Semiconductors and Semimetals v. 4

SEMICONDUCTORS AND SEMIMETALS VOLUME 4 Physics of III-V Compounds

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

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

VOLUME 4 Physics of 111-V Compounds

1968

ACADEMIC PRESS New York and London

COPYRIGHT @ 1968, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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United Kingdom Edition published bv ACADEMIC PRESS INC. (LONDON) LTD Berkeley Square House, London. W.1

LIBRARY OF

CONGRESS CATALOG CARD

NUMBER : 65-26048

PRINTED IN THE UNITED STATES OF AMERICA

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

L. W. AUKERMAN, Aerospace Corporation, Los Angeles, California (343) R . T . BATE,Central Research Laboratories, Texas Instruments, Incorporated, Dallas, Texas (459) A. S . BORSHCHEVSKII, A . F. Ioffe Physico-Technical Institute, Academy of Science of the USSR, Leningrad, USSR (3) A. G. CHYNOWETH, Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey (263) N . A. GORYUNOVA, A . F. Ioffe Physico-Technical Institute, Academy of Science of the USSR, Leningrad, USSR (3, 413) DONL. KENDALL, Texas Instruments, Incorporated, Dallas, Texas (163) F . P. KESAMANLY, Institute of Physics, Academy of Science of the Azerb. SSR. Baku, USSR (413) ROBERT W . KEYES, IBM Watson Research Center, Yorktown Heights, New York (327) D . N . NASLEDOV, A . F. Ioffe Physico-Technical Institute, Academy of Science of the USSR, Leningrad, USSR (413) N. N . SIROTA, The Institute of Solids and Semiconductors, The Belorussian Academy of Sciences, Minsk, Podlesnaya, USSR (35) D . N . TRETIAKOV, A. F. Iofe Physico-Technical Institute, Academy of Science of the USSR, Leningrad, USSR (3)

V

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Preface The extensive research devoted to1 the physics of compound semiconductors and semimetals during the paist 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 enableld 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, certain chapters will also be useful in graduate study. 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 IIILV 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 caririer phenomena, and photoelectronic effects. The present volume includes thermodynamic properties, phase diagrams, diffusion, hardness, and phenomena in solid solutions as well as the effects of strong electric fields, hydrostatic pressure, nuclear irradiation, and nonuniformity of impurity distiributions on the electrical and other properties of 111-V compounds. Subsequent volumes of the series will be vii

viii

PREFACE

devoted to further fundamental phenomena such as lattice dynamics, galvanomagnetic effects, luminescence, charge-carrier injection, and nonlinear optical phenomena, as well as to major applications which exploit properties of semiconductors. The latter category will include bulk negative resistance as well as junction devices, high-temperature diodes and power rectifiers, and a two-volume treatment of infrared detectors. 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 and Howell Company and the Battelle Memorial Institute for providing the facilities and the environment necessary for such an endeavor. Thanks are also due to the U.S. Air Force Offices of Scientific Research and Aerospace Research and the U.S. Navy Office of Naval Research and the Corona Laboratory, whose support has enabled the editors to study many features of compound semiconductors. The assistance of Rosalind Drum, Martha Karl, 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.

February, 1968

R. K. WILLARDSON ALBERTC. BEER

Contents LISTOF CONTRIBUTORS . . . MACE . . . . . CONTENTS OF PREVIOUSVOLUMES.

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xi11

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PHYSICAL PROPERTIES Chapter 1 Hardness N . A . Goryunova, A . S. Borshchetskii, and D . N . Tretiakov I. Introduction . . . . 11. Methods of Determining Hardness . 111. Experimental Results and Discussion IV. Concluding Remarks . . .

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3 4 11 32

Chapter 2 Heats of Formation and Temperatures and Heats of Fusion of Compounds A"'BV N . N . Sirota 1. Introduction . . . . . . . . . . 11. Structure of Compounds A"'BV and Some Crystallochemical Relations 111. Phase Diagrams . . . . . . . . . .

IV. V. VI. VII. VIII. IX .

Vapor Pressures . . . . . . . . , Heats, Free Energies, and Entropies of Formation . . . Bonding . . . . . . . . . . Melting . . . . . . . . . . Thermodynamic Properties, Energy Bands, and the Periodic System Conclusion . . . . . . . . .

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36 41 47 14 86 106 132 147 159

Chapter 3 Diffusion Don L . KendaN I. Introduction

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11. Defect Equilibria in Compounds 111. Diffusion in Compounds .

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IV. Self-Diffusion in 111-V Compounds. . V. Impurity Diffusion in 111-V Compounds . VI. Summary and Conclusions . . . ix

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189

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CONTENTS

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EFFECTS OF ELECTRIC FIELDS, PRESSURE, AND NUCLEAR RADIATION

Chapter 4 Charge Multiplication Phenomena A . G . Chynoweth Introduction . . . . . . Theories of the Ionization Rate . . . AvalancheBreakdowninP-NJunctions . . Methods of Measuring Charge Multiplication . . . . . Experimental Results . Miscellaneous Phenomena Associated with Charge tions . . . . . . . . VII. Breakdown in Bulk Semiconductors . . VIII. Impact Ionization of Impurities . . . I. 11. 111. IV. V. VI.

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263 268 286 293 . . . . . 300 Multiplication in Junc. . . . . 307 . . . . . 320 . . . . . 323

Chapter 5 The Effects of Hydrostatic Pressure on the Properties of 111-V Semiconductors Robert W . Keyes I. lntroduction

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11. Effect of Pressure on Energy Bands 111. Optical Absorption Spectrum .

IV. V. VI. VII. VIII.

. Electroluminescence Electrical Conductivity Other Electrical Properties . Phase Transitions . Elastic Properties .

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327 328 329 33 1 . 332 . 336 . 338 .341

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Chapter 6 Radiation Effects L. W . Aukerrnun I. General Discussion

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11. Threshold Experiments . . . . . 111. Radiation Effects in Various 111-V Compounds IV Radiation Damage in Devices . . .

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SOLID SOLUTIONS AND IMPURITY EFFECTS

Chapter 7 Phenomena in Solid Solutions N . A . Goryunouu, F. P.Kesamanly, and D . N . Nusledou I. Introduction

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11. Substitutional Solid Solutions in Systems Involving 111-V Compounds, and

111. IV. V. VI.

. . . . Their Equilibrium Phase Diagrams Preparation and Characterization of Equilibrium Alloys . Phenomena in Solid Solutions with Isovalent Substitution . Phenomena in Solid Solutions of Heterovalent Substitution Conclusions . . . . . . . .

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CONTENTS

Chapter 8 Electrical Properties of Nonuniform Crystals R. T. Bate .

I. Introduction

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11. Origin and .Nature of Inhomogeneities Usually Encountered in Crystals

Grown from the Melt . . . . . . . 111. Typical Effects of Inhomogeneity on Electrical Properties . IV. Calculation of Isothermal Transport Effects in Inhomogeneous V. Detection of Inhomogeneities . . . . . AUTHORINDEX .

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SUBJECT INDEX .

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. 460 . 464 Conductors . 471 . . . 474 . .

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Semiconductors and Semimetals Volume 1 Physics of 111-V Compounds C. Hilsum, Some Key Features of 111-V Compounds Franc0 Bassani, Methods of Band Calculations Applicable to Ill-V Compounds E. 0. Kane, The k p Method V . L. Bonch-Bruevich, Effect o f Heavy Doping on the Semiconductor Band Structure Donald Long, Energy Band Structures of Mixed Crystals of 111-V Compounds Laura M . Roth and Petros N . Argyres, Magnetic Quantum Effects S. M . Puri and T . H . Geballe, Thermomagnetic Effects in the Quantum Region W . M . Becker, Band Characteristics near Principal Minima from Magnetoresistance E. H . Putley, Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in InSb H . Weiss, Magnetoresistance Bersy Ancker-Johnson, Plasmas in Semiconductors and Semimetals

Volume 2 Physics of 111-V Compounds M . G. Holland, Thermal Conductivity S . I . Nouikoua, Thermal Expansion U. Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J . R . Drabble, Elastic Properties A . U . y a c Rae and G. W . Gobeli, Low Energy Electron Diffraction Studies Robert Lee Mieher, Nuclear Magnetic Resonance Bernard Goldstein. Electron Paramagnetic Resomnance T . S. Moss. Photoconduction in 111-V Compounds E . AntonEik and J . Tauc, Quantum Efficiency of the Internal Photoelectric Effect in lnSb G. W . Gobeli and F. G. Allen, Photoelectric Threshold and Work Function P. S. Pershan, Nonlinear Optics in 111-V Compounds M . Gershenzon, Radiative Recombination in th'e 111-V Compounds Frank Stern, Stimulated Emission in Semiconductors

Volume 3 Optical Properties of 111-V Compounds Marvin Huss, Lattice Reflection William G. Spitzer, Multiphonon Lattice Absorption D. L. Stierwalt and R . F. Potter, Emittance Studies H . R . Philipp and H . Ehrenreich, Ultraviolet Optical Properties Manuel Cordona, Optical Absorption above the Fundamental Edge Earnest J . Johnson, Absorption near the Fundamental Edge John 0. Dimmock, Introduction to the Theory of Exciton States in Semiconductors B. Lax and J . G. Mavroides, Interband Magnetooptical Effects H . Y . Fan, Effects o f Free Carriers on the Optical Properties Edward D . Palik and George B. Wright, Free-Carrier Magnetooptical Effects Richard H . Bube, Photoelectronic Analysis B. 0. Seraphin and H . E. Bennett, Optical Constants xiiii

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SEMICONDUCTORS AND SEMIMETALS VOLUME 4 Physics of 111-V Compounds

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Physical Properties

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CHAPTER 1

Hardness N . A . Goryunova A . S. Borshchevskii D.N . Tretiakov 1. INTRODUCTION . . . . . . . . . . . 11. METHODS OF DETERMINING HARDNESS. . . . 1 . Testers . . . . . . . . . . . . 2 . Indenters . . . . . . . . . . . . 3 . Preparation of the Surface of a Sj)ecimen . . . 4. Factors Infiuencing Microhardness Measurements 111. EXPERIMENTAL RESULTSAND DISCUSION. . . 5. Compounds . . . . . . . . . . . 6 . Solid Solutions . . . . . . . . . . IV. CONCLUDING REMARKS . . . . . . . .

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1. Introduction To obtain a better understanding of phenomena in semiconductors, investigations of their physical, chemical, electrical, and other properties, as well as variations in these quantities, are useful. Hardness is one of the physicochemical properties which not only characterizes the state of the material under test-with dependence on the previous history of the test specimen-but also gives information on some deeper specific features of the material as, for example, the character of the chemical bonding. This will be illustrated later in connection with the 111-Vcompounds. A strictly scientific definition of hardness has not yet been given. The reasons for this are that hardness is dependent on many factors, and a great number of widely different methods of hardness measurement have been developed, most of which require a particular hardness definition. Hardness is generally related to the elastic and plastic deformation characterististics of the solid material. All the measurement techniques may be separated into two groups, according to the method of load application : static indentation tests (e.g., by use of a diamond pyramid, cone, or steel ball) and dynamic indentation tests (e.g., scleroscopic tests). Those most widely used in determining the hardness of semiconductors are static methods. For these, hardness can be defned as the resistance of the material to the formation of an indentation in its surface. 3

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N . A . GORYUNOVA, A. S. BORSHCHEVSKII, AND D. N. TRETIAKOV

Such an indentation, or impression, is formed on a specially prepared surface of the semiconductor specimen by means of an indenter. Inasmuch as hardness and brittleness of the 111-V compounds are relatively high, it is almost impossible to carry out hardness measurements by the ordinary methods used for metals. Such methods make it necessary to apply high loads (over 5 kg) on an indenter and to produce indentations of a considerable volume (more than 640,000 p3). The 111-V compounds are fractured under such conditions of testing. Hardness in the 111-V compounds therefore must be measured by the method of microhardness, at which an indenter is loaded with a weight not exceeding 200 g, and the volume of the impression ranges between 1 p3 and 3000p3. It should be noted that some authors have reported the use of ordinary methods for hardness measurements of the 111-V compounds (e.g., the This was, however, only for the softest and most plastic of Brine11 test the group. Hardness in the 111-Vcompounds was also measured by one of the dynamic tests, namely, the scratch m e t h ~ d Hardness .~ measurements by the scratch method are carried out with the help of a number of different types of testing machines (e.g.,Bierbaum’s microcharacter testing m a ~ h i n e ~ . ~ ) , and apparatus for microhardness measurements by static methods ( e g , the PMT-3 testing machine). So far, this method has seldom applied to semiconductors. However, it appears that for certain purposes it can be successfully used-for determination of hardness anisotropy by plotting the so-called hardness “rosettes,” for instance. The method most widely used for measuring hardness in the 111-V compounds is a microhardness measurement, which utilizes, as a rule, some form of the diamond pyramid as the indenter. ‘7’).

11. Methods of Determining Hardness 1. TESTERS

A considerable number of hardness testing machines have been developed in different The Bergsman testing machine’ and the PMT-3,9,‘0 as made by Khrushchov and Berkovich, are currently used for investigation of semiconductors. A drawing of the latter apparatus is shown in Fig. 1, T. S. Liu and E. A. Peretti, Trans. Am. SOC. Metals 45,677 (1953). T. S. Liu and E. A. Peretti, Trans. Am. SOC.Metals 44,539 (1952). V. M. Gol’dshmidt [Goldschmidt] Usp. Fiz. Nauk 9, 81 1 (1929). C. Bierbaum, Iron Age 105,211 (1920). C. Bierbaum, Trans. AIME 69,972 (1923). B. W. Mott, “Microindentation Hardness Testing.” Butterworths, London and Washington, D.C. [Russian Transl. : “Ispitanie na tverdost microvdavlivaniem.” Metallurgizdat, Moscow, 19601.

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HARDNESS

5

FIG.1. The principal scheme of the PMT-3 hardness testing machine: 1-rack and pinion adjustment ; 3-+olumn ; 4-indentation mechanism ; 5-test-piece ; movement ; 2-fine b b a s e ; 7-measurement eye-piece; 8-body-tube; 9-adjusting screws: 10-illuminator: 1l-objective; 1 2 4 i a m o n d pyramid indenter; 13-testing table which is movable along orthogonal axes. b””Metodyispytaniya na mikrotverdost,” Sbornik statei [“Methods of Microhardness Testing”, Collected works] (M. M. Khrushchov, E. S. Berkovich, V. M. Glazov, V. K. Grigorovich, and D. A. Sarkisyan, eds.). Izd-vo “Nauka,” Moskva, 1965. B. I. Philipchuk, Sovremennoe sostoyanie teklhniki opredeleniya tverdosti metallov. Moskva, Gos. Izd-vo Standartov, 1960 (“Present Stage of Techniques for Metal Hardness Measurements”). * G. A. Wolff, L. Toman, N. I. Field, and J. C. Clark, in “Halbleiter und Phosphore” [“Semiconductors and Phosphors”] (M. Schon and H. Welker, eds.), p. 463. Wiley (Interscience), New York, 1958. V. M. Glazov and V. N. Vigdorovich, Mikrotverdost metallov (“Microhardness of Metals”). Moscow, gos. Nauchn.-Tekhn. Izd. Lit. PO Ch(ernoii Tsvetn. Met., 1962. l o M. M. Khrushchov and E. S. Berkovich, PMT-2 and PMT-3, 1950.

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N . A . GORYUNOVA, A .

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BORSHCHEVSKII, AND D. N . TRETIAKOV

with its indenting mechanism illustrated in Fig. 2. The PMT-3 is suitable for most studies and relatively simple in its operation. Many different automatic devices have been designed for load application by the PMT-3 indenter, but of particular interest is one based on the use of the deformation of a bimetallic plate under heating.'' This device eliminates completely the deleterious effects of mechanism operation obtained when the indentation is produced by means of motors.

FIG 2. Indenting mechanism of the PMT-3 : 1-rod : 2-nut for raising the indenting mechanism: 3 and 4-upper and lower elastic plates; 5-housing: 6-one of two (right) adjusting screws; 7-handle; 8-diamond indenter point; 9-test-piece; I-distance between the piece and the objective (for objective aperture A - 0.65 it is equal to I mm); e-necessary gap between the rod flange and the catcher.

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2. INDENTERS The indenters most widely used for microhardness measurements on semiconductors are the diamond pyramids on square" or rhombic13 bases. The first one is widely known as the Vickers diamond pyramidal indenter and the second one as the Knoop diamond pyramidal indenter. The two types of

l3

E. S . Berkovich, Industry Laboratory 29, No. 10, 1250 (1963). R. L. Smith and G . F. Sondland. J . Iron Steel Ins?. 111. 285 (1925). F. Knoop, C . G. Peters, and W. 'B. Emerson, J . Res. Natl. Bur. Std. 23, 39 (1939)

1.

HARDNESS

7

indenters of proper configurations yield hardness numbers very nearly equal to each other and approximately identical to those obtained on the same material from Brine11 hardness tests [the angle of the indentation of a ball being equal to 44”, which is considered to be optimal). We shall use microhardness values obtained with both K.noop and Vickers indenters. It is well to remember, however, that a complete identity in results does not seem to exist. In addition, it should be noted that with the Knoop indenter the hardness of very hard and brittle solids can be measured. This is because the diagonal is much longer and penetration is much shallower than is the case for a square impression, all other conditions being the same. Also, the value of the elastic aftereffect obtained with the Knoop indenter is considerably less than that obtained with the square-base indenter. On the other hand, the Knoop test has some disadvantages. A major one is that the shallowness of the Knoop indentation provides a means of examining the hardness of the uppermost surface layer only and not of the specimen as a whole. Hardness measured by the above method is defined as the ratio of the load applied (in kilograms) to the projected area of the indentation (in mm’) and can be expressed by the following formulas : H, = 1854P/d2

kg/mm’

H, = 14,230P/d2 kg/mm’

(for the pyramid on the square base), (for the Knoop pyramid, the longer diagonal),

where P is the load in grams and d the impress diagonal (in microns). As was mentioned before, the PMT-3 or some analogous instrument can be used for microhardness testing by the scratch method. Scratches are produced by a diamond pyramidal indenter under a small load, which is generally 10 g. For this, the tester table is moved along one of the coordinate axes and the appropriate drum is rotated smoothly by hand. The microhardness is computed on the basis of no less than 20 measurements and expressed as the ratio of the load (in kilograms to the square of the scratch width (in millimeters). 3. PREPARATION OF THE SURFACE OF

A

SPECIMEN

The state of the surface to be indented is a very important consideration among the numerous factors influencing the ultimate results of microhardness measurements. Therefore this problem will be discussed in detail. The preparation of the surface consists usually of lapping with special powders or papers, and of polishing and etching. The lapping is carried out in the following order. The powder of certain grain size (or the emery paper) is applied to a smooth surface of glass, wetted with water (or some other liquid), and the specimen is then uniformly ground

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N . A. GORWNOVA, A .

s. BORSHCHEVSKII, AND

D. N. TRETIAKOV

on the glass. It is important not to round the specimen edges. The specimen surface can also be ground on a special motor-driven disc or commercial lapping machine. In the study of semiconductors, we are often dealing with small-sized specimens. For the successful lapping of such specimens it is more convenient to pot them in some mass, as, for example, into polystyrene. In this first stage of specimen treatment, the surface undergoes the most important changes. Owing to high deformations, there appears a stressed condition of the surface layer (so-called work hardening) which often can influence microhardness measurements considerably. Specimen polishing is done with a cloth, stretched on the rotating disc, and impregnated with special pastes containing solids (such as Cr,03) or by water suspensions of the finest grained A1,0,. The specimen is polished to obtain a mirror surface. This stage is less critical with respect to work hardening of the surface layer, because now material deformations are much lower compared to those in the lapping stage. The specimen is now etched in a special solution. The etching is necessary for determination of the microstructure of the material. This stage is of particular significance for polycrystalline specimens. The method based on the combination of electrolytic polishing and etching has many advantages, but as of the present time, this method has not been sufficiently developed for the 111-V compounds. Surface hardening due to special preparation of the surface is likely to increase markedly the hardness to be measured. Reliable results were obtained only on semiconductor single crystals. Mil’vidskii and Liner14 have established that microhardness values for etched and nonetched silicon specimens are very different (950 and 1670 kg/mm2, respectively), which fact has been attributed by the authors to significant hardening of the surface layer during mechanical treatment of the specimen. In the case of silicon carbide, however, microhardness of the natural face and that of a machined surface were very approximately equal (2550 and 2520 kg/mm2, re~pectively).’~ Hence the conclusions of Mott6 should be valid, in that the greater the hardness or, in other words, the less the plastic deformation of the specimen, the less is the surface hardening during its mechanical treatment. The surface hardening of the specimen is undoubtedly affected by defects in the crystal structure of the surface layer. For example, the above-mentioned work on ~ i l i c o n ‘indicated ~ that an increase in the dislocation density over the area of measurement from 1 x lo3 to 2 x lo4 cme2 was followed by an increase in the microhardness l4

M. G. Mil’vidskii and L. V. Liner, Nauchn. Tr. Gos. Nauchn.-lssled. i Proektnogo in-ta Redkometallich. prom-sti “Giredmet” VI, 149 (1962). N. W. Thibault and H. L. Nyguist, Trans. Am. SOC.Metals 38,271 (1947).

1.

HARDNESS

9

from 830 to 1250 kg/mm2. It is reasonable to assume that the mechanical treatment and polishing of the specimen induce dislocations in the surface layer, which increase microhardness values. One should note that the best method for removal of the surface layer after mechanical treatment is chemical and electrolytic etching, which sometimes might even improve the quality of the surface (as is the case with electrolytic polishing). The use of an anneal for removal of the work hardening is not suitable for the III-V compounds. At elevated temperatures the more volatile BV component (P, As, Sb) evaporates from the polished surface; the surface layer composition and reflecting power then change, and measurement results are not correct. 4. FACTORS INFLUENCINGMICROHARDNESS MEASUREMENTS

The main factors affecting microhardness values of the III-V compounds and their solid solutions can be divided into two groups : factors due to instrumental errors and factors due to the properties of the material. Factors of the first group depend on probable deviation of the loading from the nominal value, some deformation of the indenter or its lateral movement, which might be induced by incorrect profiles of an indenter or poor quality of manufacturing of individual elements. These factors are common for hardness measurements of any material and have at various times been described in the They will not be discussed further here. Factors of the second group depend on the properties of the material, in this case on the bulk properties of the III-V compounds. (The role of the preparation of the surface was discussed above.) The anisotropy of semiconductor crystals was revealed when the microhardness of different crystallographical faces of indium antimonide was measured by producing the indentation by means of the square base pyramid.I6 This phenomenon is particularly prominent on the crystals of germanium and silicon. Germanium microhardness in the (1TO) direction is equal to 780 79 and in the (1 12) direction, 845 f 25 kg/mm2. Silicon microhardness values in the same directions are 1150 _+ 110 and 1330 _+ 112 kg/mm2, respectively.' The anisotropy of the III-V crystals was revealed in microhardness measurements by the scratch method. l 7 Special attention must be paid to the nature of the specimens. To exclude the influence of grain boundaries on the microhardness values, it is desirable to make measurements on monocrystalline specimens or on polycrystalline ones with several large grains. Furthermore, the indentation must be produced l6

M.S. Ablova and N . N.Feoktistova, Fiz. Tiierd. Tela 5, 364 (1963) [English Transl.: Soviet Phys.-Solid State 5,265 (1963)l. V. N. Lange and T. I. Lange, Fiz. Tverd. Tela 5, 2029 (1963) [English Transl.: Soviet Phys.Solid State 5, 1483 (1964)l.

10

N. A . GORYUNOVA, A . S. BORSHCHEVSKII, AND D. N. TRETIAKOV

at the largest possible distance from the grain boundary. The optimal distance is eight times the diagonal, and the minimum distance is twice the diagonal of the indentation. The indentation depth must not exceed one-tenth of the thickness of the grain, otherwise the measurement will be affected by the structure used to support the grain specimen. This substructure effect, which is hardly controlled, seems to be present only in the study of polycrystalline materials. The hardness of polycrystalline specimens is greatly influenced by the conditions of crystal growth and the strain in the crystal.I8 The role of the homogeneity of specimens must be emphasized, particularly for solid solutions, because in the process of their crystallizing the phenomena of segregation and coring appear. In these cases, microhardness values for the central and boundary sections of a single grain, as well as of different grains, can be different. So the main requirement in the structure of solid solutions (after the size of crystals) is the elimination of chemical inhomogeneity, both within a single grain and for adjacent grains. The effect of the elastic recovery of an indentation on microhardness measurements has not been carefully studied for the 111-V compounds. The 111-V compounds and their solid solutions, as well as a number of other semiconductors of the diamond-type structure, are characterized by high brittleness. Because of this fact we must pay special attention to the indentation quality. To obtain microhardness values close to the correct ones, only clearly defined indentations without cracks or other defects should be examined. Using only this type of indentation, the hardness/structure diagram was plotted for homogeneous solid solutions of the Ge-Si system.” According to the Kurnakov rules (to be discussed later) this diagram has a maximum, although results that appear to be different have been reported.*’ Evidently, to load the indenter properly it is necessary to take into account the specific properties of the material to be tested. The harder and more brittle the specimen, the smaller the load must be. It is much more difficult, however, to measure the indentation diagonal with sufficient accuracy at small loads. In addition, specific features of the surface layer and the indenter vibration will be of greater influence at small loads. It would undoubtedly be of interest to apply the method developed for studying microhardness variation with composition to the solid solutions InAs-InP.20aThe method of taking into account the effect of the material A. S. Borshchevskii, N. A. Goryunova, and N. K. Takhtareva, Zh. Tekhn. Fiz. 27. 1408 (1957) [English Transl.: Soviet Phys.-Tech. Phys. 2. 1301 (1957)l. V. M. Glazov and Lyu Chzhen’-Yuan’, l z v . Akad. Nauk. SSSR, Otd. Tekhn. Nauk, Mer. i Toplivo No. 2, 99 (1961). C. C. Wang and B. H. Alexander. Acta Met. 3, 515 (1955) [“Hardness of Germanium-Silicon Alloys at Room Temperature,” Russian Collection, edited by D. A. Petrov. Izd. Lit., p. 427 (1960)l. *‘*See Ref. 9. p. 212.

1.

4

11

HARDNESS

5

FIG.3. Five standards of crystal brittleness.

brittleness is the following. Each impress can be characterized by one of five standards of brittleness (see Fig. 3). Of course, the most valid microhardness values will be obtained on the indentations of “0” standard. In cases when these indentations are not obtained by measurements, they can be obtained by extrapolation of the curve of microhardness versus standard of brittleness. This dependence is often linear. The effect of small concentrations of impurities on semiconductor microhardness will be discussed later.

111. Experimental Resiults and Discussion

5 . COMPOUNDS a. Measurement Results

Systematic microhardness measurements on the 111-V compounds have been carried out since 1957.18 The most reliable results are summarized in Table I, together with hardness values a.fter Mohs scale.4 When considering the data given in Table I, it is necessary to keep in mind the effect on microhardness values of the different factors discussed in Section 4. Reference to Table I shows that in the great majority of cases there is good agreement between microhardness values obtained from the Knoop’s pyramid and those obtained from the square base pyramid. Microhardness values for indium arseriide reported by Vigdorovich and N a ~ h e l ’ s k i were i ~ ~ obtained with due account for microbrittleness, whereas this factor was not considered in other works. The technique used was

’’ Ya. K. Syrkin, Usp. Khim. 31, 397 (1962) [English Trans/.:Russ. Chem. Rev. 31, 197 (1962)l.

’’ N. A. Goryunova, N. K. Takhtareva, and D. N. Tretiakov, Thesis of the report on the IVth .2’

Conference on Crystallochemistry (in Russian), p. 148 (1961). V. N. Vigdorovich and A. Ya. Nashel‘skii, Poroshkocaya Me?., Akad. Nauk Ukr. SSR No. 2(14), 43 (1963) [English Transl.: Soviet Powder Metallurgy and Metal Ceramics No. 2(14), 123 (1963)l.

12

N . A. GORYUNOVA, A. S . BORSHCHEVSKII, AND D . N. TRETIAKOV

TABLE I HARDNESS OF 111-V COMPOUNDS DETERMINED BY EXPERIMENT Microhardness (kg/mm2) at load (g)

Compound

Hardness values after Mohs

BP AlAs AlSb GdP GaAs GaSb

5 4.8 5 4-5 4.5

20

Knoop's pyramid

50

100

25

32OOK

InP InAs InSb

Square base pyramid

3.8

394 k 23b 932 k 59[111] 1106 50 [ I 111 482 25' 520J 434J 238 8'

"After Borshchevskii and T r e t i a k ~ v . ' ~ bAfter Borshchevskii e f a!." 'After Wolffet al.' "After G o r y u n ~ v a . ~ ~

505" 413 k ISb 940 & 35' 700 20d 450 22'

233 [ l l I ] 231 [112] 224 [loo] 222[1101h

359 945 750 448 535 381

223

34'

i-155' & 42' & 27' 5 47' & 26'

20'

'After Syrkin2' and Goryunova et a / . 2 2 /After Vigdorovich and Na~hel'skii.'~ gAfter Stone and After Ablova and Feoktistova.16

described above. Other microhardness measurements were made on coarsecrystalline specimens. 18*24-25a It has been established by Borshchevskii et ~ 1 . 'that ~ the conditions of crystal growth are of great importance in the microhardness of polycrystalline specimens. For instance, microhardness values for gallium arsenide crystals grown in a narrow ampule (820 10) are much higher than for those grown in a wide ampule (700 k 20). The microhardness values given in Table I were obtained on large crystals in polycrystalline specimens. A. S. Borshchevskii and D. N. Tretiakov, Sb. Fizika Dokl. na 20 Nauchm. Konferentsii Leningr. engh.-Stroit. in-t (Physics Report to the XX Conference of LISI) Leningrad, 1962. 2 5 N. A. Goryunova, "Khimiya almazopodobnykh poluprovodnikov," Leningrad, Izd-vo Leningradskogo Universiteta, 1963 [English Trans/. : "The Chemistry of Diamond-Like Semiconductors" (translated by Scripta Technica, J. C. Anderson, ed.). Chapman & Hall. London, 19651. 25aG. V. Samsonov, L. N. Bazhenova. and A. A. Ivan'ko, Izv. Akad. Nauk SSSR, Neorgan. Materialy 2, 1 194 ( I 966) [English Trans/.: Inorganic Materials 2, 1018 ( I 966)]. 26 B. Stone and D. Hill. Phys. Rev. Letters 4.282 (1960). 24

1.

HARDNESS

13

The difficulties in making hardness comparisons for single crystals may be explained by the lack of data on dislocation density and impurity concentration. Microhardness values reported by Ablova and Feoktistova16 were obtained on a single crystal of In!Sb with a resistivity p = 0.006 ohm-cm and a carrier density n = 3 x 10l6cmP3 at 300°K. As the microhardness/ orientation relationship was observed to vary within the limits of the scatter,

N

7t

FIG.4.Microhardness values on different crysta.llographic planes of InSb. The ordinate gives, out of a total of one hundred measurements, the number of measurements N per unit of micro2.5 kg/mm'. (After Abllova and Feoktistova.") hardness in the range H

100 impresses were made by the authors on each plane for measuring microhardness, and the data were evaluated statistically. In this way a statistically significant variation in microhardness was detected on different crystallographic planes of indium antimonide (see Fig. 4). A maximum difference of approximately 5 % was detected by the indentation method between the (111) plane and the (110) plane.16 The maximum microhardness value was found on the (111)plane, and the minimum microhardness value on the (110) plane. This fact can be explained presumlably in a manner similar to that for

14

N . A . GORYUNOVA, A .

s. BORSHCHEVSKII, AND

D . N. TRETIAKOV

germanium, where the same kind of anisotropy was found.*' The (111) plane of both indium antimonide and germanium is the slip plane, while the (1 10) direction is the direction of slip.'* A study of the principal planes and directions of slip for germanium shows that the (110) plane is more plastic than is the (111) plane. This is in fair agreement with the results obtained by Patel and Alexander29 on the effect of orientation of the germanium specimen on the form of the deformation/ load curves for compression along one of three directions : (1 1 l), (loo), or (1 lo). Data" on the anisotropy of germanium microhardness agree with the deformation/load curves. Germanium displays the least plasticity in the (1 11) direction [which indicates the maximum microhardness on the (111) plane] and the maximum plasticity in the (1 10) direction [which indicates the minimum microhardness on the (110)plane]. Data on the microhardness anisotropy for other 111-V compounds are not available. Exceptions are gallium phosphide crystals, the properties of which were described by Borshchevskiiet aL3' Microhardness was measured on the (11l)planeof these crystals and on some arbitrary transverse planes (see Table I). Microhardness anisotropy of other 111-V compounds is expected to be determined later. Wolff et 0 1 . ~ report that no difference in direction was detected for the microhardness on the (111) planes by the indentation method in the 111-V compounds studied by them. With the help of the scratch method, however, the microhardness anisotropy of indium antimonide was determined as a function of the direction on one and the same plane." The curve plotted in Fig. 5 shows the relationship between the scratch hardness and the direction on the octahedral (111) plane of indium antimonide crystals. The scratch hardness variation is of periodic character along the rhombododecahedral (110) plane also. The qualitative results obtained can be explained by assuming the scratch hardness to have its maximum value when the specimen is scratched perpendicularly to the direction of the strongest bonds. For example, the scratch on the (111) plane, made in the direction perpendicular to (lTO), has an angle of 145" or 35" to the cleavage (110) plane, depending on whether the pyramid moves along the direction (112) or (112). In the first case the hardness value is higher and the scratch more narrow ; in the second case scratching is easier and the hardness value lower. 27

** 29

30

M. S. Ablova. Fiz. Tilerd. Te[a 3, 1815 (1961) [English Trans/.: Souier Phys.-Solirl Stute 3, 1320 (1961)J J. W. Allen, Phil. Mag. 2, 1475 (1957). J. R. Patel and B. H. Alexander, Acta Met. 4,385 (1956). A. S. Borshchevskii, K. A. Kalyuzhnaya, A. D. Smirnova, N. K. Takhtareva, and D. N. Tretiakov (Tretyakov), Izv. Akad. Nauk SSSR, Ser. Fiz. 28, 985 (1964) [English Transl.: Bull. Acad. Sci. U S S R . Phys. Ser. 28. 887 (196411.

1.

15

HARDNESS

100 r

I

t

,

3

60 90 120 f50 180 FIG.5. Dependence of scratch hardness on the direction on the ( I 1 1 ) plane of the InSb crystal. (After Lange and Lange.”)

30

Table 1 gives microhardness values of the 111-V compounds, but only in one case were the measurements made on a specimen of known carrier concentration. l 6 Presumably the concentration of free carriers might be correlated to the plasticity of a substance (and consequently to the microhardness). In a general form, this idea was introduced into the study of semiconductors by Ablova and Regel’.31 ‘The role of free current carriers ini semiconductor microhardness was established by several investigator^,^^-"*^ who observed considerable softening of the surface layer of indium antimonide under the action of illumination. The experiments with germanium of different conductivity3’ showed that microhardness decreases with increasing conductivity. This confirms the assumption that charge carriers are responsible for softening. Evidently due to these charge carriers the chemical bond!ing becomes more metallic. Such experiments on the 111-V compounds miay reveal new special properties of interest based on their differences fromi elements such as germanium.

b. Type of Chemical Bonding and Microhardness (1) Analog Series. It was early recognized by Gol’dshmidt, who compared hardness of crystals with different structures, that hardness is closely related to the type of chemical bonding between the participating atoms.3 Gol’dshmidt tried to specify this chemical honding by different characteristics, such as coordination number, atom valence, atomic volumes of particles, M. S. Ablova and A. R. Regel’, Fiz. Tverd. Tela 4, 1053 (1962) [English Transl.: Soviet Phys.Solid State 4, 775 (1962)l. 32 G. C . Kuczynski and R. F. Hochman, Phys. Rev. 108,946 (1957). 3 Z a Y ~Kh. . Vekilov, M. G. Mil’vidskii, V . B. 0sve:nskii. 0. G. Stolyarov, and L. P. Kholodnyi, Izv. Akud. Nauk SSSR, Neorgan. Materialy 2,6386 (1966) [English Transl. :Inorganic Materials 2, 549 (196611. 3ZbF.L. Edel’man, Izv. Sibirsk. Otd. Akad. Nauk SSSR, Ser. Tekhn. Nauk No. 6, vyp. 2. 124 (1966).

31

16

N . A . GORWNOVA, A. S. BORSHCHEVSKII, AND D. N. TRETIAKOV

etc. However, definite conclusions on the relation between these factors and hardness could not be formulated until compounds with different structures were compared. If, however, we know the character of the relation between chemical bonding and hardness for elements and compounds comprising a crystallochemical group, we can better understand the rules of this bonding. A crystallochemical group includes substances of the same or very similar structures and chemical compositions which vary in a sequence. The classical example of such groups is that of tetrahedral structure compounds or semiconductors of the diamond type structure. This includes the elements which crystallize in the diamond structure C, Si, Ge, cr-Sn, as well as binary or more complex compounds which are formed in accordance with the necessary requirements of retaining the certain valence type and the certain electron concentration with variation of the chemical c o m p ~ s i t i o nThis . ~ ~ crystallochemical group is characterized by a high symmetry of composition and bonding. It often serves as a model in studies of different phenomena. The 111-V compounds, members of a crystallochemical group, exhibit a regular variation of all the properties with chemical composition. Both elements and compounds are characterized by the regular variation of properties with increasing atomic weight. The relationship between the properties and the atomic weight is determined by the electron distribution in atoms of compounds in accordance with their position in the periodic system. The phenomenon of nonmonotonic, stepped variation in properties of elements and compounds with increasing atomic weight is observed in analog g r o ~ p s .In~the ~ ,groups ~ ~ of diamond-type analogs this phenomenon, termed “the secondary periodicity,” is accompanied by “metallization” of chemical bonding. In stepped nonmonotonic variation of properties, there is reflected the periodic variation in energetic characteristics of individual atoms and ions composing the main subgroups. As Shchukarev reported, 34 this phenomenon is effected by the filling of the d- andf-electron levels in the shell, and results in a relative strengthening of the bonding between electrons in s and p states and the nucleus of the atom. Nonmonotonic variation of properties is therefore particularly apparent in energetic characteristics of free atoms. The secondary periodicity in a somewhat smoothed form can be observed in the nonmonotonic form of the curves of property as a function of atomic number. It is reflected in the variation of such parameters as the microhardness, the energy gap, e t ~ . ~ ~ * ~ ~ ~ It may be observed from Table I that the hardness of 111-V compounds decreases with increasing atomic weight, both of metals and nonmetals. 33 34

E. V. Byron, Z h . Russ. Phys. Chim. Obshch. 47,946 (1915). S. A. Shchukarev, Z h . Obshch. Khim. 24,582 (1954).

1. HARDNESS

17

Exceptions are the aluminum complounds. There seem to be at least two possible reasons for this deviation, namely : (i) All the diamond-type compounds containing aluminum are extremely unstable in the open air, which fact causes great difficulties in the microhardness determination. (ii) The ionic contribution to the bonding between aluminum and the group V element is larger than generally assumed. The influence of the ionicity will be discussed subsequently. It appears that in the case of the analog series, the decrease of hardness with increasing atomic weight is caused by no other factor than the difference in the type of chemical bonding due to different structures of atom shells. It was supposed by Hilsum and R ~ s e - l ' n n ethat s ~ ~the lowering of the melting temperature with increasing atomic number is the cause of the decrease in hardness. This supposition was based on the fact that, in compounds of the zinc-blende structure, hardness decreases with approach to the melting point. Allen confirmed this on indium antimonide.z8 However, even when the influence of this factor is taken into account through the introduction of "reduced microhardness," the microhardness continues to decrease with increasing atomic weight. Probably it would be more nearly correct to compare hardness of different substances at one and the same Debye temperature, i.e., at a similar degree of excitation of lattice vibration. As these data are not available, the reduced microhardness can be calculated in relation to the Debye temperature. Such an approach also indicates that microhardness tends to decrease with increasing atomic number (see Table 11). In the last two columns of Table I1 we have given the reduced hardness. I n the first case it is the microhardness of a substance multiplied by a factor equal to room temperature divided by the melting point. In the second case it is the microhardness of a substance multiplied by a factor equal to room temperature divided by the Debye temperature. (2) Zsoetectronic Series. The substantial variation in the ionic character of the bonding and less substantial vairiation in the metallic component are observed for the diamond structure giroup in so-called isoelectronic series, for instance, at transitions from A" to A"'BV and then to A"BV1and A'B'". In this case, the strength of ionic bonding indicates the general electron distribution in the crystal, which might be characterized by means of effective charges of atoms." Undoubtedly the microhardness variation, more and more abrupt from AlV to A"'BV, from A"'BV to A"BV1,and from A"BV' to AIBV",is connected 35

C. Hilsum and A. C. Rose-lnnes, "Semiconducting Ill-V Compounds." Pergamon, London, ( 1961).

18

N. A . GORYUNOVA, A . S . BORSHCHEVSKII, AND I). N . TRETIAKOV

TABLE I1 REDUCEDHARDNESS VALUES

Compound GaP GaAs GaSb InP InAs InSb

I N THE

ANALOG SERIES

Debye Reduced Melting point temperature hardness ( O K ) (“K) HT 1800 1537 1012 1362 1242 825

Reduced hardness HD

161 142 131 119 103 79

315“ 270 249 a 228

712 500 516 288

“After Sirota and P a ~ h i n t s e v . ~ ~ bAfter Gul’tyaev and P e ~ r o v . ~ ’

TABLE 111 HARDNESS A N D ENERGY GAPIN ISOELECTRONIC SERIES OF DIAMOND-TYPE SEMICONDUCTORS

Substance Ge GaAs ZnSe CuBr Ge+;-Sn

H , after Mohs 6 4-5 H, kg/mm* 1000 700 AE 0.7 1.5

GaSb ZnTe Cul

X-Sn

InSbCdTe Agl

3-4

2.5

-

4.5

3.0

2.4

-

3.8

2.8

1.5

137 2.8

21 3.0

-

469 0.7

82 2.2

19 3.0

-

220 0.2

56 1.4

2.8 2.8

-

-

with the change in general electron distribution caused by the variation in atom valence and polarization. The characteristic feature of any isoelectronic series is the common number of electrons for each member of the series (under the condition of the very similar structures). Gol’dshmidt, in measuring hardness of substances after Mohs scale, concluded that, in such series as Ge-GaAs-ZnSe-CuBr, the stronger the ionic character of the bonding the lower the hardness. The results of the above-mentioned work’* obtained by the microhardness method confirm this conclusion. In Table I11 are shown the variation of hardness and energy gap in isoelectronic series of diamond-type semiconductors. 3b

37

N. N . Sirota and Yu. 1. Pashintsev, Dokl. Akad. Nauk SSSR 127.609 (1959) [English Transl.: Proc. Acad. Sci. USSR, Phys. Chem. Sect. 127,627 (1959)l. P. V. Gul’tyaev and A. V. Petrov, Fiz. Tuerd. Tela 1, 368 (1959) [English Transl.: Soviet Phys.-Solid State 1. 330 (1959)l.

1.

HARDNESS

19

Microhardness in more complete isoelectronic series was the next step considered by the same author^.'^ Greater variations in bonding type with chemical composition were obtained. Results substantiated the general trend for microhardness in these series, namely, to decrease with increasing ionic character of the bonding. Thus the decrease in the covalent bonding, caused by metallization in the analog series or by the ionicity in the isoelectronic series, results in a decrease in microhardness. c. Correlation between Microhardness and Other Properties

As was already stated, in the 111-V compound series as well as in the series of elements C, Si, Ge, a-Sn the chemical bonding decreases and becomes metallic with increasing atomic weight. Metallization influences over-all physicochemical and physical properties of substances, which fact indicates the existence of different correlations between the properties. Borshchevskii et were the first to consider the existence of correlation between hardness and electronic properties in semiconductors. Metallization of the chemical bonding in the analog series influences, though slightly, the ionicity. Using the correlation between energy gap and hardness, on one side, and the correlation between energy gap and ionicity, on the other side, Wolff et aL8 came to the conclusion that in the following series of compounds there is a progressive decrease in the ionic bonding : Alp, Gap, AlAs, AISb, GaAs, InP, GaSb, InAs, InSb. Metallization increases much more rapidly in the above sequence and so the hardness decreases. The work cited previouslyz5 was based on the assumption of a correlation between different properties, including hardness, defined with precision for certain crystallochemical groups. The investigations established the existence of the correlation between hardness and interatomic distance, originally noted by Gol'dshmidt, in the 111-V compounds. In the formula H = const Frn,where H is Knoop hardness number, r the interatomic distance, and m a constant, the constant is equal to 9 for all the 111-V compounds. Figure 6, taken from the work of Wolff et a1.,8 shows the variation of hardness with interatomic distance in the 111-V compounds. The authors conclude that in first approximation the indentation hardness is directly related to the lattice energy, as both obey a relationship of the form const'r-". Inasmuch as microhardness is directly proportional to the energy of volume deformation, H = const" U r P 3 , it follows that m=n+3. 0

=

20

N. A . GORYUNOVA, A. S. BORSHCHEVSKII, AND D . N. TRETIAKOV 1100 -

BOO 700 a, S

p

500

0 0

da

400

t

FIG.6. Microindentation hardness of the 111-V compounds plotted on a log-log scale vs interatomic distance. (After Wolff ct ~ 1 . ’ )

Hardness must undoubtedly be dependent on the energy of interaction between atoms. For covalent 111-V type substances, the latter is better characterized by the energy of atomization than by the lattice energy. Rebinder38 and K ~ z n e t s o showed v ~ ~ that the hardness of a solid is dependent on its surface energy, which is related to the energy of a t o m i ~ a t i o n .The ~~,~~ dependence of microhardness on the number of atoms in the volume under deformation was studied for glasses.42Goryunova et aL2*and Borshchevskii et ~ 1 used . the ~ concept ~ of a reduced microhardness for the 111-V compounds. 38

39

40

41

42 43

P. A. Rebinder, “Hardness”, Technical Encyclopedia (in Russian), Vol. 22, 703 (1933), ed. OGlZ RSFSR. V. D. Kuznetsov, Poverkhnostnaya energiya tverdykh tel. (Surface Energy of Solids). Moscow, gos. Izd-vo Tekhniko-teoret. Lit-ry, 1954. B. F. Ormont. Voprosy metallurgii i fiziki poluprovodnikov (Problems of Metallurgy and Semiconductor Physics) : poluprovodnikovye soedineniya i tverdye splavy. Trudy 4-go soveshchaniya. Akad. Nauk SSSR. 1961. B. F. Ormont, Dokl. Akad. Nauk S S S R 124. 129(1959)[English Transl.:Pro<,.Acad. Sci. U S S R . Phys. Chem. Sect. 124, 17 (1959)l. R. L. Miiller, Physics, Reports to the XX Conf. of LISl (in Russian). p. 18. A. S. Borshchevskii, N. K. Takhtareva, and D. N. Tretiakov, Tr. Tret’ei Konf. Molodykh. Uchenykh Moldavii, Estestv-Tekhn. Nauki (Trans. of the Third Conference of Young Scientists of Moldavia), Akad. Nauk Moldavsk. SSR, p. 10 (1964), Kishinev.

1. HARDNESS

21

The reduced microhardness was defined as the load related to some certain constant number of atoms on the suirface of indentation. It is taken to be equal to the quantity of atoms on a 1 imm2 surface area of the compound. 6 . SOLIDSOLUTIONS a. Introduction

Kurnakov developed general rules for the variation of hardness with composition for solid solutions. Based on hardness investigations of metal solid solutions, he has formulated a number of rules which are still valid.44 These are as follows: (i) The formation of metal solid solutions is accompanied by an increase in hardness. (ii) The variation in hardness over a continuous compositional range of metal solid solutions of type ABo-m yields a continuous curve possessing a maximum. (iii) The differential coefficient dH/d'x ( H is hardness and x concentration) suddenly changes with composition of the solid phase of A"'-"' solid solutions in limited concentrations. On the basis of these rules, Kurnakov determined the dependence between hardness and composition of binary systems for the following cases : (a) Continuous range of solid solution throughout the system. (b) Limited miscibility or complete immiscibility in the solid phase. (c) Presence of certain compounds. Later Kurnakov showed that the re1,ation between hardness and chemical composition, as it was previously establlished for metal alloys, exists for other types of r n a t e r i a l ~Correlating .~~ hardness to composition, t e m p e r a t ~ r e , ~ ~ electrical c o n d ~ c t i v i t y ,yield ~ ~ ~ t r e n g , t h elastic , ~ ~ m o d ~ l u s , ~and ' other properties, Kurnakov showed that hardness was a direct function of composition, and, therefore, hardness measiurements could be useful for physicochemical analysis. The use of the microindentation test as a modified method of physicochemical analysis enables one to broaden hardness investigations. The method has become more precise, and the identification of phases in multiphased specimens, easier. Pogodin et aL4* have reported the principal types of composition/microhardness diagrams for solid solutions of systems involving two components. 44 45

46

4 ' 48

N. S. Kurnakov, "Selected Works" (in Russian), Vol. 2, p. 36. Publ. by Acad. Sci. USSR. Ref. 44, Vol. 1, p. 217. Ref. 44, Vol. 1, p. 116. Ref. 44, Vol. 2, p. 166. S. A. Pogodin, L. M. Kefeli, and E. S. Berkovich, Izu. Sektora Fiz.-Khim. Analiza, Insf. Obshch. Neorgan. Khirn., Akad. Nauk S S S R 17, il93 (1949).

22

N. A . GORYUNOVA, A . S . BORSHCHEVSKII, AND D. N. TRETIAKOV

Using the isotherms and polytherms of Glazov et applied the microhardness method to the study of solid solutions with limited miscibility. The results of recent microhardness investigations on solid solutions confirm the Kurnakov rules. The relationships known as “the additivity law” were established for dilute solid solution^.^^^^^ The variation of hardness and microhardness with composition for semiconductor solid solutions was originally reported by Wang and Alexander.” These authors obtained an approximately linear dependence of hardness on composition in Ge-Si solid solutions over the whole composition range. They failed, however, to explain the absence of a maximum, which suggested that the hardness of pure silicon decreases upon addition of germanium. Later on, Glazov and Lyu-Chzhen’-Yuan’,’’ investigating microhardness in the same system, obtained the usual curve containing the maximum. For this they used indentations without cracks. These studies, and others which will be discussed in the next part, showed that, in investigations of semiconductor materials, the brittleness characteristic is of great importance. ~

1

.

b. Measurement Results and Discussion Microhardness data are now available for more than twenty systems of solid solutions containing at least one 111-V compound. The majority of the systems investigated were those with a heterovalent replacement. In the discussion of experimental data, reference is made to more precise values of microhardness, which have been recently obtained. These data were already reported in Refs. 22 and 43. (1) ArrrBV-A“’BV Systems. The investigations carried out so far have been concerned mainly with antimonide systems of solid solutions with “cation” replacement . InSb-GaSb. Measurements were made on coarse-crystalline specimens with reasonable homogeneity, which were produced with the help of zone leveling technique, by Ivanov-Omskii and K ~ l o m i e t sMicrohardness .~~ data H. Buckle, Seissnachr. 5, 93 (1944). H. Buckle, Z . Metallk. 34, 130 (1942). 5 1 H. Buckle and J. Descamps, Rev. M e t . (Paris)48, 569 (1951). 5 2 H. Buckle and J. Descamps. Compt. Rend. 230, 752 (1950). 53 H. Buckle, Mefnllforsch. 1 , 4 3 (1946). 5 4 V. M. Glazov, V. N. Vigdorovich, and G. A. Korol’kov, Zh. Fiz. Khirn. 31, 1891 (1957). 5 5 V. M. Glazov, V. N.Vigdorovich, and G. A. Korol’kov, Sbornik trudov Mintsvetmetzoloto i VNITOM (Technology of non-ferrous metals). No. 29, Metallurgizdat, 1958. 5 6 N. N. Glagolevaand V. M. Glazov. Izv. Akad. Nauk SSSR, Otd. Tekhn. Nauk. No. 1,130(1958). 5 7 V. N. Vigdorovich and V. M. Glazov, Izu. Vysshikh Uchebn. Zavedenii, Tsuetn. M e t . No. 3. 122 (1958). 5 8 V. I. Ivanov-Omskii and B. T. Kolomiets, Fiz. Tverd. Tela 1, 913 (1959) [English Transl.: Soviet Phys.-Solid State 1, 834 (1959)l. 4y

50

~

~

3

~

1.

23

HARDNESS

obtained for three compositions at loads 20 g. 50 g, and 100 g are shown in Fig. 7. The results indicate that microhardness values for compositions of 50 % and more GaSb differ significant1:y with applied load. This is explained by the fact that, at a load of 50 g, indentations fracture due to high brittleness of specimens and, as a result, somewhat lowered microhardness values are obtained. 500

400

3 00 2 00

J nSb

20

80

40

80

GaSb

mote % Ga Sb FIG. 7. Dependence of microhardness on composition of solid solution in the InSb-GaSb system. The loads are 20,50, and 100 g. 2

r

200

3nSb

20

40

60

60

mote%,

At%

-

AtSb

FIG.8. Dependence of microhardness on composition of solid solution in the InSb-AISb system : I-InSb-rich solid solution, as-grown specimen : 2-AISb-rich solid solution, as-grown specimen; 3-homogenized specimens. (After Goryunova et a1.,*’ and Baranov and Goryunova.’ 9,

Z n S b - A M . The results of microhardness r n e a s ~ r e m e n t sare ~ ~ shown ~~~ in Fig. 8. For comparison, the curves for homogeneous and nonhomogeneous specimens are presented. Complete hlomogeneity of the system InSb-A1Sb is 59

B. V. Baranov and N. A. Goryunova, Fiz. Turrd. Trla 2. 284 (1960) [English Transl.:Soviet Phys.-Solid Sfate 2, 262 (1960)l.

24

N. A. GORWNOVA, A. S. BORSHCHEVSKII, AND D. N. TRETIAKOV

very difficult to achieve. The specimens require prolonged annealing at high temperatures after etching to reach some degree of homogeneity. The two maxima on the microhardness-composition curve might also be attributed to the lack of complete homogeneity in the specimens investigated. This assumption is confirmed by the fact that the maximum microhardness values obtained on homogeneous specimens were lower than those obtained on nonhomogeneous specimens. AESb-GaSb. Measurements were made on the specimens produced by synthesizing and zone-refining.60 The results are given as Fig. 9. It is seen that for this system the microhardness-composition curve shows less deviation from linearity than that for the InSb systems discussed above.

N

t \ 0-

I

400

flESb

20

40

60

mole % GaSb

60

-

GaSL

FIG.9. Dependence of microhardness on composition of solid solution in the AISb-GaSb system. (After Burdiyan and Borshchevskii.60)

ZnAs-AlAs. Results of microhardness measurements for three compositions have been given by Borshchevskii and T r e t i a k ~ v . 'It ~ is shown that, in the solid solution of InAs in AIAs, microhardness increases and attains a value higher than that of the binary components. These measurements were probably carried out on specimens of insufficient homogeneity. InP-Gap. Microhardness measurements on specimens in which equilibrium was not attained have been reported.61 More recently, results have been given on specimens in an equilibrium condition.62 The variation of microhardness with composition for the InP-GaP system can be plotted as a reasonable curve with the maximum under 1140 kg/mm2. The maximum is at the Gap-rich end of the diagram. It should be noted that microhardness values for GaP reported in this work are lower than those reported by others. 6o

61

62

1. I. Burdiyan and A. S. Borshchevskii, Zh. Tekhn. Fiz. 28, 2684 (1958) [English Transl. Soviet Phys.-Tech. Phys. 3, 2451 (195S)l. N. A. Goryunova and V. I. Sokolova, Izu. Moldausk. Filiala Akad. Nauk S S S R No. 3 (69), 97 (1 960). N. N. Sirota and V. V. Rosov, Dokl. Akad. Nauk Belorussk. SSR 7,446 (1963). '

1.

25

HARDNESS

InSb-lnAs. A considerable increase in microhardness was observed for specimens of InSb-3InAs after annealing for 700 hours-namely, values of 430 kg/mmz. These values were much higher than those of the components, as initially measured. For example, for InSb, H = 220 10 kg/mmz; for InAs, H = 330 f 12 kg/mm2.63 InAs-lnP. Results of microhardness investigations are shown in Fig. 10.23964 It is seen that curve 1 does not possess a maximum.64 These same data were analyzedz3 with account taken of brittleness (see Fig. 3). The influence of cracks was excluded by extrapolation to the “0” standard of brittleness. The result was that a dependence war; established which could be represented by curve 2 with the maximum.

+

FIG. 10. Dependence of microhardness on composition of solid solution in the InAs-lnP system: 1-arithmetic mean value of all tests at the load of 40 g ; 2-value obtained by extrapolation to “0“ standard brittleness at the load of 20 g. (After Vigdorovich and other^.*^.^^)

ZnP-GaAs. The results of studies by Sirota and Makovet~kaya~’ show that the variation of microhardness with composition might reasonably well be plotted as a curve with the maximum at a concentration about 50%. It is of interest to note that this system is the quarternary one. Deviations of short range order from statistical particle distributions are possible for this system, as bonds Ga-P are considerably stronger than those In-As. ( 2 ) Other Systems. Not much information is available on microhardness of the A1llBV-A’lBV‘ systems. Some initial work was reported by Goryunova and Fedorova.66Microhardness was used to indicate the process of specimen 63

64

66

N. A. Goryunova and N. K. Takhtareva, Izu. Moldausk. Filiala Akad. Nauk. SSSR. No. 10(88), ( 1 960). V. N. Vigdorovich, A. Ya. Nashel‘skii, V . 2. Ostrovskaya, and G . I. Bugrova, Nauchn. Tr. Gos. Nauch.-lssled. i. Proektnyy in-t Redkometallich. Prom-sti “Giredmef” V1, 180 (1962). N. N. Sirota and L. A. Makovetskaya, Dokl. Akad. Nauk Belorussk. SSR 7,230 (1963). N. A. Goryunova and N. N. Fedorova, Fiz. Tuerd Tela 1, 344 (1959) [English Transl.: Soviet Phys.-Solid State 1, 307 (1959)l.

26

N. A. GORYUNOVA, A.

s.

BORSHCHEVSKII, AND I>. N . TRETIAKOV

homogenization. Microhardness of the InSb-CdTe system was investigated?’ This work established that the microhardness increased when solid solutions of limited composition range are formed. Microhardness investigations on the InAs-CdTe system in a limited composition range at the InAs-rich end revealed an increase in microhardness for the solid solution of CdTe in I~As.~~-’O Goryunova et aL71 investigated the variation of microhardness with composition in the InAs-HgTe system. Nonhomogeneous alloys were used ; nevertheless they showed characteristics similar to those of homogeneous solid solutions. The curve plotted for the 3InAs. 2HgTe alloy possessed a maximum. Data are available on the variation of microhardness with composition for four systems of the type A“’BV-A”B’VCV. In Figs. 11 and 12 the microhardness composition curves are given for the InSb-CdSnSb, and

300 r

CdSnSB, 20

I

40

60

80 2JnSb

-

mole% ( 2 ~ t - 1 ~ 9

FIG. 1 1 . Dependence of microhardness on composition of solid solutions in the InSb-CdSnSb, system. (After Goryunova and Prochukhan.72)

N. A. Goryunova, G . K. Averkieva, P. V. Sharavskii, and Yu. K. Tovpentsev, Sb. Fizika Dokl. na. 19 Nauchn. Konferentsii Leningr. engh.-Stroit, in-t. Leningrad (Coll. Works “Physics and Chemistry”, Reports to the Nineteenth Scientific Conference of LISI) 1962, p. 22. ’* A. V. Voitsekhovskii, “The All-Union Conference of Semiconducting Compounds.” Thesis of the report, p. 13 (in Russian). 6 9 A. V. Voitsekhovskii and N. A. Goryunova, Sb. Fizika Dokl. na. 20 Nauchn. Konferentsii Leningr. engh.-Stroit, in-t. Leningrad (Coll. Works “Physics”, Reports of the Twentieth Scientific Conf. of LISI), p. 12, 1962. 7 o L. I. Kleshchinskii, E. N. Khabarov, and P. V. Sharavskii. Sb. Fizika Dokl. na 22 Nauchn. Konferentsii Leningr. engh.-Stroit. in-t. Leningrad (Coll. Works “Physics”, Reports of the Twenty-Second Conf.) 1964, p. 12. N. A. Goryunova, V. S. Grigorieva, P. V. Sharavskii, and L. A. Osnach. Sb. Fizika Dokl. na 20 Nauchn. Konferentsii Leningr. engh.-Stroit. in-t. Leningrad (Coll. Works “Physics”. Reports of the Twentieth Scientific Conf. of LISI) 1962, p. 7. 67

1.

27

HARDNESS

mole

./o CdSnCls,

-L

FIG.12. Dependence of microhardness on composition of solid solutions in the InAs-CdSnAs, system. (After Goryunova and Prochukhan.72)

InAs-CdSnAs, systems.72Both exhibit maxima. The point of interest here is the suggestion that in the InSb-CdSnSb, system, solid solutions are formed by the stable compound InSb and the hypothetic compound CdSnSb,. More recently, however, a differing opinion has been Microhardness investigations in the InAs-ZnGeAs, system were carried out on specimens with a higher degree of homogeneity than in the two previous case^.^^,^^ The microhardness-composition curve has two maxima. The maximum that has the higher microhardness value lies near the harder component, namely, the ZnGeAs, . The InSb-ZnSnSb, system is analogous to the InSb-CdSnSb, system. The microhardness-composition curve has its Also of interest are the results of microhardness maximum at 60% InSb.68*69 investigations for the quaternary systems I n S b - A g h ~ T e , ~and ~ InAsCuInTe, .68*69 The microhardness-composition curves are shown in Figs. 13 and 14. Preliminary data on microhardness in solid solutions of the 111-V compounds with some other ternary compounds have been obtained.74 Considerable experimental data are available on microhardness in solid solutions of the 111-V compounds with defect compounds of the AY’B:’ type. Investigations of solid solutions based on InSb75-77indicate the complex character of the variation of microhardness with composition because the In,SbTe, compound is formed. In systems with a limited range of solid solution. an increase of microhardness is observed. In the InAs-In,Te, and

’*

N. A. Goryunova and V. D. Prochukhan, Fiz. Tuerd. Tela 2. 176 (1960) [English Trans/.: Soviet Phys.-Solid State 2. 161 (1 960)]. -2”N. Kh. Abrikosov, E. V. Skudnoba, L. V. Poretskaya, and N.G. Pavloba, I z t . Akad. Nauk S S S R , Nuorgan. Maturialy 2, 1416 (1966)[English Transl. ;Inorgarlic Materials 2, 1209 ( 1 966)]. 73 Specimens were produced and measured by Prochukhan. 74 N. A. Goryunova. A. V. Voitsekhovskii, and V. D. Prochukhan. Vestn. Leningr. Unia., Ser. Fiz. i K h i m . 10, 156(1961). 7 5 N.A. Goryunova, S. 1. Radautsan. and G. A. Kiosse, F i z . Tverd. Tela 1. 1858 (1959) [English Transl. : Sovie! Phys.-Solid State 1. 1702 (196O)l. 16 S. I. Radautsan and I. P. Molodyan. Izu. Mo/davsk. Filiala Akad. Nauk S S S R No. 3 (691%37 (1960). ” S. I. Radautsan, V. V. Negreskul. and I. A. Madan, Izc. Moldacsk. Filiala Akad. Nauk S S R No. 10(88), 57 (1961).

28

N. A . GORYUNOVA, A. S . BORSHCHEVSKII, ANT? D N. TRETIAKOV

mote ”/.

(2JnSg)

-

FIG. I Dependence of microhardness on composition of solid solutions in system. (After Prochukhan.’ 3,

I

CuJnTe2zo

1

1

40

60

mole”/.(23nAs)

...3

iSb-AgInTe,

I

80

-

2JnAs

FIG.14. Dependence of microhardness on composition of solid solutions in the InAs-CuInTe, system. (After Voitsekhovskii and other^.'^,^^)

InAs-In,Se, investigations established that the variation of microhardness with composition could be plotted as a curve with the maximum displaced to the InAs-rich end of the diagram. Analogous results were obtained from investigations of the systems InP-In,Se,, GaAs-Ga,S,, and GaP-Ga,S,.8’-83

’*

N. A. Goryunova and S . I. Radautsan, Dokl. Akad. Nauk S S S R 12,818 (1958).

79

N. A. Goryunova and S. I. Radautsan, Zh. Tekhn. Fiz. 28,2917 (1958).

’’ ’* 83

N. A. Goryunova, S. 1. Radautsan, and V. 1. Deryabina, Fiz. Tuerd. T e h 1. 512 (1959) [English Transl. : Souiet Phys.-Solid State 1, 460 (1959)l. S. I. Radautsan, I. A. Madan, I. P. Molodyan. and R. A. Ivanova, Izu. Moldavsk. Filiala Akad. Nauk S S S R No. 3(69), 107 (1960). I. I. Kozhina, S . S. Tolkachev, A. S . Borshchevskii, and N. A. Goryunova, Vestn. Leningr. Uniu., Ser. Fiz.i Khim. No.4(1), 122 (1962). V. V. Negreskul, Tr. Tret’ei Konf. Molodykh. Uchenykh. Moldavii. Estestv. Tkhn. Nauki (Trans. of the Third Conf. of Young Scientists of Moldavia) Akad. Nauk Moldavsk. SSR, p. 35 (1964).

1.

HARDNESS

29

Only in the case of the GaP-Ga,Se:4 system is there no apparent maximum in the microhardness composition curve. This fact can presumably be explained by the probable microsegregation reported by the authors. c. Rules of Hardness Variation in Semiconducting Solid Solutions In most cases, microhardness values in semiconducting solid solutions were obtained from investigations of the systems InP-In,Se3, GaAs-Ga,S3, of a considerable inhomogeneity. Nevertheless, the experimental data available enable one to formulate some rules common for solid solutions of

111-V corn pound^^^,^^ : (i) Formation of a solid solution is accompanied by an increase in hardness. This is true both of systems with limited range of solid solution and of systems in which solid solution occurs throughout the whole range of composition. Thus the first Kurnakov rule discussed in Section 6a is satisfied for all the homogeneous systems studied. (ii) There is a maximum in the diagrarn for microhardness in solid solutions at all compositions, i.e., these solutions agree with the second Kurnakov rule.

A detailed study of microhardness variation with composition in the continuous range of solid solution indicates that the most curves have one hardness maximum ; however, in some other curves (plotted exactly through the points obtained) there are several extrema, as, for example, in the systems InSb-AlSb, InAs-ZnGeAs, , InAs-CdSnAs,, and InAs-CdSnAs, . This might be explained by the presence of some new transition phases in the compounds, but the results obtained thus far from the study of other properties do not confirm such an assumption. The daita on the decomposition of these solutions are not available. The special experiments on the solid solutions Si-Ge and InSb-GaSb failed to disclose any d e c o r n p ~ s i t i o n .It~ ~might be assumed that the transition ZnS(wurt2ite) and ZnS(zinc blende) structures present in some solid solutions affect microhardness variation. But this has not yet been established. The problem is complicated by the observation that, in the systems with transition structures, the character of the microhardnesscomposition curve does not change. At the same time there are systems without any transitions, but the curves for these systems have several extrema. The deviations from the Kurnakov rules can be explained mostly by (i) imperfect measuring techniques and (ii) some degree of inhomogeneity of specimens. 84

85

S. I. Radautsan, I . A. Madan. and R . A. Ivanova. Izc. Moldavsk. Filiala Akad. Nauk S S S R No. 3(69), 107 (1960). V. N. Romanenko and V. I. Ivanov-Omskii, Dokl. Akad. Nauk 129, 553 (1959) [English Transl. : Soviet Phys. “Doklady” 4, 1342 (1960)l.

30

N. A. GORYUNOVA, A.

s.

BORSHCHEVSKII, AND D. N . TRETIAKOV

(i) The main reason of the lack of precise measuring methods is the very high brittleness of the 111-V compounds, which is still higher in their solid solutions. This latter fact was established by the investigations of the Ge-Si system and confirmed by measuring microhardness in the InAs-InP system, where taking into account the brittleness changed the form of the curve completely. In addition, as is shown by the curves in Fig. 7 depicting microhardness variation with increasing load in the InSb-GaSb system, the GaSb end of the diagram under the high load has apparent deflection due to considerable specimen spalling. (ii) Insufficient homogeneity of the specimens contributes to a change of the form of the composition-microhardness curve. For example, the specimens might include two or more solid solutions in comparable quantities, but without apparent variation in composition. In this case, somewhat more scattering in microhardness values will be obtained, and the mean value of microhardness for this specimen will be lower (in Fig. 15 the hatched area is that of microhardness values for nonhomogeneous specimens). The decrease in microhardness value here is easy to understand : two-phase specimens that are a mix of initial components provide a limiting case. The linear dependence of hardness on composition for such a system was originally shown by Kurnakov.

A Na

mole”/o

N B El

FIG. 15. Schematic representation of dependence of microhardness on composition for homogenized solid solutions. The hatched area is the area of microhardness values of nonhomogenized solid solutions. If N A = N , then AHA = AHBfor homogenized solid solutions.

The significant role of homogeneity might be shown with one more example. In the process of annealing, the microhardness of solid solutions increases with time of annealing even in the specimens providing distinct x-ray lines. Recently some authors have reported higher microhardness values

1.

HARDNESS

31

than those published earlier. So, as the homogeneity improves, microhardness curves should approach those (obtained for InSb-GaSb, InAs-InP, InAs-CuInTe, and, hence, the ideal curve of Fig. 15. More detailed analysis of the curves enables the following relationships to be established for most system^^^*^^ : (i) The microhardness maximum is generally displaced to the harder component end of the diagram. The displacement of the maximum appears to be dependent on the difference in hardness of the initial components. The wider the difference, the larger the displacement. (ii) Relative hardening for low- and high-component concentrations is generally of the same order (AHA % AH,) (see Fig. 15) when NA = NB. (iii) The study ofa number of systems showed that microhardness variation with composition can be expressed by the following empirical equation

or

where HA and HB are microhardness of components composing the solid solution ; NA and NB are concentrations of components in molar fractions ; His the microhardness of the solid solution with the compositions NAand NB; K is a constant. The best agreement of theoretical and experimental microhardness values was observed in the systems InSb-GaSb and InAs-CuInTe, (see Figs. 16 and 17). Satisfactory agreement was obtained for the systems InSb-AlSb, InAsCdSnAs,, InSb-CdSnSb,, InSb-AgInTe, . In these cases, the curves estimated theoretically lie within the range of scattering of microhardness values. The equation, in its first approximation, might be assumed to satisfy most systems. Deviations between theoretical and experimental data would presumably be explained by the lack of precise measuring techniques and inhomogeneity of specimens. A11 the albove-mentioned relationships, as well as the law of additiveness for low concentrations, agree with the equation. Of course, it is a rather rough approximation and, in addition, K , which can be regarded as the coefficient of hardening is dependent on the concentration to some extent. But it has been found that, for the systems considered, K changes only slightly with concentration, and the influence of short-range order on hardening for statistical solid solutions can be taken into account by the term KNA(l - NA),which is widely used in physicochemical equations.

32

N. A. GORWNOVA, A. S. BORSHCHEVSKII, AND D . N. TRETIAKOV

cy

500

t

/-

/.

- zoo I 3% 20

40

60

mote % I n S b

80

-

GnSb

FIG.16. Comparison of theoretical curve with experimental data on microhardness in the InSb-GaSb system. The load is 20 g.

r

I

400 t

CuJnTe, 20

40

60

80

(2lnAs)

mole% CuJnTe2FIG. 17. Comparison of theoretical curve with experimental data on microhardness in the InAs-CuInTe, system.

IV. Concluding Remarks The general trend observed now in the development of the science of solid state is to consider phenomena from the point of view of molecular or atomic structures. In the problem under discussion, this trend is reflected in an attempt to connect hardnessto the type of bonding, with consideration of the crystallographic planes. Although at the present time considerable experimental data have been obtained on hardness or microhardness of the 111-V compounds and their solid solutions, all the relationships are of an empirical nature. Future progress must involve the theoretical justification of

1 . HARIDNESS

33

these relationships from the standpoint of modern chemical and physical concepts concerning the solid state. Here it is possible to follow two closely connected directions. The first direction involves the deterrnination of quantitative dependencies between hardness and such “primary” basic properties as the character of chemical bonding in the compound, the place of elements composing it in the periodic system, and so on. The second direction is to determine quantitative dependencies of hardness on the “secondary” properties-first of all, on mechanical properties generally very close to hardness, as well as on other physical and physicochemical properties. The second direction will probably be more effective. For successful research in both direlctions, much more experimental data and data of higher quality will be needed. Now it is evident that additional information on microhardness for 111-V compounds, and for some new compounds of similar chemical bonding and analogous crystal structure, will be of utmost interest. In addition, it is very important to investigate the variation of microhardness witlh temperature. To obtain valuable quantitative experimental data, it is necessary to make measured hardness values dependent to a larger extent om the properties of the material itself. To solve this problem it is required, h-st of all, to develop improved techniques of microhardness testing and to use more widely other hardness test variants (such as scratching test, etc.). Automatic loading apparatus, the choice of optimal loads, due account for brittleness and elastic recovery of indentations, the use of special indenters, etc., can contribute considerably to the development of the techniques. In this connection it seems reasonable to formulate some general rules based am all the above factors and to follow them in measuring microhardness. Secondly, it is necessary to use malterial of higher quality. Use of highpurity, strain-free, monocrystalline specimens, with measurements carried out on certain crystallographical planes, will provide more precise results. These requirements as to materials must also be included in the abovementioned rules. When it is impossible to carry out measurements which meet all the requirements, more frequent use is to be made of the method of extrapolation. Examples are extrapolation to “0” standard of brittleness or to the smallest possible standard concentration of impurities. The solution of these problems might be simplified due to the fact that some work connected with these requirements has already been done on the compounds of other classes and on the substances similar to the 111-V compounds, namely, Ge and Si. Consideration of all the methods available is needed in the investigation of the 111-V compounds. A somewhat different field of investigation involves the use of microhardness testing as a method of characterizing the state of a substance.

34

N . A. GORYUNOVA, A . S. BORSHCHEVSKII, A N D D . N. TRETIAKOV

Very little such work has been carried out on the 111-V compounds, but it is closely connected with the necessary improvement of techniques mentioned above. These investigations would be of value in obtaining more precise data on hardness and vice versa. Hardness measurement as a method for determining the state of a substance is particularly useful for solid solutions. This method of physicochemical analysis will be of progressively greater value, as all the requirements formulated for compounds will be met and the experience obtained for solid solutions of metal and salt systems will be utilized. Improvements in materials and measuring techniques will enable microhardness variation with composition to be determined quantitatively. Microhardness will then become a more quantitative method for physicochemical analysis, which should be of value in elucidating the role of shortand long-range order for solid solutions involving substances with the covalent type of bonding and tetrahedral structure.

CHAPTER 2

Heats of Formation and Temperatures and Heats of Fusion of Compounds A111B v N . N . Sirota 36 I . INTRODUCTION . . . . . . . . . . . . . . . . 11. STRUCTURE OF COMPOUNDS A I ~ I B ' ~ AND SOMECRYSTALLOCHEMICAL 41 RELATIONS . . . . . . . . . . . . . . . . 47 . . . . . . . . . . . . . . . 111. PHASEDIAGRAMS 1. Phase Diagrams of Binary Systems A"'BV and Melting Points of 47 A1"BV Compounds. . . . . . . . . . . . . . 62 2. Remarks on the Character of the A1"Bv Phase Diagrams . . . 3. Thermodynamic Properties of the A"'BV Compounds from Phase 69 Diagram Data . . . . . . . . . . . . . . . 74 IV . VAPORPRESSURES. . . . . . . . . . . . . . . 74 4. Vapor Pressures of the A"'BV #Compoundsat their Melting Points. 86 AND ENTROPIES OF FORMATION . . , , V . HEATS,FREEENERGIES, 5 . Heats, Free Energies, and Entropies of Formation of A1"BV Com86 pounds . . . . . . . . . . . . . . . . . 106 VI. BONDING . . . . . . . . . . . . . . . . . 6. The Distribution of Electron Density in Crystals of Compounds 106 A"'BV Due to Enerxy ond Noture of Atomic Interocrion . . . 7. The Effective Ionic Charges of Arsenides and Antimonides of 120 Aluminum, Gallium. rind Indium . . . . . . . . . . 8. Heats of Atomization and Heats of Formation of III-V Compounds 122 . . According to the Data ofthe Electron Density Distribution 123 9 . Ionic Component of Energy and Electronegativity . . . . . 10. Change in the Speci$c Volumes of Components in Formation of' 125 A"'BV Compounds. . . . . . . . . . . . . . 130 11, Characteristic Temperatures . . , . . . . . . . . 132 VII. MELTING . . . . . . . . . . . . . . . . . 132 12. Change of Specific Volume of III-V Compounds upon Melting. . 133 13. Heats of Fusion (Experimental Data) . . . . . . . . 136 14. Effect of Pressure on the Melting Point of III-J' Compounds . . 15. Mean-Square Dynamic Ionic Displacements in III- V Compounds 139 near the Melting Point. . , . . . . . . . . . . VIII. THERMODYNAMIC PROPERTIES, ENERGYBANDS, AND THE PERIODIC 147 SYSTEM. . . . . . . . . . . . . . . . . . 16. Thermodynamic Properties of A"'BV Compounds in Connection 147 with the Position in the Mendeleeo Periodic System. . . . . 17. Width of the Forbidden Zone and Thermodynantic Properties of Compounds A"'Bv. . . . . . . . . . . . . . 154 159 IX. CONCLUSION. . . . . . . . . . . . . . . .

36

N . N . SIROTA

I. Introduction Heats of formation and atomization and temperatures and heats of fusion are the properties of compounds A"'BV that are related to their crystallochemical properties, thermodynamic and physical properties, and the mechanism of atomic interaction. There exist pronounced correlations among atomization heats, formation heats, temperatures and heats of fusion as well as between these quantities and various physical properties. For example, explicit relations are found involving the width of the forbidden zone, lattice energy, atomization energy, temperature and heat of fusion, interatomic distances, elastic and mechanical properties, etc. Such correlations between structural, thermal, and thermodynamic properties of 111-V compounds and their semiconductor and other physical properties show the existence of basic quantitative relationships. These relationships are based on the fact that all the foregoing characteristics and properties depend on the structure of electronic shells of ions, their actual sizes, and polarization properties, as is postulated by the first principle of crystallochemistry.'-' Crystallochemical structure of semiconducting 111-V compounds corresponds to the sp3 covalent ionic bond. In semiconductor elements of group IV having the diamond structure and in compounds A"'BV with sphalerite structure, the coordination number exactly follows the rule of HumeRothery and is equal to four. The conservative aspect of crystal structure should also be taken into account. Thermal motions and structural possibilities of filling the space with given effective shapes and sizes of ions restrict the tendency of these ions to take positions as close to each other as possible to achieve a maximum energy gain4 Taking into account the above-mentioned thermodynamic principle of crystallochemistry is very important in studying structural, thermodynamic, and physical properties of 111-V compounds. The compromise between energy tendency and structural possibilities. energy and entropy, is pronounced in many cases. Thus the type of bond and the appropriate structure corresponds to the condition of the free energy minimum of the system. The analysis of the correlations between physical properties, energy, and the mechanism of interatomic interaction in these compounds as well as the place of the components in the periodic table is essential for comprehension of the physicochemical nature of semiconducting 111-V compounds. In this respect, a direct study of electron distribution (electron density) in V. M. Gol'dshmidt [Goldschmidt], Kristallokhimiya, ONTI, 1937. [Goldschmidt], Naturwiss. 14, 477 (1926); Usp. Fiz. Nauk 9, 81 1 (1929). A. F. Kapustinskii, Zh. Piz. Khim. 5, 59 (1934). N. N. Sirota, Termodinamika i Kinetika fazovykh perekhodov, (Doct. Dissertatija) Moscow, 1948.

* V. M. Gol'dshmidt

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

37

stable and slightly excited states, of lattice dynamics in crystals, and their dependence on physical properties. is certainly of great interest. The actual distribution of electron density in the compounds is the most important factor which determines the type and the energy of the chemical bond, the crystal structure and physical properties in an assigned environment. Recently, studies of electron density distribution in 111-V compounds have been carried out which allow one to dlatermine semiquantitative estimations of the bond. These should allow possibilities for making evaluations of a large number of properties from diagrams of electron density distribution and atom scattering factors. A number of peculiarities of compounds A”’BVmay be shown which reveal both the specific mechanism of the chemical bond and the relatively large energies ofatomic interaction. The crystal structure, shape of the phase diagram, electric and magnetic properties, peculiarities of fusion, etc., are examples. Most semiconducting 111-V compounds have the sphalerite structure in which the first coordination number is known to be four and each ion of the same kind is therefore surrounded by four nearest neighbors-ions of different kind. The x-ray studies show that the dominant factor (in a system with tetrahedral coordination of which the s,p3 bond is characteristic) is the interaction of two neighboring atoms. The extent of sp overlapping of the electron clouds depends on the distances between A”’ and BV. Interaction with the second coordination sphere is howeve:r also of certain significance. As is known, in addition to the sphalerite structure, under certain conditions particular 111-V compounds exist in tlhe wurtzite structure for which the tetrahedral coordination is also characteristic. The first coordination spheres in wurtzite and sphalerite structures are practically identical. In transition from sphalerite to wurtzite, the effect of the second coordination sphere becomes pronounced. The fact that the melting points of compounds A“’BVare higher than those of the appropriate components is also a characteristic feature. It should be noted that melting points of 111-V compounds are higher than those of the elements of group IV with a similar “mean atomic n ~ m b e r . ” ~ Compounds A“’BV dissolve only slightly in component elements and fail to form noticeable solutions with them. In a binary system A”’BV only one semiconductor compound A”’BV is formed. Exceptions are the systems bismuth-thallium and antimony-thallium, in which no semiconducting compounds form. Fusion of 111-V compounds is accompanied by a transition from a crystal state to a liquid-crystal one. and then, with increasing temperature. by a transition from a liquid-crystal state with a distinct short-range order to a C. Hilsum and A. C. Rose-Innes,”Semiconducting 111-V Compounds.” Macmillan (Pergamon). New York. 1961.

38

N. N. SIROTA

rather disordered liquid This specific mechanism of melting affects the value of the heat of fusion and the temperature range in which thermal effects due to changes in short-range order in the melt are pronounced.8 That the specific volume of the liquid phase is smaller than that of the solid phase in fusion is also a peculiarity of compounds A"'BV with sphalerite and wurtzite s t r u ~ t u r e s . ' ~ ' ~ It should be pointed out that, in 111-V compounds, fusion and polymorphic transitions-due to, say, pressure changes-involve changes in the character of interatomic bonding from covalent (semiconductor) toward metallic bonds.'-' At present few reliable experimental data are available in which heats of formation and atomization as well as heats and even temperatures of fusion of compounds A"'BV are found directly. In many cases indirect or approximate estimations of the above quantities, which are applicable to numerous practical aims or theoretical predictions, are made. The present paper is mainly a review of experimental work on determination of heats of formation and atomization and temperatures and heats of fusion of 111-V compounds, and their correlation with particular semiconductor properties. Some special features of components A"' and BV should be discussed. Elements of groups 111 and V of the periodic table on the curves of atomic volumes as a function of the number in the periodic system lie on ascending branches for each period and approximately in the same region bounded by straight lines crossing B, Ga, In and P, As, Sb (see Fig. 1). Elements of group IV-C. Ge, Sn-lie almost on a straight line. It should be pointed out that the first element in group 111 is boron, a semiconductor with a relatively wide forbidden zone. The other elements of group 111, namely, Al, Ga, In, TI, are metals and s u p e r c ~ n d u c t o r s . ' ~Thus, ~ ' ~ in a neutral state. all elements of group 111 have two s electrons and one p electron in the external portion of the electronic hell.'^,'^ Their temperatures and heats of fusion and atomization and characteristic temperatures increase with decreasing period numbers upwards in the periodic table (see Fig. 2).

' H. Krebs, M. Haucke, and H. Weyand, in "The

Physical Chemistry of Metallic Solutions and Intermetallic Compounds," Vol. 11, Paper 4C. Her Majesty's Stationery Office, London, 1959. A. R. Regel' Soveshchanie PO Poluprovodnikovym Materialm, 1954. Vaprosy. Teorii i issledovaniya Poluprovodnikov i Protsessov Poluprovodnikovoy. Metallurgii, p. 12, Moskva, Akademii Nauk SSSR, 1955. V. K. Grigorovich and N. A. Nedumov, Sborn. "Mekhanizm i Kinetika Kristallizatsii," p. 297, Minsk, Nauka i Tekhnika, 1964. N. P. Mokrovskii and A. R. Regel, Zh. Tekhn. Fiz. 22, 1281 (1952). l o N. H. Nachtrieb and N. J. Clement, J . Phys. Chem. 62,747 (1958). " R. E. Hanneman, M. D. Banus, and H. C . Gatos, J . Phys. Chem. Solids 25,293 (1964). l 2 N. N. Sirota. Chistye Metal i Poluprovod Trudy I-oi [Pervoi] Mezhvuz Konf, Moscow 1957, 22-45 (1959). l 3 N. N. Sirota. Sborn. Fizika i fiziko-khimicheskii analiz. Moskva. 1957. p. 117.

