq

(25)

becomes an appropriate second-order elastic constant. The derivative

is clearly a component of the permittivity tensor, measured for conditions of zero strain. The “cross” derivatives of Eq. (24) are clearly related via Eq. (23). Thus, writing (aDJaqpq),,

=

eipq 7

(27)

it then follows from Eq. (23) that These latter quantities are related to the components of the piezoelectric tensor, which are usually defined in terms of the derivatives of the electrit polarization components Pi with respect to the stress components (tmn in

5.

87

ELASTIC PROPERTIES

the present case) by5 (29) Since D = cOE+ P, the piezoelectric components taken at constant electric field and entropy are dimn

dyif

=

(aPi/at,.).

(30)

= (dDi/atmn)S.E.

Consideration of a thermodynamic potential defined as

u - tmnVmn/Po - E i D i / P o

1

in which the independent variables are t,,, E i , and S , leads to the relation

ss:pq

is an elastic compliance coefficient, measured for each i, m, n, where at zero electric field and constant entropy, defined by the relation =(

s2:pq

~ ~ m n / ~ t p q ) ~ , E -

(32)

Finally, therefore, Eqs. (24) may be written for small isentropic displacements about a state of zero strain and zero electric field as tmn

=

Cmn,pq?lpq

D i

=

eipqVpq

f

- ekmnEki E&k.

(334 (33b)

There are the equations which form the basis of the theory of wave propagation. 4. ATOMISTIC THEORIES OF

THE

ELASTICPROPERTIES

Although, as mentioned in Section 1, there has been considerable theoretical development in the atomistic theories of solids in recent years, it is probably fair to say that very little of this is directly relevant to the theoretical calculation of the elastic constants. All theories concentrate on the calculation of the lattice vibration spectra, starting from various physical models and containing a number of empirical atomistic parameters. The theoretical results are then compared with experimental parameters, including the second-order elastic constants as a particular case. Since the lattice dynamics of III-V compounds is being dealt with fully elsewhere in this series (see the chapter on Lattice Dynamics by D. Kleinman appearing in a subsequent volume) there is little point in discussing any aspects in detail. However, it seems worth while to review briefly some of the more significant papers.

88

J. R . DRABBLE

Merten,I5*l6in two papers, applied the Born method to the calculation of the lattice vibrations in crystals with the zinc blende structure. This work gives a full account of the relevant geometrical aspects and, in particular, describes the effects of crystal symmetry on the second-order coupling parameter tensors for first, second, and third nearest neighbors. The equations of motion were set up and solved for the approximation of nearest neighbor interaction only. This involved only two second-order coupling parameters, denoted by B and C. In terms of these, Merten found that the elastic constants were given by ~ 1 = 1 B/u, ~ 1 = 2 (2C - B)/u, ~ 4 = 4 (B2 - C2)/Ba, (34) where a is the lattice constant. Thus, to this degree of approximation, there should be a relation between the three second-order elastic constants of the form - ~44). (35) (cii + ciz)’ = 4~11(~11 At the time of these calculations, there was little experimental data available. Merten calculated the values of B and C for indium antimonide and used these to obtain theoretical estimates of the optical mode frequencies, obtaining rather poor agreement with experiment. Recently, there has been a large output of papers on the lattice dynamics of 111-V compounds using the more general theories which take the atomic polarizability into account. Of these, the work of Kaplan and Sullivan4 provides a very thorough discussion of the general position and presents a comprehensive calculation of a number of experimental quantities including the elastic constants. The theory contains twenty-eight parameters, and it was not possible to obtain unique values of these from a comparison with the available experimental data. The same difficulty arose in a paper by Waugh and Dolling” on the crystal dynamics of gallium arsenide. For further details, reference should be made to the chapter on Lattice Dynamics in a subsequent volume of this series. From the point of view of this chapter, the position is that there are no calculations of the elastic constants of the 111-V compounds, starting from first principles.

III. The Propagation of Elastic Waves 5. INTRODUCTION Reference has already been made to the fact that the experimental determination of the elastic constants is almost invariably based on the L. Merten, 2. Naturforsch. 13%662 (1958). L. Merten, 2. Naturforsch. 13% 1067 (1958). ” J. L. T.Waugh and G . Dolling, Phys. Reu. 132, 2410 (1963) l5

l6

5.

89

ELASTIC PROPERTIES

determination of the velocity of sound. In this section, some of the more important aspects of wave propagation will be considered in the two contexts discussed in Section 3, viz., propagation in initially strained media, involving the third-order elastic constants but ignoring possible piezoelectric effects and, secondly, propagation in unstressed media. In the latter case, piezoelectric effects are included but may of course be omitted if desired by putting the appropriate constants equal to zero. The basic equation of wave propagation obtained by considering the equation of motion of a small volume element is p a2xi/at2 = atij/axj,

(36)

where xi is the ith component of the position of the volume element, p is the density, and t i jare the stress components. 6.

SMALL STRAIN THEORY OF WAVE PROPAGATION-THE SECOND-ORDER ELASTICCONSTANTS

MEASUREMENT OF

For a wave propagating in an initially unstrained medium, Eq. (36) reduces to po a2ui/at2= atij/aaj,

(37)

where ui is the ith component of the displacement associated with the wave and the significance of t i j has been discussed in Section 3. In particular, Eqs. (33a) and (33b) hold for this situation. The general theory of wave propagation under the conditions postulated has been given, for example, by Hutson and White.I3 The procedure is first to use Eq. (33a) in Eq. (37) to obtain, with the use of Eq. (5), po a2ui/at2= cijlmd2u,pajdam - ekijaEJaaj.

(38)

It is understood that the elastic and piezoelectric coefficients are adiabatic and are measured for zero electric field. Further relations between the strain and electric field components are obtained from Maxwell’s equations and Eq. (33b). For a solution corresponding to the propagation of a plane wave along the l-direction of an arbitrary reference system, all field quantities vary as expj(ka, - ot), where k is a propagation constant and o is the frequency. For this situation it is simple to show that (curl H),= ( d D , / d t

+ J1)= 0,

(39)

where J is the current density. This is defined in terms of the electric field by J, =

a,,E,,

(40)

90

J . R . DRABBLE

where gpq is the electrical conductivity tensor. The use of this relation together with Eq. (33b) in Eq. (39) gives el

a2u,/ataal

+ El, aE,/at + o l I ~=, 0.

(41)

Two further relations are obtained from the general equations governing the propagation of electromagnetic waves (for a nonmagnetic medium with B = poH grad div E - V 2 =~ -po a(aD/at

+ J}/at.

(42) For the plane wave solutions sought, the 2- and 3-components of this equation reduce, with the help of Eq. (33b), to

+

+

aZE,,/aalZ= poeulla3~l/atzaal p O ~ ,a2E,/atZ l p o ~ vaEJat. I

(43)

In Eq. (43) and the following, the subscripts u, w take on only the values 2 and 3. Writing specifically ui = Ai expj(ka,

- cot), Ei = Ciexpj(ka, - at)

(44)

and using Eq. (41) to eliminate Ci leads to the set of equations derived from Eqs. (38) and (43):

+ jkeLliCw, k2C, = jpoko2e:11AI+ p0co2~:,C,.

p a 2 A l = kZc;,,,A,

(45)

In this set of five equations, the primed coefficients are “effective” constants. The effective elastic constants are defined by c;ilt

= Cliii

+ elliellJ(E11 + - b l l / ~ )

(46)

and therefore depend on the frequency directly and also indirectly through the frequency dependence of the electrical conductivity and the dielectric constant. Equations (45) give rise to a fifth-order secular determinant for the possible wave velocities. In the limit where the piezoelectric coefficients vanish, the equations are decoupled, and the solutions correspond to three acoustic waves and two electromagnetic waves. For the coupled equations, it was shown by Hutson and White that, to a high degree of accuracy, they can be solved by solving only the thirdorder acoustic determinant, neglecting the coupling to the electromagnetic waves but using the modified elastic constants ckill. Specifically, they showed that if the solutions of the set of equations poco2Ai = k2C;ilIA,

(47)

5.

ELASTIC PROPERTIES

91

are determined and used to obtain the acoustic velocities ui2 = W 2 / k 2 = 1, 2, 3), then the actual solution of the complete set of equations would introduce corrections to these velocities which are of the order of the ratio of the velocity of sound to the velocity of light in the medium. Thus, to a high degree of approximation, it is possible to define effective elastic constants which govern the propagation of elastic waves in the medium, these being defined by Eq. (46). Tracing this result back physically, each acoustic wave may create an electric field with transverse and longitudinal components due to the piezoelectric effect. The transverse electric field with the accompanying magnetic field corresponds to an electromagnetic wave which is forced to travel at the velocity of sound. It is thus very small and has very little effect on the acoustic wave. The longitudinal electric field, however, has an effect. When these results are applied specifically to the 111-V compounds, the directions of propagation are usually the [ 1001 and [ 1101 directions. For the former case, the piezoelectric tensor components, referred to the cubic axes, are such that eijkis zero unless i , j , and k are all different. Thus, from Eq. (46), there is no modification of the elastic constants for waves propagating parallel to the cube axes. The solution of the appropriate secular equation corresponds to a longitudinal wave traveling with a velocity u = (c11/po)1’2and to transverse waves, polarized in any direction normal to the direction of propagation, having velocities of (c44/p0)”2. For propagation in a [ 1101 direction, however, there is a different situation. Taking a set of reference axes with the 1-direction parallel to [110], the 2-direction parallel to [TlO], and the 3-direction parallel to [Ool], the nonzero elastic stiffness coefficients of the form clill referred to these axes are c 1 2 1 2 = (‘11 - c12)/2? ‘1111 = ( c l l + c 1 2 + 2c44)/29 c 1 3 1 3 = c44. (48) The only nonzero piezoelectric components of the type e l l l is e l I 3 . Second-order tensor components reduce to scalars. Thus, according to Eq. (46), only the component c 3 1 is modified by the piezoelectric coupling to give (49) c;313 = c 1 3 1 3 + .:13/(& +jdW).

(i

Experimentally, this situation corresponds to a wave propagated in the [ 1101 direction with polarization parallel to [Ool]. It must be stressed that the components of the permittivity tensor and of the electrical conductivity tensor, which appear in Eqs. (46) and (49), for example, are themselves effective quantities under the conditions postulated. The conductivity tensor, for example, is defined by Eq. (40), and this in general will be a complex quantity.

92

J. R. DRABBLE

The velocities determined by the solution of the secular determinant of Eq. (47) will also in general be complex. The real part of the solution manifests itself as a change in the velocity of propagation and the imaginary part as an attenuation. The factors which may affect the velocity and its frequency dependence of sound waves in semiconductors have been discussed in detail by Hutson and White.13 The discussion takes a different form for intrinsic and extrinsic semiconductors. For an extrinsic semiconductor, the 1-component of current density is given by J 1 = 4(n + fns)PEI + (P/S)f(waai). (50) The first term is due to drift and the second to diffusion. In this equation q is the electronic charge, /3 is Boltzmann’s constant multiplied by the absolute temperature T, n is the mean density of electrons, (n fn,) is the instantaneous local density, and p is the mobility. The quantityfis the fraction of total space charge which is mobile, i.e., not bound to impurity levels. For plane wave propagation conditions with angular frequency o and propagation constant k, the relation between J and E reduces to (51) J 1 = gOE1/[1 + j(o/oD)l, where g o = nqp is the normal conductivity and oD is a “diffusion” frequency defined by W D = qpW2/fpkZ. (52) This represents, in physical terms, a frequency above which diffusion smooths out carrier density fluctuations associated with the acoustic wave. Equation (51) defines an effective conductivity which may be used, for example, in Eq. (49) to give the effective elastic constant

+

c‘ = ,{l

<[ +

+ C&

1

+

j ( o / +j(o/wD) oD) j(°C/o)

11.

(53)

where ocis a conductivity frequency defined by o c = no/&.

(54)

For simplicity, subscripts have been omitted in Eq. (53). This equation leads to a dependence of the wave velocity on the frequency given by

where vo = (c/p0)’/’ is the low frequency velocity. The case of intrinsic semiconductors has also been discussed by Hutson and White. They distinguish between the two situations where on the one

5.

93

ELASTIC PROPERTIES

hand the recombination is fast compared with the acoustic frequency and on the other hand it is slow. In the former case, the propagation of waves is the same as the extrinsic case with an effective diffusion frequency where p n and pp are, respectively, the mobilities of electrons and holes. In the second case, the expression for the effective elastic constant is

where wDp and wDn are diffusion frequencies for holes and electrons, respectively. TABLE I SECOND-ORDER ELASTIC CONSTANTS FOR VARIOUSMODESOF PROPAGATION IN CUBICMATERIALS ~

Mode

Propagation direction

Polarization direction

POV2

Affected by piezoelectric coupling.

The most direct and usual method of determining the second-order elastic constants for a material with cubic symmetry is to measure the velocity of waves propagating in the [OOl J and [110] directions with various polarizations. Table I gives the appropriate elastic constants which enter into the expressions for pou2 for such modes. The only one of these which is affected by the piezoelectric coupling effects discussed above is mode 3. For the highest precision of measurement, a number of possible sources of error have to be taken into account. The effect of surface finish has been found to be significant.18 There are also effects associated with the diffraction of waves which depend on the frequency and on the ratio of transducer to specimen dimension^.'^*^^ The bond between the transducers and the specimen is also important. For a more detailed discussion of the experimental aspects involved, see the article by McSkimin.zoa H. J. McSkimin and P. Andreatch, J . Appl. Phys. 34, 651 (1963). H. J. McSkimin, J . Acoust. Soc. Am. 32, 1401 (1960). *OH. J. McSkimin, J . Acoust. Soc. Am. 33, 606 (1961). H. J. McSkimin, in "Physical Acoustics" (W. P. Mason, ed.),Vol. I/A, p. 271. Academic Press, New York, 1964. l9

94

J.

R.

DRABBLE

There are very little data available to allow a determination of the possible effect of piezoelectric coupling in the determinations of the elastic constants which have been made so far for the III-V compounds. Insofar as these have been determined by using only modes 1, 2, 4,and 5 of Table I, there would be of course no direct effect. Some data for gallium arsenide given in Ref. 13 lead to the values wc = 10’20

(0

in ohm-’ cm-’),

wD = 7 x lo8 at 300”K, e2/2c&= 1.2 x These values in turn lead to the conclusion that, even for rather impure material, the frequency dependence of the effective elastic constant for mode 3 of Table I would be negligible below about lo8 cps. The piezoelectric constants of gallium arsenide have been determined recently by Charlson and Mott.20b 7. PROPAGATION OF WAVES IN A STRESSED MEDIUM-THEMEASUREMENT OF THIRD-ORDER ELASTIC CONSTANTS At present there appear to be no reported measurements which provide information on the third-order elastic constants of III-V compounds. However, in the author’s view, such measurements would be of considerable interest, and work is currently proceeding on gallium arsenide and indium antimonide.20cThe interest derives from the fact that the values of third-order constants which have been reported for germanium and silicon show a considerable degree of correlation. This, of course, is to be expected on the basis of the similar structure and bonding. It is therefore relevant to ask to what extent the introduction of a partial ionic bonding component in the III-V compounds affects the pattern of the third-order constants. The experimental methods which have so far been used to determine the third-order elastic constants are based on the propagation of ultrasonic waves in a medium which is subjected to a static strain. The theoretical analysis of the experimental results involves the use of Eq. (21) for a situation in which an infinitesimal alternating strain is superimposed on a finite static strain. This analysis was given by Seeger and Buck2’ for crystals possessing cubic symmetry. They considered the propagation of waves in various directions for different polarizations and conditions of applied stress. The stress conditions included both uniaxial and hydrostatic pressure. Expressions were given for the “effective” elastic constants pu2, where p was the actual density of the medium under stress and u the actual velocity. Zab

2ac 21

E. J . Charlson and G. Mott, Proc. I.E.E.E. 51, 1239 (1963). See Section 11 for a recent determination of c456for gallium arsenide. A. Seeger and 0. Buck, Z. Narurforsch 15% 1056 (1960).

5.

95

ELASTIC PROPERTIES

These expressions involved the stress, the three second-order elastic constants, and the six independent third-order constants. From the analysis it emerged that the most suitable type of singlecrystal specimen on which to carry out the measurements was a rectangular prism having two opposed (001) type faces and the remaining four faces of (110) type. Such a specimen allows, in all, nine modes of propagation to be investigated. These modes involve different combinations of the direction of applied stress, of propagation, and of polarization, summarized in Table 11. The nine experimental values of pu2 for these modes may be used to determine the six third-order elastic constants using independently determined absolute values of the second-order constants. Thus, this particular set of measurements provides checks on the derived values of the third-order constants. TABLE I1 MODESOF PROPAGATION FOR CUBICCRYSTALS UNDER UNIAXIAL COMPRESSION Mode

Compression direction

Propagation direction

Polarization direction

Measurements of the elastic properties as a function of hydrostatic pressure may also be used to provide information on the third-order constants, although they do not by themselves lead to a complete derivation. Under hydrostatic pressure the material retains its original cubic symmetry and, at any pressure, the elastic properties are defined by three effective elastic constants, c;,, ci2, and ck4. These are given by"

G l = c l l - a[2(cll + c12)+ c l l l + 2clI2l, c;2 = c12 - d-(c11 + c12) + 2c11, + c1231,

(58b)

ck4 = c44 - a[c44 + c l l

(584

+ clZ + c144 + 2c16,1,

(584

where a = P / ( c , , + c,,), P being the pressure. In practice the quantity that has actually been measured is the change of the transit time or the frequency of the waves associated with a particular mode of propagation as a function of the applied stress. This latter is

96

J . R. DRABBLE

defined in terms of a single scalar quantity p tepresenting either the magnitude of a uniaxial compressive stress or, alternatively, the magnitude of a hydrostatic pressure. Using the Seeger-Buck analysis it is then necessary to calculate the value of pu2 by translating the measured transit time into a velocity, allowing for a change in the dimensions of the specimen as a result of the applied stress. Similarly, the change in density has to be calculated. The dimensional changes involved can be calculated with sufficient accuracy knowing the original dimensions and the second-order elastic constants. In a recent paper by Thurston and Brugger,I4 a method is developed which eliminates the necessity for these latter operations. Expressions are given, for different modes, for a quantity d(poW2)/dpat p = 0 where p , as before, represents the magnitude of an applied stress. W is defined as a natural velocity, i.e., the velocity referred to natural dimensions. Thus, if T is the transit time under stress in a direction which has a path length Lo in the unstressed condition then W = Lo/T. The quantity po is the density in the unstressed state. Thus, d(p,W2)/dp at p = 0 is determined directly from the measured change of transit time or frequency with stress. The appropriate expressions for the nine conditions of Table I1 are given in Table 111. It should be noted that the expression for mode 3 (compression along [Ool], propagation along [ 1lo], polarization along [Ool] TABLE I11 =0

EXPRESSIONS FOR d(p, W2)/dpAT p Mode

w = (poV2),=o

3 4 5

c44 c11 c44

6

f44

F,O

FOR

MODESOF TABLE 11" d ( p o W 2 ) / d pat p = 0

5,

ELASTIC PROPERTIES

97

differs from that given by Thurston and Brugger, which has been given erroneously. Experimentally, the magnitude of the changes in velocity which have to be determined are of the order of a few parts in lo5 for typical stresses, and a high degree of precision of measurement is necessary. Two methods have been used so far for measurements of the higher-order constants of germanium and silicon. Bateman et al.” used a pulse overlap method in which out-of-phase wave frequencies were determined as a function of stress (see also the paper by McSkimin and Andreatch’*). Drabble and Gluyas” used the improved “sing-around” system designed by Forgacs,” in which the frequency of the ultrasonic waves is kept constant and the changes in transit time are followed as a function of stress. Both methods lead to values of the third-order elastic constants which are accurate to a few percent. Recently, Blume”” has described a coherent pulse method which may prove to be more accurate than either of the above methods under comparable conditions. The discussion of the preceding section leads to the tentative conclusion that it is unlikely that piezoelectric effects will lead to any serious errors in the determination of the third-order constants of III-V compounds, provided that the operating frequency is kept in the 10-Mc/sec range. IV. Effects of Carrier Concentration on the Elastic Constants

8. GENERAL In the preceding section, the effect of the carrier concentration on the frequency dependence of the elastic properties arising from piezoeIectric coupling was considered. In addition to this effect, there are other effects of carrier concentration which may influence the static elastic properties. Some of these have already been observed in germanium, and it may be anticipated that they will also be observable in the III-V compounds. The effects to be discussed may be divided into two groups. First, it has been shown that the elastic properties may be defined in terms of the derivatives of the free energy with respect to the strain. In semiconductors, the free energy contains a contribution from the carriers in the band and impurity levels. This contribution depends on the distribution of the carriers over these levels, and if this distribution changes with strain there will be a contribution from this source to the derivatives of the free energy with respect to the strain components, i.e., a contribution to the elastic constants. It is already well known that the distribution of carriers over the energy levels in the elemental semiconductors germanium and silicon can be considerably affected by strain, as is indicated by the marked 2 2 R. L. Forgacs, I R E Trans. Instr. 1-9, 359 (1960) 2 2 n R .J. Blume, Rev. Sci. Instr. 34,1400 (1963).

98

J . R . DRABBLE

piezo-resistance effects in these materials.23924 Similar, though less marked effects, are to be expected in the 111-V compounds. The second possible effect of carrier concentration has so far, to the author’s knowledge, only been investigated on a theoretical basis. This effect is associated with the interaction between electrons and phonons, leading to a renormalization of both the electron and phonon distribution. The former leads, in the simplest case, to a change in the effective mass and the latter to a change in the sound velocity, both being functions of the carrier concentration. The theory predicts observable effects for germanium. Both effects are critically dependent on the temperature for a given carrier concentration and hence can influence the temperature dependence of the elastic properties. In addition, since the carrier concentration depends on the temperature, there will be a variation of the elastic properties with temperature due to this source. Although none of these effects has yet been isolated in the 111-V compounds, it seems worth while to discuss the sort of effects that are likely to happen. In the author’s opinion there is a rich experimental field to be explored here, and some of the possible effects are being investigated in his laboratory. 9. EFFECTSASSOCIATED WITH THE CONTRIBUTIONS FREEENERGY

OF CARRIERS TO THE

(a) General Principles The basic principles associated with the contribution to the free energy have been outlined by Keyes.2s The expression for the free energy of a system of Fermi particles isz6 F

=

pN - kTxIn[l

+ exp{@ - Ei)/kT}],

(59)

i

where the E j characterize the energy levels over which the particles are distributed, k is Boltzmann’s constant, and T is the absolute temperature. The quantity p is the electrochemical potential (Fermi level) of the system and is related to the total number of particles N by the relation N =

C [ l + exp{(Ei - p)/kq]-’. j

When these relations are applied to the case of a semiconductor the summations range over the band levels and the impurity levels. For the C. S. Smilh. fhys. Reo. 94, 42 (1954). F. J. Morin, T. A. Geballe, and C. Herring, Phys. Reo. 105, 525 (1957). 2 5 R. W.Keyes, I B M Res. Deoelop. 5, 266 (1961). 26 L. D. Landau and E. M. Lifshitz, in “Statistical Physics.” Macmillan (Pergamon), New York 23 24

1959.

5.

ELASTIC PROPERTIES

99

former, they are replaced by integrals containing the appropriate density of states function. It is then necessary to obtain F as a function of the strain components. This problem reduces to finding how ji, N , and E j vary with strain, subject to the normalizing condition, Eq. (60). The simplest case to discuss is one in which it may be assumed that the impurity levels play no part in the rearrangement of the carriers under strain. This will be the case at sufficiently high temperatures, and also when the density of impurities is sufficiently high since, in this case, the carriers remain in the band levels at all temperatures. The appropriate summations may then be expressed in terms of integrals over the band states only. The further simplification, that N remains constant during strain, can also be used. TABLE IV CONDUCTION BANDSTRUCTURES

Compound

Other minima

Lowest energy ~~

InSb lnAs InP GaSb

k=O k=O}

GaAs

k=O Along

k=O k=O

No evidence of other close minima

Along [ 1 1 I ] directions, 0.09 eV higher Along [lo01 directions, 0.3 eV higher Along [ 1001 directions, 0.36 eV higher At (OOO), approximately 0.4 eV higher

AlSb

The problem then reduces to the specification of how the band levels depend on the strain. Once this is specified, Eq. (60) allows the Fermi level fi to be expressed in terms of the strain (since N is constant). Then F in Eq. (59) can also be expanded about the strain-free state in terms of the strain components. Using the definitions given in Eq. (1l), the contribution to the various isothermal elastic moduli can be obtained. The dependence of the band levels on strain takes a number of different forms. It will be useful here to remind the reader of the various types of band structures which are encountered in the III-V compounds (see, e.g., Ehrenreich*’). It is generally accepted that the valence band structures are likely to be similar for most of these compounds, being of the same type as for germanium and silicon. This structure consists of two degenerate bands having maxima at the center of the Brillouin zone with a third band

’’H. Ehrenreich, J . Appl. Phys. 32, 2155 (1961).

100

J . R. DRABBLE

separated from these by spin-orbit splitting having a lower maximum. The amount of spin-orbit splitting varies throughout the series. For the conduction band there is, however, no common type of structure, although certain general statements can be made. It is now accepted that three bands may be involved to a varying extent throughout the series. One such band has a minimum energy at k = 0 (k is a wave vector in the Brillouin zone). The second type of structure has a number of equivalent minima in [loo] directions and the third a number of equivalent minima in [ l l l ] directions. The relative positions of the minima vary from one material to another. Table IV summarizes the information given in Ehrenreich’s paper. These band structures may be classified as being of the “many-valley” type, although it should be noted that in the usual context this term is restricted to the situation of a single band, having a number of equivalent minima at points in the Brillouin zone which are related by crystal symmetry. The theory will be discussed separately for these two situations.

b. Conduction Band Egects The characteristic feature of the band structure which is of interest is that there are a number of nondegenerate minimum energies at various points of the Brillouin zone. For the compounds of interest one may classify these as follows: (1) Band structures with a single minimum at k = 0 and no other minima having energies close to these, e.g., indium antimonide. (2) Band structures with a single minima at k = 0 and other bands in the energy vicinity which are of the many-valley type, e.g., gallium antimonide. (3) Band structures of the “pure” many-valley type, with minima along [loo] or along [ l l l ] directions. (4) Combinations of the above. To a good approximation, in view of the small spread of energies involved, it may be assumed that the effect of strain is to shift all the states in the vicinity of a particular minimum by the same amount. Thus, the distribution of states is assumed to be the same in the presence of strain but referred to a minimum energy which is affected by the strain. The variation of the minimum energy of the ith group of states is given by an equation of the form A E ~= ~

i =:

gqylpq, ~

(61) where Ei is a second-order tensor called the deformation potential tensor. For a material with cubic symmetry this has the general form25 ~i

=

si1 + Euiaiai,

(62)

5.

101

ELASTIC PROPERTIES

where ai is the unit vector from the center of the Brillouin zone to the origin of valley i, and 1 is the unit tensor of order 3. Edi and Suiare constants and are the same for all valleys which are related to each other by crystal symmetry operations. Thus, for a valley at k = 0, the shift of energy is characterized by a single scalar quantity. For valleys in [lo01 or [ l l l ] directions, two scalar deformation potentials are involved, and the different valleys may shift differently in energy when strain is applied. Expressions are required for the density of electrons in each valley and for the contribution to the free energy of these electrons. If the band structure is taken as parabolic about the valley minima, these expressions are n"' = 2n'/2N9,/,(6' F")

(63)

2n-"2N~)kT{BF,12(t3 - d)) - $F,,,(Q- d’).

(64) Here N c ) = 2(27cmi*kT/hZ)where mi* is a density of states effective mass for valley i, F, is a Fermi-Dirac integral of order r, 6' is defined as p/kT, and w(i) = E'/kT, where E’ is the energy of the minimum in valley i. In the presence of strain we write =

8 = 0, + 6,

m(i)

=

mg) + ~

~

(

i

)

(65)

and expand the expressions (63) and (64) in a Taylor series about the unstrained values 8, and The condition that is constant with strain I

allows 6 to be expressed in terms of the Am"). The total free energy F = C,F('), similarly expanded about the zero strain position, can then be expressed in terms of the Aw'". Finally, use of Eq. (61) allows the A d i ) to be expressed in terms of the strain components, and thus the free energy may be obtained as a function of strain. The analysis is simplified when, as is the case, for example, for the conduction bands of silicon and germanium, the only minima to be considered are all equivalent, being related to each other by simple symmetry operations. For this situation, all the Na) are the same and so are all the cog). The condition that the total density of carriers remains constant then leads to the condition

102

J. R. DRABBLE

where the various derivatives are evaluated for an argument (0, - oo) and hence, for this case, are the same for all valleys. The above equation can then be solved for 6 as a power series in the Adi). This solution is conveniently put in the form

6 =

61

+ + 63,

(67)

62

where 6, is of the order (Am)’. For the case under consideration

where dw denotes the average value over the valleys of (Ad))”. For the more general case, where minima with different energies are involved, the average values have to be weighted with factors which are different for different minimum energies. The solution for 6 to any desired order can now be substituted in the expression for the total free energy. The resultant expression, for the simple many-valley model case, is

___ -

F3 = NkTFyl2{2do3- 3 ACOAm2

+ Ao3}/6F1/2. (694

In these expressions use has been made of the identity 2 F ; p = 3 F112,

(70)

and all integrals are evaluated at argument (60 So far, we have been following closely the arguments given by Keyes” in his discussion on n-type germanium. At this stage, Keyes proceeded by using the expression (61) to evaluate the Am‘” and hence the various averages in terms of the strain components. For the particular case of valleys on

5.

ELASTIC PROPERTIES

103

[ l l l ] axes, the result emerges that the only second-order elastic constant to be affected is c44. The change in c44 is given by 6c4,/c44

= - NEU2F;/2/9c44kTF1/2.

(71)

Graphs from which 6c,,/c4, can be obtained as a function of N and Tare given for n-type germanium by Keyes. In general the effect is of the order of a few percent in the range of carrier concentrations of 10’’ to 1019~ m - ~ . The third-order terms were also investigated for n-type germanium by Keyes. The conclusion was that the only third-order constant that would be affected was c456. However, in contrast to the second-order case, it was predicted that this parameter was extremely sensitive to the concentration of carriers, and, for example, its value could be an order of magnitude different to the value for a pure specimen, when the carrier concentration was loi9 ~ m - ~ . A similar argument can be readily carried out for valleys on [lo01 axes. For this situation, typified by n-type silicon, it is readily found that no shear strain components enter into the expressions and ~-

Am2 - Aw’

=

2Eu2{(1;1

+ 1i2 + ~ i 3 )

- ( 1 1 1 1 2 2 + q 2 2 1 3 3 -t 433111)}/9(kT)~.

(72)

This indicates that the only second-order elastic moduli affected are c l l and c12.The changes again would be a few percent for carrier concentrations 10”. Einspruch and Csavinszky” have reported results for n-type silicon which are in good agreement with theory. The third-order constants to be affected would be c l l l , c l l z , and ~ 1 2 and, on the basis of the above arguments, these would be extremely sensitive to the carrier concentration. While the above procedure is the best that can be done with existing knowledge and probably gives the right order of magnitude, it appears to the present author that it is not logically correct. If expressions for the free energy to the third order in the strain are required, the correct procedure would be to express the shift in the valley energies to the same degree of approximation and to replace Eq. (61) by the more general condition

-

A’“’

= zpqqpq

-

+ z p q r s q p q q r s + =pqrstuqpqtlrstltu

(73)

in which higher-order deformation potential tensors occur. The use of this equation would lead to the appearance of other terms in the free energy expressed in powers of the strain with a consequent modification of the expressions for the changes in the elastic constants.

’* N. G . Einspruch and P. Csavinszky, A p p l . Phys. Lefters 2, 1 (1963).

3

104

J . R. DRABBLE

In the absence of any information about higher-order deformation potentials and the scarcity of experimental evidence, it does not appear to be worth while to speculate on this too much in this chapter. The theory is simple to set up, and a considerable amount of experimental work will be required to sort out the parameters involved. The discussion so far has been confined mainly to the simple m a n y valley model, where the various minima in the Brillouin zone are related by crystal symmetry operations. We may now turn to consideration of the actual band structures of the 111-V compounds to see to what extent the elastic properties are likely to be affected by electron concentration. The compounds InSb, JnAs, and InP all have a single conduction band minimum at k = 0. Magnetoresistance and piezoresistance measurements have shown no indication of any other close-lying conduction band minima. Thus, for these compounds, the primary effect will be that the whole band moves with strain and there is no effective redistribution of the electrons. In these materials, therefore, any changes in the elastic constants with electro:. concentration are likely to be very small. It is perhaps possible that the strain may affect the shape of the band to the second or third order in the strain and give rise to some small effect. Gallium arsenide has a principal minimum at k = 0, and evidence has been found for the presence of [lW] minima lying 0.36 eV in energy above this. This separation is such that most of the electrons are in the [OOOl minima and will remain so in the presence of strain. Again, the influence of electron concentration on the elastic constants is likely to be negligibly small. Gallium antimonide, while possessing a k = 0 minimum, has Subsidiary minima in [ l l l ] directions lying only about 0.09 eV in energy above the [OW] minimum. For this material some influence on the elastic constants is to be expected, and according to the discussion given previously this would manifest itself principally in the constants c44 and c456. The effect would not be as large as for the comparable case of n-type germanium because the presence of the [OW] minimum would tend to freeze out some of the electrons. The remaining two materials in Table IV, gallium phosphide and aluminum antimonide, appear to be the ones which might show the largest effects. The conduction band structure here has the lowest minima along [lo01 directions with a subsidiary [OOO] minimum some 0.4 eV above. The latter is too far away to play much part in the electron redistribution under strain, and the theory would be similar to that for n-type silicon, the main effects being in c l l , c12, c l l l , c l I 2 ,~ 1 2 3 .

5.

ELASTIC PROPERTIES

105

c. Degenerate Valence B a d Efects The principles involved in estimating the effect of holes on the elastic constants are the same as those discussed previously. The calculations are, however, rendered more difficult as a result of the complex form of the valence band structure. In principle, there are three bands involved. Two of these, the heavy and light hole bands, are degenerate at k = 0, and the third band is split off from these by the spin-orbit splitting A. The bands are known to be nonparabolic and to be warped. TABLE V SPIN-ORBIT SPLITTING FACTORA Compound

4eV)

InSb InAs InP GaSb GaAs GaP AlSb

0.90 0.43 0.28 0.86 0.33 0.12 0.78

Discussions of the effects to be expected have been given by several authors with special reference to germanium and silicon. Keyes” restricted the discussion to the heavy hole band only and concluded that the effect of hole concentration on an elastic shear constant would be about two orders of magnitude less than for n-type material with the same carrier concentration. Bir and T u r s ~ n o vtook ~ ~ both heavy and light hole bands into account and also the warped nature of these bands. They showed that the influence of the split-off band could be neglected for p-type germanium provided the carrier concentration was less than lozoC I I - ~ . The value of A for germanium is approximately 0.30 eV. Table V gives the values of A for the 111-V cornpo~nds.~’For a more detailed discussion, consult the chapter by Cardona in Vol. 3 of this series. With the possible exception of gallium phosphide, it appears that the Bir-Tursunov theory should be applicable to most of these compounds. Csavinszky and E i n s p r ~ c hhave ~ ~ considered the theory for p-type silicon (A = 0.044 eV), taking all three bands into account but neglecting warping and nonparabolicity. They obtained experimental results which were interpreted in terms of deformation potentials for the bands. An experimental effect has also been reported by Mason and Baternan.j’ G. L. Bir and A. Tursunov, Fiz. Tverd. Tela 4, 2625 (1962) [English Trans: Sooief Phys.Solid State 4, 1925 (1963)l. 30 P. Csavinszky and N. G. Einspruch, Phys. Rev. 132, 2434 (1963). 3 1 W. P. Mason and T. B. Bateman, Phys. Rev. Letters 10, 151 (1963). 29

106

J. R. DRABBLE

The theory of Bir and Tursunov will be outlined here since it should be applicable to most of the 111-V compounds. They used the expression for the energy of holes in a deformed crystal given by Pikus and Bir :32

E = Ak2

(Ek

+ E q k + J!$)'/~,

(74)

where Ek Eqk

=

+ k,Zkz2 + kz2kx2),

B2k4 + (0' - 3B2)(kx2k,Z

=

+ r/22ky2 + 1 3 3 k z 2 ) + 2Dd{?12kxky + 1 2 3 k y k z + ?31krkx}? - 12212 + ( 1 2 2 - 1 3 3 ) 2 + ( 1 3 3 - 1 1 1 ) 2 ) + d " d 2 + 1& + 1:lh

3Bb{?llkx2

E, = +b2{(?ll

A, B, and D are constants which characterize the band structure of the undeformed crystal and b and d have the nature of deformation potential constants for holes. The effect of the strain is to alter the effective density of states in both the light and heavy hole bands and also to cause these bands to separate from each other. The calculation then proceeds along the standard lines. The results, expressed in a form directly comparable with Eq. (71), are

where

B2 = B2 + (D2 - 3B2)/5 and

m1 and m2 being the effective masses of light and heavy holes. In contrast to the case of a simple many-valley model, all the secondorder elastic constants are affected. The extension to the third-order constants has not been considered in detail. However, subject to the same restriction that was noted for n-type material, viz., the validity of Eq. (74) E. Pikus and G. L. Bir, Fiz. Tuerd. Tela 1, 1642 (1959) [English Trans.; Soviet Phys.Solid State 1, 1502 (1960)l.

32 G.

5.

ELASTIC PROPERTIES

107

up to cubic terms in the strain, it would be anticipated that the effects would be large, since in the theory, as pointed out by Keyes, the expansion is one in which the strain is multiplied by the ratio of a deformation potential to the Fermi energy. This ratio is large, and hence one obtains a large coefficient of the third-order terms leading to large changes in the thirdorder constants. Looking through the series of 111-V compounds, it is to be anticipated that their elastic constants will be affected by hole concentration to an extent depending on the relevant band parameters and the unknown deformation potentials. 10. ELECTRON-PHONON INTERACTION EFFECTS

In the discussion so far it has been implicitly assumed that the electrons and the lattice can be regarded as being independent of each other for the purpose of determining the elastic properties. It has been pointed out however, by K h a ~ h a t u r y a nfollowing ,~~ earlier work by Bon~h-Bruevich,~~ that the electron-phonon interaction may also play a part. The calculations of these authors indicate that the dispersion law for phonons in semiconductors is sensitive to the carrier concentration. This result clearly has a bearing on the measurement of the elastic constants. The theoretical treatment starts with a Hamiltonian function X p+ X e , representing, respectively, the Hamiltonians for the phonon and electron systems in the absence of any electron-phonon interaction. In particular X P represents the phonon spectrum in the absence of any electrons. The introduction of an interaction term produces a perturbation which leads to a modification and renormalization of the phonon spectrum (and also, of course, of the electron spectrum). The results of the calculation of this effect by Khachaturyan were expressed in the form (77) &k,v = Ok,v + Ak,v, where 1 5is~the~ “renormalized” ~ frequency in the presence of electronphonon interaction of a phonon with wave vector k in the vth branch of the spectrum (only acoustic branches were considered) and Ok,v is the corresponding unperturbed frequency. The perturbation AkSvwas evaluated for various directions of k for various branches in the limit of small k values for a cubic crystal possessing a band structure corresponding to the conduction band of germanium, i.e., with energy minima on the [ l l l ] axes of the Brillouin zone. For a nondegenerate electron gas, this had the form A. G. Khachaturyan, Fiz. Tverd. Telu 3, 2540 (1961)[English Transl.: Soviet Phys.-Solid State 3, 1848 (1962). 34 V . L. Bonch-Bruevich, Fir. Tverd. Tela 2, 1857 (1960)[English Transl.: Soviet Phys.-Solid State 2, 1678 (1961)l. 33

108

J. R. DRABBLE

where K is Boltzmann’s constant, T is the absolute temperature, p is the density, and n is the average concentration of electrons. The expression is a summation over the various energy minima, specified by a, of a function which depends on the deformation potentials for the minima and on the wavelength and polarization of the phonon. For example, for a longitudinal phonon with a wave vector in the [lo01 direction

Xalfrlz

c lfFIZ

=

8A2J;;k/J.ll.

(79)

a

Here A is a deformation potential defined by A = E, + Ed3 [Eq. (62)], and k is the magnitude of the phonon wave vector. For other directions and branches, similar expressions are obtained, all of which are proportional to p’” and to k but containing different deformation potentials and elastic constant combinations. For small wavelengths, the expression for the appropriate sound velocity change would be Ak,”/k. Using values typical of germanium it was estimated by Khachaturyan that the change for a longitudinal sound wave propagating in the [lo01 direction would be of the order of a few percent at room temperature for a carrier concentration of lo” ~ m - It~ should . be noted, however, that the deformation potential combination for this mode is not known accurately. It is difficult to predict the magnitude of this effect in the III-V compounds since it depends on the deformation potentials and the band structure. It is to be presumed that it will be present in both n- and p-type materials. It may be noted that the effect is of a very similar form to that considered in Section 9, b, Eq. (71). The main difference is that the latter effect is, in principle, confined to particular elastic constants, for example, c44 in n-type germanium, whereas the electron-phonon interaction effect affects all the elastic constants. It is perhaps relevant to note that Bruner and Keyes3’ have compared tl,e second-order elastic constants of pure germanium and of heavily doped germanium containing 3.5 x 10 arsenic donors cmP3. They observed a change in ca4 which could be attributed to the electronic effect but no significant changes were observed in cl1 and clz. This result seems to be contrary to the predictions of Khachaturyan’s the0ry.3~” L. J. Bruner and R. W. Keyes, Phys. Reo. Letters 7 , 55 (1961). Fomin has shown recently35bthat there was an error in Khachaturyan’s paper and that the: expected effect is much smaller than he predicted. 35bN. V. Fomin, Fiz. Tverd.Tela 6,3469 (1964) [English Trans/.:Sooiet Phys.-Solid State 6 , 2774 (1965)]. 35

35a

5.

109

ELASTIC PROPERTIES

V. Experimental Results 1 1 . GENERAL SURVEY

A search through the literature has revealed only a few measurements of the actual values of the elastic constants of 111-V compounds. None of the investigations has attempted anything but a direct measurement of the second-order constants.35cThus, none of the effects discussed in the preceding sections has been considered. This, of course, is not particularly surprising since the control of crystal composition and structure in these compounds has not yet reached the same high level in most cases as that for germanium and silicon. This may possibly account for some of the discrepancies in the reported values of the elastic constants. It may be noted that all measurements have been made at frequencies at which any of the effects discussed in Section 6 would be expected to be negligible. TABLE V1 MEASUREMENTS OF SECOND-ORDER ELASTIC CONSTANTS

Compound

Technique

Temperature

Ref.

(OK) InSb

71-673 77-300 4.2-300 300

InAs

4.2-300

Mechanical resonance Ultrasonic phase contrast up to 175 Mc/sec Ultrasonic pulse echo; 10 Mc/sec Ultrasonic pulse echo; 5 Mc/sec

36 37 38 39 40,44a

GaSb

300 300

Ultrasonic phase contrast, 50 Mc/sec Ultrasonic resonance, 10 and 30 Mc/sec

41 42

GaAs

298 77-300

Ultrasonic phase contrast, 6C180 Mc/sec Ultrasonic pulse echo; 10 Mc/sec

43 44

AlSb

300

Ultrasonic resonance; 10 Mc/sec

42

Table VI summarizes the information on the measurement of the secondorder elastic constants of 111-V compounds which has been found in the literature. 12. ROOMTEMPERATURE VALUES

OF THE SECOND-ORDER CONSTANTS

Table VII summarizes the information on the room temperature values of the three second-order elastic constants. In each case, the figures have been quoted as they appear in the original reference. It should again be 35c

A recent investigation by A. J. Brammer in the author’s laboratory has led to a value of the third order constant c456 of GaAs of -0.36 +_ 0.09 in units of 10’’ dynes/cmz.

110

J . R . DRABBLE

remarked however that, in the absence of any information about composition, it would not be surprising to find future measurements disagreeing with those given in Table VII by a few percent.

VALUES

OF

Compound

TABLE VII SECOND-ORDER ELASTICCONSTANTS AT 300°K" c1 I

c12

c44

Ref.

3.07 1 3.018 3.020 3.0

36 37 38 39

InSb

6.472 6.717 6.669 6.6

3.265 3.665 3.645 3.8

InAs

8.329

4.526

3.959

40

GaSb

8.849 8.840

4.037

4.325 4.319

41 42

GaAs

11.88 11.81

5.38 5.32

5.94 5.94

43 44

AlSb

8.94

4.42

4.15

42

"In units of 10" dyne cmT2.

13. TEMPERATURE DEPENDENCE OF THE ELASTIC CONSTANTS

Values of the elastic constants at temperatures other than room temperature have been reported for only two of the 111-V compounds, indium antimonide and gallium arsenide. Indium antimonide has been investigated on three occasions. The dependence of the elastic constants on temperature as reported in Refs. 36: 37, and 38 is shown in Figs. 1, 2, and 3 for cli, ci2,and c44, respectively. The results show some divergence as between different workers. This may be due in part to differences in composition and quality of the specimens used. It may also be associated with the result reported by McSkiminj' that there was an appreciable variation of velocity with frequency below about 100 Mc/sec. This was attributed to diffraction and bond effects. It is a well-known thermodynamic result" that the slope of the curves of the elastic constant versus temperature should approach zero at the absolute zero of temperature. This is observed in the results of Slutsky R. F. Potter, Phys. Rev. 103, 47 (1956). H. J. McSkimin, IRE Trans. Ultrasonics Eng. PGUE 5, 25 (1957). 38 L. J. Slutsky and C. W. Garland, Phys. Reo. 113, 167 (1959). 39 L. H. DeVaux and F. A. Pizzarello, Phys. Rev. 102, 85 (1956). 36

37

5.

ELASTIC PROPERTIES

111

Absolule temperature O K

FIG.1. c l l for indium antimonide vs temperature.

and Garland3' for c I 1 and cIz, but the trend is not very well defined for c44. Garland and Park44 measured the adiabatic elastic constants of gallium arsenide over the temperature range 77" to 300°K. The variation with temperature for each of the constants was small, and values were given at T = 0" (extrapolated), 77.4" and 300°K. These values are given in Table VIII. Recent data for indium arsenide at 4.2", 77.4", and 293°K were reported by G e r l i ~ h . ~ ~ " 14.

RELATIONS BETWEEN THE

ELASTICCONSTANTS

It has been shown by Keyes4' that the second-order elastic constants of the 111-V compounds fall into a fairly well defined pattern. On the basis 40 41

D. Gerlich, J . Appl. Phys. 34, 2915 (1963). H. J. McSkimin, W. L. Bond, G. L. Pearson, and H. J. Hrostowski, Bull. A m . Phys. SOC. 1, 111 (1956).

D. I. Bolef and M. Menes, J . Appl. Phys. 31, 1426 (1960). T. B. Bateman, H. J. McSkimin, and J. M. Whelan, J . Appl. Phys. 30, 544 (1959). 44 C. W. Garland and K. C. Park, J . Appl. Phys. 33, 759 (1962). 44a D. Gerlich, J . Appl. Phys. 35,3062 (1964). 45 R. W. Keyes, J . Appl. Phys. 33, 3371 (1962). 42

43

112

J . R . DRABBLE

TABLE VIII ELASTICCONSTANTS OF GALLIUM ARSENIDE AT DIFFERENT TEMPERATURES Temperature (“K)

c I l (10” dyne cm-*)

cI2

c44

0” 77.4 300

1.226 1.221 1.181

0.571 0.566 0.532

0.600 0.599 0.594

Extrapolated.

of a dimensional analysis an elastic constant co is defined for each compound by the relation c0 = q2/b4, where q is the electronic charge and b is the nearest neighbor distance:, given, in terms of the cubic lattice constant a, by b

= ,,ha/4.

It is then found that, to within a few percent, the reduced elastic constants 111-V compounds that have so far been measured. These reduced values are shown in Table IX.

c$ = cij/co have the same value for a given cij for all the

TABLE IX REDUCEDELASTIC CONSTANIX

GaAs InAs GaSb AlSb InSb

0.641 0.486 0.474 0.463 0.372

Average GaP InP

1.85 1.71 1.87 1.93 1.79

0.840 0.930 0.853 0.952 0.981

0.925 0.814 0.910 0.895 0.812

1.83

0.911

0.871

0.742 0.553

They differ slightly from those given by Keyes and later by G e r l i ~ h , ~ ~ presumably through the use of slightly different values of lattice constants and elastic constants. In the present case, values of co were obtained using the lattice constants determined by Giesecke and P f i ~ t e r . ~ ~ 46

G. Giesecke and H. Pfister, Acta Cryst. 11, 369 (1958).

113

FIG.2. c,* for indium antimonide vs temperature.

Absolute temperature O K

FIG.3. c+, for indium antimonide vs temperature.

114

J. R. DRABBLE

Inspection of Table IX shows that the departure from the average value is always small. Although there appears to be no basic physical reason governing this behavior it is probably justifiable to use the average values of the reduced elastic constants to provide an approximate estimate of the constants for gallium phosphide and indium phosphide which do not appear to have been measured so far. TABLE X VALUESOF 4c,,(cl, - c44)/(cll

Compound

+c,~)~

- c44)Acii + ~ 1 2 ) ’

4~11(~11

GaAs InAs GaSb AlSh InSh

0.95 0.88 0.98 0.96 0.85

Another feature worth noting is that the elastic constants of the 111-V compounds do not depart drastically from the relation given in Eq. (35). It will be recalled that this so-called Born relation is based on the very simplest atomistic model of nearest neighbor interaction. Table X shows the values of 4cl,(c,l - c44)/(cI1 c12)’. In all cases, the value is less than unity but by only a comparatively small amount. It is, however, not possible to attach too much significance to this result, since the Born model, with nearest neighbor interaction only, certainly does not lead to good agreement with experiment in other respects.

+

VI. Conclusion This chapter has attempted to summarize some of the possible effects which are likely to be encountered in the measurement of the elastic properties of the 111-V compounds. Attention has been focused on the velocity of acoustic waves traveling in the medium under particular conditions since this experimental technique offers the maximum precision, in the foreseeable future, for the determination of the effects. In view of the scarcity of experimental data for the 111-V compounds, considerable emphasis has been placed on the effects and results already obtained for the group IV semiconductors on the one hand and the 11-VI compounds on the other. Comparison with these of the effects in the 111-V compounds, when obtained, should certainly be of interest. In addition, however, it will be somewhat disappointing if studies of the 111-V compounds d o not reveal new effects. It is the author’s hope that this chapter will provide a basis for the discussion of such future measurements.

CHAPTER 6

Low Energy Electron Diffraction Studies A . U . Mac Rae and G. W. Gobeli I. INTRODUCTION. . . 1 . General Discussion . 2. Substrate Structure . 3. Surface Unit Mesh . 4. Analytical Techniques

. . . . . . . . . . . . . . . . . , . . . . . . . . . . . . 11. EXPERIMENTAL TECHNIQUES . . 5. Surface Preparation . . . . . . . 6 . Diffraetion Equipment . . . . . . 111. RESULTS. . . . . . . . . . . 7. Cteaaed (110) Surfaces . . . . . . 8 . Heat Treatment of Cleaved Surfaces . . 9. Ion Bombardment and Anneal . . . . 10. O.Yygen Adsorpiion . . . . . . IV. CONCLUSIONS..

. .

.

.

. . . . . . . . .

. .

. .

. .

.

. .

. . . . . . .

. . . . . . , . . . . . . . .

. . . . . . . . . ,

.

. .

.

. .

. . . . . . . . . . .

115 115 117 119

120 122 122 123

. .

. . . . . , . . , .

.

.

,

.

.

.

124 124 131 131 135

. . . . . . .

,

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.

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136

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.

,

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.

I. Introduction 1. GENERAL DISCUSSION

Many of the properties of semiconductors are influenced by phenomena whose origin is at the surface of the material. A number of reviews and conference proceedings have been published which discuss many aspects of semiconductor surfaces. The surface properties of germanium and silicon have received much more attention than those of the 111-V compounds, but during the past few years interest has broadened to include these photoelectric emission,4Ts field electron compounds. Gas

’ R. H. Kingston (ed.), “Semiconductor Surface Physics.” Univ. of Pennsylvania

Press, Philadelphia, Pennsylvania, 1957: J. W. Zernel (ed.),J . Phys. Chem. Solids 14,1960; R. H. Kingston, J . Appl. Phys. 27, 101 (1956); M. Lax, Proc. Inrvrn Coizf: Semicond. Phys., Prague, 1960 Czech. Acad. Sci., Prague, 1961; Conference on Clean Surfaces and Surface Phenomena i n Semiconductors, Ann. N . Y Acad. Sci. 101. 585 (1963). ’ D. Haneman, J . Phys. Chem. Solids 14. 162 (1960); Phys. Rev. 121, 1093 (1961). A. U. Mac Rae and G . W. Gobeli, J . Appl. Phys. 35. 1629 (1964). D. Haneman, J . Phys. Chem. Solids 11, 205 (1959). D. Haneman and E. W. J. Mitchell, J . Phys. Chem. Solids 15, 82 (1960). G. W. Gobeli and F. G. Allen, this book. Chap. 11.

115

116

A. U. MAC RAE AND G . W . GOBELI

emission,6 work functi0n,4,~and s t r ~ c t u r e ~ , ~ studies .’ of the atomically clean surfaces have received the most attention. The mathematical treatment of solids is based to a large extent on a knowledge of the arrangement of atoms in crystals. It seems only logical that a quantitative understanding of surfaces would require a knowledge of surface atom arrangements. Certainly the chemical reactivity, the energy level scheme, and the wave functions at the surface should be based in part on the symmetry and spatial arrangement of these atoms. For this reason, low energy electron diffraction (hereafter, abbreviated as LEED) has attracted considerable interest, especially in the past few years. Electrons having an energy of 5 to 500eV are ideal for surface structure determinations, since most of the interaction with a crystal is confined to the first two or three layers. This is just the region that is normally considered as the surface. X-rays and the conventional 50-keV electrons penetrate too deeply into a crystal to be effective for work of this type. The advent of recently perfected LEED equipment, the relative ease of obtaining pressures in the ultra-high vacuum range (lo-’’ Torr), the availability of pure single crystals, and improved surface cleaning techniques have been factors in the recent resurgence of interest in surface structures. There have been several intensive LEED studies of germanium and silicon surfaces.899In contrast, the study of the 111-V compound surfaces is still in an early stage of development. Two groups of workers have made the main contributions to this field. Haneman, when at Brown University, investigated the (111) and (Ui) surfaces of GaSb and InSb and the (100) surfaces of InSb.2*7His results indicate that there is a rearrangement of the positions of the atoms at these surfaces. Mac Rae and Gobeli at Bell Telephone Laboratories have investigated the { 1lo} surfaces of InSb, InAs, GaSb and G ~ A s and , ~ AlSb.’’ Their results indicate that the arrangement of atoms on this surface plane is quite similar to that of atoms on (110) planes in the bulk of the crystals. These workers have also made preliminary surfaces of GaAs.” The results of the investistudies of the (111) and gations of these surfaces as well as an outline of the type of analysis required for an interpretation of LEED for compounds is presented here.

(ni)

W. R. Savage, J . Appl. Phys. 34,732 (1963).

’D. Haneman, “Compound Semiconductors,” p. 423, (R. K. Willardson and H. L. Goering eds.). Reinhold, New York, 1962; Proc. Intern. ConJ Setnicond. Phys., Prague, 1960 p. 540. Czech. Acad. Sci., Prague, 1961. * R. E. Schlier and H. E. Farnsworth, J . Chem. Phys. 30, 91 7 (1959); R. E. Schlier and H. E. Farnsworth, in “Semiconductor Surface Physics,” p. 3. Univ. of Pennsylvania Press, Philadelphia, Pennsylvania, 1960. J. J. Lander and J. Morrison, J . Chem Phys. 37, 729 (1962); J. Appl. Phys. 34, 1403 (1963). l o A. U. Mac Rae and G. W. Gobeli, unpublished. A. U. Mac Rae, “Surface Science,” 1966 (in press).

6.

117

LOW ENERGY ELECTRON DIFFRACTION STUDIES

2. SUBSTRATE STRUCTURE

The 111-V compounds crystallize in the zinc-blende structure, which consists of two interpenetrating face centered cubic sublattices. The origin of the first lattice, consisting of the group 111 elements, is at 000 and the This origin of the second, consisting of the group V elements, at structure is identical to that of the diamond lattice except that each group I11 atom has four tetrahedrally arranged group V neighbors and vice versa. The zinc-blende structure, unlike the diamond structure, does not possess a center of inversion, and opposite directions in the crystal are not necessarily equivalent. This leads to an interesting arrangement of the atoms in the {llO}, {loo), and (111) substrate planes. Models of these planes are shown in Fig. 1. These models are for the purpose of showing the symmetry and the arrangement of atoms in the substrate planes only and are not meant to represent the atomic configuration at the surface. The lattice constants and other pertinent properties of these materials are listed in Table I.

+a+.

TABLE I Lattice constant" Compound

40

lonicityh

(A)

GaAs GaSb InAs InSb AlSb

5.654 6.095 6.058 6.479 6.136

0.51 0.33 0.56 0.42 -

Melting point ("C)

Max. heat Est. anneal treatment temp. temp. (110) surfaces (1 10) surfaces ( "C) ( "C)

1238 706 942 530 1065

1000

600 600 450

700

550 600 500 450 ~

"G. Giesecke and H. Pfister, Acta. Cryst. 11, 369 (1958). hM. Hass and B. Henvis, J . Phys. Chrm. SolitlP 23, 1099 (1962).

The presence of a slight ionic component in the bonds (see Table I ) and the fact that there are equal numbers of group 111 and V atoms on the (110) planes result in the (110) cleavage of the 111-V semiconducting compounds." In contrast, germanium and silicon, which have the same structure, but with all atoms identical, cleave on { 11l} planes. For the { 1lo} substrate plane each atom has two of its tetrahedral bonds in the plane. A third bond is directed downward at an angle of 35"16' from the normal and the fourth is directed upward at the same angle. One consequence of this arrangement is that the crystal, when viewed in the [110] direction, has a right and left side, as can be seen in Fig. l(a). H. Pfister, Z . Naturforsch. lOa, 79 (1955).

118

A. U. MAC RAE AND G. W . GOBELI

FIG.1. Schematic drawings of the zinc-blende structure as viewed along three principal directions of the crystal, namely, (a) { 1lo}, (b) {loo}, (c) { 11I}. The shaded circles represent atoms of one kind, say the group A atoms, and the unshaded circles those of the other kind, say group B atoms. The relative sizes of the two types of circles are an indication of depth into the crystal; the small circles being situated in deeper layers. The substrate unit mesh, its dirnensions, and principal directions in the substrate planes are marked.

6.

LOW ENERGY ELECTRON DIFFRACTION STUDIES

119

The { 100) substrate planes are composed of alternate, equally spaced, layers containing group I11 or group V atoms [see Fig. l(b)]. Two bonds are directed upward at an angle of 54'44' to the normal, and the other two bonds are directed downward at the same angle. The { 11 1) planes are also composed of alternate layers of group 111 and group V atoms. For the model shown in Fig. l(c), three of the tetrahedral bonds are directed downward at an angle of 70'32' to the normal and the fourth bond is directed upward. As a result of this configuration, 111) surfaces of compounds having the zinc-blende structure can be prepared with either group 111 or group V atoms on the surface, since it is more probable for a series of single bonds to be broken than a series of triple bonds when a (111) surface is prepared. One would expect that these two classes of surfaces would exhibit striking dift'erences in their chemical, electrical, and mechanical proper tie^,'^ which is indeed true. The convention used here to describe the orientation of the two types of { I l l ) surfaces is based on the crystal structure designation of WycoH.14 In this way the [ill] direction is from a group 111 atom, located at 000, to a group V atom, located at The (1 11) plane is composed of group 111, or A, atoms and the plane is composed of group V, or B, atoms. To prevent confusion, an A or B will be used before the designation of the Unfortunately, no planes and directions in the manner A( 111) or €%(Iff). standard convention is used in the literature, although the majority of workers follow the above convention.

A$+. (in)

3. SURFACE UNITMESH For a given crystal plane the surface atom arrangement is not necessarily the same as the bulk atom configuration. The surface atomic arrangement is designated as the surface structure, and there is invariably a simple relationship between the periodic bulk atom configuration and the periodic surface structure. Normally it is found that surface structures have repeat distances that are equal to or are integral multiples of the dimensions of the substrate unit mesh. It is therefore convenient to describe the difl'raction patterns and the structures in terms of these substrate meshes. The mesh for the { 1 ll}, { 1001, and { 110)- surfaces is a 120" rhombus, a square, and a rectangle, respectively. These units are outlined on the models in Fig. 1. The type and dimensions of these meshes determine the positions of the difraction beams with integral order Miller indices hk. "J. F. Dewald. J . Electrochem. SOL.. 104, 244 (1957); J. D. Venables and R. M. Broudy, J . Appl. Phys. 29, 1025 (1958); M. C. Lavine, A. J. Rosenberg, and H. C. Gatos, J . Appl. Phys. 29, 1131 (1958). R. W. G. Wycoff, "Crystal Structures." 2nd ed.. VoL I, p. 108. Wiley, New York, 1963.

120

A . U. MAC RAE AND G . W . GOBELI

FIG.2. Schematic drawing of diffraction tube. Two grids have not been included.

If electrons are incident on a crystal, as shown in Fig. 2, back dif’racted beams will appear at angles that satisfy the plane grating formula i, = dhk)sin‘pi - sin q r ( .

(1)

Here, h and k are the two-dimensional Miller indices appropriate to the row spacings d,, of the meshes shown in Fig. 1, and the subscripts i and r refer to the angle that the incident and reflected beams make with the normal to the crystal. The wavelength 1. is determined by the deBroglie relationship and reduces to i = (150/V)1’2,with the wavelength in angstroms and I/ in volts. Generally the crystal inner potential must be considered when determining I/: When the surface structures have repeat distances that are integral multiples of the unit mesh dimensions, the diffraction beams from these structures can be designated by fractional order Miller indices by referring to the substrate unit mesh. Thus if dif’raction spots are observed at the 9, @, ti, etc., positions in the pattern from a [ 100)-oriented crystal surface, the surface mesh has twice the linear dimensions of the substrate unit mesh.

4. ANALYTICAL TECHNIQUES

The determination of surface structures from LEED data is in an embryonic stage, both from the standpoint of the amount of work done and also in the development of the analytical techniques. Fairly reliable nontrivial structures have been proposed on the basis of experimental data only

6.

LOW ENERGY ELECTRON DIFFRACTION STUDIES

121

in a few instances, notably for the clean {loo} and { 111) surfaces of silicon and g e r m a n i ~ m ’ ,and ~ for adsorbed species on Basically, surface structure analysis follows two steps. The first involves the determination of the dimensions of the surface mesh. This is done by observing the position of the ditfraction beams in the pattern. This step is usually trivial since the beams due to the known substrate structure provide a built-in calibration for additional beams from the surface structure. The surface mesh dimensions can be calculated from a knowledge of the positions of the beams originating from the surface by the plane grating formula, Eq. (1). The position and number of the diffraction beams reveal only the mesh dimensions. The actual arrangement of atoms in the mesh has to be determined from the analysis of the wavelength dependence of the intensity of the beams. Many problems are encountered in the analysis of intensity data. This is due, primarily, to the strong interaction of the incident low energy electrons with the atoms in the crystal. High absorption of the electron beam, multiple scattering, shadowing of one atom by another, a strong angular dependence of the scattering factor, and the existence of an inner potential are some of the factors that need to be considered in the analysis. The difficulties have been discussed in some detail by Lander and Morrison” for elemental crystals. The presence of two or more atomic species in the crystal, as for the 111-V compounds, introduces additional analytical complications. The atomic scattering factorf, is not the same for all atoms, and needs to be evaluated before precise structure determinations are possible. In addition, fi contains an imaginary part since the scattering process introduces a nonnegligible phase change between the incident and scattered beams. This phase change can be included in the expression for the single scattering structure factor n

Fhk =

1 fioAi exp 2ni([hx,+ kyi + pi]+ Z,aJ I=

(2)

1

at normal incidence, for the beam having Miller indices hk. The summation is over all atoms in the crystal that scatter appreciably,f,.’ is the real part of the scattering factor, A , is a transmission factor that includes both the amplitude of the incident beam on atom i as well as absorption of the scattered beam. x i and yi are the fractional atomic co-ordinates parallel to L. H. Germer. A. U. Mac Rae. and C. D. Hartman, J . Appl. Phys. 32, 2432 (1961): L. H. Germer and A. U. Mac Rae, J . Appl. Phys. 33. 2923 (1962). H. E. Farnsworth and .I. Tuul, J . Phys. Chem. Solids 9,48 (1959); H. E. Farnsworth and H. H. Madden, Jr., J . Appl. Phys. 32, 1933 (1961). ” J. J. Lander and J. Morrison, J . Appl. Phps. 34, 3517 (1963).

l5

122

A . U. MAC RAE A N D G . W. GOBELI

the substrate plane, pi includes the imaginary part off,, Z , is the absolute atomic coordinate normal to the surface, and a,which contains the wavelength as a variable, is equal to (1 cos cp,)/A. The intensity, which is proportional to the square of Fhk,can be calculated as a function of voltage for a given structure and compared with the experimental results. In principle, it should be possible to obtain agreement between the calculated and experimental results for the correct surface structure. In actuality there are so many unknown factors that perfect agreement is rare. This is especially true for the compounds, sincef,’ and pi are not known very accurately. Some approximate values are tabulated in Ref. 17 for atoms in free space. The tables illustrate that bothf,’ and are strongly dependent on the type of atom, the initial electron energy, and the scattering angle. The lack of knowledge of the actual potential environment around the atoms as well as problems that are introduced in including factors such as polarization and exchange into the calculation make it difficult to obtain reliable values of these parameters. It may be possible to evaluate some of these factors experimentally by investigating the diffraction effects due to the same type of atom in many similar compounds. Since pi is simply additive to hxi k y , in the structural factor in Eq. (2), a structure obtained from an intensity analysis is not necessarily unique. There may be several values of pi and thus several structures with difTerent values of hx, ky, that yield agreement between the calculated and experimental results. The calculations of the intensity of the diffraction beams discussed here were performed with these limitations clearly in mind.

+

+

+

11. Experimental Techniques

5. SURFACE PREPARATION The clean surfaces used in these investigations were obtained either by cleavage or by ion bombardment and anneal. Mac Rae and Gobeli’ used the Gobeli-Allen’* cleavage technique for their studies of the { 110) surfaces. This technique utilizes an “L” shaped sample and produces a surface of unusual perfection. Examination of the cleaved { l l O } surfaces with an optical microscope of 1000 x magnification failed to resolve any imperfections in the best cleavage areas. Electron microscope examinations were made on direct cast replicas, with a limit of resolution of 30 to 40A. The cleaved surfaces showed some terraced structure, with the flat areas of 3000 to 10,OOOA in extent being separated by steps of 50 to 200A in height. No cracks or other indications of disorder were detected and only very G. W . Gobeli and F. G. Allen, J . Phys. Chem. Solids 14. 23 (1960).

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minor areas of brittle fracture, as distinct from cleavage, were found in an area of some of the cleaved samples. Unfortunately, surfaces produced by cleavage are restricted to the { 1lo} faces of the 111-V compounds. Other crystal planes must be prepared by careful mechanical and chemical techniques. It is necessary to cut the crystal to within a fraction of a degree of the desired orientation, polish the resultant surfaces with successively finer grinding material, and then etch the surfaces. Surfaces prepared in this manner are not clean, and a method of cleaning in the experimental apparatus is necessary. Since heating alone rarely yields a clean surface, it is usually necessary to resort to ion bombardment with 100- to 200-eV argon ions followed by an anneal at an appropriate temperature.” It is not at all obvious that a surface produced in this manner is clean however, especially in the case of these materials. It is entirely possible that preferential sputtering of one of the atomic constituents can occur, leaving behind an excess of the other on the surface. The ion bombardment-anneal procedure has been applied to the cleavage surfaces of Bi,Te,,*’ Ge and Si,” and the 111-V c o m p o ~ n d s ,and ~ these surfaces yield the same diffraction data as the cleaved ones if the proper annealing technique is used. Thus ion bombardment and anneal is probably a valid surface cleaning technique when certain precautions are exercised. The diffraction results on the noncleavage surfaces tend to support this viewpoint. If an excessive preferential sputtering of one component occurred, one might expect to see dilt’raction beams characteristic of the elemental structure of the other component, which is contrary to the experimental results. Of course the annealing, if done at too high a temperature, can cause an out-diffusion of impurities from the bulk of the crystal to the surface or the migration of impurity atoms to the surface from the crystal supports. Apparatus design and experimental procedures must be such as to minimize these problems. 6. DIFFRACTION EQUIPMENT

Two types of diftiaction equipment are currently in use. The first, which in many respects is similar to that used in the historic Davisson-Germer experiment of 1927, utilizes a Faraday cage to detect the difl‘racted electrons. While this type is useful for absolute intensity measurements, it suffers the drawback that it is extremely time consuming to determine an entire dift‘raction pattern by moving the Faraday cage over the complete 2n solid angle subtended by the pattern. As a result, only the diffraction beams along H. E. Farnsworth, R. E. Schlier, M. George, and R. M. Burger, J . Appl. Phps. 26, 252 (1955). D. Haneman, Phys. Rev. 119, 563 (1960). 2 1 J. J. Lander, G. W. Gobeli, and J. Morrison, J. Appl. Phys. 34, 2298 (1963).

*’

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A . U. MAC RAE AND G. W . GOBELI

principal crystallographic directions are usually detected. Thus details in the pattern may be missed completely. The second technique, originally developed in 1934 by Ehrenberg22and recently perfected at the Bell Telephone L a b ~ r a t o r i e s is , ~currently ~ in use by the majority of the LEED groups. The diffraction patterns are displayed directly on a fluorescent screen by post-accelerating the diffracted electrons after they have passed through a series of fine mesh grids, which suppress the unwanted inelastically scattered electrons. This technique offers the advantage that the entire diffraction pattern is displayed and that changes in any part of the pattern can be monitored continuously. Intensity measurements, which must be calibrated by a separate current measurement, can be made by determining the light output of a particular spot on the screen with a photometer containing appropriate optics. Haneman's results were obtained using the Faraday box detection scheme, while those of Mac Rae and Gobeli were obtained in a postacceleration tube. The tube, used by the present authors, is mounted on a conventional bakable ultra-high vacuum system, which uses commercial sputter-ion pumps to reach a pressure of 2-4 x lo-'' Torr. At this pressure a clean surface can usually be maintained for the time necessary to complete the desired measurements.

In. Results 7. CLEAVED { 1lo} SURFACES

The diff'raction patterns obtained from L.e cleave1 (110) surfaces of InSb, InAs, GaAs, GaSb, and AlSb contained only spots that could be indexed with integral two-dimensional Miller in dice^.^.'^ This implies that the repeat distances of the two-dimensional unit mesh on these surfaces are identical to those of the parallel substrate unit mesh of the appropriate material. No spots having fractional Miller indices were detected in the 5- to 400-eV energy range. It is estimated that such spots would have been seen if their intensities were 2 0.1 % of the average intensities of the integral order beams. A reconstruction of this surface is insignificant for these materials. At most there can be only a slight displacement in the positions of the surface atoms on the 1lo} face. Photographs of typical diffraction patterns from the (110) surfaces of these materials are shown in Fig. 3. These patterns were observed within "W. 23

Ehrenberg, Phil. Mag. 18, 878 (1934). E. J. Scheibner, L. H. Germer, and C D. Hartman, Rev. Sci. Instr. 31, 112 (1960); L. H. Germer and C. D. Hartman, ibid. 31, 784 (1960); J. J. Lander, J. Morrison, and F. G. Unterwald, ibid. 33, 782 (1962).

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FIG.3. Photographs of diffraction patterns from the { 110) cleaved surface of (a) InSb at 38 V, (b) InSb at 100 V, (c) InAs at 100 V, (d) GaAs at IOOV, and (e) GaSb at 72 V.

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A. U. MAC RAE AND G . W. GOBELI

one minute of the actual cleavage, and they all have the same general appearance. Measurements on the photographs confirmed that these spots were produced by a unit mesh having dimension a 0 l a x a. where a, is the appropriate lattice constant. Note the extreme sharpness and brightness of the spots relative to the background, which is indicative of a surface of unusual perfection. The bright spots in the pictures are broadened by overexposure of the photographic film. There are two factors that immediately suggest that the atomic configuration of the group 111 and group V atoms at the (110) surface of these compounds is not the same as that found in the substrate of the crystal. They are the existence of the extreme asymmetry in the intensity of hk and hk beams and the strong intensity of the 10 and TO beams from both InSb and GaAs. If there were no displacement in the positions of the atoms in these materials and if their scattering factors were quite similar, which one might expect for atoms of comparable atomic number, the 10 and TO spots would be extremely weak, since the scattering from the rows of atoms defined by the A and B atoms would be out of phase. The 10 beam thus must be mainly the result of a change in the surface structure while the 01 and 01 beams contain effects due to both the surface and substrate structures. The asymmetry in the intensity of the hk and hk beams also indicates that the group 111 and group V atoms in the surface layer are displaced with respect to each other either in a horizontal or vertical direction. For this to occur, however, it is necessary to maintain the dimensions of the unit mesh of the bulk plane. Any ordered displacement of the top layer of atoms from their equilibrium positions which did not preserve this unit mesh would produce a structure with repeat distance of some integral multiple of the { 1lo} unit mesh. Thereby “fractional” order spots would be produced in the diffraction patterns corresponding to the ndhk repeat distance of the surface structure, but they were not observed. In addition to the difficulties outlined in a previous section on structure determination, the analysis of these results was complicated by the lack of fractional order spots. While this may appear to be an advantage because the possible positions of the atoms are minimized, it is actually a great disadvantage since the information about both the surface and bulk structures are contained in the intensities of the same beam in all cases. It is not at all obvious how the bulk and surface contributions can be separated to yield surface structures. With these limitations in mind, we proceeded to analyze the intensity vs voltage curves in an attempt to determine the surface structure or to at least gain some insight into the possible arrangements of the surface atoms. Figures 4, 5, and 6 contain a few of the curves that the authors obtained from these four compounds. These curves were taken with the electron beam

6. LOW ENERGY

ELECTRON DIFFRACTION STUDIES

127

at normal incidence since a slight departure from this condition resulted in large changes in the intensity vs voltage curves for all the beams. The strong asymmetry in the intensities of the 01 and 01 beams is clearly seen in these figures. Note that very few of the intensity maxima of these beams coincide, which would be the case if all the atoms were identical. 35,

140,

loot

0

II

100 VOLTAGE d

200

FIG.4. Intensity vs voltage from the cleaved { 110) surface of InSb at normal incidence for the (a) 10 beam, (b) 20 beam. and (3) 01 beam, solid curve and Of beam, dashed curve.

The hypothetical positions of the peaks are easily obtained under the assumption that there are no distortions of the surface layer and no differences in the phases of the atomic scattering functions. Maxima would occur whenever the factor within the brackets of Eq. (3)is equal to an integer, i.e., whenever [ h x , + k y , + (1 + cos cp)Z,/i] = n, with Zi= ao/2$ the spacing between the layers of the substrate. Eliminating the angle cp by the

128

A. U. MAC RAE AND G . W . GOBELI

plane grating formula i. = dhksin cp, yields the relationship for the wavelength necessary to produce a peak as

where the factor N

=

n

+ 3 whenever the sum h + k of the Miller indices of

t

30 > 25

t-

y,

z

20

: z

15 10

5

0 35

I

0

100 VOLTAGE

2 00

FIG.5. Intensity vs voltage for the 01 and Oi beams from the cleaved { l l O } surfaces of (a) InAs, (b) GaSb, and (c) GaAs. Solid curve is for the 01 beam and dashed curve for the OT beam.

the beam is odd and N = n whenever the sum h + k is even. The integer n identifies the successive maxima. The positions of these maxima, as calculated by Eq. (3), are marked on the curves of Figs. 4,5,and 6 by arrows. In some instances the peaks are specified quite closely by the calculated values, if an inner potential of about 14V is assumed. This agreement is

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fair for the 01 and 01 beams but almost nonexistent for the 10 beam. The numerous other curves that we obtained also exhibit good agreement for some peaks, with an inner potential of approximately 14 V, but poor agreement for others. An examination of these results does not indicate whether the lack of agreement for the diff'erent beams is due to the dilferences in phase changes between the two atoms or due to a change in the structure at the surface. Interaction of the incident electrons with the substrate 35 r

25

(C 1 20 -

"i

J

10 O

0

L

100 VOLTAGE

-

FIG. 6. Intensity vs voltage for the 10 bean? from (a) InAs, (b) GaSb, and (c) GaAs.

structure will produce peaks, in the 01 and 01 beams the positions of which are in agreement with those calculated by Eq. (3) and are indicated by the arrows. Since the undistorted substrate structure, assuming identical scattering factors for all the'atoms, would not produce a 10 beam, the best place to start in the determination of the surface structure is with the 10 beam. Crystal structure analysis consists of finding a structure whose calculated

130

A . U. MAC RAE AND G . W . GOBELI

diffraction spectra match the observed set. This involves setting up a structure and then making a calculation of the intensity vs voltage using this structure. Various structures were assumed and then these curves were calculated using an analog computer designed especially for this purpose. One would expect that there would be no displacement of the surface atoms parallel to the [TlO] direction on the surface. Such a displacement would probably result in fractional order spots, but none were detected. The existence of domains on the surface, where the top layer atoms were shifted to the [I101 direction in some areas and to the [liO] direction in other areas, does not appear likely. A displacement parallel to the [Ool] direction and/or normal to the surface is conceivable, however. Gatos and Lavine have developed a model of the polar (111) faces of the 111-V compounds24that should be partially applicable to this problem. According to this model the B surface atoms have a dangling bond with two unshared electrons and thus retain the sp3 or tetrahedral configuration of the bulk. The A surface atoms, on the other hand, have no unbonded electrons and presumably try to acquire a sp2 configuration. The resultant configuration on the A face is probably intermediate between planar and tetrahedral due to the forces exerted by the next layer of atoms. A significant surface distortion should occur, however, and this has apparently been The bonding confirmed in a series of experiments on these { 11l} ~urfaces.’~ of atoms on these { 110) surfaces is similar in some respects to that found on the (111) substrate. This is illustrated in Fig. l(c) where the bonds between the atoms of one layer to those of the underlying layer are drawn. Both the A and B atoms are triply bonded. If the Gatos-Lavine model is appropriate to this surface, one might expect that the position of the A, or group 111 atom, might be depressed toward the substrate and also displaced toward the [OOl] direction in an attempt to attain the planar configuration. Various displacements in the coordinates of the atoms were introduced in the calculations. As was mentioned in a previous paragraph, these displacements need not lead to a unique structure since a variation in the difference between the phase changes is capable of yielding alternate structures that satisfy the data equally well. Calculations, which agree reasonably well with the data, were made with the following assumptions: (1) A vertical displacement of the topmost In atoms of 0.2a0/2$ toward the substrate and a vertical displacement of the Sb atoms 0.2a0/2$ away from their normal substrate position; (2) a difference in the phase change between the In and Sb atoms of (0.1)2n; (3) an intensity contribution from three layers and an inner potential of 14V. No attempt was made to introduce a correction that would account for the observed over-all decrease in the 24 25

H. C . Gatos and M. C . Lavine, J. Electrochem Soc. 107, 433 (1963). H. C. Gatos, private communication.

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intensity of the beams with increasing voltage. There is reasonable agreement between the positions of the calculated and observed intensity maxima. Calculations of this type, using difYerent parameters, were also made for GaAs, and similar agreement was obtained with the experimental results. The reliability of these structures is open to question due to the phase change problem and the possibility of multiple scattering. The phase change might be evaluated by obtaining intensity data as a function of the angle of incidence and trying to match it with the calculations. It is not such a simple matter, however, since the shadowing factor will have a strong dependence on the angle of incidence, and it would be difficult to separate these two effects. The main conclusion that can be reached as a result of this analysis is that the atoms at the { 110) surfaces of the 111-V compounds do not have the same configuration as the atoms in the substrate even though the dimensions of the surface and substrate unit meshes are equal and that dynamic diffraction efTects need to be considered in the analysis of the data. 8. HEATTREATMENT OF CLEAVED SLJRFACES

After the patterns were photographed and the intensity distributions were obtained, all the cleaved crystals were heat-treated at successively higher temperatures for 30-sec time periods to determine the effects on the diffraction patterns. The maximum temperatures employed were close to the melting points, and the estimated values are given in Table I. In no case was there any observable efl'ect due to the heat treatment on the diffraction pattern. The spot locations themselves as well as the intensity vs electron energy distributions were the same both before and after the heat treatment. This is in striking contrast to the same experiments which were performed on the {ill} cleavage faces of Ge and Si, where great changes in the pattern were observed.2' The obvious conclusion to be drawn from the result of these experiments is that no metastable configuration of the surface atoms exists after cleavage at room temperature in the IIILV compound surface, nor is there any appreciable degree of strain or disorder which can be relieved by heat treatment. 9. ION BOMBARDMENT AND ANNEAL a. { 1101 Surfaces

All samples, including the cleaved ones, were bombarded with 150-V argon ions for times and currents which were calculated to remove a minimum of 1000 layers of atoms from the surface (assuming a sputtering efficiency of 0.1, which also includes a geometrical factor). In all cases it

132

A. U. MAC RAE AND G . W. GOBELI

was observed that such treatments completely obliterated the diffraction patterns and that only a heavy background indicative of a disordered surface was observable. The samples were then again annealed at successively higher temperatures for 30-sec intervals. At the temperatures listed as anneal temperatures in Table I a large increase in the pressure was observed, which lasted several seconds and was presumably due to the removal of the argon that was embedded in the surface layers. Following this anneal the dirraction patterns were found to be regenerated both in the positions of the spots in the patterns and the spot intensity vs wavelength distribution. There was a slight decrease in the intensity of the spots, however, compared to the intensity of the spots obtained from the cleaved surface. This decrease was no greater than 30% in any of the cases investigated. Only a very minor improvement in the patterns was observed upon going 50°C above the anneal temperature, and none whatever was observed upon annealing at still higher temperatures. The InSb crystal was finally heated for 3 min at the maximum temperature with no noticeable change occurring in the pattern. The anneal of the last sample, GaSb, was interrupted abruptly at the instant an increase in the pressure was detected at the anneal temperature, and it was found that the pattern was only partially restored. A subsequent anneal for 30sec at this same temperature fully restored the pattern. This result suggests that the relieving of surface damage by annealing the clean but disordered surface above the anneal temperature is accomplished in very short times, of the order of seconds. Long time anneals, of the order of hours, may lead to changes in the diffraction pattern, which may possibly be due to contamination of the clean surface by diffusion of impurities from the bulk or by surface diFusion from adjacent unclean surfaces such as the crystal holder. Of course, long time anneals at temperatures appreciably below the anneal temperatures of Table I would be required to produce the clean surface and thus would yield a distinct long time improvement in the pattern.

-

-

b. GaAs (111) Preliminary LEED results from the A( 111) and B(Tfi) surfaces of GaAs have been obtained by the present authors." The measurements taken in this investigation were obtained from the two surfaces on the same sample. Crystals were cut in the form of rectangular bars, with the front surface having the (111) orientation and the back surface having the (TIT) orientation. In this way it was possible to view the diffraction patterns alternately from one surface and then the other by rotating the crystal 180" in the vacuum chamber. Such a crystal holder has been described elsewhere."