'

'

cm3

vA @i&%

70 65

60 55

50

45 40 33

JO 23 20

f5

/O 5

0

FIG. I . Atomic and ionic volumes of elements a s a function of atomic number in the periodic table. The straight lines, discussed in the text, connect the following sets of elements: -B-In and P-Sb, - - S-Te.

-

FIG.2. Heats of atomization AH"' in kcal/mole, melting point T, in "C, boiling point T,. atomic weights A , ionization potentials I, in eV, and electronic structures as functions of atomic number Z .

2.

HEATS OF FORMATION AND TEMPE:RATURES AND HEATS OF FUSION

41

Elements of group V are metalloids N, P and semimetals As. Sb, Bi. In the neutral state they contain two s electrons and three p electrons in the external portion of the electronic hell.'^.'^ Their melting temperatures and heats of atomization pass through a rnaximum with decreasing period number. Arsenic has the highest melting point among the elements of group V. All elements of group V are characterized by high vapor pressures. Particularly, arsenic sublimes at normal pressure before reaching the melting point. Fusion of Ga, Sb, Bi of groups 111 and V as well as Si and Ge of group IV is accompanied by decrease of volume, while the great majority of elements melt with an increase in volume. As EIochvar and Kuznetsov have shown,’6 entropy of fusion is an explicit periodic function of the atomic number. Absolute values of fusion entropy of Ga. Si, Ge, As. Sb. P, Bi are very large. Table I contains the main thermodynamic properties of elements of groups 111 and V and, for comparison, of group IV of the periodic

II. Structure of Compounds A”’BVand Some Crystallochemical Relations At least twelve compounds formed of elements of group 111 [B, Al, Ga, In (TI)] with the elements of group V [IV. P. As. Sb (Bi)] have the sphalerite structure and are semiconductors (see Table II).29*30From twenty-five combinations involving five elements of group 111 with five elements of group V, there are compounds with the sphalerite structure in twelve cases, with the wurtzite structure in four cases, with the structure of other types in three cases. and no compounds for six of the combinations. Table 111 gives values of lattice constants, closest interionic distances, and calculated ionic sizes.31-39 Ionic sizes for the coordination number z = 4 are calculated by proceeding from the concept of sphalerite and wurtzite structures as those of the tightest packings of large ions (mainly, of metalloids), in half tetrahedral sites in which smaller ions are placed (mainly, ions of The concepts of ionic radii in structures of sphalerite or wurtzite are certainly conditional,, but in many cases they are useful. l4

Is l6

l9

C. Kittel, “Introduction to Solid State Physics.” Wiley, New York, 1956. F. Seitz, “The Modern Theory of Solids.” McGraw-Hill, New York, 1940. A. A. Bochvar and G. M. Kuznetsov, Dokl. Akad. Nauk SSSR98,221(1954). M. Hansen and K. Anderko, “Constitution of Binary Alloys.” McGraw-Hill, New York, 1958. A. Schneider and G. Heymer, in “The Physical Chemistry of Metallic Solutions and Intermetallic Compounds,” Vol. 11. Paper 4A. Her Majesty’s Stationery Office, London, 1959. D. R. Stull and G. C. Sinke, “Thermodynamic Properties of the Elements,” Advances in Chemistry Series. Am. Chem. SOC..Washington. D.C., 1956. 0. Kubaschewski and E. L. Evans, “Metallurgical Thermochemistry.” Pergamon, New York, 1958.

42

N. N . SIROTA TABLE 1

Element No.

Element

Atomic weight

Melting point T, (“C)

Boiling point Tb (“C)

5

B

10.811

21002200

13

A1

26.9815

660.1

2480

Volume change m fusion (%)

(2550)

8.296

0.33

+6.0

2.73

2.55

9.99

2.70

5.984

0.52

-3.2

4.41

1.335

11.79

5.91

6.00

1.82

0.780

15.71

7.31

5.785

-

17.24

11.85

29.8

2070

114.82

156.4

1730

2.0

81

TI

204.37

303

1460

2.2

7

N

14.0067 -210.0

15

White P Red Yellow

30.9738

44 41.1

280

33

As

74.9216

(817)

616

51

Sb

121.75

630.5

1640

-0.95

83

Bi

208.98

271.3

1560

-3.35

14

Si

28.086

($1

2.54

In

(5000)

g-atom

Density Ionization d potential Electron (eV) affinity

4.26

Ga

Diamond C 12.01 I15 Graphite

)(

Atomic volume

5.3

31

6

AHr kcal g-atom

2.23

49

69.72

1

ASr e.u. -~ gatom

-

- 195.8

0.150

0.47 -

-

13.19

2.34

6.07

662

13.07

5.73

5.25 ~

Ge

50

Sn

72.59

White

10.484

0.71

9.81

18.24

6.68

8.639

2.0

21.32

9.80

7.287

0.7

(5000)

3.41

3.52

5.36

2.25

11.256

1412

2600

6.58

11.1

7.24

12.1b

-9.6

12.11

937 231.9

2700

2.32

(2200)

6.27

7.6

13.64

5.32

+2.8

3.41

1.78

16.26

7.30

20.61

5.76

1.24

8.149 ~

-5.0

_

7.88

1.46 _

_

_

~

7.342

Gray

‘From Seitz.15

0.05

4.74

118.69

Reference

2.1

14.53

-

~

32

6.106

0.172

3.5

--

17

17

18

19

17. 21

17

22

22

’From Kubaschewski and Evans.’”

W. B. Pearson, “A Handbook of Lattice Spacings and Structures of Metals and Alloys.” Pergamon, New York, 1958. 2 2 V. I. Vedeneev, L. V. Gurvich, V. N. Kondrat’ev, V. A. Medvedev, and E. L. Frankevich, Energii razryva khimicheskikh svyazey. Potentsialy ionizatsii i srodstvo k elektronu. Spravochnik. Moskva, Izd-vo Akademii nauk SSSR, 1962. ” W. Gordy, Phys. Rev. 69,604 (1946). 2 4 R. S. Mulliken, J . Chem. Phys. 2 , 7 8 2 (1934); J . Chim. Phys. 46,498 (1949). 2 5 0. G. Folberth, 2. Nnturforsch. 13a, 856 (1958). 26 A. N. Nesmeyanov, Davlenie para khimicheskikh elementov. Moskva, Izd-vo Akademii Nauk SSSR, 1961 [English Transl. : “Vapour Pressure of the Elements.” Academic Press, New York. 19631. 2 7 T. L. Cottrell, “The Strengths of Chemical Bonds.” Butterworths, London and Washington, D.C., 1958. 21

2.

43

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

THERMODYNAMIC PROPERTIW OF GROUP111-V ELEMENTS

1.9

1.53

1.15

138.0f9

141

90.00

1.40

36.65

130.044

2.63

418

1.5

1.05

77.5

76.2t4.0

75

74.97

6.77

39.30

67.80

5.82

125"

1.4

1.055

65.0

66.5C3.0

66

65.00

9.82

40.38

55.888

6.23

109

1.4

1.02

57.0

59.1 k2.5

-

57.40

13.82

41.51

48.744

6.39

89

1.3

1.45

43.0

43

42.775

~~

-

113.0

2.1

1.862

2.00

36.61

79.8

75.00

9.80 5.46

108.76 5.63

38-98

69.805 4.98

-

2.0

1.725

1.8

1.525

117

1.8

(2000)

2.55

658

1.8

366

1.7

189 212

I4

2.23

60.ff 79 189

62.7

50.8f2.0

1.42

47 5

49.5

2.19

170.89

1.7

24

25

61

k 1.5

58.00

8.40

41.61

59.098

5.90

63.00

10.92

43.06

53.117

6.03

1.37

186.532

2.07

4.53

94.388

4.80

49.000

170.9

171.70

105

105

90.00

90

89

78.40

72

1.7

23

60

19

20

72

70.00

27

28

7.43

40.10

80.259

5.59

12.29

4024

63667

6.30

19

19

19

19

From Nesmeyanov."

'' L. Brewer in "The Chemistry and Metallurgy of Miscellaneous Materials : Thermodynamics" 29

3"

3' 32

(L. L. Quill, ed.), p. 26-27. McGraw-Hill, New York, 1950. G. V. Samsonov, L. Ya. Markovskii, and A. F. Zhigach, M. G. Valyashko, Bor, ego soedineniya i splavy. Kiev, Izd-vo AN UkrSSR, 19680 [English Transl.: "Boron, Its Compounds and Alloys." Office of Technical Services, Dept. of Commerce, Washington, D.C.]. E. Kauer and A. Rabenau, Z. Narurfbrsch. 12a, 942 (1957). R. S. Pease, Acta Cryst. 5,356 (1952). M. V. Stackelberg, E. Schnorrenberg, R. Paulus, and K. Spies, Z. Physik. Chem. A175, 127 (1935).

33

N. N. Sirotaand N. M. Olekhnovich, Dokl. Akad. Nuuk SSSR 148,71 (1963); 151,1079 (1963) [English Transl. : Soviet Phys. "Doklady" 8, 34 (1963);8. 810 (1964)l.

44

N. N. SIROTA

There is a relative similarity of lattice constants (identical periods) of 111-V compounds with the same anion in antimonides, arsenides, phosphides, and, particularly. in isoelectric sequences, respectively-as was shown by TABLE I1

AVERAGENUMBEROF

STRUCTURE, MOLECULAR W E I G H T , AND

Z

Atomic number

P A 30 9738

A

Z 5

14 0067

B

51

A

749216

Hexagonal sphalente 24.8177 10" 3.8b 6

Sphalerite 41 7848 5.9"

Sphalerite 85.7326

10

19

Structure Mol. wt. A€. eV iiZ, + 2")

Wurtzite 40.9882 3 8'

Sphalerite 57.9553 3.0" 14

Structure

Wurtzite 83.7267 3.3" 19

Sphalerite 100.6938 2.25' 23

Sphalerite 144.6416 1.4" 32

Wurtzite 128.8267

Sphalerite 145.7938 1.29" 32

Sphalerite 189.7416 0.36" 41

+

- -

A

A

Structure Mol. w t A€. eV &Z, Za)

B,

Sb

As

AIIIBV

Z 83

2

33

15

N

Atomic weight

z

Z

15

Element

COMPOUNDS

I21 75

208 980

Sphaleritc 132 561

28

A 10.81 I

Z I3

Al

I0

Sphalerite 101.9031

2.16" 23

Sphalerite 148.7315 1.62" 32

4

269x15

Z 31

Mol. wt.

Ga

A€, eV kiZA Z,I

+

Sphalerite 191 47

0.67* 41

A

69 72

Z 49 Ill

Structure Mol. wt. AE. eV

:cz, + Z")

28

Sphalente 236 57 0 17" 50

Tetrapon 323.800 Metal 66

4 114.82

Z XI

Structure TI

CSCl

326 12

Mol. wt. A€. eV

tiz,

+ 2,)

413 310 Metal

66

xz

A

204 31

* From Hilsum and Rose-lnnes

34 35

36

From Samsonov

PI

'From Kauer and Rabenau."

R. Juza and H. Hahn, Z . Anorg. Allgem. Chern. 239,282 (1938). S. Rundquist, Congr. intern. Chim. pure et appl. 16' Paris, 1957, Mem. sect. Chim. minerale, 539 (1958). V. M. Gol'dshmidt [Goldschmidt], Skrifer Norske Vidensknps Akad. Oslo. I . Matemar.Naturv. Klasse 1926 No. 8, 7 (1927).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

45

TABL.E 111

LATTICECONSTANTS, DISTANCES BETWEENNEIGHBORING ATOMS,RADII OF IONS IN THE LATTICE, AND COVALENT RADII ACCORDING TO MAGNETIC SUSCEPTIBILITY (DETERMINED BY X-RAY DATA) Element a, c.

A

a

6. A rA.rB.A

B

a, A c, A

6. A

Al

r A . rB. A

N

P

AS

2.504” 6.661 1.446

4.53i’d

4.177’ 2.058 0.37 1.68

~~

1.9h4 0.36 1.60

3.104b 4.965 1.888 0.36 1.53

5.46‘ ~2.36 043 1.93

3.180‘ 5.160 1.943 0.36 1.58

5.4505’

r A . r g ,A(from‘x,) a. A

c. A

&A

Ga

rA. rB. A

~~

a,

A

c. A 6. A rA.rB3A r A , rB, A (from’ xd) ~~~

3.53‘ 5.690 2.152 0.40 1.75

~

~

2.6547 0.49 2.17 1.14 1.57

5.6534’

6.0954’ ~

2.3581 0.43 1.93

2.4492 0.45 2.0 1.21 1.29

2.639 0.48 2.15 1.26 I 51

5.86315’

6.0584, 2.6213 0.48 2.14 1.48 1.38

6.47871’

~~

2.5409 0.47 2.07

B,

6.1355’

2.4497 0.45 2.00 1.22 1.41

~

r h , r a ,A(from‘x,)

In

5.662Zh

Sb

~

2.8052 0.51 2.29 1.36 1.51

~

‘From Pease” ’From Stackelberg et al.” ‘From Juza and Hahn A From Rundquist ‘From Gol’d~hmidt.’~

’From Giesecke and Pfister.” From Perri et 0 1 . ’ ~ ’From Pashintsev and Sirotd ’“ ’ Determined by diamagnetic susceptibility calculated from electron density data given by Sirota and Olekhno~ich.’~

Gol’dshmidt 1*2*44-46 in his early work. on the crystallochemistry of phases with the sphalerite structure. This fact may be considered as evidence of the validity of the concept of tightest packing for the 111-V compounds.

G. Giesecke and H. Pfister, Acta Cryst. 11,369 (1958). J. A. Perri, S. La Paca, and B. Post, Acta Cryst. 11, 310 (1958). 39 Yu.I. Pashintsev and N. N. Sirota, Dokl. Akad. Nauk BSSR3,38 (1959); Inzh. Fiz..Zh. No. 12. 38 (1958). 40 W. H. Bragg and W. L. Bragg, “The Crystalline State,“ Vol. I . Bell & Sons, London. 1933. 4 1 N. V. Belov, Struktura lonnykh kristallov i metallicheskikh faz. Moskva, 1947, Akad. Nauk SSSR. lnstitut Kristallografii. 4 2 N. N. Sirota, Editor, Ferrity : fizicheskie i fiziko-khimicheskie svoystva. Doklady soveshchaniya PO fizike, fiziko-khimicheskim svolystvam ferrotov i fizicheskim osnovam ikh primeneniya. 3d, Minsk, 1959. Minsk, Izd-vo Akad. Nauk BSSR, 1960. 4 3 E. S. Makarov. Stroenie tverdykh foz s peremennymchislom atomov v elementarnoi yacheyki, Moscow. Akad. Nauk SSSR, 1947. 4 4 See pages 33,34, 110 of Ref. 36. 45 V. M. Gol’dshmidt [Goldschmidt], Trans. Faraday SOC. 25, 253 (1929). 46 V. M. Gol’dshmidt [Goldschmidt], %. Physik. Chem. 133,397 (1928). 37 38

46

N. N . SIROTA

However the strong dependence of ionic size on the coordination number and the considerable differences in sizes of metal ions in various types of lattices show that the concept of tightest packing is insufficient and undesirable for a profound study of the problems of crystallochemistry of compounds A"'BV. Discussion of the structure of diamond, silicon. germanium, or grey tin in terms of the theory of tightest packing is also somewhat difficult. It should be pointed out that the lattice constants of 111-V compounds depend on the positions of the components in the periodic table, including the position in isoelectronic sequences. It should also be pointed out that large differences exist between ionic covalent radii calculated by electron density diagrams and by the diamagnetic component of magnetic susceptibility and the radii of tetrahedral sites in the sphalerite structure of 111-V compounds found according to the theory of tightest packing. The values given in Table 111 show that a discussion of the loose sphalerite structure in terms of the theory of tightest packing is inadequate. It may be seen that the ratios of ionic radii in compounds A'I'B" do not correspond to the Magnus n ~ m b e r . ~ ' , ~ ' At normal pressures under equilibrium conditions. compounds A"'BV as a rule do not undergo polymorphic transformations. At high pressures. however, and in certain nonequilibrium crystallization processes, allotropic and monomorphic transformations take place which are accompanied by modifications, mainly in the structure of wurtzite and body-centered tetragonal lattices. This fact shows that the structures of wurtzite and possibly of white tin are similar to the structure of sphalerite with respect to energy relationships. According to the theory of tightest packing, the difference between the structures of sphalerite and wurtzite is in the different arrangements of tetrahedrons which essentially do not change in transitions from one type of structure to the other. In sphalerite structures, tetrahedrons are arranged so that the apex of one tetrahedron touches the base of another, etc. The sphalerite structure may be also considered as two face-centered cubic lattices inserted into each other. In wurtzite structures. tetrahedron apexes are conjugated. In Fig. 3 wurtzite and sphalerite structures are described by means of Pauling tetrahedron^.^^ In Fig. 4, (a) represents the arrangement of tetrahedral and octohedral sites in an elementary cell of a facecentered cube, and (b) represents the sphalerite structure corresponding to the filling of one-half of the tetrahedral sites.

47

48

R. C. Evans, "An Introduction to Crystal Chemistry." Cambridge Univ. Press, London and Ncw York. 1939. 0. Hassel, "Kristallchemie." Th. Steinkopff, Leipzig, 1934.

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

47

FIG.3. Sphalerite (a) and wurtzite (b) structures represented by means of Pauling's tetrahedrons (After be lo^.^^)

FIG.4. (a) Arrangement of tetrahedral and octahedral sites in the fcc lattice. (b) Structure of sphalerite (one-halfof the tetrahedral sites are filled with the component A"').

111. Phase Diagrams 1. PHASE DIAGRAMS OF BINARYSYSTEMIS AIIIB~

AND

MELTINGPOINTSOF

COMPOUNDS

A phase diagram is the principal geometric representation of conditions of thermodynamic phase equilibrium as a function of composition and temperature.

48

N. N . SIROTA

In many cases a precise experimental determination of melting points of the A1"BV compounds and the liquidus curves involves difficulties. These result from high melting points which considerably exceed the melting points of the components; high vapor pressures of the components and their high reactivity, partial dissociation of the compounds, their tendency to supercooling, and, particularly, the flat shape of the liquidus curve near the melting point of the compounds. All these as well as a considerable gap between the beginning and the end of crystallization, i.e., between the liquidus and solidus curves on the diagram, diminish the accuracy in determination of the shape of the liquidus line. In many A"'BV systems, the melting points of compounds can be defined accurately only under equilibrium conditions with volatile component vapors at the controlled pressure. This is especially true in the case of nitrides, phosphides, and arsenides of the group I11 elements. In the first approximation one can be satisfied with measurements at normal pressure, i.e., with plotting the phase diagram in X - T coordinates only for some antimonides and certain arsenides. Only in a few cases49 have attempts been made to plot phase diagrams in X-T-P coordinates for the A"'BV systems. Investigations carried out by the methods of x-ray, thermal, and metallographic analyses as well as those using radioactive isotopes show that the A"'BV compounds of the sphalerite structure possess a very narrow region of homogeneity. However it is found that in some cases, at least at higher temperatures, the A1"BVcompounds deviate from the stoichiometric composition to such a degree that account must be made of this fact when considering the semiconducting properties. In spite of some deviations from the stoichiometric composition. all A"'BV semiconducting compounds with sphalerite structure belong to daltonides, i.e., to compounds of a constant composition as classified by K~rnakov.~~.~ The principal experimental data available on the phase diagrams and melting points of compounds A"'BV are considered in the following subsections. a. Borides of BV Elements

At present there are no phase diagrams of the systems boron-BVelements. The melting points of some borides are known only tentati~ely.'~ The boron-nitrogen system includes probably two compounds that are stable at normal pressure. These are the boron nitrides, BN-although 49

J. van den Boomgaard and K. Schol, Phillips Res. Rept. 12, 127 (1957). N. S. Kurnakov. Vvedenie v fiziko-khimicheskii analiz, Akad. Nauk SSSR, Moskva, 1940. V. Ya. Anosov and S. A. Pogodin, Osnovnye nachala fiziko-khimicheskogo analiza. Moscow, Akad. Nauk SSSR, 1947.

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

49

B3N is also possible. Under equilibrium conditions boron nitride has a hexagonal lattice which is similar to the structure of graphite with the following constants : a

=

2.504

c = 6.661

A, A,

cJa = 2.6601 . 3 1

The x-ray density is d = 2.278 g/cm3.

Recently a cubic modification of boron nitride, BN, with the sphalerite structure was discovered. It is b o r a z c ~ nwith ~ ~ ,the ~ ~lattice constant a = 3.615

A

and x-ray density d

=

3.488 g/cm3.

Borazon is produced both at high pressures, exceeding 62,000 atm, and at a temperature of 1360°C as well as by interaction between nitrogen and cubic boron p h ~ s p h i d eAt . ~ present ~ there is no information on the melting points of the boron nitrides. In the boron-phosphorus system there is the known boron phosphide BP54-59of the sphalerite structure with the lattice constant a = 4.537 and . ~ ’existence of B,P or B,,P, compounds x-ray density d = 2.97 g / ~ m ~ The has not yet been established. No data on melting points of boron phosphides are available in the literature. Boron phosphide BP is a s e r n i c o n d ~ c t o r . ~ ~ Boron a r ~ e n i d e ~is’ .formed ~~ by heating arsenic with boron at a temperature of 800°C. It has the sphalerite structure with a lattice constant a = 4.777 and x-ray density d = 5.22 g/cm3. The melting point of boron arsenide is not known. Boron antimonide.The existence of BSlbis expected, but not

’* 53

R. H. Wentorf, J . Chem. Phys. 26,956 (1957). A. Neuhaus and H. Menger, 2. Angew. Chem. ‘69,556(1957).

N. A. Goryunova. Khimiya almazopodobnyk h poluprovodnikov. Leningrad, Izd-vo Leningradskogo Universiteta, 1963 [English Trun.sl.: “The Chemistry of Diamond-Like Semiconductors” (translated by Scripta Technica, .I.C. Anderson, ed.) Chapman & Hall, London, 19651. 5 5 N. N. Moissen, Compt. Rend. 113,624,726 (1891). 5 6 P. Popper and T. A. Ingles, Nature 179, 1075 (1957). ” F. V. Williams and R. A. Ruehrwein, J . Am. Chem. SOC.82, 1330 (1960). B. Stone and D. Hill, Phys. Rev. Letters 4,282 (1960). 5 9 V. 1. Matkovich, Actu Cryst. 14,93 (1961). 54

’’

50

N. N . SIROTA

b. Nitrides of Elements of Group 111 The existence of aluminum nitride with the wurtzite structure in the system of aluminum-nitrogen has been reliably e~tablished.'~ The lattice constants are u = 3.104 & 0.005 A , c

=

4.965 & 0.008

C/U =

A,

1.600.32

The x-ray density calculated according to these data is d = 3.28 g/cm3. The melting point of aluminum nitride is reported to be 2200"C,54at a pressure of nitrogen equal to 4 atm. Renner6' points out, however, that at a pressure of 12.5 atm up to a temperature of 2450°C there is no observed fusion of aluminum nitride, and that the melting point of aluminum nitride considerably exceeds 2400°C regardless of the data in the literature.61 The width of the forbidden zone of AlN was determined as AE 3.8 eV.30The preceding works are reviewed by Hansen and Anderko." In the system gallium-nitrogen only one compound, namely, gallium nitride (GaN), is known, having the crystal lattice of ~ u r t z i t e " ~ with ~ ~ , the ~~,~~ constants a = 3.180 0.004 A,

-

A,

c

=

5.160

C/U

=

1.625 3 4

and an x-ray density of d = 6.11 g/cm3. The melting point of gallium nitride is 1300" to 1500°C. It is possible that it slightly exceeds the above value at equilibrium pressure of nitrogen vapor.60 In the indium-nitrogen system there is only one known compound, indiumnitride, InN, having the wurtzite lattice with constants a

=

3.53

0.004 A ,

c = 5.69 & 0.004 C/U

A,

= 1.611

and an x-ray density of d = 6.98 g / ~ m ~Renner6' . ~ ~ *assumes ~ ~ the melting point of indium nitride equal to 1200°C. Since it starts dissociating in vacuum at a temperature of 620°C it must have a high equilibrium vapor pressure at the melting point. 6o 61

63

T. Renner, Z. Anorg. Chem. 298,22 (1959). K. M. Taylor and C. Lenie, J . Electrochem. SOC. 107,308 (1960). J. V. Zirman and H. S. Zhdanov, Acta Physicochem. USSR6,306 (1937). H. Hahn and R. Juza, Z. Anorg. Allgem. Chem. 244,111 (1940).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

51

c. The Phosphides of the Group I I I Elements

In the system of aluminum-phosphorus there is only one aluminum phosphide compound AIP with the sphalerite structure. According to Gol’dshmidt the lattice constant of aluminum phosphide is a

=

5.46

A36

or

5.43

A6‘

and the x-ray density is d = 2.36g/cm3. The melting point of aluminum phosphide is not known p r e ~ i s e l y . ~According **~~ to R a b e n a ~ the~ melting ~ point of AlP is above 2000°C. The gallium-phosphide system includes only one compound, Gap, with the sphalerite structure. According to Gol’dshmidt4’ the lattice constant is

a

=

5.447 :k 0.006

A.

a

=

5.450 :t 0.001

A,68

More recent values are and

a = 5.4505 f 0.0001 A at 18 f 0.1”C.37 Under nonequilibrium conditions, on cold substrates and with high supersaturation, G a P may apparently crystallize into the wurtzite structure. F ~ l b e r t hhas ~ ~tentatively ,~~ determined the melting point to be at 1350°C. With reference to a private communication by Richman, Frosch and Derick7 give the maximum melting point of G aP as 1470”. However, R i ~ h r n a n ~ ~ gives the value of 1467°C at the equilibrium pressure of phosphorus vapor. The melting point of G aP is assumeld equal to 1525°C by Marina et ~ 2 1 . ~ ~ and to 1522°C by Vigdorovich et aL7’j h4

L. Passerini. Go;;. Chim. Ital. 58.655 (1928). H. G . Grimmeiss. W. Kischio. and A. Rabenau. J. Phys. Chc,ni. Solids 16. 302 (1960). 6 6 W. E. White and A. H. Bushey, in “1norgan:ic Syntheses,” Vol. 4,p. 19. McGraw-Hill, New York, 1958. 6 7 A. Rabenau in “Compound Semiconductors” (R. K. Willardson and H. L. Goering, eds.), p. 181. Reinhold. New York, and Chapman & Hall. London, 1962. 6 8 N. N. Sirota and N. N. Koren. Dokl. Akad. Nauk. BSSR No. 6. 373 (1963). b9 0. G. Folbcrth, J. Phys. Chem. Solids 7,295 (11958). ’* 0.G. Folberth, “Halbleiterprobleme,” Vol. 5, p. 40.Vieweg and Son, Braunschweig, 1960. ” C. J. Frosch and L. Derick, J. Elrctrochem. Soc. 108,251 (1961). ’* K. Weiser in “Compound Semiconductors” (R. K. Willardson and H. L. Goering, eds.), p. 471. Reinhold, New York, and Chapman &: Hall, London, 1962. 7 3 D. Richman, J. Phys. Chem. Solids 24, 1131 (1 963). 74 L. I. Marina, A. Ya. Nashcl’skii, and S. V. Yakobson, Zh. Fiz. Khim. 36, 1086 (1962) [English Transl. : Russian J . Phys. Chem. 36, 575 (1962)tl. 7 5 V. N. Vigdorovich. A. Ya. Nashel’skii, A. 1. Marina. and N. Ya. Zakharova, Nauchnye trudy. Gosudarstvennyy Nauchno-issledovaflskl~ i proekrnyy institut redkometallicheskay promiphlennosti 10, 1963. 65

52

N. N. SIROTA

The microstructure of alloys in the region of the Ga-GaP compositions shows that the eutectic of Ga-GaP is actually fully degenerated. The GaP compound does not form noticeable regions of solubility with its components. At higher temperatures, G a P is probably somewhat dissociated, and it follows that some deviation from the stoichiometric composition is expected. This deviation is however very small from the viewpoint of conventional physicochemical representations. In Fig. 5 a part of the phase diagram of the Ga-P system is presented on which experimental points obtained by R~benstein’~, Hall,77and R i ~ h m a n ~ ~ are shown. The diagram is plotted according to the data of thermal analysis and the determination of the composition of saturated gallium-phosphorus melts at given temperatures. According to the shape of the liquidus curve on

Ga FIG.5. Ga-P phase diagram. 0-After 76

”

at % A-After

M. Rubenstein, J . Electrochem. SOC.109, 69C (1962). R. N. Hall, J . Electrochem. SOC. 110,385 (1963).

P

Weiser.” R u b ~ n s t e i n .x~ -After ~

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

53

the phase diagram, from the data of R ~ b e n s t e i nand ~ ~ Hall,77 the melting point of G a P is about 1540°C in contrast to 1467°Cas determined by Richman.73Evidently the true melting point of gallium phosphide exceeds 1520°C and ranges from 1520" to 1550°C. In accordance with the liquidus curve shape near the melting point on the phase diagram, we report a melting point of 1540°C. In the equilibrium system indium-phosphorus there is only one known chemical compound indium phosphide, InP, with the sphalerite strucThe lattice constant a = 5.87 A.79 It should be noted, however, that the alloys studied79 were made using indium which contained 2 % germanium. According to Giesecke aind P f i ~ t e r ~ ~

-

a = 5.86875 & 0.0001

A

at

18 & 0.1"C.

Values close to this have been obtained by others.80v81The x-ray density is d = 4.79 g/cm3. The solubility of the components in the compound cannot be determined by x-ray analysis.82 The phase diagram of In-InP was studied by van den Boomgaard and S c h 0 1 at ~ ~pressures up to 40 atm. The imelting point has not been determined precisely since the equilibrium pressure at the melting point was supposed to attain 60 atm. According to their extr,apolated estimates, the melting point T,= 1062 7°C. The In-InP phase diagram has also been studied by Shafer and W e i ~ e r ~ ~ and by Koster and U l r i ~ h The . ~ ~liquidus line within the regions of low concentrations has been determined by Hall77 from the composition of saturated melts at a constant temperature. R i ~ h m a nhas ~ ~estimated the melting point of indium phosphide as 11058°C at the equilibrium pressure of phosphorus vapor. In Fig. 6, a phase diagram of In-InP is presented. The liquidus line is the projection of T-P-X three-phase equilibrium onto the T-X plane. The melting point on the diagram was assumed to be 1060°C after taking into account the liquidus curve shape according to the data by R i ~ h m a n , ~ ~ , ~ ~Hall.77 van den Boomgaard and Sch01,"~Koster and U l r i ~ hand In the system In-InP the eutectic is, almost completely degenerated, as and by metallographical analysis.68,80 is proved by the data of Thiel and H. Koelsch, Z . Anorg. Chem. 66,319 (1910). A. Iandelli. Gazz. Chim.Ital. 71, 58 (1941). N. N. Sirota and L. A. Makovetskaya, Dokl. Aiiad. Nauk B S S R 7,230 (1963). A. S. Borshchevskii, N. A. Goryunova, and N. I<. Takhtareva, Zh. Tekhn. Fiz. 27, 1408 (1957) [English Transl. : Soviet Phys.-Tech. Phys. 2,1301 (1957)l. 8 2 N. A. Goryunova, N. N. Fedorova, and V. I . Sokolova, Zh. Tekhn. Fiz. 28, 1672 (1958) [English Transl. : Soviet Phys.-Tech. Phys. 3, 1542 (1957)J 8 3 M. Shafer and K. Weiser, J . Phys. Chem. 61,1424 (1957). 84 W. Koster and W. Ulrich, 2. Metallk. 49, 365 (1958). " A. 79

54

N. N. SIROTA

Jn

at %

P

FIG. 6. In-P phase diagram. 0-After Hall." A-After van den Boomgaard and S c h 0 1 . ~ ~ m-After Shafer and W e i ~ e r . ' x~ -After Koster and U I r i ~ h . ~ ~

d . Arsenides ofthe Group I I I Elements In the aluminum-arsenic system there is also only one compound with the sphalerite crystal structure.36 The lattice constant of AlAs has been determined by Natta and Passerinia5as a = 5.622 kx units, by Sirota and Pashint, ~ ~by Koster and sev86,39as a = 5.6622 A, by Sirota and O l e k h n o v i ~ hand Thomaas as a = 5.63 A. 86

''

G. Natta and L. Passerini, Gazz. Chiin. Ital. 58,458 (1928). N. N. Sirota and Yu. I. Pashintsev, Dokl. Akad. Nauk S S S R 127,609 (1959) [English Transl.: Proc. Acad. Sci. U S S R , Phys. Chem. S c ~ t 127, . 627 (1959)l. N. N. Sirota and N. M. Olekhnovich, Dokl. Akad. Nauk S S S R 136,660 (1961) [English Transl. : Proc. Acad. Sci. U S S R , Phys. Chem. Sect. 136, 97 (1961)l. W. Koster and B. Thoma, 2. Metallk. 46,293 (1955).

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

55

Phase diagrams have been partially studied by Koster and Thoma.'8 According to their data both Al-AIAs and AlAs-As eutectics are fully degenerated. Moreover, in the system studied, Koster and Thoma could not detect any stratification at a temperature above the melting point of the AlAs compound, as was assumed by M a n ~ u r i . ' ~Nevertheless Hansen and AnderkoI7 do not consider this problem completely settled. Koster and Thoma have roughly determined the melting point of aluminum arsenide to be 1600°C. In Fig. 7 a tentative phase diagram is presented according to Koster and Thoma.88

at%,41 FIG.7. 41-As phase diagram. (After Koster and Thoma.")

There is one chemical compound, gallium arsenide (GaAs), in the galliumarsenic system that is crystallized in the sphalerite s t r u c t ~ r e . ~ ~ The ,~' lattice constant has been determined bsy G ~ l ' d s h m i d tand ~ ~ other investig a t o r ~ . ~ ~According , ~ ~ , ' ~to Giesecke and P f i ~ t e ra, ~=~5.6534 k 0.0002 A at 18 O.l"C,and the x-ray density d = 5.31 g/cm3. 89 90

Q. A. Mansuri, J . Chem. SOC.121.2272 (1922). V. M. Gol'dshmidt [Goldschmidt], Ber. deut. chem. Ges. 60, 1263 (1927).

56

N. N . SIROTA

The phase diagram of the Ga-As system was thoroughly studied by Koster and Thoma88 using thermal and microstructural analyses. The GaAs-Ga and GaAs-As eutectics are wholly degenerated. The solidification end temperature on the arsenic side corresponds to 810°C and that on the gallium side to 29.5”C. This is in agreement with the results of previous studies.I7 Actually when the alloys are being cooled, no supercooling is observed except for a slight supercooling of arsenic. Koster and Thoma88 call attention to the fact that the thermal analysis and the determination of the melting point of gallium arsenide were carried out at some unknown pressure that should be taken into account when analyzing the phase diagram. They also point out that this pressure is considerably lower than was previously assumed. The melting point of gallium arsenide was determined by Koster and Thoma to be 1238°C. Further studies made by van den Boomgaard and S c h 0 1 ~confirmed ~ Kiister and Thoma’s phase diagram. According to them4’ the melting point of GaAs under an equilibrium pressure of the arsenic vapor is 1237 3°C; according to R i ~ h m a n under ~ ~ these same conditions it is 1238°C. At small concentrations of arsenic, has determined the lines of the liquidus from the composition of melts saturated under isothermal conditions. In Fig. 8 a phase diagram for the Ga-As system is given according to the data of and of Koster and Thoma.88From the shape of the liquidus curve, the melting point of GaAs is indicated to be 1240°C. In the indium-arsenic system only one chemical compound has been found. Indium arsenide (InAs) crystallizes in a sphalerite lattice.79 The lattice constant has been determined by a number of investigators : according to Iandelli79 it is 6.036 kx units; Liu and Peretti’’ give a value of 6.058 A ; and a more precise value of 6.0584 & 0.0001 A at 18 ?c 0.1”C is given by Giesecke and P f i ~ t e rThe . ~ ~indium used by Iandelli7’ contained 2 % germanium. The phase diagram was studied by Liu and Peretti” by the methods of microstructural and thermal analyses. It is shown that the eutectic on the In-InAs side is close to degeneration. They report the eutectic contains 0.03 at. arsenic and has the melting point 155.2 k 0.2”C. On the InAs-As side, the eutectic point corresponds to 81.5 k 1.5 at. % arsenic. The melting point of the eutectic is determined to be 731 k 1°C. The melting point of the indium arsenide was found by Liu and Peretti to be 942 3°C.9’ The phase diagram of In-As was also studied by van den Boomgaard and S c h 0 1 , ~who ~ confirmed the results of Liu and Peretti. According to their data the melting point of indium arsenide is 943 f 3°C under equilibrium pressure of arsenic vapor. In Fig. 9 the phase diagram is plotted from the data of Liu and Peretti”’ and Hall.” We have assumed the melting point of T. S. Liu and E. A. Peretti. Truns. Am. Soc. Metois 45, 677 (1953).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

57

1200

{OOO

800

600

400

206

1 245 a

OO

20

60

7

Ga

FIG.8. Ga-As phase diagram. 0-After

80

at

Hall." A-After

f00 As Koster and Thoma."

lnAs to be 943 f 3°C on the basis of an analysis of the data available and the shape of the liquidus curve.

e. Antimonides of the Group I I l Elements Apparently al~minum-antimony'~.''~-~~ was the first binaryphase diagram of the groups I11 and V elements to be thoroughly studied. The diagram in 92

93 94 95 96

M. Hansen, "Aufbau de Zweistofflegierungen.'.' Springer, Berlin. 1936 [English Edition : M. Hansen and K. Anderko, "Constitution of Binary Alloys," 2nd ed. McGraw-Hill, New York, 19581. G. G. Urasov, Zh. Russ. Fiz. Khim. Obshchestua51,461(1919). G. G. Urasov, Izu. Inst. Fiz. Khim. Anal. I, 1921. V. M. Glazov and I).A. Petrov. Izo. Akad. Nauk S S S R , Otd. Tekhn. Nauk No. 4. 125 (1958). G . Tammann, Z . Anorg. Alleg. Chem. 48,53 ( I 905).

58

N . N . SIROTA

813”

7 FIG.9. In-As phase diagram. 0-After

Hall.” A-After

Liu and Peretti.”

Fig. 10 is mainly that from the data of U r a ~ o v . It~ also ~ , ~contains ~ some results from recent studies.95 A review of other earlier works is made by Hansen and Anderko.I7 In the aluminum-antimony system there is one known compound, aluminum antimonide, AISb, crystallizing in the sphalerite l a t t i ~ e . ’The ~ lattice constant a has been determined to be 6.1368,.98 6.09598,\,996.1368, at 20°C*00and 6.1355 f 0.0001 8, at 18 f 0.1°C.37According to Giesecke and P f i ~ t e r the , ~ ~x-ray density is d = 4.27 g/cm3. The methods of thermal, metallographic, and x-ray analyses were used to determine the phase diagram. The shape ( X . T )of the liquidus curve near the ordinate of the compound was established more exactly by Glazov and Petr~v.~~ E. A. Owen and G. D. Preston, Proc. Phys. Soc. (London) 36,341 (1924). K. Willardson. A. C. Beer. and A. E. Middleton, J . Electrochem. SOC.101. 354 (1954). 9y R. F. Blunt, H. P. R. Frederikse, J. H. Becker, and W. R. Hosler, Phys. Rec. 96, 578 (1954). l o o N. N. Sirota and E. M. Gololobov, Dokl. Akad. Naitk S S S R 144,398 (1962) [English Trans/.: Proc. Acad. Scr U S S R , Phys. Chrm. Sect. 144, 405 (1962)] ”

” R.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

AL FIG. 10. AI-Sb phase diagram. 0-After Glazov and P e t r ~ v . ~ ~

59

a t 7:

S6

U r a ~ o v . ~ A-After ~.'~

Veszelka.los 8-After

Tammann96 and U r a ~ o v note ~ ~ ,a ~certain ~ difficulty in aluminum antimonide formation in a liquid state and extreme slowness in establishing equilibrium in the system under considerakion, both in the homogeneous and heterogeneous regions of the diagram. The liquidus temperatures were determined from the thermal cooling curves of alloys formed by the direct combination of aluminum with antimony. Reproducible results are obtained only after a rather prolonged holding of the melt above the liquidus lines by 10Ck150"C. Urazov has noted the necessity of taking into account the time in a liquid state required for the compound formation when the alloys are being synthesized from their compontmts. In his opinion the slowness of the process is attributed to a peculiar inertness of antimony when forming compounds (polymerizations).At present, especially after the work of Glazov, Chizhevskaya, and Vertman (see Ref. 101), it has become clear that the slow

60

N. N. SIROTA

nature of the process of compound formation in a liquid state is typical for A%b and other A"'BV compounds. Glazov has shown that formation of a compound in a liquid state is of the nature of a second-order reaction."' A logical application of the methods of physicochemical analysis made it possible to predict a certain number of characteristic features of AISb, among which are its semiconducting proper tie^.'^^-'^^ According to Urazov's data, aluminum antimonide does not form any noticeable region of solubility with its components. Despite an earlier opinion,lo5further studies on this s u b j e ~ thave ~ ~ confirmed ,~~ this fact. The diagram of AlSb is notable for the presence of essentially degenerated eutectics with the components AI-A1Sb and AlSb-Sb. This is easily ascribed to the fact that the melting point of a compound is considerably higher than the melting point of its components if the latter are insoluble in each other in a solid state. The melting point of the Al-AlSb eutectic differs from the melting point of aluminum by approximately 3°C. The eutectic contains about 0.25 at. % Sb, and is almost unobserved metallographically. The melting point of the AlSb-Sb eutectic is essentially the same as that of antimony. The difference lies within the range of experimental error. It should be noted that among the A"' antimonides, the AlSb compound is characterized by a lack of corrosion resistance, which is to a considerable extent dependent upon the degree of purity. The corrosion resistance of the compound increases with purity.lo6 The melting point of AlSb is 1050~ ~ , ~ according ~ to Welker,"' and 1080°C according to U r a ~ o v .1060°C 1080°C according to other^.^^,'^* Hansen and Anderko17 give a melting point of 1065°C. In Fig. 10 the melting point of AlSb is 1080"C, corresponding more to the liquidus line of Urazov and other authors. I n the gallium-antimony system there is one compound, gallium antimonide, GaSb, with the sphalerite structure. The lattice constant a = 6.0954 0.0001 8, at 18 & 0.1"C.37which is close to the data of other a u t h ~ r s . ' ~ ~ ~ ' ' ~ V. M. Glazov, Autoreferat dissertatsii, Moscow, 1959. G. G. Urazov and N. N. Sirota, Sb. Fiziko i fiziko-khimicheskii analiz. MOSCOW, 1957, p. 3. Io3 W. P. Allred, W. L. Mefferd, and R. K. Willardson, J . Electrochem. Soc. 107, 117 (1960). D. N. Nasledov and S. V. Slobodchikov, Zh. Tekhn. Fiz. 28, 715 (1958) [English Transl.: Soviet Phys.-Tech. Phys. 3, 669 (1958)J l o 5 W. Guertles and A. Bergmann, Z . Metallk. 25,82 (1933). G. N. Nikolaenko, Voprosy metallurgii i fiziki poluprovodnikov; trudy 3-go soveshchaniya PO poluprovodenkovyam materialam, 1957. Moscow, Izd-vo Akad. Nauk SSSR, 1959, p. 80. lo' H. Welker, Z . Naturforsch. 8a, 248 (1953). J. Veszelka, Mitt. berg-u. hiitttnmann. Abt. kgl. ungar. Hochschule Berg-u. Forstw. Sopron 3, 193 (1931). Io9 N. A. Goryunova and N. N. Fedorova, Zh. Tekhn. Fiz. 25,1339 (1955). I l o N. N. Sirota and E. M. Gololobov, Dokl. Akad. Nauk S S S R 138, 162 (1961) [English Transl.: Proc. Acad. Sci. U S S R 138,405 (1961)l.

lo*

lo'

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

61

The phase diagram of the gallium-antimony system has been plotted by Greenfield and Smith"' and almost at the same time by Koster and Thomas8 In general both diagrams are in agreement. A critical review has been given by ~ Ga-GaSb eutectic is completely Hansen and Anderko'? and by V O ~ . "'The degenerated."," 13113 The GaSb-Sb eutectic is reported to contain 88.2 %I1' or 87% Sb.88 The melting point of the eutectic is given as 589.8""' or 583"C.88 The melting point of the compound has been determined to be 705.9"C,"' 72OOC,' l 3 725"C,'14 702°C,'07*"5 and 712°C.95,1'6The shape of the liquidus lines is typical for the A""BVcompounds. The presence of a singular point in the ordinate of the compounds has been pointed The compound and its components do not form any noticeable regions of solid solution. In Fig. 11, the phase diagram of Ga-Sb is given, plotted mainly from the data of Greenfield and Smith' '' and Hall.77The melting point of the compound is thus assumed to be 712°C in accordance with the shape of the liquidus curve. I n the indium-antimony system one compound, indium antimonide, InSb, has been found. It crystallizes in the sphalerite lattice and has a lattice constant of a = 6.470&"' a = 6.47877 jk 0.00005b; at 18 5 0.1"C.37 For InSb, a rather narrow homogeneity region is t y p i ~ a l . " ~ . " ~Pogodin and Dubinskii'*' were the first to study the phase diagram of indium and antimony by the methods of thermal, mi'crostructural, and x-ray analyses. Studies carried out by Liu and Peretti' 18," l 9 have confirmed their results.'21 The In-InSb eutectic is practically degenerated. Its melting point is below or 154.8"C.' l 9 The In-InSb that of indium by 1°C and is given as 15~5"C,'20 eutectic contains 0.66 at. % Sb; the InSb-Sb eutectic is reported to contain 68.3 at. % Sb' l 9 or 70.4 at. % Sb.'20 The melting point of the eutectic is given approximately as 508-507°C'20 or 494°C.' l 9 Solubility of components in each other is very small. The melting :point of the compound is reported as 536"C,'20 525°C,"9 or 540"C.95It was noted that there is a singular point on the liquidus curve in the ordinate of the compound.95

-

''I

"* 'I4

'" 'I8 ''O

''I

1. G . Greenfield and R. L. Smith, J . Metals 7,351 (1955). A. E. Vol, Stroenie i svoystva dvoynykh metallicheskikh sistem, Moscow Fizmatiz, Vol. 1. 1959, Vol. 11, 1962. H. Welker, Physica 20, 893 (1954). F. A. Cunnell, J. T. Edmond, and J. L. Richards, ,Proc. Phys. Sor. (London)B67,848 (1954). R. F. Blunt, W. R. Hosler,and H. P. R. Frederikse, Phys. Rev. 96,576(1954). J. Bednar and K. Smirous, Czech. J . Phys. 5, 546 i(1955). T. S. Liu and E. A. Peretti, Trans. A l M E 191,791 (1951). T. S. Liu and E. A. Peretti, J . M e t a l s 3 , 791 (1951). T. S. Liu and E. A. Peretti, Trans. Am. Soc. Metals 44,539 (1952). S. A. Pogodin and S. A. DuSinskii, Izv. Sektora F i z . Khim. Analiza, Inst. Obshch. i Neorgan. Khim. Akad. Nauk SSSR 17,204 (1949). N. N. Sirota and E. M. Gololobov, Dokl. Akad. Nauk SSSR 143,156 (1962)[English TransE. : Proc. Acad. Sci. USSR 143, 215 (1962)l.

.

62

N. N. SIROTA

'630.5'

3

Ga

at %

1

FIG.11. Ga-Sb phase diagram. 0-After Greenfield and Smith."'

x -After

Hall" a-After Koster and Thoma.88 A-After Glazov and P e t r o ~ . ~ ~

In Fig. 12 a phase diagram shows the data of Pogodin and Dubinskii'20 and Liu and Peretti."' The melting point of InSb is assumed to be 533°C in accordance with the shape of the iiquidus curve based on their data.' 19*120 In Table TV are listed melting points T, of compounds A1"BV from data of various authors, and the values preferred by this author.12'

2. REMARKSON THE CHARACTER OF THE A1"BVPHASE DIAGRAMS The A"'BV compounds with the sphalerite and wurtzite structures are compounds of constant composition, or daltonides according to Kurnakov's classification. For details see Kurnakov's outstanding work "Solutions and alloy^".^^^'^^" The daltonide phases are characterized by an ordinate of the 122

I. E. Campbell, C. F. Powell, D. H. Nowtcki, and B. W. Gonser, J . Ektrochem. Soc. 96, 318 (1 949).

lLZaN. S . Kurnakov, Sobranie izbranich rabot. Leningrad ONTl 1, 1938.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

63

500"

I

Jn

at

%I

/oo S6

FIG. 12. In-Sb phase diagram. 0-After Hall." A-After Pogodin and Dubinskii."' v-After Liu and Peretti.'I9 x -After Glazov and P e t r ~ v . ~ ~

compound with a certain composition which lies within the phase concentration limits. The ordinate of the compound is characterized by the highest degree of orderliness, namely, singular points on the curves of the change of different properties as functions of composition. [n general, daltonide phases may or may not form solid solutions with the components. The concentration region on the phase diagram for the existence of the A"'BV compounds is very narrow and practically shrinks into a point. These compounds are typical examples of daltonides that do not form solutions with components. On the solidus curve in the phase diagram there must be a singular point in the ordinate of the compound of constant composition-the daltonide. If melting of the compound occurs without dissociation of the compound, the singular point should be on the liquidus curve. There are no singular points on the liquidus curve in cases where in a liquid phase complete dissociation of the compound occurs. In intermediate cases of partial dissociation of the compound, the shape of the liquidus curve near the ordinate of the compound is of an intermediate nature, with notable indications of deviation from additivity (see Fig. 13). The compounds of variable composition, berthollides as designated by Kurnakov, in contrast to daltonides, have no definite ordinate-the composition of which is invariant with the temperature and other external equilibrium

TABLE 1V MELTING POINTS BN 3000b.bb

BP

AIN

AIP

2200"

OF

COMPOUNDS A1"Bv ACCORDINGTO DATAOF VARIOUSAUTHORS AlAs

AlSb

1600"

105010800.' 1060'

2400'

T. "C

GaN 13001500'

ZOO@ 1080q.' 1065'

1 W C 'From Hansen and Anderko." *From Samsonov et ~ l . ' ~ 'From van den Boomgaard and Schol."q 'From Goryunova.'4 'From Renner." From Taylor and Lenie." a From Folberth.69 'From F o l b e r ~ h . ' ~ ' From Frosch and Derick." 1 From Weiser."

' From Marina rf al." From Vigdorovich et 0 1 . ' ~ From landelli." From Liu and P e r e t t ~ . ~ ' From L i r a ~ o v . ~ ~ * F r o m Glazov and P e t r o ~ . ~ ' ' From Blunt er a/.99 'From Welker.'" ' From Vesrelka."' " F r o m Greenfield and Smith ' ' I I

GaAs

GaSb

1525'

GaP

1237'

712','

1522' 1350K,k 1500' 147w 1467'

1238'

1540

1240

705.9"

lnN

1200'

InP

InAs

lnSb

1062'

942"

533".'

1058'

943'

536"" 525' 5404

720" 125"

70Py 712

1060

943

' From Welker

'I3

" From Cunnell

rt a/.'"

533

From Bednar and Smirous."D From Liu and Peretti."8 ' F r o m Liu and Peretti."9 '' From Pogodin and Dubinski1 ' 2 0 "From Campbell er "' Values in bold lace type are preferred vdlues. A

? z

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

B

P Y

f

0

R

mm mm

1 Y

s

r,

B &

D

C

A6

A6

1 1

AB

AB

65

E

L

A8

FIG.13. Daltonide and berthollide phases according to Kurnakov. A, B, C, and D are daltonides and E is a berthollide phase; in B and D the liquid phase compound AB is dissociated.

parameters-within the concentration range of phase existence on the constitutional diagram. The Hume-Rothery phases, y and 6 phases, in the system Bi-T1'7,50*51,92 can be identified with such berthollides. In the case of berthollide phases, no singular points are observed on the liquidus and solidus curves, or on the curves of composition-property in the region of phase existence. This is proven by the shape of the compositionproperty curves in the Bi-TI system (Fig. 14). The thermodynamic theory of daltonide and berthollide phases has been developed by Lipson and Wilson,123S i r ~ t a ,and ' ~ ~others. Of all the A1"BVsystems, only in the Bi-TI system, and possibly in TI-Sb, are berthollide phases formed (Fig. 14). In the remaining systems, phases of the daltonide type exist. According to Glazov and P e t r ~ v the , ~ ~liquidus curves of antimonides of the A"'BV comjpounds in the region of the ordinate of the compound have a singular point which corresponds to the stoichiometric composition. With melting of the A"'BV compounds, no ideal solutions are formed. This is shown in the previously cited works of ,Regel and co-worker~,',~ Glazov,"' and Grigorovich and Nedumov.* This comment also refers to the Ga-Sb system for which Schottky and B e ~ e rhave ' ~ ~found an insignificant deviation of the liquidus curve from an ideal one. These solutions can be called quasiideal. At temperatures near the melting point, we would rather consider solutions in the concentration ranges to the right and left of the ordinate as quasi-ideal solutions of component and compound, since in a liquid state

lZ5

H. Lipson and A. J . C. Wilson, J . fron cmd Steel Inst. 142, 107 (1940). N. N. Sirota, Compt. Rend. (Doklady) Acad. Sci. U.RS.S. 4 4 , 3 3 1 (1944)(in English) W. F. Schottky and M. B. Bever, Acta M e t . 6,320 (1958).

66

N. N . SIROTA

TL /O

20 30 40 50 60 70 d0 00 Bi

at % FIG. 14. Bismuth-thallium system (example of berthollide phases) : (a) Phase diagram. (b) Variation of such physical properties as Brine11 hardness ( H s ) . temperature coefficient of resistivity (ao),and electrical conductivity (u)as functions of composition. (After V O I " ~; also see Kurnakov et a1.122a,126a )

the compound is only slightly dissociated. In case of ideal and quasi-ideal solutions in daltonide and berthollide phases, the shape of the liquidus curve in the coordinates composition-temperature is of exponential character and can be approximated by equations of the type'26 A logx = - + B . (1)

T

At the ordinate of the compounds which melt without dissociation, there is an J. J. van Laar, "Sechs Vortrage iiber das Thermodynamische Potential und Seine Anwendungen auf chemische und Physikalische Gleichgewichtsprobleme, angeleitet durch zwei Vortrage iiber nicht Verdiinnte Losungen und uber den Osmotischen Druck." Braunschweig, Vieweg, 1906 [Soviet Transl. : Shest lektsii o termodinamicheskom Potentsiale ONTI, 19381. lZh'N. S. Kurnakor, S. F. Zhemchuzhni, and V. Tararin, Zh. Rus. Fizikochirnichesckogo obschestva 45, 300 (1 91 3).

lZ6

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

67

intersection of the right and left bra.nches of the liquidus curve. In Fig. 15 the liquidus curves of the A"'BV systems are given in the coordinate variables :

4

(a) log x and 103/T for the left (A"') branches up to x = ; (b) log(1 - x) and 103/Tfor the right (BV)branches up to 1 - x = i .

FIG. 15. Logarithmic variation of concentrations X, and X , in left- and right-hand liquidus branches of A"'BV systems versus inverse temperature.

The figure shows that a deviation from the rectilinear shape is observed at small concentrations of the BVcompoinents, especially in the In-Sb and In-As systems, as well as near the ordinate of the compound. The values obtained for A and B and the calculated heats of fusion of the A"'BV compounds are shown in Table V. The values obtained differ from those reported by Hall, which are given in Table VI in the next section. One of the peculiarities of the phase diagram of the A"'BV systems is that the compositions of the eutectics are close to the components at small and sometimes at vanishingly small con tent of the second component. This peculiarity, which Koster and Thoma8' have noted, is attributed to the large difference between the melting points of the compound and those of its components. The greater the difference in the melting points of the components and of the compound, the lower are the concentrations of the second component at which the eutectic exists. Degeneration of the eutectic begins when the thermodynamically possible supersaturations (with respect to the saturation of the liquid by a component) are insufficient for the origin of nucleation of crystals of the components, and the process consists of the

TABLE V VALUES OF

CONSTANTS A

AND

B IN

THE

EQUATIONS OF THE RIGHT-A N D LEFT-HAND BRANCHESOF SYSTEM A"'BV LIQUIDUS CURVES, A N D HEATSOF FUSION

A111

BV

As

Sb

A B AH, from A AH, from B

A

Al

P

B AH, from A AHf from B

12,708 8.80 25.4 23.8

A H , ave

T,, "C A

Ga

A

B AHf from A AHi from B AH, aye AH,,exp,l Tm,'C

12,268 6.07 24.6 22.0 23.3

10,282 6.18 20.6 18.6

10,354 6.232 20.6 18.6

I540

In

A

B A H , from A AH, from B

B AHf from A AH, from B AH,.., AHf,,,,, Tm,"C

*From Schottky and Bevrr.'"

"

From Richman and Hockings."'