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This technique has the advantage that the patterns from both surfaces could be monitored as a function of identical heat treatment and gas exposure. The temperature of the crystal was increased by Joule heating. Initially some very weak diBraction beams were observed from the A( 1 1 1 ) surface and none from the B(TTT) surface of the crystal etched in a 1070 bromine in alcohol solution. I t was not possible to make any measurements on these patterns, however. Ion bombardment with 125-eV argon ions for a time corresponding to the removal of 10l8 atoms from each surface in one experiment and 1017 atoms in another experiment resulted in the complete annihilation of any dill'raction features. A gradual increase in the

-

FIG.7. Photograph of diffraction pattern from the A( I 1 1 ) surface of GaAs at 52 V. The intense spots are those expected from a { 11 11 substrate such as shown in Fig. I(c) while the rest, which are weaker, are due to the reconstructed surface. This type of structure is referred to as a (2 x 2 ) structure.

temperature to approximately 600°C, which is approximately the same temperature used in the anneal of the [ 110) surface, resulted in the appearance of a weak diffraction pattern from the A(111) surface but none from the B(TiT) surface. Increasing the temperature about 50°C resulted in an increase in the intensity of the hk spots and the appearance of weak streaks between the hk positions from the A( 111) surface. This additional heating also caused a weak pattern having f-order spots to appear on the B(TTT) surface. For one of the crystals, the experiment was terminated at this point at a pressure of 4 x l o - ' " Torr 15 h later, the pattern from the B(Tii) surface weakened to the point where it was possible to ascertain only integral order spots. The pattern from the A(111) surface was not aRected perceptibly, however. Presumably this was due to preferential adsorption of the residual gas in the vacuum system on the B(7i-l) surface. Additional heating of the crystals produced strong half-order patterns on the A(111) surface and weak integral order patterns on the B(TT1) surface. A photograph of the dif'raction pattern from the A(111) surface at this

134

A . U. MAC RAE AND G. W . GOBELI

state of development is shown in Fig. 7. These half-order spots indicate that the linear dimensions of the surface mesh are twice those of the substrate unit mesh shown in Fig. l(c). While these results should be considered as preliminary, they indicate that the diffraction patterns from the A( 111) and B(Tii) surfaces of GaAs are not identical. While it was possible to obtain intense patterns from the surface composed of Ga atoms it was never possible to obtain good patterns from the surface composed of As atoms. It also appears that the gas adsorption characteristics of these surfaces are dissimilar. Momentary heating to 800°C caused facet spots26 to appear in both patterns." These facet spots, whose angular position did not obey the plane grating formula for normal incidence on the (111) surfaces, signified the existence of well-developed microplanes that were inclined to the original surface. An analysis of the change in the positions of these spots with a change in the energy of the incident beam revealed that these facets were composed of (110) surfaces, the most stable surface arrangement of atoms for this material. In view of these results, extreme caution must be exercised in the heat treatment of ( 11l} surfaces of the 111-V compounds. { 111) faces of GaAs, on which (110) facets have been produced by excessive heating, exhibit kinetic energy distributions of electrons ejected by ions (Auger effect) that are significantly different from the ( 111) surfaces not subjected to temperatures above 800°C.'' We suggest that this may be a property that is common to all 111-V compound (111) surfaces and perhaps the {loo} surfaces as well.

c. InSb (111) Following ion bombardment, using 100-400 eV argon ions, and annealing at 350°C for several hours, Haneman obtained diffraction patterns from both the A(111) and B(Tn) surfaces of InSb that contained diffraction beams at the h/2 k/2 positions, with h and k being integers.' He found that it was not possible to obtain good patterns by heating alone in vacuum, without sputtering. Also, a temperature of 300°C was not sufficient to anneal out the damage introduced by the sputtering process. Annealing at 400"C, on the other hand, produced hillocks on the surfaces. It was found that the intensity of the nonintegral order or half-order spots was comparable to the intensity of the integral order spots. The intensity maxima of the beams from the A(111) surface did not occur at the same voltage as the intensity maxima from the B(m)surface. This is the only observed difference in the patterns. 26

A. U. Mac Rae, Science 139, 379 (1963). H. D. Hagstrum, private communication.

'' D. D. Pretzer and

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The presence of diffraction beams at the Oq, ff, $0, etc., positions in the patterns implies that the arrangement of atoms at the surface is not the same as that in the bulk and that the repeat distances, or dimensions, of the surface structure is twice that of the substrate unit mesh in both the [I211 and [if21 directions on the surface. It also implies that the sputtering and annealing procedure did not preferentially remove one of the constituent atoms to the point where the structure of indium (fcc, a, = 3.84A) or antimony (rhomb., uo = 4.51 A) alone was observed. Since the intensity vs voltage measurements were not given in this publication it is virtually impossible to arrive at a reliable surface structure on the basis of these data. The one conclusion that can be drawn about the structure is the size of the surface mesh. Haneman has proposed a model to account for the half-order beams that involves the raising and lowering of every other surface atom.’ Since it is highly speculative, it will not be discussed further. d. GaSb { l l l } Diffraction beams having half-order Miller indices were also observed in the diffraction patterns from both the A(111) and B(iT1) surfaces of GaSb cleaned by a 400-eV argon ion bombardment and an anneal at 350-400°C for 30 min.2*7The half-integral order beams had an intensity that was from one-fourth to one-half the intensity of the integral order beams. In addition, the patterns from the A(111) surface were somewhat more intense than those from the B(Tn)surface. The same conclusions that were drawn for the structures on the InSb { 1 1 l} surfaces apply to these surfaces. e. lnSb (100) Beams having Miller indices of h k / 2 and h/2 k, (h, k = 1 , 2 , 3 , . . ,) were observed from InSb { 100) surfaces cleaned by the ion bombardment and anneal technique.2 The half-order beams had an intensity that was of the order of 1/10 the intensity of the integral order beams. No beams having __ ii, . . indices were observed, however. Similar patterns have been ~ ~were ~ observed from the {loo} surfaces of germanium and ~ i l i c o nand interpreted in terms of a rectangular unit mesh at the surface.’

$4,.

10. OXYGENADSORPTION

The effects on the diffraction pattern of exposure of the cleaved-sputtered. ~ was established annealed { llO} surfaces to oxygen were also ~ t u d i e d It that the surfaces were extremely inert with respect to the adsorption of Torr for an exposure of 0.6 Torroxygen. An oxygen pressure of 1 x sec produced only a barely detectable increase in back-ground intensity

136

A. U. MAC RAE AND G . W . GOBELI

relative to the spot intensity and a slight decrease in the sharpness and clarity of the spots of InSb. Estimating that such changes could correspond to no more than a 10% coverage of the surface with adsorbed oxygen and using the usual definition of sticking coefficient, s, one obtains from the data that sI There is always the possibility that impurities in the oxygen are responsible for the slight decrease in intensity. In this case the sticking probability for oxygen actually would be less than this value. This low value of the sticking coefficient is accurate, of course, only if the oxygen were adsorbed on sites that would produce a change in the surface unit mesh or if it were adsorbed in an amorphous form. No changes in the intensity vs voltage curves or an increase in pressure due to heating the crystal were observed, however, indicating that this value of s < lo-’ is probably quite reliable for this process. All (110) surfaces were inert with respect to residual gases in the vacuum system except AlSb. The diffraction pattern from the (110) AlSb cleavage had completely disappeared after 450 min exposure to background gases at a pressure of I 1 x l o p 9 Torr. Assuming that a monolayer coverage would destroy the pattern implies that the sticking coefficient for the background gases of the apparatus on AlSb is greater than lo-’. This is orders of magnitude higher than for InSb, InAs, GaAs, and GaSb and is consistent with the fact that AlSb tarnishes more rapidly in room air than the other compounds. The adsorption of oxygen on all the clean surfaces of the 111-V compounds including the ( l l l ) , (TIT), and (100) caused a gradual decrease in the intensity of the diffraction beams7 No new patterns were observed to form indicating that the oxygen presumably was adsorbed on the surface in a random arrangement or that a polycrystalline oxide was formed. The and initial sticking probability of oxygen on the A(111), B(TTT), (110)3 of InSb was of the order of lo-’. The adsorption of oxygen on the than on the S b ( l l l ) Ga(ll1) surface proceeded at a faster rate (s of GaSb (s

-

-

IV. Conclusiom The outstanding feature of the results of the LEED studies of the

111-V compounds is the apparent relative stability of the {110} surfaces. This conclusion is based on a summary of the above results, which can be stated briefly as follows. The absence of fractional order spots in the diffraction patterns from the (110) surfaces indicates an absence of reconstruction in the positions of the surface atoms and shows that the dimensions of the surface atom mesh are the same as those of the parallel substrate mesh. Furthermore there is no observable reconstruction,

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relaxation, or rearrangement of the cleaved surfaces upon heating to the temperatures listed in Table I, such as might be expected if a more stable atomic configuration existed. This is in contrast to cleaved { 11 1j silicon and germanium surfaces which undergo extensive reconstruction upon heating to 700" and 450°K respectively. This stability of the {llO} surfaces is further implied by the gross formation of (110) facets on both the A(111) and B(Tn)GaAs surfaces upon heating to approximately SOO'C, a temperature so low that special care must be taken in preparing atomically clean surfaces having this orientation. It should also be noted that the same surface atom arrangements occur upon the { l l o } faces of the III-V compounds which are prepared either by cleavage or by the ion bombardment and anneal technique. This extreme stability is probably due to the slightly ionic character of the atoms that compose these compounds. The ( I 10) surface is that surface which has an equal number of group I11 and group V atoms, and thus tends to be the surface of minimum free charge. In contrast to these results, the A(111) and B(Tfl) surfaces of GaAs, InSb, and GaSb and the { l o o ) surface of InSb, the only other surfaces investigated, exhibit reconstruction in the positions of surface atoms in such a way as to produce a surface mesh that is larger than the bulk or parallel substrate mesh. In addition, the patterns from the A(111) and B ( i f l ) surfaces of GaAs are distinctly different, which clearly indicates that these surfaces are polarized by a preferential exposure of the gallium or arsenic atoms. This result is substantiated by the different behavior of the two types of surfaces upon identical heat treatments and the different adsorption characteristics upon exposure to the residual gas in the vacuum system. The state of theoretical and analytical understanding of the mechanisms of low energy electron difl'raction is so embryonic and uncertain (in the opinion of the authors) that any proposals as to the exact details of the surface atom arrangements for specific cases are too speculative to be of reasonable validity at this time. Therefore, details of the surface structure, especially for the surfaces having other than the (110) orientation, from existent LEED data remain for a more complete understanding of the physics of low energy electron diffraction.

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Magnetic Resonances

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CHAPTER 7

Nuclear Magnetic Resonance Robert Lee Mieher I.

INTRODUCTION .

11.

NMR ABSORPTION LINE . . , 1. Perfect Crystals . . . . . 2. Static Strain-Quadrupole Egects 3 . Electric Field-Quadrupole Efterrs 4. Impurity-Quadrupole Broadening

,

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141

. . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 152 155 159

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165 165 173 176 178

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183

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111. RELAXATION,SATURATION,AND POLARIZATION 5. Nuclear Quadrupole Spin Lartice Relaxation . 6. Oscillaring Elecrric Fields . . . . . . I . Ultrasonic Saturation . . . . . , . 8 . Nuclear Po1nri:ation bv Hot Electrons . . .

APPENDIX.

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I. Introduction Nuclear magnetic resonance has become a standard laboratory technique for studies of chemical and solid state physics problems. The 111-V compounds are especially suited for NMR studies because the 111-V elements possess nuclei with abundant isotopes having spins I , large magnetic moments p, and electric quadrupole moments Q. Excellent texts on NMR’,’ and review articles of both introductory3 and specialized 4-8 nature are readily available. For this reason this chapter will concentrate on the specific NMR experiments that have been performed on the 111-V compounds. This brief introduction is intended to aid in the understanding of

’ C. P . Slichter, “Principles of Magnetic Resonance.” Harper. New York.

1963

’ A. Abragam, “The Principles of Nuclear Magnetism.” Oxford Univ. Press. London and New York. 1961. G. E. Pake, Solid State Phys. 2 (1956). * W. D. Knight, Solid State Phys. 2. 93 11956). M. H. Cohen and F. Reif, Solid State Phys. 5, 322 (1957). T. P. Das and E. L. Hahn, Solid Stare Phvs.. Suppl. 1 (1958). ’L. C. Hebel, Jr., Solid State Phys. 15, 409 (1963). * T. J. Rowland, Progr. Mater. Sci. 9 (1961).

141

142

ROBERT LEE MIEHER

the experimental results by those physicists and engineers not directly involved in NMR work. When a nucleus with a magnetic moment p and spin 1 is placed in a magnetic field H o , the resulting Zeeman splitting produces 21 + 1 energy levels [see Fig. l(a) for 1 = 3/21. These levels may be labeled by the component of spin, ni,, along the magnetic field. If only magnetic effects are important, then the levels are equally spaced with a separation

AE

=

( E m - Em+,)= pH/Z

=

hyH

= l$L,

(1)

wheref; is the Larmor frequency and y is the nuclear gyromagnetic ratio.

4

(0)

lbl

AE. hfL

(C 1

FIG.1. Energy levels and absorption lines foi I = 3/2. (a) Zeernan splitting and dipole line width. (h) Uniform electric field gradients. (c) Random electric field gradients.

Because the nuclei are surrounded by electrons, the effective field Heff at the nucleus may be different than the applied field H o . These shielding effects are proportional to the applied field and may be written

Heff= Ho(1 - c),

(2)

where CJ is the shielding constant. In insulating crystals, CJ is referred to as the chemical shift and there are two contributions, a diagmagnetic part oD and a paramagnetic part oP = -]cp].The magnitude of c is in the range of i O P to There is always some interaction between the nuclei and the lattice vibrations, conduction electrons, or paramagnetic impurities. These interactions provide a "thermal contact" so that a Boltzmann population distribution corresponding to the lattice temperature TLmay be established for the 21 1 energy levels. If the sample is placed quickly in the magnetic field or if the equilibrium distribution is briefly perturbed by a pulsed rf magnetic field at the Larmor frequency, then some time is required to

+

7.

143

NUCLEAR MAGNETIC RESONANCE

establish the equilibrium magnetization Mo

=

N x ~ H = -?/’h21(l 3kTL

+ l)H

(3)

The approach to equilibrium is usually exponential with a time constant Tl that is referred to as the spin-lattice relaxation time:

dM/dt = ( M o - M)/T,. to lo4 sec. Typical spin-lattice relaxation times are in the range of The nuclei are placed in an rf coil between the poles of a magnet with the coil axis perpendicular to the magnetic field. Either the magnetic field or the radio frequency is varied through the resonance condition 2zjL= yH. The interaction between the rf magnetic field and the nuclear magnetic moments induces transitions, Am, = k 1, between the energy levels. There will be a net absorption of energy from the rf circuit due to the population difference between the energy levels. Because the energy level separation is small compared to the thermal energy kT, (typically hf/kT lop6),the resonance absorption of energy is small compared to normal rf circuit losses. Therefore, it is usually necessary to reduce the bandwidth of the detection circuit to about 1 cps. The most convenient way to do this is to audio-modulate the radio frequency or the magnetic field and to follow the rf detector by a phase-sensitive audio detector. The resulting dc signal (usually displayed on a millivolt chart recorder) is then proportional to the derivative of the nuclear absorption, df/i?j or ? f / d H , where x is the imaginary part of the nuclear magnetic susceptibility. It is also possible to study the real part of the susceptibility, x (dispersion), but we shall not be concerned with it in this chapter. Since the 21 + 1 energy levels are equally spaced, all of the Am = 5 1 transitions will occur at the same frequency. However, there is always a breadth, AH or AJ for the absorption. The lifetime broadening due to the spin-lattice relaxation transitions would give A j l/Tl. Nevertheless, the magnetic dipole-dipole interaction between the nuclear spins usually results in much larger values of Af for solids at temperatures somewhat below their melting points. The dipole-dipole line width is about Aj jDDp2/hR3, where R is the interatomic distance. Typical dipole line widths are between lo3 and lo5 sec- The dipole-dipole line width is also frequently expressed in units of time, T2 l / A j , commonly referred to as the spin-spin relaxation time. The spin-spin relaxation time appears most dramatically in the second major technique of NMR, the pulsed resonance technique. Whereas the measurements of NMR absorption, hereafter called the C.W. technique, use rf voltages in the range of microvolts to volts across the coil, the pulse

-

-

- -

’.

-

144

ROBERT LEE MIEHER

technique uses hundreds or thousands of volts of rf applied for a few microseconds (Ref. 3 contains a good introduction to the pulse technique). The nuclei precess about the large rf magnetic field H , with a precession frequency 27$= yH,. If the rf pulse length is adjusted to be on for a time At = 7r/2yH, (called a 90” pulse), then after the pulse the nuclear magnetization will be perpendicular to the dc magnetic field H , . The nuclear magnetization will now precess about H , at the Larmor frequency and will induce an rf voltage in the coil, and this voltage may be amplified and detected. This precession signal will decay in about the spin-spin relaxation time T,. It has been shown’ that the precession signal is the Fourier transform of the C.W. absorption signal. The pulse technique is useful for measuring the nuclear magnetization when TI is long because the C.W. absorption is then very weak. Also, the pulse technique is easier to calibrate because no external rf voltages are applied to the system during the measurement of the precessing magnetization, so the observed signal depends only on the amplifier gain and the magnetization. A very useful concept in NMR is the spin temperature’ T,. The magnetic dipole-dipole interactions between the nuclei are very effective in maintaining a Boltzmann population distribution for the Zeeman energy levels. Therefore, the interacting nuclei may be considered to form a spin system with its own temperature & for the 21 + 1 energy levels. Any deviation from a Boltzmann distribution will not persist longer than the spin-spin relaxation time. The spin temperature characterizing the population distribution will depend on the perturbations applied to spin system (i.e., the heat input) and the spin lattice relaxation time (i.e., the thermal contact to the lattice). If the nucleus has a spin greater than f, then it will also have an electric quadrupole moment. This quadrupole moment will interact with electric field gradients and destroy the equal spacing of the energy levels.’ The energy level shifts are [Fig. l(b)]

where a

=

Z(Z

+ 1) and VQ =

3eQ V,, 21(21 - 1)h’

where V,, is the electric field gradient along the magnetic field (a more detailed discussion of quadrupole effects is given in the Appendix), Q is I. J. Lowe and R. E. Norberg, Phys. Rev. 107, 46 (1957).

7.

NUCLEAR MAGNETIC RESONANCE

145

the nuclear quadrupole moment, and i n is the quantum number. If the energy level shifts are greater than the line width, then the C.W. resonance will consist of 21 separate absorption lines. In cubic crystals the electric field gradients at the nuclear sites are zero for reasons of symmetry. Of special importance for the 111-V compounds is the fact that the electric field gradients are zero at sites of tetrahedral symmetry, Td = 43in. However, anything that destroys this symmetry will produce electric field gradients. If these field gradients vary in magnitude and orientation for different lattice sites, then the quadrupolar interaction will contribute to the N MR line width [Fig. l(c)]. If the field gradients are oscillatory they may produce AmI = 1 or + 2 spin transitions, just as the magnetic dipole interactions produce Am, = k 1 transitions. Sources of electric field gradients in cubic crystals are, roughly speaking, of two types, strain effects and electric field effects. Examples of strain effects are dislocations, impurity size effects, and elastic strain produced by applied stresses. Examples of electric field effects are charged impurities and externally applied electric fields. The following experiments are divided into two groups: Part 11. Continuous wave absorption measurements of the amplitude, shape, position, and width of the N M R line and studies of the effects of strain, electric fields, impurities, etc., on the absorption line. Part ZII. Pulsed measurements of the spin lattice relaxation time and studies of the effects of oscillating strains (ultrasonics), electric fields, and other interactions which produce spin transitions and change the spin temperature. 11. NMR Absorption Line

1. PERFECT CRYSTALS

a. Van Vleck Second Moment In a perfect cubic crystal the shape and width of the C.W.absorption line is determined by the interactions between the nuclear magnetic dipoles. In addition to the applied magnetic field, each nucleus is in a local magnetic field due to the surrounding nuclei. Obviously this local field will vary from site to site and there is a very large number of possible local fields. These local fields depend on the spins and magnetic moments of the nuclei, the crystalline structure and lattice constant, and the orientation of the crystal in the magnetic field. A general expression for the line shape has not been obtained, but Van Vleckl' has calculated the second moment lo

J. H. Van Vleck, Phys. RPC.74, 1 168 ( 1948).

146

ROBERT LEE MIEHER

of the absorption line

where g ( o ) is the shape factor of the absorption line. The second moment of the absorption line of nuclear species I due to magnetic dipole interactions with like spins I and unlike spins S is

+ +yr2ys2h2S(S+ 1) C (1 - 3 coszOj)2/rj6,

(8)

j

where rk is the distance from the I site taken as the origin to another I site, and rj is the distance to an S site, and !&, B j are the angles between the magnetic field and the vectors rk,r j . If the sample is polycrystalline or powdered then Eq. (8) is averaged over 8 to give

c rL6 + 1) c r,:

( A W ~ ’ )=~ $yr4h21(I ~~ + 1)

k

+ &yr2ys’h2S(S

6.

(9)

j

The expressions for the second moment are important because they provide exact theoretical results to compare with experiments. The experimental second moment is frequently greater than the Van Vleck dipoledipole second moment. When there are contributions to the line width from interactions other than the simple dipole-dipole interaction, the additive nature of the second moment is very convenient. The second moment may then be written = (AW2)dipole

+ (AW2)quadrupole

-k

(AW2)other*

(10)

Because of the importance to the second moment of the “wings” of the resonance, a good signal-to-noise ratio is necessary if an accurate experimental determination of is to be obtained. For this reason the second moment is sometimes determined by fitting a Gaussian curve to the experimental results if the signal-to-noise ratio is low. We shall see in the next section (1, b) that this is a good approximation for the 111-V compounds because the absorption line shapes in pure 111-V compounds are very nearly Gaussian.

7.

147

NUCLEAR MAGNETIC RESONANCE

6. Pseudoexchange Broadening in 111- V Compounds In addition to the direct dipole-dipole interaction between the nuclei there are other indirect interactions due to perturbations of the electrons by the nuclear magnetic moments. The details of the calculation of these interactions are complicated, and the reader is referred to the original articles. The indirect exchange process was first discussed for metals by Ruderman and Kittel" and by Bloembergen and Rowland." An expression for semiconductors has been derived by Anderson, and is repeated by Shulman and coauthor^.'^ TABLE 1 LINE WIDTHSA N D SECONDMOMENTS".~ Nucleus measured Ga69Y Ga71Y Gap3'' Ga69As Ga7'As G~As'~ In"SAs I~As'~ Ga69Sb Ga'ISb GaSb GaSbIz3 In'I5Sb InSb'"

''

6H (gauss) 2.26 f 0. I5 2.26 f 0.15 2.9 f 0.3 2.43 i 0.2 2.43 0.2 2.86 0.2 3.30 f 0.2 8.4 k 0.3 5.1 f 0 . 1 5 5.6 f 0.15 4.7 If: 0.2 5.1 f 0.2 9.0 f 0.2 17.5 i 1

**

AHz' (gaussZ)

I .28 1.28 2.10 1.47 1.47 2.04 2.72 17.6 i 2.0 6.5 0.4 6.2 f 0.4 6.7 0.4 8.4 f I 24.0 I 65.0 f 4

*

+

*

Calculated (AHzz)dipole

(AHZZLh

1.08 1.09 1.70 0.79 0.80 1.34 1.68 3.60 1.20 1.10 1.05 0.93 1.65 2.52

0.20 & 0.2 0.19 f 0.2 0.40 f 0.4 0.68 k 0.3 0.67 f 0.3 0.70 f 0.3 1.0 0.4 14.0 f 2.0 5.3 0.4 5.0 0.4 5.6 f 0.4 7.3 f 1 22.0 f 1 62.0 i 4

* **

"For InAs, GaAs, and GaP the values listcd in the third column are derived from the experimental results listed in the second colunin by the relation 6 H Z = 4 A H z 2 (6H is the peak-to-peak distance for the absorption derivative and the relation assumes a Gaussian shape). For lnSb and GaSb the third column lists measured values of AHz'. The last column lists the difference between the third and fourth columns which are attributed to exchange etrects. See Ref. 15 for additional measurements of 6 H . R. G . Shulman et a/., Refs. 13 and 14. except as noted. ' M . J. Weber. Ref. 16.

Since the exchange mechanism is a second-order effect involving the electron-nuclear contact hyperfine interaction, the contributions to the line widths are greater for the heavier elements. Indeed, the pseudoexchange M. A. Ruderrnan and C . Kittel, Phys. Reo. 96, 99 (1954). N. Bloembergen and T. J. Rowland, Phys. Rec. 97, 1679 (1955). 1 3 R. G. Shulman, J. M. Mays, and D. W. McCall. Phps. Reo. 100. 692 (1955).

'I

148

ROBERT LEE MIEHER

contributions to the second moment, may be an order of magnitude greater than the regular dipole-dipole second moment in some materials. Schulman and c o - ~ o r k e r s ' ~ ,have ' ~ measured the line widths and shapes of several 111-V compounds. The line shapes are all very nearly Gaussian, and the second moments (see Table I) are all greater than predicted by Eqs. (8) and (9). Also, the line shapes and second moments of single crystals were independent of crystalline orientation in the magnetic field, contrary to the prediction of Eq. (8). These results may be explained by Anderson's theory of the nuclear exchange interaction for semiconductors. The direct dipole-dipole interaction which leads to Eqs. (8) and (9) is Xij(dipole-dipole)

(Ii* rij)(Ij* r i j ) r '2J

h2y.y.

=2 7 3. 'J

Assuming Bloch wave functions, 4k(r) = pk(r)eik.r, the pseudoexchange interaction is m

%",(exchange)

=

-(S* Ii)(S*Ij)AiAj

m

1 1

k = O k'=O

exp[ - i(k - k'). rij] E , -k (h2k'2/2me)+ (h2k2/2mh)'

(12)

where S is the electron spin, E , is the energy gap of the semiconductor, and Ai is the matrix of the hyperfine interaction for S electrons with atomic wave functions pk(r). It is difficult to obtain an absolute theoretical value of the exchange interactions because Eq. (12) involves a sum over all momentum states. However, if spherical energy surfaces are assumed, the dominant term is

z.. = A..I.*Ij, 'J

IJ I

where A , . = 3.36 x V

1 o - ~ n 2 ~ g ~ g ~ t , b ~ ~ ~seto ~ ', t,b,z~o~~~~~ (13)

IiIjr$

0 is the atomic volume, m = 4(memh)3'2/(me + n ~ , ) t,bi2(0) ~, is the probability of finding the valence s electron of atom i at its nucleus, and

5 . = [~i(O)holet,bi(O)electronlsolid [t,bi2(0)Iatom l4

R. G. Shulman, B. J. Wyluda, and H. J. Hrostowski, Phys.

Reu. 109, 808 (1958).

7.

NUCLEAR MAGNETIC RESONANCE

149

The energy gap does not appear in A i j because the most important contributions come from the high momentum states. Shulman et al. discuss various choices of m and ti and show that reasonable assumptions lead to the correct order of magnitude for A i j . Since the interaction AijIi* Ij is isotropic, the exchange second moment will be independent of crystalline orientation. If ( A w ~ ) , , , ~ is ~ ~it ~ is~ apparent ~ , why the experiment's C.W. larger than ( A w ~ ) ~then absorption lines are insensitive to rotations of single crystals. Also, the same results will be obtained for polycrystalline samples.

FIG.2. Typical recorder trace of the derivative of the In"' resonance. Crosses are plotted from the Gaussian shape function and circles froni the Lorentzian shape function. (R. G . Shulnian et a/., Ref. 13.)

Pseudodipole also exist and may be comparable to the regular dipole-dipole interaction for large atomic numbers. However, the pseudoexchange interaction would be larger than the pseudodipolar by roughly the ratio of the atomic hyperfine interactions for s and p electrons times the ratio of s to p character for the valence and conduction bands. The nearly Gaussian shape of most of the 111-V absorption lines is due to the large values of spin for most of the nuclei. The large values of spin make possible large numbers of different nuclear spin orientations which leads to a Gaussian distribution of local fields. For the 1n"'Sb resonance (see Fig. 2), there are 475 possible values of Zimi for just the nearest neighbor interactions. Figures 2, 3, and 4 show the experimental absorption derivatives compared to Gaussian derivatives for several materials. Although quadrupole effects can change the line shape, the deviation from Gaussian behavior in GaP is probably due to the low value of spin (f) for phosphorus nuclei.

150

ROBERT LEE MIEHER

-EXPERIMENTAL CURVE

FIG.3. Ga7' in pure GaSb at 90°K. The full line is the high frequency half of the experimental record which is proportional to the derivative of the absorption. The circles are obtained from the Gaussian curve of best fit. (D. J. Oliver, Ref. 23.)

c. Chemical Shifts

The preceding discussion was concerned with electron-nuclear couplings which broaden the NMR line. Other electron-nuclear couplings result in shifts of the resonance frequency because the effective magnetic field at a nucleus is different from the applied field. The difference is proportional ln

k

3 z

OEXPERIMENTAL

>

K R 4

t m

-az 4

0 6 L

a ln 0

m

a

LL

0 W

5 I-

a

2

W R

n

0

1.0

2.0

3.0

4.0

MAGNETIC FIELD (GAUSS)

FIG.4. Line shape of typical G a 7 ' resonance in GaP compared to Gaussian and Lorentzian shape functions. (M. J. Weber, Ref. 16.)

7.

NUCLEAR MAGNETIC RESONANCE

151

to the applied field H , and may be written

Heff= H,(1 - a).

(2)

The largest shifts (known as the Knight4*’ shift) occur in metals and are due to the contact interaction with the electrons at the Fermi surface. The metallic Knight shift increases the effective field (i.e., a is negative) Although it is possible in principle for the to and 101 is about conduction electrons in the III-V compounds to produce a Knight shift, attempts to detect such shifts have been unsuccessful, and they are expected to be very small. In insulators a is referred to as the chemical shift,’*l and there are two contributions of opposite sign. The diagmagnetic contribution a,, decreases the effective field. This shielding is of the form

where (llr) = 3 Jp(x2 + y Z ) / r 3lit and p is the electron density. There is also a paramagnetic contribution gp which increases the effective field and is of the form ap =

where (i/r3) = J p / r 3 dr and A is an electronic excitation energy. The oDis essentially a Lenz’s law effect and a, is due to the magnetic interaction of the nucleus with the angular momentum of the electron which is “unquenched” by the applied magnetic field ; therefore, both effects are proportional to the applied field. Since the two contributions are of opposite sign the chemical shift may be either positive or negative. In some simple molecules (e.g., the hydrogen molecule) aD may be larger than a,. Nevertheless, in most solids aD is about lo-’, whereas apis to l o p 3 .Although detailed theoretical calculations have been made on chemical shifts for alkali halide crystals and many types of molecules, apparently no calculations have been carried out for the III-V compounds. A calculation of the important paramagnetic part is especially complicated because it requires a detailed knowledge of the wave functions and energies for all values of k. The problems involved are similar to those of the evaluation of Eq. (12) for the pseudoexchange effect. Experimental measurements of the chemical shifts in several III-V compounds have recently been made Liitgemeier. l 5 Since the term H. Lutgemeier, 2. Naturjorsch. 19a, 1297 (1964).

152

ROBERT LEE MIEHER

( T/r3)in Eq. (15) may depend on the effective charge of the compound, the chemical shifts are plotted in Fig. 5 so as to show a correlation with the effective charge. The relative shifts for In, Ga, and Sb in different compounds are shown. The relative shifts for As and P were too small to detect. Since the nuclei are always surrounded by some electrons, only relative shifts can be given. Weber16 has reported that AHIH = +0.55 x for gallium nuclei in GaP with respect to a solution of GaCl,. x 10-3 A

In Sb

\-

\

\ \

0.5WSb

1

!&p

-0.5

-_

0.5

’

-EFFECTIVE

I

1.0

CHARGE

FIG. 5. Chemical shifts of Ga, In, and Sb as a function of the effective charge. (H. Liitgemeier, Ref. 15.)

2. STATICSTRAIN-QUADRUPOLE EFFECTS Because all 111-V nuclei except phosphorus (see Table 11) have quadrupole moments, it is possible to use the C.W. absorption line to detect deviations from the perfect zinc-blende lattice symmetry. Fortunately, both single crystals and polycrystalline 111-V compounds of readily available purity are free of noticeable quadrupole broadening of the resonance line. This is not the case, for instance, in the alkali halides where it is often difficult to obtain crystals perfect enough to be free of random quadrupole interactions. The first successful experiment to produce a uniform quadrupole interaction in a single crystal was made by Shulman et al.” They detected a broadening of the 1n”’Sb NMR line upon applying stresses to a single crystal of InSb. The effects were completely reversible, indicating that the strains had not exceeded the elastic limit of the crystal. Therefore, the l6 l7

M. J. Weber, J . Phys. Chem. Solids 21, 210 (1961). R. G. Shulman, B. J. Wyluda, and P. W. Anderson, Phys. Rev. 107, 953 (1957).

7.

153

NUCLEAR MAGNETIC RESONANCE TABLE I1

NUCLEARPROPERTIES A N D ATOMICFIELDGRADIENTS q Nucleus

% Abundance Larmor freq.“

Spin

Q(lO-’“ ~ r n . ’ ) ~4(104 cm-.’rtd

for lo4 gauss 100.0 100.0 60.2 39.8 7.6 100.0 95.8 51.2 42.8

~ 1 2 ’

P31 Ga7’ Ge7 As’ 1~115

Sb’” SblZ3 ~_______

~

~~

1 1.094 17.235 10.218 12.984 1.485 1.292 9.329 10.19 5.518 ~______

512 112 312 312 912 312 912 512 712

0.15 f 0.01 -

0.19 0.12 0.20 0.30 1.16 0.53 0.68

f 0.01 f 0.01 f 0.10 f 0.20

i 0.05 f 0.10 f 0.10

~_______

7.23 18.9 18.9 31.2 40.8 22.3 70.4 70.4 ~_______

“ G . E. Pake, Ref. 3. b C . H. Townes, in “Encyclopedia o f Physics” (S. Flugge, ed.), Vol. 3811. Springer, Berlin. 1958. ‘T. P. Das and E. L. Hahn, Ref. 6. R. G . Barnes and W. V. Smith, Phys. Rev. 93, 95 (1954).

measured broadening could be used to determine the components of the electric field gradient stress tensor, c I 1 and c4& (See Appendix for a more detailed discussion of this tensor.) Ideally, since In”’ has a spin of 9/2, sufficient deviation from cubic symmetry would result in 9 lines for the C.W. absorption spectrum. However, for strains within the elastic limit of the crystal, the quadrupole shifts were less than the line width of the normal material. Therefore, only a line broadening could be produced. Nevertheless, by careful measurements of the absorption line it was possible to determine cll and c4., with an accuracy of about 20%. Measurements were made on two cylindrical single-crystal samples, one with a [110] axis and one with a [lo01 axis. Stresses of 3.5 x lo7 dyn/cm2 (500 lb/in.’) were applied along the axis of the sample in a direction perpendicular to the magnetic field. The first-order quadrupole shift5 for a m -, m - 1 transition is

3(2m - 1)eQ v,, = YJ, 41(21 - 1)h 27l’ where V,, is the field gradient at the nucleus along the magnetic field and 6, is the splitting Avm expressed in gauss. For a given applied stress, V,, will depend on the orientation of the crystal in the magnetic field. If the crystal is stressed along [110] and rotated about [110] the field gradient is (see Appendix for a discussion of the angular dependence) Av,

= V , , , + . ~- I y,H =

~

154

ROBERT LEE MIEHER

where 8 is the angle between the magnetic field and the [Ool] crystalline direction and X is the applied stress. If the crystal is stressed along [loo] the field gradient is

v,, = - XC,J2

(18)

and is independent of rotation about [loo].

10 .

0.9

+ Y

a x

a

0.8 0.7 0.6

0.5 0.4

FIG.6 . Plot of [dl/dH]peak-,~penk vs 0,. Open circles are normalized experimental values of [dl/dH]peak.,o-peat for [110] compression while solid curve is best fit of (1 + ~,A,S,2/AH22(0)]~1. The suninlation over m was taken as the average value of the sun1 from m = - 5/2 to 7/2 and the sum froni m = - 7/2 to 9/2. The straight line is the average of 25 experiniental values of dl/dH,,,,,,,,,, for [OlO] compression. (R.G. Shulnian er al., Ref. 17.)

+

Since the line width in the unstrained crystal is determined primarily by the indirect exchange interaction with the nearest Sb nuclei, it may be assumed that the shape of the individual components ( v , . + , - ~ ) does not change with strain. Then the additive property of the second moment may be used to obtain

AH12

=

AH12(0)+

1A,bm2, m

where AHZ2(0)is the second moment of the unstrained crystal and A , is thenormalized relativeintensity ofthem -+ ( m - 1) transition (i.e., C A , = I ) given by A,,,

=

Z(Z + 1) - m(m - 1) . +Z(I + 1)(21 + 1)

Because of a 12% reversible loss of intensity upon application of the stress, the m = 3-g and -$++ -4 transitions were omitted from the summation in Eq. (19). If the line shape remains Gaussian when the crystal is stressed, then the peak-to-peak recorder deflection of the absorption

7.

NUCLEAR MAGNETIC RESONANCE

155

derivative should be proportional to the integrated intensity. Figure 6 shows a plot of

for the two cases discussed above. The best fit of the theoretical curve to the experimental points for strain along [110] gives eQ1/Zz(6 = 0) = 36 kc/sec and eQT/r,(e = 90’) = f52 kc/sec (only relative signs can be determined). The values of C , , and C,, are now determined by Eq. (17) as eQC,, = f 2 x eQC,,

=

(kc/sec)(cm2/dyn),

+ 2 x l o p 6 (kc/sec)(cm2/dyn),

with an estimated accuracy of +20%. Taking the quadrupole moment of In115 as 1.16 x 10-24cm2the values of C,, and C,, are

C,,

=

( 7 2 . 4 f 0.5) x lo4 stat V/dyn,

C,,

=

( T 2 . 4 f 0.5) x lo4 stat V/dyn.

An estimate of the electric field gradient can be made by approximating the lattice sites by point charges and calculating the field gradients for the 350 stat V/dyn. The measured strained crystal. This calculation gives C, value is about 70 times greater. Although changes in the bonding orbitals are probably a major contribution to V,,, it is well known that electric field gradients are usually much larger than predicted by a point charge model. This is due to an antishielding effect of the electrons which usually greatly increases the magnitude of the field gradient at the nucleus. The effect was first studied by Sternheimer, and a good discussion is given by Cohen and Reif.’

,-

FIELD-QUADRUPOLE EFFECTS 3. ELECTRIC The application of an electric field to a crystal should produce quadrupole effects. Nevertheless, early attempts to detect such effects in alkali halides were unsuccessful. It was pointed out by Bloembergen” that linear Stark effects would occur for nuclei at sites lacking inversion symmetry. Linear Stark effects have since been observed in several materials, and the subject has been reviewed by Bloembergen.” We shall discuss here the electric field-quadrupole experiment of Gill and Bloembergen” on GaAs. l9

N . Bloembergen, Science 133. 1363 (1961). N . Bloembergen, in “Magnetic and Electric Resonance and Relaxation” (1. Smidt, ed.). p. 39. North-Holland Pub., Amsterdam. 1963. D. Gill and N. Bloembergen, Phys. R w . 129, 2398 (1963).

156

ROBERT LEE MIEHER

Just as in the elastic strain experiment on InSb, the 21 transitions will have different frequencies when the cubic symmetry is removed by application of the electric field. However, unlike the elastic strain experiment, the splitting is now large enough compared to the normal line width to actually separate the 21 transitions. Gallium arsenide was chosen because of its narrow line width (see Table I) and because it is available in the high resistivity form ( l o 9 R-cm at 77°K) necessary for the application of large electric fields (20,000V/cm). The electric field gradient produced by the electric field is

The Rijktensor is discussed in the Appendix and has only one component, R I b The first term is the field gradient produced by the electric field for constant lattice strain. The last term is the field gradient produced by the piezoelectric strain (dlmkEk)where S i j f mis the electric field gradient-strain tensor. The Sijrm tensor may be obtained from the Cijfm(i.e., C,, and C44in this case) and the elastic constants. However, in an elastic strain measurement similar to the one discussed above for 1n"'Sb no detectable line broadening was produced in GaAs. Therefore, only an upper limit S, < 4.5 x 10" statC/cm3 could be determined. The piezoelectric tensor has only the component d l b The constant d,, was measured (using apparatus calibrated with NaC103) to be d,,(GaAs) = (2.0 f 0.5) x lO-'cgs units. Since the product d14S44is less than 1% of the measured Ri4, only the first term in Eq. (22)is important. If the electric field and magnetic field are both along the [l 1 11 crystalline direction, then the field gradient along the magnetic field is (see Appendix)

2R 14E

V,z =

7.

The values of R , , for the nuclei in GaAs are listed in Table 111. The splitting AvE is given by Eq. (16). A simple model was used by Gill and Bloembergen to make an interesting estimate of R14 for GaAs. The electric field gradient is caused partly by a distortion of the valence orbitals and partly by the relative displacement of the Ga sublattice with respect to the As sublattice. It is possible to relate these two effects to the low frequency dielectric constant, E = 12.5 0.2, and to the infrared index of refraction, n = 3.3. Using a point charge model to represent the lattice, the contribution to the field gradient from the displacement Ar of a nucleus from the center of its tetrahedron toward one of the four corners is V,, = (40/3)eeffAr a- 4,

7.

NUCLEAR MAGNETIC RESONANCE

157

TABLE 111 DATAFOR ELECTRIC SHIFTOF THE MAGNETIC RESONANCEI N GaAs" Nucleus

AvE in kc/secb for lo4 V/cm

Rl, (cm - ')

Gah9 Ga7' As7'

3.5 i 0.6 1.9 f 0.2 6.5 f 1.5

1.05 x 10" 0.9 x 10" 1.55 x 10''

" D. Gill and B. Bloembergen. Ref. 20. ' Av, is the splitting between each satellite and the central component if the electric field and magnetic field are applied along [ 1 111.

where a is the Ga-As distance and eeff is the effective charge at a lattice site. The product eeffAr is related to E and n by the expression for the ionic polarization

NeerfAr

= [(E -

n2)/4n]E,

(23)

where N = 2.2 x is the number of GaAs molecules per unit volume. Introducing the Sternheimer antishielding factor5 (1 - y,) for the distortion of the core electrons, the ionic displacement contribution to V,, is

40

v., = -(1 3

-

c - n2 y )-4nNa4 Ell1

Therefore, the ionic contribution to R I 4 is Rion 14

=

-

5

E

- n2

(1 - YaJnNa4 v/i -

'

= 24 for Ga and 30 for As are used, then

If the values (1 - y,) Ry",Ga)

-

0.4 x 10"

cm-'

and

R';"",As) = -0.5 x 10" cm-'.

(E has been taken as positive if it points from Ga toward As.) The contribution from the distortion of the valence orbitals by the electric field was treated as a perturbation of the undistorted orbitals. The tetrahedral orbitals made up of 4s and 4p atomic orbitals of Ga and As are h e 1

=W 4 S

+ $4px + $ 4 p y + $ 4 p d

(26)

Neglecting overlap of atomic Ga and As orbitals, the valence band may be

158

ROBERT LEE MIEHER

approximated as the bonding molecular orbitals =

(A$::

+ $;:)(I + AZ)-'l2,

(27)

and the conduction band may be approximated as the corresponding antibonding orbital $cond

=

(-$::

+ A$;i)(1 + A’)-’".

(28)

The parameter A is chosen to give an effective charge in agreement with a modified Szigeti formula eeff = u O [n (c- nz)/4xN]''z,

(29)

where uo(GaAs) = 5.04 x IOl3 sec-' is the transverse fundamental optical mode frequency. Using second-order perturbation theory, the change in energy of the ml nuclear spin level may be written

-

(30) where ZQis the quadrupole Hamiltonian discussed in the Appendix and W, - W, is the average separation between the valence and conduction bands. Equation (30) shows the necessity of a lack of inversion symmetry for a linear Stark effect. Because ;XQis an even function of electron coordinates and eE*r is odd, Eq. (30) will vanish unless or have mixed parity." The electric field gradient for E along [111] may now be written as AE'2'(m) =

2($val'

mlxQl$cond'

m)($condleE

rl$val)(K

vzy = 2 e E ~ l l ( $ v a l ~ ~ = ~ $ c o n d ~ ( $ ~ o n d ~ r ~ $ v-a ~K))(~

>

*

(31)

Using the above expressions for and $cond, writing eqa, for the field gradient produced by an atomic p , orbital, and including the contribution of all four bonds, the covalent contribution to R 1 4 is

R?l=

* f f i A ( l + nZ)-'(K - Wv)-'eqat($condlerl$val).

(32)

To complete the estimate of R?: the dipole matrix element is determined from the electronic polarizability, i.e., from the infrared index of refraction. The electronic polarizability of all four bonds is Pel

=

(n’

-

1)E/4n = ( 8 / 3 ) N E ~ ( $ c o n d ~ e r ~ $v aW)-'. l)~2(~

(33)

~ , = Using n = 3.3, Wc - Wv = 2eV, qa,(Ga) = - 18.9 x loz4~ m - q,,(As) -40.8 x 1024cm-3,and 1 = 0.4, the covalent contributions are R?;(Ga) = 0.53 x 10" cm-' and R"(As) = - 1.1 x 10" cm-'. Adding RYI and Rf": gives R,,(Ga) = 0.93 x 10'Ocm-' and R,,(As) = 1.6 x 10'Ocm-', in very good agreement with the experimental results. Considering the difficulties usually encountered in calculating quadrupole effects in solids,

7.

NUCLEAR MAGNETIC RESONANCE

159

this simplified model is interesting because it has been used essentially to introduce independently determined experimental parameters into the calculation of R 1 4 . 4. IMPURITY-QUADRUPOLE BROADENING

Impurities introduced into the lattice will produce electric field gradients at the lattice sites of the host nuclei. Since the gradients will vary in orientation and magnitude from site to site, the NMR line of the host nuclei will broaden. There will be a contribution to the field gradients from both the lattice strains due to impurity size effects and the electric fields if the impurity introduces a different charge into the lattice. That is, we shall now deal with a random distribution of the elastic strain and electric field effects of the preceding discussions. Studies of the quadrupole broadening of the NMR lines have been made in InSb and GaAs by Rhoderick2”22and in GaSb by Oliver.23As discussed in Section 1,b the line shape of the pure material is usually Gaussian. As the concentration of impurities is increased the line may simultaneously change its shape, broaden, and decrease in intensity. Each of these effects yields information but may be difficult to interpret individually. It simplifies the discussion to consider only the change in the peak-to-peak amplitude of the absorption derivative (dx”/d~),,,~,,since this is a quantity that can easily be taken directly from the experimental curve (note that it depends on shape, breadth, and intensity of the line). Figures 7, 8, and 9 show the changes in (i?f/i?~),~, for increasing concentrations of various impurities in InSb and GaAs. The most striking feature of these curves is the difference in slope for small concentrations and for larger concentrations. Similar curves are also obtained for cold worked metals and dilute metal alloys, and the subject is discussed in detail in a review article by Rowland’ on NMR studies of metals. In the perfect lattice with no field gradients at the nuclear sites the 21 transitions (m + m - 1 ) all contribute to the absorption line. However, in the presence of the random field gradients all of the transitions will transition. This broadening is broaden except the f

- -+

ave

=

3rQ(2m - 1) 41(21 - 1)h ( Y z A a v e .

-*

(34)

When AvaVebecomes somewhat larger than the original line width only transition will be detected. The initial rapid the absorption of the $ + ”

E. H. Rhoderick, Phil. Mag. 3. 545 (1958).

’’ E. H. Rhoderick, J . Phys. Chern. Solids 8, 498 (1959). ” D.

J. Oliver, J . Phys. Chern. Solids 11, 257 (1959).

160

ROBERT LEE MIEHER SINGLE, CRYSTAL

t 1

.

In115in InSb

m

POWDER

“0

*

L

0.5

I-

: CENTRAL:COMPONENT

0

0

,.

5 ~ 1 0 ~ ~ Te CONCENTRATION (ATOMS c ~ n - ~ )

FIG.7. Variation with Te concentration of derivative of In’ l 5 resonance in InSb. Also shown are results for two particle sizes of the pure powdered material (290°K). (E. H. Rhoderick, Ref. 21.)

decrease in (ax”/av),,, is due to the first-order broadening. For higher concentrations of impurities (a~ /~v),,, will decrease due to second-order effects on the central transition. The exact expression5 for the second-order shifts is rather complicated and is not necessary for this discussion. The

I

0

lX1020 Go CONCENTRATlON(AT0MS

2x11020

FIG.8. Variation with Ga concentration of derivative of I n X X 5resonance in InSb-GaSb mixed crystals (290°K). (E. H.Rhoderick, Ref. 21.)

7.

NUCLEAR MAGNETIC RESONANCE

1.0

161

in GaAs

-w W

I-

U d

' -E" . W

0.5

P co

Lo X

C

I I I n CONCENTRATION

I 2 (ATOMS cm-3 XIO-20)

FIG.9. Variation with In concentration of derivative of crystal. (E. H. Rhoderick, Ref. 22.)

resonance in GaAs-InAs mixed

second-order broadening is proportional to

Although the shape and breadth of the line for random second-order quadrupole interactions can be calculated, it is observed experimentally that the resonance line shape of the central transition usually does not change appreciably and that the intensity of the central component for low impurity concentrations is given simply by

where C is the impurity concentration and n is determined by the slope of the curve in the slowly decreasing region. This is interpreted as due to the second-order effects being large enough to remove n host nuclei about each impurity from the detected absorption line. If this region of gentle slope is extrapolated back to the zero-concentration axis, then the intercept should give the normalized contribution, A1/2+-1/2

=

+ +t + 1)'

I(I 1) +I(I + 1)(21

(37)

of the central transition if all of the transitions contributed to the resonance in the "pure" material. Comparison of Figs. 7 and 8 shows that the broadening effects of the charged impurities are an order of magnitude greater than the effects of neutral impurities. For an electric charge e in a medium of dielectric constant

162 E

ROBERT LEE MIEHER

the potential is V ( v )= e/Er.

and the electric field gradient is

a2V/dr2 = 2 e / ~ r ~ However, if there are sufficient carriers to produce a screened Coulomb field the potential is5 e V(r) = - exp( - qr) , Er

where q2 = 16712e2(2nz*)312k1i2TFI , 2 ' ( q ) / ~ hm* 3 , is the effective mass. and Fl,2(u)= F,,,(C/kT) is the Fermi-Dirac integral. This results in a field gradient

azv = -exp(-qr)[l 2e 82 &r3

~

+ qr + 3q r 1. 1 2 2

Rhoderick found that such screening effects were negligible for electrons in InSb (presumably because of the small nz*) but detectable for holes if the concentration was as high as Table IV shows that there is little TABLE IV RELATIVE INTENSITIES

OF IN

In'Is N M R ABSORPTION DERIVATIVE DOPEIISAMPLES"

Sample

Relative intensity

Intrinsic (single crystal) ri-type (lid = 10" C I ~ - ~ p-type (11, = 10" C I ~ - ~ ri-type (q, = i 0 i 9 c m - 3 ) p-type (17, = l o i 9 cm-') Compensated ~

~~

~

lid

+ 19,

I .oo 0.23 k 0.01 0.25 f 0.01 0.073 k 0.005 0.130 0.008

)

)

=

cm-'

nd - n, = 2 x 10''c1n-~

0.075

0.005

~

E. H . Rhoderick, Ref. 21.

difference in the broadening effects of donors and acceptors for concentrations of 10l8 cmP3 but that the effect for l O I 9 acceptors was less than for l O I 9 donors. The heavily doped, compensated sample had little screening because of ti;: low concentration of holes and behaved like a regular n-type sample of the same impurity concentration. Again the antishielding factor enhances the field gradients by two or three orders of magnitude. Cohen and Reif' have shown that the effect

7.

NUCLEAR MAGNETIC RESONANCE

163

of the polarization of the dielectric medium should be given by replacing

1 / in ~ the above equations by (2.5 + 3)/5.5. Using this correction the resulting empirical values of (1 - ym) are listed in Table V. TABLE V EMPIRICAL ANTISHIELDING FACTORS FOR CHARGED IMPURITIES

Nucleus

Antishielding factor

175 32 1 49 1000

Ga69Sba Ga' Sb" GaSb'z"' In"'Sbh

D. J. Oliver, Ref. 23.

' E. H. Rhoderich. Ref. 22. The effects of neutral impurities are shown in Fig. 8 for InSb-GaSb mixed crystals. Rhoderick found good agreement with experiment (within a factor of 2) by representing the lattice strain as a contraction of a sphere of radius a, to a radius of a, in an isotropic elastic medium. This strain is radial and is given by

The radial strain, a, - a,, is taken as the difference in the InSb and GaSb nearest neighbor distance a , - a, =

(2.80 - 2.65) A ,

and a, is taken as the "tetrahedral covalent radius" of a Ga atom, which according to Pauling is 1.26A. The relation between the strain and the field gradient is taken from the elastic strain experiment of Shulman on InSb (Section 2) Kz =

1

Cll(C11 - c,,)[e,, - A e x x

+ eYY)l,

(43)

where cl,, c I 2are the elastic constants and C , , is the field gradient-stress tensor component. These approximations predict that field gradients due to neutral G a impurities in InSb would be about 1/60 as strong as the charged Te impurities and the experimental ratio is about 1/30. Knowing the behavior of charged and neutral impurities, Rhoderick investigated the effects of Si impurities in GaAs. It had been shown24 24

J. T. Edmond, Proc. Phys. Soc. (London) 73,622 (1959).

164

ROBERT LEE MIEHER

that Si acts as a donor in GaAs with Si replacing Ga up to a concentration of about 4 x 1 0 " ~ m - ~donors. However, X-ray measurements2' of the lattice parameter of GaAs-Si alloys had indicated the presence of neutral Si up to 1.1 x 1020atoms/cm3.Figure 10 shows a plot of ( ? ~ " / ? v ) , , , ~vs~ electron concentration as determined by Hall measurements up to 4 x 10" cm-3 for Te- and Si-doped GaAs. The more rapid decrease in Sf/2v for the Si-doped material is interpreted as due to large numbers of neutral silicons in the lattice, presumably pairs of Si atoms replace GaAs molecules and do not contribute to the carrier concentration. Since the same value 1.0

G O ~ ~ I GoAs II 0

-I

01 0

I

Te DOPED SI DOPED

I

2

( R"~)-I(ELECTRONS

I

3

1

4

cm-3,10-1*1

FIG. 10. Maxiniuni value of derivative of Ga69 resonance in Te- and Si-doped GaAs plotted against reciprocal of Hall constant (290°K). (E. H. Rhoderick, Ref. 22.)

of (dx"/av),,, in Si-doped samples corresponds to about one-half the carrier Concentration of a Te-doped sample, the strain and charge effects must be about equal in the Si-doped material. However, we have seen that the effects of neutral impurities in InSb are about an order of magnitude less than the effects of charged impurities. The effect of neutral impurities for GaAs was investigated for GaAs-InAs mixed crystals and is shown in Fig. 9. Therefore, the concentration of neutral defects must be about an order of magnitude greater than the concentration of charged defects. Since the relative decrease in intensity is about the same for 1.3 x 10'' In atoms/cm3 and for 1.1 x lo2' Si atoms/cm3 and since the charge and strain effects are about equal in the Si material, the neutral silicons are 25

C. Kolm, S. A. Kulin, and B. A . Averbach, Phys. Rev. 108, 965 (1957).

7.

NUCLEAR MAGNETIC RESONANCE

165

about half as effective as indiums. This indicates that the silicons may indeed be present as pairs in the lattice. (However, other experiments2sa indicate that a random distribution of Si on the two sub-lattices may occur.)

III. Relaxation, Saturation, and Polarization 5. NUCLEAR QUADRUPOLE SPIN LATTICERELAXATION

The importance of the nuclear quadrupole moment in determining the spin lattice relaxation was first discussed and demonstrated by Pound.26 The first detailed theory of the effect was developed by Van K r a n e n d ~ n k . ~ ~ Using the approximations that the lattice vibrations could be described by a Debye spectrum and that the lattice could be represented by an array of point charges located at the lattice sites, he derived expressions for the quadrupole transition probabilities for the NaCl lattice. The dominant mechanism is the Raman process in which one quantum of lattice vibration is absorbed, another quantum is emitted, and a nucleus makes a Am = i-1 or f 2 transition. The difference in energy of the two lattice quanta is equal to the energy exchanged in the process with the nuclear spin system which is very small in comparison with the energy of a typical lattice vibration quantum. The effects of the Sternheimer polarization of the electrons surrounding the nucleus and any other field gradient contributions, such as the changes of the electron wave function discussed in Section 3, are described by a simple multiplication factor for the point charges. Van Kranendonk also derived the temperature dependence for this model. Since the detailed calculation is very long, only an order of magnitude estimate will be given here in order to demonstrate the main physical points involved. If the displacements of the nuclei from their equilibrium positions are small compared to the interatomic distance, the quadrupole Hamiltonian may be expanded in a power series of the displacements. The total displacement of a nucleus by a given mode of lattice vibration is given by the amplitude of vibration of that mode; however, it is the relative displacements of two lattice sites that determine the local distortion of the lattice. This relative displacement r is given by r 2nRq/l,where q is the amplitude of vibration of a normal mode, l is the wavelength of the mode, and R is the interatomic distance. The amplitude of vibration at high temperatures

-

25a

26

J. M. Whelan, J. D. Struthers, and J. A. Ditzenberger, Proc. Intern. Con& Semicond. Phys., Prague, 1960 p. 943. Czech. Acad. Sci., Prague, 1961. R. V. Pound, Phys. Rev. 79, 685 (1950). J. Van Kranendonk, Physira 20, 781 (1954).

’’

166

ROBERT LEE MIEHER

may be approximated by considering a classical oscillator whose energy is

Mw’q’

=

kT,

(44)

where M is now taken as the mass of the crystal. [The crystal mass M appears in the final expression, Eq. (50), as the mass density, m = M / V , because the volume of the crystal, V, appears in the density of states expression, Eq. (49).] The relative displacement is now given by

(

r N -2nR kT) I Mw’

‘I’

-

ti)

‘I2

(45)

where v is taken to be the velocity of sound in the crystal. The quadrupolar Hamiltonian may be written

where XQ(0) is the static interaction and is taken to be zero, and the effects of terms of higher order than XQ(’) are assumed to be negligible. For the order of magnitude calculations these terms are approximated as follows :

The term y is the multiplication factor that is introduced to account for the contributions to the field gradient due to the Sternheimer effect, covalent effects, etc. The important mechanism is the Raman process that results from a first-order perturbation treatment of the second-order term of the The transition probability W for this Raman process Hamiltonian is

2d’).

The integral counts all of the possible ways in which a phonon of lattice mode i can be absorbed and a phonon of the mode .j can be emitted. The integral is over all lattice frequencies up to w,, the maximum allowed frequency for a Debye distribution. The 6 function requires that the energy difference of the initial and final lattice states is just the energy exchanged with the spin system. p(w) is the density of lattice modes per unit frequency interval. Since the Larmor frequency wL is very small compared to most of the lattice frequencies, it may be assumed that p ( o i ) = p ( o j ) . For a

7.

NUCLEAR MAGNETIC RESONANCE

167

Debye distribution

where V is the volume of the crystal. The integration of Eq. (48) gives

This order of magnitude term gives the T 2 dependence of the high temperature transition probabilities and shows the dependence on the square of the quadrupole moment. Using the values Q = cm2, m = 3 gm/cc, u = 2.5 x lo5 cmlsec, 0, = 1013sec-', R = 2.5 x lo-* cm, T = 300"K, and y = 100 in Eq. (50) gives a relaxation time of 1.0 sec. Because of the high powers of several factors in Eq. (50), a different choice of constants would vary the predicted Tl by an order of magnitude. However, typical experimental values at room temperature are in the range of 0.1 to 10 sec. The multiplication factor y is one of the most difficult points of a quantitative comparison between theory and experiment. Similar order of magnitude calculations show that the direct process is negligible at high temperatures by a factor of lo-*.The Raman process is more effective because all of the lattice modes can contribute to relaxation and only modes at the Larmor frequency can contribute to the direct process. Also, the Larmor frequency is low on the lattice frequency scale and, therefore, the density of states is small at this frequency. Another Raman process results from the second-order perturbation treatment of in the Hamiltonian. This process is also negthe first-order term, XQ('), ligible by a factor of lo-'' because it involves the ratio of the quadrupolar Hamiltonian to the much larger energy of a lattice phonon. Using the Van Kranendonk model, Mieher28929has calculated the transition probabilities for the zinc-blende lattice. The h a 1 expression for the probability of a transition from the nuclear spin state m to state m + p ( p = f 1, f 2 ) is given by

W m , m + p ) = ~JQ, I’(T/~)~E(T/O)E,(~ ),

(51)

where C = 24/nd2u3a3,d is the mass density, u is the velocity of sound, a is the lattice constant, T is the lattice temperature, 0 is the Debye temperature, and the Q,, are the quadrupole operator matrix elements [see Appendix, Eq. (A2)]. The term E(T/O)is a numerical function and has been R. L. Mieher, Phys. Rev. Letters 4, 57 (1960). 29

R. L. Mieher, Phys. Reo. 125, 1537 (1962).

168

ROBERT LEE MIEHER

evaluated by Van Kranendonk” for the NaCl lattice for a Debye vibration spectrum. The terms E,(oo) depend on the details of the calculation, but refers to the fact that the temperature dependence is contained in the (a) E(T/O)and E,(co) is temperature independent. The E,(co)’s for the point charge zinc-blende lattice are (re)’ E,,(oo) = ,,[723.4

-

E*2(w) = -p-[645.4

+ 78a2],

r

+

312a2],

+

where a2 = a,2a22 a12a32 ~ 1 2 and ~ ~ a~, , a32 , a3 ~ are the direction cosines between the magnetic field and the [lOO], [OlO], and [Ool] crystal directions, and y is again the multiplication factor for the field gradient. TABLE VI SPINDEPENDENCE OF THE RELAXATION TIM@

[Q]’

= 1.63,

f(7’2) - 0.425 f (5/2)

~

’’

=

’

Tl(AISb Tl(AISb’

[%I2

I)

= 0.75

f 0.10 (experiment)

3,

= 4.28,

Tl(Rb8’) T1(RbE5)

=

T1(Rb8’C1) Tl(Rb8’C1)

=

a

0.69 (theory)

__ (5’2’ - 0.24 f (3’2)

(i)’;

=

1.23

1.027 (theory)

0.40 (experiment)

R. L. Mieher, Refs. 28 and 29.

Since it is the relaxation time Tl that is observed experimentally instead of the transition probabilities, it is necessary to express Tl in terms of W ( m , m + p). Using the spin temperature concept discussed in the introduction. it can be ~ h o w n ~ ’that .~~

30 31

C. J. Gorter, “Paramagnetic Relaxation.” Elsevier, Amsterdam, 1947. L.C. Hebel and C. P. Slichter, Phys. Rev. 113, 1504 (1959).

7.

169

NUCLEAR MAGNETIC RESONANCE

where Em = -yhHrn. When W,, as determined from Eq. (51) are substituted into Eq. (53) the relaxation time is given by

T;

= [(21

+ 3)/401’(21

- l)](eQ) C(T/o) E(T/o)[E,(oo) + 4E,(a)]. (54)

The term f(1) = (21 + 3)/Z2(21 - I) contains the entire dependence of TI on the nuclear spin. Since the sum involves the quadrupole operators Qfim, the spin factor f(1) is perfectly general for any cubic lattice and is independent of assumptions about the details of the coupling mechanism.

10

30

100

300

TEMPERATURE (OEGREES KELVIN)

FIG.11. Relaxation times of In“’ in InP. The dashed line is calculated using Van Kranendonk’s function E(T*), [T* = T/O]. The solid line shows the derivation froni the high temperature region where TI cc T - z for the Ranian process (R. L. Mieher, Refs. 28 and 29.).

Table VI gives a comparison of experimental and theoretical results on the spin dependence of Tl. If Eq. (52) is substituted into Eq. (54) the final expression for Tl is

1 Tl

-=

~

3305 f(l)[(ezyQ)Z/r o]C(T/O)ZE( T/O). 40

(55)

Therefore, the point charge model combined with Eq. (53) predicts an isotropic relaxation time for the zinc-blende lattice because the angular

170

ROBERT LEE MIEHER

o EXPERIMENTAL DATA6'Go

EXPERIMENTAL DATA 7 1 ~ 0

- THEORY (8.400

I

I

I

200

'K)

I

I

400 TEMPERATURE I'KI

I b] 600

FIG.12. Spin-lattice relaxation tinie TI of the Ga69 and Ga" resonances in Gap. Van Kranendonk's theoretical teniperature dependence for a two-phonon relaxation process is shown for a Debye teniperature 0 = 400°K. The points 0 and mare the T,'s for quadrupoles relaxation after correcting for paramagnetic impurity relaxation. (M. J. Weber, Ref. 16.)

1.0-

0.8

-

0.6

-

2- 0.4 -

-

0.2 -

0

0.2

0.4

0.6

0.8

1.0

Tl€tD

FIG. 13. Van Kranendonk's function E(T/O).The experimental data for 1n"'P is plotted as E(T/O)= TRZE(TR/0)Tl(TR)/T2Tl(T) where TR is room temperature and 0 = 400 K.(R.L. Mieher, Ref. 29.)

7.

171

NUCLEAR MAGNETIC RESONANCE

dependence of the transition probabilities exactly cancel in the tinal expression. It has r e ~ e n t l ybeen ~ ~ ,shown ~ ~ that this isotropy of quadrupole relaxation is independent of the detailed assumptions about the coupling dependence was observed for mechanism. E ~ p e r i m e n t a l l y ~ no ~ . ’angular ~ Tl for the In115resonance in a single crystal of InSb. TABLE VII EXPERIMENTAL TI’S MEASITRED Nucleus and compound

TI (s4

1n’”Sb 0.80 k 0.05 InSb’” 0.51 0.10 In’ ‘As 1.35 2 0.05 I~As?~ 1.45 k 0.10 In1’’P 2.93 0.05 1n~31 165.00 2 15 Ga7’Sb 12.0 i 0.5 GaSbtz3 1.55 k 0.1

* *

Debye 0 ( O K )

200 200 240 240 400 400

265 265

AT

77.4”K

AND

Nucleus and compound Gab9As Ga7’As GaAsT5 Ga69Pb G a 7 Pb GaP3Ib AI2’Sb AISb’” AISblz3 Ge7’

DEBYETEMPERATURES~

TI (set)

14.5 k 1.0 33.0 3.0 5.0 k 0.3 16.0 & 1.0 41.0 k 2.0 110.0 2 20.0 400.0 k 20.0 2.0 2 0.1 2.65 0.15 375.0 i 40.0

Debye 0 ( O K )

355 355 355 400 400 400 340 340 340 360

The experimental temperature dependence of Tl follows the prediction of the Van Kranendonk model quite closely between 77” and 300°K. Figure 11 shows the temperature variation of Tl for In”’ in InP. Weber16 has obtained similar results for GaP (Fig. 12). The function E(T/O) is given in Fig. 13 and the experimental points are for In”’P. If the temperature dependence is given accurately by (T/O)’E(T/O), then it is only necessary to know an experimental value of at one temperature and the Debye temperature in order to predict Tl at any temperature. Table VII lists the experimental T, sfor liquid nitrogen temperature and “best” values of the Debye temperatures. The Debye temperatures for InP, InAs, InSb, AISb, and G a P were determined by fitting the experimental data to the function E(T/O)for temperatures between 77” and 300°K. Since these measurements were made, P i e ~ b e r g e nhas ~ ~made extensive measurements of the specific heats of several 111-V compounds. Although the Debye A. Abragam, private comrnunicdticin J. Zak, Physrca 30, 401 (1964). 34 U. Piesbergen, Z. Naturforsch. 18a. 141 (1963). 32

33

172

ROBERT LEE MIEHER

temperatures listed in Table VII agree fairly well with the average of the specific heat Debye temperatures between 77" and 300"K, it appears that the effective Debye temperature for Tl is less sensitive to the details of the lattice vibration spectrum than is the specific heat Debye temperature. Since the functional form of the dependence on temperature and density of states is different for TI and the specific heat, agreement between the Debye temperatures for the two phenomena is to be expected only to the extent that the Debye spectrum is a good approximation. The Debye temperatures for GaAs and GaSb were calculated from the elastic constants (they agree well with the high temperature specific heat 0 ' s as measured by Piesbergen).

Nucleus and compound

(;)2E(;)

InSb"* GaSb' ' AISbI2' Ga7'Sb Ga7'As Ga' P

'

0.054 0.050 0.041 0.56 0.58 0.58

Nucleus and compound

(:)2E(;)

1n115~

1n115~s

1n'"Sb 1

~

~

GaAs7'

~

7

5

0.032 0.086 0.084 0.092 0.088

R. L. Mieher, Ref. 29.

The experimental data on In115Sb fit the Van Kranendonk theory very well between 77" and 300°K for a Debye temperature of 200°K. However, recent measurements by Clark35 on 1n"'Sb below 77°K indicate that in the temperature range of 10" to 20"K, Tl is a factor of 1.5 to 2 longer than would be predicted using the high temperature Ti's and the function E ( T / @ ) for the high temperature Debye 0.Weber36 has discussed the deviation of the TI temperature dependence from the Van Kranendonk theory for the alkali halides. A comparison of Tl for a given nucleus in different compounds is shown in Table VIII. The second column is a "normalized" relaxation time because the temperature dependence has been removed by multiplying the experimental T,'s by (T/O)'E(T/@).It is apparent that a very strong correlation exists for Tl of a given nucleus in different compounds once the temperature dependence is removed. A further comparison of the strength 35

36

W. G. Clark, to be published. M. J. Weber, Phys. Rev. 130, 1 (1963).

7.

173

NUCLEAR MAGNETIC RESONANCE

of coupling to the lattice can be made if the dependence on the nuclear spin and the quadrupole moment is normalized. Table IX shows two comparisons; in the second column the average values of Table VIII have been multiplied by Q2f.(I), in the third column the factor (Qq)2f’(Z) was used where q is the field gradient due to a p-valence electron of the free TABLE IX COMPARISON OF RELAXATION TIMES”

AI(Sb) Ga(P, As, Sb) In(As, Sb) (Ga, 1n)As (Ga, 1n)Sb Ge a

5.9 1.1 0.85 1.1 4.I 1.8

3.0 4.0 4.2 18.0 23.1 17.5

R. L. Mieher, Ref. 29.

atom (see Table IT). The latter comparison appears to give a better correlation of the data, indicating that covalent effects are important. Nevertheless, an accurate comparison of the ionic contributions would require consideration of the Sternheimer antishielding factor. Since both the antishielding factor and (l/r3) increase with increasing atomic number, it would be difficult to separate the two contributions to Tl in such a comparison. The calculations of Gill and Bloembergen (Section 3) show that the ionic and electronic contributions are about equal for first-order electric field effects. The table includes Ge, which is purely covalent, for comparison. A detailed theoretical calculation including covalent effects has not been made on relaxation times for the 111-V compounds. 6. OSCILLATING ELECTRIC FIELDS

The field gradients at a nuclear site produced by applying a static electric field to a crystal were discussed in Section 3. If these field gradients are made to vary at the Larmor frequency, y N H o , or at twice this frequency, 2yNH0, then the nuclear spin transitions Am = k1 or k2 will be induced. Such an experiment has been performed by Brun et ~ 1 . ~ ’ * in ~ * GaAs, and they obtained an independent measurement of the electric field gradientelectric field tensor R14. As in the static electric field experiment, GaAs was used because it is available as high resistivity material. 37 3R

E. Brun, R. Hann, W. L. Pierce, and W. H. Tanttila, Phys. Rev. Letters 8, 365 (1962). E. Brun, R. J. Mahler, H. Mahon, and W. L. Pierce, Phys. Rev. 129. 1965 (1963).

174

ROBERT LEE MIEHER

The GaAs single crystal served as the dielectric between two copper electrodes, and an rf coil was wound around the sample. Radio frequency voltage at twice the Larmor frequency was applied to the electrodes. The equilibrium nuclear magnetization was measured by the Bloch decay following a 90” pulse at the Larmor frequency applied to the rf coil. The Am = f2 transitions induced by the oscillating electric field begin to reduce the equilibrium magnetization when the transition probability is W l/Tl. A common spin temperature is maintained between the different spin levels because of the dipole-dipole interaction, i.e., T2 4 l/ W TI. The equilibrium magnetization for a spin of 3/2 subject to Am = 2 transitions is2 N

-

MO

M -

+ (8/5)WT1’

- 1

where M o is the equilibrium magnetization of Eq. (3). The transition probability is W = 71l x m n l 2g(m)/2h2,

(57)

where Zmn is the quadrupole matrix element between the states m and n and for I = 3/2 is given by 1xm,m*212

= eZQ2[(Kx -

J‘yyI2

+ 4Vxy21/48.

(58)

If the line shape is taken to be Gaussian, then

where Sw is the half-width of the line for a decrease of l/e. At the center of the line

w,,, = n

(60)

21Xm,21/2hdo

and

’

[(V’, - VYy)’

+ 4VXy2]

(61)

As discussed in the Appendix and in Section 3, the electric field gradients are I/. 1J =

1RijkEk k

since the piezoelectric effects are negligible. The transition probability will depend on the relative orientations of the electric field, magnetic field, and crystal axes. However, since there is only one independent term, R,,, if

7.

175

NUCLEAR MAGNETIC RESONANCE

RF AMPLITUDE = 36.1 VOLTS

O

I b ; ; : h ; : ; ; L b

FREQUENCY DIFFERENCE, KILOCYCLES FIG. 14. Resonance curve for Ga6' for electrically induced quadrupole saturation. (E. Brun et al., Ref. 38.)

both the electric field and magnetic field are parallel to the [Ool] axis, then

M , = M , 1 + n"'l.'Q'TlR,,'E,2]-1 15h’ 60

[

In order to determine R,, it is necessary to know Q , T,, 60, and E , . The line width for quadrupole saturation, 60, may not be the same as determined by the ordinary Zeeman resonance absorption curve. Therefore, the line width 6w was determined by measuring M , for electric fields of various frequencies near 2yH. A typical line width plot is shown in Fig. 14.

"0

10

20 30 40 RF AMPLITUDE,VOLTS

50

60

FIG.15. Saturation vs rf amplitude. Both electric and magnetic field are parallel to the [ 1001 axis. (E. Brun et al., Ref. 38.)

176

ROBERT LEE MIEHER

It will be noted in Eq. (63) that M , K [l + aV2]-'; a typical saturation curve of M , vs Vis shown in Fig. 15. Also, the angular dependence of the saturation for constant rf voltage (see Appendix) was observed to deviate slightly from the simple theory [Eq. (A18)] because of variations in 6w with orientation. For E and H parallel to [Ool], the experimental line widths were 6 ~ ( A s =~ 9.1 ~ ) f 1.3 kc/sec, b ~ ( G a =~ 12.0 ~ ) f 2.0 kc/sec, 6w(Ga7')

=

14.5 f 2.0 kc/sec.

The interaction parameter R14 as determined by Eq. (63) was

R ~ ~ ( A=s (2.0 ~ ~ &) 0.2) x 10’O

cm-',

(1.5 k 0.2) x 10"

cm-',

R,,(Ga69)

=

RI4(Ga7') = (1.5 k 0.2) x 10"

cm-

’,

which agree reasonably well with the values obtained in the static field experiment (Table 111).

7. ULTRASONIC SATURATION Saturation of the nuclear magnetization via quadrupole transitions may also be produced by ultrasonic excitation. If a quartz transducer is attached to a crystal and driven by rf voltages at yH or 2yH, then Am = & 1 or f 2 transitions will occur just as discussed above for oscillating electric fields. The interaction is the same as the strain (stress)-electricfield gradient effect discussed in Section 2. Such experiments were first reported by Proctor and Tanttila.39 Although most of the work has been done on alkali halides, the important points of both theory and experiment are similar for the 111-V compounds. A good detailed treatment of the problem has been given by Kraus and Tanttila.40 Equations (56) through (61) of Section 6 still apply where y j is now produced by the strains of the ultrasonic vibrations. However, an additional complication occurs because of a nonuniform distribution of the ultrasonic strain due to standing waves in the crystal. For a velocity of lo5 cm/sec and a frequency of 10 Mc/sec the wave length is cm or about lo5 lattice constants between nodes and antinodes. The saturation effects of these hot and cold spots could in principle be smoothed out if the spin-spin 39 40

W. G. Proctor and W. H. Tanttila, Phys. Rev. 98, 1854 (1955). 0. Kraus and W. H. Tanttila, Phys. Rev. 109, 1052 (1958).

7.

177

NUCLEAR MAGNETIC RESONANCE

interaction could produce a rapid energy diffusion through the crystal and thereby maintain a uniform nuclear spin temperature. This diffusion may be estimated by considering a random walk process in which energy is transported one lattice “step” in T, seconds and the total effective number of steps in T,/T,. Since T, sec and it becomes difficult to work with Tl > 100 sec, a very optimistic estimate of the spin-spin diffusion distance is (Tl/T2)1’2 lo3 lattice constants, which is much less than the distance between the nodes and antinodes. Therefore, there is a spatial variation of both the strain and spin temperature. Also, the amplitude of vibration is typically only cm, but this must be known accurately if the strain field gradient tensor C,,, is to be determined. Consequently, an ultrasonic saturation experiment requires very careful work if a quantitative value of C i j is to be obtained. On the other hand, the technique is more widely applicable than the static strain experiment because the magnitude of the h/J(T,T,), as compared to QK, h/T, interaction need be only QK, for the static strain case. Some of the above difficulties are removed if only relative saturation effects of two different nuclei in the same crystal are determined (although it is still necessary to determine the quadrupole saturation line widths as discussed in Section 6). Such measurements on GaAs and InSb have been reported by Denison and M a h ~ n . ~The l ratios of the electric field gradients were determined to be

-

-

-

-

Since the only positive results for static strain are for In”’ in InSb, it is not possible to compare results of the two techniques at the present time. A variation of the ultrasonic saturation experiment has been reported by Mahler et u E . ~ , By simultaneously applying ultrasonic vibrations at two different frequencies via separate transducers, a saturation was produced when the frequency difference was twice the Larmor frequency. resonance in The ultrasonic frequencies were 10 and 25 Mc and the GaAs was observed at 7.5 Mc. The Am = f 2 transitions could be produced either by 15-Mc phonons or by a Raman process. The presence of either 10- or 25-Mc vibrations alone produced no change in the nuclear magnetization. The Raman process was discussed in Section 5 and was shown to be the spin lattice relaxation mechanism. This would result in this experiment in 41

42

A. B. Denison and H. P. Mahon, Bull. Am. Phys. SOC. 7, 482 (1962). R. J. Mahler. H. P. Mahon, S. C. Miller, and W. H. Tanttila, Phys. Rro. Letters 10. 395 (1963).

178

ROBERT LEE MIEHER

a simultaneous absorption (emission) of a 25-Mc phonon and emission (absorption) of a 10-Mc phonon while the nucleus makes a Am = f 2 transition. Nevertheless, this mechanism was ruled out by simultaneous application of 15- and 25-Mc vibrations. The partial saturation due to the 15-Mc phonons was enhanced upon application of the 25-Mc phonons. The enhancement was independent of the exact frequency of the rf voltage applied to the 25-Mc transducer. Since the Raman process is not possible for this combination of phonons, the only possible interpretation is that 15-Mc phonons are produced by some nonlinear effect. Mahler et al. attribute the production of 15-Mc phonons to the anharmonicity of the elasticity of the GaAs crystal. When 10- and 25-Mc phonons are present they mix to produce 15-Mc phonons; application of 25-Mc phonons when 15-Mc phonons are present results in preferential conversion of the 25-Mc phonons to 10- and 15-Mc phonons, thereby enhancing the saturation. Just as the NMR C.W.absorption signal is detected by the additional absorption of rf energy at the Larmor frequency, it is possible to detect the additional absorption of ultrasonic energy due to the quadrupole transitions by the change in accoustical Q of the sample. Resonance absorption of ultrasonic energy may occur at either yH or 2yH. This has been observed by Menes and B01ef~~ for In"' in InSb. This technique could be useful in experiments where a conducting sample must be thick compared to the rf skin depth. 8. NUCLEAR POLARIZATION BY HOT ELECTRONS

It was first pointed out by O ~ e r h a u s e that r ~ ~ the nuclear polarization in a metal could be enhanced by rf saturation of the electron spin resonance of the conduction electrons. The nuclear spin system is thermally coupled to the electrons at the Fermi surface via the scalar contact interaction, A1 S (the same interaction that produces the Knight shift), and this coupling is much stronger than the quadrupole couplings to the lattice vibrations discussed in Section 5. The Z,S, terms of the contact interaction produce the mutual electron-nuclear spin flips and the energy difference is taken up by the kinetic energy of the electron, i.e., it moves to a slightly different k state.'.' The electron spin system is in thermal contact with the lattice via a spin-orbit coupling mechanism. Figure 16 shows the energy levels for a nucleus with spin and a positive magnetic moment and an electron with a positive g factor. Although the electron-nuclear hypefine interaction is responsible for the mutual spin flips, the Knight shift is so small that it is negligible on the scale of Fig. 16.

4

43

44

M. Menes and D. I. Bolef, Phys. Rev. 109, 218 (1958). A. W. Overhauser, Phys. Rev. 92, 41 1 (1953).

7.

179

NUCLEAR MAGNETIC RESONANCE

The vertical transitions W(a b) and W(c * d) are produced thermally by the electron spin relaxation mechanism and the cross transitions W(a-d) by the I , S , terms. These are both thermal processes which produce the normal Boltzmann population distribution for all of the levels f--,

where A

=

iky,Ho, 6

=

ihy,Ho, and it has been assumed that A/kT < 1.

r n s = + lmr’ ~=-1/2 T ~ m ~ : ~ ~ z l ; A

mS= -112

2s

d

4)

FIG.16. Energy level diagram for an electron and a nucleus ( I = with a weak hyperfine interaction. The electron spin temperature is T,, the cross transitions are determined by either the lattice temperature TLor the reservoir temperature TR.

The nuclear polarization is just that of a Boltzmann distribution for the lattice temperature TL.

N(m, N(m,

= -+)

-

= 4-i)

+ Nd

N a + N b =

N,

[L-g].

(65)

An Overhauser polarization is produced by rf saturation (i.e., T, -+ 00) of the vertical transitions, Ams = k1, Am, = 0. However, the cross transition W(a t,d) will still maintain a Boltzmann population ratio between states a and d. Therefore, the level populations are now

and the nuclear polarization is

N(m, = -$) N , + Nb N(m, = -ti) N , -t Nd =

[I-

1’

2(A + 6) kTL

(67)

The nuclear polarization of Eq. (67) now corresponds to the normal electron polarization (T, = TL) and is about three orders of magnitude greater than Eq. (65). Other polarization mechanisms involving saturation

180

ROBERT LEE MIEHER

of the electron spin resonance of paramagnetic impurities in insulators have been widely s t ~ d i e d . ~ ’ , ~ ~ Feher47 has proposed a polarization mechanism that does not require a saturating rf field. This mechanism requires only a difference between the electron spin temperature and the mean kinetic temperature of the electrons. Such a temperature difference can be obtained for “hot electrons” in semiconductors. It was stated above that the energy change of the electronnuclear spin system for a cross transition is taken up by the kinetic energy of the electrons. If the electron kinetic energy is not in thermal equilibrium with the lattice vibration, i.e., the electrons are hot, then the effective temperature for the cross relaxation process is no longer TL. Feher calls this effective temperature the reservoir temperature TR. If T R is very large but T, is considerably cooler, then the level populations are

and the nuclear polarization is

Note that Eq. (69) gives a negative polarization; either sign of polarization may occur depending upon the signs of y e , yN, and TR- T,.If the resulting polarization is given in terms of an enhancement factor a, defined as the ratio of the lattice temperature TL,to the nuclear spin temperature TN and if relaxation mechanisms other than that of conduction electrons (e.g., paramagnetic impurities or quadrupole coupling) are taken into account, then the enhancement factor is given by48

where z is the total nuclear spin lattice relaxation time, 7 R is the conduction electron part, and zL is due to all other mechanisms. Since the other relaxation processes “short circuit” the polarization because they tend to maintain the normal populations corresponding to TL, the polarization is performed W. A. Barker, Reo. Mod. Phys. 34, 173 (1962). C. D. Jeffries, “Dynamic Nuclear Orientation.” Wiley (Interscience), New York, 1963. 47 G. Feher, Phys. Rev. Letters 3, 135 (1959). 48 W. G. Clark and G. Feher, Phys. Ren. Letters 10, 134 (1963): 12, 717 (1964). 46

7.

NUCLEAR MAGNETIC RESONANCE

181

at liquid helium temperatures where quadrupole relaxation is negligible and very pure materials are used to avoid paramagnetic impurities. One technique for establishing a difference between T, and TRis acceleration (or deceleration) of electrons in an electric field gradient. If there are different relaxation times for establishing TR and T,, then the two temperatures will be different if they are always changing. Clark and Feher48

FIG. 17. Enhancement of the nuclear resonance signal of in InSb as a function of the applied electric field. The magnitude of M , represents the value of the magnetization in a magnetic field of 12.7 kOe after an elapsed time interval of 10 min. Similar plots with larger enhancements are obtained for SbI2' and Sb'13. (W. G. Clark and G. Feher, Ref. 48.)

have used a wedge-shaped sample of InSb to produce an electric field gradient for the hot electrons. Since the term (l/&- 1/TR) appears in Eq. (70), the sign of a will depend on whether the kinetic temperature is increasing or decreasing. Such a change in sign of a was observed when the current direction was reversed for the wedge-shaped sample. Similar effects are to be expected if the electric field gradients are produced by impurity concentration gradients. Weger49 has discussed a polarization mechanism that can occur in homogeneous material with a uniform electric field. This mechanism requires a different kinetic temperature TR for spin up and spin down electrons. This situation may occur for hot electrons if the spin resonance frequency 49

M. Weger, Phys. Rec. 132, 581 (1963).

182

ROBERT LEE MIEHER

we = y,H and the cyclotron frequency w, = ehH/m*c are comparable

-

to the kinetic energy kT, and the Fermi energy EF. These conditions occur for InSb because y,(InSb) -507, (free electron) and m* is very small.

TIME 1 HOURS) FIG. 18. Enhancement a of the nuclear resonance signal for In"', Sb'", and SbIZ3as a function of time. The applied field was 0.9 V/cm. (W. G. Clark and G. Feher, Ref. 48.)

Figures 17 and 18 show some typical polarizations observed by Clark and Feher in InSb which apparently correspond to the mechanisms discussed by Weger. The experimental polarizations shown are rather small because tL < tR. Recent experiments" in higher purity materials have yielded better results. These experiments are still in progress because there are several ways to establish T, # TR or TR (spin up) # TR (spin down), and, therefore, carefully controlled samples and experimental conditions are necessary in order to separate the various mechanisms. Such work will undoubtedly yield much information about hot electrons in magnetic fields in addition to providing a simple way to polarize nuclei.50a W. G. Clark and G. Feher, private communication. "'An interesting ESR study of conduction electrons in InSb has recently been made by G u e r ~ n This . ~ ~experiment ~ is related to the above discussion and gives direct evidence of the homogeneity or inhomogeneity of the nuclear polarization for the different techniques of producing hot electrons. Also, the density of conduction electrons at the nuclei - ~ ~ $ ( R 3 , , ) ~ z / ~ $ ( R=, n1.70 ] ~ z _+ 0.1. is determined to be I$(R,,)l2 = (9.35 k I ) x 10" ~ r n and M. Gueron, Phys. Rev. 135, A200 (1964).