10,380 7.078 20.8 18.6 19.7 1 2h 1060

8636 8.16 17.2 16.0

19.6 21h 1245 6504 4.666 13.0 8.6

? 10,754 10.312 21.4 20.4

18.8 I 2" 712 ~~

A

7308 4.748 14.6 12.6

19 14.2' 1080

AHiexptl

B AH, from A A l l , from B

CAL('ULATED

~~~~~~

7386 3.38 14.8 6.4 10.7 26h 945

From Glazov and Czhen'-Yuan'. 1 2 *

2178 2.098 4.4 3.4

8472 9.904 16.8 16.0 10.2

12.2"

9.jh 533

.4

$

3>

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

69

formation of nuclei of the compound. Thus in a criterion for degeneration of eutectics, not only thermodynamiic but kinetic factors as well must be considered.

3. THERMODYNAMIC PROPERTIES OF DIAGRAM DATA

THE

A1"Bv COMPOUNDS FROM PHASE

An accurately plotted phase diagrarn is a quantitative geometric expression of thermodynamic conditions of phase equilibrium in a physicochemical system. Additional information is needed to determine the basic thermodynamic properties of phases of a given physicochemical system from the phase diagram.'26,' 29-141 For systems similar to the A"'-BV systems where in each there is only one chemical compound A"'BV with the sphalerite structure, which does not form solutions with components, Wagner'42 has derived formulas permitting the calculation of heats of formation of the compounds with the help of phase diagrams when heats of fusion are At the melting point T, of the compound, its standard molar free energy Gg,T,,, is equal to the free molar energy GT,,xe of the liquid phase of the same composition, G;,=,,, = GT,,xe. The free energy of the liquid phase is the sum of the free energies of the liquid components A(1) and B(1), the free energy 127

D. Richman and E. F. Hockings, cited according to M. B. Bever in "Compound Semiconductors" (R.K. Willardson and H. L. (Goering, eds.), p. 500. Reinhold, New York, and Chapman & Hall, London, 1962. 12' V. M. Glazov and Lyu Czhen'-Yuan', Zlz. Neorgan. Khim. 7 , 582 (1962) [English Trans/.: Russian J . Inorganic Chem. I,296 (1962)l. See Rudolf Vogel, "Die Heterogenen Gleichgewichte," 2nd ed. Akademische Verlagsgesellschaft, Geest und Portig, Leipzig, 1959. 1 3 0 N. M. Vittorf, Teoriya splavov, SPB 1909. 1 3 ' 1. F. Shreder, Corn. Zh. 11,272 (1x90). 1 3 2 R. Becker, Proc. Phys. SOC.(London)52, 71 (1949). 1 3 3 U. Dehlinger, "Chemische Physik der Metalle und Legierungen;. Akademische Verlagsgesellschaft m.b.h., Leipzig, 1944. 134 W. elsen, E. Schiirmann, and G . Heynert, Arch. Eisenhiittenw. 26, 19 (1955). 1 3 5 A. B. Mlodzeevskii, Teoriiafaz (sprimenenkm K tnerdym i zhidkim sostoianuam) Moskva. Obedeninnde nauchno-tekh. izd. vo, 1937. 13' B. Ya. Pines, Izc. Sektora Fiz. Khini. Analiza, Inst. Obshch. i. Neorgan. Khim. 16, 64 (1943); Zh. Eksperim. i. Teor. Fiz. 13,411 ( 1 943). 137 V. 1. Danilov and D. S. Kamenetskaya, Zh. Fiz. Khim. 22.69 (1948). C. Wagner, "Thermodynamics of Alloys," Addison-Wesley, Reading, Massachusetts, 1952. 139 J. Lumsden, "Thermodynamics of Alloys." Inst. of Metals, London, 1952. I4O L. S. Darken, R. W. Gurry, and M. B. Bever, "Physical Chemistry of Metals," McGraw-Hill, New York, 1953. 14' N. 1. Stepanov, U s p . Khim. 5 , No. 7-8 (1936:) 1 4 2 C . Wagner, Acta M e t . 6, 309 (1958). 1 4 3 L. J. Vieland, Acta M e t . 11, 137 (1963).

70

N. N. SIROTA

of mixing GUTg,xB-formation of ideal solutions-and of an additional term characterizing deviation from the ideal condition, i.e., GIT,,xB = x A ~ k , , , , ,

+ xBGL,,,,,

+ G:,xB + GETm,xB.

(2)

Under the assumption of ideal solubility of the liquid components in one another we have

G",g,xB = T, AS:

= RT,(xA

+

In XA

XB

In xB)

(24

or using the aforegiven, the free energy of formation of the compound from liquid components will be AGS,Tm3E

= Gi.T,,,

- xAG;I,A,T,,,

-

XBGY,,.T,,,,~ = Tm,c

AS:&

+ G7,,,,,cxB. (3)

The standard free energy of formation of the compound of pure solid components at Tm,cis determined by taking into account the entropies of fusion of the components and the differences between the melting points of the compound Tm,cand the components T m . A , Tm,B : AGg,C,Tm,c

= G;,C,Tm,c = Tm,cAs% ASPA

- xAGS,A,T,,,

- XBGS,B,T,,,

+ G : m , c ~ ~- xT,,,(Tm,c - xB(Tm,c

- Tm,B)

As&?

-

'm,A)

(4)

where ASPA,ASPB are entropies of fusion of the components A and B, respectively; and it is supposed that Tm,A < Tm,c> Tm,Bof A"'BV at XA

= XB =

4,

AS:&

=

0.7.

Within the limits of the theory of regular solutions, the value of the additional free energy G:B is equal to the heat of mixing. Wagner'42 has developed a method for the determination of the excess free energy GEE,using the phase diagram. According to his equation (61) for the case of the AB compound when XA = XB = f :

In the theory of regular solutions, the function Z,(x) containing entropy additionally to the ideal solutions may be neglected without any great error : Z2(x) = 0. In this case

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

71

where

A S is the entropy of fusion, and

is the temperature of the ideal liquidus curve for ideal solutions of the components in liquid state. The function cp is defined by cp = 1

R + -1 __In 2 AS:c

1 4 ~ , ( 1- xB)'

(7)

The shape of the liquidus curve for the case of the ideal solution may also be calculated by the formulas

T=

_____ 'm,c

1 - f(RTm,,/AHfc)In 4x(1 - x)'

and, for case of the regular solution,

=

1 - (K/4AHf,)(2x - 1)2 _____ . Tm*c 1 - (RTm,,/2AH,,) In 4x(1 - x) '

AHIE= Kx(1 - x).

(7~)

Schottky and Bever, * 25 using Wagner's equations mentioned above, have determined heats, entropies, and free energies of formation for the InSb and GaSb compounds from the data of the phase diagrams and the experimental values of entropies of fusion of the components. Their values are in good agreement with those obtained by direct calorimetric measurements. According to Schottky and Bever,"' the liquidus curves in the system Ga-Sb are close to the calculated ideal (curves,while in the system In-Sb the experimental liquidus curve is considerably lower than the ideal curve, which corresponds to the large negative values of GE. Using the concepts of the theory of regular solutions, they'25 assume that GE is wholly determined by the energy of mixing which may be represented, as usual, in the form GE = - xA).They found for the InSb system GE = (- 7.92 & 1.6)(1 - xSb)xSb kcal/mole ,

(8)

72

N . N. SIROTA

for the GaSb system

These authors125assume that, in the GaAs and AlSb systems, liquid phases are close to the ideal solutions. This follows from studies of phase diagrams and estimates concerning the entropies of fusion. In Fig. 16 the liquidus curves of a number of systems A"'-BV are compared with the liquidus curves calculated to the approximation of regular solutions, and with the ideal liquidus curves at GE = 0.

A

a

20

40

60 80 at %

loo

6

FIG.16. Ideal and experimental liquidus curves of A"*BVsystems.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

73

For ideal solutions the shape of liquidus curves is satisfactorily described by the equation of Shreder13' and vain Laar,'26 that is a quantitative expression of the Raoult law

This equation results also from the theory of regular solutions as xB --+ 0 and is essentially valid in the region of concentrations in which the liquidus curve, represented by the coordinates In x and l/T, is expressed by a straight line. Hall,77 analyzing his own experimlental data on the shape of the liquidus curves in some A"'BV systems at small1 concentrations (Fig. 17), showed that for the Ga-Sb and Ga-P systems in gallium solutions and for the In-Sb, In-As, In-P systems in indium solutions there are negative deviations from the ideal condition. The highest deviation from the ideal condition is observed in the In-As and In-Sb systems. This conclusion agrees with Fig. 16, where the liquidus curves of the A"'BV system were determined experimentally from the data of various authors and where the ideal liquidus curves were calculated by the present author.

FIG. 17. A"'BV system liquidus lines in dimensionless form (after - x)] versus relative temperature difference (T,, - T ) / T :

Variation of

log[4x( I

I

(a) *-GaSb A - G ~ A s -After V -Gap V-Gap-After Rubenstein."

A-GaAs

After Koster and Thoma.'"

(b) *-lnSb

A-InAs

I

-After

V-InSb V-InP-After

van den Boomgaard and Sch01.~~

-After Liu and Peretti." A-InAs The dashed lines correspond to an ideal solution with AS, = 14 cal/deg-mole.

74

N . N . SIROTA

According to Hall,77 the curves of solubility can be described by the two-term formulas of the theory of regular solutions GE = Wx(1 - x) - ln[4x(1

-

x)]

=

where T, is the melting point of the compound, ASf* is the apparent effective entropy of fusion, and W is an interaction parameter. (See also Ref. 143.) Hall has shown that the curves of solubility may be reproduced accurately if the values given in Table VI are a s ~ u r n e d . ' ~ TABLE VI

Compounds

Tm ( O K )

GaSb GaAs GaP InSb InAs InP

985

IS10 1743 79 8 121s 1323

AS,* WI4 T, (cal deg-' mole-')(caldeg-'mole-') 19.8 22.2 20.1 29.9 30.0 21.1

4.4 13.4 7.8 31.0 29.2

8.1

However the values of AS," given in Table VI differ considerably from the entropies of solution measured experimentally. For ideal solutions, when W = 0, the expression of In 4x(1 - x) versus T,/T - 1 corresponds to the dashed line in Fig. 17, the slope of which depends on ASf*/R. It should be noted that application of the theory of regular and ideal solutions to the melts of the A"'-BV systems is not quite substantiated, and the results of Table VI obtained by Hall should be considered as qualitative estimates rather than as quantitative data. Additional analyses of the phase diagrams of 111-V compounds are also given by Vieland.'43

N. Vapor Pressures 4.

VAPOR PRESSURES OF THE

COMPOUNDS AT

THEIR MELTING POINTS

For thermodynamic calculations and correct analysis of the phase diagrams it is very important to know the temperature dependence of the vapor pressure over the compound and the composition of the vapor and its equilibrium temperature change, especially in the region of the melting point. The data available are scanty and contradictory. Examples involving 111-V

2.

HEATS OF FORMATION A N D TEMPERATURES AND HEATS OF FUSION

75

compounds reveal the obvious necessity of considering the external pressure and the vapor pressure of the components when studying phase diagrams. There are sporadic data on the vapor pressure of nitrides of the group 111 elements. Hexagonal boron nitride, B N , begins to dissociate at a temperature of about 1000”C.35The vapor pressure over the nitride at 1220°C is 9 mm, at 2045°C it is 158 mm, and at 2500°C it is ‘760mm.29 A change in nitrogen pressure over the boron nitride is dexribed by the e q u a t i ~ n ’ ’ * ’ ~ ~ 6450 logpN2,mm = - _ _+ 4.0; T

logPN2,atm=

6450 + 1.12. (12) T

-__

Measurements of vapor pressure over the hexagonal BN have been carried out by Hildenbrand and Hall’“4 (Fig. 18). According to their results log PN2, atm

23530 T

= --__

+ 9.09.

(1 2 4

From this equation, P = 4.1 atm at 2500°C and P = 2 x atm at 2000”K . Aluminum nitride, AlN, begins to dissociate noticeably at 1750°C. At temperatures close to the melting point ( - 3 22OO0C),the equilibrium pressure is 4 atm, according to Renner.60The results of Hildenbrand and Hall144 are given in Fig. 18. According to their dlata 25900 log PN2,atm = - _ _ T

+ 8.92.

From this equation, P = 0.38 atm at T = 2500°C and P = 9.10-5 atm at 2000°K. Gallium nitride, GaN, begins to dissociate in vacuum at 1050°C according to Renner.60 Indium nitride, InN, begins to dissociate noticeably in vacuum at 620°C according to Renner.60 When boron phosphide, BP, is heated in1 vacuum, dissociation o c c ~ i r s , ~ ’ - ~ ~ ~ which is accompanied by formation of phosphorus hexaboride in the reaction 6 BP -+ B,P + 5 p,. log Pdis,atm 144

14’

= -

13700 + 7.22 T

D. L. Hildenbrand and W. F. Hall, J . Phys. Chmr. 67.888 (1963). F. V. Williams, in “Compound Semiconductors” (R. K. Willardson and H. L. Goering, eds), p. 171. Reinhold. New York, and Chapman & Hall, London. 1962.

76

N. N. SIROTA

Qatn

10”

a

10 -4

to-G

4.6

2

FIG.18. Temperature dependence of BN and AIN dissociation vapor pressures. (After Hildenbrand and

There are no data on dissociation pressure of aluminum phosphide, AIP. It is noted that in a vacuum the compound dissociates at temperatures above 1 0 0 0 ~146 ~. Gallium phosphide, Gap. According to Folberth’s data,70the phosphorus vapor pressure over G a P at 1350°C is higher than 13.5atm and perhaps attains 50 atm.69 It is supposed that vapor consists mainly of P,. Frosch and Derick” give the equilibrium vapor pressure over G a P as 20atm at 1500-1550” (in the range of which, in the authors’ opinion, there is a melting point of the compound). Marina et aL7, and Vigdorovich et al.” assumed the equilibrium pressure at the melting point of 1525°Cto be 13 atm. J o h n ~ t o n ’ ~studied ’ the change in vapor pressure over G a P in the range of temperatures from 1000 to 1300°K by the Langmuir method and by the Knudsen effusion method. Under the assumption that at these temperatures the gas phase consists of 99 % P, and only of 1 % P, he has found an expression 146

14’

John R. Van Wazer, “Phosphorus and Its Compounds,” Vol. 1. Wiley (Interscience), New York, 1958 [Societ Transl. : “Fosfor i ego soedineniya”. Moskva, 19621. W. D. Johnston, J . Electrochum. SOC. 110, I17 (1963).

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

77

for vapor pressure in the form log Pz, atm

=

18870 --__ + 10.72. T

Richman 7 3 has studied the temperature dependence of vapor pressure over G a P and over some other A"'BV compounds with the help of a Bourdon manometer made of quartz. Measurements were carried out up to 11 atm at 1455°C. The results given in Fig. 19 permit, in Richmond's opinion, extrapolation up to the melting point of 14167°Csuggested by him.73At this temperature his estimate for the extrapolated equilibrium pressure of phosphorus vapor over GaP is 35 f 10 atm, which approximately agrees with the estimation made by F ~ l b e r t h . ~Richman's ' extrapolation can lead to overestimated values of dissociation pressure of GaP near the melting point. et and of Golodushko On the basis of the data of R i ~ h m a nof, ~Marina ~ and Sirota'48 the total pressure in the interval of the linear change of log P versus 1/T is given by the equation

16600 + 9.83 ,r Vapor pressure over indium phosphide was studied by van den Boomgaard and S c h 0 1 , ~F~~ l b e r t hW , ~e~i ~ e r , Drowart '~~ and Goldfinger,' and others. Van den Boomgaard and S c h 0 1 ~constructed ~ a three-phase equilibrium line (P-T-X) in the system InP. By extrapolation up to the composition of the InP compound, the melting point was estimated to be 1062 k 7°C under an equilibrium pressure of phosphorus vapor of 60atm. This value of vapor pressure at the melting point of InP is undoubtedly overestimated-apparently because of inaccurate estimations of the vapor pressures over phosphorus at the experimental temperatures. Investigations were carried out by the following methods. An evacuated quartz vessel was placed in a twotemperature furnace. The indium was in the high-temperature portion while the phosphorus was in the low-temperature portion. The pressure in the vessel was determined by the temperature of the volatile component (phosphorus) with the help of standard tables. The violet modification of phosphorus was used. W e i ~ e r has ' ~ ~used the "dew point" method to determine the pressure of phosphorus vapors over InP. Assuming that the vapor consists mainly of P, molecules, he has described the temperature dependence of pressure by log P =

log PP2,mm= -__ 23000 T 14'

+ 21.0;

----

logcp2,atm= -__ 23000 + 18.12. (16) T

V. Z. Golodushko and N. N. Sirota in Sborn. "Khimicheskaya svyaz v Poluprovodnikakh i tverdykh telakh" (Chemical Bonds in Semiconductors and Solids), p. 125, Minsk, 1965.

78

N . N. SIROTA

FIG.19. Temperature dependence of GaP dissociation vapor pressure. x -After Ri~hrnan.'~ 0-After Marina rt and Vigdorovich er A-After Golodushko and Sirota.'48

At the melting point of InP, accepted to be 1O7O0C, the vapor pressure, is 10.5 atm. according to Wei~er,',~ According to Drowart and Goldfinger's data, partial pressures of phosphorus P, and P, over the InP are given by the equations logPp4,atm

=

21300 -__ + 16.84; T

logPp,,atm

=

14850

-__ + 10.32. (17) T

The ratio of partial pressures of four- and two-atom molecules of phosphorus is thus given by PP log?=

PP,

6450 T

+ 6.52.

-__

(18)

According to this equation, near the melting point the equilibrium vapor over InP consists approximately of 97.75% of four-atom molecules of P, and 2.25% of two-atom molecules of P,. However, these results are only 149

K. Weiser, J . Phys. Chem. 61,513 (1957). J. Drowart and P. Goldfinger. J. Chim.Phvs. 55. 721 (1958).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

79

tentative. The pressure of phosphorus P, vapor calculated by Renner' according to thermochemical data is expressed by the equation

51

22600 logP,,,atm = --__ T

+ 21.94,

which, in Renner's opinion, gives overestimated values. According to the simplified equations, the pressure of phiosphorus vapor over InP at the melting point is 11 atm. However, as Richman has shown, direct extrapolation to the melting point without taking into account possible changes in the shape of the curve near the melting point does not give very accurate results. The total pressure of P, and P, according to Wei~er,',~Drowart and Goldfinger, 50 and van den Boomgaard and Scho14' in the range of temperatures of the linear change of log P versus 1/T is given by the equation

'

log(P,,

+ Pp,),atm = --17780 + 13.72. T

In Richman's the total pressure near the melting point was measured from 1026 to 1056°C. At the latter temperature, it was 10atm. To determine the equilibrium pressure at the melting point of 1058 & 3"C, he resorts to extrapolation and obtains the total pressure of phosphorus vapor over InP to be 21 f 5 atm. It should be noted, however, that as a rule quartz ampules are not destroyed when indium phosphide is being melted, that is the total pressure over the InP apparently does not exceed 5-12 atm. In Fig. 20, experimental data are given on the change of log P versus the inverse temperature ( Boron arsenide, BAS, with the sphalerite structure dissociates at 10001100°C under arsenic pressures of less than 1 atm to form the orthorhombic phase B,, There are only meager data on the vapor pressure of aluminum arsenide, AIAs. Hoch and Hinge151ahave studied the dissociation pressure of AlAs by the Knudsen method in vacuum up to 1500°K. Disks of AlAs, 6.25 mm in diameter, were heated in graphite Knudsen cells. Their results at 1455°K are 1/T).493733'4',150

PAsz= 41 x lo-,

atm,

PA,= 34.7 x lo-'

atm.

(20)

It was assumed that in the range from 1500 to 1600"K, As, is the main component of the vapor over AlAs. Vapor pressure of gallium arsenide, GaAs, was studied by van den Boomgaard and S c h 0 1 , ~Drowart ~ and G~ldfinger,"~F ~ l b e r t h Rubinshteiin ,~~ T. Renner, Solid State Electron. 1, 39 (1960). 15''M. Hoch and K. S. Hinge, J Chrm. Phvs. 35,451 (1961)

80

N. N . SIROTA

FIG.20. Temperature dependence of InP dissociation vapor pressure. x -After R i ~ h m a n . ' ~ A-After van den Boomgaard and S c h 0 1 . ~0-After ~ W e i ~ e r . '8-After ~~ Drowart and Goldfinger.'"

and K ~ z l o v s k a y a , 'and ~ ~ others. Data from mass-spectrometric analysis' 5 0 have been used to develop equations for the temperature dependence of the partial pressures of four- and two-atom molecules of arsenic vapor in the range from 950 to 1200°K :

19320 atm = -__ log PAs4, T

+ 11.43;

logP,,,,atm

17340 T

= -__

+ 9.86. (21)

According to these equations, at 1518°K near the melting point, PASS = 0.048 atm, PAs2= 0.028 atm. The total pressure is P = 0.076 atm. The ratio of partial pressures of four- and two-atom molecules is given by the equation 1980 log-PAS4 = - _ _ 1.54, T PAS2

+

R. N. Ruhinshteiin and V. M. Kozlovskaya, Voprosy radiorfektroniki, ser. l l . ,vyp. 1.94(1959).

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

81

according to which, at 1510°K in the equilibrium gas phase over the GaAs, there are 63 % of four-atom and 37 % of two-atom molecules of arsenic. According to Richman’s data,73at a melting point of 1238°C (Ref. 88), the equilibrium pressure of arsenic over GaAs is 1 & 0.02atm. He thinks the experimental value of 740 Torr is somewhat underestimated. At low temperatures, Richman’s data are in good agreement with the results of Drowart However, near and Goldfinger,’” as well as those of Lyons and Silve~tri.’’~ the melting point of GaAs (Fig. 21), a change in the shape of the curve of log of vapor pressure versus inverse temperature was found. Figure 21 gives a summary of results taken from the works ofvarious investigator^^^^^^^'^^^^'^ as well as those of Golodushko and S i r ~ t a , ’and ~ ~ Rubinshteiin and Kozl o v ~ k a y a . ”The ~ total pressure PI and PI!,according to the data of Fig. 21, in the range of temperatures of the linear change of log P versus 1/T is given by the equation 19230 logP, atm = 11.81. T

+

Vapor pressure over indium arsenide, InAs, versus temperature in the range from 911 to 1159°K is defined by equations obtained on the basis of the mass spectrometric data of Goldfinger and J e ~ n e h o m r n e ”:~ 22725 18840 togP,,,,atm = -__ 12.12. log PAs4, atm = -__ f 15.57; T T (24) According to these data, near the melting point of indium arsenide at 1216°K the partial pressures of the two- and four-atom molecules of arsenic and PAs4 7.76 x lop4 atm. Thus the total presare PAsl 4.26 x sure is P 12.0 x 10-4atm. The ratio of partial pressures as a function of temperature is described by the equation

+

--

-

(25) According to this equation, near the melting point of 1218°K the vapor phase consists of 64.6% four-atom molecules and 35.4% two-atom molecules of arsenic. Some different values of pressures at tlhe melting point are given by other authors. According to van den Boomgaard and S c h 0 1 , ~the ~ total pressure at the melting point is equal to 0.33 atm; according to Folberth7’ it is approximately 0.25 atm. 15’

V. J. Lyons and V. J. Silvestri, J . Phys. Chem. 65, 1275 (1961). P. Goldfinger and M. Jeunehomme, in “Advan’cesin Mass Spectrometry” (J. D. Waldron, ed.) p. 534. Pergamon, London, 1959.

82

N. N. SIROTA

r ici. L I . I emperarure oepenoence 01 clans oissuciarion vapor pressure. x -niter nicnman. A-After van den Boomgaard and Sch01.~' I-After Lyons and S i l v e ~ t r i . ' 8-After ~~ Drowart and G ~ l d f i n g e r . ' ~ ~

-

On the basis of the preceding data it is possible to give tentatively the pressure of arsenic vapor over InAs near the melting point as being in the range from 2 x to 0.2-0.3atm. According to the data of van den Boomgaard and S c h 0 1 , ~Goldfinger ~ and J e ~ n e h o m m e , and ' ~ ~ Golodushko and S i r ~ t a , the ' ~ ~equation for the total pressure in the linear range of inverse temperature as a function of the logarithm of pressure is of the form log P , atm

= -___ 28800

T

+ 22.30.

(26)

A plot of the relationship is given in Fig. 22, using the data of various authors.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

83

P,atm

\0-'

10-2

to"

-5

1U6

10'~

FIG.22. Temperature dependence of I d s dissociation vapor pressure. A-After van den Boomgaard and SchoL4' 8-After Goldfinger and J e ~ n e h o r n r n e . ' 0-After ~~ Golodushko and Sirota.148

Indium antimonide, InSb. Figure 23 gives the vapor pressures of InSb in the range from 45&520"C, which were determined from the mass spectrometric data of K o z l ~ v s k a y a . 'According ~~ to these data, the temperature dependence of the vapor pressure of Sb above InSb is described by the equation log P , atm

+ 23.559

= -__20519

T

(27)

These data are to be preferred to the results given by Nesmeyanov et 155 156

V. M. Kozlovskaya, Voprosy radioelektroniki, Seir. 11, V. 3, str. (1961). An. N. Nesmeyanov, B. Z. Iofa, and A. S. Polyakov, Zh. Neorgan. Khim. 5,246(1960) [English Transl. : Russian J . Inorganic Chem. 5, 119 (1960)l.

84

N . N. SIROTA

FIG.23. Temperature dependence of lnSb dissociation vapor pressure. (After Kozlovskaya.155)

For all A"'BV compounds near the melting point one observes a deviation of the logarithm of the dissociation vapor pressure as a function of the inverse temperature from the linear relation. The dissociation vapor pressure rises sharply at T -+ T,. R i ~ h m a n 'has ~ extrapolated the value of the dissociation vapor pressure up to the melting point using the following relations. At the temperature 17; with dissociation of A"'BV, the pressure of a four-atom vapor of BVelement is in equilibrium with a liquid phase of xB composition, so that = PXixxy. Therefore 1 dInP,

4 dl/T

-

1 dln Po,4

4 d1JT

1 dx +--+xdl/T

dlny

dl/T'

(28)

where Po,, is the pressure over liquid melt of pure Bv element and y is the activity coefficient of this element in the melt. It follows from these relations that at the melting point the slope of log P vs 1/T tends to infinity. (Also see Refs. 72 and 143.) In Table VII are given coefficients A and B from the equation for the temperature dependence of vapor pressure, log P = A/T + B, by different authors; also values of the vapor pressure at the melting points, according to that formula.

N TABLE VII

VALUESOF CONSTANTS A AND B

BN

BP

I N THE

EXPRESSION log Pa'= - A / T -k B

FOR DISSOCIATION V A P O R PRESSURE OF AND CALCULATED PRESSURES AT T,

AIP

AIN

AlAs

AlSb

GaN

GaP

GaAs

GaSb

InN

COMPOUNDS A"'BV,

InP

InAs

lnSb

560-680" 670-1060'

638-886'

45&520"

14,850"

18,840*

-

Temperature range. "C

7W1000' 650-900"

16% IYW

1220-2550y~h 1500-1700'

A,'

645Wh 23,530'

B,

1.12'J 9.09'

13.7OWj 18,870'

25.900'

17.34P

5,

Z 7.92"

10.72k

8.92'

9.86"

10.32"

U

12.12'

23,0042'

19.320"

21,300" 22,600"

i 1.43"

52

A

16,600

B

9.83

Linear extrapolation of P,ot,lat T,

4.22

Experimental determination of pressure at 7 . log PB, =

- A , / T + B, :

'

11.81 0.138

206

0.76"

13'.*

I'

35'

log P., = AJT

+ B,:

From Samsonov er 'From van den Boomgaard and S ~ h o l . ~ ~ 'From Williams and Ruehrwehs' From Frosch and Derick." 'From Richman." From Marina er a

19,230

log P,.,,, = - A l l

From 'From ' From From li From

+ B.

Vigdorovich PI n1." Campbell er ul."' Hildenbrand and Hall.'44

!8.!2' 16.84' 21.94" 17,780

20.5 I9a 22,725' 15.57'

23.559'

28,800

13.72

22.30

5,

2.39

0.04

u

60b

10.5' 11" 21*

z

0.0012 0.33h 0.25b

51

' From Weiser.'"' From Drowart and G ~ l d f i n g e r . " ~ " F r o m Renner.'" From Goldfinger and J e ~ n e h o m m e . " ~ From Kozlovskaya."'

p

Johnston."'

00

cn

86

N . N. SIROTA

V. Heats, Free Energies, and Entropies of Formation 5. HEATS, FREEENERGIES,AND ENTROPIESOF FORMATION OF A1”BV COMPOUNDS

Reviews of thermodynamic properties of the most important III-V compounds were made by Bever,16’ Brewer,28 W e i ~ e r , ’ ~ G ~ l d f i n g e r , ’and ~ ~ other^.^',^^ We shall summarize the basic thermodynamic properties of the A“’BVcompounds, using mainly recent works and partially those covered by the reviews mentioned. Until 1958 there was essentially no thermodynamic data on the III-V compounds. Note, for example, in the well-known monographs of Kubaschewski and Evans.20 Kubaschewski and Catterall,’58 C ~ t t r e l l ,only ~~ limited data are given on heats of formation of aluminum, gallium, and indium nitrides and indium antimonide. a. Boron Nitride, B N

Standard heats of formation of boron nitride, BN, out of crystallic boron and gaseous nitrogen N, were determined by direct measurements in a calorimetric bomb,‘ 5y,28 AH”,,, = -60.7 ? 0.7 kcal/mole. as well as by nitriding boron’”

AH”,,,

= - 60.3

kcal/mole .

On the basis of effusion measurements of vapor pressures above hexagonal boron nitride,’44 a value of

AH”,,, = -59.8

0.6 kcal/mole

was obtained. The problem of the standard heat of boron nitride formation should not be considered to be settled. It is possible that its true value exceeds considerably the reported data. b. Aluminum Nitride, A1N

Among the original investigations of the aluminum nitrides are the works of Neumann et a1.16’ and Roth.’62 (See also Refs. 20 and 163.)The reviews of P. Goldfinger, in “Compound Semiconductors” (R. K. Willardson and H. L. Goering. eds.), p. 483. Reinhold, New York, and Chapman & Hall, London, 1962. 0. Kubaschewski and J. A. Catterall, “Thermochemical Data of Alloys.” Pergamon. London, 1956. A. S. Dworkin, D. J. Sasmar,and E. R. Van Artsdalen, J. Chem. Phys. 22,837 (1954). I b 0 G. L. Gal’chenko. A. N. Kornilov, and S. M. Skuratov, Zh. Neorgun. Khim. 5, 2651 (1960) [English Transl. : Russian J. Inorganic Chem. 5, 1282 (1960)l. 1 6 ’ B. Neumann, C . Kroger, and H. Haebler, 2. Anorg. Alleg. Chern. 204.81 (1932). 1 6 2 W. A. Roth, 2. Elektrochem. 48, 267 (1942). 1 b 3 F. Weibke and 0. Kubaschewski, “Thermochemie der Legierungen.” Springer, Berlin, 1943. 57

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

87

K e l I e ~ , Brewer,” ’~~ and Bever’6s are well known. For the standard heat of formation, K e l l e ~ gives ’ ~ ~ the value of

AH2,, = -57.4

kcal/mole,

and Brewer,, gives

AH,,, = - 64 kcal/mole, while Neugebauer and Margrave’66 have obtained a heat of formation of

AH,,* = -76.47 & 0.2 kcal/mole by nitration of aluminum in a calorimetric bomb. In mass spectrometric studies involving evaporation of aluminum nitride, Schissel and Williams 167 determined the standard heat of aluminum nitride formation

AH,,,

=

- 63 kcal/mole.

A detailed investigation into thermodynamic properties, entropies, and free energies of aluminum nitride formation at various temperatures was ~ amount of earlier data was obtained carried out by Mah et ~ 1 . A’ ~limited by sat^.'^^ Heats of formation were obtained by burning aluminum nitride with an admixture of paraffin wax in a calorimetric bomb. The heat of the reaction AlN,,)

+ 3 02(g) = f A1,03 + f N,

(29)

was found to be

AH,,,.l5

= - 124.62 k 0.37

kcal/mole

Using the earlier value17’ of the heat of formation of A1,0,,

AH,,, = - 100.4 1 0.3 kcal/mole, K. K. Kelley, “Contributions to the Data on Theoretical Metallurgy-XIII. High-Temperature Heat-Content, Heat-Capacity, and Entropy Data for the Elements and Inorganic Compounds,” Bull. 584, Bur. of Mines, p. 1 1 . U.S. Gov. Printing Office, Washington, 1960. 165 M. B. Bever in “Compound Semiconductors” (R. K. Willardson and H. L. Goering, eds.), p. 498. Reinhold, New York, and Chapman & Hall, London, 1962. C. A. Neugebauer and J. L. Margrave, Z . Anorg. Alleg. Chem. 290,82 (1957). 16’ P. 0. Schissel and W. S. Williams, Bull. Am. Phys. Soc. 4,139 (1959). 1 6 * A. D. Mah, E. G. King, W. W. Weller, and A. U. Christensen, Bur. Mines, Rep. Investigations 5716 (1961). 169 S. Sato, Sci. Papers Inst. Phys. Chem. Res. Tokyo 29.19 (1936). A. D. Mah, J . Phys. Chem. 61, 1572(1957). 164

88

N. N. SIROTA

one obtains for the reaction of aluminum nitride formation

Al,,,

+

N2(g)= AlN,,

A H z g 8 = - 75.6

t- 0.4

kcal/mole.

(30)

In the same work,'68 the heat capacity of aluminum nitride was found in the temperature range from 51 to 298.15"K and the enthalpy of AlN was obtained in the temperature range from 298 to 1800°K. At low temperatures, changes of heat capacity with the temperature follow the Debye law

c, = 1.66 x

10-6773.

(31)

Over the whole temperature range from 51 to 298.15"K, heat capacity is fairly well described by the formula containing two terms-the Debye and Einstein terms

c, = D(?)+

.(y)

The standard entropy found from these data is s;98

= 4.80

k 0.02 e.u./mole.

Using the tabulated entropy and enthalpy values of the component^'^' and experimental data,16' heats and free energies of aluminum nitride formation were found in the temperature range up to 2000°K. The following standard values are obtained :

AH",,, = -75.6 k 0.4 AGig8 = -68.15 +_ 0.4 s i 9 8 = 4.80 _+ 0.02 AS = - 11.1 1.4

kcal/mole, kcal/mole, e.u./mole, e.u./mole.

Hildenbrand and Hall'44 have found the standard heat of formation AH",8

=

-76.1 f 2.1 kcallmole

from the determination of AlN vapor dissociation pressure. According to the above values, standard heats, free energies, and entropies of aluminum nitride atomization, with account for the data for the components given by Anosov and P ~ g o d i nare ,~~ as follows: AH;t98 = -209.7 & 2 kcal/mole, AG;t9,, = -190 & 2

kcal/mole,

AS",b,

e.u./mole .

=

K. K. Kelley, Ref. 164, pp. 10, 132.

-64

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

89

c. Aluminum Arsenide, AlAs

Hoch and Hinge’”” have found the heat of aluminum arsenide formation from the temperature relation of vapor pressures by the effusion method. Knudsen cells with apertures of various diameters were made of graphite. Evaporation took place in vacuum at 1455 and 1457°K. The dissociation heats found by the method of “the third law” and by the slope of the curve of the logarithm of dissociation pressure versus the inverse temperature were AHT = - 136.9

t 1.4

A H T = -130

kcal/mole,

kcal/mole

and

respectively. Assuming the evaporation heats of aluminum54

AH,,, = - 77.5 kcal/mole and for

As,

AHT

= - 24

kcal/mole,

the authors have Ldundthe standard heat of aluminum arseniL2 formation to be AH;9s

= - 35.4 jI3.1

kcal/mole.

Using this value and adding the sum of heats of arsenic and aluminum atomi z a t i ~ n , ’ ~the , ’ ~heat ~ of aluminum arsenide atomization may be taken as = - 182 -1- 3.5

kcal/mole.

d . Aluminum Antimonide, AlSb Kubaschewski and Evans” give the heat of aluminum antimonide formation as AH,,s

= - 23.0

2.5 kcal/mole.

Attempts made by Schottky and B e ~ e r ’ , to ~ find heats of aluminum antimonide formation by tin solution calorimetry failed because aluminum antimonide could not be dissolved in liquid tin in reasonable time at temperatures up to 380°C. Piesbergen’” has found the standard entropy of AlSb from calorimetry data to be S;98 = 15.36 0.1 e.u./mole. Renner15’ used the approximate I

’’ U. Piesbergen, Z . Narur/orsch. 18a, I41 ( I963 I; Semiconductors and Semimetals 2.49 ( 1 966).

90

N. N . SIROTA

formula by Eastmann S = R In

“4”’ 1 V

-

+ 12.5

e.u./mole

(33)

( A is the mean atomic weight, V is the mean atomic volume, and T, is the melting temperature) to temperature determine the standard entropy of aluminum antimonide S”,,, = 14.1 e.u./mole.

In view of the above data, we may assume at present the following values for heats, free energies, and entropies of formation : AH”,,,

= - 25

AGi98

=

3

kcal/mole ,

- 24.4 & 3 kcal/mole ,

= -2 & 0.5

e.u./mole.

The heats, free energies, and entropies of atomization from the data obtained by various authors are given in Table VIII. Some of these results are unpublished.’ 7 3 TABLE VIII HEATS,FREEENERGIES. AND ENTROPIES OF ATOMIZATION OF AlSb

- 150.9

- 68.3

- 130.5

173 14.1 15.36

- 165.2

- 145

- 67.8

151

172 (Preferred values)

e. Gallium Nitride, GaN

Hahn and J ~ z a ”have ~ .reported values for the standard heat of gallium nitride GaN formation of AH,,, = -24.9

0.9 kcal/mole,

which were obtained by combustion in oxygen in a calorimetric bomb. The heat of atomization was found to be N 173 174

- 203

kcal/mole .

Unpublished data of Folberth, cited by Renner.’” H. Hahn and R. Juza, Z . Anorg. Alleg. Chern. 244,I l l (1940).

2.

91

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

f. Gallium Phosphide, GaP Marina et aL7, and Vigdorovich et give the value for the standard heat of gallium phosphide GaP formation from the reaction Ga(,) + P4(,) = Gap(,, as AH”,,, = -17.2kcal/mole; the entropy of formation as AS;98 = - 12.0 e.u./mole; and the free energy as AG2,,, = - 14.5 kcal/mole. By burning in oxygen in a calorimetriic bomb, a heat of formation of AH,,,, = 29 kcal/mole was found.’75 On the basis of the available data, standard heats, free energies, and entropies of gallium phosphide atomization may be determined as follows :

AH;;,, = - 162

kcal/mole,

AG;;8 = - 124.3 kcal/mole, AS;t,,,

= - 126.5

e.u./mole.

However, these numbers must be improved. It is possible that the heat of formation determined by combustion data is closer to the true value. In that case, the heat of atomization is

AH;t,8 = - 171 kcal/mole. g. Gallium Arsenide, GaAs

From mass spectrometry data, Drowart and Goldfinger’ have found the heat, free energy, and entropy of gallium arsenide formation for the reactions: Ga,

+ i As,(,,

Ga,

i As,(,,

F? GaAs,

AH = -21.8

kcal/mole at

1080°K

* GaAs,

AH

kcal/mole at

1080°K. (33b)

=

-39.4

In this case, AS”,,, = - 38.0 e.u./mole at the total pressure of As, atm. The standard formation heats at 298°C are AH;98 = - 12.3 kcal/mole

and

AH”,,, = -25.8

+ As,

(33a)

=

1

kcal/mole,

respectively.

G ~ t b i e r ’has ~ ~used mass Spectrometry for the analysis of the vapor composition over gallium arsenide in the temperature range 1040-1200°C and confirmed the assumption that molecules As, and As, alone evaporated. He found the heats of vaporization of gallium arsenide in the above temperature range were

AH;,,,4 = 90 k 3 kcal/mole

’’’

E. N. Ermolenko and N. N. Sirota in Sborn. “Khimicheskaya svyaz v poluprovodnikakh i tverdykh telakh” (Chemical Bonds in Semicondutztors and Solids), p. 128, Minsk, 1965. H. B. Gutbier, Z . Naturforsch. 16a, 268 (1961).

92

N. N. SIROTA

and AH;,AS2= 80 k 3 kcal/mole. The values of standard heats of formation of

(34)

AH;98 = - 18.0 _+ 1.0 kcal/mole

were calculated (from the data for As,). This figure should not however be considered reliable” since the ratio As,/As, changed within the experiment and thus there was no adequate equilibrium. Weiser7, refers to unpublished data of Johnson who used “the dew-point method” for determination of heat of gallium arsenide formation over the temperature range 1000-1280°K for the reaction GaAs(,, = Ga(,, + As,(,,,

4

AH,

= - 35

kcal/mole.

(35)

Rubinshteiin and K o z l o v ~ k a y a ‘have ~ ~ found sublimation heats of gallium arsenide by mass spectrometry. In the temperature range 100& 1300°K the values AH,, = -84.2

4 kcal/mole,

AH,,, = -88.5

4 kcal/mole

are obtained by the slope of log I*T versus l/T. A b b a ~ o v ’found ~ ~ thermodynamic functions of gallium arsenide by the method of electromotive forces in the cell (-)Ga(,J(KCI,NaC1) + GaCI,J(GaAs

+ As)‘+).

(36)

The electrolyte was at a melting point of 206°C. The experiments were carried out in the range 497463°C. The relationships involving the electromotive force are given b y the equation E

=

(301.6 - 0.1639T) millivolts.

(37)

0.6 kcal/mole,

(38a)

Hence, AGT = - 2 F E = - 14.2

= -20.8

177

I 1.8 kcal/mole,

A. S. Abbasov, Autoreferat dissertatsii, Moscow, 1964.

(38b)

2.

HEATS OF FORMATION AND TEMPERATURES A N D HEATS OF FUSION

93

In the calculation, the charge of a gallium ion in a chloride melt was assumed 2 = 3.’’,, With the use of reported data on heat capacities of compounds

and parent elements and fusion heats of gallium, standard values of AH”,,,

= - 19.4 k

1.8 kcal/mole,

AG;,,, = - 17.4 k 0.6 kcal/mole , AS;,,,

S;,8

=

-6.0 5 1.8

e.u./mole,

=

11.4

e.u./mole

1.8,

were calculated.’ ” Piesbergen has found that S;,,, = 15.34 k 0.1 e.u./mole.

Sirota and Yu~hkevich”~ have determined thermodynamic properties of gallium arsenide from the measured elelztromotive force of the cell (-)Ga(,J(KCl,LiCl)

+ GaC:l,I(GaAs + As)“).

(39)

In the temperature range 683-743°K the following values were obtained :

AH,

=

-22.12

kcal/mole,

ACT = - 13.04 kcal/mole, AS,

=

-13.37

e.u./mole;

i.e., the standard values at 298°C are AH;,,, = -20.96

AG;,,,

=

:k 1 kcal/mole3

-18.18 :_t 1 kcal/mole,

AS”,,, = -9.32

:? 2 e.u./mole.

These data are in agreement with the values of heats of gallium arsenide formation found by the method of burning in a calorimetric bomb. At present, standard thermodynamic quantities for gallium arsenide formation lie in the following ranges : AH;,,, = -20

2.0

kcal/mole,

AG”,,, = - 18.5 5 2.0 kcal/mole,

AS”,,, = -5.5 & 1.0

e.u./mole,

S”,,, = 15.0 f 6.0

e.u./mole.

H. A. Laitinen, C. V. Liu,and W. S. Ferguson, Anal. Chem. 30, 1266(1958). N. N. Sirota and N. N . Yushkevich in Sborn. “Khimicheskaya svyaz v. poluprovodnikakh i tverdykh telakh,“ (Chemical Bonds in Semiconductors and Solids), p. 122, Minsk, 1965.

94

N . N . SIROTA

By proceeding from these values, heats, free energies, and entropies of gallium arsenide atomization can be found. The results are given in Table IX. TABLE IX HEATS,FREEENERGIES, A N D ENTROPIES OF ATOMIZATION OF GaAs ~

AH&

(kcal/mole)

~~

ASXS

AG'98

(kcal/mole)

(e.u./mole)

- 159.4 - 140.9

- 133.8 - 122.4

- 62.1

- 146

- 131

- 50.5

~

%98

Reference

(e.u./mole) 11.4 15.34

177 173 172

(Preferred values)

h. Gallium Antimonide, GaSb Schottky and B e ~ e r 'have ~ ~ obtained the heat of gallium antimonide formation by tin solution calorimetry. The tin was maintained at temperatures in the range 24&355"C, and the gallium antimonide or antimony and gallium, which were added, had a temperature of 0°C. They obtained a value of A H , , , = -9.94 & 0.44 kcal/mole. This value was adopted by these authors'25 as a standard heat of formation at 298°K. From the phase diagram and with the aid of equations derived by Wagner,'42 a free energy of gallium antimonide formation of AGT,s = - 1.81 f 0.2 kcal/mole was found at the melting point for the reaction Ga(,, + Sb,,, e GaSb. This allowed the calculation of gallium antimonide formation at 298"C, AH;98

=

-9.94

kcal/mole.

Goldfinger and J e u n e h ~ m m e have ' ~ ~ obtained the heat, free energy, and entropy of gallium antimonide formation at 900°K from mass spectrometry data. For the reaction GaSb = Ga,,, + Sb,(,,, they obtain :

+

AH

= - 41.0

kcal/mole ,

AG

= - 13.4

kcal/mole,

AS

=

-

30.7 e.u./mole ;

(40)

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

95

a

for the reaction GaSb FI Gael, + Sb,,,,, the results are :

AH

=

-25.5

kcal/mole,

AG

=

-9.6

kcal/mole,

(41)

AS = - 17.7 e.u./mole . For the standard heat of formation at 298"K, these authors' 5 4 give mean values of

AH298= - 10 kcal/mole. From the data on evaporation of diatomic molecules Sb,, AH,,,

= - 13.5

kcal/mole,

bG29, = - 12.3 k ~ a l / m o l e . ' ~ '

have found thermodynamic properties of gallium Abbasov et al.' antimonide over the temperature range 360-560°C by measuring electromotive forces of the cell (-)Ga(,,l(KCl,LiCI)

+ GaCl,I(GaSb + Sb)")

(42)

for the reaction

Ga(l,+ Sb,,, = GaSb,,,.

(43)

On the basis of experimental data, the equation for temperature variation of the electromotive force of a cell E = (161.1 - 0.095T) millivolts

(44)

was obtained. Hence

z)

(

AH633-8330K = -ZF E - T -

i?E AS633-8330K= - Z F -iiT

=

= -11.2

2 1.0 kcal/mole. (44a)

-6.6 5 1 e.u./mole.

(444

A. S. Abbasov, A. V. Nikol'skaya, Ya. 1. Gerasimov, and V. P. Vasil'ev, Dokl. Akad. Nauk S S S R 156, 1399 (1964) [English Transl.: Proc. Acad. Sci. USSR, Phys. Chem. Sect. 156, 638 (1964).]

96

N. N. SIROTA

Using the reported data on heat capacities of the constituent elements as well as the c o m p ~ u n d , ’ ~ these , ’ ~ ~ authorsL8’ have found the following standard values at 298°K : AH”,,, = -9.4 k 1 kcal/mole,

AG”,,,

=

- 9.0

0.6 kcalimole ,

AS”,,, = - 1.4 f 1.4 e.u./mole. Ermolenko and Sirota17’ found the heats of formation to be

AH”,,,

2 kcalimole (45) by combustion in a calorimetric bomb. By measurement of the electromotive force of an electrolytic cell such as that depicted in (42) above, the values A H ; = - 12.02 kcal/mole, AG; = - 6.52 kcalimole,

AS;

= - 11

= - 7.37 e.u./mole

(46) were obtained by Sirota and Y~shkevich.”~ From these data, the standard values at 298°K were calculated : AH,,, = -9.8 +_ 1 kcal/mole,

AG”,,,

=

-9.4 f 1

kcal/mole,

1 e.u./mole.

AS”,,, = -1.34

From heat capacity rneasurements,l7, the value of standard entropy of GaSb has been determined to be

S;,,, = 18.18

0.1 e.u./mole

The presently available values of standard thermodynamic properties of gallium antimonide are within the following ranges :

AH”,,,

= - 10.5 5 1.5 kcalimole.

AG”,,, = -9.0 f 1

kcal/mole:

AS”,,, = - 1

e.u./mole,

Si9,,= 18

0.7 0.2

e.u./mole .

The appropriate values of GaSb atomization energies are found by using the above thermodynamic properties and the tabulated values of heats and free energies of component atomization.’1 Results are given in Table X. ’*I

N. M. Kochetkova and T. N. Rezukhina, Voprosy metallurgii i fiziki poluprovodnikov; poluprovodnikovye soedineniya i tverdye splavy. Trudy 4-go soveshchaniya pa poluprovodnikovym materialam, 1960. Moskva, Izd-vo Akademii Nauk SSSR, 1961, p. 34 [English Transl. : “Proc. 4th All-Union Conference on Semiconductor Materials” (N. Kh. Abrikosov, ed.), p. 26. Consultants Bureau, New York, 19631.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

97

TABLE X

HEATS,FREEENERGIES, AH";,, (kcal/mole) ~~~~

~

- 137 - 138.4 - 138

AND ENTROPIES OF

AS%* (e.u./mole)

AG%J8

(kcal/mole)

ATOMIZATION OF GaSb

S",8 (e.u./mole)

Reference

18.18

172 180 173 (Preferred values)

~

- 64.2

-118 - 118.7 - 118.0

- 66.1 - 64

i. Indium Nitride, InN

According to Hahn and J ~ z a ' ' the ~ heat of indium nitride formation found by combustion is

AH;98 = -4.6 5 0.5 kcal/mole. The heat of atomization in this case was

AHy9,, = - 175 kcal/mole . These values however may be underestimated. j . Indium Phosphide, InP

The heat of formation of indium phosphide has been determined by Gadzhiev'82 by the method of thermal decomposition of indium phosphide in a calorimetric bomb.183 A weighted amount of indium phosphide was placed in a quartz tube which had a resistance heating winding on it. In the calorimetric bomb some external pressure was produced opposing the pressure developing inside the tube. The tube was heated for 1OOsec. Sixty seconds after initiating the experiment, the temperature in the test tube had increased to 1100°C. At this temperature, the indium phosphide is partially dissociated. The degree of dissociation was determined from the weight loss, which was 35 %. During cooling, a reduced pressure was present in the tube and the calorimetric bomb. This decreased the heat conduction and prolonged the experiment. The accuracy of determination therefore proved to be unsatisfactory. According to this e v a l ~ a t i o n , ' ~the ~ ' 'standard ~~ heat of formation of indium phosphide is

AH",,, 18*

=

-21

2 kcal/mole.

S. N. Gadzhiev,Autoreferat dissertatsii, ruk. K. A. Sharifov. K. A. Sharifov and S. N. Gadzhiev, Zh. Fir. Khim. 38,2070 (1964) [English Transl. : Russian J . Phys. Chem. 38,1122 (196411.

98

N. N. SIROTA

Using the experimental data on the determination of indium phosphide vapor pressure obtained by the dew-point method, W e i ~ e r ’ ,has ~ calculated the heat of formation of InP from liquid indium and gaseous phosphorous P4 at 1273°K to be AH,,,, = -22 kcal/mole. According to Weiser, the heat of indium phosphide atomization can be determined from the sum of indium formation and evaporation heats (56 kcal/mole) and 3 the heat of molecular dissociation P, of 288 kcal/mole). The heat of indium phosphide atomization is thus estimated by him as

(4

AH;;,,

‘v

- 150 kcal/mole.

(47)

According to the data of mass spectrometric measurements by Drowart and G ~ l d f i n g e r , ’ ~ ’ ,the ’ ~ ~heat of formation of indium phosphide at 1000°K is the same for the reactions

+ & P4(2)T’, InP,,,, s InP,,,, In(l)+ 3 PZf2)

AH,

=

AHT

= - 39.4

-25.4

kcal/mole

(47a)

kcal/mole.

(47b)

Hence, using the tabulated data for heat capacities and enthalpies of the components, one obtains, respectively,

AH,,,,

=

-21.6

kcal/mole

and

AHz9,,= -22.1

kcal/mole;

wherei5‘

AS”,,,

= - 11.3

e.u./mole,

AG”,,,

= - 12.7

kcal/mole.

Thermochemical determination of the InP heat of formation by combustion in a calorimetric bomb gives the

AH”,,, = -21.5 & 1.5 kcal/mole. At present, standard thermodynamic values for indium phosphide are within the following ranges :

AH”,,,

=

-22 & 2 kcal/mole,

AG”,,,

=

-13 k 2 kcal/mole,

AS“,,,

=

-3 f 1

e.u./mole.

The standard entropy of the compound according to P i e ~ b e r g e n ”is~ S”,,, = 14.28 5 0.1 e.u./mole.

In Table XI are given the values of heats, free energy, and entropy of indium phosphide atomization calculated from the above values, as well as other data from the literature.

2.

99

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

TABLE XI HEATS,FREEENERGIES, AND ]ENTROPIES OF ATOMIZATION OF InP ~~

AH;'98 (kcal/mole)

(kcal/rnole)

ASa,L,, (e.u./mole)

- 154.6

- 134.0

- 69.0

4'9*

s;9* -

13.4 14.28

- 159

-92.5

- 131.5

Reference

(e.u./mole) 173 151 172 (Preferred values)

k. Indium Arsenide, InAs

Schottky and BeverlZ5have determined the heat of formation of indium arsenide, InAs, with the aid of tin solution calorimetry. According to their measurements, the heat of formation of indium arsenide at 273°K is =

-14.8 & 1.28 kcal/mole.

Sharifov et ~ 1 . ' ' ~have obtained the heat of formation of InAs by direct synthesis in a calorimetric bomb. The method is described by Sharifov and G a d ~ h i e v . ' In ~ ~ the calorimetric bomb, an ampule which contained a stoichiometric mixture of powders of indium (99.999% pure) and arsenic (99.99%pure) was heated for 3 min up to 1000°C. During this period the electrical energy was carefully measured with a standard meter, and then it was subtracted from the thermal effect determined by calorimetry. The heat of indium arsenide formation was thus found by the difference between the observed thermal effect and the energy due to the current. The degree of the transformation of the powder mixture components into the compound InAs was estimated by subsequent subliimation of the arsenic in vacuum. These authors found the standard heat of formation of indium arsenide to be

AH",,, = - 13.8

0.2 kcal/mole.

Gutbier''' has determined the standaLrd heat of formation of indium arsenide from the data on mass spectrometry investigations of InAs evaporation at 1100°K. For the reaction InAs(,, -+ In(,, + 4As,(,, the evaporation heat is AHlloo = -4(88 k 5) kcal/mole.

(48)

K. A. Sharifov, S. N. Gadzhiev, and 1. M. Garibov, Izv. Akad. Nauk Azerb. S S R , Ser. Fiz. Mat. i Tekhn. Nauk No. 2,53 (1963). H. B. Gutbier, Z. Narurforsch. 14a, 32 (1959).

100

N . N . SIROTA

The reaction of solid indium arsenide evaporation is represented by Gutbier as follows: InAs(,) 1%)

InAs

-+

+

-+

In(s) +

AS(,)

- ED,

(484

h,)- E ,

>

(48b)

t

Esubl >

(484

As4(g)

In,,,

+

-

As,(,, - ( E D

+ E , + Esubl).

(484

He used 0.8 kcal/g-atom as the value for the heat of fusion of InAs, a value of Esubl= 7.7 kcal/g-atom, and thus the standard heat of formation of the compound InAs is AH;,,,

=

-ED = -(22 - 0.8 - 7.7) = -13.5 f 0.15 kcal/mole. (49)

The values of dissociation heats of indium arsenide obtained by Goldfinger and J e ~ n e h o r n m e ' ~ ~at~ 1000°K '~' for the reactions InAs(,, 3 In(,) + 1 As,(,,, InAs,,,

-+

AH,,,,

In(,) + $As,(,,,

= -43.1

kcal/mole,

(49a)

AH,,oo = -26.0

kcal/mole,

(49b)

yield larger values of the standard heat of formation. In particular, on using the value of the evaporation heat for the reaction (49b) we get the standard heat of formation

(50)

AH",,, = - 17 kcal/mole.

Abbasov et ~ 1 . have ' ~ ~ studied the thermodynamic properties of indium arsenide by the method of the electromotive forces of the cell, (-)In(,,

+

(a) (LiCI, KC1) InCl or (b) (KCI, NaC1,ZnClJ

+ InCl

(InAs

+ As)"),

(a) within the range of 360-550°C and (b) between 220-380°C. An equation describing the variation of electromotive force with temperature over the range from 513-783°K has been obtained using the experimental data :

E = ( - 586.3 - 0.232T) millivolts. A. S. Abbasov, A. V. Nikol'skaya. Ya. 1. Gerasimov, and V. P. Vasil'ev. Dokl. Akad. Nauk

SSSR 156, 118 (1964) [English Transl. : Proc. Acad. Sci. U S S R , Phys. Chem. Sect. 156, 439 ( I 964)].

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

101

These authors have found the heat of fcrmation of indium arsenide in the above temperature range to be

AH, = -13.6

* 0.8

AS, = - 5.2 f 1.2

kcal/mole: e.u./mole,

AS,,* = - 10.2 _+ 0.8 e.u./mole. Applying these data and assuming ACp constant, they have calculated the standard values of the heat, entropy, andl free energy of formation of indium arsenide. The data for indium and arsenic heat capacities were taken from Stull and Sinke.” Heat capacity at temperatures higher than 273°K has been evaluated from the data of Pie~bergen‘~’on the assumption of the same temperature variation as with InSb.I8’ In particular, ACp = -0.41 cal/g-atom in the temperature range from 298 to 429°K and ACp = -0.76 cal/g-atom for liquid indium in the temperature range from 429 to 648°K. Abbasov, Nikol’skaya, Gerasimov, and Vasil’ev have obtained the following standard values : AH”,,, = - 12.4 f 10.8 kcal/mole,

AG”,8 = -11.6 AS;,,

= - 2.6

0.8 kcal/mole,

* 1.2

e.u./mole .

The standard entropy is

s”,, = 19.6 f 1.2

e.u./mole.

According to Piesbergen’72 S;9g,

= 18.10 f 0.1 e.u./mole.

At present, the main thermodynamic properties of indium arsenide are thus covered by the following ranges :

+1

AH”,,,

= - 13

AG”,,, AS”,,,

=

- 12 & 1

=

- 4 f 1.5

kcal/mole, kcal/mole, e.u./mole.

(51)

The value of the standard entropy of the compound reported by Piesbergen”’ is presumed to be the most exact : Si98 = 18.1 & 0.1 e.u./mole. On the basis of these and other values, the results for heat, free energy, and entropy of atomization are calculated and compared with those from the literature in Table XII.

102

N. N. SIROTA

TABLE XI1 HEATS,FREEENERGIES, AND ENTROPIES OF ATOMIZATION OF InAs

- 130.4

- 110.3

- 67.3

173 172 151

18.1 k 0.1 15.9

- 130

-111.8

- 61

(Preferred values)

1. Indium Antimonide, f n S b The heat of formation of indium antimonide was determined by Kleppalg7 by the tin solution calorimetry method. A sample weighing 1 g was dissolved at 450°C in 100g of liquid tin in a calorimeter. Similarly the heats of solution of indium and antimony, respectively, were evaluated. The heat of InSb formation from solid In and Sb was found from the difference between the heats of indium antimonide solution and the calculated heats of solution of the appropriate mixture of components, namely, = -8.64 kcal/mole. In this, the heat of fusion of indium AH,,,, = -0.78 & 0.02 kcal/mole was used. The recalculation for a temperature of 298°K yieldsIB7

AH 298

=

- 8.0 kcal/mole .

Schottky and BeverI2' have also determined the heat of formation,

AH273= -6.94 ? 0.22 kcal/mole, by the method of tin solution calorimetry in the range from 240 to 355"C, by dissolving in the liquid tin the indium antimonide, antimony, and indium, which were at a temperature of 0°C. The value for the standard heat of formation of indium antimonide can be taken as AH298 = -6.94 -t 0.22 kcal/mole, assuming that within the range of 25°C the enthalpy changes considerably less than the possible errors of the experiment. Upon analyzing phase diagrams with the aid of the equations obtained by Wagner, the values of standard free energy and entropy of indium antimonide formation have been calculated as AGZP8= -5.76 0.36 kcal/mole .Iz5

"' 0. 1. Kleppa, J . Am. Chem. Soc. 77,897 (1955).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

103

These values are to be compared with the standard free energy of indium antimonide formation obtained by summing the heat of formation, AH,,,

= -6.94

~t0.22 kcal/mole,

the entropies of the components, and the entropy of the compound calculated according to Debye. Assuming the characteristic temperature of InSb to be

0 = 200°K S&,

=

201.84 e.u./mole,

s;,,,,

=

13.88 & o.1,'88

si98sb

=

10.92 f 0.05,'89

one obtains AG,,, = -5.76

0.36 kcal/mole,

As298 = - 3.98 3: 0.52 e.u./mole . Terpilowskii and T r z e b i a t o w ~ k i ihave ' ~ ~ determined the heat, entropy, and free energy of formation of indium antimonide by the electromotive force method. Lithium and potassium bromides served as the electrolyte. The measurements were carried out within the range from 643 to 763°C in the cell (-)In1(0.6InBr

+ 0.4 KBr) + 0.1 InBrl(1nSb + Sb)").

These experiments yieldedlgo" the following results : AH",, AS;,,

= -8.22 =

3: 0.44 kcal/mote,

-5.00 3: 0.46 e.u./mole,

(52)

AG",,, = - 6.74 3: 0.58 kcal/mole. In the work by Nikol'skaya et a1.I9l heat, entropy, and free energy of formation of indium antimonide have: also been determined by the method of electromotive in the teinperature range from 635 to 650°C. Eutectic mixtures of lithium and potassium chlorides with an addition of 0.1 % indium monochloride, InCl, served as electrolytes. The results of measuring the electromotive forces in the chain (-)In(,J(KCI,LiCl) + InClI(1nSb + Sb);:) K. Clusius and L. Schachinger, Z . Angew. Phys. 4,442 (1952). W. DeSorbo, Acta M e t . 1,503 (1953). J. Terpilowskii and W. Trzebiatowskii,Bull. m a d . polon. sci., SPr sci. chim. 8,95 (1960). lgo"Theconversion factor (Faraday's number) was assumed to be equal to 23,066.19' 19' A. V. Nikol'skaya, V. A. Geyderikh, and Ya. 1. Gerasimov, Dokl. Akad. Nauk S S S R 130, 1074 (1960) [English Transl.: Proc. Acad. Sci. USSR, Phys. Chem. Sect. 130, 163 (196011. 19' P. V. Gul'tyaev and A. V. Petrov, Fiz. Tuerd. Tela 1, 368 (1959) [English Transl.: Souiet Phys.-Solid State 1, 330 (1959)l.

104

N . N . SIROTA

FIG.24. Variation of electromotive force versus temperature for the cell ‘-’Ino,J(KCI,LiC1) + InClI(1nSb + Sb){$). x -After Terpilowskii and Trzebiatow~kii.‘~~ 0-After Nikol’skaya et ~ 1 . ‘ ~V-After ’ Sirota and Yu~hkevich.’’~

are shown in Fig. 24. On the basis of these data, using the method of least squares, these authors obtained an equation for emf variation with temperature, E = (0.3455 - 0.241 x T ) volts. Hence it is found that, within the temperature range of the measurements, AH,,, = -7.96 & 0.4 kcal/mole,

AS723= -5.56 k 0.5 e.u./mole. By using heat capacities of the components51 and of indium antimo,,ide 1 7 2 , 1 9 2 and the heat of fusion of indium,’, Nikol’skaya et have determined the standard values of thermodynamic functions to be AG,,, = -6.14 0.4 kcal/mole,

AH,,,

=

-7.34

0.4 kcal/mole,

AS29B = -4.02 f 0.5 e.u./mole.

Sirota and Yu~hkevich”~ obtained by electrode potential determination values of AG,,, = -6.44 k 0.5 kcal/mole, AH,,,

=

-7.84 k 0.45 kcal/mole,

AS,,,

=

-4.68 f 0.5 e.u./mole.

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

105

Schneider and K l ~ t z ' , have ~ determined the heat of formation of indium antimonide by direct melting to be

AH,,, = - 6.8 kcal/mole . Gadzhiev and S h a r i f ~ v have ' ~ ~ found the heat of formation of indium antimonide with the aid of a synthesis reaction in a calorimetric bomb as AH,,,

=

-7.78 _+ 0..08 kcal/mole

From data obtained from mass spectrometric measurements, Kozlovskaya found the heat of indium antimonide formation to be's2*'55

AH,,,

=

- 10 & 2 kcal/mole

At present, the most reliable values are within the following ranges

AH,,, = -8.0 f 1

kcal/mole,

AS,,, = -4.0 f 0.5 e.u./mole, kcal/mole.

AG,9, = -6 f 1

On the basis of these results, values of heat, free energy, and entropy of atomization have been calculated and are compared with the data from the literature in Table XIII. TABLE :XI11 HEATS,FREEENERGIES, AND ENTROlPlES OF ATOMIZATION OF InSb

(kcal/mole)

AG%* (kcal/moIe)

(e.u./moIe)

- 128.1

- 109.1

-63.1

W ' g n

W ' q

n

Sign

(20.84) 20.60

- 127

- 64.0

173 125 172

(Preferred values)

Schneider and H. Klotz, Naturwiss. 46, 141 (1959). S. N. Gadzhiev and K. A. Sharifov, Dokl. Akad. Nauk S S S R 136,1339 (1961)[English Transl. : Proc. Acad. Sci. U S S R , Chem. Sect. 136, 227 (1961)].

l YA. 3

Iy4

- 108.3

Reference

(e.u./moIe)

106

N . N . SIROTA

VI. Bonding 6. THEDISTRIBUTION OF ELECTRON DENSITY IN CRYSTALS OF COMPOUNDS A I * I B ~ DUETO ENERGY AND NATURE OF ATOMIC INTERACTION

At present the energy and nature of interatomic binding in compounds AfL1BV can be evaluated at least semiquantitatively from experimental data on distribution of electrons in the crystal lattice. From the charts of electron density distribution in a crystal constructed experimentally, one can judge the type of chemical bond and, as is shown below, make quantitative conclusions about physical properties of crystals, including their unit cell sizes and heats of formation and a t 0 m i ~ a t i o n . l ~ ~ In Fig. 25 the charts are shown of electron density distribution in ionic crystals of NaCI, in silicon with covalent bonding, and in aluminum with metallic bonding. In the structure of NaC1, ions with different charges are seen; in silicon, covalent bridges can be observed. One can estimate the number of free electrons and the character of the electron gas distribution in the lattice in aluminum.196 The study of electron density distribution in compounds AfflBVis very fruitful. From the atomic scattering factors and from the charts of electron density distribution, the dia- and paramagnetic susceptibility, effective ionic charges, heats of atomization and formation of compounds can be determined. Other properties can also be estimated. Dorfmant9' utilized the separation of magnetic susceptibility into dia- and paramagnetic components for the study of the nature of the chemical bond. Elucidation of the possibility of determination of thermal properties of compounds A"'BV from x-ray data analysis shows new perspective. Bragg and his co-workers198and then subsequent investigator^'^^^'^"^^^ demonstrated the possibility of determination of the electron distribution in N. F. Mott, Proc. Roy. SOC.(London) A146,465 (1934). N.V.Ageev and L. N. Guseva, Izv. Akad. Nauk SSSR, Ord. Khim. Nauk No.4,289(1945). '97 Ya. G.Dorfman, Diamagnitizm i khimicheskaya svyaz, Moskva. Fizmatgiz, 1961. l Y 8 W. H. Bragg, Trans. Roy. Soc.(London)A215,253(1915);W. L. Bragg,R. W. James,andC. H. Bosanquet, Phil. Mag. 41,309 (1921). 1 9 9 W. Duane, Proc. Natl. Acad. Sci. U.S. 9,158 (1923); 11,489(1925). zoo R.J. Havighurst, Proc. Natl. Acad. Sci. U.S. 11,502(1925); Phys. Rev. 29,1 (1927). '01 A. Sommerfeld, Naturwiss. 28,769(1940). 2 0 2 H.G.Grimm, R. Brill, C. Hermann, and C. L. Peters, Naturwiss. 26,479 (1938); Ann. Phys. 34,393 (1939). '03 R. Brill, C. Hermann, and C. L. Peters, Ann. Phys. 41,37(1942); Naturwiss. 32,33(1944). '04 C.Hermann, 2. Elektrochem. 46,425(1940). ' 0 5 N.V. Ageev and L. N. Guseva, Dokl. Akad. Nauk S S S R 59,65 (1948). 2 0 6 N. V. Ageev and L. N. Guseva, Izv. Akad. Nauk S S S R Otd. Khim. Nauk No. 5,470 (1948). No. 3,225( 1949),No.1,31 ( 1952). 20' N. V. Ageev and D. L. Ageeva, Izv. Akad. Nauk SSSR Otd. Khim. Nauk No. 1, 17 (1948). * 0 8 N. V. Ageev and D. L. Ageeva, Izv. Akad. Nauk SSSR Otd. Khim. Nauk No.3,213(1948). 195

196

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

107

the lattice of crystals by synthesis of the Fourier series, where the structural amplitudes related to the volume of an elementary cell are the coefficients 4

t n

The square of the structural amplitude is proportional to the rate of hkl response on the x-ray pattern, that is

Thus to determine the electron density distribution, measurements of response intensity are necessary. The intensity of response for a given structure depends on the distribution of electron density, which is characterized by the atomic scattering factorf, of th'e i-type atom, on temperature which is accounted for by the temperature Eactor M , and on some other factors (Lorentz recurrence, absorption, etc.). On the other hand, the value of the squared structural amplitude (structural factor) is related to the factorf, of atomic scattering and the temperature factor M as follows:

In Eq. (55), the atomic scattering factor at the temperature T can be expressed bv the relation in terms of the atomic scattering factor fo at T -+ 0 and the temperature factor M . The temperature factor depends on the value of the mean square dynamic displacement of ions from thle equilibrium state, M,

=

8r2iiT2.

Experimental determination of electron density distribution is thus associated with the necessity to determine the atomic-scattering fo and the temperature factor M . In the case of spherical symmetry, the atomic scattering factor is determined by the distribution of the electron density p, and is a function of the angle of the Bragg reflection at the given wavelength A,

dr, where sin 9 p = 471-

1 .

Grimm, Brill, Hermann and Peters, Ageev with Guseva and Ageeva, Cochran, Witte, Wolfel, Hosemann, van Reijen et al. have contributed significantly to

108

N. N. SIROTA

i

x N

d

% 3 30 25

k

E

AC

5

s

z

-

20

FIG. 25. Electron density distributions in crystals : (a) NaCl in (100) plane (after Witte and Wo1fe12’6). (b) Ce in (110) plane (after Sirota and Shelegzz7). ( c ) A! in [OOl] direction (after Hume-Rothery and RaynorZ8’).

/5 .

FIG.25(c)

110

N.

N. SIROTA

the development of methods for experimental determination of electron density distribution in various crystals and to critical discussions of the results.33.87,196,202-234 To determine the electron density, one must sum the slowly convergent three-dimensional Fourier series ; therefore it is necessary to measure the intensity of a great number of responses. To eliminate the effects of series truncation, the temperature factors were introduced ear1y.’96,202-208The calculated temperature reached 5,00Ck10,000”K, which naturally caused some revision of the actual picture of electron distribution in a crystal, though ’09

A. J . Snow, Acta Cryst. 4,48 I (1951);J . Chem. Phys. 19, I 124 (1951).

’” N. V. Ageev, Izv. Akad. Nauk SSSR Otd. Khim. Nauk No. 1, 176 (1954) [English Transl.:

Bull. Acad. Sci. U S S R , Division of Chemical Sciences No. 1, 147 (1954)l. W. H. Taylor, “The Physical Chemistry of Metallic Solutions and Intermetallic Compounds,” Vol. 1, Paper ID. Her Majesty’s Stationery Office, London, 1959. A. Kh. Breger and G. S. Zhdanov, Compt. Rend. (Doklady)Acad. Sci. U R S S 28,629 (1940). ’ I 3 R. W. James and E. M. Firth, Proc. Roy. Soc. (London)A117,62 (1927). ’I4 R. Hosemann and S. N. Bagchi, Nature 171,785 (1953). L. L. van Reijen, Physica 9,461 (1942). ’I6 H. Witte and E. Wolfel,Z. Phys. Chem. N.F.3,296(1955). H. Bensch, H. Witte, and E. Wolfel, Z. Phys. Chem. N.F. 4,65 (1955). ’ I 8 S. Gottlicher and E. Wolfel, 2. E/ektrochem. 63. 891 (1959). ’ I 9 W. Cochran, Rev. Mod. Phys. 30.47 (1958). ’’O G. B. Carpenter, J . Chem. Phys. 32,525 (1960). N. N. Sirota, N. M. Olekhnovich, and A. U. Sheleg, Dokl. Akad. Nauk S S S R 132, 160 (1960) [English Transl.: Proc. Acad. Sci. U S S R , Phys. Chern. Sect. 132,393 (196011. N. N. Sirota, Dokl. Akad. Nauk SSSR 150,781 (1963)[English Transl. : Societ Phys. “Doklady” 8, 573 (1963)l. 2 2 3 N. N. Sirota, N. M. Olekhnovich, and A. U. Sheleg, Dokl. Akad. Nauk Belorussk. 4,144 (1960). 224 N. N. Sirota and N. M. Olekhnovich, Dokl. Akad. Nnuk S S S R 136, 879 (1961) [English Transl.: Proc. Acad. Sci. USSR, Phys. Chem. Sect. 136, 137 (1961)l. 2 2 5 N . N. Sirota and N. M. Olekhnovich, Dokl. Akad. Nauk SSSR 143, 370 (1962) [English Tmnsl.: Proc. Acad. Sci. U S S R , Phys. Chern. Sect. 143,228 (196231. 2 2 6 N. M. Olekhnovich, Vestsi Akad. Nauk Belorusk. SSR., Ser. Fiz. Tekhn. Nauk N o . 1 , 35 (1964). N. N. Sirota and A. U. Sheleg, Dokl. Akad. Nauk S S S R 135, I176 (1960) [English Trans/.: Proc. Acad. Sci. U S S R , Phys. Chem. Sect. 135,1165 (1960)l. N. N. Sirota and A. U. Sheleg, Dokl. Akad. Nauk S S S R 147, 1344 (1962) [English Trans/.: Soviet Phys. “Doklady” 7, 1146 (1963)l. 2 2 9 A. U. Sheleg, Vestsi Akad. Nauk Belorusk. SSR., Ser. Fiz. Tekhn. Nauk No. 2,51 (1964). 2 3 0 N. N. Sirota and E. M. Gololobov, Dokl. Akad. Nauk S S S R 156, 1075 (1964) [English Transl.: Soviet Phys. “Doklady” 9,477 (1964)l. 2 3 E. M. Gololobov, Opredelenie teplop atomizatsii i effektivnykh zaryadov ionov soedinenii 111-V PO dannym rentgenovskogo analiza, Minsk, 1964 (autoreferat dissertatsii). 232 N. N. Sirota, Dokl. Akad. Nauk SSSR 142, 1278 (1962) [English Transl. : Soviet Phys. “Doklady” 7, 143 (1962)l. 2 3 3 N. N. Sirota and A. U. Sheleg, Dokl. Akad. Nauk S S S R 152, 81 (1963) [English Transl.: Soviet Phys. “Doklady” 8, 887 (196411. 2 3 4 A. E. Attard and L. V. Asiroff. J. Appl. Phys. 34.774(1963). ’I1

’”

’”

’” ’”

”’

2.

HEATS OF FORMATION A N D TEMPERATURES AND HEATS OF FUSION

111

the perturbation in the intermediate region between ions was considerably less than supposed by many investigators. Although the summing of onedimensional Fourier series simplified the problem of summation, it at the same time impeded interpretation of the results and did not eliminate the difficulties due to the series truncation.?12 At present, some methods are available for eliminating the effect of series truncation, which do not require introduction of the calculated temperatures. The method of separation and ~ ~ ~ , ~ approximation of the atomic-scattering functionfis widely ~ s e d .Out of the total number of electrons, a considerable fraction is chosen, which is placed mainly in the middle part of an atom and is distributed according to a definite law, for which the atomic-scattering functionx is known or can be easily calculated. The distribution of the remaining ion electrons is determined by synthesis of a triple Fourier series. This approximation turned out to be very convenient, either with the help of the exponential function of the first order or with the Gaussian functions. For instance, the following can be used as approximating functions : f = Be-@"',

.f

(564

Be-BH, (5W 8 = (1 + p 2 H 2 ) 2 ' where H is the vector of the reciprocal lattice. We can also use other types of function^.^'^,^ 1 5 * 2 2 2 The distribution of the electron density in the inner part of an atom can be expressed by a Gaussian function of the form =

p1 = ~ , e - ~ l r '

(57)

and in the outer part of the atom, by p2 = A2e-azr2.

(574

The atomic scattering factors of the group IV elements and semiconductor compounds A"'BV are systematically determined in the works by Sirota, Gololobov, Olekhnovich, and Sheleg ;33,87,100,1 1,22 also the possibility of determination of thermodynamic and physical properties of crystals from the charts of electron density distribution and directly from fcurves is shown for the first time. The experimentally determined curves (fcurves) of atomic scattering factors of A"'BV elements in antimonides and arsenides of aluminum, gallium, and indium can be satisfactorily approximated by two Gaussian curves. This facilitates the calculation of the electron density distribution. Figures 26 and 27 are charts of the electron density distribution in arsenides and antimonides of aluminum, gallium, and indium in a (1 10) plane. In Figs. 28 and 29 the electron density distribution in the ( 1 11) direction between the atoms A"'BV of these compounds is shown.

112 N. N . SIROTA

i

e 8

>

zU FIG.26. Electron density distributions in the lattices of arsenides of aluminum, gallium, indium in ( 1 10) planes at - 100°C. (a) AlAs-after (b) GaAs-after Sirota and O l e k h n o v i ~ h . ~ ~ ~ Sirota and Olekhn~vich.~' (c) 1nAs-after Sirota and O l e k h n ~ v i c h . ~ ~ ~

FIG.26(c).

114 N . N. SIROTA

d

r;

az FIG.27. Electron density distributions in the lattices of antimonides of aluminum, gallium, indium in (110)planes at - 100°C.(a)AlSb-after Sirota and Gololobov. loo (b) GaSb-after Sirota and Gololobov.' (c) InSb-after Sirota and Gololobov.'*'

5.

z

U

FIG.27(c).

A1

e

As

A1

u,L/1

-2 05

a

0

Ga

As

Ga

,

e

I

2

4

6

8

05

A

FIG.28(a).

FIG.28(b).

Ga

I

As

e

.A".&

t/5

0

"I

FIG.28. Electron density distributions in the lattices of arsenides of aluminum, gallium, indium in the [ 1 1 11 directions. (a) AlAs-after Sirota and O l e k h n o v i ~ h(b) . ~ ~GaAs -after Sirota and Olekhnovich."" (c) InAs-after Sirota and Olekhnovich.zzs

05

05

FIG.28(c).

118

ct

N. N . SIROTA

D N m

2. HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

f

119

120

N . N . SIROTA

7. THE EFFECTIVE IONIC CHARGES OF ARSENIDES AND ANTIMONIDES OF ALUMINUM, GALLIUM, AND INDIUM The values of the ionic charges can be determined from charts of electron distribution and directly from f curves. The fact that, to a sufficiently good approximation, ions of compounds A“’BV can be considered spherical and the so-called “bond bridges” can be regarded as overlapping of spheres, facilitates the problem. By integration over the volumes of ions,

z=

s

pdv,

ionic charges can be calculated. Ionic charges determined from charts of electron density230and the values of ionic charges determined fromfcurves are given in Table XIV.

EFFECTIVEIONICCHARGES F”P 200

A”LBY

AIP GaP InP AlAs GaAs lnAs AlSb GaSb lnSb

-

TABLE XIV A%”

OF COMPOUNDS

FROM

ERective charges

FZ”

__

Fr; (HFY

F;;“,1 (TFY

1.798 0.906 1.061 1.514 -

1.315 0.922 0.964 1.041 1.353 0.957 1.023 1.043 1.181

ANALYSIS OF X-RAY DATA

From charts of electron density .~

1.00 0.80 0.49 0.67 0.58 0.38

f 0.25 0.22 ? 0.15 0.2F 2 0.17 2 0.10

+

Using F-factor IHF) 0.80 i 0 20 0.75 0.20 070)~ 0.60 0.15 0.51 2 0.10 (0.50)’ (0.63)’ (0.59~ (0.27)’

+

Using F-factor ITF) 0 32 f 0.06

0.62 k 0.20 0.58 i 0.15 0.40 0.08 0.36 i 0.07 0.35 f 007 0.45 5 0.08 0.43 _+ 0.07 0.18 k 0.05

Calcul. from Folberth’s data

0.52 0.50 0 55 0.44 0.43 0.46 0.32 0.31 0 35

H F denotes Hartree-Fock method. T F denotes Thomas-Fermi method Values obtained by correlating the charges calculated from the T F and H F F-factors

‘ Effective charge determined from the Sb ion.

It is well known that the structural amplitude for the pair of atoms in the sphalerite lattice having even indices that can be divided by four is equal to the sum of atomic scattering factors of the component ions. For example, IF440/ = f A

+fB.

The structural amplitudes are equal to the difference of atomic scattering factors of the component ions for the planes with four indices which cannot be divided by four. For example, In the case of even-even indices, the structural amplitude is not sensitive to different ionic charges of the components. When the sum of indices squared is zero, the sum of atomic scattering factors is equal to the sum of the atomic

2.

HEATS OF FORMATION A N D TEMPERATURES A N D HEATS OF FUSION

121

numbers of A1"BV.However, the difference of atomic scattering factors is sensitive to the variation of ion charges. In Table XIV the values of the ratio FZ%/F;Y,ral a'om are given for the compounds A1"BV. On using the apparent relation

one can find the effective charge of ions. The effective number of electrons per ion is

hence the effective charge is

To find

the curves of the atomic scattering factors according to Hartree-Fock and Thomas-Fermi were used. The calculated values of ion charges are presented in Table XIV. The following conclusions can be made on the basis of the data obtained. In compounds A"'BV the charges of ions A"' are positive and those of BVare negative. The values of the ionic charges found for the compounds agree to a certain degree with optical data.235-237However, optical measurements do not show the signs of the charges. The magnitude of the charge (the degree of ionicity) decreases with increase in the atomic number and the periodic number of the components. It should be pointed out that in the literature there is no accurate definition of the concept of the effective ionic charge. which has been mentioned in the works by Mooser and P e a r ~ o n , *C~~~c h r a n T, ~o ~l p~y g ~ and , ~ ~others. ~ The ionic charges shown above are those determined by integrating the density of the electron distribution over the ion volumes, although their boundaries pass through the regions of overlapping. The boundaries 235 236 237

238 239 240

M. Hassand B. W. Henvis,J. Phys. Chem. Solids 23, 1099 (1962). W. G. Spitzer and H. Y. Fan, Phys. Rev. 99, 1893 (1955). F. Oswald and R. Schade, Z . Nnturforsch. 9a..61 1 (1954). E. Mooser and W. B. Pearson, J . Electron. I, 629 (1956);J . Chem. Phys. 26,893 (1957). W. Cochran, Nature 191,60 (1961). K . B. Tolpygo. Zh. Eksperirn. i Teor. Fiz. 20,497 (1950).

122

N. N. SIROTA

between ions were determined from the chart of electron density in the directions where there is no overlapping between atoms, and by the atomic scattering factors of the ions (Fig. 27).

8. HEATSOF ATOMIZATION A N D HEATSOF FORMATION OF 1II-v COMPOUNDS ACCORDING TO THE DATA OF THE ELECTRON DENSITY DISTRIBUTION It was convenient to assume the statistical atomic theory as a basis of the bond energy calculation.241In this case the energy of interaction between two atotns can be represented as the sum of energies of interaction between ions ( U J , between ions and electrons in the overlapping region ( U J , energy of mutual repulsion of electrons in the region of overlapping ( U J , kinetic and exchange energies of electrons ( U , and U,) in the region of overlapping, and interaction of electrons with antiparallel spins (U,,,). The energy of interaction between the electrons, with total charge q = q A + q B , in the region of overlapping and the ions which are beyond the region of overlapping, Q1 and Q 2 ,can be described roughly by the expression

where

RA + R B = b

is the interatomic distance ;

u,

xk

![(PA +

PB)5’3

do

- Pi’3 -

V

is the kinetic energy; and

u,

-X,

lo

[(PA

+

&3)4’3

-

- p i ’ 3 ] dv

is the exchange energy. The integration is carried out over the volume of overlapping; the calculation can be performed both by numerical integration according to the charts of electron density distribution over the annulus in the region of overlapping, and analytically by using the analytical expression for the electron density at a given point. In the following approximation of the statistical atomic theory, the energy of interaction between two atoms is given by the relation e2 U = ( Z , - N , ) ( Z 2 - N 2 ) -6 + eZ2y,(6)+ eZIy2(4 -$[N2yl(fi)

+ N1y2(6) +

1

YlPZ dv

+ Jo

y2PI

”1 + Uk + u,,

(62) 241

P. Gombas, “Die Statistische Theorie des Atoms und ihre Anwendungen.” Wien, Springer, 1949.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

123

where

eZ eN(r) r r Z is the atomic number, N is the number of electrons per ion, and 6 is the distance between ions. Substituting the experimental expressions for the electron density p into the given expression, one can find the energy of interatomic interaction in 111-V compouncls. Although at present this method of determination of the energy of interatomic interaction by x-ray analysis is inferior in its accuracy to direct thermochemical measurements, it allows realistic values of atomization heats to be obtained, and in addition the relative role of various components of the interaction energies can be estimated. At the same time, this method shows new possibilities for application of quantum mechanical techniques and calculation methods of energy of interatomic interaction, using experimentally determined wave functions instead of theoretical ones, since($12= p . Table XV shows the calculated values of bond energies, i.e., atomization, heats of formation, and their components from the data on electron density distribution. y=---i

9. IONICCOMPONENT OF ENERGY AND ELECTRONEGATIVITY The concept of electronegativity of tlhe component elements is often used for the determination of the ionic component of interaction energy in 111-V compounds. According to Pa~ling,'~'the ionic component of bond energy is equal to the squared difference of electronegativities of each component:

Eion= 23,060(~, - ,K~)' kcal/mole . TABLE XV HEATSOF ATOMIZATION AND FORMATION, AND THEIR COMPONENTS, FOR COMPOUNDS A"'BV FROM THE ANALYSIS OF X-RAY DATA

UAe=

L',

,kcal/%olel AlAs GaAs

- 285 - 277

InAs

-212 - 182 -170 - 137

AlSb GaSb lnSb

(kcallmole) 215 216 147 I I4 128

61

L',

(kcallmole)

U. + ti, + ti, (kcal/malel

___-

- 125 - 99 - 81) - XX -93 -47

- 195 - 160 145 -- 155 -- 135 -- 123 .-

0 , + u. Sum of atomization

energies"

- I53 4 - 143.5 - 135.8 -137.1 - (27.2 - 119.5

Heat of formation AH (kca1:mole) From electron From Ilteradensity Lure -41.6 - 16.5 - 9.2 - 17.2 - 7.8

-3.5

- 20.4 -14P - 23.ff -9.94h - 6.94h

From Nesmeyanov.'" 'From Schottky and Bever."5 From Kubaschewski and

242

L. Pauling, "Nature of the Chemical Bond." Cornell Univ. Press, Ithaca, New York, 1940 (2nd ed.) or 1960 (3rd ed.).

124

N . N. SIROTA

It should be noted that there is no well-established definition of the concept and value of element electronegativity, and for the estimation of the potentiality and advantage of its usage. In spite of the wide use of the term, and the explanations of various relationships by means of electronegativity concepts used in the literature, there exist diverse definitions of these quantities. Highly critical remarks are given in some cases, which suggest that the general usage of the concept of electronegativity is not advisable. A number of papers by Tatevskii and c o - w o r k e r ~ the , ~ ~discussions ~ by S ~ r k i and n ~ ~ ~ B a t ~ a n o vand , ~ ~other ~ works should be noted in this respect. Spiridonov and T a t e ~ s k icompare i ~ ~ ~ critically the methods for calculation of numerical values of electronegativity by various authors. They particularly state that a uniform approach to this problem is missing since the empirical formulas of various authors cannot be considered as equations defining a certain new physical quantity characterizing the ability of an atom in a molecule and crystal to attract electrons, as follows from the definition of electronegativity according to Pauling. It is our opinion that both the overestimation of the potentiality of the concept of electronegativity and its complete neglect are undesirable. However, for an adequate quantitative usage, further development of the theory is required, especially with respect to applicability to III-V compounds. The most popular concept of electronegativity is according to Gordy,23,246who defines electronegativity as (64) where n is the number of valence electrons, r* is the covalent element radius. Attention should be paid to the fact that electronegativity according to Gordy is very similar to generalized moments, introduced originally by and defined as Semenchenko in 1927247,248 ne m=r where n is the number of valence electrons, r is the crystallographic radius of the ion, and e is the electron charge. 243

244

245

24b 247 248

V. P. Spiridonov and V. M. Tatevskii, Zh. Fiz. Khint. 37, 994. 1583 (1963) [English Trans!.: Russian J . Phys. Chem. 37, 522,848 (1963)l. Ya. K. Syrkin, Usp. Khim. 31, 397 (1962); Zh. Fiz. Khim. 37, 1422 (1963) [English Transl. . Russian J . Phys. Chem. 37,764 (196311. S. S . Batsanov, Zh. Fiz. Khim. 37, 1418 (1963) [English Transl.: Russian J . Phys. Chem. 37,

761 (1963)l. W. Gordy and W. J . 0.Thomas, J . Chem. Phys. 24,439 (1956). V. K. Semenchenko, B. P. Bering, and N. L. Pokrovskii, Zh. Fiz. Khim. 8,364 (1934). V. K. Semenchenko, Poverkhnostnye yavleniya v metallakh i splavakh, Moskva, Gos. izd-vo tekhniko-teoreticheskoy lit-ry, 1957. 491 p. [English Transl. : “Surface Phenomena in Metals and Alloys.” Pergamon, London, 19621.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

125

Folberth" and W e l l ~ e r , 'discussing ~~ the theory of chemical bond, used the value of electronegativity that is defined by the Mulliken rule as the difference between the energy of the electron affinity and that of the first ionization potential divided by 130.24,2"0*25 Table XVI gives the values of electronegativity according to and Folberth," the values of the ionic components of the interaction energies in 111-V compounds calculated from these electronegativity values, as well as the values of the ionic components calculated by the magnitude of ionic charges determined by x-ray methods. It may be seen from the data presented that the ionic component of the interatornic binding increases with decreasing periodic number of the component in the periodic table. That Folberth's estimations of electronegativity of ionic charges are sometimes close to the values determined by x-ray methods is a noteworthy fact.

10. CHANGE IN THE SPECIFIC VOLUMES, OF COMPONENTS I N FORMATION OF A"'BV COMPOUNDS The connection between heats of formation and atomization and the change of specific volumes at formatioln of chemical compounds has been As a rule there exists a distinct proportionnoted more than once.20,163*252 ality between the decrease in interatomic distances. the change of the specific volume in compound formation, and hieats of formation.20 In the majority of cases exceptions can be reasonably explained. In Table XVII the values of specific (molar) volumes V , of 111-V compounds are given according to data from x-raLymeasurements, as well as relative deviations of V, - ZV, 6 = ____ - At' CVi

zq.

from the additive value, i.e., from the sum of specific atomic volumes of components CV,. It follows from the data given in the table that the molar volume of 111-V compounds is greater than the sum of atomic volumes of the components, and hence formation of cornpourids A1"BVwith the sphalerite structure is accompanied by an increase in the atomic volume, i.e., the relative change of the volume upon formation of a cornpound is a positive value, namely,

249

250

25

'

252

H. Welker, Z . Naturforsch. 7a, 744 (1952). H. Preuss, "Die Methoden der Molekulphysik und ihre Anwendungsbereiche." Akad. Verlag, Berlin, 1959. H. Welker, Z . Naturforsch. Sa, 248 ( 1 953). G. Tammann and A. Rohmann. Z. Anorg. Chern. 190.227 (1930).

126

N. N . SIROTA

IONIC COMPONENTS OF THE

TABLE XVI BINIIING ENERGYFROM ELECTRONEGATIVITY AND THE DATA OF X-RAYANALYSIS Electronegativity

%"

According to Gordy"

According 10 Folberthb

Al

15

1.05

P

2 19

1.862

Ga

1.4R

1.055

P

2.19

1.862

I"

I36

1.02

P

2.19

1.862

A1 P

GaP

1"P Al

1.5

1.05

As

2.04

1.725

Ga

1.48

1.055

As

2.04

1.725

In

I36

1.02

AS

2.04

1.725

Al

IS

1.05

Sb

1.82

1.525

Ga

1-48

1.055

Sb

1.82

1.525

In

1.36

1.02

Sb

1.82

1.525

AIAs

GaAs

lnAs

AlSb

GdSb

1nSb

%"

U.."

from data by Gordy

(kcal/mols) from data by Folberth

43.9

60 8

145.4

46.5

60.07

129.6

63.5

65.4

104.9

26 9

42.0

1430

?a 9

41.4

93 9

42.6

45

R

4Y.X

9.4

20.R

91 9

10 7

20.4

69.4

195

23 5

28.0

(kcal/moleJ

Elem

(kcallmole) from etreective charge'

* F r o m Gordy" and Gordy and tho ma^."^ 'From Folberth." ' From Sirota and Gololobov '''

In order to correlate the relative decrease of volume in formation of compounds for structures of various types, it is n e ~ e s s a r y ~ to ' . ~introduce ~~ the factor e

into the expression for the relative volume change. In particular, for the structure CsCl with the coordination number z = 8, this factor is 0.95; for the structure NaCl ( z = 6), 6 = 0.825. Introduction of 6 is not reasonable when isostructural series, i.e., compounds with one and the same type of crystal lattice, are considered. By analyzing the volume changes in connection with the heat of formation of 111-V compounds, one can establish that the smaller the relative increase 253

W. Biltz. "Raurnchemie der festen Stoffe."Leipzig. L. Voss, 1934.

TABLE XVII

MOLARVOLUMESOF COMPOUNDS A ~

~

Compound Structure V-. cm'/mole

d. density x-ray gjum' ZVcm" AV,cm'b AV

T" x

c

A"'.

in

A

From Pearson."

~~~~

AIP

AlAs

AlSb

GaN

GaP

GaAs

GaSb

InN

InP

InAs

lnSb

GraDh. Sphal.

SDhal.

Wurtr.

Sohal.

Sohal.

Sohal.

Wurtz.

SDhal.

SDhal.

SDhal.

Wurlz.

Sphal.

SDhal.

Swhal.

C,

Si,

Gel

27.28

10.89

14.04

16.42

12.47

24.51

27.34

34.78

13.61

24.39

27.21

34.10

18.49

30.44

33.49

40.96

6.82

24.22

2.27

2.97

5.22

3.28

2.36

3.72

4.27

6.11

4.13

5.31

5.61

698

4.79

5.66

5.17

3.52

2.32

5.32

21.9R

17.45

17.3.1

27.71

23.18

23.06

28.23

29.51

24.98

2486

30.03

33.43

28.90

28.78

1395

6.82

24.22

17.28 -

-741 -19.0

1.75

1.75

1.06

E"-B" in A"'B~,A A"1-BY *, i n A"'BY,A

~~~~~~~~~~~~

AIN

A

BY-BvcmEV,A

~~~

BAS

-11.09

{n

~~

BP

BN

-50.5

100%

X-RAY ~ B DATA ~ A N D DEVIATIONS FROM THE SUMOF ATOMIC VOLUMESOF COMPONENTS; ~NTERATOMIC DISTANCES BETWEEN VARIOUSSITES

FROM ~

1.44

-0.91

-15.23

!.34

4.2:

-5.3

-54.9

5.8

14.2

1.75

2.86

3.21

3.37

3.06

3.86

218

2.51

1.06

2.18

3.21

3.37

3.06

3.86

1.96

2.06

1.89

2.36

'From Hansen and Anderko."

2.86

2.86

6.55 23.2

-15.90

-0.60

2.35

-53.9

-2.4

9.4 2.44

2.44

-14.94

1.54

-44.8

5.3

3.25

3.25

4.71 16.3 3.25

7.01 26.2

2.86

2.44

4.00

4.34

3.17

3.85

3.99

4.31

350

4.15

4.28

4.58

2.51

2.90

1.06

2.18

2.51

290

1.06

2.18

2.51

2.90

4.00

4.34

3.17

3.85

3.Y9

4.31

3.50

4.15

4.28

4.58

2.45

2.65

1.94

2.36

2.45

2.64

215

2.54

2.62

2.80

'Interatomic distances

2.44

4.10 13.2

3.25

-

-

.-

-

-

-

-

-

8

128

N. N. SIROTA

of the volume in compound A'*'BVformation, the higher the absolute value of the heat of the reaction of compound formation (Fig. 30). In the first approximation -AH K(c - 6)". (67) More specifically, the following empirical relation has been established254: -AH

= 0.56AV-

0.014(AV)2

+3 x

1 0 - 4 ( A V ) 3 - 40

+ 5z.

(68)

where A V is the relative change, in percent, of the volume upon formation of the compound, z is the coordinate number in the lattice of the compound, and the constant 40 is in units of kcal/g-atom. The increase in molar volume in comparison with the sum of atomic volumes in formation of compounds with the sphalerite structure can be attributed to a certain extent to the fact that the sphalerite structure is rather loose, since in its formation only half the tetrahedral sites in close-packed arrangements of ions of either component are filled. Spheres of equal size

60 50 LO -

' +20

0

-20 -40

60

%L?%

FIG. 30. Heats of formation of compounds A"'BV due to specific volume variations at compound formation. 254

See F. Weibke and 0. Kubaschewski, "Thcrmochcmie dcr Legicrungen," p. 339. SpringerVerlag. Berlin. 1943; 0.Kubaschcwski, 2. Elekrrochem. Angew. Physik. C h m . 47,623 (1941).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

129

fill 34% of the whole space, in contrast to 74% for tighter cubic and hexagonal packings. If the changes of interatomic distances A1"-A1",BV-BV,and AII1-BV(Table XVII) are examined, the following regularity may be noted : with increasing heat of formation of the 111-V compourids. the interatomic distances A"'-BV decrease. In compounds, distances A1I1--A"'and BV-BV exceed appropriate distances in elemental solids. It may thus be considered that the main contribution to the energy of atomic interaction of a 111-V compound is a result of a convergence of atoms A"'BV. This energy gain not only compensates the loss of the atomic bond energy of the components due to increase of the distances A"'-A"' and BV-BV, but also conditions the exothermal heat of formation of the compounds. Despite a larger molecular volume of 111-V compounds compared to the sum of atomic volumes of A"' and BV,the entropy of the compounds appears to be less than the sum of atomic entropies of the components. This follows, for example, from the data of Stull and Sinke19and Piesbergen. 17' Standard entropy and, perhaps, a relative deviation of the compound entropy from the additive value, decrease with increasing heat of formation of the compounds and decreasing number of the group and period which determine the position of the componeints in the periodic table (see Table XVIII). Changes of the deviations of entropies of compounds from the additive values depend on the ratios of the effective characteristic temperatures of the components and the compounds and thus on the energy of interatomic binding, i.e., on the deviation of a.tomization heats from the additive values. TABLE X'VIlI STANDARD ENTROPIES OF THE COMPOUNDS A"'B~,SUMSOF STANDARD ENTROPIES OF EACHCOMPONENT, AND

AlSb GaAs GaSb InP lnAs lnSb Ge,

15.36 15.34 18.18 14.28 18.10 20.60 14.92

CALCULATED ENTROPYOF COMFQUND FORMATION

(7.69 18.22 20.74 19.28 22.22 24.74

-

- 2.33 - 2.88

-2.56 - 5.00

-4.12 -4.14

130

N. N . SIROTA

1 1 . CHARACTERISTIC TEMPERATURES That the temperature variation of heat capacities of 111-V compounds could not be adequately described by the Debye expression follows, for example, from Piesbergen's work. 1 7 2 The Debye characteristic temperatures obtained from heat capacity curves change with temperature over a wide range, as is shown in Fig. 31.

,

3 . 50 O

ca

O

k

l

2501TzDl I : i i -

2000

100

200

JUU

280

TOK

I

1250

ca

I

zw

450

/oo

f 250

0

100

200

280

FIG. 31. Variation of characteristic temperatures in compounds A"'BV versus temperature. (After Piesbergen.'72)

The frequency spectrum of ion vibrations in the sphalerite lattice is quite different from the Debye spectrum, and these vibrations cannot be considered as vibrations of anisotropic spherical oscillators. In first approximation, the dimensions of elastic wave propagation in a crystal should be considered. Particularly, for Si and Ge, heat capacities as a function of temperature in the range of low temperatures are better described by the relation2s5 C,

F)'

=I(

at

n<3

than by the Debye expression for isotropic spherical oscillators for which II = 3. The temperature dependence of heat capacity of compounds A"'BV is also better described by the above type of equation at n < 3. 255

N. N. Sirota, Compt. Rend. (Doklady)Acad. Sci. URSS 47,39 (1945).

2.

131

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

Irrespective of the above reservations, characteristic temperatures of 111-V compounds are of considerable interest as quantities that characterize thermodynamic properties of the compounds and their temperature dependencies. Expediency of usage of characteristic temperatures is somewhat justified by the fact that, in the region of room temperature and slightly higher, the characteristic temperature changes are relatively small, and thermodynamic functions predicted by the characteristic temperatures under normal conditions are close to experimental values. As is known (see, e.g., Herbstein2"'), characteristic temperatures obtained by various methods (such as elastic constants, values of the Debye-Waller temperature factor, heat-capacity curves, atomic scattering factor curves, curves of linear expansion coefficients) are sometimes very different. Characteristic temperatures obtained by x-ray methods are as a rule somewhat higher than those found from heat capacity curves. This difference is due to basic physical causes. It should be pointed out, however, that characteristic temperatures determined by x-ray methods are closer to the values calculated from thermodynamic functions-nthalpy and entropy (HZ9,- Hoo) and S",,,. For example, the characteristic temperature may be found from the TABLE XIX CHARACTERISTIC TEMPERATURES OF COMPOUNDS A"'BV FROM HEATCAPACITY, DYNAMIC

DISPLACEMENTS. ELASTICCONSTANTS,

A N D STANDARD

ENTROPIES

AIP

AlAs

AlSb

GaP

GaAs

GaSb

InP

InAs

lnSb

Si

Ge

a-Sn

From heat capacity'

-

-

370

-

362

240

420

280

161

-

348

-

From dynamic displacements

-

395".'

32od

500'

295*,'

240d

4w'

230".'

2lod

680'

390'

150'

-

-

314' 346' 349

233'

249'

2OOh 208'

322

251

253

204

660

334

142

Q,,

( O K )

From elastic constants

-

-

From standard entropy S",,,'

321

'After Piesbergen.'" 'After Sirota and Pashintsev

353

'After Joshi and Mitra."' 'Alter Philiips.l'q

'After Pashintsev and Strota."

'After Gerhch.'"

dAlter Sirota and Gololobov.'" *From data by Rozov and S~rota.'b'' 'After Shelep.2"

'After Potter.'" 'After Garland and Park.'"

'" F. H . Herbstein. Adcun. Phys. 10, 313 (1961). Roy F. Potter, Phys. Rev. 103,47 (1956). S. K. Joshi and S. S. Mitra, Proc. Phys. Soc. i[london)76,295 (1960). 2 5 9 J. C . Phillips, Phys. Rev. 113,147 (1959). 260 C . W. Garland and K. C . Park, J . Appl. PhJ6.33.759(1962). 2 6 t D. Gerlich, J . Appl. Phys. 34, 2915 (1963). 26'aV. V. Rozov and N. N. Sirota, in Sborn. "Khimicheskaya svyaz v poluprovodnikakh i tverdykh telakh" (Chemical Bonds in Superconductors and Sollds), p. 180, Minsk, 1965. 257

*'*

132

N.

N.

SIROTA

approximate relation

where

.f (@/TI is the Debye function. In Table XIX some characteristic temperatures of Ill-V compounds are listed as calculated from elastic constants, heat capacity curves, x-ray methods, and from values of the standard entropy S i g 8 .In Table XXII, which appears in a later section, are listed values ofcharacteristic temperatures, as determined from melting points of A"'BV compounds.

VII. Melting 12. CHANGE OF SPECIFIC VOLUME OF I1I-v COMPOUNDS UPON MELTING Specific volumes of 111-V compounds with the sphalerite structure and those of group IV elements of the periodic table having the diamond structure decrease upon melting while specific volumes of most other crystal bodies increase upon melting. Wartenberg'" and Logan and Bond263 have shown that the specific volume of silicon decreases by 9-10% in melting. Schneider and Heymer18 give the value AV ~- - 9.6% (see also Massing264). Yolid

The decrease of specific volume in melting of germanium is 4.9 ~ ~ which ;,'"' is obtained by Wartenberg's method.262Values determined by other methods are -5.5 _+ 0.5%. Picnometric measurements by Sangster and Carman266 have yielded -5.4 k 0.4%, which is probably the most reliable value. Mokrovskii and Regel' give A V V = -5.08%. In Ref. 18. a value of -5.5 i 0.5% is given. Mokrovskii and Regel' were the first to do systematic measurements of specific volume in melting by metering the level in cylindrical ampules. The density change

i"+L in melting for GaSb was found to be 7.5 "4 and for InSb. 12.9(':,. or the change 262 2h3

264 265 2b6

H. Wartenberg, Naturwiss. 36, 373 (1949). R. A. Logan and W. L. Bond, J . Appl. Phys. 30,322 (1959). G. Massing, "Lehrbuch der Allgemeinen Metallkunde." Springer, Berlin, 1950. W. Klemm, H. Spitzer, W. Lindenberg, and H. J. Junker, Monutsh. Chcm. 83.629 (1952) R. C. Sangster and J. N. Carman. J . Chem. Phys. 23.206 (1955).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

133

in specific volume related to that in the solid state was 7.0% and 11.4% for GaSb and InSb, respectively (Fig. 32). Nachtrieb and Clement" determined the change in the specific volume for indium antimonide by measuring argon pressure in a constant volume system. They gave the change of the InSb specific volume

A?,

- = -13.7

K

L:CM:

0.5%

Jn S6

gal

l/

200 19.0

18.0

GaS f710

k

/6U 15.5.0

Ge

j4.0

FIG.32. Specific volume variation upon fusion of GdSb, InSb, and Ge. (After Mokrovskii and RegeL9)

13. HEATSOF FUSION(EXPERIMENTAL DATA) A number of direct experimental determinations of heats of fusion of 111-V compounds have been carried out. Richman and H o c k i n g ~ ' ~found ' the heats of melting of InP. InAs, InSb, GaAs, and GaSb from the peak areas of the

134

N . N . SIROTA

differential thermal analysis. They determined the following values, measured in kcal/mole : AH,,,, = 12.0 f 3,

AHfGaAs = 21 f 5 . These data are noteworthy. It should be pointed out however that determination of thermal effects such as heats of transition and fusion from peak areas on plots of thermal analysis curves is rather ~emiquantitative,~~' and these results should be refined. The most precise values of heats of fusion of indium antimonide were obtained by Nachtrieb and Clement268in a dropping calorimeter. Samples of indium antimonide containing 51.44 f 0.16% Sb (the theoretical value is 51.48% Sb) were inserted into a Vycor bulb and then heated in a tube furnace to a temperature approaching the melting point. The temperature was controlled within fO.5"C. The bulb was dropped at temperatures somewhat higher and lower than the melting point but very close to it. The calorimeter was placed in a 1-liter Dewar flask containing 825g of water. The water equivalent was found by mixing water of different temperatures. Evaporation losses were measured by weighing water. They were below 0.2 g. Two series of experiments using 0.2104 and 0.162 moles of indium antimonide were conducted. The latent heat of fusion

AH,

=

11.2 f 0.4 kcal/mole

was determined from six runs, and the appropriate entropy of fusion ( T , = 798°K) was found to be AH AS, = = 14.1 + 0.5 e.u./mole. 171) T, The mean heat capacity was determined simultaneously with the latent heat of fusion in the temperature ranges of 20-90, 90-170, 170-350, and 350-500°C. Schottky and B e ~ e r found ' ~ ~ heats of fusion of gallium antimonide by quantitative thermal analysis according to Oelsen et In this method, the heat liberated during the cooling of a sample was transferred to a water bath, and the temperature changes of this bath and of the sample were measured simultaneously. Thus the heat quantity transferred was determined ~

'" L. G. Berg, Termografiya; krivye Nagrevaniya : okhlazhdeniya. Moskva, lzd-vo akademii 268 26y

Nauk SSSR, 1944. 175 p. N. H. Nachtrieb and N. J. C1ement.J. Phys. Chem. 62.876 (1958). W. Oelsen, K. H. Rieskamp, and 0. Oelsen, Arch. Eisenhuttenwesen 26,253 ( 1955).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

135

(Fig. 33). The calorimeter temperature was measured by a Beckman thermometer with 0.01"Cdivisions. The calorimeter consisted of a Dewar flask containing 500ml of water covered by a movable top with openings for a mechanical stirrer and a copper tube. A sample of 15 g was placed in a Vycor bulb which was then inserted in the copper tube of the calorimeter.

200 460 720

FIG. 33. Calorimeter temperature variation on cooling of InSb and GaSb samples. (After Schottky and Bever.lZ5)

These authors determined the latent heat of fusion of indium antimonide (T', = 798°K) to be

AHflnSb = 12.2 k 0.7 kcal/mole, entropy of fusion

ASflnSb = 15.2

1

e.u./mole,

latent heat and entropy of fusion of gallium antimonide ( T , = 976°K)

AHfGaSb = 12.0

0.7 kcal/mole,

ASfGaSb = 12.3 & 1

e.u./mole.

In the method used by Nachtrieb and Clement,268the main uncertainties of measurements are attributed to difficulties of estimation of heat losses for radiation during dropping of the bulb with the sample and for water evaporation. In the method of Schottky and Bever, they are caused by difficulties in measuring simultaneously the temperatures of the sample and calorimeter with equalizing the temperature of the calorimeter.

136

N. N. SIROTA

At present, the measured calorimetric heats of fusion of 111-V compounds are within the ranges presented in Table XX. Expressing liquidus curves by the ShrederI3'-van Laar 126 equation, namely, A Inx = -- + B , (72) T and with the variables defined by the functions given in (a) and (b) following Eq. (1) in Section 2, mean values of fusion heats were calculated by the data from the phase diagrams in Fig. 15. Results are listed in Table XX. Glazov et ~ 1 . give ' ~ ~the heat of fusion of AlSb as determined from the liquidus curve in a quasibinary section of the AISb-Ge and the AlSb-Si systems with the aid of Shreder's equations. A heat of fusion AHf for aluminum antimonide of 14.2 1 kcal/mole was estimated by these authors. From the analysis of the reported data, a probable conjecture follows that the entropy of fusion is approximately the same for all the compounds. In Table XX is also a listing of values of heats of fusion, which are based on the assumption that ASf of all compounds A"'BV is a constant value equal to 12.5 e.u./mole. TABLEXX HEATSOF FUSION AND ENTROPIES OF 111-V COMFQUNDS ACCORDING TO EXPERIMENTAL DATA AlSb

GaP

GaAs

GaSb

InP

lnAs

Melting point T,. "K

AIP

AlAs

1353

1813

1518

985

1333

1218

Heat of fusion AH,,,,,,. kcalJmole

14.2+ 1' I9.W

21 i - 5 b 19.6"

12.0f0.72d 12.0f3' 18.8" 19.7'

26+3' 10.7"

28.3'

InSb 806 12.2k0.7d 11.2+0.4' 9*3b

10.2"

AHr,ca,c(assuming constant AS^ of 12.5 e u /mole) ASc, e LI :mole

16.9

22.6

1o.?' 14'

12.7'

18.9 13.R5' 12.9"

12.3 12.3d 19.1"

16.6 9.0' 14.7"

IS2 21.6b 8.7"

101 15.Zd 14.1' 1I . @

1265"

Ratio of heats of fusion and atomization' AHr z d A H a < "Averaged over liquidus branches (Table V). 'From Sharifov and Gadrhiev."' 'From Glazov and Czhen'-Yuan'."s

14. EFFECTOF PRESSURE ON

THE

From Schottky and Bever.'2s From Nachtrieb and Clement.'b8 'Values from Table X X I I I .

MELTING POINTOF 111-V COMPOUNDS

The P-T phase diagrams of 111-V compounds provide vast information on their thermodynamic properties. In particular, the investigation of the effect of pressure on the melting points makes it possible to draw a number of important conclusions on volume change with melting, values of entropy, and heats of fusion of these compounds.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

137

The shape of the curves in the phase diagram is described satisfactorily by the Clausius-Clapeyron equation, from which the slope of the curve of the melting temperature versus pressure is proportional to the ratio of the change of molar volume to that of molar entropy during fusion :

Since. in fusion of solids, the entropy of the system increases and S , - S , always has a positive sign, the sign of the shape of the melting temperature curve versus pressure is determined by the sign of the change in specific volume upon fusion. For all 111-V compounds having the sphalerite structure, which have been investigated,

In accordance with the le Chatelier principle and the Clausius-Clapeyron equation, this corresponds to a decrease in the volume upon fusion. The value of the slope of the melting teimperature curve versus the pressure

1 laT, T, ap

- .-

is proportional to the ratio of the volume change in fusion to the heat of fusion. Thus, knowing the value of the: relative slope of the curve T,(P),one can determine heats of fusion if values of volume changes in fusion are known, and vice versa, if heats and entropies of fusion are known, volume changes in melting can be determined. ' ~ determined for InSb that For example, Jayaraman et ~ z l . ~have aT dP

-=

10 kbar/deg,

and assuming"

AV

-=

V

13.7%,

they have found the value of latent heat of fusion AHrInsb= 45.4 cal/g or 10.7 kcal/mole Ponyatovskii and Peresada" ' have estimated

(E)+ =

- 9.1 kbar/deg .

1

''O

"'

A. Jayaraman, R. C. Newton, and G. C. Kennedy, Nature 191, 1288 (1961). E. G. Ponyatovskii and G. I. Peresada, Dokl. Akad. Nauk S S S R 144, 129 (1962) [English Transl. : Proc. Acad. Sci. U S S R , Chem. Sect. 144,408 (1962)l.

138

N . N. SIROTA

0 02

0.8 1. a 0

60

80

{OO

i20

P,(~barl

FIG.34. Variation in melting temperature of compounds A"'BV as a function of pressure. (After Jayaraman ef ~ 1 . ~ ' ~ )

Assuming a relative volume change in fusion of 13 %, they found the heat of fusion AHllnSb= 46.5 cal/g or 11.0 kcal/mole. Assuming"

AV V

-=

13.7%,

a heat of fusion of 11.7 kcal/mole is obtained. These values of AHf agree closely with the data of Nachtrieb and Clement268 and Schottky and Bever.I2' In Table XXI are given the main results of presently known experiments on the effect of pressure on melting temperatures of 111-V compounds, Si and Ge. They are used to estimate relative values of volume changes in fusion and latent heats of melting. In Fig. 34 curves showing the relative change in melting temperature versus pressure are plotted from the data of Jayaraman et al.272 (also see Refs. 266, 271, and 274). 272

A. Jayaraman, W. Klement, Jr., and G. C. Kennedy, Phys. Rev. 130,540(1963).

274

R. Hultgren, cited according t o Ref. 272. H. A. Gebbie, P. L. Smith, 1. G. Austin, and J. H. King, Nature 188, 1095 (1960).

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

139

15. MEAN-SQUARE DYNAMIC IONIC DISPLACEMENTS IN 111-V COMPOUNDS NEAR THE

MELTINGPOINT

Fusion of solids is fairly well described by Lindemann’s theory,275 especially in its modern form given by G i l ~ a r r y ~(also ’ ~ see Ref. 277). According to the Lindemann-Gilvarry theory, melting sets in at a temperature at which the amplitude of atomic oscillations u is a certain fraction of the distance 6 between neighboring atoms in the crystal lattice. The theory assumes that this relative amplitude of oscillation should be equal for all isotropic solids. In the simplest case of Lindemann’s theory, ion vibration in a lattice is considered as a system of linear oscillators. Assuming that at the melting point T, the ratio of the mean amplitude U, to the distance between neighbors 6 is a constant value urn

const,

-=

6

the characteristic temperature 0 of a nnonoatomic crystal of atomic weight A will be

Selecting germanium as the reference element, for which

A,

a

=

5.6574

6

=

0.433~1 = 2.480

A,

T, = 1210.4,

we obtain the following value for y~ : y~ =

197.7.

The above relation shows in particular the existence of isotopic effects for the melting and characteristic temperatures. To estimate the amplitude of the ionic oscillations in a crystal lattice, it is more reasonable to use the DebyeWaller theory of the temperature factor of x-ray scattering by lattice

oscillation^.^ 78,2 F. A. Lindemann, 2. Physik 11,609 (1910). J. J. Gilvarry, Phys. Reo. 102, 308 (1956). *” D. McLachlan and L. L. Chamberlain, Actu Met. 12,571 (1964). 278 1. Waller, Z . Physik 17,398 (1923);51,213 (1928). 278aI. Waller and R. W. James, Proc. Roy Soc. (London) A117,214(1927)

275

276

140 N . N . SIROTA

c

z

w

2 k

4

Q

0

-'?

A H , , kcal/mole

AH,'

11.7

16.9

22.6

18.9

12.3

16.6

15.2

8.9, 9.9e.f,k 10.1, 11.8f.' 10.7' 9.5" 11.6"

16.4

10.1

21.1

10. I 11.1

1 l.lk.e

AS,, e.u./mole

1 1.9k,h

13.3',' 11.9" 14.4" 12.3f.' 14.7f.I 14.86p

Preliminarj information. from Jayaraman ('t a / . 2 7 2 "Calculated from (7Tm/(7Pdata. assuming AS, = 12.2 e.u./mole for all compounds (see last paragraph, Sec!los !5). Calculated, assuming AS, = 12.5e.u./mole. 'After Jayaraman et 'After Jayaraman et al.270 'After Ponyatovskii and Peresada.'" gAfter H ~ l t g r e n . ~ ' ~ After Wartenberg.'" (I

After Logan and Bond.z63 'After Schneider and Heymer." 'After Mokrovskii and Regel.' 'After Nachtrieb and Clement." After Nesmeyanor.'6 "After Hanneman et al.' After Sangster and Carman.266 4After Klemm et ~ 2 1 . ' ~ ~

9.7"J

10.5 11.4 15.1 8.44.d 9.2p,d 8.7'.' 9.4','

P

3 5

v1

e

0"'p

5el

5 z t

z

tl el

m

5

5E m

v)

't

zU

z

F G!

142

N. N . SIROTA

To determine the mean square displacements at which melting occurs, G i l ~ a r r y ~uses ' ~ the Debye-Waller expression. In this case ~

urn2 _ -c

a2

where

=-[-+a].

6h @(O/T) 8n2rnk0 @IT

@(@IT) is the Debye function, O/T

0 xdx

m is the atom mass, k is the Boltzmann constant, 0 is the characteristic temperature. It is easy to see that the Debye-Waller expression is derived under the assumption of a constant shape of the frequency spectrum. In Table XXII are listed the characteristic temperatures of elements and 111-V compounds as calculated from relations (74a) and (74b).The coefficients were determined by using germanium as a standard. In connection with the rms displacements, a ratio of the averaged oscillation amplitude to the closest interatomic distance of ~

(U,2) 1'2 ___ --6.05%

6 was assumed. This corresponds to

TABLE XXII CHARACTERISTIC TEMPERATURES (a) CALCULATED FROM MELTING POINTS ; (b) DETERMINED FROM THE CONDITION OF CONSTANCY OF THE RATIO = C, WHERE C IS RELATED TO THE MEANSQUARE DYNAMIC DISPLACEMENTS OF THE IONS AT THE MELTING POINT

as2

Q, "K

AIP

AlAs

AlSh

GaP

GaAs

GaSb

512

317

503

371

240

493

310

499

373

195"

320'

500

2Y5'*

InP

InAs

lnSh

Si

Ge

0-Sn

From 0 = 333

270

241

330

240'

400

1x3

653

328

I25

274

184

642

328

125

230"*

21ff

6806

328

IS06

('I= 197.7)

-

From uk/Sz = C ( C = 366 x lo-') QW

-

"After Sirota and Pashintsev '* "After Pashintsev and Sirota 39 'After Sirota and Gololohov '" *After Sheleg '19

In Fig. 35 the mean square displacements are plotted versus the atom temperature in the lattices of Si, Ge, gray tin, and several 111-V compounds. In Fig. 36, the mean square displacements are given both for compounds

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

143

FIG.35. Mean square dynamic displacements in compounds A"'Bv as a function of temperature

I

0

500

! I 1000

1500

7 ,k

FIG.36. Mean square dynamic displacements of In. Cia, and P ions in InP and GaP as a function of temperature.

144

N. N . SIROTA

A"'BV and ions A"'BV, respectively. The x-ray determinations of rms displacements of A"'BV ions in the sphalerite lattice show that both the mean frequencies (characteristic temperatures) and the rms displacements are somewhat different for each type of ion. For example, according to Rozov and Sirota's corrected data,26l a at 300°K the characteristic temperature of gallium in gallium phosphide is OGa= 240"K, phosphorus is about 565"K, while the general characteristic temperature of the compound is approximately 500°K. The characteristic temperatures of indium and phosphorus in indium phosphide are 0,"= 135°K and Op = 550"K, respectively, and the general characteristic temperature of the compound is approximately 400°K. These characteristic temperatures correspond to mean displacements at room temperature : in gallium phosphide -

-

(u&a)l/2 = (up2)'/2,

and in indium phosphide -

-

(u&)1'2 N (UpZ)1'2.

Similar behavior is observed in other compounds. For example, according to Sirota and Yanovich's data,27yat 360°K in the compound ZnTe, mean displacements are

(zn)1/2 0.13 A =

(z)l= I2 0.09

and

A.

Waller and James278ahave found, for NaCl,

-

(u&) = 0.22

A

-

and

(u;,)

= 0.24

A

at

290°K.

For the physics and thermodynamics of 111-V compounds, values of rms displacements are undoubtedly of considerable interest. Although initial progress has been made, the values quoted above should be considered tentative. The value of rms displacements may be a direct measure of the entropy of a compound. Particularly, L ~ r n s d e n 'has ~ ~ paid attention to this aspect of the problem. The Lindemann-Gilvarry theory developed for monoatomic solids may be applied to compounds of the A"'BV type only for approximate evaluations, In the case of indium and gallium antimonides, averaged values of 0 estimated by fusion temperatures in terms of Lindemann and Gilvarry's theory appear to be relatively close to experimental values determined, for example, by calorimetric analysis. However, for arsenides, and especially phosphides, calculated values exceed calorimetric ones. This difference may be attributed to the difference in atomic weights 279

N. N. Sirota and V. D. Yanovich, in Sborn. "Khimicheskaya svyaz v poluprovodnikakh i tverdykh telakh" (Chemical Bonds in Semiconductors and Solids), p. 21 1, Minsk, 1965.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

145

of the components, in the binding energy of A-A, B-B, A-B atoms in A"'BV compounds and, therefore, in the frequency spectra of the ion oscillations in the lattice. At the melting point of a compound, the free energy of the liquid phase equals the free energy of the solid phase in the melt, namely,

G;, = H ; , - TrnSSTm= H;, - TrnS,'. (75) Thus the entropy of fusion AS, is equal to the ratio of the heat of fusion to the melting point

Since the transition from solid to liquid state may be considered as a certain polymorphous transition, in this case the entropy of fusion may be roughly assumed to be

c; - c,' dT

ASf = or

J O

In the case of an Einstein spectrum for the oscillations, it follows from the above relation that the fusion temperature is approximately proportional to the ratio of the heat of fusion to the logarithm of the ratio of characteristic temperatures of solid and liquid phases,

Similar relations have been obtained by €;renke1280(also see Ref. 281). Assuming similar temperature dependencies of heat capacities of liquid and solid phases, it is possible to estimate the discontinuities in the changes of the Debye characteristic temperature and the rms displacements. Figure 37 shows variations of mean square dynamic displacements of group Ixelements and A"'BV compounds at fusion as well as variations in the ratio u2/d2. 280

*"

Ya. 1. Frenkel, Kineticheskaya teoriya zhidkostey, SSSR, Moskva, 1945. Moskva, 1958 [English Transl.: Kinetic theory of liquids]. K. F. Gertsfel'd, "Kineticheskaya teoriya materii," Moskva, 1935 [K. F. Herzfeld, "Kinetische Theorie der Warme." Braunschweig, 19251.

146

N. N. SIROTA

-

't

35

GaAs

6

30

Gas6

25

20

15 10 5

0

I

400

800 /200 JPK

1200 1600 TPK

400 800

FIG.37. Variation of mean square dynamic displacements at fusion of compounds A"'BV

I

I

2000 .I

i n 1

I

II

I

I

I

I

I

I

I

1

I

I

I

I

I

I

I

I

I

I

I

\ 1 \ 1 1

FIG.38. Variation of scattered x-ray intensity from liquid InSb as a function of sin O / i . (After Krebs et aL6)

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

147

Despite a considerable increase in the rms dynamic displacements, the decrease of the specific volume in the melting of Si, Ge, a-Sn and 111-V compounds with the sphalerite lattice is attributed to the fact that melting is accompanied by a change in the coordination number. As the experiments of Krebs et d 6have shown, in the fusion of InSb, the coordination number changes from z = 4 in the solid phase to z = 6 in a liquid phase-the latter corresponding to the NaCl structure (Fig. 38). This change of the structure of the short-range order explains the decrease of the specific volume of InSb in fusion. According to Krebs et aL6 solid state interatomic distances in the sphalerite lattice of InSb are 2.8 A, and in a liquid state with short-range order similar to the NaCl structure, distances between neighboring atoms are on the average 3.17A. In accorda.nce with these data, the volume of atomic oscillations in the solid state is 27 A3 and in a liquid state, 32 A3. These values give an idea of the factors which cause considerable change in entropy during fusion of A"'BV compounds.

VIII. Thermodynamic Properties, Energy Bands, and the Periodic System 16. THERMODYNAMIC PROPERTIES OF An'BV COMPOUNDS IN CONNECTION WITH THE POSITION IN THE MENDELEEV PERIODIC SYSTEM In Table XXIII we have compiled basic data on thermodynamic properties of the A"'BV compounds. Despite their incompleteness and, in some cases, lack of accuracy, the data available at present do permit some conclusions to be derived regarding the thermodynamic properties of the whole group of semiconducting compounds A1"BV.In particular, these properties depend pronouncedly on the position of the components in the Mendeleev periodic table. Melting points of the A1"BVcompounds with the sphalerite structure are higher, the lower the period number of the components and the lower the average atomic number of the compound Z,m

+

%,v

2

Heats and free energies of atomization and formation also pronouncedly increase with decreasing period number and average atomic number of the constituent elements. K a p ~ s t i n s k i i has ~ ~ formulated ~ , ~ ~ ~ an empirical rule for establishing a semiquantitative correlation between the positions of the constituent elements in the Mendeleev periodic table and the heat of formation of the znz A. F. Kapustinskii, Izo. Akad. Nauk S S S R , Otd. Khim. Nauk No. 6,568 (1948). 2 8 3 A. F. Kapustinskii and 0. Ya. Samoylov, Izv. Akad. Nauk SSSR, Otd. Kkim. Nauk No. 4,

337 (1950).

TABLE XXIII

THERMODYNAMIC PROPERTIES OF BN AfGga.

kcallmole

60.7" 59.Xb 60.3'

BP

199d

AIN 57.4' 64* 76.47* 64h 75.6',' 76.1'

AIP

AIAF

COMPOUNDS A"%"

ACCORDING TO DATAOF

AlSh

GaN

Cap

23.0'

24.9"'

17.2",p

35.4'

29"

GdAs

VARIOUS

CaSb

InN

AUTHORS* InP

InAs

lnSb

12.3(A~,)~ 9.94',' ZS.X(AS,)~ lov 18.00'"" 13.5(Sb2)"" 4.6" 9.4" IY.4-' 11.1' 20.96sR 9.WR

21' 22" 21.6(P4)q 22.1(P,)hb 21.5'

14.8' 13.8" 13.5' 17(A~,)",'~ 12.4hh

8.0" 6.94' 8.22" 7.3466 7.84mK 6.8" 7.78"

20

10 5

22

13

80

17.418,1ER8

12.7""

I 1.6hh

123"-

5.76' S.76kL."" 6.74'' 6.14'" 6.44$'

90

130

12

6

1.4JJ 1,34KR

1 I 3""

~OII."'

25

14.Y"

68. IS'

9.01' 9.4'8

AG;,,, kcalimole

I8 5

24.4

12",n

I l.l*

3.V

6.0" 9 32Kx

AS&, e.u./mole 12 0

2

150.9'"

10

159.4" 140.9" 145.96"

137'' 138.4'" 137.SgZ

162

146

124.3

133.8" 122.4"' 133.11" 131

203'"

209 7

182

1652

130.5'" AG;$,. kcal/mole 190

145

3.98'*."" s.o(yc

4.02" 4.6Sg8

55

AH&. kcal/mole

2.6hh

3

4

4.0

154.6""

130"" 1384""

128"" 12754'*

138

159

130

127

II8" 118.7'" 118.4*' 118

134""

110.3""

109.1' 108.3'9

131.5

111.8

1083

150s

175"

r r E

N

68.3"

AS;ba, e.u./mole 64

4.8' S",,,.e.u./mole

67.3

126.5

15.36" 14.1"

62 1" 42.9ca

64.2l' 66.1" 64.1a8

69.W'

67.3""

63.7" 64.Y'

505

64

92.5

61

64

11.4" 15.34''

18.18"

14.28" 13.4"'

20.84" 20.60"

15.0

18.0

19.6hh 18.1" 15.9'' 18.1

20.6

si

8z P

z 14 2""

AH,, kcal;molc

19.v-

21

23.3"

1 5"

12.0+0.72'

12013" 19.7"

19.6"" 18.8""

* T111c L---._ 1 . piciciicu VLIUCS, I.c., ---I-

. l

.I.

~

LIIC

"After Gal'chenko ef ~ 1 . " ' ~ 'Alter Hildenbrand and 'After Dworkin et aI.'59 Alter Williams. '" 'Alter Kelly. After Brewer.101 After Neugebauer and Margrave.'6b After Schissel and Williams.'b' 'After Mah el af.'68 'Alter Sato.'69 XAfterHoch a n d Hinge."'* 'After Kubaschewski and Evans " "After Hahn and J ~ z a . ' ' ~ 'After Marina et of." 'Alter Vigdorovich el 0 1 . ' ~ q Afler Drowart and Goldfinger.'SU 'Alter Schottky and Bever.'?'

26+3" 10.7"

12.210.7' 11.2~0.4'" 9 1 3" 10.2"

U

.

d u e si adupieci in this ariicie, appear in boiaface type. Alter Wagner."' 'After Gadzhiev.'8' "After Ermolenko and Sirota."5 ' After Kleppa.lR' "Alter Goldfinger and Jeunehomme."' After Weiser.'49 ?After Sharilov et 'After Gutbier."' "After Renner. "' OnAfter Goldfinger.'" '
"After Schneider and K l o t ~ . ' ~ ' "Alter Gadzhiev and Sharilov."' After Clusius and Schachinger.'88 " After Gol'dshmidt.'o Alter Kozlovskaya. ""Alter DeSorbo.lsg ppAlter Gadzhiev and Sharifov.'" 'Iq Alter Rubinshteiin and K ~ z l o v s k a y a . ' ~ ' " After Piesbergen."' "After Molt. 19' "After Richman and Hockings."' ""After Nachtrieb and Clement.'b8 "' Alter Glazov and Cihen'-Yuan'."" *' Mean values from liauidus branches i n Table V. 1111

"'

>

z tl

si

150

N . N . SIROTA

compounds. According to this rule the heat of formation of the compound is a linear function of the logarithm of the atomic number in the range of both groups and the periods in the periodic table. In application to the 111-V compounds, it is only partially fulfilled (Fig. 39). At present however it is difficult to judge to what extent the discrepancy is accounted for by the inaccuracy of experimental data on heats of formation and atomization of the compounds. From the viewpoint of the Kapustinskii rule, the experimental values of heats of formation of compounds BN, Gap, and certain others are objectionable. For example, the experimental value of 17 kcal/mole for the heat of formation of gallium phosphide seems to be underestimated by a factor of two. The value 26.5 kcal/moie obtained in experiments on burning G a P in a calorimetric bomb in an oxygen atmosphere seems to be more nearly valid. A Hi98 7 K C O 1. mole

80

70 60

50 40

30 20

to 0 FIG. 39. Heat of formation of compounds A"'BV versus logarithm of BV element atomic number.

Heats of formation and atomization and temperatures and heats of fusion of the A"'BV compound are also functions of the distances 6 between the nearest neighbors of the crystalline lattice (Fig. 40)47,284,285 and the ratio between the distance 6 and the sum of ionic radii of the components A f 3 and B+' (Fig. 41). The results given in Fig. 40 indicate that the change in 284

285

A. F. Kapustinskii and Yu. M. Golutvin, Izu. Akud. Nauk SSSR, Otd. Khim. Nauk No. 1.3; No. 2, 192 (1951). N. N. Sirota, VIII Mendeleevskii s'ezd. Sektsiya metallov. Problemy polyprovod. Referaty dokl. Moscow Akad. Nauk SSSR, 1959.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

151

FIG. 40. Variation of melting points, heats for formation, BE, and @'A as functions of distance between neighbors in A1"BVcompouncls.

FIG.41. Atomization energies of compounds A"'B" versus ratio of interatomic distances to ionic radii.

152

N. N. SIROTA

T,, AE, AH;\8 due to the change in 6 should be considered not as single-valued strict relationship, but rather as the tendency of the properties mentioned to vary with 6. The values of interatomic distances in 111-V compounds as well as in crystals of group IV elements are satisfactorily described by the Hume-Rothery relation 6

n

-

a 2"'

where n is the principal quantum number, Z is the atomic number, and the exponent x depends on the period number of the element in the periodic table286,287(Fig. 42). A relation of this type may be used to describe the dependence of thermodynamic properties of elastic, thermal, and other physical constants on the atomic number and the principal quantum number (Fig. 43). As is evident from the foregoing there is quite a definite relationship between many physical properties of the 111-V compounds and the energy of the lattice (heat of atomization). For example, heats of formation, and of fusion, microhardness, moduli of compression, inverse values of expansion coefficients, surface energy, and certain other quantities are essentially linear functions of the energy of the lattice (heat of atomization) (Fig. 44).

FIG.42. Logarithm of ratio of interatomic distance to mean principal quantum number versus logarithm of sum of atomic numbers of elements in compounds A"'BV. 286

287

W. Hume-Rothery, Proc. Roy. SOC.(London)A197, 17 (1949). W. Hume-Rothery and G. V. Raynor, "The Structure of Metals and Alloys." Inst. of Metals, London, 1962.

2. HEATS OF FORMATION AND TEMPBIRATURES AND HEATS OF FUSION

153

A\ $9

100 75

50

25

IL

I

25

50

9

FIG.43a. Variation ofatomization heats, temperatures ofmelting, heats of formation, and @*A of compounds A"'Bv versus the mean atomic number of the component elements.

/w(%) b4 12

to 08

0.6 0.4 0.2 13

1.4

l5

16

FIG.43b. Logarithm of ratio of energy of formation to mean principal quantum number for compounds A"'Bv versus logarithm of the sum of atomic numbers of components.

154

N. N. SIROTA

Ga P

FIG.44. Relations between heats of formation, melting points, and O2 A of 111-V compounds and atomization energies.

THE FORBIDDEN ZONEAND COMPOUNDS A I I I B ~

17. WIDTHOF

THERMODYNAMIC PROPERTIES OF

As the width of the forbidden zone AE is characterized by the work expended on the transition of an electron from a valence band into the conduction band and, hence, on the transition of a system from one thermodynamic state to another, it is evident that there must be a relation between thermodynamic properties of compounds and the energy of activation, i.e., width of the forbidden zone. Welker249.251called attention to the relation between the width of the forbidden zone, the effective ionic charges and the energy value of the ionic component of the bond. Sirota13 pointed out the proportionality between the width of the forbidden zone of the group IV elements, as well as semiconducting compounds, and the heat of sublimation (energy of the lattice, atomization heat). The width of the forbidden zone increases with the energy of interatomic bindings, i.e., energy of atomization and energy of the lattice.

2.

HEATS OF FORMATION AND TEMPER.ATURES AND HEATS OF FUSION

155

Linear proportionality may be established between the heat of atomization (energy of the lattice) and the width of the forbidden zone, namely, AE

=

K(AHa' - A ) ,

(79) where A and K are constants13 (Fig. 45). Manca288has confirmed this relation. (See also Ref. 301 .) In Fig. 46 temperatures of fusion, heats of formation, and @ A are related to the width of the forbidden zone. There are similar relations among numerous properties of A"'BV compounds, which in one way or another characterize the energy of interatomic binding in the crystal lattice of the compound, and the moduli of volume compression or the inverse value of the squared dielectric permeability, as pointed to by I ~ f f and e ~ Mott ~ ~ and Jonesz9'

A€,eL

2.0

15

10

0.5

FIG. 288 289

290

45. Variation of the forbidden zone width in IIIl-V compounds with atomization energy.

P. Manca, J . Phys. Chem. Solids 20,268 (1961). A. F. loKe, "Fizika poluprovodnikov Moskva" Izd. vo Akademii Nauk SSSR, 1957. [English Transl. : "Physics of Semiconductors." Infosearch, London, 19601. N. F. Mott and H. Jones, "The Theory of Propierties of Metals and Alloys." Clarendon Press. Oxford, 1936.

156

N. N. SIROTA

@A

75

50

h,oK ,Hpmn & l 9

mole

60 2:

40 20 I

0

1

n E, eV

FIG.46. Relationship between formation heat, melting point, and @'A of 111-V compounds and the forbidden zone width.

b00/4 0.012 fl.fli0

OD08

0.006 0.004

0.002

FIG.47. Variation of the square of the inverse dielectric permeability with the width of the forbidden zone.

2.

HEATS OF FORMATION AND TEMPEFlATURES AND HEATS OF FUSION

157

(Fig. 47), the inverse value of the linear expansion coefficient, the dissociation pressure, etc. S i r ~ t a , ~ ~ who , ' ~ ' pointed ~ out the: relation between AE and AH"' and between physical properties and AH"', did not distinguish between the concepts of the lattice energy, atomization energy, and sublimation energy of elements of group iV and A"'BV compounds; these energies were accepted as the H," resulting from the relation

G

=

Hoo

+ AH,

-

TS,,

where if T -+ 0, G -+ H,".

(80)

Later on, O r r n ~ n temphasized ~~' the necessity of comparing the width of the forbidden zone with the energy of atomization. It should be noted, however, that the lattice energies of covalent coinpounds and monoatomic substances are at the present time practically identical concepts and correspond to the concept of atomization energy. In application, however, to ionic compounds, these concepts must be differentiated. It should be also noted that there is a certain correlation between the width of the forbidden zone and the energy of the lattice (according to Born) of ionic compounds. S e r n e n k ~ v i c h ' has ~ ~ put forward the proposition that it is expedient to compare the width of the forbidden .zone and the standard free energy of atomization of A"'BV compounds, i.e., with the energy of dissociation into vaporous components. SubsequentIy a similar point of view was expressed by Sharifov and A b b a ~ o v Goodman295 .~~~ has related the widths of the forbidden zone of the group IV elements and of A"'BV compounds to the melting point, length of bond, and interatomic distances. M i y a ~ c h has i~~~ pointed out the existence of a proportionality between the width of the forbidden zone and the ratio of distances d between nearest atoms in the lattice of the A"'BV compound and the sum of radii of positive ions of the A + 3 and B+' components. For example in the case of InSb

=

2.80

A

the ionic radii ~ n =+0.81 ~ Sb+'

.u 0.62

A, A,

290"Unfortunately in Table 2, p. 127 (of Ref. 13), conversion factors were omitted in the calculation of the energy of the lattice (energy o f atomization), as pointed out by S a r k i s o ~ . ~ ~ ~ '" B. F. Ormont, Zh. Neorgun. Khirn. 5 , 255 (1960); Zh. Fiz. Khirn. 33, 1455 (1959) [English Transl. : Russian J . Inorganic Chem. 5,123 (1960); Russian J . Phys. Chem. 33,4 (1959)l. 2 9 2 E. S. Sarkisov, Zh. Fiz. Khim. 28,627 (1954). 2 y 3 S. A. Semenkovich, Tezisy Vsesoyuznogo soveshchaniya PO khimicheskoi svyazi v poluprovodnikakh, Minsk, 1962. 2 9 4 K. A. Sharifov and A. S. Abbasov, Dokl. Aka'd. Nauk S S S R 157,430 (1964) [English Transl. : Proc. Acad. Sci. U S S R , Phys. Chem. Sect. 157,739 (1964)l. 2 9 5 C. H. L. Goodman, J . Elecrron. 1, 115 (195.5). 296 T. Miyauchi, J . Phys. SOC.Japan 12, 308 (19.57).

158

N. N . SIROTA

yield the ratio

6/(rIn+ + rSb+ 5 )

=

1.95.

Here we should also cite the work of S a i d ~ v , ~S ~~ 'c h e t , * P ~ *r e ~ n o v , ~ ~ ~ Z h ~ z e , ~and " other^.^" The width of the forbidden zone at absolute zero is proportional to the energy of the lattice (energy of atomization at T + 0, U , = H,"). It decreases with increasing temperature in most semiconductors. We may point out the relation between the temperature dependence of the width of the forbidden zone and the enthalpy. In Figs. 48 and 49 a comparison is made between changes in the width of the forbidden zone with enthalpy and with temperature for InSb and GaSb, using published data,' 7 3 3 3 0 2 , 3 0 3 as well as the temperature variation of the enthalpy. It follows from these figures that changes with temperature of H , and AE as well as of C, and

a AE aT

are of similar character (Fig. 50) : AE=KAH and

a AE

-- -

aT

KC,

In these expressions if AE is measured in eV and AH in kcal/mole, in parfor InSb and K = -7.5 x for GaSb, ticular, K = -4.7 x according to published data.' 7 3 , 3 0 2 , 3 0 3 The widths of the forbidden zones of A"'BV compounds change regularly with the position of the elements in the Mendeleev periodic table (Fig. 51). The width decreases with increase in the number of the period in which the components A1"BV are located. In this respect there is qualitative similarity with changes in heats of atomization and formation, temperatures and heats of fusion, e 2 A ,and compression moduli with the position of the components in the periodic table (as is partially indicated in Fig. 52). The values decrease with increasing numbers of the period (the principal quantum number) where the element is located. 29' 298

299

300

301

M. S . Saidov, Dokl. Akad. Nuuk U z S S R 20, 17 (1963). J. P. Suchet, Compt. Rend. 255, 1444 (1962): J . Phys. Chem. Solids 21, 156 (1961). V. A. Presnov, Fiz. Tverd. Tela 4, 548 (1962) [English Transl. : Soviet Phys.-Solid State 4, 399 (1962)l. V.-P. Zhuze. Zh. Tekhn. Fiz. 25.2079 (1955). H. C . Gatos and A. J. Rosenberg, in "The Physics and Chemistry of Ceramics" (C. Klingsberg, ed.), p. 196. Gordon and Breach, New York, 1963.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

0

200

400

600

159

800 1000 1200 H,-H,,Caf/mo~e

FIG.48. Width of the forbidden zone in GaSb as a function of (a)enthalpy and (b) temperature. (Data from Roberts and Q ~ a r r i n g t o nquoted , ~ ~ ~ also by Welker and we is^.^^^ 0-AE determined by extrapolation from steepest part of absorption curve [(Khv)"' vs h v ] ; @-AE determined from frequency at which K - K O = 100 cm-.').

IX. Conclusion The 111-V compounds practically are very important and scientifically very interesting objects for investigation of thermodynamic and physical properties. Having a simple crystallographic structure, the compounds A"'BV with the sphalerite structure constitute a homologous group in which there is a transition of physical properties, corresponding to very narrow 302 303

H. Welker and H. Weiss, Solid State Phys. 3, 1 (1956). V. Roberts and J. E. Quarrington, J . Electron. 1. 152 (1955).

160

N. N. SIROTA

Y-Hv

cal

mXe

0 400 800 1200

i6cO

0

40

80

120

160

200

240

280

TaK

FIG 49 Width of the forbidden zone In lnSb as a function of (a) enthalpy and (b) temperature (For source of data and meaning of symbols, see Fig 48 )

forbidden zones AE up to relatively broad ones where the energy of the interatomic binding changes noticeably and where the semiconducting and other physical properties vary over a wide range. Good correlation exists among the position of the components in the periodic table, the structure of electronic shells, the change in the energy of interatomic binding, and other physical properties. At the same time, the analysis of the available thermodynamic data on heats of formation and atomization and temperatures and heats of fusion of these compounds has revealed that the reported thermodynamic properties are scattered, are incomplete, and lack agreement. Systematic studies of thermodynamic and physicochemical properties of the 111-V compounds would be very useful.

2.

HEATS OF FORMATION AND TEMPERATURES AND HEATS OF FUSION

161

FIG.50. Temperature derivative of the width of the forbidden zone as a function of the heat capacity C , (in cal/g-atom-deg) for GaSb (0)and InSb ((3).

FIG.51. Dependence of the width of the forbidden zone in Ill-V compounds on the position of the components in the periodic table.

162

N. N . SIROTA

FIG.52. Dependencies of energies of atomization and formation, melting points, and e 2 A of compounds A"'Bv on the position of the components in the periodic table.

ACKNOWLEDGMENTS The author gratefully acknowledges the assistance of N. M. Olekhnovich and E. K. Stribuk in compiling tables, plotting graphs, and in the preparation of the manuscript.

CHAPTER 3

Diffusion Don L . Kendall I . INTRODUCTION .

. . . . . . . . . . . . . . . . . . . . . 111 . DIFFUSION IN COMPOUNDS . . . . . . . 3. Self-Diffusion in Pure Compounds . . . . 4. Self-Diffusion in Impure Compounds . . . 5 . Impurity Diffusion in Cornpounds . . . . 6 . Parallel Mode Diffusion . . . . . . . 7. biterstitial-Substitutiorlal Diffusion . . . . Iv . SELF-DIFFUSION IN I1I-v COMPOUNDS . . . . 8.AlSb. . . . . . . . . . . . 9 . GaAs . . . . . . . . . . . . 10. GaSb . . . . . . . . . . . . I 1 . InP . . . . . . . . . . . . 12. InSb . . . . . . . . . . . . V . IMPURITYDIFFUSION I N 111-V COMPOUNDS. . 13. Zn and Cu in AlSb . . . . . . . . 14. Zn in GaP . . . . . . . . . . 15. S. Se. Te. and Sn in GaAs . . . . . . 16. Zn, Cd. Hg, and Mg in GaAs . . . . . 17. Mn, Cu, Tm, In, Ag, A u , and Li in GaAs . . 18. Summary of Diffusion in GaAs . . . . . 19. In, Sn, Te, and Li in GaSb . . . . . . 20 . Z n i n I n P . . . . . . . . . . . 21. S, Se, Te, Sn, Ge, Zn,Cd, Mg, arid Cu in InAs 22 . Te and Sn in InSb . . . . . . . . 23 . Zn, C d , andHgin InSb . . . . . . . 24 . Cu, Au, Co, Ag. Fe, and Pb in InSb . . . 25 . Summary of Diffusion in lnSb . . . . . VI . SUMMARY AND CONCLUSIONS . . . . . . 11. DEFECT EQUILIBRIA IN C0MPOUND:I 1. Disorder in Pure Compounds . 2. Disorder in Impure Compounds .

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

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

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

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

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163 164 165 173 181 182 183 183 184 185 189 190 190 191 191 192 193 196 196 197 205 227

234 235 236 236 237 239 253 256 256

I. Introduction The compounds formed by combining atoms from group IIIA with those from group VA are very interesting from both a scientific and a technological standpoint . Their electrical and other physical properties bridge the gap between the purely covalent materials in group IVA and the predominantly ionic I-VII compounds, the alkali halides . Over the last 10 to 15 years, single 163

164

DON L. KENDALL

crystals of most 111-V compounds have become available. The electronic band structure, the attendant transport and optical properties, and the effect of impurities on these have been studied in considerable detail. As a result, the physics of perfect crystals of this family is relatively well understood. However, little is known about the defect structure of any of the 111-V compounds. The pronounced tendency of this family to combine in nearly exact stoichiometric proportion leads to very small native defect concentrations. This complicates the study of such defects by electrical and optical means, since quite small impurity concentrations will mask any contributions of the native defects. Also, electron spin resonance techniques are of much less use in 111-V compounds than in 11-VI and group IV semiconductors. This is due to nuclear spin contributions which lead to a sharply reduced signal-to-noise ratio for electron spin resonance originating at the defects. Thus, self-diffusion and impurity-diffusion studies in these compounds can provide information on defect structure when more direct methods may not be applicable. This follows from the fact that native defects must be involved in any diffusion process occurring in a crystal (with the exception of simple impurity interstitial motion). The aim of this chapter is to review the literature regarding diffusion in 111-V compounds, particularly as the results relate to defect behavior and mode of diffusion. As will be seen, very few experiments have been undertaken that relate directly to these. These data are of considerable technological importance since diodes, infrared emitters, injection lasers, infrared detectors and transistors of several types have been made using diffusion techniques. Much of the data suffers from large experimental errors, due primarily to ignoring surface reactions during the diffusion anneal. The ambient conditions during diffusion are often neither controlled nor specified, even though this is an important factor in most experiments. Also, very large variations in the diffusion coefficient with impurity concentration have often been overlooked. Further, the significance of the ternary phases formed between an impurity and the group I11 and V elements has also been ignored.

IT. Defect Equilibria in Compounds Defect equilibria in both chemically pure and impure compounds will be discussed in this section. A notation similar to that of Kroger and Vink’.’ will be used throughout with some modification to be consonant with most of the semiconductor solid state literature.

’ F. A. Kroger and H. J. Vink, Solid State Phys. 3,311 (1956). F. A. Kroger, “The Chemistry of Imperfect Crystals.” Wiley, New York, 1964.

3.

DIFFUSION

165

1. DISORDER IN PURECOMPOUNDS The atomic defects possible in a pure compound AB are basically of three types, namely, interstitial, vacancy, and antistr~cture.~ The interstitial type consists of metal (or cation) interstitials Ai and anion interstitials Bi. These can each exist in two different kinds of sites, one being an interstice surrounded by A atoms and the other by B atoms. Of the vacancy types, the possible defects are metal vacancies V, and anion vacancies V,. The two antistructure defects are the metal atom (ion) on an anion site, A,, and an anion on a metal site, B,. In addition, any of the above may become ionized by accepting or donating electrons. Also, association of two or more of the above defects may occur. Parenthetically, it should be pointed out that the nomenclature of metal (or cation) and anion used here for A and B, respectively, should not be thought to have the same physical meaning in 111-V compounds that it has in true ionic compounds. In fact, the rigorous acceptance of the terms would lead one to believe that the metal atom is generally electron deficient and the so-called anion is generally electron affluent. In actual fact, 111-V compounds are predominantly covalently bonded with an ionic contribution of only a few per ~ e n t . This ~ , ~implies that the electrons are shared between the 111 and V atoms, so physically the A site really is electron rich and the B site is electron poor. With this observation made, however, the metal-anion nomenclature will be retained since it allows a common treatment of all twocomponent compounds irrespective of their type of bonding. The basic types of disorder possible with the defects mentioned are threefold if defects of the same nature are involved (i.e., all interstitial, all vacancy, or all antistructure). These are termed symmetrical by Kroger,6 whose treatment is followed here. Three additional types of disorder, termed asymmetric, can be formulated involving combinations or hybrids of the symmetric types. The symmetric types are :

(1) Schottky (or vacancy) disorder involving V, and V,.7-’0 (2) Interstitial disorder involving Ai amd BP7-’ (3) Antistructure disorder involving AB and B,.

’ The dislocation is not thermodynamically stable and is thus not included in this discussion. Its effects on the interstitial-substitutional diffusion process are treated in Section 7. H . Welker and H . Weiss,SoiidStutePhys.3,1(1956). L. Pauling, “The Nature of the Chemical Bond”, p. 95R.Cornell Univ. Press, Ithaca, New York, 1960. Ref. 2, p. 406ff. W. Schottky and C. Wagner, Z. Physik. Chem. BI1, 163 (1931). C. Wagner, 2. Physik. Chem., Bodenstein Festband 177 (1931). C . Wagner, Z. Physik. Chem. B22,181 (1933). l o W. Schottky, Z. Physik. Chem. B29.335 (1935).

166

DON L. KENDALL

The asymmetric types are : (4) Frenkel (or vacancy-interstitial) disorder involving V, and Ai, or alternatively VB and Bi. ( 5 ) Vacancy-antistructure disorder, involving V, and A, or alternatively V, and B A . I 2 (6) Interstitial-antistructure disorder, involving Ai and B,, or alternatively Bi and A,."

These types of disorder will be discussed individually with more detailed attention given to those that may be involved in 111-V compounds. In order that the point ofview adopted may be understood more fully, several characteristics of 111-V compounds relevant to their native defects are listed :

(1) Narrow phase fields, i.e., extremely small deviations from stoichiometry. (2) Very small concentrations of electrically active native defects. (3) Small self-diffusion coefficients near the melting point (relative to metals, for example). (4) Diffusion coefficients of the components of the same order of magnitude. (5) Large values of the pre-exponential Do in the diffusion equation. a. Schottky (or Vacancy) Disorder

Formation of vacancies on the two sublattices may be represented by the transfer of A and B atoms from normal positions in the solid (A, and B,) to new sites at the surface.'-1° The most direct (though not physically apparent) way to formulate this is by a reaction whereby vacancies are formed at normally occupied atom sites, designated by N.O., namely, * N.0. $ VAO + V,O, (1) where VA0 and VBo are un-ionized A and B vacancies, respectively. One way of attaining this state will be treated with Eqs. (6) and (7). For the cases of interest, the vacancy concentrations, (V,) and (V,), will be very small fractions of the concentrations of normally occupied sites, so (N.O.) will be treated as constant in the mass action relation for reaction (l),namely, (vAo)(vBo)

=

KS

7

(2)

where K , is the Schottky disorder constant, and the parentheses denote concentrations. J. Frenkel, 2. Physik 35, 652 (1926).

'* F. A. Kroger, J . Phys. Chem. Solids 23, 1342 (1962). ""Ref. 2, p. 407.

3.

DIFFUSION

167

Now each of these vacancies may become ionized by secondary reactions such as

where VAo and VB0 are shown as accepting and donating electrons, respectively. Although the above are the charges generally assigned to vacancies in ionic compounds, it should not be assumed a priori that predominantly covalent materials such as 111-V compounds necessarily behave the same way. If no substantiative information is available concerning the state of ionization of the vacancies in a given 111-V compound, it should be assumed that each vacancy may either donate or accept electrons. This implies that each vacancy may have more than one electronic state in the forbidden band. The important point from the defect equilibria standpoint is that oppositely charged vacancies on the two sublattices can react similarly to those of reaction (1)and follow a mass action relation similar to Eq. (2),namely,

( v ~ - ) (+)v=~Ks'.

(5)

The concentration of each of the charged defects will in general be a function of the impurity concentration and hence the Fermi level, whereas to first order the uncharged vacancies, V 2 and VBo, will not be affected. Hence the reactions of the uncharged or neutral species will be regarded as primary here. The concentration of the neutral vacancies of either type may be modified by controlling the vapor pressure of one of the components surrounding the crystal. If A exists monatomically in the vapor, and B exists as a tetramer, the reactions of the gaseous species with the crystal are

+ Ft A,, B4(gas) + 4VB+ 4BB. A(gas)

'JA

(6) ( 7)

These lead to the relations

KA (VA )=---, *'f

where again small vacancy concentrations are assumed, and further it is are proportional to their thermoassumed that the vapor pressures PAand PB4 dynamic activities in the vapor. These equations may be related by noting

168

DON L. KENDALL

the reaction of the gases to form the compound, which is

4A (gas)

+ B, (gas) e 4AB,

(10)

which leads to

Using this relation, the metal vacancy concentration can be obtained in terms of the anion vapor pressure, namely,

In fact, it will be noted that the Schottky disorder equation, Eq. (2), can be justified using the product of Eqs. (9)and (12). The group V gases exist primarily as tetramers when in equilibrium with their pure elements. When in equilibrium with their respective 111-V compounds, however, dimers and monomers will also exist in significant quantities in some temperature ranges. At a given temperature there is a finite range over which the pressure of the tetramer B, can vary. There is a similar limitation on the excursion of pressure of the dimer and the monomer. As pointed out by Thurmond,' this restricts the range over which the neutral vacancy concentration can be varied. For example, in GaAs at 900°C the monatomic As pressure in equilibrium with GaAs can vary by only a factor of 50, which restricts the neutral vacancy concentration variation of both A and B vacancies to the same factor. This is obviously relevant to diffusion mechanisms in these systems. The vacancies on the two sublattices may tend to associate and form vacancy pairs, or divacancies. The pair is denoted with square brackets. The pairing reaction and mass action relations are v.4

+ vB

@ [vAvBl

(13)

=

(15)

and

This can be written ([vAvBl)

with the use of the Schottky disorder equation (2). As seen in Eq. (15), a feature of this type of disorder is that the divacancy concentration is a function of temperature only (through the equilibrium constants) and is not l3

C . D. Thurmond, J . Phys. Chem. Solids 26,785 (1965).

3.

DIFFU!IION

169

affected by changing the component vapor pressure. Kendall l 4 has suggested that the AB divacancy is the defect responsible for self-diffusion of In and Sb in InSb. An argument was presented which suggested that at temperatures approaching the melting point the AB divacancy may be present in larger concentrations than either of the simple vacancies. He attributed this primarily to the fact that the vibrational modes of a large number of atoms are affected by the divacancy. Thus there is a large vibrational entropy contribution’’ to the pre-exponential term in the expression relating divacancy concentration to temperature. A directly related but more complex defect may be formed from the AB divacancy by movement of a B atom into the A end of the divacancy. This reaction and its counterpart involving the motion of an A atom can be stated

where the + and - indicate the most probable charge state of each complex when ionized. The donor form [VBBAVE~]+ is more likely to be found in p-type material where its solubility would be enhanced. The acceptor form [VAABVAI-is expected in n-type material for a similar reason. Thus these may become important as metastable defects on cooling a doped crystal, but may be less important at high temperatures where the intrinsic electrons and holes dominate. It should be noted thlat the AB divacancy and its modifications are stoichiometric defects and as such they introduce no width to the phase fields of the binary phase diagra.ms. Another modification of Schottky disorder involves association of like vacancies to form species such as [VAVA] or [VBVB]. Using the normally applied arguments, these should be acceptor and donor, respectively. Each of these may also have a modified version similar to those of [VAVB]mentioned above. For example, [vAv,] could become [VAVBBA]by simple exchange of one of its common nearest neighbors BB with V,, and similarly, [VBVB] could rearrange to [VBVAAB].Tjhese modifications are probably donor and acceptor, respectively, though weak ones (deep states). Note that each of the modified versions yields a defect of opposite type from the original. These complexes, as well as those mentioned above, may be observable l4

lS

D. L. Kendall, PhD Dissertation, Stanford University, 1965. N. F. Mott and R. W. Gurney, “Electronic Processes in Ionic Crystals,” 2nd ed., p. 26ff. Oxford Univ. Press, London and New York, 1946.

170

DON L. KENDALL

under conditions such as those obtainable in internal friction experiments such as those of Chakraverty and DreyfusI6 and the thermally stimulated current measurements of Blanc et ~ 1 . l ~ ~ b. Interstitial Disorder

This is similar to Schottky disorder except that interstitials of both types are involved instead of For this reason, the interstitial pair of atoms is sometimes called the “anti-Schottky d e f e ~ t . ” ’The ~ reaction can be formulated as

which leads to

(Ai)(Bi) = K , .

(19)

The above assumes small degrees of disorder, and for simplicity no distinction is made between different types of interstitial sites. Reaction (18) is actually not directly realizable without the simultaneous formation of vacancies, but these can be eliminated from consideration by assuming that the vacancies quickly anneal by diffusion to dislocations or free surfaces. The concentrations of Ai and Bi will be affected by component vapor pressure similar to the case of simple vacancies, albeit in the opposite sense. For example, the concentration of Bi will be proportional to the fourth root of the anion pressure if B is a tetramer in the vapor. A modification of this type of disorder is interstitial-pair disorder, where interstitials of each type tend to associate with each other into dumbbell arrangements. In its purest form the concentration of interstitial pairs is expected to be independent of external vapor pressure. c. Antistructure Disorder

The remaining type of simple disorder involving defects of similar nature is antistructure disorder,’-’ namely,

which for low degrees of disorder leads to

(Ad @A)

=

KA.

(21)

The similarity between this product relation and that of Schottky (vacancy) disorder and interstitial disorder is immediately apparent. The concentration of anions on the metal sublattice, (BA),will be proportional to the square root of anion pressure (for a tetramer in the vapor). l6

B. K . Chakraverty and R. W . Dreyfus, J . Appl. Phys. 37,631 (1966). Blanc, R. H. Bube, and L. R. Weisberg, J . Phys. Chem. Solids 25,225 (1964).

16=J.

3.

DIFFUSION

171

A modified form of antistructure disorder occurs by association of antistructure defects to give an antistructure pair, [ABBA],namely,

which for small amounts of disorder gives (lABBAI)

=

(23)

Note again that this modified form of disorder involving a paired species is not affected by component vapor pressure, the concentration of pairs being a function of temperature only. The antistructure defects will genertally be electrically active centers with B, donating at least one electron to become B A + ,and AB accepting one to become AB-. These defects may then be electrostatically attracted to form the antistructure pair of Eq. (22). This pair is expected to be essentially neutral with perhaps a deep lying donor state in the bottom half of the forbidden gap and an acceptor state in the upper half. d . Frenkel Disorder (Vacancy-lnterstitid)

Of the asymmetric types of disorder, the most important is probably vacancy-interstitial or Frenkel disorder.' It occurs by the dissociation of an atom on a lattice site into an interstitial and a vacancy as in

or

which for low degrees of disorder leads t o

These defects are usually ionized, the most likely charge states being A,', VA-, B,-, and VB+ in ionic compounds. In covalently bonded compounds, as mentioned earlier, the vacancies may have different charges than these. Also, the anion interstitial Bi may even donate an electron to become B i + so that its size becomes more compatible with the interstitial site. Paired defects are also possible for Freinkel disorder. Two possibilities are

172

DON L. KENDALL

and a hybrid form Ai

+ VB

[AiV,].

(29)

The pair of reaction (28) may be thought of as a substitutional metal atom in an excited state. Similarly, the AiVB pair resembles an excited antistructure defect. The possibility of transitions from this state to the true antistructure state under various conditions should not be ignored. There is little evidence that either of the above pairing mechanisms are important as equilibrium types of disorder, but Blanc et ~ 1 . ' ' ~have proposed the existence of a GaiVAs pair in GaAs as a possible explanation for the large concentration of defects introduced by certain annealing treatments. e. Schottky-Antistructure Disorder

Another hybrid type, which involves vacancies and antistructure defects, has recently been proposed by Kroger.6,'2 It involves the following reaction for the metal ion :

A,

A,

2vA,

(30)

(AB)(VA)2= K s A .

(31)

which leads to

This reaction is difficult to visualize, but it can be arrived at in two steps by contemplating how A, could become ABby reaction with a vacancy, namely, AA

+ VB * [ABVA]* AB + VA,

(32)

leading to

Using the Schottky vacancy equilibrium, Eq. (2), this can be rewritten as (AB)(VA)2

= K33KS

==

KSA,

(34)

which corresponds to Eq. (31). Note also the metastable defect [ABVA] of Eq. (32) which is probably an acceptor (derived from a donor).

1: Interstitial-Antistructure

Disorder

A type similar to the above is a hybrid of interstitial and antistructure disorder62l2and is given by the reaction A,

+ 2BB

AB + 2Bi,

(35)

which for small degrees of disorder is specified by (AB)(Bi)'= K I A .

(36)

3.

DIFFUSION

173

This and the previous type of disorder have not yet been recognized in any system. Before leaving this section, it should be pointed out that in actual systems several types of disorder may be present simultaneously. Kroger makes the additional point that “the simultaneous occurrence of two symmetrical types is identical with two versions of their hybrids.”’2a For example, simultaneous Schottky and interstitial disorder is equivalent to simultaneous anion and metal ion Frenkel disorder. 2. DISORDER IN IMPURECOMPOUNDS

Disorder in impure compounds is generally of the same nature as that in pure compounds, although there are additional types of defects involving associates of native defects and impurities. One of the commonly observed effects involves the solubility product of oppositely charged This will be illustrated with a rather complex example of the possible defects in a Zn-doped 111-V compound, which will serve to illustrate both the common-ion effect and the effects ofasslociation. Following this will be a brief discussion of the effects of an amphoteric impurity that can occupy either an A or a B site. Finally, the effects of incorporating a large concentration of higher order associates by donor-vacancy association will be treated. As a simple example of the common-ion effect, which is defined as an interaction between ionization reactions, consider the ionization of an anion vacancy in an AB compound, namely,

V,O

s VB+ + e - ,

(37)

the mass action expression for which is

where n is the electron concentration. Using this relation in combination with the charge neutrality condition,

and the hole-electron equilibrium relatio,n, n p = n,’ , H. Reiss, C. S. Fuller,and F. J. Morin, BellSystem Tech.J . 35,535(1956). Ref. 2, p. 158. R. L. Longini and R. F. Greene, Phys. Rev. 102, 992 (1956). 2o W. Shockley and J. L. Moll, Phys. Rev. 119,1480 (1960). Is

(40)

174

DON L . KENDALL

it is possible to specify the positively charged vacancy concentration, (VB+), in terms of either the electron or hole concentration, n or p . The symbol n i represents the electron concentration in intrinsic material. The relationship of (VB+)to the hole concentration is

As discussed in Section l a , the concentration of neutral anion vacancies, (VBo),is not affected (to first order) by the hole or electron concentration,'9920 so the term in brackets is constant at a given temperature. However, (VBo) can be affected by anion vapor pressure [see Eq. (9)], so the bracketed term will be a function of this parameter. The bracketed term can also be equated with the concentration of VB+ in intrinsic material, ( V B + ) i . u. Acceptor Impurity Incorporation

As an extension of the previous example, consider the effect of adding an impurity which is predominantly a substitutional acceptor on the A sublattice, but can also exist, although at lower concentrations, as an interstitial The impurity. This dual nature of an impurity is one type of ampl~oterisrn.'~ acceptor Zn is chosen since it can be utilized later in this work. The major types of disorder assumed will be Schottky and modified Schottky (AB divacancy). The reactions involved are based on the principles just discussed for B vacancies, so they will merely be listed here :

Zn (gas)

+ VAo+ ZnAo,

Zn (gas) + Znio, ZnAo= ZnA-

+ e+ ,

Znio ~ ) Zni+ t + e- ,

VAoe V,-

VBo

and

+ e+ ,

+ VB+ + e- ,

N.O. @ e+ + e - ,

3.

DIFFUSION

175

where the meaning of various symbols is obvious or has been previously defined. The notation N.O. here meains a filled (normally occupied) valence band state. Additional reactions may involve association among the defects listed above. Three of the more likely pairing reactions are

and

The neutral species are shown in the reactions, even though the actual reaction will generally be more likely between oppositely charged defects. Formulation via the charged defects leads to the same results, but with modified equilibrium constants. Each of the above defect pairs may also become ionized, but this is ignored. The charge neutrality condition, including all the charged defects, is

n

+ (ZnA-) + (VA-) = p + (VB+)+ (Zn,+).

(52)

Due to the power-law dependencies of defect concentration on Pz,, the neutrality condition can usually be simplified so that in any given range of pressure only two defects are involved. This is shown in Fig. I, where for simplicity the concentration of each charged defect bends sharply at the boundaries between ranges. This simlplification is known as Brouwer's approximation.21 In range I, intrinsic conditions prevail, namely n = p . In range 11, the prevailing neutrality condition is p = (Zn,-). In range 111, the neutrality condition is dominated by (Zn,') = (ZnA-), and the hole concentration becomes constant. The dependence of the various defect concentrations on the Zn partial pressure, which is here assumed to be proportional to the thermodynamic activity ofthe Zn, can be expressed by ( N ) = K P;",

(53)

where ( N ) is the concentration of any defcct, m is a simple fraction exponent, and K is a constant of proportionality. The value of m for each defect in any range is easily derived from the equations given, and these are shown in Table I.

*' G . Brouwer, Philips Rex Rept. 9, 366 (1954).

176

DON L . KENDALL

LOG Zn PRESSURE FIG.1. Variation of concentration of defects with Zn pressure over a 111-V(AB) compound.

The effect of component vapor pressure on the incorporation of Zn or other impurities can be treated in the same manner as the above with explicit account taken of the effect of anion pressure. The effect of anion vapor pressure on the vacancy equilibria has already been discussed [Eq. (9)], with the result that increasing P,, causes an increase in (V,) and a decrease in (VB). The efflecton the Zn concentration occurs through the vacancy concentration in Eq. (42), i.e., for a tetramer anion in the vapor, the expected dependence is (ZnAo)K PA',". An additional factor which complicates the issue is the dependence of P,, (or more correctly the activity of Zn) on anion pressure (activity). This may occur, for example, by the formation of an intermediate

3.

177

DIFFUSION

TAB'LE I VALUESOF THE EXPONENT I N ( N ) = K e n RELATINGDEFECT CONZn PRESSURE OVER AN AB COMPOUND. CONSTANT ACTIVITY OF A AND B ASSUMED

CENTRATIONS TO

Defect

Range 1 n = p

Range I1 p = (ZnA-) 1

0

-5

0 0 0 0 0

+f

1 1 1 1 2 1 0

1

0 -T

I

Range 111 (Zn,') = (ZnA-) 0 0 0 0

0

0

I -

0 1

2

1

T

1 1 1

2

2

5

1 3

1

1

0

0

phase such as Zn,B,, or by formation of various molecular species in the vapor. The effect of increasing PBqwill generally be to decrease P,, ,but this may be negligible at high enough temperatures such that none of the interfering compounds or molecular species is stable. As will be discussed in a later section, the effects introduced by the Ga-As-Zn ternary phase diagram are also very important in this regard.

6. Amphoteric Impurity Incorporation There are certain impurities which when added to AB compound may substitute on either the A or B sublattice. This type of amphoteric behavior" is obeyed by several group IV atoms in 111-V compounds. Observations of this type behavior have been made for Ge,23 and SiZ4in GaAs. For example, at low concentrations under normal crystal growth conditions, Si shows a strong tendency to occupy Ga sites where it acts as a donor. However, at higher Si concentrations (where Si& begins to be important in the charge neutrality condition), the Si,, species becomes important. Finally, at still higher concentrations of Si, equal concentrations of the two types are incorporated, and the electron concentration levels off at some value which may be considerably greater than the intrinsic electron concentration. 22

23 24

G. R. Cronin, G. B. Larrabee, and K. G. Heinen, to be published. L. J. Vieland and T. Siedel, J . Appl. Phys. 33,2414 (1962). Ref. 2, p. 703; J. M. Whelan et a/., Proc. Intern. CoiyC Sernicond. Phys., Prague, 1960 p. 943. Czech. Acad. Sci., Prague, and Academic Press, New York, 1961.

178

DON L. KENDALL

Increasing the As vapor pressure increases (VGJ and concomitantly increases (Si&) and lowers (SiLJ. This is discussed in detail by Kroger and Whe la^^.^^ c. Donor-Vocuncy Associution The last case of disorder that will be discussed is that of group VI incorporation in III-V compounds with attendant impurity-vacancy associa t i ~ nObserved . ~ ~ ~ examples of this are found in the systems InSb Te,25325a InAs + Te,26 InAs + Se,27 and GaAs Se.28-29a These systems are exemplified by InSb + Te, where InzTe3 is added to InSb in the melt and after solidification the sample is annealed below the melting point. At low In2Te, concentrations, normal donors are formed, namely, Tegb and its ionized form Telb. At higher concentrations considerably fewer active donors are formed than one would expect from the amount of Te added. With the addition of still more Te, the net donor concentration decreases. In at least one case, (GaAs Se), the sample actually changes from n-type to p-type at very high Se concentrations.28 We will return to this particular aspect in Section 15, as well as certain other anomalies associated with Se and Te doping. There are several ways of explaining these unusual effects, but they all have in common the enhancement of solubility of an acceptor by the Te donor. KrogerZ4”assumed that the In vacancy is a triply charged acceptor and that it tends to pair strongly with the Te donor by the reaction

+

+

+

Te& f V i 3 [TesbV~,]-~. (54) He then assumed that this associate pairs with still other Te,f, atoms until a neutral “conglomerate” is formed, the symbol for which is [3TeSbVI,]. Schottkyz9”has treated these results in a similar fashion, as have Vieland and K~idman.’~ C a ~ e and y ~Woolley ~ ~ and Williamsz5”present alternative models. This model, though artificial in some respects, does account for the observed effects, namely, the formation of one donor per Te atom a t low Te concentrations and one vacancy per three Te atoms at high concentrations. However, as Kroger points out, the existence of VG3 in the pure crystal is Z4”Ref.2, p. 618ff. 2 5 J. C. Woolley, C. M. Gillett, and J. A. Evans, 1.Phvs. Chem. Solids 16, 138 (1960). zsaJ. C. Woolley and E. W. Williams, Can. J. Phys. 44, 1200 (1966). 2 6 J. C. Woolley, B. R. Pamplin, and J. A. Evans, J . Phys. Chern. Solids 19, 147 (1961). ” J. C. Woolley and P. N. Keating, Proc. Phys. SOC.(London)78, 1008 (1961). 2 8 D. N. Nasledov and 1. A. Feltin’sh, Fiz. Tuerd. Tela 1,565 (1959); 2,823 (1960)[English Transl.: Soviet Phvs.-Solid State 1, 510 (1959); 2, 755 (1960)l. 2 9 L. J. Vieland and 1. Kudman, J . Phys. Chem. Solids 24,437 (1963); Ref. 2, p. 618. z9aG.Schottky, J . Phys. Chem. Solids 27, 1721 (1966). 2ybH.C. Casey, Jr., J . Electrochem. Sot. 114, 153 (1967).

3.

DIFFUSION

179

quite unlikely, and it probably owes its existence to ion-pairing or other effects. Other parameters, which neither he nor the original authors took into account, such as Fermi-Dirac degeneracy, band structure changes, and ion-interaction effects, are such that calculations using simple mass action principles at these very high concentrations are tenuous at best. In the following section some of these effects will be discussed.

d . Effects Due to High Concentration At high defect concentrations certain interaction effectsbecome important. In this work “high concentrations” means simply concentrations where the law of mass action is not strictly applicable, at least without modification. This may occur at quite low chemical concentrations. For example, concentrations as low as mole fraction lo-’ impurities in InSb can cause noticeable deviations in the np product relation, Eq. (40). A restriction for this discussion is that only high temperature equilibria are considered (where defect motion is possible). Also, dissociation into new phases is not considered. The effects of interest are Fermi-Dirac degeneracy ; ion interaction, which includes ion pairing, Debye-Huckel effects, and ionization energy variation ; and energy band structure variations. (1) Fermi-Dirac Degeneracy. In the normal formulation of an ionization reaction in a semiconductor, it is usually implicitly assumed that the Fermi level lies in the band gap and that it is not too near either the conduction band or the valence band. When this is so, the Fermi function denoting the electron or hole concentration as a function of Fermi energy may be replaced by a simple exponential relation. Boltzmann statistics can then be applied instead of the more exact Fermi-Dirac statistics. Under these circumstances, simple mass action expressions are adequate for describing the various defect ionization reactions. At high enough impurity concentrations such that the Fermi level approaches either band edge, the simple expressions are no longer valid. This effect, unlike most of the ones to follow, is amenable to direct calculation if the electron or hole concentrations are known. A useful quasichemical approach has been introduced by Rose3’ and Rosenberg3’ and reviewed by K r O g e ~ -which , ~ ~ uses an “electron (hole) activity coefficient” to account for variations from the simple mass action relationships at high electron or hole concentrations. However, the author has found that a graphical plot, similar to that of Shockley and of the various defect concentrations, but utilizing Fermi-Dirac statistics for the electron and hole concentrations, is generally easier to use than the activity coefficient treatment. This is 30 3’

32

F. W. G. Rose, Proc. Phys. Soc. (London)71,699 (1958). A. .I. Rosenberg, J . Chem. Phys. 33, 665 (1960). Ref. 2, pp. 212-213. Discusses an error in Ref. 31.

180

DON L. KENDALL

especially so when Fermi-Dirac statistics are required anyway to account for deionization of the impurity as the Fermi level moves through the impurity energy level. This will be illustrated in Section 23 (see Fig. 16). The activity coefficient concept is quite useful from a qualitative standpoint, however, and furthermore greatly simplifies calculations if complete ionization of the impurity can be assumed. However, when this is the case, the band edge must change as discussed in Section 3. (2). Ion Interaction. When a defect in a crystal is affected in any way by the presence of nearby impurity atoms or other defects, that defect is undergoing “ion interaction.” In the broadest sense, this includes physical effects such as interaction through strain fields and even interactions between neutral defects, which in the strict sense are not ions at all. Interactions through the charge neutrality condition such as the common-ion effect are not included, however, since these usually operate over large distances. In the following, only interactions between charged defects will be discussed. In crystals containing charged defects, certain of these may tend to be surrounded by imperfections of opposite sign, or similarly charged ions may tend to avoid each other. In this case, the Debye-Huckel theory for the interaction of ions in solution can be applied to calculate interaction parameters. Harvey33 pointed out the need for these corrections when high concentrations of electrons, holes, and other charged defects are present. When oppositely charged defects closely approach each other and tend to be bound together by electrostatic and/or strain interactions, it is generally simpler to treat the associated defects as an ion pair. These pairs will have characteristic properties of their own and in fact can be treated thermodynamically as separate species. Examples of this have already been given. An additional ion-interaction effect which likely occurs is a modification of the ionization energy of a defect near which another charged defect is present. As an example, consider a shallow donor impurity or defect D’ to which is bound an electron in a hydrogen-like orbit. An estimate of this radius is given by34

where rH is the Bohr radius for a hydrogen atom (0.49 A), E is the dielectric constant of the semiconductor, and m,*/mo is the effective mass ratio of the electron relative to that of a free electron. The size of this radius varies from 10 to lOOOA in various covalent semiconductors. With the larger electron 33

34

W. W. Harvey, J . Phys. Chem. Solids 23, 1545 (1962). H. J. Hrostowski, in “Semiconductors“ (N. B. Hannay, ed.), p. 465. Reinhold, New York, 1960.

3.

181

DIFFUSION

(or hole) orbitals, interaction between defects may be pronounced at quite low concentrations. For example, orbital overlap of the electrons from ~ . shallow donors in InSb occurs at a concentration of about loi5~ m - As the donor concentration is increased into the range where orbital overlap occurs, the ionization energy is observed to decrease.35 Similarly, an oppositely charged defect (acceptor) within the electron orbital of a donor can also decrease the ionization energy of the donor. This is an ion-pairing effect of a sort, but can be significant even when the defects are separated by a considerable distance. (3). Band Structure Changes. At high impurity concentrations the band structure itself may be affected. For example, Bernard et have postulated that new states are introduced into both the conduction and valence bands as a result of heavy doping with either donors or acceptors. These new states manifest themselves as small effective mass (high curvature) “tails” in crystal energy momentum diagrams. They also suggest that both bands are depressed in energy by the attractive potential of the ionized donors at high donor concentrations. Similarly, the repulsive potential of ionized acceptors causes have recently explained both bands to be raised in energy. Casey et the dependence of Zn surface concentration on Zn partial pressure (Section 16, Fig. 7) using the theoretical methods of Halperin and The significant feature of this work is that it predicts that the electron (or hole) activity coefficients are less than unity when band tailing occurs (which is opposite to the Fermi-Dirac correction mentioned earlier). Physically this means that, as the donor (acceptor) concentration increases, the Fermi level does not move as far into the conduction (valence) band as one would expect without band tailing. The above considerations apply to semiconductors doped with only one type of impurity (uncompensated). It might be imagined that similar considerations apply to heavily doped compensated material, but this has not been treated theoretically in any detail. ~

1

.

~

~

3

~

~

~

111. Diffusion in Compounds

Atomic diffusion in compounds is fundamentally related to the defect equilibria discussed in the previous chapter. In this respect, the mechanisms of atomic motion in compounds exhibit considerably more variety than in 35

G. L. Pearson and J. Bardeen, Phys. Rev. 75, 86.5 (1949); P. P. Debye and E. M. Conwell,

Phys. Rev. 93, 693 (1954). W. Bernard, H. Roth, A. P. Schmid, and P. Zeldes, Phys. Rev. 131,627 (1963). 37 M. B. Panish and H . C. Casey, Jr., J . Phys. Chem. Solids 28, 1673 (1967). ”*H. C. Casey, Jr., M. B. Panish, and L. L. Chang, Phys. Rev. 162, 660 (1967). B. I. Halperin and M. Lax, Phys. Rut.. 148, 722 (1966). 36

182

DON L. KENDALL

the case of simple elemental diffusion. The basic concepts, however, are similar. Reviews of these fundamentals are found in several Diffusion in compounds and semiconductors is treated by B i r ~ h e n a l l , ~ ~ B ~ l t a k sand , ~ ~others.2,44-46 3. SELF-DIFFUSION IN PURECOMPOUNDS

In compound self-diffusion, stoichiometry plays an important role. At equilibrium, this is a function of the activity of the components (or less precisely the vapor pressure) in the external phase. For example, if Schottky (vacancy) disorder is present alone, the diffusion of the metal-ions and anions will be enhanced and depressed, respectively, with increasing anion vapor pressure. For pure interstitial disorder, the opposite dependence on pressure will be observed. These statements follow from Eqs. (9), (12), and (19). On the other hand, Frenkel disorder involves interstitials and vacancies of a given type. For anion Frenkel disorder, increasing anion pressure causes the anion interstitial concentration to increase and the anion vacancy concentration to decrease. This may lead to a shift from interstitially controlled to substitutionally controlled diffusion with a minimum effective diffusion coefficient at some intermediate vapor pressure. The effects of anion pressure on metal-ion diffusion when metal-ion Frenkel disorder is present are similar in nature, though opposite in direction. For diffusion controlled by one of the modified forms of Schottky or interstitial disorder (involving AB divacancies or AB interstitial pairs, respectively), no dependence on component vapor pressure is expected. Diffusion involving antistructure defects may occur by direct atom exchange, some sort of ring exchange mechanism, or by simple nearestneighbor vacancy exchange. The direct and ring exchange processes have been shown to require inordinately large energies and so are believed to be highly unlikely.46 In a completely neutral or metal compound whose atoms are chemically similar, antistructure defects may be quite coniinon, and increasing the vacancy concentration on either sublattice should lead to enhanced diffusion of both components. In this case, a minimum in the R. M. Barrer, “Diffusion in and Through Solids.” Cambridge Univ. Press, London and New York, 1951. 40 W. Jost, “Diffusion in Solids, Liquids, and Gases.’’ Academic Press, New York, 1960. 4 1 P. G . Shewman, “Diffusion in Solids.” McGraw-Hill, New York, 1963. 4 2 C. E. Birchenall, M e t . Rev. 3,235 (1958). 43 B. I. Boltaks, “Diffusion in Semiconductors.” Academic Press, New York, 1963. 4 4 H. Reiss and C . S. Fuller, in “Semiconductors” (N. B. Hannay, ed.), p. 222. Reinhold, New York, 1960. 4 5 F. M. Smits, in “Ergebnisse der Exakten Naturwissenschaften” (S. Flugge and F. Trendelenberg eds.), Vol. 31, p. 167. Springer-Verlag, Berlin, 1959. 46 D. Lazarus, Solid State Phvs. 10, 71 (1960). 39

3.

DIFFUSION

183

diffusion coefficient of each component might be observed near the stoichiometric composition. This kind of behavior is probably more likely in certain ordered alloys than in true compounds, since the requirement of chemical similarity is seldom, if ever, met in a normal compound.

4. SELF-DIFFUSION IN IMPURE COMPOUNDS Impurities can influence self-diffusion of the component atoms whenever the native defect concentrations are aEected. This may occur through the common-ion effect, in which case the concentration of ionized donor defects will be enhanced by the addition of acceptor impurities and vice versa. This will be significant only when the impurity concentration exceeds the intrinsic electron concentration. For example, adding an acceptor like Zn to a 111-V compound should increase the solubility of VB+ (see Fig. l),and at the same time increase the diffusion rate of B. A positively charged (donor) divacancy would be similarly affected by the addition of Zn, but would lead to enhanced diffusion of both components. Likewise, the diffusion rate of an A, atom might be decreased by the Zn if V,- is the predominant defect responsible for diffusion. On the other hand, the diffusion rate of A might be increased by the same impurity if Ai+ is involved in the diffusion process. Self-diffusion of the components might also be affected if large concentrations of impurity-vacancy complexes were present. This should be quite pronounced in the case of Se and Te doping in 111-V compounds if the mechanism proposed in Section 2c is corirect. This effect, as well as the others mentioned above, can cause order of magnitude changes in the self-diffusion coefficients.

5 . IMPURITY DIFFUSION IN COMPOUNDS

The diffusion of the impurity atoms may also be affected, of course, by the effects noted in the previous section. For example, if the Zn acceptor above moves via exchange with vacancies on the A sublattice, its rate of diffusion should increase with Zn concentration if a donor divacancy or donor A vacancy is involved. It should decrease if the above defects are acceptors. If a small fraction of the Zn atoms are interstitial donors, the diffusion coefficient may be enhanced at high Zn concentrations. This is an example of a problem often encountered in impurity diffusion in elemental and compound semiconductors, namely, diffusion by parallel modes. This subject will be developed in the following section. This will then be illustrated by a generalized discussion of the interstitial-s,ubstitutional diffusion process.

184

DON L. KENDALL

6. PARALLEL MODE DIFFUSION Consider an impurity atom which can diffuse by several different modes ; for example, interstitially, substitutionally via simple vacancy exchange, substitutionally via divacancy exchange, etc. Labeling these modes of transport as j = 1,2,3,. . . ,n, the total flux J of the impurity in one dimension will be given by

(56) where D j is the diffusion coefficient of the atom in the jth mode, N j is the concentration of the atoms diffusing by this mode, and x is the distance. For diffusion by two modes this can be written47a

By manipulation this becomes

where N = N , + N,. By analogy with the normal form of Fick’s first law, the bracketed term can be called the effective diffusion coefficient 0,namely,

dN, ?N

0 = D , __

N, + D 2 pdSN .

(59)

An adequate approximation for D is often

N , 2 N This applies when the fraction in each mode is not a function ofconcentration ( N , / N = constant). It is often used as a qualitative guide, however, even when this is not the case. For the particular case of simultaneous interstitial and substitutional diffusion, the above equation becomes -

D

=

D,-

Nl N

+D

where (I) and (S) are interstitial and substitutional concentrations and Di and Ds are the interstitial and substitutional diffusion coefficients, re~pectively.~’ 4’ F. C. Frank and D. Turnbull, Phys. Rev. 104; 617 (1956). 47”Forsimplicity, interactions between different diffusion modes (cross terms), as well as variations of D, and D, with concentration and distance, are not included in the discussion. Correlation effects are also ignored.

3.

DIFFUSION

185

This equation is exact only when vacancy and other defect equilibria are maintained throughout the crystal and when the impurity concentration is less than the intrinsic electron concentration at a given temperature. It has been applied to the case of Cu diffusion in Ge44,45,47 with high dislocation density. Other cases of interest are treated in the ensuing section. For the case of Zn diffusion in 111-V compounds, Eq. (61) is valid only at quite low concentrations. This follows from the fact that the defects that control the diffusion process are primarily donors and their solubility is grossly affected at high acceptor concentrations (see Fig. 1). One case of interest is the simultaneous diffusion of an atom that exists in different states of ionization. An example of this is the parallel diffusion of a neutral and a positively charged interstitial, or the parallel diffusion of a substitutional atom existing in two or more ionization states. Allen proposed such a mechanism to explain the Zn diffusion results in GaAs, but he did not derive an expression for the effective diffusion ~ o e f f i c i e n t Kendall .~~ and Huggins have treated this in more detaiL4’ Diffusion of an ionized impurity by a vacancy mechanism suffers still another idiosyncrasy. This is the fact that an atom-vacancy exchange can only occur when the impurity is next to a vacancy. This impurity-vacancy pair must be thought to have ionization and other properties of its own. Thus it is the solubility and motional properties of this paired defect which will dominate the diffusion behavior of the i m p ~ r i t y . ~ ’

7. INTERSTITIAL-SUBSTITUTIONAL DIFFUSION As already mentioned, some impurities in semiconductors can exist in significant quantities in both interstitial and substitutional positions in the crystal lattice. An interstitial atom can become substitutional by entering a vacant site, or it may even become so by displacement of one of the host atoms by a collision process (interstitialcy mechanism). The available evidence suggests that in semiconductors the interstitial-vacancy reaction is more important than the latter, at least under equilibrium conditions. It should also be pointed out that a substitutional atom may dissociate into a vacancy and an interstitial atom. Thus the interstitial-substitutional diffusion process is often called “dissociative diff~sion.”~’ The reaction of an interstitial I with a vacancy V to form a substitutional atom S can be written as

48 49

J. W. Allen, J . Phys. Chem. Solids 15, 134 (1960). D. L. Kendall and R. A. Huggins, to be published Ref. 2, p. 801.

186

DON L . KENDALL

When a crystal is raised to elevated temperatures in the presence of an external supply of certain impurities, several processes occur. Vacancies enter at surfaces and also are formed at internal sites such as dislocations. They further may be released from previously condensed vacancy clusters. The interstitial impurities generally diffuse much more rapidly than vacancies and may establish themselves more or less uniformly throughout the crystal, perhaps at some very low level of concentration. The reaction of these interstitial atoms with the vacancies diffusing in from the surface and with those generated at internal sources will determine the apparent “diffusion” properties of the impurity. The special cases of diffusion into low, intermediate, and high dislocation density material are discussed in the following. In each case, pure substitutional diffusion by impurity atom-vacancy exchange is assumed to be negligible compared to diffusion caused by the interstitial-substitutional process. For a more detailed treatment, which takes into account the kinetics of vacancy generation at dislocations and the subsequent reaction with interstitials, see S t ~ r g e . ~ ‘ Case i. Low Dislocation Density. Here the vacancies are assumed to be introduced only at the surfaces of the crystal. Reiss and FullerS2show that with certain simplifying assumptions the effective diffusion coefficient under these conditions is related to the vacancy diffusion coefficient Dv by

where (V) and (S) are the vacancy and substitutional impurity concentration at any point. The interstitials are assumed to be mobile enough to maintain their equilibrium concentration throughout the crystal. Case ii. Intermediate Dislocation Density. Here a modest number of vacancies is created at dislocations throughout the bulk of the sample. Reaction (62) builds up a substitutional impurity concentration in the vicinity of dislocations and at other vacancy sources. The important features of Cases i and ii are illustrated in Fig. 2. Note that substitutional atoms can build up at discrete points in the center of the crystal. Accumulation of the substitutional atoms at the back surface of a sample is also possible, even though the impurity diffusion source may be restricted to the front surface. This reaction of interstitial atoms originally introduced at the front surface with vacancies coming in from the opposite surface forms the basis for testing whether an interstitial diffusion process is operative for a given 51 52

M. D. Sturge, Proc. Phys. Soc. (London) 73,297 (1957). H. Reiss and C. S. Fuller, in “Semiconductors“ (N. B. Hannay, ed.), p. 241. Reinhold, New York. 1960.

3.

DIFFUSION

187

x+ FIG.2. Illustration of interstitial-substitutional mechanism in low-dislocation density material. where interstitial source is restricted to the front surface.

d i f f ~ s a n t . In ’ ~ this method, which the author calls the “figure test,” a radiotracer of the diffusant is deposited in the form of a geometric figure on the front of a thin, low-dislocation density slice. Following a brief diffusion cycle, autoradiographs of the front, back, and central regions (the latter after lapping off the diffusion zones on front and back) indicate the nature of the diffusion process. If a trace of the geometric figure is seen on the back of the slice, but not in the middle (or is much weaker), the interstitial-substitutional process is strongly indicated. Kendall‘4s49showed that such a mechanism was not operative for the self-diffusants In and Sb in InSb, although a vapor transport mechanism did move the diffusants from front to back. Au diffusion in Si has been shown to accumulate in a region near the back of the slice, presumably by such a proce~s.‘~ The apparent “uphill diffusion” that occursnear the back of such a slice does not violate thermodynamic principles since the process is not occurring at equilibrium, the reaction, Eq. (62), going almost completely to the right. Such behavior is probably better described as an “accumulative” process than as a dissociative diffusion process. Kendall, Kanz, and Reed have shown that a similar process occurs during In and Zn diffusion in G~AS.’~,’’ These will be discussed in Sections 16a and 17d. 53 54

55

G . J. Sprokel and J. M.Fairfield, J . Elecirochern. SOC.112,200 (1965). D. L. Kendall, Appl. Phys. Letters 4, 67 (1964). D. L. Kendall, J. A. Kanz, and B. S. Reed, to be published.

188

DON L. KENDALL

Case iii. High Dislocation Density. When the crystal contains enough dislocations, it may be possible to quickly attain the equilibrium vacancy concentration throughout the sample. In such a case the effective diffusion coefficient is given by

which is just Eq. (61) under the assumption that (I)DiS (S)D,. Frank and T ~ r n b u linvoked l ~ ~ this simple relation to explain their results for Cu in Ge. A more recent study of Cu diffusion in Si, Ge, and GaAs has been made by Hall and Racette?‘ but they emphasize the pure interstitial diffusion process. Implicit in the above arguments is the assumption that a single type of vacancy is involved in the transfer of an interstitial to a substitutional site. However, in compounds there are two types of simple vacancies, as well as more complex species such as AB divacancies. These latter defects are particularly important since they are probably more mobile than single oacancies and may even be present in larger concentration^.'^ For illustration, consider the reaction of an interstitial impurity such as Zn with an AB divacancy. This may occur as

where the complex on the right may or may not dissociate readily. Such a reaction must be involved in the vicinity of a dislocation in III-V compounds, since for geometric reasons a dislocation in this type structure (or the diamond structure as well) can emit a [V,V,] much more easily than either v, or vB.57It is likely that a defect such as [Zn,V,] will be almost immobile, and that the accumulation of such debris in the vicinity of a dislocation might quickly “pin” it from further climb (which would stop vacancy generation). This reaction probably accounts for the metastable phase that Gershenzon and M i k ~ l y a kobserved ~~ during dislocation decoration with Zn in Gap. It could also explain the puzzling lack of Zn in significant quantities in the interior of GaAs and InSb after diffusion, even though Zn has been shown unambiguously to diffuse by the interstitial-substitutional processs5 (see Figs. 12 and 20 and related discussion in Sections 16 and 23). 56

57 58

R. N. Hall and J. H. Racette, J . Appl. Phys. 35,379 (1964). J. N. Hobstetter, in “Semiconductors” (N. B. Hannay, ed.), p. 525. Reinhold, New York. 1960. M. Gershenzon and R . M. Mikulyak, J . A p p l . Phys. 35.2132 (1964).

3.