7.

183

NUCLEAR MAGNETIC RESONANCE

Appendix

Several of the experiments discussed in this article involve electric quadrupole interactions that are produced by strains or electric fields. Therefore, some relations and transformations of second, third, and fourth rank tensors are discussed here for the case of 43 m tetrahedral symmetry. The quadrupole Hamiltonian is discussed by Cohen and Reif’ and may be written X Q=

(‘41)

Qpv-p

c

where

+ l)],

Qo

=

A[3IZ2- I(I

Qki

=

A(IkIz + IZI,),

Q i 2

= AI,’,

vo = Vz, v, 1 = vx, k ivy, v* 2 = ft v,, - vyy, ivxy

*

(A21

in which A = eQ/41(21 - I), I ,

=

I,

il,,

and the terms V,,, Vxy,etc., are the electric field gradients at the nucleus, Q is the nuclear quadrupole moment, and I is the nuclear spin. The operators I , and I , have the following matrix elements

(m)LJm> = m (m

k 111,Im)

=

[ ( I ‘F m)(I -t m

+ l)]”’.

(A3)

The matrix elements of the quadrupole Hamiltonian are ( m / M Q l m )= A[3m2 - 1(1

+ l)]VO

( m i-l)MQlm)= A(2m _+ I)[([ T m)(I (m

* m + l)]1i2VT,

21%‘Qlm)= A[(I T m)(I T m - 1 ) ( 1 f m

(A4)

+ 1)(1 + m + Z)I”2Vv,2.

The electric field gradient tensor at the nucleus, F j , has only 5 independent components because it is symmetric and the trace is zero (because V’V= 0 at the nucleus for the charge distributions that produce field gradients). The tensor is diagonal in its principal axes coordinate system and has only two independent components,

184

ROBERT LEE MIEHER

In general it is necessary to know the Vp's for other than the principal axes system because the magnetic field is seldom along one of the principal axes and it is often desirable to make an angular dependence study of the interactions. It is possible to obtain a general expression in terms of Euler angles for transforming the %'s like second-order spherical However, the general expression is more complicated than is necessary for this discussion. The electric fields and stresses are usually applied along some simple crystalline direction, and the magnetic field may be rotated in a principal plane (or a plane having a simple relation to the principal axes) of the electric field gradients. Therefore, the transformation of I/ij to the new coordinate system may be quite simple. The transformation is V = RV,R-',

('46)

where R is the rotation matrix that rotates the principal z axis of the electric field gradient away from the magnetic field. This is obtained by considering the transformation necessary to express some vector r in the new coordinate system Rxx Rx, Rxz

(:)

r' = y'

=

:$(!).

Rr = R,,

(Rzx

('47)

For example, a rotation of 8 about the x axis is given by

where C = cos 8 and S for this simple case by

=

R-ye)

sin 8. The inverse of the matrix, R-', is given

=

R(-8)

vj

=

i' : 0

".

0

-s c

Since is a second-rank tensor, a third-rank tensor will determine the electric field gradients produced by an applied electric field, and a fourthproduced by an applied stress. With one rank tensor will determine the exception these third- and fourth-rank tensors will have the same symmetry properties as other tensors51 of the same rank, e.g., piezoelectric and elasticity tensors, for the same crystalline symmetry. The exception is that

vj

51

C . J. Smith, Solid State Phys. 6, 175 (1958).

7.

NUCLEAR MAGNETIC RESONANCE

185

an additional reduction of the independent components of the tensors sometimes occurs because the trace of K j is zero. It is quite common to use Voigt’s subscript convention for the higher rank tensors. For example, the relation

is written in Voigt’s notation

c

x, =

(ni,n = 1 , . . . , 6)

c,,x,

(AJ1)

n

where

1

= xx,

4

2

= yq’,

5 = zx,

3

= ZZ,

6

=

yz,

= XY.

The following examples are chosen to correspond to the experiments discussed in the text. The third-rank tensor for the case of 43 m symmetry has only one independent c ~ m p o n e n t , ~ R I 4 .Therefore, the relation V{ J. =

c RijkEk k

has the very simple form

W

=

c. :::I

R14 E,

0

Ex ,

6413)

where the x, y, z axes are the (100) directions of the zinc-blende lattice. In the Stark experiment of Gill and Bloembergen” the electric field is applied along a [ l l l ] body diagonal. Therefore, in the coordinate system of the crystal axes the field gradients are

The measurements are made with the magnetic field also parallel to [ l l l ] (a principal axis for so the field gradients are

vj).

186

ROBERT LEE MIEHER

Since the quadrupole splitting measures the relative shifts of the energy levels, it is seen from Eqs. (A2) and (A4) that V,, = 2 R 1 4 E / J 5 is the quantity measured. In the experiment of Brun et ~ l . , ~ ' an alternating electric field parallel to the [Ool] axis of the crystal produces Am = + 2 spin transitions. The resulting field gradients for rotation of the crystal about the [ 1001 direction and the [ 1101direction are

V([100] rot. axis)

=

Rl,E

e -:a

(A 16)

0

V([110] rot. axis)

=

R14E 0

-Cz

-CS

,

-cs

where 8 is now the angle between the electric field and the magnetic field. In this experiment the measured quantity is the saturation which depends on

I<m

f 21xQlm>IZ

a $Vxx

- %y)'

+Ii:y.

(A171

Therefore, the expected angular dependence of the saturation would depend on

The fourth-rank tensor for the case of 43 m symmetry has three independent components, C11, CIz, C44.Therefore, the field gradient-stress relations are given by Vxx = C,lXXX

+ C12(X, + X,,)

VXy= C44Xxy(cyclic),

(cyclic),

(A191

where Xij are the stress components. However, this relation is simplified by the requirement Xi Fi = 0. This requires that C1, = -$Cll. Therefore, the Vj are

7.

187

NUCLEAR MAGNETIC RESONANCE

x(i a a),

In the experiment of Shulman et al.,” the stress is applied to single crystals of InSb along [lo01 and [110] directions. For stress along [lo01 the stress tensor is

x

=

where X is the applied stress (the sign of X depends on whether compression tensor is or extension is used). Therefore, the

cj

Cll

0 (A22) 0

and this is independent of rotations about [loo]. For stress along [110] the stress tensor in the crystalline coordinate system is (A231 where X is the stress along [110]. Therefore, the

Kj tensor is (‘424)

Rotating the x axis into [ 1101 gives ClJ2

w,

=

XI2

+ c44 0 0

0 c11/2

- c44 0

-C11

1

(A251

.

Rotations about the [110] axis give the field gradient along the magnetic field as V,, = X,[(Cll/2 - C44)sin28 - C l l cos28], where 8 is the angle between the magnetic field and the [OOl] crystalline direction. The relation between the electric field gradients and strain may be found using C , , and C44if the elastic constants are known.

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CHAPTER 8

Electron Paramagnetic Resonance Bernard Goldstein I.

INTRODUCTION .

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

11. THE PARAMAGNETIC RESONANCE CONDITIONA N D THE SPIN HAMILTONIAN . . . . . . . . . . . . . . . .

. . . 111. GALLIUM ARSENIDE 1 . Manganese . . . . . . . . . . 2. Iron 3. Zinc, Cadmium. and Nickef 4. Conduction Electrons .

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

. . . IV. INDIUM ANTIMONIDE.. . . V. GALLIUM PHOSPHIDE . VI.

Rksud

AND

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

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

CONCLUDING REMARKS .

.

.

.

.

.

.

.

.

189 190 191 193 194 196 198 199 200 200

I. Introduction The initial report of the observation of electron paramagnetic resonance' has opened up a field of solid state physics which continues to provide knowledge of a unique character. Some of the more important of the specific properties measured by EPR techniques are the magnitude of the splitting of electronic energy levels by crystalline fields, the identification of different nuclear species by their nuclear magnetic moments, the charge state of impurities, and the spatial orientation and local symmetries of lattice defects. Over the past ten years, paramagnetic resonance phenomena have been extensively studied in elemental semiconductors.' More recently, as single crystals have become available, work has expanded into the compound semiconductors composed of elements from columns I11 and V of the periodic table. In this chapter, we will present and discuss briefly the existing results of EPR studies on some of these compound semiconductors.

' E. J. Zavoisky, J . Phys. ( U S S R ) 9, 21 1 (1945).

* G. W. Ludwig and H. H. Woodbury, Solid

State Phys. 13, 223 (1962).

189

190

BERNARD GOLDSTEIN

11. The Paramagnetic Resonance Condition and the Spin Hamiltonian

If a free paramagnetic ion having angular momentum J is placed in a uniform magnetic field H , its energy levels are given by E = gBHM. Here, M , the magnetic quantum number, is the spatial projection of J on H and ranges in integral steps from - J to +p.I is , the Bohr magneton eh/4nmc, and g is the so-called “spectroscopic splitting factor.” The “g-value” is given by the Lande formula g = 1

+ [J(J + 1) + S(S + 1) - L(L + 1)]/2J(J + l),

where S and L are the spin and orbital angular momentum vectors, respectively. In the presence of rf quanta of energy hv = gBH, transitions will be induced between adjacent levels governed by the selection rule for magnetic dipole transitions AM = +1. Both upward and downward transitions have equal a priori probability. However, since in thermal equilibrium the lower levels are more densely populated the net result is that energy is absorbed by the electronic system from the rf field. It is this loss in energy (for example, from a resonant cavity) which is detected as the EPR signal. If the nucleus of the paramagnetic ion also possesses a net angular momentum, and there is interaction between the electrons and the nuclear spin I, then each electronic level characterized by an M value is itself split into 21 + 1 equally spaced levels. The allowed transitions now have an added selection rule, Am = 0, where m is the projection of the nuclear magnetic moment on H . Electron paramagnetic resonance spectra are almost always taken at fixed radio frequencies and with a slowly varying dc magnetic field H . This is by far the simpler experimental situation to achieve and the most facile one to work with. The displayed signal is almost always the derivative of the power absorption rather than the absorption itself. This makes for better sensitivity, resolution, and accuracy. General experimental considerations are treated in some detail in a review article by Bleaney and Stevens. When the paramagnetic atom is in a crystal its spin has, of course, many more interactions than those of the free ion. The spin Hamiltonian is the analytical expression which describes the interactions of the spin of the paramagnetic atom3a with its environment. In addition to interactions with the external magnetic field and the nuclear magnetic and quadrupole B. Bleaney and K. W. H. Stevens, Rept. Progr. Phys. 16, 108 (1953). 3”We can speak now of either a paramagnetic atom, ion, or center. A vacancy complex in a crystal, for example, has dangling bonds and unpaired electrons which can result in a net spin. This can be treated in exactly the same way as is the spin of a free atom or a free radical.

8.

ELECTRON PARAMAGNETIC RESONANCE

191

moments, these include the crystalline fields and the other paramagnetic centers. A representative spin Hamiltonian 2‘ can be expressed in the form Z s= p S . g . H

+ V ( S )+ A S . 1

(1)

The first term on the right is the interaction between the spin and the external magnetic field (the Zeeman term); the second is between the spin and the crystalline field which together with the Zeeman term describes the fine structure of the resonance spectrum ; the last is the interaction between the electron spin S and the nuclear spin I . There may be other terms in the spin Hamiltonian describing the direct interactions between the nuclear spin I and the magnetic field H, between the nuclear spin I and the nuclear quadrupole moment, and between the electron spin S and the nuclear and electron spins of neighboring atoms. These latter terms are small and often need not be taken into account. It should be pointed out that in Eq. (1) the g-value for an atom in a crystal is now an experimentally determined quantity and a tensor, its value being determined in part by the spin-orbit coupling and the effects of the crystal fields on the orbitals. The values and behavior of the g-tensor are often used for the identification of paramagnetic centers in the same host crystal. A schematic representation illustrating the ways in which the various interactions split electronic energy levels is shown in Fig. 1 for the case of the M n + + 6S,,, ion in a cubic crystalline field. A much more detailed and complete treatment of the general and spin Hamiltonian can be found in a review article by LOW.^ 111. Gallium Arsenide

Gallium arsenide, by virtue of the advanced state of the technology of its controlled crystal growth, has long been the most common III-V compound host lattice for several fields of study. To these has now been added electron paramagnetic resonance. Almeleh and Goldstein5 and Bleekrode et d 6have reported on manganese in GaAs, presenting essentially identical results. DeWit and Estle’ and Bleekrode et al.* have measured the electron paramagnetic resonance properties of iron in GaAs, while Goldstein and Almeleh’ have reported a resonance spectrum which appears to be iron except that it is introduced W. Low, Solid State Phys. Suppl. 2, 39 (1960). N. Alrneleh and B. Goldstein, Phys. Reu. 128, 1568 (1962). R. Bleekrode, J . Dieleman, and H. J. Vegter, Phq’s. Letters 2, 355 (1962). M. DeWit and T. L. Estle, Phys. Reo. 132, 195 (1963). R. Bleekrode, J . Dieleman, and H. J. Vegter, Philips Res. Rept. 17. 513 (1962). B. Goldstein and N . Almeleh, Appl. Phys. Letters 2, 130 (1963).

’

192

BERNARD GOLDSTEIN

by heat treatment anneals and the density of spins is orders of magnitude greater than the iron density. DeWit and Estle have also reported on nickel in GaAs." Title" has detected the paramagnetic resonance absorption of neutral zinc and cadmium by applying a uniaxial stress to the crystal. Finally, Duncan and Schneider" have detected the spin resonance of the

rn

M

5/2 3/2 ,'_ I

112 -1/2

-3/2 -5/2 512 1/2

-3/2

3/2

-1/2

-5/2

!!

\' \\ '\ \ - 3 / 2 \

\

\ \-5/2

Zero magnetic field

High rnognetic field

Hypertine interaction

FIG. 1. Energy levels of manganese in a cubic crystalline field. The crystal field first splits the 6S,,, level into two levels, one quadruply degenerate and the other doubly degenerate. At high external magnetic fields, these degeneracies are lifted as shown. The hyperfine interaction then splits each of the six M-levels into the six hyperfine levels. The transitions, illustrated for the two highest M-levels, are governed by the selection rules AM = il,Am = 0. The separations between levels do not correspond to the actual energies of the transitionsfor the actual separations see W. Low, Phys. Reu. 105, 793 (1957). lo I'

M. DeWit and T. L. Estle, Bull. Am. Phys. SOC.7 , 449 (1962). R. S. Title, IBM J. Res. Develop. 7, 68 (1963). W. Duncan and E. E. Schneider, Phys. Letters 7 , 23 (1963).

8.

ELECTRON PARAMAGNETIC RESONANCE

193

conduction electrons in GaAs. In the following paragraphs these results are discussed in some detail. 1. MANGANESE

Manganese is an acceptor in GaAs and substitutes for Ga in the lattice. Its energy level obtained from Hall and optical data was found to be about 0.09 eV above the valence band.5 Since the valence electrons are involved in the mostly covalent crystal bonds, the ground state is an S-state and its paramagnetic resonance behavior stems from the five 3d electrons.

31 10

H (GAUSS)-

3433

FIG.2. Electron paramagnetic resonance spectra at 77°K of manganese in gallium arsenide for different directions of magnetic field H in the (110) plane. (After N. Almeleh and B. Goldstein, Ref. 5.)

Typical spectra at 77°K are shown in Fig. 2 for various magnetic field orientations in the (110) plane. They consist of six hyperfine lines, characteristic of the Mn55nuclear spin I of 3. These lines are quite broad, having a half-width of about 28 G."" The five fine-structure lines ordinarily seen In silicon, for example, resonance lines are typically from 1 to 3 G in half-width.

194

BERNARD GOLDSTEIN

for this atom with its electron spin S of 3 are not resolved. Both the g-value and the hyperfine interaction constant A are isotropic; their values are 2.003 and 57 G, respectively. The value for A lies between that found for manganese in germanium (48 G)13 and that found for manganese in zinc sulfide (68 G).14 Since the bond of GaAs is less ionic than that of zinc sulfide but more than that of germanium, this behavior is consistent with the general observation that the more ionic the bond, the greater the hyperfine interaction." This has been interpreted in terms of the greater localization of the electron cloud in the case of the ionic bond.16 The cubic field splitting parameter /a( as deduced from fitting the angular variation of the EPR spectrum to the spin H a m i l t ~ n i a n ' ~ L#

= g@H*

S

+ u/6[Sx4 + SY4+ SZ4- (S/5)(S+ 1)(3S2 + 3s - l)] + AS - I (2)

is 15.5 G. No change in the spectrum was found at 4°K. No fine structure or change in line widths was observed. This suggests that the effect of spin-lattice interactions on line width, which should be strongly temperature dependent, is negligible. It has been suggested' that the unusual breadth may be due to hyperfine interactions with nearest neighbor arsenic and gallium atoms, all of which have nuclear spins of 3. 2. IRON

Iron can be incorporated into GaAs either by diffusion or by introduction from the melt. It, too, is most likely substitutional, replacing gallium in the lattice. It is a p-type impurity and introduces a level 0.37 eV above the valence band." Its spin resonance spectrum is that of a 6Ss,z state in a cubic lattice, consistent with the loss of its valence electrons to the crystal bonds and leaving 5 unpaired 3d electrons to determine its EPR behavior as they do for manganese. The results of DeWit and Estle' and Bleekrode et aL8 are in good agreement. A typical spectrum at 1.3"K is shown in Fig. 3. The five-line fine structure shows symmetry in the (001) direction. The half-widths of the lines are about 54G, and again this rather large value appears to be due to hyperfine interactions with nearest neighbor gallium and arsenic G. Watkins, Bull. Am. Phys. Soc. 2, 345 (1957). L. M. Matarrese and C. Kikuchi, J. Phys. Chem. Solids 1, 117 (1956). l 5 J. S. Van Wieringen, Discussions Faraday SOC.19, 118 (1955). l 6 0. Matumura, J. Phys. Soc. Japan 14, 108 (1959). I' W. Low, Phys. Rev. 105, 793 (1957). '* F. A. Cunnell, J. T. Edmond, and W. R. Harding, Solid-state Electron. 1. 97 (1960). l3

l4

8.

ELECTRON PARAMAGNETIC RESONANCE

I .-A

195

1. 2

2

i

3-4

1-5

J

5-6

FIG.3. Electron paramagnetic resonance spectrum at 1.3"K of iron in gallium arsenide with the magnetic field H in the (091) direction. The M-levels involved in each transition are indicated at the top of each line. (After M . DeWit and T. L. Estle, Ref. 7.)

2ooo0 [OOll

10

20

30

40

50

60

I t I 11

70

80

90

" 101

t?(DEG)

FIG. 4. Angular dependence of resonance line position for iron in gallium arsenide. The solid lines give the theoretical variation predicted by Eq. (2) with g = 2.046 and (uI = 374 G. (After M. DeWit and T. L. Estle. Ref. 7.)

196

BERNARD GOLDSTEIN

nuclei. The g-value is isotropic and is 2.046. The fine structure splitting la1 is 374 G. The theoretical angular dependence of the lines calculated by inserting these values into the spin Hamiltonian equation (2), using S = 2, is given by the solid lines of Fig. 4.The very good agreement with experimental data is clearly evident. In addition to the ‘‘allowed’’ transitions which give rise to the five finestructure lines, DeWit and Estle have also detected nine of the ten possible “quasi-forbidden” transitions between states for which AM is greater than 1. They have studied this spectrum in some detail and have found that the positions and intensities of the transitions are in good agreement with the values calculated from the spin Hamiltonian, except that the over-all intensity of the spectrum is 5 to 10 times weaker than expected. Goldstein and Almeleh’ have reported an EPR spectrum in GaAs which has been heat treated, but not purposely doped. The spectrum grows with heat treatment temperature from 900” to 1200°C. It is identical in all essential details to the one shown in Figs. 3 and 4. The g and la1 values also agree with those quoted above. The spectrum appears only when the GaAs is p-type. The interesting part of their work is the fact that they have gone to some lengths to measure the spin densities and the iron concentrations and have found a total lack of correspondence between them on both an absolute and a relative basis. The total iron concentrations were measured with a mass spectrograph with a sensitivity of about 3 parts per billion and an uncertainty of less than 50%. The spin densities were measured by comparing the spectrum with those of pink ruby, manganese in GaAs, and the conduction electron line in silicon at 4°K. After first establishing internal consistency among the ruby, manganese, and silicon samples, they find that the spin density in their samples is between 2 x 1019/cm3and 5 x 1019/cm3,while the iron concentrations vary only between 2 x 10’6/cm3 and 2 x lOI7/cm3. Not being able to account for this discrepancy, they suggest the possibility that these high spin densities may be due to latent or intrinsic crystal defects, and suggest some simple defects which could have five unpaired electrons. It is worth noting in this regard that there have been three independent reports of latent or intrinsic defects in GaAs which have been detected approximately in these concentrations. 9--2 3. ZINC, CADMIUM, AND NICKEL If a hole that is loosely bound to a shallow acceptor is considered as free, then its ground state at k = 0 is fourfold degenerate and can be characterized J . Blanc, R . H. Bube, and L. R. Weisberg, J . Phys. Chem. Solids 25. 225 (1964). ’OC. S . Fuller, and K. 9. Wolfstirn, J . Appl. Phys. 34, 1914 (1963). B. Chakraverty and R. W. Dreyfus, Bull. Am. Phys. SOC.9. 49 (1964).

’’

8.

197

ELECTRON PARAMAGNETIC RESONANCE

0 KG/CM2

749

1

I 796

843

890

H IN GAUSS

FIG.5. Electron paramagnetic resonance of the bound hole in zinc-doped gallium arsenide under various uniaxial stresses. (After R. S. Title, Ref, 11.)

by J = $. If a uniaxial stress is applied, then this fourfold degenerate level will split into two doubly degenerate levels. Their separation is proportional to the applied stress, and transitions are now possible between these two doublets. Title has used this effect, first reported by Feher et ~ 1 . ~ ’ for acceptor resonance in silicon, to detect the EPR absorption for zinc and cadmium.l 1 22

G. Feher, J. C. Hensel, and E. A. Gere. Phys.

Reti.

Letters 5, 309 (1960).

198

BERNARD GOLDSTEIN

Figure 5 shows the appearance of the resonance signal as a uniaxial stress is increased on a sample of zinc-doped GaAs. Here, the applied stress and the external magnetic field H are perpendicular. A quite strong dependence of the magnitude of the resonance signal on the applied stress was found. The g-value is 8.1. For cadmium in GaAs, stresses above 2 x lo3 kg/cm2 were required, apparently because of the lower doping level. The comparable g-value is 6.7. The fact that this is somewhat lower than the value for zinc indicates a more tightly bound hole for cadmium. The EPR spectrum reported for nickel consists of one line about 130 G wide, and agrees with a 3d’ configuration in a tetrahedral crystal field environment with a spin of $. The g-value is 2.106 and is isotropic. Note, however, that with a line width of 130 G small changes in g might not be detectable.

ELECTRONS 4. CONDUCTION Unbound electrons in the conduction band ( S = J = $) can give rise to a paramagnetic resonance absorption, but it is often more difficult to observe than that of either bound electrons or atoms. There may be several reasons for this. For example, at appropriately low temperatures the electron concentration may be too small; conversely if the concentration is too large, the resonance line may broaden and ultimately disappear. Most important, however, the effective magnetic moment of the electrons may be much different from the Bohr magneton. This can be seen from the expression for the effective magnetic moment which has been calculated by Roth et aLZ3to be perf= p[1 - (mo/m* - 1) A/(3E,

+ 2A)]

(3)

where p is the Bohr magneton, m* is the effective mass, A is the spin-orbit interaction, and E , is the band gap of the semiconductor. Since it is not uncommon to have m* = 10-2rno, perfcan be very different from B, and the g-value (g = 2 p / p ) can be very much different from 2. This has the effect of locating the resonance line at magnetic fields far removed from those at which resonance lines are more customarily observed. For GaAs, E, = 1.53eV, A is about 0.33eV,24 and m*/m, is about 0.074.25 Due to uncertainties in the last two of these values, the expected g-value would lie somewhere in the range from 0.28 to 0.56. Duncan and Schneider” have isolated an EPR line at 4.2”K for n-type GaAs having a g-value of 0.52, and have attributed this to the conduction electrons. 23

*’

L. Roth, B. Lax, and S. Zwerdling, Phys. R w . 114, 90 (1959). R. Braunstein, J . Phys. Chem. Solids 8, 280 (1959). T. S. Moss and A. K. Walton, Proc. Phys. SOC.(London) 74, 131 (1959)

8.

ELECTRON PARAMAGNETIC RESONANCE

L

I

I

-

I

I

-

I

I

1

199

II

I

I

-

-

-

-

2+

i----1

( b 1 - 1

I

I

I

1

1

I

I

I

FIG.6. Electron paramagnetic resonance of conduction electrons in indium antimonide. The line broadens and ultimately disappears at higher electron concentrations. (After G. Bemski, Ref. 26.)

In their work they also report the observation of a low-field resonance with a g-value of 1.2, well outside the range for conduction electrons. They suggest that this may be due to deep donor states.

IV. Indium Antimonide

In indium antimonide, only the conduction electron resonance has been reported.26 The g-value is isotropic and depends on the conduction band electron concentration. As the concentration increases from 2 x 1014/cm3 to 3 x 10”/cm3 the g-value decreases from 50.7 to 48.8. These values are in good agreement with those calculated from Eq. (3) in which m* varies 26

G. Bemski, Phys. Rev. Letters 4, 62 (1960).

200

BERNARD GOLDSTEIN

with the Fermi level from 0.0131m0 to 0.0136m0, E , = 0.25 eV, and A = 0.9 eV. The resonance line broadens with increasing electron concentration until finally no line is observed at concentrations over 1016/cm3.The shape of the line and the broadening effect are shown in Fig. 6. V. Gallium Phosphide Two resonances have been reported for gallium phosphide. One is mangane~e,'~and the other is ironZ7Although the manganese spectrum consists of only one line, it is nonetheless presumed to be an S-state ion with J = S = 3 as has been found for GaAs. It is thought that the other lines are broadened beyond detection by crystalline strains. The g-value is 2.002, and the hyperfine interaction A is given as 60.5 G. The iron spectrum has five fine-structure lines, showing directly that J = S = 3. Again, iron is presumed to substitute for gallium in such a way that five 3d electrons determine the EPR behavior, with the valence electrons contributing to the crystal bonds. The g-value is 2.023 ; the fine structure splitting constant (a(is 429 G. VI. R6sumC and Concluding Remarks

We show in Table I a summary of the electron paramagnetic resonance behavior reported for the III-V compound semiconductors. The information is given in terms of host lattice, paramagnetic atom, net spin S, the g-value, the fine structure splitting constant ( a [ ,the hyperfine interaction constant A , and a brief pertinent comment. In conclusion, some general observations may be in order as the EPR work in the compound semiconductors is compared to the work in the elemental semiconductors. In the first place, no one has yet reported a sharp resonance line in any of the compounds. In the case of silicon, for example, lines as narrow as 1 or 2 G have been reported, while for the compound semiconductors they are typically from 30 to 60G in halfwidth. Of the 10 resonances reported here, the fact that not one is sharp would suggest that there may be some generic reason for this. It may be that local strains and/or high nuclear spin numbersZ7" prevent the appearance of sharp resonance lines. If this were so, it would be unfortunate, because many of the detailed properties of paramagnetic impurities, including their charge states and interactions, depend critically on the resolution of fine and hyperfine structure.2 Secondly, manganese and iron are most probably substitutional. As a consequshce these atoms, often 27

H. H. Woodbury and G. W. Ludwig, Bull. Am. Phvs. SOC.6. 118 (1961). Of the elements Al, Ga, In, P, As, and Sb, all except P have nuclear spins of $ or greater.

27a

8.

201

ELECTRON PARAMAGNETIC RESONANCE

TABLE I SUMMARY OF ELECTRONPARAMAGNETIC RESONANCE RESULTSI N COMPOUND SEMICONDUCTORS

Spin S

Fine Hyperfine structure interg-value constant la1 action A (gauss) (gauss)

Host lattice

Paramagnetic atom

GaAs

Manganese

2.003

GaAs

Iron

2.046

374

-

GaAs

Iron or defects

2.046

314

-

GaAs

Nickel

2.106

GaAs

Zinc

GaAs

Cadmium

GaAs

Conduction elec.

InSb

Conduction elec.

GaP GaP

Manganese Iron

50.7-48.8

2.002 2.023

15.5

-

~

-

429

51

-

-

60.5 -

Remarks

Fine structure not resolved Hyperfine structure not resolved Spin density lo2 times greater than iron density One very broad line, about 130G in halfwidth Bound hole detected by uniaxial stress Bound hole detected by uniaxial stress About lOOG in halfwidth g-value and line width change with concentration Only one line observed Hyperfine structure not resolved

used as paramagnetic “probe” atoms, probably could not be used to study impurity or impurity-defect interactions for which fast diffusing interstitials are usually required. However, there are other good possibilities for such “probe” atoms, such as copper, cobalt, and chromium which have yet to be tried. Finally, there have been essentially no detailed defect studies in the compound semiconductors, as there have been for silicon. This is very probably due to the lack of single-crystal material (excepting perhaps GaAs). The crystal structure presents a very interesting host lattice’ 7 b for such studies, and electron paramagnetic resonance could provide a most powerful research technique. The structure of the Ill-V semiconductors consist of two interpenetrating cubic lattices, with each sublattice composed of the same column Ill or column V elements.

2’b

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Photoelectric Effects

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CHAPTER 9

Photoconduction in 111-V Compounds T. S . Moss I . INTRODUCTION . .

.

.

.

.

.

.

.

.

.

I1 . THEORY . . . . . . . . . . . . . 1. Photostatic Effects in High Resistivity Materials . 2 . Low Resisriaity Materiul . . . . . . . 3. Photovoltaic cYr.cts . . . . . . . . . 4 . Photoconductivity . . . . . . . . . 5 . p-n Juncrions and Solar Butteries . . . . . 6 . Recombination . . . . . . . . . . 111. EXPERIMENTAL RESULTS . 7 . Indium Antimonide . 8. Indium Arsenide . . 9 . Indium Phosphide . 10. Aluminium Antimonide 11. Aluminium Arsenide . 12. Aluminium Phosphide 13. Gallium Antimonide . 14. Gallium Arsenide 15 . Gallium Phosphide . 16. Other Compounds

LISTOF SYMBOLS . .

.

.

.

.

.

.

205

. . . . .

206 209 . . . . . 210 . . . . . 211 . . . . . 21 2 . . . . . 215 . . . . . 222

.

.

.

.

.

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

. .

. .

. .

. .

. . .

. . .

. . .

. . .

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

225 225 231 233 234 235 236 236 237 241 242

.

.

.

.

.

243

. .

.

. .

.

. .

.

. .

.

. .

.

. .

.

. .

.

. .

.

Introduction

The basic process of photoconductivity is the production of “free” charge carriers in a semiconductor by optical excitation . In the simplest case of an intrinsic semiconductor. optical excitation raises electrons to the conduction band where they are free to conduct. at the same time leaving holes in the valence band which further add to the conductivity. In an impure semiconductor. in addition to the intrinsic effect which will still be present. it is possible to excite electrons from a bound state at donor levels into the conduction band . Equally. in a p-type semiconductor. electrons can be excited from the valence band to acceptor centers. leaving mobile holes. In both cases of impurity photoconductivity only one type of free carrier will be produced .

206

T. S. MOSS

It follows from the above description that photoconductivity can result only if the exciting radiation is of sufficiently short wavelengths for the photon energy hv to be large enough to exceed the energy gaps involvedi.e., between valence and conduction band extrema for intrinsic effects and between extrema and donor or acceptor levels for impurity effects. The wavelength of the onset of photoconductivity and the spectral dependence of the photosensitivity curve are thus determined mainly by the optical properties of the material. Conversely, study of photoeffects provides information about these optical properties. A virtue of this method of studying absorption is that only active absorption is included-incidental losses such as scattering or free carrier absorption are ignored. After excitation, photoconductivity continues until the carriers recombine, that is, throughout the lifetime of the excited carriers. Thus the time dependence of the photoeffects and to an important degree the magnitude of the effects depend on recombination processes. Study of photoeffects provides one of the most important ways of obtaining information about recombination. During their lifetime the carriers move under the influence of internal and external fields, concentration gradients, etc. As these fields are largely under the control of the experimenter a variety of different effects can be observed. These will depend in different ways on the fundamental properties which it is desired to study, and by combining the results of several experiments some parameters can be eliminated and others evaluated. The types of internal and external fields which are present characterize the observed effects in the following way : (i) No applied fields, homogeneous materials. Photovoltaic effects at point contacts and photodiffusion effects in the direction of irradiation can be observed even in this very simple system. (ii) Applied electric field, homogeneous material. This is the “standard” form of photoconductivity. (iii) Applied magnetic field, homogeneous material. This is the photoelectro-magnetic (PEM) effect. (iv) Internal p-n junctions, no applied fields. This gives rise to photovoltaic effects and to power conversion in solar batteries.

11. Theory

The symbols used in the theoretical section are given on page 243. There are three main types of equations which govern internal photoeffects in semiconductors, namely :

9.

PHOTOCONDUCTION IN III-V

207

COMPOUNDS

(i) Continuity equations. These cover the generation and recombination of excess charge carriers. (ii) Transport equations which describe the motion of carriers under the influence of applied electric or magnetic fields and diffusion. (iii) Space charge or neutrality conditions. The following assumptions will be made in the theory: (i) Bulk recombination is linear. In the bulk the recombination rate is proportional to the excess carrier density, i.e.,

U,

U p= {(n

=

+ An)@ + Ap) - np}/(n + ph.

(1)

(ii) Surface recombination is directly proportional to the excess concentration at the surface, so that the current flow into the surface is

J = esAp.

(2)

(iii) Only small-signal conditions considered, so that the equations are linear, i.e., Ap < p , and An < n.

(iv) The quantum efficiency is unity, so that the rate of carrier generation g equals the rate of photon absorption, namely : g

=

I K exp( - K y ) .

(3)

We will consider a rectangular block whose area is very large compared with its thickness t, with its upper surface normal to the illumination as shown in Fig. 1. We shall ignore end effects and not yet consider any applied fields, so that the problem is essentially one dimensional, the only charge flow and the only field E being in the y direction. The equations are

dJ dY

+

Ap - An

=

+ n Ap) (P + n b '

- e(p An

(4)

(E/e)dE/dy.

(7)

2e K I exp(-Kv)

From Eqs. (4), (5), and (6) we obtain, writing for brevity J

=

J,+/e

208

T. S . MOSS

Now the ambipolar diffusion length is given by

L2 = DT(p

+ n)/(n + p / b f ,

(9)

so that d2J/dy2 - J/L2 = -K21exp(-Ky).

(10)

It may be noted that this equation for the charge flow through the photoconductor does not involve the dielectric constant E and is only slightly dependent on the carrier concentrations through the ambipolar diffusion length L. Photon flux I

FIG.1. Schematic diagram of photoconductor.

Solving Eq. (10) we obtain J = A sinh y / L

+ B cosh yfL - IK2e-Ky/(K2- L-’).

(11)

In order to make this equation informative we must put in boundary conditions, which will be the surface recombination condition

y

=

0,

J

=

-sAp(O);

y

=

t, J

=

sAp(t).

(12)

To obtain general solutions of the equations we must find Ap, by using Eq. (7), although it will be shown that for the majority of semiccmductors the assumption of charge neutrality throughout the material is a good approximation, so that An and Ap can be equated and Eq. (7) ignored. However, for materials of very high resistivity, use of Eq. (7) is necessary, and it will show that significant charge separations or what may be called “photostatic” effects occur. We shall therefore first solve using Eq. (7), with simplifying assumptions as to boundary conditions in order to illustrate the form of these “photostatic” effects, and then solve Eqs. (4), (51 and ( 6 ) with the assumption of charge neutrality but more general boundary conditions to study “photoconductive” effects.

9.

PHOTOCONDUCTION IN 111-V

COMPOUNDS

209

1 . PHOTOSTATIC EFFECTSIN HIGH RESISTIVITY MATERIALS

We will consider a thick well-etched specimen, so that t / L % 1 and s = 0, with radiation which is absorbed so strongly that K t % 1. Then Eq. (1 1 ) becomes

J

=

J,+/e

=

f [ K 2 L 2 / ( K 2 L-2 l)][e-YIL- e - K Y ] .

(13)

For the case where K L % 1, the above reduces to J = I(e-YIL -

(13a)

c - ~ Y ) .

Using Eqs. (7), (6),and (5) we find d2E dY2

~-

+

e p ( p n)E DE

-

b-1 -~ DEb

Jy+*

(14)

Putting d 2 = D&/ep(p+ n) and using the boundary conditions y = 0, E = 0 ; y = t, E = 0 (i.e., zero external field) we obtain-again for Kt 9 1, t / L 9 1: E

=

E,

K ~ L L~ ~ K2L2- 1

={

,-YlL

d2 K ~ -L l e -~p , d -L2 K2d2 - 1

L2 - d 2 + L2(K2d2- 1) where the effective field 15,= (b - l)f/bpO, + n). When the radiation is so strongly absorbed that Kd % 1 and K L % 1, this becomes

E z E f [ L 2 / ( L 2- dZ)][e-YIL - Cyid]. From Eq. (7), Ap - An = [(b

-

(15)

1)Id2/bD][L2/(Lz - d2)][(l/d)e-y’d- ( l / L ) e - y i L ] , (16)

so that charge neutrality is seen to occur at a distance below the surface where

where

d2

=

D&/pe(p+ n ) .

Thus d is seen to be a diffusion length appropriate to the diffusion constant of holes and a “dielectric” time constant, -&/a. For intrinsic InSb, for

210

T. S . MOSS

example, this time would be only sec, but for semi-insulating GaAs it could reach several microseconds. Between yo and the surface there is a net positive charge, and beyond yo, a net negative charge-assuming as usual that the electrons are the more mobile carrier. The net positive charge in the upper region is AP

=syo

(Ap - An) dy

=

{E(yo) - E(O)}c/e

0

=

[I(b - l)dz/bD][ L 2 / ( L 2- d 2 ) ][(d/L)d’(L-d) - (d/L)L’(L-d’]

x (Idz/D)(d/L)d‘Lx I d Z / D

for L 9 d

z (IL2/D)(L/d)L‘d x IL2/D

for d

$.

(19)

L,

+

since b 1 for the 111-V compounds. The total charge separation thus depends very strongly on d for the case where L $. d and thus on the dielectric time constant. For semi-insulating GaAs, where the resistivity can reach lo7 t2 cm and the carrier concentration falls to lo8 cm-3, one finds that d 9 L and that A P z I L 2 / D x lo-“ electrons/cm2 z C/cmZ, for D = 20 cm2/sec, z = loT7sec, and I = 10l8 photons/cm2-sec. Depending on whether L d (or d $. L), the “center of gravity” of the electropositive region lies approximately d (or L) below the illuminated surface, while that for the electronegative region is - L (or d) from the surface. Hence the dipole produced is

-

+

Dipole

=

Id2(L - d ) / D x I L d 2 / D ,

x IdL2/D,

if d

+L.

if L

$.

d

(20)

No observations of such “photostatic” dipoles seem to have been described in the literature, but it is clear that semi-insulating GaAs could be a promising material in which to study these phenomena. 2 . Low RESISTIVITY MATERIAL In the majority of semiconductors the space charge effects considered above can be ignored, and we may use Eqs. (4), (5), and (6) with An = Ap. From Eq. (4), Aplz = K l e - K Y- d J / d y . Thus from Eq. (11) with the boundary conditions of Eq. (12) we can obtain Ap. The expression is cumbersome, and to simplify it we will treat the case of a relatively thick specimen where we may put exp( - K t ) = 0 and

9.

PHOTOCONDUCTION IN III-V

COMPOUNDS

21 1

cosh tlL = sinh t / L 9 1. In this case Ap

=

KIt((cr

+ KL)e-"L - (1 + a)e-"')/(K2C - l)(l+ a),

(21)

where the dimensionless parameter ct = ts/L. Thus at the illuminated surface (y = 0), the excess carrier concentration is

3. PHOTOVOLTAIC EFFECTS The simplest photoelectric measurement which can be made is to place a metal probe on the illuminated surface and measure the potential difference between this probe and some distant, unilluminated, part of the specimen where conditions are essentially those of thermal equilibrium. Basically this experiment measures the difference in the Fermi level in the undisturbed region and the pseudo-Fermi level immediately under the probe. In the equilibrium region the densities of holes and electrons are, POIN,

=

noPc

exp(e, - V O ) ,

=

exp

-

(e, - V O ) ?

(23)

where N , and N , are the densities of states in the conduction and valence bands whose extrema lie at energies e, and e, and qo is the Fermi level (measured in units of kT). Hence log(nolpo)= 2q0 - e,

-

e,

+ log NcIN,.

(24)

For the illuminated region, where 0 is the pseudo-Fermi level, we have, similarly, log(n/p) = 2q - e, The photovoltage, which is Aq

Av Putting p - po

=

n

-

-

e,

+ log N , / N , .

= q - qo, is =

(25)

therefore given by

t ~og(nPolPn0).

(26)

no = ( A P ) ~we obtain AV = ( A P ) O ( P O - no)/2ni2.

(27)

Thus we see that for the small signal conditions assumed, the photovoltage Aq is proportional to the excess carrier concentration at the surface, ( A P ) ~ , which was obtained in Eq. (22). For strongly absorbed radiation, K L 9 1 and we have

( A P )=~ I t / ( L + ZS),

(28)

212

T. S. MOSS

which for a well-etched specimen reduces to ( A P ) ~= l z / L FZ l(z/D)'"

(29)

where D is the appropriate diffusion constant. It will be seen from this expression that the photovoltaic sensitivity falls off only as the square root of the carrier lifetime, so that this type of experiment is favorable where lifetimes are inconveniently short. As shown by Eq. (22), for any value of the surface recombination velocity, the spectral dependence of the photovoltage is determined by the term KL/(l + KL). Thus throughout the short wavelength, high absorption region the quantum efficiency should be constant, falling in the neighborhood of the absorption edge to half its short wavelength value when KL = 1. Now in general the diffusion length does not vary over an excessively wide range. For the 111-V compounds, in general, z lies between lo-' and sec and D between 10 and 1000 cm2/sec, so that the range to 3 x of diffusion length is from cm. As the absorption constant rises very steeply over the range 30 to lo4 cm-' for many of these materials (those whose absorption edge is determined by vertical transitions) the determination of this wavelength of half-sensitivity gives quite a good approximation to the energy gap of the material. In GaAs, for example, the absorption constant rises from 30 cm-' at 1.37 eV to 3000 cm-' at 1.42 eV.' Alternatively, when the absorption curve is known accurately, L and z can be found by study of the photosensitivity curve. This technique has the virtue that the measurement is made inro the material, so that it is relatively independent of the state of the surface. 4. PHOTOCONDUCTIVITY The excited carriers in the illuminated specimen discussed above will give rise to photocurrents on applying end electrodes and an electric field-for example, in the x direction. Provided the field is small it will not have any significant effect on the carrier distributions governed by Eqs. (4), (51, and (6). The photocurrent density will be determined by the total excess carrier concentrations and their mobilities, namely, Ai

=

p(b

+ l)eE, AP,

(30)

so that the photocurrent is directly proportional to the total number of excess holes, given by *f

A P = J0 A p d y .

' T. S. Moss and T. D. F.Hawkins, Infrared Phys. 1, 111 (1961).

(31)

9.

PHOTOCONDUCTION IN III-V

213

COMPOUNDS

Solving Eqs. (4),(5), and (6) with the boundary conditions of Eq. (12) and integrating we obtain the following expression for AP :

AP(l

-

K’L’)/IT

=

1 - e-Kt

+ K L ( K L e w K ‘- K L - c.te-Kt - ci)/(l +

ci

coth t / 2 L ) .

(32)

This expression may be evaluated for various particular cases : (i) Negligible surface recombination, i.e., ci

AP

=T x

=

-,0. We then have simply

(total absorbed photons)

(33) (33a)

IT

for a thick sample or high absorption. This expression is the form obtained from simple photoconductivity theory and with measurements of the absolute magnitude of the photocurrent is widely used to obtain estimates of carrier lifetimes. Equation (33) shows that study of the spectral dependence of the photocurrent may be used to obtain the absorption coefficient for the material. As this method measures only the photoelectrically active part of the absorption and ignores incidental losses due to scattering, free carrier absorption, etc., it is thus particularly valuable in initial studies of new materials or those which can only be obtained in poor quality. It has been used recently by Mead and Spitzer’ to obtain the first comprehensive absorption data on AlAs. It may also be used to study absorption at levels much lower than those which would be possible by direct transition methods. Moss and hawk in^,^ working with germanium, succeeded in measuring K values down to cm-’. Such measurements may be used to obtain absolute values of absorption coefficient without the necessity of doing absolute measurements of photoconductivity. Consider the configuration shown in Fig. 2 in which a specimen is relatively thick in the direction of the radiation with the electrodes placed at distances t , and t , from the illuminated surface. Then for welletched material, the photosensitivity S is proportional to the total absorbed photons, i.e.,

S

=

C r ’ exp( - K y ) d y

=

C ( e - K t l- e - K f 2 ) ,

(34)

fl

where C is a constant of proportionality. On differentiating with respect to K this expression is seen to have a maximum when

K ( t , - t l ) = log(t,/t,). C. A. Mead and W. G. Spitzer, Phys. Rev. Letters 11, 358 (1963). T. S. Moss and T. D. F. Hawkins, Phys. Reo. Letters 1, 129 (1958).

(35)

'r. s.

214

MOSS

I

FIG.2. Geometry of specimen for measuring absorption coefficient.

where =

t1/(t2- t l )

and

y = t2/(tz- t i ) .

(37)

Thus the relative photosensitivity curve yields the absolute absorption curve, assuming that f 1 and t, are known. (ii) Thick sample (t 9 L) and high absorption such that K L 9 1. Putting coth f / 2 L = 1, Eq. (32) gives

AP

=

IT(^

+ K L + a)/(l + KL)(1 + a).

139)

When both a and K L are large this becomes

AP

=

Zz(l/KL

-

+ l/a).

(40)

As K L and a can both reach values 100, the short wavelength sensitivity can fall to only 1% of the full value given by Eq. (33). However, the photosensitivity rises to considerable values in the neighborhood of the absorption edge-even if s (and thus a) approach co. (iii) Thin sample (t < L) with high absorption ( K L 9 1, K t 9 l), and moderate surface recombination. We find from (32) AP = I t / 2 ~ ,

(41)

showing that the short wavelength sensitivity is directly proportional to thickness and inversely proportional to surface recombination velocity.

9.

PHOTOCONDUCTION IN Ill-V

COMPOUNDS

215

For thin samples with highwrface recombination the spectral sensitivity curve will peak very sharply in the region of the absorption edge. For example, if t = L, Eq. (32) shows that the peak occurs at an absorption level K L = 2.6. TABLE I CALCULATED PHOTOSENSITIVITY OF

GaAs

SAMPLE

I (microns)

K * (cm-')" APlIr ( 7 ~

) ~

0.87

0.89

0.92

4000 2.5

260 25

3 3

See Ref. I. *t= L= lop.

As an illustration, Table I shows the calculated photosensitivity of a GaAs sample having t = L = 100 p, for which a very high surface recombination velocity has been assumed. It will be seen that the curve is so sharply peaked that it rises and falls by a factor of LO:1, all within a spectral range of only 0.05 p. Such a peaked curve has been observed for GaAs4 as shown in Fig. 3.

-

5 . p-n JUNCTIONS

AND

SOLARBATTERIES

Large area p-n junctions can be made, and they are used in infrared detectors and solar batteries. The former are photovoltaic devices while the latter are power convertors. They consist of a thin slice of material which has a p-n junction parallel to the illuminated surface and at a very small depth below it (see Fig. 4). From the theoretical point of view all flow can again be assumed to be normal to the surface in the y direction. We shall consider the behavior of minority carriers in the p and n regions in turn. The surface layer is assumed to be p type, although there may be advantages in the opposite configuration in some cases. Both regions are heavily doped, so that p % n at the surface and n %- p in the base region. a. Electrons in the p Region Instead of Eq. (4) we have dJ-jdy where

t, is

= dn/t,

- K l exp( - K y ) ,

the electron lifetime.

H. P. R. Frederikse and R. F. Blunt, Proc. I R E 43, 1828 (1955).

(414

216

T. S . MOSS

1.2 Wavelength (microns)

FIG.3. Photoconduction in GaAs. (After Frederikse and Blunt, Ref. 4.)

Now as shown by Eq. (14a), the internal field set up by the illumination is very small in highly conducting material such as is being considered at present. We will therefore ignore the field term in Eq. (5) and put

J-

=

D,d(An)/dy,

where D, is the diffusion constant for the electrons. I Photonslsec

FIG.4. Diagramatic representation of solar battery.

(42)

9.

PHOTOCONDUCTION IN 111-V COMPOUNDS

217

The solution of these equations is

An

=

A cosh y / L + B sinh y / L - (FL/D,) exp( - K y ) ,

(43)

where F

I K L / ( K 2 L 2- 1)

=

and the diffusion length

L

=

(D,Z~)''~.

The boundary conditions are

y

=

J-

0,

y

=

t,

=

D , d(An)/dy = s A n ,

n,

=

n,expeV/kT,

(44) (45)

where no is the electron concentration in the dark, and V the voltage across the junction. Using (44) and (45) to evaluate A and B, we find the current at the junction ( y = t ) to be J;= F K L exp( - K t ) - F(a

+ [ N + F exp( where we have put tanh a' = a

NLID,

=

=

+ KL)/(cosh t / L + a sinh t / L ) K t ) ]tanh(a' + t/L ),

-

(46)

t s / L and

n,(expeV/kT - 1).

At short wavelengths where K is large (i.e., KL 9 1) and for typical conditions where a < t / L < 1, the short-circuit current (V = 0 = N ) becomes - JsJI =

(cosh t / L

+ a sinh t / L ) - '

-

exp( - K t ) .

(47)

In order to obtain good efficiency it is thus clearly necessary to make a as small as possible, and t/L as small as is compatible with the requirements of absorption in the layer and low transverse-layer resistance. b. Holes in the n Region

The primary equations are similar to (41) and (42), namely,

dJ'/dy

=

J'

K I exp( - K y ) - A p / t h ,

(48)

= - Dh d ( A p ) / d y ,

(49)

giving

Ap

=

R cosh y / l

+ S sinh y / l

-

(Gl/D,) exp( - K y )

(50)

218

T. S. MOSS

where G

=

K11/(K212- 1)

and the hole diffusion length is 1 = (0,2,)”2 The boundary conditions are y

=

t,

pt

y=d,

=

po exp eV/kT

Ap=O

assuming a relatively thick n layer. With this latter assumption we may put exp( - Kd) = 0 and coth d/l = 1 and obtain for the current at the junction due to holes J,’ = P - G(K1 - l)exp(-Kt), (53) where Pl/D, = po(exp eYkT - 1). It may be noted that J,’ remains significant for the condition t -+ 0, i.e., when the p layer is so thin that there is negligible absorption in it. The short-circuit current (i.e., V = 0 and hence P = 0) becomes - J:

IKI/(l

=

+ KZ),

(54)

which shows that the full current of I electrons per second will be obtained if K1 & 1, i.e., if the absorption takes place well within a diffusion length in the n region. In the practical solar battery, photoholes produced in the n region will give an important contribution to the total current, particularly at the longer wavelengths where the absorption is such that the p layer is relatively transparent ( K t < 1) but K1 > 1. For typical parameters, i.e., a = 0.1 and t/l = t/L = 0.3, the current contribution of the n layer exceeds that of the p layer when K L > 1.5. c. Current at the Junction Writing for convenience exp (- Kt) [Eqs. (46) and (53)] become J,-= FKLX - F(a

J,’ = -GX(K1

=

X , the currents at the junction

+ KL)/(cosh t/L + a sinh t/L)

+ ( N + XF) tanh (a’ + t/L) - 1) + P = P - KIIX/(l + K1).

(55)

(56)

The total short-circuit junction current (i.e., N = P = 0) is thus : - J,JF = a + KL - X(sinh t/L + u cosh t/L) - X(KL + 1/15) . (57) cosh t/L a sinh t/L 1 + Kl

+

9.

PHOTOCONDUCTION IN 111-V COMPOUNDS

219

In discussing this expression, it will be seen that the useful terms on the right-hand side are the positive ones, the negative terms representing losses. For reasonably high absorption conditions the last term is negligible, showing that radiation transmitted through the p layer is used effectively in the n layer, provided that it is absorbed within a diffusion length. Clearly the positive part of the first term should be maximized and the negative part minimized Reducing c1 does both these things, so it is clearly advisable to keep the surface recombination small; although the effect is not pronounced. Increase in L is always advantageous as it reduces t/L, while increases in either diffusion length extend the band of effective wavelengths. It will be seen from Eq. (57) that there is only slight dependence on the relative magnitudes of 1 and L. It is not, therefore, automatically preferable to have the higher mobility minority carriers in the surface layer, as is common design practice, but on the contrary it may be advantageous to have a surface n layer with more mobile majority carriers in order to obtain lower transverse resistance. Since the relative magnitudes of 1 and L have little effect, it is convenient to put them equal in Eq. (57) and obtain -J J I =

+

K L [ a 4- K L - (1 a)exp (t/L- K t ) ] ( K 2 L 2- l)(cosh t/L c1 sinh t/L)'

+

(58)

This expression, which may be shown to be finite even when K L = 1, is plotted in Fig. 5 as a function of the dimensionless parameters K L and t/L. It has a maximum (for any absorption level such that K L > 1) when t

Surface layer thickness

FIG.5. Calculated collection efficiencies for various absorption levels

220

T. S . MOSS

is given by 2expKt(sinht/L)

=

1 - 2/KL+ exp2t/L

(59)

which approximates to Kt(exp K t - 1)

or exp K t

=

1 + L/t

=

KL - 1

(60)

if K L % 1.

(61)

-

If the useful range of K is thus 2500-10,000 cm- and L 8 p for example, the surface layer should be 1-3p thick. The curves however are fairly flat, and a value of t/L = 0.25 will be near the optimum for values of K L between 2 and 8, the “collection efficiency” given by Eq. (58) exceeding 92% if KL > 5. More generally there is also a maximum when L # I, i.e., for Eq. (57). In the practical design of a solar battery it may be advisable to have the surface layer somewhat thicker than this optimum in order to reduce the transverse resistance through which the output current has to flow. The total short-circuit current ( J o ) will be obtained by integrating Eq. (58) over all wavelengths. d. Voltage Output The general expression for the junction current, derived from Eqs. (55) and (56) may be written in the form

J, = jo(exp eV/kT - 1) - J o ,

where

j , = poDh/l

(62)

+ (noDe/L)tanh (a’ + t/L).

(63)

Hence the open-circuit photovoltage is given by eT/,, =

kT log (1 + Jo/jo) = kT log Jo/jo,

(64)

as Jo/jo is extremely large compared with unity. e. Power Output From Eq. (62) the power delivered by the junction is V J , = V(j,expeV/kT - J o ) , since J o >> j,. This can readily be shown to have a maximum value when

V,,,

r(kT/e) and where the parameter r is given by =

(Y

(JJopt = -Jor/(r

+ l)er = J o / j o .

+ l),

(65)

(66)

9.

PHOTOCONDUCTION IN 111-V

COMPOUNDS

221

The optimum power is then W,,,

=

( r - l)J,kT/e

(67)

inasmuch as r 1. The efficiency (at optimum power) is given by 'I =

(I

-

l)kTJ0/1.4Qo,

(68)

where Qo is the total rate of arrival in the semiconductor of solar photons, whose average energy is readily shown to be 1.4 eV. f. Conditions for High Eficiency For optimum matching conditions the solar battery is seen to behave as though each photocarrier reaching the junction provided ( I - 1)kT joules of energy to the external circuit. Ways of increasing J,, and (hence J , ) in order to have the maximum number of carriers reaching the junction have already been discussed. To increase the energy yield per carrier it is necessary to make r as large as possible and hence, since I logJ,/j,, to make j, as small as possible. Now, from Eq. (63),

-

j , = poD,/I

+ ( n , D J L ) tanh (a' + t/L)

or j , = Pol/%

+ not/%?

(69)

as it should be possible to make ci' small compared with t/L. The densities of minority carriers, p o and no, are given by,

po

=

niz/nd,

no = n,2/p,,

(70)

where ni is the intrinsic concentration and nd and pa are the concentration of ionized donors and acceptors, respectively. Hence

The most important feature of this equation is the dependence on ni and thus on activation energy. By contrast the other parameters will probably not differ very much from one semiconductor to another, particularly if the semiconductors are at similar states of technological development. As the efficiency will be seen to depend roughly on the logarithm of j,, and only fall about 3% for 3 : 1 increase in j,, it is not necessary to estimate j o with high accuracy. In order to reduce j , it is clearly necessary to use high doping levels. With increasing doping, however, lifetimes will fall, and at high concentrations there will be a tendency for thnd and zep, to become constant and

222

T. S. MOSS

roughly equal. Also, as it is necessary to maintain long diffusion lengths in order to keep J , large, the doping cannot be too heavy. Usually the first term in Eq. (71) is found to be several times smaller than the second, so that the latter is the important one to minimize. In selecting an optimum semiconductor for solar battery use, the prime requirement is to make J , large while keeping j , small. This is essentially a compromise between reducing the energy gap to increase the number of absorbed photons, and increasing the energy gap to reduce the concentration of intrinsic carriers. Making the broad generalization that for all photoconductors all parameters except energy gap are near enough the same, we may plot a curve of calculated efficiency versus energy gap. Two such curves are shown in Fig. 6, assuming first that the effective absorption

0.6

1.0

1.4

1.8

2.2

2.6

Energy gap, eV

FIG.6 . Dependence of efficiency of solar cell on energy gap.

edge (A*) is equivalent to the energy gap and secondly that it is at a somewhat shorter wavelength than that corresponding to the energy gap. (Doping concentrations were taken as 10'8cm-3, and L/z = lo3.) In either case it will be seen that optimum efficiency should occur for an energy gap in the region 1.3-1.5 eV-values which can readily be realized using 111-V compounds.

6. RECOMBINATION One of the assumptions at the beginning of the theoretical section and in Eq. (4) was that the recombination between carriers was direct. For this case the lifetimes of the two carriers are readily related, for the recombination rates must be equal, giving Apfzp = Anfz,.

(72)

9.

PHOTOCONDUCTION IN 111-V

COMPOUNDS

In most cases neutrality will exist, and so we have An Direct recombination can occur in two ways :

=

223

Ap and sP = T,.

(i) Radiative recombination-which is important in semiconductors where both band extrema are at K[000]. This applies in GaAs, InP, InAs, and InSb, and it is the basic reason why these materials can be operated as injection lasers. (ii) Auger recombination-where third carrier.

the excess energy is removed by a

However, for many materials the predominant recombination mechanism is indirect, the process occurring via various centers of imperfection in the crystal. These centers are termed recombination centers if a carrier which reaches such a center has a high probability of recombination with a carrier of opposite sign rather than returning to the nearby band. If the converse is true, and the probability of returning to the band is high, the imperfection is known as a trapping center. In the case of trapping, although the total t h e between excitation and recombination will be the same for electrons and holes (with the usual proviso of charge neutrality) the amount of time spent out of traps will differ, and it is only during this time that carriers can contribute to the photoconductive processes. In such cases the carrier lifetimes 7, and zP which we use mean the trap-free periods of the carrier's total excitation-recombination cycle. If the number of electrons which become trapped after illumination is An,, the neutrality condition becomes Ap

=

An

+ An,,

(73)

with a corresponding equation for hole trapping. As the recombination rates are still equal, we have tp/7, =

1

+ AnJAn,

(74)

so that clearly the lifetimes of holes and electrons are no longer equal. For the case where Ap P An, Eq. (4) becomes dJ/dy = e [ K I exp ( - K y ) - Ap/t,],

(75)

giving Ap

=

7,[Kf exp ( - K y ) - dJ/dy].

Integrating over the specimen thickness we obtain AP

=s'

0

Apdy

= z,1[1-

exp(-Kt)]

+ (J,

- J,)7,.

(76)

For the simple case where the surface recombination is negligible we can

224

T. S. MOSS

put J o = 0. If we also assume a thick specimen, then J, = 0 and exp ( - K t ) is negligible, so that AP = I t , .

(77)

The photocurrent for a small transverse field Ex will differ from Eq. (30) and become Ai = ep(AP + b A N ) E , = e p AP (1

+ bzJzp)Ex

(78)

from Eq. (72). Hence from (77)

Ai = epZ(zp + bz,)E,.

(79)

If we define an effective photoconductive lifetime zPcsuch that the number of photoelectrons and photoholes are each l z p cand the resulting photocurrent is the same as in Eq. (79) we have Ai = epI(b + l ) E x z p c , so that

zPc = ( z p

(80)

+ bz,,)/(l + b).

Thus in general the lifetime determined from the magnitude of the photocurrent will give a function of the two carrier lifetimes, and only in the particular case when they are known to be equal (because trapping is negligible), i.e., zP = t, = zpc = T can a specific lifetime be obtained. Otherwise at least two independent measurements are required. The second measurement which is generally used is the short circuit PEM current. This is treated by Bube in volume 3 of this series. For the present it is sufficient to note that for the simplifying assumptions used in deriving Eq. (79), the PEM current is proportional to the square root of an effective lifetime zpem which has been shown by Zit ter5 to be zpem = (Porn + n ~ z p ) / ( P+ ~ no). (82) This expression also reduces to the simple value for zpemwhen the lifetimes are equal, i.e., Tp =

z, = tpem = z.

The popular method of evaluating the lifetime by taking the ratio of the PC and PEM currents-which has the experimental attraction of avoiding absolute measurements of light intensity-assumes that zpc and zpem are R. N. Zitter, Phys. Rev. 112, 852 (1958).

9.

PHOTOCONDUCTION IN III-V

COMPOUNDS

225

equal and thus cannot be used in general. On the contrary these two effective lifetimes must be determined separately and the simultaneous equations (81) and (82) solved for zp and 7,. As stated above, these effective lifetimes are equal when T~ = z,. By equating (81) and (82) it is clear that they are equal in another specific case, namely if p

=

(83)

bn.

This will be recognized as the condition for maximum resistivity of a semiconductor when p

=

b“’ni

and

n

=

b-”’ni.

In infrared detectors made of high conductivity materials, such as InSb, it is desirable to keep the resistance of the detector as high as possible. In order to achieve this the material is often doped deliberately to about this level. For such a case therefore it may be useful to realize that the two effective lifetimes will be equal-irrespective of the presence of trappingand so the ratio method of determining zpc = zpemcan be employed. 111. Experimental Results

The various compounds will be treated separately and generally in order of importance. Indium antimonide, which has been the subject of most basic research and which is also the most important photodetector, will thus be treated first. 7. INDIUM ANTIMONIDE For many years it has been possible to produce InSb of high purity and of good crystal perfection in large ingots. It is thus an excellent material for experimental work. It can be doped readily to the required level by group I1 elements (usually Zn or Cd) and group VI elements (Se or Te). a. Spectral Sensitivity Typical curves for the spectral sensitivity of an InSb photoconductor are shown in Fig. 7. There is comparatively little variation at short wavelengths-showing that the quantum efficiency is roughly constantalthough there is always some peaking just prior to the long wavelength drop at the absorption edge. At the shortest wavelengths there is evidence of the onset of multicarrier generation, which is treated more fully by AntonCik and Tauc in the next chapter of this volume. At room temperature the sensitivity has fallen to half-value at ,Il,’ = 7.7 p . With cooling the edge

226

T. S. MOSS

moves to shorter wavelengths, as is the case with all the III-V compounds which have been studied. The curves shown give a spectral shift, d E / d T - -2 x

eVf°C.

The average of a variety of measurements of the temperature dependence of the energy gap is rather higher than this,6 namely, 3 x eV/OC. Analysis shows that most of the effect is due to lattice dilatation, the broadening effect being small.’ 50

20 0”

” z $ W .-

c

10

c 0

u 3

-

w 0 -

5

d

2

I

2

3

4

5

6

7

6

3

Wavelength (microns)

FIG.7. Photoconductivity in InSb.

Assuming that the fall in sensitivity in the short wavelength, high absorption region is due to surface recombination, the value of a = zs/L can be found. For the specimen used, t / L = 5, so that the theory gives the ratio of peak to short wavelength sensitivity to be AP,,JAP,, 0.5. Hence at 20°C, a = 4.7 and s = 1.5 x lo5 cm/sec. At very low temperatures the curves are more sharply peaked in the region of the absorption edge’ as shown in Fig. 8. At 15 K the ratio of peak to short wavelength sensitivity is 5 :1 and the half-width of the peak is only 0.0064eV. Laff and Fan conclude from an analysis of their data that in addition to surface recombination there is enhancement of the peaking eZect due to surface trapping.

-

T. S. Moss, “Optical Properties of Semiconductors.” Butterworth$ London and Academic Press, New York, 1961. T. S. Moss, Proc. Phys. SOC.(London) B67, 175 (1954). * R. A. Laff and H. Y. Fan, Phys. Rev. 121, 53 (1961).

9.

PHOTOCONDUCTION IN 111-V

COMPOUNDS

227

o 15°K A 27OK

59'K

I

Wovelength (microns)

FIG.8. Low temperature photoconductivity in InSb. (From Laff and Fan, Ref. 8.)

A spectral sensitivity curve for the photovoltaic effect at a metal point contact is shown in Fig. 9. It will be seen that the sensitivityfor constant incident power is fairly constant at short wavelengths, so that the quantum efficiency is falling as the wavelength decreases. This is very similar to the results found by Laff and Fan for the PEM effect. and reflects the fact that

287'K

1 4

7

Wavelength (microns)

FIG.9. Photovohaic sensitivity at a metal probe on InSb.

228

T. S . MOSS

both these effects vary only as the square root of carrier lifetime whereas photoconductivity is directly proportional. The sensitivity does not extend to such long wavelengths as the photoconductivity, the power sensitivity falling to half-value at Al,2 = 7.15 p. Detailed measurements of carrier lifetimes have been made by various workers.*-'" At room temperature it is found that the lifetimes of the two carriers are equal, but at liquid air temperatures trapping effects are important and z p and z, differ widely. Results for p-type material are shown in Fig. 10. It will be seen that on cooling below 170°K the lifetimes begin

Zitter eta/.

I$

10

2

14

1000/TaK

FIG.10. Carrier lifetimes in InSb. (See Refs. 8 and 9.)

to differ, z p increasing as z, continues to fall, and that at liquid nitrogen temperatures the hole lifetime is a thousand times as large as the electron lifetime. Laff and Fan find that their results can be fitted with a theory based on two recombination levels situated 0.055 and 0.12 eV above the valence band. R. N. Zitter, A. J. Strauss, and A. E. Attard, Phys. Rev. 115, 266 (1959). C. Hilsum, Proc. Phys. SOC. (London) 74, 81 (1959).

"O

9.

PHOTOCONDUCTION IN 111-V COMPOUNDS

229

At temperatures above 250°K the carrier lifetimes fall rapidly with increasing temperature due to direct recombination. Both radiative and Auger recombination contribute to this, the latter being the more important. l 1 The room temperature lifetime decreases with increasing hole concentration. Results of Zitter et a!.' for material with up to lo'* holes/cm3 showed that z varied approximately inversely as concentration, with

-

pz

5 x lo8 cm-3 sec.

b. Impurity E f e c t s Photoconductive effects in doped InSb have been investigated by Engeler et a l l 2 These workers found that, at liquid helium temperatures, samples doped with Cu, Ag, and Au all exhibited photoconductivity at long wavelengths. The oscillatory nature of the spectral response curves is shown in /

.--. .

0

5

,*--.

0

W

c n

?

P

f

n

-W

1 1

0

$

Doshed curves are corrected for lattice absorption

I

I \

I

1 4

I

12

I 20

I

I 28

I

Wovelength (microns)

FIG.1 1 . Impurity photoconduction in InSb. (From Engeler et al., Ref. 12.)

Fig. 11. The separation, in terms of energy, of adjacent minima is the same at all wavelengths and for all dopants, being 0.0244 eV, which is identified with the energy of the longitudinal optical phonon. Because of the constant T. Landsberg and A. R. Beattie, Proc. Roy. SOC.(London) 249, 16 (1959). W. Engeler, H. Levinstein, and C. Stannard, Phys. Rev. Letters 7 , 62 (1961); J . Phys. Chem. Solids 22, 249 (1961).

" P.

l2

230

T. S. MOSS

energy separation, plots of energy of minima against the number of the minima are straight lines. These can then be extrapolated to zero to obtain accurate values of the impurity activation energy-provided that numbers can be correctly assigned to the minima. In the case of Au this is easy, since the long wavelength threshold of sensitivity lies in the range of measurements at -29 p. For the other two dopants the assignment was made by correlating the energy with activation energies found from Hall effect measurements. The results obtaiqed are given in the accompanying tabulation. If these lower energy levels are filled, upper levels become Activation Energies of Impurities in InSb

Copper Silver Gold

0.026 eV 0.028 eV 0.043 eV

active since these dopants, being monovalent, each give two energy levels as they replace the trivalent In atoms. In these cases the spectral sensitivity curve has a second plateau region beyond the normal absorption edge, about lo3 times lower in sensitivity. From the long wavelength threshold beyond the plateau it is possible to determine these higher energy levels. The values are: Cu, 0.056 eV; Ag, 0.039 eV; and Au, 0.066 eV. c. Infrared Detectors The factors governing various types of detector have been discussed previously by the author.13 It is found that for a photoconductor which is limited by Johnson noise-which will be the case for the best possible detector at room temperature-the minimum quantum signal which can be detected’3a is given by Qmln

= (4kTt’ Af/H)”’(bn

+ p)/z(b + I),

where the square root term includes the engineering and environmental parameters, and the other terms are material factors. Considering only the latter we have, Qmln 0~ (bn + e)/r(b + 1). If variation of z with carrier concentration is ignored, this expression is seen to have an optimum when p = bn, giving Qmln cc bl/’ni/z(b + 1 ) or ni/zb’/’ l3

T . S . Moss, Progr. Semicond. 5, 191 (1960). This is taken as the condition that signal = noise.

13a

as b > 1.

9.

PHOTOCONDUCTION IN Ill-V

COMPOUNDS

231

This expression shows that it is advisable to have large values of z-as expected. It shows also that a large mobility ratio is advantageous so that 80, is very favorable in this respect. Finally, the intrinsic InSb, where b carrier concentration ni should be small. Now for a given operating wavelength the energy gap may be assumed fixed, and as the temperature is specified as room temperature, the only remaining terms to consider are the densities of states terms. Two factors enter here:

-

(i) Degeneracy of band extrema. For InSb both extrema lie at the center of the Brillouin zone, so that this factor is as small as it can be. (ii) Effective masses. Here the very low value of the electron mass is particularly favorable. As a result of these factors the term outside the exponent is many times less than in PbTe, for example, and is an important reason why InSb is the better room temperature detector although the exponent term is actually smaller for PbTe. When good quality detectors are cooled it should be possible to reach the condition that the thermally generated carriers are fewer than the photocarriers produced by room temperature radiation from the environment. This is the so-called “background limited” condition. Again the lower density of states in intrinsic InSb is particularly favorable, and it should be possible using the best material to reach this condition at temperatures as high as 150 K,and cooling below this temperature will not improve the sensitivity significantly. By comparison, gold-doped germanium-which has roughly the same wavelength limit of sensitivityhas to be cooled to 65°K to reach maximum sensitivity. Measured absolute sensitivities of various liquid air cooled InSb detectors are shown in Fig. 12. Reviews of infrared detectors, which enable InSb to be compared with other materials, have been given by Bratt et a l l 4 and Potter and Eisenmann.” Use of high speed InSb p-n junctions as mixer detectors for reception of coherent light is discussed by Lucovsky et ~ 1 . ’ ~ 8. INDIUM ARSENIDE Photoconductive, photovoltaic, and PEM effects have all been observed in InAs, although the studies have not been as extensive as in InSb. This may be partly due to the fact that it is not possible to prepare InAs of anywhere near the purity of InSb, the best material at the present time being an order of magnitude worse than intrinsic at room temperature. l4 l5

l6

P.Bratt, W. Engeler, H. Levinstein, A. Mac Rae, and J. Pehek, Infrured Phys. 1, 27 (1961). R. F. Potter and W. L. Eisenmann, Appl. Opt. 1, 567 (1962). G. Lucovsky, R. B. Emmons, and H. Altemose, Infrared Phys. 4, 193 (1964).

232

T. S . MOSS

FIG.12. Sensitivities of liquid air cooled InSb detectors. (Curve A, Ref. 43; curve B, Ref. 41; curve C, Ref. 42.)

A spectral sensitivity curve for photoconductivity at room temperature is shown in Fig. 13. The quantum efficiency vanes little from 1 to 3 . 6 ~ . From the curve we find Al,2 = 3 . 8 ~ . 10-

Room temperoture

I I

2

3

4

Wavelength (microns)

FIG. 13. Photoconduction in InAs. (After Hilsum, Ref. 17.)

Photovoltaic effects have been observed in both grown junctions and at naturally occurring junctions in ingots-probably at grain boundaries. A spectral sensitivity measured at liquid air temperatures, is shown in Fig. 14. Measurements of lifetime have been made, by study of magnitude of PC and PEM effects as well as transient response, by various l7

C. Hilsum, Proc. Phys. SOC. (London) B70, 101 1 (1957).

J. R. Dixon, Phys. Rev. 107, 374 (1957).

9.

PHOTOCONDUCTION IN 111-V

233

COMPOUNDS

At room temperature lifetimes are usually between lo-' and lO-'sec, and there is little evidence of trapping. Lifetimes below sec have been observed however in p-type material, and on the other hand values up to lO-'sec have been obtained by heat treatment.

4 Wavelength (microns)

FIG. 14. Photovoltaic sensitivity of lnAs junction. (After Talley and Enright, Ref. 44.)

-

Dixon" finds that the surface recombination velocity is lo5 cm/sec for a ground surface, falling to < lo3 cm/sec on etching. Use of InAs in PC and PEM infrared detectors has been discussed by Lee and Roberts" and Bratt et a l l 4 Lucovsky et a l l 6 have compared the sensitivity of p-n junction cells as detectors of incoherent radiation and coherent, laser, radiation. In the latter case it should be possible to detect W (referred to 1-cps bandwidth). < 9. INDIUMPHOSPHIDE Photovoltaic effects have been studied in this material by Reynolds u ~ . , ~ and O both PC and PEM effects have been measured by Mikhailova

et et

d 2 1

Spectral response curves by the former workers are shown in Fig. 15. At room temperature the response is down to half-value at hv = 1.32 eV, l9

*'

D. H. Lee and D. H. Roberts, Opt. Acta 7, 271 (1960). W. N. Reynolds, M. T. Lilburn, and R. M. Dell, Proc. Phys. Soc. (London)71,416 (1958). M. P. Mikhailova, D. N. Nasledov, and S. V. Slobodchikov, Fiz. Tuerd. Tela 4, 1227 (1962) [English Transl.; Soviet Phys.-Solid State 4, 899 (1962)].

*'

234

T. S. MOSS

and the spectral shift of the half-sensitivity point is given by hv

=

1.44- 4.5 x 10-47-OK.

These measurements were made using a metal point contact on p-type material.

Photon energy(eV)

FIG.15. Spectral response of InP photodiode. (From Reynolds et al., Ref. 20.)

Mikhailova et dZ1 studied n-type material. Strong trapping effects were present, even at room temperature. The lifetimes found, at 300”K, were

-

Holes,

zp

Electrons,

z, = 2 x

sec sec.

Rappaport” made solar batteries by alloying zinc-indium dots onto the InP. Power generation efficiencies up to 2 % were obtained, although the cells showed evidence of high internal resistance. Bube et aLZ3found large photo-Hall effects in InP, the Hall mobility varying as much as 30: 1 on illumination. 10. ALUMINUM ANTIMONIDE

Photoconductivity has been observed in AlSb by Abrahamz4 and K ~ v e r . ’ More ~ detailed measurements have been made by Mead and P. Rappaport, R C A Rev. 20, 273 (1959). R. H. Bube, H. E. Macdonald, and J. Blanc, J . Phys. Chem. Solids 22, 173 (1961). 24 A. Abraham, Czech. J . Phys. 6, 624 (1956). 2 5 F. Kover, Solid State Phys. Electron. Telecommun., Proc. Intern. Conj Brussels, 1958 Vol. 2. p. 768. Academic Press, New York 1960. 22

23

9,

PHOTOCONDUCTION IN III-V

COMPOUNDS

235

Spitzer’ using surface barrier photovoltaic effects. Their results, taken at room temperature, are shown in Fig. 16, where the square root of the photoresponse has been plotted in order to relate the results directly to the expected absorption law and hence permit extrapolation to determine

Photon energy(eV)

FIG. 16. Photovoltaic response of a surface barrier AlSb detector. (From Mead and Spitzer, Ref. 2.)

the energy gap. The authors interpret their results as indicating energy gaps of 1.5, 1.85, and 2.1 eV, which they suggest correspond to transitions from the valence band to 100, 111, and OOO conduction band minima, respectively. In many cases, KoverZ5found a peak of sensitivity at a photon energy of 1.27 eV. He concludes that this is an impurity effect. 1 1 . ALUMINUM ARSENIDE

Mead and Spitzer’ have also used their surface barrier photovoltaic technique to study AIAs. This material deteriorates rapidly on exposure to air, but good results for the spectral sensitivity curve were obtained as shown in Fig. 17. The response at low photon energies is attributed to photo-injection from the metal used to make the surface barrier. On subtracting this from the response at shorter wavelengths the straight line (shown dashed) is obtained. This extrapolates to cut the axis at 2.1 eV, which is thus the room temperature energy gap for this material. The 8 can be interpreted in like very steep curve at photon energies ~ 2 . eV manner to yield an energy gap of 2.9 eV, which is assumed to be the direct, vertical, transition.

236

T. S. MOSS

1.8

14

22

26

3.0

Photon energy(eV1

FIG.17. Photovoltaic response in AlAs. (From Mead and Spitzer, Ref. 2.)

12. ALUMINUM PHOSPHIDE This material has been little studied, the only work known being that of Grimmeiss et dZ6 Spectral sensitivity curves for both photovoltaic and photoconductive effects are shown in Fig 18, These workers conclude that these curves are due to impurity excitation and that the forbidden energy gap is approximately 2.42 eV., i.e., only a little greater than in 3

-Photovoltage

-_-__ Photoconductiwty

g2

5: P

gl t

/ n

-

0.4

0.5

06

0.7

Wavelength (microns)

FIG.18. Photoresponse in AIP. (From Grimmeiss et a/., Ref. 26.)

13. GALLIUM ANTIMONIDE

Little work has been carried out on photoeffects in this material Spectral sensitivity measurements of photoconductivity were carried out on one 26

H. G. Grimmeiss, W. Kischio, and A. Rabenaq J . Phys. Chem. Solids 16, 302 (1960). G. Grimmeiss and H. Koelmans, Philips Res. Rept. 15, 290 (1960).

" H.

9.

PHOTOCONDUCTION IN III-V

COMPOUNDS

237

p-type specimen by Frederikse and Blunt.4 Their results, shown in Fig. 19, indicate energy gaps of 0.65 eV at room temperature and 0.77 eV at 85 K. Habegger and Fanz8 have reported observations of long wavelength photoresponse due to impurities. They conclude that there are levels situated 0.034, 0.062, 0.076, and 0.103 eV above the valence band edge.

I

2 Wovelengt h (microns)

FIG. 19. Photoconductivity in GaSb. (From Frederikse and Blunt, Ref. 4.)

LukesZsa found that the spectral response fell rapidly as the wavelength was decreased, presumably due to high surface recombination. It is estimated that the lifetime is near 1 p e c from the ratio of PC and PEM effects.’

14. GALLIUM ARSENIDE This material has now been studied fairly extensively-second only to InSb. Again much of the work has been stimulated by its potentialities for device application. Some spectral sensitivity curves29 are shown in Fig. 20. The curves fall very steeply in the neighborhood of 1 p, a consequence of the very steep absorption edge found in GaAs3’ As usual the edge for the PEM effect lies at somewhat shorter wavelengths than the PC curve. M. A. Habegger and H. Y. Fan, Bull. Am. Phys. SOC.8, 245 (1963). F. Lukes, Czech. J. Phys. 6, 359 (1956). 2q C. Hilsum and A. C. Rose-Innes, “Semiconducting III-V Compounds.” Macmillan (Pergamon), New York, 1961. ”T. S. Moss, J . App. Phys. 32, 2136 (1961). 28

28a

238

T. S. MOSS

In p-type material an impurity response is often found in the 1.5-p region (as shown in Fig. 20). This is absent in n-type material. The long wavelength tail in high resistivity material has been examined in more detail by

Wavelength (microns)

FIG.20. Spectral sensitivity of GaAs. (From Hilsum and Rose-lnnes, Ref. 29.)

Holeman and H i l ~ u m , ~whose ' results are shown in Fig. 21. It will be seen that the data for both specimens give straight lines when is plotted against photon energy, thus indicating that the absorption coefficient follows the law

K a (hv - E ) 5 / 2

where E = 0.63 eV.

Carrier lifetimes have been deduced from photo effect^.^' For various samples of material these authors obtained the following results : p type:

~ , 1 0 - ~to

n type:

z, lo-''

to

t,10-12 to lo-'' ~ , 1 0 - ~to

sec sec.

In all cases therefore the majority carrier lifetimes are greater than 1p e c while the minority carriers are usually several orders less. On some specimens it was possible to eliminate the effects of trapping by flooding the material with white light, rendering rP and z, equal. The most important application of GaAs in the photoelectric field is in solar batteries. For this use its basic characteristics are so favorable that 3’

B. R. Holeman and C. Hilsum, J . Phys. Chem. Sotids 22, 19 (1961).

32C.Hilsum and B. R, Holeman, Proc. Intern. Con5 Semicond. Phys, Prague, 1960 p. 962. Czech. Acad. Sci., Prague, 1961.

9.

PHOTOCONDUCTION IN 111-V

COMPOUNDS

239

it seems unlikely that any other material will be found capable of giving as high a conversion efficiency as GaAs. It has already been pointed out that the main compromise in the design of a solar battery is between materials of low energy gap, where most of the solar photons are sufficiently energetic to produce photoelectrons so that J o and the short-circuit current are large, and high energy gap materials where q and j , are low so that the open-circuit voltage is high. For the simple assumption that all photons of energy greater than the energy gap produce useful photocarriers and all other parameters are taken to be the same for all materials, it is shown that there is an optimum energy gap, which is about 1.4 eV (see Fig. 6}, i.e., very close to the value for GaAs.

~

7

08 Photon energy (eV)

09

0

FIG.21. Long wavelength photoconductivity in high resistivity GaAs. (From Holeman and Hilsum, Ref. 31.)

It is clearly very important that in a material of given energy gap (and hence given ni) the absorption should be high for wavelengths right up to those equivalent to the energy gap, i.e., the material should have a steep absorption edge, and in this respect GaAs is markedly superior to silicon for example. The edge in GaAs is very steep, the fundamental reason being From that the transitions are vertical as both band extrema are at K[000].

240

T. S. MOSS

the absorption curve the spectral distribution of the short-circuit current can be computed. On multiplying the results by the distribution of solar radiation and integrating, we can determine the collection efficiency in terms of short-circuit electron flow per photon, i.e., Jo/Qo. In order to do this we must estimate diffusion lengths in material of suitable doping concentrations. We will take the two sides of the junction to be equally doped, so that in the surface layer p = 5 x 1017 cm-3 and in the bulk n = 5 x 1017cm-’. Assume that the lifetime is determined by direct recombinations (radiative or Auger) such that q,nd = rep,

=

10”

~ m - ~ ,

i.e., both lifetimes are 2 x lo-* sec. We thus have diffusion lengths of lop for electrons in the surface layer, and 4 p for holes in the bulk. With these values the computed collection efficiency is Jo/Qo

=

0.4 electrons per solar photon.

The other parameter which is important isj, [Eq. (63)] which should be as small as possible. This parameter depends mainly on the intrinsic carrier concentration, which is given by

n,’

=

6 ( 2 7 c ~ n * k T / hexp ~ ) ~( - E / k T ) .

For GaAs the degeneracy parameter 6 is as small as it can be-namely unity-as the extrema are in the middle of the Brillouin zone. The energy bands are spherical, so that the density of states mass term becomes (rn*/rno)3

=

(me*rnh*/m,2)3/2

=

0.01.

This is a very low value, and it is unlikely that values significantly lower than this will occur for materials with an energy gap 1.4 eV. Hence n,’ 4 x 10” cm-6 and j, = 0.22. Thus the calculated efficiency for a GaAs solar cell with the above parameters, ignoring losses, is

-

-

q

=

28%.

Assuming losses by surface reflection of 10% and a junction efficiency such that Power out Short-circuit current x Open-circuit voltage a realistic figure for achievable efficiency is q = 20%.

=

80%,

9.

PHOTOCONDUCTION 1N Ill-V

COMPOUNDS

241

An early curve of spectral response is given in Fig. 22. Later gave better spectral response and a collection efficiency up to about half of theoretical. With open-circuit voltages > 0.8 V these workers achieved efficiencies of 7:4. Progress is continuing and the most recent laboratory samples have given efficiencies as high as 13 %.

FIG.22. Response of a GaAs solar cell. (From Welker and Weiss, Ref. 45.)