189

DIFFUSION

IV. Self-Diffusionin 111-V Compounds

Self-diffusion data in 111-V compounds have been reported for AlSb, GaAs, GaSb, InP, and InSb. The variation between observers is extremely large and will be discussed in the ensuing section. All the data were reported to fit the standard Arrenhius type

D

=

Do exp( - Q / k T ) ,

where D is the diffusion coefficient, Do is a pre-exponential factor, Q is the activation energy for diffusion, k is the Boltzmann constant, and T is the absolute temperature. These data are summarized in Table 11, with what are believed to be the more reliable values italicized. The significant features of the italicized data in Table I1 are the high values of D o , the relatively low values of the diffusion coefficient D at the melting point (compared to metals, for example), and the relatively high values of Q, the activation energy for diffusion. TABLE: 11

REPORTEDSELF-DIFFUSION MEASUREMENTS IN 1II-V COMPOUNDS, WITH THE MORERELIABLE VALUESITALICIZED Compound

AlSb A1 Sb GaAs Ga As As GaSb Ga Sb Sb

InP In

P InSb In In I I1 Sb Sb Sb

DO (cm' sec-')

2 x 100 1 x 10' 1 x 107 4 x 102l 7 x lo-'

Q (ev)

1.88 1.70 5.6 10.2 3.2

D at M . P . (cm2 sec-')

1.6 x

4 x 10-7

Ref,

59 59

2.4 x lo-'' 3.8 x 1.3 x lo-"

60.98 60.98 14

3.2 x 103 8.7 x lo2 3.4 x lo4

3.15 1.13

2.0 x

66

3.45

5.9 x

67 66

I x lo5 7 x 10"

3.85 5.65'

3.6 x lo-'' 4.4 x lo-"

60.98 60,98

1.8 x lo-' 5 x 1.8 x

0.28 1.82 4.3 0.7s 1.94 4.3

1.4 x 5 x lo-' 3.1 x l o L 3

69 66 1.3 x

14

2.3 x 1 0 - 1 ~

69 66 14

190

DON L. KENDALL

8. AlSb The self-diffusion coefficients of A1 and Sb in AlSb have not yet been measured directly. They were estimated, however, by Pines and Chaikovskii* from “reactive diffusion” results. In these experiments the thickness of an AlSb layer being reactively formed between cylinders of elemental Sb and A1 was monitored as a function of time and temperature. By making several assumptions regarding the growth mechanism, the values of Do and Q shown in Table I1 were estimated. These should be regarded as quite tentative due to the experimental uncertainties, the most important of which was probably the difficulty of avoiding vapor phase mixing of the two elements. 9. GaAs Goldstein6’ published the only values for self-diffusion in GaAs, although Harper6 and K e n d a l P made measurements a t isolated temperatures. The D values were quite low in all cases and the experimental difficulties formidable. A particularly serious problem is vapor transport of the GaAs from the sample during diffusion anneal. This can be reduced significantly by eliminating temperature gradients across the ampoule during diffusion. A still better technique would involve the use of sputtered SiO, to eliminate surface d e t e r i ~ r a t i o n It . ~ is ~ likely that the values for the self-diffusion coefficients obtained by Goldstein are too low due to vapor transport. Goldstein also reported a temperature insensitive diffusion coefficient for As at temperatures below 1200°C. He attributed this to diffusion via short circuiting paths such as dislocations. From the magnitude of the D value (4 x 10-’4cm2/sec), it is likely that this merely represented the precision limit of his sectioning technique. Harper observed that the surface concentration of radioactive As (at a pressure of 1 atm) was very slow in building up to its equilibrium value at 1200”C, reaching a mole fraction of only 10- radioactive As after 4 hours.6 This suggests the existence of a severe rate-limiting process for entrance of the As from the gas phase into the crystal.64 He obtained a D value for As of about 9 x 10-’2cm2sec-’ at 1200°C. The author obtained very apcm2 sec-’ for both Ga and As at 1013°C proximate values of 3 x using irradiated GaAs as a diffusion source.62These values are 1 to 2 orders of magnitude higher than Goldstein’s values for Ga and As at equivalent

’’B. Y . Pines and E. F. Chaikovskii, Fiz. Tueid. Tela 1. 946 (1959) [Engiish Trans!.: Sorief 6o

62 h3 64

Phys.-Solid State 1, 864 (1959)l. B. Goldstein, Phys. Rev. 121, 1305 (1961). J. A. Harper, unpublished data. D. L. Kendall. unpublished data. S . R. Shortes, J. A. Kanz, and E. C. Wurst, Jr., Truns. A l M E 230, 300 (1964). F. M. Smits and R. C . Miller, Phys. Rev. 104. 1242 (1956).

3.

DIFFUSION

191

temperatures. The Do and Q values for As diffusion calculated from the isolated data points of Harper and Kendall are 7.0 x lo-’ cm2 sec-l and 3.2 eV. The latter value compares favorably with an estimate of 3.0 eV for Q by Pearson et on the basis of lattice constant measurements of GaAs following quenching of the diffusion ampoule from elevated temperatures. However, several assumptions were involved in this estimate, the most fundamental one being that the native defects would be homogeneously dispersed a t their high temperature equilibrium concentration after quenching. Condensation of the vacancies and/or interstitials into clusters or loops during cooling may have influenced the results. The point to be realized is that no well-substantiated self-diffusion data exist for GaAs. This is regretable in view of the technological importance of this material. 10. GaSb Eisen and Birchenall reported the self-diffusion coefficients of G a and Sb in GaSb,66 and Boltaks and Gutorov6’ measured that of Sb in the same material. The latter work predicts much higher D values for Sb than do Eisen and Birchenall. It is likely that Boltak’s values are not typical of bulk diffusion in GaSb, but are due instead to a short circuiting diffusion effect similar to that observed by Eisen and Birchenall in the same material. The latter observed a “tail” in the diffusion profile which they showed by autoradiography to be due to diffusion of some sort in localized regions of the crystal. The number of these spots per unit area was much lower than the grown-in-dislocation density, so they postulated that the spots might be dislocations induced by the surface preparation which acted as diffusion pipes. In view of work by Pugh and Samuels,68 who noted crack-like dislocation arrays running deep into the crystal after certain lapping procedures, this seems plausible. Diffusion from molten pits in the surface is an even more likely possibility (see Section 12). 11. InP The only self-diffusion measurements in InP have been made by Goldstein.60 He noted severe vapor etching during P diffusion unless P,O, was somehow removed from the P diffusion source. He did not report the extent to which this problem was eliminated. As seen in Table 11, the calculated D values at the melting point are considerably higher than the corresponding values in other 111-V compounds. G. L. Pearson, H. R. Potts, and V. G. Macres, in “Radiation Damage in Semiconductors” (Proc. 7th Intern. Conf.), p. 197. Dunod, Paris, and Academic Press, New York, 1965. 66 F. H. Eisen and C. E. Birchenall, Acca Met. 5,265 (1957). 67 B. I . Boltaks and Yu. A. Gutorov, Fiz. Tvrrd. E l a 1, 1015 (1960) [English Trans/.: Soviet Phys.-Solid Srate 1 , 930 (1960)l. E. N. Pugh and L. E. Samuels, J . Appl. Phys. 35, 1966 (1964). 65

192

DON L . KENDALL

12. InSb The first measurements of self-diffusion in InSb were made by Boltaks and K ~ l i k o v . ~Their ’ values of Do and Q are shown in Table 11. These lead to D values much higher than later results. It is likely their measurements were grossly affected by grain boundaries, or alternatively they measured the diffusion behavior of a radioactive contaminant. Eisen and Birchenal166 measured these same parameters in InSb and carefully reported their experimental techniques and results. Their Do and Q values are much higher than Boltak’s and lead to much lower D values. However, they mentioned two puzzling aspects of their diffusion profiles, namely (1) some of the radioactive In and Sb seemed to be “held-up” near the surface and (2) the extrapolated apparent surface concentration of the more deeply penetrating activity was much lower than expected. To explain the latter observation, they suggested that the In and Sb atoms could only enter the crystal at discrete surface sites. Millea et aL7’ as well as Williams and Slitl~in,~’ using a precision lapping technique, were not able to reproduce this work, and thus questioned whether Eisen’s values were typical of true bulk diffusion. Also, Stocker72inferred from Cu diffusion results in InSb that Eisen’s values for the self-diffusion coefficients were somewhat high. In view of the above questions, Kendall and H ~ g g i n s ’ ~decided ,~’ to redetermine the self-diffusion coefficients in InSb. They used a hand lapping procedure which maintained optically flat surfaces throughout the diffusion anneal and analysis. This procedure produced and maintained much flatter surfaces (as monitored by interference techniques) than they were able to produce with so-called precision lapping machines. Autoradiographs of the sample before and after each lap indicated that the diffusants were distributed uniformly only in the first few microns. The more deeply penetrating activity was found to be associated with small pits that formed during the diffusion cycle. Some of these pits had well-defined threefold symmetry suggesting a vapor etching process at dislocations intersecting the [ l l l ] surfaces. However, the number of such pits was much lower than the dislocation density. The pits presumably resulted from local melting and subsequent reactions with impurities on the surface. These are probably similar to the melting patterns observed on InSb by Millea and T ~ m i z u k aand ~ ~on Si and InSb by Pearson and T r e ~ t i n g . ’The ~ latter attributed the melting to localized B. I . Boltaks and G. S. Kulikov, Z h . Tekn. Fiz. 27, 82 (1957), rEnniish Transl.:Sovier Phys.Tech. P h y s . 2, 67 (1957)l. M. F. Millea, C. T. Tomizuka, and L. Slifkin, unpublished data discussed in Ref. 71. ” G. P. Williams, Jr., and L. Slifkin, A r t a Met. 11, 319 (1963). ’* H. J. Stocker, Phys. Rev. 130,2160 (1963). 73 M. F. Millea and C . T. Tomizuka, J . Appl. Phys. 27, 96 (1956). 7 4 G. L. Pearson and R. G. Treuting, Acta Cryst. 11, 397 (1958). 69

’”

3.

DIFFUSION

193

impurities on the surface which lo,wered the melting point. The apparent diffusion coefficients obtained from ihe activity associated with the pits were in rough agreement with Eisen’s results but with much more scatter. His method of sample preparation using polishing grade sandpaper evidently resulted in many more preferred points for pit nucleation and hence more consistency in this anomalous branch. Kendall and H ~ g g i n also s ~ ~showed that the shallow regions near the surface of each profile were due to bulk diffusion rather than any “hold back” near the surface as suggested by Eisen. The self-diffusion coefficients were found to be very low, reaching cm2 sec-’ just below the melting point. They also determined that the self-diffusion coefficients of both In and Sb were unaffected by changes in ambient conditions from In rich to Sb rich. Diffusion of each component by exchange with AB divacancies was invoked to explain this result. The very large pre-exponential Do was thought to result primarily from the large vibrational entropy effects associated with a divacancy. This calculation is based on a simple argument by Mott and Gurney,” and has been applied to Ge by Tweet.75It will be noted in Table I1 that several 111-V compounds are characterized by large values of Do. The authors estimated the AB divacancy and single-vacancy concentrations. This was done by estimating the binding energy of the divacancy and the various entropy factors and by Stocker’s estimate of 10l6cm-3 native defects (divacancies?) at the melting point of InSb. The numbers of simple vacancies obtained are consistent with estimates of deep trap concentration (Sb vacancies?) in InSb of about l O I 4 cm-3 by Bullis and H a r r a ~ ’ ~ as well as Laff and Fan.77All of the estimates should be considered as quite tentative due to the several assumptions required. However, they illustrate an approach that should more often be attempted for a given compound, namely, the analysis of all the existing data in order to arrive a t a self-consistent scheme for the defect equilibria. V. Impurity Diffusioii in 111-V Compounds

In this section an attempt will be made to review all the published work concerning impurity diffusion in 111-V compounds. This should be helpful as a collection of data and references., but probably more significantly it should outline areas that need further development. The review will be limited to compounds of the elements A.1, Ga, and In in permutation with P, As, and Sb. Of the A1 compounds, impurity diffusion has been studied only in AISb, and this to a very limited extent. Impurity diffusion in all three ’5

76

77

A. G . Tweet, J . A p p l . Phys. 30,2002 (1959). W. M. Bullis and V. Harrap, in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 847. Dunod, Paris, and Academic Press, New York, 1964. R. A. Laff and H. Y. Fan, Phys. Rec. 121, 53 (1961).

194

DON L. K E N D A L L

TABLE 111 REPORTED IMPURITY Do AND p VALUESIN 1 I I - v COMPOUNDS. VALUESIN ITALICSARE TYPICAL OF DIFFUSION AT THE LOW CONCENTRATION LIMIT(SEE END OF TABLE FOR MEANING OF SYMBOLS) Compound

and

diffusant

Q

Ref.

3.3 x 10-1 3.5 x 10-3

1.93 0.36

78 79

Apparently independent of Zn conc., T

1.0 x loo

2.1

80

T,

4.0 x 103 2.6 x 10-5 1.2 x 10-4

4.04 1.86 1.8 1.63 4.16 2.5 2.7 2.49 3.0 0.6 1.O 2.43 2.8 1.89 2.5 2.7 2.49 1.7 0.53 0.53

5.3 x lo-’

0.33 1.5 0.80 I .0 1.0

60 84 85,87 83.14 60 94 93 97a 14 83 123 97a 123 129 62 14.130 132 83 56 56.14 137 139 140 140 141 142

1.2 x

0.53

67

DO (cm2 sec-

1)

Comments

AlSb Zn

cu GaP Zn GaAs S S S S Se Sn Zn

Cd

1.6 x 3.0 x lo3

6.0 x 3.8 x 1.5 x 2.5 x 6.0 x 3.0 x

10-4 lo-’ 10’ lo-’

10-7 lo-’ 5.0 x lo-’

5.0 x

Mg

Mn

cu

1.4 2.1 2.6 6.5 8.5 3.0

x 10-4

x lo-’ x x lo-’ x 10-3 x lo-’

1.0 x 10-3

Tm Ag

Au Li GaSb In

2.3 x 3.9 x 10-11 2.5 10-3 4.0 x I O - ~ 1.0 x 10-3

( - ) 1.0

T

+

Large vaporization losses, T AI,S, source with powdered GaAs, pn 2 atm As pressure, p n Minimized vapor loss, ISR, T Ga’Se, layer formed, T T, + Average D,T T, + Estimate for intrinsic GaAs, no excess As, T Estimated at 3 x 1019~ r n - T~ , Average D,I S R T, Average D,ISR ISR I SR Purest Mg, SR 8.5 x atm As Average B,no excess As, I S R Interstitial D, T estimate for int-sub mechanism Retrograde with temperature, T Q,Do seem low, T Artifact due to vapor reaction, T T T Interstitial-substitutional pairing

++

T

3.

195

DlFFUSION

TABLE 111-continued Compound and diffusant Sn Te InAs Mg Zn Cd Ge Sn S

Se Te cu InSb Te Sn Zn

Cd

Hg

cu Au

co Ag Fe

T pn

+

Q (eV)

Ref.

2.4 x 10-5 3.8 x lo-'

0.80 I .20

67 67

1.98 x 3.11 x 4.25 x 3.74 x 1.49 x 6.78 x 1.26 x 3.43

1.17 1.17 1.17 1.17 1.17 2.20 2.20 1.28 0.52

146 146 146 146 146 146 146 146 149

Pn pn, Pn pn Pn Pn Pa pn, surface erosion Interstitial D. T

0.57 0.75 1.35 2.3 0.86 0.7 0 2.61 2.61 1.60 1.2 0.52 1.10 1.75

69 149a 150 151

T

DO (anZsec- ')

1.7 5.5 5.0 1.6 1.4 8.7 9.0 6.3 5.3 5.5 1.3 1.23 1.0 1.26

10-3 10-4

10' 10' 10-5

x 10-7 x 10-8 x lo-' x 10'

x10-7 x 10-10 x lo-" x lo8 x 107 x loo x 10-4 x 10-9 x10-5 x 10' 4.0 x 9.0 x 10-4 3.0 x 1 0 - 5 7.0 10-4 1.0 x 1 0 - 7 2.7 x lo-" 1.0 x 10-7 1.0 X w 7

1.17

I .08 0.37 0.32

0.25 0.39 0.25 0.25

Tracer. p-n junction depth measurements. Concentration dependence not taken into account.

Comments

T T

+

T T, +, + Vapor diffusion, pn, f 149a at lower conc., T 152,14 B a t low conc., T 152,14 B a t 2 x 1020cm13Zn,T High Zn conc., T 154 154 Lower Zn conc., T Vapor diffusion, pn, 153 153 T, 156 Average D,T, 157 T, + 155a T, + 160 T Surface vacancy controlled, T, ISR 72 161 Dislocation controlled, T 161 T T 163 T, low Do 164 T 163 T 163

+

++, +

+

+

ISR Incremental sheet resistance. SR Sheet resistance and assumed distribution. + + Average typical of high concentration.

196

DON L . KENDALL

Ga compounds has been reported, but only GaAs has been extensively studied. Even here, however, there is little agreement between observers, which is evidence that experimental difficulties are still dominant. Finally, impurity diffusion in all the In compounds has been studied and these will be analyzed and discussed. When values of Do and Q are available for each system, they are tabulated in Table 111. 13. Zn

AND

CU IN AlSb

Shaw et a 1 . I 8 reported that B for Zn in AlSb was apparently not dependent

on Zn concentration, but that it was depressed somewhat in n-type material. Their values of Do and Q are shown in Table 111. The constancy of b with Zn concentration is in sharp contrast with Zn diffusion results in several other 111-V compounds, as will be discussed in succeeding sections. They inferred that the diffusion coefficients were constant by the good fit of the radiotracer data with a complementary error function distribution, but they pointed out that the p-n junction depth was from 1&25% shallower than their tracer profiles would have predicted. It is possible that this was due to nonparallel lapping and the profile was actually steeper than an erfc distribution. A sensitive test of the constancy of D would be provided by reducing the Zn vapor pressure during diffusion. As discussed in Section 2a, this would lower the Zn surface concentration, and a change in the diffusion coefficient could easily be detected. The complex problem of Zn diffusion in the 111-V compounds is treated in most detail in Section 16a for GaAs and in Section 23a for InSb. There are strong similarities in the Zn diffusion behavior in several 111-V compounds. Wieber et ~ 1 . ' measured ~ the diffusion coefficient of Cu in AlSb using radiotracer techniques. The activation energy for diffusion of 0.36 eV shown in Table I11 is quite low but is typical of diffusion of Cu and other impurities whose diffusion properties are dominated by the interstitial form of the impurity. Cu evidently is mainly a substitutional acceptor, however, after cooling to room temperature, as shown by the p-n junctions obtained on diffusion into n-type AlSb. 14. Zn

IN

GaP

Zinc is the only impurity whose diffusion properties have been reported in any detail in Gap. AllisonBoused a fractional uptake method for estimating the diffusion coefficient at various temperatures. Successful application of this method requires that the diffusion coefficient at a given temperature be independent of composition. Hence, the strong concentration dependence of noted by Chang and Pearson" was not seen. Allison's values of Do and Q,

'' D. Shaw, P.Jones, and D. Hazelby, Proc. Phys. SOC.(Loadon)80, 167 (1962). '' R. H. Wieber, H. C. Gorton, and C . S. Peet, J . A p p l . Phys. 31,608 (1960). H. W. Allison, J . Appl. Phys. 34, 231 (1963).

*' L. L. Chang and G. L. Pearson, J . Appl. Phys. 35,374 (1964).

3.

DIFFUSION

197

which should yield some sort of “mean” diffusion coefficient, are shown in Table 111. On the basis of autoradiographic evidence, Allison suggested that interstitial Zn was probably involved in the diffusion process. Gershenzon and M i k ~ l y a kdecorated ~~ dislocations in G a P with Zn by diffusing from the vapor at elevated temperatures. Various cooling rates and annealing cycles at lower temperatures did not significantly modify the “precipitate” along the dislocation as viewed by visible light. Further, the zinc seemed to “pin” the dislocations, thereby inhibiting their motion by either climb or glide. The decorated region around the dislocation was not elemental Zn, but seemed to be some other second phase. This latter “phase” may have consisted of Zn,V, complexes formed by the reaction of Zn interstitials with divacancies emitted by the dislocations, as discussed earlier with regard to reaction equation (65). Chang and Pearson8lP8’ showed that the diffusion coefficient of Zn in Ga P is strongly dependent on Zn concentration. At temperatures below 9OO0C,data from a single profile indicated that D is proportional to the square root of the Zn concentration. At higher temperatures, the concentration dependence is larger, reaching the square of the Zn concentration between 1019 and 1020cm-3. Further, analy.sis of a single profile diffused at these higher temperatures leads one to the conclusion that D reaches a maximum and then decreases at higher concentrations. This is probably a nonequilibrium effect due to the large concentration gradients. To eliminate these gradients, they applied the isoconceritration diffusion technique of Kendall and Jones82,83which involves diffusing radiotracer Zn into a sample already doped with nonradioactive Zn. Using this technique at several concentrations at 9OO0C, they observed a concentra.tion dependence of D on (Zny with n varying between 2 and 3. They explained this with an interstitial-substitutional model with the Zn interstitial being a doubly ionized donor and the Zn substitutional being a well-behaved acceptor (i.e., with a well-defined energy level even at high Zn concentrations). The significant difference they observed between isoconcentration measurements and profile measurements of has also been noticed in the Zn :G a A s system. An explanation of this is given in Section 16a.

15. S, Se, Te, A N D Sn

IN

GaAs

Considerable effort has been expended in measuring and reporting impurity diffusion coefficients in Ga14s. The experimental difficulties are formidable with some impurities, and it is not unusual for reports between ‘ldL. L. Chang and G. L. Pearson, J . Appl. Ph,ys. 35, 1960 (1964). ‘lbL. L. Chang and G. L. Pearson, J . Phys. Chem. Solids 25, 23 (1964). 8 2 D. L. Kendall and M. E. Jones, Sol. State Device Res. Conf., Stanford (1961), unpublished. 8 3 Final Report, Texas Instruments Incorporated. “Research and Development of High Temperature Semiconductor Devices,“ Contracts NObsr-77532, 85424 (March 1963).

198

DON L . KENDALL

observers to differ by several orders of magnitude. In the following, donor diffusion will be treated first, then simple acceptors, and finally other impurities. The various impurity diffusion coefficients in GaAs are shown in Fig. 3. lOOO/T

(OK)

4 I

U

w

[I]

.. H

U

H E

cu W

8 Z [I]

2E

H

n

12oooc

1100~C

lO0O~C

9oooc

FIG.3. Diffusion coefficients in GaAs at low concentration limit

u. S in GuAs

The diffusion of S in GaAs has been reported by Goldstein,"' Kenda11,83 F r i e ~ e rVieland,85 ,~~ and Yeh.86 The values of Do and Q obtained by these authors are shown in Table 111. The various data are also portrayed in Fig. 4.

-3.

DIFFUSION

199

1000/T (OK)

\

-

\

\

4

I

CJ

m w N

B

w

H H

E

E ral 0

u

a H 0)

2

b a

ll0O~C

lO0O~C

9oooc

800°C

FIG.4. Diffusion coefficient of S in GaAs. Circles from incremental sheet resistance measurements, and the triangle represents the tracer run of Fig. 6.

Goldstein used elemental S and a small amount of excess As in a sealed ampoule and obtained B values that are much lower at a given temperature than those obtained by the other a ~ t h o r s . ~However, ~ ~ ~ ~he*has ~ since ~ , ~ ~ found under similar diffusion conditions that the GaAs slices lose 1-5 % of their weight during the diffusion run due to a vapor-etching process.88 This would account for the anomalously low D values. F r i e ~ e r *used ~ sources of Al,S, and other sulfur compounds as diffusion sources along with a large quantity of crushed GaAs in a sealed ampoule. 84 85

86

R.G. Frieser, J . Electrochem. Soc. 112,697 (1965). L. J. Vieland, unpublished data quoted in Ref. 87; L. J. Vieland, J . Phys. Chem. Solids 21, 318 (1961). T. H . Yeh, J . Electrochem. SOC. 111,253 (1964). L. R.Weisberg, Trans. A I M E 230,291 (1964). B. Goldstein, private communication.

200

DON L. KENDALL

He noted serious vapor transport if the temperature gradient across the ampoule was greater than 3°C. He estimated the maximum electrically active S to be about 4 x 10" cmP3. His values for 4 are lower than those of the author and Vieland (see Fig. 4). He probably suffered surface losses similar to those of Goldstein. His Q value is similar to Vieland's, but Do is markedly lower. Vieland8' measured 4 values for S with a very small amount of S in the ampoule (3 p g cm-3) as a function of As pressure. Over the range 1-5 atm of As (assuming As, is the gaseous species), he found the p-n junction depth practically constant. He recorded a sharp decrease in one run at lower As pressures, but this was likely an anomalous result due to the vapor-etching process mentioned earlier. Additional data of Vieland are referenced by Wei~berg,'~ who commented on the wide disparity between Vieland's data based on p-n junction measurements and the radiotracer work of Goldstein.6" Using radiotracers and incremental sheet resistance techniques8' the ,'~ author obtained B values for S at several temperatures in G ~ A s . ' ~ The results of one of these runs are shown in Fig. 5. The maximum electron con-

FIG.5. Sheet resistance R, data showing resistivity p and electron concentration n following S diffusion in GaAs at a pressure of 0.1 atm for 4 hours at 900°C. 89

H. Reiss and C. S. Fuller, in "Semiconductors" (N. B. Hannay, ed.), p. 230. Reinhold, New York. 1960.

3.

201

DIFFUSION

centration found near the surface in any of these runs was 1.6 x lo'* ~ m - ~ . A complementary error function distribution fits the data reasonably well. This distribution was assumed in the (diffusioncoefficient calculations. In several runs a sulfur compound was formed on the surface during ' Goldslein6' showed this to be GaS by x-ray diffusion. Osborne et ~ 1 . ~ and diffraction analysis. When this was observed, the weight loss of the sample was usually quite marked. However, in the 900°C run shown in Fig. 5 and in other runs used to calculate the D values of Fig. 4, the average thickness loss from each surface as determined from weight measurements was 2 microns or less. The vapor reaction was greatly depressed by reducing the temperature gradient across the ampoule to less than 3°C. Reducing the amount of S in the ampoule was also helpful in this regard. The vaporization loss was also very small in the radioactive S 3 5 run shown in Fig. 6 although there is evidence in the data points near the surface that excess S in some form is present. There is an uncertainty of several percent in concentration in each data point of Fig. 6 due to absorption of the low energy beta emitted from S35by the polishing compound (American Optical 309W) used to lap the sample.

LL

2!c

2

z

L

U

t

0

U

10

0

-*

1

2

3

4

c

5

6

DEPTH IN MICRONS

FIG. 6. Radioactive S35distribution in GaAs after 115hours at 880°C. ( S , pressure of0.03atm.) 90

J. F. Osborne, K . G. Heinen, and H. Riser, unpublished data.

202

DON L. KENDALL

An additional complication observed in the incremental sheet resistance measurements was the formation of a very high resistivity layer between the diffused S layer and the bulk p-type material. This was generally of the same order of thickness as the diffused n-layer. Rather than being associated with S, this anomalous layer is probably due to contamination by a fast diffusing impurity such as Cr, Mn, or Fe, all of which have deep-lying energy levels in GaAs.” Because of the question this introduces as to the true bulk concentration near the p-n junction of the sheet resistance data and other uncertainties mentioned above, rather large limits of error have been placed on the diffusion coefficients in Fig. 4.

h. Se in GnAs Several a ~ t h o r s ~ ~ have - ~ discussed ~ , ~ ~ -the ~ ~ doping properties of Se in GaAs. They all observed low donor doping efficiency for Se concentrations ~ . of the above assume that Se tends to complex with a above lo’* ~ m - Most vacancy to form a neutral or acceptor associate. In fact, Nasledov and Feltin’sh28 noted that, a t Se concentrations approaching the compound Ga,Se,, the conductivity changed to p-type. This is in agreement with the fact that Ga,Se3 is a p-type defect semiconductor. Abrahams et ul.92a recently reported that Ga,Se, precipitates in GaAs coherently (crystallographic coincidence) when the Se concentration exceeds 2 x 10’8cm-3. This observation supports the possibility of going from n-type GaAs to the p-type defect semiconductor Ga2Se3without rupturing C K - ~ , one either lattice. At average Se concentrations of order 10’8-1020 might obtain a patchwork of p- and n-type material. If electrons can tunnel from the n-type GaAs into the (probably) heavily p-type Ga,Se,, the carrier density and mobility measured by Hall measurements might behave anomalously. This may explain Strack’s unusual observation that the measured electron concentration exceeded the donor concentration in GaAs crystals grown in the (100) directi~n.~’’ The dopant in his case was Te rather than Se, but the same arguments may apply. The anomalous mobility measurements of S t r a u ~ and s ~ ~Williams92d ~ on Se and Te doped GaSb and GaAs, respectively, may be explicable using similar considerations. R. W. Haisty and G. R. Cronin, in “Physics of Semiconductors” (Proc. 7th Intern. Conf.) p. 1161. Dunod, Paris and Academic Press, New York, 1964. 9 2 C. S. Fuller and K. B. Wolfstirn, J . A p p l . Phys. 34,2287 (1963). 92aM.S . Abrahams, C. J. Buiocchi, and J. J. Tietjen, J . Appl. Phys. 38,760 (1967). 9ZbH. Strack, in “Gallium Arsenide” (Proc. Intern. Symp., Reading, 1966). p. 206. Inst. Phys. and Phys. Soc., London. 1967. ”‘A. J . Strauss. Ph,w R ~ T .121. 1087 (1961). 92dF.V. Williams, J . Elecrr-oc,/irrti.S o . . 112, 876 (1965). y 3 R. W. Fane and A. J . Goss. Solid-State Elec./,on. 6, 383 (1963).

3.

DIFFUSION

203

As an alternative suggestion, Fuller and W01fsth-n~~ proposed that Se and Te may form polyatomic molecules in GaAs. They noted that annealing a crystal doped with 10'9cm-3 Se or Te for several days at temperatures ranging from 60CL1100"C caused the electron concentration to drop until an equilibrium value was established. They estimated an effective D of order lo-'' cm2 sec- for the diffusing species at 700°C. They further derived an activation energy for diffusion of 0.3 eV. These values are not consistent with simple vacancy diffusion, but in fact seem to be more typical of an interstitially controlled process. However, formation of a more mobile species, such as a Se : vacancy conglomerate, and subsequent diffusion cannot be excluded (see Section 2c). Goldstein6' reported the Do and Q values for Se in GaAs shown in Table 111. He noted the formation of a thick layer of "selenium glass" on the surface of the GaAs during the diffusion. He did not take into account the fact that the boundary was in motion during the diffusion process due to the progressive formation of the layer. Fane and G O S Sreported ~~ an unusual distribution of Se in GaAs using tracers in which the apparent went through a maximum ofabout 1 x cm2 sec- ' at a concentration of about IOl9 cm-3 at 1100°C. They also reported a dependence of the p-n junction depth on As pressure, with the depth increasing weakly with increasing As pressure (approximately as Pi;''). This is in the wrong direction to be explained by simple As sublattice diffusion for the Se atoms. Also, the p-n junction depth was shallower than expected from their tracer profiles.

'

c. Tc in GaAs

Fuller and W ~ l f s t i r nnoted ~ ~ practiically identical behavior for Te in GaAs as that for Se discussed in the prekious section. Another published result regarding diffusion of Te in GaAs is that of Yeh.86 He obtained a p-njunction depth of 4.5 microns after 120 hours at 1040°C.This gives a B a t this temperature on the order of cm2 sec-'. He found that a film of SiO of 200CL 3000A was generally sufficient to inhibit the formation ofGa2Te3on the GaAs surface. He also stated that S, Se, and Te could diffuse through much thicker films of SiO (up to at least 20,000 A) and still form n-layers on p-type GaAs. obtained radiotracer profiles using Te'27 diffused from Osborne et the vapor into wafers protected with sputtered S i 0 2 films.63They obtained B valuesofabout cm2 sec- at 100WCand2 x 10- 12cm2sec-' at 1100°C. d . Sn in GaAs Goldstein and Keller94 reported the first values for of Sn in GaAs, and their values of Do and Q are shown in Table 111. The value of 2.5 eV for Q is in 94

B Goldstein and H Keller, J A p p l Phys 32, 1180 (1961)

204

DON L. KENDALL

apparent agreement with the Q for Zn and Cd in GaAs. Goldstein also pointed out that the value of 2.5 eV was much lower than the value o f 4 eV he obtained for S and Se diffusion, He used this in support ofa sublattice vacancy diffusion model, where Zn, Cd, and Sn were purported to diffuse on the Ga sublattice and S and Se on the As sublattice. More recent measurements, however, bring this conclusion into question. In particular, the dependence of the diffusion coefficients of several impurities on As pressure is in the wrong direction to be explained by simple sublattice diffusion.85 It has been shown recently that Sn is not a well-behaved diffusant, i.e., its distribution in a diffused sample indicates an unusual concentration dependence. This has been noted by Larrabee and O ~ b o r n e , ~as’ well as by Fane and G o s s . Of ~ ~ even more significance are the autoradiographs obtained by Larrabee and Osborne at a depth of about 200 microns after a diffusion run of 6 hours at 835°C. They observed many discrete spots of Sn1I3in the autoradiographs (about lo4 spots cm-’) in lightly doped p-type material (4 x 1OI6 cm-j). In lightly doped n-type material (1 x 10’’ ~ m - ~ ) , the density of spots was similar, but less Sn1l3was present in each spot and the spots were also more diffuse. The spot density observed in the autoradiographs was of the same magnitude as the dislocation density in these crystals. This suggests that the Sn is preferentially situated at dislocations. A plausible mechanism to explain this result is represented by the quasichemical reaction 2Sn‘

+ V,,V,,

+ 2e+,

F? SncaSnAs

(67)

where two Sn interstitials react with a divacancy (which is emitted by a dislocation) to form the electrically inactive substitutional paired Sn defect shown. From their data, an apparent D of order to cm2 sec-’ at 835°C can be estimated for the Sn interstitial if the above model is valid. The interstitial solubility of Sn in t.he lightly p-type material above would be approximately that expected in intrinsic material. In the n-type material, however, the interstitial solubility should be slightly depressed, especially if the Sn, is a multiply ionized donor. TMs may explain the lower Sn113 density in the vicinity of dislocations in the n-type material, since fewer interstitials would be present in the vicinity of the dislocations when they emit divacancies. The effective D values reported by Fane and G O S Sfor~ the ~ slow diffusion near the surface ranged from 5 x cmz sec-’ at 1100°C to cm2 sec-’ at 900°C. The Do and Q values calculated from these two values are shown in Table 111.They also mentioned that an interstitial mechanism might account for the tails observed in their tracer profiles at low concentrations. 95

G. B. Larrabee and J. F. Osborne, unpublished data

3.

DIFFUSION

205

Fane and Goss further noted a slowly increasing p-n junction depth with As pressure up to about 1 atm, above which the junction was constant. 16. Zn, Cd, Hg, and Mg

IN

GaAs

a. Zn in GaAs

Turning now to acceptors in GaA:s, perhaps the most interesting is the diffusion of Zn in GaAs. The first report of this unusual system was made by Allen and C ~ n n e l l who , ~ ~ suggested that the diffusion coefficient of Zn appeared to be discontinuous at a Zn concentration of about 10" ~ m - ~ . They attributed this concentration dependence to the parallel diffusion of ionized Zn and un-ionized Zn, with the latter having a much higher diffusion coefficient. According to this model, they predicted a rapid change in D as the Fermi level moved through the Zn energy level in the forbidden gap. Allen48 later discussed this problem in more detail and presented theoretical estimates of the fraction of the Zn atoms in each ionization state. The next report on this complex problem was made by Kendall and Jones,97 who showed that D changed continuously by several orders of magnitude over the concentration range 10'8-1020~ 1 3 in1 both ~ ~ GaAs and InSb. They proposed a substitutional diffusion model whereby positively charged vacancies generated in the bulk of the crystal were caused to flow toward the surface layer by the large: Zn concentration gradients. These excess vacancies in the Zn diffusion zone were held responsible for the rapid diffusion. KendallS4 later presented data which support the concept that vacancies flow toward the surface during acceptor diffusion, but, as will be shown, this is likely a second-order effect with regard to explaining the concentration dependence of the D of 2,n in GaAs. G ~ l d s t e i n ~ ~also " , ~noted * the sharp concentration dependence of the diffusion coefficient of Zn in GaAs. From Hall measurements on a sample which had been diffused throughout with Zn, he demonstrated that the hole concentration was approximately equal to the total Zn concentration. He presented this as evidence against Allen's proposal regarding the parallel diffusion of un-ionized and ionized Zn. Goldstein attempted to surmount the difficulties of concentration dependence by diffusing from a thin electroplated source. The penetration curves under these conditions appeared to follow a Gaussian distribution. The D values obtained by such a method are some sort of "average" as a function of J. W. Allen and F. A. Cunnell, Nature 182, 1158 (1958). Kendall and M. E. Jones, recent news p,sper at Electrochem. SOC.Meeting, Chicago (1960), unpublished. 97aB. Goldstein, Phys. Rev. 118, 1024 (1960). 9 8 B. Goldstein, in "Compound Semiconductors" (R.K . Willardson and H. L. Goering, eds.), Vol. 1, p. 34%. Reinhold, New York, 1963. 96

'' D. L.

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DON L. KENDALL

concentration, and probably vary widely with source thickness and diffusion time. Cunnell and Gooch next published a more complete set of data which are quite useful from an empirical ~ t a n d p o i n tThey . ~ ~ used a two-temperature diffusion technique with the Zn held a t a lower temperature to control the vapor pressure and hence the Zn surface concentration. Their profiles indicated an apparent maximum D at a concentration somewhat below the surface concentration. They also showed that diffusion in the presence of excess As depressed D near the surface. They further discussed stoichiometric equilibria, but advanced no theoretical arguments to explain the data other than the theory of Allen.48 presented data over a much wider range of Zn Kendall and concentration in which the Zn vapor pressure was controlled using a Zn-Ga alloy source. The relationship of the surface concentration to Zn pressure, assuming ideal behavior of the Zn-Ga alloy, is shown in Fig. 7. Also shown are several points calculated from the data of Cunnell and G ~ o c as h ~ well ~ as

I?

2

U

U

z N

1

10

100

lono

IDEALIZED Z I N C PRESSURE IN TORR

FIG.7. Effect of Zn partial pressure on Zn surface concentration in GaAs with theoretical curves assuming Zn ionization energy of 0.08 eV. Ideal conditions assumed for alloys and vapor. 99

F. A. Cunnell and C . H. Gooch, J. Phys. Chem. Solids 15,127 (1960)

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207

from Shortes et ai.63and Chang and Pearson.81b The agreement between observers is very good, especially considering the different experimental techniques used. The solid theoretical lines shown are based on a straightforward application of the law of mass action as discussed in Section 2a. The theory assumes that the Zn exists in both ionized and un-ionized forms at diffusion t e ~ p e r a t u r e sChang . ~ ~ and Pearson’ l b came to the same conclusion regarding the charge state of Zn in GaAs at high temperatures. Their treatment also extended to the calculation of solubility and distribution coefficients of Zn in GaAs and Gap. Panish and C a ~ e recently y ~ ~ suggested that the Zn is fully ionized at high Zn concentrations. They fit the data of Fig. 7 using the theory of Halperin and Lax,38 which takes account of the “bandblending” or impurity tails that develop in the valence and conduction bands at high Zn concentrations [see Section 2d(3)]. Kendall and J o n e ~ ~also ~ , ’introduced ~ an “isoconcentration diffusion technique” which eliminated the large Zn concentration gradients and allowed the determination of D that should be much closer to an equilibrium value. The method consists of diffusing radioactive Zn at a prespecified surface concentration into a sample already uniformly doped to the same concentration with non-tracer Zn (either grown-in or by a long prediffusion). Chang and Pearson’ l a later applied {his technique at several Zn concentrations at 900°C and to the Zn in G a P system. The first isoconcentration run is shown in Fig. 8 along with other profiles selected from Ref. 83. The excellent fit of the isoconcentration run with the complementary error function distribution is good evidence that (1) the radiotracer Zn surface concentration closely matched the nonradioactive Zn bulk concentration (which was outdiffusing to exactly complement the in-diffusing tracers) and (2) that defect equilibria were maintained throughout the diffusion zone. The tracer profiles shown in Fig. 8 were analyzed for the effective diffusion coefficient 6, using a variation of a standard technique in which the total acceptor concentration was replaced by the un-ionized acceptor concentration in the Boltzmann-Matano analysis. l o o In the standard BoltzmannMatano analysis the effective diffusion coefficient at a given Zn concentration N is given by 1

rN 1

where t is the diffusion time and x is the distance. This eqcation is valid when

D is a function of concentration only, ie., it is not applicable if D depends on t or x in some anomalous manner. Also, the values of the extremes of concentration may be in considerable error, especially if the concentration loo

Ref. 41, p. 95.

208

DON L. KENDALL

c

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I

I

I

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I

i

200

220

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z

z

0 t

2 10’~

ISOCONCENTRATION RUN

F-

2 W 0 2 0 0

Io‘*

0

20

40

60

80 100 120 140 DEPTH IN MICRONS

160

180

FIG.8, Diffusion profiles of Zn in GaAs at 9OO0C,showing diffusion time and idealized Zn partial pressure.

dependence is large. The variation”’ utilizes the concept that the thermodynamic “activity” is the driving force for diffusion, and that this is proportional to the concentration of the neutral species (rather than total c o n c e n t r a t i ~ n ) . ‘ ~For ~ ~ ’an ~ ~impurity with a well-behaved energy level, this variation compensates for the effect of the concentration gradient (built-in electric field) on the diffusion coefficient. Thus, in principle, the “gradient free” value of D should be obtainable. The values obtained for D by applying this variation to the profiles of Fig. 8 are plotted in Fig. 9. Using the total concentration instead of the un-ionized concentration in the analysis leads to apparent values of b about two times higher than those shown. This factor-of-two enhancement is precisely what is predicted for the effect of an ionized impurity concentration gradient on its own diffusion coefficient.’03345However, in view of the work mentioned above regarding the band-blending of Zn with the valence band,37 the above calculation is probably of academic interest only. The attainment of a meaningful D from the profiles is complicated considerably by the band-blending phenomena, although Casey et al.37ahave recently made such an attempt, as will be discussed later. D. L. Kendall, unpublished calculation. H. Reiss, J . Chem. Phys. 21,1209 (1953). ‘03 Ref. 17, p. 593.

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209

DIFFUSION

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MODIFIED BOLTZMANNMATANO METHOD ISOCONCENTRATION RUNS KENDALL CHANG I

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1

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1 I IIII

10" ZINC CONCENTAL\TION IN CM3

-Am

0

:

A 1

-

-

1

-

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FIG.9. Analysis of concentration profiles of Fig. 8 using modified Boltzmann-Matano method. Normal B-M method leads to values 2 times larger. Solid line through the isoconcentration and assuming simple points is based on the interstitial-substitutional process using Zn; de-ionization of Zn,, at high concentrations. +

At this point a somewhat detailed discussion of the Zn diffusion profile will be undertaken. The interstitial-substitutional model originally proposed by LonginiIo4 for Zn diffusion in GaA$ will be utilized throughout. Other models based on substitutional diffusion (vacancy exchange) can explain the concentration dependence of B, but experiments by Kendall et to be discussed later, have shown that the interstitial-substitutional model is much more likely. First, it is important to note that the measured concentration dependence of is dependent on the experimental conditions. For example, the concentration dependence as measured by the Boltzmann-Matano analysis of a single diffusion profile should be weaker than that obtained from the isoconcentration technique applied at several concentrations from the following simplified argument. Consider, as Longini did l rig in ally,'^^ that the domiand Zn,,. The reaction whereby the interstitial nant forms of Zn are Zn: becomes substitutional is given by +

zn: Io4

+

+ Vga * Zn,, + 3e+,

R L Longmi, Solrd-State Electron. 5, 127 (1962)

(684

210

DON L . KENDALL

the mass action expression for which may be written

Since the effective diffusion coefficient for an interstitially controlled dissociative process is to first order controlled by the fraction of the atoms that are interstitial [Eq. (61)], then as long as Zn+ + 6 Zn,, it is clear that

Thus B should be proportional to the cube of the hole concentration if all of the species involved in the reaction of Eq. (68a) are under local equilibrium throughout the diffusion zone. This may be the case under isoconcentration conditions, but this is probably not fulfilled under the conditions of Zn diffusion into undoped GaAs (profile conditions). The most likely cause for this deviation from local equilibrium under profile conditions is the high diffusion coefficient of the Zn interstitial relative to that of the vacancy. This will enable the interstitial to maintain equilibrium with the external Zn vapor phase over much greater distances than the Ga vacancies and the substitutional Zn atoms. For example, as a limiting case, assume that the interstitials maintain equilibrium with the external vapor phase throughout the whole crystal. Then the neutral interstitial concentration is constant throughout the diffusion zone and the interior of the slice regardless ofdoping. By a simple ionization reaction like that shown in Eq. (49, the concentration of Zn: + can be shown to be proportional to p2. We can estimate the effective diffusion coefficient at any point in the diffusion zone by again assuming that it is controlled by the fraction of the atoms that are interstitial, namely, (Zn: ')/(Zn&). However, in this case, since the reaction of Eq. (68a) is not assumed to be at equilibrium, we must evaluate (Zn: + ) and (Zn;,) separately, setting Zn: K p 2 and Zn& = pin the range of interest (10'' to 10'' ~ r n - ~ ) . Thus the D under profile conditions is proportional to thejirst power of the hole (or Zn) concentration in this limiting case. Therefore under these two different experimental conditions, namely, isoconcentration and profile conditions, the measured is expected to vary with the cube and the first power of the Zn concentration, respectikly. This difference is evident in Fig. 9 where the isoconcentration data can be fit by a cube law in the 101s-1019cm-3 range, and the profile data are better fit by a linear dependence. A similar result was obtained by Chang and Pearson'' for Zn diffusion in G a P (Section 14). Another anomaly is the maximum in D and a decreasing D near the surface as determined from the profiles (I and I1 from Fig. 8). This is either a non+

3.

211

DIFFUSION

equilibrium effect due to the concentration gradient, or it may be explicable in terms of the band-tailing model for Zn in GaAs proposed by Casey et which will be discussed later in this section. The apparent decrease in D at high concentrations could also be due to ion pairing of the Zn: and the Zn,, similar to that of Li and O in Si.'05 Implicit in all the discussions so far is the fact that V&,is not affected by Zn concentration. Stated in another way this means the Zn in the external phase does not affect the As or Ga activity in the external phase. This is probably a valid assumption in the lower Zn pressures used in the work discussed. However, if excess As is added to the ampoule quite a different situation will exist as will be discussed later. Note also in profile 111 of Fig. 8 the unusual behavior near the diffusion front. The data points shown are accurate to better than lo%, and hence the maximum near the p-n junction must be considered as real. Mehta and Pearson'06 under very similar diffusion conditions obtained the same effect.A possible explanation may be provided by Kendall's observation that vacancies apparently flow from the bulk of the sample to the surface when a p-layer is present.54 During the long diffusion anneal of run 111, this could cause a localized build-up of substitutional Zn atoms via the reaction of incoming interstitials with out-d@sin;: vacancies (see Section 17d for other evidence regarding this outward vacancy flux). If the vacancies generated in the bulk are actually diffusing out of the crystal, then the vacancies in the diffusion zone must be depressed from equilibrium. One would expect to find a tail on the diffusion profiles at low concentrations with a very high D typical of the pure interstitial. This has not been unambiguously observed, evidently because the concentration of interstitials in the undoped crystals is too low to be measured with radiotracers. A lower limit of 5 x cm2 sec-' can be placed on the interstitial diffusion coefficient at 900°C by noting the highest effective D measured in heavily p-type material where the interstitial concentration is highest. Another surprising result is the apparently negligible concentration of substitutional Zn atoms in the vicini1.y of dislocations in the interior of a slice of GaAs following a diffusion cycle. The dislocations no doubt emit divacanciesS7 which should react with the interstitial Zn atoms to form Zn substitutionals. As discussed earlier with respect to Eq. (65),the dislocations must be quickly blocked by the collection of immobile species such as [Zn,,V,,] and divacancies can no longer be emitted. For a crystal with a dislocation density of lo5cm-', the atomic site density along the dislocations is about 1 0 1 3 ~ m - If~ one . arbitrarily assumes that about 100 [V,V,]'s are emitted per dislocation site before the dislocation becomes locked, the +

lo5 'Oh

E. M . Pell, J . A p p l . Phys. 32, 1048 (1961). R. Mehta and G. L. Pearson, unpublished d,ata; also Ref. 125.

212

DON L. KENDALL

average Zn concentration in the crystal reaches only 10” cm-3. These might be detected by autoradiography or indirectly by hardness measurements. The author has observed such an effect by tracer measurements during In and Cd diffusion in GaAs (Sections 16b and 17d) and Larrabee and Osborne have seen it during Cu and Sn diffusion (Sections 17b and 15d). Casey et ~ 1 . ~have ’ ~ recently analyzed the Zn diffusion profiles in GaAs by the Boltzmann-Matano method [Eq. (68)]. They derive an expression for D which extended previous analyses to include both the built-in field due to the substitutional Zn gradient and the nonideal behavior of holes which occurs when the impurity level broadens into an impurity band and merges with the valence band. They assume that the Zn interstitial is a singly ionized donor and that the substitutional species, ZnGa,is completely ionized at all concentrations. The expression they use for the flux of Zn by the interstitialsubstitutional process is a(Zn; ) qDi(Zn )E J,, - D i p kT 8X

’

where Di is the interstitial diffusion coefficient of Zn, q the electronic charge, k the Boltzmann constant, T the absolute temperature, and E the built-in field. This is analogous to Eq. (59) except for the field term, with diffusion by the substitutional mode being negligible. Although not stated, Di must be independent of concentration for this expression to be precisely true. With this model, they obtain an effective diffusion coefficient under extrinsic conditions ( p % ni) given by

where K , is the reaction constant for the interstitial-gallium vacancy reaction, PAs4is the pressure of As, [which reduces D by increasing the Ga vacancy concentration, see Eq. (68c)],and y p is the “hole activity coefficient.” The hole activity coefficient is unity at Zn concentrations low enough so that the hole concentration can be expressed by Boltzmann statistics. As discussed in Section 2d(l), y p would be expected to become greater than unity when the Fermi level approaches the valence band edge and if FermiDirac statistics were applicable. However, when the impurity band merges with the valence band, y p is less than unity, and Casey et al. believe this to be the case for Zn in GaAs. To obtain values of y p at various Zn concentrations, they assume Eq. (68e) is an accurate representation of B over the extrinsic range of Zn concentrations (> 1 O I 8 ~ m - ~They ) . use values taken from profiles like those of Fig. 8 and calculate y p using Eq. (68e). Their calculated y p at 900°C deviates from unity in the region where D goes through an apparent maximum in Fig. 9, namely, at a Zn concentration of about

3.

DIFFUSION

213

6 x l O I 9 cm- 1 3 . At concentrations above this, y p decreases steadily to about 3 x 1020cm-3 where it assymtotes to 0.4. This model accounts for the decrease in at 6-8 x 1019cm-3 and also predicts an increase in b in the profile data at 900°C at higher Zn concentrations to bring b to meet the isoconcentration point at 3 x lo2' C I I ~ - ~ . As further support for the above, they explain their Zn solubility as a function of Zn concentration in the liquidus in terms of the band-tail model. Since the y p variation with Zn concentration is consistent with two independent experiments, they feel strongly justified in the approach. In this regard, it will be noted that calculations based on the much simpler model of simple deionization of Zn at high concentrations will also explain two independent experiments, namely, the concentration dependence of the isoconcentration diffusion data over a wide range of Zn concentrations (see Fig. 9 and Chang and Pearson"") and the Zn solubility as a function of Zn concentration in Zn-Ga alloy diffusion sources (see the theoretical curves of Fig. 7 and Chang and Pearson8 "). The band-tailing model has much to recommend it since it appears to explain the shape of the profile at hig,h concentrations. However, the increase in expected from profile data in the range 1 to 3 x 1020cm-3 is not very apparent in the data. This region is near the surface where outdiffusion on the cooling cycle occurs and compromises the BoltzmannMatano analysis. Isoconcentration data in this range of concentration would be of much value in this regard. The model does not explain the results of the analysis of profile I1 of Fig. 8, which goes through a maximum in b at a significantly lower Zn concentration. Cunnell and G ~ o c show h ~ ~the same effect in their profiles. The model also fails to fit the isoconcentration data, which is better matched with a cubic dependence in the lo" cm-3 range. In spite of these objections, the importance of one aspect of this work should not be overlooked, namely, the inclusion of the band-tailing phenomena in calculations of this nature. This is particularly significant since it predicts that y p is less than rather than greater than unity, as the FermiDirac correction would imply. The temperature dependence of Zn diffusion under isoconcentration conditions has recently been studied by Malkovich and Ma1ysh.l"" They used a Zn doped GaAs powder diffusiqn source doped to about 1.5 x 10'' cm-3 with nonradioactive Zn for a long prediffusion at 900°C. Using a similar source made with radioactive Zn., they made so-called "isoconcentration" runs with the diffusion source m.aintained at 900°C and the GaAs samples held at temperatures varying from 400 to 1100°C. This method should give true isoconcentration conditions only a t 9 W C , and one should 106aR. Sh. Malkovich and G. K . Malysh, Fiz. Tuerd. Tela 9, 553 (1967)[English Transl.: Soviet Phys.-Solid State 9, 423 (I 967)].

214

DON L. KENDALL

expect some mismatch between the tracer surface concentration and the nonradioactive Zn concentration introduced during the prediffusion cycle. Their measured D’s under these conditions were exponentially dependent on T-’ below 700°C with an activation energy of 1.2eV. Over the range 8O0-110O0C, however, was almost independent of T. The significance of this is not easily evaluated because of the variation in surface concentration, the higher temperatures showing lower surface concentrations. In an appendix they discuss several theoretical points which are of interest. These include “cross terms” and also the conditions necessary to meet the requirements of local defect equilibrium. Black and Jungbluth1O7 observed several defect structures in Zn-diffused GaAs by x-ray topography and infrared transmission microscopy. They used a limited amount of Zn in a sealed ampoule and obtained a surface ~ 800°C . and lower they found no concentration of about 1 x lo2’ ~ m - At evidence of damage occurring below the Zn diffusion zone, but at 900°C and above they noted a large increase in dislocation density as monitored by a dislocation etchant. They also noted a very large increase in dislocation density in the Zn diffusion zone in all cases. In diffusion runs into already Zn-doped GaAs (from the melt) in the l O I 9 cm-3 range, the Zn diffusion cycle definitely decorated the dislocations. The relative ease of decoration in heavily p-type GaAs is consistent with the increased interstitial concentration in this material, where decoration probably occurs by the mechanism represented by the reaction of Eq. (65). They also suggested that Zn precipitated on dislocations in the diffusion zone, but their evidence was mostly circumstantial. Schwuttke and R ~ p p r e c h t ” ~also ” observed the dislocation structure of Zn-diffused GaAs by x-ray topography. They noted the introduction of dislocations along (1 10) directions. These were associated with the LomerCottrell reaction whereby these immobile dislocations were spontaneously nucleated in the diffusion zone in order to relieve stress. As mentioned earlier, L ~ n g i n i ”was ~ the first to suggest that an interstitial-substitutional mechanism similar to that proposed for C u diffusion in Ge by Frank and Turnbul14’ might be involved in the diffusion of Zn in GaAs. However, in the latter case he proposed the solubility of the interstitial form of Zn should be a sharp function of the concentration of the predominant acceptor form of Zn (see Section 2u). He suggested that the effective diffusion coefficient b of the Zn under this mechanism should be proportional to the ratio of interstitial to substitutional concentration. Assuming that the interstitial Zn is present as ZnT2 and the acceptor form of Zn is completely ionized as Zn&, Longini showed that B should be pro-

’’’

J. F. Black and E. D. Jungbluth, J . Electrochmi. SOC.114,181,188,297 (1967) 107rG. H . Schwuttke and H. Rupprecht, J . Appl. Phys. 37, 167 (1966).

3.

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portional to the cube of the Zn concentration. This is in good agreement with the experimental results, and data will be shown later that strongly support such a model. Weisberg and Blanc"' discussed the interstitial-substitutional equilibrium further. They formulated the problem so they could in principle distinguish the charge on the interstitial ion and found that a singly charged Zn interstitial, which leads to a D proportional to the square of the Zn concentration, gave a best fit to the data of Cunnell and G ~ o c hBecause . ~ ~ of the various anomalies associated with the Zn profiles, however, this conclusion must be accepted with reservation. They also suggested that the low activation energy at high concentrations was consistent with an interstitially dominated process. Weiser"' also preferred a choice of a singly ionized donor Zn interstitial on theoretical grounds. By using a combination of theoretical arguments along with the experimental results of Hall and RacetteS6on another interstitial diffusant in GaAs (Cu), Weiser calculated an effective diffusion coefficient for Zn at several temperatures. These estimates are 1&100 lower than the measured values, but in view of the uncertainties involved in the calculation, the agreement has to be considered as adequate. It at least demonstrates the plausibility of the interstitial-substitutional argument. Rupprecht and LeMayl" studied the diffusion of Zn in GaAs at 850°C using a standard amount of ZnAs, as a diffusion source and enough excess As to provide an As pressure of about 1 atm. They found a depression of D of about one hundredfold at high concentrations over the D with no excess As added. At low Zn concentrations (10'7-10'9~ m - ~they ) , reported an apparent increase in B of about a hundredfold above the data of Kendall and Jonesa2in this concentration range. The apparent increase at low concentration is questionable since they did not reduce their Zn pressure enough to obtain the surface concentrations necessary for an accurate measurement of D at low concentrations. As discussed earlier with respect to Fig. 8, the resultant D's are much higher than the equilibrium D values when measured in the low concentration range of a Zn profile with a high surface concentration. They attributed both the decrease in D at high Zn concentrations and the apparent increase in D at low concentrations under excess As pressure to an increase in the Ga vacancy concentration [see Eq. (12)]. At high concentrations, the dominant term affected in D is the ratio of interstitial concentration to substitutional concentration, and this is decreased by As pressure (through an increase in V,, primarily). The factor of 100 depression of is in reasonable agreement with Thurmond's estimate of 50 as the maximum allowed lo*

'09 'lo

L. R. Weisberg and J. Blanc, Phys. Rev. 131, 1548 (1963). K. Weiser, J . Appl. Phys. 34,3387 (1963). H. Rupprecht and C. 2. LeMay, J . A p p l . Phys. 35,1970 (1964).

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DON L. KENDALL

increase in (V;,) under conditions of excess At low concentrations they explained the apparently enhanced on the basis of substitutional diffusion via the increased number of Ga vacancies under excess As pressure. These are both interesting observations. Accurate corroborative data for the effect of As pressure on the for Zn at low concentrations would be especially helpful in deciding whether substitutional diffusion is important for Zn at the low concentration limit. Pilkuhn and Rupprecht"' discussed the use of ZnAs, as a diffusion source for GaAs. This was said to result in more planar p-n junction than obtainable with elemental Zn in the same range of concentration. By incremental sheet resistance measurements they estimated the surface concentration of Zn to be about 2 x 10" cm-3 at 850°C using a ZnAs, source with a m / V ratio of 1.0 mg/cm3, where n7 is the mass of the diffusion source and V is the ampoule volume. Use of the same amount of elemental Zn actually leads to an even higher Zn concentration of about 2.3 x 10" cm-3 using the data of Fig. 7. They also discussed various factors that can cause junction nonplanarity such as impurity striations and crystal defects. The behavior of B under the conditions of excess As pressure has been studied extensively by Shih et ~ 1 . ' ~ They ' obtained Zn diffusion profiles over a wide range of As pressure at 900 and 1050°C. They used measured quantities of Zn and As in a sealed ampoule, with the amount of Zn being adequate to assure that the Zn vapor reached saturation. They interpreted the results in terms of the Ga-As-Zn ternary phase diagram.Il2" At 1050"C, the diffusion profile changed continuously when extra As was added, with D being proportional to PiL4 at a given Zn concentration. At 900°C the profiles did not change significantly until the As pressure reached about 0.1 atm at which point D decreased abruptly by an order of magnitude. At still higher As pressure D decreased as Pi:," while the Zn surface concentration increased up to an As, pressure of 2.5 atm, beyond which it decreased. They showed that the abrupt decrease in occurred when the As pressure was large enough to cause the formation of Zn3As, in the external phase. At Zn concentrations greater than 10" cm-3 under all conditions the profiles exhibited many of the anomalies mentioned with respect to the profiles of Fig. 8. Shih also measured the As vapor pressure by optical absorption as a function of As in the external phase and discussed the behavior in terms of the Ga-AsZn ternary phase diagram. With regard to measurement of B at low concentrations, Becke et al.' l 3 diffused from a pyrolytically deposited S O 2 film with no excess As and M. H. Piikuhn and H. Rupprecht, Trans. AIME 230, 296 (1964). K. K. Shih, J. W. Allen, and G. L. Pearson, to be published. 'lZaM.B. Panish, J. Phys. Chem. Solids 27, 291 (1966); J . Electrochem. Soc. 113,861 (1966) H. Becke, D. Flatley, W. Kern, and D. Stolnitz, Trans. A I M E 230, 307 (1964). ''I

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obtained a final Zn surface concentration of IOI7 cm-3 using radioactive Zn. The B value at 900°C extracted from their curve is 3.2 x 10-'4cmZ sec-' assuming a Gaussian distribution (limited source), with an upper limit of 6 x cm2 sec-' using an erfc distribution (constant surface concentration). Shortes et ~ 1 diffused . ~ throug,h ~ a sputtered SiO, film using a Zn-Ga alloy source and also obtained a Zn surface concentration of about 10'' ~ m - From ~ . this profile, a D value at 1000°C of about 5 x cm2 sec-' can be calculated. Using the above viilues for D,the Do and Q for Zn at the low concentration limit in GaAs without excess As are calculated as 2.5 x lo-' cm2 sec- and 3.0 eV, respectively. These values should be accepted with reservation, but they are probably the 'best available at low Zn concentrations. observed that the surface concentation of Zn was reduced Shortes et by a factor of four by an SiO, film of 6500-A thickness. They reported more uniform p-n junctions using this technique, as well as much less surface deterioration. They also showed a useful set of data showing the relationship of Zn surface concentration to Zn pressure (using some of the data in Fig. 7). Chang' l 4 published a discussion of the p-n junction depth for Zn diffusion in GaAs and other 111-V compounds. He suggested that the p-n junction depth obtained under saturated Zn vapor conditions in several 111-V compounds is exponentially dependent on T - * , where T is the temperature in degrees Kelvin. He discussed this in terms of the interstitial-substitutional mechanism, but his arguments apply equally well to any mechanism where B is proportional to the Zn concentration raised to some arbitrary power. Kendall and Bartning"4a reported on the p - n junction depth xi in GaAs under saturated conditions over a much wider range of temperature. They found that the normalized x j deviated somewhat from an exponential dependence on T - ' over the 450-1000°C range. They also estimated the surface concentration over this same range from sheet resistance measurements and suggested that diffusion under saturated conditions at lower temperatures could be used to control the surface concentration. The results on junction depth and surface concentration under these conditions are shown in Fig. 10. Kendall and Bartning' 14' also discuss other methods of controlling the Zn surface concentration, all without adding excess As. These methods have in common the control of the Zn vapor pressure which then determines the Zn surface concentration (see Fig. 7). In one of these methods the GaAs is maintained at the warmer end of a sealed ampoule and elemental Zn is placed at the other end.99 The Zn vapor pressure is determined by the temperature of the cold end. In another method, the alloy-source method,97 L. L. Chang, Solid-Stare Electron. 7 , 8 5 3 (1964). ""D. L. Kendall and A. M. Bartning, to be published.

'I4

DON L. KENDALL

< ‘02’

hour)

7

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08

0.9

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I

1100” 1000” 900”

I I

BOOo

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10 II 1000/T(”K ) 700°

I 12 I

600° 550°

13 I

500° ( “ C )

FIG.10. Estimated surface concentration of Zn in GaAs using enough elemental Zn to saturate n-type GaAs for one hour (normalized) the vapor. Also shown is the junction depth in 10” diffusion time under these conditions.

an alloy of Zn :In, Zn :Ga, Zn :Sn, etc., is used as a diffusion source. The In, Ga, or Sn act as “dilutants” for the Zn vapor pressure. The alloy source and diffusion sample are maintained separate throughout the diffusion anneal, and the ampoule is kept at a constant temperature across its length. They show expressions relating the mole fraction, mass of the alloy, and

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DIFFUSION

219

ampoule volume to the partial pressure of Zn. As a special case of the alloysource method, they suggest using a limited amount of elemental Zn as a diffusion source. This method results in little surface damage and good surface concentration control down to about 10l8~ m - but ~ , the mass of Zn becomes inconveniently small for lower concentrations. The surface concentration increases significantly during the cooling cycle using the limited Zn method. This increase on cooling is explained in terms of the data of Fig. 7. This upturn near the surface is evident in profile I1 of Fig. 8 which was diffused under limited Zn conditions. Slumping, or out-diffusion near the surface, is observed in profiles I and IV of the same figure. These were done under saturated Zn and alloy-source conditions, respectively. The efficacy of each of these methods for producing GaAs light emitters was also discussed. Various anomalies associated with the p-n junction obtained by Zn diffusion into n-type material were discussed by Marinace.' l 5 In particular, on chemically staining the junction he observed two distinctly stained lines, the deeper of which corresponds to the p-n junction and the other corresponding approximately with the shoulder in the Zn concentration distribution. He also discussed the possible effects of oxygen and structural defects on junction nonflatness. Pilkuhn and Rupprecht commented further on the same effects.' The degradation of Zn-doped tunnel diodes under forward bias conditions has received attention from several authors. This effect is quite pronounced at room temperature, with the degradation rate, as monitored by the peak current decrease, increasing sharply with Zn concentration. Various diffusion mechanisms have been proposed to explain this unusual effect. These include a Zn interstitial diffusion m 0 d e 1 , ' ~ " ~ a' ~ Cu ~ interstitial contamination model,56,116-1 19 a Zn-vacancy associate model," and an electron injection dissociation model. '22 None of the above present strong enough arguments to decide the issue. Kogan et al.123reported on both Zn and Cd using a two-temperature . ~ ~used the incremental method similar to that of Cunnell and G ~ o c hThey 'Is 'I6

"' 'Ix 'I9

12'

IZz

J. C . Marinace, J . Electrochem. Soc. 110, 1153 (1963). H . J. Henkel, Z . Naturforsch. 17a, 358 (1962). R. D. Gold, B. Goldstein, L. R. Weisberg, and R. M. Williams, Bull. Am. Phys. Soc. 6, 312 (1961). N. Holonyak, ScientificRept. No. 3b, Dec. 1960, AF Contract 19(604)-6623. A. Pilor, G. Elie, and R. Glicksman, J . Electrochem. Sac. 110, 178 (1963). A. Shibata, Solid-state Electron. 7, 215 (1964). A. S. Epstein and J. P. Caldwell, J . A p p l . F,hys. 35,2481 (1964). G. M. Glasford and R. L. Anderson, Tech. Documentary Rept. No. RADC-TDR-64-313, under contract A F 30(602)2778. Syracuse University, Sept. 1964. L. M. Kogan, S . S. Meskin, and A. Ya. Gomikhman, Fiz. Tverd. Tela 6, 1145 (1964) [English Trans/.: Souier Phys.-Solid Stute 6, 882 (1964)l.

8 7 7 V a N m l ‘7 N O a

ZINC CONCENTRATION AT THE SURFACE C, (cm-’)

ozz

w U FIG.11. Average conductivity of Zn-diffused layers in GaAs a s 5 a function of Zn surface concentration.
$

2,

AVERAGE CONDUCTIVITY b (a) (ohm-crn)-'

FIG.!!(c).

222

DON L. KENDALL

sheet resistivity technique for their measurements. They also reported that changes in As pressure did not measurably influence the form of the penetration curve nor the surface concentration, but they did not show the data. Their values of Do and Q are included in Table 111. Boltaks et ~ 1 . ' ' ~reported some interesting results using an electric field superimposed across a slice of GaAs. They observed the drift of the p-n junctions on opposite faces of a diffused sample and found that the junctions were displaced toward a cathode. This suggests that Zn diffuses as a positive species, which is in accordance with the interstitial-substitutional model. However, motion of negatively charged vacancies to the anode could also explain the results since this would cause an anomalously large flux of Zn atoms toward the cathode. They also found that the effective mobility of the diffusing species went through a maximum at about 1000°C and then decreased at higher temperatures. They attributed the decrease above 1000°C to ion drag by the free electrons moving toward the anode. Mehta'25 made careful sheet resistance measurements of Zn diffused layers under a variety of diffusion conditions (elemental Zn and alloy source method). Using the experimental results as a guide, he developed a set of graphs that relate the average conductivity of the layer to the surface concentration of the diffusing Zn atoms. He did this for bulk donor concentrations ranging from 10l6 to 10l8 cm-3. These curves, which are reproduced in Fig. 11, were found to be adequate for predicting the surface concentration when it was below lo2' cm-3, but they failed at higher concentrations. He thought this might be due to the fact that the hole concentration was often smaller than the Zn concentration in these samples. However, it was probably due to the anomalous effects associated with D at high concentrations discussed with regard to Fig. 9. One final experiment involving Zn in GaAs needs to be discussed, since the results bear directly on the diffusion mechanism. This is the two-step Zn diffusion experiments of Kendall et In Step I, a limited amount of elemental Zn (0.07mg/cm3) was diffused into 10'' cm- n-type GaAs for various lengths of time. Several were diffused for 10 minutes, removed from the ampoule and rinsed in HCI, resealed in a clean ampoule without Zn, and rediffused. The results are shown in Fig. 12. The sheet resistance of about 5 ohms per square remained almost constant throughout the heat-treat steps. The important effect, of course, is the apparent cessation of diffusion during the heat-treat steps (as monitored by the p-n junction depth xi).This 124

125

B. I. Boltaks, T. D. Dzhafarov, V. I. Sokolov, and F. S. Shishiyanu F i i . T r o d . T d u 6, 151 1 (1964) [English Transl.; Soviet Phyx-Solid Stute 6, 1181 (1964)l. R. Mehta, P h D dissertation, Stanford University, 1964. Also Tech. Rept. No. 5103-1, U S . Army Research Office, Durham, Contract DA31(124)-ARO(D)-155(Stanford Electronics Laboratories Rept. 64-062).

3.

01

223

DIFFLJSION

I0

10

100

1000

Diffusion time (niinutes )

FIG.12. The effect of diffusion time on p-n junction depth in GaAs at 900°C (0.7 mg of Zn in a 10cm3 ampoule). After 10 minutes several samples from one run were removed, cleaned, and reinserted into separate ampoules witliout Zn. Subsequent heat-treatment at 900°C showed little forward diffusion. (After Kendall, Kanz, and

unusual result has been directly observed in InSb during similar two-step diffusions of radiotracer Zn. This is shown in Fig. 20. The major effect is seen to be the more or less uniform decrease in Zn concentration during Step 11, with iittle forward difluusion. To explain the above results with an interstitial-substitutional diffusion process, the following steps are proposed. During Step I above, interstitial Zn atoms and Ga vacancies enter the crystal logether from the surface and react to form substitutional Zn acceptors on Ga sites, namely, Zn+

+

+ V,,

4

Zn,,

+ 3e+.

(69)

This reaction goes predominately to the right during the short first step. During Step 11, there is no Zn in the external phase, hence no supply of interstitials, so the main effect in the diffused region is dissociation of the substitutional Zn into an interstitial Zn and a Ga vacancy, which is just the reverse of the reaction in Eq. (69).The buill-in electric field due to the Zn concentration gradient (on the order of lo3 volts cm-' in the above example)

224

DON L . KENDALL

causes the positively charged Zn interstitials to be swept predominantly toward the surface. This follows from a simple calculation that takes into account both diffusion and drift due to the concentration gradient. These experiments are probably the best evidence to date that an interstitial-substitutional mechanism is operative for Zn diffusion in GaAs and InSb. Other results regarding the concentration dependence of 0, the effect of As pressure on 4, the degradation of tunnel diodes and other experiments have always been ambiguous in that a vacancy diffusion model of some nature could generally explain the observations equally well. For example, the decrease of with As pressure can be explained by Robert's for Mn diffusion which assumes the diffusion step consists of MnGaexchanging with [ v A , v A , ] . This model can also explain the concentration dependence since [ v , , v A , ] is probably a donor and its solubility would be enhanced in heavily p-type material. It would be very difficult for this model, however, to explain the two-step Zn diffusion experiments above. Specifically, in order to explain the apparent cessation of diffusion during Step 11, one must account for a large reduction in concentration of the defect responsible for Zn diffusion when the Zn pressure is reduced in the ampoule. If [V,,V,,] is indeed the defect promoting Zn diffusion, this implies that the removal of excess Zn from the ampoule causes the As pressure (activity) to increase by several orders of magnitude. There is no evidence to support such an increase. Parenthetically, one should note that Fig. 1, relating the effect of Zn pressure to the various defect concentrations, is not strictly applicable during Step I1 since the Zn diffused crystal is no longer in equilibrium with its external vapor. In particular, (Vi,) and (VLJ in the Zn diffusion zone do not decrease when the Zn pressure is suddenly decreased. (VLJ will decrease only when (Zn;,) in the diffusion zone decreases by precipitation or out-diffusion. A similar argument applies to [ v A , v , , ] +. The important findings regarding the diffusion properties of the Zn in GaAs are: (1) 0 varies basically as under equilibrium conditions with no excess As. (2) The dependence is less than cubic when measured from a single diffusion profile. (3) B varies as Pi? at a given Zn concentration under certain conditions. (4) Substitutional Zn is completely ionized (Zn,,) at high concentrations because the Zn level becomes part of the valence band. (5) The diffusion is dominated by the interstitial-substitutional process. (6) The Zn interstitial is probably doubly ionized (Zn"). (7) The interstitial B is greater than 5 x lo-' cm2 sec-' at 900°C. lZ5*F. E. R. Roberts, unpublished calculation discussed in Ref. 132.

3.

DIIWJSION

225

Some of the important questions yet to be answered for the system are : (1) Is substitutional diffusion a significant factor at low Zn concentrations and/or at high As pressure? (2) What is the cause of the unusual behavior of D measured under profile conditions at high Zn concentrations? (3) What role does the ~ , V , ] divacancy play in determining the diffusion profile? (4) Can dislocations introduced during diffusion at high concentrations explain some of the anomalies? (5) Is the divacancy-interstitial reaction responsible for blocking the Zn decoration of dislocations in the interior of GaAs sample during vapor diffusion? (6) What is the effect of dislocation density in the starting material? (7) Where do the various defects involved go on the cooling cycle and how do they manifest themselves?

b. Cd in GaAs Several investigators have reported on the diffusion of Cd in GaAs. G ~ l d s t e i n diffused ~~” radioactive Cd from the vapor and obtained B values assuming the concentration distribution was that of a complementary error function. This implies that D is independent of Cd concentration. However, Kendal154 has shown that the B for Cd is concentration dependent. Goldstein’s values are more or less typical of the values obtained at the highest Cd vapor pressures (see Table 111). Cunnell and Gooch’26 reported a value of D for Cd at 1000°C about three times lower than Goldstein’s. But, in addition, they reported uphill digusion deep in the interior of the sample, i.e., diffusion prcgressed toward a region of higher concentration. The apparent concentration of the deeply diffusing species reached a value that varied in an unaccountable manner between . concentration was observed at a 1OI6cmU3and 4 x lo’* ~ m - A~minimum depth slightly beyond the steeply falling surface branch (just in front of the expected p-n junction). They claimed that the half-life of the deep diffusing species was the same as Cd1I5, but, from Kendall’s results,54 this is very likely not correct. Cunnell and Gooch offered no explanation for this very unusual result, although they did several experiments under widely varying conditions in an attempt to elucidate the mechanism. For example, variations in conductivity, orientation, and dislocation density did not noticeably affect the process. KendallS4 studied the diffusion of Cd using the same tracer as Cunnell and Gooch (Cd’ 15), He found that the deeply penetrating activity was in fact Iz6

F. A. Cunnell and C. H. Gooch, Nature 188,1096 (1960)

226

DON L. KENDALL

the daughter product of Cd115, namely, In“5 with a half-life of 4.5 hours (instead of 54 hours for the Cd1I5).This radioactive decay product diffused deep into the sample independently of its parent. He further demonstrated this by diffusing long-lived In114 (50 days) into GaAs simultaneously with nonradioactive Cd. He found the In indeed diffused uphill under these circumstances and that the actual In concentration deep in the interior of the sample ~ . explanation given for the reached a maximum of about 2 x 10l6~ m - The uphill diffusion was based on an interstitial-substitutional model. This will be discussed in the section on In diffusion (Section 17d). Kendall also found that B was a function of Cd concentration, His data are tabulated in Table IV. From these data, B is seen to be approximately proportional to the first power of Cd concentration at the upper concentrations at 1000°C. This manifests itself in a concentration distribution which is somewhat steeper than an erfc, but not as steep as the Zn profiles shown earlier. Bendik et ~ 1 . ’ ~ ’reported an effect of dislocations on the location of the p-n junction following Cd diffusion in GaAs. They noted two stained “junctions,” with the shallow one being more or less independent of dislocation density. The deeper one, which was the actual p-n junction, increased markedly with dislocation density above a density of about lo4 cm-2. They attributed this deep diffusion to a fast diffusing component of Cd. However, the radiotracer results of Kendal154 showed no evidence of a fast diffusing branch at lower concentrations. Further, Bendik’s shallow “junction” corresponded very well with the tracer results. Thus, it is likely that the deep dislocation dependent junctions they observed were due to a fast diffusing impurity such as Cu or Mn. Diffusion of this type impurity has been shown to be a sensitive function of dislocation d e n ~ i t y . ’ ~ , ~ ~ TABLE IV DIFFUSION COEFFICIENT OF Cd I N GaAs NEAR SURFACE AND SURFACE OF Cd vs Pcd CONCENTRATION

T (“C)

lo00 1000 lo00 1000 1100

Cd pressure Surface concentration (~m-~) (atm)

0.07 0.36 0.75 2.20 0.20

7 2.5 4 7 2

x 1017 x 10“ x x lo’* x 1018

Diffusion coefficient (cm’ sec-’) 8x

1 x 10-12 1.4 x lo-‘* 3 x 10-12 1.4 x lo-”

Kogan et published diffusion data for Cd which they analyzed using sheet resistance techniques. They used a two-temperature diffusion technique and assumed the profile to follow an erfc distribution. They inferred the sur-

3.

DIFFUSION

227

face concentration from sheet conductivity measurements to be in the range of 1019~ m - They ~ . did not see the dependence on dislocation density pre” they observed a wide high resistivity region on dicted by Bendik et ~ l . , ~but then-side of the junction at the highest temperatures( 105&1 lOOOC).The latter is consistent with the interpretation given here of Bendik’s results, where the deep high resistivity region can also be attributed to contamination by a fast diffusing acceptor such as Cu or Mn. Kogan’s Do and Q shown in Table 111 give B values in good agreement with Kendall’s data at low concentration. c . H g in GaAs

The only report of Hg diffusion in GaAs is that of Kanzlz8 who diffused Hg from the vapor in a sealed ampoule using enough HgZo3to give I atm pressure. He obtained a surface concentration of about 5 x 10’’ cm-3 and a of 5 x 10- l 4 cm2 sec-l a t 1000°C. This is about an order of magnitude lower than the other simple acceptors at this temperature (see Fig. 3). d. Mg in GaAs

Diffusion of Mg in GaAs has been studied by Gupta and Shortes’29and Moore et They all used sheet resistance and p-n junction measurements for the B estimates. The values seemed to be a function of the purity of the to cm2 sec-’ Mg, with the apparent B values ranging from 5 x at 1000°C. The highest values were found to be associated with the presence of Mn. The lowest values reported are those of Moore et al., who used probably the purest Mg available (99.999%). They had difficulty, however, with the Mg reacting with the quartz ampoule. Their lowest values are similar to Zn at very low concentrations (Fig. 3). 17. Mn, Cu, Tm, In, Ag, Au,

AND

Li

IN

GaAs

a. Mn in GaAs

Mn diffusion in GaAs has been studied by Kenda11,83Larrabee et aE.,131 Seltzer,13* and Vieland.” Vieland observed that the p-n junction depth obtained by Mn diffusion decreased with increasing As pressure. The apparently radiotracer profiles taken by Larrabee et al. indicated that decreased with increasing Mn concentration. Seltzer sometimes observed this as two distinct profiles. According to Seltzer, both the fast and slow M. A. Bendik, R. L. Petrusevich, and E. S. Sollertinskaya, Fiz. Twrd. Trla 5, 3247 (1963) [English Trans/.:Souiet Phys.-Solid State 5, 2375 (1964)l. ’** J. A. Kanz, Ref. 83, p. 34. D. C. Gupta and S. R. Shortes, unpublished data. R. G. Moore, Jr.. M. Belasco, and H. Strack, Bull. Am. Phys. SOC.10, 731 (1965). 1 3 ’ G. B. Larrabee, R. W. Haisty, and J. F. Osborne, unpublished data. M. Seltzer, J . Phys. Chem. Solids 26,243 (1965). IZ7

228

DON L. KENDALL

branch B values were proportional to Pi;’’, although there was considerable scatter in the data. His “fast” values were about an order of magnitude higher than the slow ones over a wide range of As pressures. He explained the dependence on As pressure with an As divacancy (substitutional)diffusion model originated by R~berts.’~’“ However, the effect of As pressure on the Mn activity (through formation of Mn,As,) was not taken into account. When this is done, the data can be explained with an interstitial-substitutional model similar to that proposed earlier for Zn. The interstitial form of Mn must be predominantly neutral though, to explain the various observations. Larrabee et ~ 1 . l measured ~ ‘ a B ofabout 1.5 x lo-’’ cm2 sec-’ at 825°C at Mn concentrations between 10l8 and 1019cm-3 (with no added As). 6 the measured B From the same profile at low concentrations ( l o L cmP33, was in the 10-8-10-’ cm2 sec-’ range. Larrabee also observed that a large fraction of the Mn in the vicinity of the p-n junction was electrically inactive. The latter effect was significantly reduced when excess As was present during the diffusion. This inactive Mn can be attributed to the neutral interstitial mentioned above. The depression of the inactive Mn by excess As (enhancement of active Mn) indicates that the As overpressure caused the Mn activity to decrease, thus lowering the neutral interstitial concentration relative to the substitutional form. Kenda1183reported B values using incremental sheet resistance techniques to determine the electrically active distribution. Assuming an erfc distribution, he obtained the Do and Q shown in Table 111. In view of the work reported above using tracers, these values cannot represent true D values, but should be useful for estimating p-n junction depths under the condition of no excess As. Peart et al.’j3 noted that radioactive Mn diffused out ofa region of the Zn diffusion zone when Zn was diffused into a Mn-doped crystal. This is very similar to the Cu :Zn results of Larrabee and O ~ b o r n e to ’ ~ be ~ discussed in the following section and is also related to the In :Cd results of the author.54 The Mn goes through a minimum very similar to that shown in Fig. 13 for Cu diffused along with Zn. They explain the results by assuming the Mn is amphoteric, that is, it can be either a donor or an acceptor. In the Zn p-region the two latter forms are enhanced and depressed, respectively. At nearly intrinsic concentrations the Mn concentration is expected to go through a minimum. They also observed some interesting “structure” in the Mn distribution near the Zn diffusion front. This may be explained by assuming that a significant quantity of neutral Mn species (interstitials or Mn-vacancy pairs) are present in this region of the diffusion zone. 133

134

R. T. Peart, K. Weiser, J. Woodall, and R. Fern, Appl. Phys. Letters 9, 200 (1966); K. Weiser, M. Drougard, and R. Fern, J . Phys. Chem. Solids 28, 171 (1967). G. B. Larrabee and J. F. Osborne, J . Electrochem. SOC. 113,564 (1966).

3.

' \

229

DIFFUSION

ZnGa (ESTIMATED)

\ I I I I I I

I

- - I- - _ - - - BULK - _ DONORS _____

-----

I

I

I

I 0

20

I I 40

I

60 DEPTH IN MICRONS

I

I

80

100

120

FIG.13. Effect of Zn player o n distribution of Cu in GaAs; Zn and Cu64 diffused simultaneously for 6 hours at 835°C. (After Larrabee and 0 ~ b o r n e . l ~ ~ )

b. C u in GaAs

Fuller and Whelan'35 showed that Cu diffuses rapidly into GaAs in a manner very similar to its behavior in Ge. They observed two and perhaps three branches in the concentration. profiles, D values at 1000°C and 1100°C were and 2-3 x l o p 5 cm2 siec-l, respectively. An even faster 135

C. S. Fuller and J. M. Whelan, J . Phys. Chern. Solids 6, 173 (1958).

230

DON L . KENDALL

branch at 1100°C with a D of 2-3 x lOP4cm2sec-' was also observed. Fuller and Wolfstirn 13' also noted certain electrical anomalies associated with Cu diffusion and reactions with AB or AA divacancies and some other defect or impurity. Hall and RacetteS6 published a more detailed study of the diffusion and solubility of Cu in GaAs. Their solubility measurements agree with those of Fuller and in addition show the expected maximum in solubility at about 1100°C. Below 700°C the Cu solubility leveled off at about 1.5 x 1016cm-3 instead of continuing to decrease as expected. They attributed this to complex formation with some unidentified defect. They also found that the Cu solubility was greatly enhanced in p-type material. This suggests that the donor interstitial form of Cu is dominant in heavily p-type GaAs. From a simple theoretical argument, they estimated the interstitial Cu solubility in intrinsic GaAs. They measured the interstitial D for Cu in heavily p-type material and obtained the Do and Q values shown in Table 111. These D values, along with an estimate from their data of the interstitial-substitutional B for Cu, are shown in Fig. 3. The latter is in good agreement with Fuller's values at elevated temperatures. Hall and Racette also demonstrated that the Cu interstitial is a singly charged donor in GaAs. They did this by drift measurements in an electric field. D values calculated from these experiments using the Einstein relation were consistent with their own measurements. In still another experiment, they heat-treated n-type GaAs crystals in the presence of Cu and found that one Cu atom compensated approximately two donors. On the basis of this, they suggested that substitutional Cu in GaAs is a double acceptor. However, formation of a Cu :donor pair which acts as an acceptor will also explain the results. Larrabee and O ~ b o r n erecorded '~~ diffusion profiles using C U ' in ~ GaAs under a wide variety of conditions at a temperature of 825°C. They generally of about saw a relatively slow diffusing branch near the surface with a 2-5 x l o - ' ' cm2 sec-' and a deeper branch with a D of 1-5 x lo-' cm2 sec-'. From Hall's data, an estimated D for Cu in high dislocation density cm2 sec- at this temperature. Thus the latter material is about 3 x values of Larrabee are evidently typical of the modest dislocation density used (104-105 cm-2). The low D values observed near the surface are probably due to the reaction of the Cu interstitials with incoming vacancies. Larrabee and Osborne also noted a depression in the Cu concentration of 10-100 in the vicinity of a p-n junction when the Cu was diffused simultaneously with Zn or Mn. An example of this unusual effect is shown in Fig. 13. The magnitude of the depression, as well as the zone over which it 136

C . S. Fuller and K. B.Wolfstirn, Solid State Commun. 2,87,277 (1964);J . Phys. Chum. Solids 27, 1889 (1966).

3.

DIFFUSION

231

extended, was very irreproducible. The depression and subsequent uphill diffusion is very similar to the effect noted by Kendall for the simultaneous diffusion of In and Cd into G ~ A s In. ~explanation ~ of this, Kendall proposed an interstitial-substitutional mechanism similar to that of Cu in Ge4'. The uphill branch, which was obtained only when a p-layer was present, was ascribed to a nonequilibrium effect whereby the rapidly diffusing interstitials served to label the vacancy distribution in the interior of the crystal. However, in Larrabee's case a simple solubility argument is more reasonable. The dominant form of Cu in the interior of the GaAs is the acceptor CU&,and this is depressed to very low concentrations in the p-type Zn diffusion zone. In the p-layer the dominant form of Cu is the interstitial C h i + . The solubility enhancement factor for the interstitial is p / n , , where p and ni are the hole and intrinsic carrier concentrations, respectively. This factor is about 100 near the surface in Fig. 13, so the bulk concentration of Cu, should be depressed by 100 at the surface. Similarly the surface concentration of Cu,' should be enhanced by the same factor relative to the interstitial concentration in the bulk. By this reasoning the equilibrium interstitial concentration in the bulk at 825°C should be about 5 x 1013 ~ m - This ~ . is in reasonable agreement with the data of Hall and R a ~ e t t e , ~ ~ who predicted about cm-3 for this quantity at 825°C. c. Tm in GaAs

Casey and P e a r ~ o n ' ~studied ' Tm in the GaAs system. They reported to be higher at lower temperatures, and the values were given by

DTm= 2.3

x 10-'6exp(+l.0eV/kT),

D

(70)

where the + sign is inserted as a reminder of the retrograde diffusion behavior. They explained the inverse temperature dependence of B using an interstitial-substitutional model, wherein the effective D was given by

where Di is the interstitial D of T m i + .On'this model, the substitutional Tm solubility was said to increase more rapidly with temperature than does the product (Tmi+)Di.Casey also made measurements in heavily p - and n-type crystals and noted enhancement and depression of B, respectively, as expected on the above model. In many of Casey's profiles, there was an unexplained fast diffusing branch at low concentrations with a B value of about 2 x 10- l o cm2 sec-' over a wide range of temperatures (8OCrlOOO"C). 13'

H. C. Casey, Jr., and G . L.Pearson, J. A p p l . Phys. 35.3401 (1964).

232

DON L. KENDALL

d. In in GaAs

Kendall diffused radioactive In into GaAs simultaneously with the acceptor Cd.54 Data were also obtained diffusing In alone into GaAs by Kendall, Larrabee, and H a i ~ t y . ' ~ 'These ' ~ ~ data are shown in Fig. 14. The latter measurement suggested that In diffusion proceeds with a B of 7 x 10- ' cm2 sec- * at 1000°C.These data are shown in Fig. 14. Deep in the

0

50

100

150

200

DEPTH IN M I C R O N S FIG.14. Diffusion of In into GaAs with and without the simultaneous diffusion of Cd, normalized to a diffusion time of 24 hours. Pulse height analysis (P) and total integrated intensity (T) as noted. (After Kendd11.54)

crystal, In is seen to be uniformly distributed at a concentration of about 10" cm-3. The surface branch and the uniform region in the interior are attributed to reaction of incoming In interstitials with surface and bulk generated vacancies, respectively. Much less In diffused into a GaAs sample when Cd was introduced simultaneously. Thus, the Cd diffused layer evidently retarded the diffusion of In into the wafer. Kendall attributed this to the built-in electric field associated with the acceptor gradient. As seen in Fig. 14, the major portion of the 13*

D. L. Kendall, G. B. Larrabee, and R. W. Haisty, unpublished data quoted in Ref. 54

3.

DIFFUSION

233

In is restricted to the Cd diffusion zone. In fact, the In concentration is approximately proportional to the cube of hole concentration in this region. This was given as evidence that the reactants had a net charge of + 3, as in the reaction

which leads to

The uphill branch was also attributed to the above reaction. In this case, however, the rapidly diffusing interstitials reacted with the dislocation generated vacancies as they,fEowed toward the surface. This implies that the local equilibrium vacancy concentration in the Cd diffusion zone is sharply depressed by the incoming interstitials in the reaction above or by some mechanism associated with the Cd gradient. On this model, the diffusion coefficient of the out-diffusing species (either a G a vacancy, an AA or AB divacancy) was estimated to be about cm2 sec-‘. However, it should be noted that complex formation between the In interstitials and an outdiffusing impurity can also explain the uphill diffusion.

e. Ag in GaAs Ag diffusion in GaAs has been reported by Rybka et ~ l . , ’and ~ ~their values of Do and Q are shown in Table 111. They diffused radioactive Ag from a thin electroplated film and assumed a limited source type solution was obeyed. Boltaks and S h i s h i y a n ~ reported ’~~ D values for Ag in GaAs which are considerably higher than those above. Two branches were apparent in the diffusion profiles, the slower one near the surface. The surface concentration of this branch gave an apparent Ag solubility limit of about 5 x lo2’ cm-3 and an apparent of about 10-9cm2 sec-’ at 1000°C. This “solubility” is much higher than expected for an atom with the size and valence of Ag, so it is possible that the shallow branch was an artifact due to a vapor or surface reaction. In fact, they reported surface evaporation of about 100 microns during their runs. If this is the case, the B values in this range are of questionable significance. The B values derived from the deeply penetrating activity are more likely associated with true Ag diffusion (see Table 111). 13’

I4O

V. Rybka, N. Yoseli, and M. Aoki, J . Phys. Soc. Japan 17,1812 (1962). B. I. Boltaks and F. S. Shishiyanu, Fiz. 7uerd. Tela 5, 2310 (1963) [English Transl.: Soviet Phys.-Solid State 5, 1680 (1 964)].

234

DON L. KENDALL

f: ALLin GaAs

Sokolov and ShishiyanuI4l measured D for Au in GaAs. Their values of Do and Q are given in Table 111. An apparent D of 2.7 x lo-" cm2 sec-' at 1000°C was observed in a surface branch, but the surface concentration of greater than 1020cm-3 suggests that this was a surface effect of some nature. Larrabee and O ~ b o r n ehave ~ ~ recorded Au profiles at 835°C. An apparent D in the surface region of 3 x cm2 sec-' and an apparent surface concentration of about 2 x 1020cm-3 at 835°C are consistent with Sokolov's results. A very deep branch leads to a D of approximately 8 x lo-* cm2 sec- I , which compares with a calculated D of Sokolov's of 3 x l o p 8cm2 sec-' at the same temperature. Extrapolation of this branch to the surface leads to a solubility of 1.5 x 10'' c m P3at 835°C. g. Li in GaAs

Fuller and W ~ l f s t i r n 'reported ~~ on the diffusion and solubility of Li in GaAs. The Do and Q values reported for this impurity are shown in Table 111. The Li was present both as a donor interstitial and in an acceptor form. The acceptor was tentatively described as a pair of Li atoms in the configuration [LiiLiGJ. This species evidently caused the diffusion to be non-ideal as evidenced from out-diffusion studies. The two oppositely charged species also strongly compensated each other, with the acceptor form being dominant after various treatments at low temperatures. Hayes'43 and Lorimor and S p i t ~ e rhave ' ~ ~ noted the tendency of Li to compensate crystals of GaAs doped with either donor or acceptor impurities. Various infrared-active vibrational modes of Li-doped GaAs were identified as unassociated Lii+ and LiC, ions, [Znc,Li,+], [Cd&Li,+], and [TeLsLi&,] pairs,'43 and [LiSi,,] p a i r ~ . ' ~ ~ - L o r i m and o r Spitzer report Li bands can be associated with four distinct Li-Zn c o m p l e x e ~ ' ~and ~ " three bands each for Mn and Cd doped material. They see Li,+ in the tetrahedral site surrounded by As atoms, but not by Ga atoms. 18. SUMMARY OF DIFFUSION IN GaAs

The reported impurity diffusion coefficients in GaAs as functions of temperature are portrayed in Fig. 3. The ones shown are believed to be the best reported to date at the low concentration limit of each impurity. The latter point is emphasized since the values at higher concentrations may be 14'

V. I. Sokolov and F. S. Shishiyanu, Fiz. Turrd. Tela 6, 328 (1964) [English Trunsl.: Souirt

Phys.-Solid State 6, 265 (1964)l. C. S. Fuller and K. B. Wolfstirn, J . Appl. Phys. 33,2507 (1962). 143 W. Hayes, Phys. Rev. 138, A1227 (1965). 144 0. G. Lorimor and W. G. Spitzer, J . Appl. Phys. 37, 3687 (1966). 14411 0.G. Lorimor and W. G. Spitzer, J . A p p l . Phys. 38, 3008 (1967). 142

3.

DIFFUSION

235

larger by several orders of magnitude, as in the case of Zn, for example, which is shown only at the lower limit in Fig. 3. The self-diffusion results for Ga and As of Goldstein6' are also shown for comparison. However, as discussed in Section 9, these values should not be taken too seriously until the conflict with other measurements is resolved. 19. In, Sn, Te, AND Li in GaSb Eisen and BirchenalP published self-diffusion results for Ga and Sb in GaSb, and Boltaks and G u t ~ r o vsubmitted ~~ tracer data on In, Sn, Te, and Sb. As discussed in Section 10, the precision of the latter measurements may be low due to some unknown experimental difficulty. They showed no diffusion profiles, so the work is difficult to evaluate. Their Do and Q values for In, Sn, and Te are shown in Table 111. The electrical properties of Li diffused GaSb have been investigated by Baxter et The defect acceptor level often found in pure GaSb disappeared after the Li treatment, presumably by ion pairing with Li. They suggested that the unknown defect was a double acceptor of the form Gasb. Yep and B e ~ k e r came ' ~ ~ to ~ a similar conclusion from Li diffusion studies of Te-doped GaSb, but van ma are^^'^.^^ preferred a choice of V,, on the basis of his experiments. However, the latter defect cannot explain the stoichiometry studies of Effer and Etter'44e and Reid et which require G a enrichment. Van Maaren thus suggested that the complex [V,,G%,,] might be the actual acceptor involved. This is not unreasonable sincethis is just a "rearranged VSb,"which satisfies the stoichiometric results [see reaction (3211. This argument has been extended by van der M e ~ l e n . ' ~ ~ g report on the apparent diffusion behavior of Li in Bougnot et p-type GaSb. They assume that each Li atom behaves as a donor which compensates the native defects present. They infer the Li distribution from incremental sheet resistance data using the Boltzmann-Matano method [Eq. (68)] to obtain the effective D of the Li. They suggest that D at 655°C varies from lo-" cmz sec-' to 2 x cm2 sec-l at Li concentrations of 10l6 and 7 x l O I 7 ~ m - respectively. ~ , There are several errors in interpretation and mathematics in this work, so these conclusions should not be taken too seriously. Nevertheless, careful analysis of sheet resistance data after Li diffusion at several temperatures should provide useful information on the defect reactions in this system. '44bR.D. Baxter. R . T. Bate, and F. J . Reid, J . P h j x C ~ U Solirls X 26. 41 (1965). 144cT. 0. Yep and W. M. Becker, J . Appl. Pliys. 37, 456 (1966). 144dM.H. van Maaren, J . Phys. Chem. Solids 27,472 (1966). '44eD. Effer and P. J. Etter, J . Phys. Chew. Solids 25,451 (1964). 144fF.J. Reid, R. D. Baxter, and S. E. Miller, J . Elwtrochem. Soc. 113,713 (1966). 144EY.J. van der Meulen, J . Phys. Chrm. Solids 28, 25 (1967). 144hJ. Bougnot, E. Monteil, and C. Llinares, P h j Stat. ~ Solidi 21, K31 (1967).

236 20. Zn

DON L. KENDALL IN

InP

Chang and Caseyi4’ reported on the diffusion and solubility of Zn in InP. They found a very high solid solubility for Zn of about 2 x 10” cm-3 at 950°C. The effective diffusion coefficient was found to be concentration dependent in a manner similar to that of Zn in GaAs. Near the surface the D obtained from the diffusion profiles was lower than expected. They suggested that the reduction of D in this region was due to the attraction between the interstitial and substitutional species. A similar effect has been noted for Zn diffusion in GaAs, however, and attributed to possible nonequilibrium effects associated with the large acceptor gradient (see Figs. 8 and 9 and relevent discussion). The recent work of Casey and Panish on band tailing at high concentrations needs also to be taken into account here for a detailed understanding of the

21. S, Se, Te, Sn, Ge, Zn, Cd, Mg, AND Cu I N InAs S ~ h i l l m a n reported ’~~ on the diffusion properties of several impurities in InAs. Following each diffusion run, he measured the p-n junction depth using a thermoelectric probe. His estimates of Do and Q for S, Se, Te, Sn, Ge, Zn, Cd, and Mg are shown in Table 111. The high D for Zn of l o p 8cm2 sec-’ at 800°C is reminiscent of its behavior in Ga-As. In Schillman’s view, the similar values of Q for the normal acceptors and the group IV elements suggested a common diffusion mechanism. Contamination from the quartz and other sources may have played a role in some of Schillman’s measurements. Chen measured radiotracer profiles of Zn and Sn diffusion in polycrystalline 111As.l~~ He obtained a D for Zn of lo-’* cm2 sec-’ at 736°C which is about 4000 times lower than the D obtained by Schillman at the same temperature. This is probably evidence of the strong concentration dependence of for Zn, with Chen’s measurements being typical of a lower average Zn concentration. In fact, Bueh1e1-l~~ obtained tracer profiles for Zn in InAs following vapor diffusion and found impurity distributions similar to those of Zn in GaAs. Data regarding p-n junction depth from this work have been published by Chang.’ l 4 Buehler also obtained incremental sheet resistance and Hall data on the diffused layers. Following a diffusion cycle at 5OO0C, the hole concentration was 2 to 3 times less than the Zn concentration in the 1018-10’9cm-3 region. He also observed a diffusion tail in 145

146

14’ 148

L. L. Chang and H. C. Casey, Jr., Solid-Stare Electron. 7,481 (1964). E. Schillman, Z . Naturforsch. lla, 472 (1956); in “Compound Semiconductors” (R.K . Willardson and H. L. Goering, eds.),p. 358. Reinhold, New York,1963. W. S. Chen, unpublished data given in Ref. 149a. M. G. Buehler, Quarterly Research Rev. No. 6, Stanford Electronics Lab., Stanford University, 1963.

3.

DIFFlJSION

237

his radiotracer profiles in the 1016-1017cm-3 range. Near the surface a much lower apparent D was found. The apparent surface concentration was about lo2’ ~ m - ~ . Boltaks and RembezaI4*” have determined the diffusion coefficients as well as the electrical transport properties of Zn in InAs. They obtained profiles similar to those of Buehler, except no diffusion tail was detected at low concentrations. By diffusing in an electric field of the order of 0.4 volts cm- they deduced that Zn diffused as a positive ion with an effective charge of + 2 below 660°C. At higher temperatures the effective charge of the Zn was decreased because of the drag by electrons. They assumed that the diffusing species was the doubly charged interstitial ZnT2, but the results can also be explained by motion of the complex [VAsZnGaVAs]+2 toward the cathode, or by motion of V& toward the anode. Motion of the latter could contribute to enhanced diffusion of substitutional Znca toward the cathode. Arseni et a1.148bmeasured the solubility and diffusion coefficients of Cd in InAs using radiotracer and p-n junction measurements over the range 65(r90O0C. The solubility reached a maximum of 3.5 x 1019cm-3 near 800°C. Their values of Do and Q were close to those of S ~ h i 1 l m a n . Since l~~ their profiles followed erfc distributions, they assumed was independent of Cd concentration and that diffusion was predominantly by a vacancy exchange mechanism. However, the magnitude of Do and Q appear to be more compatible with an interstitial-substitutional process. As mentioned in Section 13 for Zn in AlSb, a more sensitive test for the constancy of would be provided by reducing the Cd vapor pressure during diffusion. Fuller and W o l f ~ t i r n ’have ~ ~ recently measured the diffusion coefficient and 0.52 eV, both of of Cu in InAs. The values of Do and Q were 3.6 x which appear to be typical of interstitial diffusion. It was found to be almost exclusively a singly ionized interstitial donor, although a small substitutional acceptor concentration was found after quenching from 85(r9OO0C.

22. Te AND Sn IN InSb Impurity diffusion has not been studied in InSb to the same extent that it has in GaAs, but a considerable amount of data exists nonetheless. Only two donors, Te and Sn, have been studied with the remainder of the work having been devoted to the acceptors Zn, Cd, and Hg, and the fast diffusing species Cu, Au, Co, Ag, and Fe. P b diffusion in InSb has also been studied 148aB.I. Boltaks and S. I . Rembeza, Fiz. Tuerd. Tela 8, 2649 (1966) [English Transl.: Soviet Phys.-Solid State 8, 2117 (1967)l. 148bK. A. Arseni, B. I. Boltaks, and S. 1. Rembeza, Fiz. Tverd. Tela 8,2809 (1966) [Eng[is/iTrans/. : Soviet Pltys.-Solid State 8, 2248 (1967)l. 149 C. S. Fuller and K. B. Wolfstirn, J . E/ectror/iem. Sor. 114, 856 (1967).

238

DON L. KENDALL

briefly. The values of D for each impurity believed to be the most accurate reported to date are shown in Fig. 15. Boltaks and Kulikov6’ reported on the diffusion of Te in InSb. Their D values for the self-diffusion of In and Sb in InSb were several orders of magnitude higher than those of later workers (see Section 12). The Te results were obtained using the same methods, so the Do and Q shown in Table 111 should be accepted with caution. A radioactive contaminant or grain-boundary diffusion may have influenced their results. 1OOO/r(°K)

**

+ Dislocation free At 2 x 10’

* l o 4 dislocations crn -2

FIG.15. Diffusion coefficients in InSb at low concentration limit except as noted.

3.

239

DIFFUSION

Sze and Wei149ndiffused radioactive Sn113 from a thin plated film and obtained penetration curves typical of limited source diffusion except for a few high points very near the surface. These latter cast some doubt as to whether their values are typical of true bulk diffusion, especially in view of the author’s work on self-diffusion in InSb. In the latter, which was discussed in Section 12, profiles very similar to the Sn profiles of Sze and Wei were found to consist of a bulk diffusion branch with very low D values, and a deeper branch associated with surface pitting during the diffusion run. The very shallow points on Sze’s profiles correspond to D values cm2 secat 512°C. This is similar to the values for self-diffusion of In and Sb at this temperature. Their values of Do and Q for the deeper branch are shown in Table 111. Sze and Wei also showed that Sn diffused rapidly along grain boundaries in InSb. The apparent D along grain boundaries was of order 10-7-10-8 cm2 sec- over the temperature range studied (390-512°C).

-

’

’

23. Zn, Cd, AND Hg I N InSb a. Zn in InSb

Several workers have reported on the diffusion properties of Zn in InSb, their results varying widely. These variations are discussed briefly in this section and then results of the author on this system are presented in considerable detail. G ~ l d s t e i n ’ ’diffused ~ Zn65 from an evaporated film and reported that the concentration profiles were typical of limited source diffusion. He did not observe the extreme concentration dependence of later workers. His values are probably typical of some high average Zn concentration and are expected to bea function of the thickness of the initial evaporated layer. Sze and Wei,149”for example, used the same method and obtained D values 102-103 lower than Goldstein’s. The Zn layer thickness was evidently the only difference between the two experiments, with Sze using the thinner ones. Hulme and Kemplsl diffused Zn from the vapor into Te-doped InSb at various concentrations and calculated D values assuming an erfc distribution was obtained. They noted large run to run variations which were probably due to the concentration dependence of D.Their Do and Q values shown in Table 111 are applicable only under specific experimental conditions. Kendall and Jones97 reported that changed continuously over several orders of magnitude at a single temperature over the Zn concentration range 10’s-1020 ~ m - They ~ . proposed a qualitative model based on diffusion 149aS.M. Sze and L. Y. Wei, Phys. Rev. 124, 84 (1961 ). B. Goldstein, in “Properties of Elemental and Compound Semiconductors,” p. 155ff.

Wiley (Interscience), New York, 1960. K. F.Hulme and J. E. Kemp, J . Phys. Chem. Solids 10,335 (1959).

240

DON L. KENDALL

via positively charged vacancies, but strongly influenced by the built-in field of the Zn acceptor gradient. They also introduced a method for controlling the Zn vapor pressure during diffusion with a Zn-In alloy. The same authors82 later reported that the concentration dependence of D could be explained by a D which was proportional to the concentration of positively charged vacancies (of unspecified nature). The positive vacancy concentration was derived from the ionization reaction e+ + V ' + V + ,

(74)

which leads to

(V + ) = K(VO)P

1

(75)

where (V') is the neutral vacancy Concentration. The latter was assumed to be independent of impurity concentration [see Eq. (5) and discussion]. Equation (75) does not apply in the case of heavily doped material, but as discussed in Section 2d(l), the defect concentrations may still be calculable if Fermi-Dirac statistics are used instead of Boltzmann statistics. A graphical solution of (V') is shown in Fig. 16 for the case at hand. The calculation of the Fermi level E , is shown for a Zn concentration of 5 x 1019cm-3, in

FIG. 16. Graphical solution of positively charged defect solubility in InSb at 450°C using Zn completely FermiLDirac statistics; Fermi level E , calculated assuming 5 x 1019 ionized. Fermi level is at A if deionization of Zn, occurs.

3.

241

DIFFUSION

which case (V+)isabout3 x lOI7~ m -Note ~ . that &isdetermined where the positive species (predominantly p ) are equal to the negative species (predominantly Zn;). The Zn is assumed to be completely ionized. With this assumption, the variation of (V') with Zn concentration is in excellent agreement with the change in D with Zn concentration, as shown in Fig. 24. However, as will be shown, this agreement is entirely fortuitous. The dotted line labeled by (Zn,) in Fig. 16 illustrates how the ionized acceptor concentration would vary if the Zn were not completely ionized, but had instead a well-defined energy level in the forbidden band. If this were the case, E , would be shifted to point A where p = (Zn,)', which would ~ . particular vacancy concentration predict (V') to be about 1OI6 ~ m - 'The chosen for this illustration is arbitrary. The diffusion profiles for Zn obtained'52 over the range 400-507"C are shown in Figs. 17-22. The experimental conditions for each run are given in Table V. The profiles were analyzed for D as a function of Zn concentration using the standard Boltzmann-Matano method [see Eq. (68)]. The results of these analyses are shown in Figs. 23-25. The variation of the BoltzmannMatano method, which takes into account the presence of un-ionized Zn (Section 16a), was not used, since the Zn is probably completely ionized at

u__ 8

12 16 ZO 24 28 DEPTH I N MICRONS

32

36

0

FIG.17. Zn diffusion profiles in InSb at 400°C (Series I). Is2

Texas Instruments Incorporated, "An Operational Transistor From InSb," Contract NOrd 18902 Final Report, March 1962.

242

DON L . KENDALL

DEPTH I N MICRONS FIG.18. Zn diffusion profiles in InSb at 453°C (Series 11).

R U N 8B:H I G H DISLOCATION D E N S I T Y ( 1 0 5 / c m 2o)

16 20 24 D E P T H I N MICRONS FIG.19. Diffusion profiles of Zn in lnSb showing negligible effect of varying dislocation density; 8 hours at 454°C (Series 11).

3.

0

I

2

3

4

5

243

DIFFUSION

6

7

DEPTH

8

91011

121314

(pl

Frc. 20. Two-step Zn diffusion experiment in InSb. Note that the concentration decreases during the second step, but little forward diffusion occurs. (After Kendall, Kanz, and Reed.55)

D E P T H IN M I C R O N S

FIG.21. Zn diffusion profiles in InSb at 453°C. low surface concentrations (Series 11)

244

DON L . KENDALL I

I

I

I

I

I

I

I

I

I

DEPTH IN M I C R O N S

FIG.22.Zn diffusion profiles in InSb at 507°C (Series Ill).

all concentrations in InSb. This can be inferred from the dependence of the Zn surface concentration on Zn partial pressure shown in Fig. 26. The exact form of these curves is not understood, but they would be quite different if simple deionization at high concentrations had occurred (see Fig. 7 for Zn in GaAs, for example). The diffusion behavior ofZn in lnSb was not affected by changes ofdislocation density, at least over the range studied. This is shown in Fig. 19 for a run at 454°C. The result appeared at first to eliminate the possibility that an interstitial-substitutional diffusion mechanism was operative for Zn in InSb, since other interstitial-substitutional diffusants are grossly affected by varying dislocation d e n ~ i t y . However, ~ ~ , ~ ~ a more likely explanation is that the dislocations in InSb are not good vacancy sources, at least compared to the surface (see Fig. 2 and discussion). The interstitial-divacancy reaction mentioned earlier may also act to block vacancy generation at dislocations [reaction (65)]. The apparent D’s derived from the profiles of Figs. 17, 18, and 22 were observed to be time dependent. That is, for runs made under similar diffusion conditions, the measured D’s from long runs were higher than those of shorter runs (e.g., compare runs 2 and 3 at 507°C). This is believed to be evidence that the defects in the diffusion zone are not at equilibrium. In support of this is the fact that the surface concentration at short times is higher than at long times (507°C data). This can be explained by the changing

3.

245

DIFFUSION

TABLE V

EXPERIMENTAL CONDITIONS FOR Zn DIFFUSION RUNS

Run

Source

1-1 1-2 1-3 1-4 1-5 11-1 11-2 11-3" 11-4 11-5 11-6 11-7 11-8Ab 11-8B' 11-9 11-10 111-1 111-2 111-3 111-4 111-5 111-6 111-7 1II-P

15.0% Zn 10.0% Zn 5.0 % Zn 2.0 % Zn 1.0% Zn 100.0% Zn

a

10.0% Zn 100.0% Zn 3.2 % Zn 1.0% Zn 0.1 :4 Zn 0.01 % Zn 10.0%Zn 10.0% Zn 100.0% Zn 100.0 % Zn 3.2 % Zn 2.0% Zn 2.0 % Zn 1.0% Zn 0.50% Zn 0.1 % Zn 0.01 % Zn 10.0% Zn

Source temp. ("C)

Sample temp. (T)

398 397 398 397 398 324 453 369 452 450 453 453 447 447 308 270 498 498 498 497 498 497 490 490

400 400 400 400 400 44 5 456 450 457 453 450 456 454 454 453 453 507 507 507 507 507 507 495 495

IN

InSb

Surface Junction Diffusion time concentration depth (hr) ( ~ m - ~ ) (microns) 24.0 24.0 24.0 24.5 72.0 64.0 24.0 87.0 24.0 24.0 236.0 236.0 8.0 8.0 112.0 160.0 4.0 48.0 4.0 8.0 49.0 115.0 115.0 8.0

2.2 x 1.8 x 1.0 x 3 4 x 2-3 1.9 x 2.1 x 1.5 x 1.2 x 3.5 x 1.3-1.5 x 5-lox 1.5 x 1.5 x 1-1.5 x 6-9 x 2.6 x 1.3-1.4 x 1.4-1.6 x 1.0 x 6.5 x 3-3.5 x 8-9 x 2.65 x

1020 1020 1020

10" 1019

10'' 1020 lozo lozo

1oi4 10" 1Ol8 lozo 10'' 10''

loL8 10'' lozo lozo 1020 1014

1019 lo'* 10''

35 20 5-6 3-3.5 3 4 130 80 135 22 6-7 12 2 4 25 25 5 4 3-5 19 43 10-11 11-12 16 9-1 1 3 4 8&90

[113] orientation ; precipitate near junction.

Low dislocation density.

' High dislocation density. Surface melting.

surface composition of the InSb as the In-Zn diffusion source absorbs Sb. Thus the InSb surface becomes "In-rich'' as time progresses, which implies that the Sb vapor pressure in the ampoule steadily decreases until a steady state situation exists. The In vacancy concentration near the surface decreases along with the Sb pressure. This in turn causes the Zn surface concentration to decrease through the incorporation reaction [Eq. (42)] and the effective diffusion coefficient to increase by decreasing the ratio of (Zn") to (Zn,,). The two-step Zn diffusion e ~ p e r i m e n t ~shown ~ * ' ~ ~in Fig. 20 provides critical evidence regarding the diffusion mechanism of this impurity in InSb. In the first diffusion step, the diffusion front advanced to a depth of about

246

DON L. KENDALL

1

E -1st

-151

10

I

I

1

1 1 IIIII

I

I

I I Ill11

1

1o2O

z I N C c o N CENT RATIO N ( crn-3)

FIG.23. Effective diffusion coefficients for Zn in InSb at 400°C. Estimated curves at 453°C and 507°C also.

13 microns. During the second diffusion step of much longer duration with no Zn in the ampoule, little,forward difusion occurred. Instead, the Zn concentration decreased by a factor of about three throughout the diffusion zone. An explanation for this unexpected result based on an interstitialsubstitutional mechanism has already been presented for a similar experiment in GaAs, so only a brief summary will be made here. In the first step, Zn,' (or Zn+2)and Vpn enter at the crystal surface together and react to form substitutional Zn,, throughout the diffusion zone [see reaction (69)]. During the second step, there is no external supply of Zn, so the main effect is dissociation of the Znlninto Zni+ and V,: . Following dissociation, the built-in electric field due to the Zn concentration gradient sweeps the Zn,' toward the surface and subsequently out into the large evacuated ampoule. It would be difficult, without several artificial assumptions, to explain the above twostep diffusion result using the positive vacancy model proposed earlier. Thus

3.

247

DIFFUSION

-8

10

-9

10 0 W

m -10 \I0

N

E Q k

z

-11

+I0

0 THEORETICAL CURVE

LL LL

w -12

",lo

2

2 -13 LI. LL

-

10

T H E O R Y USING

BOLTZM A N N

n

STAT I S T I C S

-15

10

l0l8

d9

ZINC C O N C E N T R A T I O N

lozo

(~rn-~)

FIG.24. Effective diffusion coefficients for Zn in InSb at 453°C. Theoretical curve assumed proportional to positive vacancy concentration (as in Fig. 16).

the interstitial-substitutional model seems assured in spite of the fact that the concentration dependence of B agrees very well for the positive vacancy model. A detailed solution of the interstitial-substitutional problem has not been attempted. This would be difficult in the case of InSb because of the large hole concentrations involved relative to the normal density of states in the valence band. As discussed in Section 2d, the accompanying band structure changes are quite significant at these concentrations. It is not possible, because of the aforementioned time dependence of D, to obtain accurate measures of Do and Q for Zn at various concentrations from the present data. However, a curve for D vs Zn concentration believed to be typical of very long diffusion time can be drawn for each temperature studied. These curves are collected in Fig. 23. At high Zn concentrations, the activation energy for diffusion appears to essentially vanish. By extrapolation to low Zn concentrations, an approxiinate value for Q of 0.7 eV is obtained.

248

DON L. KENDALL -8

10

-

-9

10

CADMIUM 50Oo-51O0C A

V W v)

-10

"10 E &a I-

z

-11 W -I O

2

LL LL

gw 1-12 0 z

2

2 -13 LL LL

-

10

n

I-15 1 0 1018

6

1

0l

~

~

4

1o2O

6

Z I N C A N D C A D M I U M C O N C E N T R A T I O N( ~ r n - ~ )

FIG.25. Effective diffusion coefficients of Zn and Cd in InSb at 507°C

These values along with measurements of other workers are shown in Fig. 27. The other reporters did not take account of the extreme concentration

dependence, although all their values fall between the low and high concentration extremes for shown in Fig. 27. The values of Wilson and H e a ~ e l lfor ' ~ Do ~ and Q are also shown in Table I11 and Fig. 27. To obtain these values, they assumed an erfc distribution and then doubled their measured D values to take account of the time dependence of the diffusion process. They observed this as a p - n junction depth that progressed at instead of as expected. They attributed this to a rate limitation process.64 This is similar to the explanation given earlier with regard to the time dependence of D in the 507°C data. However, rate limitation generally predicts that the surface concentration should increase with time rather than decrease as observed. lS3

R. B. Wilson and E. L. Heasell, Proc. Phys. SOC.(London)79,403 (1962).

3.

DIFFUSION

249

10

Id8

16’

lo-6 IDEALIZED Z N C PRESSURE IN ATM

FIG.26. Zn surface concentration in InSb related to Zn partial pressure (idealized)

also observed the concentration dependence of D Gusev and for Zn in InSb. They assumed that D is an exponential function of Zn concentration. Combining this with measurements at various temperatures, they obtained D, = 6.3 x lo9 exp[-2.61eV/kT + 2.47(C/C0 - I)], (76) where C, is the Zn surface concentration and Dc is the effective diffusion coefficient in cm2 sec- at concentration C. At any temperature, this accounts for a variation of only a factor of 12 between low and high concentrations. They also measured the Zn distribution in samples with widely different dislocation densities. In agreement with the profiles shown in Fig. 19, they observed no difference attributable to the presence of dislocations. They further reported the maximum solid solubility of Zn in InSb to be 3.5 x 10’’ ~ m - This ~ . value is about ten times higher than the data in Fig. 26 would suggest. Pumper and Prostoserdova,’ 5 5 using incremental sheet resistance techniques, reported on the steep profiles o f the ionized Zn atoms in InSb

’

lS4

1. A. Gusev and A. N . Murin, Fiz. Tuerd. Tela 6, 1208 (1964) [English Trans/.:Soviet Phys.-

lS5

Solid State 6,932 (1964)l. E . Ya. Pumper and I. V. Prostoserdova, Fiz. Tuerd. Tela 6, 899 (1964) [English Transl.: Soviet Phys.-Solid State 6, 692 (1964)l.

250

DON L. KENDALL 500°C

\I'

I

400°C

450°C I

300" C I

I

1

I

'\

\

x. '

10-l~ I .2

1.3

1.4

1.5

\

\

€3 (INTRINSIC)

Q = 0.7 eV + 0.2

1.6

1.7

103b (OK) FIG.27. Effective diffusion coefficientsat high and low Zn concentrations in fnSb,along with b values of other workers.

following diffusion over the range 370-440°C. They proposed that the diffusion of Zn in InSb is accompanied by the capture of the diffusing atoms by traps of some nature. They proposed that above a certain concentration the traps were filled and below this they were empty. They thought that the traps might be somehow related to vacancies.

3.

251

DIFFUSION

b. Cd in InSb Cd diffusion in InSb has been studied by the author,152 Wilson and Watt and Orth,lS6and by Boltaks and SokoH e a ~ e 1 1 , Gusev ' ~ ~ et 1 0 v . l ~Wilson ~ and Heasell used p-M junction techniques and assumed an erfc distribution was obtained. Their values of Do and Q, as shown in Table 111, should be considered as average b values over the range ofconcentrations encompassed. The author studied Cd diffusion in the InSb using radioactive Cd and obtained profiles at a temperature of about 505°C. His experimental conditions are shown in Table VI and the resulting profiles in Fig. 28. The calculated b values as a function of Cd concentration are shown in Fig. 25 along with results for Zn at a similar temperature. The value of 2 x cm2 sec-' at 505°C calculated from Wilson and Heasell agrees with our B values at a Cd concentration of 2-3 x l o t 9cm-'. From runs 5A and 5B, it was concluded that Cu does not significantly affect the Cd diffusion. This was done in response to a suggestion of HennekelS8that Cu interstitials might keep the vacancy concentration depressed. TABLE VI

EXPERIMENTAL CONDITIONS FOR Cd DIFFUSION RUNSI N InSb

Run

1 2" 3* 4 5A' 5Bd 6

Source

100.00% Cd lO.OO%Cd lO.OO%Cd O.lO%Cd O.Ol%Cd O.Ol%Cd I.OO%Cd

Source temp. ("C)

Sample temp.

501 501 500 502 505 505 502

502 502 500 510 508

("C)

508 510

Diffusion time (hr) 62 62 41 62 64 64 62

Surface concentration

co

Xi (microns)

(~m-~) 3.5-4 3-4 44.5 6-9 6-8 8-10 1.3

x lOI9 x 10i9 x 1019 x 10lS

x 10" x 10" 1019

15-17 16-22 12-14 3 4 6-8 7-9 8-11

Wedged during lapping.

' Both sides analyzed. Ultraclean. Copper plated.

155al. A. Gusev, A. N. Murin, and P. P. Seregin, Fiz. Titerd. Tela 6, 1895 (1964) [English Trans/.: 15"

15'

15'

Societ Phys.-Solid State 6, 1491 (196411. L. A. K. Watt and R. W. Orth, Bull. Am. Phys. So(,.8,472 (1963). B. I. Boltaks and V. I. Sokolov, Fiz. T w r d . Tda 5, 1077 (1963) [English T'cinsl.: Socier Phys.-Solid State 5. 785 (1963)l. H. Henneke, private communication.

252

DON L. KENDALL

D E P T H IN M I C R O N S

FIG.28. Radioactive Cd profiles in InSb at 505°C

Watt and Orth'56 reported Do and Q values that differ considerably from the above. They used an electroplated source similar to that used by Goldstein for Zn,15' and thus their values are subject to the same reservations. However, at 505°C their value of 5 x cm' sec-' is close to the average B for Cd at this temperature shown in Fig. 25. Boltaks and S o k o 1 0 v ' ~reported ~ the Do and Q values for Cd diffusion in InSb shown in Table 111. They used a novel technique which involved dragging a diffused radioactive sample along a piece of emery cloth. They then took an autoradiograph of the cloth and a densitometry trace of this. They assumed uniform material removal. The uncertainties of such a method are cm' secno doubt formidable, nevertheless their valueat 505°C of7 x is also close to the values of Fig. 25. They quoted a solubility maximum for Cd of 2.5 x lo2' cm-3 at 400°C but their data at neighboring temperatures indicate that this is probably an anomalous point. A more likely value at this temperature is about 4 x 1019cm-3. Gusev et ~ 2 1 . ' ~also ~ " reported on the diffusion of Cd in InSb. Their values of Do and Q shown in Table 111 lead to a B value of 5 x lo-'' cm2 sec-l

3.

253

DIFFUSION

at 505”C,which is about two times higher than that shown jn Fig. 25 at the highest concentrations. c. H g in InSb

Kanz and L~velace’~’ measured radiotracer profiles for Hg diffusion in InSb at 500°C. With a Hg pressure of 0.2 atm, the surface concentration was about 5 x lo’* cm-3 and the b was 5 x cm2 sec- This D is similar to the value for Zn at this same concentration and temperature (see Fig. 15). Gusev and Murin16’ reported b measurements for Hg in InSb over the range 425-500°C. A calculated D from their Do and Q values shown in Table 111 is about twice that of Kanz and Lovelace.

’.

24. Cu, Au, C o , Ag, Fe, AND Pb IN InSb a. Cu in InSb

The diffusion of Cu in InSb has been studied by Stocker72and Boltaks and Sokolov.16’ Stocker, using both radiotracer and incremental sheet resistance techniques, found that the apparent was highly structure sensitive as well as being dependent on the history of the sample. His findings are discussed in some detail in the following : (1) A rapid diffusion branch was observed at low concentration consisting of electrically inactive Cu. He assumed this was interstitial Cu with a B of about cm2 sec-’ at 480°C. (2) A much lower b was calculated from a region near the surface. He attributed this to reaction of interstitial Cu with vacancies diffusing in from the surface. (3) He reported values for the solid solubility of substitutional Cu, and estimated the interstitial solubility to be about 1/50 of this. (4) A branch with an “apparent” D intermediate between the vacancy limited D of (2), and the interstitial D was observed by incremental sheet resistance techniques. This apparent D increased approximately linearly with dislocation density. (5) For unannealed samples, the apparent D was insensitive to dislocation density up to lo3 crnp2.This was attributed to interaction of the interstitial atoms with vacancies trapped in the crystal during growth. (6) Accordingly, he suggested that the Cu concentration in the interior region of dislocation free InSb was equal to the number of vacancies ‘59

lb0

16’

J. A. Kanz and K. Lovelace, unpublished data. 1. A. Gusev and A. N. Murin, Fiz. Tuerd. Tela 6. 1563 (1964) [Enxlish Transl.: Socier P h p Solid State 6, 1229 (1964)l. B. I . Boltaks and V. I. Sokolov, Fiz. Tuerd. Tela 6,771 (1964) [English Trunsl.: Sovier Phys.Solid State 6, 600 (1964)l.

254

DON L. KENDALL

incorporated during growth. On this basis, he estimated that the equilibrium vacancy concentration at the melting point of InSb was about 1 x 1016crnp3. (7) Assuming the above were In vacancies, he estimated the enthalpy of formation of the vacancy to be 1.02 eV. For simplicity, the entropy of formation for the vacancy was taken as zero. This assumption may have introduced a rather large error into the enthalpy estimate (see Section 12 for comments on the magnitude of the entropy of formation for the simple vacancy and the divacancy). (8) Utilizing an expression that relates the self-diffusion coefficient to the vacancy diffusion coefficient,45346

4, = DVI,(Vd,

(77)

where (V,,) is the In va’cancy concentration in mole fraction, he estimated the self-diffusion coefficient of In to be about 4 x cm2 sec-’ at the melting point of InSb. He also predicted that the Q for self-diffusion of In should be about 1.8 eV. These predictions are subject to the same reservations mentioned above. (9) To explain the faster than expected annealing of the excess vacancies at 175”C, he postulated that the excess In and Sb vacancies paired to form the more mobile divacancy species, which then diffused rapidly to free surfaces. However, condensation of the excess vacancies into “collapsed vacancy discs” to form dislocation loops’62 throughout the crystal will also explain the annealing behavior. The above observations provide valuable information on the behavior of vacancies in InSb, although there remain several questions regarding quantitative interpretation. Also, if the divacancy [V,,Vsb] is dominant at high temperatures, as was proposed to explain the self-diffusion in InSb,I4 the results need to be reinterpreted. However, the estimate of the vacancy concentration at the melting point discussed in (6) should be valid whether a V,, or a [V,,Vsb] has been labeled by the interstitial Cu. The reaction with the divacancy to form the neutral complex, [Cu,,V,,], may even explain Stocker’s observation of electrically inactive Cu in the interior of InSb. Boltaks and Sokolov16’ presented data for the diffusion of Cu over the range 23&490”C. The dislocation density of their samples was stated to be approximately lo4 cm-2. Their Do and Q values shown in Table I11 lead to a value of 3 x lo-* cm2 sec-’ at 350”C, which is about 10 times higher than Stocker’s value at this temperature and dislocation density. The value coincides with Stocker’s at a dislocation density of 4 x 104cm-2, which is probably within the accuracy of the dislocation count. 162

H. G. Van Bueren, “Imperfections in Solids,” p. 72. North-Holland Publ., Amsterdam, 1960.

3.

DIFFUSION

255

b. Au in InSb

The diffusion of Au in InSb has been studied by Boltaks and Sokolov.16’ The Do and Q values shown in Table 111 are of the expected order of magnitude for an interstitial-substitutional diffusion process. The solid solubility of Au was said to vary from 1 to 4 x 10” cm-3 over the range 300-500°C. However, they showed a profile at 35WC with an extrapolated surface concentration of 10l6cm-3, so the above figures may be somewhat low. They also observed a slow “diffusion” branch near the surface with a quasisolubility of 2 x l O ” ~ m - ~This . was likely an artifact caused by a surface reaction. c.

Co in InSb

and more recently Gusev and M ~ r i n , studied ’ ~ ~ the Watt and diffusion of radioactive Co in InSb. The values of Do and Q reported by the two groups are shown in Table 111. Watt’s values lead to a B at 500°C of cm2 sec-‘ at the same 2.4 x cm2 sec-’ and Gusev’s to 7.5 x temperature. The reason for the wide divergence is not apparent. However, Gusev’s value of Do of 2.7 x 10- l 1 cni2 sec- is much lower than that for other impurities. Also, Gusev’s reported solubility of 2 x 1019cm-3 is higher than one would expect, so it is possible that they have reported on an artifact associated with a surface reaction similar to that discussed for Au in InSb above. d. Ag and Fe in I n S b

Watt and C h e r ~ reported ’~~ Do and Q values for Ag and Fe diffusion identical with those for Co above. All their profiles could be fit better with an exponential distribution than with an erfc. Boltaks and Sokolov16’ quoted cm2 sec- at 390°C, which is in rough agreea D value for Ag of 4 x cm2 sec-’ ment with the calculated value of Watt and Chen of 1.3 x at this temperature.

e. Pb i n I n S b D e ~ b n e r ’made ~ ~ a careful, though limited, study of the diffusion of Pb in InSb. He evaporated thin layers ofradioactive Pb onto one side of optically flat polished wafers of InSb. A surface reaction during the diffusion anneal often led to the formation of pits, some of which showed threefold symmetry on the [ l l l ] oriented slices. Autoradiographs showed that these pits (of several microns depth) consisted of a high concentration of radioactive Pb, ‘63

165

L. A. K. Watt and W. S. Chen, Bull.Am. Phys. SOC.7,89(1962). I. A. Gusev and A. N. Murin, Fiz. Tterd.Tela 6,2859 (1964)[English Transl.: Soviet Phys.Solid State 6,2274 (1965)l abstracted in Solid State Commun. 2, No. 11, p. V (1964). D. C. Deubner. unpublished data.

256

DON L. KENDALL

and that very little true bulk diffusion, even after several weeks at about 500°C, occurred except perhaps in the first 1-3 microns. The pits formed more readily on the Sb side than on the In side of a [ 1113 wafer. The D value at 500°C for cm2 sec-', which is of the same the very shallow branch was 2.7 x order as the self-diffusion coefficients of In and Sb reported by the author (see Fig. 15).

25. SUMMARY OF DIFFUSION IN InSb The values of Do and Q for the various impurities in InSb are tabulated in Table 111 along with brief comments. The values believed to be the most reliable to date at the low concentration limit are in italics. The calculated D values from these data are shown graphically in Fig. 15. The self-diffusion coefficients of In and Sb are also shown for comparison. Note that the diffusion activation energies Q are of order 1.0eV for Sn, Cd, Hg, and Zn as well as for Cu in dislocation free material. This suggests diffusion by a common process, the most likely possibility being the interstitial-substitutional (surface vacancy limited) process. The much higher Do and Q for In and Sb are more consistent with substitutional diffusion. By contrast, the very fast diffusants Au, Fe, Co, and Ag are probably dominantly interstitial, as evidenced by their high values of D and low values of Q of order 0.3 eV. The pre-exponentials, D o , for the impurities diffusing by the interstitial-substitutional and the pure interstitial processes are of the same order of magnitude, namely, 10- 5-iO- '.

VI. Summary and Conclusions In order to bring into perspective the work reviewed in this article, the significant features of each section, as the author interprets them, will be summarized. The major points of Parts I1 and 111 on defect equilibria and diffusion in compounds as related to 111-Vcompounds are the following : (1) These compounds have very narrow phase fields, that is, extremely small deviations from stoichiometry. (2) The concentration of electrically active native defects is generally very small after cooling to room temperature. (3) Under equilibrium conditions at high temperatures, AB divacancies may be present in higher concentrations and are expected to be more mobile than simple A or B vacancies in these compounds. (4) Physically rearranged defects may be important in these compounds. Examples of this are

V B f + [V,AB]-

+ 2e+

[VAVB]' * [VAA,VA]-

+ e+

VA- + [VBBA]++ 2e[VAVB]'

--*

[VBBAV,]'

+ e-

3. [vAvA]-

A,-

+

--f

--f

+

[vBvB]+

+ 2e-

BA'

[VAVBBA]' 2e[Aiv~]'

257

DIFFUSION

[ABVAVBI- + 2e+ -+

[ B i v ~ ] --I-2e+

and - signs suggest the most likely charge states when the where the defects are ionized. The most stable arrangement at room temperature may be different from the higher temperature equilibrium configuration, especially in doped crystals where the neutrality condition may change from intrinsically to extrinsically controlled on cooling. On solubility arguments the donor (acceptor) forms should be favored in p-type (n-type) material. ( 5 ) The possible effects of high impurity concentrations on defect ionization energies, energy band structure, and mass action calculations are manifold and cannot be ignored. In Part IV on self-diffusion in 111-V compounds the extreme variation between observers is stressed. The most reliable data appear to have been taken in the GaSb and InSb systems. The significant features of these two systems are the following : (1) High values of the pre-exponential, D o . (2) Rather high values of the diffusion activation energy Q. (3) Low values of the diffusion coefficient at the melting point (compared to metals, for example). (4) Diffusion coefficients of the two components of similar magnitude. ( 5 ) No observed dependence of the diffusion coefficient on composition in InSb. (6) Substitutional diffusion via AB divacancies can explain the lack of dependence on composition. (7) The high values of Do are also consistent with a divacancy process, since a large vibrational entropy contribution is not unexpected for such a defect. Several of the above comments could also be made about InP self-diffusion, especially with regard to the high values of D o . However, in this case the diffusion activation energy of the B atom is much larger than that of the A atom, and further the reported diffusion coefficients of both components at the melting point are much larger than those of GaSb and InSb. In the section on impurity diffusion, most of the published literature is discussed and several sets of previously unpublished data are presented in some detail, namely, data on S, Cd, and Zn in GaAs and Cd and Zn in InSb. The conclusions reached are listed below ( 1 ) The diffusion coefficient of Zn is very dependent on Zn concentration in all of the 111-V compounds, with the possible exception of AlSb. (2) Diffusion effectively ceases during the second step of a two-step diffusion process in both G a k s and InSh. An explanation of this unusual

258

D O N L . KENDALL

result was presented which strongly favors an interstitial-substitutional mechanism over a vacancy diffusion mechanism for this impurity. (3) Changes in the valence band edge at high Zn concentrations must be taken into account to explain the concentration dependence of D in these compounds. (4) Concentration gradients can seriously modify the diffusion characteristics of impurities and defects. This is especially evident during diffusion of Cu and In into GaAs when a high concentration p-layer is present near the surface, causing each impurity to apparently diffuse uphill. (5) Most of the impurities in these compounds have low values of D o . Taken with the high values of Do expected and observed for self-diffusion by a substitutional process this suggests that the diffusion properties are often dominated by the interstitial, even though the concentration of these may be much lower than that of the substitutional. (6) Lithium is dominantly (though not completely) an interstitial diffusant in GaAs. Cu, Ag, and Au diffuse rapidly by the interstitial-substitutional process. Manganese also probably diffuses by the interstitial-substitutional process, but an arsenic divacancy mechanism can also explain the results. (7) A surprising diffusant by the interstitial-substitutional process is In in GaAs. (8) Sulfur diffuses much more rapidly than expected in GaAs. If this is due to an interstitial-substitutional diffusion process, the interstitial is electrically neutral or an acceptor with a level near the conduction band edge. (9) Sn, Mg, Cd, and Zn all have approximately the same rate of diffusion in GaAs at the low concentration limit. This suggests a common diffusion mechanism, most probably the interstitial-substitutional process with the rate of diffusion controlled by the vacancies entering at the surface. (10) The slowest impurity diffusants in GaAs are Se and Te (and perhaps Hg). Selenium has a reported Do and Q much higher than the aforementioned impurities. Thus a vacancy mechanism may be responsible for diffusion of Se and Te. (11) A much more rapid diffusion mechanism must be invoked to explain the electrical changes that occur during annealing treatments of heavily Se- and Te-doped GaAs. This may be due to diffusion of vacancy-Te (or vacancy-Se) agglomerates or perhaps an interstitial (neutral) form of these impurities. (12) No reliable self-diffusion data have been taken for GaAs. (13) Diffusion of most impurities in InSb is typified by much higher D values and much lower D o and Q values than are observed for selfdiffusion in this material. Thus, the diffusion of most of the impurities measured to date in this compound is probably of the interstitial-substitutional type.

3.

DIFFUSION

259

The above conclusions demonstrate the wide range of effects that have been observed during diffusion studies in 111-V compounds. It is hoped that the questions raised will stimulate more meaningful research in this field. The pressing need is for accurate and well-reported experimental data. When this becomes available, it should be possible to elucidate in considerable detail the defect and diffusion mechanisms operative in this family of compounds.

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Effects of Electric Fields, Pressure, and Nuclear Radiation

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CHAPTER 4

Charge Multiplication Phenomena A . G . Chynoweth

I . INTRODUCTION . . . . . . . . . . . 1 . General Discussion . . . . . . . . . . . . . . 2 . Fundamental Parameters of Charge Multiplication Mechanisms . OF THE IONIZATIONRATE . . . . . . . . . . I1 . THEORIES 3 . General Discussion . . . . . . . . . . . . . . 4 . Collective Electron Breakdown (n > ni) . . . . . . . . 5 . Breakdown at Low Electron Densities (n < ni) . . . . . . BREAKD~WN IN P-N JUNCTIONS . . . . . . . 111. AVALANCHE 6 . Analysis of Current Build-Up . . . . . . . . . . . I . Avalanching in NonunSform Fields . . . . . . . . . CHARGE MULTIPLICATION . . . . . IV . METHODSOF MEASURING 8 . Elements of a Well-Designed E..;p eriment . . . . . . . 9 . Experimental Techniques . . . . . . . . . . . . V . EXPERIMENTAL RESULTS . . . . . . . . . . . . . 10. Ionization Rates in Silicon . . . . . . . . . . . 1 1 . Ionization Rates in Gallium Arsenide . . . . . . . . . 12. Ionization Rates in Gallium Phosphide . . . . . . . . 13 . BarafSCuruesfor Other Materials . . . . . . . . . VI . MISCELLANEOUS PHENOMENA ASSOCIATED WITH CHARGE MULTIPLICATION I N JUNCTIONS . . . . . . . . . . . . . . 14. Light Emission . . . . . . . . . . . . . . . 15 . Microplasmas . . . . . . . . . . . . . . . 16 . Average Energy for Electron-Hole Pair Production by High Energy Particles . . . . . . . . . . . . . . . . VII . BREAKDOWNIN BULK SEMICONDUCTORS . . . . . . . . 17 . General Discussion . . . . . . . . . . . . . . VIII . IMPACTIONIZATIONOF IMPURITIES . . . . . . . . . . 18 . General Discussion . . . . . . . . . . . . . .

263 263 264 268 268 269 272 286 286 289 293 293 294 300 300 302 305 306 307 307 313 319 320 320

323 323

.

I Introduction 1. GENERAL DISCUSSION As electrons are drifted through a crystal by an electric field they gain energy from the field only to lose it again in various types of collisions. At relatively low or moderate field strengths. a steady state is reached where 263

264

A.

G. CHYNOWETH

the energy gained from the field is balanced by the energy lost by collisions with acoustical and optical phonons. The field dependence of the mobility in this regime has been extensively studied and there exists a considerable literature on this subject.' In this chapter we will be concerned with the ionization phenomena that occur at higher fields where electrons (we shall frequently refer to electrons synonymously with holes) gain energy from the field faster than they can lose energy to phonons. Under these conditions the electrons experience a steady acceleration to higher energies. This acceleration does not go on indefinitely because eventually the electrons achieve sufficient energy for an additional type of energy-losing collision to occur, namely, pair production. In pair production, an energetic electron gives up a considerable fraction of its kinetic energy to a valence electron, thereby raising the latter to the conduction band and leaving a positive hole in the valence band. In this case, the original electron has created two additional carriers. Another type of collision that an energetic electron can undergo is an ionizing collision with an impurity. In this collision, the original electron loses energy in raising an electron to the conduction band from an impurity level in the forbidden gap. Only one additional carrier results from this collision, the positive hole being left immobile on the impurity site. We begin this chapter by considering some of the fundamental facts and concepts behind the multiplication process. For convenience, this discussion will be biased toward a consideration of the events leading up to pair production though much of the material will apply directly to the impact ionization of all but the shallowest of impurity levels. 2. FUNDAMENTAL PARAMETERS OF CHARGE MULTIPLICATION MECHANISMS

Obviously, to cause pair production, an electron must be accelerated to an energy a t least equal to the energy gap and probably greater when the need for simultaneous conservation of energy and momentum in the collision process is taken into account. For all the semiconductors with which we need be concerned here, this energy is considerably greater than any of the phonon energies and it is also very much greater than the thermal energy at normal temperatures. At low energies and lattice temperatures, electrons interact primarily with acoustical phonons, but as they gain energy they become able to interact with the optical phonons, i.e., excite optical phonons. Ultimately, their energy becomes sufficient that they can produce pairs by collision. Let us consider the effect that each of these types of collision has on the motion of the electron. First, the acoustical phonons. The effect of a collision with an acoustical phonon can be established as follows: Consider an electron of energy E, = +m*u2, where m* is the effective mass of the electron and u is its velocity.

' See, for example, E. M. Conwell, Solid State Phys. Suppl. NO. 9 (1967).

4.

CHARGE MULTIPLICATION PHENOMENA

265

The electron momentum is then p , = m*u. In a collision in which the electron generates an acoustical phonon, the change in momentum of the electron must equal the momentum of the created phonon, pq. Now the maximum change of momentum that the electron can suffer is 2p, = 2m*u and this equals pV.But plP = hk, where k is the phonon wave vector, and in turn, the phonon energy, E~ = uspq, where us is the sound velocity. Thus, the energy lost by the electron is E~ = u s . 2m*u. Alternatively, the fractional loss of energy is E ~ / E , = 4u,/v. Now at normal temperatures, the thermal velocity alone of the electrons is of the order o f lo7 cm sec- while us lies between 10’ and lo6 cm sec-’. Thus E,/E, < 1. That is, a collision with an acoustical phonon can be a very efficient means for changing the electron momentum but its energy is only slightly affected. We can therefore regard the acoustical phonon collisions as elastic and serving mainly to randomize the electron velocity distribution in space. Turning to the optical phonons, clearly in order to lose energy to an optical phonon of energy gR, the electron energy must be at least equal to E ~ We note that the optical branch of the phonon spectrum is roughly of constant energy from k = 0 to the zone boundary so that, for scattering to occur, E, = E ~ Then . conservation of momentum requires that the k vector of the created phonon be PR/h 2 (2~~m*)’’’/h.Choosing typical values for and m* leads to values of k of the order of lo6 cm- which is considerably smaller than the value of k at the zone boundary. Again, the collisions serve to completely randomize the velocity distribution but, in addition, the energy lost by the electron, E ~ is, substantial. In collisions with acoustical phonons, since they are primarily momentumchanging, it matters not whether the electrons absorb or emit phonons. On the other hand, for optical phonons it is important to consider whether absorption or emission occurs. The ratio of the probability of phonon absorption, Pa, to that of phonon emission, P,, is given by (at least for nonpolar interactions) pa N -

P, N $- 1’ where N is the normalized phonon density given by N = { e x p ( s ) - I}-: where k is Boltzmann’s constant and T is the absolute temperature. Therefore, PJP, = exp( - ER/kT).Now for many semiconductors under normal operating conditions, the optical phonon energy is somewhat greater (perhaps by as much as a factor of 2 or more) than kT, so we see that in optical phonon collisions the electron is much more likely to emit a phonon than to absorb one. Thus, optical phonons act as a drag on the electrons, opposing the accelerating action of the electric field.

.

266

A . G. CHYNOWETH

In considering the frequency of the electron-optical phonon collisions it is customary to assume that the scattering matrix element is independent of energy.2*28 Using the usual argument that the scattering rate depends on the available phase space leads to the conclusion that the mean free path between optical phonon collisions, 2, is essentially independent of energy. This cannot be regarded as a rigorous result, particularly for the higher energy electrons which generally have a much more elaborate band structure available to them than low energy ones. However, its virtue is that it makes the mathematical analysis tractable, and, as yet, there has been no theoretical indication of any more accurate way of describing the general behavior of 2 as a function of electron energy. The effect of the constant mean free path is to provide a threshold electric field for the average electron below which it is never accelerated to energies much greater than the optical phonon energy. In other words, if the energy , as gained from the field in traveling a mean free path A is less than E ~ then soon as the electron reaches energy cRit is returned to about zero energy in the next collision, after which it is accelerated again, and so on. Thus, the time average energy of the electron is approximately &R/2, implying a time average constant velocity. This is the origin of the well-known velocity saturation effect in semiconductors. However, if the energy gained between , electron energy steadily increases on the collisions is greater than E ~ the average. We thus see that a threshold field can be defined by ER/e2',where e is the electron charge and A' is the mean distance traveled by the electron in the direction of the electric jield between collisions. For random velocity distributions, clearly l' < 1, but as we shall see later, the motion of the electrons can often be thought of as more or less rectilinear in the field direction, in which case 2' _N 1.The threshold fields are usually somewhat greater than lo4 volt cm-'. As the electron energy increases it can eventually attain the threshold energy for pair production, zi. In the case of a simple collision with a valence electron, energy conservation alone requires that ei 2 cG, the energy gap. But when momentum conservation is included, the theoretical minimum pair. example, for a production energy will be somewhat greater than E ~ For simple billiard-ball type collision represented by parabolic bands and with the hole mass equal to the electron mass, it is easy to show that the minimum energy the hot electron must have is (3eG/2), corresponding to the equal sharing of energy between three particles after the collision. If, as is usually the case, the hole and electron masses are different and the energy bands depart appreciably from parabolicity, the value of ci could take on virtually any value greater than cG. Even so, the threshold energy need not be sharply W. Shockley, Bell System Tech. J . 30,990 (1951). *'F. Seitz, Phys. Rev. 73,550 (1948).

4.

CHARGE MULTIPLICATION PHENOMENA

267

defined since higher order processes are possible in principle in which, for example, two hot electrons cooperate in creating an electron-hole pair, a form of Auger process. Clearly, the two hot electrons can have energies considerably less than ci if their combined energy is available. Thus, we cannot regard E~ as a sharply defined quantity but rather as a representative parameter in the theory of the ionization rate. Once an electron has achieved the energy E; it will have a mean free path ,Ii for ionization. The cross section for ionizing is, of course, zero for E equal to ei, but it will increase rapidly (and quadratically) with increasing energy for simple band structure. In addition to electron-phonon collisions and ionizing collisions the electrons may also suffer collisions with other electrons. Energy and momentum are exchanged in such collisions which therefore serve to randomize the velocity distributions. Clearly, the frequency of such collisions depends on the electron density. The density needed for the velocity distribution to be maintained roughly spherically symmetric by this means at prebreakdown fields has been estimated by S t r a t t ~ n For . ~ electron-electron collisions, the rate of energy loss experienced by an electron of momentum p passing through a Maxwellian distribution of carriers at temperature Tis4 :

(z)..

=

4m1e*~

---’ P

where e* is an effective electronic charge which takes into account the dielectric polarization of the medium so that (e*/e)*NN 1 / ~with , K a suitable dielectric constant which depends on the velocity of the electrons. At high temperatures (kT 9 ho)the rate of energy loss to polar modes is approximately hw P, - Pa hw 1 (:)opt= = z. (2N0 + 1)’

- y ( m ) -7

where hw is the optical phonon energy and N o is the normalized optical phonon energy density. It can be shown that these two rates of energy loss are comparable when the density n is

where Fo is a constant with the dimensions of an electric field; it is given by

R. Stratton, Proc. Roy. SOC.(London)A246,406 (1958). Equation (10.13) of D. Pines, Solid Stutr Phys. 1,441 (1955).

268

A , G. CHYNOWETH

where a, is the Bohr radius. Typically (e.g., using the parameters appropriate to InSb) ni loi4~ r n - If~ n. is greater than this value, interelectronic collisions will control the energy distributions. Again for high temperatures, the critical density for momentum changes is ni* n,(kT/Aw) because momentum changes in both interelectronic and electron-phonon collisions are of the same order. it can be shown that the critical density At low temperatures ( k T < ho), is given approximately by

-

-

and

ni* z

ni(g) g) {exp(

- $}-I

The low temperature value of n, increases with increasing T and approaches the constant high temperature value given above. Also, at low temperatures, ni* is much less than n. The theoretical approach to charge multiplication differs in the two regimes: n > ni and n < ni . In the former, it is assumed that the distribution function for the electrons is simply a displaced Maxwellian function ; this is substituted into expressions for ( d f / d t ) arising from polar scattering, the solutions to which give a relation between any two of the three quantities ; field, mobility, electron t e m p e r a t ~ r eWhen .~ n < n,, the distribution function has to be determined by solving the steady-state Boltzmann equation :

In practice, the condition n < ni will generally apply in the field-swepl regions of reverse-biased p-n junctions. The condition n > ni is more likelj to apply to high field effects in bulk crystals. 11. Theories of the Ionization Rate

3. GENERAL DISCUSSION In the previous section most of the basic parameters were introduced and discussed. In this section, the various theoretical attacks on the problem of deriving the field dependence of the ionization rate will be outlined. These theories have been developed principally with experimental data on the traditional semiconductors silicon and germanium in mind, but, with minor modifications of the appropriate parameters, the same theories are applicable to the compound semiconductors.

4.

CHARGE MULTIPLICATION PHENOMENA

269

In the absence of an electric field, the electron gas is in equilibrium with the lattice through electron-phonon collisions and its average energy, or temperature, is obviously given by the ambient temperature T. When a field is applied, the energy input into the electron gas is increased so that its temperature rises until a new equilibrium is established. This new equilibrium can be described by an effective electron temperature T,, given by the energy balance condition that the rate at which the electrons gain energy from the field is equal to that at which they lose it to the lattice. The relation between T, and the electric field E is of fundamental interest and there have been many theoretical studies of this problem, particularly at relatively low fields in the so-called “warm-electron’’ regime. In this regime, the field is still too low for the electron to have any significant chance of being accelerated to the high energies required for ionization, and interest centers around determining the field dependence of the mobility. However, as the field is increased, a critical field is eventually reached beyond which the electrons gain energy from the field faster than they can lose it in lattice collisions; this obviously leads to a breakdown condition. Stratton3v5has given a detailed theoretical treatment of this critical breakdown field which we shall follow for the case n > n,. It perhaps should be emphasized that this breakdown field does not define a catastrophic runaway condition ;rather, the distribution function is affected by the onset of the ionizing collisions so that the treatment appropriate at lower fields where only phonon collisions have to be considered is no longer valid. In short, there will still be a steady-state distribution function even in the presence of ionizing collisions. 4. COLLECTIVE ELECTRONBREAKDOWN (n > n , )

In this condition, the distribution function is assumed to be a displaced Maxwellian :

f ( P ) = a expup

- Po12/2mkT,l 7

(1)

where a is a normalization constant. This is expanded in spherical harmonics :

where 8 is the angle between jj and the electric field. Thus, the rate of change off due to optical phonon collisions is

R. Stratton, J . Phys. SOC.Japan 17, 590 (1962).

270

A. G . CHYNOWETH

Rather involved expressions have been derived for Frohlich and Paranjape6 and Stratton.’ From (l),

(afo/at) and ( d J J d t )

by

and

We also have jF =

IEp)

d3p

and eFn

=

s p cos 8

($1

d3p,

(7)

where the current density is

j = nepF

=

nepo/m.

These integral expressions have been solved by Stratton to yield the following expressions :

and

with

where K Oand K1 are Bessel functions and y = ( h o / k T , )and y o = (hw/kTo), To being the lattice temperature and T, the electron temperature. The quantity a, is the Bohr radius. These two equations give for the zero field mobility NOFO

YY

1 exP(hJo)Kl(+Yo).

H.FrGhlich and B. V. Paranjape, Proc. Phys. Soc. (London) B69,21 (1956).

(10)

4.

CHARGE MULTIPLICATION PHENOMENA

271

There are two cases to consider: (a) high lattice temperatures, y o G 1, and (b) low lattice temperatures, y o % 1. a. High Temperatures ( y o 6 1)

At high temperatures, Eqs. (8) and (9) reduce to

and

where In D is a pure number of about unity. The right-hand side of Eq. (1 1) has a maximum at y = y c = h o / k T , where

The corresponding value of the field, F , , is the collective breakdown field. For small yo, the solution of (13) is approximately

y, x +yo[ 1 - 4{ln(2D/y0))- . . .] .

(14)

Hence, by substitution, the breakdown field is F,

=

For(;)

1n(3]1’2>

which is a slow function of To.The mobility is given by

Thus, the mobility should rise slowly with increasing field up to breakdown.

b. Low Temperatures ( y o 9 1) At high fields, exp(yo - y) % 1 since yo % 1. Then Eqs. (8) and (9) become F2

=

F o ~ ( ~ ) y ’ e x P ( - y ) K o ( ~ l ) [ I ( o ( ~+ y ) f(l(4Y)l

(171

and Po2

=

3mfw[l

+ { Kl(tY)/f(o(ty)ll’l.

(18)

272

A . G. CHYNOWETH

The right-hand side of Eq. (17) has a maximum at a value of y with a maximum value of the field

F,

=

y , !z 0.46

= 0.5 F , .

Eliminating y from Eqs. (17) and (18) leads to the drift velocity or mobility as a function of F. It is found that the drift velocity rises only slowly with field so that the mobility drops; the magnitude of the drop is far greater than the corresponding rise at high temperatures.

5. BREAKDOWN AT Low ELECTRON DENSITIES ( n < ni) The acceleration of electrons to ionizing energies under the assumption that electron-electron collisions can be ignored has been treated by several authors.'-'' The method in each case is to solve the Boltzmann transport equation under the assumption that electrons experience only two types of collision : collisions with optical phonons and ionizing collisions. With these assumptions, the distribution function is derived and from this it is a relatively simple matter to calculate the ionization rate a, the average number of ionizing collisions experienced by a single electron in traveling unit distance in the direction of the electric field. The first theoretical treatment of this problem was given by W01ff.~He argued that the mean free path for ionizing collisions would be considerably less than that for phonon collisions and that this would result in a considerable attenuation in the distribution function in the high energy tail, for energies E > E ; . More recent work by Moll and colleagues9~' has thrown doubt on the validity of this starting assumption. These authors have repeated the Wolff theory under less restricting assumptions and while the results are qualitatively similar to Wolff's, there are differences of detail. Their theory will be briefly outlined here. As will be discussed afterwards, this theory applies particularly to very high fields. a. Ionization Rates for Very High Fields

As discussed earlier, at very high fields the various collisions serve to keep the velocity distribution almost spherically symmetric so that , f ( P , 0) 2 f o

+fl

cos 0

(19)

'

The electron density in the momentum range p to p

+ d p is

N P ) &J= 47cP2f0 4

' P. A. Wolff, Phys. Rev. 95, 1415 (1954). W. Shockley, Solid State Electron. 2, 35 (1961). J. L. Moll and R. van Overstraeten, Solid State Electron. 6, 147 (1963). l o G. A. Baraff, Phys. Rev. 128,2507 (1962). ' I J. L. Moll and N. Meyer, Solid State Electron. 3, 155 (1961). D. J. Bartelink, J. L. Moll, and N. Meyer, Phys. Reu. 130, 972 (1963).

''

(20)

4.

213

CHARGE MULTIPLICATION PHENOMENA

and the current in the x-direction is

It is assumed that the electron gas is in the steady state under a field E and that the scattering is isotropic. To remain in the steady state, the: net electron flux across the surface of a sphere in momentum space is zero. That is, the rate at which electrons move into the sphere because of energy lost in phonon or ionization collisions is equal to the rate at which they are accelerated out of the sphere by the electric field. Moll and van Overstraeten’ have shown that, in this steady state, f

eEl 1 -

u

t?fo

dp’

where

and

where

e2E2iR2 3&R

E l =--

and E2

=

e2E211R

3&It The parameter E~ is the optical phonon energy and AR is the mean free path for optical phonon scattering; ci is the threshold energy for ionization and /zi is the corresponding mean free path. The function U ( E )represents the reentry of ionizing electrons into the distribution and is the fraction of ionizing electrons that end up with energy less than E . For E < ei, they let U ( F )= Then, from Eq. (24),

fo

=

A exp[ - (E -

EJ/E~]

AR + -Ei{exp[

- (F -

- 1i

E~)/F~]

lZiER

for

E

< ci.

(26)

274

For

A . G. CHYNOWETH E

> ci, they assume that

~ l R / & R l i%

1 and

fo = A exp[ - Z(E= A exp[ - ( E

E / E ~%

1 ; then

E~)/E~]

- ci)/kT2].

where

kT2 = c2/Z and

[

z = - 1 + ( 1 + - 4- 2 2

2)';'] ,

From Eqs. (26) and (27),

fo

=A

& k T2 - 11, exp[ - ( E - &,)/kTI] B _ _ ki kT,) (exp[ - (E - &,)/kT1] IZiEREi (28)

+

+

where kT, = c l . Matching at E = ci requires A The ionization rate a can be expressed as

=

B.

(No. of electrons with energy > E ~ )x a=

U -

J-i

Total no. of electrons x o,,

where the drift velocity vD is obtained from J(P) = e w Y p ) .

Thus, from Eqs. (20) and (21)

4.

CHARGE MULTIPLICATION PHENOMENA

275

Hence HiR

= 3(c,/eE&)(k T2h,)(1

+ k TJe,)

(33) This expression for the ionization rate is rather elaborate, but we note that, at high fields, it will be dominated by the exponential term, tl

- (

exp -

= expi -

~

s). e E AR2

(34)

Thus, the ionization rate has an exponential 1/E2 dependence 011 the electric field. A similar result was obtained by Wolff under the starting assumption Izi 6 I z R . A qualitative understanding of this exponential factor can be obtained by considering the average rate of energy gain by electrons in a spherically symmetric velocity distribution. After being accelerated for a time z, an electron with initial velocity u cos 0 along the field will have a velocity (u cos 9 + eEz/m) in this direction. The average energy gain in time z, obtained by squaring this expression and averaging over 9, is then (eEIzR)2/ 2mv2. On the other hand, an electron loses energy Ftw in this time. At equilibrium the rates of energy gain and loss must be equal, which gives the condition

(-__ e W 2 ,” 4E

ho .

(35)

Now the ionization rate will be proportional to the number of carriers with energy E ~ and , for a Maxwellian distribution this will be given by a - exp

( - %) - ( -

4EiER exp - ___

)

e2E2&’ ’

which, except for a slight difference in numerical factor, is identical to the above conclusion.

b. Ionization Rates for Low Fields The theory presented in the previous section is essentially a theory of electron “diffusion” in energy-momentum space, in which electrons undergo many collisions in transport from one energy to another. The distribution function of the electron gas can be described by a temperature or energy . low fields, on the other hand, which is not negligible compared with E ~ At

276

A. G . CHYNOWETH

the effective electron temperature is so small that electrons which suffer their normal share of phonon collisions would never achieve an energy e i . Instead, one has to conclude that those electrons that have reached E~ are those “fortunate” ones that have managed to escape any phonon collisions while being accelerated. The electrons which do the ionizing therefore reside in a high velocity spike in the otherwise spherical velocity distribution, directed along the field direction. This interpretation of low field ionization was first given in detail by Shockley’ who showed that this simple concept was remarkably successful in accounting for diverse breakdown phenomena measured in silicon p-n junctions. Shockley’s ionization theory has been elaborated upon by Moll and Meyer,” whose results are presented here. An electron starting at zero energy has to travel a distance d in the field direction before it achieves energy ei without suffering phonon collisions, where d = Ei/eE. The probability that it does so without suffering any phonon collisions is Po

exp( - d//z,).

=

(37)

In the simplest model, the electron, having achieved energy c i , must ionize in its next collision. Otherwise, it is supposed that the first phonon collision it suffers would return it to an energy less than e i . The probability that the next collision is an ionizing one is simply &/,Ii = l / r , assuming r >> 1. The total probability of ionizing per try in this single process is then P , = (l/r) exp(-ei/eEAR).

(38)

Now for every start from zero energy that an electron makes leading it without collisions to E ~ it, makes (P;’ - 1) starts which end in a phonon collision with a loss of energy E ~ If. it is assumed that after each of these collisions the electron is returned to zero energy, then the total energy expended on the electron to produce one ionization is eE

_ -

Ei

+ (PF1 -

c(

Hence, eE

(39)

In the low field limit, this becomes a = - eeE xp TER

(-&).

4.

CHARGE MULTIPLICATION PHENOMENA

211

The above treatment is oversimplified in two respects, as Moll and Meyer have discussed. It does not take into account (a)processes in which an electron experiences one phonon collision somewhere in mid-flight between zero energy and ci and (b) processes in which electrons achieve energy sufficiently greater than E~ that they can experience a phonon collision before ionizing. Shockley ignored (a) on the grounds that an electron which has been scattered into a random direction by a phonon collision at some intermediate energy must on the average travel farther to reach ei than a carrier starting at rest, and so the exp( - d/&) factor weighs heavily against it in determining the overall ionization rate. Moll and Meyer, however, have made the appropriate corrections to Shockley's theory and they find these effects are not negligible. Of course, these arguments can be extended into a series to cover two, three, four, etc., phonon collisions. When this is done, it is found that the probability PI should be replaced by PI

PI

+

1 -

[Y/(

1

+ r) exp( -&R/eEAR)]'

(41)

c. General Theory of Ionization Rates

In the preceding sections, an almost spherically symmetric velocity distribution was used for the high field case, whereas, in the low field case, it was argued that the effective part of the distribution was spike-shaped. Clearly, to obtain a general expression for the ionization rate it would be necessary to solve the Boltzmann equation using an infinite series expansion for the distribution function. This is mathematically intractable. However, it turns out that it is not a bad approximation to assume a distribution function which is a mixture ofa spherical part and spike-shaped part, namely, f(U,

cos Q) = A(u) + B(u)6(1 - cos 0).

(42)

The solution of the Boltzmann equation using this approximate distribution function has been obtained by Baraff.10,13The technique is to expand Eq. (42)as an infinite series of spherical harmonics

f (v, cos 13) = C n,(u)P,(cosO),

(43)

1

1

In such an expansion, no is proportional to the density of electrons at velocity u. This expansion converts the integro-differential Boltzmann equation into l3

G. A. Barafl, Phys. Rev. 133, A26 (1964).

278

A. G. CHYNOWETH

an infinite set of coupled differential equations for the coefficients nl(u).The assumption of the form (42) for f leads to n 2 = ($)n,.

(45)

This relation may be used to decouple the first two differential equations from the rest, whereupon they may be combined to give a single equation for the density n,(u). For constant mean free path. this solution is no =

exp( - & I T ) ,

with

2

For

E

= ti and eE2, =

eEi,

(47)

E,.

no

=

-

1 exp( -

A),

Ei

t?E/.,

(49)

which is basically the shape of the spike proposed by Shockley for the low field condition [Eq. (40)]. For eE, >> E,, the solution becomes

which is basically the Maxwellian distribution derived by Wolff’ and Moll and van Overstraeten’ for the high field condition. The criterion for the validity of the Shockley spike is that eEA, < E,. In practice, fields are generally of intermediate strength so that neither the high field nor low field limits are good approximations. It is necessary, therefore, to have a more general solution to the Boltzmann equation, which makes no starting assumptions concerning the shape of the velocity distribution. A method for obtaining a general solution has been devised by Baraff.“ (1) Exact General Solution. Baraff’s iheory makes the following starting assumptions : (a) There are no electron-electron interactions. (b) Electrons emit or absorb acoustic phonons without a change in energy. (c) Electrons emit optical phonons and lose a fixed amount of energy. (d) The probability that an electron is scattered from momentum p to p’ depends only on its initial and final energy.

4.

CHARGE MULTIPLICATION PHENOMENA

279

(e) An electron causing ionization gives up all of its energy in the ionization process. (This implies the electron does not travel far once it has sufficient energy to cause ionization, an assumption which is somewhat questionable in the light of experimental data on ionization rates in silicon.) (f) Scattering is spherically symmetrical. (g) The hot electron has a constant mean free path. This iast condition makes the mathematics more tractable but it is not necessarily a good assumption, particularly for the 111-V compounds in which several energy-dependent scattering mechanisms operate. It would be more realistic to make A energy-dependent in some way, with A decreasing as electron energy increases, but such refinements to the theory have not yet been attempted. The distribution functionf(z, p , t ) describes a packet of electrons released at the plane z = 0 at time t = 0 with zero energy. Then at t = 0,

The distribution function develops in time in a way described by the Boltzmann equation :

a

f ( z , p , t ) = j d p ' f ( z , p', t)F(p -+ p')/z(p'),

(52)

where z(p) is the mean free time between collisions for an electron of momentum p i t includes all collisions, those resulting in ionization as well as those involving phonons. The function F ( p 4 p') is the probability that an electron will be scattered from p to p' by phonons. Excluding the ionizing collisions from the right-hand side of the equations is equivalent to treating these collisions as though they absorb the electrons. Hence, f describes electrons which have not yet caused a single ionization. Now, consider a collision density, M(z, E), defined so that the number of collisions of all types which occur in the slab of thickness dz at z involving electrons of energy between E and E + 6~is M ( z , E ) Sz 88.The total number of collisions in volume 6z 6 p is the time integral of the collision rate f/z(p). Another integration over all momenta whose energy ~ ( pis) equal to E gives the collision density

Define @ion(&)

r ( ~=) ___ 01total(E)

=

ioniz,ation probability,

(54)

280

A. G . CHYNOWETH

where uionand utotalare the ionization collision and total collision cross sections, respectively. Then the number of ionizing collisions in the slab at z is

The ionization rate a is the reciprocal of the average distance ionizing collisions, namely,

jmdz zN(z, a-1 =

(2)

=

between

E)

0

Imdz N(z,

(2)

(56) E)

0

(since it was assumed that the electron starts off at zero energy again after the ionizing collision). The collision density can be regarded as a sum of partial densities :

where the subscript n denotes the number of optical phonons the electron has emitted prior to the collision counted in M(z, E). An electron which has energy E after having emitted n phonons must have acquired this energy by having drifted a distance

z=

(E

+ eE

down the field. Hence, the z dependence of the nth partial density must be contained in a delta function guaranteeing this relation, and M , is given by M,(z,

E)

=

(

M,(E)6 z - ___ EL,...).

(59)

Once the set of functions M,(E)is known, IXcan be calculated by means of the equations given above. To obtain M,(E),a function M(q, E ) is defined, being a modified Laplace transform of the collision density :

where

( 2)

q=exp - - .

4.

CHARGE MULTIPLICATION PHENOMENA

281

This ensures that M has the form of a power series in q whose coefficients are just the desired functions M,(E):

Hence, if we have M(q, E), we do not have to invert the Laplace tr,ansform ;all that has to be done is to pick off coefficients in a power series. Now the Laplace transform was obtained by an integral over A4(z, E ) which was in turn an integral over the distribution function f: These very same integrations can be performed on the Boltzmann equation itself, and the result is an integral equation for M(q, E ) :

M(v, 4

=

Q(4 +-

Jfd ~V‘ q ,

6,

&’)M(q,6’) ,

(63)

where the kernel T and the inhomogeneous term Q arise just by performing these integrations. Both T and Q can be evaluated numerically so that M(q, E ) can be obtained. Having obtained these coefficients, the collision density M ( z , E ) can be calculated which leads, in turn, to the ionization rate a. The results of this theory are shown in Fig. 1 in the form of universal plots of log(al) versus (Ei/eEIR)for various values of the parameter ( E ~ / E J . Clearly, this parameter is a constant for a given material, so that there will be a particular curve applicable to that material. Experimentally, a is determined as a function of field E ; thus, by finding the curve that gives the best fit, the two adjustable parameters 1 and ci are found. The curves were calculated with the assumption that r = 0 for E < E~ and r = 0.5 for E > E ~ The . abrupt rise in r approximates the behavior of the ionization cross section near threshold. The choice of r = 0.5 is quite arbitrary, however, though other calculations in which Y was varied showed that the ionization rate was only weakly dependent on the choice of r, at least for r 2 0.25. (See Fig. 2.) The parallel straight lines in the lower portion of Fig. 1 correspond in form and slope to the predictions of the low field ballistic theory given earlier. The concave downward curves in the upper part of the figure have a form approximating that given by the high field diffusion theory. Baraff’s theory therefore embraces the previous theories as limiting cases. (2) Almost Exact General Solution. At the beginning of this section, mention was made of an approximate solution of the Boltzmann equation that can be obtained by inserting an approximate distribution function made up of spherical and spike-shaped parts. It is clearly of interest to inquire as to how accurate a solution can be obtained iff is represented by an expansion to a sufficient number of terms of a spherical harmonic series. B a r a P 3 has been

282

A. G . CHYNOWETH

&;/eEX,

FIG. 1. Universal Baraff curves of the ionization rates versus the reciprocal of the electric field in normalized units for various values of the ratio of the phonon energy to the ionization energy. (After Baraff.'O)

able to show that this technique is feasible, and the results compare very closely with the mathematically exact but much more tedious procedure described above. As before, we write m

where p = cos 8 and 0, as before, is the angle between p and the electric field. The Boltzmann equation gives the rate at which electrons are scattered into the element at ( z ,p ) :

4.