15. GALLIUM PHOSPHIDE The energy gap in Gap is 2.25 eV, so that the main absorption edge is in the green. A room temperature photovoltaic response curve, taken at a metal point contact by Grimmeiss and K ~ e l m a n s is , ~shown ~ in Fig. 23. The intrinsic response peaks fairly sharply at 0.42p, and there is an impurity response extending well beyond the band edge. The same shape of curve was obtained when the light was prevented from illuminating the metal probe, so that the curve is characteristic of the GaP alone. The spectral response of a p-n junction measured by Grimmeiss et ~ 1 at -170°C is shown in Fig. 24. The intrinsic edge appears clearly at approximately 0.55 p, but again there is an impurity response at long wavelength. For such cooled specimens these workers obtained photovoltages as high as 1.7 V from a single junction. Allen and Cherry35find that very high resistivity material can be produced by diffusing Cu into n-type Gap. In such material the photoconductivity is superlinear in the visible, intrinsic, region and is sublinear in the infrared, impurity region. E. G. Bylander, A. J. Hodges, and J. A. Roberts, J . Opt. SOC. Am. 50, 983 (1960). H. G. Grimmeiss, A. Rabenau, and H. Koelmans, J. Appl. Phys. 32, 2123 (1961). 35 J. W. Allen and R. J. Cherry, J. Phys. Chem. Solids 23, 509 (1962).

33

34

.

~

~

242

T. S. MOSS

Wovelength (microns)

FIG.23. Spectral response of metal probe on Gap. (From Grimmeiss and Koelmans, Ref. 27.)

16. OTHERCOMPOUNDS

Other 111-V compounds exist-particularly those of the lighter elements-but their electronic properties have been little studied. Most of the work which has been done has concentrated on luminescence. Grimmeiss et al. 36 have studied GaN, including some observations of photoconductivity. The activation energy is approximately 3.3 eV.

J Liquid oir temperature

I0 4

0.5

0.6

07

8

Wovelength (microns)

FIG.24. Spectral response of GaP p-n junction. (From Grimmeiss et al., Ref. 34.) 36

H. G. Grimmeiss, R. Groth, and J. Maak, 2. Naturforsch. 159, 799 (1960).

9.

PHOTOCONDUCTION IN 111-V

COMPOUNDS

243

The energy gap in AlN is quoted as 4.6 eV.37 Stone and Hill3* have studied BP, giving an energy gap of 6 eV from absorption data. Band theory calculations on BN39 indicate an energy gap 10 eV. InN has an estimated energy gap of 2.4 eV.40

-

3'H. F. Ivey, Advan. Electron. Electron. Phys. Suppl. 1, 169 (1963). 38 B. Stone and D. Hill, Phys. Rer. Letters 4, 282 (1960). 39 L. Kleinman and J. C. Phillips, Phvs. Rev. 117, 460 (1960). 40 B. F. Ormont, Zh. Neorgan. Khim. 4, 3176 (19591, (English Transl.: Russ. J . lnorg. Chrm. 4, 988 (1959)l. D. G. Avery, D. W. Goodwin, and A. E. Rennie, J. Sci. Instr. 34, 394 (1957) 4 2 M. E. Lasser, P. Cholet, and E. C. Wurst, J. Opt. SOC.Am. 48, 468 (1958). 43 D. W. Goodwin, J. Sci. Instr. 34, 367 (1957). 44 R. M. Talley and D. P. Enright, Phgs. Rrri. 95, 1092 (1954). 4 5 H. Welker and H. Weiss, Solid State Phys. 3, 65 (1956). 41

List of Symbols Constant of integration Constant of integration Mobility ratio, pc/ph Constant of proportionality Diffusion constants Space charge layer depth, solar battery thickness Electron charge Energies Electric fields Bandwidth Af Intensity parameter F G Intensity parameter Generation rate g H Heating parameter I Photon intensity Planck's constant h J , J,, J ,J -,J , , J , , Currents or charge flows Junction parameter lo Absorption coefficient K Boltzmann's constant k Diffusion lengths L,1 Electron mass m0 Effective masses m*, me*, mh* Junction parameter N Densities of states N " , Nc Electron concentrations n, ndrno, n,,n, A

B b C

+

P P. P a , Po AP, AP Anj An,

Po, Qmin r R

S S

Junction parameter Concentrations of holes Increases in hole concent ration Increases in electron concentration Intensities of quanta Junction parameter Integration constant Integration constant or sensitivity Surface recombination velocity Absolute temperature Thickness Recombination rates Voltage Wattage exp ( - Kt) Coordinates Recombination parameter Relative thicknesses Degeneracy parameter Dielectric constant Efficiency, Fermi level Wavelengths Mobility Carrier lifetimes

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CHAPTER 10

Quantum Efficiency of the Internal Photoelectric Effect in InSb E. AntonGk and J . Tauc I. INTRODUCTION. . . 11. EXPERIMENTAL . . . 1. Methods . . . . 2. Results . . . . 111. THEORY . . . . . 3. Introduction . . . 4. Band Structure of InSh 5. Spectral Dependence of 6. Impact Ionization . . I . Results . . . . . IV. DISCUSSION . . . .

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245 247 241 248 249 249 250 251 253 259 261

I. Introduction This paper deals with the dependence on the photon energy of the number of electron-hole pairs created by the absorption of a photon in a semiconductor. The quantum efficiency q defined in this way is to be distinguished from the different concepts of quantum efficiencies used in the theory of photoconductivity where the introduction of additional carriers through the contacts plays an essential role. The quantum efficiency discussed in this chapter determines the short-circuit current in p-n junctions or in photoelectromagnetic effect which depends on the existence of both electron and hole as a pair. Quantum efficiency is a useful concept if the mean lifetime of pairs is longer than the duration of the processes leading to their generation and is a characteristic of the material itself. The experimental determination of q is unfortunately not straightforward, and several complications arise which we shall discuss later. It has been known for a long time that the quantum efficiency in lightly doped semiconductors' is equal to one (within the precision of

' For heavily doped semiconductors, see V.

K. Subashiev and G. B. Dubrovskii, Fiz. Twrd. Tela 6, 512 (1964) [English Transl.; Sooirt Phys.-Solid State 6, 403 (1964)].

245

246

E. A N T O N ~ ~ AND K J. TAUC

measurement which is rather small) near the absorption edge and proportional to hv at photon energies hv much larger than the gap (e.g., in Ge the energy needed for the creation of an electron-hole pair in the X-ray region is 2.5 eV 2 ) , In this chapter we discuss the transition region. Here the first measurement of q was performed by Koc3 in Ge by means of the photovoltaic effect in p-n junctions; the measurement was repeated by Vavilov and B r i t ~ y n Both . ~ authors found qualitatively similar results, that is, there exists a certain threshold photon energy at which q starts to rise above the value one observed near the absorption edge; Koc obtained 2.2 eV for this threshold, Vavilov and Britsyn 2.9 eV. This discrepancy was thought’ to be due to the different reflectivities determined by the respective authors. However, Thiessen6 repeated these measurements and found other important difficulties inherent in the determination of q in Ge. They appear to be connected with the dependence of the diffusion lengths on the wavelength of incident light. This means that the lifetime of the pair is not constant in the spectral range considered and may cause a structure in the spectral photoresponse which is not caused by the spectral dependence of r]. Similar difficulties may exist in the case of Si for which q was determined by Vavilov and Britsyn7 and by Tuzzolino.8 Recently, Loh’ measured the spectral photoresponse in GaAs, Si, and Ge; his results are very different from those given in Refs. 3, 4, and 7. The present chapter deals with InSb, where the difficulties mentioned appear to be less important at room temperature. From the experimental standpoint InSb has an almost constant reflectivity in the region where the ~ ( h vcurve ) shows a structure. Therefore we consider the results obtained on InSb more reliable. The theory of the quantum efficiency q in semiconductors is based on the idea that the absorbed photon creates an electron and a hole with excess energies which under certain circumstances may be used to produce additional carriers by impact ionization. The study of quantum efficiency contributes to the understanding of the behavior of high energy, or hot, current carriers in solids.

* J. Drahokoupil, M. Malkovski, and J. Tauc. Czech. J . Phys. 7, 57 (1957). S. Koc, Czech. J . Phys. 7, 91 (1957). 4V. S. Vavilov and K. I. Britsyn, Zh. eksperim. i Teor. Fiz. 34, 521 (1958). [English Transl.: Soviet Phys. J E T P 34(7),359 (1958)l. J. Taw, J . Phys. Chem. Solids 8, 219 (1959). K. Thiessen, private communication. ’V. S. Vavilov and K. I. Britsyn, Zh. eksperim. i Teor. Fiz. 34, 1354 (1958). [English Transl.: Soviet Phys. JETP 34(7), 935 (1958)l. * A. J. Tuuolino, Phys. Rev. 134, A205 (1964). E. Loh, J . Phys. Chem. Solids 24, 493 (1963).

10.

INTERNAL PHOTOELECTRIC EFFECT I N

InSb

247

11. Experimental

1 . METHODS A convenient way to determine the quantum efficiency q is to measure the short-circuit current of a p-n junction. If the radiation is absorbed within the barrier and the lifetime of excess carriers is longer than the time the carriers spend inside the barrier then the short-circuit current density divided by e is equal to the number of pairs generated by photon absorption per second per cmz of the barrier cross section. In practical cases it is usually possible to make corrections for deviations from this ideal arrangement (cf., e.g., Ref. 2). However, for InSb this method is hardly applicable at room temperature as it is difficult to prepare p-n junctions with satisfactory properties. Therefore another method" was used, the simultaneous measurement of the photoconductive ( P C ) and photoelectromagnetic ( P E M ) effects. The latter effect is proportional to the number of free electron-hole pairs. The photoconductivity can be influenced by creation and trapping of single carriers. In both cases changes in the state of impurity centers at illumination may influence substantially the results. It is known from the work of Zitter et al.", that at room temperature trapping centers do not manifest themselves. Our observation was that both the PEM and PC effects gave the same photon energy dependence of the signals; at very low temperatures often inconsistent results were found. Also, an additional illumination of the sample with an incandescent bulb did not change the PEM and PC signals measured with interrupted light from a monochromator. We think it therefore safe to assume that at room temperature the curves to be described below correspond actually to the spectral dependence of q. The simultaneous measurement of the PEM and PC effects allows us also to estimate the influence of the spectral dependence of the absorption constant K , as described in Ref. 10. In the conditions of measurement, K > L-' where L is the effective diffusion length of carriers. Under such conditions, the PEM effect is independent of K , and, provided that the surface recombination velocity is low, this is also true for the PC effect. The measurement was carried out with interrupted light from a monochromator with a beam divided in two parts: one part was falling on the sample and the signal was measured using a narrow-band amplifier, the other part on a Hilger-Schwarz thermocouple and the signal measured with the same amplifier. For reasons discussed above, the ratio of both signals was measured in the photon energy range where they are lo

J. Tauc and A. Abrahim. Czech. J . Phys. 9. 95 (1959).

'' R. N. Zitter. A. J. Straws, and A. E. Attard. Phys. Rrc. 115. 266 (1959).

248

E.

ANTONC~KAND

J. TAUC

proportional to q. From other work (in particular from Ref. 12) it is known that near the absorption edge q = I, and this was used to normalize the q(hv) curve. Such a simple procedure was possible because the reflectivity of InSb in the spectral range considered was found to be almost constant (cf. Ref. 10). Otherwise the absolute value of the reflectivity R must be determined by measurement and the factor (1-R) used when calculating q from the currents. Precise determination of the correct value of R is difficult. The reflectivity depends on surface conditions, and one should therefore measure R on the actual sample; in this case the surface is etched and nonflat and part of the light is scattered. All these difficulties may be significant in those parts of the spectrum where R has a sharp structure, and this may introduce an erroneous structure into the ~ ( h vcurve. ) We assume the reflectivity of the thermocouple to be constant. Details of the experimental arrangement are described in Ref. 10. 2. RESULTS A typical result is shown in Fig. 1. The measurement was performed on an etched sample cut from a single crystal of InSb and containing an 1.5

17

I .o

I

I

0.6

I

I

0.8

I

I .o

FIG. 1. The quantum efficiency of the internal photoelectric effect in InSb as a function of the photon energy (measured at room temperature.).(After J. T a w and A. Abraham. Ref. 10.)

acceptor concentration of 6.4 x lo1’ cm-3 as determined by measurement of the Hall coefficient. The mobility of electrons at room temperature was D. W. Goodwin, Rep. Meeting Semicond., Rugby, 1956 p.137. Phys. SOC.,London, 1956.

10.

INTERNAL PHOTOELECTRIC EFFECT IN

InSb

249

4.6 x 104cm2V-'sec-'. The curve shown was obtained from the PEM effect at 18°C. The q(hv)curve shows a typical structure. The quantum efficiency begins to increase at an energy of 0.47 eV. The increase of q shows a saturation, and a new break is observable at 0.6eV. The positions of the breaks were well reproducible on various samples and with various surface conditions but the slopes of the curves differed within rather broad limits. It was possible to measure down to - 64°C and estimate the temperature dependeV/grad. ence of the breaks, which is negative and of the order It is difficult to estimate the error of these measurements as we do not know the errors involved in the assumptions used. The measurement itself was done with a precision better than 5 % ; the reproducibility of the slopes measured on different samples and different surface conditions was much worse. The general features of the q(hv) curve, are believed to be correct.

111. Theory

3. INTRODUCTION If the energy hv of photons absorbed in a semiconductor is greater than the energy gap E , , the excess energy (h\l-Eg) is divided in some way between the electrons and holes produced. When studying the mechanism of dissipation of this excess energy in connection with the quantum efficiency two types of processes have to be considered separately : processes which do not change the number of pairs created during the absorption and processes which do change this number of electron-hole pairs. Of course, we are not able to consider all possible processes, and we shall confine ourselves to those which seem to play the dominating role. In the first case the electron-phonon interaction has to be considered the most effective one. In the second case the generation of additional electron-hole pairs by the across-the-gap impact ionization (interband Auger effect) will be shown to be the most important process ; generally the recombination as an inverse process should be considered as well. However, it can be shown that the probability of recombination is much smaller than the probability of impact ionization in the temperature range in which the spectral dependence of the quantum efficiency has been measured, so that the process of recombination may be neglected in this case. The quantum efficiency is defined as the number of electron-hole pairs produced per absorbed photon. As long as the excess energies of electrons and holes are lower than a certain limit the dissipation mechanism of this energy is determined by the electron-phonon interaction only, and no additional electron-hole pairs are generated. The quantum efficiency in

250

E.

ANTONE~KAND

J . TAUC

this energy range is unity. For higher excess energies, above a certain threshold, impact ionization takes place, and the quantum efficiency curve rises with increasing energy of photons absorbed. The detailed form of the spectral dependence is determined both by the primary absorption process and by the secondary process including the electron-phonon interaction and impact ionization. We see that a satisfactory knowledge of the three processes mentioned is needed for calculation of the spectral dependence of the quantum efficiency. Unfortunately none of them is known with necessary accuracy, especially the absorption process and the impact ionization. As we shall see the probabilities of both processes are strongly dependent on the accuracy of the band structure calculations, and these are not satisfactorily known for InSb at the present time. It will be shown that an approximate treatment is necessary for this case which will enable us to evaluate the probabilities of both processes in an approximate way. The results make it possible to calculate the spectral dependence of the quantum efficiency. 4. BAND STRUCTURE OF InSb

In the present paper we use the energy band structure of InSb as calculated by Kane’ by means of the k * p perturbation approach. (See chapter on the k . p method in Volume 1 of this series.) In the first approximation Kane’s theory treats the mutual interaction of the conduction, heavy and light mass hole, and the split-off bands exactly. This approximation seems to give good approximation for the energy and the wave functions for all bands except for the heavy hole band which is found to have the wrong curvature. Kane shows that the interaction with higher bands can be taken into account by a second-order perturbation treatment ;unfortunately this approximation includes several not very well known constants. Therefore we shall limit ourselves to the simple Kane model modified by Ehrenreich14 for finite temperature :

+

[

E C = . R 2 k 2 f E;2(T) + 8 p32 k 2 ] ’ 2 - )E,*(T) 2m0 h2k2 E,, = - -, 2m, 1

+ E,(T),

where E , , E,,, and EV2are the energies of states in the conduction, heavy I’

l4

E. 0. Kane, J . Phys. Chem. Solids 1, 249 (1957). H. Ehrenreich, J . Phys. Chem. Solids 2, 131 (1957).

10.

INTERNAL PHOTOELECTRIC EFFECT IN

InSb

25 I

hole, and light hole bands, respectively; rn, is the free electron mass; rn,, is the effective mass of the heavy hole band, E,(T) is the energy gap; E,*(T) is the value the energy gap would have if its temperature dependence were determined by lattice dilatation only. P is the matrix element which couples the p functions corresponding to the valence bands with the s functions corresponding to the conduction band. The value of P z can be estimated from the experimentally observed conduction band masses at

k

0. The nonparabolic character of the conduction and light mass hole bands which is typical for InSb presents some difficulties, especially when calculating the Auger transition probabilities which can be done more simply for parabolic bands. In order to have a standard of comparison for the more complicated band structure (1) we shall calculate all quantities for the parabolic bands too; in this case we shall use the following notation for the effective masses : rncr rn,,, and mv2,respectively. Using the calculated wave functions and the experimental value for the matrix element P, Kane was able to calculate the absorption constant corresponding to transitions for electrons into the conduction band from both valence bands. The accuracy of the optical absorption calculations is determined by the accuracy of the band structure calculations. Kane’s theoretical absorption curve agrees with the experimental curve except that it is about a factor of 1.5 too low in absolute value. In our case we are interested in both types of optical transitions separately because of the different ways in which they influence the subsequent relaxation process. We denote a1 and a2 as the absorption constants corressponding to the transitions of electrons from heavy mass hole and light mass hole bands, respectively, to the conduction band. As the probability of absorption of a photon per unit time is given as ac/n, the relative probability of photons being absorbed by electrons in the heavy mass hole or light mass hole bands is a l / ( a l a z ) and az/(al a2), respectively. Both these expressions are functions of the frequency v and can be calculated using the Kane’s theory. Unfortunately, these calculations are affected at least by the same inaccuracy as Kane’s calculations of the absorption constant (al az) or even greater. A more precise calculation of a1 and az would require a better knowledge of the corresponding wave functions, especially of the heavy mass hole wave functions. =

+

+

+

5 . SPECTRAL DEPENDENCE OF QUANTUM EFFICIENCY During the primary absorption process electrons from both valence bands are excited into the conduction band. The energy of these excited electrons and holes depends on the energy of the photon hv and can be

252

E.

ANTONC~KAND

J . TAUC

easily calculated for both the parabolic bands and Kane band model. If electrons are excited from the valence band E,,, most of the excess energy (hv-E,) is passed predominantly to the conduction electrons because of the large difference between the effective masses of both bands. On the other hand, if electrons are excited from the valence band EV2,the excess energy of electrons and holes is almost equal. As long as the excess energies of electrons or holes are lower than a certain limit, the dissipation mechanism of this energy is determined mostly by the electron-phonon interaction. For higher energies another process takes place in the energy dissipation mechanism-the across-the-gap impact ionization-by which additional electron-hole pairs are generated. It can be easily seen in our case that the electrons excited from the valence band E,, will be the first to have the necessary energy for this impact ionization process. In fact, it can be shown that processes of this type determine the initial energy dependence of the quantum efficiency. We shall confine ourselves to this simple case although it is possible to take into account more complicated processes with several types of different ionizing carriers. However, it is assumed that over the corresponding range of hv “third generation” pairs can be neglected. We shall suppose in conformity with the Kane band model of InSb, that the distribution function of the excited electrons (or holes) N(E) and the impact ionization probability per second Pi@) are functions of the excess energy of electrons only. Denote by E , the excess energy of the ionizing electrons just after the absorption of the photon with the energy hv and Ei the smallest excess energy for which the impact ionization can take place. The probability for an electron with excess energy E , to reach this threshold energy Ei without causing impact ionization is expressed by

Consequently the number of additional electron-hole pairs generated by ionizing electrons in slowing down by electron-phonon interaction from E , to below Ei is given as

where N is the number of photons absorbed in the bulk of the semiconductor and a,N/(u, + aZ) is the corresponding part of these photons absorbed in the valence band E,,. The spectral dependence of the quantum efficiencycan be written in this case in the following form “

q=1+a1

+ a2

[I

- exp

(-

Pi ($)-I

dE)].

(3)

10.

INTERNAL PHOTOELECTRIC EFFECT IN

InSb

253

Until now we have considered only one type of transition. In fact expressions similar to (2) can be derived for other transitions as well. In the case that different ionizing particles take part in this process, the right-hand side of Eq.(3) will contain a sum of expressions similar to (2). Note that for Pi = 0 the quantum efficiency q is equal to unity.

6 . IMPACT IONIZATION In this section we shall calculate the probability of impact ionization both for the Kane band model and for the equivalent parabolic band structure. Let us now consider a particular Auger transition of the following type. Suppose that a single energetic particle (electron or hole) with original energy E,,(k,) and wave vector k , is scattered to a state of energy E,,.(kl‘) and wave vector kl’, and in the process produces an electron-hole pair characterized by energies En2,(k2’) and E,,(k,) and by wave vectors k,’ and k,, respectively. We seek the total transition probability per unit time for processes of this type, in which a pair is created. We shall treat the problem using time-dependent perturbation theory assuming that the perturbing Hamiltonian is the difference, due to the Coulomb interaction of electrons, between the complete Hamiltonian for the crystal and the Hamiltonian which is used in the one-electron approximation. In the latter, the Coulomb interaction is replaced by a self-consistent field, containing exclusively terms which depend on single-electron coordinates. The Coulomb interaction terms in the complete Hamiltonian are explicitly taken into account only for electrons of the conduction and valence bands. Electronic interactions in which the remaining electrons participate give rise to polarizibility, and their effects are represented by the use of an effective dielectric constant E. In order to be in a position to estimate the effect of electron screening on the impact ionization we shall assume the perturbing Hamiltonian is given in a more general form as

H’

1

=

Here rii = Ir, - rilr where ri and rj are the position vectors of the ith and jth electrons, E is an effective dielectric constant to be used in our problem, 1 is the screening constant, and the summation is over all pairs of electrons. In accordance with the one-electron approximation the electron wave functions for the conduction and valence bands of the crystal are described by Bloch function, and the electrons which d o not take part in a transition are supposed to have their states unaltered by the transition. When evaluating the matrix elements we do not consider the antisymmetrization of the

254

E.

ANTONC~KAND

J. TAUC

wave functions for simplicity because of the relatively larger uncertainty caused by other approximations used in this problem. We shall not consider the phonon-assisted Auger transitions because Eagles’ has shown that at room temperatures these processes in InSb are less probable by a factor than those without phonons. The transition probability per unit time Pi due to our perturbation from an initial state t+bn1,k1is given by

The factor 2 in the right-hand side of this equation arises from the spin degeneracy of the energy levels in our band model. The transition amplitude akllk.k21(t) is related to the matrix element of the perturbation

in which o is defined by the condition hw

=

En,(kl)+ En2(kz)- Eni,(k~’) - ~%2@2’)*

(6)

As the last factor in the right-hand side of the Eq. (5) considerably differs from zero for hw = 0 only, energy must be conserved in the process of impact ionization. Let the electrons taking part in a transition be labeled 1 and 2. Under the conditions stated, only the term involving e2 exp(-Lr1,)/~rl2 of the perturbation operator H‘ can have a nonzero matrix element
)

eikl’*rl

un I ’,k 1 (rl)eik~’.r2 un2‘,k2’(r2)

d3r1 d3r2.

(7)

The evaluation of this matrix element is very difficult and in general has not yet been done; it is very often taken as a constant16 because of the lack of more detailed information. Such an approximation will be shown to be rather crude. Fortunately, Beattie and Landsberg” have shown that in InSb the matrix element can be simplified by expanding the periodic parts of the Bloch functions in terms of plane waves. The matrix element is then obtained as a multiple sum over reciprocal lattice vectors. Many D. M. Eagles, Proc. Phys. SOC.(London) 78, 204 (1961). I’D. L. Dexter, Proc. Intern. Con$ Semicond. Phys., Prague, 1960 p.122. Czech Acad. Sci., Prague, 1961. A. R. Beattie and P. T. Landsberg, Proc. Roy. SOC. (London) A249, 16 (1959)

10.

INTERNAL PHOTOELECTRIC EFFECT IN

InSb

255

of the terms which occur in this sum can be shown to be small and may be neglected. In this case the matrix element (7) becomes

where g

= k,

-

k,'

=

k2' - k,. F , and F , are overlap integrals such that

The Kronecker delta function in Eq. (8) expresses the law of conservation of momentum in the electron system during the transition. Inserting the matrix element (8) into Eqs. (5 and 6) we obtain

Before proceeding further in the calculations it is necessary to calculate the overlap integrals F , and F , of the periodic parts of Bloch functions. If sufficiently precise wave functions are available we can estimate these integrals directly from Eq. (9). On the other hand if the wave functions are not known with great precision, the estimate obtained from Eq. (9) may be considerably in error. Unfortunately, the latter case seems to occur with the heavy hole band of InSb as mentioned in Section 4. It then becomes of interest to find another method which would enable us to estimate those integrals with sufficient accuracy both for Kane band model and the parabolic bands without explicit use of the wave functions." For the sake of simplicity we shall write the overlap integrals (9) in the form I(kX2' = (U,',k+q'*U,,k+q d3r, (11) where the integration is over a unit cell. Suppose that in (11) q and q' are small enough for the results of first-order k - p perturbation theory to be applicable. Expanding Un,k+q in terms of the wave functions tdn,,k the following expressions can be easily derived. If n’ = n Il(k):flZ

1.

(12)

On the right-hand side of the equation the quadratic and higher terms in q,q' have been neglected. For overlaps between different bands n' # n the following expressions can be written as

E. AntonEik and P. T. Landsberg,

Proc.

Phys. SOC.(London) 82, 337 (1963).

256

E. A N T O N ~ ~AND K J . TAUC

wherefnrnt[) is the oscillator strength defined as

K,,, are the matrix elements of the velocity operator including the spin-orbit effects. It is known that the oscillator strengths satisfy an f-sum rule,lg so that for the overlap integrals a similar sum rule can be easily derived:

By picking out the most important terms in the sum on the left-hand side of (14), using effective mass determination or oscillator strength data, one can obtain information concerning the overlap integrals (11). Concluding, let us remark that the value of overlap integrals corresponding to the intraband transitions goes to unity for q, q' approaching zero, while for interband transitions it goes to zero. To proceed further in the calculations it is necessary to choose a special type of transition. Beattie2' has considered all possible Auger transitions in InSb which could determine the quantum efficiency curve in the energy range considered; they are shown in Fig. 2. In this connection we are interested in two problems. Firstly, it is the minimum energy E,,(k,) of the ionizing particle (or the corresponding threshold energy of the absorbed photons) for which the Auger transition of the type considered takes place. Secondly we are interested in the absolute value of the probability Pi. Beattie has shown that of ten possible transitions only two (transitions 1 and 7) need to be considered; the others are either much less probable (transitions 2 , 5 , 6 , 8) or are beyond the range of the experimental data (transitions 3,4, 9, 10). Actually it can be shown" that the probability of the transition 7 is of the same order as of the transitions 2,5, 6, 8, so that spectral dependence of the quantum efficiency in the range investigated seems to be mainly determined by the transition 1. In further calculations we shall confine ourselves to these transitions only, although it is easy to calculate the other transitions as well. l9 2o

E. I. Mount, Solid State Phys. 13,305 (1962). A. R. Beattie, J . Phys. Chem. Solids 23, 1049 (1962). E. Anton&, in "Physics of Semiconductors" (Proc. 7th Intern. Conf.), p. 473. Dunod, Paris and Academic Press, New York, 1964.

''

I21

photon

(71

photon

2’

(bl (C)

FIG.2. The ten types of phononless collision processes which can produce an electron-hole pair are shown. In each diagram the energy bands are shown schematically by curve (a) for the conduction band, curve (b) for the heavy hole band, and curve (c) for the light hole band. The states of the primary carriers initially created by the incident photon are shown by a filled in circle for the electron and an open circle for the hole. The arrows represent the transitions of electrons from their initial states to their final states by the appropriate collision process, in which energy and momentum are conserved. (After A. R. Beattie, Ref. 20.)

g 4

258

E.

ANTONC~KAND

J . TAUC

Transition 1 is characterized by the initial states k , and kz in the conduction (n, = c) and the valence band (nz = vl), respectively, and the final states k,‘ and k2‘ in the conduction band (n, = n2‘ = c). When evaluating the integrals in expression (10) a further simplification is made by taking 1= 0 since it has been shown by Beattie and Landsberg” that the effect of screening in Auger transitions is negligible in InSb at room temperature. Using the scalar effective mass mv,, we can write according to Eqs. (12) and (14) for small vectors g the overlap integrals in the form

where the constant @’ can be estimated using some empirical parameters of the semiconductor. For too large g the expression (15) cannot be used. Beattie and Landsberg” and recently H a l ~ e r nhave ~ ~ shown for special cases that, for large vectors g, the expression IF,F212 may be approximately taken as a constant, the value of which can be eventually estimated from experiment. It can be shown that for energies E , which are not much above the threshold energy Ei the probabilities Pi calculated using IF,F2I2 either in the form (15) or as an equivalent constant corresponding to the threshold value of the vector g differ very little. The probabilities of the other transitions can be calculated in similar manner, and a similar conclusion holds for them as for the transition 1 except for transitions in which the direct transitions (g = 0) or nearly direct transitions (g FZ 0) take place. In this case only expression (15) has to be used because otherwise if the approximation of IF,F,12 as a constant is used integral (10) will diverge at the lower limit for vanishing 1(e.g., this problem arises when calculating Pi for Beattie’s transition 7 shown in Fig. 2). To proceed further in the calculations of the spectral dependence of the quantum efficiency it is necessary to evaluate the rate of energy dissipation through the emission of phonons. Since in InSb the interaction of electrons with acoustical phonons is much smaller than that one with optical phonons14 the expression dE/dt is determined mainly by the emission of optical phonons. The rate of energy loss is expressed in this case as

where ho, is the energy of optical phonons and is assumed to be independent of the wave vector of phonons q and wk,k.-,, is the emission rate ”

k R . Beattie and P. T. Landsberg, Proc. Roy. SOC. (London) A258, 486 (1960). Halpern, J . Phys. Chem. Solids 24, 1495 (1963).

23 V.

10.

INTERNAL PHOTOELECTRIC EFFECT I N

259

InSb

and wk,k+q is the absorption rate of a phonon, respectively. CallenZ4 has shown that for a parabolic band the rate of energy loss is given by

where M is the reduced mass of the ions, Q is the volume of an ion pair, and e* is the effective ionic charge. For the Kane band model the nonparabolicity of the bands has to be taken into account, and an expression similar to (16) can be easily derived. 7. RESULTS

In this section we give some numerical results for InSb at room temperature using the values given in Table I for the parameters concerned. Before numerical values can be obtained from the theoretical expression for the spectral dependence of quantum efficiency, it is necessary to know the value of the constant a2in expression (15) for overlap integrals ( F , F , ( * . This constant can be estimated from Eq. (14) as follows. Using the effective mass m,, in place of m, in the expression (14) and neglecting all other overlap integrals which seem to be small'6 the minimum value for cDz for the transition 1 can be roughly estimated as

which gives approximately cDmin2 = 5 x lo', a.u. so that the value a12 = 7.5 x 10' a.u. seems to be quite a reasonable one. TABLE I PARAMETERS USEDI N THE NUMERICAL CALCULATION OF THE SPECTRAL DEPENDENCE OF QUANIU M EFFICIENCY m, = 0.03 m,

a

mv,= 0.18 tn,

m, = 0.012 m o c m,' = 0.02 mo

E,* (300°K) = 0.20 eV E,(300"K)

=

0.18eV

P2 = 0.48a,u.b t; =

16"

D . M . Eagles, Ref. 15. A. R. Beattie, Ref. 20. 'G. W. Gobeli and H. Y . Fan, Phys. Rat,. 119, 613 (1960)

"

H. Ehrenreich, Ref. 14. 24

H. B. Callen, Phys. Re11 76. 1394 (1949).

R A4

= 6.8 x =

0.99 x 1 0 - " g d

e* = 0.18

hw,

=

cm'

0.025 eV

260

E . A N T O N ~ ~ AND K J . TAUC

The evaluation of the rate of energy dissipation dE/dt is quite straightforward. Unfortunately the numerical values of this function are affected by a certain discrepancy concerning the value of the effective ionic charge e*. We have used in our calculation the value estimated by Ehrenrei~h’~ as e* = 0.18 although another value e* = 0.13 is quoted by Spitzer and Fan.” When using the latter value the values of the quantum efficiency would be approximately doubled in the energy range considered. There is a certain difficulty when calculating the spectral dependence of the absorption constants a1 and az. Actually we could use the Kane calculations of the absorption constant, but they are unfortunately affected by the incomplete knowledge of the wave functions of the heavy hole band. As it is not clear what is the effect of this inaccuracy on the shape of the spectral dependence of the quantum efficiency, we have used for simplicity the absorption constant formula as calculated for simple parabolic bands so that the factor depending on the absorption constant a1 and a2 is constant for all energies considered. In Fig. 3 the right-hand side of Eq. (3) has been plotted as a function of photon energy for both Kane band model and parabolic bands.

03

04

05

06

hu

FIG. 3. The spectral dependence of the quantum efficiency plotted according to Eq. ( 3 ) : Kane band model, - - - - - - - parabolic bands. 25

W. G. Spitzer and H. Y. Fan, Phys. Rea. 99, 1893 (1955).

10.

INTERNAL PHOTOELECTRIC EFFECT IN

InSb

261

IV. Discussion

A comparison of experiments and theory must be done cautiously as both experimental results and theoretical calculations may be influenced by errors the magnitude of which it is difficult to evaluate at the present stage. Although TJ is a simply defined parameter, its experimental determination is not straightforward. A severe difficulty appears to be the presence of traps the state of which varies with illumination. Other experimental difficulties exist, among them the determination of the correct reflectivity as discussed in Section 1. Small errors in the determination of reflectivity can lead to fictitious structure of q curves.26 All these complications explain the fact that the determinations of q curves by various authors do not agree very well. One should expect that with InSb at room temperature the situation is better in the above mentioned respects for reasons explained in the text. Therefore the general features of the curve are considered to be correct. It is, of course, desirable that the measurements be repeated. From the theoretical point of view the calculation of the spectral dependence of quantum efficiency is a rather difficult problem because a relatively good knowledge is needed not only of the primary absorption process but also of both types of interactions determining the secondary relaxation process. The shape of the enhanced quantum efficiency curve of germanium and silicon has been discussed previously by several author^.^*-^^ The explanation of the quantum efficiency curve of InSb has been suggested by Beattie.” Assuming that there exists a direct correlation between the shape of the spectral dependence of TJ and of Pi,Beattie has shown that initial part of the curve is caused by transitions 1, while the subsequent monotonic increase corresponds to transition 7. Moreover the corresponding threshold energies seemed to be in good agreement with the experimentally observed breaks in Fig. 1. Unfortunately the present more detailed calculations show that this explanation appears not to be correct for the following reason. Beattie’s statement concerning the role of transition 7 is based on an estimation of Piin which the overlap integrals IF,F,I2 are assumed to be constant in the From this standpoint, one should be cautious with the curves reported by Loh.’ Particularly since the mechanism suggested by Loh and Phillips2’ for the explanation of structure is obviously not applicable to lnSb (and also not to the results obtained on p - n junctions). E. Loh and J. C. Phillips, J . Phys. Chem. Solids 24, 495 (1963). E. Anton€ik, Czech. J . Phys. 7 . 674 (1957); 8, 492 (1958). 2 9 W. Shockley, Czech. J . Phys. B11, 81 (1961). 30 R. J. Hodgkinson, Proc. Phys. Soc. (London)82, 58 and 1010 (1963).

26

’’ ’*

262

E.

ANTONC~KAND

J . TAUC

energy range considered. Actually this assumption does not hold as the direct Auger transitions are predominant in this case and, as mentioned in Section 6, expression (15) has to be used, rather than the assumption of a constant IF,F21. Generally speaking the role of transition 7 has been greatly overestimated in Beattie’s discussion, while in fact their contribution to the quantum efficiency is of the same order as of the other transitions (except transition 1) in this energy range which are negligible. The comparison of Figs. 1 and 3 shows that the present theory explains satisfactorily both the general dependence of the quantum efficiency on photon energy and the correct order of magnitude without using any adjustable parameters. Figure 3 also shows that the Kane band model is more suitable than the parabolic band approximation. The fact that the theoretical curves do not show any fine structure found experimentally appears to be the consequence of our rough approximation of the factor C ( ~ / ( C I+ ~ u2). Unfortunately, the evaluation of this factor within the Kane band model is not satisfactory. This problem cannot be solved at the present time without more detailed calculation of the band structure of InSb and more precise estimation of some important parameters (such as e* etc.).

CHAPTER 11

Photoelectric Threshold and Work Function G . W. Gobeli and F. G. Allen I. INTRODUCTION AND DISCUSSION . . 11. MEASUREMENT TECHNIQUES . . . 1. Kelvin Contact Poterrrial Difference 2. Thermionic Methods . . . . 3. Photoelectric Techniques . . 4 . Electron Retardation Reflection . 111.

.

.

.

.

.

.

.

.

.

.

263

.

.

.

. .

. .

. .

. .

. .

. .

269 269 211 212 274

Technique

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

RESULTSAND DISCUSSION . . . . . . . . . 5. Comparison with Preuious Results in the Literature . 6 . Changes in cp u t u l m Due to Anneuling Cleaaed Surfaces 7. Changes in cp and @ with Doping . . . . . . 8. Faces Other Than the ( 1 10) . . . . . . . .

.

.

.

.

. . . . , , . . . . . . . . . .

214 211 219 219 280

I. Introduction and Discussion The amount of data in the literature concerning work function and photoelectric properties of 111-V compounds is very limited because it is only recently that these compounds have become available in well-defined single crystals and their importance realized. The time when such photoelectric measurements were fashionable and when considerable work was being reported preceded the Ill-V compounds by about 15 years. Although the definitions of both work function and photoelectric threshold are precise in themselves, the interpretations of the results of various methods of measurement are subject to some error. Specifically: the true work function cp is the difference in energy of an electron at the Fermi level just inside the surface and of an electron at rest just at the maximum of the potential barrier in the vacuum [see Fig. l(a)]. The latter energy will be called the vacuum energy Eva=. The work function of a surface is unarnbiguous only for atomically clean, single-crystal surfaces which are in thermodynamic equilibrium. Otherwise the value obtained by a measurement can well depend upon the physics of the measurement technique itself, as is discussed in detail in Section 1. For example, for nonhomogeneous

263

264

G . W . GOBELI AND F . G . ALLEN

surfaces, the work function determined by the Kelvin method is the areal average of the local work function while a photoelectric or thermionic determination weights more heavily the lower work function patches.' The photoelectric threshold is defined as being equal to the minimum photon energy hv, needed to raise an electron in the solid to the vacuum level, Eva,, where escape into the vacuum is energetically possible. The objective in determining the threshold value hv, is to use it to establish the quantity CD illustrated in Fig. 1 which is the energy difference

k

(a1

(b)

FIG. 1 (a) Energy levels at a surface and definition of terms. (b) E vs k diagram, to same energy scale as (a), for a hypothetical semiconductor.

between an electron in the highest valence band state at the surface and Eva,.From CD, the electron affinity x may be determined as where E , is the energy gap. When both CD and the work function cp are known, the energy separation between the valence band maximum and the Fermi level, evaluated at the surface, is given by Unfortunately, the determination of CD from photoelectric measurements is quite complex and requires considerable knowledge of the photoelectric process itself as will be discussed below. Photoelectric emission is considered as a two-step event : the absorption of the incident photon and the escape of the excited electron. First consider See, for example, C. Herring and M. Nichols, Rev. Mod Phys. 21, 185 (1949)

11.

PHOTOELECTRIC THRESHOLD AND WORK FUNCTION

265

the absorption of the incident photon raising an electron into an upper energy state. Early theoretical work, based on the free electron model of conductors, required that absorption of the light by conduction electrons occur within a few angstroms of the surface barrier, since the interaction of a “third body” was required to satisfy both energy and momentum conservation. (The photon brings in far less momentum than the electron carries out.) Thus photoelectric emission was intrinsically a surface effect. Today, the band theory of solids, with the specification of a crystal momentum in the lattice, does not place such a restriction on the electronic absorption processes. An electron in the periodic potential of the lattice can absorb a photon and be raised from a lower to an upper state without interaction with a barrier. We now know that a volume excitation of this sort is usually dominant in external photoelectric emission. However, the existence and importance of purely surface-dependent phenomena of the sort analyzed so intensively in the 1930’s as well as the more recently considered surface state emission must be carefully considered. Since we are dealing with emission of electrons from a surface and since the path length of excited electrons is finite, the depth beneath the surface at which electrons are excited is of obvious importance. Hence, a priori, the optical absorption constant c( is important, and the emission yield spectrum will be expected to mirror in some detail the optical absorption spectrum structure. Also, band structure details of the particular material being studied will determine what fraction of the light absorption at a particular photon energy yields electrons in final states lying above the vacuum energy E,,,. Since only such electrons can escape, the band structure may have strong effects on the magnitude and structure of the spectral yield. Now consider the escape of excited electrons into the vacuum. Electrons excited just at the surface, such as from surface states, can escape directly provided only that their energy is sufficient and their velocity is so directed that they can overcome the barrier. However, for the most important excitation mechanism, absorption within the crystal volume, the attrition of the excited electrons by various scattering mechanisms as they traverse the volume from their point of creation to the emitting surface must be considered. In the present work the excited electrons in the volume which can surmount the barrier E,,, are from 4.5 to 6.0 eV above the Fermi level E,. For such electrons the scattering processes are usually strong, and the resultant escape depths are quite small. Values of 25 A have been reported by two groups for clean silicon surface^.^^^ Two different kinds of scattering are normally important. In electron-lattice collisions, each phonon

-

G. W. Gobeli and F. G. Allen, Phys. Reti. 127, 141 (1962); F. G. Allen and G. W. Gobeli, Phys. Reo. 127, 150 (1962). J. van Laar and J. J. Scheer, Philips Res. Rept. 17, 101 (1962).

266

G . W. GOBELI AND F . G . ALLEN

created can change the electron’s momentum vector by an arbitrary amount but can decrease its energy by only a few hundredths of an electron volt. In electron-electron scattering, the excited electron can lose an energy greater than the band gap E , whenever creation of electron-hole pairs results, and this normally makes subsequent escape impossible. The two types of scattering are expected to have different mean free paths with different energy dependence, but unfortunately values are not yet well known for either type. Both are believed to lie in the range from twenty to a few hundred angstroms for the energy range considered here. The above considerations can be expected to govern the magnitude of the quantum yield and the dependence of the yield on photon energy, Y(hv). The value of the yield threshold itself, hv,. as defined above is determined by the material, the crystal plane defined by the emitting surface, the structural properties of the surface, and the presence or absence of contaminants adsorbed upon it. It is known that the work function of atomically clean metals can vary by several tenths of a volt depending only upon the crystal plane being studied, and that the adsorption of a fraction of a monolayer of such species as N, and 0, can change cp by a full volt or more. Thus details of the atomic surface structure must be specified along with a measured value of work function if this value is to be reproduced later. Tiius, the preceding chapter on Low Energy Electron Diffraction (Chap. 6 ) of the (110) surfaces of 111-V crystals supplies this necessary information since the measurements were performed on surfaces identical to those for which the work function and photoelectric measurements were made. The works are thus complementary and attempt to provide a fairly complete specification of the surfaces and their photoelectric properties. To determine the photoelectric threshold hv, experimentally one measures the yield as it approaches zero with decreasing photon energy. Since one cannot measure yields below some experimental limit, and since the experimental yield plot usually does not meet the axis sharply at threshold, an operational definition of the threshold is needed. Either a knowledge of the physical processes themselves or an empirical formula is needed in order to establish the analytical dependence of yield Y upon photon energy hv so that the proper extrapolation to Y = 0 can be made. Kane4 has discussed the general theory of photoelectric emission from semiconductors and has given the spectral dependence of Y(hv) for a variety of excitation and escape conditions. In general it was found that Y(hv) = C(hv - hv,)n ,, where n is an integer and could assume values from 2 to 7. 4E.0. Kane, Phys. Rev. 127, 131 (1962).

(3)

1 1.

PHOTOELECTRIC THRESHOLD AND WORK FUNCTION

267

Work on Si2 and Ge5 demonstrated the existence of a minimum of two types of emission process near threshold. Just above threshold the yield followed an approximate cube law [n in Eq. (3) lies between 5 and 7 with a best fit for n = 61 while at higher photon energy a linear dependence set in. This behavior can be expressed in the form of a sum of two processes

Y(hv) = C,(hv - hvJ3

+ C,(hv

-

hv,),

(4)

with C , > C, and hv, > hv,. To fit the data, the cubic term must drop out a few tenths of an electron volt above hv,. For both silicon and germanium the linear term can be associated with a direct optical excitation of the electrons (no change in k vector from lower to upper state) followed by escape from the solid without subsequent scattering of the emitted elect r o n ~ Direct .~ excitation is generally accepted as the cause of the very high optical absorption levels ( a lo6 cm-’) in these materials at hv above hv,. The linear dependence on hv which is observed can be accounted for on a theoretical basis only by assuming a direct optical excitation followed by escape without scattering. A conclusive proof of this hypothesis of emission without scattering has since been provided by means of a polarization experiment: in which the direction of emission of electrons outside the surface was shown to depend on the direction of polarization of the incident light. Scattering would have destroyed the electrons’ “memory” of the excitation polarization. The identification of hv, with @ cannot be made although the emission process associated with it is clearly due to excitation of valence band electrons. This is seen more clearly with the aid of Fig. l(b) which is an E vs k diagram for a hypothetical semiconductor on the same energy scale as Fig. l(a). For this situation the direct transition threshold hv, would exceed @ by the amount of kinetic energy AE, given to the hole at the k vector k,. AEv can in general be of the order of tenths of a volt since the energy of valence band states varies by about 1 V from k = 0 to k values at the Brillouin zone boundary for the cases considered. The low efficiency component characterized by the threshold value hv, corresponds to the uppermost filled electronic states of the semiconductor. In principle, filled surface states lying between the valence band edge and

-

G. W. Gobeli and F. G. Allen, i n “Solid Surfaces” (Proc. Intern. Conf. on Phys. and Chem. of Solids) (€€ C. Gatos, ed.), p. 402 North-Holland Pub., Amsterdam, 1964. G. W. Gobeli, F. G. Allen, and E. 0. Kane, Phys. Rev. Letters 12, 94 (1964). Also in “Physics of Semiconductors” (Proc. 7th Intern. Cod), p. 937. Dunod, Paris and Academic Press, New York, 1964.

268

G . W. GOBELI A N D F. G . ALLEN

the Fermi level at the surface can yield photoelectrons. Thus hv, must correspond either to CD or to a value between CD and cp which marks the position of such filled surface states. In the polarization results for Ge referred to above6 it was found that the correlation between emission direction and polarization direction persisted smoothly down to photon energies hv below the direct threshold hv,. This showed that a major fraction of the “tail” emission in that material was not randomized by phonon scattering, as would be the case for the indirect transition process first postulated.2 Instead, it suggested the presence of a quasidirect transition with take-up of normal momentum by the barrier in the act of absorption, i.e., in a process analogous to the classical surface photoelectric effect. In this case the tangential component of the momentum vector in the initial and final states is still conserved, but the normal component is not. Hence the initial state becomes arbitrary in one dimension of k space (that normal to the surface), so that excitations can occur from states up to the highest in the valence band. For this case, when surface state emission is not important the relation hv, = @ becomes valid. In any case we shall define the threshold resulting from the extrapolation of the cube law plot as hv,. For the remainder of this chapter we shall assume that hv, = CD for the (110) surface of the 111-V compounds although the above reservation must be held in mind. A final complication in the interpretation of photoelectric measurements on semiconductors comes from the fact that appreciable bending up or down of the energy bands can occur in the x dimension normal to the surface. As is shown in Fig. l(a), emission due to a volume excitation can have a threshold which varies with the depth of excitation beneath the surface, CD(x). This presents no obstacle to the identification of CD when the bands bend up in going from the volume to the surface since in that case the highest filled valence band states are just at the surface, and the identity of hv, = CD is unambiguous. However for the converse, bands that bend down in going from the volume to surface, excitation in the bulk at a depth x can well yield emission for hv < CD. This difficulty is surmounted by using samples of sufficient purity that the dimension x over which appreciable band bending (of the order of 0.1 V) can occur is long compared to the absorption depth of the incident light near threshold, or the escape depth of the excited electrons, whichever is shorter. For the 111-V group, where thresholds are between 4.5 and 6eV, the absorption depth of the light is always I 5 0 0 A. The escape depth of excited electrons is at present an unknown factor. However, it is quite likely that they do not exceed greatly that determined for Si of 25A.2,3 This means that band bending effects should be unimportant for samples with doping levels less than about 2 x lo” ~ m - In ~ .any case, high resistivity or n-type samples will be a

11.

PHOTOELECTRIC THRESHOLD AND WORK FUNCTION

249

preferable choice for determination of since the bands will either be flat or will only bend upward and hence will yield unambiguous results.6a 11. Measurement Techniques

There are four basic methods for the measurement of the work function of a surface:

(1) The Kelvin contact potential difference or vibrating capacitor technique (2) Thermionic emission (3) Photoelectric threshold measurements, and (4) Electron beam retardation measurements. These will be discussed in order. 1. KELVINCONTACT POTENTIAL. DIFFERENCE TECHNIQUE

This technique can be used reliably on any conductor of reasonable conductivity, say greater than lo-' (ohm-cm)-'. In this measurement a capacitor is made by bringing a reference surface very close to the sample surface. One of the surfaces, usually the reference, is vibrated mechanically so that the capacitance C varies sinusoidally with the vibration frequency. The circuit then consists of a battery V, applied between ground and the reference surface, the capacitance C between reference and sample surface, and the input impedance R of an amplifier connected from sample to ground. The voltage V across the capacitor is made up of two dc terms-the work function difference of the two surfaces, (pR-qS,and the applied potential V,-minus the small ac error voltage E developed across the impedance R by the vibrating capacitor. Thus

From the relation Q

=

CV

(5)

we have

6a

-

Only in the most heavily doped n-type crystals, with N , 10" can the band bending & sufficiently rapid that filled conduction band states exist within 50 A of the surface where they could give a component of emission with a threshold energy equal to the work function. All work reported here will be on crystals where this cannot be a problem.

270

G . W. GOBELI AND F. G . ALLEN

For small capacitor vibrations (ACjC 4 1) we set

C

=

C,

+ AC e i d ,

and since E has the same frequency, we can set

E

=

E , eid.

Then

and from (6) we have

dV dC dV - = - i o ( l R ) = -ioRC-- iwRV-. dt dt dt Thus

Hence for oRC > 1 we have

AC E z -V C

eim.

By properly adjusting V, a null position can be found where E requires V = qR - qS + V, = 0 and thus

(7) =

0 which

(PS - (PR = VA.

In practice the experiment is capable of null accuracies ranging from a few millivolts to a few tens of millivolts depending chiefly on the area of sample surface available. Several types of systematic errors must be considered. First, if plane surfaces are used very accurate parallelness compared to the dimension of separation of the surfaces is difficult to achieve. Thus edge effects, where the work functions may be different from the desired average value, can lead to uncertainty. This edge effect uncertainty is minimized by rounding the reference surface slightly near edges and corners. Stray capacitance to undesired portions of the sample surface, such as dirty edges of a cleaved surface, are eliminated by keeping the reference surface small compared to the sample surface. Second, parasitic vibrations of tube parts can produce spurious error signals. That portion of these spurious signals that is in phase with the desired error signal leads to a direct error in null determination, while the quadrature component reduces null sensitivity. A careful check of this condition must always be

1 1.

PHOTOELECTRIC THRESHOLD A N D WORK FUNCTION

271

made. Third, except in the case of atomically clean single-crystal surfaces, local areas of either reference or sample surfaces can have slightly different work functions. It is obvious that the effective, or average work function q k measured by this technique has the form

i.e., it is the areal average of the various work function values of the patches. Since these values can vary by many tenths of a volt depending on the exact crystal ?lane or the presence of an adsorbed gas, the properties of the material itself are reliably obtained only for atomically clean crystal surfaces. Finally, since this measurement technique gives only the difference between reference and sample, it is actually a secondary standard type of measurement, i.e., the problem of establishing the proper value of the work function of the reference surface remains. In the past the reference surface was usually platinum or some similar relatively inert metal surface treated by a rigidly prescribed “cook book“ recipe which was assumed to yield a fixed value of qR.In light of present technology and knowledge concerning the extreme sensitivity of q to any small surface impurities, this is hardly acceptable today. There are two acceptable methods: (1) Use as the reference surface a single-crystal metal ribbon the surface of which can be made atomically clean and establish its work function absolutely as described in Section 3 ; or (2) use any arbitrary reference surface (that is fairly uniform and fairly stable on at least a short term basis), and establish its work function relative to an atomically clean single-crystal metal surface, within a short time of the contact potential measurement. The metal surface work function must then, in the same experiment, be determined absolutely as described in Section 3. The practice of using a single-crystal tungsten ribbon as reference and assuming its work function after flashing to be that given in a reference table is not reliable. The authors have found that the true work function of a cleaned single-crystal (113) tungsten surface can vary from ribbon to ribbon by as much as 0.15 V. Similar variations have also been found on the same ribbon as a function of its thermal history. Such variations are presumably due to thermal faceting, or possibly, the presence of residual carbon contamination.

2. THERMIONIC METHODS The work function q can be determined for a conductor (or semiconductor) by measuring the thermionic electron emission as a function of

272

G. W. GOBELI AND F. G. ALLEN

temperature and analyzing the results according to the Richardson equation

J is the current density, T is absolute temperature A

= 4nmek2/h3= 120 A/cmz/degz, r is the reflection coefficient representing the probability that an electron with sufficient energy to exceed the surface barrier will be reflected back, cp is the work function at the temperature T , and J(Ee) is the reduction in the potential maximum just outside the surface due to the presence of an electric field E at the surface. This method is generally satisfactory for refractory type metals but is generally unsatisfactory for semiconductors. For the usually encountered range of work functions, cp z 4.5 eV, the temperature at which measurable emission occurs ( T > 1500°K) is such that appreciable evaporation of the semiconductor occurs if not outright melting. Also the movement of the Fermi level in the forbidden gap adds to the uncertainties of interpreting the results even though the true work function cp is obtained. In view of these fundamental objections this method will not be considered further here.

3. PHOTOELECTRIC TECHNIQUES

The photoelectric threshold @ and the work function cp of an atomically clean single-crystal metal surface are identical in all respects. This is true only because the Fermi level of a metal resides in a region of high density of allowed electronic states. Therefore a simple measurement of the photoelectric threshold, assuming

Ymeta,= A(hv

- hv,)2

(10)

is sufficient for the determination of OM= cpM = hv,. Note that Eq. (10) is the low temperature limiting form for the Fowler expression, applicable for a free electron metal near threshold. A theoretical explanation of why relation (10) works so well for a single-crystal metal like tungsten, even several volts above threshold, has not yet been offered. The restriction to atomically clean single-crystal surfaces is necessary because of the different effect the patch areas have on work functions as determined photoelectrically and those determined by the Kelvin method. If there are local areas of a surface having different work functions, the yield will be made up of a series of terms such that Ymeta,=

c A,(hv i

- hVIi)2.

11.

PHOTOELECTRIC THRESHOLD AND WORK FUNCTION

273

The lowest extrapolated threshold will thus approach that of the lowest work function patch area and will not be a true areal average over the surface. Finally, the thermal history of the surface being examined should be specified since this determines the extent to which faceting can expose unwanted crystal planes. As shown in the preceding chapter on Low Energy Electron Diffraction (Chap. 6) a strong thermal faceting of GaAs (111) and surfaces occurs in only a few seconds at 1100°K. This exposes (1 10) planes and would make any photoelectric or Kelvin measurements on such surfaces invalid as characterizing (111) and (TIT) faces. It has been pointed out that hv, = cp does not prevail for semiconductors, since the Fermi level can and usually does reside in a region of low (surface) state density or of zero state density (the forbidden gap). In this case the identity of hv, with CD and/or cp cannot be made, so that a direct photoelectric determination of the work function of a semiconductor is not possible. However, a “contact” potential difference measurement between the semiconductor and a collecting surface can be made using photoelectrically emitted electrons.’ Instead of modulating the capacitance between sample and reference as in the Kelvin method, this technique studies the kinetic energy distribution of emitted electrons reaching the collector and is known as the retarding potential method. Once the electrons are outside the semiconductor they are acted on by electric fields determined by the potentials just outside the surfaces of the various electrodes of the ~ y s t e m . ’As ~ in the Kelvin method, if the electrodes are in thermal equilibrium and if no battery voltages are applied, the Fermi level will be the same for all and hence the potential difference outside any two surfaces will be just the difference in their work functions,

(In)

VA-B

= -((PA

-

(PB).

(12)

If the semiconductor is surrounded by a metallic collector which has a known work function qRthen the work function cps of the semiconductor is determined by the voltage applied to the collector K, which first starts to retard photoeleetrons, i.e., the saturation current point, or the zero field point.

The precision of this method is usually limited by the homogeneity of the work function over the large, complex shaped collector surface. In practice, carefully prepared graphite collectors or gold evaporated collectors L. Apker, E. Taft, and J. Dickey, Phys. Reti. 74, 1462 (1948). Note that there is no single “vacuum level” E,,, of the system, but rather a different value just outside each plane of each conductor.

7a

274

G . W. GOBELI AND F. G . ALLEN

can give work functions which are uniform to a few hundredths of a volt. The collector work function can then be established either by measuring its photoelectric threshold directly, or better, by determining the saturation point using a metal photoelectric emitter of measurable work function t o replace the sample. Work functions of semiconductors established by the authors with this method have invariably checked within limits of experimental accuracies (-0.1 eV) with those determined by the somewhat more accurate Kelvin technique. Provided that the proper precautions are taken to insure against spurious emission from sample supports and unwanted portions of the sample itself, the photoelectric method is very powerful in that two experiments are combined in one apparatus; that is, one can determine both rp and hv,.

4, ELECTRON RETARDATION REFLECTION This method of measurement of work function is almost precisely the inverse of the photoelectric retarding potential experiment. In this case a collimated beam of electrons of a known energy is directed toward the sample surface, and the applied voltage between the emitter and the collecting (sample) surface which will just reflect the incident beam is determined. Knowing the work function of the emitter rpR allows 'ps to be determined from E q . (13). In practice, the electrons are produced by thermionic emission from a hot cathode, the work function of which is determined at the emitting temperature by Eq. (9). A series of grids and apertures focus the electrons into a beam and accelerate them to a definite energy. Collimation at the sample surface where they are being retarded and/or reflected can be maintained with an axial magnetic field. See Refs. (8) and (9) for applications. This method is in principle capable of accuracy of a few hundredths of a volt although it has been used so seldom as to preclude a precise assessment of its reliability. To date no results on 111-V surfaces using this method have been reported, and hence it will not be considered further. 111. Results and Discussion

The available results of measurements of work function rp = Eva, - E , , photoelectric thresholds hv, = 0’(cube law extrapolation), and hv, = Od (direct transition threshold, linear extrapolation) are summarized in Table I. The experimental uncertainty for both work function and threshold values is f0.05 eV. Results are taken from work of the authors (a), preliminary work of T. Fischer (b), and from published work of D. Haneman (c). The H. Shelton, Phys. Rev. 107, 1553 (1957). P. Kisliuk, Phys. Rev. 122, 405 (1961).

TABLE I WORK FUNCTIONS AND PHOTOELECTRIC THRESHOLD VALUESFOR 111-V COMPOUNDS Compound and piane

E, (eV)

cp

Q,

(eV)

(eV)

x=

Qd Qt

- EG

6=

(EF -

EJB

(EF- Evk

Sample resistivity, band bending, surface state density

AlSb( 1

1.5

4.86

5.15

5.75

3.65

0.29

GaAs( 110)”

1.40

4.71

5.47

5.15

4.07

0.76

GaSb( 1lo)”

0.70

4.76

1.35

0.080-cm n-type, ND = 2 x 1OI6 cm-’. Bands bend up to surface by -0.6eV, surface near intrinsic N,, 2 cm-2.

4.76

5.24

4.06

InAs( 110)”

0.41

4.90

5.31

5.58

4.90

InSb(11O)a

0.18

4.75

4.77

5.26

4.59

-0

0.41 -0

1.2

10i7cm-’. Surface de-

0.08

0.07R-cm p-type. N ,

0.31

generately p-type. 0.01 R-cm n-type, N , = 2.4 x 1Ol6 ~ r n - ~Surface . degenerately n-type.

0.12

0.019R-cm n-type, N ,

=

=

x

5.5 x 10i4cm-’ at 77°K.

Surface probably strongly p-type. 4.57

(1lOY InP( 1

1.3

4.45

s 4

2

2 n

0.30

4.69

(1 10y

0.10R-cm n-type, N , = 1.8 x loi7cm-’, surface strongly p-type. Bands bend up to surface by -1.0eV.

L

2

z

v)

LZ

PU >

2 W

0

5.68

5.94

4.38

1.23

0.03 to 3 R-cm, N , = 1.3 to 80 x 10“ cm-’. Surface strongly n-type.

For Comparison Si(111)”

1.09

4.83

5.10

5.45

4.01

0.27

200R-cm p-type [flat bandsb N,, 2 2 x lOI4cm-2.

Ge(ll1)” (111)’

0.67

4.80 4.75

4.80

5.22

4.13

0

0.2 a-cm p-type [nearly flat bands], N ,

Present authors. T. E. Fischer, to be published. ‘ D. Haneman and E. W. J. Mitchell, Ref. 13.

I, 2 x

loi3cm- ’.

$ cl 2 2

Yl u

276

G . W. GOBELI A N D F . G . ALLEN

assignment of hv, = E,,, - E , = @ is made as being the most probable situation. However, in light of the present status of theoretical understanding, it must be recognized that @ could have a somewhat higher value if the identity of hv, is confused by occupied surface states just above the valence band edge, i.e., @ could have an intermediate value satisfying the condition Ia) 5

@d,

adis a firm upper bound to CD since it corresponds to transitions originating somewhat below the valence band maximum. Similarly the electron affinity x = 0 - E , is subject to the same possible error of a few tenths of a volt, and the given value thus represents a minimum value that x could have. Again it should be emphasized that it is, in the authors' estimation, the most probable value. The results from Refs. (a) and (b) listed in Table I for the (110) plane were measurements made on surfaces freshly cleaved in a vacuum of I - 2 x lo-" T ~ r r . ' The ~ samples were cut to within 2" of the (110) axis, lapped and etched prior to insertion in the experimental tube. The cleavage was performed by manipulating a scribe and cleavage chisel through a sylphon bellows.2,'0 Pressure rises during cleavage never exceeded 1-2 x 10-'Torr for 3-5sec. They were sometimes as low as 2-3 x 10- l o Torr, so that they can be ascribed principally to movement of tube parts. Even for unity sticking coefficient, such pressure rises would have led to contamination of cleaved surfaces of less than 0.01 monolayers, and Low Energy Electron Diffraction studies (preceding chapter) have shown that sticking coefficients are very low for these surfaces. The surfaces were of excellent optical quality over areas 2 mm x 8 mm. While no firm differentiation of cleavage quality from material to material could be established, particularly good surfaces resulted on InAs and GaAs and poorer ones on InSb, for the small number of samples (one to three) used of each. The vacuum bake-out at 380°C for several hours caused no visible change in any of the samples except for InSb and GaSb, which developed a grey skin, presumably through evaporation. The number of surface states N,, entered in the last column of Table I for GaAs is derived roughly from the observed band bending and known doping level. Thus, using the Schottky exhaustion layer assumption, the charge held in surface states, QSs, to balance the equal and opposite charge in the space charge layer, is given approximately by

-

9a

lo

See chapter on Low Energy Electron Diffraction by A. U. Mac Rae and G. W. Gobeli for description and assessment of quality of these surfaces. G. W. Gobeli and F. G. Allen, J . Phys. Chem. Solids 14, 23 (1960).

11.

PHOTOELECTRIC THRESHOLD AND WORK FUNCTION

277

where K is the static dielectric constant, L! the electronic charge, N the density of donors or acceptors, and (V, - V,) is the total band bending from bulk to surface. This value was -0.6eV for the n-type sample used (see Table I). The actual number of surface states must be greater than Q,, depending upon their location relative to the Fermi level. The AlSb crystals also exhibit strong band bending at the surface for the bulk dopings chosen, and hence must possess high surface state densities. Derivations of the more exact surface state distributions given for Si and Ge are given elsehere.^,^ Figure 2 ( a 4 ) presents the plots of the yield in electrons per absorbed quantum for InSb, GaSb, InAs, and GaAs. (Reflectivity data for InSb, GaAs, and InAs were taken from the literature" while a constant reflectivity of 0.50 was assumed for GaSb.) These yield curves are representative of the data obtained for 111-V compound (110) faces and illustrate the quality of the cube law and linear extrapolations. They also illustrate that the two antimonides are alike and similar to Ge while the two arsenides are alike and similar to Si in several respects: Ge, InSb, and GaSb all have photothresholds equal to their work function, @, z rp, and all show three different processes in the yield curve over the present energy range-a cube law tail near threshold and two distinct linear portions of different slopes at higher energies. Si, GaAs, and InAs on the other hand all have photothresholds significantly higher than their work functions, and they show a cube law tail and only one linear process at higher energies (up to 6.3 eV).

5. COMPARISON WITH PREVIOUS RESULTSI N

THE

LITERATURE

Haneman'2,'3 has published photoelectric and work function results on surfaces of InSb, GaAs, Bi,Te,, and Ge which were cleaved or broken in His results on work function are indicated on vacua of -1O-'Torr. Table I and are seen to agree with the present work for GaAs and Ge but not for InSb. Since he did not use a power law extrapolation of his photoelectric yield, it is difficult to make comparisons here. His value of 6 = (EF - Ev& for GaAs of 0.30eV is in clear disagreement with the present value of 0.76 eV. The great improvement in quality of available 111-V crystals since 1959 together with the excellent surfaces provided by present cleavage techniques in ultrahigh vacuum probably make the present values more reliable than those in Haneman's early work. H. R. Philipp and H. Ehrenreich, Phys. Rev. 129. 1550 (1963).

'*D. Haneman, J . Phys. Chem. Solids 11, 205 (1959). l3

D. Haneman and E. W. J. Mitchell, J . Phys. Chem. Solids 15, 82 (1960).

278

G . W. GOBELI AND F. G . ALLEN ,6

x 10-4

I

-

GaSb (B-1)

n

I

-

I

I

I

GaAs

-

-

4.6

5.0

5.4

5.0

6.2

4.6

5.0

5.4

5.0

6.2

hv(ev) FIG.2. Photoelectric yield in electrons per absorbed photon for cleaved (110) surfaces of (a) InSb, (b) GaSb, (c) InAs, (d) GaAs. Inserts show cube law threshold extrapolations.

1 1.

PHOTOELECTRIC THRESHOLD AND WORK FUNCTION

6. CHANGES I N cp

AND @

279

DUETO ANNEALING CLEAVED SURFACES

It has been f o ~ n d ' ~ that * ' ~ the cleaved (111) surfaces of silicon and germanium both undergo a conversion of surface structure upon heating briefly in high vacuum to 1000" and 450"K,respectively. This rearrangement of surface atoms into a presumably lower energy form is accompanied by changes in both work function and p h o t ~ t h r e s h o l d . ~For ~ ' ~both germanium and silicon the work function decreases by 0.2 to 0.3eV, while the photothreshold drops by -0.4eV for silicon but remains unchanged for germanium. For these semiconductors, one should then use the cleaved and annealed values in comparing with surfaces cleaned by heating or sputtering and annealing. This is not the case for the 111-V (110) cleavage surfaces. As discussed in the preceding chapter on Low Energy Electron Diffraction, these surfaces do not change their surface atom structure upon annealing up to the melting point. Changes in cp and @ are not to be expected, therefore, upon heating, although this point has not yet been checked experimentally.

7. CHANGES IN cp

AND @ WITH

DOPING

If surface states are not present in sufficient density to clamp the Fermi level position at the surface, the work function of a semiconductor is expected to increase by one energy gap in going from extreme n- to extreme p-type. For both cleaved Ge and Si almost complete clamping occurs, the work function difference due to doping being at most 0.2eV for Si and perhaps 0.05 eV for Ge with realizable doping level^.^^^ While only one of the present 111-V crystals was studied at different doping levels (GaAs at 0.12 and 0.08 R-cm n-type), no significant difference was found in that case. Furthermore, the GaAs crystals (and probably also the AISb) do give definite evidence of surface state densities of at least 10'2/cm2 to produce a bending of bands upward toward the surface of -0.6eV. Hence it is probably safe to predict high surface state densities and thus no appreciable doping dependence of cp values for the 111-V crystals discussed here. (At extreme n- and p-type doping levels, one always expects the photothreshold to become equal to the work function.2) Although there is strong evidence for surface state densities of close to one per surface atom on cleaved silicon surfaces' derived from surface Fermi level clamping, any photoemission from these states is small compared to that originating from the upper edge of the valence band. The fact that 6 = (EF - Ev)s measured here for GaAs is -0.76eV indicates that l5

l6

H. E. Farnsworth, Ann. N . E Acad. Sci. 101, 658 (1963). J. J. Lander, G. W. Gobeli, and J. R. Morrison, J . Appl. Phys. 34, 2298 (1963). F. G . Allen and G. W. Gobeli, J . Appl. Phys. 35, 597 (1964).

280

G. W. GOBELI AND F.. G . ALLEN

any emission from filled surface states lying in this range of energy beneath E , is again small compared to that originating from the top of the valence band. Since the measured band bending indicates a surface state density of at least 2 x 10I2 cm-2, again it appears that even though present, the surface states are a very inefficient source of photoelectrons compared to the valence band in the bulk.

8. FACESOTHER THANTHE (110) Since it is impossible to obtain good cleavage on faces other than the (110) due to the crystal structure of the 111-V compounds, no values of cp and on other cleaved faces are available. One might expect fairly large differences in cp between the (111) and the (TTT) surface, for example, due to the fact that one terminates in type A atoms, the other in type B atoms, and the broken surface bonds are strongly ionic. One preliminary experiment looking for this difference on partially cleaved InSb (1 11) surfaces has been carried out,” and a difference in cp of -0.5 eV was found. In any case, it will probably be unwise to predict that the work functions and photothresholds of the (100) and (111) or (TIT) faces of the 111-V compounds are very close to those reported here for (110) surfaces. This is in contrast to the case for Si and Ge where such differences are limited to 0.1 or 0.2 eV.18*19 ACKNOWLEDGMENTS The authors wish to thank Dr. E 0. Kane for many helpful discussions, Dr. T. E. Fischer for allowing us to publish recent values obtained by him in collaboration with us, and k A. Studna and F. R. Eyler for much technical assistance.

S. Kawajii and M. Nakatsukasa, private communication. F. G. Allen and A. B. Fowler, J . Phys. Chem. Solids 3, 107 (1957). l 9 J. A. Dillon and H. E. Farnsworth, J . Appl. Phys. 29, 1195 (1958).

Photon Emission

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CHAPTER 12

Nonlinear Optics in 111-V Compounds P. S . Pershan* I. 11.

INTRODUCTION

GENERAL DISCUSSION.

111. THEORY .

IV.

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283

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286

EXPERIMENT . .

V. CONCLUSXON . .

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286

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I. Introduction The development of extremely high power sources of monochromatic optical radiation (i.e. lasers) has made it possible to observe nonlinear effects such as second harmonic generation at optical frequencies. Since there are a number of comprehensive reviews’-3 available in this field, after a brief general discussion, this article will deal only with the particular features of nonlinear optics that are specific to 111-V compounds. 11. General Discussion

Whether one deals in linear or nonlinear phenomena, optics can be divided into two separate problems. Taking J(o) and E(w) as the Fourier amplitudes, at angular frequency o,of the current density and electric field respectively, one can write Ji(O)

= cj @ i j ( o ) E j ( O )-k i 0 x j k X i j k ( O , $w,$o)Ej($o)Ek(&O),

or J(o)

= c ( w ) .E(o)

+ iw X(W,&D,$o): E(+w)E($w).

(1)

(2)

The first term is the linear conductivity and the second is a nonlinear term

* Alfred P. Sloan Research Fellow.

‘ P. A. Franken and J. F. Ward, Ren Mod. Phys. 35, 23 (1963). N. Bloembergen, “Non-Linear Optics.” W. A. Benjamin, New York, 1965. P. S. Pershan, Progr. Opt. V, 85 (1965).

283

284

P. S. PERSHAN

describing second harmonic generation. Other nonlinearities can also be introduced, but this is the most common and we will restrict ourselves to only this. The particular form of this term with the “iw” arises since most often one discusses nonlinearities in ionic crystals for which the polarization rather than the current is used [J(w) = iwP(w)]. The first problem in knowing J(w) is to calculate the tensors a(w) and X(o, fw, *w). From simple parity considerations X(w, f w , 30) vanishes for crystals which remain invarient under spatial inversion. Thus in the 111-V compounds which do not have a center of inversion nonlinearities are observable, while in pure crystals like Ge and Si, they are not. Given Eq. (1) or (2), one must now calculate the effects of these currents on the electromagnetic field. This is an exceedingly complicated problem in linear optics since the current produces a field that interferes with the field that produced it. This leads to the phenomena of refraction and reflection. Further discussion of this point is beyond the scope of this review, but it is thoroughly discussed by Born and Wolf? For the nonlinear problem the polarization (at o) cannot interact strongly with the fields that produced it (at fo), and one has simple inhomogeneous partial differential equations

V

iw x H(w) = +o). E(w) C

71 + 4-JNLS(w),

(3)

C

where JNLS(o), given by the second term in Eq. (2), can be treated most simply as an inhomogeneous source term arising from the nonlinearity. This approximation is possible since due to dispersion the phase velocity for the harmonic wave C [ E ( O ) ] - ” ~ is different from the phase velocity for the fundamental C[E(~O)]A well known trick of nonlinear o p t i ~ s , ~ . ~ possible in some anisotropic crystals, is to choose an orientation and polarizations for which these two are equal ~ ( o=)E($w).This is known as “phase matching” and under these conditions one must consider the back . ~ cubic 111-V compounds the linear optical reaction of J(o)on E ( ~ w )For properties are isotropic and phase matching would be possible only in a fortuitous case of anomolous dispersion. In all known cases it is not observed. M. Born and E. Wolf, “Principles of Optics,” Section 2.4, pp. 98-104. Macmillan (Pergamon), New York, 1959. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Reo. Letters 8,21 (1962). J. A. Giordmaine, Phys. Rev. Letters 8, 19 (1962). J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).

12. NONLINEAR

OPTICS IN 111-V

285

COMPOUNDS

General solutions to Eq. (3) have been obtained with the proper boundary conditions for a variety of geometries.' To illustrate some typical features of the solutions consider a nonlinear material filling all space with z > 0, for z < 0 there is vacuum. Consider also a plane wave coming from the vacuum of the form E(fw) = E , exp[ - ifiozc- ')I incident on the nonlinear dielectric ; it is partially transmitted and partially reflected by the surface. The transmitted wave will generate a nonlinear polarization of the form P,(w) expi -iwzc-' x [ E ( ~ w ) ] ~or/ ~equivalently }, a current

JNLS(o) = ioPNLS(m) = ioP,(o) exp{ - iozc-

'[E(+W)]~/~}.

(4)

On substitution of Eq. (4) into Eq. (3) it is possible to solve for the transmitted and reflected waves. The solutions for the reflected wave

ER(u)=

-

+

4nP0(o) exp[ iozc(1 f [E(~)]'"}{[&(~~)]''2[E(W)]'/2}

+

'1

(5)

and for the transmitted wave

x [exp( +iwzc- '{[E(O)]'/~- [ ~ ( + o ) ]-' / 11 ~}) x exp{ - iozc- '[&(w)]"~}.

(6)

For 111-V compounds E ( W ) is often complex, causing the transmitted wave to be attenuated. Note, however, that so long as the material transmits the fundamental, the transmitted second harmonic is at least of comparable magnitude to the reflected harmonic. Even if E(&o) is complex corresponding to the fundamental also being attenuated, harmonic generation is predicted to occur in reflection. For other than normal incidence the directions for the reflected and transmitted harmonics are given by Bloembergen and Pershan8 Taking the incident wave as coming from a linear material with dielectric constant at the fundamental of ci($o) and at the harmonic of ei(o), the angle of incidence, as measured from the normal to the boundary is @(&) and the angle of reflection for the harmonic is OR(w);then one can show [ E ~ ( + W )sin[@(+)] ]'~~ = [ ~ ' ( w ) ] 'sin[OR(w)]. '~

(7)

If the primary wave is ihcident from a nondispersive medium, the reflected harmonic is parallel to the reflected fundamental, otherwise they are related as described by Eq. (7). N. Bloembergen and P. S. Pershan, Phys. Reo. 128, 606 (1962).

286

P. S. PERSHAN

HI. Theory A number of authors have treated the nonlinear conductivity (or polarizability) for semiconductors in terms of one electron nonlocalized By treating the radiation field as a perturbation, one calculates the wave functions to second order in the field amplitudes E(&). Using these wave functions one calculates the expectation value of the current density operator and obtains those terms proportional to [E(&0)]2. One must take care to select only those terms with time dependence expiwt and not take the time independent terms. The coefficient of Ei(;w)E,~(h in) the expression for Jk(w)is simply iwXkjj(w, $A $m) (except possibly for a factor of two depending on how one sums over the indices i, j”). The most meaningful result is obtained in the “dipole approximation” in which one takes all optical wave vectors as zero. One then finds the nonlinear conductivity is made up of a sum of terms like

+ other terms, where Pk is the kth component of the momentum operator ( k = s , y . z ) and In), In’), In”} are Bloch states. The first observation is that for a lattice with a center of inversion the sum over matrix elements must be zero. This follows from parity considerations. Secondly, one expects a significant increase in Xkij(w,+w,+) when w approaches the band gap. From the usual symmetry considerations one can show for cubic 111-V compounds with the crystal point group 43 m the only nonvanishing tensor elements are

IV. Experiment The unique contribution of 111-V compounds to nonlinear optics has been in the observation of reflected harmonics. Mainly this is because many of the 111-V compounds are opaque to the second harmonic of ruby

’R. Braunstein, Phys. Rev. 125, 475 (1962). R. Loudon, Proc. Phys. SOC.(London) 80, 952 (1962). P. N. Butcher and T. P. McLean, Proc. Phys. SOC. (London) 81, 219 (1963). P. L. Kelley, J . Phys. Chem. Solids 24, 607 (1963). l 3 B. Lax, A. L. McWhorter, and J. G. Mavroides, “Quantum Electronics” (Proc; 3rd Intern. Congr.), p. 1521. Columbia Univ. Press, New York, 1964. l4 P. N. Butcher and T. P. McLean, “Quantum Electronics” (Proc. 3rd Intern. Congr.), p. 1620. Columbia Univ. Press, New York, 1964. lo

12. NONLINEAR

OPTICS IN 111-V

287

COMPOUNDS

laser light and thus experimentally there is no confusion between a weak reflected harmonic and scattered radiation from a much stronger transmitted harmonic. Ducuing and Bloembergen" were the first to detect the reflected harmonic. They used single-crystal GaAs with a polished (110) surface. The incident light was polarized normal to the plane of incidence and thus in the (110) crystal plane. As the crystal was rotated about the normal to this plane the nonlinear polarization varied with angle. Thus when the (001) axis was parallel to the incident light polarization there was no second harmonic generation since

On the other hand for the (110) axis parallel to incident polarization, the nonlinear polarization was in the (001) direction, and thus the reflected harmonic radiation was polarized in the plane of incidence of the fundamental. The expected dependence on crystal orientation was completely verified. Ducuing and Bloembergen16 also verified some of the other laws of reflection.' In particular they verified Eq. (7) by immersing the GaAs in benzene and measuring the angular spread between the reflected fundamental and harmonic waves. Measurement of the absolute value for the nonlinear Xijk requires very specific knowledge of the mode structure of the laser and its power output. Since this can only be obtained with the relatively weak gas lasers, absolute measurements of X, are only possible on crystals for which phase matching is possible, for example KDP (i.e. KH,PO,). On the other hand measurements of xijk of one material relative to another are possible with the high power solid state lasers in which the mode structure is not known. Bloembergen et a!.'' used a beam splitter to simultaneously generate harmonics in KDP and some other crystal. In this way they observed the relative values for X,,,, see Table I.17a Chang et al." have separately measured relative values of x3.6 for avariety of incident wavelengths in the vicinity of the absorption (i.e., $w E J h ) and

-

l5 l6

J. Ducuing and N. Bloembergen, Phys. Reo. Lerters 10, 474 (1963). J. Ducuing and N. Bloembergen. Phys. Letters 6, 5 (1963).

"N. Bloembergen, R K. Chang, J. Ducuing and P. Lallemand, Bull. Am Phys. SOC. 9, 269 (1964). ""The values reported in Ref. 17 were incorrect. The numbers in Table I are the corrected ones obtained by the same authors (private communication). R. K. Chang, J. Ducuing, and N. Bloembergen, Phys. Reo. Lefters 15, 415 (1965).

'*

288

P. S. PERSHAN

TABLE I VALUES OF X,,6 RELATNETO X,,6 IN KDP (= 6 x lo-' e s ~ ) ' ' ~ Neodymium laser I = 1.06~

GaAs InAs InSb GaSb"

580 k 70 770 f 110 1100 f 100 1300

Ruby laser

I = 0.694~ 490 k 50 340 k 40 590 k 60 400

These values were obtained by taking the linear dielectric constants of GaSb, which are not known. to be the same as for InSb.

critical points of the Brillouin Zone for a number of 111-V compounds and found some structure in the frequency dependence of X 3 . 6 . Aside from this work of Chang et al., which was not completed when this article was written, the most comprehensive study of X,, in the vicinity of an absorption edge is not on 111-V compounds but 11-VI compounds. Soref and Moos19 measured Xi, in ZnS-CdS and CdS-CdSe monocrystals. By varying the relative concentrations of Zn :Cd or S :Se they could vary the band gap by a factor of two, from 1.5 times the harmonic frequency to 0.7 times it. The tensor element Xz,x,y(w, &o, &o) increased by nearly an order of magnitude as o passed through the band gap.

V. Conclusion In principle, nonlinear optical effects should provide more information than linear optical effects about the electronic structure of 111-V compounds. Practically however, nonlinear optics experiments are more difficult, and in order to explain them theoretically, one must do calculations to a higher order than is necessary for the linear effects. It is not clear that the potential for extra information arising from this higher order effect will be realized in the face of a concomitant increase in complexity. Since this article was prepared, Chang et ~ 1 . ~ have ' measured the phase of the nonlinear tensor X 3 , 6 in GaAs, InAs, and ZnTe relative to the nonlinear tensor in KDP. l9

R. A. Soref and H. W. Moos, J . Appl. Phys. 35, 2152 (1964). R. K. Chang, J. Ducuing, and N. Bloembergen, Phys. Reo. Letters 15,6 (1965).

CHAPTER 13

Radiative Recombination in the III-V Compounds M . Gershenzon I . INTRODUCTION .

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1 . General Introduction . . . 2 . Excitation . . . . . 3 . Radiatiue Recombination . 4 . Nonradiatioe Recombination 5 . Phonon Cooperation . . . 6 . Hot Carriers . . . . .

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14. Introduction . . . 15 . Radiative Transitions 16. Absorption . . . 1I . Photoluminescence . 18. Cathodoluminescenre 19. Junction Luminescence

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ADDEDIN

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. . . . . PROOF . . .

26 . GaP . . . . 21 . GaAs . . . . 28 . Other Compounds

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IV . OTHER COMPOUNDS. . . 20 . InSb . . . . . . 21 . InP . . . . . . 22 . InAs . . . . . . 23 . GaSb . . . . . . 24 . Miscellaneous Compounds 25 . Alloys . . . . . .

v.

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I1 . GaP . . . . . . . . . . I . Introduction . . . . . . . 8 . Bound Exciton Decay . . . . 9 . Donor-Acceptor Pair Recombination 10. Donor-Acceptor Edge Emission . . 1 1 . Recombination at Deep Leoels . . 12. Cathodoluminescence . . . . 13. Electrolurninescence . . . . . 111. GaAs .

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290 290 292 295 299 302 303 303 303 305 309 314 316 318 318 325 325 326 330 333 336 338 351 357 359 360 361 362 363 366 366 361 369

290

M . GERSI-ENZON

I. Introduction 1. GENERAL INTRODUCTION Radiative recombination in the 111-V compounds was reported almost immediately (1954) after the discovery that these compounds were Despite reports that the radiative efficiency was quite high, notably for InSb,&’ little attention was given to this field. Most of the subsequent work was confined to the high band-gap compounds with left emission in the visible spectrum, and the low quantum yields (7 the impression that radiative recombination in these compounds was inherently inefficient, despite the early InSb results. It was not until 1962 that quantum efficiencies near unity were found for GaAs*-” and for Gap,’’ and the almost immediate announcements of the GaAs quickly led to a vast increase of effort in this field. This review, written barely two years after these developments, attempts to assess the current status and the unsolved problems associated with radiative recombination in the 111-V compounds. With the current rate of progress in this field, it is realized that much of what can be said now wil! be obsolete, or perhaps, even wrong, by the time of publication. Bearing this in mind, we stress the results that will probably not undergo revision, and carefully note the working hypotheses that are almost certainly unproven at the time of writing. Several of the basic problems that are not properly understood at the present time are the following: (1) The mechanisms of nonradiative recombination which compete with the radiative processes and limit the luminescence efficiencies. R. Braunstein, Phys. Rev. 99, 1892 (1955). G. A. Wolff, R. A. Hebert, and J. D. Broder, in “Semiconductors and Phosphors” (Proc. Intern. Colloq., Garmisch-Partenkirchen, 1956). p. 547. Wiley (Interscience), New York. 1958.