283

CHARGE MULTIPLICATION PHENOMENA

E , /eEX,

FIG.2. Baraff curves for a given value of E ~ / and E ~ various values of r, the ratio of the ionization cross section to the total cross section above threshold. (After Baraff.")

since it is possible to expand (;lf/at),, into a similar Legendre series. After substituting the series expansion for f in this equation, relations between the Legendre coefficients can be picked off:

2 5 3

e E . -7 ( n i + : n 3 )

+e

~ :(nl' - :nl)

+ 3z = s,,

etc.,

(68)

284

A. G . CHYNOWETH

where the prime denotes the differentiation

Any finite number N of these equations contain in general ( N + 1) unknown coefficients. Since the information of physical interest is usually contained in the expansion coefficients of low 1, the problem is to truncate the set so that the first N equations contain only the first N unknown coefficients. The generalized diffusion theory approximation follows from the assumption that the distribution is nearly isotropic. In that case, the coefficients will decrease rapidly with increasing 1. The natural truncation is then to set nN equal to zero, which allows the first N equations to be solved. In the other extreme of maximum anisotropy, where the electrons are streaming in the field direction, the distribution function is then

f ( z , P>PI = g(z, P ) 4 1 - P 1.

(70)

With the use of Eq. (44), this distribution function leads to the relation

The convergence of the series expansion for f is obviously slowest for this situation of maximum anisotropy, or streaming, but rather by analogy with the diffusion case. it is reasonable to use the relation

to truncate the series.

Baraff has pursued this latter truncation scheme and finds that it holds even in the case of an isotropic distribution. Clearly, the equations truncated in this way would lead to the exact distribution function only if the exact distribution were really of the form of Eq. (70). This scheme would seem at first to become less valid as the distribution becomes more isotropic. In the limit of near isotropy, however, the last retained expansion coefficient is so small that replacing it by something of order unity times itself makes negligible change in the equations. This means that the truncation technique for the maximum anisotropy condition can lead to exact results again in the limit of complete isotropy. Some of Baraff's results are shown in Figs. 3 and 4. Figure 3 shows a typical comparison between the distribution functions obtained by the exact method and the maximum anisotropy method. In Fig. 4, universal ionization rate

4.

285

CHARGE MULTIPLICATION PHENOMENA

-

10-1. 0

I

I

I

I

I

0.5

1.0

1.5

2.0

2.5

1

1

3.0 3.5

E

FIG.3. Comparison of the analytical distribution functions obtained by the maximum anisotropy method with the numerically calculated exact distribution for various values of the electric field (after BaraffI3). The ordinates are defined by the relation ?no(&)= &n,(p).

curves are shown as obtained by the two methods for various values of (ER/EJ. The overall agreement of the curves is within 15 % over a three and a half decade range of CI. It is clear. therefore, that the maximum anisotropy method offers a relatively straightforward technique for deriving distribution functions and ionization rates to a very satisfactory degree of accuracy ; certainly to an accuracy more than sufficient for comparison with experimental data obtained to date.

286

A. G . CHYNOWETH

0

2

4

6

8

1 0 1 2 1 4

E , / e E A,

FIG.4. Comparison of the Townsend a coefficient calculated using the analytical distribution functions with the exact numerical calculations. (After Baraff.13)

111. Avalanche Breakdown in P-N Junctions

6. ANALYSIS OF CURRENT BUILD-UP In the preceding part of this chapter we have reviewed the theories of hot electrons which are concerned with the field dependence of the ionization rate by electrons or holes. In this and subsequent sections, we shall survey the experimental manifestations of charge multiplication, most of which revolve around the most obvious consequence of multiplication, namely, avalanche breakdown. In avalanche breakdown, an electron is accelerated to an energy sufficient to produce ionization, thereby creating an electron-hole pair. As a result there are two electrons to be accelerated, and these subsequently can each produce additional pairs, and so on. In addition, the holes are accelerated in the opposite direction and they too can achieve energies sufficient to ionize. The holes thus provide a positive feedback mechanism which injects fresh electrons “upstream” in the electron flow. Clearly if the field extends over an infinite volume, the current will build up to infinity. In a finite volume, however, the chance that some carriers will pass right through the high field

4.

CHARGE MULTIPLICATION PHENOMENA

287

region without ionizing is sufficiently great that the multiplication is restrained from running away, and, instead, the current is held at a definable value. We will now present the relations between the experimentally measured quantities, current and field, on the one hand and the “fundamental” material parameters, the ionization rates for electrons and holes, on the other. These relations result from a simple application to solids of ideas first developed long ago for gas discharges by Townsend. They were developed first by McKayI4 to explain the breakdown characteristics of silicon and germanium p-n junctions. Following McKay, it will be assumed that, for a first approximation, the ionization rates for electrons and holes can be taken as equal. Assume the high field to be directed along the x-axis and to exist only in the region 0 < x < W. Let the number of electrons injected into the high field region at x = 0 be no. Let these electrons produce n , additional electron-hole pairs between x = 0 and x and n, pairs between x and W (See Fig. 5.) Then the number of electrons produced between x and x + dx is

dn,

=

(no + n,)adx

+ n,crdx

= na dx,

(73)

where n = no + n , + n , is the number of electrons collected at the anode. Integrating with the boundary conditions n , = 0 at x = 0 and n = n , + no at x = W we obtain

where M = n/no is the charge multiplication factor. This is the basic formula for avalanching, relating the experimentally measurable multiplication, M ,

JNS c

JP

JPS c

-

“2

“0 Electrons

x

tax

FIG.5. Diagram defining the symbols used in deriving the expression for the amount of charge multiplication occurring in a high field region of width W l4

K. G. McKay, Phys. Rev. 94,877 (1954).

288

A. G . CHYNOWETH

to the ionization rate by means of measurable parameters. When the righthand side achieves unity, M = 00, which represents the breakdown condition. It is obvious that the positive feedback represented by the n2a d x term is necessary for M to approach infinity at a finite field. Without this term, we have W M = exp[J 0 U ( E ) , (75)

&j

and this is a simple exponential function without any singularity driving M rapidly to infinity. There were some sweeping assumptions made in the derivation of Eq. (74), and while these are admissible for rough comparisons between experimental results and theory, it is usually desirable to consider the effects of (a) the ionization rates for electrons and holes not being equal and (b) mixed injection of carriers in which both electrons and holes are injected into the high field region from outside and, in addition, other primary carriers are generated in the high field region itself. MillerI5 was the first to extend McKay's theory14 to unequal ionization rates. Let a(E) and B(E) be the ionization rates for electrons and holes, respectively. Then Eq. (73) becomes

dn, = (no + n , ) a d x =

(no

+ n2pdx

+ nl)(a - p) dx + (no + n, + n2)P dx.

(76)

Integrating with the boundary condition n, = 0 at x = 0 and n2 = 0 at x = W we obtain 1--= 1 j w a(E) exp[ - (a - p) dx' d x . (77) M o 0 This expression applies specifically to the case where the avalanche is initiated by the injection of no electrons at the cathode. If we consider the avalanche to be initiated by holes at the anode, we obtain

jx

1-

1

=

1 p(E)exp[ -j:(p w

0

1

1

- a)dx' d x .

Note that besides interchanging a and 8, a change in the limits of integration of the second integral has to be made, a point which has been emphasized recently. For the general case of simultaneous injection of electrons and holes, Lee et a l l 6 have shown that if the injected electron and hole currents are Jn, and J p s ,respectively, and k = J p , / J , where J , = J,, + J p s , I5 l6

S . L. Miller, Phys. Rev. 105, 1246 (1957); 99, 1234 (1955). C. A. Lee, R. A. Logan, R. L. Batdorf, J. J. Kleimack, and W. Wiegmann, Phys. Rev. 134, A761 (1964).

4. I - - =1

CHARGE MULTIPLICATION PHENOMENA

-lo l-k+kexp[-{o(a-P)dx] W

W

a exp

289

[-j: (a - B) dx'] dx

When the injected current is purely electrons or holes (k = 0, 1, respectively), Eq. (79) reduces to Eqs. (77) and (78), respectively. The effect of primary carriers injected within the high field region is easily taken care of. If v electrons and holes are generated per unit distance in the x-direction, the term vxadx v(W - X ) U ~ X= ~ W a d x (80) has to be added to the right-hand side of Eq. (73). The net result is that Eq. (74) still holds exactly provided that A4 is now taken to be (no n, n, vW)/(no vW). When the ionization rates for electrons and holes are not equal, the expression for the multiplication due to carriers injected within the space charge region becomes much more complex. Expressions for various such cases have been given by Howard.17 In addition to injection in the high field region there can be recombination. This case has not been worked out in detail, though it would seem reasonable to represent this effect as a weighting factor applied to the quantity v.

+

+ + +

+

7. AVALANCHING IN NONUNIFORM FIELDS In experiments on bulk crystals, the field can be taken as uniform provided the injected (primaries and secondaries, etc.) charge density is small compared with the charge density in the absence of injection. In this case, the integrals in Eqs. (74), (75), (77), and (78) are trivial; from Eq. (74), for example, the breakdown condition becomes simply

a w = 1. (81) However, it is generally necessary to use the high fields attainable only in the space charge layers ofp-njunctions to get avalanche breakdown, and, in such layers, the spatial variation of the field has to be taken into account in the integration^.'^ There are two cases to consider: (a) step junctions and (b) linear gradient junctions. In the former, there is an abrupt change from p type to n-type at x = 0. In the latter, the net charge density due to the donors and acceptors varies linearly with x. The field distributions corresponding to these two cases are indicated in Fig. 6. l7

N. R. Howard, J . Electron. Control 13, 537 (1962).

290

A . G. CHYNOWETH

I

I

I

I

0

W

we X *

- w_

2

I

I

-we _ 0 2

I

w

2

2

(b)

(a)

FIG.6. Diagrams representing the actual field distribution (solid-line) in the case of (a) a step junction and (b) a linear gradient junction. The field profiles represented by the broken lines indicate the approximations used in order to make the multiplication analysis simple.

a. Step Junctions The field is described by

where EM is the maximum field in the junction (= 2 V / W )and where V is the total potential drop across the junction, equal to the sum of the applied and the built-in bias. (The contribution due to the built-in bias is appreciable for narrow junctions.) We also have the useful definition

w = w1vl12,

(83)

where W, is the width of the space charge region for a total potential drop across it of one volt. By means of Eq. (82),the variable of integration in Eq. (74) can be changed, so that we obtain

Differentiating, we obtain

This relation offers two ways of determining x ( E ): (i) If M is measured versus EM on a single junction, W is then constant and

4.

CHARGE MULTIPLICATION PHENOMENA

291

(ii) If EB, the maximum field in a junction at breakdown, (:M = co) is measured for various junction widths,

b. Linear Gradient Junction

In this junction, N, - N,

=

ax,

(88)

where N,, N , are the local donor and acceptor concentrations, and a is a constant for a given junction. Solving Poisson’s equation with this charge distribution leads to the field distribution indicated in Fig. 6(b).The following relations hold : E = EM[1 - ( ~ x / W ) ~ ] , (89)

E M = 1.5 V / W ,

(90)

w = w,v”~= [ + w ~ ~ E ~ ] ~ / ~ . (91) Again changing the integration variable it follows that

By use of Abel’s integral theorem, this can be transformed to

This can be solved numerically if M is measured as a function of EMin a given junction.

It is obvious that the case of different ionization rates for electrons and holes is much more complicated. However, in the special case where multiplication data are obtained on a given junction for both hole injection and electron injection separately, Lee et a1.16 have shown that the following pair of equations results for the linear gradient junction [from Eqs. (77) and (78)]. 1 1 - - = 2 exp (a - p) djq a cosh JEEM(a - B) dy‘] dy , (94) M,

[-

0

-

IEM 0

[

292

A . G . CHYNOWETH

where dy = d[-(W/2)(1 - E/EM)’’2].The experimental refinements necessary to justify the use of these expansions have so far only been developed for studies of avalanching in silicon junctions. The reader is referred to the literatureI6 for a prescription for solving this pair of simultaneous integral equations.

c. Approximate Solution of Basic Equation For many purposes, it is sufficiently accurate to approximate the field distribution in the p-n junction as indicated by the dashed outlines in Fig. 6. The field is taken as equal to EM but extending over an effective width Weless than the actual width W We can write

We = rnW,

(96)

where 0 < m < 1. Equation (74) is then simply 1 1 - - = a(E,)rnW.

M

(97)

Using the values of LX(EM)obtained from the exact solution of Eq. (74), McKay was able to evaluate m for various junction^.'^ For most junctions, rn lies somewhere between 0.3 and 0.5 and is relatively insensitive to field. i t is worth noting that, if Eq. (97) is used for determining a(E) from experimental data, a slight uncertainty as to the true value of rn will only affect, to a corresponding degree, the absolute value of a ;it will not affect the form of the field dependence of ct and this is usually of greater interest. Instead of replacing the actual field distribution by a square one, some authors have chosen to find approximate ways of solving the integral in Eq. (93). Miller15 established that experimental data on M versus reverse bias Vcould be fitted by the empirical relation

where V, is the breakdown voltage and p and q are constants. This relation gives a good fit over most of the bias range of interest except for biases when M is only slightly greater than unity; for low M , a better approximation is to replace E by ( E - Eo), where E , is a threshold field below which no multiplication can be obtained (see below). The integral can be solved analytically if Eq. (98) is substituted in Eq. (93). By obtaining data on pairs of junctions in which the multiplication was initiated by electrons and holes, separately, Miller” was able to extend his method of analysis to the case of unequal electron and hole ionization rates for silicon and germanium junctions.

4.

CHARGE MULTIPLJCATION PHENOMENA

293

Moll and van Overstraeten’ employed yet another technique for obtaining a and 8. They were able to evaluate the integrals and solve the two simultaneous equations by assumingat the outset that (i) the form of the field dependence of a and 8 was given by Eq. (40), with different parameters for the electrons and holes, and (ii) that = ya over the field range of interest, where y is a constant. These assumptions rather beg the question though for many studies, including perhaps the silicon junctions to which they applied their analysis, they may not lead to serious error. IV. Methods of Measuring Charge Multiplication

8. ELEMENTS OF

A

WELL-DESIGNED EXPERIMENT

At low reverse biases across an ideal p-n junction, the current flow is equal to the saturation current, which is made up of electron and hole components. As the bias is increased, these saturation currents will begin to multiply, leading to breakdown. However, the reverse current is a notoriously unreliable measurement of the true diode current because of additional currents, usually ascribed to surface leakage effects. Consequently, it has been the invariable practice to inject a few extra carriers into the junction by some means and to study the current due to these extra carriers as a function of reverse bias.‘ In the earliest experiments on Si and Ge,14*19carriers were injected into the junctions by means of light or a-particles. These experiments, analyzed with the assumption a = 8, sufficed to map out the field dependence of the ionization rate, and these results were subsequently fitted reasonably well by .~ injection methods, unless Wolff s high field diffusion-type t h e ~ r y These special precautions are taken, are generally crude, however, as they inject both electrons and holes simultaneously. Miller” was the first to employ techniques whereby pure electron or pure hole currents were injected into a reverse-biased junction by using the forward-biased emitter junctions in pnp and npn transistor structures. He was able to show that the ionization rates for electrons and holes were appreciably different in both Si and Ge though his techniques restricted him to obtaining results at only a few particular high field strengths. Subsequent refinements in the sensitivity of the multiplication measuring techniques enabled the ionization rate to be studied over a wide field range in a single junction. From such studies” it was established that the low-field formula [Eq. (40)] was generally a better fit to experimental data in silicon junctions, and this led to the further theoretical developments described earlier. l8

l9

A. G. Chynoweth and K. G. McKay, Phys. Rev. 108, 29 (1957). K. G. McKay and K. B. McAfee, Phys. Rev.91, 1079 (1953). A. G. Chynoweth, Phys. Rev. 109, 1537 (1958).

294

A . G . CHYNOWETH

The considerable refinement in the theory achieved by Baraff l o called for yet more refined experimental studies. For such measurements, the following conditions should be met : (i) It should be possible to initiate multiplication with pure electron and pure hole injection, separately, in the same junction rather than by using complementary pairs of junctions. (ii) The junction width should be large enough to avoid having to make end corrections (see below). Conversely, these should be made if the junctions are narrow. (iii) The measurements should be made on junctions free from microplasma-inducing defects (see below) or, alternatively, by using local injection techniques, on parts of junctions free from such defects. (iv) The most useful measurements for the purpose of solving the simultaneous integral equations are those where M lies in the range 1 < M < 2. (The multiplication curves are appreciably different for electron and hole currents in this range but, as is obvious, at higher multiplications, the curves approach each other and become indistinguishable; clearly, when M + GO at breakdown, it is immaterial whether the avalanche was initiated by electrons or holes.) (v) The diffusion lengths of the carriers should be long compared with the width of the space charge region in order to avoid having to include the variation of the collection efficiency with junction width. (vi) The field profile in the junction should be accurately known. The above conditions have been substantially met by Lee et ~ 1 . in ' ~recent studies of silicon junctions with V, ranging from 6 to 95 volts, and ionization rates for electrons and holes have been obtained which are good fits to Baraff's theory' to within experimental error. The technology of junction fabrication in group 111-V semiconductors being somewhat behind that in Si and Ge, similarly refined experiments have not yet been possible in these materials. However, some preliminary multiplication studies have been made, and it appears that Baraff's theory is equally applicable to these compound semiconductors, as will be discussed later. 9. EXPERIMENTAL TECHNIQUES

T o measure the multiplication, it is the usual practice to generate electrons and holes at a constant rate by means of light though, as mentioned above, other injection schemes are also possible. The resulting photocurrent arises both from carriers generated within the space charge region and from those that diffuse to it from the field-free end regions of the junction. Measuring the photocurrent I , as a function of reverse bias gives the multiplication

4. CHARGE

MULTIPLICATION PHENOMENA

295

characteristic directly since

where IQ( V ) is the photocurrent at reverse bias I/: Two injection geometries have been employed, shown in Fig. 7. In Fig. 7(a), the light approaches in a direction parallel to the junction field; in Fig. 7(b), the light direction is perpendicular to the junction field.

LirI

I c x

I

LirI

Surface

Surface

i

"

P

I

W

(a)

(b)

FIG.7. Diagrams illustrating the two types of geometry commonly used for measuring charge multiplication in p-n junctions: (a) the junction lies parallel to and beneath the surface on which the light impinges and (b) the junction lies perpendicular to and intersects the surface o n which the light impinges.

In the parallel configuration, used typically with diffused junctions where the junction lies parallel to the crystal surface and at a depth d below it, light of low penetrating power (short wavelengths) is absorbed within the diffused layer so that only the minority carriers specific to the diffused layer are injected into the junction [these would be holes in the case shown in Fig. 7(a) of an n-type diffused layer]. Thus injection purity is easily achieved in this case. For more penetrating radiation, however, both electrons and holes will be injected so that the condition of injection purity is n o longer met. The amount of admixture of the two components has been determined by Terman21 in connection with the spectral response of solar cells. He finds that the number of holes reaching the junction due to pairs generated in the n-region is given by

where N o is the surface density of photons, a is the absorption coefficient for the wavelength of light used, and L,, is the diffusion length for holes in

'*L. M. Terinan, Stanford Electronics Laboratories,Tech. Rept. No. 1605-1, September 21,1959.

296

A . G . CHYNOWETH

the n-type material. Similarly, the number of electrons reaching the junction due to the creation of hole-electron pairs in the p-region is

where L, is the diffusion length for electrons in the p-type material. We therefore have the ratio

R=

Electron injection current Hole injection current

- (a

- Lp1)

(a + L;

I)

exp( - Wa) [exp( - dL; + da) - 11

'

Numerical evaluation of this expression establishes that R increases as a decreases, eventually approaching a constant value for highly penetrating radiation as is to be expected. The greater the value of R, the greater the injection purity for the case of penetrating radiation, but the actual value of R that can be tolerated in an experiment will depend on the relative magnitudes of the ionization rates a and /I. For silicon, a (electrons) > (holes), so that it is best to use an n-type layer diffused into a p-type material, as shown in Fig. 7(a). If the configuration were reversed, then even if R is not changed, the more effective (in terms of ionization) electrons injected from the surface layer could significantly distort the effect of the holes injected from the bulk of the crystal. The situation would be reversed in materials where B > a, germanium, for example. In the perpendicular configuration of Fig. 7(b) it is impossible to avoid mixed injection unless the diffusion lengths are sufficiently long (not generally true for compound semiconductors) to allow use of a well-collimated light beam for pinpointing the site for the primary injection. With mixed injection, some simple relation between c i and p has to be assumed in order to analyze the multiplication data from a single junction. In Fig. 8 is shown a typical circuit arrangement used for determining the multiplication characteristic. The light beam is modulated by a chopper so as to give rise to an ac photocurrent. The photocurrent is amplified by a phase sensitive amplifier and then applied to one axis of an X-Y recorder, the other axis of which is driven by the bias across the junction. As the bias is swept, the recorder traces out the photocurrent which is a direct plot of the multiplication characteristic if the constant value of the photocurrent at low biases is taken as unity. A typical set of good quality multiplication curves, obtained with silicon junctions, is shown in Fig. 9. Multiplication curves showing extensive horizontal segments at low biases are not obtained if conditions are such that the collection efficiency

a

4.

-

297

CHARGE MULTIPLICATION PHENOMENA

-

X-Y Recorder FIG.8. Schematic diagram of the circuit arrangement used for measuring c irge mt iplication in a p-n junction as a function of reverse bias. (After Chynoweth and McKay.")

7-

6-

Unit 12H h

5

4-

I

0

20

30

40

50 60 70 80 90 100 Bias (volts) FIG.9. Curves representing the photoresponse versus reverse bias for a graded silicon p-n junction and progressively decreasing light intensity. (After Lee et d L 6 ) 10

298

A. G . CHYNOWETH

of the junction varies with bias through the widening effect. Such a multiplication curve, which was obtained with a G a P junction and using the injection geometry of Fig. 7(b), is shown in Fig. 10. The sharp rise at high biases is probably due to microplasma breakdown, thus making this part of the characteristic of dubious value for determining the ionization rate. At lower biases, however, as the bias is increased from zero, instead of being flat the curve rises but is concave toward the bias axis at first. This curvature is due to the change in collection efficiency. At somewhat higher biases, multiplication sets in and the curve starts to bend away in the opposite direction. The curvature in the low bias range provides a novel means for determining carrier lifetime. Referring to Fig. 7(b) it is clear that the photocurrent

where L, is the sum of the diffusion lengths of the minority carriers in the end regions of the junction. Since W C - *, where C is the measured junction capacitance, a plot of I,, against C - is a straight line, and the intercept at I,, = 0 gives a measure of L,. In practice, quite good straight lines are indeed obtained, and this technique proves useful for measuring short diffusion lengths or small lifetimes.22

-

't

01

0

I

I

I

I

2

4

6

8

I

10

I

l

l

I

12

14

16

18

Reverse bias (volts)

FIG. 10. Photoresponse versus reverse bias for a GaP junction showing the effect of the increase in the space charge width with bias. (After Logan and Chynoweth.22)

a. Ionization Threshold and the Correction for Threshold Voltage in Narrow Junctions All theoretical treatments of ionization rates assume that the steady state has been reached in which the distribution function is independent of position. This is strictly true for an infinite body, but the high field regions 22

R. A. Logan and A. G. Chynoweth, J . Appl. Phys. 33, 1649 (1962)

4. CHARGE

299

MULTIPI.ICATION PHENOMENA

of p-n junctions are anything but infinite in extent. In particular, for narrow junctions quite sizable end corrections may be necessary. The need for these can be seen as follows: In narrow junctions, the field E may be enormous but the voltage drop across the junction, = EW may be quite small. If it is less than (Ei/e), then clearly the electrons can never get up to ionizing energy in spite of the high fields, and, instead, such junctions break down by tunneling m e c h a n i s r n ~ .For ~ ~ slightly ,~~ higher values of ER electrons injected at one edge of the junction may achieve ionizing energies, but only after passing through a substantial fraction of the junction width. Stated differently, electrons injected into the high field region with thermal velocities have to travel through a certain "dead-space,'' xo, before they achieve the energy distribution characteristic of the magnitude of the field in the junction. Likewise, once these electrons have enough energy to ionize, the secondaries they create also have to travel a distance xo before they can ionize, and so on. As a result of these dead-space corrections, the effective junction width is somewhat less than the full width, or, alternatively, the ionization rates calculated in the usual way have to be corrected upward. A rigorous theoretical treatment of this dead-space correction would be extremely difficult. However, the correction takes a calculable and simple form in the highly idealized case of a uniform field and equal ionization rates. For this special case, Baraff2s has shown that the usual multiplication expression is modified by a factor F, i.e., 1 1 - -= F M

W

ctdx, 0

where

In all experiments to date, this factor is not seriously different from unity, even for the narrowest junctions. Moll and van Overstraeten9 adopted a different approach to this endcorrection problem. They again assumed that the ionization rate would vary ct d x would be with field as exp( - b/E) so they could extrapolate to what for narrow junctions based on their measurements on wide ones. Thus, they

50"

23 24

z5

A. G. Chynoweth and K. G. McKay, Phys. Rev. 106,418 (1957). A. G. Chynoweth, W. L. Feldmann, C. A. Lee, R. A. Logan, G. L. Pearson, and P. Aigrain, Phys. Rev. 118,425 (1960). G. A. Baraff. private communication.

300

A. G. CHYNOWETH

derived F for the narrow junction from

I t was found that F112plotted against reverse bias gave straight lines and the extrapolation to F = 0 gave the total bias equivalent to the ionization threshold, ti. Other authorsI8 have chosen to obtain ci from experiment by making use of the empirical relation

after Eq. (98), where V, is the threshold reverse bias above which multiplication occurs. For V >> V,, p is readily evaluated and is approximately 0.5. Therefore, a plot of (1 - 1/M)’12 against V gives an approximately straight line with intercept V,. Now at this bias we can assume that an electron has just achieved sufficient energy to ionize on having passed completely through the junction. Thus, eV0 = t i

+E~W~,),

(108)

where .sLis the energy the electron lost in phonon collisions in traversing the width W,, . The dependence of E~ on W is not necessarily known but obviously tL = 0 for W = 0. So plotting V, against W,, as obtained for a series ofjunctions of different widths and extrapolating to what the threshold bias would be for a junction of zero width yields .si. It is difficult to say that one technique for obtaining ci is any better than another ;they all involve some simplifying assumptions. Such measurements have been made most extensively for silicon where the first technique gave ei = 1.8 0.1 eV for electrons, 2.4 t- 0.1 eV for holes, and the second gave 2.25 t- 0.1 eV for electrons. These values scatter around an average, which is approximately equal to twice the energy gap (gap = 1.1 eV). V. Experimental Results

10. IONIZATION RATES IN SILICON

The exponential ( E - ’ ) dependence for CI was first indicated by experiments on silicon,” and, as noted above, this led to considerable refinements in the theory. However, the slopes of ln(cl) versus E-‘ plots for different junctions

4.

CHARGE MULTIPLICATION PHENOMENA

301

varied significantly, implying errors in the field determinations or variation in mean free path between phonon collisions, or both. In the more recent experiments of Lee et al.,I6 where the field conditions were perhaps better defined, the slopes of such plots still scatter but in a way which can now be formally understood by means of Baraff’s theory.‘O The silicon results are shown in Fig. 11 where the data obtained from three different junctions of widely different breakdown voltages have been fitted to within experimental error to a single Baraff curve of u;l plotted aginst (q/eE/Z,). This multiple fitting can be achieved if 1, is regarded as an adjustable parameter. The , determines a particular Baraff curve. other adjustable parameter is E ~ which The sensitivity of the fitting technique will be discussed in the following sections on the 111-V materials where Baraff’s theory was first tested in detail.

E , /e E 1,

FIG.11. Fit of the experimentally measured electron ionization rate to one of Baraffs curves for silicon junctions with breakdown voltages ranging from 6 to 95 volts. (After Lee et a l l 6 )

302

A. G . CHYNOWETH

The silicon data, however, cover a wider range of junction widths and demonstrate how data from different junctions fall on the same curve provided LR is not taken to be a constant for the material. For the junctions shown, AR apparently varies from 44 to 68 A ; this suggests that hot electronoptical phonon scattering events are influenced by the presence of impurities or imperfections. This result is rather surprising and bears further investigation. In particular, it would be interesting to follow the change (if any) in ionization rate as imperfections are introduced by radiation damage, for e ~ a m p l e . ’ ~It”is worth emphasizing that LR enters Baraff’s theory purely in a formal way as the mean free path between energy-losing collisions. We have chosen to identify these energy losses with optical phonons and have thereby derived fits of experimental data to Baraff curves. It is not inconceivable that a variety of energy-losing collisions occurs. Baraff’s theory would apply equally well in a formal way to this situation; AR would still represent the mean free path between energy-losing collisions but ER would have to be redefined as the average energy lost per collision. For present purposes we shall persist in putting E~ equal to the optical phonon energy. Before leaving the silicon results, we note that even for the narrowest (6-volt)junction used, the data would still fall very well on a In CI - E - plot. Apparently, no experiments have yet been done in silicon where the product EL has been high enough for the ionization behavior to be in the exp( - E - ’ ) region. In the next two sections it will be seen that this region has apparently been reached in gallium arsenide and gallium phosphide. No dead-space correction was made for the 6-volt junction data, but the fact that the data lie within experimental error on the same curve used for the wider junctions indicates that this correction is negligible for the 6-volt junctions. 11.

IONIZATION

RATES IN GALLIUM ARSENIDE

Multiplication and ionization rate studies have been made on microplasma-free diffused junctions in GaAs with breakdown voltages of about 8 volts.26 These are therefore narrow junctions with the consequent deadspace errors noted above. Also, the circumstances of the experiment were such that both electrons and holes were injected by light into the junction, but it is probably safe to assume that the multiplication curve is heavily dominated by the carrier species with the higher ionization rate. The data were also analyzed using the simple square field approximation though this is probably sufficientlyaccurate, as was verified originally in silicon. First, the ionization rate is plotted in two ways in Fig. 1 2 ; for one curve (Shockley theory - E - region) it is plotted as log(ct/EM)against E , and for the other (Wolff theory - E-’ region) it is plotted as log CI against E i ’ . A 25aJ. L. Moll, Phys. Rev. 137, A938 (1965). 26 R.A. Logan, A. G. Chynoweth, and B. G. Cohen, Phys. Rev. 128,2518 (1962).

4. CHARGE

EG'(cm 6

-5 -?

7

303

MULTIPIJCATION PHENOMENA

8

9

volt-')

10

I1

12

x lo-' 13

-I

lo-' -

105

-z rn

(u

0

.-

C .-0 0

II

N .-

W

C

U

0

lo4 4

10-2

5

6

7

8

EM' (cmZ volts-')

9

1011 X I O - ~ ~

FIG.12. The field dependence of the experimentally measured ionization rate in GaAs plotted in ways appropriate to the Shockley theory and the Wolff theory. (After Logan et d z 6 )

linear plot implies agreement with theory and it is clear that the results, unlike those for Si, are in favor of Wolff's theory. The data are shown again in the form of a Baraff plot in Fig. 13. The fitting procedure was as follows : The optical phonon energy is known for GaAs to be 0.036 eV. Guessing a value for ei then determines the particular Baraff curve to be fitted. (For Fig. 13, E~ was chosen to be 1.8 eV.) Various values of AR are then tried to obtain the best fit. Three such tries are shown in Fig. 13, and clearly 1, = 15 8, is the best fit. The sensitivity of the fitting to the value of AR is also clearly evident. The whole procedure must then be repeated starting with different choices for E~ until the best two-parameter lit results (Fig. 13). The effect of varying E~ is shown in Fig. 14. Starting with gi equal to 2 eV or 1.35 eV (= F ~ leads ) to worse fits than with E; = 1.8 eV, though not severely so. It is clear that ci cannot be determined very accurately by this , required fitting procedure. On the other hand, for this wide range of E ~ the value for E,, ranged only from 12.5 to 16.5A. It was concluded that, for GaAs, ci = 1.7 f 0.3 eV and I,, = 15 & 2 A. Earlier it was noted that Baraff obtained a criterion for Wolff's theory to apply, namely, eEE.R/&R2 3 .

(109)

304

A . G. CHYNOWETH

L

1

0

2

I

I

4

6

8

1 0 1 2 1 3

Ei/eEXR

FIG.13. Same data as in Fig. 12 fitted to a Baraff curve. The value of &,Jciwas selected initially and the three fits of points correspond to the three choices for /I, indicated. (After Logan et dZ6)

r

t

t 10-4

0

2

4

6

8

1

0

1

2

14

V E X R FIG.14. The same data as Figs. 12 and 13 fitted to two Baraff curves representing two different initial choices for the ionization energy 8,.(After Logan et

4. CHARGE MULTIPLICATION PHENOMENA

305

Over the field range studied this factor ranges from 3.3 to 6.5, so that the result for A is consistent with the Wolff plot shown in Fig. 12. 12. IONIZATION RATES IN GALLIUM PHOSPHIDE

Experiments under conditions similar to those for GaAs have been performed on G a P junctions (containing narrow i-regions so as to make them roughly p-i-n junctions).22 More recently, Logan and White26ahave made multiplication measurements over a wider range of junctions and again find that the ionization rate data fit Wolff plots better than Shockley plots (Fig. 15). As in the case of the silicon results, the data from all the junctions can ~ ~ be satisfactorily fitted to the same Baraff curve (Fig. 16) with E~ ;= 1 . 5 and A, = 39 3A.

*

I

103

0.5

I

I

I

I

I

I

1.0

1.5

2.0

2.5

3.0

3.5

EW2 (CM2 V o l t - z )

x

10

FIG.15. The experimentally measured ionization rates in various GaP junctions plotted as a function of the field in a way appropriate to the Wolff theory. (After Logan and White.zb”)

26aR. A. Logan and H. G. White, J. Appl. Phys. 36,3945 (1965). The author is indebted to R. A. Logan for bringing these more recent results to his attention.

306

A. G. CHYNOWETH

FIG.16. The same data as in Fig. 15 fitted to a single Baraff curve. (After Logan and White.26")

13. BARAFFCURVESFOR OTHERMATERIALS The author is not aware of any publications to date of detailed studies of the multiplication and ionization rates in any of the other 111-V compounds. However, from the work described above a pattern begins to emerge. In all cases, it was found possible to fit the experimental data by choosing E~ to be close to 1.56,. Since the optical phonon energies are known for a good many materials, it is possible to make a rough prediction of the Baraff curve appropriate to each material simply from the ratio &,J1.5.zG. Such curves are shown in Fig. 17 for a number of materials. They make an interesting study, but, unfortunately, they cannot be used directly for predicting the multiplication in various materials under various operating conditions since a vital key is missing, the value of I, for each material. The author is not aware of any formal theoretical treatments of this quantity, so that, for the time being, the only guide is the experimental determination, and it follows that little comment, one way or the other, can be made about the significance of these. The curves for ZnO and CdS are hypothetical but provocative; if Baraff's theory applies to these materials, they would appear to have relatively high

4. CHARGE MULTIPLICATION PHENOMENA

r

307

ASSUMPTIONS

E , /eE X FIG.17. Predicted universal Baraff curves for various materials. In predicting these curves it was assumed that the ionization energy was equal to 1.5 x the energy gap. Multiplication has been studied in only a few of the materials included in this figure; the rest are included for illustrative purposes only at the present time and their inclusion should not be taken to imply that charge multiplication will necessarily occur in these materials.

ionization rates for 2, and E conditions equivalent to other materials, yet, as far as the author is aware, n o hot electron effects and certainly no multiplication phenomena have ever been identified in these materials,26beven at fields in excess of lo6 volt cm- '. VI. Miscellaneous Phenomena Associated with Charge Multiplication in Junctions

14. LIGHTEMISSION

When the field is high enough to produce charge multiplication with acrossthe-gap excitations, the result is an electron-hole plasma. Inevitably, some of the recombinations between the electrons and holes occur radiatively, the inverse of photoelectric excitation. In addition, hot electrons and holes can 26bA. Rose, private communication.

308

A. G . CHYNOWETH

undergo radiative transitions while remaining in the same energy band ; this is the inverse of free carrier absorption. Another possible mechanism whereby hot carriers can emit photons is the bremsstrahlung process, which can occur when a carrier is rapidly decelerated in the Coulomb field of a charged impurity center. Light emission from junctions in avalanche breakdown was first observed using silicon and it has been studied in detail by numerous authors. Light emission has also been observed from breakdown in germ a n i ~ m silicon , ~ ~ carbide,30 gallium a r ~ e n i d e , ~gallium ' ph~sphide,~~,~' and indium phosphide3* junctions. The total light output varies linearly with breakdown current, and the efficiency of light production is generally in the range 10- to lo-' photons per electron crossing the junction. For all materials the spectrum is very broad, extending to energies both considerably greater than and less than the energy gap. The spectrum for gallium arsenide is shown in Fig. 18. O n the high energy side the spectrum is monotonic and extends to photon energies equal to at least twice the energy r

Photon energy ( e V )

FIG.18. The spectrum of the radiation from a reversed biased GaAs p-n junction at 77°K. The solid curve is proportional to the photomultiplier current. The dashed curve is corrected for instrument response. (After Michel et ~ 1 . ~ ' ) 27

R. Newman, Phys. Reu. 100, 700 (1955).

** A. G. Chynoweth and K. G. McKay, Phys. Rev. 102,369 (1956). 29 30

31 32

A. G. Chynoweth and H. K. Gummel, J . Phys. Chem. Solids 16, 191 (1960). L. Patrick, J . Appl. Phys. 28, 765 (1957). A. E. Michel, M. I. Nathan, and J. C. Marinace, J . Appl. Phys. 35,3543 (1964). M. Gershenzon and R. M. Mikulyak, J . Appl. Phys. 32,1338 (1961).

4.

CHARGE MULTIPLICATION PHENOMENA

309

gap (up to three times the energy gap has been observed in silicon and germanium). For energies slightly less than the band gap, the spectrum shows some structure (this has not been reported in silicon and germanium) superimposed on a broad spectrum extending to lower energies. Clearly, photon energies much less than the band gap must arise from intraband transitions. The highest energy photons, on the other hand, most probably arise from interband (recombination) transitions. As regards the structure observed close to the band gap, this is very similar to that observed in recombination luminescence in forward-biased GaAs junctions, strongly suggesting that it originates in transitions via impurity levels.31This could occur even in reverse-biased junctions if some of the hot electrons in the high field region diffuse back and thermalize in the p-type end region, or vice versa. A theoretical treatment of the spectrum to be expected from interband and intraband transitions has been given by W ~ l f fwith , ~ ~particular reference to data obtained from germanium junctions.29To calculate the spectrum it is necessary to consider the band structure of the material and the distribution functions for the hot electrons and holes. To take the band structure into account exactly would be an extremely difficult, tedious, and unrewarding undertaking, and so rather sweeping simplifications have to be made while retaining the essence of the calculation. In the case of direct gap materials, such as most of the group 111-V materials, Wolff argues that the hot electron cloud can be considered as populating the conduction band minimum at k = 0 provided that there are no other energy minima close by, energywise, and the hot hole cloud as populating the vaience band maximum at k = 0. The valence band will consist of a light hole band and a heavy hole band, effectively degenerate at k = 0 because spin-orbit splitting effects can be neglected in comparison with the hole energies. The-maximum electron and hole energies attained within their respective bands will be roughly equal to the ionization energy ci, as indicated in Fig. 19. Clearly, for optical transitions only vertical transitions need be considered since indirect transitions involving phonon cooperation will always be less likely, other factors being equal. The various possible direct transitions are indicated in Fig. 19. Arrow I indicates direct intraband transitions of hot holes between the heavy and light mass bands. Arrows I1 and 111 represent recombinations between electrons and the heavy and light holes, respectively. Wolff was able to confine his attention to these transitions even for the indirect gap material, germanium, since he could argue that because the conduction band at k = 0 was only 0.14eV above the conduction band minimum it would be substantially populated at avalanche breakdown field strengths where the average carrier energy is a few tenths of an electron 33

P. A. Wolff, J . Phys. Chem. Solids 16, 184 (1960).

310

A . G. CHYNOWETH

-

k=, -

(111) Direction

Crystal momentum

-

Direction

FIG.19. Diagram illustrating the various types of direct radiative transitions that can occur in Ge in avalanche breakdown. (After W ~ l f f . ~ ~ )

volt. In addition, the curvatures of the energy bands near other minima in the conduction band make optical transitions less likely than in the higher curvature region around k = 0. Wolff notes that type I11 transitions should enable the production of photon energies of up to (2q E ~ ) which , is considerably greater than the highest observed energies. His calculations show, however, that as one proceeds away from the point k = 0 in the Brillouin zone the optical transition between the conduction and split-off valence bands becomes forbidden because, for energies comparable to or larger than the band gap, the transformation which diagonalizes the Hamiltonian (which then contains a large k . p term) tends also to diagonalize the momentum operator. As a result, the optical matrix element for process I11 is sufficiently reduced that it should not be observable. From simple arguments, the maximum possible photon energy kv, for the intraband process is given by

+

4.

CHARGE MULTIPLICATION PHENOMENA

311

where m,*, m,* are the masses of the light and heavy holes, respectively. This predicts energy thresholds which are in good agreement with the lower thresholds found in the emission from germanium and silicon junctions. The maximum photon energy possible in process I1 is given by

(1 10a) where me* is the effective mass in the conduction band at k = 0. This likewise predicts threshold energies that are in good agreement with those observed in Ge and Si. For the spectra emitted by processes I and I1 Wolff derives the following ,expressions :

and

where m is the free electron mass,f, andf, are the distribution functions for hot holes in the split-off valence band and the unperturbed valence band, respectively, and f, is the distribution function for electrons In the conduction band. Choosing an approximate analytical form for the distribution functions, Wolff was able to obtain a reasonably good fit to the measured spectrum of light emitted by germanium in avalanche breakdown (Fig. 20). Two parameters were involved in the fitting process: the electron temperature and the ratio of the interband to intraband transition rates. A best fit was obtained for an electron temperature of 0.25eV, which is a reasonable value. The intensity ratio of the two types of transition depends (because one is a unimolecular and the other a bimolecular process) on the density of carriers in the avalanche region. The needed density turned out to be lo" cm-3, which is not too unreasonable for the carrier concentration present in microplasmas. Wolff's theory has also been applied to spectral data from Si junctions, and again a good fit has been obtained to the total spectrum by invoking both interband and intraband processes.

312

A. G . CHYNOWETH

“ t

101 1.0

1

1.2

I

I

I

1.4 1.6 1.8 2.0 Photon energy ( e V )

I

I

2.2

2.4

FIG.20. Comparison between experimental and theoretical curves for the radiation spectrum from germanium p-n junctions in avalanche breakdown. (After W ~ l f f . ~ ~ )

T o date, the most extensive work on the light emission from avalanche breakdown in group 111-Vmaterials has been reported for GaAs by Michel et d 3 ’As we noted earlier, in connection with Fig. 18, they find a broad emission spectrum, qualitatively similar to that in Ge in that it shows a hump at lower energies. The authors conclude that the high energy tail must arise from recombinations between energetic carriers in the high field region. A new feature of the spectrum, not previously reported in other materials, is the occurrence of some structure in the neighborhood of the band gap energy. This structure appears identical in energy and temperature variation with the emission spectrum obtained in forward-biased junctions where the radiative processes certainly involve impurity states. They are forced to conclude that, in avalanche breakdown, some hot carriers diffuse out of the high field region and enter the end regions as minority carriers. Here they first thermalize and then recombine via impurity states. The spectrum of light emitted from avalanche breakdown in G a P junctions has been measured by Gershenzon and M i k ~ 1 y a k .Again j~ a broad spectrum was observed extending from energies considerably below the energy gap to energies well above it. A shoulder in the emission curve in the neighborhood of the band gap was attributed to a change in the internal absorption from

4. CHARGE MULTIPLICATION PHENOMENA

313

indirect transitions (at lower energies) to direct transitions at k = 0 (at higher energies). The authors quote a total efficiency of photons per carrier crossing the junction. While recombination and intraband transitions seem capable of accounting for the observed spectra, Figielski and T o r ~ have n ~ suggested ~ an alternative explanation for at least the high energy end of the spectrum, namely, bremsstrahlung of hot carriers in the Coulomb field of charged centers. They derive the following expression for the emitted spectrum :

where K is the optical dielectric constant, K* is an effective dielectric constant defined by a = e2x*m*r2 where a is the electron acceleration in its hyperbolic orbit at distance r from the charged impurity center, and ell, dv is the energy radiated in the frequency range v to v dv per unit time by all the electrons. The simple expression on the right-hand side can be made to fit the experimental data for silicon quite well using an electron temperature T, of about 4500"K, which is a reasonable value. Assuming that the breakdown current flow is concentrated into microplasmas (see below), Figielski and Torun derive the following expression for the emission efficiency :

+

nid q = B --, 11

where

ni is the density of charged centers in the microplasmas, d is the length of the ionizing region, and u is the saturation drift velocity. With reasonable choices for the various quantities involved, the efficiency is predicted to lie between 3 x and 5 x lo-* photons in the visible range per electron crossing the junction. This value compares well with that found experimentally2' for the visible emission efficiency, 7 x lop9. 15. MICROPLASMAS

When the reverse bias applied to a p-n junction is in the neighborhood of the onset of avalanche breakdown, the current passing through the junction often exhibits a characteristic form of noise arising from the fact that the current is rapidly switching back and forth, in a random manner, between two 34

T. Figielski and A. Torun, Proc. Intern. Con$ Semicond. Phys., Exeter, 1962 p. 863. Inst. of Phys. and Phys. SOC., London, 1962.

314

A . G. CHYNOWETH

distinct levels. This noise was first observed in silicon junction^'^ but was subsequently observed in other materials as well, including g e r m a n i ~ r n , ~ GaAs,26and GaP.j6 In the work on silicon it was also shown that, in general, when a junction is in avalanche breakdown, the current is not flowing uniformly through the junction area but rather is concentrated in a number of highly localized sites scattered over the junction area.“ I f the junction is sufficiently close to the crystal surface these spots can be seen on account of the recombination radiation generated at them. In a series of studies on silicon junctions37it was established that, as the avalanche current increases, the number of light spots increases in proportion while their individual intensities were essentially unaltered. Furthermore, it was frequently observed that the formation of each new spot was accompanied by generation of the characteristic current noise mentioned above. That essentially all the current was conducted by way of these spots was indicated by the fact that dividing the total current by the number of spots gave an average current per spot of the order of 100 PA, and this was about equal to the current amplitude of the noise, The term m i c r o p l a ~ m awas ~ ~ coined for these local breakdown spots. A number of authors have studied the behavior of r n i c r o p l a s m a ~ . ’ ~ ~ ~ ~ , ~ ~ As the average direct current through the junction is slowly increased, the first sign of microplasma noise is a few narrow rectangular pulses of current of amplitude of roughly 100 PA, randomly distributed in time. As the average current is increased those pulses become both longer and more numerous until the point is reached where the microplasma current is on as much as it is off. Further increase in the current achieves the situation where the microplasma current is on for a greater fraction of the time than it is off, and an oscilloscope display gives the impression of “negative” pulses of current becoming steadily fewer and more widely separated as the net current is increased. Eventually, the current becomes essentially quiet again, corresponding to the condition where the microplasma is on all the time. As the T. Tokuyama, Japan. J . Appl. Phys. 1, 324 (1962). M. Kikuchi and T. Izeika, J . Phys. SOC.Japan 15,935 (1960). 3 7 A. G. Chynoweth and K. G. McKay, J . Appl. Phys. 30. 1811 (1959). 38 D. J. Rose, Phys. Rev. 105,413 (1957). 39 K. S. Champlin, J . Appl. Phys. 30, 1039 (1959). 40 M. Kikuchi and K. Tachikawa, J. Phys. SOC.Japan 15,835 (1960); M. Kikuchi, ihid 15, 1822 (1960). R. J. McIntyre, J . Appl. Phys. 32, 983 (1961). 42 R. H. Haitz, A. Goetzberger, R. M. Scarlett, and W. Shockley, J . Appl. Phys. 34,1581 (1963). 43 R. H. Haitz, private communication. 44 K. Maeda and K. Suzuki, Japan. J. Appl. Phys. 1, 193 (1962). 4 5 I. Ruge and R. Conradt, Z. Naturforsch. Ma, 1016 (1963). 35

36

4.

CHARGE MULTIPLICATION PHENOMENA

315

current is increased further, the whole cycle can be repeated as another microplasma is formed. Very often, the sets of noise for different microplasmas show considerable overlap, i.e., for a given average current there may be several microplasmas present at various stages of “noisiness,” so that an oscilloscope display of the current shows a noisy staircase effect. The factors affecting the turning on and turning off of the microplasmas in silicon junctions have also been s t ~ d i e d . ~ ~ *I t ~has ’ *been ~ ~ found - ~ ~ that under steady ambient conditions both the turn-on and turn-off are random. On the other hand, the probability of turn-on increases with temperature , ~ ~ within the vicinity of the or when light, a-, p-, or y - p a r t i ~ l e s ’ ~strike microplasma. The turn-off probability is not affected by these external agents. Multiplication measurements have been made at microplasma sites by using fine light probes for injecting the carriers.’6940342 It is found that the multiplication is considerably greater at a microplasma than in adjacent areas of the junction free from microplasmas. Microplasmas clearly indicate local preferred sites for breakdown, and in studies in silicon junctions it has been shown that dislocations can promote the occurrence of micro plasma^.^^ Conversely, when dislocation-free material is used and care about chemical cleanliness is taken during the junction fabrication, it is possible to obtain junctions that do not exhibit microplasma spots and the usual noise. These have been called uniform

junction^.'^'^^'^^ It is significant that microplasma phenomena are very similar in all materials where they have been observed. The light spots have been observed in all these materials and the current noise amplitude is generally in the range 25 to LOOPA. The generality of this phenomenon, in widely different materials, indicates that the cause of a microplasma is probably similar in each case and that it is most likely an electrostatic or space charge effect. In fact, the picture that is gradually emerging is that a microplasnia occurs at a spot in the junction where the space charge width is somewhat narrower than elsewhere due to a local statistical fluctuation in the impurity concentration and that there exist also traps which can accept charge of appropriate sign. Furthermore, the state of charge of the traps can fluctuate depending on whether or not carriers are trapped there. This model stems from original proposals by Shockley,’ and, for more detailed information, the reader is referred to the extensive literature on this subject that he and his colleagues have built up. O 4h 47

48 49

50

1. Ruge and G. Keil, Rev. Sci. Instr. 34. 390 (1964). A. G. Chynoweth and G. L. Pearson, J. Appl. Phys. 29, 1103 (1958). R. L. Batdorf, A. G. Chynoweth, G. C. Dacey, and P. W. Foy, J . Appl. Phys. 31, 1153 (1960). A. Goetzberger, B. McDonald, R. H. Haitz, and R. M. Scarlett, J . Appl. Phys. 34, 1591 (1963). R. H. Haitz, J. Appl. Phys. 35, 1370 (1964).

316

A. G . CHYNOWETH

Consider by way of an illustrative example an n'p step junction so that the space charge region is populated by ionized acceptors (i.e., negatively charged centers). Suppose also that a number of deep trap levels exist in the space charge region which, say, become negatively charged when electrons are trapped on them. Electrons can reach these levels by being trapped from the conduction band though this is highly improbable in the high field region of the junction. More likely, electrons reach these trap levels by thermal or other excitation from the valence band, thereby releasing a hole which is immediately swept out by the field. For similar reasons to those just given it is unlikely that this trapped electron will recombine with a hole in the high field region but, more likely, will eventually leave the trap by thermal or other excitation up to the conduction band. The trap is therefore functioning purely as a charge generation center. Additional mechanisms that can occur because of the high field are that trapped electrons can be impact ionized by hot electrons or again electrons can be excited from the valence band to empty traps by the same means. As the net state of charge in these traps fluctuates, so does the width of the space charge region; the greater the density of trapped electrons in our example the narrower the space charge width and hence the lower the breakdown voltage (see Fig. 21). Because the number of charges determining the local space charge width is not all that large (lo3 to lo4, typically) a fluctuation of only a few electron charges would suffice to trigger on a local avalanche breakdown if the applied bias is very slightly below the average breakdown voltage. Having started a local avalanche, it will continue until a

x c r )

FIG.21. Diagram illustrating the variation in the local field profile in a p-n junction brought about by space charge inhomogeneities and trapping effects : (A) the average field distribution in the junction with traps empty; (B) the field distribution in those parts of the junction which are locally narrower due to locally increased concentrations of fixed charge; ( C )the field profile brought about in B regions by charge trapping of the appropriate sign.

4.CHARGE MULTIPLICATION PHENOMENA

317

fluctuation of the amount of trapped charge in the opposite direction (brought about by impact ionization or thermal excitation) causes the breakdown voltage to increase again5’” While detailed corroboration of this model is still needed, it appears to be qualitatively consistent with many of the observed properties of microplasmas. In particular, it fits the known turn-on and turn-off properties of the microplasma, and, specifically, it is consistent with the fact that microplasmas can be started by thermal excitation44 (the activation energy for this is consistent with the postulate of deep traps) and by light. In connection with the latter it is of interest to note that it has been demonstrated in some very pretty experiments that the light generated when one microplasma turns on can cause others at some distance away to switch on by photon coupling along the space charge region (which probably acts as a light ~aveguide).~~.~~ With the reasonable assumption that these traps will be most effective in triggering a microplasma where the space charge width is locally narrower (due to statistical fluctuations in the acceptor impurity concentration) the current saturation in the microplasma and its relative invariance from junction to junction and in different materials can be understood. This follows from considerations by Shockley of the effects of fluctuations in the local impurity concentration on the breakdown voltage of a junction.’ Shockley divides up the space charge region of a junction into an array of cubes of average cube dimension equal to the junction width at breakdown WB. If the average impurity concentration is N , then the average cube contains N WB3ions. The average fluctuation in this quantity from cube to cube The average breakdown field in a step will be approximately (NwB3)1’2. junction is E , = 2VB/wB. Also, WB2 ‘v rcVB/2zeN,where K is the dielectric constant. Hence

Also, V,

-

’

N - since E B varies only slowly with N ; it follows that

where 6 N is the root mean square fluctuation in the local impurity concentration. Hence,

50aR, H. Haitz, Phys. Rev. 138, A260 (1965).

318

A. G. CHYNOWETH

and therefore, SVB = (E,7Ce/U)"2,

(1 19)

implying that the fluctuation SV, is relatively constant and independent of impurity concentration and crystal material, since E , is of roughly the same order of magnitude, lo5 volt cm-', over a wide variety of junction widths 1 volt. and materials. Substituting this value for E , yields SV, This simple model has led to the conclusion that if the junction area is broken up into an array of squares of area WB2,the average fluctuation in the breakdown voltage of these squares about the mean is about 1 volt. Thus, typically, the first spots to break down will do so when the bias reaches (V, - 6VB). As this spot draws current, a local series resistance R, will set up which will limit the ultimate value of the current passing through the spot to I SVJR,. This series resistance will include the spreading resistance from the breakdown spot into the end regions of the crystal, but Shockley has shown that a much more important contribution to this resistance stems from the current itself. As described earlier in this article, multiplication studies have shown that for analysis purposes it is adequate to consider the breakdown as being confined to an effective junction width roughly equal to a third of the total junction width. Let this be the middle third of the junction. Then, because of the avalanche process, an electron space charge will build up in the junction third lying on the n-side, and a hole space charge will build up in the junction third lying on the p-side of the space charge region. The carriers drift through most of the junction at their saturation velocity uD, given by solving the energy balance relation

-

-

~EUD = ER(Ur/&)

(120)

and the momentum conservation relation

That is, uD = ( ~ ~ / m * = ) ' 2/ ~x lo7cm sec-' for most materials. The magnitude of the hole space charge on the p-side or the electron space charge on the n-side will be

where the carrier density n is given by I = WB2nevD

4.

CHARGE MULTIPLICATION PHENOMENA

319

Hence

These positive and negative space charges are equivalent to sheet charges of density Q/ WB2.These sheet charges set up a space charge field

opposing the junction field. The maximum value that this field can reach is clearly given by about 6V/WB. We thus have an effective series resistance R, due to space charge : Rs

6V I

47c 3K

= -= -v

D

- lo4

ohms.

Thus, the current flowing through the microplasma should be approximately 6VR, which is 100 PA, in excellent agreement with what is observed. Moreover, the microplasma current is predicted by this model to be relatively insensitive to junction width and material, as observed.

16. AVERAGE ENERGYFOR ELECTRON-HOLE PAIR ENERGYPARTICLES

PRODUCTION BY

HIGH

An interesting side result of the studies of avalanche breakdown is an understanding of the experimentally measurable quantity, the mean energy required by a high energy particle (or photon) in order to produce an electronhole pair. This quantity can be determined in a semiconductor provided all the electron-hole pairs created by the primary particle are collected and their charge measured. Knowing the amount of charge released by the primary of known energy leads directly to the mean energy per pair produced by the primary. This energy has been measured for a number of materials by several authors using a variety of primary high energy excitations. It has been most thoroughly measured for silicon, for which the average value is about 3.6 eV per pair. Shockley’ has given a prescription for deriving the mean pair-production energy cp based on his model for ionization processes described earlier. A high energy particle loses nearly all its energy by ionization processes, giving rise first to high energy secondaries, then tertiaries, and so on, thus leading to a cascade of electrons. The incident energy is used up in three ways :(i) every pair-production collision consumes an energy ci ;(ii)in general, as we have seen, for every ionizing collision a hot electron suffers, an amount of energy rcRis converted to phonon energy before the ionization is produced ;

320

A. G. CHYNOWETH

(iii) after a carrier falls to an energy E~ less than ci, as a result of an ionizing collision, it dissipates this energy to phonons. Thus, on the average, Now the residual kinetic energy E~ can lie anywhere between 0 and E, .Crudely then, the average value of Ef could be taken to be 4 2 . For parabolic energy surfaces, however, it can be shown that Ef should be 0.68,. Thus,

The quantity r, the ratio of the mean free paths for ionizing and phonon collisions, is known from analysis (by Shockley's model) of the multiplication measurements. This yields estimates for E, in good agreement with those found experimentally. The simple cascade model raises a statistical problem in regard to Ef that cannot be solved exactly without further information on how the residual energy E - E~ is distributed among the carriers subsequent to pair production. This problem has been pursued in some detail by van Roosbroeck5' who concluded that the simple model is not seriously in error when compared with the more detailed treatment.

VII. Breakdown in Bulk Semiconductors 17. GENERAL DISCUSSION Nearly all experimental studies of breakdown to date have been confined to p-n junctions, as these offer by far the most practical means of achieving the high electric field strengths necessary for the effect. Indium antimonide is a notable exception ; in bulk crystal form it is found that fields of only a few hundred volts per cm will suffice to produce breakdown. It is therefore possible to create electron-hole plasmas over a sizable volume of crystal, and, hence, indium antimonide is an excellent vehicle for studies of effects in solid state plasmas, a topic which is covered by Ancker-Johnson in Volume 1. Here, we are concerned solely with the breakdown mechanism itself. The approach to breakdown in bulk crystals of indium antimonide is studied experimentally by measuring the current-field characteristic and the Hall effect of a crystal. Pulsed conductivity and Hall effect studies have been made by Kanais2and by Glicksman and Hi~inbothem.'~ Typical examples53 of the current and the Hall coefficient plotted versus electric field for n-type InSb at 77°K are shown in Fig. 22. The shape of the current characteristic 52

W. van Roosbroeck, private communication. Y. Kanai, J . Phys. SOC.Japan 14, 1302 (1959).

53

M. Glicksman and W. A. Hicinbothem, Phys. Rev. 129, 1572 (1963).

4.

CHARGE MULTIPLICATION PHENOMENA

321

E (V/cm)

FIG.22. The current density and the Hall coefficient at 77°K of a bar of InSb as a function of the electric field, in a magnetic field of 260 gauss. (After Glicksman and H i c i n b ~ t h e m . ~ ~ )

may be interpreted as follows : At low fields, the crystal behaves in an ohmic manner and the current varies linearly with field. When the field exceeds about 20voltcm-' the current begins to rise less rapidly, indicative of carrier heating effects with a consequent drop in mobility. A t fields somewhat greater than 100 volt cm- ', the current rises rapidly denoting breakdown. That carrier multiplication is taking place in this high field range is evident from the accompanying drop in the Hall coefficient. It should be pointed out that once the avalanche breakdown currents exceed a few amperes, the pinch effect can occur (provided there are no stabilizing magnetic fields present), and thus carrier concentrations and drift velocities deduced from conductivity and Hall measurements in this range must be suspect. Even at somewhat lower currents, the measurements may sometimes be complicated by the occurrence of plasma instabilities, for example, the oscillistor effect.

322

A . G . CHYNOWETH

The general shape of the current-field characteristic is usually interpreted in terms of Stratton’s theory3 of breakdown in polar crystals, reviewed earlier. As the field is increased, the electron temperature rises, and, consequently, the electrons interact more and more strongly with the polar modes of the lattice with the result that their mobility drops. This accounts for the departures from Ohm’s law at moderate field strengths. At sufficiently high fields, the electrons are gaining energy from the field faster than they can lose it to the lattice, and this runaway condition leads to breakdown. Stratton’s theory in the case where kT, 4 kT 4 ho (not strictly applicable to most experiments) indicates that this runaway condition will occur when the drift energy of the electrons becomes comparable with the phonon energy. In InSb this will occur at drift velocities approaching 10’ cm sec-‘. In Kanai’s experiments5’ the maximum velocities achieved were 2-3 x lo7 cm sec-’ and in Glicksman and Hicinbothem’ss3 velocities up to 9 x lO’sec-’ were deduced though the cautions noted above might be relevant to both of these results. Nevertheless, these velocities (and the fields required to generate them) are not inconsistent with the predictions of Stratton’s theory. Kanai also studied breakdown in a variety of n-type samples of varying electron concentrations and low-field mobilities. It is significant that the greater the low-field mobility of the sample, the lower the breakdown field strength, again consistent with the critical drift velocity criterion. At the highest fields used, all the experiments show a drop-off of the velocity, the velocity actually decreasing as the field increases. It is not clear whether this is a reliable observation since the current levels at which these experiments were performed should be sufficient to start pinching. If the drop in velocity is real, however, it suggests the onset of new mechanisms for the dissipation of energy by the electrons. One intriguing possibility is that twostream instabilities in the electron-hole plasma are being generated.53 It is interesting to note that the fields present at avalanche breakdown in InSb p-n junctions can be two orders of magnitude or more higher than those required for bulk breakdown.54 The most likely cause of this difference is that a fundamental requirement for Stratton’s theory, that electron-electron collisions are sufficiently frequent to randomize the velocity distribution, does not hold in the narrow, high field region of a junction. For current densities of the order of 1 A cm-’ and assuming that the electrons travel through the junction at about their saturation velocity 10’cm sec- *,the resulting ~ . estimates that carrier concentration is of the order of 10” ~ m - Stratton the density should be at least of the order of 10’4cm-3 for his theory to apply to InSb, and so it is reasonable to approach breakdown in this material from the Baraff point of view.

-

54

M. T. Minamoto and C . M. Allen, Solid State Electron. 5, 263 (1962).

4.

CHARGE MULTIPLJCATION PHENOMENA

323

VIII. Impact Ionization of Impurities 18. GENERAL DISCUSSION

Up to this point, this chapter has been concerned with impact ionization across the energy gap with the consequent formation of electron-hole pairs. Another important class of ionizations that can occur is the impact ionization of impurity levels. In these processes, an energetic carrier can raise an electron from an impurity level to the conduction band or an electron from the valence band to an impurity level. In either event, the net result is one additional free carrier and one immobile charge site. Thus, impact ionization of impurities leads to two important effects: (a) the current is increased, thereby exhibiting a breakdown type of behavior, and (b) the space charge field of the charged impurity centers will distort the field distribution in the crystal. The conditions necessary for the onset of breakdown have received considerable attention in connection with low temperature breakdown in silicon and germanium ; the implications and consequences of the ensuing field distortion have been largely ignored. One of the earliest observations of this form of breakdown was reported by Sclar and B ~ r s t e i nfor ~ ~germanium at 4.2"K. Subsequently, Koenig et u1.56,57formulated the behavior of the free electron concentration using a conventional rate equation approach: The rate processes that need to be considered are thermal ionization AT and impact ionization A , of the neutral donors by energetic electrons, and the corresponding inverse processes, single electron recombination BT and Auger recombination B,. If n is the carrier density and the donor and acceptor concentrations are ND and NA, the rate equation becomes

dn dt

- = AT(T)(ND - N A -

n) + nA,(f)(N, - NA - n )

where ?: f indicate whether the coefficients are functions of the lattice temperature T or the electron temperature represented by the distribution function f. In practice, n < NA. (ND- NA).Also, for small n, the Auger recombination term in n2 may be ignored. In the steady state, electrons are being ionized as fast as they recombine, i.e., dnldt = 0. The solution to this equation is then

55 56

''

N. Sclar and E. Burstein, J . Phys. Chem. Solids 2, 1 (1957). S. H. Koenig. J . Phys. Chenz. Solids 8, 227 (1959). S . H. Koenig, R. D. Brown, and W. Schillinger, Phys. Reo. 128, 1668 (1962).

324 Clearly n

A . G . CHYNOWETH

-+

cc and breakdown occurs when BTNA=

- NA)

or &/A, = c - 1 ,

where C is the comparison factor N,/NA. Now we can expect A , to increase as the field increases and B T to decrease. Thus, the left-hand side of this expression decreases as the field increases which leads to the prediction that the breakdown field will increase as the compensation factor decreases. Such a variation has been observed in low temperature breakdown studies in germanium,57 but a similarly comprehensive study has not yet been reported for a group 111-Vcompound. Low temperature breakdown due to impact ionization of impurities has been observed in GaAs by Oliver.58 Current-field characteristics are shown in Fig. 23 as obtained on a sample of relatively pure n-type GaAs at various low temperatures. At 4"K, the current-voltage curve is linear for small fields, but at fields in the vicinity of 24 volt cm- the current increases rapidly by a factor of 100. For fields greater than this the rate of rise of the current drops. The results for the highest temperature (26°K) show no steep rise in the current but a steady increase in conductivity over the whole range of applied fields above about 1 volt cm- '. For intermediate temperatures the curves exhibit a gradual transition between these two extremes. At the highest fields all the curves merge so that the conductivity is then independent of the lattice temperature. Resistivity and Hall measurements showed that, at low fields, carriers were freezing out on shallow donors as the temperature was reduced, and, in this regime, conduction occurs by phonon-assisted hopping processes in the impurity band. As the field is increased, the first effect is a reduction in the recombination rate (BT),thereby causing the current to increase superlinearly, but ultimately, impact ionization sets in resulting in the steep rise in the current. At sufficiently high fields, all of the impurity levels are empty, and the continuation of the current-voltage characteristic reflects then the field dependence of the mobility of the warm electrons. Theoretical studies of impact ionization and low temperature breakdown concern calculations, whenever possible, of the generation and recombination coefficients. This in turn requires a knowledge of the carrier distribution function f. There are numerous theories of the distribution functions of warm electrons (i.e., electrons whose temperature is slightly above the lattice temperature but not nearly so hot as to lead to electron-hole pair production),

'

58

D. J. Oliver, Proc. Intern. Conf: Semicond. Phys., Exeter, I962 p. 133. Inst. of Phys. and Phys. SOC.,London, 1962; Phys. Reo. 127,1045 (1962).

4.

CHARGE MULTIPLICATION PHENOMENA

325

10

I1

NI

E

-

a '0 ) .

c .-

v)

C

U W

.C L W

5 10V

10-

10 -

I

I

1.0

10

I00

Electric field ( V cm-'1

FIG.23. Current density versus electric field in GaAs at low temperatures illustrating the occurrence of impact ionization. (After Oliver.58)

one of the best known being that by S t r a t t ~ n referred ,~ to earlier. Similar theoretical studies have been made by YamashitaS9 who, in particular, has applied his results to the low temperature breakdown problem. Theoretical calculations of Auger recombination at impurity levels have been pursued by Landsberg and colleagues,6o though this work is not strictly applicable since it is primarily concerned with carriers in thermal' equilibrium with the lattice. These various theoretical studies fall somewhat outside the scope of this chapter and the reader is referred to the literature or to the chapter on the transport properties of warm electrons for further information. The author greatly appreciates having had the benefit of many useful discussions with Dr. G. A. Baraff and he is also indebted to him for his reading of the manuscript. 59

6o

J. Yamashita, J . Phys. Soc. Japan 16, 720 (1961). D. A. Evans and P. T. Landsberg, Solid State Electron. 6, 169 (1963); P. T. Landsberg, IXA. Evans, and C . Rhys-Roberts, Proc. Phys. SOC.(London)83,325 (1964).

This Page Intentionally Left Blank

CHAPTER 5

The Effects of Hydrostatic Pressure on the Properties of 111-V Semiconductors Robert W . Keyes INTRODUCTION . . . . . . . . . . . . . EFFECTOF PRESSURE ON ENERGY BANDS . . . . . OPTICAL ABSORPTION SPECTRUM . . . . . . . . ELECTROLUMINESCENCE . . . . . . . . . . ELECTRICAL CONDUCTIVITY . . . . . . . . . OTHERELECTRICAL PROPERTIES . . . . . . . . 1. Piezoresistance . . . . . . . . . . . . 2. Hall Effect and Thermoelectric Power of GaSb . . . 3. Microwaae Oscillations . . . . . . . . . VII. PHASETRANSITIONS. . . . . . . . . . . VIII. ELASTICPROPERTIES . . . . . . . . . . . I. 11. 111. IV. V. VI .

. . . .

. . . .

. . . .

. . . . . . .

. . . . . . .

. . . . . . .

321 328 329 33 1 332 336 336 337 338 338 341

I. Introduction Many years ago P. W. Bridgman inaugurated the study of the electrical properties of semiconductors under high hydrostatic pressures. Bridgman’s work pre-dated the development of modern semiconductor technology and science, and only a few semiconducting materials of relatively low purity were available to him. Nevertheless, significant results were achieved, and the high pressure laboratory of Harvard University was established as a leader in the investigation of semiconductors at high pressure. This leadership has been maintained by the work of W. Paul and his students. The more recent era of highly developed semiconductor technology and science has made a large number of semiconducting materials available to the physicist and has made possible intensive study of the effect of pressure on the properties of semiconductors. Many additional high pressure laboratories have come into existence and made notable contributions to this field of study. Two laboratories have stood out, however. That of Harvard University, as already mentioned, has produced a continuing series of important results under the leadership of Professor Paul. Professor Drickamer’s laboratory at the University of Illinois has extended the range of a variety of physical measurements to nearly hydrostatic pressures of a few hundred thousand kg/crn2. The high pressure investigations have played an important

327

328

ROBERT W. KEYES

role in the growth of semiconductor science, and, in particular, in the formulation of models of the band structures of the 111-V semiconducting compounds. General features of the picture that has emerged will be described in the next part. Succeeding parts will describe the changes in various electronic properties which result from the changes in the energy bands with pressure. Finally, the phase transitions which have been found in many diamond and zinc-blende semiconductors at very high pressures will be described and the last part will treat elastic properties.

11. Effect of Pressure on Energy Bands Many measurements of the changes of the electronic properties of the 111-V semiconductors with pressure have been carried out since these compounds became prominent in the semiconductor field in the early 1950’s. It has been found that the pressure variation of electronic properties can be understood as straightforward consequences of the change in the electronic band structure of the compounds with pressure. The changes in electronic band structure have, of course, been derived by interpretation of the pressure dependence of electronic properties. The picture of the band structures of the 111-V semiconductors and the way in which they are affected by pressure which will be used in the present work has been largely derived by W. Paul and H. Drickamer and their collaborators from the work in their laboratories at Harvard University and the University of I l l i n ~ i s . ’ ~ ~ The maximum energy of the valence band of the 111-V semiconductors occurs at the point [OOO] of k space.4 This maximum has a complex structure which exhibits large changes when the cubic symmetry of the crystal is destroyed by shear strains. The complex structure need not concern us here, however, since hydrostatic pressure does not destroy the cubic symmetry, and experiments have shown that pressure produces very little effect on the properties of the valence band. The lowest energy of the conduction band may occur at three different points in the k space.4 These are (apart from a factor with dimensions of inverse length) : [OOO], the same point as the maximum of the valence band. [OOl], apoint on thefourfoldaxis.Therearesixequivalent pointsofthis type. [ l l l l , a point on the threefold axis at the Brillouin zone boundary. There are four equivalent points of this type.

’ A. L. Edwards, T. E. Slykhouse, and H. G. Drickamer, J . Phys. Chem. Solids 11, 140 (1959). W. Paul, J . Appl. Phys. 32,2082 (1961). W. Paul and D. M. Warschauer, in “Solids Under Pressure” (W. Paul and D. M. Warschauer, eds.), p. 179. McGraw-Hill, New York, 1963. For a review of the band structures of the Ill-V compounds, see H. Ehrenreich, J . Appl. Phgs. 32, 2155 (1961).

5 . EFFECTS OF HYDROSTATIC PRESSURE ON 111-vSEMICONDUCTORS

329

The energy differences between the various conduction band extrema are frequently small in 111-Vsemiconductors, so that the effects of extrema other than the lowest may appear in physical phenomena. Many of the most striking high pressure effects are produced by proximity in energy of two kinds of band extrema. Work of the laboratories of Paul and Drickamer has shown that most of the effects of pressure on the electronic properties of the 111-Vsemiconductors can be explained by a model in which pressure changes the energies of the three types of conduction band extrema relative to the valence band and to one another without affecting the nature of the extremum.'--3 The same TABLE I PRESSURE DEPENDENCE OF THE ENERGY GAP BETWEEN THE VALENCEBAND A N D THE VARIOUS CONDUCTION BANDMINIMA'

Conduction band minimum [0001 [1111 [loo1

dE,ldP in 10- 6eV/(kg/cm2)

+ 12 + 5 - 1.5

After Zallen."

work has further established that the rate of change of the energies of the various extrema is about the same for all of the diamond and zinc-blende type semiconductors. The rates of change of the energy of the three types of conduction band minima with respect to the valence band are shown in Table I. The effects of pressure on the electronic properties of all of the 111-V semiconductors can be almost completely understood in terms of the simple model which is quantitatively characterized by Table I. 111. Optical Absorption Spectrum

The most important feature of the optical absorption spectrum of a semiconductor, and the only one which has been very extensively studied, is the intrinsic absorption edge, the energy at which absorption of light by excitation of electrons from the valence band to the conduction band sets in. The absorption constant increases very rapidly with increasing photon energy in the vicinity of the energy gap of the semiconductor. In the simplest model case, the absorption spectrum shifts in energy with the gap as the pressure is varied. Then the change in absorption spectrum with pressure can be

330

ROBERT W . KEYES

immediately inferred from the facts presented in the foregoing section if the location of the conduction band minimum in k space is known. The simplest behavior, a shift of the absorption spectrum with pressure without change in shape, is not always found, h ~ w e v e r . ’ , ~If , the ~ shape changes, more complex and less general interpretations, which have been discussed at some length for the example of germanium, are r e q ~ i r e d . ~ , ~ ~ ~ These more complicated cases usually result from the participation of more than one conduction band minimum in the absorption process. Nevertheless, a useful way to summarize measurements of the effect of pressure on optical absorption is to plot the photon energy at which the absorption constant has a fixed value versus the pressure. A plot of the photon energy at which the absorption constant of InP is 30 cm- as a function of pressure is given in Fig. 1, for example. It closely resembles a plot of the energy gap in InP against pressure. The minimum

’

1.3 -

20

0

40

60

I

0

P (lo3 kg/cm2) FIG.1. The pressure dependence of the absorption edge of InP, defined as the energy at which the absorption constant is 30 cm- (After Edwards and Drickamer.’) At low pressures the [OOO] conduction band minimum is lowest and at high pressures the [OOI] is lowest.

’.

energy of the conduction band is at the [OOO] point of k space at low pressure. The energy gap increases with increasing pressure, as explained in Part 11. Since the energy of the [loo] minimum decreases with pressure, a certain H. Y. Fan, M. L. Shepherd, and W. G. Spitzer, in “Photoconductivity Conference” (R. G. Breckenridge, B. R. Russell, and E. E. Hahn, eds.). Wiley, New York, 1956. D. M. Warschauer and W. Paul, quoted by Professor Brooks in “Photoconductivity Conference” (R. G . Breckenridge, B. R. Russell, and E. E. Hahn, eds.), p. 201. Wiley, New York, 1956.

’ W. Paul and D. M. Warschauer, J . Phys. Chem. Solids 5, 89 (1958). L. J. Neuringer, Phys. Rev. 113, 1495 (1959).

5.

EFFECTS OF HYDROSTATIC PRESSURE ON 111-V SEMICONDUCTORS

331

pressure exists at which the [loo] minimum becomes the lower one. The energy gap decreases with increasing pressure above this transition pressure. Of course, the shape of the absorption spectrum varies in a complicated way in the transition region. The use of plots such as that of Fig. 1 to investigate the changes of energy gaps to very high pressures has been extensively exploited by Edwards, Drickamer, and Slykhouse. 1,9 In addition to InP, the pressure dependence of the optical energy gaps of A1Sb,9 GaP,',''"' GaAs,' GaSb,'393'2InSb,I2" and InAs'','2 has been measured. The measurements of Zallen and Paul on GaP are of especial interest in that several features of the optical absorption and reflectivity spectra which are not close to the gap in energy were also studied."." These other features apparently represent transitions involving band edges other than those which bound the gap. The study of many such transitions illustrates a potential for determining the pressure variation of a large part of the energy band structure rather than just that part which is close to the gap.

IV. Electroluminescence The effects of pressure on the electroluminescent spectrum and on the optical absorption spectrum are closely related. The recombination of an electron and a hole, the process which produces luminescence, is the inverse of the absorption process, in which a photon creates an electron and a hole. Photons resulting from recombination of an electron in the conduction band with a hole in the valence band have an energy close to the gap energy. This radiation near the gap energy constitutes the main emission in 111-V semiconductors with [OOO] conduction band minima, and can be very efficient. Injection lasers operating in the main line can be made by passing large currents through a p-n junction incorporated in a Fabry-Perot laser structure. The discovery of efficient p-n junction electroluminescence and of injection lasing in gallium arsenide stimulated several investigations of the effect of pressure on electroluminescence.lo~l 3,14 It has been found, as anticipated, that the energy of the main line shifts in the same way as the gap energy with pressure in GaAs. The emission of GaAs lasers shifts at the same average rate as the peak of the main line.'3,'4 However, since the laser emission is

' A. L. Edwards and H. G. Drickamer, Phys. Rev. 122, 1149 (1961). R. Zallen and W. Paul, Phys. Rev. 134, A1628 (1964).

' R. Zallen, Thesis, Harvard University, 1964 (unpublished). '' J. H. Taylor, Bull. Am. Phys. Soe. 3, 121 (1958). C. Bradley and H. A. Gebbie, Phys. Letters 16, 109 (1965). J. Feinleib, S. Groves, W. Paul, and R. Zallen, Phys. Rev. 131, 2070 (1963). G. E. Fenner, J . Appl. Phys. 34,2955 (1963).

l Z T . l3

l4

332

ROBERT W. KEYES

restricted to certain distinct modes, the shift occurs by jumping of the emission from mode to mode rather than continuously. The frequency of any given mode changes slowly with pressure because of changes of the dimensions of the laser and of the dielectric constant. Similarly, the “edge emission” of G a P has been found to vary in the same way as the energy gap with pressure.” The electroluminescent spectra of G a P diodes, and, also, of most GaAs diodes, contain lines of considerably less energy than the gap. These lines arise from recombination of a hole (electron) with an electron (hole) which has been trapped in some deep donor (acceptor) state. The dependence of the position of the low energy lines on pressure usually is weaker than the dependence of the gap on pressure. The weaker dependence is believed to be a result of the tendency of levels deep in the forbidden gap to maintain a fixed position with respect to the valence band when pressure is applied, and suggests that the low energy photons come from the recombination of a hole with a trapped e1ectr0n.I~ Electroluminescence spectra provide a very convenient way to study the pressure dependence of impurity levels in the gap, as the impurity transitions are frequently much more prominent in electroluminescence than in optical absorption.

V. Electrical Conductivity Electrical conductivity in the 111-V compounds at the temperatures at which high pressure experiments are ordinarily performed may be divided into two general types, intrinsic and extrinsic, which are conveniently discussed separately. At ordinary temperatures, 111-V semiconductors most often show extrinsic conductivity, in which the number of carriers is determined by the concentration of impurities and is independent of pressure and temperature. Effects of pressure on conductivity then arise from effects of pressure on mobility. The mobility of holes in 111-V semiconductors is essentially independent of pressure. An example is shown in Fig. 2. The mobility of electrons may be affected by pressure in two ways. If the [OOO] minimum of the conduction band is lowest, the effective mass of the electrons is changed by pressure because the effective mass is roughly proportional to the gap between the [OOO] minimum and the valence Effective mass and mobility are, however, closely related physical properties, low effective mass generally being associated with high mobility. Thus, since the [OOO] gap increases with pressure, the mobility and the conductivity of extrinsic n-type semiconductors with [OOO] conduction band l5

l6

D. Long, Phys. Rev. 99, 388 (1955). R. W. Keyes, Phys. Rev. 99,490 (1955). R. W. Keyes, Solid State Phys. 11, 149 (1960).

5.