G. A. Wolff, R. A. Hebert, and J. D. Broder, Phys. Rev. 100, 1144 (1955). 4T. S. Moss and T. H. Hawkins, Phys. Reo. 101, 1609 (1956). T. S. Moss, T. Hawkins. and S. D. Smith, Rept. Meeting Semicond., Rugby, 1956, p. 133. Phys. Soc., London, 1956. ‘T. S. Moss and T. H. Hawkins, J . Phys. Radium 17, 712 (1956). T. S. Moss, Proc. Phys. SOC. (London) B70, 247 (1957). J. I. Pankove and M. Massoulie, Electron. Div. Abstr., Spring Meeting Electrochem. Soc., Los Angeles, 1962, p. 71, Abstr. No. 48. R . J. Keyes and T. M. Quist, I R E Trans. Electron Devices EM, 503 (1962). l o R. J. Keyes and T. M. Quist, Proc. I R E SO, 1822 (1962). M. Gershenzon, I R E Trans. Electron Devices ED9, 503 (1962). D. N. Nasledov, A. A. Rogachev. S. M. Ryvkin, and B. V. Tsarenkov, Soviet Phys.-Solid State (English Transl.) 4, 782 (1962) [Fiz. Tverd. Tela 4. 1062 (1962)]. l 3 R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. 0. Carlson, Phys. Rev. Letters 9. 366 (1962). l4 M. I. Nathan, W. P. Durnke, G. Burns. F. H. Dill, and G. Lasher. Appl. Phys. Letters 1.

’

62 ( 1962).

I5T. M. Quist, R. J. Keyes, W. E. Krag, B. Lax, A. L. McWhorter. R. H. Rediker, and H. J. Zeiger, Appf. Phys. Letters 1, 91 (1962).

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(2) A theoretical formulation of the density of states near the band edges in heavily doped (and compensated) materials of low effective masses, such as GaAs, wherein impurity states cannot be separated from the normal band states. (3) A quantitative description of deep, tightly bound states in semiconductors, together with the transition probabilities to such states. (4) The detailed mechanisms of phonon cooperation during radiative transitions. (5) The effect of a high electric field, such as exists in the depletion layer of p-n junctions, upon radiative transition probabilities. (6) The mechanisms of injection and recombination in p-n junctions not dominated by diffusion-controlled recombination past the spacecharge layer. We shall consider first, very briefly, some concepts basic to, but not necessarily peculiar to radiative recombination in the 111-V compounds : the excitation processes in photoluminescence and in forward bias, p - n junction electroluminescence, the mechanisms and kinetics of radiative recombination, and some miscellaneous topics, such as phonon cooperation and nonradiative recombination. Aspects of some of these topics will be covered more fully in the later discussions under the individual compounds where they are directly relevant. Some of the earlier work in electroluminescence has been reviewed in monographs by IveyI6 and by Henisch,” and some more recent work, with emphasis on stimulated emission, is covered in a review article by Burns and Nathan.” In the following chapter Stern reviews the subject of stimulated emission in the 111-V compounds. In the following sections, we shall be almost entirely concerned with recombination of minority carriers in thermal equilibrium with one of the bands, but not in thermal equilibrium with the majority carriers, so that we can describe the distribution of majority carriers in the valence (conduction) band by a well-defined Fermi level and of minority carriers in the conduction (valence) band by a quasi-Fermi level. At worst, both carrier distributions will require quasi-Fermi levels, as, for example, during recombination in the space-charge layer of a junction. Tunneling processes fall under this category provided the injected carriers therinalize faster than they recombine. Hot carrier recombination, such as results from the injection of nonthermal carriers in high field regions (for example, at reverse bias breakdown of p-n junctions or surface barriers), does not fit this picture and will have to be considered separately. I’

’

H. F. hey, “Electroluminescence and Related Effects,” Chap. 4.Academic Press, New York, 1963. H. K. Henisch. “Electroluminescence,” Chap. 4. Macmillan (Pergamon), New York. 1962. G. Burns and M. I. Nathan, Proc. I E E E 52. 770 (1964).

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2. EXCITATION We start with a crystal containing a distribution of free carriers at thermal equilibrium, usually dominated by the majority carriers, electrons or holes, if the crystal is not close to intrinsic. Excess minority carriers, which may in turn decay radiatively or nonradiatively by recombination with the majority carriers, must first be created by the excitation process. This may consist of irradiation with photons (photoluminescence) or electrons (cathodoluminescence), or by injection in a forward-biased p-n junction (junction electroluminescence), at a heterojunction, over a surface barrier, or through an insulating layer by tunneling. In photoluminescence the exciting photon energy must be greater than the band gap for each photon to create an electron-hole pair (although see below for two-photon processes). The absorption constant rises very rapidly to l o f 4 to 10” cm-’ above the edge for a direct gap material, and somewhat less rapidly to to cm- for an indirect substance. Thus, the recombination takes place very close to the surface, depending upon the absorption constant at the exciting wavelength and the diffusion length of the minority carriers created. Competition with surface recombination may then be very severe and a surface pretreatment may be necessary to improve the radiative efficiency. Two photon-two step excitation involving an intermediate deep level can be used to obtain more penetrating excitation in the 111-V c o m p ~ u n d s ’ ~ as* has ~ ~ been used in the 11-VI compounds,21*22although the recombination rate per unit volume is thereby greatly reduced. To obtain high injection levels one is usualIy constrained to use one of the limited number of strong emission lines available in a gas discharge, often a mercury lamp. Sometimes a broad spectrum, as from a xenon arc, is used. The flux densities from currently available discharge lamps limit the excitation density to about lot8 to lo2’ photons/sec/cm2 for continuous excitation, or to about to for pulsed operation. Enough energy is available from currently available lasers, pulsed or cw, to burn holes in most materials, if the beam is focused. These figures compare with several times obtainable by electrical injection in forward biased junctions, assuming unit injection efficiency in each case. Photoluminescence, however, permits the study of homogeneous crystals, whereas the doping profiles vary drastically. D. G. Thomas, M. Gershenzon, and J. J. Hopfield, Phys. Rev. 131, 2397 (1963). F. Gross and D. S. Nedzvetskii, Souiet Phys. “Dokrady” (English Transl.) 8, 989 (1964) [Dokl. Akad. Natrk SSSR 152, 1335 (1963)J. 2 1 R. E. Halsted, E. F. Apple, and J. S. Prener, Phys. Reu. Letters 2, 420 (1959). 2 2 R. E. Halsted, E. F. Apple, J. S. Prener, and W. W. Piper, Proc. Intern. Con$ Semicond. Phys., Prague, 1960, p. 116. Academic Press, New York, 1961. 2o E.

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Cathodoluminescence, although studied intensively in the 11-VI compounds, mostly because of its technological importance, has only recently begun to be investigated in any detail in the 111-V compounds. Here, the penetration depths depend upon the electron energies. High energies simultaneously produce radiation damage. Since the impinging electrons have many times the band gap energy, hot carriers are created and the quantum efficiency (which may be greater than unity) loses its significance in terms of a thermalized minority carrier distribution. Some remarks on hot carriers will be made below. Injection across forward biased p-n junctions has become the most popular way of exciting luminescence. In the simplest model of such a junction (applicable to Ge), the steady-state current increases with the bias V as loexp(eV/kT) for eV kT, with minority carrier injection occurring on both sides of the junction, the ratio depending upon minority carrier lifetimes and mobilities.23 The process is controlled by diffusion, which in turn depends upon lifetime. The latter is determined by the competition for the minority carrier between various capture and recombination processes. The majority carrier densities are assumed constant in this region. At high injection levels, saturation of some of the recombination centers can alter the dominant recombination routes and increase the effective diffusion lengths. Recombination must usually be considered on both sides of the junction, the relative importance of each depending upon the injection ratio, which can be determined for simple step-junction, or linearly graded junction models.23 In wider gap semiconductors (Si), much of the recombination occurs in the junction depletion layer. With some grossly simplifying assumptions, this current has been shown to depend upon bias as I,exp(eV/nkT) with 1 7 n 7 2.24325In this case, both types of carrier must be injected into the space-charge layer, and the steady-state populations in the various recombination centers can vary considerably in traversing the depletion layer from the n side to the p side. At high bias, simple injection past the spacecharge layer begins to dominate. Recombination inside the space-charge layer occurs in the presence of the junction field ( - 104-106V/cm). Very little is known about whether such processes are radiative. Generally, this current component, which is not understood in detail, is regarded as an excess current in parallel with the normal injection current which does result in radiative recombinatiori. In tunneling junctions, carriers can tunnel through the junction potential barrier and recombine on the opposite side of the junction. In normal

+

23 W. Shockley, Bell System Tech. J . 28, 435 (1949). 24C.T. Sah, R. N. Noyce, and W. Shockley, Proc. I R E 45, 1228 (1957). D. A. Evans and P. T. Landsberg, Solid-state Electron. 6, 169 (1963).

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Esaki diodes, electrons tunnel horizontally in energy-space coordinates into empty valence band states, so that the recombination does not release energy, and is therefore not radiative. At higher bias, when the full and empty bands are no longer in juxtaposition, tunneling to deep states may occur, giving rise to an excess current in the valley between normal tunneling current and thermal injection current.26327 Subsequent recombination from these deep states can be radiative. Photon-assisted tunneling is basically the inverse of the Franz-Keldysh effect in absorption.” The overlap of the tails of the electron and the hole wave functions in the transition region of narrow junctions permit direct (tunneling) transitions which conserve momentum.29 Such transitions will be considered in greater detail in the discussion about GaAs. Tunneling between impurity states, i.e., impurity band conduction, can be important in heavily doped diodes, for example in GaAs. When the impurity band is merged with the conduction (valence) band, the composite band will be filled with electrons (holes) up to the Fermi level. States near the Fermi level will have a higher density of states than will the deeper states, and the effective mobility in such states will also be higher. The injection current, therefore, will be greatest near the Fermi level. On the opposite side of the junction these injected minority carriers can recombine directly, perhaps radiatively, in which case the shape of the emission band will reflect the density of states, mobility and Fermi level cutoff as a function of energy. Or else, they can thermalize first, depending on radiative lifetimes and scattering lifetimes, and reflect in emission the density of states and radiative lifetime as a function of energy of the band at the position in space from which they recombine. Again, this process will be discussed in greater detail under GaAs. For the sake of completeness, we mention several other injecting mechanisms, some of which might be useful for creating minority carriers in the 111-V compounds. Metal surface-barrier contacts can inject in forward bias, provided the metal work function and the semiconductor electron affinity (or the surface states that pin the Fermi level at the interface) result in a small thermal barrier for minority carrier injection. A thin insulating film between the metal and the semiconductor gives one more degree of freedom in juxtaposing the Fermi level in the metal opposite the minority carrier band of the semiconductor, but tunneling through the insulator is then needed.30p31 Injection at p-n heterojunctions between dissimilar 26

A. G. Chynoweth, W. L. Feldmann, and R. A. Logan, Phys. Reu. 121, 684 (1961).

28

J. I. Pankove, Phys. Rev. Letters 9, 283 (1962). R. C. C. Leite, J. C. S a m e , D. H. Olson, B. G . Cohen, J. M. Whelan and A. Yariv, Phys. Rev.

’’R. A. Logan and A. G. Chynoweth, Phys. Rev. 131, 89 (1963).

29

137, A 1583 (1965).

”A. G. Fischer and H. I. Moss, J . Appl. Phys. 34,1112 (1963). 3 1 R. C. Jaklevic, D. K. Donald, J. Lambe, and W. C. Vassell, Appl. Phys. Letters 2 , 7 (1963).

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semiconductors is another po~sibility.~’ Finally, in compensated or nearintrinsic regions, injection may become drift controlled rather than diffusion controlled, and the methods of space-charge limited currents must be used.33-3s

3 . RADIATIVE RECOMBINATION After the minority carriers are produced, presumably in quasi-thermal equilibrium, in either the conduction band or the valence band, they may decay by recombining with a majority carrier, but they may proceed along either of several different paths. Hall has reviewed the implications of recombination via several of these paths.36 Each minority carrier can recombine with a free majority carrier directly [Fig. l(a)] or through an

FIG. 1. Various recombination mechanisms: (a) band-to-band, (b) donor-to-valence band, (c) conduction band to acceptor, (d) donor to acceptor, (e) typical three particle nonradiative Auger process.

intermediate free exciton state, or it can be captured onto a local defect site from which it may eventually recombine with a majority carrier [Fig. 1(M)]. The different mechanisms are competitive and the dominant processes result from an optimization of many parameters. The kinetics of the competitive processes are most easily analyzed in terms of ShockleyRead-Hall s t a t i s t i ~ s . In ~ ~these , ~ ~ terms, band-to-band recombination is proportional to the product of the minority carrier and the majority carrier concentrations. Recombination via a localized level involves the consecutive A. G. Fischer, Solid-Srate Electron. 2, 232 (1961). M. A. Lampert, Phys. Rev. 103, 1648 (1956). 3 4 M . A. Lampert, A. Rose, and R. W. Smith, J . Phps. Chem. Solids 8, 464 (1958). ” P. N. Keating, Phys. Rev. 135, A1407 (1964). 36 R. N. Hall, Proc. Inst. Efec. Engrs. (London) 8106, Suppl. 17, 923 (1960). W. Shockley and W. T. Read, Phys. Reu. 87. 835 (1952). R. N. Hall, Phys. Rev. 87. 387 (1952).

32 33

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M. GERSHENZON

capture of both types of carrier. At steady state, capture is proportional to the number of such centers ready to receive carriers, to the concentration and thermal velocity of the free carriers and to a capture cross section. Thermal release of these carriers after capture, in competition with the recombination, can be deduced by considering thermal release and capture at thermal equilibrium, where they must be equal. In a direct band-gap semiconductor, the transitions which involve no momentum change are allowed, so that only vertical transitions in E-k space occur-the momentum of the photon is usually negligible [Fig. 2(a)].

i-; (a1 (bl FIG. 2 Schematic representations of radiative band-to-band recombination for a direct edge (a) and an indirect edge, (b) in energy-momentum space. Electron transitions are shown.

Since energy is a single-valued function of k, in any direction, in each of the bands, the energy of the transition corresponds to a specific point in k space. The shape of the spontaneous emission peak as a function of energy then depends on the matrix element for the transition, the joint density of states, and the Fermi functions. The recombination rate constants can be calculated from the matrix elements, and the actual rate of recombination is then given by the product of this rate constant, the majority carrier density, and the minority carrier density. For an indirect semiconductor, the purely radiative transitions between the normally occupied states are forbidden by the k-selection rule [Fig. 2(b)]. Transitions can only occur with the simultaneous emission or absorption of one or more phonons to conserve momentum. (Electron-electron and

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RADIATIVE RECOMBINATION IN THE 111-V

COMPOUNDS

297

electron-impurity interactions can also play this r0le.j') Again the transition probabilities can be calculated in terms of each phonon absorption or emission process, and the shapes and recombination rates can be predicted. The principle of detailed balance4' has often been used to bypass the calculations, since it can predict the rate constant for the radiative process and the shape of the emission band from the experimentally measured absorption edge.7s40-48In each energy interval the total emission rate per unit volume is a sum of the spontaneous emission and the stimulated emission. At thermal equilibrium the net emission is zero, and the stimulated portion is negative and corresponds to the absorption of the blackbody radiation at the equilibrium temperature. From a knowledge of the Planck function and the absorption edge, the spontaneous emission can be deduced from this condition. The total recombination rate can then be written as B(np - n,p,), where B is the rate constant referred to earlier, n and p are the arbitrary free carrier densities, and nope = ai2 is the product of the carrier densities at eq~ilibrium.~'The limits of applicability of detailed balance have been discussed re~ently.~' In any structure with a given concentration of free majority carriers, bandto-band recombination provides the limiting (maximum) lifetime that a minority carrier enjoys. Usually, capture into various impurity levels proceeds more rapidly, and therefore the efficiency of band-to-band recombination may be small. The dependence of band-to-band processes on current in forward biased diodes, taking into consideration the spatial distribution of both minority and majority carriers has been discussed by several authors. '*' A second major recombination process involves the consecutive capture of both an electron and a hole at an isolated point defect [Fig. l(b, c)]. The kinetics of such processes have been treated successfully by ShockleyRead-Hall mentioned above. The Shockley-Read-Hall formulation yields the recombination current through each type of center present and the resultant minority carrier lifetime, provided one knows the

'

C . Haas, Phys. Rev. 125, 1965 (1962). W. van Roosbroeck and W. Shockley, Phys. Rev. 94, 1558 (1954). 4 ' I. M. Mackintosh and J. W. Allen, Proc. Phys. SOC.(London) B68, 985 (1955). 4 2 J. W. Allen and I. M. Mackintosh, J . Electron. 1, 138 (1955). 43 P. T. Landsberg and T. S. Moss, Proc. Phys. SOC. (London) 869,661 (1956). 44 J. R. Haynes, Phys. Rev. 98, 1866 (1955). 45 G. K. Wertheim, Phys. Reu. 104, 662 (1956). 46J. C. Sarace, R. H. Kaiser, J. M. Whelan, and R. C. C. Leite, Phys. Rev. 137, A 623 (1965). 47 W. P. Dumke, Phys. Reu. 105, 139 (1957). 48 P. H. Brill and R. F. Schwarz, J . Phys. Chem. Solids 8, 75 (1959). 49 D. E. McCumber, Phys. Rev. 136, A 954 (1964). 5 0 S . Mayburg and J. Black, J . Appl. Phys. 34, 1521 (1963). 39

40

298

M. GERSHENZON

majority and the minority carrier densities and the capture cross sections for each type of center (for both electrons and holes), as well as the densities of the recombination centers. Either the capture of an electron, or of a hole, or both can be radiati~e.’’-’~ The photon emission rate thus corresponds to the current through the center. These currents can saturate as a function of minority carrier concentration, depending upon the appropriate densities of the recombination centers, their capture cross sections, and the majority carrier concentration. In junctions these processes become sublinear in injection current in this range.” Recombination through centers having multiple levels have also been treated in the Another important recombination mechanism involves the recombination of an electron trapped on a donor with a hole on an acceptor [Fig. l(d)]. The donor and acceptor may be quite far apart in the lattice. The spectra consist of a series of sharp lines, each corresponding to an allowed pair separation in the lattice, and merging at lower energies into a broad band.58 This process has a low transition probability because of the small overlap of the separated wave functions, and hence a long radiative lifetime. At low temperatures, however, all carriers are frozen out or captured quickly by donors or acceptors, so that few free carriers remain, and the donoracceptor transitions have little effective competition. Even at higher temperatures, deep levels can behave this way.59Radiative donor-acceptor pair recombination was predicted a number of years ag0,60+61 but it is only recently that it has been studied in detail in Gap, where both the sharp lines and the broad band Accordingly, this mechanism will be considered more fully in the discussion under Gap. It has become Ya. E. Pokrovskii and K. I. Svistunova, Souiet Phys.-Solid State (English Transl.) 5, 1373 (1964) [Fiz. Tuerd. Tela 5, 1880 (1963)l. 52 Ya. E. Pokrovskii and K. I. Svistunova, Souiet Phys.-Solid Stnte (English Transl.) 6, 13 (1964) [Fiz. Tuerd. Tela 6, 19 (1964)]. 53 Ya. E. Pokrovskii (3’. E. Pokrovsky), in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 129. Dunod, Paris and Academic Press, New York, 1965. 5 4 C. T. Sah and W. Shockley, Phys. Reu. 109, 1103 (1958). 55 M. Nagae, J . Phys. SOC.Japan 17, 1677 (1962). 5 6 M. Nagae, J . Phys. SOC. Japan 18, 207 (1963). 5’ G. Giroux, Proc. Intern. Con$ Sernicond. Phys., Prague, 1960, p. 275. Academic Press, New York, 1961. D. G. Thomas, M. Gershenzon, and F. A. Trumbore, Phys. Rev. 133, A269 (1964). 5 g M. Gershenzon, F. A. Trumbore, R. M. Mikulyak, and M. Kowalchik, J . Appl. Phys. 36, 1528 (1965). 6 o J. S. Prener and F. E. Williams, Phys. Reo. 101, 1427 (1956). 6 ‘ F. E. Williams, J . Phys. Chem. Solids 12, 265 (1960). 6 2 D . G. Thomas, J. J. Hopfield, and K. Colbow, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 67. Dunod, Paris and Academic Press, New York, 1965.

51

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299

apparent that this process is rather common in other material^,^^-^^ and the conditions necessary to obtain this type of radiation have been discussed r e ~ e n t l y . ~ ~ ? ~ ~ Transitions involving correlated electron-hole pairs, excitons, become particularly significant at low temperatures, where most of the minority carriers exist as free excitons in thermal equilibrium with the free carriers (via a mass-action law). Their formation may be observed in absorption and their decay in e m i ~ s i o n . ~The * * ~excitons ~ may decay only after first being captured by isolated point defects (bound excitons) giving rise to emission lines whose positions depend upon the binding energy of the exciton to the center. Exciton transitions are discussed elsewhere in this series and will not be dwelt upon here. Other possible types of radiative recombination include transitions between crystal field states, involving the d and the f electrons in transition metals and in rare earths, respectively70-72 and recombination at extended defects, such as dislocation^.^ 3-76 These processes remain to be explored in the 111-V compounds. 4. NONRADIATIVE RECOMBINATION

It has been tacitly assumed that the processes we have been considering are all radiative, that is, the energy released in each step on the recombination path is emitted as a photon. For the energies under consideration (depending on band gap, recombination center depths, or exciton binding A. Honig and R. Enck in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 113. Dunod, Paris and Academic Press, New York, 1965. 6 4 W. J. Choyke, D. R. Hamilton, and L. Patrick, Phys. Rea. 117, 1430 (1960). 6 5 B. S. Razbirin, Souid Phys.-Solid Stare (English Transl.) 6, 256 (1964) [Fiz. Tuerd. Tela 6, 318 (1964)l. 66 J . J. Hopfield in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 725. Dunod, Paris and Academic Press, New York, 1964. 6 7 J. Shaffer and F. Williams in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 811. Dunod, Paris and Academic Press, New York, 1964. R. J. Elliott, Phys. Reu. 108, 1384 (1957). 69 T. P. McLean, Progr. Semicond. 5, 53 (1960). 70 D. S. McClure, Solid State Phys. 8, 1 (1958). 7 ’ D. S. McClure, Solid State Phys. 9, 400 (1959). 7 2 J. W. Allen in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 781. Dunod, Paris and Academic Press, New York, 1964. 73 R. Newman, Phys. Reu. 105, 1715 (1957). 7 4 A. A. Gippius and V. S. Vavilov, Souier Phys.-Solid State (English Transl.) 4, 1777 (1963) [Fiz. Tverd. Tela 4, 2426 (1962)]. ” A. A. Gippius and V. S. Vavilov, Soviet Phys.-Solid State (English Transl.)6,1873 (1965) [Fiz. Tuerd. Tela. 6, 2361 (1964)l. 76A. A. Gippius and V. S. Vavilov in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 137. Dunod, Pans and Academic Press, New York, 1965. 63

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M. GERSHENZON

energies), these transitions should appear in an easily accessible portion of the spectrum-the visible and near infrared. Yet the measured quantum efficiencies are most often far below unity, implying either that some of these processes are nonradiative or that there are other competing processes which are nonradiative. “Nonradiative” here simply means recombination without emission of photons within the specified spectral range. Unfortunately, although experimental studies of luminescence can observe the radiative processes directly, the nonradiative mechanisms can only be deduced indirectly. Several such mechanisms have been postulated, but none have been shown to have general validity for the 111-V compounds. Multiphonon nonradiative processes are probably significant in compounds with small band gaps, or for a transition between a band edge and a localized level lying very close to it. If there are no real states between the initial and final states of the transition, the phonons must be emitted simultaneously. The probability for such an event depends upon the phonon energies and a lattice coupling constant. The oscillator strengths for such transitions become negligibly small when more than a few phonons are involved (however, some recent work in emissionSS and in photoconductivity7’ does not support this contention), so that such processes cannot dissipate more than several tenths of an electron volt in any one step. In order for one electron to recombine with one hole without involving any other particles, it is necessary then to have a ladder of real states spanning the forbidden gap. Consider a hydrogenic donor in the effective mass approximation. The capture cross section of such an ionized center for a free electron into the ground state can be c a l c ~ l a t e dThe . ~ ~ same is true for the capture of holes into the ground state of an acceptor. The cross sections in Ge and in Si measured experimentally are usually much larger than these values.79It is necessary to consider the spatially extended hydrogenic excited states to explain these large numbers. Thus it can be shown that for this case capture is into an excited state and the ground state may be reached only after a series of consecutive phonon emission steps.79 This is the simplest kind of ladder, but it still only covers a very small part of the forbidden gap. For nonhydrogenic deep levels, the lack of a well-defined model makes calculation difficult. Again measured cross sections are much larger than can be accounted for by capture into the ground states, which are spatially very compact in this case. Several attempts have been made to explain these “giant traps” by capture into the ionized state via a phonon-emitting C. Benoit B la Guillaume and J. Cernogora, J . Phys. Chem. Solids 24,383 (1963). W. Kohn, Solid State Phys. 5, 258 (1957). 79 M. Lax, J . Phys. Chem. Solids 8, 66 (1959). 77

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cascade down the excited states, or into a neutral center via a similar cascade down the states originating in the attractive potential due to the polarizability of the Such calculations have not been in quantitative agreement with experiment.' Thus the possibilities for nonradiative recombination are here, but no satisfactory theory exists. Auger processes, in which the recombination energy is transferred to another carrier, may be extremely important at high doping levels [Fig. 1(e)].82Here the energy, which would normally be released as a photon, is absorbed by a bound or a free carrier as kinetic energy, raising it well above the edge of the conduction (or valence) band. This carrier can of course thermalize nonradiativeiy. Such processes depend upon the proximity of this third particle to the recombination site, and, therefore, if the normal recombination rate varies linearly with carrier density, the Auger probability will vary quadratically with carrier density : thus, its importance at high doping levels. This mechanism has been implicated in InSb" and has been considered in some detail as a dominant recombination process in heavily doped p-n junction^.^^^^^ The interaction of nonisolated impurity states to form impurity bands with a tail of states extending well into the forbidden gap has already been mentioned and will be considered at length under GaAs. These states provide a carrier with another mechanism for penetrating deep into the forbidden gap while emitting only phonons. Again the theoretical treatments are not sufficiently refined to permit comparison with experiment. Other nonradiative processes might include recombination at dislocations, crystal interfaces and other defects,83 and Auger processes at multicharged centers.84 From the practical standpoint of increasing the luminescence efficiencies, it is, paradoxically, the details of the competing nonradiative processes that are required. Efficiencies can be increased by either increasing the strengths of the radiative processes, or by eliminating the competing nonradiative mechanism. It is quite clear then, that a vital gap remains in our understanding of luminescence. V. L. Bonch-Bruevich, SoLIiet Phys.-Solid State (English Transl.) 4, 215 (1962) [Fiz. T w r d . Tela 4, 298 (1962)l. " V. L. Bonch-Bruevich and V. B. Glasko, Souiet Phys.-Solid State (English Transl.) 4, 371 (1962) [Fiz. Tuerd. Tela 4, 510 (1962)J '* P. T. Landsberg and A. R. Beattie, J . Phys. Chem. Solids 8, 73 (1959). See also the chapter in this volume by AntonCik and T a w entitled *'Quantum Efficiency of the Internal Photoelectric Effect in InSb." W. Schultz, Solid State Phys. Electron. Telecomnzun., Proc. Intern Conj.. Brussels. 1958 Vol. I, p. 456. Academic Press, New York, 1960. 84 M. K. Sheinkman, Souiet Phys.-Solid Stare (English Transl.) 5, 2035 (1964) [Fiz. Tuerd. Tela 5, 2780 (1963)l.

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M. GERSHENZON

5 . PHONON COOPERATION

Although the theory of phonon cooperation, particularly for transitions involving deep levels, remains to be developed from first principles, certain crude generalizations can be made. If the transition is momentum-allowed, no change in k (e.g., a band-toband transition in a direct gap material), phonons are not needed to conserve momentum. A transition from a shallow impurity level in a direct gap material falls into this category. As the ionization energy of the donor or acceptor increases, the ground state becomes more tightly bound, and the transition from the excited state involves a change in the configuration coordinates. This leads to phonon cooperation. The phonons that may participate are those whose wavelengths are greater than the extent of the ground-state wave function. Thus, only lattice phonons with k vector less than the corresponding (maximum) value are used. The phonons can come from the several branches of the phonon spectra and the phonon energies are thus defined. The transition is then accompanied by phononwing satellites, corresponding to the different phonon branches, the shapes and energies governed by the phonon densities of state. The LO phonon is generally unique in that it can couple to the transition very strongly because of the polarization field associated with it. In the case of forbidden transitions-in an indirect material-involving band-to-band recombination or transitions through very shallow levels, momentum conservation requires the use of one or more phonons from very specific positions in k space. Thus, fairly discrete phonon energies are required. As the level becomes deeper, the momentum can be transmitted directly to the localized impurity, and finally, in the limit of very tightlybound states, the emission of broad phonon wings proceeds for the reasons outlined above. Thus, for such a transition, with increasing depth, we start with the well-defined momentum-conserving phonons and no zero phonon line, then the zero phonon line gains strength as the phonon lines disappear, and finally the zero phonon line loses strength to the broad phonon wings. Thus, very tightly bound states in both cases (direct and indirect gaps) and very shallow states in an indirect material need not exhibit a zero phonon line. These phonons are either absorbed or emitted, their ratio depending upon temperature. At low temperatures only phonon emission is possible and the over-all transition observed in emission may mirror the transition as seen in absorption, about the position of the zero phonon line, the phonon satellites falling below in emission, and above in a b ~ o r p t i o n . ~ ~ 85

R. E. Dietz, D. G. Thomas, and J. J. Hopfield, Phys. Rev. Letters 8, 391 (1962).

13.

RADIATIVE RECOMBINATION I N THE

III-v COMPOUNDS

303

6. HOT CARRIERS There are many ways of creating free carriers with kinetic energies greatly exceeding thermal energies. Usually these result from an electric field. In reverse-biased p-n junctions these can be by field ionization, by avalanche multiplication, or by the acceleration of thermally generated carriers. These processes can also occur in reverse-biased surface-barrier junctions. In all these cases the hot carriers are majority carriers. Nonthermal minority carriers can be created by certain tunneling structures, or by impact ionization or pair production due to energetic majority carriers, or by direct optical injection. The maximum energy of these hot carriers is the pair-production threshold. above which they rapidly lose energy by producing new electron-hole pairs. These energetic carriers may lose some of their energy radiatively as they fall back to the band edge or as they undergo interband transitions.86 Usually a broad emission band results from intraband de-excitation, although structure observed in the case of Ge has been interpreted in terms of transitions between well-defined points in the band structure.87 For a detailed discussion of charge multiplication phenomena, the reader is referred to the chapter by Chynoweth in this series. 11. GaP

The discussion in Parts 11-IV are aimed at presenting the pertinent literature in a logical order rather than historically, and merely pointing out the references to work that is not on the direct line of the discussion or else has been superseded. The judgment is entirely that of the reviewer. We will consider GaP first, despite the fact that it is an indirect material, because some, but not all, of the recombination is well understood in terms of current theories. GaAs will come next. Although, in principle, it is a simpler direct-gap compound, and its band structure is better understood, the present status of the radiative mechanisms is not completely defined, despite the enormous amount of labor in the past two or three years. Finally, the other 111-V compounds and the mixed crystal systems will be considered. 7. INTRODUCTION GaP has an indirect band gap and closely resembles Si, as shown in Fig. 3. The lowest minima in the conduction band lie in the (100) 86

A. G. Chynoweth and K. G . McKay, Phys. Reo. 102, 369 (1956).

P.A. Wolff, J . Phys. Chem. Solids 16, 184 (1960).

304

M. GERSHENZON

corresponding to a band gap (2.20eV at directions in k space 298°K88,92) which varies quadratically with temperature at low tem~~ there peratures,” and linearly near room t e m p e r a t ~ r e . ~In~ .addition, are a similar set of minima in the conduction band (xg), 0.3 eV above the

0.33

5

3.7

I 2.893

Y

I

2.325

FIG.3. Schematic band structure of Gap. Energy separations are for 0°K (After D. F. Nelson et al., Ref. 96).

lowest minima.90~92~9s Finally, the direct gap at k = 0 is 2.89 eV at room temperature 92*96-100 with a temperature dependence close to that of the indirect gap.92*96The valence band consists of light and heavy mass bands degenerate at k = 0, and a spin-orbit split off band lying 0.127 eV below. O 1 A. L. Edwards, T. E. Slykhouse, and H. G. Drickamer, J. Phys. Chem. Solids 11, 140 (1959). W. G. Spitzer, M. Gershenzon, C. J. Frosch, and D. F. Gibbs, J . Phys. Chern. Solids 11, 339 (1959). 90 W. Paul, J. Appl. Phys. 32, 2082 (1961). 9 1 M. Gershenzon, D. G. Thomas, and R. E. Dietz, Rept. Intern. Conj Phys. Semicond., Exeter, 1962, p. 752. Inst. of Phys. and Phys. SOC.,London, 1962. 92 R. Zallen and W. Paul, Phys. Rev. 134, A1628 (1964). 93 0. G. Folberth and F. Oswald, Z. Naturforsch. 9a, 1050 (1954). y 4 F. Oswald, 2. Nnturforsch. 10a, 927 (1955). J. W. Allen and J. W. Hodby, Proc. Phys. SOC. (London)82, 315 (1963). 9 b D. F. Nelson, L. F. Johnson, and M. Gershenzon, Phys. Rev. 135, A1399 (1964). 97 E. F. Gross, N. S. Kochneva, and D. S. Nedzvetskii, Souiet Phys. “Doklady” (English Transl.) 8, 1103 (1964) [Dokl. Akad. Nauk SSSR 153, 574 (1963)]. 9M W. G. Spitzer and C. A. Mead, Phys. Rev. 133, A872 (1964). 99 D. A. Cusano, G. E. Fenner, and R. 0. Carlson, Appl. Phys. Letters 5, 144 (1964). l o o M. Pilkuhn and H. Rupprecht, J . Appl. Phys. 36, 684 (1965). l o ’ J. W. Hodby, Proc. Phys. SOC. (London) 82, 324 (1963). 89

’’

13.

RADIATIVE RECOMBINATION IN THE 111-V

COMPOUNDS

305

From the measured absorption coefficient near the band edge at room temperature,88 an application of the principle of detailed balance40 yields a rate constant for free electron-free hole recombination of 3 x 10- cm3/ sec. For a majority free carrier concentration of 10’’ to loL8carriers/cm3 this results in a limiting minority carrier lifetime of 3 x to 3 x sec. Since the measured minority carrier lifetimes in GaP at room temthe expected perature have ranged between lo-’ and lo-’’ sec.,L02,103 to efficiencies of band-to-band radiative recombination are 3 x 3 x lo-*, a range in which they could be detected.

8 . BOUNDEXCITON DECAY

A number of sharp lines observed in photoluminescence or in absorption near the band gap can be attributed to the recombination of an electronhole pair, bound to a point defect, either a neutral, or an ionized, donor or The many-valleyed conduction band of GaP ac~eptor.’~.~ could produce a rather complicated spectrum. Certain simplifications, however, can be made.” For a donor binding one electron, with valleyorbit coupling, the lowest valley-orbit state will be a singlet (2-fold spin degenerate), as it is for Si. For two electrons bound to a donor, as in the case of an exciton bound to a neutral donor, with an attractive central cell correction, the lowest state is the symmetric singlet state, which can hold both electrons with spins paired. Thus, if the temperature is low enough the only levels which participate in optical transitions are these lowest states, and the usual atomic selection rules predict the dipoleallowed transitions and the expected splittings in a magnetic field. For acceptors, similar simplifications cannot be made and the spectra may be more complicated. In particular, transitions between the lowest states will be forbidden and therefore lines in absorption need not be identical with those in emission. A sharp line 5 x eV wide has been observed in absorption at 77°K and below, and at exactly the same frequency in photoluminescence at 20°K and below (see Fig. 4),and called the “C” line” or the sulfur boundexciton line.” This is therefore a zero phonon line. Very weak lattice phonon replicas of this line can be detected in absorption.” (In Ref. 91 the zero phonon line was incorrectly identified as the TA2 replica. The 1997*104-106

R. A. Logan and A. G. Chynoweth, J . Appl. Phj’s. 33, 1649 (1962). R. A. Logan, H. G. White, and R. M. Mikulyak, Appl. Phys. Letters 5, 41 (1964). Io4E. F. Gross, G. K. Kaluzhnaya, and D. S. Nedzvetskii, Soviet Phys-Solid State (English Transl.) 3, 2573 (1962) [Fiz. Tuerd. T C ~3, Q 3543 (1961)]. loS E. F. Gross and D. S. Nedzvetskii, Sooiet Phys. “Doklady” (English Transl.) 7. 908 (1963) [Dokl. Akad. Nauk SSSR 146, 1047 (1962)l. l o 6 A. T. Vink and C. Z. van Doom, Phys. Letters 1, 332 (1962). lo*

306

M. GERSHENZON

TA2 phonon energy should therefore be subtracted from all the peaks.) The main line itself lies 0.005eV below the free exciton energy. Thus the state responsible for this line is bound strongly enough not to require lattice phonons of the proper k values to conserve momentum, but it is bound weakly enough so that most of the oscillator strength appears in the zero phonon line rather than in broad phonon spectrum bands. The line is not intrinsic, but varies from crystal to crystal, and since its binding energy is much lower than the ionization energies of normal donors and

4

PHOTOLUM I NESCENCE

I

I

I 2.10

I

I

I

I

212 I

I

I

I

I

I

I

I

2.24 I

I

2.28 I

I

2 20 1

1.8OK

1

2.32 I

1

FIG.4. Photoluminescence and absorption of the “A,” “B,” and “C” bound exciton transitions in GaP and the phonon (LO, TO, X) replicas of the “A” line. (After D. G. Thomas er al., Ref. 19.)

acceptors in Gap, it cannot be due to an exciton bound to an ionized impurity. Thus the exciton must be bound to a neutral impurity. The frequency of the transition is identical in absorption and in emission. As mentioned above, this would probably not be true for an acceptor in Gap. Thus, the neutral impurity is a donor. This was confirmed by the Zeeman effect.” The ground state of a neutral donor, as discussed above, can be split into two Zeeman states because of the twofold spin degeneracy for the electron in its lowest valley-orbit state [see Fig. S(a)]. The excited state, containing an additional electron and a hole, can be split into four states, due to the hole spin only, since the two electrons must be paired. Of the eight possible transitions, two are forbidden and the polarizations and relative intensities of the remaining six are predicted by the usual selection rules, and the actual splittings can be written in terms of the g values of the spin f electron and the f and $ hole. At very low temperatures and high fields, thermalization in the excited state should populate only the lowest

13.

RADIATIVE RECOMBINATION IN THE 111-V

COMPOUNDS

307

Zeeman level, from which only one allowed transition to the ground state can occur. All of these predictions were confirmed by the experimental results shown in Fig. 6(a), proving that the line is due to the decay of an exciton bound to a neutral donor. NEUTRAL DONOR

I

IONIZED CENTER

m

$-

J

0

(a)

FIG. 5. Expected splittings of bound exciton transitions in GaP in a magnetic field (H). The structure of the ground state and the excited states are shown, with circles specifying bound charge and crosses zero charge. The structure for the ionized center transition would apply to any center with no unpaired spins in the ground state. Polarizations of the allowed transitions are indicated and the forbidden transitions are labeled f. (After D. G. Thomas ef al., Ref. 19.)

In doped floating-zone crystals, this line in absorption was correlated with neutral sulfur donors from Hall effect data.9' This line also occurs in some undoped crystals which contain sulfur as a c ~ n t a m i n a n t , ' ~ . ~lo'~ * ' ~ ~ * and it has also been observed in photoconductivity spectra in sulfur-doped crystal^.'^ A line associated with the donor Se, whose ionization energy is very close to sulfur,58 has been observed to lie very close to the sulfur bound-exciton line and probably arises from the same type of c ~ m p l e x . ~ ' Another narrow extrinsic line, the "A" line," has been observed at low temperatures in both emission and in absorption by several investigators (see Fig. 4).'9,97,'04-'06It 1'ies very close to the free exciton energy (2.317 eV at 20°K and below), and is reproduced very strongly by simultaneous emission of optical phonons, both in emission (Fig. 4)'9,97,'05and in lo'

H. Flicker, B. Goldstein, and P. Hoss, J. Appl. Phys. 35, 2959 (1964).

308

M. GERSHENZON

absorption, 04*lo5 and it exhibits broad acoustical phonon wings, implying that it is a tightly bound state, strongly coupled with the lattice. At low temperatures 7 4°K the emission strength shifts to another line (“B”) lying 0.8 x lop3eV below the “A” line. The phonon replicas shift accordingly.

(b)

FIG. 6. Zeeman splittings of the “C” line and of the “A” and “B” lines. The splittings correspond to the energy level diagrams of Fig. 5 for an exciton bound to a neutral donor and to an ionized center, respectively. The two inner peaks corresponding to the “C” line are unpolarized, and each consists of a pair of lines of opposite polarization. (After D. G . Thomas et al., Ref. 19.)

The Zeeman data [Fig. 6(b)J correspond to the case of an exciton bound to a defect with no net spin in the ground state (J = 0), for example, an ionized donor [Fig. 5(b)]. j . j coupling between the electron and the hole in the excited state then yields a 3-fold degenerate J = 1 state, with allowed transitions to the ground state, the “A” line, and a lower lying 5-fold degenerate J = 2 state, whose transitions to the ground state, although forbidden, can nevertheless be seen in emission at the lowest temperatures, the “B” line, when these excited states only, can be thermally populated. The degeneracies are lifted in a magnetic field and the observed splittings, polarizations, and intensities [Fig. 6(b)] confirm this simple model.” The phonon cooperation implies a tightly bound state, yet the transition occurs very close to the band edge. It was proposed that the center was an ionized deep donor, the electron in the bound-exciton complex being derived from a higher lying minimum in the conduction band, in particular, the k = 0 minimum, which would impart a high oscillator strength to the

13.

RADIATIVE RECOMBINATION IN THE

111-vCOMPOUNDS

309

transition. l 9 However, the pressure shift of the line, as observed in electroluminescence, was that appropriate to the normal indirect minimum, not the k = 0 r n i n i m ~ m . ~It' is conceivable that the X 3 minima lying above the normal minima are involved, since their pressure coefficient is close to that of the normal minima. However, the Zeeman data merely show that the exciton is bound to a defect with no net unpaired spin. Thus, a second possibility is that the center is a neutral doubly-ionizable donor with paired spins. The chemical nature of the defect is not known although a simple donor or acceptor is unlikely. The "A" line is present in crystals but so far has not been grown from excess gallium found in crystals grown near stoichiometry by the floating-zone techniquealthough the sulfur bound-exciton line appears." Thus, a native defect, vacancy, interstitial, or some complex thereof, might be involved. This, however, does not resolve the paradox of a tightly bound state lying close to the band gap. A number of other sharp lines, with and without phonon cooperation, have been observed at low temperatures and may be due to exciton complexes with other impurities, but no physical or chemical identifications have as yet been made.'9320,97*'05 These include a hydrogen-like series of lines."* 9. DONOR-ACCEPTOR PAIR RECOMBINATION As many as 300 sharp lines have been observed simultaneously in photoluminescence of GaP at low temperatures within 0.2eV of the band gap (Fig. 7)~58.91,105,106,109-1~1 These were explained as arising from the recombination of an electron on a donor with a hole on an Allowing a simple coulombic interaction between donor and acceptor, the ionization energy of a donor (acceptor) is reduced by the coulombic field of the neighboring acceptor (donor). The recombination energy (hv) is then increased by this effect and depends upon the donor-acceptor pair separation Y : e2 hv(r) = E , - ( E A E D ) -, (1)

+

+ Er

where E , is the band gap, ( E A + ED)the sum of the normal ionization energies of donor and acceptor, e the electronic charge, and E the static losE. F. Gross and D. S. Nedzvetskii, Souiet Phys. "Doklady" (English Transl.) 9, 38 (1964) [Dokl. Akad. Nauk SSSR 154, 64 (1964)l. E. F. Gross and D. S. Nedzvetskii, Soiliet Phys. "Doklady" (English Transl.) 8, 896 (1964) (Dokl. Akad. Nauk SSSR 152, 309 (1963)]. ""J. J. Hopfield, D. G. Thomas, and M. Gershenzon. Phys. Rev. Letters 10, 162 (1963). '" F. A. Trumbore and D. G . Thomas, Phys. Reo. 137, A 1030 (1965).

310

M. GERSHENZON

dielectric constant. Since donors and acceptors must lie on discrete lattice sites only, r is not continuous. The spectrum will then be discrete and depend upon the separations allowed by the lattice, and upon the statistical distribution of donors and acceptors.

218

2.19 2.20 221

2.22 2.23 2.24 2.25 226

227

228

2.29

50

Photon energy ( e V )

FIG.7. Photoluminescence spectra of GaP crystals at 1.6"K, showing the isolated pair lines Si-Te (IB) and Zn-S (MA) and the broad pair edge emission band corresponding to Si-S (IA), acceptor-donor combinations. Some of the shell numbers are given. (After D. G. Thomas et al., Ref. 58.)

If a donor (acceptor) is assumed to lie on a lattice site at r = 0 then the number of available lattice sites for the acceptors (donors) may be derived as a function of r, assuming either that both donor and acceptor lie on the same type of lattice site, e.g., both on Ga or both on P, or that they lie on opposite sites, e.g., one on Ga, the other on P. These two cases are very different, and in fact in each case the distribution of allowed r values and the numbers of states for each allowed r appear quite complex. If a random distribution of donors and acceptors is assumed, the emission intensity of each line corresponding to a given r should be proportional to the number of states for that separation. Over a wide range of r the monotonically varying transition probability, capture cross sections, and line widths will govern the over-all level of the spectrum, but in a small range of r the

13.

RADIATIVE RECOMBINATION I N THE

111-v

311

COMPOUNDS

rapidly fluctuating distribution of allowed states should predominate. The separations r can be converted into energy differences referred to the energy at r = 00 from Eq. (l), and the distribution of intensity vs energy, omitting the long range variations, can be compared to the measured spectra. Because of the wildly fluctuating nature of the distribution, little doubt remains when a fit is found. Figure 8(a) shows such a fit to the spectrum (01 type

I

I

type I,

I.

,I -r

61

50

41

34

29 25

21

15

18

z

9

12

C

BA

m

I50 100 50

IllllIlllll,lIl I

0

(b) type I

1

I

I

1

I

223

224

225

226

I

I

I

1

I

,

I

I

227 228 Photon energy (eV1

II

I

I

I

I

229

230

231

I 232

FIG.8. The predicted distribution of pairs as a function of pair separation for donor and acceptor on the same lattice sites (Type I) and on opposite lattice sites (Type 11). The shell numbers (m), measuring these separations, are given. The distribution is converted to one in energy by Eq. (1) and is related to the real photon energy of the pair lines by the arbitrary additive term Eg-(E, + E,). This term is deduced by sliding the energy scale to fit the measured Si-S pair data (Type I,, upper spectrum) and the measured Zn-S pair data (Type 11,, lower spectrum) thereby identifying these lines and determining these parameters for the two sets of pairs. (From D. G. Thomas et al., Ref. 58.)

of an undoped crystal, assuming donor and acceptor on the same lattice site (Type I) (S, I, of Fig. 7), and Fig. 8(b) is the fit for donor and acceptor on opposite sites (Type 11) to the spectrum of a crystal doped with zinc (Zn, 11, of Fig. 7). Once the lines have been identified with the pair spacings r, the actual photon energies of the lines can be plotted vs r as in Fig. 9 and compared with Eq. (1). Extrapolation to r = co yields E , - (ED+ E,) from which ED + E , is deduced. Note that the observed lines correspond to a wide range of separation from 10 to 40 A and not to near neighbors. This is very different from previously proposed donor-acceptor recombination models.60.6' In Fig. 9 the fit to the simple coulombic law can be improved at the closer separations by inclusion of a van der Waals term, the dashed curve.

312

M . GERSHENZON

From such spectra two important results are deduced : (1) (ED+ EA) and (2) whether the donors and acceptors are on the same or on opposite lattice sites. Table I illustrates the deductions that can be made from such data from a variety of crystals.58*”’It is noted that the occurrence of sulfur as a residual impurity is deduced from these data (the “C” line due to sulfur also appears), and that the acceptor (in undoped crystals) must lie on a P site, most probably Si, with an ionization energy of 0.04 eV. 2.30C

>, 2.275 z >

0 [L

w

z

w 2.25C

z

*0

0

r

2.225

2.2N

I 15

I

I 1

I

20

25

30

I

r

IN

H

1

FIG.^. The pair-line energies from Fig. 8 plotted against the separation of the pairs. The solid lines show the coulombic variation from Eq. ( 1 ) and the dashed curves add a van der Waals correction term. (From Thomas et al., Ref. 58.)

Hopfield has considered the binding of a free carrier to an ionized donor-acceptor pair and concludes that for donor-acceptor separations less than the normal Bohr radius of the isolated bound carrier, binding cannot occur.66 Thus for donor-acceptor separations less than the normal radius of the more tightly bound particle, neither an electron nor a hole can be bound to the donor-acceptor pair. For such transitions the relatively unlikely capture of a free exciton is necessary, whereas for greater separations the subsequent capture of a free electron and then a free hole (or conversely) makes the process much more efficient. The intensity of the pair lines therefore cut off abruptly at separations less than some critical value, and from the measured cutoff values Hopfield has deduced the ionization energies of some of the individual donors and acceptors, independent of their sums as given in Table I.

TABLE I IONIZATION ENERGYSUMS(eV) DERIVED FROM PAIRSPECTRA IN Gap” AND COMPARED WITH HALLDATA Hall results‘ Dopant None I

s

J

Te Se

Zn I Zn+SI Zn Te Zn + Se Cd Cd Cd

Typeb

EG (ED + EA)C ED + EAd

Donor

Donor Site

Acceptor

Acceptor site

ED

EA

I

2.1856

0.1394

S

Si

P

0.1 1

(0.03)l

I I

2.2007 2.1867

0.1243 0.1383

Te

Si

P

(0.04)J

Se

Si

P

0.08 (0.llV

ED

+ EA

I1

2.1701

0.1549

S

Zn

Ga

0.11

0.04

0.15

+

I1 I1

2.1848 2.1718

0. I402 0.1532

Te Se

Zn Zn

Ga Ga

0.08 (0.10)’

0.04 0.04

0.12

+S + Te + Se

I1

2. I369

0.1881

S

Cd

Ga

0.11

0.08

0.19

I1 I1

2.1517 2.1383

0.1733 0.1867

Te Se

Cd Cd

Ga Ga

0.08 (0.11)’

0.08 0.08

0.16

~~

After D. G. Thomas et al., Ref. 58 and F. A. Trumbore and D. G. Thomas, Ref. 111. Type I means donor and acceptor on the same lattice site, Type I1 on opposite sites. Energies derived for r = tci. IZ Band gap EG taken as 2.235 eV at 1.6°K.y1 ‘From M. Gershenzon and R. M. Mikulyak, Ref. 114 and unpublished data of H. C. Montgomery and M. Gershenzon. Derived from the spectroscopic sum.

314

M. GERSHENZON

10. DONOR-ACCEPTOR EDGEEMISSION

-

Whenever the many sharp donor-acceptor pair recombination lines appear in emission, a broad band, 0.02eV wide, appears on the low energy side of the spectrum (Fig. 7).20~58~62,91~’05~’06~109~111~112 This band is replicated by the simultaneous emission of optical phonons (Fig. 10)2 0 , 5 8.6 2.9 1,i0 5,109.1 12 and may be fairly efficient at low temperatures

~0=0.048+ 0.W2eV

1.80

1.90

2.00 2.10 PHOTON ENERGY IN

220

ev

:

FIG.10. The Si-S broad pair band at low resolution at 20°K. showing the simultaneous emission of multiple LO phonons. The sharp pair lines are unresolved, but lie in the highenergy shoulder above the 0 LO peak. (From D. G. Thomas et al., Ref. 58.)

-

’

(photoluminescence quantum efficiency 60 % at 20°K’ 3). The band shifts with the pair lines as (ED+ EA) is altered by varying the doping elements (Fig. 7), and therefore the band must be associated with donoracceptor recombination.58362This emission may be understood in terms of several simple assumption^.^^^^^ The band is due to donor-acceptor pair recombination, as are the sharp lines. The band correspoRds to separations of 40-2OOA. As shown in Table 11, the peak always occurs near a donor-acceptor separation of 50A.In this range the individual lines become very closely spaced owing I2

E. F. Gross and D. S. Nedzvetskii (Nedzvetsky) in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 81. Dunod, Paris and Academic Press, New York, 1965. K. Maeda, J . Phys. Chem. Solids 26, 595 (1965).

13.

RADIATIVE RECOMBINATION IN THE

111-v COMPOUNDS

315

TABLE I1 POSITION OF THE PEAKOF THE BROAD PAIRBANDIN GaP AND THE CORRESPONDING PAIR SEPARATION"

Donor

Acceptor

S Se Te S

Si Si Si Zn Zn Zn Cd

Se Te S

Peak position (eV)

Pair separation

2.213 2.213 2.226 2.200 2.202 2.215

53 55 57 47 47 47 47

2.167

(A)

After F. A. Trumbore and D. G. Thomas. Ref. 111.

to the r - ' coulombic dependence, and as r approaches the average donoracceptor separation, three (and higher) body interactions may become important, also increasing the density of lines, so that they merge into a structureless band. As r increases the transition probability decreases, but the number of available states per unit energy increment increases. This defines the shape of the band. These postulates have recently been proven.62 The decay following pulsed excitation appears quite complex, approximating a power law in various ranges of time after excitation. These curves, however, can be fit by assuming a random distribution of donors and acceptors, with one type predominating, and an exponential fall-off of the transition probability with pair separation. Pulsed excitation fills all levels. The close pairs recombine first, followed at a later time by the more separated pairs in which the many-body interactions become important. The experimental decay curves agree quite well with the predicted decay of the total light emission, covering all photon energies. Furthermore, the excited states (filled states) can be split in a magnetic field, and at sufficiently low temperatures the bound carriers populate only the lowest magnetic substates. A donor-acceptor transition between such states is forbidden, thereby decreasing the transition probabilities. The predicted effect of the magnetic field upon the decay characteristic has been verified.62 The shape of the pair emission band varies with excitation intensity and is different for continuous and for pulsed excitation, and in the latter case changes with time during the d e ~ a y . ~ ~l 2, 'These ~ ~ , ' effects can be explained in terms of progressive saturation of pairs of different separation as a function of excitation, and in terms of the more rapid decay of the closer pairs, provided the emission is assumed to contain a broadening process due to phonon cooperation.62 This effect limits the width of the

316

M.

GERSHENZON

band in cases where the emission should be very narrow, and if this experimentally deduced phonon function is used instead of the sharp pair lines, the shapes of all the other curves can be generated. Although these edge emission bands are very efficient at low temperatures, where once a minority carrier is trapped at a shallow donor or acceptor it cannot be thermally reionized, the efficiency drops rapidly with increasing temperature. During the long radiative lifetimes for these recombination processes the trapped carriers are thermally re-excited to produce free carriers, whose eventual fate is recombination by other means. Maeda has explained the dependence of the emission intensity of the Si-S pair edge emission band in GaP both upon temperature and upon excitation intensity in terms of a simple m0de1.l'~ He considers an ionized donoracceptor pair, of separation corresponding to that at the measured maximum of the broad pair band emission. The pair may capture either an electron or a hole first and then the other free carrier. After each capture step the bound carriers may be released by thermal ionization. The various possible sequences of steps leading to the radiative recombination are analyzed in terms of Shockley-Read-Hall kinetics, leading to the predicted behavior as a function of temperature and of pumping level. By comparison with the experimental data for the Si-S band, he concludes that, for every recombination, the pair captured an electron first. At low temperatures thermal ionization is insignificant. As the temperature is raised however, first the shallow Si acceptors, and later the deeper S donors begin to lose their captured carriers, thus quenching the luminescence. These pair bands have also been excited by light of energy much less than the band gap, presumably by a two-step excitation process involving a deep l e ~ e l . ~Since ~ ' ~ *the oscillator strengths for these distant pair processes are very small, these bands, as well as the sharp, isolated pair lines have not been detected in absorption. 11. RECOMBINATION AT DEEP LEVELS

Ever since electroluminescence was first observed in much of the interest centered around certain relatively broad emission bands in the red, appearing in electroluminescence at room temperature with moderately high efficiencies. Although these bands have given rise to a variety of sometimes contradictory interpretations, the experimental data of the various investigators do fall into a consistent pattern, which becomes clear if the photoluminescence results are considered first. At 20"K, the photoluminescence is dominated by the broad donoracceptor pair bands lying within 0.2 eV of the band edge. In addition, one or more broad bands appear in the At 77°K thermal ionization of the shallow states, producing the pair bands, severely attenuate these

13.

RADIATIVE RECOMBINATION IN THE 111-V

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processes and then the red bands dominate the radiative recombination spectrum in the visible region with quantum efficiencies of 10-4-10- 1.59 The peaks are generally 0.2 eV wide. In p-type crystals containing the acceptor zinc as well as the donor sulfur, photoluminescence at 20°K is dominated by the Zn-S sharp pair lines as well as the corresponding broad pair band.59 A broad red band at 1.83 eV also appears. At 77°K the sharp Zn-S pair lines disappear, but the Zn-S band remains although with reduced intensity. The red band remains with about its original intensity. When the zinc is replaced by cadmium (another acceptor), the Cd-S pair lines and the Cd-S pair band replace the Zn-S pair emission. Cadmium as an acceptor is deeper than zinc by 0.04eV. Accordingly, the pair emission band and the individual pair lines are shifted to lower energies by this amount. The red band, however, also shifts to lower energies by this amount. The acceptor magnesium produces similar results in both the pair band with sulfur and the red band, based, this time, on the ionization energy of magnesium. Thus the red band involves a shallow acceptor in each case. Since its position lies very far below the band gap, it must involve another and deeper level-a deep donor, and therefore the red band must be due to donor-acceptor pair recombination at deep levels.59 Chemical doping correlations showed that the 1.83-eV red band (77°K) was due to both the acceptor zinc and the deep-donor oxygen.59In fact, oxygen had already been shown to behave as a deep donor from electrical measurements' l4 and oxygen is a very common contaminant in GaP.'I4 Because the oxygen is a deep donor (several tenths of an electron volt), minority carriers trapped on it are not thermally ionized even near room temperature, and therefore photoluminescence quantum efficiencies as high as 1.5% could be measured at 298°K for the Zn-0 pair band.59 The width of the band is probably due to phonon cooperation. Other evidence that oxygen is a donor and probably a moderately deep one comes from a correlation with two bound-exciton transitions in high resistivity p-type crystals.59These may correspond to bound-exciton decay with each of the ionized states of oxygen if oxygen is a double donor. If this is true, oxygen may also be implicated in the "A" line which may be due to an exciton bound to a neutral doubly ionizable donor. Thus oxygen, a common contaminant, may be responsible for much of the radiative recombination in Gap. Several other bands in photoluminescence of p-type crystals have been identified as donor-acceptor pair bands by their shifts as the shallow acceptor is varied, but the donors have not been identified. In n-type

-

l4

M.Gershenzon and R. M.Mikulyak, Solid-State Electron. 5, 313 (1962).

318

M. GERSHENZON

crystals a strong red band appears in photoluminescence at low temperatures at 1.96 eV.114-117This may be a donor-acceptor transition but this remains to be shown. Other deep level transitions have been reported but not identified.’ 14,118-122 The description of these and other deep bands in electroluminescence will be deferred until later.

12. CATHODOLUMINESCENCE Although Wolff and his co-workers observed luminescence from GaP during electron-beam excitation a number of years his results were not followed up until very recently.123.’24van der Does de Bye examined p-type crystals with a 600-kV beam.lZ4 At 77°K he observed (1) very fast emission near the edge (perhaps the “A” line), (2) a slow emission band at 2.195eV, probably due to shallow donor-acceptor pairs, (3) a very slow red band at 1.9eV (perhaps a deep donor-acceptor pair), and (4) some bremmstrahlung emission. From the temperature dependence of the decay time, he obtained thermal activation energies of 0.07 and 0.09eV, due presumably to shallow acceptor traps. He noted that the red band increased in intensity at first, while the green bands decayed, immediately following the end of the excitation pulse. This, he believed, was due to thermal ionization out of the green centers and recapture into the red centers. This is precisely the effect seen in the temperature dependence of the photoluminescence intensities of the Zn-S and Zn-0 bands in zincdoped crystal^.'^

13. ELECTROLUMINESCENCE Electroluminescence in GaP has been observed from a wide variety of structures, some p-n junctions, some not. These include alloyed junctions H. G. Grimmeiss and H. Koelmans, Philips Res. Rept. 15, 290 (1960). F. G. Ullman, Nature 190, 161 (1961). 11 F. G. Ullman, J . Electrochem. SOC.109, 805 (1962). M. Gershenzon and R. M. Mikulyak, Electron. Diu. Abstr., Spring Meeting Electrochem. SOC.,Pittsburgh, 1963 p. 13, Abstr. No. 21. M. Gershenzon, R. A. Logan, D. F. Nelson, and D. G. Thomas Bull. Am. Phys. Soc. 9, 236 (1964). lZo E. E. Loebner and E. W. Poor, Bull. Am. Phys. SOC.4, 45 (1959). J. Starkiewicz and J. W. Allen, J . Phys. Chem. Solids 23, 881 (1962). l Z 2 H.G. Grimmeiss and H. Scholz, Phys. Letters 8, 233 (1964). G. Mayer and G. Bisson in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 251. Dunod, Paris and Academic Press, New York, 1965. I z 4 J . A. W. van der Does de Bye in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 243. Dunod, Paris and Academic Press, New York, 1965. 15

16

’”

”’

13.

RADIATIVE RECOMBINATION IN THE

111-v

COMPOUNDS

319

to n-type crystals [In-Zn or Au-Zn,12' Ag-Zn,lZ6 AglZ7] and to p-type crystals [Sn, ',12'-' 32 Ag-Te126]; diffused junctions [Zn into n103.114,119.126,~27,~30,133,134 Cd into n-type,126 and Si into p type, type'35] ; intentionally doped melt-grown junctions [S-Mg, S-Cd 14]; and epitaxial junctions deposited from the vapor phase [S-Mg, S-Cd' 14]. In addition, some junctions are unintentionally produced during crystal groWth,2,3,115-1 17.1 36,137 particularly by segregation at grain boundaries.' 38*139 Other structures have included point-contact and surfacebarrier devices,2.3.1 16.1 17.1 3 1,140- 1 4 2 and powders inserted between the plates of a c a p a c i t ~ r . ~ * ~ * ' ~ ~ The literature in this field is very confusing because most of the reports have interpreted the emission as due i o recombination, following minority carrier injection at a forward biased p-n junction. As we will see, many of the structures were not p-n junctions at all. The grown junctions, the epitaxial junctions, the diffused junctions, and the built-in junctions were p-n junctions. The alloyed diodes may or may not have been. All the rest were not. In interpreting the results, it is important to separate the injection process from the recombination process. In most diode structures the forward current is carried by more than one process. Only the components that inject minority carriers play a significant role in the luminescence. Thus in Gap, the total forward diode current may be dominated by other mechanisms and may not be correlated with the luminescence process, J. W. Allen and P. E. Gibbons, J . Electron. Control 7, 518 (1959). M. Gershenzon and R. M. Mikulyak, J . Appl. Phys. 32, 1338 (1961). A. Pfahnl, Bell System Tech. J . 43,333 (1964). lZ8 J. W. Allen, M. E. Moncaster. and J. Starkiewicz, Solid-State Electron. 6, 95 (1963). W. Glasser, H. G. Grimmeiss, and H. Scholz, Philips Tech. Rev. 25, 20 (1963/64). 1 3 0 H. G. Grimmeiss, A. Rabenau, and H. Koelmans. J . Appl. Phys. 32,2123 (1961). 1 3 ' H. G. Grimmeiss and H. Koelmans, Phys. Reu. 123, 1939 (1961). R. Bolger and H. Koelmans in "Radiative Recombination in Semiconductors" (7th Intern. Conf.), p. 95. Dunod, Paris and Academic Press, New York, 1965. 1 3 3 R. S. Ricks and M. D. Pope, Japan. J . Appl. Phys. 2, 520 (1963). 134 S. Iizima and M. Kikuchi, Japan. J. Appl. Phys. 1, 303 (1962). 135 F. M. Ryan and W. Stickel, Electron. Div. Abstr., Spring Meeting Electrochem. Soc., Pittsburgh, I963 p. 75, Abstr. No. 28. 136 M. Gershenzon, R. M. Mikulyak, R. A. Logan, and P. W. Foy, Solid-State Electron. 7, 113 (1964). 13' E. E. Loebner and E. W. Poor, Phys. Rev. Letters 3, 23 (1959). 13' D. B. Holt, G. F. Alfrey, and C. S. Wiggins, Nature 181, 109 (1958). I3'G. F. Alfrey and C. S. Wiggins, Solid-state Phys. Electron. Telecommun., Proc. Intern. Conf., Brussels, 19.58 Vol. 11, p. 747. Academic Press, New York, 1960. 140 M. Kikuchi and T. Iizuka, J . Phys. SOC.Japan 15, 935 (1960). S. Iizima and M. Kikuchi, J. Phys. Soc. Japan 16, 1784 (1961). 14’ H. C. Gorton, J. M. Swartz, and C. S. Peet, Nature 188, 303 (1960). 143 L. J. Bodi, J . Electrochem. SOC. 109, 497 (1962).

320

M. GERSHENZON

at least in the low current region where meaningful current-voltage relations can be measured. Thus, the over-all kinetics of injection and radiative recombination in GaP can be fairly complex. In carefully fabricated zinc-diffused p-n junctions, it was shown that the dominant current component (at currents low enough so that series resistance does not make it impossible to measure the true bias) was due to thermal injection and recombination within the depletion layer of the junction."' This current varied with bias as exp(eV/nkT) with 1 < n 2. The exponent n can change abruptly at certain critical voltages. Recombination from this process was not radiative. Instead, in parallel with this process was the injection of minority carriers beyond the depletion layer into both the n and the p side of the junction. These carriers did recombine radiatively. This current varied with bias as exp(eV/kT). Shockley-ReadHall statistics could be applied to the recombination process only when this current component was used. Logan, et al. have shown that degradation of such diodes by neutron bombardment decreases the minority carrier lifetimes beyond the space-charge region, and that the radiative efficiency depends quantitatively upon these lifetimes.'03 In thermal injection the barrier to the flow of forward current is essentially the band gap, as decreased by the applied bias. Thus, normal thermal injection at a p-n junction only becomes significant as the forward bias approaches the band gap. This is true in the p-n junctions tabulated above. For many of the alloyed structures, however, current flows at very low voltages, implying that this current is not due to injection across a p-n junction. At higher biases injection may, however, predominate.

-=

a. p-n Junctions in Forward Bias In this section we will concentrate on the radiative recombination of the injected minority carriers, and so we may include some results from structures other than p-n junctions, where minority carriers are formed. We start first with the emission peaks close to the band edge. While studying the photoluminescence due to bound excitons and donor-acceptor pairs described above, a number of accidental, as-grown ~ ~ forward bias these junctions appeared in some of the ~ r y s t a 1 s . lAt junctions exhibited the same peaks that were seen in photoluminescence (see Fig. 11). These peaks were then followed as the temperature was increased. Up to 77"K, the dominant emission was the broad donoracceptor pair band (Si-s). Above 77', only the "A" line and its phononassisted satellites remained. Bolger and Koelmans have also observed the "A" line as well as pair bands in forward emission of alloyed diodes at low temperatures.' 3 2

13.

RADIATIVE RECOMBINATION IN THE

111-v COMPOUNDS

321

H

I

.

I

2.28

.

I

I

I

I

2 26 2 24 Photon energy

I

I

2.22

I

I

2.20

I

2.1

FIG. 11. Comparison between photoluminescence and junction electroluminescence of a GaP diode at 20°K. The bound exciton “A” line (and its phonon emission counterparts) and the “C” line appear in both spectra as do the many sharp pair lines (labeled with their shell numbers) and the broad pair band due to the acceptor Si and the donor S (Type I*). The asterisks indicate calibration lines omitted from these tracings. (From. M. Gershenzon et al., Ref. 136.)

It was stated that although the green emission started out as the “A” line at lower temperatures, smearing of the phonon structure, presumably by thermal broadening at the higher temperatures, made it difficult to rule out the possibility that simple band-to-band recombination was becoming the dominant emission near room temperature.’ 36 However, the radiative lifetime for this process (deduced from the detailed balance argument given earlier, together with the majority carrier concentrations), compared to the measured minority carrier lifetimes in a group of diffused junctions, predicted a radiative efficiency which was about 10 times less than that actually measured. Hence the process may not be band-to-band recombination and thus may remain the “A” line even at room temperature.“’ As noted earlier, the “A” line is possibly due to the recombination of an electron-hole pair, bound to a completely neutral, but doubly ionizable,

322

M. GERSHENZON

donor, perhaps oxygen. Therefore, in a p-n junction, such recombination could only arise from recombination on the n side of the junction, outside the depletion layer. In a group of diffused junctions, it was shown that the intensity of this emission varied simply with the normal thermal injection current, but did not depend upon the space-charge recombination current."g Moreover, by visual inspection, the emission clearly originated from the n side of the junction ( a red band being generated on the p side). Both observations agree with the model. This green peak, 2.20 eV at room temperature, has been seen by a number of investigators, and interpreted differently. Starkiewicz and Allen have correlated its appearance with the presence of zinc in tin-alloyed diodes on zinc-doped crystals."' They note, however, that a small amount of oxygen improves the efficiency. Furthermore as the zinc concentration is increased, injection into the n side becomes more important, in agreement with the neutral oxygen donor model. In addition, Pfahnl observes this peak when silver alloys containing no zinc are used to make alloyed junctions to n-type crystals (which incidentally exhibit the Zn-0 red band in zinc-diffused str~ctures).''~ Thus at 77°K and below, emission near the edge, with external quantum efficiencies as high as is due primarily to donor-acceptor pairs, e.g., S-Mg, S-Cd, S-ZII,'~~ whereas above 774 the "A" line-phonon complex predominates. This change of dominant mechanism has led some authors to the erroneous conclusion, after measurements at only two or three temperatures, that the so-called green emission does not shift with temperature as the band gap ~hifts."~*'~O,' 31 The low energy emission bands which appear in forward-bias junction luminescence are those deep levels already described, in particular the fairly common deep oxygen level which produces a donor-acceptor pair band in conjunction with the various shallow acceptors, and which occurs predominantly by injection into the p region, followed by recombination past the depletion layer.llg In such a region the donor is always ionized and offers a large cross section for capture of minority carrier electrons. As discussed earlier, the level is deep enough so that thermal release is slow, even up to room temperature. Thus, recombination by this mechanism can be very efficient, both at low temperatures (-lo-' quantum efficiencies at 77°K11*122) and at room temperature (1.5 x lo-' at 298"KlZ2). Starkiewicz and Allen have examined the electroluminescence spectrum from alloyed junctions prepared on solution-grown crystals containing various impurities."' They found that the simultaneous presence of both zinc and oxygen in the crystals is necessary to produce the low energy "zinc" red band (1.75 eV at 295°K). This agrees with the photoluminescence

13.

RADIATIVE RECOMBINATION IN THE

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323

results cited above, that the emission is due to recombination at oxygendonor : zinc-acceptor pairs. Ryan and Stickel studied some diodes prepared by diffusion of silicon into p-type as-grown crystals.' 35 At 77" the electroluminescence peaks at 1.97 eV, agreeing with the normal photoluminescent peak in n-type material cited earlier. The diodes were damaged by irradiation with electrons of various energies and dosages. They showed that they could produce predominantly either Ga vacancy-interstitial pairs, or P vacancyinterstitial pairs, depending on the energy of the beam. Furthermore, they deduced that these were probably equivalent to the corresponding thermally generated defects. In both cases the luminescent efficiency decreased, which they interpreted as indicating that the 1.97eV band was not due to an inherent G a or P defect. In general, for samples containing zinc and probably oxygen, the room temperature electroluminescence always exhibits the 1.77-eV Z n - 0 pair peak,2.3.116.1 17.1 1 9 , 1 2 t , 1 2 2 , 1 2 7 , 1 2 8 , 1 3 0 , 1 3 1 whereas the type" band (1.96 at 77") appears at -1.9eV in other samples."4~1'6~"7~'26~135 In addition to these deep levels, a number of broad peaks have been seen at longer wavelengths, and are probably associated with other deep ievels.l t 4 , t 18-122

In attempting to interpret the kinetics of the recombination process, it has already been pointed out that the total current may not be a proper measure of the level of injection. Thus, some models based upon such an interpretation should be re-examined. 8 s 1 30,1 3 1 It was also deduced from the shift of a shallow donor-shallow acceptor pair band that there was an excess "nonradiative" current component at low bias in some j ~ n c t i 0 n s . l ~ ~ This component was probably due to space-charge layer injection in parallel with the normal injection current. This was proved in some later work on zinc-diffused diodes. l 9

'

b. Reverse Bias Light Emission Light is often generated at reverse-biased p-n junctions in Gap. In this case the large electric fields of the depletion layer inject energetic majority carriers into each side of the junction. The carriers may arise from any of several mechanisms, but in each case the net result is carriers with energies greatly exceeding thermal energies. In each case these carriers decay partly by radiative intraband relaxation, producing a broad (orange) structureless spectrum"s~120~'26~'42 which extends well above the band gap at shorter wavelengths, exhibiting a cut-off which mirrors the absorption edge and the length of crystal through which the light travels, and extending well into the infrared at the other end of the spectrum.120.'26 As expected,

324

M. GERSHENZON

minority carrier effects (interband transitions) are usually absent. The and quantum efficiency, in the visible range only, is between Three distinct means of carrier generation have been postulated, all three leading to identical spectra: (1) Carriers are produced by thermal generation in the depletion layer at recombination-generation sites at reverse bias below breakdown, and accelerated and swept out of the junction by the junction field. (2) In very narrow alloyed diodes (1W 200 A), breakdown occurs by internal field emission across the depletion layer, producing a uniform light distribution.'26 (3) In most other diodes breakdown occurs by avalanching through localized microplasmas.'25~'26,133~134~140-142~144,145 In the latter case, the light occurs only from a multiplying r e g i ~ n ' ~ and ~ * is ' ~emitted ~ at the discrete microplasmas which turn on reversibly, one by one, at breakd~wn.’~~~’~~~’~~*’~’ Turn-on bias is about 0.2V,'33 and the bistable nature of each microplasma results in the expected microplasma noise in the reverse characteristic.' 25 p1337140The light always increases linearly with current,' 25,’ 26,141 even though, just at breakdown, the number of visible microplasmas increases linearly with current (each corresponding to about 0. 1 l z 6 or 1 mA134), whereas at higher bias the brightness of each spot increases'34 and corresponds to an increasing current through each spot. Near room temperature, the microplasmas correspond to gross crystallographic defects in the junction, whereas at 77°K they occur at dislocations or at segregated sec have been measi m p ~ r i t i e s . 'Radiative ~~ decay times of < 5 x ured. 12 5,126,144 c. Emission from "Nonohmic" Contacts

Light is also produced when current flows through a nonohmic contact to Gap, such as is formed with low melting metals or with silver paste.2.3.115-117 These are probably surface-barrier junctions. Evaporated metals form normal metal-semiconductor barrier junctions with Gap, with the Fermi level at the interface pinned in the forbidden gap, 5 of the way down from the edge of the conduction band147*'48due to surface states, as it is for most of the other III-V corn pound^.'^^ At forward bias these contacts allow thermal majority carriers to flow into or out of the semiconductor and can therefore produce no light emission. At sufficiently large reverse bias, breakdown can occur, permitting energetic majority J. W. Allen and P. E. Gibbons, Nucl. Insrr. & Methods 14, 355 (1961). J. Mandelkorn, Proc. IRE 47, 2012 (1959). 146 M. Gershenzon and R. M. Mikulyak, J . Appl. Phys. 35, 2132 (1964). 14’ H. G. White and R. A. Logan, J. Appl. Phys. 34, 1990 (1963). 14* M. Cowley and H. Heffner, J. Appl. Phys. 35, 255 (1964). C. A. Mead and W. G. Spitzer, Phys. Rev. 134, A713 (1964). 144

13.

RADIATIVE RECOMBINATION I N THE 111-v COMPOUNDS

325

carriers to enter. These can relax radiatively yielding a spectrum akin to normal p-n junction reverse-breakdown radiation. In either bias condition, built-in junctions, such as can occur naturally at grain boundaries, may emit light,'38,'39 but these are in no way associated with the surfacebarrier junctions. When light is emitted close to the contact it only occurs when that contact is in reverse b i a ~ . ~ - ~ , " ~ However, *"' very often the spectrum reveals the presence of bands which can only appear during minority carrier r e c ~ m b i n a t i o n , ~1 6,v~1 -l '7 in addition to a broad breakdown spectrum. These minority carriers might be generated by impact ionization of various centers by the energetic majority carriers. Such recombination might also occur in reverse breakdown if the carriers are trapped on recombination sites before the field sweeps them out of the depletion layer. Another means of introducing minority carriers, involving tunneling through a thin dielectric between the metal and the semic o n d u ~ t o r can ~ ~ .be~ ruled ~ out because it occurs with the wrong polarity. 111. GaAs

14. INTRODUCTION Luminescence processes have been investigated extensively in GaAs, because it is relatively easy to prepare diodes in GaAs, having both high quantum efficiencies, and low thresholds for stimulated emission. This is apparently a result of the direct edge in GaAs, leading to short radiative lifetimes for band-to-band or near band-to-band transitions, thereby competing favorably with nonradiative recombination processes. Most of the literature in this field (much of it reviewed recently by Burns and Nathan") has been aimed exclusively at the stimulated emission aspects of luminescence, and this phase will be covered thoroughly in the next chapter. However, too often, the emission characteristics below the laser threshold were only examined cursorily. It will be evident from below that there are a number of distinct mechanisms giving rise to light emission in GaAs, but many of their characteristics, such as spectra, and current and bias dependences, are very similar. Thus great care is needed in comparing the results of different experiments. GaAs is a direct-gap semiconductor with spherically symmetric, parabolic bands, with an effective mass of 0.08 in the conduction band and light and heavy hole masses of 0.20 and 0.68 respectively.' 5 0 From absorption measurements the band gap is 1.521, 1.511, and 1.435eV at 21", 90" and 294°K respectively, and the free exciton binding energy is 0.0034eV.1s1 Is’

H.Ehrenreich, Phys. Reti. 120, 1951 (1960). M.D. Sturge, Phys. Rev. 127. 768 (1962).

326

M. GERSHENZON

In very lightly doped crystals (- 1016cm-3), sharp lines in photoluminescence, electroluminescence, and absorption can be ascribed to such expected transitions as those involving free and bound excitons, and also, perhaps, donor-acceptor pairs. However, in more heavily doped crystals, two effects caused by the very low effective masses, particularly in the conduction band, lead to broad-band transitions. First, the density of states is fairly low, so that at thermal equilibrium, degeneracy sets in and the bands fill rapidly for fairly modest free carrier concentrations. Thus, the free carriers can exist in a broad range of k values. The recombination spectrum will depend upon these distributions as well as upon the appropriate selection rules. Second, the hydrogenic donors (S, Se, Te, C, Si, Ge, Sn) and acceptors (Zn, Cd) are characterized by very small ionization energies: 0.001-0.003 eV for donors’52 and 0.014.04 for acceptor^.'^^ Free carrier freezeout should not occur, therefore, except at very low temperatures. Furthermore, because of the large Bohr radii of these bound states, mutual interactions, resulting in the formation of impurity bands, begin at fairly low donor and acceptor densities, about 8 x 10l6~ r n for - ~donors.’53 Moreover, because the levels are so shallow, these impurity bands merge with the conduction or valence bands, again at very modest impurity concentrations, 5 x lo1’ cm-3 for donors.’52 Thus, in this case also, carriers are not confined to discrete energy ranges.

15. RADIATIVE TRANSITIONS

a. Low Doping Levels The absorption edge and the spontaneous emission spectrum can be calculated for intrinsic material involving transitions between the filled valence bands and the empty conduction band, which are both parabolic and spherically symmetric. The allowed transitions must conserve momentum. This permits only vertical transitions in E-k space [Fig. 12(a)l, since the momentum of a photon in this energy range is much less than that of the free carriers. Thus, the energy is a single-valued function of k, and the transition probability in any energy range depends upon the joint densities of state of conduction and valence bands corresponding to that energy range, and upon the matrix element for the transition. The latter C. Hilsum and A. C. Rose-Innes, “Semiconducting III-V Compounds,” pp. 77-78. Macmillan (Pergamon),New York, 1961. R. Broom, R. Barrie, and I. M. Ross, “Semiconductors and Phosphors” (Proc. Intern. Colloq., Garmisch-Partenkirchen, 1956), p. 453. Wiley (Interscience), New York, 1958.

lS2

13.

RADIATIVE RECOMBINATION IN THE 111-v COMPOUNDS

327

can be calculated.4791 54*1s5 The absorption edge shape-absorption increasing as the square root of the difference between the energy and the gapfollows directly from the variation of the densities of states of the bands with energy. When allowance is made for the formation of free excitons, a rapidly rising peak is predicted just below the band gap in absorption, and the remainder of the absorption curve is somewhat m~dified.~'.~'

(a )

(b)

(C)

FIG. 12. Band-to-band transitions in a direct gap material such as GaAs. (a) simple radiative recombination or absorption, (b) absorption (upward transition) in degenerate n-type material (the Burstein-Moss Effect). The lowest energy photon that can be absorbed raises an electron to the lowest empty level in the conduction band which is at the Fermi level. Recombination of electrons with injected holes in quasithermal equilibrium in the valence band can occur as shown. A momentum selection rule is assumed. (c) absorption and recombination when the conduction band is modified by a tail of impurity states. The spread of holes in k-space, determined by temperature or by hole degeneracy, together with the distribution of electrons in k-space in the modified conduction band governs the energy distribution of the recombination radiation, again assuming the k-selection rule.

The spontaneous emission can now be deduced, either from the measured absorption (using detailed balance), or directly from the predicted transition probabilities. This also yields the rate constant for recombination, from which the radiative lifetime may be determined as a function of the majority carrier concentration. Next we consider transitions between one of the bands and a hydrogenic impurity level associated with the opposite band. [We will not discuss transitions between the ground state and the excited states of the hydrogenic donors (acceptors)." For GaAs such transitions occur in the far infrared.] Transitions from the conduction band to hydrogenic acceptors were first considered by Eagles,lS6 and later by others.'55,'57,158In the

lS6

15’

H. J. Bowlden, Phys. Rev. 106, 427 (1957). H. J. Zeiger, J. Appl. Phys. 35, 1657 (1964). D. M. Eagles, J. Phys. Chem. Solids 16, 76 (1960). J. Callaway, J . Phys. Chem. Solids 24, 1063 (1963). W.P. Dumke, Phys. Rev. 132, 1998 (1963).

328

M. GERSHENZON

effective mass approximation, the acceptor states are composed of valence band wave functions, and therefore the transition probabilities may be derived using the k-selection rule together with the matrix elements for band-to-band transitions. The predicted absorption curves look like the normal band edge, but moved to lower e n e r g i e ~ , ” ~and , ~ ~a ~radiative lifetime of 2.3 x 10-9sec at 77°K has been calculated by Dumke for recombination in 1OI8 p-type Transitions between donors and acceptors may occur as in the case of Gap. However, the pair separations corresponding to isolated pair lines observed in GaP would all lie above the band gap in GaAs for the hydrogenic donors and acceptors in GaAs, because their binding energies are much smaller than those in Gap. Thus, such pair lines could not be observed. The broad unresolved pair band could appear, however. Hopfield66 and Shaffer and Williams6’ have considered the conditions for pair binding and Callaway has briefly described such processes in GaAs.’ 5 7 Finally predictions concerning transitions involving deep, nonhydrogenic levels are not possible, because the wave functions for such levels are unknown.

b. High Doping Levels In highly doped crystals the situation is far more complex, because the carriers in the conduction (or the valence) band become degenerate, and the simple donor and acceptor levels broaden into impurity bands, which eventually merge with the conduction (or the valence) band. The latter effect leads to a blurring of the normal band edges and can be described by a tractable model only in the limit of very high doping. Unfortunately, the doping levels corresponding to the most efficient luminescent diodes (with the lowest laser thresholds) are not quite in this range. Thus, most of our understanding of the active radiative mechanisms in such diodes is not predictable but must be derived from experiment. The shift of the absorption edge to higher energies as degeneracy sets in-the Burstein-Moss effect- is due to the filling of states in one of the Since the electron mass in GaAs is much less than the hole mass, the density of states in the conduction band is lower than in the valence band, and the conduction band fills rapidly as the donor concentration increases beyond the point of degeneracy. This is indicated in Fig. 12(b). Since there are no empty states in the conduction band below the Fermi P. Dumke, Proc. Symp. Opt. Masers, New York, 1963 p. 461. Polytech. Press, Brooklyn, New York, 1963. I 6 O E. Burstein, Phys. Rev. 93,632 (1954). 16 T. S. Moss, Proc. Phys. SOC. (London) B67, 775 (1954).

l S 9W.

13.

RADIATIVE RECOMBINATION IN THE

111-v COMPOUNDS

329

level, interband absorption can only begin at energies greater than the band gap plus the height of the Fermi level above the edge. Thus the effective absorption edge has moved to higher energies. The exact shape of the absorption can be predicted.'62 Radiative recombination, however, does not have the same form. Suppose, as in Fig. 12(b), that the Fermi level lies well above the edge of the conduction band, and that minority carriers (holes) are thermalized (with an average energy of $kT) in the valence band. The distribution in k space is predetermined, and the use of the k selection for the interband transition shows that only electrons well below the Fermi level can be used. Thus the luminescence originating from a degenerate band resembles that below degeneracy, despite the difference in absorption. At the donor and acceptor densities present in the usual efficient diodes, several times loL8per cm3, both donors and acceptors have formed impurity bands and the donor band, certainly (and perhaps also the acceptor band), has merged with the conduction band (and the valence band) [Fig. 12(c)]. The net result is an apparent shrinkage of the band gap, if that term still has any meaning. This process has been studied optically in Ge,39*'63*'64 in I I I S ~ , 'and ~ ~ in InAs.'66-'68 A number of theoretical approaches have been taken to describe the resultant states, taking into account electronimpurity and electron-electron interaction^.'^^*'^^^'^^-^^^ The reader is also referred to the chapter by Bonch-Bruevich entitled "Effect of Heavy Doping on the Semiconductor Band Structure" in Volume 1 of this series. Results are generally applicable only in the region of extreme degeneracy, W. Kaiser and H. Y. Fan, Phys. Retr. 98, 996 (1955). J. I. Pankove a n d P. Aigrain, Phys. Rev. 126. 956 (1962). 164 H. S. Sommers, Phys. Rec. 124, 1101 (1961). 1 6 5 P. Aigrain and J. des Cloizeaux, Compt. Rend. 241, 859 (1955). 16' F. Stern and R. M. Talley, Phys. R m 100. 1638 (1955). 16' F. Stern, J . Appl. Phys. 32, 2166 (1961). F. Stem and J. R. Dixon. J . Appl. Phys. 30,268 (1959). 1 6 9 R. H. Parmenter, Phys. Reti. 97, 587 (1955). 170 M. Lax and J. C. Phillips, Phys. Rev. 110, 41 (1958). P. A. Wolff, Phys. Rev. 126. 405 (1962). "'E. 0. Kane, Rpt. intern. Conf Phys. Semicond., Exeter, 1962 p. 252. Inst. of Phys. and Phys. SOC.,London, 1962. 1 7 3 E. 0. Kane, Phys. Rev. 131. 79 (1963). 174V. L. Bonch-Bruevich, Rpt. Intern. Cotzl. Phys. Semicond., Exeter, 1962 p. 216. Inst. of Phys. and Phys. SOC.,London, 1962. 17' V. L. Bonch-Bruevich, Sooiet Phys.-Solid State (English Transl.) 4. 1953 (1963) [Fiz. Tuerd. Tela 4, 2660 (1962)l. 17' V. L. Bonch-Bruevich, Sot'iet Phyx-Solid Stare (English Transl.) 5, 1353 (1964) [Fiz. Tcerd. Telu 5, 1852 (1963)l. 1 7 7 V. L. Bonch-Bruevich, and R. Rozmdn, Souiet Phys.-Solid State (English Transl.) 5, 21 17 (1964) [Fiz. Twrd. Tela 5, 2890 (1963)l. 16'

163

330

M. GERSHENZON

and none of the theories have been completely confirmed experimentally. The situation is worse in the region of weak degeneracy. The statistical theory of Kane which is probably fairly accurate at high d ~ p i n g ' ~ ~ , ' ~ ~ predicts a Gaussian shape to the density of states, tailing off into the forbidden gap. Tails appear on both the conduction band and the valence band in material containing only donors or only acceptors. A perturbationmoment calculation showed that these results were surprisingly accurate at much lower doping levels.'78 The Gaussian shape has been roughly confirmed by the observation of the excess (valley) current in tunnel diodes in Si26*27 and in G ~ A s , which ' ~ ~ is due to tunneling into band tail states at biases higher than those where the normal bands are juxtaposed. Redfield has taken another approach, considering tunneling transitions (the Franz-Keldysh effect'80*'81) between conduction and valence band states under the influence of the large local electric fields due to ionized impurities leading to an absorption edge shift to lower energies.182-'85 Morgan has pointed out that, for GaAs, the higher hole effective mass must shrink the band gap more in p-type material than in n-type material (up to 0.1 eV).lS6 Therefore the barrier for injecting electrons into the p side of a p-n junction is reduced, thus favoring injection and recombination on the p side. Lasher and Stern have by-passed some of the above difficulties by noting that the deeper states in the impurity band tail, corresponding to the closer spaced impurity clusters of Kane,'73 are described by wave functions containing a wide range of k values.'87 For such states the k-selection rule cannot be applied, and transitions between all states are allowed. 16. ABSORPTION

In relatively pure crystals (less than several times 10l6 shallow impurities per cm3), Sturge first showed clearly that the measured absorption edge agrees with the theory of E l l i ~ t , for ~ ~interband ,~~ absorption in a directgap material, with allowance for free exciton formation. 5 1 A binding

'

"'E. 0. Kane, Phys. Rev. 131, 1532 (1963). R. P. Nanavati, Proc. I E E E 52, 869 (1964). W. Franz, Z . Naturforsch. 13a, 484 (1958). ''I L. V. Keldysh, V. S . Vavilov, and K. I. Britsin, Proc. Intern. Conf Semicond. Phys., Prague, 1960 p. 824. Academic Press, New York, 1961. D. Redfield, Phys. Rev. 130,914 (1963). l a 3 D. Redfield, Phys. Rev. 130, 916 (1963). l a 4 D. Redfield, Solid State Commun. 1, 151 (1963). D. Redfield, Trans. N . Y Acad. Sci. 26, 590 (1964). T. N. Morgan, Bull. Am. Phys. Soc. 9, 77 (1964). G. Lasher and F. Stern, Phys. Rev. 133, A553 (1964).

"'

13.

RADIATIVE RECOMBINATION IN THE

111-v

COMPOUNDS

331

energy of 0.0034eV was deduced for the free exciton. These results have been confirmed by other w o r k e r ~ . ~ ~ , ' ~ ~ - ' ~ ' In doped samples, there is some disagreement about the behavior of the absorption edge as a function of doping level. This effect has been studied by Hill,'** by Braunstein et al. (see Fig. 13),le9by Kudman and Vieland,'92 by Turner and Reese,'" and by Luc~vsky.'~'Hill's purest sample almost reproduces Sturge's data. At 77°K the edge moves to higher energies with

I30

1.35

1.40

145 ev

FIG. 13. Room temperature absorption curves near the band edge for GaAs crystals doped with donors and with acceptors as shown. (From R. Braunstein et al., Ref. 189.)

'*

19’

D. E. Hill, Phys. Rev. 133, A866 (1964). R. Braunstein, J. 1. Pankove, and H. Nelson, Appl. Phys. Letters 3, 31 (1963). W. J. Turner and W. E. Reese, J. Appl. Phys. 35,350 (1964). G. Lucovsky, Appl. Phys. Letters 5, 37 (1964). I. Kudman and L. Vieland, J. Phys. Chem. Solids 24, 967 (1963).

332

M. GERSHENZON

shallow donor doping (Te, Se, Sn, Si, but S is anomolous) and roughly fits a Burstein-Moss shift, but with an unrealistically high effective mass.'" The fit is worse at higher doping levels, due perhaps to the nonparabolicity of the bands, but more likely to the effective shrinkage of the gap discussed above, which occurs simultaneously. At room temperature the latter is important even at low donor densities. In p type material the gap shrinkage predominates at room temperature and competes with the Burstein-Moss effect at 77" in Hill's data, but in the other data (Fig. 13) the Burstein-Moss shift still dominates. Lucovsky has pointed out that the absorption edge is also dependent upon the degree of compensation in the crystals, compensation increasing the absorption coefficient up to a factor of five, as seen in Fig. 14.19' I- n

2- n An

144

145

146

147

148

- 8x10"

--

149

7x10'' 10l6

150

T

'

151

Photon energy (eV)

FIG. 14. The absorption edge at 77°K in compensated GaAs samples. (From G. Lucovsky, Ref. 191.)

The shapes of the curves in Fig. 13 merit comment. At high absorption they resemble band-to-band transitions, and presumably arise from such unperturbed states far above the band minima. Below the effective edge

13.

RADIATIVE RECOMBINATION IN THE 111-V

COMPOUNDS

333

there is additional absorption which is partly free carrier absorption, but probably includes some band-tail and some deep state transitions. Such data remain to be quantitatively interpreted, but some remarks will be made below concerning the exponential behavior of the absorption coefficient with energy in Lucovsky's data in Fig. 14. Sturge observed a kink in the absorption edge at low energies which he attributed to absorption by some unknown i m p ~ r i t y . ' ~Gutkin ' et al. observed a similar kink in the photovoltaic response of a diode (which should reflect the absorption spectrum).1 9 3 eV wide) has been A series of about 25 very sharp lines (2 x observed in absorption in very pure n-type material (7 x 10'5cm-3) within several hundredths of an electron volt of the band gap at 2.1 K.’94 These were interpreted as due to donor-acceptor pair transitions as seen in Gap. In view of the remarks made earlier, this interpretation is not quite clear, particularly since the lines were observed in absorption despite the fact that they should have very low oscillator strengths. 1 7.

PHOTOLUMINESCENCE

Nathan and Burns examined the photoluminescence of relatively pure n-type crystals at 4.2 K.’95 A sharp emission line, 0.003 eV wide at 1.5143eV, was equated by them to the free exciton transition. This is the satellite just above the line labeled "A" in Fig. 15, which is a spectrum of a crystal doped with oxygen. Benoit a la Guillaume and Tric probably observed the same line at 20°K at 1.5180eV.'96 Nathan and Burns attributed the discrepancy in energy (0.0027 eV) between this line and the free exciton peak at the absorption edge observed by Sturge (1.517 eV at 21°K) to a questionable surface damage correction used by Sturge. (The band gap increases only slightly between 21" and 4.2"K). However, as Sarace et aZ. indicate, proper use of the principle of detailed balance might remove this d i ~ c r e p a n c y The . ~ ~ emission line might also conceivably be a boundexciton transition involving some very shallow states. However, the line appears in all samples and is, therefore, in all probability, the intrinsic exciton. The position of the line closely follows the absorption edge as a function of temperature, at least to 200"K, indicating that the transition does not turn into free electron-free hole recombination at the higher temperatures. A. A. Gutkin. M. M. Kozlov. D. N. Nasledov. and V. E. Sedov, Soriet Phys.-Solid State (English Transl.) 5, 2654 (1964) [Fi;. Tcerd. Tela 5, 3617 (1963)l. 194 R. F. Schaufele, H. Statz, J. M. Lavine, and A. A. Iannini, Appl. Phys. Letters 3, 40 (1963). l g 5 M. I. Nathan and G. Burns, Phys. Reo. 129, 125 (1963). 196 C. Benoit a la Cuillaume and C. Tric, J . Phys. Chem. Solids 25, 837 (1964).

193

334

M. GERSHENZON

eV wide, lying 0.0015 eV below the A very sharp emission line, 5 x free exciton line (line A in Fig. 15) varies in intensity relative to the free exciton line, as the square root of the oxygen pressure under which the crystal was grown, and, therefore, linearly with the oxygen content in the crystal. Presumably, this is the decay of an exciton bound to an oxygen center. In crystals grown under high oxygen pressures, an additional weak line appears 0.0016 eV above the free exciton. This is not understood.

-

6

5 4 c h c

’Gu 3 e, >

;02 -

B I

0 1.40

I42

I44

1.46

1.48

1.50

1.52

Photon energy (eV)

FIG. 15. Photoluminescence of GaAs doped with oxygen at 4.2”K. The high energy satellite above the line labeled “A” is the free exciton. (M. I. Nathan and G. Burns, Ref. 195.)

Nathan and Burns have also observed a band, 0.022 eV below the free exciton line, at 1.492eV at 4 K which is replicated by emission of LO phonons (the B lines of Fig. 15).19’ This is correlated with a bump on the absorption edge and may be a donor-acceptor recombination band. Benoit a la Guillaume and Tric have reported bands at 1.4937 and 1.4580eV at 20°K.’96 These are presumably the “B” band reported by Nathan and Burns and its first LO phonon replica (phonon energy: 0.0364 f 0.0005 eVi97, although Benoit A la Guillaume and Tric believe they are transitions between the conduction band and two unknown acceptor levels at 0.024 and 0.060 eV.196 It is interesting to note here that, although exciton processes play an important role in these fairly pure crystals, they cannot be important in heavily doped material. Using the criterion that, when the Debye shielding length due to the free carriers becomes shorter than the unperturbed exciton (bound or free) Bohr radius, excitons cannot exist, Casella has shown

13.

RADIATIVE RECOMBINATION IN THE 111-V

335

COMPOUNDS

that, in the usual moderately heavily doped GaAs p-n junctions, excitons cannot form.'97 In addition, the maximum exciton concentration that can exist in GaAs is less than that needed to observe a condensation of these Bose-Einstein particle^.'^'-'^^ The dependence on doping of photoluminescence near the edge at 77" has been examined by Nathan and Burns,200by Nathan et ~ l . , by ~~' and by Leite et ~ 1Nathan . et ~ al. ~(Fig. ~16) find a weak luminescence band

Te Doped Eg

r.

D

I

77°K

0

d

1.50 0

Diodes

E

-

r\

X

Z n Doped

0 0

a 1.48 \ 1

1.46

1.44 I

Iot5

I IIIllil

lo=

I

I1111111

I

I1111111

10"

I

lo'*

I I111111

I

1 IIIll

toi9

lozo

Carrier concentration (cm-3)

FIG. 16. Position of the photoluminescence edge emission peak at 77°K in n-type Te and in p-type Zn doped crystals as a function of doping. The diode electroluminescence peak from Zn-diffused diodes prepared on Te doped substrates is also shown as a function of the Te concentration (From M. I. Nathan et al., Ref. 201.)

at 1.510eV, fairly close to the normal band edge in n-type material (Te) which begins to shift to higher energies with doping, at a donor concentration of 1OI8cmV3.Hill's results are similar and parallel the shift of the absorption edge at a constant absorption level, indicating that the transition is from the valence band to very shallow donor states or to the conduction band itself. At higher doping levels the Burstein-Moss shift dominates although the emission band does not shift as rapidly as the edge. As discussed earlier, the shift in luminescence should lag the absorption edge shift.

-

19'

R. C. Casella, J . Appl. Phys. 34, 1703 (1963).

19'

R. C. Casella, J . Phys. Chem. Solids 24, 19 (1963).

J. M. Blatt, K. W. Boer, and W. Brandt, Phys. Rev. 126, 1691 (1962). M. I. Nathan and G. Burns, Appl. Phys. Letters 1, 89 (1962). '01 M. I. Nathan, G. Burns, S. E. Blum, and J. C. Marinace, Phys. Rev. 132, 1482 (1963). ' 0 2 R. C. C. Leite, J. E. Ripper, and P. A. Guglielmi, Appl. Phys. Letters 5, 188 (1964). 199

336

M . GERSHENZON

In Zn-doped p-type samples, the emission is much more intense.”’ At low doping levels its energy is 1.48 eV (Fig. 16) corresponding to a transition from the conduction band to the discrete zinc acceptor level. Above 5 x l O ” ~ m - the ~ peak shifts rapidly to lower energies (1.44eV at 7 x loL9cm-3).200*20’Hill’s data are similar but not identical. In this case the emission peak does not follow the corresponding absorption curve at 77O, but is rather closer to the absorption behavior at room temperature. Hill does not observe the increase in efficiency of the p-type emission peak over that in the n-type material. In any case, the merging of the impurity band with the valence band, and consequent depression of states into the gap, is the probable explanation of these shifts. The doping level at which merger occurred was noted by Nathan and co-workers as the density above which carrier freezeout no longer occurred. This was 8 x 1OI6 ~ r n for - ~ Te and 2 x lo’* cm-3 for Zn.’O’ The photoluminescence data of Leite et ~ l . , ’ ~ ’ indicating that the emission bands shift to higher energy with increasing excitation in compensated material, will be discussed in a later section describing the bandfilling model as applied to junction electroluminescence.

-

18. CATHODOLUMINESCENCE

A number of reports appeared in the summer of 1964 pertaining to excitation of GaAs by means of electron beams. Presumably, most of these were aimed at attaining laser threshold. In fact, Hurwitz and Keyes did observe stimulated emission at 1.47 eV, at 4.2”K, in heavily doped, p-type material.’03 Most workers, usually using beam energies below 100 keV to avoid radiation damage, found emission peaks very close to the band edge, with decay times of lo-’ sec or less, and peaks at energies of several tenths of an electron volt lower, with decay times in the 10-6-10-7sec range.’23*’24y204 With electron energies sufficient to cause extensive damage (30 MeV), only the slow, low energy luminescence appeared.205,206 Using 15-keV electrons, Cusano examined the emission peaks at 77” and 298°K as a function of donor and acceptor doping.’” The results, shown in Fig. 17, strongly corroborate the photoluminescence data of Fig. 16. At 77°K in lightly doped crystals, two bands appear at 1.507 and 1.483 eV, the former dominant in the n-type crystals and the latter dominant in p-type crystals. Their half-widths are about $ kT, as expected for carriers C. E. Hurwitz and R. J. Keyes, Appl. Phys. Letters 5, 139 (1964). H. Flicker and J. A. Baicker, Bull. Am. Phys. SOC.9, 446 (1964). ’05 D. M. J. Compton, G. T. Cheney, and J. F. Bryant. in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 235. Dunod, Paris and Academic Press, New York, 1965. * 0 6 J. F. Bryant, G. T. Cheney, and D. M. J. Compton. Bull. Am. Phys. SOC.9, 446 (1964). 07 D. A. Cusano, Solid State Commun. 2,353 (1964). 03

204

13.

RADIATIVE RECOMBINATION IN THE

III-v

337

COMPOUNDS

that are free in one band. Cusano interpreted the 1.507-eV transition in n-type crystals as arising from transitions between donors ( - 6 meV below the conduction band) and free holes in the valence band. At high donor densities, the donors form a tail on the band edge, and in addition,

0

Cd,Zn p-Type

2' I 0 a, a

)

1.36

I

1016

I 10"

I 1018

I

1ot9

,

I

I I oZ0

Donor (or acceptor) concentration in cm-3

FIG. 17. Peak positions in cathodoluminescence at 77 and 300°K of the edge emission and of a deeper level, as a function of doping in n-type and in p-type GaAs crystals. (From D. A. Cusano. Ref. 207.)

degeneracy must lead to a Burstein-Moss shift. Again, as discussed under photoluminescence, the emission peak did not shift as rapidly as the absorption edge. In p-type crystals the 1.483eV emission is believed to arise from the recombination of a free electron with a hole bound to an acceptor lying 30 meV above the valence band. Again, as in photoluminescence, the peak shifts to lower energies as the acceptor density is increased. In a novel approach to cathodoluminescence, Wittry and Kyser have used the electron probe to excite luminescence in GaAs crystals, utilizing a photodetector as the output for the electron probe display.208 This allows the luminescence to be studied in microscopic detail on any crystal face, a technique which should prove quite useful in determining the spatial origin of the luminescence. 208

D. B. Wittry and D. F. Kyser. J . Appl. Phys. 35, 2439 (1964).

338

M. GERSHENZON

19. JUNCTION LUMINESCENCE a. Preliminary Comments As we will show, there appear to be several different emission mechanisms contributing to the luminescence close to the band edge in p-n junction electroluminescence. Many reports in the literature contain insufficient information to permit proper classification. One emission band, however, appears to dominate the recombination spectrum at 77°K in moderately heavily doped diodes (1017-1019~ m - ~ This ) . band appears in a wide variety of junctions-diffused, alloyed, epitaxially grown ; prepared from the donors S, Te, C, Si, Ge, Sn, and the acceptors Zn, Cd, Be. Most common are diodes prepared by diffusion of Zn into crystals containing 5 x 10" to 5 x 10" Te atoms/cm3. It is this emission band which appears to be responsible for most of the stimulated emission described at 77" and below. We shall show below that this emission is probably due to a band-filling mechanism. We will then assume, somewhat arbitrarily, that unless known otherwise, all diodes prepared in the above manner emit light by this mechanism. We shall be describing emission bands lying between 1.47 and 1.51eV at 77°K and below. An examination of Fig. 14 shows that, for material doped with shallow impurity concentrations greater than 10l8 ~ m - ~ , selective reabsorption of the emitted light within the diode structure can be a serious problem. The simple Burstein-Moss shift to higher energies in n-type material actually reduces the absorption coefficient in the spectral range of interest. However, the edge moves to lower energies in p-type crystals, making absorption a serious problem in this region. The problem is compounded because the index of refraction of GaAs is very large, 3.6 at 1.48 eV and 103"K,209and, therefore, the critical angle for total reflection at a GaAs-air interface is small, -16". Thus, depending on the diode geometry, an average emitted ray may traverse the crystal many times before emerging. This problem is most severe at room temperature, and for the thermal injection mechanism (discussed below) where the emission occurs at 1.50 to 1.51eV at low temperatures. Both Carr and Biard'" and Sarace et ~ 1 have . shown ~ ~ conclusively that the emission peak becomes grossly distorted, and new "ghost" peaks can emerge. Carr and Biard put a quarter-wave dielectric film on the back of a diode to eliminate reflection at the back surface. They observed the light emitted through the front surface. When an external mirror was used to reflect the,light emerging from the rear back through the structure, the spectrum (observed from the front) was completely altered, as shown in Fig. 18. The altered spectrum

-

209

D. T. F.Marple, J . Appt. Phys. 35, 1241 (1%4). N. Carr and J. R. Biard,J . Appl. Phys. 35, 2776 (1964).

* l o W.

13.

RADIATIVE RECOMBINATION I N THE

132

1.34

136

138

140

III-v

1.42

COMPOUNDS

144

339

146

Photon energy (eV)

RG. 18. Direct, single pass, emission spectrum from a GaAs diode (A) compared with the transformed spectrum (B) obtained when the light emitted in the back direction is reflected back through the diode. (From W. N. Carr and J. R. Biard, Ref. 210.)

resembles that obtained when no antireflection coatings were used. Sarace et al. performed essentially the same experiment and with similar results. They used a heavily doped p-region on the back of the diode structure to absorb the light emitted toward the rear. When this layer was removed, the light was reflected back through the diode.

b. Quantum Eficiencies External quantum efficiencies in “good” GaAs diodes have generally ranged from 0.5 to 55 % at 77°K and below, and up to about 1% at room

340

M . GERSHENZON

temperature.211-216 In some of these cases, stimulated emission has already begun, and these numbers will be considered more thoroughly in the next chapter. Radiative cw power outputs have been reported as high as 1 W (77°K s p o n t a n e ~ u s ) ~and ' ~ 3.2 W (20"K, stimulated).216When stimulated emission begins there may not be any major change in internal efficiency, although the shorter radiative decay times accompanying stimulated emission could lead to more favorable competition with nonradiative processes. In some cases, such as the four-sided laser structure, there is little change in external quantum efficiencies.'4*218In others, an increase has been attributed to the fact that trapping in modes lying in the junction plane keeps more light from traversing the absorbing p region.2' Since much of the internally generated light is absorbed in the diode structure because of the high reflection at the surfaces and the high absorption coefficient, particularly on the p side, external efficiencies can be increased by reducing either of these two effects ; physically removing most of the p layer,2' coating with antireflecting quarter-wave dielectric films,2179219,220 or using geometries which minimize the number of internal reflections2 20-222 792

c. Injection Mechanisms There are several ways of injecting minority carriers across a p-n junction in forward bias, and at least three distinct mechanisms have been shown to exist in luminescent GaAs diodes. These processes may occur simultaneously in the same diode, their relative importance depending on current density and temperature. Thus, the recombination current as a function of applied bias is a sum of several parallel, independent components, and it is therefore often more useful to plot light output against bias rather than against total current.223In fact, a nonradiative current component, which changes with time and may be due to surface leakage, can dominate the G. Cheroff, F. Stern, and S. Triebwasser, Appl. Phys. Letters 2, 173 (1963). S. V. Galginaitis, J . Appl. Phys. 35, 295 (1964). 2 1 3 J. I. Pankove and J. E. Berkeyheiser, Proc. IRE 50, 1976 (1962). *I4 R. C. C. Leite, J. C. Sarace, and A. Yariv, Appl. Phys. Letters 4, 69 (1964). 215 C. Hilsum, Brit. Commun. & Electron. 10, 450 (1963). 216 W. E. Engeler and M. Garfmkel, J . Appl. Phys. 35, 1734 (1964). ’I7 W. N. Carr and G. E. Pittman, Appl. Phys. Letters 3, 173 (1963). 218 G. Burns and M. I. Nathan, Proc. ZEEE 51,471 (1963). 219 0. A. Weinreich, J . Electrochem. SOC. 110, 1124 (1963). 220 W. N. Carr and G. E. Pittman, Proc. I E E E 52, 204 (1964). 2 2 1 A. R. Franklin and R. Newman, J . Appl. Phys. 35, 1153 (1964). 2 2 2 K. M. Arnold and S. Mayburg, J . Appl. Phys. 34, 3136 (1963). 223 R. J. Archer, R. C. C. Leite, A. Yariv, S. P. S. Porto, and J. M. Whelan, Phys. Rev. Letters 10, 483 (1963). 211

212

13.

RADIATIVE RECOMBINATION IN THE

111-v COMPOUNDS

341

total current, particularly at low bias.224,225Because of series resistance in the diodes, it is often difficult to measure the true junction bias in the range where the light emission is measured, and too often the available data only show the dependence upon total current. Such data are sometimes misleading, since the current may be almost entirely due to another, independent injection mechanism. Petree found that the current-voltage characteristics of several diodes did not change upon neutron bombardment (although Millea and Aukerman found that they do change226), but that the radiative efficiency diminished.227 This is most easily explained by postulating that radiation damage introduced nonradiative recombination sites, which reduced the minority carrier lifetime in the bulk, and therefore reduced the radiative efficiency. However, the dominant injection (or leakage) current was not affected. For most injection mechanisms the current varies exponentially with applied bias. In one of these, thermal injection, the exponential reflects the thermal probability of crossing the junction barrier as reduced by the applied bias, and therefore contains a factor nkT, where n, usually ranging between one and two, depends upon the details of the process. Too often, values of n have been deduced from observations made at a single temperature, when, in reality, the factor nkT did not depend upon temperature at all, and thermal injection was not the dominant injection process. We will broadly divide our discussion of forward-bias emission near the band edge into several parts, based upon the dominant injection mechanism (thermal injection, band filling, and tunneling), and later consider recombination from deep levels and from reverse-biased junctions. We point out again that many experiments will be included under “band filling” because of the doping levels reported, even though the observations, in themselves, do not allow us to determine the injection mechanism independently.

d. Thermal Injection Simple radiative recombination, following the thermal injection of minority carriers, has been described by several ~ b ~ e r ~ e R. J. Archer and R. W. H. Engelmann, ZEEE Solid State Device Res. Conf..,Boulder, 1964, Paper VII-8. 2 2 5 J. R. Biard, J. F. Leezer, and B. S. Reed, I E E E Solid State Device Res. Conj., Boulder, 1964 Paper VII-7. 226 M. F. Millea and L. W. Aukerman, Appl. Phys. Letters 5, 168 (1964). M. C. Petree, Appl. Phys. Letters 3, 67 (1963). W. N. Carr and J. R. Biard, J . Appl. Phys. 35, 2777 (1964). 229 R. L. Anderson, Proc. I E E E 51,610 (1963). D. K. Wilson, Appl. Phys. Letters 3, 127 (1963).

224

’” ’*’

r ~ .

342

M. GERSHENZON

Sarace et al. have prepared step junctions by zinc diffusion in very pure n-type crystals (n = 2 x loi6cm-3).46The emitted light varied with voltage as exp eV/l.l kT, indicating thermal injection and probable recombination on the n side of the junction. The emission peak occurred very close to the band gap, and so great care was taken in the construction of the diodes to avoid internal absorption. As Fig. 19 shows, the shape of the emission 2o

I Diode 35-3A No.23 J.18.7 omps/crn* T=77"K

-

18

'

16

.-

1.48

I49

I50

1.51

1.52

1.53

1.54

1.55

photon energy (eV)

FIG. 19. Junction luminescence at 77°K from a diode designed to prevent multiple internal reflections (solid curve), and the predicted emission spectrum, derived from the measured absorption edge by the principle of detailed balance (dashed curve). (From J. C. Sarace et al., Ref. 46.)

peak was predicted very successfully, at both 77" and 298°K from the measured absorption edge on the n side (almost identical with Sturge's results), using the principle of detailed balance. The peak did not shift with current density. The results also agree with photoluminescence on such n type crystals (Fig. 16). In general, this mechanism dominated in the more lightly doped diodes, and at the higher temperatures, the latter effect

13.

RADIATIVE RECOMBINATION IN THE

III-v

COMPOUNDS

343

probably due to the increase of normal thermal injection with increasing temperature. Carr and Biard228 used somewhat more heavily doped substrates (n = 1-3 x l o t 7cm-3X but also took pains to reduce absorption. Their emission also increased as exp eV/kT, again implying simple thermal injection with recombination beyond the depletion layer. Again the peak did not shift with current, but here the emission corresponded rather well to the photoluminescence on the p side of the junction, indicating electron rather than hole injection. Anderson has shown that the current in some diodes increased as exp eV/nkT with n > 1 implying space-charge r e c ~ m b i n a t i o n The . ~ ~ ~light output, however, was proportional to exp eV/k7', i.e., to a parallel simple thermal injection current which was dominated by the "excess" spacecharge current. Millea and Aukerman have reported similar behavior between 78" and 298"K.226 Wilson has studied the radiative recombination at 77°K and below in step junctions prepared on very lightly doped n-type (1 x 10'6cm-3) material.230 The forward current varied with bias and temperature as exp(eV/2kT) implying thermal injection followed by recombination in the depletion layer for the dominant current component. Two peaks appeared near the band edge at 1.48 and 1.505 eV. The higher lying peak was interpreted as a bound exciton transition involving a shallow impurity state. At low bias the electric field in the space-charge layer is high-1O3-1O4 V/cm. A second-order Stark effect should shift the line to lower energies; the inhomogeneity of the field should broaden the line; and the radiation should be polarized parallel to the field. At low bias the emission was 60% polarized. As the bias increased the polarization ratio decreased, the peak shifted to higher energies, and the width decreased, all as expected as the built-in field diminished with increasing forward bias. The limiting position at high bias was very close to the free exciton line of Nathan and Burns.'95 The emission band at 1.48 eV was broader and occurred with simultaneous this band emission of phonons. At higher donor concentrations (begins to dominate the spectrum and it may be the precursor of the major emission band observed in the heavily doped diodes. It is possible that Wilson's two peaks are simply the peak observed by Sarace et a!., due to recombination on the n side, and the peak observed by Carr and Biard due to p-side injection, although the peak shifts with current reported by Wilson were not observed in the other two cases. Additional experimental information is obviously needed. Several other emission bands near the band gap have been ob~ e r v e d . ~Th ~ese , ~bands ~ ~ ,also ~ ~do ~ not shift with bias. Leite et al. 231

M. I. Nathan, G. Burns, and A. E. Michel, B d l . Am. Phys. SOC.9, 269 (1964).

344

M. GERSHENZON

observed one weak band at 1.49eV at 4.2"K at low currents in diodes prepared on crystals of low S or Te donor concentration^.^^.^'^ The peak was repeated at lower energies by lattice phonons. The peak was easily obscured by the normal band-filling emission at high bias. Nathan and co-workers have reported a similar peak.23 Leite et al. observed another peak in similar diodes at 1.511 eV, also replicated by phonons. This band seemed to agree with photoluminescence on the n side, and may be the peak reported by Sarace et al.46 The peak grows relative to the normal emission band as the temperature increases, and may be the dominant emission at room temperature. This peak is easily absorbed by the p region of the diode. Nasledov and his co-workers also noted this band becoming important near room temperature.232 They interpreted it, together with the normal peak, as arising from simple band-to-band transitions involving a valence band whose maxima are shifted away from k = 0. From the photovoltaic effect, Gutkin et al. deduced the shape of the absorption edge, which they related to the emission peaks, presumably by detailed b a 1 a n ~ e . IThe ~~ deduced absorption edge did not have the form expected for indirect interband transitions, and so they concluded that Nasledov's mechanism was incorrect. More likely, these are the same two peaks considered above in the studies by Sarace et ~ 1 . : ~and by Carr and Biard"' which result from the deformation due to excessive absorption and multiple internal reflections. The peak at 1.505 eV at 77°K was also observed by Ing et al. in doubleinjection negative resistance diodes having a semi-insulating zone between the p and the n regions.233 Again the peak does not shift with current. Emission in similar diodes was reported by Y a r n a m ~ t o . ~ ~ ~ It is worth noting that the current-voltage characteristics and the spectra at room temperature often indicate thermal injection, even in diodes where simple tunneling and impurity band tunneling predominate at low temperatures. Since the dominant radiative mechanism may change between 77" and 300°K in many diodes, any theory for the temperature dependence of the threshold for stimulated emission should take this into account.

-

e. Band Filling At 77°K and below, an efficient band appears in diodes prepared by the diffusion of Zn into 1017-1019cm-3 Te-doped crystals. The same band D. N. Nasledov, A. A. Rogachev, S. M. Ryvkin, V. E. Khartsiev, and B. V. Tsarenkov, Souiet Phys.-Solid State (English Transl.)4, 2449 (1963) [Fiz. Tuerd. Tela 4, 3346 (1962)l. "'S. W. Ing, H. A. Jensen, and B. Stern, Appl. Phys. Letters 4, 162 (1964). 234 T. Yamamoto, Proc. I E E E 52, 409 (1964). 232

13. RADIATIVE

RECOMBINATION IN THE

III-v COMPOUNDS

345

occurs when other shallow donors (Se, S, Sn, Si) are ~ s e d , or’ ~ ~ ~ when an acceptor similar to Zn (Cd) is employed.”’ Visual observation indicates that the emission originates near the junction, but on the p side rather than the n side.237 At 77” the emission occurs at -1.48eV with a half-width of 0.014.02 eV. The band center shifts to higher frequencies as the current or forward boltage is increased, before superradiance sets in.28~29~201~223~236.238248 The shape of the band, Fig. 20, is exponential both on the low energy side and on the high energy side. As the current increases, the low energy tail saturates (Fig. 20), and the high energy cutoff moves to higher energies, leading to the shift of the peak.’87.242As the temperature is reduced, the high energy tail falls off more rapidly, and the peak at constant current moves to higher energies, corresponding to the band gap shift with ternperat~re.’~’The position of the peak (hv) follows the forward potential ( V ) as hv = eV k eV,, with I.b equal to several times l o p 3eV (Fig. 21).223,242-244*249 A shift ofup to 0.1 eV with bias has been o b ~ e r v e d . ’ ~ ~ , ’ ~ ~ In this region the forward current depends upon bias as I = I . exp eV/E where E is 8-25 meV (Fig. 21).2233242 The current-voltage characteristic is independent of temperature ( 7 774 except that the voltage at constant current follows the band gap dependence on temp e ra t~ r e .’ ~ ~ *The ’ ~ ’ light output is usually linear in current.8.’ 5 ~ 2 8 , 2 1 8 ~ 2 2 3 , 2 4I2n some cases, at low current (Fig. 21), the light becomes superlinear in current, but in this region, although the emission peak continues to show the same dependence on J. C. Marinace, J . Electrochem. SOC.110, 1 1 53 (1963). G. C. Dousmanis, C. W. Mueller, and H. Nelson, Appl. Phys. Letters 3, 133 (1963). 2 3 7 A. E. Michel, E. J. Walker, and M. I. Nathan, ZBM J . Res. Develop. 7 , 70 (1963). 238M. I. Nathan, Solid-state Electron. 6, 425 (1963). 239 R. N. Hall, Solid-state Electron. 6, 405 (1963). 240 W. E. Engeler and M. Garfinkel, J. Appl. Phys. 34, 2746 (1963). 241 H. Nelson, R C A Reo. 24, 603 (1963). 2 4 2 D. F. Nelson, M. Gershenzon, A. Ashkin, L. A. D’Asaro, and J. C. Sarace, Appl. Phys. Letters 2, 182 (1963). 2 4 3 J. 1. Pankove, J . Appl. Phys. 35, 1890 (1964). 244 J. I. Pankove, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 201. Dunod, Paris and Academic Press, New York, 1965. 245 H. Nelson and G. C. Dousmanis, Appl. Phys. Letters 4, 192 (1964). 246V. S. Bagaev, Y. N. Berozashvili. B. M. Vul, E. I. Zavaritskaya, L. V. Keldysh, and A. P. Shotov, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 149. Dunod, Paris, and Academic Press, New York, 1965. 2 4 ? V. S. Bagaev. Y. N. Berozashvili, B. M. Vul, E. I. Zavaritskaya, L. V. Keldysh, and A. P. Shotov, Soviet Phys.-Solid State (English Transl.) 6, 1093 (1964) [Fiz. Tuerd. Tela 6, 1399 (196411. M. I. Nathan and G. Bums, in “Quantum Electronics” (Proc. 3rd Intern. Congr., Paris. (1964), p. 1863. Columbia Univ. Press, New York, 1964. 249G.C. Dousmanis, C. W. Mueller, H. Nelson, and K. G. Petzinger, Phys. Reo. 133, A316 ( 1964). 235

236

346

M. GERSHENZON

bias as before, the current-voltage characteristic is altered, indicating an “excess” parallel current which is negligible only at high bias.223.242To avoid such difficulties, the light output itself has been used as the measure of “radiative”

PHOTON ENERGY IN

ev

FIG.20. Injection luminescence spectra of a GaAs diode at 20°K as a function of forward bias current in the band-filling range. Note that the intensity scale is logarithmic. Dotted portions of curves are resolution limited. The spike on the spectrum at the highest current marks the onset of stimulated emission. (From D. F. Nelson et al., Ref. 242.)

These data are consistent with a band-filling model indicated schemaInjection is considered into the p tically in Fig. 22.’87~201*223~242~250,251 side of the junction. The residual donors present there have completely merged with the conduction band, forming a band tail extending into the forbidden gap. From the Thomas-Fermi statistical model,’ 73 Kane has calculated that for 3 x lo’* donors/cm’, the density of states 0.05 eV below the conduction band is -lo1* states/cm3/eV, the density of states falling off as the Gaussian of the energy.242The radiative transition is taken between these states and an acceptor impurity band, which may be merged with the valence band. The high quantum efficiency and the linear relation between light and current indicate that this radiative mechanism dominates the recombination. Since the current-voltage characteristic is independent of temperature, thermal injection is not important. Injection is by tunneling, in this case ’”J.

I. Pankove, Phys. Rev. Letters 4,20 (1960). G. Lucovsky, Bull. Am. Phys. SOC.8, 110 (1963).

13.

RADIATIVE RECOMBINATIONS

r~

THE

111-vCOMPOUNDS

347

1.50

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B C

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1.35

IO-~

IO-~

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FIG.21. (a) The emission peak (in eV) for a diode at 20°K is plotted against the diode current (on a logarithmic scale) (circles) together with the current-voltage characteristic as measured directly (squares) and as corrected for series diode resistance (triangles). The emission peak stops shifting at the highest currents when stimulated emission begins. (b) The corresponding emission intensity for different viewing directions and with different polarizations as a function of current. The superlinearity at the Lowest currents is due to an excess current component which dominates the total current in this range and does not lead to radiative recombination. This also causes the peak of the emission in (a) to shift very rapidly in this range. (From D. F. Nelson et al., Ref. 242.)

presumably by an impurity-band hopping process, electrons tunneling between donor sites in the band tail across the junction (Fig. 22). The radiative lifetime has been estimated from several models as 10- l o sec and has been measured from the build-up time to invert the population as a

-

348

M. GERSHENZON

function of current as 2 x lop9sec252or more directly as 2 x lo-’’ s ~ c . ~ ’ ~ Thermal equilibration times within the band tail have been estimated by Kane as lo-” sec, much shorter than the radiative lifetime.242Thus the injected carriers are thermally equilibrated in the band tail as they await recombination. The band states are therefore filled to a quasi-Fermi level, which for a tunneling mechanism is simply a projection of the Fermi level from the n side of the junction (Fig. 22).

-

V

hv>

X SPACE

I

h SPACE

FIG. 22 GaAs diode at high forward bias. Schematic representation of injection from degenerate n-type material into a p-region characterized by an impurity tail on the conduction band. The applied bias (V) is the difference between the electron Fermi level (5,) (extended into the p-region) and the hole Fermi level (t,,).The latter may lie below the top of the valence band, and the acceptors have probably formed an impurity band, perhaps already overlapping the valence band. The band structure and the filled states in the recombination region on the p-side are shown to the right.

If the radiative lifetime is independent of energy in the band tail, the shape of the spectral band is given by the increase of the density of states in the band tail with energy, and is cut off at high energies by the Fermi function. The maximum density of states occurs near the Fermi level, and therefore the peak of the emission, which arises from these states, should have energies close to the separation between this electron quasi-Fermi level and the hole Fermi level. This separation is just the forward bias. Thus hv equals eV, as observed. The low energy tail should saturate with

’’’ K. Konnerth and C. Lanza, Appl. Phys. Letters 4,120 (1964). G. Winstel and K. Mettler, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 183.Dunod, Paris and Academic Press, New York, 1965.

253

13.

RADIATIVE RECOMBINATION R\I THE 111-V COMPOUNDS

349

bias as the states are filled and should be independent of temperature, as observed, and the high energy, Fermi cutoff should fall off exponentially with a slope of about kT, also as ~bserved.’~’The exponential low energy tail implies an exponentially decreasing density of state^'^^*'^' as opposed to the predicted Gaussian fal1-0ff.l~~ Note that the absorption edge in compensated material (Fig. 14)191 may also imply an exponential tail. The exponential slope of the low energy tail should be the same as the exponential slope of the current as a function of bias in the linear lightcurrent range, since both measure the filling of the band. A small discrepancy between the slope of the spectrum and the slope of the shift has been attributed to the failure to include the spatial fall-off of the electron quasi-Fermi level. 8 7 . 2 4 2 The peak position depends more accurately on bias as hv = eV + V,. V, arises from the fact that the peak is the maximum of the product of the density of states and the Fermi functions (although the acceptor-valence band structure may contribute). Lasher and Stern have pointed out that V, may be positive or negative depending on the relation between the slopes of the density of states and of the Fermi f ~ n c t i 0 n . ISince ~ ~ the latter depends upon kT, V, should also depend upon kT, and when the two slopes are equal, the emission band should become very broad, an effect that may already have been observed.lS7 At high current densities, stimulated emission begins, the radiative lifetime decreases, and the band no longer fills to higher levels, so that the spontaneous emission becomes stationary.242 Dousmanis and co-workers have studied the shift of the peak with current as a function of donor doping at 77” and below, and find that the rate of shift and the width of the emission both increase with donor doping density, in agreement with a band-filling Archer et al. made the same ob~ervation.’’~These results were also corroborated by Braunstein et al., who measured the shift of the emission peak at constant current density to avoid the bias-dependent shift, as a function of donor and acceptor densities.’89 The shift roughly paralleled the shift of the absorption edge in homogeneous material. Nathan and Burns compared the position of the band in electroluminescence with their photoluminescence results (see Fig. 16), and in fact with photoluminescence from angle-lapped junctions, showing that the electroluminescence corresponded with the more efficient p-side photoluminescence, rather than with higher energy n-side luminescence, confirmHowever, the expected shift ing that injection was into the p of the emission band with fluorescence excitation was not observed.20’ Leite et al., however, found the expected shift in p-type material, provided that the sample was compensated.’”

’

350

M. GERSHENZON

Lucovsky and Repper also studied the shift as a function of doping, but at high current levels, where they claimed that the band positions no longer depended upon bias.z51*254 At low doping levels the peak does not depend upon concentration, and is interpreted as a transition from the unmodhed conduction band to a discrete acceptor. As the doping increases the transition goes from a donor impurity band tail to the same final state, and at the higher doping levels the transition follows the hole Fermi level into the valence band. These data are not entirely consistent with those of Braunstein and of Dousmanis, in particular, with the cessation of shifting. We have tacitly assumed that band filling was due to rapid equilibration within the deep band-tail states. It is also possible that the states are filled up to the quasi-Fermi level because these are the energies available for electrons tunneling through from the n side at constant energy. In this model, thermal injection into higher levels might yield a spectral peak of higher energy if the electrons do not equilibrate in the deep tail states before they recombine. This might account for the higher lying peak, that does not shift with current, but only occurs at high current densities, and is probably the thermal injection peak discussed previously. Lasher and Stern have noted that the lack of a k-selection rule applies best to these deep band-tail states.‘” They have attempted to fit the experimental data of Nathanz3’ and of Nelson et to such a model using an exponential band tail. Leite et at. have noted that the slope of the shift of the peak with bias is not related to the effective width of the junction, as it is for the photonassisted, one-step tunneling model considered in the next ~ection.’~ Bagaev et at., with very similar data, concluded that band filling involves acceptor states on the valence band, rather than a tail on the conduction They also fit a Gaussian distribution for the density of states in the tail to a small region of the peak shift with bias Over a wide range, an exponential is the better Winogradoff and Kessler have clearly shown that the efficient emission band under discussion arises from the p side of the junction and that the p layer must be heavily compensated.255Using heavily doped epitaxial step junctions they were able to observe injection into the n side, into an uncompensated p region, and into a compensated p region, by varying the doping levels. The efficient band appeared during recombination in a region containing 5.2 x 1019 Zn acceptors per cm3 and 2.6 x 10’’ compensating Te donors per cm3. Galeener et a l have investigated the effect of a magnetic field upon the 254

*”

G. Lucovsky and C. J. Repper, Appl. Phys. Letters 3, 71 (1963). N. N. Winogradoff and H. K. Kessler, Solid-state Commun. 2, 119 (1964).

13.

RADIATIVE RECOMBINATION IN THE

111-v COMPOUNDS

351

diode luminescence at 4.2°.256A linear shift with field was expected due to the splitting of the Landau levels if the transition were band-to-band, and a quadratic effect at low fields if the ground state of a hydrogenic donor

0.005 c

>

9-4 0.004 +

.c r

* 0003

h

;

5 0.002 C

9

a'

0.001

0.000

I 1000 2000 300 4000 5000 6000 7000 8ooO 9Ooo Square of the magnetlc field 8' (kilogauss)2 (b)

0.005 c

>

4 0.004 w

4.2OK

-4

c

'c

-5

0.003

h

e

g 0.002 0

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3a 0.001 0.0001

# 1000 2000 3000 4000 5000 6000 7000 8000 9000 Square of the magnetic field B2,(kilogauss)2

FIG.23. The quadratic shift of the edge emission band in GaAs diodes at 4.2"K with magnetic field for stimulated emission (a) and for spontaneous emission (b). (From F. L. Galeener et al., Ref. 256.)

were involved. The observed shift with magnetic field, as shown in Fig. 23, was quadratic at constant current, implicating shallow donors or excitons, although no detailed analysis could be made. F. L. Galeener, G. B. Wright, W. E. Krag, T. M. Quist, and H. J. Zeiger, Phys. Rev. Letters 10, 472 (1963).

256

352

M. GERSHENZON

The shift of the spontaneous emission with hydrostatic pressure has been , ~ ~by ~ Stevenson and coexamined by Feinleib et aZ.,257by F e n n e ~ and workers, 259 who found that the peak shifts with pressure to higher energy, the pressure coefficient being close to that of the normal band gap.88 The calculated change of the pressure coefficient for a shallow level is within the experimental error. The dependence of effective mass upon pressure should, however, lead to a predictable increase in the density of states of the conduction band and therefore to a relative drop in the Fermi level and a consequent change in the shape of the emission. This change was not observed, and Feinleib et al. concluded that a simple conduction band is not involved.257They also noted that the emission peak position continued to follow the applied bias, independent of pressure. Uniaxial stress measurements260-261 agreed with the hydrostatic experiments, although Meyerhofer and Braunstein have noted some nonlinear stress coefficients in some diodes.260 Miller et al. interpreted the splitting of the emission band upon application of an oriented uniaxial stress as due to the splitting of hydrogenic acceptor levels, thereby again implicating the acceptor levels as the end points of the radiative transition.262 Two peaks near the band edge were observed by Kibler et al., in formedpoint diodes.263 They exhibited some unexplained shifts in energy, and superlinearities in intensity, as the current and the direction of observation were altered. Nelson and Dousmanis have observed two peaks from an n-p-p+ structure, the p region being very narrow.245A 1.47-eV band at 77” originates at the p-n junction and a 1.41-eV band at the p-p+ interface. Both bands shift to higher energies with current. Whenever a forward biased junction emits radiation whose photon energy is greater than the applied bias, heat must be supplied by the lattice to conserve energy, and the possibility of refrigeration The limitations on such a process can be derived from the second law of thermodynamics, or they can be deduced directly from the Fermi-Dirac statistics. It was noted earlier that the peak of the band-filling emission sometimes occurred at energies greater than the applied bias. Dousmanis et al. have applied the first and second laws of thermodynamics to such a process

”’J. Feinleib, S. Groves, W. Paul, and R. Zallen, Phys. Rev. 131, 2070 (1963). G. E. Fenner, J . Appl. Phys. 34, 2955 (1963). M. J. Stevenson, J. D. Axe, and J. R. Lankard, IBM J . Res. Develop. 7, 155 (1963). 260 D. Meyerhofer and R. Braunstein, Appl. Phys. Letters 3, 171 (1963). 261 F. M. Ryan and R. C. Miller, Appl. Phys. Letters 3, 162 (1963). 2 6 2 R. C. Miller, F. M. Ryan, and P. R. Emtage, in “Radiative Recombination in Semiconductors’’ (7th Intern. Conf.), p. 209. Dunod, Paris and Academic Press, New York, 1965. 263 L. V. Kibler, C. A. Burrus, and R. F. Trambarulo, Proc. IEEE 52, 850 (1964). 264 J. Taw, Czech. J . Phys. 7 , 275 (1957). 259

13.

RADIATIVE RECOMBINATION IN THE 111-V COMPOUNDS

353

(neglecting all other parallel recombination processes) in terms of the lattice temperature (the heat source) and the effective photon temperature (the sink), the latter defined by the photon populations of the electromagnetic modes.249 They predicted that the energy discrepancy between the emitted radiation and the bias can increase linearly with temperature, but must always approach zero at high bias. They presented data on some diodes which behaved in just this way, and they therefore concluded that the process was indeed being limited by the second law. have concluded that the nonand Nathan et Both Leite et shifting, thermal injection emission peaks that they observed at photon energies well above the applied bias (at low bias) violated this condition. The excess energy could not come as thermal energy from the lattice, but might perhaps arise from the junction field or from a three-particle (Auger) process.

f: Tunneling To explain the bias dependent shift of the normal (band-filling) emission considered above, Pankove suggested a photon-assisted tunneling process, wherein electrons and holes near the Fermi levels in their respective bands can, because of the junction field, tunnel into the junction into virtual states from which they recombine radiatively." This is the inverse of the FranzKeldysh effect for absorption in an electric field.i64 This model does not predict many of the properties of the slow-shifting, band-filling emission. However, another band which appears at 77" and below can be explained by this type of m e ~ h a n i s m . ~ ~ , ~ The band appears at low bias and shifts rapidly with voltage to higher energies (Fig. 24). A shift of up to 0.3 eV has been At higher bias the normal slow-shifting (bandfilling) band appears and completely dominates the spectrum. Both peaks can be resolved at intermediate bias. This rapidly moving band is very broad and occurs at a bias where several injection mechanisms compete, and so the integrated light intensity serves as a measure of the current carried by this process.223 Leite et al. calculate the probability of tunneling through the field for carriers at the Fermi level (Fig. 25).29 The probability depends on the barrier height at mid-junction [ ( E - hv)/2 in Fig. 251, the junction field, and the junction width. The light yield, proportional to this probability, is then expected to increase as expSV. S should be independent of temperature, but inversely proportional to the junction width. The energy at the peak should always lie close to the applied bias, and the band should not saturate on the low energy side. These predictions were all confirmed L472239243,265

*''

R. J. Archer and J. C. Sarace, Bull. Am. Phys. SOC.8, 310 (1963).

354

M. GERSHENZON

(see Fig. 24). Moreover, from an observation of interference between the light generated at the junction and that portion of it reflected back from the surface of the diode, it was clearly established that the effective width of the recombination region was < 500 A.223 The normal slow-shifting 107-

0.3a7

-

0

Io6

7 o

-

hv v s intensity

v

A V

v s intensity v s current

-

lo5 -

-

lo2

10

E I

.-c F

E 3

0

10-I

I.4

1.3

1.5

V or hv (eV)

FIG.24. The integrated intensity of the edge emission of a GaAs diode at 77 K vs the photon energy at the peak and vs the applied forward bias. The current-voltage relation is also shown. At low bias a large nonradiative excess current dominates. The current-voltage behavior at high bias is used to determine the voltage drop due to diode series resistance and the bias is correspondingly corrected in the upper curves. The low current, rapidly shifting peak is due to photon-assisted tunneling and the curve at higher currents above the break is due to band-filling. (From R.J. Archer er a/., Ref. 223.)

13.

RADIATIVE RECOMBINATION IN THE 111-V COMPOUNDS

355

band in the same diode did not show such interference; it exhibited saturation on the low energy side, and its exponential slope did not vary linearly with the zero bias depletion layer width, thereby clearly distinguishing it as a different mechanism.

FIG.25. Direct photon-assisted tunneling at a narrow p-n junction, degenerate on both sides. Electrons near their Fermi level, and holes near theirs, have the largest spatial extension of electron and hole wave functions, beyond the n-side, and the p-side, respectively. These have their greatest overlap near the center of the transition region. (From R. C. C. Leite et al., Ref. 29.)

Leite et al. have demonstrated that virtual states are not required for this transition, since the wave functions for electrons and holes overlap in the transition region, and the transition conserves crystal momentum.29 P a n k o ~ also e ~ ~correlated ~ the slope of the shift with the junction width (his peak hv,), but in contrast with Archer et al.223and with Leite et ~ l . , ~ ~ his peak always occurred 30 meV below the applied bias. This he believed was due to termination at an acceptor level of that energy. Hoover has observed light emission, although of unknown wavelength, from forward-biased tunnel diodes (Sn alloys on Zn-doped crystals).266 No light appears in the pure tunneling region since the transitions are horizontal. Light appears at higher bias in the injection region, as expected. It is interesting to note that an excess current appears in the valley and it is accompanied by light emission. This supports one model for the excess valley current, attributed to tunneling to deep band-tail states with subsequent r e ~ o m b i n a t i o n . ~ ~Nanavati .’~ has indeed shown that the excess valley current in GaAs is due to such states.’79 Apropos the band-filling model of the last section, it is interesting to note that when this model was first proposed for the excess current in Si tunnel diodes, the band tails 266

G. J. Hoover, Proc. IEEE 51, 1237 (1963).

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M. GERSHENZON

derived from the current-voltage characteristic depended exponentially on energy,26 but later, to agree with theory, Gaussian tails were invoked.27

g . Deep Levels A number of bands have been observed at energies well below the band 0~ gap during recombination in diode electroluminescence,1~9~2,29*2017214* 230,238.2 54.25 7,26 7-2 7 3 in photoluminescence,2 0 1923 8,27492 7 5 during electron irradiation, 3, 124*204-’ 7s276 by absorption, 90,2 and by ph otoconduct i ~ i t y . ~Recombination ’~ peaks at 1.3, 1.0, and 0.95 eV are commonly observed. Such bands are probably due to recombination at deep levels, occurring perhaps as donor-acceptor pair recombination bands. Nathan and Burns have shown that the deep levels appearing in electroluminescence of Zn-diffused diodes correspond in position and shape with photoluminescence peaks in the n-type substrate, implying injection into the n side.201,238Pure n-type crystals do not show these peaks either in photoluminescence or in diodes prepared on them, nor do they appear in alloyed junctions onto p-type material. Peaks at 0.95 and 0.65eV were correlated with Cy269and with oxygen,274respectively, and at 1.29, 1.06, and 0.98 eV at 77” again with C U . ~ ~ ’ A 0.80-eV band was caused by Cr.275A band at 1.39eV, due to Mn accept o r ~ was , ~ used ~ ~ by Weiser et al., to study the recombination process in negative resistance Zn-Mn double-diffused diodes, above and below a space-charge limited regime.27’-273*278 The pressure shift of some of these peaks is substantially less than the band gap shift.257 Lattice phonon replicas of one such band at 1.35eV have been observed where the peak itself seemed to occur in S, but not in Te-doped diodes.214

’

J. Black, H. Lockwood, and S. Mayburg, J . Appl. Phys. 34, 178 (1963). W. F. J. Hare, M. Gershenzon, and J. M. Whelan, I R E Trans. Electron Deuices ED9, 503 (1962). 269 T. L. Larsen, Appl. Phys. Letters 3, 113 (1963). 270 M. H. Pilkuhn and H. Rupprecht, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 195. Dunod, Paris and Academic Press, New York, 1965. 2 7 ’ K. Weiser and A. E. Michel. in “Radiative Recombination in Semiconductors” (7th Intern. Conf.). p. 177. Dunod, Paris and Academic Press, New York, 1965. K. Weiser and R. S. Levitt, J . Appl. Phys. 35, 2431 (1964). 273 R. S. Levitt, K. Weiser, A. E. Michel, and E. J. Walker, IRE Trans. Electron Devices ED10, 333 (1963). 274 W. J. Turner, G. D. Pettit, and N. G. Ainslie, J . Appl. Phys. 34, 3274 (1963). 2 7 5 W. J. Turner and G. D. Pettit, Bull Am. Phys. Sac. 9, 269 (1964). 276 N. G. Basov and 0. V. Bogdankevich, Soviet Phys. J E T P (English Transl.) 17, 751 (1963) [Zh. Eksperim. i Teor. Fiz. 44, 1 1 15 (1963)l. 2 7 7 A. E. Michel, W. J. Turner, and W. E. Reese, Bull. Am. Phys. SOC. 8, 215 (1963). 278 W. P. Dumke, Bull. Am. Phys. SOC.9, 217 (1964). 267

268

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h. Reverse Bias Emission Michel and Nathan have observed light emission from reverse-biased junctions.279The light comes from breakdown microplasmas and consists of two components. One is a broad band, probably characteristic of intraband radiative relaxation of the hot carriers. The other, however, is a peak of lop4 quantum efficiency, which is linear in current and does not shift with bias. It occurs very close to the normal forward emission band corresponding to high forward bias. It corresponds then to an interband recombination process, necessitating the creation of minority carriers. The efficiency appears quite large if one considers impact ionization or Auger processes as the source of these carriers. However, pair production, during multiplication in an avalanching junction, does produce both types of carrier. Normally, the junction field sweeps them out in opposite directions before they can recombine. However, in a direct gap material, like GaAs, the radiative recombination time may be short enough to permit extensive recombination before the field separates them. IV. Other Compounds 20. InSb Although the details of the band structure of InSb are better known than those of any other III-V compound, the small band gap, 0.236eV at 0°K places the recombination radiation in a spectral region where detectability is poor. As a result, InSb has been somewhat neglected in the study of radiative recombination. This is unfortunate, because the few careful investigations that have been made indicate that the recombination results from transitions already well documented from absorption and from magnetooptic data. As a simple, direct band-gap crystal (although the maxima in the valence band lie slightly away from k = 0280),with small effective masses, the recombination spectra should be similar to GaAs. The band-to-band radiative lifetime for intrinsic material at room temperature (ni = 2 x 10'6cm-3) as deduced from the measured absorption edge and the principle of detailed balance is 4-8 x lo-' sec.41-43*281Wertheim showed that in certain samples the minority carrier lifetime near room temperature was close to the radiative lifetime.45 The detailed balance calculations also yield the shape of the emission. This agrees with the observed ~ p e c t r a . ~ *Photoluminescence '.~~ at room temperature yields the band-toband peak with a quantum efficiency of 20%.4,5*7*16 E. Michel and M. I. Nathan, Buif. Am. Phys. Soc. 9, 269 (1964). E. 0. Kane, this series, Volume 1. D. W. Goodwin and T. P. McLean, Proc. Phys. Soc. (London) B69,689 (1956).

'''A.

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Below 77°K radiative recombination spectra have been investigated in photoluminescence,282 in cathodoluminescence,283 and in forward bias injection, both in diffused diodes284*285 and in alloyed The spectra are dominated by an emission band lying at 0.235 eV at 4°K very close to the edge.282,283*286 The line narrows somewhat as the excitation intensity is increased.282This was explained as due to non-equilibration of free carriers at low excitation intensity, leading to a broad range of initial and final states for the transition. At higher excitation levels, however, when more carriers are created, the carriers quickly reach thermal equilibrium because of increased electron-hole scattering, and so the band should be of normal thermal width (narrower).282 These A magnetic field splits the main band into two are, presumably, the two spin substates of the lowest Landau level of the conduction band.282,284*285 The extent of the splitting yields the correct g valuez84*285 and agrees with the magnetooptical absorption of Zwerdling et ~ 1 . ~Both ~ ’ peaks move to higher energy with increasing field, but not with the excitation intensity, and the emission intensity increases linearly with the field, as expected for allowed (direct) transitions between spherical bands (or excitons derived from such bands).285 The relative intensities of the peaks may,284or may not,282be explained in terms of thermalization between the excited states. The magnetic field not only bunches the conduction band states into narrow, discrete Landau levels, lowering the thresholds for stimulated e m i s s i ~ n , but ~ ~also, ~ . ~when ~ ~ perpendicular to the direction of current flow, it shortens the free carrier diffusion length, again lowering the t h r e s h ~ l d . Ho ~ ~wever, ~ . ~ the ~ ~opposite effect, lowering of the efficiency with magnetic field, has also been ~bserved.~” Since the density-of-states effective mass in the conduction band is low, 0.013, degeneracy occurs at fairly low free carrier concentrations. Its effect has been considered on the band-to-band radiative lifetime41-43 and on ”’C. Benoit la Guillaume and P. Lavallard. in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 53. Dunod, Paris and Academic Press, New York, 1965. 283 C. Benoit a la Guillaume and J. M. Debever, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 255. Dunod, Paris and Academic Press, New York, 1965. 284 R. L. Bell and K. T. Rogers, Appl. Phys. Letters 5, 9 (1964). 2 8 5 R. J. Phelan and R. H. Rediker, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 47. Dunod, Paris and Academic Press, New York, 1965. 2 8 6 C. Benoit a la Guillaume and P. Lavallard, Rept. Intern. Conf: Phys. Semicond., Exeter, 1962 p. 875. Inst. of Phys. and Phys. SOC.,London, 1962. C. Benoit il la Guillaume and P. Lavallard, Solid-State Commun. 1, 148 (1963). S. Zwerdling, W. H. Kleiner, and J. P. Theriault, J. Appl. Phys. 32, 2118 (1961). ”’R. J. Phelan, A. R. Calawa, R. H. Rediker, R. J. Keyes, and B. Lax, Appl. Phys. Letters 3, 143 (1964). R. J. Phelan and R. H. Rediker, Proc. IEEE 52.91 (1964).

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extrinsic recombination processes.43 The shape of the band-to-band emission in degenerate diodes at low temperatures clearly reflects the distribution of electrons in the band, from the sharp Fermi level cutoff, well above the bottom of the band, to the tailing-off of states, deep into the normally forbidden gap,286 effects already discussed for GaAs. The peak which produces stimulated emission by electron bombardment lies 0.010-0.015eV above the normal band edge emission.283 This was interpreted as arising from transitions from states near the Fermi level, which is well above the conduction band minimum, in degenerate crystals.283 Near band-gap radiation has also been seen in tunnel diodes at forward bias,291 in p + p n + structures, in which recombination occurred in a large volume,292 and by recombination following impact ionization resulting from the application of high electric fields to uniform ~ r y s t a l s . ~ ~ ~ , * ~ ~ 21. InP Like GaAs and InSb, InP is a direct-gap semiconductor, with a band gap of 1.42eV at 0°K. Electroluminescence near the gap is easily ~ b s e r v e d . ' , ~37, ~ , ~ A number of peaks have been seen in photoluminescence at 6" to 77°K in lightly doped, n-type crystals.295A sharp line, 0.002 eV wide, at 1.416 eV and 6"K, was ascribed to the free e ~ c i t o n , 'and ~ ~ it coincides with the free exciton peak observed in a b s o r p t i ~ n .Another ~ ~ ~ * line, ~ ~ ~0.006 eV wide at 1.374 eV, is replicated several times with decreasing intensity by the LO p h ~ n o n 'and ~ ~ by a TA p h ~ n o n . ' In ~ ~addition, two broader bands at 1.04 and 0.72 eV appear.295 In p-type material only a line at 1.382 eV and a band at 0.95 eV were detected. In Zn-diffused diodes, a band appears at 1:38eV at 4.2", with a halfwidth of 0.01 eV.298-301Th'is follows the band gap to higher temperatures. B. M. Vul, A. P. Shotov, and V. S. Bagaev, Soviet. Phys.-Solid State (English Transl.) 4, 2689 (1963) [Fiz. Tuerd. Tela 4, 3676 (1962)j. 292 I. Melngailis, R. J. Phelan, and R. H. Rediker, Appl. Phys. Letters 5, 99 (1964). 293 N. G. Basov, B. D. Osipov, and A. N. Khvoshchev, Soviet Phys. J E T P (English Transl.) 13, 1323 (1961) [Zh. Eksperim. i Teor. Fiz. 40, 1882 (1961)l. 294 N . G . Basov, B. D. Osipov, and A. N. Khvoshchev, lntern. Symp. Luminescence, Balatonuilrigos, I961 p. 145. Akad. Kudo, Budapest, 1962. *"W. J. Turner and G. D. Pettit, Appl. Phys. Letters 3, 102 (1963). 296 W. J. Turner and W. E. Reese, in "Radiative Recombination in Semiconductors" (7th Intern. Conf.), p. 59. Dunod. Paris and Academic Press, New York, 1965. '"W. J. Turner, W. E. Reese, and G. D. Pettit, Phys. Rev. 136, A 1467 (1964). K. Weiser and R. S. Levitt, Bull. Am. Phys. Soc. 8, 29 (1963). 299 K. Weiser, R. S. Levitt, M. I. Nathan, G. Burns, and J. Woodall, Trans. A I M E 230, 271 (1964). 300 K. Weiser and R. S. Levitt, Appl. Phys. Letters 2, 179 (1963). G. Bums, R. S. Levitt, M. I. Nathan, and K. Weiser, Proc. IEEE 51, 1148 (1963). 291

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It is very similar to the broad edge emission in GaAs. It falls just below the band gap. It is superlinear at low currents and linear at high currents, and has a quantum efficiency of 2-3% at 774 and 40 times less than this at room temperature. The peak shifts to higher energy with increasing current density, shifting more rapidly in the superlinear region. The origin of this emission is presumably the same as that in GaAs (band-filling).

22. InAs InAs is another direct-gap semiconductor with a band gap of 0.41 eV at 0°K. An emission band at 0.410eV, observed at P K , in photoluminescence of both n- and p-type crystals, shifts linearly to higher energy with magnetic field, and has been ascribed to the free exciton.302*303 At slightly higher energies, a shoulder, which can be resolved by a magnetic field, may be due to free electron-hole recombination.302 At 20°196*304 and 77°K,302,303 the emission is a single band, whose shape reflects the filling of the degenerate conduction band, as in InSb, with the peak shifting to higher energies with the free electron concentration. This band also appears in cathodoluminescence.283.305In addition, peaks arising from deeper levels appear in n-type’96,302-304and in p-type302*303 crystals. Zinc-diffused diodes in more heavily doped n-type material exhibit a band, 0.020 eV below the gapY3O6which also shifts linearly with magnetic field to higher energies,3o693o7and is linear, or slightly sublinear, with current.306 Again, the transition probably starts from the lowest Landau level in the conduction band, but probably terminates at a discrete acceptor level, 0.020 eV above the valence band.306 Deeper lying transitions, replicated by the LO phonon, were also observed, and the main peak got broader and shifted to lower energies as the shallow donor concentration was increased.306 The accompanying change of the magnetic field dependence, from linear to quadratic, indicated that donor states (very likely with impurity bands and conduction band-tailing) were now being used.306 Near band-edge emission was also observed from Cd-diffused diodes.308 A. Mooradian and H. Y. Fan, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 39. Dunod, Paris and Academic Press, New York, 1965. 303A.Mooradian and H. Y. Fan, Bull. Am. Phys. Soc. 9, 237 (1964). ’04 C. Tric, C. Benoit a la Guillaume, and J. Debever, Compt. Rend. 255, 3152 (1962). ’O5 C. Benoit B la Guillaume and J. M. Debever, Solid-State Cummun. 2, 145 (1964). jo6 I. Melngailis, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 33. Dunod, Paris and Academic Press, New York, 1965. 307 I. Melngailis, Appl. Phys. Letters 2, 176 (1963). 308 1. D. Anisimova, V. M. Yungerman, and A. V. Kulymanov, Soviet Phys.Solid State (English Transl.) 6, 2036 (1965) [Fiz. Tuerd. Tela 6, 2555 (1964)l. jo2

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23. GaSb GaSb is also a direct-gap semiconductor with a band gap of 0.81 eV at 0 K but here the next lowest conduction band minima, in the (111) directions, lie fairly close,3o9 only 0.05eV above the k = 0 minimum at 4"K.3l o Johnson et al. have studied absorption, photoluminescence, and injection luminescence as a function of t e m p e r a t ~ r e . ~In~ undoped p type crystals (1017~ m - at ~ 4.2"K, ) the direct free exciton peak is observed in absorption, as well as two bound-exciton lines. One of the boundexciton lines also occurs in photoluminescence, in addition to two lower lying bands involving deeper levels. At 77°K one of these deep levels appears (also observed in photoconductivity), together with band-to-band recombination. The latter also occurs at room temperature, as does the second deep level. The band-to-band peak increases with the free hole concentration. In p-type samples, compensated with Se or Te, several new bands appear, some of which are characteristic of pure Se or Te. At high donor doping levels the band-to-band peak shifts to higher energies due to the Burstein-Moss effect. cm-3), Calawa has In alloyed diodes prepared on p-type crystals observed, at 77" and at room temperature, what are apparently the same deep-level bands noted above.312At 77" the band shifts neither with current nor with magnetic field, clearly indicating that it is not a band-to-band transition. The peak is also linear in current, becoming sublinear at high current densities, and the external quantum efficiency is 8.5% at 77". Another deep-level peak occurs in junction regrowth layers containing tin. The peak observed by Braunstein in electroluminescence occurs at a lower photon energy than any of these bands.' Diodes fabricated by the diffusion of zinc into moderately heavily doped n-type crystals behave like equivalent GaAs junctions.313 At 77" a band at 0.70-0.73 eV shifts to higher energy, and narrows as the current increases, and at high currents a new peak appears at 0.79 eV. The shifting peak was attributed to a transition from an acceptor impurity band to a donor level on the n side of the junction, and the high current peak to a valence banddonor level transition.313 Peak shifts with current were also observed in heavily doped Cd :Te grown junctions that apparently lased at higher

current^.^ l4 A. Sagar, Phys. Reo. 117,93 (1960). H. Pillar, J . Phys. Chem. Solids 24, 425 (1963). ’I1 E. J. Johnson, I. Filinski, and H. Y. Fan, Rept. Intern. Con$ Phys. Semiconduc., Exeter, 1962 p. 375. Inst. of Phys. and Phys. SOC.. London. 1962. '"A. R. Calawa, J. Appl. Phys. 34, 1660 (1963). 3 1 3 T. Deutsch, R. C. Ellis, and D. M. Warschauer, Phys. Stat. Solidi 3, 1001 (1963). 'I4C. Chipaux, G. Duraffourg, J. Loudette, J.-P. Noblanc. and M. Bernard, in "Radiative Recombination in Semiconductors" (7th Intern. Conf), p. 217. Dunod, Paris and Academic Press, New York, 1965. 309

310

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M. GERSHENZON

The low density of states in the conduction band and the close proximity of the next higher lying minima (which can become populated at high carrier concentrations, or at high injection levels) should lead to profound changes in the recombination statistics as the Fermi level rises. An attempt to sort these two processes by studying the effect of hydrostatic pressure upon the luminescence-the two types of minima having different pressure coefficients-produced results which could not be interpreted easily.31 24. MISCELLANEOUS COMPOUNDS Luminescence has been studied in some of the remaining 111-V compounds. Their band structures are generally unknown, not even whether they are direct or indirect, and in some cases the band gaps have never been determined. Most of them have not been available in single-crystal form and some, the nitrides, do not even crystallize (normally) with the zinc-blende structure. Several emission peaks below the band gap (2.42eV at room temperature) have been observed in photoluminescence and point-contact electroluminescence of Alp.316 An emission band, replicated at lower energies by optical phonons, has been observed317 close to the recently revised, indirect band edge (2.0eV3'7,318)in BP. Its similarity to the broad donor-acceptor pair edge emission in GaP (also indirect) may imply an identical recombination mechanism. The nitrides are quite ionic, with large gaps, and the tendency has been to interpret luminescence in terms of similar processes in ZnS. Photol u m i n e s ~ e n c eand ~~c ~ a*t~h~o d ~ o l u m i n e s ~ e n c eha ~ ~ve~ been ~ ~ ~ ~studied in GaN (band gap 3.26 eV320).A number of lines and broad bands, near and below the band gap, have been interpreted in terms of crystal d ~ p i n g , ~ ' ~ . ~ ~ ~ and in terms of the temperature and the chemical reaction by which the crystals were p r o d ~ c e d . ~ ~El~ectroluminescence, -~~' photoluminescence, and cathodoluminescence were studied in sintered masses of A1N,322J23 T. Deutsch and B. Kosicki, Bull. Am. Phys. SOC.9, 60 (1964). H. G. Grimmeiss, W. Kischio, and A. Rabenau, J . Phys. Chem. Solids 16, 302 (1960). jl' R. J. Archer, R. Y. Koyama, E. E. Loebner, and R. C. Lucas, Phys. Rev. Letters 12, 538 (1964). j1*C. C. Wan& M. Cardona, and A. G. Fischer, R C A Rev. 25, 159 (1964). 319 H. G. Grimmeiss, R. Groth, and J. Maak, Z . Naturforsch. 15a, 799 (1960). jZo H. G. Grimmeiss and H. Koelmans, Z . Naturforsch. 16, 264 (1959). 3 2 1 M. R. Lorenz and B. B. Binkowski, J . Electrochem. SOC.109, 24 (1962). 322 G. A. Wolff, I. Adams, and J. W. Mellichamp, Phys. Rev. 114, 1262 (1959). 323 I. Adams, J. W. Mellichamp, and G. A. Wolff, Electron. Din Abstr., Spring Meeting Eiectrochem. SOC.,Chicago, 1960 p. 55, Abstr. No. 53. 315

316

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and in mixtures of AlN with A1,03.324,32s A number of sharp emission lines between 4000 and 5000 A were due to molecular nitrogen, presumably excited at crystal-gas interfaces, but at lower energies emission bands Photoluminescence and highcharacteristic of AlN were M any field, dielectric-sandwich luminescence were observed in BN.326*327 lines, independent of the means of excitation, were detected between 2950 and 6500 A.326,327Lehmann has suggested an interpretation based upon the close analogy between the condensed B-N-B . . . rings in the layered graphitic structure and the condensed rings of aromatic hydrocarbons.328 25. ALLOYS Radiative recombination from mixed crystals of the III-V compounds was observed a number of years ago by Wolff ( G a A Q - , and In,Ga, -xP).2*3More recently, the desirability of varying injection laser frequencies, by varying the band gap of the lasing crystals, has led to studies of a number of mixed crystal systems. In the simplest case, in which both binary end members of the ternary phase diagram have the same type of band structure, the band gap varies linearly with composition,328aretaining the same band structure throughout. Examples of these are InPxAs,-,,329 and I ~ , G ~ , - , A S . In ~ ~both ~~~~’ cases zinc-diffused diodes on heavily tellurium-doped crystals produce an emission band at forward bias which closely resembles that of the individual end members (InP, InAs, GaAs), except that the peak position corresponds roughly to the appropriate intermediate band gap. The situation is different for GaP,As,_,, because G a P has indirect (100) minima, whereas GaAs is direct. Both minima shift linearly but at different rates with x (see Fig. 261 and they cross at x = 0.44,3320.43,99 0.3898 at room temperature, and at x = 0.40,990.45, 00at 77°K. On the As side (x < 0.4) the luminescence is similar to that of pure GaAs, but the I. Adams, T. R. AuCoin, and G. A. Wolff, Electron. Din Abstr., Spring Meeting Electrochern. SOC., Los Angeles, 1962 p. 83, Abstr. No. 50. 3 2 5 I. Adams, T. R. AuCoin, and G. A. Wolff, J . Electrochem. SOC.109, 1050 (1962). 3 2 6 S. Larach and R. E. Shrader, Phys. Rev. 102, 582 (1956). 327 S. Larach and R. E. Shrader, Phys. Reo. 104,68 (1956). 3 2 8 W. Lehmann, Electron. Din Abstr., Spring Meeting Electrochem SOC.,Indianapolis, I961 p. 16, Abstr. No. 29. 3 2 8 a Exceptions to strictly linear variations are discussed by Hilsum in his chapter “Some Key Features of 111-V Compounds,” which appears in Vol. 1 of this series. 329 F. B. Alexander, V. R. Bird, D. R. Carpenter. G. W. Manley, P. S. McDermott, J. R. Peloke, H. F. Quinn, R. J. Riley, and L. R. Yetter, Appl. Phys. Letters 4, 13 (1964). 330 I. Melngailis and R. H. Rediker, IRE Trans. Electron. Devices ELHO, 333 (1963). 3 3 1 I. Melngailis, A. J. Strauss, and R. H. Rediker, Proc. IEEE 51, 1154 (1963). 332 G. E. Fenner, Phys. Rev. 134, A1 113 (1964). 324

364

M. GERSHENZON

peak position depends linearly on x,99*100%333-336the peak occurring near This emisthe band gap as determined from absorption mea~urernents.~~' sion band shifts to higher energies with current at 77O, as it does for pure GaAs, presumably for the same reason.334

FIG.26. The position of the emission peak near the band edge in Ga(P,As,-,) crystals at 77°K as a function of composition (x): bathodoluminescence in p-type cr y ~ t al s?~ x -junction eIectrol~minescence,~~ 0-junction electroluminescence.'" *junction elect r o l ~ m i n e s c e n c e ,A~ -junction ~~ electrol~minescence,~~~ +-junction electroluminescence, stimulated e m i s ~ i o n ?A-junction ~ electrolurninescence, at reverse bias,'" For pure GaP the k = (100) band gap is at 2.32 eV 91 and the k = 0 gap is at 2.89?6 and for GaAs the k = 0 gap is at 1.51 l S 1 and the k = 100 gap is at 1.87.'50

An additional broad band is often seen at lower energies.99*'009335*338 The fact that it always occurs 0.40 to 0.47eV below the band gap, as x N. Holonyak and S. F. Bevacqua, Appl. Phys. Letters 1, 82 (1962). N. Ainslie, M. Pilkuhn, and H. Rupprecht, J . Appl. Phys. 35, 105 (1964). 3 3 s J. W. Allen and M. E. Moncaster, Phys. Letters 4, 27 (1963). 336 S. M. Ku, J. Electrochem. Soc. 110, 991 (1963). 337 H. F. Lockwood, S. M. Ku, and J. F. Black, Bull. Am. Phys. SOC. 7 , 537 (1962). 338 S. M. Ku and J. F. Black, Solid-State Electron. 6, 505 (1963). 333

334

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RADIATIVE RECOMBINATION IN THE 111-V COMPOUNDS

365

varies between 0 and 1, has led Pilkuhn and Rupprecht to suggest that it is due to the same type of transition at both ends of the alloy sequence.'" This band is linear in current at low currents, and sublinear at high currents, as opposed to the band edge emission, which is quadratic at first, becoming linear at high currents.338 This distinction, together with the relative increase of the edge emission band over the broad peak at low temperatures, has been taken as evidence that the first is band-to-band recombination, and the second involves an impurity state.338 If it is assumed that the highest energy emission band always lies very near the edge, it is possible to map the band gap variation with composition. This has been done for injection l u r n i n e s c e n ~ e , and ~ ~ ~for ' ~ ~cathodol u m i n e ~ c e n c eFigure . ~ ~ 26 is a summary of peaks, as a function of composition, that have been reported by several investigators. (It also includes some weak peaks from reverse biased junctions that were found by Pilkuhn and Rupprecht to lie well above the energy gap on the P-side of the diagram. They believe them due to recombination involving electrons from the direct minimum of the conduction band. l o o ) The pattern, showing the linear variation of the direct edge with composition on the As side, and the linear variation of the indirect edge on the P side, agrees with that derived from the photoresponse of surface barrier diodes,98 and extrapolation allows the k = 0 edge in Gap, and the k = (100) edge in GaAs, to be determined. At the cross-over point between direct and indirect energy gaps the properties of the crystals can change violently : (1) The normal donor and acceptor ionization energies in GaP are much greater than those in GaAs. Thus, at low temperatures, carrier freeze-out is far more important in Gap. In addition, electron and hole mobilities are much lower for Gap. Both effects lead to a large change in resistance, particularly at low temperatures, when the critical point is crossed, the resistance being much higher when the indirect minima ~ o n t r o 1 . ~ ~ ~ , ~ (2) The pressure coefficient for the k = 0 minimum is positive, while that for the (100) minima is negative.92 Application of hydrostatic pressure to a luminescing diode, with x chosen to be just on the As side, will therefore change its band structure from direct to indirect, with several interesting consequences.340 Below the transition the edge emission peak shifts to higher energy with pressure, following the shift of the direct minimum, whereas past the transition, the peak follows the indirect edge down with pressure. Below the transition the luminescence efficiency is high and the laser threshold is low, whereas at the transition from direct to indirect, the efficiency drops abruptly and lasing ceases. In addition the series resistance N. Holonyak, S. F. Bevacqua, C. V. Bielan, F. A. Corranti, B. G. Hess, and S. J. Lubowski, Proc. ZEEE 51, 364 (1963). 340 T. A. Fulton, D. B. Fitchen, and G. E. Fenner, Appl. Phys. Letters 4, 9 (1964).

339

366

M. GERSHENZON

increases abruptly at the transition"' for the reasons referred to above. (3) The temperature dependences of the two types of minima are also different, so that a change in temperature can invert the lowest lying minimum, a decrease in temperature lowering the direct minimum, relative to the indirect gap. The current-voltage characteristic of a Zn-diffused diode on a crystal just on the P-side was studied as a function of temperature. 3 4 1 As the temperature was reduced below room temperature, the series resistance increased, reflecting the normal freeze-out of carriers in the indirect structure. However near 77" the threshold for direct transitions was crossed, whereupon the series resistance decreased abruptly and the luminescence efficiency rose very rapidly. The emission wavelength here at 77', 6400 A'' 6380 A,''' is presumably very close to the shortest wavelength that can be emitted from Ga(P-As) alloys whose band gap is still direct.

V. Notes Added in Proof In the following, we briefly summarize the pertinent literature that appeared too late to be included in the main body of this article.

26. GaP The decay of shallow donor-shallow acceptor pair recombination has been studied at low temperatures in cathodoluminescence, and the kinetics have been derived and confirmed for several limiting cases.342As expected, radiation damage destroys emission preferentially from distant pairs while leaving close pairs and point defect recombination unaffected.343 An electric field applied during the course of donor-acceptor pair decay increases the overlap between pairs oriented parallel to the field and with the proper sense, thereby enhancing the emi~sion.~"This process might account for some of the electrophotoluminescence effects observed in other semiconductors. The photoluminescence decay of the shallow-acceptor zincdeep-donor oxygen pair band has been studied in detail.345 The nonexponential decay, independent of temperature and photoconductivity at low temperatures, the shift of the pair band to lower energies during the course of the decay, and the dependence of the lifetime only on the more concentrated member of the pair (zinc) provide further proof that the Zn-0 band is a pair band. The competition between distant pairs, close pairs and bound excitons at low temperatures has been discussed.346 A broad band (probably the one at 1.96 eV sometimes observed in electroluminescence) has been observed in photoluminescence along with several sharp lines.'47 An emission peak at 1.65 eV at 103°K has been attributed to

N. Holonyak, S. F. Bevacqua, C. Bielan, and S. J. Lubowski, Appl. Phys. Letters 3, 47 (1963). 342 D. G. Thomas, J. J. Hopfield, and W. M. Augustyniak, Phys. Reo. 140, A202 (1965). 343 J. D. Cuthbert, Bull. Am. Phys. SOC. 10, 595 (1965). 344 K. Colbow, Phys. Reu. 139, A274 (1965). 345 D. F. Nelson and K. F. Rodgers, Bull. Am. Phys. SOC. 10,720 (1965). 346 E. F. Gross and D. S. Nedzvedskii, Souiet Phys.-Solid State (English Transl.)6, 1721 (1965) [Fiz. Tuerd. Tela 6, 2180 (1964)l. 347 E. F. Gross and D. S . Nedzvedskii, Vestn. Leningr. Uniu., Ser. Fiz. i Khim. 19,7 (1964). 348 H. G. Grimmeiss and H. Scholz, Philips Res. Rept. 20, 107 (1965).

341

13.

RADIATIVE RECOMBINATION IN THE

III-v

COMPOUNDS

367

The current-voltage characteristics of diodes fabricated by the out-diffusion of zinc from p-type crystals containing zinc, tellurium and oxygen have been shown to be due to nonradiative space-charge layer recombination, but the Z n - 0 pair band is emitted by parallel injection into the p-side past the depletion layer.349 Another class of diodes, where both the current-voltage characteristics and the light emission (the Z n - 0 pair band) are due to tunneling injection, can be prepared by simply evaporating metals, or applying silver paste to crystals of similar composition350 and may be related to the nonohmic structures discussed on p. 324. Foster and Pilkuhn have detected shallow donor-shallow acceptor pair radiation in p-n junctions at room temperature and have also observed a green emission band at 77" that shifts with the applied bias, reminiscent of the photon-assisted tunneling or the band-filling mechanisms of G ~ A S . ~ ~ '

27. GaAs Although some progress has been made in defining the conditions under which different emission bands are observed, it is still difficult to sort out most of the reported luminescence in terms of the individual distinct bands that can be observed. The problems of band shrinkage,352 band tails,353 and impurity bands354 have received continued treatment from a theoretical point of view. Additional absorption data on heavily doped samples have been interpreted in terms of exponential band tails.355However, careful absorption edge measurements covering about five decades in absorption constant have been shown by Lucovsky to be in agreement with the theoretically expected Gaussian band tails rather than with exponential tails.356 Lucovsky has followed this up by showing that the photoluminescence emission can be directly predicted from the absorption data using detailed b a l a n ~ e . ~ Thus ~ ' . ~the ~ ~positions of the photoluminescence peaks (e.g. Fig. 16) as a function of doping in both n-type, p-type, and compensated samples can be interpreted in terms of the expected impurity bands (or band tails) and the Burstein-Moss shift. The implications to electroluminescence become obvious. With the bands nominally filled to a bias-dependent quasi-Fermi level this leads to a quantitative explanation for the band-filling model. Rieck has shown that the effective lifetime in such a band at high pumping levels involves both the spontaneous and a stimulated lifetime and is still longer than the lifetime for thermal relaxation within the band, so that the distribution is still fixed by the position of the quasi-Fermi level even at high forward bias.359 Band tails, in particular Gaussian tails, have also been deduced

M. Gershenzon, R. A. Logan, D. F. Nelson, and F. A. Trumbore, Bull. Am. Phys. SOC.10, 388 (1965). 3 5 0 R. A. Logan, M. Gershenzon, F. A. Trumbore, and H. G. White, Appl. Phys. Letters 6, 113 (1965). 3 5 1 L. M. Foster and M. Pilkuhn, Appl. Phys. Letters 7 , 65 (1965). 3 5 2 V. L. Bonch-Bruevich and R. Roman, Soviet Phyx-Solid State (English Trans/.) 6, 2016 (1965) [Fiz. Tuerd. Telu 6, 2535 (1964)). 3 5 3 B. I. Halperin and M. Lax, Bull. Am. Phys. SOC.10, 302 (1965). 3 5 4 T. N. Morgan, Phys. Reu. 139, A343 (1965). 3 5 5 J. I. Pankove, Electron. Div. Abstr., Spring Meeting Electrochem. SOC.,San Francisco, 1965 p. 83, Abstr. No. 63. 3 5 6 G. Lucovsky, Solid-State Commun. 3, 105 (1965). 3 5 7 G. Lucovsky, A. J. Varga, and R. F. Schwarz, Solid-state Commun. 3 , 9 (1965). 3 5 8 G. Lucovsky, Phys. Quantum Electron. Conf, San Juan, June 1965 Paper D-7. 3 5 9 H. Rieck, Solid-state Electron. 8, 83 (1965). 349

368

M. GERSHENZON

from current-voltage and light-voltage measurements by Vul et al.360and by Bagaev et al.361 Aukerman and Millea have concluded that band-filling emission occurs within the depletion layer from radiation damage Tunneling mechanisms have received considerable attention. In Esaki diodes, light emission corresponds to the excess current range.363 The emission shifts rapidly with bias364 and the low energy tail does not saturate as it does for the band-filling mechanism.365In other diodes, photon-assisted tunneling can still occur. Ripper and Leite have shown that while photonassisted tunneling-recombination should occur within the depletion layer at low bias it should occur in the p-region at higher bias.366 Morgan and Nathan have reinterpreted some “bandfilling” data in terms of photon-assisted tunneling.367 Various peaks have been observed near the edge and interpreted in terms of band-to-band recombination or recombination involving discrete levels in lightly doped diode^^^',^^' or involving impurity bands at higher doping levels.370The similarity between simple donoracceptor pair band recombination and recombination between two band tails or impurity bands has been noted.”I The theory of the splitting of a shallow acceptor level by strain has been derived by EmtageJ7’ to confirm the previously observed splitting from which the participation of an acceptor level was deduced.z6z An acceptor level has also been implicated by the shift of the emission line in a magnetic field.373 Hill has come to the interesting conclusion that the high quantum efficiencies measured at low temperatures in electroluminescence persist to room temperature but are effectively . ~ ~has ~ been substantially confirmed by reducing reduced by self absorption in the n - l a ~ e rThis the thickness of the n-region, then reinserting an equivalent thickness in the path of the beam.375 However, by studying the over-all coupling efficiency in p-n-p opto-transistors, Minden and Donahue come to the opposite conclusion, that it is the quantum efficiency of the emitter that falls off at high temperatures.376 Diffusion lengths near a p-n junction have been determined by measuring the current B. M. Vul, E. I. Zavaritskaya, and A. P. Shotov, Soviet Phys.-Solid State (English Transl.) 6, 1146 (1964) [Fiz. Tuerd. Tela 6, 1465 (1964)l. 361 V. S. Bagaev, Yu. N. Berozashvili, B. M. Vul, E. I. Zavaritskaya, and A. P. Shotov, Souiet Phys.-Solid State (English Transl.) 6, 959 (1964) [Fiz. Tuerd. Tela 6, 1235 (1964)]. 3 6 2 L. W. Aukerman and M. F. Millea, J. Appl. Phys. 36, 2585 (1965). 363 A. N. Imenkov, M. M. Kozlov, S. S. Meskin, D. N. Nasledov, V. N. Ravich, and B. V. Tsarenkov, Soviet Phys.-Solid State (English Transl.) 7, 504 (1965) [Fiz. Tuerd. Tela 7, 634 (19631. 364A. N. Imenkov, M. M. Kozlov, S. S. Meskin, D. N. Nasledov, V. N. Ravich, and B. V. Tsarenkov, Soviet Phys.-Solid State (English Transl.) 7, 618 (1965) [Fiz. Tuerd. Tela 7, 775 (1965)l. 3 6 5 A. A. Rogachev and S. M. Ryvkin, Soviet Phys.-Solid State (English Transl.) 6, 2548 (1965) [Fiz. Tuerd. Tela 6, 3188 (1964)l. 366 J. E. Ripper and R. C. C. Leite, Proc. IEEE 53, 160 (1965). 367 T. N. Morgan and M. I. Nathan, Bull. Am. Phys. SOC.10,389 (1965). 368 0. Ohtsuki, T. Kotani, Y. Iwai, and I. Isurumi, Japan. J. Appl. Phys. 4,314 (1965). 369 M. H. Norwood, H. Strack, and W. G. Hutchinson, Appl. Phys. Letters 6, 71 (1965). G. Lucovsky and A. J. Varga, J. Appl. Phys. 35, 3419 (1964). 3 7 1 J. C. Shaffer and F. E. Williams, J. Appl. Phys. 36, 860 (1965). 372 P. R. Emtage, J. Appl. Phys. 36, 1408 (1965). 3 7 3 G. B. Wright and F. L. Galeener, Bull. Am. Phys. SOC. 10, 369 (1965). 3 7 4 D. E. Hill, Bull. A m . Phys. SOC.10, 97 (1965). 3 7 5 G. Lucovsky and A. J. Varga, Proc. IEEE 53,491 (1965). 3 7 6 H. T. Minden and J. A. Donahue, Solid-State Electron. 8, 613 (1965).

360

”’

13. RADIATIVE RECOMBINATION IN THE III-V COMPOUNDS

369

collected across the junction upon the approach of an electron m i ~ r o b e a r n .Emission ~~~ and the theory of the negative resistance region from p-i-n structures has been due to photogeneration has been extended.379 Further results on the observation of the “forward bias emission peak” in reverse bias have been published.3s0 Emission has been Cathodoluminescence associated with each of three levels due to Cu’” and with a Mn peaks have been measured.3s3 Nonradiative decay induced by radiation damage has been s t ~ d i e d ’ ~ ~and . ’ ~it ~was shown that the dominant nonradiative center introduced is a displaced As atom.’84

28. OTHERCOMPOUNDS Band-to-band recombination has been examined in InSb at low temperatures by observing the Landau level splittings in a magnetic field.38b.387Although an extrinsic band appears in zinc-doped samples, it is the band-to-band peak that eventually lases.386 In InP as well as in GaAs the reverse bias spectrum is dominated by a “forward bias peak” at low temperature^.^'^ In InP a Bose-Einstein condensation of free excitons is expected at a concentration of 1015cm-3,where the excitons may not yet be dissociated by the Coulombic fields present.388 Baryshev has calculated the band-to-band radiative lifetime in InAs by detailed balance, and by comparison with Auger lifetimes shows that band-to-band recombination may be dominant in n-type material at room temperature.389 From Zeeman data, Galeener et al. have deduced the presence of a conduction band-acceptor level transition in InAs diodes at 77“, as well as band-to-band recombination at higher injection levels.390 In Ga(As-P) alloys it has been shown that the direct radiative transitions persist even when the alloy energy gap is already several kT on the indirect side because the shorter direct lifetime can compensate the Boltzmann fa~tor.’~’Cathodoluminescence across a p-n junction in a Ga(As-P) alloy has shown that the most efficient radiative recombination occurs on the p-side close to the junction as it does for pure G ~ A s . ’ ~ Galginaitis ’ has shown that external quantum efficiencies from p-n junctions on the As-rich side of these alloys can be as high as in pure GaAs by placing the junction at the focus of a paraboloid so that the light emerges from the diode structure on the first pass, and with the rest of the paraboloid richer in P the higher band gap reduces the ab~orption.~”

-

377

D. B. Wittry and D. F. Kyser, J. Appl. Phys. 36, 1387 (1965).

’” M. Pilkuhn and H. Rupprecht, Bull. Am. Phys. SOC.10, 96 (1965). K. Weiser, Bull. Am. Phys. SOC. 10, 389 (1965). ’*‘A. E. Michel, M. I. Nathan, and J. C. Marinace, J. Appl. Phys. 35, 3543 (1964). T. N. Morgan, M. Pilkuhn, and H. Rupprecht, Phys. Rev. 138, A1551 (1965). 382 T. C. Lee and W. W. Anderson, Solid-State Commun. 2, 265 (1964). R. V. Babcock, J. Appl. Phys. 35, 3354 (1964). ’“G. W. Gobeli and G. W. Arnold, Bull. Am. Phys. SOC. 10, 321 (1965). 3 8 5 J. J. Loferski, H. Flicker, and M.H. Wu, Bull. Am. Phys. SOC.10, 321 (1965). R. J. Phelan and R. H. Rediker, Bull. Am. Phys. SOC. 10, 389 (1965). 3 8 7 A. Mooradian and H. Y . Fan, Bull. Am. Phys. SOC.10, 369 (1965). 3 8 8 R. C. Casella, J. Appl. Phys. 36, 2485 (1965). 3 8 9 N. S. Boryshev, Soviet Phys.-Solid State (English Transl.) 6, 2410 (1965) [Fiz. Tuerd. Tela 6, 3027 (1964)l. F. L. Galeener, I. Melngailis, G. B. Wright, and R. H. Rediker, J . Appl. Phys. 36, 1574 (1965). 3 9 ’ T. L. Larsen, E. E. Loebner, and R. J. Archer, Bull. Am. Phys. Soc. 10, 388 (1965). 392C.M. Wolfe, M. D. Sirkis, C. J. Nuese, N. Holonyak, 0. L. Caddy, 0. T. Purl, and W. E. Kunz J. Appl. Phys. 36, 2087 (1965). 393 S. V. Galginaitis, J . Appl. Phys. 36, 460 (1965). 379

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CHAPTER 14

Stimulated Emission in Semiconductors Frank Stern I . INTRODUCTION.

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371

I1 . RELATION BETWEEN STIMULATED AND SPONTANEOUS EMISSION 111.

LASERSTRUCTURES. .

374

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

1 . Chemical Structures . 2. Geometrical Structures

AND COHERENCE. IV . MODES.DIRECTIONALITY, 3. Mode Structure . . . . . . . .

. . . . . . . . . . . . . . 4 . Directionality: Experiment . . . . . . . . . . . 5 . Directionality : Theory . . . . . . . . . . . . 6 . Directionality: A Simple Model 7 . Coherence . . . . . .

EFFICIENCY. V . QUANTUM 8. Theoretical . . . 9 . Experimental Results

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

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

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10. Temperature Dependence und Healing Effects

VI . RADIATION CONFINEMENT. THRESHOLD. AND Lass 1 1 . Modes in a Dielectric Slab . . . . . 12. Results for Multilayer Structures . . . . 13. Comparison with Experiment . . . . . 14. Temperature Dependence . . . . . .

VII . LASERMATERIALS .

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VIII . EFFECTS OF AMBENTS AND EXTERNAL FIELDS . . . . . . . . I5. Temperature . . . . . . . . . . . . . . . 16. Pressure . . . . . . . . . . . . . . . . 17 . Uniaxial Strain . 18 . Magnetic Fields

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

376 376 379 380 380 383 384 386 387 389 389 392 394 396 396 398 400 403 403 407 407 409 409

410

.

I Introduction

'

The relation between spontaneous and stimulated emission. and the existence of spontaneous emission. ie., recombination radiation. in semi-

' A. Einstein. Physik . Z . 18. 121 (1917).

372

FRANK STERN

conductors,’ have both been known for many years, but it was not until 1962 that the discovery of the gallium arsenide injection l a s e F made stimulated emission in semiconductors a reality. Stimulated emission in other solids and in gases had been known for a number of years previously, and has an extensive In this chapter we concern ourselves principally with those aspects of stimulated emission which are new or different in the case of semiconductor lasers. Because most work has been done on gallium arsenide injection lasers, we generally use GaAs as the example and model. The excitation of the electronic system in an injection laser is achieved by passing current through a p-n junction in the forward direction (electrons flow from the n-type side to the p-type side). As electrons and holes cross the junction, they become minority carriers, or excited states, and recombine within a distance of the order of a diffusion length from the junction. The radiative fraction of the recombination is the source of the lasing. The density of minority carriers per unit volume can be of the order of lo” cmor more, and the power gain per centimeter can reach values of the order of 100 cm-’ or more. These values exceed those generally found in conventional lasers, and the injection laser therefore requires less feedback in order to lase; it can be shorter, or have poorer reflectors, or both, than a conventional laser. An important advantage of the injection laser is the simplicity of excitation by an electric current compared to the more cumbersome and less efficient optical excitation required for most conventional lasers. This advantage is partly offset by the need for pulsing the current when an injection laser is operated near room temperature. If a forward current is passed through a laser diode, the electronic excitation increases with increasing current. But the diode also is heated by Joule heating and by that part of the recombination energy which does not leave the crystal. The resulting temperature rise increases the threshold current required to reach the electronic excitation necessary for lasing. In most diodes it is not possible to reach the lasing threshold by gradually

* See the chapter o n Radiative Recombination by M. Gershenzon in this volume for a comprehensive review.

’R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. 0.Carlson, Phys. Rev. Letters 9, 366 (1962). M. I. Nathan, W. P. Dumke, G. Burns, F. H. Dill, Jr., and G. J. Lasher, Appl. Phys. Letters 1, 62 (1962). 5T.M. Quist, R. H. Rediker, R. J. Keyes, W. E. Krag, B. Lax, A. L. McWhorter, and H. J. Zeiger, Appl. Phys. Letters 1, 91 (1962). B. A. Lengyel, “Lasers.” Wiley, New York, 1962. G. Bimbaum, “Optical Masers.” Academic Press, New York. 1964. * W. V. Smith and P. P. Sorokin, “The Laser.” McGraw-Hill, New York, 1966.

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373

increasing the current. Pulsed operation with a sufficiently low duty cycle and with short pulses avoids the heating problem in most cases. Continuous operation has been achieved at 77°K,9 but continuous operation at room temperature will be difficult to attain. Heating effects are discussed more fully in Section 10. Keyes" has pointed out that an understanding of injection laser operation requires a chemical model, an electrical model, a thermal model, and an optical model. These consider, in turn, the impurity profile and its production ; the flow of electrons and holes ; the heat flow ; and the electronic transitions involved in the recombination, the quantum efficiency, and the electromagnetic modes of the laser. We shall touch on all of these models, but concentrate on the optical model. In Part I1 we discuss the relation between stimulated and spontaneous emission in semiconductors. In Part I11 we describe the structure, both chemical and geometrical, of some of the many types of injection lasers that have been made and proposed, and some of the characteristics of their emission. Part IV briefly considers the mode structure. directionality, and coherence of the radiation emitted by injection lasers. Quantum efficiency, both above and below threshold, is discussed in Part V. Part VI considers the effect of the optical properties of the semiconductor on the threshold current and other experimentally observed properties of GaAs lasers. Part VII discusses laser materials other than GaAs, and refers to some recent work on excitation of stimulated emission by electron beams. The last section briefly describes the effects of temperature, pressure, strain, and magnetic field on semiconductor lasers. Several comprehensive summary and review articles on the injectior. laser have been ~ r i t t e n . ~ . * . ' ' - 'The ~ reader is referred to these for aspects of the subject that are omitted here, and for some of the historical background. The paper of Burns and Nathan12 has a particularly comprehensive bibliography. An interesting report on the status of the subject rather early in its development was presented in three invited papers at the March, 1963 meeting of the American Physical Society."-" There have been a number J. C. Marinace, IBM J . Res. Develop. 8, 543 (1964). See also Ref. 94, and M. Pilkuhn, H. Rupprecht, and J. Woodall, Proc. IEEE 51, 1243 (1963). l o R. W. Keyes, unpublished (1963). l 1 B. Lax, Science 141, 1247 (1963). '' G. Burns and M. I. Nathan, Proc. IEEE 52, 770 (1964). l3 W. P. Dumke, Adoan. Lasers. Marcel Dekker, New York, to be published. l4 E. Haken and H. Haken, 2. Physik 176. 421 (1963); H. Haken, in "Festkorperprobleme" (F. Sauter, ed.). Friedr. Vieweg and Son, Braunschweig, to be published. l5 R. N. Hall, Solid-state Electron. 6, 405 (1963). l6 A. L. McWhorter, Solid-state Electron. 6, 417 (1963). l 7 M. I. Nathan, Solid-state Ekctron. 6, 425 (1963).

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FRANK STERN

of other papers which discuss injection lasers from a fairly general viewpoint.

’*-**

II. Relation between Stimulated and Spontaneous Emission In most injection lasers the upper state or the lower state involved in the radiative recombination lies in a continuum. Thus it is not possible to characterize the system by giving the number of excited and unexcited atoms as in conventional lasers. It is convenient instead to use the absorption coefficient of the medium and the spontaneous recombination rate per unit energy interval to describe the system. These quantities are related to each other, and both of them depend on the degree of excitation. If the rate of electron-hole recombination is much slower than the rate with which electrons come to equilibrium among themselves and holes come to equilibrium among themselves, then it is possible to assign quasiFermi levels2’ to the electrons and to the holes separately. For a system in thermal equilibrium, these two quasi-Fermi levels become the Fermi level, which is constant throughout the sample. When the system is excited, the difference AF between the electron quasi-Fermi level F , and the hole quasi-Fermi level F,, is a measure of the degree of excitation. The total radiative recombination rate per unit volume in a photon energy interval dE and in solid angle dR can be written

r(E)dE (dW4n) = [rspon(E) + JYi.stim(E)I dE (dQ/47~), (1) where N i s the number of photons per mode, given for thermal equilibrium by N b ( E ) = [exp(E/KT) - 11(2) K is Boltzmann’s constant, and T is the absolute temperature. The first term on the right in Eq. (1) is the rate of spontaneous downward transitions of the electronic system, and the second term is the difference between the stimulated rates of downward and upward transitions. The stimulated and spontaneous emission functions are related by rstim(E) =

rspon(E){l - e x ~ [ ( E- W / K T I ) ,

(3)

a result which can be proved quite g e n e r a l l ~ . Finally, ~ ~ . ~ ~the absorption

‘*J. R. Biard, W. N. Carr, and B. S. Reed, Trans. A I M E 230,286 (1964). G. Winstel, Z . Angew. Phys. 17, 10 (1964). T. Peten);, Phys. Status Solidi 6, 651 (1964). ” W. Shockley, “Electrons and Holes in Semiconductors,” p. 302. Van Nostrand, Princeton, New Jersey, 1950. 22 G. J. Lasher and F. Stern, Phys. Rev. 133, A553 (1964). 23 D. E. McCumber, Phys. Rev. 136, A954 (1964). l9

2o

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STIMULATED EMISSION IN SEMICONDUCTORS

375

coefficient c@), which is often more convenient to use than the stimulated function rstim(E),is related to it by24

a(E) =

- (sc2c2h3/n2~2)r,,im(~),

(4)

where n is the index of refraction. The minus sign appears because rstimis positive when radiation is emitted, while ct is positive when radiation is absorbed. Lasing can only occur at photon energies for which there is enough amplification to overcome the losses. Thus a necessary condition for lasing is that a(E) be negative. From Eys. (3) and (4)we see that this require^^^-^' AF > E. Near the p-n junction AF is equal to the magnitude of the electron charge times the forward voltage across the junction; it falls off to zero within a few diffusion lengths on either side. The layer near the junction in which the radiative recombination takes place is called the active layer, as indicated schematically in Fig. 1, and plays an important role in models of laser operation. p-TYPE LAYER

i

n-TYPE LAYER

FIG.1. Simplified model of an injection laser. The y direction is toward the reader.

When the gain in the active layer is sufficient to overcome the bulk losses, which arise because of free carrier absorption in the active layer itself and because part of the electromagnetic field penetrates into adjacent absorbing layers, then the wave will be amplified. If the net power gain is G , the intensity will increase by a factor exp Gz on traversing a distance z along the active layer. If the ends of the crystal are made nonreflecting by suitable coatings, then the structure acts as an a r n p l i f i e ~ - . ~As * . ~the ~ reflectivity of the ends is increased, we can either say that the loss of power out the ends is decreased, or that the amount of feedback is increased. In either See, for example, F. Stern, Solid State Phys. 15, 299 (1963). G. A. Bernard and G. Duraffourg, Phys. Sratus Solidi 1, 699 (1961). “ E . 1. Adirovich and E. M. Kuznetsova, Fiz. Tiwd. Tela 3, 3339 (1961) [English Trunsl.; Soviet Phys. Solid State 3, 2424 (1962)]. ” N. G. Basov. 0. N. Krokhin, and Y. M . Popov. Zh. Eksperirn. i Teor. Fiz. 40, 1879 (1961) [English Trunsl.: SoOkt PhYS. J E T P 13, 1310 (lY61)] require that the diRerence in quasiFermi levels be greater than the energy gap. 2 8 M. J . Coupland, K. G. Hambleton, and C. Hilsum. Phys. Letters 7, 231 (1963). *’J. W. Crowe and R. M. Craig, Jr., A p p l . Phys. Letters 4, 57 (19M). 24

” M.

376

FRANK STERN

case, we find that the threshold gain Gth for lasing or self-sustaining oscillation is given by the condition R eXp G,&

=

1,

(5)

where L is the distance covered in a one-way traversal between the two ends, whose geometric-mean reflectivity is R. We assumed that the end faces are flat, perpendicular to the active layer, and perpendicular to each other. Configurations in which the ends make other angles with the active layer can also lase, but in most cases they would have higher thresholds because they do not make optimum use of the gain in the active layer. Although the “a” in “laser” denotes amplification, the word ‘‘laser’’ is generally used to refer to a laser oscillator, one which has enough gain to fulfill condition (5). Where confusion is likely to arise, we shall try to distinguish between laser amplifiers and laser oscillators. If we know the rate of spontaneous radiative recombination and its spectrum, Eqs. (3) and (4) give the corresponding gain or loss coefficient. If we know this and the index of refraction as a function of energy for each part of the sample, then the theory of Part VI can in principle give the net gain G for waves propagating in any laser structure, and can also predict the photon energy of the lasing peak.

III. Laser Structures In this part we briefly describe some of the structures that have been used to make injection lasers. The variety of these structures is growing rapidly, so the examples given here should be considered to be only a partial list of the possible or interesting structures, based on what has been published as of the end of 1964. It is convenient to distinguish between the chemical structure, i.e., the composition of the laser at each point, and the geometrical structure, by which we mean both the over-all size and shape of the unit, and the preparation of its surfaces. 1. CHEMICAL STRUCTURES

The first GaAs injection lasers3-’ were made by diffusing zinc30931 into an n-type substrate, and this procedure is still widely used, with substrate donor concentrations that generally range from lo” cm-3 to 5 x loi8 ~ m - The ~ . zinc concentration in a typical diffusion rises approximately linearly from a value below 1019 cm-3 to a value above 10’’ cm-3 within about 5 p, giving a concentration gradient in this range of 2 x C. Marinace, J . Electrochem. Soc. 110, 1153 (1963). M. H. Pilkuhn and H. Rupprecht, Trans. A I M E 230, 296 (1964).

3a J. 31

14. cm-4

STIMULATED EMISSION IN SEMICONDUCTORS

377

.3 1 In short diffusions the gradient is even steeper. At the junction

the gradient is quite sensitive to the diffusion conditions and to the substrate doping. Capacitance measurements indicate a gradient which varies over about 5 orders of magnitudes from an upper limit of the order of 5 x cm-4 for high substrate do ping^.^^ The thickness of the active layer in an injection laser is of the order of magnitude of the diffusion length. We give here a very simple way to estimate this distance in a linearly graded junction. In homogeneous material the diffusion length, here called d, the diffusion constant D,and the average carrier lifetime t are related by33

(DT)”’. (64 We further assume that the average lifetime of injected electrons is inversely proportional to the average hole concentration P , provided P is not too large,34 and that P is given by product of the impurity concentration gradient A and the average distance the carriers diffuse. Then d

=

7

x (BP)-

’,

P z Ad.

(6b) (6 4

Combining these equations, we find

d x (D/AB)’’3.

(64

This estimate of the average diffusion length d is uncertain within a numerical factor of the order of unity. If we estimate the concentration gradient A near the junction (in a typical zinc-diffused diode) to be 4 x 10” cmP4, the rate constant B for the recombination of electrons in GaAs at 77°K to be 4 x 10- l o ~ m ~ / s e c , ~ ~ and the electron diffusion constant to be 15 cm2/sec, we find that d x 1 p and 7 z 1 ns. These values agree quite well with the active layer width of 1.5 p deduced from an analysis of several experimentally observed properties of typical zinc-diffused gallium arsenide lasers,3s and with the lifetime deduced from laser time delays,36 suggesting that our model for the relation between the chemical structure and the carrier recombination is at least qualitatively correct. Lasers with lower substrate doping than the 10” cmW3which we assume for our typical unit may be expected to have N. Morgan, private communication. R. A. Smith, “Semiconductors,” p. 241. Cambridge Univ. Press, London and New York, 1961. See also p. 314 of Ref. 21. 34 W. P. Dumke, Phys. Rev. 132, 1998 (1963). 3s F. Stem, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 165, Dunod, Paris and Academic Press, New York, 1965. 36 K. Konnerth and C. Lanza, Appl. Phys. Letters 4, 120 (1964) and unpublished work, 1964. 32 T. 33

378

FRANK STERN

wider active layers. The substrate doping level affects other properties of zinc-diffused lasers, particularly the wavelength at which lasing occurs.374o Diffusion of donors into a p-type substrate has also been used to make 1asers;l but no advantage has yet been demonstrated for this procedure. A P-Po-N structure in which first Mn and then Zn are diffused into an n-type substrate has been made; it has negative resistance at low current and lases at high current levels.43The Po layer is a high-resistivity layer dominated by Mn. This laser has negative resistance; it can be made to switch from a nonlasing state to a lasing state, or the reverse. A similar structure can be made by vapor growth of n-type material onto a homogeneous Mn-doped substrate, with subsequent diffusion of Zn into the substrate. This structure also gives clear evidence for the presence of highorder modes in the electromagnetic field perpendicular to the junction plane, both from near-field and far-field radiation patterns.44 A P-V-N structure, in which the v layer is a lightly doped ( x 1 O I 6 cm-j) vapor-grown n-type layer and the P layer is made narrower and more abrupt by means of shorter times, lower temperatures, and higher surface concentrations than in ordinary zinc diffusions, has been made by Wilson?’ It differs from the other structures we have described in a number of respects. The recombination radiation is thought to originate primarily on the n-type side of the junction, whereas there is good evidence that in the structures we described previously it originates on the p-type side. The spontaneous emission line which lases lies at higher photon energy, and has a smaller spectral width, than in conventional lasers. Wilson identifies the lasing transition as recombination of excitons bound to a shallow d ~ n o r . ”Such ~ a process cannot occur in the conventional structures, where the high carrier concentrations screen the Coulomb attraction between electron and hole. and make the exciton unstable.47 Wilson has M. I. Nathan and G. Burns, Appl. Phys. Letters 1, 89 (1962). R. Braunstein, J. I. Pankove, and H. Nelson, Appl. Phys. Letters 3, 31 (1963). 39 G. C. Dousmanis, C. W. Mueller, and H. Nelson, Appl. Phys. Letters 3, 133 (1963). 40 M. I. Nathan, G. Burns, S. E. Blum, and J. C. Marinace, Phys. Rev. 132, 1482 (1963). 4 1 C. E. Kelly, Proc. IEEE 51, 1239 (1963). 42 K. Weiser and R. S. Levitt, J . Appl. Phys. 35, 2431 (1964). 43 K. Weiser and A. E. Michel, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 177. Dunod, Paris and Academic Press, New York, 1965. 44 K. Weiser and F. Stem, Appl. Phys. Letters 5, 115 (1964). 45 D. K. Wilson, Appl. Phys. Letters 3, 127 (1963); in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 171. Dunod, Paris and Academic Press, New York, 1965. 46 M. H. Pilkuhn and H. Rupprecht (to be published) identify the transition as recombination of a hole to a shallow donor level. 47R.C. Casella, J . Appl. Phys. 34, 1703 (1963). 3

38

14. STIMULATED EMISSION IN SEMICONDUCTORS

379

deduced the presence of antisymmetric modes from the appearance of a dark stripe in the near-field radiation pattern of his diodes. Diodes made by solution growth, i.e., epitaxial growth from a doped melt onto a substrate:' have properties somewhat different from those of diffused diodes. The chemical structure of these diodes is not yet completely characterized. Other laser structures have been made4' and prop~sed,~'-~ and l ~ it seems likely that still others will be conceived and built in the future. 2. GEOMETRICAL STRUCTURES The conventional injection laser crystal has the Fabry-Perot structure, in which two faces of a rectangular parallelepiped, parallel to each other and perpendicular to the junction plane, are made optically flat either by polishing or by cleaving. The other two faces perpendicular to the junction plane are generally roughened to suppress competing modes traveling between these faces. The remaining two faces provide the electrical contacts to the n-type and p-type sides of the diode. For some purposes the ends of the Fabry-Perot diodes may be coated with reflecting or antireflecting coatings; usually they are not coated. Another structure with very interesting properties is similar to the Fabry-Perot structure except that all four sides are cleaved. If the length-towidth ratio is not too far from 1, such a diode can support modes which travel around the perimeter and are almost totally internally reflected. The loss due to transmission out of the crystal is, therefore, very small, and these four-sides-cleaved units have very low threshold current densities for lasing. The direction of the emitted beam can be controlled in a laser with a triangular structure.52s53A cylindrical laser has also been made.54 Other special structures whose principal purpose is to increase the electroluminescent light output in the current range in which lasing is not achieved are described by Gershenzon.2 All the injection lasers referred to in this chapter have the Fabry-Perot G. C. Dousmanis, H. Nelson, and D. L. Staebler, Appl. Phys. Letters 5, 174 (1964). H. Nelson and G. C. Dousmanis, Appl. Phys. Letters 4, 192 (1964). 'ON. G. Basov, 0. N.Krokhin, and Yu. M. Popov, Usp. Fiz. Nauk 72, 161 (1960) [English Transl.: Soviet Phys. Usp. 3, 702 (1961)l. 50a H. Kroemer, Proc. IEEE 51, 1782 (1963). 5 1 G. Diemer and B. Bolger, Physica 29, 600 (1963). 'l'G. Wade, C. A. Wheeler, and R. G. Hunsperger, Proc. IEEE 53, 98 (1965). ’lb H. Hora, Phys. Status Solidi 8, 197 (1965); Z . Naturforsch. ZOa, 543 (1965). 5 2 M. Garfinkel, W. E. Engeler, and D. J. Locke, J . Appl. Phys. 35, 2321 (1964). 5 3 J. C. Marinace, A. E. Michel, and M. I. Nathan, Proc. IEEE 52, 722 (1964). 54K. M. Arnold and S. Mayburg, J . Appl. Phys. 34, 3136 (1963).

48

49

380

FRANK STERN

structure, with uncoated ends, and are made by the conventional zinc diffusion, unless some other structure is specified. IV. Modes, Directionality, and Coherence 3. MODESTRUCTURE

A rectangular parallelepiped with perfectly reflecting walls admits standing waves in each of its three directions. The Fabry-Perot structure lasers which we consider have the shape of rectangular parallelepipeds, but usually have only one pair of cleaved or polished faces. The distance between these faces is the length L of the laser, and the direction perpendicular to these faces is the z direction in our model, shown schematically in Fig. 1. Variation of the field along the length of the laser is called a longitudinal variation, and variation in the other two directions a transverse variation. We therefore speak of longitudinal and transverse modes” in cases in which only one or the other variation enters in the phenomenon being discussed. Variation of the electric field along the width of the junction, the y direction of Fig. 1, gives rise to the sharp and variable angular structure found in the far-field radiation pattern in the junction plane, described below, and to the spots seen in the near-field pattern on the junction In most laser diodes the front face is relatively uniformly illuminated at currents below threshold, indicating a relatively uniform recombination rate across the width. Above threshold, however, only a few filaments lase, the exact configuration being determined by inhomogenities along the junction plane or on the junction face. Whether the filaments are independent or represent some complicated mode across the width of the lasers6 shall not concern us here, since in most cases the physical parameters which determine the actual light configuration are unknown or uncontrolled. Thus the only transverse variation we consider in detail is a variation in the x direction, the direction perpendicular to the junction plane. The easiest of the field variations to discuss is the longitudinal variation. Our diode of length L admits a standing wave of order m when kL= mr,

(74

By “mode” we mean one of a discrete set of field distributions that can be excited with relatively large amplitude. See, for example, A. Kastler, Appl. Opt. 1, 17 (1962). Unlike the modes in an ideal cavity with perfectly reflecting walls, such modes d o not constitute a complete set for the expansion of an arbitrary field that satisfies the boundary conditions. 56 G. E. Fenner and 3. D. Kingsley, J . Appl. Phys. 34,3204 (1963). 57 E. J. Walker and A. E. Michel, J . Appl. Phys. 35, 2285 (1964). 55

14.

STIMULATED EMISSION IN SEMICONDUCTORS

38 1

where k is the z component of the propagation vector of the radiation. The mode number may not be an integer, but its differences between successive modes are very close to unity, and are assumed to equal unity. As we shall see in Part 6, the propagation vector k depends on the optical properties of the active layer and the adjacent layers. We simplify the treatment here by writing k = 27tii/E,, (7b) where I is the vacuum wavelength of the radiation, and 6 is an average index of refraction in the region through which the radiation extend^.^' Then (7a) takes the more familiar form ~nzAm/ii(I,)= L .

(74

The wavelength difference between the modes of order m and n? + 1 is

A,

A,, where I is the average of I , and Am+ -

=

12/2Ln', and

n' = Ti - A(dii/dI).

(8)

(9)

We neglect higher derivatives in (8) because the modes are very closely spaced in the cases we consider. Wavelength peaks corresponding to the longitudinal modes are often observed in the stimulated emission spectrum of GaAs diode^,'^.^^ and in some diodes with especially uniform structure and good surfaces, the oscillations are visible over much of the spontaneous emission spectrum,60*61as shown in Fig. 2. The values of n' have been found in this way over a range of wavelengths,60*61and are shown in Fig. 3. The substantial dispersion near the absorption edge is expected from the rapid rise in the absorption coefficient there.'* These results can be used to extend the prism measurements of the index of refraction made by M a r ~ l e ' into ~ the difficult region of the absorption edge. A maximum in the index of refraction is expected near a sufficiently steep absorption edge62 and would be accompanied by a sharp decrease in n' with increasing energy, but this has not been observed, probably because it occurs at energies above the When several transverse modes are considered, each will have a different dependence of k on L, as discussed in Part VI. "P. P. Sorokin, J. D. Axe, and J. R. Lankard, J . Appl. Phys. 34,2553 (1963). 6o M. I. Nathan, A. B. Fowler, and G. Burns, Phys. Rev. Letters 11, 152 (1963); 12, 41 (1964). 6 1 M. I. Nathan, G. Burns, and A. B. Fowler, in "Radiative Recombination in Semiconductors" (7th Intern. Conf.), p. 3305. Dunod, Paris and Academic Press, New York, 1965. 6 2 F. Stern, Phys. Rev. 133, A1653 (1963). 6 3 D .T. F. Marple, J . Appl. Phys. 35, 1241 (1964). 58

FRANK STERN

I

0350

8400

8450

8500

8550

X(A)

FIG.2. Oscillations in the spontaneous and stimulated emission spectrum of a zinc-diffused GaAs injection laser at 2°K. (After M. I. Nathan et a[., Ref. 60.)

4 C

FIG. 3. Variation of n’ = n - I(dn/dl) with wavelength, as determined at 2'K for GaAs diodes with the indicated substrate donor concentrations. (After M. I. Nathan et al., Ref. 61.)

14.

STIMULATED EMISSION IN SEMICONDUCTORS

383

peak of the spontaneous emission curve, where the oscillations are obscured by the strong internal absorption. If more than one transverse mode is present, the emission spectrum has several groups of Fabry-Perot modes. The linewidth of an individual mode above threshold depends on the power level. Measured l i n e ~ i d t h s ~ ~ *as~ ~ small " - ~as ~ "150 kc have been reported. s89s9

4. DIRECTIONALITY : EXPERIMENT

The o b s e r ~ e dfar-field ~ ~ ~ ~radiation ~ patterns of lasing GaAs diodes exhibit considerable variety, but a typical pattern has a principal maximum about 3" wide in the junction plane, and 15" wide perpendicular to the junction plane. The structure of the far-field pattern in the junction plane is often quite complex, indicating the prcsence of high-order modes in this direction, or combinations of several such modes. One of the diodes studied by Michel and WalkeP4 gave a diffraction pattern with regularly spaced peaks whose spacing indicated that the entire width of the front face of the junction was contributing to their pattern. In conventional zinc-diffused GaAs lasers the half-power width of the far-field angular distribution in the plane perpendicular to the junction plane and to the junction face (the x-z plane of Fig, 1) has been found to be of the order of 10'-15" by Fenner and K i n g ~ l e yand , ~ ~in the range 20'-30" by A detailed model which can account for such beam widths is presented in Part VI. In some diodes, particularly those with special chemical structures, there is evidence for the existence of higher order modes perpendicular to the junction plane. This is deduced by Wilson45 from near-field patterns, and by Weiser and Stern44 from near-field and far-field patterns. Both of the structures involved in these experiments have a thick (5-10 p ) inactive layer adjacent to the active layer. The index difference between the two layers is thought to be small, so that light generated in the active layer can penetrate into the adjacent layer before meeting a barrier imposed by a material with either a smaller index of refraction or a high absorption coefficient. These conditions favor high-order modes, since such modes are better able to concentrate their energy in the layer where there is gain. "' J. A. Armstrong and A. W. Smith, Appl. Phys. Letters 4, 196 (1964). W. Crowe and R. M. Craig, Jr., Appl. Phys. Letters 5, 72 (1964). M. Ciftan and P. P. Debye, App2. Phys. Letters 6, 120 (1965). 64 A. E. Michel and E. J. Walker, Proc. Symp. Opt. Masers, N a v York, April 1963 p. 471. Polytech. Instit. of Brooklyn, New York, 1963. 65 C. Lanza, unpublished work, 1964. I am indebted to Mr. Lanza for permission to refer to these results. 63b J. 63c

384

FRANK STERN

Figure 4 shows both the two-peaked far-field pattern obtained by Weiser66 with a diode having a high-resistivity Mn-doped layer about 5 p thick between the n-type layer and the active layer, and the single-peaked pattern obtained on a typical GaAs laser made by diffusion of zinc into an n-type substrate with lo1*carriers/cm3. II-

t a711 K

0’

-35

I I

-25

I I

-15

I I

-5

I I

5

I I

I5

I I

25

35

ANGLE FROM NORMAL(DEGREES)

FIG.4. Angular dependence of the far-field intensity in the plane perpendicular to the junction face and to the junction plane. The zero is determined by reflection of a light beam from the junction face. Positive angles are toward the p-type side of the junction plane. Curve (a) shows the radiation pattern of a conventional diode made by zinc diffusing into a substrate with 10l8 electrons/cm3. The threshold current is 0.87 A, and the curve was obtained at a pulse current of 1 A. Curve (b) is the pattern of a diode with a 5-p Mn-doped high resistance layer between the n-type and p-type layers. The threshold is 1.3 A, and the pulse current used was 1.8 A. Both curves were obtained at 77°K. (After K. Weiser, Ref. 66.)

5. DIRECTIONALJTY: THEORY~~~

The simplest form of propagating wave which we can consider in discussing angular dependence is one whose electric field is polarized in the y direction of Fig. 1 and has the form

E, = u(x) exp(iGz + ikz

- id).

(10)

Here G is the power gain per unit distance traversed, k is the z component K. Weiser, unpublished work, 1964. I am indebted to Dr. Weiser for providing these curves. 66aI am indebted to A. C. Beer for a comment and to G. J. Lasher for a suggestion which strongly influenced the treatment given here. 66

14.

STIMULATED EMISSION IN SEMICONDUCTORS

385

of the propagation vector, which enters in Eq. (7a), and w is the angular frequency. The amplitude function u(x) is determined by the optical properties of the laser, as discussed in Part VI. If there is more than one transverse electric field variation, then each transverse mode has its own values of G, k , and u(x). Usually we need consider only the mode with the largest gain. Note that we have omitted any dependence of the field on y, for reasons already discussed. With the particular choice made in (lo), the magnetic field has components along the x and z directions. When a propagating mode strikes the end of the crystal, there will be both a reflected and a transmitted field, whose amplitude is determined by the boundary conditions that the electric field and its normal derivative be continuous. To fit these conditions we must include in the reflected field modes other than the incident mode, which leads to a rather difficult problem that has until now not been solved for cases relevant to injection lasers. We are forced, for the present, to assume that the reflected field has the same form as the incident field, but with the signs of G and k changed. If the effective amplitude reflection coefficient is r, and if the end of the crystal is the plane z = 0, then at t = 0 the field at the surface is (1 + r)u(x). If we retain only the boundary condition that the field be continuous at the surface and note that the field in the outer medium must be a superposition of solutions of the wave equation, we find the transmitted field to be

Et,y= (1 + v )

Ern

u(q)exp(iqx + iq'z - i w f ) d q ,

(1 la)

where

I

m

u(q) = (2n)-

u(x) exp( - iqx) dx ,

Ulb)

m

and, if the index of refraction of the outer medium is n,

q'

=

[(no/c)' - q y .

(1 1 4

To find the far-field radiation pattern associated with the field (ll), we expand the plane waves in cylindrical waves, keeping only the outgoing part. The outward flux is given by the Poynting vector. Then, if the radiating area has width W, the time averaged far-field intensity in the angular interval between 9 and 9 + d6' is

Z(6’) d% = nn2~,,wTWl~(Q)12 CO$% d o ,

(12a)

where 9 is measured from the normal to the surface (the z direction of Fig. l),

386

FRANK STERN

Q

=

(nw/c)sin 0 ,

(12b)

and T = 11 + rI2 is the transmissivity of the surface. Equation (12a) is in mks units; to convert it to Gaussian units, replace the factor e0 by (471)When the outer medium is air or vacuum, the index of refraction n in (12) will equal 1. The only factor in (12) which vanes rapidly with angle in most of the cases of interest is the factor Iv(Q)I2, and this has been given for the angular dependence by Fenner and K i n g ~ l e yand ~ ~ by AntonoK6’ We note in passing that when u(x) is real, v(-Q) = u*(Q), so that I( -0) = I(0) even though u( - x) might not equal u(x).

’.

6. DIRECTIONALITY: A SIMPLE MODEL We consider here a very simple model which allows some qualitative conclusions about directionality to be drawn. A more detailed and more realistic analysis is presented in Part VI. Our model is a symmetrical three-layer model in which the inner layer has thickness d and a real index of refraction ni, and the two outer layers are infinitely thick and have a real index of refraction no. If we consider a mode whose electric field has the form of Eq. (lo), then the amplitude u(x) for an even mode is u(x) = uo cos(qx),

1x1 I i d

= uo cos(iqd) exp(4sd - SIX]) ,

(13) 1x1 > i d . A similar expression gives the amplitude for the odd modes. The values of q and s are determined by the requirement that the electric field satisfy the wave equation in the inner and outer layers, and by the continuity condition6* s = q tan(+qqd).

(14)

The values of q and s in terms of ni ,no, and d can be obtained numerically or graphically. Our purpose here is to examine the angular distribution, as given by (12), which results from various types of solutions. One simple limiting form of the amplitude (13) is obtained when qd 4 1 and sd 4 1. In this case most of the energy in the mode is located in the outer layers, and a negligible amount is located in the inner layer. Then the Fourier transform v(Q) which enters in the angular dependence is v(Q) w C 1 u 0 s ( q 2 67

68

+ s2)- ’,

sd 4 1.

(154

M.M.Antonoff, J . Appl. Phys. 35,3623 (1964). R. E. Collin, “Field Theory of Guided Waves,” Section 11.5. McGraw-Hill, New York, 1960.

14.

STIMULATED EMISSION IN SEMICONDUCTORS

387

It is instructive to look at both the angular half-power width of the far-field pattern and the half-power width of the near-field pattern, given approximately by uz(x). For the case in which all the energy is in the outer layers we find

AX = l0g,2/s = 0.693/~,

(15b)

A0 = R-'"'''

(15c)

-

1)"'~l = 0.205d,

where we used n = 1 in (12b) and assumed that A0 is small. Thus for this case we have A9

=

0.142L/Ax.

(154

The second limiting case is obtained if sd 9 1, so that there is very little energy in the outer layers. In that case the amplitude goes to zero at 1x1 = 9, and the q in Eq. (13) assumes one of the values QH =

(M

+ 1)R/d,

(16)

where M is the number of nodes in u ( x ) in the open interval (-&f,*d). For the even functions of Eq. (lo), M is an even integer. For these solutions, we have For the second case, if we choose the lowest-order transverse mode, with M = 0, we find that

AX = )d, All = l.l89L/d

(17b) =

0.5941/Ax.

(174

The wide disparity between (15d) and (17c), neither of which agrees with the value A0 x 0.886A/Ax obtained for a slit illuminated with uniform amplitude, shows how difficult it can be to estimate the width of the lightemitting region from the width of the far-field angular peak. The simple model of this section can also be used to obtain information about the peak positions and half-widths for the higher-order modes. For example, the magnitude of the peak angle is roughly given by Mi/2d radians for large values of sl, and moves to smaller angles as sl decreases. These results are a rough guide in interpreting far-field patterns such as those found by Weiser and Stern.44 7. COHERENCE

Evidence for coherence in the light output of injection lasers was obtained from the sharp structure3 of the far-field radiation pattern of these units,

FRANK STERN

FIG.5. Interference pattern produced by superimposing the radiation from opposite ends of a GaAs injection laser. (After A. E. Michel and E. 3. Walker, Ref. 69.)

which showed that the radiation was confined to a small number of modes of the crystal. This is particularly evident in the diffraction pattern of Fig. 5, which shows the interference between light from two ends of a lasing diode.69 Such interference effects are absent when the diodes are operated 69

A. E. Michel and E. J. Walker, J . Appl. Phys. 34, 2492 (1963)

14.

STIMULATED EMISSION IN SEMICONDUCTORS

389

at currents below threshold, since the spontaneous emission involves many modes.

V. Quantum Efficiency 8. THEORETICAL The external quantum efficiency qeXtof an electroluminescent diode is the number of photons emitted from the crystal in a specified band of wavelengths for each carrier crossing the p-n junction. In all the cases we consider, there is only one important band of radiation near the energy gap, but the theory could easily be extended to cases in which several wavelength bands are emitted. The internal quantum efficiency q is the number of photons emitted internally for each carrier crossing the junction. If re is the fraction of the current carried across the junction by electrons, then one can write 7

=

rebe

+ (I -

re)bh,

(18)

where be is the fraction of electrons injected into the p-type side of the junction which recombines radiatively, and b, is the corresponding fraction for holes injected into the n-type side. In an ideal abrupt junction which has negligible recombination in the depletion region, the injection ratio for classical statistics is given by” Ye/(l -

Ye) = n p D e 1 i 2 T p 1 ’ 2 / p n ~ h 1 ’ 2 T ~ 1 i 2 ,

(19)

where n p , D,, and T , are, respectively, the electron concentration, diffusion constant, and total recombination lifetime on the p-type side of the junction, and the remaining three quantities in Eq. (19)give the corresponding values for holes on the n-type side of the junction. In a junction with equal doping levels on both sides, the dominant factor in (19) will be the ratio (D,/Dh)”2, which favors electron injection because the electron mobility in GaAs and most other III-V semiconductors is substantially higher than the hole mobility. When the electron and hole populations become degenerate, the barrier for electron injection into the p-type side will be smaller (for an abrupt junction with the same doping level on both sides) than the barrier for hole injection into the n-type side, because the smaller electron effective mass leads to a higher electron Fermi level. Coulomb interactions also lead to differences in barrier height which favor injection of the lighter-mass carrier.71 No complete treatment of these effects for the doping profile found in diffused diodes is available, but it seems reasonable to expect ’O

See Section 12.5 of Shockley” or p. 270 of Smith.33

’’ T.N. Morgan, Bull. Am. Phys. SOC.9, 77 (1964).

390

FRANK STERN

that in most cases the barrier for hole injection will exceed the barrier for electron injection, and that re therefore approached 100% at low temperatures. Only in cases in which the p-type doping level is very high, the n-type doping level low, and the junction nearly abrupt, as in Wilson’s struct~re,4~ is hole injection into the n-type side of the junction expected to dominate. If the probability that an internally emitted photon escapes from the crystal is F , then the external quantum efficiency is vex1

=

Fv *

(20)

We present here a very simple model for F , based on the assumption that radiant energy is uniformly diffused throughout the crystal. This assumption is satisfied provided that T < 1,

(214

{ G ( ( s ) ~ s4 1,

(2W

where T is an effective transmission coefficient for escape of radiation through the surface, o! is the absorption coefficient of the medium, and the integral in (21b) includes all representative light paths from the place where photons are generated to points on the surface. The conditions in Eq. (21) are fairly well satisfied for semiconducting diodes of rectangular or random shapes. In such cases the effective transmissivity is the average transmissivity T,, for isotropic radiati~n,’~ which is less than 0.1 for crystals whose index of refraction exceeds 2.8. Equation (2la) does not hold for that fraction of the radiation which strikes the diode surface at nearly normal incidence, since the transmissivity is then w 0.7. In most diode geometries the fraction of such radiation is small. An exception is the hemispherical structure of Carr and Pittman,73 to has described a procedure applicable which our results do not apply. when (21b) is not valid. With uniform energy density p = J p ( E )d E inside the crystal, the power absorbed is

[

Zabs = p(E)u,cr(r, E ) dr dE

= pij,tiV/,

where i?, is the average group velocity of photons in the wavelength region of interest,74 V is the total volume, and 8 is the effective absorption F. Stern, A p p l . Opt. 3, 111 (1964). W. N. Carr and G. E. Pittman, Appl. Phys. Letters 3, 173 (1963). 73a D. E. Hill, J . Appl. Phys. 36, 3405 (1965). I am indebted to Dr. Hill for a preprint of this paper. ’4 If some of the absorbed energy is reradiated into the spectral range of interest, the corresponding fraction of the absorption coefficient of the medium does not contribute to d. 72 73

1 4 . STIMULATED EMISSION IN SEMICONDUCTORS

391

constant for loss of these photons. The transmitted power lext =

J p(E)v,T dA dE = pij,T,,A/4,

(23)

where T,, is the effective transmissivity, and the integration is carried out over the available surface, with total area A. The escape probability F for the photons is thus found to be75 where the subscript 3 is used because this emission is approximately isotropic or "three-dimensional." Similar considerations can also be used for the lasing modes in a FabryPerot cavity and give F,

=

[l

+ (a"L/T,)]-l ,

(25)

where a" is an effective loss coefficient for the "one-dimensional" case, and T, is the transmissivity at normal incidence. In this case, however, condition (21a)is no longer satisfied, since T, is not small compared to 1. A calculation of the escape probability which assumes that the radiation is generated uniformly along the length of the diode but avoids the assumptions in (21) gives, for the total fraction of the light escaping after 0,1,2,. . . reflections at the ends: F,

=

(a"L[exp(a"L) - 11-l

+ (a"L/T,)]-'.

(264

- +)]-I.

(26b)

For small values of a"L this gives F, x [I

+ a"L(T,-'

The efficiency expression given by Biard et al." is essentially equivalent to (26b). An alternative formalism for efficiency is that of S ~ o t t . ' ~ The effective absorption coefficient a" which enters in Eqs. (25) and (26) is the average absorption coefficient seen by the lasing mode. It arises because the mode extends into absorbing layers adjacent to the active layer, and because of free-carrier absorption in the active layer itself. In terms of the model to be given in Part VI, we can write a" = a'

+ ruff.

(27)

If the constant J o of Eq. (37) vanishes, then a" is equal to the loss term a of Eq. (39). The effective loss is discussed further in Part VI. The escape probability goes to unity if the absorption goes to zero, and increases with increasing surface transmissivity for nonzero values of

''G. Cheroff, F. Stern, and S. Triebwasser, Appl. Phys. Letters 2, 173 (1963). l6

A. C. Scott, Proc. IEEE 52, 325 (1964).

392

FRANK STERN

absorption. It is generally higher for the lasing modes than for the isotropic emission for two reasons. First, the lasing modes strike the surface at normal incidence, for which the t ransmissivity is substantially higher than the average transmissivity for isotropic radiation from high-index media. Second, the effective absorption coefficient for the lasing modes, which are confined to the immediate neighborhood of the active layer, may be smaller than for the modes which sample the whole crystal, particularly if heavily doped p-type regions are present. We shall see below that it is not uncommon to have the fractional escape probability higher by a factor of 10 for the lasing modes than for the isotropic emission. 9 . EXPERIMENTAL RESULTS

External quantum efficiencies of electroluminescent diodes have been measured using c ~ n v e n t i o n a lor ~ ~modified7* integrating spheres. Typical curves of light emission versus current for Fabry-Perot diodes are given in Fig. 6. There is a linear variation at low currents followed by a fairly sharp increase in slope at a current close to the threshold current density estimated from line narrowing or increasing directionality. The observed behavior can be accounted for if we assume that, below the threshold current, the injected carriers recombine with internal quantum efficiency q”, and that the current above threshold recombines with internal quantum efficiency q’. Then we expect the rate of photon emission, $ext, to be given by75

e4ex, = F3q”I, &xt

=

F3q”1th

+ Flq’(z

-

‘I,),

I < Ith,

(284

> Ith,

(28b)

where F , and F3 are the escape probabilities of the “one-dimensional” lasing emission, and of the isotropic spontaneous emission, respectively. It is convenient to introduce the differential external quantum efficiency, y]d = e(d4ext/dI),to discuss experimental results. Equation (28) shows that q d equals F3q” below threshold and F,q’ above threshold. We expect that q’ > q”, since nonradiative processes will saturate once the lasing threshold is reached and since lasing should lead to a more favorable injection ratio. Furthermore, as discussed above, Fl > F3 in most cases. Thus we expect a sharp rise in the differential external quantum efficiency above threshold, as illustrated in Fig. 6. When currents far above threshold are used, the diode temperature rises, and qd usually falls.77 . the ~ dependence ~ on sample length of the differential Cheroff et ~ 1 used external efficiency above threshold to deduce the absorption coefficient ” S.

V. Galginaitis, J . Appl. Phys. 35, 295 (1964).

’* G. Cheroff, C. Lanza, and S. Triebwasser, Rev. Sci. Instr. 34, 1138 (1963).

14.

STIMULATED EMISSION IN SEMICONDUCTORS

393

FIG.6 . Light output vs current in three GaAs injection lasers at 77°K. The diodes were made by diffusing zinc into n-type inaterial with about 1.5 x 10” carriers/cm’. (After S. V. Galginaitis, Ref. 77.)

a” of Eq. (26) and the internal quantum efficiency q’. They found q’ = 0.7 and a” = 56 cm-’, but these values must be treated with some caution, since differences between units of nominally identical properties are often large enough to obscure the dependence of external quantum efficiency on length. The value 56 cm-’ is rather higher than one would expect from the considerations which led to Eq. (27). But in any case the internal quantum efficiency at 77°K must be greater than 0.5 since differential external quantum efficiencies of this magnitude have been reported.77 At very low current densities the linear dependence of light output on current predicted by (28a) is no longer observed for GaAs diodes.79 Other 79

R. J. Keyes and T. M. Quist, Proc. 1 R E 50, 1822 (1962).

394

FRANK STERN

processes compete with the radiative process, but become less important at higher currents. It is interesting that in diodes made from a number of 11-VI the linear relation between light output and current persists down to the lowest currents for which light can be detected. External quantum efficiencies up to 0.12 at 77°K have been reported for spontaneous emission from these compounds.*' The fractional escape probability for the spontaneous emission is expected to increase as the transmissivity of the surface increases, and this has been verified in experiments in which electroluminescent diodes have been coated with or imbedded in materials with lower dielectric con~tant.'~,'~

10. T E M P ~ T U DEPENDENCE RE AND HEATING EFFECTS The external quantum efficiency for both the spontaneous and the lasing emission decreases monotonically with increasing t e m p e r a t ~ r e . ~ ~ . ~ ~ . ' ~ Much of the decrease is associated with a decrease in escape probability due to increasing absorption of the emitted radiation as the temperature r i s e ~ . ' ~ - ' ~Hill73a ~ has deduced room-temperature internal quantum efficiencies of about 50 % from analysis of spontaneous emission from diffused diodes. The highest reported differential external quantum &ciency at room temperature is about 0.4 for a lasing diode operated with 120 A, 100-ns pulse^.'^ That fraction of the input power which is not emitted as radiation from the crystal, or conducted as heat through the contacts, leads to heating of the diode. A number of authors have considered the heat flow problem, both for steady state and for pulsed conditions, to estimate the temperature The calculations are uncertain to some extent rise near the because it is not accurately known what fraction of the heat generated is localized near the junction, as it would be if produced by phonon-assisted G. Mandel and F. F.Morehead, Appl. Phys. Letters 4,143 (1964). F.F.Morehead and G. Mandel, Appl. Phys. Letters 5, 53 (1964). 0. A, Weinreich, J . Electrochem. SOC.110,1124 (1963). R. S.Levitt, unpublished work, 1963. 84 W. N.Carr and G. E. Pittman, Proc. IEEE 52,204 (1964). 85 W. G. Spitzer and J. M. Whelan, Phys. Rev. 114, 59 (1959). 86 W. J. Turner and W. E. Reese, J . Appl. Phys. 35, 350 (1964). 86aD.E.Hill, Phys. Rev. 133, A866 (1964). 87 H. Nelson, J. I. Pankove, F. Hawrylo, and G. C. Dousmanis, Proc. IEEE 52, 1360 (1964). S. Mayburg, J . Appl. Phys. 34, 3417 (1963). J. P. Quine, K. Tomiyasu, and C. Younger, Proc. IEEE 51, 1141 (1963). W. E. Engeler and M. Garfinkel, J . Appl. Phys. 35, 1734 (1964). 91 G. J. Lasher and W. V. Smith, IBM J . Res. Develop. 8, 532 (1964). R. W. Keyes, IBM J . Res. Develop. 9,303 (1965). 82

83

14.

STIMULATED EMISSION IN SEMICONDUCTORS

395

nonradiative recombination there, and how much is generated throughout the diode by absorption of the isotropic part of the radiation emitted at the junction and by Joule heating. Nevertheless, one can obtain useful estimates of the temperature rise near the junction. If the temperature of the active layer rises during the operation of the diode, the emission peak moves to longer wavelength with increasing temperature, and can either be followed in time if suitable techniques are USed,87.92-93a or gives rise to an apparent broadening of the emitted spectrum. More important, the threshold current density rises with increasing temperatureY4 as discussed in Section 14. Thus, continuous operation at a given ambient temperature may not be possible if the temperature rise caused by increasing the current through the laser makes the threshold current rise faster than the applied current. The steady-state temperature rise at the junctiong4’ is determined by the dimensions and the thermal c o n d ~ c t i v i t y ~of~the ” diode, and by the heat sink to which it is attached. Continuous operation with an output power of W has been reported at temperatures up to 90°K,9 and can undoubtedly be extended further, but room temperature continuous operation will be difficult.88 Higher continuous power outputs can be obtained at lower temperature^.^.^^ To avoid heating effects, injection lasers are usually operated with short pulses. The thermal analysis in this case is somewhat different from the steady-state analysis, since the heat generated at the junction is confined to a layer of thickness z (9t)’” after a time t , where 9 is the thermal diffusivity. The treatment is further complicated by the time delay required to build up the population inversion necessary for lasing. Konnerth and L a n ~ have a ~ ~shown that if trapping effects are neglected the delay is

4

where z is the carrier recombination lifetime in the active layer, and I and It,, are the current and its threshold value, respectively. Thus lasing can be achieved with pulsed operation if at the end of the delay time the pulse current exceeds the threshold current for the temperature the active layer has reached. J. D. Kingsley and G. E. Fenner, in “Quantum Electronics” (3rd Intern. Congr.), Vol. 2, p. 1883. Dunod, Paris, 1964. 93 K. Konnerth, Proc. IEEE 53, 397 (1965). g3aC.H. Gooch, Phys. Letters 16, 5 (1965). 94G. Burns, F. H. Dill, Jr., and M. I. Nathan, Proc. IEEE 51, 947 (1963). 94a M. H. Pilkuhn and H. Rupprecht, IBM J . Res. Develop. 9, 400 (1965). 94bSee,for example, R. 0. Carlson, G. A. Slack, and S. J. Silverman, J . Appl. Phys. 36, 505 92

(1965).

3%

FRANK STERN

VI. Radiation Confinement, Threshold, and Loss 11. MODESIN A DIELECTRIC SLAB

In this section we describe some approximate solutions for the electromagnetic modes which are generated in injection lasers with Fabry-Perot structure. If we know the index of refraction n and the absorption coefficient M: for the medium at each point, and for each angular frequency w,we can calculate the modes of the crystal. The spontaneous emission rate rSpn at each point x and angular frequency w is the driving term which determines the energy present in each of the modes. We shall assume that the medium is optically isotropic. Because we do not know n, a, and rSponfor all points and for all relevant w, and because the complete solution of the problem is quite formidable even with this information, we make a number of approximations. In particular, we work only at a single value of w ,generally the value for the lasing mode or modes. Doing the calculation for a range of values of w would give a theoretical prediction of the angular frequency at which there is greatest gain but would increase the labor required, since each value requires a separate solution. We consider only the case of electric fields polarized in the plane of the p-n junction, the TE modes, because the problem is somewhat simpler for these, and because we do not expect the results to be substantially different for the TM If the junction plane is the y-z plane, as in Fig. 1, and the waves propagate in the z direction, we can write the electric field of a mode in the form E,(x, y, z, t ) = u(x) exp(ikz

+ )Gz

- iwt)

(10)

given previously, where k is the z component of the propagation vector and G is the power gain per unit distance traversed.95 G, k, and u(x) will vary from mode to mode, and are determined by the solutions of the wave equation dZU/dX2

+ [(iG + ik)’ + ( O ~ / C ~ ) K ( X ) ] U ( X=) 0,

(30)

where K(X)

=

[n(x)+ +iCCr(X)/W]2

is the complex dielectric constant, which depends only on the distance No consistent polarization is found experimentally. See, for example, Y. Nannichi, Japan. J . Appl. Phys. 3, 360 (1964). 95 We assume that k is positive, corresponding to a wave moving in the direction of increasing z. Then G > 0 indicates a growing wave. Changing the sign of both k and G reverses the direction of the wave.

94c

14. STIMULATED

EMISSION IN SEMICONDUCTORS

397

perpendicular to the junction plane. The corresponding equation for TM modes is given by B r e k h o v ~ k i k hand ~ ~ by Cooley and Stern.97 Equation (30)is a complex eigenvalue equation, in which - ~ ( xplays ) the role of the potential energy in the one-dimensional Schrodinger equation, and ( $ 3 + ik)’ is the complex eigenvalue. Because the potential is complex, or non-Hermitian, Eq. (30) is not a self-adjoint differential equation, and many powerful results for solutions of such equations do not apply. We can, however, conclude that for the dielectric constants K ( X ) which arise in our work, the solutions contain both a discrete spectrum, for which lu(x)12 can be normalized to unity, and a continuous spectrum for which u(x) is bounded but not normalizable. If we label the limiting values of the optical constants as x goes to - a and + a with the subscripts L and R, respectively, then we find that the continuous spectrum has two sets ofeigenvalues: Gk

=

-wnLciL/c,

Ikl < w n J c ,

(324

Gk

=

-wnRciR/c,

jk( < o n , / c ,

(32b)

A solution u(x) for one of the eigenvalues (32a) will be a linear combination of incoming and outgoing plane waves for large negative values of x ; the relative amplitude of the two plane waves is determined by the requirement that the solution for large positive values of x be bounded. For two eigenfunctions ui(x)and uj(x) belonging to different eigenvalues, at least one of which is in the discrete spectrum, it is easy to show that a,

ui(x)uj(x)d x

-m

=

0.

133)

For an eigenfunction u(x) of the discrete spectrum, normalized to give s(u(x)12dx = 1, we have98 Gk

=

-

j

m --oo

wc- ‘n(x)a(x)lu(x)I2 dx

.

( 34)

When a(x) is identically zero, this becomes the familiar result that the eigenvalues of an Hermitian Hamiltonian are real. In the cases of interest to us, the fractional variation of n(x) is small in the regions where lu(x)12 is appreciable, and we have k x wiilc, where ii is the average value of the index. Then (34) becomes m

G

%

cx(x)lu(x)l’ d x ,

-

(35)

-0c

96L.M. Brekhovskikh, “Waves in Layered Media,” p. 168. Academic Press, New York, 1960. 97

J. W. Cooley and F. Stem, IBM J . Res. Develop. 9, 405 (1965). F. Stern, Bull. Am. Phys. SOC.9, 270 (1964).

398

FRANK STERN

which shows that the gain G is an average of -tl(x), weighted by the probability distribution for the field intensity. A numerical procedure for finding the discrete eigenvalues and the corresponding eigenfunctions for complex ~ ( x )has been developed by C ~ o l e ybased , ~ ~ on his method for solving the one-dimensional Schrodinger equation for real potential^.^^ In general there will be more than one discrete solution, and it is not possible to use simple methods like counting nodes, which are available for real potentials, to make sure that no solutions have been overlooked. Procedures for counting and locating the solutions will be described elsewhere.97 12. RESULTS FOR MULTILAYER STRUCTURE^

In an injection laser the index of refraction n(x) and the absorption coefficient a(x) vary continuously with distance. Our present knowledge of these quantities and their variation with distance is so uncertain, however, that it is more instructive to work with a multilayer dielectric slab in which the optical constants are constant within each layer, but vary from layer to layer. The simplest such problem, a layer with a real dielectric constant imbedded in a medium which has a different, but also real, dielectric constant, is a standard problem in the propagation of electromagnetic waves6* Solutions of three-layer dielectric slab problems in the context of injection lasers have been given by a number of authors, usually with one or more approximations. Lasherioo and Hall and Olechna"' considered a symmetrical three-layer slab (in which the two outer layers have the same optical constants) in which the index of refraction was the same in all three layers. It became apparent soon after the discovery of injection lasers that a higher index of refraction was required in the active layer than in adjoining layers to explain the observed threshold current densities and internal losses, which were much smaller than those predicted by the theory without index differences. A number of author^'^^-'^^ considered the symmetrical case with unequal indices of refraction, making the assumption that the inner layer is thin. A general three-layer slab was considered with the assumption of a thin inner layer by McWhorter,16 and without that assumption by Antonoff,6' and Stern.35 Where both TE and TM modes have been J. W. Cooley, Math. Comp. 15, 363 (1961). G. J. Lasher, IBM J. Res. Deoelop. 7, 58 (1963). l o l R. N. Hall and D. J. Olechna, J. Appl. Phys. 34, 2565 (1963). l o * A. L. McWhorter, H. J. Zeiger, and B. Lax, J . Appl. Phys. 34, 235 (1963). ’03 A. Yariv and R. C. C. Leite, Appl. Phys. Letters 2, 55 and 161 (1963). R. C. C. Leite and A. Yariv, Proc. IEEE 51, 1035 (1963). 99

loo

14.

STIMULATED EMISSION IN SEMICONDUCTORS

399

considered, the differences have been found to be small for the ranges of parameters applicable in injection lasers. The most important layer in a multilayer slab model of an injection laser is the active layer, the layer in which the recombination of electrons and holes takes place, and in which a is negative at sufficiently high current densities. In three-layer slabs, the active layer is always the middle layer. We shall assume that in conventional zinc-diffused GaAs injection lasers the active layer lies mainly on the p-type side of the p-n junction. This is supported by direct observation of the light emission, both above and below thresh01d.l’~ It is consistent with the wavelength of the emitted light,3740 and with the observed invariance of the quantity n of Eq. (9) to the donor concentration of the substrate over a wide range.6’ Calculations of the flow of electrons and holes in linearly graded junctions with parameters representative of GaAs have shown that the bulk of the recombination, as measured by the product of the electron and hole concentrations, takes place on the p-type side of the junction.106 If the thicknesses and optical constants of all the layers in a multilayer slab model are given except for the absorption aactin the active layer, we can find the gain G as a function of aaCt.For the experimentally accessible range of values of G , this function is often linear, and we can then write

G

=

- raact -

a’,

where the dimensionless coefficient r and the loss term a’ depend parametrically on the dimensions and optical constants other than aact.If there are several modes for which solutions exist, then each mode will have its own values for r and a‘. We can use Eq. (35) as a guide to the interpretation of the constants r and a’. If the relation between G and aaCtis quite closely linear, then r is simply the fraction of the energy of the wave which lies within the active layer. Our calculations show that r z 0.9 for the lowest-order mode in a typical zinc-diffused GaAs injection laser. The loss term a’ can be thought of as the absorption which arises because part of the energy of the mode lies in the absorbing layers of the multilayer slab. It has sometimes been called the “diffraction ~oss,”’’~although the term “penetration loss” or “leakage loss” might be more appropriate. The absorption constant in the active layer of a n injection laser will depend on the current density J , and we shall assume that over a certain A. E. Michel, E. J. Walker, and M. I. Nathan, IBM J . Res. Deoelop. 7, 70 (1963). L. Esaki and J. W. Cooley, unpublished work, 1963. I am indebted to Dr. Esaki for permission to refer to these results. lo’ Lasher’s’’’ a, is defined as the value of -aae, required to give G = 0. It is equivalent to our a‘/T when Eq. (36) applies.

400

FRANK STERN

range this relation is linear. We write it in the form a t,

=

-b(J

- Jo)

+ aft,

(37)

where afc is the free-carrier coefficient absorption in the active layer. At very low temperatures, Jo vanishes, and one can show that"' b = 1.58 x 10-5q(n2E2d,,,AE)-',

(38) where E and AE are the energy and width"* of the emission line below threshold in electron volts, q is the internal quantum efficiency, n is the index of refraction, and d,,, is the active layer thickness in centimeters. If we substitute (37) into (36), we find that

G = /?J

- a,

(39)

where

/? = Tb, a=

13.

COMPARISON WITH

+ raft + rbJo.

(40) (41)

EXPERIMENT

Both G and J in Eq. (39) cz. be measured directly, and one can in principle measure a and by varying the current density.29 At threshold the gain saturates at the value G,, = L- log(R-') given by Eq. (5). For GaAs lasers immersed in liquid nitrogen, R = 0.25, and log R - ' = 1.39. All of the reported values of a and /? have been determined from the relation between G,, and the threshold current density J t h . The gain required at threshold has been varied by varying either the length of the or the reflectivity of the Typical values"' of a and p are given in Table I. The experimentally determined internal loss a for conventional zincdiffused GaAs lasers at 77°K is typically 15 cm-' or less, and one of the major goals of the optical model of laser operation is to account for the magnitude of this loss, which is much smaller than predicted by models without a variation in the index of refraction.'OO*'O' Of the three terms in (41) which contribute to a, the free-carrier term is easily estimated. From the spontaneous lifetime of about 1 ns found in The width AE used here is the integral of rspon(E)over the spontaneous line, divided by the peak value of r.pon. M. H. Pilkuhn and H. Rupprecht, Proc. IEEE 51, 1243 (1963). ' l o M. H. Pilkuhn, H. Rupprecht, and S. E. Blum, Solid-state Electron. 7, 905 (1964). M. H. Pilkuhn and H. Rupprecht, in "Radiative Recombination in Semiconductors" (7th Intern. Cod.), p. 195. Dunod, Paris and Academic Press, New York, 1965. ""Y. Nannichi, Japan. J . Appl. Phys. 4, 53 (1965).

14.

STIMULATED EMISSION IN SEMICONDUCTORS

401

GaAs diodes at 77°K by Konnerth and L a n ~ a and , ~ ~Dumke's theory34 for spontaneous recombination of injected electrons in p-type material, one can deduce an average hole concentration of about 2 x 1018cm-3 in the active layer. The data of Turner and Reeses6 on free-carrier absorption then give afc = 4 cm-'. If, as we shall find to be the case, r is close to 1, the remaining two terms in (41) can add to only about 10 cm-'. TABLE I THEINTERNALLoss a A N D THE GAINFACTOR /3 OF EQ. (39)"

Temperature

a

B

(OK)

(cm- I)

(WA)

4.2

13 15

5.1 x lo-* 2.5 x 1.8 10-3 4.9 x 10-4

77 195 296

16 30

' As measured at several temperatures by Pilkuhn et al.llo on lasers made by zinc diffusion into GaAs with 5 x 10" donor atoms per cm'.

We have already stated that at low temperatures J o will vanish. The same is true at any temperature if the transitions responsible for the gain arise from electrons injected into a band tail in which the density of states varies exponentially with energy. Such tails have been observed in many diodes,' although the steepness of the exponential rise varies from diode to diode. The theory of Lasher and Stern," which assumes a parabolic conduction band, gives a dependence of gain on current which is more nearly quadratic than linear at 77°K. If their curves are approximated by Eq. (37), with afc = 0, near typical threshold values of J , the resulting value of J , gives a loss much higher than the observed values. Thus we conclude that the tail in the density of states is important in determining the dependence of gain on current, and tends to make J , much smaller than it would be for the abrupt onset of the conduction band density of states which one has for a parabolic band. In the considerations which follow we shall assume that J , vanishes, although the experiments d o not rule out J o values whose magnitude is small compared to Jth, and whose sign may be either positive or negative. The remaining term in the loss is the penetration loss a'. This term can be made small by widening the active layer, by reducing the absorption in the adjoining layers, and by increasing the difference between the index

'*

"*

See, for example, D. F. Nelson, M.Gershenzon, A. Ashkin, L. A. D'Asaro, and J. C. Sarace, Appl. Phys. Letters 2, 182 (1963).

402

FRANK STERN

of refraction in the active layer and the index in the adjoining layers. Stern35 attempted to find the combination of these factors which was consistent with the values of c i deduced from experiment, and with other experiments. It was found that the index difference between the active and the adjoining layers must be of the order of 0.02, and the active layer thickness about 13p, if the loss and the width of the far-field pattern perpendicular to the junction plane56965are to be accounted for. If the active layer width and the loss do not vary strongly with temperature between 77" and 2"K, then the threshold current density predicted at low temperature using Eqs. (37) and (38) is in agreement with experiment. The index difference between the active layer and the adjacent layers which is required by our analysis is only partly accounted for by the freecarrier contribution to the index.' O3 Index changes directly associated with the population inversion'" are probably and appear experimentally to have the wrong sign for radiation ~onfinement.~~.' l 3 The remaining effect appears to come from the sensitivity of the index of refraction near the absorption edge to the position and steepness of the edge,62 and to the variation of the absorption edge with impurity concentration.86.86aThe experiments show that the absorption edge in uncompensated samples lies at lowest energies for acceptor concentrations near 1018~m-3,86a and that compensation moves the edge to lower energies.l14*' In a typical diffused p-n junction the impurity content goes from n type, to compensated n type, to compensated p type, to heavily doped and slightly compensated p type. The minimum absorption edge will occur in the compensated p-type material near the junction, and this is where we believe most of the recombination takes place. At the wavelength of the emission from such a junction, the relation between the index of refraction and the absorption edge6' suggests that the index also has its peak on the p-type side of the junction, which favors radiation confinement. Also favorable is the fact that the higher absorption edge in the layers adjacent to the active layer leads to lower absorption for the light that does penetrate into these layers. The lowering of the threshold by compensation noted by Winogradoff and Kessler'16 may well be due to the same effects. A similar index rise is required to explain the light-channel effect found in GaP p-n

'

’14

"* 'I'

S. Iida and T. Kushida, Japan. J . Appl. Phys. 3, 162 (1964). F. Stern and J. R. Dixon, J . Appl. Phys. 30,268 (1959). G. Lucovsky, Appl. Phys. Letters 5, 37 (1964). N. N. Winogradoff and H. K. Kessler, Solid State Commun. 2, 119 (1964). A. Ashkin and M. Gershenzon,J . Appl. Phys. 34, 2116 (1963). D. F. Nelson, and F. K. Reinhart, Appl. Phys. Letters 5, 148 (1964).

14. STIMULATED EMISSION

403

IN SEMICONDUCTORS

We conclude that a multilayer model with reasonable constants can account for the observed properties of conventional zinc-diffused GaAs injection lasers. Some of the assumptions of the model, particularly the assumptions about the index of refraction, must await confirmation by independent experiments. The same theory can be applied to other chemical structures, like those which we describe in Section 1, but it is desirable to have experimental values for the threshold current density, the internal loss a, and the far-field angular dependence for these structures. 14. TEMPERATURE DEPENDENCE

The threshold current density of conventional zinc-diffused GaAs injection lasers is lowest at low temperatures, rises with increasing steepness (on a log-log plot) as the temperature increases, and varies approximately as T 3 between liquid nitrogen temperature and room temperat~re.’~ Pankove“’ has found that the threshold for some lasers rises approximately exponentially with temperature. The internal loss a, on the other hand, varies very little with temperature.”.’ The main factor which contributes to the increase of threshold current density with temperature is the spreading of the Fermi-Dirac distribution function which decreases the occupation of the electron and hole states at energies near the band edge. Thus the degree of excitation, as measured by the difference AF between electron and hole quasi-Fermi levels, must be increased to maintain the same peak gain. Decreases in internal quantum efficiency with rising temperature also lead to an increase in threshold. Although there have been some attempts to explain the temperature dependence of the threshold current density,22*’20they have ignored the temperature dependence of the active layer width d,,, and of the internal quantum efficiency. Lasher and Stern,22 who assumed that there is no selection rule for the recombination near the band edge, found I,,, T2, while Mayburg’” found Ith T’.’ . Further work seems warranted, particularly if more accurate descriptions of the wave functions and energy levels of states near the band edges in impure semiconductors become available.

’’*’‘’

-

-

VII. Laser Materials In this section we summarize some of the stimulated emission results that have been reported for materials other than GaAs, which has been J. I. Pankove, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.),p. 201, Dunod, Paris and Academic Press, New York, 1965. ‘*‘S. Mayburg, J . Appl. Phys. 34, 1791 (1963).

404

FRANK STERN

the prototype semiconductor laser in our discussion so far. In principle any semiconductor can be used to make an injection laser, provided that an appreciable fraction of the recombination of excess electron-hole pairs is radiative, that the available gain exceeds the losses, and that p-n junctions which admit sufficiently large forward injection currents can be made. The second of these conditions tends to eliminate semiconductors with indirect transitions at the energy gap, since the available gain from a population inversion is quite small in such materials.’” The last condition eliminates most ionic materials with large band gaps, since these usually can be made only n type or p type, rarely both.lZ2 A number of mechanisms available for exciting luminescence in semiconductors are summarized by FischerlZ3and I ~ e y . ’ ’Of ~ these, excitation by an external electron beam has proved to be a successful means of inducing lasing. Cathodoluminescence, the excitation of light in solids by Excitation of fast charged particles, predates the semiconductor field. lasing by this method should have substantial interest quite apart from the lasing itself. For example, experimental and theoretical investigation of quantum efficiency and energy transfer mechanisms is of considerable importance. The details of laser operations will differ from one mode of excitation to another, but much of the discussion presented in this chapter will apply to stimulated emission no matter what the mode of excitation. In the brief listing below we give first some results for GaAs not previously mentioned, and then mention other materials in which stimulated emission has been reported or sought.

GaAs. Stimulated emission in GaAs has been obtained not only in p-n junctions, but also in homogeneous samples excited by light’ 25a.1Zsb and by electron beams.’2s”’2se .ZnP. The energy gap of InP is only slightly smaller than that of GaAs, W. P. Dumke, Phys. Rev. 127, 1559 (1962).

”’G. Mandel, Phys. Reu. 134, A1073 (1964).

A. G. Fischer, Solid-state Electron. 2, 232 (1961). H. F. hey, “Electroluminescence and Related Effects.” Academic Press, New York, 1963. Cathode rays were discovered by observation of the luminescence they induce in the glass walls of a cathode ray tube. See, for example, J. J. Thomson, in “Encyclopedia Britannia,” 11th ed.,Vol. 6, p. 887,1910, or P. Pringsheim and M. Vogel, “Luminescence of Liquids and Solids,” Interscience, New York, 1946. lZsa J. J. Schlickman, M. E. Fitzgerald, and R. H. Kingston, Proc. IEEE 52, 1739 (1964). lZsbN.G. Basov, Physics of Quantum Electronics Conference, San Juan, 1965. McGraw-Hill, New York, 1965. C. E. Hurwitz and R. J. Keyes, Appl. Phys. Letters 5, 139 (1964). D. A. Cusano and J. D. Kingsley, Appl. Phys. Letters 6, 91 (1965). lZ5’P.D. Coleman and G. E. Bennett, Proc. IEEE 53,419 (1965).

lZ3

lZ4

14.

STIMULATED EMISSION IN SEMICONDUCTORS

405

and InP lasers 126-128 have properties quite like those of GaAs. An InP laser with four cleaved sides’28 demonstrates very clearly the expected22 shift of the lasing peak from the peak of the spontaneous emission at very low temperature to lower energies as the temperature is raised, an effect which has also been observed in G ~ A s . ’ ~ ~ ZnAs. Lasing in InAs has been excited by forward injection in p-n junctions’30 and by a beam of 20-keV electron^.'^' Magnetic field effects were found in some of the experiments, and will be discussed below. ZnSb. Of the 111-V semiconductors, InSb has been one of the most intensively studied. It has the smallest electron effective mass, the highest mobilities, and can be prepared with the highest purity. Lasing in InSb has been attained in a number of laboratories using p-n junction^,'^^-'^^^ electron beam e ~ c i t a t i o n , ’ ~and ~ optical Magnetic fields lower the threshold substantially, an effect we shall discuss in the following section. GaSb. Although the difference between the direct and indirect gaps is quite small, lasing has been achieved in GaSb at 77°K by injection’36 and by electron bombardment. but there are a number of Sic. Lasing has been reported in reasons to doubt the report.’39 No further results have been published as of the summer of 1965.

’

K. Weiser and R. S. Levitt, Appl. Phys. Letters 2, 178 (1963). I Z 7 G .Bums, R. S. Levitt, M. I. Nathan, and K. Weiser, Proc. IEEE 51, 1148 (1963). K. Weiser, R. S. Levitt, M. I. Nathan, G. Burns, and J. Woodall, Trans. AIME 230, 271 (1964).

G. Burns and M. I. Nathan, Proc. IEEE 51, 471 and 860 (1963). I. Melngailis, Appl. Phys. Letters 2, 176 (1963). 131 C. Benoit a la Guillaume and J. M. Debever, Solid State Commm. 2, 145 (1964). 1 3 * R. J. Phelan, A. R. Calawa, R. H. Rediker, R. J. Keyes, and B. Lax, Appl. Phys. Letters 3, 130

143 (1963).

Benoit a la Guillaume and P. LaVallard, Solid State Commun. 1, 148 (1963). M. Bernard, C. Chipaux, G. Duraffourg, M. Jean-Louis, J. Loudette, and J.-P. Noblanc, Compt. Rend. 257, 2984 (1963). 134a I. Melngailis, R. J. Phelan, and R. H. Rediker, Appl. Phys. Letters 5, 99 (1964). C. Benoit A la Guillaume and J. M. Debever, in “Radiative Recombination in Semiconductors’’ (7th Intern. Conf.), p. 255. Dunod, Paris and Academic Press, New York, 1965. 1 3 ” R. J. Phelan, Jr., and R. H. Rediker, Appl. Phys. Letters 6, 70 (1965). 135bR.J. Phelan, Jr., Physics of Quantum Electronics Conference, San Juan, 1965. McGrawHill, New York, 1965. 136C. Chipaux, G. Duraffourg, J. Loudette, J.-P. Noblanc, and M. Bernard, in “Radiative Recombination in Semiconductors” (7th Intern. Conf.), p. 217. Dunod, Paris and Academic Press, New York, 1965. 1 3 7 C. Benoit fi la Guillaume and J. M. Debever, Compr. Rend. 259, 2200 (1964). I3’L. B. Griffiths, A. I. Mlavsky, G. Rupprecht, A. J. Rosenberg, P. H. Smakula, and M. A. Wright, Proc. IEEE 51, 1374 (1963). R. N. Hall, Proc. IEEE 52, 91 (1964). 133 C. 134

406

FRANK STERN

GaAs,P,-,. Soon after the first GaAs lasers were reported, lasing was There has been considerable interest found in alloys of GaAs and in this s y ~ t e m ' ~ ' - 'because ~ ~ ~ it offers the possibility of getting a laser with visible light output. Unfortunately, the transition from a direct to an indirect gap occurs at x x 0.55, and the shortest wavelength at which lasing has been observed is 6380A at 77°K.146 There is evidence that fluctuations in composition can lead to large internal losses and high threshold current densities in these alloys. Other alloys. In alloy systems in which the gap remains direct, it should be possible to make lasers over the entire composition range. That has been done for 1nAs-GaA~'~'and for I~AS-IIIP.'~~ ZI-VI compounds. Line narrowing in CdS excited at 77°K by fast electrons The 11-VI compounds have so far resisted has been attempts at exciting lasing by forward injection in p-n junctions, in spite of the high quantum efficiency found in the electroluminescence of CdTe and CdTe-ZnTe alloys.80*81 The relatively high contact and bulk resistance of these materials and their low thermal conductivity lead to severe heating when high currents are sent through the samples. Lead salts. PbS, PbSe, and PbTe all have direct optical transitions at the band edges, which lie at the ends of the (111) directions in the Brillouin zone for both conduction and valence bands. Lasing would therefore be expected in all three compounds, and has been found both in diodes'51-152~ and by electron beam pumping'52d in all three. N. Holonyak, Jr., and S. F. Bevacqua, Appl. Phys. Letters 1, 82 (1963). N. G. Ainslie, M. H. Pilkuhn, and H. Rupprecht, J . App!. Phys. 35, 105 (1964). "'N. Holonyak, Jr., S. F. Bevacqua, C. V. Bielan, and S. J. Lubowski, Appl. Phys. Letters 3, 47 (1963). N. Holonyak, Jr., Trans. AIME 230,276 (1964). M. H. Pilkuhn and H. Rupprecht, Trans. AIME 230,282 (1964). 145 T. A. Fulton, D. B. Fitchen, and G. E. Fenner, Appl. Phys. Letters 4, 9 (1964). 146 M. H. Pilkuhn and H. Rupprecht, J . Appl. Phys. 36, 684 (1965). 14’ D. A. Cusano, G. E. Fenner, and R. 0. Carlson, Appl. Phys. Letters 5, 144 (1964). 14" J. J. Tietjen and S. A. Ochs, Proc. IEEE 53, 180 (1965). 148 I. Melngailis, A. J. Strauss, and R. H. Rediker, Proc. IEEE 51, 1154 (1963). 14' F. B. Alexander, V. R. Bird, D. R. Carpenter, G. W. Manley, P. S . McDermott, J. R. Peloke, H. F. Quinn, R. J. Riley, and L. R. Yetter, Appl. Phys. Letters 4, 13 (1964). 150N. G. Basov, 0. V. Bogdankevich and A. G. Devyatkov, Zh. Eksperim i Teor. Fiz. 47, 1588 (1964) [English Transl.: Soviet Phys. J E T P 20, 1067 (1965)l. nO’ N. G. Basov and 0. V. Bogdankevich, in "Radiative Recombination in Semiconductors" (7th Intern. Cod.), p. 225. Dunod, Paris and Academic Press, New York, 1965. 15' J. F. Butler, A. R. Calapa, R. J. Phelan, Jr., T. C. Harman, A. J. Strauss, and R. H. Rediker, Appl. Phys. Letters 5, 75 (1964). 15'F. A. Junga, K. F. Cuff, J. S. Blakemore, and E. R. Washwell, Phys. Letters 13, 103 (1964). 152* J. F. Butler, A. R. Calawa, R. J. Phelan, Jr., A. J. Strauss, and R. H. Rediker, Solid State Commun. 2, 303 (1964). 141

14.

STIMULATED EMISSION IN SEMICONDUCTORS

407

Tellurium. The only elemental semiconductor in which lasing has so far been achieved is Te, which has been excited by electron beams.'52e We see that lasing may be found in many semiconductor materials, over a wide range of band gaps. One important goal for the future is to extend this range toward higher energies, to make possible semiconductor lasers with wavelengths shorter than their present limit in the red part of the visible spectrum. VIII. Effects of Ambients and External Fields

In this section we describe some of the ways in which the properties of' lasers are affected by changes in temperature, pressure, uniaxial strain, or magnetic field. Effects of temperature and pressure on the spontaneous emission depend primarily on the change of the energy gap and will not be described in detail. Each of the effects we describe can be used in principle as a fine tuning scheme for the lasing energy. 15. TEMPERATURE

As the temperature of an electroluminescent diode is changed, the peak photon energy of the emission shifts approximately at the same rate as the energy gap. For lasing diodes with Fabry-Perot structure there is in addition a shift of individual modes with temperature, usually at a smaller rate. Thus, as the temperature is varied, any particular mode will change its wavelength continuously, but will have its amplitude diminish and become unobservable when it no longer falls within the spontaneous emission envelope. The temperature shift of an individual mode is found from Eq. (7c) to be dI/dT = (I/n')[(dfi/dT)+ ( E / L ) ( d L / d T ) ] , (43) where n’ is defined in Eq. (9). For the materials we consider, the thermal expansion term in (43) is negligible. The temperature dependence of the lasing modes has been measured for GaAs'2~94*'29~'53 and for InP'27s'2S at a number of temperatures, and the temperature dependence of the index of refraction at the wavelength of the lasing modes has been deduced. The results are summarized in Table 11. J. F. Butler, A. R. Calawa, and R. H. Rediker, ZEEE J. Quantum Electron. 1,4 (1965). J. F. Butler and A. R. Calawa. Physics of Quantum Electronics Conference, San Juan, 1965. McGraw-Hill, New York, 1965. 1f2dC.E. Hurwitz, A. R. Calawa, and R. H. Rediker, IEEE J. Quantum Electron. 1, 102 (1965). C. Benoit a la Guillaume and J . M. Debever, Solid State Commun. 3, 19 (1965); Physics of Quantum Electronics Conference, San Juan, 1965. McGraw-Hill, New York, 1965. l S 3 W. E. Engeler and M. Garfinkel, J. Appl. Phys. 34, 2746 (1963). 15"

TABLE I1 OBSERVED VALUESOF n’ = E - A(dii/dA) AND

EFFECTSOF TEMPERATURE, PRESSURE,AND MAGNETIC INDUCTIONCHANGES ON THE ENERGY LASINGMODESAND ON THE INDEX OF REFRACTION OF SEVERAL INJECTION LASERS

OF THE

OR WAVELENGTH OF INDIVIDUAL

~

Semiconductor

Ambient

@m)

n’

Energy change

0.8400 0.9000 0.8395 0.9030

5.4 4.4 5.5 5 k0.5

-8.1 x -6 x -7.4 x 10-5 -8 x

0.46 0.4 0.42 0.55 f 0.1

2.9 x 10-4 2 x 10-4 2.8 x 10-4 3 x 10-~

12,94 94,129 153 127,128

0.8390 0.8675 0.7990

5.4 5.2

3.5 x 3.06 x (4 _+ 1) x

- 0.020 - 0.0186 -0.021 f 0.005

-1.1 x 10-5 -9.6 x

154 156 155

67°K

1.476 1.38 1.477 1.37

GaAs GaAs GaAs

77°K 200"K, lo00 atm 77"K, 7000 atm

1.478 1.429 1.55

77°K 300°K

50 K

Magnetic induction effects

(eV/kG)

2°K 1.7"K, 27kG 2"K, 52kG 4.2"K, 6kG

0.234 0.239 0.245 0.398 ~

~

Wavelength

GaAs GaAs GaAs InP

InSb InSb InSb lnAs

~

Energy (eV)

~~

1 atm = 1.013 bar = 1.033 kg/crn2. Calculated from data in the paper cited.

5.30 5.18 5.07 3.12 ~

5.22 4.1 5.9 4.5' ~

~

7 x 10-5 3.3 x 10-5 ~

~~~~~~~~

Wavelength change

(&kG 1

- 15 - 2.6

Index change

References

(G-1)

-1.7 x -3.8 x

133 132 164 163

;1

w

14.

STIMULATED EMISSION IN SEMICONDUCTORS

409

The temperature dependence oft he index of refraction near the absorption edge tends to be considerably larger than at longer wavelengths. Engeler and Garfinkells3 obtained good agreement with their data over a temperature range from 20" to 60°K by relating the change in the index to the temperature dependence of the energy gap. Stern62 found good agreement with the observed value at 77°K by taking the temperature dependence of the index to be a sum of two terms, one related to the shift of the gap, and the other related to the temperature dependence of the index at long wavelengths, which arises primarily from absorption processes in the visible and ultraviolet. Both models emphasize the importance of the rapid dispersion of the index associated with the presence of a steep absorption edge.

16.

h€?SSURE

The effect of pressure changes on the wavelength of individual lasing modes is completely analogous to the temperature effect we discussed above, and Eq. (43) applies if temperature derivatives are replaced by pressure derivatives everywhere. The results of three measurements'54-' 5 6 on GaAs are given in Table 11. The pressure coefficient is bigger near the edge than at longer wavelengths, as one expects from the considerations of the previous paragraph. STRAIN 17. UNIAXIAL When GaAs diodes are subjected to compressive uniaxial stress perpendicular to the junction plane, a number of rather striking effects are found. The spontaneous emission peak shifts with the stress,'57~'58but not always linearly or reproducibly from diode to diode. The light output in the junction plane increases, and the threshold current for lasing decreases.' 5 7 * 15 9 The variation from diode to diode is not completely understood, and may be due to internal strains, bur in at least some diodes the effects can be quantitatively analyzed in terms of the change from cubic symmetry which results from uniaxial strain. The resulting change in the valence band wave functions is reflected in the wave functions of impurity levels, which M. J. Stevenson, J. D. Axe, and J. R . Lankard, IBM J . Res. Deoelop. 7, 155 (1963). J. Feinleib, S. Groves, W. Paul, and R. Zallen, Phys. Rev. 131, 2070 (1963). '"G. E. Fenner, J. Appl. Phys. 34, 2955 (1963). 15’ F. M. Ryan and R. C. Miller, Appl. Phys. Lerters 3, 162 (1963). "* D. Meyerhofer and R. Braunstein, Appl. Phys. Letters 3, 171 (1963). R. C. Miller, F. M. Ryan, and P. R. Emtage, in "Radiative Recombination in Semiconductors" (7th Intern. Conf.), p. 209, Dunod, Paris and Academic Press, New York, 1965. I am indebted to Dr. Ryan for a preprint of this paper. 154

410

FRANK STERN

in turn makes the emission anisotropic and produces changes in the absorption for light of different polarizations. The results for GaAs diodes at 77°K have been fitted fairly well using reasonable values for the deformation potential parameters.'59*'60 The effects of pressure and uniaxial strain on diode lasers of the lead salts have been considered by Pratt and Ripper.l6O" 18. MAGNETIC FIELDS

Up to magnetic fields of the order of 10 kG, changes in the emission of GaAs laser diodes are relatively small and difficult to characterize. Fowler and WalkerI6' found changes in the near-field emission pattern of GaAs lasers which they attribute to the effect of the magnetic field in the presence of the inhomogeneities which are responsible for the filamentary nature of the lasing in many such diodes. At higher fields, the spontaneous and stimulated emission peaks in GaAs lasers have both been found to shift quadratically with magnetic field.'62 At 90 kG the observed shift at liquid helium temperature is 4meV. The observed shift is in good agreement with the expected shift for a hydrogenic donor, if reasonable values of electron effective mass and other parameters are used. But this agreement is a little puzzling, since the quadratic dependence would not be expected to continue to fields as high as 90 kG. At the donor concentrations used in these experiments, of the order of 3 x lo" cmT3, the donor levels are expected to be merged with the conduction band. Detailed analysis of the observed magnetic field effects must await a more complete theory of the properties of levels near the band edge in this difficult impurity range. In I ~ A s and ' ~ ~InSb,'32.'64*'65 magnetic fields have been used to shift the energy of the dominant lasing mode. This effect is analogous to the temperature and pressure effects we discussed above, and arises from the decrease in the index of refraction, which in turn is caused by the increase in the energy of the optical absorption edge with increasing magnetic induction. Equation (43) again applies, with B substituted for T, except that the variation of length with magnetic induction is negligible. The observed values of n' and dA/dB, and the value of dii/dB which has been deduced from these, are given in Table 11. P. R. Emtage, J . Appl. Phys. 36, 1408 (1965). I am indebted t o Dr. Emtage for a prepnnt of this paper. 160nG. W. Pratt, Jr., and J. E. Ripper, J . Appl. Phys. 36, 1525 (1965). A. B. Fowler and E. J. Walker, J . Appl. Phys. 35, 727 (1964). 162 F. L. Galeener, G. B. Wright, W. E. Krag, T. M. Quist, and H. J. Zziger, Phys. Rev. Letters 10,472 (1963). I. Melngailis and R. H. Rediker, Appl. Phys. Letters 2, 202 (1963). 164R.J. Phelan, Jr., and R. H. Rediker, Proc. lEEE 52, 91 (1964). l C sR. L. Bell and K. T. Rogers, Appl. Phys. Letters 5, 9 (1964). 160

14.

STIMULATED EMISSION IN SEMICONDUCTORS

411

The emission of InSb in a strong magnetic field shows two lines which move at different rates with magnetic field.'32*'64*'65 The higher energy line is the first to lase, but as the field is increased the lower energy line becomes the lasing line. The two lines result from the spin splitting of the lowest Landau level in the conduction band. Bell and Rogers'65 explain the transfer of lasing from one line to the other by supposing that the larger matrix element is associated with the higher energy line, which thus lases first. At high enough fields, however, the upper level rises above the quasi-Fermi level in the conduction band, and the lower level becomes the dominant one. For magnetic fields high enough that the conduction band is in the quantum limit, but low enough that acceptor states are unaffected, Beleznay and Pataki'66 find that the interband matrix element is unchanged by the field. Perhaps the most dramatic effect of magnetic fields on the operation of injection lasers is the rapid drop in threshold current density which has been observed in InAs and InSb as the field is i n c r e a ~ e d . ' ~ ~ * 'F~or ~-'~~ example, the threshold current density in InSb is reduced from above lo4 A/cm2 to below 2 x lo3A/cm2 as the magnetic induction is increased from 10 to 90 kG.13' The field in this case is parallel to the current. At the higher fields, continuous operation of I ~ A s and ' ~ ~InSb'64 is possible. The reduction in threshold is associated with the increase in the density of states as a magnetic field is applied. InSb and InAs can, however, lase in the absence of a magnetic field.'31*'33*'34,'35

ACKNOWLEDGMENTS I am indebted to my colleagues G. Bums, W. P. Dumke, R. W. Keyes, K. Konnerth, C. Lanza, G. J. Lasher, J. C. Marinace, A. E. Michel, T. N. Morgan, M. I. Nathan, M. H. Pilkuhn, H. S. Rupprecht, R. F. Rutz, W. J. Turner, E, J. Walker, and K. Weiser for valuable discussions of injection laser phenomena, and to many of them, particularly R. W. Keyes and M. I. Nathan, for critical comments on the manuscript.

*'

F. Belemay and G. Pataki, Phys. Status Solidi 8, 805 (1965).

This page intentionally left blank

Author Index Numbers in parentheses are footnote numbers and are inserted to enable the reader to locate those cross references where the author's name does not appear at the point of reference in the text. Abeles, B., 10, 11,18, 20, 30 Abragam, A., 141, I49 (2). I51 (2). 171. 174 (2), 178 (2), 194 Abraham, A., 234,247, 248 Abrikosov, N. Kh., 37, 38 (10). 39 Adams, I., 362, 363 (322, 323) Addamiano, A., 68, 69, 73 (17) Adirovich, E. I., 375 Aigrain, P.,329 Ainslie, N. G., 356, 364, 406 Alexander, F. B., 363,406 Alfrey, G. F., 319, 325 (138, 139) Allen, F. G., 115, 116 ( 5 ) , 122, 265, 267, 268 (2, 6), 276 (2), 277 (2, 5 ) , 279 (2, 5 ) . 280 Allen, J. W., 241, 297, 299, 304, 318, 319 (121), 322 (121), 323 (121, 128), 324 (125). 357 (41, 42), 358 (41, 42), 364 Almeleh, N., 191, 193, 194 ( 5 ) . 196 Altemose, H., 231, 233 (16) Ambegoakar, V.,27 Amirkhanova, D. Kh., 5, 13,21 (14), 22 (14), 23 Amith. A,. 11, 17, 25 Anderson, P.W., 152, 154 (17), 187 (17) Anderson, R. L., 341, 343 Anderson, W. W.. 369 Andreatch. P.. 93. 95 (18). 97 Angstrom, A. J., 11 Anisimova, 1. D., 360 AntonEik. 255, 256. 259 (18), 261 Antonoff, M. M., 386, 398 Apker, L., 273 Apple, E. F., 292 Archer, R. J., 340, 341, 345 (223), 346 (223), 349 (223). 353 (223). 354. 355 (223). 362. 369 Armstrong, J. A,, 284, 383 Arnold, G. W.. 369

Arnold, K. M., 340, 379 Ashkin, A., 345, 346 (242), 347 (242). 348 (242), 349 (242), 350 (242), 401,402 Attard, A. E., 228, 229 (9), 247 Au Coin, T. R.,363 Augustyniak, W. M.,366 Aukerman, L. W., 341, 343, 368 Averbach, B. A., 164 Averkieva, G. K., 70, 73 (26) Avery, D. G., 243 Axe. J. D., 352, 381. 383 (59). 408 (154). 409 Babcock. R. V.. 369 Bagaev, V. S., 345, 350 (246, 247). 359, 368 Baicker, J. A., 336, 356 (204) Barker, W.A., 180 Barnes, R. G., 153 Barrie, R., 69, 326 Barron, T. H., 35 Bashirov, R. I., 5, 13, 21 (14), 22 (14), 23 Basov, N. G., 356, 359, 375, 379, 404, 406 (125b) Bateman, T. B., 60,84, 95 (ll), 97 ( I l ) , 105, 109 (43), 110 (43), 11 1 Beak. R. J., 42, 43 (20). 44 Beattie, A. R., 229, 254, 255. 256. 257. 258. 159. 261. 301 Becker, J. H., 72 Beer, A. C., 384 Beers, D. S., 10, 11, 18, 30 Belemay, F., 41 1 Bell, R. L.,358, 410, 41 1 (165) Bemski, G., 199 Bennett, G. E., 404 Benoit A la Guillaume, C., 300,333,334, 358, 359 (283, 286), 360 (119, 283). 405, 407, 408 (133),411 (131, 133, 135) Berger, L. J., 15 Berkeyheiser, J. E., 340

413

414

AUTHOR INDEX

Berman, R., 7, 8, I 1 Broder, J. D., 290, 316 (2, 3), 318 (2, 3), 319 Bernard, M., 361, 375,405,411 (134) (2, 3), 323 (2, 3), 324 (2, 3), 325 (2, 3). 359 Bernstein, L., 42, 43 (20) (2, 31, 363 (2, 3) Berozashvili. Y. N.. 345. 350 (246. 247). 368 Broom, R., 326 Betts, D. D.. 54 Bross, H., 4 Bevacqua, S. F., 364, 365, 366, 406 Broudy, R. M., 119 Bhatia, A. B., 54 Brugger, K., 78, 80, 81, 82, 85, 96 Biard, J. R.,338,339, 341,343, 344, 374, 391 Brun, E.,173, 175, 186 Bielan, C. V., 365, 366, 406 Bruner, L. J., 108 Binkowski, B. B., 362 Bryant, J. F., 336, 356 (205,206) Bir, G. L., 105, 106 Bube, R. H., 196,234 Birch, F., 84 Buck, O., 94 Bud, V. R., 363, 406 Burger, R. M., 123 Birnbaum, G., 372,373 (7) Bums, G., 290, 291, 325, 333, 334, 335, 336 Bisson, G., 318, 336 (123), 356 (123) (200, 201). 340 (14). 343, 344 (231). 345 Black, J., 297,298 (50), 356, 364, 365 (338) (201. 218), 346 (201), 349 (200, 201), 353 Blackman, M., 28, 36, 53 (231), 356 (201), 359, 372, 373, 376 (4). Blakemore, J. S.,406 378, 381, 382 (60,61), 395,399 (37,40,61), Blanc. J., 196,234 400 (12), 402 (a), 403 (12, 94), 405, 407 Blatt, J. M., 335 (12, 94, 127, 128, 129), 408 (12, 94, 127, Bleaney, B., 190 128, 129) Bleekrode, R., 191, 194 Burrus, C. A., 352 Bloembergen, N., 147, 149 (12), 155, 157, Burstein, E.,328 162 (19), 185, 283, 284, 285, 287 (8), 288 Busch, G., 13, 14, 25 Blount, E. I., 256 Bushey, A. H., 68 Blum, S. E., 335, 336 (u)1),345 (201). 346 Butcher, P. N., 286 (201), 349 (201), 356 (201), 378, 399 (a),Butler, J. F., 406,407 400 (IIO), 403 (110) Bylander, E.G., 241 Blume, R. J., 97 Blunt, R. F.,72,215, 216, 237 Cady, W. G., 78 Bodi, L. J., 319 Calawa, A. R., 358, 361, 405, 406,407, 408 Bolger, R., 319, 320 (132) (132), 410 (132), 411 (132) Boer, K. W.,335 Callaway, J., 8, 27, 327, 328 (157) Bogdankevich, 0. V., 356,406 Callen, H. B., 80, 259 Bolef, D. I., 60, 109 (42), 110 (42), 111, 178 Cardona, M.,105, 362 Bolger, B., 379 364 (99), Carlson, R. O., 290, 304, 363 (W), Bonch-Bruevich, V. L., 107, 301, 329. 367 365 (99),372, 376 (3), 387 (3), 395, 406 Bond, W. L., 60,109 (41), 110 (41), 111 Carpenter, D. R., 363,406 Born, M., 76,284 C a r , W.N., 5, 338, 339, 340, 341, 343, 344, Boryshev. N. S.,369 374, 390, 391 (18), 394 (73) Carruthers, P., 4, 6 (4), 7, 9, 14 (24, 36), 18 Bowlden. H. J., 327 Brarnmer, A. J., 109 (36), 22, 26 (24), 27 (36), 30 Brandt, W., 335 Casella, R. C., 335, 369, 378 Bratt, P., 231, 233 Casimir, H. B. G.. 7. 8 Braunstein, R.,198,286,290, 331,345 (189), Cernogora, J., 300 349 (189), 352, 356 (I), 359 (I), 361, 378, Chakraverty, B., 196 399 (38), 409 Challis, L. J., 13, 14, 21 (51) Breckenridge, R. G., 72 Chang, R. K., 287. 288 Charlson, E. J., 94 Brekhovskikh, L. M., 397 Brill, P. H., 297 Chasmar, R. P., I 1 Britsyn, K. I., 246, 330 Cheeke, J. D. N., 13, 14 (51), 21 (51)

AUTHOR INDEX

Cheney, G. T., 336, 356 (205,206) Cheroff, G., 340, 391, 392 Cherry, R. J., 241 Chipaux, C., 361, 405,411 (134) Cholet, P., 243 Choyke, W. J., 299 Chynoweth, A. G., 294, 303, 305, 324 (102), 330 (26, 27), 355 (26, 27), 356 (26, 27) Ciftan, M., 383 Clark, W. G., 172, 180, 181, 182 Cochran, W., 53 Cody. C. D.. 10. 1 1 . 18. 30 Cohen, B. G., 294, 343 (29). 344 (29). 345 (291, 346 (29), 350 (29), 353 (29), 355 (29), 356 (29) Cohen, M. H., 141, 144 (5), 153 ( 5 ) . 155, 160 (5). 162, 183, 184 (5) Colbow. K., 298, 314 (62). 315 (62). 366 Coleman, P. D., 404 Collin, R. E., 386, 398 (68) Compton, D. hl. J., 336, 356 (205, 206) Cooley, J. W., 397, 398 (97), 399 Corranti, F. A., 365 Coupland, M. J., 375 Cowley, M., 324 Craig, R. M., Jr., 375, 383,400 (29) Crowe, J. W., 375, 383,400 (29) Csavinszky, P., 103, 105 Cuff, K. F., 406 Cunnell, F. A., 69, 194 Cusano, D. A., 304, 336, 337, 356 (207), 363 (99), 364 (99), 365 (99),404,406 Cuthbert, J. D., 366 Danielson, G. C., 11 Das, T. P., 141, 153 D’Asaro, L. A., 345, 346 (242), 347 (242), 348 (242), 349 (242), 350 (242), 401 Davydov, B. I., 10 Debever, J. M., 358,359 (283), 360 (283), 405, 407, 411 (131, 135) Debye. P.. 3. 49 Debye. P. P..383 DeLaunay, J., 54 Dell, R. M., 233, 234 (20) Denison, A. B., 177 des Cloizeaux, J., 329 Deutsch, T., 361, 362 DeVaux. L. H., 109 (39), 110 Devyatkov, A. G., 406 Dewald, J. F., 119

415

DeVAt, M., 191, 192, 194, 195 Dexter. D. L.. 254 Dickey, J., 273 Dieleman, J., 191, 194 (8) Diemer, G., 379 Dietz, R. E., 302, 304, 305 (91). 307 (91). 309 (91), 314 (91), 316 (91), 364 (91) Dill, F. H., 290, 340 (14), 372, 376 (4). 395, 403 (94), 407 (94), 408 (94) Dillon, J. A., 280 Dismukes, J. P., 30 Ditzenberger, J. A,, 165 Dixon, J. R., 232, 233, 329, 402 Dolling, G., 88 Donahue, J. A., 368 Donald, D. K., 294, 325 (31) Dousmanis, G. C., 345, 349 (236), 352, 353 (249), 378, 379, 394, 395 (87), 399 (39) Drabble, J. R., 10, 84, 85 (12), 97 Drahokoupil, J., 245 Dreyfus, R.W., 196 Drickamer, H. G., 304, 305 (88), 352 (88) Dubrovskii, G. B., 245 Ducuing, J., 284, 287, 288 Dumke, W. P., 290, 297, 327 (47), 328 ( I S ) , 340 (14), 356, 372, 373, 376 (4), 377, 401,

404 Duncan, W., 192, 198 Duraffourg, G., 361, 375, 405,411 (134) Eagles, D. M., 254, 259, 327, 328 (156) Edmond, J. T., 69, 163, 194 Edwards, A. L., 304, 305 (88), 352 (88) Ehrenberg, W., 124 Ehrenreich, H., 15, 99, 105 (27), 250, 258 (14), 259, 260, 277, 325, 364 (150) Einspruch, N. G., 103, 105 Einstein, A., 49, 371 Eisenmann, W. L., 231 Elliott, R. J., 299, 327 (68), 330 (68) Ellis, R. C. 361 Emmons, R. B., 231, 233 (16) Emtage, P. R.,352 (262). 409. 410 (159). 368 Enck, R., 299 Engeler, W., 229, 231, 233 (14). 340, 345, 379, 394, 395 (90),407, 408 (153). 409 Engelmann, R. W. H., 341 Enright, D. P., 232 (M),233, 243 Erfling, H. D., 39 Esaki, L., 399 Estle, T. L., 191, 192, 194, 195

416

AUTHOR INDEX

Evans. D. A., 293, 297 (25), 301 (25) Fan, H. Y.,226, 227, 228 (8), 237, 259, 260, 329. 360. 361, 369 Farnsworth, H. E., 116, 121 (S), 123, 135 (8), 279, 280 Fedorova, N. N., 71 Feher, G., 180, 181, 182, 197 Feinleib, J., 352, 356 (257),4IB (155), 409 Feldman, W. L., 294, 330 (26), 355 (26), 356 (26) Fenner, G. E., 290, 304, 352, 363 (99). 364 (99),365 (99, 332), 366 (332), 372, 376 (3). 380, 383, 386, 387 (3), 395, 402 (56), 406, 408 (156), 409 Filinski, I., 361 Fischer, A. G., 294,295, 325 (30), 362,404 Fischer, T. E., 275 Fitchen, D. B., 365, 406 Fitzgerald, M. E., 404 Flicker, H., 307, 336, 356 (204), 369 Folberth, 0. G., 304 Fomin. N. M. V., 108 Forgacs, R. L., 97 Foster, E. L., 7 (22), 8 Foster, L. M..367 Fowler, A. B., 280,381,382 (60,61), 399 (61), 402 (60),410 Foy, P. W.,319, 320 (136), 321 (136), 322 (136), 323 (136) Franken. P. A,, 283 Franklin, A. R., 340 Franz, W., 330 Fray, S.J., 53 Frederikse, H. P. R., 15, 72, 215, 216, 237 Frosch, C. J., 304 Fuller, C. S., 196 Fulton, T. A., 365. 406 Caddy, 0.L., 369 Gartner, W. W., 5 Galeener. F. L., 351, 368, 369, 410 Galginaitis, S., 14, 340, 369, 392. 393. 394 Garfinkel, M., 340, 345, 379, 394, 395 (go), 407, 408 (153), 409 Garland, C. W.,60, 109 (38, 44),110. I 1 I , I13 Gatos, H. C., 119, 130 Geballe, T. A., 98 Genzel, L., 10 George, M., 123 Gere, E. A,, 197

Gerlich, D., 60, 109 (40),110 (40),I 1 1, 112 Germer, L. H., 121, 124 Gershenzon, M., 290, 292, 298, 300 (58), 304, 305 (19, 91), 306 (19). 307 (19, 58, 91, 96), 308 (19), 309 (19, 58, 59, 91), 310 (58), 311 (58), 312 (58), 313, 314 (58, 91), 316 (58, 59, 91), 317 (59), 318 (59, 114), 319 (114 119). 320 (119, 136). 321 (119, 136),322(11,114, 119,136),323(114,118, 119, 126, 136), 324 (126), 345, 346 (242), 347 (242), 348 (242), 349 (242), 350 (242). 356, 364 (91. 961, 367, 312, 379, 401, 402 Gibbons, D. F., 37 (1 I ) , 38, 39 Gibbons, P. E., 319, 324 (115) Gibbs, D. F., 304 Giesecke, G., 69, 70, 71, 72, 73 (22), 112, 117 Gill, D., 155, 157, 185 Gilvarry, J. J., 46 Giordmaine, J. A,, 284 Gippius, A. A., 299 Giroux, G., 298 Glasser, W., 319 Glasko, V. B., 301 Glassbrenner, C. J., 9, 10 (29). 12, 25 Gluyas, M., 84,85 (12), 97 Gobeli, G. W., 115, 116 (3, 5), 122, 123 (3), 124 (3, lo), 131 (21), 132 (111, 134 ( I I ) , 135 (3), 136 (3), 259, 265, 267, 268 (2, 6). 276 (2), 277 (2, 5 ) , 279 (2, 5), 369 Goldschmidt, V. M., 68, 69, 70, 72 Goldsmid, H. J., 10 Goldstein, B., 191, 193, 194 (3,196, 307 Gooch. C. H., 395 Goodwin, D. W., 243,248, 357 Gorjunova, N. A., 68, 70, 71, 73 (20, 26) Gorter, C. J., 168 Gorton, H. C., 319, 323 (142), 324 (142) Griffin, A., 7, 9, 14 (36), 18 (36), 22, 27 (36) Griffiths, L. B., 405 Grimmeiss, H. G., 236, 241, 242, 318, 319 ( 1 1 9 , 322 (122, 130, 131), 323 (115, 122, 130, 131). 324 (115). 362. 366 Gross, E. F., 292, 304, 305 (97). 307 (97, 104, 1051, 308 (104, 105), 309 (20, 97, 104, IOS), 314 (20, 105, 109), 315 (109, 112). 316 (20). 366 Groth, R., 242, 362 Groves, S., 352, 356 (257), 408 ( I S ) , 409 Griineisen, E., 34 Gueron, M., 182 Guglielmi, P.A., 335, 336 (202), 349 (202)

AUTHOR INDEX Gul’tyaev. P. V., 51 Gutkin. A. A,, 333, 344

Haas, C., 297, 329 (39) Habegger, M. A., 237 Hagstrum, H. D., 134 Hahn, E. L., 141, 153 Hahn, H., 66, 73 (8) Haken, E., 373 Haken, H., 373 Hall, R. N., 290, 295. 297, 301 (36). 345. 372. 373. 376 (3). 380 (15). 381 (15). 387 ( 3 ) . 398. 400 (101). 405 Halperin. B. I.. 367 Halpern, V., 258 Halsted, R. E., 292 Hambleton, K. G., 375 Hamilton, D. R., 299 Haneman, D., 115, 116 (2, 4), 123, 134 (2), 135 (2, 7), 136 (7), 275, 277 H am , R.,173 Harding, W. R., 194 Hare, W. F. J., 356 Harman, T. C., 406 Harness, J. B., 13, 14 (51), 21 (51) Hartman, C. D., 121, 124 Hass, M., 117 Hawkins, T. D. F., 212, 213, 215 (1) Hawkins, T. H., 290, 357 (4, 5) Hawrylo, F., 394, 395 (87) Haynes, J. R., 297 Hebel, L. C., Jr., 141, 144 (7), 168 Hebert, R. A., 290, 316 (2, 3), 318 (2, 3). 319 (2, 3), 323 (2, 3), 324 (2, 3), 325 (2, 3). 359 (2, 3), 363 (2, 3) Heffner, H., 324 Henisch, H. K., 291 Hensel, J. C., 197 Henvis, B., 117 Herman, F., 76 Herring, C., 7, 9, 27 (26), 98, 264 Hess, B. G., 365 Hill, D., 243, 331. 332 (188), 335 (188). 368. 390. 394, 402 (86a) Hilsum, C., 228,232, 237,238, 239,326, 340, 315 Hodby, J. W., 304 Hodges, A. J., 241 Hodgkinson, R. J., 261 Holeman, B. R., 238, 239

417

Holland, M. G., 4, 5, 7, 8, 9 (7, 18). 1 I , 13, 14 (7, 13), 17, 18 (7). 19, 21. 22 (13). 15 (7, 18), 26 (7, 53), 27 (18), 28 (18) Holonyak, N., 364, 365, 366. 369,406 Holt, D. B., 319, 325 (138) Honig, A., 299 Hoover, G. J., 355 Hopf, L., 54 Hopfield, J. J., 292, 298, 299, 302, 305 (19). 306 (19X 307 (19), 308 (19), 309 (191 312 (66). 314 (62). 315 (62). 328, 366 Hora, H., 379 Horton, G. K., 54 Hosler, W. R., 72 Hoss, P. 307 Houston, W. V., 54 Hrostowski, H. J., 60,109 (41), 110 (41 ), 111, 147 (14), 148 Huang. K., 76 Huggins, M. L., 65, 67, 73 (2) Hunsperger, R. G., 379 Huntington, H. B., 60,77 Hurwitz, C. E., 336, 404, 406 (152d). 407 Hutchinson, W. G., 368 Hutson, A. R., 85, 89, 92, 94 (13) Iandelli, A., 71, 72 Iannini, A. A,, 333 Ievin’sh, A. F., 70, 73 (26) Iida, S., 402 lizima, S.,319, 324 (134, 141) lizuka, T., 319, 324 (140) lmenkov, A. N., 368 lng, S. W., 344 Ingles, T. A,, 67 Isurumi, I., 368 Ivey, H. F.,243, 291, 357 (16), 404 Iwai. Y.. 368 Jaklevic, R. C., 294, 325 (31) Jean-Louis, M., 405, 411 (134) Jeffrey, G. A., 66, 73 (6) Jeffties, C. D., 180 Jensen, H. A,, 344 Joffe, A. F., 5 Johnson, E. J., 361 Johnson, F. A., 53 Johnson, L. F., 304, 307 (96), 364 (96) Jones, R. H., 53 Joshi, S. K., 54 Junga, F. A., 406 Juza, R., 66, 73 (8)

418

AUTHOR INDEX

Kaiser, R. H., 297, 331 (46),333 (46), 338 (46),341 (46), 342 (46),344 (46) Kaiser, W., 329 Kaluzhnaya, G. K., 305,307 (1041,308 (104), 309 (104) Kane, E. O., 250, 266, 267 (4), 268 (6), 329, 330 (172, 173), 346, 349 (173), 357 Kaplan, H., 78, 88 Kastler, A., 380 Kawajii, S., 280 Keating, P. N., 295 Keldysh, L. V., 330, 345, 350 (246, 247) Kelley, P. L.,286 Kelly, C. E., 378 Kennedy, A. J., 54 Kessler, H.K., 350, 402 Keyes, R. J., 290, 336, 345 (15), 356 (9, lo), 358, 372, 376 (5), 393, 404, 405,408 (132), 410 (132), 411 (132) Keyes, R. W., 9, 14, 18, 22, 27, 54, 98, 100 (25), 102, 105, 108, 111, 373 Khachaturyan, A. G., 107 Khartsiev, V. E., 344 Khvoshchev, A. N., 359 Kibler, L. V., 352 Kikuchi, C., 194 Kikuchi, M., 319,324 (134, 140, 141) Kingsley, J. D., 290, 372, 376 (3), 380. 383, 386, 387 (3;, 395, 402 (56), 404 Kingston, R.H., 115,404 Kischio, W., 236, 362 Kisliuk, P., 274 Kittel, C., 147 Klein, M. J., 42 Kleiner, W. H.. 358 Kleinman, L., 243 Klemens, P. G., 4, 6 (3), 7, 8 (3), 9 (3). 10 (3) Knight, W. D., 141, 151 Koc, S., 246 Kochetkova, N. M.,51 Kochneva, N. S., 304, 305 (97), 307 (97), 309 (97) Koelrnans, H., 236, 241, 242, 318, 319 (115). 320 (132), 322 (130, 131), 323 (115, 130, 131), 324 (I 15), 362 Kohn, W., 300,327 (78) Kolrn, C., 164 Konnerth, K., 348, 377, 395, 401 Kopef, Z., 18 Kosicki, B., 362 Kotani, T., 368

Kover, F., 234, 235 Kowalchik, M., 298, 309 (59), 316 (59), 317 (59), 318 (59) Koyama, R. Y.,362 Kozlov, M. M. 333, 344 (193). 368 Krag, W. E., 290, 345 (15), 351, 372, 376 (9, 410 Kraus, O., 176 Kroemer, H.,379 Krokhin, 0. N., 375, 379 Ku, S. M., 364, 365 (338) Kudman,I.,4.9(8), 11, 13, 14(8),15, 16.17. 18. 19, 20, 25, 28.29, 51,331, 334 (192) Kulin, S. A., 164 Kulymanov, A. V.,360 Kunz. W. E., 369 Kushida, T., 402 Kuznetsova, E. M., 375 Kyser. D. F.,337, 369 Laff, R. A., 226,227, 228 (8) Lallemand, P.,287 Lambe, J., 294,325 (31) Larnpert, M. A., 295 Landau, L. D., 34,98 Lander, J. J., 116, 121 (9), 122 (17), 123, 124, 131 (21), 132 (17), 135 (9), 279 Landsberg, P. T., 229, 254, 255, 258. 259 (18), 293, 297 (25), 301 (25), 357 (43), 358 (43), 359 (43) Lankard, J. R.,352, 381, 383 (59), 408 (154). A09 Lanza, C., 348, 377, 384, 392, 395, 401, 402 (65) LaPlaca, S., 67, 72, 73 (14) Larach, S., 363 Larsen, T. L., 356, 369 Lasher, G., 290, 330, 340 (14), 345 (187). 346, (187), 349 (187), 350 (187), 372, 374. 376 (4), 384, 394, 398, 399, 400 (IOO), 401, 403,405 Laser, M. E., 243 Lavallard, Ph., 358,359 (286),405,408 (133), 411 (133) Lavine, J. M., 333 Lavine, M. C., 119, 130 Lax, B., 198, 286, 290, 345 (15), 358, 372, 373, 376 (5), 398,402 (102),405,408 (132), 410 (132), 41 1 (132) Lax, M.,115, 300,301 (79), 329, 367 Lechner, G., 54

AUTHOR INDEX Lee, D. H., 233 Lee, T. C.. 369 Leezer, J. F., 341 Lehmann, W., 363 Leibfried, G., 27 Leite, R. C. C., 294, 297, 331 (46),333 (46), 335, 336 (202), 338 (46),340, 341 (46), 342 (461, 343 (29, 214), 344 (29, 46,214). 345 (29, 223), 346 (29, 223). 349 (202, 223), 350 (29b 353 (29, 214, 2233, 354 (223), 355 (29, 223). 356 (29. 214), 368. 398. 402 (103) Lengyel, B. A., 372 Levinstein, H., 229, 231, 233 (14) Levitt, R. S., 356, 359, 378, 394, 405, 407 (127, 128), 408 (127, 128) Lifshitz, E. M., 34, 98 Lilburn, M. T., 233, 234 (20) Lirmann, G. W., 66 Locke, D. J., 379 Lockwood, H., 356,364 Loebner, E. E., 318,319,323 (120), 359 (137). 362. 369 Loferski, J. J., 369 Logan, R. A., 294, 305, 318, 319 (103, 119), 320 (103, 119, 136), 321 (119, 136), 322 (119, 136), 323 (119, 136), 324 (102). 330 (26. 27). 355 (26, 27). 356 (26, 27). 367 Loh, E., 246, 261 Lorenz, M. R., 362 Loudette, J., 361,405, 411 (134) Loudon, R., 286 Low, W., 191, 192 Lowe, I. J., 144 Lubowski, S. J., 365, 366,406 Lucas, R . C., 362 Lucovsky,G., 231, 233, 331, 332 (191). 346, 349 (191). 350. 356 (254). 367. 368. 402 Ludwig. G. W.. 189. 200, 201 ( 2 ) Lukes. F.. 237 Lutgemeier, H., 151, 152 Maak, J., 242, 362 McCall, D. W.. 147, 148 (13), 149 (13) McCammon, R. D., 45 McClure, D. S., 299 McCumber, D. E., 297, 374 McDermott, P. S., 363, 406 Macdonald, H. E., 234 McKay, K. G., 303

419

Mackintosh, I. M., 297, 357 (41, 42). 358 (41, 42 1 McLean, T. P., 286, 299, 327 (69), 330 (69), 357 Mac Rae, A. U., 115, 116 (3). 121, 122. 123 (3). 124 (3, 101, 132 (11). 134 (11). 135 (3). 136 (3), 231, 233 (14). 276 McSkimin, H. J.,60, 84,93, 95 (I]), 97 ( I I ) , 109 (37, 41,43), 110. 11 1, 113 McWhorter, A. L., 286, 290, 345 (15), 372, 373, 376 (5), 398 (16), 402 (102) Madden, H. H., Jr., 121 Maeda, K., 314, 316 (113) Mahler, R. J., 173, 175 (38), 177, 186 (38) Mahon, H., 173, 175 (38), 177, 186 (38) Maker, P. D., 284 Malkovskk, M., 246 Mandel, G., 394,404,406 (80, 81) Mandelkorn, J., 324 Manley, G. W., 363,406 Marcus, P. M., 54 Marinace, J. C., 335, 336 (201), 345 (201), 346 (201). 349 (201). 356 (201), 369, 373. 376, 378. 379, 395 (9), 399 (40) Marple, D. T. F., 338, 381 Mason, W. P., 84, 95 (11). 97 ( I l ) , 105 Massoulit, M.. 290, 345 (8) Matarrese, L. M., 194 Matkovich, V. I., 72, 73 (31) Matumura, O., 194 Mavroides, J. G., 286 Mayburg, S., 297, 298 (50), 340, 356, 379, 394, 395 (88),403 Mayer, G., 318, 336 (123), 356 (123) Mays, J. M., 147, 148 (13), 149 (13) Mead, C. A., 213, 235, 236, 304, 324, 363 (98), 365 (98) Mellichamp, J. W., 362, 363 (322, 323) Melngailis, I., 359, 360, 363, 369, 405, 406, 408 (163). 410, 411 (134a. 163) Mendelssohn, K., 6, 10 (15) Menes, M., 60,109 (42), I10 (42), 1 1 1, 178 Merten, L., 88 Meskin, S. S., 368 Mettler, K., 348 Meyerhofer, D., 352, 409 Michel, A. E., 343, 344 (231), 345, 353 (231). 356. 357, 369, 378. 379. 380, 383, 388. 399, 400 (57) Mieher, R. L., 167, 168, 169, 170, 171, 172, 173, 184 (29)

420

AUTHOR INDEX

Mielczarek, E. V., 15 Mikhailova, M. P.,233, 234 Mikulyak, R. M., 298, 305, 309 (59), 313, 316 (59), 317 (59), 318 (59, 114), 319 (103, 114), 320 (103, 136), 321 (136), 322 (114, 136), 323 (114, 118, 126, 136), 324 (126) Millea, M. F., 341, 343, 368 Miller. R. C., 352, 368 (262). 409, 410 (159) Miller, S. C., 177 Minden, H. T., 368 Mitchell, E.W. J., 115, 116 (4). 275, 277 Mitra, S. S., 54 Mlavsky, A. I., 405 Moncaster, M. E., 319, 323 (128), 364 Mooradian, A., 360, 369 Moos, H.W., 288 Morehead, F. F., 394, 406 (80, 8 I ) Morgan, T. N., 330, 367, 368, 369, 377, 389 Morin, F. J., 98 , 121 (9), 122 (17), 123, 124, Morrison, .I.116, 131 (21), 132 (17), 135 (9), 279 Moss, H. I., 294, 325 (30) Moss, T. S.. 198, 212, 213, 215 ( I ) , 226. 230, 237. 290, 297 (7), 328, 357 (4, 5, 7. 43). 358 (43). 359 (43) Mott, G., 94 Mountain, R. D., 42 Mozzi, R. L., 66 Mueller, C. W., 345, 349 (236), 353 (249), 378, 399 (39) Murnaghan, F. D., 78, 80 (7), 85 (7)

Nag. B. D.. 54 Nagae, M., 298 Nair. P. S., 54 Nakatsukasa, M., 280 Nanavati, R. P., 330, 355 (179) Nannichi, Y.,396, 400 Nasledov. D. N., 223, 234 (21). 290, 333, 344 (193). 356 (12), 368 Nathan, M. I., 290, 291, 325, 333, 334, 335, 336 (200,201), 340 (14). 343,344,345 (201, 218), 346 (201) 349 (200, 201), 350 (238), 353 (231). 356 (201,238). 357,359,368.369. 372. 373, 376 (4), 378, 379, 381, 382. 395, 399 (37.40.61). 400 (12), 402 (60), 403 (12, 94). 405, 407 (12, 94, 127, 128, 129), 408 (12,94, 127, 128, 129) Natta, G., 68

Nedzvetskii, D. S., 292, 304, 305 (97). 307 (97, 104, 105), 308 (104, 105), 309 (20, 97, 104, 105), 314 (20, 105, 109). 315 (109, 112), 316 (20). 366 Nelson, D. F., 304, 307 (96). 318, 319 (119). 320 (119), 321 (119), 322 (119), 323 (119), 345, 346 (242), 347 (242), 348 (242), 349 (242). 350 (242). 364 (96), 366, 367, 401. 402 Nelson, H., 331, 345 (189), 349 (189, 236). 352,353 (249), 378, 379, 394, 395 (87), 399 (38, 39) Nelson, J. B., 69, 70 Nethercot. A. H.Jr., 5 Neuringer, L. J., 14, 26 (53) Newman, R.,299,340 Nichols, M., 264 Nickle, H. H.,5 Nisenoff, M., 284 Noblanc, J-P., 361, 405, 411 (134) Norberg, R. E., 144 Nonvood. M. H.,368 (369) Novikova, S. I., 37 (8, 9, lo), 38 (8,9, lo), 39 Noyce, R. N., 293 Nuese, C. J., 369 Nye, J. F., 78, 87 (5) Ochs, S. A., 406 Ohtsuki, 0.. 368 Olechna, D. J., 398,400 (101) Oliver, D. J., 150, 159, 163 Olson, D. H., 294, 343 (29). 344 (29). 345 (29), 346 (29), 350 (29), 353 (29), 355 (29), 356 (29) Ormont, B. F., 243 Oshinsky, W., 72 Osipov, B. D., 359 Oswald, F., 304 Ott, H.,66 Overhauser, A. W., 178 Owen, E. A., 69 Ozolin’sh, G. V., 70, 73 (26) Pake, G. E., 141, 144 (3), 153 Pankove, J. I., 290, 294, 329, 331. 340, 345 (8, 28, 189), 346, 349 (189), 353 (28, 243). 355 (243), 367, 378, 394, 395 (87). 399 (38). 403 Park, K. C., 109 (a), 110 (44),111 Parmenter, R. H.,329 Parry, G. S., 66, 73 (6)

AUTHOR INDEX Pashintsev, Yu I., 37 (12), 39, 43, 44 (I.?), 51. 52 (10) Passenni, L., 68 Pataki, G., 411 Patrick, L., 299 Paul, W., 304, 309 (92), 352, 356 (257), 365 (92), 408 (1 55), 409 Pauling. L., 65, 67, 73 (2) Pearson, G . L., 60, 109 (41), 110 (41), 11 I Pease. R. S.. 64 (2), 65 Peteny, T., 374 Peet, C. S.,319, 323 (142), 324 (142) Pehek, J., 231, 233 (14) Peierls, R., 3 Peloke, J . R., 363,406 Pern, J. A., 67, 72, 73 (14) Pershan, P. S., 283, 284, 285, 287 (8) Petree, M. C., 341 Petrov. A. V., 51 Pettit, G. D., 356, 359 Petzinger, K. G., 345, 353 (249) Pfahnl, A., 319,322 (127), 323 (127) Hster, H., 69, 70, 71, 72, 73 (22), 112, 117 Phelan. R. J., 358, 359, 369, 405, 406. 408 (132, 164). 410 (132). 411 (132, 134a, 164) Philipp. H. R., 277 Phillips, J . C., 42, 243, 261, 329 Pierce, W. L., 173, 175 (38), 186 (38) Piesbergen, U.. 42. 51, 52 (5). 171 Pikus, G. E., 106 Pilkuhn, M., 304, 356, 363 (IOO), 364 (IOO), 365 (100). 366 (loo), 367,369.373, 376,377 (31). 378, 395 (9), 400. 401, 403 (110, I l l ) . 406 Pillar, H., 361 Piper, W. W., 292 Pittman, G. E., 5, 340, 390, 394 (73) Pizzarello. F. A,, I09 (39). 110 Pohl, R . O., 7 , 9 , 14 (31), 26, 27 (30) Pokrovskii, Y. E., 298 Pomeranchuk, I., 7. 9 Poor, E. W., 318, 319, 323 (120), 359 (137) Pope, M. D., 319, 324 (133) Popov, Y.M., 375, 379 Popper, P., 67 Porto, S.P. S.,340, 345 (223), 346 (223). 349 (223). 353 (223), 354 (223). 355 (223) Post, B., 67, 72, 73 (14) Potter. Roy F., 60, IOY (36). 110, 113. 231 Pound, R. V., 165 Pratt, G . W., Jr., 410

421

Prener. J . S., 292. 298, 311 (60) Preston, G. D., 69 Pretzer, D. D., 134 Price, P. J., 10 Pnngsheim, P., 404 Proctor, W. G., 176 Pryce, M. H. L., 194 Purl, 0.T., 369 Pyle, I . C., 9, 14 Quarrington, J . E., 53 Quimby, S. L., 54 Quine, J. P., 394 Quinn, H. F., 363, 406 Quist, T. M., 290, 345 (15), 351, 356 (9, lo), 372, 376 (5), 393, 410 Rabenau, A., 236, 241, 242, 319, 322 (130), 323 (130), 362 Rappaport, P., 234 Ravich, V. N., 368 Razbirin, B. S., 299 Read, W. T., 295, 297 Redfield, D., 330 Rediker, R. H., 290, 345 (151, 358, 359, 363, 369, 372, 376 (9,405, 406, 407, 408 (132, 163,164),410(132),411(132, 134a, 163, 164) Reed, B. S., 341, 374, 391 (18) Reese, W. E., 331, 356 (190), 359, 394, 401, 402 (86) Reif, F., 141, 144 ( 5 ) , 153 (5), 155, 160 (5), 162, 183, 184 (5) Reinhart, F. K., 402 Renner, Th., 52 Rennie, A. E., 243 Repper, C. J., 350, 356 (254) Reynolds, W. N., 233, 234 Rezukhina, T. N., 52 Rhodenck, E. H., 159, 160, 161, 162, 163, 164

Ricks, R. S., 319, 324 (133) Rieck. H., 367 Riley, D. P., 69, 70 Riley, R. J., 363, 406 Ripper, J . E.. 335. 336 (202). 349 (202). 368. 410 Roberts, D. H.,233 Roberts, J. A,, 241 Kodgers. K. F.. 366 Rogachev, A. A.. 290, 344, 356 (12). 368

422

AUTHOR INDEX

Rogers, K. T., 358,410, 411 (165) Rooymans, C. J . M.,63 Rose, A., 295 Rose-Innes, A. C., 237, 238, 326 Rosenberg, A. J., 119,405 Rosenberg, H. M., 6, 10 (15) Ross. I. M.,69, 326 Roth, L., 198 Rowland, T. J., 141, 147, 149 (12), 151 (8), 159 Rozman. R., 329. 367 Rubin, L. G., 11 Ruderman, M.A., 147 Ruehrwein, R.A,, 67, 72 Rupprecht, G., 405 Rupprecht, H., 304,356, 363 (loo), 364 (IOO), 365 (100). 366 (100). 369, 373,376,377 (31). 378, 395 (9), 400, 401 (1101 403 (110, I l l ) , 406 Ryan, F. M.,319, 323 (135), 352, 409, 410 (159), 368 (262) Ryvkin. S. M.,290, 344, 356 (12). 368 Sagar, A., 361 Sah, C. T., 293, 298 Sarace, J. C., 294, 297, 331 (46),333 (46), 338, 340, 341 (46),342, 343 (29, 214), 344 (29, 46, 214), 345 (29), 346 (29, 242), 347 (242), 348 (242), 349 (242), 350 (29, 242), 353 (29, 214), 355 (29), 356 (29, 214), 401 Savage, C. M.,284 Savage, W. R., 116 Schaufele, R. F., 333 Scheer, J. J., 265, 268 (3) Scheibner, E. J., 124 Schlickman, J. J., 404 Schlier, R. E., 116, 121 (S), 123, 135 (8) Schloemann, E., 27 Schneider, E. E., 192, 198 Scholz. H., 318,319,322 (122). 323 (122). 366 Schultz, W., 301 Schwarz, R. F., 297. 367 Scott, A. C., 391 Sedov, V. E., 333, 344 (193) Seeger, A., 94 Shafer, M., 71 Shaffer. J.. 299, 328, 368 Shalyt, S. S., 13, 14, 15, 21 (55) Shdanow, G. S., 66 Sheinkman, M. K., 301 Shelton, H., 274

Shmushkevitch,I. M., 10 Shockley, W., 261, 293, 295, 297, 298, 305 (40).374, 389 Shotov, A. P.,345, 350 (246. 247), 359, 368 Shrader, R. E., 363 Shulman, R. G., 147, 148, 149, 152, 154, 187 Sidles, P. H., 11 Silverman, S. J., 395 S h o n , F. E., 7 (21), 8, 50 Sinclair, H., 69 Sirkis, M.D.. 369 Sirota, N. N., 15. 37 (12), 39, 43, 44. 51. 52 (10)

Slack, G. A,, 9, 10 (29). 11, 12, 14, 25, 26 (44). 395 Slitcher, C. P., 141, 151 ( I ) , 168, 178 (1) Slobodchikov, S. V., 233, 234 (21) Slutsky, L. J., 60, 109 (38), 110. 111. 113 Slykhouse, T. E., 304, 305 (88), 352 (88) Smakula, P. H., 405 Smith, A. W., 383 Smith, C. J., 184, 185 (51) Smith, C. S., 98 Smith, R. A., 377, 389 Smith, R. W., 295 Smith, S. D., 290, 357 ( 5 ) Smith, W. V., 153, 372, 373 (8). 394 Sokolova, W. J., 71 Soltys, T. J., 290, 372, 376 (3), 387 (3) Sommers, H. S., 329, 353 (164) Soref, R. A., 288 Sorokin, P. P., 372, 373 (8), 381, 383 (159) Spiess, K. F., 66 Spitzer, W.G., 213, 235. 236, 260. 304, 324. 363 (98), 365 (98), 394 Stackelberg, M.v., 66 Staebler, D. L., 379 Stannard, C., 229 Starkiewicz, J., 318, 319 (121). 322 (121). 323 (121, 128) Statz. H.. 333 Steigmeier, E. F., 4,9 (8), 11, 13, 14 (8). 15, 16, 17, 19, 20, 25, 28, 29, 51, 54, 5 5 , 56, 57, 58, 59, 60 Stern, B., 344 Stern, F., 329, 330,340, 345 (187). 346 (187), 349 (187), 350 (187), 374, 375, 377, 378, 381, 383, 387, 390, 391 (72), 392 (75). 397. 398, 401, 402 (62), 403, 405 (22),409 Stevens, K. W.H., 190

AUTHOR INDEX

Stevenson, M. J., 352, 408 (154), 409 Stickel, W., 319, 323 (135) Stone, B., 243 Strark. H., 368 Strauss, A. J., 228, 229 (9), 247, 363, 406 Strelkov, P. G., 39 Struthers, J. D., 165 Stukes, A. D., 11 Sturge, M. D., 325, 330 (151), 333 (151), 364 (151)

Subashien, V. K., 245 Sullivan, J. J., 78, 88 Sutton. P. M., 54, 55, 56. 57, 58. 59. 60 Svistunova, K. I., 298 Swartz, J. M., 319, 323 (142), 324 (142)

Taft, E., 273 Talley, R. M., 232 (44),233,243, 329 Tanttila, W. H., 173, 176, 177 Tauc, J., 246, 247, 248, 352 Taylor, A,, 69 Terhune, R. W., 284 Theriault, J. P., 358 Thiessen, K., 246 Thomas, D. G., 292, 298, 300 (58), 302, 304, 305 (19, 91), 306,307 (19, 58, 91), 308 (19), 309 (19, 58, 91), 310, 311, 312 ( I l l ) , 313, 314 (58, 62, 91, Ill), 315 (62, 1111 316 (58,91),318,319 (1 19), 320 (1 19), 321 (1 19), 322 (119). 323 (119). 364 (91), 366 Thomson, J. J., 404 Thurston, R. N., 85, 96 Tietjen, J. J., 406 Title, R. S., 192, 197 Tomiyasu, K.,394 Townes, C. H., 153 Toxen, A. M., 30 Trambarulo, R. F., 352 Tric, C., 333, 334, 360 (196) Triebwasser, S., 340, 391, 392 (75) Trumbore, F. A., 298,300 (58). 307 (58). 309 (58, 59), 310 (58), 311 (58), 312 (58, I l l ) , 313, 314 (58, I l l ) , 315 ( l l l ) , 316 (58, 59), 317 (59). 318 (59), 367 Tsarenkov, 9. V., 290, 344, 356 (12), 368 Turner, W. J., 331, 356 (190), 359, 394, 401, 402 (86) Tursunov, A,, 105 Tuul, J., 121 Tuzzolino, A. J., 246

423

Ullman, F. G., 318, 319 (116, 117), 323 (116, 117), 324 (116, 117), 325 (116, 117) Unterwald, F. G., 124 Valentiner, S., 39 van Baeyer, H. C., 27 van der Does de Bye, J. A. W., 318, 336 (124), 356 (124) van Doom, C. Z., 305, 307 (106), 309 (106), 314 (106) Van Kranendonk. J., 165. 168. 170 van Laar, J., 265, 268 (3) van Roosbroeck, W., 297, 305 (40) Van Vleck, J. H., 145 Van Wieringen, J. S., 194 Varga, A. J., 367, 368 Vassell, W. C., 294, 325 (31) Vavilov, V. S., 246, 299, 330 Vegter, H. J., 191, 194 (8) Venables, J. D., 119 Verma, J. K. D., 54 Vieland, L., 331, 333 (192) Vink, A. T., 305, 307 (106). 309 (I&), 314 (106) Vogel, M., 404 Vook, F. L., 22, 23, 24,25 Vul. B. M., 345, 350 (246, 247), 359, 368 Wade, G., 379 Wagner, M., 9, 26 Walker, C. T., 7, 9, 14 (31) Walker, E. J., 345. 356, 380, 383, 388. 399. 400 (57), 410 Wallot, J., 39 Walton, A. K., 198 Wang, C. C., 362 Ward, J. F., 283 Warschauer, D . M., 361 Washwell, E. R., 406 Watkins, G., 194 Waugh, J. L. T., 88 Weber, M. J., 147, 150, 152, 170, 171, 172 Weger, M., I81 Weinreich, 0. A., 340,394 Weisberg, L. R., 196 Weiser. K., 71, 356, 359, 369, 378. 383, 384. 387,405, 407 (127, 128). 408 (127. 128) Weiss, H., 243 Weit, G. K., 45 Welker, H., 243 Wentorf, R. H., 67, 73 (10)

424

AUTHOR INDEX

Wertheim, G. K., 297, 357 Wheeler, C. A., 379 Whelan, J. M., 60, 109 (43), 110 (43). 11I, 165, 294, 297, 331 (46),333 (46), 338 (46), 340, 341 (46),342 (46), 343 (29), 344 (29, 46), 345 (29, 223), 346 (29, 223), 349 (223), 350 (29). 353 (29, 223), 354 (223). 355 (29, 233), 356 (29), 394 White, D. L., 5, 85, 89, 92, 94 (13) White, H. G., 305, 319 (103), 320 (103). 324. 367 White, W. E., 68 Wiggins, C. S., 319, 325 (138, 139) Willardson, R. K., 31 Williams, F., 299, 328 Williams, F. E., 298, 311 (60, 61), 368 Williams, F. V., 67, 72 Williams, N., 53 Wilson, D. K., 341, 343, 356 (230), 378. 383 390 Winogradoff, N. N., 350,402 Winstel, G., 348, 374 Wittry. D. B., 337, 369 Wolf, E., 284 Wolfe. C. M., 369 Wolff, G. A., 290, 316 (2, 3), 318 (2, 3), 319 (2, 3), 323 (2, 3), 324 (2, 3), 325 (2, 3), 359 (2, 3), 362, 363 (2, 3, 322, 323) Wolff, P. A., 303, 329 Wolfstirn, K. B., 196 Woodall, J., 359, 373, 395 (9), 405,407 (128), 408 (128) Woodbury, H.H., 189,200,201 (2)

Woolley. J. C., 65 Wright, G. B., 351. 368. 369, 410 Wright, M. A., 405 Wu. M. H., 369 Wurst, E. C., 243 Wycoff, R. W. G., 119 Wyluda, B. J., 147 (14), 148, 152, 154 (17), 187 (17) Wyman, M., 54 Yamamoto, T., 344 Yariv, A., 294, 340, 343 (29, 214), 344 (29, 214), 345 (29, 223), 346 (29, 223). 349 (223), 350 (29), 353 (29, 214, 223), 354 (223), 355 (29, 223), 356 (29, 214), 398, 402 (103) Yetter, L. R., 363,406 Younger, C., 394 Yungerrnan, V. M., 360 Zak, J., 171 Zallen, R., 304, 309 (92), 352, 356 (257). 365 (92). 408 ( 155). 409 Zavdritskaya. E. I., 345, 350 (246, 247). 368 Zavoisky, E. J., 189 Zeiger, H. J., 290, 327, 345 (15), 351, 372,. 376 (5), 398, 402 (102), 410 Zemel, J. W.,115 Ziman, J. M., 6, 7 (21, 22), 8, 9, 10 (16). 14, 18 titter, R. N., 224, 228, 229, 247 Zwerdling, S., 198, 358

Subject Index A

B

Absorption at band edge, 331, 332, see also Photoconduction, Internal photoelectric effect, Radiative recombination, and Stimulated emission Aluminum antimonide band structure, 99, 104 Debye temperature, 55, 171 elastic constants, 55, 109, 110, 112, 114 Griineisen parameter, 45, 46 lattice constant, 73 melting point, 117 NMR chemical shifts, 152 NMR relaxation times, 171-173 photoconduction, 234, 235 photoelectric threshold values, 275 specific heat, 52 surface studies, 116-1 19, 123-126, 136 thermal conductivity, 19, 20 thermal expansion, 37, 41 work functions, 275, 279 Aluminum arsenide Debye temperature, 55 lattice constant, 73 photoconduction, 235, 236 specific heat, 52 Aluminum nitride lattice constants, 73 luminescence, 362 photoconduction, 243 photoluminescence, 362 Aluminum phosphide band gap, 362 Debye temperature, 55 lattice constant, 73 luminescence, 362 photoconduction, 236 photoluminescence, 362 Ambipolar diffusion, 10, 14, 208 Anharmonicity parameter, 27-29, see also Griineisen parameter Annealing effects, 24, 117, 131, 132, 276 Auger transitions, 223, 229, 240, 249-262, 301, 369

Band structure, see specific listings of compounds Band-to-band transitions, 212, 257, 295-297, 305, 327, 368, 369, see also Photoconduction, Radiative recombination, Stimulated emission, and Internal photoelectric effect Bismuth telluride photoelectric threshold values, 277 surface studies, 123 work functions, 277 Boron arsenide lattice constant, 73 Boron nitride (cubic) lattice constant, 73 Boron nitride (hexagonal) lattice constants, 73 luminescence, 363 Boron phosphide band gap, 362 lattice constant, 73 luminescence, 362 photoconduction, 243 photoluminescence, 362

425

C

Cadmium telluride electroluminescence, 406 ionic binding, 39, 40 thermal expansion, 3 8 4 Carrier lifetimes GaAs, 238 Gap, 305 InAs, 233, 369 InP, 234 InSb, 238, 357 Cathodoluminescence, 293 GaAs, 336, 337 GaAs-Gap, 369 Gap, 3 18 Characteristic temperature see Debye temperature

426

SUBJECT INDEX

Cleavage, 117, 122, 124131 Coherent light, 387, see also Stimulated emission

D Debye temperature, 15, 29, 40, see also Thermal conductivity, Thermal expansion AIP, AIAs, 55 AISb, 55, 171 Gap, GaAs,56, 171 GaSb, 57, 171 InAs, 59, 171 InP, 58, 171 InSb, 60, 171 Defects, 31, 196, see also Thermal conductivityscattering Diffraction patterns, 125, 133, see also LEED

E Effective mass GaAs, 240 InSb, 259 Edge emission, 314, 354 Efficiency quantum, see Quantum efficiency SOIX cell, 221, 222, 239-241 Elastic constants, see specific listings of compounds Elastic properties, 75ff elastic constants, 75ff, see also specific listings of compounds band effects, 97-107 effects of carrier concentration, 97-107 electron-phonon interactions, 107, 108 inter-relations, 11 1-1 14 second order, 89-94 temperature dependence, 11&113 thermodynamic definitions, 8 M 7 third order, 94-97 elastic waves, 88-97 propagation modes, 93,95,96 finite strain theory, 7 7 4 4 Elastic wave propagation, see Elastic properties--elastic waves Electroluminescence, 29 1 GaAs, 338-357 Gap, 316, 318-325

GaSb, 361 InAs, 360 InP, 359 InSb, 357 Electron defraction, see LEED Electron paramagnetic resonance (EPR), 189ff spectra, 191-201, see also specific listing of compounds spin Hamiltonian, 190, 191 Emission peaks, see Photoluminescence, Cathodoluminescence, Electroluminescence, and Stimulated emission Excitation densities, 292 Exciton energies G A S , 325, 333, 334 Gap, 305-309 I&, 360 InP, 359 Experimental techniques elastic constants, 109 electron beam retardation, 274 electron diffraction, 123 Kelvin contact potential difference, 269 lattice constants, 65-72 linear expansion, 38, 39, 42, 44 NMR absorption, 143, 145 NMR pulsed resonance, 143 photoelectric threshold, 272 surface preparation, 122 thermal conductivity, 11 thermionic emission. 271

F Fermi level, 21 1, 264, 294

G Gallium antimonide antishielding factors, 163 band gap, 361 band structure, 99, 104 Debye temperature, 57, 171 effective mass, 240 elastic constants, 57, 109, 110, 112, 114 electroluminescence, 36 1 exciton energy, 361

SUBJECT INDEX

Griineisen parameter, 45, 46 ionic binding, ionicity, 117 laser action, 405 lattice constant, 73 luminescence, 361 melting point, 117 NMR chemical shifts, 152 NMR line widths, 147, 150 NMR relaxation times, 171-173 nonlinear polarizability, 288 photoconduction, 236, 237 photoelectric threshold values, 275, 278 photoluminescence, 361 quantum efficiency, 361 specific heat, 52 surface studies, 1 1 6 119, 123-126, 128, 129, 135-1 37 thermal conductivity, 18-20 thermal expansion, 37, 41,43 work functions, 275 Gallium arsenide band filling, 344-346, 367, 368 band gap, 325,330-333, 335, 337 band structure, 99. 104, 325, 367. 368 carrier lifetimes, 238, 367 cathodoluminescence, 336, 337, 369 Debye temperature, 56, 171 elastic constants, 56. 109-1 12, 114 electroluminescence. 338-357, 367, 368 effect of magnetic field, 350-351, 368 of pressure, 252 of stress. 252, 368 EPR results conduction electrons, 198, 199, 201 iron, 194196, 201 manganese, 191-194, 201 nickel, cadmium, 198, 201 zinc, 196-198, 201 exciton energies, 325, 333, 334 Griineisen parameter, 45, 46 heavy doping, 346-350, 367, 368 impurity ionization energies, 325-330, 333-338, 356 ionic binding, ionicity, 39, 40, 117 junction luminescence, 338-357 laser action, 382-384, 393, 401, 408410 lattice constant, 73 melting point, 117 NMR chemical shifts, 152 NMR electric field shift, 157 NMR impurity broadening, 164

427

NMR line widths, 147 NMR relaxation times, 171-173 nonlinear polarizabiiity, 288 phonon energies, 334 photoconduction, 237-241 photoelectric threshold values, 275-279 photoluminescence, 333-336, 349, 367 quantum efficiency, 238, 339. 340, 368 surface studies, 1 1 6 1 19, 123-126, 128, 129, 132-134, 136, 137 specific heat, 52 stimulated emission, see laser action thermal conductivity, 17, 18 thermal expansion, 37, 38, 40,43 tunneling, 353-355 work functions, 275279 Gallium arsenide-galhum phosphide band gap, 363, 364 laser action, 406 luminescence, 363-366, 369 Gallium nitride lattice constants, 73 luminescence, 362 photoconduction, 242 photoluminescence, 362 Gallium phosphide band gap, 304 band structure, 99, 104 carrier lifetime, 305 cathodoluminescence, 318, 366 Debye temperature, 56, 171 elastic constants, 112 electroluminescence. 316, 318-325, 366 EPR results iron, 200,201 manganese, 200, 201 exciton energies, 305309 impurity ionization energies, 305-312, 313, 3 15-323 junction luminescence, 318-325 lattice constant, 73 NMR chemical shifts, 152 NMR line widths, 147, 150 NMR relaxation times, 17&173 phonon energies, 305, 306, 314 photoconduction, 241, 242 photoluminescence, 306, 309-3 18, 32 1, 366 quantum efficiency, 290, 314, 322 radiative recombination, 303-325, 366 thermal conductivity, 19,20 thermal expansion, 43

428

SUBJECT INDEX

Germanium Debye temperature, 171 Griineisen parameter, 46 NMR relaxation times, 171 photoelectric threshold values, 275, 277,

279, 280 surface studies, 123, 131 thermal expansion, 3840 work functions, 275, 277,279, 280 Gilvarry equation, 4&48 Gray tin Griineisen parameter, 46 thermal expansion, 3840 Griineisen parameter, 27-29, 34-37,4 4 4 8

Impurity effects antishielding factors, 163 NMR absorption, 162 photoconductivity, 229,230 quadrupole broadening, 159 Impurity scattering, thermal conductivity, 13,

26

Indium antimonide antishielding factors, 163 Auger transitions, 249-262 band gap, 357 band structure, 99, 104,250, 251 carrier lifetimes, 228,357 Debye temperature, 60, 171 deep impurity levels, 229,230 effective masses, 259 H elastic constants, 60, lOITl14 electroluminescence, 357 Heat capacity, 49ff EPR results, 199-201 Debye function, 50 Gruneisen parameter, 45,46 Debye temperature, 4940 impact ionization, 246, 250, 253-359 Einstein frequency, 49,50 internal photoelectric effect, 245ff Hexagonal structure, 64,65,73 ionic binding, ionicity, 39,40, 117 Hot carriers, 179-182, 246, 303 laser action, 405,408,41 1 lattice constant, 73 luminescence, 357,358 I melting point, 117 Impact ionization NMR chemical shifts, 152 In InSb, 246,250,253-259 NMR electric field enhancement. 18 I. 182 Impurities NMR impurity broadening, 160, 162 cadmium, 17, 18, 196, 201, 313, 315, 317, NMR line widths, 147 319, 323, 326,338, 345, 361 NMR relaxation times, 171-173 chromium, 356 nonlinear polarizability, 288 copper, 229-231, 241, 356, 366 nonparabolicity of band, 250, 251 gold, 229-231, 319 photoconduction, 225-232 iron, 194, 195, 200, 201, 320, 356 photoelectric threshold values, 275-278 manganese, 17, 18, 21, 192, 193,200, 201, photoluminescence, 357 323, 334,356,378, 383, 384 radiative lifetime, 357 nickel, 196,201, 338 specific heat, 52 selenium, 307, 313, 315, 326, 337, 345 surface studies, 116119, 123-127, 134, silicon, 163, 31&312, 314-316, 320, 321, 135-137 326, 338, 345 thermal conductivity, 13, 15 silver, 229-23 I , 3 19 thermal expansion, 37, 38, 40,41,43 sulfur, 305-307, 310, 313-317, 321, 326, work functions, 275-278,280 345,356 Indium antimonide-gallium antimonide, tellurium, 17, 18, 164, 310, 313, 315, 326, NMR impurity broadening, 160 335, 337, 338, 344, 356, 361, 363, 367 Indium arsenide zinc, 17, 18, 22, 196, 201, 310, 313, 315, band gap, 360 317-320, 323,326, 335, 337, 338, 344, band structure, 99, 104 345, 355, 356, 359-361, 363, 367, 369, carrier lifetimes, 233 376,378,382-384, 393,400,401,403 Debye temperature, 59, 171

429

SUBJECT INDEX

elastic constants, 59, 109, 110, 112, I14 electroluminescence, 360 exciton energy, 360 ionic binding, ionicity, 117 laser action, 405, 408, 410, 41 1 lattice constant, 73 luminescence, 360 melting point, 117 NMR chemical shifts, 152 NMR line widths, 147 NMR relaxation times, 171-173 nonlinear polarizability, 288 photoconduction, 231-233 photoelectric threshold values, 275-278 phonon energy, 360 specific heat, 52 surfacestudies, 11&119, 123-126, 128, 129, 136, 137 thermal conductivity, 14-17 thermal expansion, 37,43 work functions, 275-278 Indium arsenide-gallium arsenide laser action, 406 NMR impurity broadening, 161 thermal conductivity, 20 Indium arsenide-indium phosphide laser action, 406 thermal conductivity, 20 Indium nitride lattice constants, 73 photoconduction, 243 Indium phosphide band structure, 99, 104 carrier lifetimes, 234 Debye temperature, 58, 171 elastic constants, 112 electroluminescence, 359 exciton energy, 359 impurity ionization energies, 359 laser action, 404, 405, 408 lattice constant, 73 luminescence, 359, 360 NMR chemical shifts, 152 NMR relaxation times, 169, 171-173 photoconduction, 233, 234 photoelectric threshold values, 275 specific heat, 52 thermal conductivity, 17 thermal expansion, 43 work functions, 275 Infrared detectors, 23C232

Injection laser, see Stimulated emission Injection luminescence, see Electroluminescence Internal photoelectric effect (in InSb), 245ff Auger transitions, 249-262 band structure of InSb, 250, 251 impact ionization, 246, 250,253259 nonparabolicity of band in InSb, 250, 251 quantum efficiency, 245-253, 256-262 Ionic binding (ionicity), 117 thermal expansion, 39-41 Ionization energies, impurity GaAs, 325-330, 333-338, 356 Gap, 305-312, 313, 315-323 InP, 359 Irradiation effects, 23, 24, 366, 369 Isoelectronic sequences, 40

J

Junction laser, see Stimulated emission Junction luminescence, see nlso Electroluminescence band filling, 344 deep levels, 316, 356 reverse bias, 323, 351 thermal injection, 341 tunneling, 353

1

Landt formula, 190 Larmor frequency, 142, 144, 153, 167, 173 Laser materials, see Stimulated emission Laser structures, see Stimulated emission Lattice constants, 63ff hexagonal structure, BN, B,P, 65, 72, 73 rhombohedral structure B,P, 72, 73 BioP, (B,P), 72, 73 wurtzite structure, AIN, GaN, InN, 66, 73 zinc blende structure AIP, MAS, AISb, 68, 69, 73 BN, BP, BAS, 67, 68,73 Gap, GaAs, GaSb, 69, 70, 73 InSb, InAs, InP, 71, 72, 73 Lead sulfide, -selenide, -telluride laser action, 406 Linear expansion coefficients, see Thermal expansion

430

SUBJECT INDEX

Lorenz number, 10 Low energy electron defraction (LEED), 115ff, see also surface studies under specific listings of compounds AlSb, 136 annealed surfaces, 132 atomic scattering factor, 121, 122 cleaved surfaces, 124131 equipment, 123, 124 GaAs, 125, 128, 129 heat treated surfaces, 131 ion bombarded surfaces, 131, 132 oxygen adsorption, 135, 136 substrate structure, 117-120 surface preparation, 122, 123

M Magnetic field, 21-23, 369 heat switch, 5 stimulated emission, 408, 410, 41 1 thermal conduction, 5, 21-23 Melting point, 117 GaAs, GaSb, InAs, InSb, AlSb, 117 Miller indices, 120, 121, 124, 128

N Nonlinear optics, 2838 reflected harmonic radiation GaAs, 287 Nonradiative recombination, see Radiative recombination, nonradiative processes Nuclear magnetic resonance (NMR), 141ff antishielding factors, 163 absorption line, 145-165 chemical shifts, 15C152 electrical shifts, 155-158 exchange broadening, 147-150 impurity-quadrupole broadening, 159-165 line widths, 1 4 H 5 0 nuclear polarization, hot electrons, 178182 oscillating electric fields, 173-176 quadrupole effects, 152-159, 183-187 second moments, 145-147 spin lattice relaxation, 165-173 spin temperature, 144, 145 strain, 152-155

ultrasonic saturation, I 7 6 1 78

P Phonon cooperative transitions, 291,302,355 Phonons amplifiers, 5 energies GaAs, 334 Gap, 305, 306, 314 InP, 359 thermal transport, see Thermal conductivity Photoconduction, 205ff, see also specific listings of compounds junction effects, 215-222 low resistivity materials, 210, 21 1 photoconductivity, 212, 215 photostatic effects (high resistivity materials), 209, 210 photovoltaic effects, 211, 212 recombination, 222-225 solar batteries, 215-222 Photoelectric effect, see Internal photoelectric effect, Photoelectric threshold and work function Photoelectric emission, 266, 267, 275, 278, see Photoelectric threshold and work function Photoelectric threshold and work function, 263ff effect of doping, 279, 280 effect of annealing, 279 measurement techniques electron retardation reflection, 274 Kelvin contact potential difference, 269271 photoelectric techniques, 272-274 thermionic methods, 271, 272 spectral dependence, 266-268, 278 surface states, 275, 276, 219, 280 tabulations, 275 threshold values, 275-278 Photoelectromagnetic Effect (PEM), 206,224, 227, 231-233, 237 Photoluminescence, 292 AIP, BP, GaN, AIN, 362 GaAs, 333-336, 349 Gap, 306, 309-318, 321 GaSb, 361 InSb, 357

431

SUBJECT INDEX

Photons, 10 Photostatic effects, 209, 210 Photovoltaic effects, 206, 21 1, 212, 215-222, 227, 231-236, 241, 242 surface barrier, 234, 235

Recomhnation, radiative, see Radiative recombination Relaxation times thermal transport, 7-10, 10-31 Rhombohedral structure, 64, 72, 73

S

Q Quadrupole effects, see Nuclear magnetic resonance Quadrupole Hamiltonian, 166, 183 Quantum efficiency, 207, 290 electroluminescence, 368. 389-395 GaAs, 238, 368 GaAs-Gap, 369 Gap, 290, 314, 322 GaSb, 361 InSb, 245-253, 256262, 357

R Radiation patterns (lasers), 383, 388 Radiative recombination, 289ff Auger processes, 301 excitation, 292-295 exciton energies, 305-309, 325, 333 GaAs, 325-357 GaAs-Gap, 369 Gap, 303325 GaSb, 361,362 heavy doping, 291, 301, 328-330, 346350 hot carrier effects, 303 I d s , 360 InP, 359, 360 InSb, 357-359 light doping, 326-328 miscellaneous compounds and alloys, 362366 nonradiative processes, 290, 299-301, 320, 323, 325, 367, 369 phonon cooperation, 291, 302, 355 quantum efficiency, 290, 314, 322. 369 radiative lifetime, 348 Recombination, 222-225, 240, 293-299, 309-313, see also Radiative recombination Recombination, donor acceptor pair, 309313 Recombination, nonradiative, see Radiative recombination

Scattering processes, see Thermal conductivityscattering Silicon Griineisen parameter, 46 photoelectric threshold values, 275, 277, 279, 280 surface studies, 123, 131 work functions, 275, 277, 279, 280 Solar batteries, 21 5-222, 239-241 Specific heat, 49ff, see Heat capacity specific values, 52 Spectra sensitivity GaAs, 238 InSb photoconductor, 225-227 Spin Hamiltonian, 191, 194, 196 Spin orbit splitting, 105 Spontaneous emission, spontaneous vs stimulated emission, 367, 371-376 Stimulated emission, 371ff coherence, 387-389 directionality, 383-387 effects of external fields magnetic fields, 408, 410, 411 pressure, 408, 409 strain, 408410 temperature, 394, 395, 403, 407, 408 laser materials, 372, 4 0 M 7 laser structures, 376380, 396-401 mode structure, 380-383 quantum efficiency, 38%395 radiation confinement, 3 9 M 3 stimulated vs spontaneous emission, 371376 Stoichiometry, 31, 196 thermal conduction, 31 EPR, 196 Surfaces. 1 ISff, see also Low energy electron defraction Surfaces states, 275-276, 279-280, 1

T Tellurium, laser action, 407 Thermal conductivity, 3ff

432

SUBJECT INDEX

ambipolar diffusion, 10, 14 anharmonicity parameter, 27-29 compounds, see specific listings of compounds Debye temperature, 4, 6, 15, 27-29 electrons, holes, 10, 30 Griineisen parameter, 27-29 heat switch, 5 irradiation effects, 22-24 lattice, 3ff Lorenz number, 10 magnetic field, 5, 21-23 measurement techniques, 11, 12 Angstrom method, 1I , 12 comparison method, 11, 12 radial heat flow, 12 phonons, 4,6-9, 27-30 scattering (thermal transport) acoustical modes, 17, 28 boundary, 8,25,26 defect, mass difference, 7-9, 13-21 electron-phonon, 9, 18, 27 impurity, 8, 13, 26 mass difference, defect, 7-9, 13-21 normal processes, 7-9, 25, 28, 30 optical modes, 15, 20, 28, 29 phonon-phonon, 4,6-9, 13, 15 relaxation times, 7-10, 13-31 resonance, 18, 26, 27 three-phonon, 9, 27, 28 Umpklapp processes, 7-9, 25, 28, 30 Wiedemann-Franz relation, 10, 14 Thermal expansion, 33ff Debye temperature, 35, 40-42 expansion coefficients, linear AlAs, 43 AISb, 37, 41 CdTe, 38-40 GaAs, 37, 38,40,43

Gap, 43 GaSb, 37,41,43 Ge, 38-40 gray tin, 3 W InAs, 37, 43 InP, 43 InSb, 37, 38, 40,41, 43 negative values, 36, 39 ZnSe, 38-40 Gilvarry equation, 4 6 4 8 Griineisen’s law, 34 Griineisen parameter, 34-37,44-48 Tunneling GaAs, 353-355, 368 Gap. 367

U Ultrasonic saturation, 176-178

V Van Vleck second moments, 145-147

W Wiedemann-Fram relation, 10, 14 Work function, see Photoelectric threshold and work function Wurtzite structure, 64,66, 73

2 Zinc blende structure, 64,67, 73, 118 Zinc selenide ionic binding, 39, 40 thermal expansion, 38-40