EFFECTS OF HYDROSTATIC PRESSURE ON Ill-V SEMICONDUCTORS

333

20 10 -

0

K

5-

\

a

0

‘

I

10 20 PRESSURE (lo3 kg/cm2)

3

FIG.2. The pressure dependence of the extrinsic resistivity of some 111-V semiconductors. (Measurements of Howard and Paul.’) The large effects in n-type GaSb and GaAs are caused by near degeneracies of two kinds of conduction band minima.

decrease with increasing pressure. The magnitude of the effect on the conductivity is approximately inversely proportional to the energy gap, as is illustrated in Table II.” TABLE I1 THEPRESSURE DEPENDENCE OF THE MOBILITY I N THE [OOO] BAND OF 111-V SEMICONDUCTORS AND ITS RELATION TO THE ENERGY GAP’

~~~~~

InSb InAs GaAs InP

60 x 35 x

0.27 0.47

9.6 x 8 x 10-6

1.51

16 16 14

1.42

11

After Keyes. * ’

The second kind of dependence of electron mobility on pressure arises when two of the conduction band extrema are at nearly the same energy, so that both contain electrons. The relative position of the two bands changes with pressure and electrons are transferred from the one whose energy is increased to the one whose energy is decreased. Since the electron mobility is different in the two bands, the average electron mobility is changed by the transfer of electrons. In addition, when the two bands are near in energy the

334

ROBERT W. KEYES

states of both bands are available as final states for scattering of electrons in either band. As the bands shift with pressure the scattering rates change and the mobilities within each band are pressure dependent. Pressure dependence of n-type extrinsic conductivity caused by these interband effects is known in GaSb and in GaAs and its alloys with phosp h o r ~ ~ . ~ Examples , ' ~ , ' ~ of the large changes in resistivity produced by the interband effects are given in Fig. 2. Most of the charge carriers in intrinsic conductivity are present because of thermal excitation of carriers across the gap. Therefore the number of carriers changes rapidly if the gap is changed by pressure. The conductivity is proportional to the number of carriers and thus also changes rapidly with pressure. The conductivity may also be affected by changes in the mobility with pressure, as discussed above, but the changes in number dominate the effects of pressure on intrinsic conductivity in the 111-V compounds. An example is given in Fig. 3, which shows the effect of pressure on a specimen of p-type InSb. At 85°C and zero pressure the InSb is intrinsic.

Pressure

(lo3 hg/crnz)

FIG.3. The pressure dependence of the conductivity of lnSb in the intrinsic regime. (After Keyes.I6) Pressure increases the energy gap and decreases the number of intrinsic carriers.

l9

A. Sagar, Phys. Rev. 117,93 (1960). G. E. Fenner, Phys. Rev. 134, A1 13 (1964)

5.

EFFECTS OF HYDROSTATIC PRESSURE ON 111-V SEMICONDUCTORS

335

The energy gap increases and the number of intrinsic carriers and the conductivity decrease with increasing pressure. Above 6000 kg/cm2 the concentration of intrinsic carriers becomes so small thpt the extrinsic carriers start to dominate the Conductivity. The rapid decrease of conductivity begins to disappear and the conductivity approaches a pressure-independent extrinsic value at high pressures. The energy gap of most of the 111-V semiconductors is so large that intrinsic conductivity occurs only at high temperatures, temperatures beyond the range of ordinary high pressure apparatus. The pressure dependence of intrinsic conductivity has been studied only in InSb’5,’6720and InAs.21 The interpretation of the changes in intrinsic conductivity primarily due to changes in the number of carriers excited across the gap has also been confirmed by measurements of the pressure dependence of the Hall coefficient in these materials.20’21 The sample of Fig. 3 contains a negligible number of intrinsic carriers at 0°C above 4000 kg/cm2. The pressure independence of the conductivity at higher pressures further illustrates the small effect of pressure on p-type extrinsic conductivity. The pressure dependence of a third type of conductivity has also been studied in GaAs, namely, the conductivity due to carriers thermally excited from traps far from the band extremum.22 If the depth of the trap depends on pressure, the number of thermally excited electrons and the conductivity will depend on pressure. It was found that the energy difference between the trap level and the conduction band increases with pressure, so that the conductivity decreases with pressure. The effect is quite large ; the trap depth increases at about the same rate as the energy gap and the conductivity decreases by almost an order of magnitude on application of 6 kilobars at 300°K. The interpretation of the conductivity change as a change in electron concentration was confirmed by measurements of the pressure dependence of the Hall constant.22 Minomura and Drickamer measured the resistance of many 111-V semi~~ conductors at pressures up to several hundred thousand k g / ~ r n ’ .They found very large increases (two to five orders of magnitude) in the resistivity of InP and InAs. These large increases are probably similar to the increases studied by Sladek in GaAs, that is, they are probably due to increases in the binding energy of deep trapping centers. 2o

22

23

J. Gielessen and K. H. von Klitzing, 2. Physik 145, 151 (1956). J. H. Taylor, Phys. Rev. 100, 1593 (1955). R. J. Sladek, Bull. Am. Phys. SOC.9, 258 (1964): “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 545. Dunod, Paris and Academic Press, New York, 1965. S. Minomura and H. G. Drickamer, J . Phys. Chern. Solids 23,451 (1962).

336

ROBERT W. KEYES

VI. Other Electrical Properties 1. PIEZORESISTANCE Piezoresistance is the effect of stress on electrical resistance. It may be large in both p - and n-type semiconductors, but for different reasons. Shear stresses destroy the symmetry of the valence band and produce complex changes in the structure of the electronic energy levels which lead to large changes in the electrical conductivity. Hydrostatic pressure has little effect on the valence band and does not change the piezoresistance coefficients of p-type semiconductors a p p r e ~ i a b l yExamples .~~ are given in Fig. 4. 1

p- lnSb

[I

B

Y

0

65c

-

0

V

"

" - 4

50C n

0

-

O

1

Pressure

W

"

n

n

p -G e

-

0

I

(lo3kg/cm2)

FIG. 4. The pressure dependence of large shear piezoresistance coefficients of p-type InSb ~ ~ )curves show the independence of the properties and p-type Ge. (After Keyes and P ~ l l a k .The of the valence band of the covalent semiconductors on pressure, a feature which was also illustrated by Fig. 2.

Multivalley n-type semiconductors have large shear piezoresistance coefficients because shear strains can destroy the symmetry which caused all valleys to have the same energy and the same number of carrier^.'^.'^ Shear strain produces a transfer of electrons between valleys and a change in the form of the resistivity tensor. This electron transfer effect does not occur in semiconductors with [0:000]conduction band minima, however. Thus, when pressure shifts the energy of the conduction band minima enough to make a new kind of minimum the lowest, a striking change in the nature of the piezoresistance tensor takes place. This effect is illustrated for the case of GaSb in Fig. 5. At zero pressure most of the electrons are in the [OOO] minimum and cause no piezoresistance. The [ 1111 minima are not much above the [OOO] minimum and contain some electrons which contribute to the piezoresistance coefficient 7 ~ Application ~ ~ . 24

l5 26

R. W. Keyes and M. Pollak, Phys. Rev. 118, 1001 (1960). C. S. Smith, Phys. Rev. 94,42 (1954). C. Herring, Bell System Tech, J . 34,237 (1955).

5.

EFFECTS OF HYDROSTATIC PRESSURE ON 111-V SEMICONDUCTORS

337

FIG.5. The pressure dependence of the piezoresistance coefficients of n-type GaSb. (After Keyes and Pollak.") The data show that electrons are being transferred from the [OOO] to the [I 1 I] minima as pressure is increased.

of pressure raises the energy of the [OOO] minimum, transferring electrons to the [ l l l ] band where they contributed to n44.The n44 increases. The process is nearly completed at 12,000 kgicm', most of the electrons are in the [ l l l ] band, and nd4 is approaching a constant value which is characteristic of the [I 113 band and is similar to that found in germanium, for example. The piezoresistance coefficients ( n l l - n12) and (nll 2n1,)/3 are also plotted in Fig. 5. The quantity (nll - n12)is the shear coefficient produced by electrons in the [loo] band. Since there are no electrons in the [loo] band in GaSb in the pressure range covered, n, - 7t12 is small throughout. The combination grill + 2nI2) is simply the derivative of the resistivity with respect to pressure. It is large whenever the effects described in Part V produce pressure dependence of the resistivity. Thus n l l+ 2x1, is large and positive in GaSb because the transfer of electrons to the low-mobility [ l l l ] bands increases the resistivity. As the transfer nears completion the pressure derivative decreases toward zero.

+

2. HALLEFFECTAND THERMOELECTRIC POWEROF GaSb

Sagar originally formulated the [OOOI-[lll] model of the band structure of GaSb from his studies of piezoresistance and the effect of pressure on resistivity.'* He confirmed the details of the model in other experiments with Miller." The Hall constant and the thermoelectric power depend on

'' A. Sagar and R. C . Miller, J . Appl. Phys. 32,2073 (1961).

338

ROBERT W. KEYES

pressure qualitatively in the expected way. The transfer of electrons from the low-mass [OOO] band to the high-mass [l 111 band increases the thermoelectric power. A value characteristic of the [I 111 band is approached at high pressure. In the case of the Hall constant, the Hall constant is larger than its normal value of about (llne) at low pressure because of the two-band effect.’* At high pressures when most of the electrons are in the [ l l 13 band, the twoband effect disappears. Thus the Hall constant decreases with increasing pressure. The exact values of the thermoelectric power and the Hall constant and the precise way in which they change with pressure depend on details of the scattering processes in the two bands and can be fitted to models of the s ~ a t t e r i n g . ’ ~ . ~ ’

3. MICROWAVE OSCILLATIONS Gunn discovered that coherent electrical oscillations of the current through a GaAs specimen occur when a steady voltage which produces a field above a certain threshold field is applied to the specimen.29The effect presumably occurs because the threshold field (which is in the hot electron range) increases the temperature of the electrons sufficiently that an appreciable number of them are in the higher energy [loo] band. The electron mobility in the [loo] band is much lower than the mobility in the [OOO] band. An increase in field can, by transferring more electrons to the low mobility [loo] band, produce a decrease in current. Gunn has shown that this kind of current instability causes periodic propagation of electric shock waves through the specimen and can lead to oscillatory behavior of the current.30 It can be seen that when the energy difference between the [loo] and [OOO] bands is decreased by the application of pressure, the threshold field for the Gunn effect should decrease. The decrease was indeed found by Hutson et al., who observed oscillations up to 25,000 k g / ~ m ’ . ~ ’The threshold field at the highest pressure was 213 of the threshold at zero pressure. The range of voltage over which the effect existed decreased in the high end of the pressure range and no instability was observed above 26.000 kg/cm2.

VII. Phase Transitions The 111-V semiconductors all have the type of crystal structure known as “zinc blende.” The zinc-blende structure is very loosely packed ; each atom has only four near neighbors. The energetic stability of such a loosely packed 28

29 30

’‘

See, for example, J.-P. Jan, Solid State Phys. 5, l(1957). J. B. Gunn, IBM J . Res. Develop. 8, 141 (1964). J. B. Gunn, unpublished. A. R. Hutson, A. Jayaraman, A. G. Chynoweth, A. S. Coriell, and W. L. Feldmann, Phys. Reu. Letters 14, 639 (1965).

5 . EFFECTS OF HYDROSTATIC PRESSLJRE ON 111-V SEMICONDUCTORS

339

structure is reduced by high pressures. The free energy of a substance contains a term PK so that phases of low volume tend to become more stable at high pressures. High pressure experiments now reach several hundred thousand kg/cm2, at which pressures the PV term in the free energy approaches 100 kcal/mole and is an important part of the free energy. It is indeed found that other crystal structures become more stable than the zinc blende when the 111-V semiconductors are subjected to sufficiently high pressure^.^^,^^-^^ Phase transitions to more closely packed structures take place. The volume of the material decreases by about 20 % in the transition to the high pressure form.34 A typical transition of this kind has been known for a long time in tin, the cr-Sn--P-Sn transition. The cr-Sn structure is closely related to the zinc-blende structure and the high pressure phase of the 111-V semiconductors is frequently nearly the P-Sn structure. Similar transitions also take place in germanium and silicon and in 11-VI compounds.9s2336-39 Although the high pressure P- (or 11) phases usually have a P-Sn-like structure, the high pressure phase has the NaCl structure in a few 111-V c o m p o ~ n d s . ~ Table ~ * ~ *I11, ~lists ~ ~known ~ high pressure transitions in the 111-V semiconductors and parameters of the transition and the high ~~ that the transition pressures pressure phase^.^^.^^ J a m i e ~ o nobserved seem to be roughly a function of the energy gaps of the semiconductors and also the remarkable fact that the approximate difference in Gibbs free energy between the low pressure and high pressure phases (transition pressure x difference in volume) is close to half of the energy gap. Ail of the physical properties of the 111-V semiconductors vary in systematic ways with progression through the periodic table, and thus all appear to be correlated with 32 33 34

35

36 3’ 38

39 40 41

42

43 44

45

46

H. A. Gebbie, P. L. Smith, I. G. Austin, and J. H. King, Nature 188, 1095 (1960). A. Jayaraman, R. C. Newton, and G. C. Kennedy, Nature 191, 1288 (1961). J. C. Jamieson, Science 139, 845 (1963). A. Jayaraman, W. Klement, Jr., and G. C. Kennedy, Phys. Rev. 130,540 (1963). J. C. Jamieson, Science 139, 762 (1963). R. G. McQueen, S. P. Marsh, and J. Wackerle, Bull. Am. Phys. SOC.7,447 (1962) C. H. Bates, W. B. White, and R. Roy, Science 137, 993 (1962). C. H. Bates, W. B. White, and R. Roy, Science 147, 860 (1965). P. L. Smith and J. E. Martin, Nature 1%, 762 (1962). A. J. Darnel1 and W. F. Libby, Science 139, 1301 (1963). M. D. Banus, R. E. Hanneman, A. N. Mhriano, E. P. Warekois, H. C. Gatos, and J. A. Kafalas, Appl. Phys. Letters 2, 35 (1963). R.E. Hanneman, M. D. Banus, and H. C. Gatos, J. Phys. Chem. Solids 25,293 (1964). T. R. R. McDonald, R. Sand, and E. Gregory, J . Appl. Phys. 36,1498 (1965). P. L. Smith, J. H. King, and H. A. Gebbie, in ”Physics and Chemistry of High Pressures.” SOC. Chem. Ind., London, 1963. H. G. Drickamer, in “Physics and Chemistry of High Pressures.” SOC. Chem. Ind., London, 1963.

340

ROBERT W. KEYES

TABLE 111 PARAMETERS OF THE PHASETRANSITIONS AND THE HIGH PRESSURE PHASESOF 111-V SEMICONDUCTORS AV3b Transition pressureb (cm3/mole) (lo3 kg/cm2)

Substance

lnSb GaSb InAs AlSb InP GaAs Gap"

1.6 5.2 5.8 5.0 5.4 5.5

22 90 100 125 133 240

High pressure crystal structureb

Superconducting trans. temp.

8-Sn 8-Sn NaCl

1.87' 4.2d

P-Sn

2.8'

( O K )

* It is worth noting that Minomura and Drickamer found no transition in G a P to 550 x lo3 kg/cm*. After J a m i e ~ o n . ~ ~ 'Refs. 47, 54-56, 59. Another high pressure phase of InSb has = 4.2"K (Ref. 50). Ref. 51. Ref. 60.

one another. A correlation between lattice parameter and transition pressure of the 111-V compounds is shown in Fig. 6. The high pressure phase diagram of InSb is actually far more complicated than suggested by the simple picture of an or-Sn-fi-Sn type transition just I

0

5 200 0 r

m

150

P

kz

50

a

LATTICE CONSTANT

(1)

FIG.6. The transition pressures of the 111-V compounds (as in Table 111) plotted against zero pressure lattice parameter. Most physical properties of the 111-V compounds show some correlation with one another.

5.

EFFECTS OF HYDROSTATIC PRESSURE ON 111-V SEMICONDUCTORS

341

described. Other transitions occur, and several studies of the phase relationships and structures have been It appears that there is a high pressure phase of InSb with the p-Sn structure (InSb II), and it is relatively easy to produce InSb I1 by quenching from high p r e s s ~ r e . ~ ’ A , ~dis~.~~ cussion of the many studies of the properties of InSb I1 and its alloys with p-Sn that have been carried out is beyond the scope of this work. There is also an orthorhombic phase, a phase InSb 111, which is formed at high temperature and pressure, and still another phase above 80 kbar.;!3*47-50The occurrence of these phases seems to be somewhat sensitive to the details of the experiment ; some of them may be metastable. The electrical resistivity of the 111-V compounds decreases by orders of magnitude at the transition.23932~43~4s*46~51~53 The high pressure p-Sn phases are metallic. Like P-Sn, the high pressure phases of InSb, AlSb, and GaSb are superconducting.23~32~43.45.46*47-~51~53-60 It has proved possible to study the high pressure forms of InSb and GaSb at atmospheric pressure by applying high pressure, allowing the transition to take place, quenching the high pressure apparatus to some low temperature, as 77”K, then releasing the pressure. The samples remain in the metastable high pressure phase for long times if they are kept at low temperature. The transition temperatures of the high pressure phases are also shown in Table 111.

VIII. Elastic Properties The volume of the 111-V compounds decreases with increasing pressure. The dependence of the volume on pressure is quantitatively described by the bulk modulus B, defined by B = - V(dP/dV),. The bulk modulus and the shear elastic constants of several 111-V compounds have been measured by ultrasonic methods. The bulk modulus turns out to have a value of about

J. S. Kasper and H. Brandhorst, J . Chem. Phys. 41, 3768 (1964). J. E. Martin and P. L. Smith, Brit. J . Appl. Phys. 16, 495 (1965). 49 D. B. McWhan and M. Marezio, J . Chem. Phys. 45, 2508 (1966). ’O M. D. Banus and M. C. Lavine, J . Appl. Phys. 38,2042 (1967). ” A. J. Darnell and W. F. Libby, Phys. Rev. 135, A1453 (1964). 5 2 M. D. Banus, L. B. Farrell, and A. J. Strauss, J . Appl. Phys. 36, 2186 (1965). 53 S. Minomura, B. Okai, H. Nagasaki, and S. Tanuma, Phys. Letters 21, 272 (1966). 5 4 H. E. Bommel, A. J. Darnell, W. F. Libby, and B. R. Tittman, Science 139, 1301 (1963). S. Geller, D. B. McWhan, and G. W. Hull, Jr., Science 140, 62 (1963). 5 6 L. F. Stromberg and C. A. Swenson, Phys. Rev. 134, A21 (1964). D. B. McWhan, G. W. Hull, Jr., T. R. R. McDonald, and E. Gregory, Sciencel47,1441(1965). ’* M. D. Banus, S. N. Vernon, and H. C. Gatos, J . Appl. Phys. 36,864 (1965). 5 9 B. R. Tittman, A. J. Darnell, H. E. Bommel, and W. F. Libby, Phys. Reu. 135, A1460 (1964). 6 o J. Wittig, Science 155, 685 (1967). 47

48

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342

ROBERT W . KEYES

B = 1.2q2/aO4,where q is the electronic charge and a, is the lattice constant, for all of the 111-V semiconductors.61 The elastic properties of GaAs have been investigated in greater Drabble and Brammer62 and McSkimin et have investigated the pressure dependence of the elastic constants of GaAs. Their results are presented in Table IV. Since there is a strong similarity between the elastic TABLE 1V THE ELASTIC CONSTANTS OF GaAs AND THEIRPRESSURE DEPENDENCE

Ba.J c8a. J

c44 dB/dP dc'ldP dcddp

GaAsb

GaAsC.d

Gee

0.747 0.326 0.594 4.61 0.1 1.1

0.749 0.326 0.595 4.73 0.3 1.8

0.744 0.402 0.668 4.76 0.4 1.4

In terms of the ordinary elastic constants, B = ( c I I 2c,,),3 and c' = (c,, - c I z )2. After McSkimin et After Drabble and Bramrner.62 dValues of B , c', and c44 have also been reported by Bateman et d 6 U and Garland and Park.65 After McSkimin and A n d r e a t ~ h ~ and ~.~' In units of 10" dynes cm*.

+

properties of the 111-V and group IV semiconductors,61 the same elastic which has nearly the same lattice parameter constants of as GaAs, are also presented for comparison. 61

63

64 65

66 67 bs

R. W. Keyes, J . Appl. Phys. 33,3371 (1962) This paper also contains references to the original measurements of the elastic constants. J. R.Drabble and A. J. Brammer, Solid State Commun. 4,467 (1966). H. J. McSkimin, A. Jayaraman, and P. Andreatch, Jr., J . Appl. Phys. 38, 2362 (1967). T. B. Bateman, H. J. McSkimin, and J. M. Whelan, J. Appl. Phys. 30, 544 (1959). C. W. Garland and K. C. Park, J . Appl. Phys. 33,759 (1962). H. J. McSkimin and P. Andreatch, Jr., J . Appl. Phys. 34,561 (1963). H. J. McSkimin and P. Andreatch, Jr., J . A p p l . Phys. 35, 3312 (1964). J. J. Hall, Bull. Am. Phys. Soc. 10,43 (1965), and to be published.

CHAPTER 6

Radiation Effects L . W. Aukerman

I. GENERAL DISCUSSION. . . . . . . . 1 . Introduction . . . . . . . . . . 2. Atomic Displacements . . . . . . . 3. Recovery of Radiation Damage. . . . . 11. THRESHOLD EXPERIMENTS. . . . . . . 4 . Experimental Results . . . . . . . 5. Discussion . . . . . . . . . . IN VARIOUS 1II-V COMPOUNDS 111. RADIATIONEFFECTS 6. Comments Regarding Energy Levels . . . 7. Indium Antimonide . . . . . . . . 8. Indium Arsenide . . . . . . . . . 9. Gallium Arsenide . . . . . . . . 10. Gallium Antimonide . . . . . . . . 1 1 . Aluminum Antimonide . . . . . . . 12. Indium Phosphide . . . . . . . . 1 v . RADIATIONDAMAGE IN DEVICES. . . . . 13. General Remarks . . . . . . . .

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343 343 34s 355 368 368 372 373 373 376 388 391 401 402 403 403 403

1. General Discussion 1. INTRODUCTION

The effects of energetic neutrons and various charged particles on the physical properties of solids have been the object of numerous investigations for nearly two decades. A review of much of the early work as well as a penetrating discussion of radiation damage theory can be found in the wellknown article by Seitz and Koehler.' Effects on the electrical properties of semiconductors have been of particular interest recently because of (a) the relatively large changes induced by small amounts of irradiation and (b) the technological importance of this class of materials. Of the available semiconducting materials, germanium and silicon have received the greatest attention ;other semiconductors have been studied on a more modest scale, if at all.

' F. Seitz and J. S. Koehler, Solid State Phys. 2, 307 (1956) 343

344

L. W . AUKERMAN

Review articles and books have appeared from time to time which include radiation effects in semiconductors’-6b but these deal mostly with silicon and germanium, since early semiconductor work was devoted almost exclusively to these substances. However, in recent years more and more work concerning radiation effects in compound semiconductors has appeared, resulting in the accumulation of fairly sizable literature. A recent review of electron-radiation damage by CorbetPa gives the most thorough discussion of radiation damage in semiconductors to date, but little is said concerning neutron or heavy particle damage. A study of radiation effects on compound semiconductors is useful from both fundamental and applied points of view. First of all, a greater variety of intrinsic defects is possible in a compound than in an element. In addition, the many compound semiconductors available provide a large variety of intrinsic properties to be studied. By subjecting various materials encompassing a wide range of intrinsic properties to controlled amounts of irradiation, new insights into the mechanisms of radiation damage may be acquired. Finally, the technological importance of compound semiconducting materials has increased to the point where a knowledge of radiation effects is of value to the design engineer. This chapter will be almost exclusively concerned with the effects of radiation damage on 111-V compounds, although examples from semiconductors outside this class or even from metals may be included occasionally if they help the reader to understand some of the basic ideas and concepts involved in radiation damage. Furthermore, electronic redistribution (or trapping effects), as contrasted to atomic rearrangement, will not be considered, except as a possible tool for the study of the latter. Of course, it must be recognized that these two categories may be difficult to separate experimentally, especially in the case of semiconductors or insulators at low temperatures, where carriers may be trapped for long periods of time. G. J. Dienes and G. H. Vineyard, “Radiation Effects in Solids.” Wiley (Interscience), New York, 1957. D. S. Billington and J . H. Crawford. “Radiation Damage in Solids.” Princeton Univ. Press. Princeton. New Jersey. 1961. Proc. liirern. Sclzool Phys. “Enrico Fermi” XVlll (D. S . Billington, ed.), 1963. H. Y. Fan and K. Lark-Horovitz, Rept. Cotit Defects Crystalline Solids, Bristol, IY54, p. 232. Phys. SOC. London, 1955. 5aH.Y. Fan and K. Lark-Horovitr, in “Effects of Radiation on Materials” (J. J . Harwood. H. H. Hausner, J . G. Morse, and W. G . Rauch, eds.). p. 159. Reinhold. New York. 1958. V. S. Vavilov, “Effects of Radiation on Semiconductors,” p. 141. Consultants Bureau, New York, 1965. “J. W. Corbett, Solid Stare Phys. Suppl. No. 7 (1966). 6bConferenceon Radiation Effects in Semiconductors, Gatlinberg, Tennessee, J . Appl. Phys. 30, 1117-1322 (1959).

‘

’

‘

6.

RADIATION EFFECTS

345

The next two sections discuss, respectively, the creation of atomic displacements by energetic particles and the recovery of this damage, I.e., how the initial properties are restored. These discussions are quite general, but special assumptions or considerations for compounds are included wherever it is feasible to do so. In the next section, the threshold experiments on all the compounds are discussed together, and, following this, energy levels and various methods of determining them are treated. The remaining radiation damage properties for bulk specimens are discussed under each compound in the following sections. The last section discusses radiation damage in devices.

2. ATOMIC DISPLACEMENTS The role of the bombarding particles is simply that of transferring enough energy to individual atoms to knock them from normal lattice positions into interstitial positions. This creates vacant lattice sites (vacancies) and interstitial atoms (interstitials). If enough energy is imparted to an atom, it may produce additional vacancies and interstitials by colliding with some of its neighbors. if the bombarding particle is charged or if it is a photon (e.g., gamma ray), most of the energy loss in the specimen will occur in the form of ionization (i.e., inelastic collisions). Radiation damage theories therefore focus attention on the relatively few elastic collisions. Nevertheless, it should be kept in mind that a high degree of ionization almost invariably is present, for the intensity of ionization may influence the displacement ~.~" ionization process or have a bearing on radiation a n n e a l i ~ ~ g .Furthermore, may affect the measured property if trapping is present. Even in the case of fast neutron bombardment, ionization is usually present because of accompanying gamma radiation, or electronic excitation resulting from highly energetic recoil atoms. Another important aspect of radiation damage is the spatial distribution of the defects. Much of the damage resulting from a very energetic primary knock-on occurs in a rather small localized region containing a high density of defects. For example, a typical recoil atom in germanium bombarded with fission neutrons has about 30 keV of energy, and its range, according to van Lint et is about 200 A. Thus, the total damage produced by such a recoil atom should, roughly speaking, lie within a region of about 200-A radius. However, the calculations of Yoshida' show that the more intense

' J. W. MacKay and E. E. Klontz, in "Radiation Damage in Solids,"

Vol. Ill. p. 27. Intern. Atomic Energy Agency, Vienna, 1963. 7aE.E. Klontz and J. W. MacKay, J . Phys. SOC.J a p a n 18, Suppl. 111, 216 (1963). V. A. J. van Lint, M. E. Wyatt, R. A. Schmitt, C. S. Saffrendi, and D. K. Nichols, Phys. Rev. 147. 242 (1966). M. J. Yoshida, J . Phys. SOC.J a p a n 16,44 (1961).

346

L. W. AUKERMAN

damage usually lies considerably closer to the primary event, for the mean free path between collisions will usually be only a few atomic spacings. The "electrical" size of these damaged regions may be considerably larger than their physical size and they are thus expected to play an important role in the transport properties of a semiconductor. Gossick l o and Crawford and Cleland"" have calculated the potential and charge distribution for defect regions which are of opposite conductivity type relative to the bulk of the specimen. Their model has been substantiated by drift mobility measurements" and observations of the initial changes in mobility and conafter irradiation with fast neutrons. ductivity in n-type More direct evidence for the inhomogeneous nature of fast neutron damage is provided by electron microscope studies, 13,14 and for deuteron damage by low angle x-ray scattering. 14a Ultrasonic attenuation and velocity measurements in neutron-irradiated silicon have also been interpreted in terms of Sputtering experiments on gold 1 6 3 1 6 a provide damaged additional evidence for defect clusters. A rather thorough discussion of damaged regions of semiconductors is included in a recent paper by Vook. l 7 In those cases where multiple displacements are very rare, as for example with electron or y-irradiation, the vacancies are distributed randomly.' However, there should be a high degree of correlation between vacancies and interstitials.Thus for elements the primordial damage, that is, the damage that exists before any annealing can occur, is imagined to consist of the following : (a) For fast neutrons, damaged regions of various sizes are superimposed on a background of Frenkel defects (vacancy-interstitial pairs). Very little correlation between vacancies and interstitials is expected in the damaged regions. (b) Heavy charged particles (protons, deuterons, etc.) create predominantly Frenkel defects with damaged regions near the end of their range. B. R. Gossick, J. Appl. Phys. 30, 1214 (1959). H. Crawford, Jr., and J. W. Cleland, J . Appl. Phps. 30, 1204 (1959). W. H. Closser, J . Appl. Phys. 31, 1693 (1960). '* H. J. Stein, J . Appl. Phys. 31, 1309 (1960). l 3 J. R. Parsons, R. W. Balluffi, and J. S. Koehler, Appl. Phys. Lefters 1, 57 (1962). l4 D. P. Miller and H. L. Taylor, Appl. Phys. Letters 2, 33 (1963). 14aF.E. Fugita and U. Gonser, J . Phys. SOC.Japan 13, 1068 (1958). l 5 R. Truell, L. J. Teutonico, and P. W. Levy, Phys. Rev. 105. 1723 (1957). I s a R . Truell. Phys. Rev. 116, 890 (1959). 15bR. Truell. J . A p p l . Phys. 30, 1275 (1959). l 6 M. W. Thompson and R. S. Nelson, Phil. Mag. 7, 2015 (1961). 16aM.W. Thompson. Proc. Intern. Scltool Phys. "Enrico Fermi" XVIII. 169 (1962). l 7 F. L. Vook, in "Radiation Damage in Semiconductors" (Proc. 7th Intern. Conf.), p. 51. Dunod, Paris and Academic Press, New York, 1964. "'Electron damage is, of course, inhomogeneous over distances comparable to the range. but over much smaller distances the vacancies are distributed randomly. lo

loaJ.

6.

RADIATION EFFECTS

347

(c) Electrons or prays create vacancy-interstitial pairs almost exclusively. The degree of correlation should be a function of energy, less correlation corresponding to a greater bombarding energy. a. Calculation of Damage Rates

Calculation of the damage rate, or the number of displacements per unit of incident flux, is carried out in two ~ t e p s . ’ - ~ , ’ ~F’,irst ’ ~ the number of primary recoil atoms N,(T) is calculated, where NP(T )d T is the number of primary recoil atoms per unit volume which received energy in the interval T to T + dT. Then the average number of displacements v(T) per primary recoil of energy T a r e calculated. Thus the total concentration of displacements is N, = NP(T)v(T - G)dT. (1)

;:1

Here the use of the lower limit G,called the threshold displacement energy, takes into account the fact that very low energy recoils do not result in a displacement. That is, a sharp threshold energy is assumed. Equation (1) also implies that the primary knock-on is degraded in energy by the amount G in escaping its lattice site. Typical values for Td lie in the range 5 to 50 eV. The upper integration limit in Eq. (1) is the maximum energy that can be imparted to the target atom in an elastic collision. For nonrelativistic elastic

where M I and M , are the masses of the projectile and target, respectively, and E is the energy of the projectile. If the bombarding particles are not monoenergetic, Eq. (1) is first integrated with respect to T and then with respect to E . For relativistic energies (e.g., electrons of a few hundred keV or greater), 2(E + 2mc’)E Tm= (3) M,c2 ’ ~~

where rn is the rest mass of the electron and c the velocity of light. Now, let (P(E)dE be the incident particles per unit area with energy in the interval dE near E, and let do = f(E,T ) d T be the cross section for producing a recoil possessing energy in the range dT near T Then the number of primary displacements for a thin specimen is m

Np(T)=J N,(P(E)f(E, T) dE >

’* D. K. Holmes, Proc. Intern. School Phys. “Enrico Fermi” XVIII, 182 (1962). l9

G. Leibfried, Proc. Intern. School Phys. “Enrico Fermi” XVIII, 227 (1962).

(4)

348

L . W . AUKERMAN

and putting this into (1) gives ... ND

=NoJ

Td

dT v(T - T,)J +(E)f(E,T )d E . 0

Reversing the order of integration gives

where C

T ~the ,

displacement cross section, is

and C, the average number of displacements per primary, is , 11

=

rTm

-

J

OD

f(E,T)v(T- T,)dT. TD

For monoenergetic bombardment at energy E o , N,

=

N~cTD(E~)C(EO)@,

where CD is the incident flux. For fast neutron irradiation f(E,T ) is frequently assumed to be constant, giving

where us, the scattering cross section, has a value in the range from 2 to 4 x cm2 for most nuclei, and A is the atomic mass. In reality this is probably a poor assumption and tends to overestimate the energy transferred to the primary. 1 8 + 1 The interaction with heavy charged particles in the 2- to 20-MeV range is essentially coulombic, i.e., the conditions for Rutherford scattering are satisfied and

where b, the classical distance of closest approach, is b

=

2 Z , Z 2 e 2 / p v 2;

(10)

Z 1 and Z , are the atomic numbers of the incident particle and target atom, respectively, e the electronic charge, v the velocity, and p the reduced mass rp = MlMZ/(Ml + '44211.

6.

349

RADIATION EFFECTS

Electron irradiation at energies sufficient to produce damage usually will require relativistic treatment. The differential cross section in the relativistic range has been calculated by McKinley and Feshbach” by the use of appropriate approximations. Expressing their result in terms of the primary recoil energy, one obtains d o , = G(E. T )doRuth, (11) where doRuthis the Rutherford cross section of Eq. (9) and G(E, T ) is a correction factor,

G(E, T ) = (1 - 8’)[1 -

P’X + ~CC$(X~’’

-

x)],

where x = T/T,,,fi = velocity of electrons divided by the velocity of light, cc = Z/137. This result is more accurate for low Z values and probably should not be relied upon for Z greater than about 30 or 40, unless the energy is near threshold. Even so, for large Z values the error is probably not greater than a factor of two.” It is seen that G(E, T ) is a weighting factor which tends to shift the energy distribution of primary recoils slightly toward lower energies. Gamma irradiation is usually considered to be a special case of electron irradiation for metals and semiconductors, since the mechanism envisioned consists first of the creation of energetic electrons via the Compton effect, photoelectric effect, or pair production, and then the generation of displacements by these internally produced electrons. Thecross sections for the various processes involved are discussed in detail by Oen and Holmes22and Cahn.23 This subject has also received excellent treatment by Dienes and Vineyard.2 However, mechanisms have been proposed whereby displacements could result directly from the ionization event as is known to happen in the case of a number of ionic crystals. Whether or not elastic collisions are necessary for the creation of displacements by gamma or x-rays in semiconductors is still an open question.

b. Secondary Displacements When the primary atom is ejected with an energy well in excess of &, it may, in colliding with other atoms, produce additional displaced atoms. W. A. McKinley and H. Feshbach, Phys. Rev. 74, 1759 (1948). E. Merzbacher and G . R. Khandelwal, Bull. Am. Phys. SOC. 7, 543 (1962). 2 2 0.S. Oen and D. K. Holmes, J. Appl. Phys. 30, 1289 (1959). 2 3 J. H. Cahn, J. Appl. Pliys. 30, 1310 (1959). 2 4 F. Seitz. Phys. Rev.80, 239 (1950). 24aJ. H. 0. Varley, Nature 174, 886 (1954). Z4bJ. H. 0.Varley, J . Nucl. Energy 1. 130 (1954). 24eR. E. Howard. S. Vosko. and R. Srnoluchowski, Phys. Rev. 122, 1406 (1961). 24dR, Smoluchowski and D. A. Wiegand, Discussions Faraday SOC.31, 151 (1961). 24eJ. Durup and R. L. Platzrnan, Discussion Faraday SOC.31, 156 (1961). 2o

350

L . W . AUKERMAN

This process, continuing until all displaced atoms have been reduced in energy to Td or lower, is called a displacement cascade. Usually the primary displacements will be spaced sufficiently far apart that displacement cascades initiated by different primary atoms will be confined to different regions of the crystal. Thus in calculating v( T ) , the average number of displacements per primary recoil of energy T, one calculates the average number of displacements in one such cascade process. If the energy of the primary is not too high (no electronic excitation), one can assume that the atoms behave as hard spheres. The differential cross section for hard sphere collisions is h ( E ,T )

=

4xR2 dT E

~

=o

for

T<E

for

T 3E,

where R is the radius of the hard sphere. Although the hard sphere assumption is not strictly validz4fit does not appear to introduce much error. The atoms are also assumed to be distributed randomly. This assumption is at present one of the most serious limitations to cascade theory. More recent calculation^^^ and experimental resultsz6 have demonstrated several mechanisms26acapable of removing energy from a cascade in a manner which decreases the overall displacement count. Nevertheless, randomness of the atoms seems to be a necessary assumption if an analytic expression for v(T) is to be feasible. Fortunately certain anisotropic effects of crystal structure can be introduced into cascade theory rather easily ( e g , channeling). The various methods of calculating v differ primarily in the manner in which the threshold energy is taken into account, and in the definition of “displacement.” Thus Kinchin and Pease2’ consider the atoms as free but take the threshold energy into consideration by postulating that no displacement will occur unless both the striking atom and the struck atom have at least the energy Td after collision. If the striking atom retains less than while the struck atom receives more than Td it is assumed that the former will replace the latter and no net displacement will occur. The method of Snyder and Neufeld,28 on the other hand, requires that the displaced atom in being ejected from its lattice site be degraded in energy by the amount &. However. the total number of displacements are counted, whether they 24‘See, for example, Ref. 2. p. 19. z 5 R. H. Silsbee, J . A p p l . Phys. 28, 1246 (1957). ” R. S. Nelson and M. W. Thompson. Proc. Roy. SOC. (London) A259. 458 (1961). 26aForexample, focusing collisions, channeling, and the dynamic crowdion ; see the following subsection. G . H. Kinchin and R. S. Pease, Rep!. Progr. Phys. 18, 1 (1955). ’* W. S. Snyder and J. Neufeld. Phgs. Rev. 97. 1637 (1955).

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RADIATION EFFECTS

351

finally come to rest on a lattice site or not. Both methods give approximately the same result which, for large energies, can be put in the form, v ( T )= R T / T , .

(13)

For the Kinchin-Pease formulation B = 3 and for the Snyder-Neufeld, B = 0.56. Both methods probably overestimate the damage slightly, but for different reasons : the former because of the failure to take into account the energy lost by the knock-on atom in climbing out of its potential well, and the latter because of the inclusion of replacement collisions. Snyder and Neufeld subsequently modified their formulation to exclude replacement^.'^ This leads to the above asymptotic form (13) with' a value of B equal to 0.35. Calculation of the number of replacements appears to be very sensitive to certain details of the collision process that are not at present well understood. Kinchin and Pease3' estimated the number of replacements by employing a model that allows replacements when the struck atom has energy less than Td but greater than a postulated replacement threshold T, and greater than the energy of the striking atom ;that is, a replacement is assumed to result if

& b Ti d Tz

b T, d

& '

where TI and T2 are, respectively, the energies of the striking and the struck atom. The argument for this model is that the energy required to displace an atom could be less than Tdif the striking atom can replace it. The quantity T, is left as an adjustable parameter. Dienes and Vineyard solve the same problem using a simpler mathematical technique ;they obtain3'= for p ( T ) ,the number of replacements per primary atom of energy T (for large T),

If T,is considerably less than Tdthis model predicts a very high concentration of replacements. Even if T, = TD the replacements and displacements are equal. This model does not provide for the possibility that fewer replacements than displacements might occur. A more realistic calculation, for example, might have taken into consideration the variation of potential with the positions of both atoms, a problem of considerably greater complexity. The Snyder and Neufeld model gives 0.6 for the ratio of replacements to vacancies at large energies. This is to be compared with the Kinchin-Pease 2y

W. S. Snyder and J. Neufeld, Phys. Rec. 99, 1326 (1955).

'' G . H. Kinchin and R. S. Pease, J . Nucl. Energy 1, 200 (1955) "'See

Ref. 2, p. 28.

352

L . W . AUKERMAN

model with T, = &, which gives unity for this ratio. The difference is due to the different treatment of the ejected atom. Each of the above treatments of the cascade process predicts that at high energies v is proportional to the energy of the moving atom divided by the threshold energy with a proportionality constant of, roughly 0.5. Experimental estimates of the damage rates tend to be considerably lower than these calculated values by as much as a factor of 5 to 10.The discrepancy is usually worse the higher the energy of the knock-on atom. Further refinements of cascade theory taking into account various probability functions3 1-33 indicate that, in order to account for the discrepancy, the distribution in threshold energies must extend over a very large range. Other factors that may have a bearing on the discrepancy between calculated and measured defect concentrations are (1) partial annealing, (2) inhomogeneous nature of the damage, (3) electronic excitation by the moving atom (inelastic collisions), and (4) the assumption of randomly spaced atoms. Both of the foregoing methods of calculating v assumed hard sphere collisions. This may be justified for low primary energies. However, if the primary energy becomes so high as to produce appreciable electronic excitation, this additional energy loss must somehow be taken into account. This can be done rather crudely by assuming that for T > (7; is called the threshold ionization energy) the atom loses energy only by ionization, and for T < T , only by elastic collisions. Seitz and Koehler’ estimate for semiconductors to be

where M is the mass of the atom, me the mass of the electron, and EG the band gap of the irradiated semiconductor. A more thorough treatment of ionization losses has been carried out by Lindhard and T h ~ m s e nTheir . ~ ~ work indicates that the fraction of the energy lost by an atom in the form of ionization does not exhibit a sharp threshold, as assumed above, but instead increases gradually with energy. Their results have been confirmed experimentally by Sattler et u1.34a,34b The effect of other scattering potentials, more realistic than the hard sphere potential, has recently been considered in calculations of replacement W. S. Snyder and J. Neufeld, Phys. Rev. 103, 862 (1956). A. E. Fein, Phys. Rev. 109, 1076 (1958). 3 3 E. M. Baroody, Phys. Rev. 116, 1418 (1959). 34 J . Lindhard and P. V. Thomsen, in “Radiation Damage in Solids,” Vol. I, pp. 65-76. Intern. Atomic Energy Agency, Vienna, 1962; Kgl. Danske Videnskab. Selskab., Mar.-Fys. Medd. 33, No. 10 (1963). 34aA.R. Sattler, Phys. Rev. 138, A1815 (1965). 34bA.R. Sattler, F. L. Vook, and J. M. Palms, Phys. Rev. 143, 588 (1966). 31

32

6. RADIATION

EFFECTS

353

collisions34c; but comparison with experiment was not made because of the difficulty of determining experimentally the number of replacements. c. EfSects of the Crystal Lattice

The foregoing methods of calculating displacements do not take into account the ordered arrangement ofatoms. Recently a number of mechanisms have been proposed that are sensitive to the lattice structure. Thus SilsbeeZ5 a focusing effect along rows of closely and 0 t h e 1 - s have ~ ~ ~demonstrated ~~~ packed atoms ;that is, an impulse directed within a certain small angle down a closely packed row is propagated with very little attenuation. The direction of the impulse approaches the direction of the closely packed row with successive collisions. The dynamic c r ~ w d i o n is~ similar. ~ . ~ ~ A crowdion is pictured as a closely packed row having an extra atom crowded in. The dynamic, or moving, crowdion transports mass as well as energy. Focusing collisions transport energy only. For metals these concepts have been fairly well established3’ by experimentz6 and by machine computation as ~ e l l . Machine ~ ~ 3 ~ computations ~ ~ are employed by setting up typical events mathematically and calculating the paths of all the atoms involved. These studies have been very valuable in helping to visualize the kinds of events that can occur. Because of the open nature of the diamond and zinc-blende lattices, focusing collisions and dynamic crowdions are not expected to be very probable. However, Robinson et a1.39-39band other^^'-^'^ have considered the possibility of an energetic atom being shot down the channel between atoms aligned in the close packed directions. Once an atom becomes channeled it loses energy rather slowly because it makes only grazing collisions with the adjacent rows of atoms. These channeling events are thought to be important in metals as well as semiconductors; however, they may be 34cP.H. Dederichs, Chr. Lehmann, and H. Wegener, Phys. Status Solidi 8, 213 (1965). 3 5 G . Leibfried, J . Appl. Phys. 30,1388 (1959). 35aC.Erginsoy, G. H. Vineyard, and A. Englert, Phys. Rev. 133, A595 (1964). 36 D. 0.Thomsen, T. H. Blewitt, and D. K. Holmes, J. Appl. Phys. 28,742 (1956). 3’ R. I. Garber and A. 1. Fedorenko, Lisp. Fiz. Nauk 83,385 (1964) [English Trurzst. :Soviet Phys.Lispekhi 7,479 (1965)l. J. B. Gibson. A. N. Goland, M. Milgram, and G . H. Vineyard, Phys. Rev. 120. 1229 (1960). 38aG.H. Vineyard, Proc. Intern. School Phys. “Enrico Fermi” XVIII. 291 (1962). 39 M. T. Robinson and 0.S. Oen, Appl. Phys. Letters 2, 30 (1963). 39aM.T. Robinson and 0. S. O m , Phys. Rev. 132, 2385 (1963). 39bM.T. Robinson, D. K. Holmes, and 0. S. Oen, Bull. Am. Phys. SOC.7 , 171 (1962). 40 J. R. Beeler, Jr., and D. G. Besco, in “Radiation Damage in Solids,” Vol. 1, pp. 43--63. Intern. Atomic Energy Agency, Vienna, 1962. 40aJ. R. Beeler, Jr., and D. G. Besco, J . Phys. SOC. Japan 18, Suppl. 111, 159 (1963). 40bJ. R. Beeler, Jr., and D. G . Besco, J. Appl. Phys. 34,2873 (1963). 4 1 G. R. Piercy. F. Brown, J. A. Davies. and M. McCargo, Phys. Rev. Letters 10. 399 (1963). 41aR.S. Nelson and M. W. Thompson. Phys. Letters 2, 124 (1962). 41bC.Lehrnann and G. Leibfried. J . Appl. Phys. 34.2821 (1963). 38

354

L. W. AUKERMAN

more probable in the latter. Although most calculations involving focusing or channeling collisions treat fcc closest packing structures, Beeler and treat channeling in the wurtzite structure, which is quite similar to the zinc-blende lattice. Oen and Robinson4' introduced channeling (and/ or focusing) collisions into cascade calculations for monatomic solids by assuming (a) a constant channeling probability and (b) that once an atom is channeled it is lost as far as the production of additional displacements is concerned. Their calculation shows that, rather than increasing with the first power of energy, v increases at a lower power as follows : v = [(l - P)y'-'P - P]/(1 - 2 P ) , (16) where y = T/2Td and P is the channeling (or focusing) probability. This offers a possible explanation of the fact that the discrepancies between experimental and theoretical estimates of the number of defects have been greater for more energetic particles. d. Displacement Cascades in Compounds

The calculation of v is considerably more complicated for compounds than for monatomic solids, although the same physical principles are involved. In compounds the energy transferred in an elastic collision is strongly influenced by the ratio of the atomic masses. The maximum energy that can be transferred in such a collision is /z times the energy of the striking atom, where

and M I and M , are the masses of the two atoms involved in the collision. Furthermore, in compounds one must consider different kinds of displacements and the fact that one type of displaced atom can, in general, engender more than one type of displacement. Finally, it is evident that replacements become much more important in compounds than in monatomic solids. The problem of calculating displacements in type AB compounds was first addressed by Harris43 and later by B a r ~ o d yand ~ ~more recently by F e l d e ~ - .Baroody ~~" was able to obtain explicit relations for the numbers of displaced atoms by extending the Kinchin and Pease model. It was shown that for the total number of displacements, the Kinchin and Pease result (13) can be extended to type AB compounds of any mass ratio, provided all atoms have the same threshold and a given atom has the same probability of colliding with a like as with an unlike atom. Furthermore, if the mass of the heavy atom is less than ten times that of the light atom the numbers of A and B atoms 0.S. Oen and M. T. Robinson. A p p l . Phys. Letters 2. 83 (1963). E. G. Harris. P hys. Rev. 98. 1 I 5 1A (1955). 44 E. M. Baroody. Phys. Rev. 112. 1571 (1958). 44aR.M. Felder, J . Phys. Clwm Solids 28, 1383 (1967). 42

43

6.

355

RADIATION EFFECTS

displaced by an A atom are nearly equal for energies ofabout twenty times the threshold energy or greater. These calculations were later generalized to include unequal thresholds, compounds of more than two types of atoms, and unequal collision p r ~ b a b i l i t i e sThe . ~ ~ special case that is of interest from the standpoint of 111-V compounds is the one for diatomic solids of equal threshold energies but different masses and collision probabilities (the thresholds for the more common 111-V compounds appear to be approximately equal ; see Section 4 on Threshold Experiments). For this case, B of (13) is approximately f except when 1 < 5 and when a moving atom has a very high probability of striking an atom of the opposite type in its next collision. Since both restrictions are very unlikely (see Table I) the Kinchin and Pease result, (13), is a valid approximation for binary compounds, provided it is understood that v refers to the average total number of vacancies in a cascade event. Table I shows A values for some common 111-V compounds. Baroody made no attempt to estimate the number of replacements, and, of course, channeling effects were not included in his calculations. TABLE I i. VALUESFOR 111-V COMPOUNDS

P A1 Ga In

0.997 0.854 0.67

As

Sb

0.78

0.592 0.925 1.0

-1.0

0.956

-

It has already been pointed out that replacement collisions in compounds might account for an appreciable fraction of the damage. However, those events in which an atom replaces a like atom are of no interest. Only those replacements involving unlike atoms need be considered, for these will result in atoms lying at wrong lattice sites. These defects (sometimes called antidefects) will be called misplaced atoms or misplacements.

3. RECOVERYOF RADIATIONDAMAGE The creation of vacancies and interstitials by energetic particles increases the free energy of the crystal and must be followed by changes which eventually restore thermodynamic equilibrium. Therefore one expects the initial properties of the crystal to be restored sooner or later, depending upon the heat treatment, if the crystal was initially in equilibrium, and il'contamination by impurities can be avoided during warming cycles. Nevertheless the path to recovery is usually indirect, for if the annealing is carried out slowly

356

L . W . AUKERMAN

the crystal will frequently experience a number of metastable states before recovery is complete. If these metastable states are well separated in energy the recovery will tend to occur in separate stages, i.e., different annealing processes will tend to occur in different temperature ranges. This section will first discuss various methods of studying annealing processes phenomenologically ; then a number of specific annealing processes and their characteristics will be discussed. a . Annealing Experiments The various annealing stages are frequently explored by means of isochronal or pulsed annealing experiments. That is, the specimen is annealed at successively higher temperatures for a definite period of time, but the measurements of the property changes are carried out at a fixed low temperature after each annealing period. The property change then, when plotted as a function of the temperature of anneal, will frequently show a number of sigmoid sections, each corresponding to a different annealing stage. Examples of isochronal annealing curves can be seen in Sections 7e, Sa, and 9c. If the property is not strongly temperature dependent, or if its temperature dependence is accurately predictable (a situation rarely encountered with semiconductors) tempering experiments are more convenient and are just as useful. In such experiments the temperature is slowly increased and the property is continuously monitored. In isothermal annealing experiments the temperature is held constant and the property changes are recorded as a function of time. When it is desirable to measure the property and perform the annealing at different temperatures, the specimen is cycled between the measuring and annealing temperatures. In this case, as in the isochronal case, the measuring temperature must be low enough that no annealing takes place during the measurement procedure. Furthermore the cycling times must be short compared to the annealing periods. If a property p is a linear function of the concentration of a particular species of defect, then the fraction cp of this species remaining after some unspecified heat treatment is cp=-

Pm

-

P

P m - Po

(18)

where p o is the initial property value and pm is the value after essentially infinite time has elapsed. In the case of an isochronal experiment where more than one well-defined stage may be present, p o and p , represent the plateau values before and after the “jog.” Ifp is taken to be the electrical conductivity, a = nep, and if the change in conductivity, ACJis small, we can write

Aa

r a[(An/n)- pA(l/p)].

(19)

6.

357

RADIATION EFFECTS

For semiconductors An and A(l/,u) will be linearly related to the concentrations ofthe various species present if the Fermi level does not shift appreciably during the anneal, i.e., if An is not too large. Usually if the fractional change in cs is 10 to 20% or less, A g can be used in (19) to estimate the fractional damage.

b. Analysis of Annealing Experiments Isochronal experiments are very useful in exploring the temperature ranges of various annealing stages. The regions of rapid annealing, or inflection points in an isochronal “curve,” are never very strongly influenced by the time interval at each temperature or the spacing between temperature points. Of course, the longer the time interval and the closer the temperature spacing, the better the resolution for separating stages close in activation energy. On the other hand, the experiments become increasingly tedious as the resolution is improved in this manner. Isothermal experiments can usually be employed to separate different processes which can occur simultaneously. The inflection point of an isochronal “curve” is intimately related to the activation energy of the annealing process. Consider a process that proceeds according to the following expression :

where cp is the fraction of the annealing stage not completed, T is the temperature, k is Boltzmann’s constant, and E, is the activation energy. The frequency factor A is a constant, and y is the order of the reaction. Perhaps the most important reactions of this class are first- and second-order reactions ( y = 1 and 2, respectively); however, other values of y, even nonintegral values, have been reported. For an isochronal experiment, in which the time interval at each annealing temperature is At, G(cp,-

-

(21)

G(y7,) = A At e-Ea‘kTi,

where ‘piis the value of cp after the ith anneal at the temperature ‘< and C(p)

=

Iq-ydp

=

In cp

-

(9l-y

for 7 -

1)(1 - y)-’

= 1

for 1’ # 1.

(22)

If the annealing temperatures are selected such that 1/T intervals are constant, ~e.,

358

L . W . AUKERMAN

and if '7;.=, , the temperature of the "zeroth" anneal, is selected to be so low that the annealing rate is negligible at this temperature, then q0 % 1 and Eq. (21) can be expressed as follows :

where

By making To sufficiently small, one can neglect the C i aterm over most of the temperature range. Then the points of Eq. (24) lie very close to the continuous curve G ( q ) = - A Ar (1 - eC6)-'e-', (26) where X = E,/kT. Equation (26) is a good representation for annealing temperatures T > T,(1 - kT,/E,)-'. T o find the inflection point, set d 2 q / d X 2 = 0. This condition and Eq. (26) give

E,

=

kT, ln[A At (1 - e-')-'].

In a tempering experiment, 6 + 0 and At E,

=

-+

(29)

0, resulting in

$),

kT, In(

where a = d(l/T)/dt. Since the change in cp occurs rather abruptly, it is not necessary that d ( l / T ) / d r be strictly constant. If the temperature changes at a constant rate, 1/T also changes at an approximately constant rate over a small temperature interval and Eq. (30)can still be applied with a fair degree of accuracy. The same can be said for Eq. (29) if 6 in Eq. (24) is not strictly constant. To apply Eqs. (29) or (30)it is necessary to know either A or E, from another source. If A is estimated to be roughly the same order of magnitude as the major lattice vibration frequency, one then has a means of obtaining at least a crude approximation of the activation energy. However, there is an alternative method of determining E and A from isochronal experiments. Referring to Eqs. (21) and (22) it is seen that a plot of In[G(cp,- ') - G(cpi)] versus 1/T should generate a straight line if the proper value of y is used.

6 . RADIATION EFFECTS

359

From the slope and intercept of such a plot, one can then obtain the quantities A and E, . The correct value of y is determined by trial and error. Meechan and Brinkman4’ have suggested a method of analyzing annealing processes which involves the comparison of isochronal and isothermal curves obtained separately from two identical samples. Their method applied to a class of processes somewhat less restricted than those included under Eq. (20), namely, those which proceed according to the following relationship :

where F(cp) is any monotonicaliy increasing function. If F(cp) is of the form (py, this method of analysis will also yield the value of y. The main objections to the Meechan and Brinkman method are the requirement of identical samples and the necessity of carefully selecting an annealing temperature for the isothermal anneal such that the recovery occurs neither too rapidly nor too slowly. The main objection to the methods based on isochronal data only is that to obtain a definitive value for y it is usually necessary to take many points closely spaced in temperature. Both methods have the disadvantage that they are based on an assumed phenomenological model ; furthermore, if more than one process occurs simultaneously, erroneous conclusions may result. Isothermal annealing experiments are valuable for studying the details of the annealing kinetics. There are very few instances where a single process obeys Eq. (20) or Eq. (31) throughout the whole range. More common are processes which are rather complicated during the early stages of annealing but follow Eq. (20) during the latter stages, usually with y = 1 or 2. In these instances isothermal experiments can be employed to analyze the latter stages of annealing. For T = const, integration of Eq. (20) gives cp = e-Af, cpl-;, =

1 + ( y - l)It,

y = l

y# 1

where A, the rate constant, is

One can determine the activation energy by studying the temperature dependence of the rate constant I , after it has been established that beyond a certain point either first- or second-order kinetics are obeyed. The isothermal curve for a reaction consisting of two first-order processes occurring simultaneously and independently can be decomposed into two curves, each obeying firstorder kinetics. 45

C . J . Meechan and J . A. Brinkman, Phys. Rev. 103. I193 (1956).

360

L. W . AUKERMAN

In diffusion controlled reactions, the form of the annealing may be determined by complicated geometrical considerations ; however, the time and temperature dependence of cp can be expressed as cp = f ( D t ) ,

(34)

where the temperature dependence is contained in D, the diffusion coefficient. Since the effect of changing the temperature is to change the time scale, one can determine the activation energy for D by comparing isothermal annealing curves carried out on identical samples at different Processes in which one species is converted to a different species that is in turn converted to a third species, e.g., APBPC cannot be analyzed by the means described in earlier paragraphs for isochronal experiments, since the steady state concentration of the B species is usually temperature dependent. In some cases the isothermal curves for reactions of this type will follow first- or second-order kinetics after an initial transient which establishes a “quasiequilibrium” condition. Then if the temperature is changed, a new transient occurs followed again by the well-behaved annealing at a different rate. Finally it is important to bear in mind that the physical property used to trace the extent of recovery may distort the picture considerably, since different defects may affect a physical property differently, and some defects possibly not at all.

c. Annealing Processes and Annealing Kinetics (1) Intrinsic Defects. Defects which do not involve impurities are called intrinsic defects. Included in this class are vacant lattice sites, interstitial atoms, misplaced atoms (i.e., atoms on wrong sites in compounds), or various combinations of these. Electron or gamma-ray irradiation (the order of 1 MeV so that V z 1) should produce randomly located vacancies in elements such as germanium and silicon. If the density of the defects is not too great, the interstitials should be highly correlated in position with their parent vacancies, resulting in vacancy-interstitial pairs or Frenkel defects. Irradiation by heavy particles, especially fast neutrons, tends to produce a high density of vacancies near each primary event with interstitials, on the average, rather far removed from their parent vacancies. Channeling, if present, would tend to remove some of the interstitials still farther from the point of their creation. R. C. Fletcher and W. L. Brown. Phys. R e r . 92. 585 (1953) 4haT.R. Waite. Phys. Reu. 107. 471 (1957). 46bV. V. Antonov-Romanovsky. P h j x R t v . 125. 1 (1962).

6.

RADIATION EFFECTS

361

The annealing of Frenkel pairs has been discussed by Fletcher and Brown46 who divide the annealing process into three stages: (1) close pair recombination, which is first order, (2) an intermediate stage in which the mobile defect either recombines with its partner or diffuses away never to return, and (3) recombination of randomly spaced vacancies and interstitials. The latter is assumed to be a simple second-order process. This model neglects the possibility that the intrinsic defects created might interact with each other to form clusters or with impurities to form complexes. These defects, especially the latter, have recently been shown to be quite important in certain materials. In compounds, even the simplest type of damage, that resulting from low energy electron or gamma irradiation, may be considerably more complicated than in elements. For, in addition to the greater variety of intrinsic defects in compounds, there is, because of misplacements, no assurance that the numbers of vacancies and interstitials of a given type are the same. For example, in 111-V compounds low energy electron or gamma irradiation would tend to create the following correlated pairs and triplets :

+ ‘I} vB

+ Bt

v - I pairs

In the above notation, the large letter designates the species, that is whether the defect consists of an A atom, B atom, or a vacancy ; the subscript designates position, for example, at an A site, a B site, or an interstitial p o ~ itio n .~ ’ Fortunately, the triplets are probably easily converted into V-I pairs, for regardless of the energy of the primary knock-on we can expect a high degree of correlation between each misplacement and an interstitial. This can be seen by considering the probability P,(y) that an atom of reduced energy y = T/T,will create a misplacement in its next collision. To do so the moving atom must strike first an unlike atom. The probability of this can be taken to be 4. Since this is an elastic collision the atom can lose at most Ay of its energy. Consequently, to create a misplacement an atom must be left with the energy y,’ lying between 1 and (1 - 1)y after collision, while the struck atom must be left with the energy y,’ = y - yl’ > 1. Or, stated differently, a misplacement is created when (1 - 1)y < y,’ < [l and ( y - l)]; whence, 47

F. A. Kroger, “The Chemistry of Imperfect Crystals,” p. 194ff.North Holland Publ., Amsterdam, 1964.

362

L. W . AUKERMAN

PAY) = 0

otherwise.

Thus, most of the misplacements will be produced by atoms of relatively low energy, in the region y z 2. This means that if a misplacement is created there will rarely be enough energy left to remove the ejected atom more than a few lattice spacings away. Then if the interstitials are the most mobile species, a reasonable assumption for the zinc-blende structure, we can expect the lowest-temperature annealing processes to consist of annihilation of close vacancy-interstitial pairs, and/or conversion of triplets into pairs (e.g., V, + AB + B, + V, + A,). The latter reaction would remove nearly all the misplacements, leaving only vacancies and interstitials in equal numbers. At very low temperatures, however, the presence of misplacements is quite probable. Annealing of radiation damage in semiconductors, in contrast to metals, is likely to be affected by the charge state of the species that interact. This is especially important for processes that are detected by changes in carrier density, for such changes imply variations in the net bound charge localized at various defects (because of the charge neutrality requirement). Although it would not be feasible to treat every reaction imaginable, even if the mathematics were tractible, it may still be instructive to consider a few simple examples. Consider, then, a negatively charged defect which disappears by a first-order (or almost first-order) chemical rate process : D-

+ annihilation

+ e-

D - could represent for example a close V-I pair whose net charge is negative, or a single defect that migrates to some sort of sink (e.g., dislocation). Two possible situations can be imagined : (a) either the annihilation process frees the electron, resulting in a simple first-order process, or (b) the defect, for some unspecified reason, must lose its electron before annihilation can proceed. The latter case is described as follows :

Do + annihilation. K3

(35)

6.

RADIATION EFFECTS

363

Setting N - and NO equal to the concentration, respectively, of the negatively charged and neutral defect species, one writes the usual chemical rate equations

dN= -K,Ndt

~

+ K,nNo,

dNo __ = K I N - - ( K 3 dt

+ nK,)No

If the total change in carrier density is small compared to its initial value, i.e., if N 4 n, n can be considered essentially constant. With this approximation, Eq. (36) becomes a set of coupled first-order differential equations which are satisfied by the following solution : ~

N - = a l e - A 1 1+ aze-l I , N o = ble-atr + bze-l-r.

(37)

where I + and 1- are roots of the following equation

The concentration of defects remaining (N-+ N o ) can be expressed as the sum of two exponential functions in time. Frequently, it may be that the electronic part of the reaction occurs at a much greater rate than the atomic rearrangement, in which case K 3 e K 1 or nK,, and the relative occupation of N - and Nu is very nearly the equilibrium value __-

(39)

Then,'A and A- can be identified, respectively, as the electronic rate constant if 3 K,

+ n K z = K, /( l - f )

and the atomic rate constant

where

f=

[+ ( I

exp

-__ CkTC)I

-

(40)

364

L . W . AUKERMAN

is the occupation probability for the electronic energy level at E . In this case the fraction cp of the remaining defects is cp

x e-A-r

for

A+ 9 ,I-,

(43)

and, if the Fermi level is sufficiently greater than E so thatf x 1,

for the nondegenerate case. Thus, the rate constant A- and the apparent activation energy are Fermi-level dependent. In a similar manner, for the reaction

Do --t annihilation

(45)

one obtains the same result as (43) for cp with

2-

=1K3,

(46)

provided N + < n, and electronic quasi-equilibrium is maintained. If, in this case, the Fermi level is well below E so thatf < 1,

In the event that the carrier density changes appreciably during annealing, the decay is no longer strictly first order. However, the equations are still soluble4*if electronic equilibrium is maintained. Other first-order processes which may occur are (a) migration to a fixed number of sinks2 or surfaces and (b) certain trapping effects.49350 Clusters, another type of intrinsic defect, are thought to be more prevalent for vacancies than for interstitials on account of the greater strain energy for the latter. Presumably the formation of divacancies from random vacancies would be a second-order process; however, since the vacancies must be mobile for this process there is also the possibility of competing reactions, which could severely complicate matters. The presence of divacancies has been demonstrated in electron-irradiated silicon.51 The formation of large clusters or voids is analogous to the precipitation problem, which has been treated in the literature.s2 F. H. Eisen. Phys. Rev. 123. 736 (1961). A. C. Damask and G . J. Dienes, Phys. Rec. 120,99 (1960). A. Sosin. Phys. Rev. 122. 1 1 12 (1961). ” G. D. Watkins and J. W. Corbett, Discussions Faraday Soc. 31, 86 (1961) 5z F. S. Ham, J . Appl. Pkps. 30, 1518 (1959).

48

49

6.

RADIATION EFFECTS

365

The annealing behavior of an irradiated material may be expected to depend on the concentration of defects and their spatial distribution. Heavy irradiation would tend to favor clustering and bimolecular recombination in the latter stages of annealing. Very small concentrations of defects would tend to favor annihilation a t sinks or the formation of complexes. If a recoil atom has a high probability of striking its nearest neighbor, the formation of divacancies as part of the primordial damage is favored. Furthermore, viewing fast neutron damage in terms of cascade theory (as contrasted to the thermal spike’ or displacement models), one anticipates a high concentration of vacancies near the primary event with the interstitial atoms lodged somewhat farther away. On this basis it is reasonable to assume that fast neutron irradiation (and to a lesser extent heavy charged particle bombardment) would be highly favorable for the formation of clusters of vacancies, or voids. Dislocation loops can result from the collapse of void regions. (2) Defect-Impurity Complexes. During the latter stages of annealing, at temperatures sufficiently high that one or more of the intrinsic defects is mobile, there may be a tendency for vacancies or interstitials to become associated with certain impurities to form defect-impurity complexes. The nature of the driving force for this reaction may be mechanical (relaxation of strain energy) or electrical (ion pairing) or a combination of these. Complex formation is known to occur in both g e r m a n i ~ m and ~ ~ .s ~i l~ i~~ o n ~ ~ , ~ ~ ” during the high temperature annealing stages when certain impurities are present. The annealing of vacancies (or interstitials) when this sort of trapping effect is present has been treated by Damask and D i e n e and ~ ~ ~by S ~ s i n . ~ ’ The model they consider is as follows :

u

-+

K3

sinks,

(49)

where L’, I , and C are the concentration of vacancies, unbound impurities, and vacancy impurity complexes, respectively. The reaction constants J. A. Brinkman. J . Appl. Phys. 25. 961 (1954). W. L. Brown, W. M. Augustyniak, and T. R. Waite, J . A p p l . Phys. 30. 1258 (1959). 54aS.Ishinio. F. Nakazawa. and R. R. Hasiguti. J . Phys. Chem. Solids 24. 1033 (1963). ” G. D. Watkins and J. W. Corbett. Phys. R w . 121. 1001 (1961). 55aJ. W. Corbett. G . D. Watkins. R. S. MacDonald. and R. M. Chrenko. Phys:. Rec. 121. 1015 53 ‘4

(1961).

366

L. W. AUKERMAN

K 1 , K,, and K 3 are the usual Arrhenius relations involving the activation energies for migration, EM,and for binding at the trap, B, K , cc K , K K , oc , - ( E M + B ) / k T

(50)

(51)

Although the calculations are made for vacancies, the mathematics are identical if, instead, interstitials are assumed to be the mobile defect. Reaction (49) implies that the vacancies eventually disappear by migration to a fixed number of sinks. Equations (48) and (49) give rise to the following set of nonlinear coupled differential equations, dC - = KllOv - K1CV - K , C , dt (52)

where I , = I + C is the total concentration of trapping centers. Machine solutions indicate two regions of interest :(1) the initial transient stage during which the concentration of complexes increases rapidly, establishing a quasiequilibrium between complexes and vacancies, and (2) the later stage after this quasiequilibrium has been established. The transient region can usually be approximated by first-order kinetics, u cc e-". If initially C is very small,50the rate constant A is i= K,Zo

+ K3.

(53)

Thus, the transient is dominated by the migration energy EM,as would have been expected. After enough time has elapsed such that KICu is negligibly small, Eq. (52) becomes linear and can be solved by the method used for Eq. (36). This yields, for the latter stages of annealing, a rate constant of

K3

A = (Kl/K2)10

+ (K3/K2) + 1.

(54)

This reduces to the value obtained by Damask and Dienes if K 3 < K I Z o . Thus, eventually first-order kinetics will be obeyed, but the rate constant will be a composite of EM and B. In either case, 1- is a function of the concentration of traps, I , . If the vacancies disappear by recombination with an equal number of interstitials rather than at a constant number of sinks, it turns out that eventually bimolecular kinetics56 will be approached. 5h

A. V. Spitsyn and L. S. Smirnov, Fiz. Twrd. Trla. 4, 3455 (1962) [English Transl.: Societ Phys.-Solid State 4,2529 (1963)l.

6.

RADIATION EFFECTS

367

The above trapping model of Damask and Dienes was devised as a means of studying radiation damage and quenching experiments in metals ; consequently reactions involving the release of electrons or holes were not considered. This model can still be applied to semiconductors if this restriction is kept in mind, or if the equations are appropriately modified. Consider, for example, the following reaction in which electrons are freed : c?

t’+

+v+

+ e-,

(55)

+ I ?K Cz + , K3

o -+

sinks,

(57)

i.e.. the vacancy must lose an electron in order to combine with the impurity to form a complex. The asymptotic solution for this reaction can be worked out quite easily if it is assumed that electronic equilibrium is always present (i.e., the electronic processes are fast compared to the other processes) and that the change in electron concentration is small compared to the total electron concentration n, so that n is essentially constant. We then obtain

A=

K3

Wl’/K2’)IO f ( K 3 / K 2 ’ )+ 1 + ? ’

(58)

where

F - ( )

? = exp(E

(59)

and E is the energy level of the defect that becomes ionized. For the nondegenerate case, q is inversely proportional to the free electron concentration. Various other models can be considered too. Instead of requiring o to be ionized, one can consider what happens if I must be ionized to bring about complex formation. One then obtained Kl

and if C can exist in two charge states (C.+and Co)one finds

368

L . W. AUKERMAN

Of course (58), (60), and (61) apply only to the asymptotic solution where v or v + is very small. In either case, an increase in n tends to increase the rate cons tan t. In the preceding paragraphs a number of relatively simple processes were considered. It is not to be expected that, in general, annealing behavior will be so straightforward. Indeed, in assigning specificmicroscopic annealing processes to experimentally observed annealing behavior it may be necessary to bring as much experimental information and imagination to bear on the problem as possible. Thus one will be helped considerably by first selecting experiments and experimental conditions in such a manner as to bring out the simpler aspects of the annealing. If this is done, carefully executed annealing experiments should prove to be a powerful tool for studying various defect configurations and their interactions with one another.

II. Threshold Experiments 4. EXPERIMENTAL RESULTS

The threshold displacement energy, or threshold, is the minimum energy a lattice atom must receive if it is to be displaced to a stable interstitial position at low temperatures. As shown in previous paragraphs, a knowledge ofthe threshold is required for the calculation of the number of displacements created. Wigner is responsible for the first theoretical estimate of threshold energy, about 30 eV for most materials. KohnS7later made a more accurate calculation for germanium, which indicated that the threshold should be considerably lower and should be orientation dependent. The first threshold determinations for germanium by Klontz and L a r k - H o r ~ v i t z ~ ’ ” - ~ ~ ” indicated a threshold at about 30 eV, but later more sensitive measurements by Loferski and R a p p a p ~ r t ’ ~ - ’and ~ ~ by Brown and Augustyniak60.60a indicated the threshold of germanium as well as silicon to be approximately 14eV, which is in better agreement with Kohn’s prediction. Brown and Augustyniak found only a slight orientation dependence of the threshold in germanium. Eisen and Bicke16’ have found the threshold of InSb to be approximately 6 eV.

’

W. Kohn. Plrys. Rev. 94, 1409 (1954). 5’aSee also Ref. 1, pp. 328-334. E. E. Klontz and K. Lark-Horovitz. Phys. Reo. 86. 643 (1952). s*aE.E. Klontz and K. Lark-Horovitz, Phys. Rev. 82. 763 (1951). ” J. J. Loferski and P. Rappaport. Phys. Reu. 98, 1861 (1955). 59a.J.J. Loferski and P. Rappaport. Phys. Rev. 100. 1261 (1955). 59hJ. J. Loferski and P. Rappaport, Phys. Rev. 111,432 (lY58). ‘O W. L. Brown and W. M. Augustyniak. Bull. Am. Phys. SOC. 2, 156 (1957). ‘OaW.L. Brown and W. M. Augustyniak. J . Appl. Pkys. 30. 1300 (1959). 6 1 F. H. Eisen and P. W. Bickel. Phys. Rev. 115. 345 (1959). ”

’*

6.

RADIATION EFFECTS

369

The results of InSb are shown in Fig. 1, which is a plot of the carrier removal rate as a function of the energy of bombarding electrons. The specimen must be very thin for this type of experiment. In this case the specimen thickness was 170 p .

ELECTRON ENERGY, MeV

FIG. 1. Damage rate dii/dN, as a function of the energy of bombarding electrons for lnSb at 78°K. (After Eisen and BickeL6')

It is usually assumed that the threshold is sharp, i.e., that the probability of creating a displacement is zero below the threshold & and jumps abruptly to unity for energies equal to or greater than Td. However, this assumption has not been experimentally verified. Various methods of generalizing cascade theory have been attempted, such as (a) considering the probability of displacement to vary linearly over a range of energies31 and (b) treating the threshold as a random ~ a r i a b l e . ~Neither * . ~ ~ method has been very successful in improving the comparison between theory and experiment, however. Experimentally, to determine thresholds one would like to start irradiating at some low subthreshold energy and gradually increase the energy until a change of some physical property is observed. This would define the electron energy E , corresponding to & = T, in Eq. (3). Unfortunately the problem is rarely this simple. A number of complications enter, one of which is the subthreshold tail, or the presence of incipient damage at energies below threshold. Thus, if one plots the damage rate, or something believed to be at least a monotonic function of the damage rate, as a function of the energy of the bombarding particle, an abrupt decrease is observed as the energy is lowered. But the damage rate, rather than dropping to zero at a welldefined E,, exhibits a low energy tail as shown in Fig. 1 for InSb. This is typical behavior for semiconductors, although, possibly because of varying

370

L. W . AUKERMAN

degrees of sensitivity to the defects, it is not always observed. The low energy tail is variously explained as (1) displacements near dislocations or other structural defects, (2) displacement of lighter impurities (hydrogen) which in turn displace host atoms,6Z and (3) surface effects. Whatever is the correct mechanism for this tail, it apparently does not represent the generation of true displacements of typical host atoms by the bombarding particle. Consequently, to estimate displacement energies, some form of extrapolation is necessary. The shapes of curves of damage rate versus electron energy are not well understood, even for silicon63 or germanium. B a ~ e r l e i n assumes ~ ~ - ~ ~that ~ the square root of the damage rate should increase linearly with electron energy E as follows: cc E - EB,

where EB is the electron energy which imparts to the atom in the most favorable collision [see Eq. (3)],fi is the average concentration of defects, and N , the integrated electron flux. This relationship can be verified, for a restricted energy range, on the basis of a highly simplified a r g ~ m e n t , ~ ~ - ~ ~ ' but a more precise theoretical treatment would appear to predict a slightly different b e h a ~ i o r . ~Nevertheless ' the experimental behavior does seem to fit Eq. (62) remarkably ell.^^^' By plotting the damage rate for various compounds on a quadratic scale, .~ able the resulting damage rate curves into B a ~ e r l e i n is ~ - ~to~ decompose ~ two straight-line sections obeying Eq. (62). This gives two values for EB which presumably correspond to the thresholds for the two constituents. Where a considerable mass difference exists between the two constituents the values of E, can be assigned unambiguously to the constituents. However, for constituents of nearly equal mass (such as InSb and GaAs) there is a certain arbitrariness in the assignment of E,. The results are presented in Table 11, most of which is taken from B a ~ e r l e i n . ~The " ~ ~compounds ~ for which the assignment of EB is not ambiguous are InP, I d s , and CdS. In each of these cases the element from Column I11 or I1 is observed to have the lower threshold. Therefore, in assigning EB values for the other compounds Bauerlein assumed that for these too the type 111 or I1 element has J. A . Naber and H. M . James. Bull. Ant. Phys. Soc. 6. 303 (1961). H. Flicker. J. J. Loferski. and J. Scott-Monk, Phys. Rec. 128, 2557 (1962). 64 R. Bauerlein, Z. Physik 176, 498 (1963). 64aR. Bauerlein. Z. Naturforsch. 14a, 1069 (1959). 64bR. Bauerlein. Proc. Znrern. School Pkys. "Enrico Fermi" XVIII. 358 (1 962). b 5 B. Ya. Yurkov. Fiz. Tcerd. Tela 2. 2710 (1960) [Eng-Lisli Transl.: Sovier Phys-Solid Srare 2. 2412 (1961)l. 63

TABLE 11 THRESHOLD DISPLACEMENT ENERGIES I N SEMICONDUCTORS ~~~

~

~~

~~~~~~

~~~

InP

Substance

SI

Ge

Displaced atom

Si

Ge

Ga

As

In

P

In

AS

In

Sb

Zn

Se

Cd

S

Threshold electron energy EB(keV) 173

355

228

273

270

110

277

236

247

286

238

325

290

115

Threshold displacement energy T,(eV) 15.9”

14Sh.‘

8.8‘

10.1’

6.6’

8.8’

5.8d.e 6.8’.‘ 6.4’ 8.5-9.9’

9.9‘”

2.9P

5.6’

10.2’

3.85’

5.65’

Self-diffusion energy (eV)

5.13‘

GaAs

InAs

6.7‘

InSb

8.5’

1.82h

ZnSe

CdS

P 11.9d3J

7.3’

8.7’

;d

z

5

1.94h

E;

2 From Ref. 63. From Refs. 59-59b. From Refs. 60, 60a.

‘From Refs. 64-64b.

From Ref. 61. From Refs. 66-66b.

B. A. Kulp and R. M. Detweiler. Phys. Rev. 129. 2422 (1963). 66aB.A. Kulp, Pliys. Rev. 125, 1865 (1962). 66bB.A. Kulp and R. H . Kelley. J . A p p l . Pliys. 31, 1057 (1960). ’’ W. M. Portnoy, H . Letaw, Jr., and L. Slifkin, Phys. Rev. 98, 1536 (1955). 67aF.H. Eisen and C. E. Birchenall, .4cfa.Met. 5. 265 (1957j. b7bB.Goldstein. Phys. Reu. 121. 1305 (1961). “‘F. H. Eisen. Pltys. Re[>.135. A1394 (1964). 6’dJ. M. Fairfield and B. J. Masters, J. Appl. Phys. 38, 3148 (1967). b* F. H. Eisen. Bull. Am. Pliys. Soc. 9. 290 ( I 964).

From Ref. 67. * From Ref. 67a. i From Refs. 67c. 68.

‘ J

From Ref. 67b. From Ref. 67d.

m -I -I

5 v1

66

Y Y

372

L . W . AUKERMAN

the lower threshold. For further justification he points out an analogy with the activation energy for self-diffusion, which is smaller for the type 111 element in those I1 I-V compounds for which diffusion s t u d i e ~ ~ ’ ~ have ’~’~ been carried out. In addition to the fact that the more metallic element has the lower threshold, it is also apparent that the partially ionic 111-V and 11-VI compounds in general have lower threshold values than do the covalent elements from the fourth column, Ge and Si. Bauerlein also points out a correlation between the difference in threshold energies of the two constituents and the ionicity or effective ionic charge for the 111-V compounds of Table 11. Thus if the assignments used in Table I1 are correct, it would appear that a greater effective ionic charge corresponds to a greater difference in the thresholds for the two constituents. More recently, E i ~ e nhas ~ shown ~ ~ , that, ~ ~although it is true that in InSb the indium atom has the lower threshold, the threshold for Sb is not represented by the kink in the curve to which Bauerlein ascribes it, but rather occurs at a higher energy. Eisen’s recent threshold estimates are included in Table 11. His method of separating the In displacements from the Sb displacements rests upon the interpretation of annealing experiments which will be discussed in Section 7e. From cathodoluminescence measurements in GaAs, Loferski et obtain two threshold values at about 300 and 350 keV (in terms of electron energy EB).These are slightly higher than Bauerlein’s values. Grimshaw and Banbury68bestimate the threshold in GaAs to be about 17 eV (EB% 390 keV) by comparing the shapes of curves of carrier removal rate and calculated displacement cross section vs energy. 5. DISCUSSION

In the threshold determinations summarized in Table I1 every effort was made to maximize the sensitivity to defects so that very low introduction rates could be detected. Now it is almost invariably true that, for energies well in excess of the threshold, the value of & most appropriate for calculating the concentration of defects is considerably larger than the threshold value listed in Table 11. Conversely, if, in a threshold experiment. the presence of defects is indicated by a property that is relatively insensitive to radiation effects, a considerably higher threshold value is usually obtained. For example the early threshold determination for Ge by Klontz and LarkHorovitz a-s 8a gave 30 eV, whereas the more sensitive method of Loferski and R a p p a p ~ r t ~ yielded ~ - ’ ~ ~about half this value. More recent studies by 68aJ. J. Loferski, H. Flicker, M. H. Wu. R. M. Esposito, W. Patterson. and J. Schreiber, Jr., “Radiation Effects in Semiconductors” (Final Report, June 1965),Brown Univ., Providence, Rhode Island. AF33(657)i3. 68bJ. A. Grimshaw and P. C. Banbury. Proc. P h y . Soc. (London)84. 151 (1964).

6.

RADIATION EFFECTS

373

Wikner et aZ.,6’ indicate that the ratios of damage rates for 5, 1 5 , and 45-MeV electrons are consistent with threshold values of & = 30eV for Ge and 25 eV for Si. For both InSb70~70a and InAs’I the carrier removal rates resulting from 4.5-MeV electron irradiation at 80°K are more consistent with a threshold energy of about 30 eV rather than the lower values listed in Table 11. Finally for InSb and GaAs, V 0 0 k ~estimates ~ effective threshold energies of about 44 eV and 45 eV, respectively, on the basis of dilatation measurements, which are relatively insensitive to structural defects. The reasons for such vast discrepancies between the measured threshold and the effective threshold needed to account for defect generation rates are not understood. Most workers in the field generally employ rule-of-thumb threshold values of about 30 eV for calculating damage rates at energies well above the experimentally determined threshold.

111. Radiation Effects in Various 111-V Compounds 6. COMMENTS REGARDINGENERGYLEVELS It is an experimental fact that in nearly every semiconductor studied both donors and acceptors are introduced by bombardment. James and LarkHorovitz7 proposed a model for silicon and germanium whereby the acceptors were identified as vacant lattice sites and the donors as interstitials. They also presented arguments indicating that these sites, if isolated, could be singly or doubly ionized depending on the position of the Fermi level. Although this model has been quite useful in interpreting experiments qualitatively, it does not appear adequate to explain quantitatively all observations. In a similar model by B10unt’~each defect can act either as a donor or as an acceptor, depending on the position of the Fermi level. Either model is capable of explaining the gross features, namely, that the major effect of radiation damage in semiconductors, as far as electrical properties are concerned, is to introduce defect states in the forbidden gap. These defect states alter the carrier density, mobility, and carrier lifetimes in much the same manner as defect states resulting from impurities. Much of the radiation damage research on semiconductor materials has been concerned with detecting bombardment-produced energy levels. Many levels have been detected in germanium and silicon, fewer in compounds. E. G. Wikner, H. Horiye, and J. W. Harrity, 1.Phys. Sac. Japan 18, Suppl. 111,222 (1963). L. W. Aukerman. Phys. Rev. 115, 1125 (1959). ’OaL. W. Aukerman and K. Lark-Horovitz. Bull. Am. Phys. Soc. 1. 332 (1956). ’*L. W. Aukerman. Phys. Rev. 115. 1133 (1959). l 2 F. L. Vook, J . Phys. SOC.Japan 18, Suppl. 11, 190 (1963). 7 3 H. M. James and K. Lark-Horovitz, Z. Physik. Chem. (Leipzig) 198. 107 (1951). 7 4 E. I. Blount. J . Appl. Phys. 30. 1218 (1959). b9

’*

374

L . W . AUKERMAN

Clear-cut explanations which relate the levels to the defects and their geometrical arrangement have been generally lacking. Many methods are availabL for locating bombardment-produced energy levels, Those most frequently employed fall into one or the other of two classes : (a) those which take advantage of the occupation statistics7' of the level and (b) those which involve optical excitation between a discrete level and a band edge. Examples of the f ~ r m e r ~are - ~temperature dependence of carrier density or minority carrier lifetime, and dependence of carrier removal rate on the position of the Fermi level. of the latter, (b), are, of course, spectral dependence of optical absorption and of photoconductivity, and other photoelectronic effects. Some of the energy levels which have been found in irradiated 111-V compounds are listed in Table 111. This list is probably not complete for any of TABLE 111 ENERGYLEVELSIN IRRADIATED II1-V SEMICONDUCTORS' InSb

(0.04-0.1)b Electron or gamma irradiation [0.12]k [0.055]'

[0.081Ib [0.048Ib

{ 0.02i d Fast neutron irradiation4

GaAs

(0.03)" (0.062) {0.083)p { 0.103i

{ 0.13 ) r,n (0.381"

[0.04]Op [0.03Ib.'

[0.35]" [0.05]'

jO.01:' (0.03)' [0.08]'

(0.1i8 {0.5ig

[0.06]' [0.0544.06]' [0.034]'

In eV from nearest band edge. Braces { 1 refer to energies below the conduction band, and brackets [ I to energies above the valence band. From Refs. 70, 70a. From Ref. 76. From Refs. 17, 71a. From Ref. 78. From Refs. 79, 79a. From Ref. 80. From Ref. 81.

InP

GaSb

(0.12-0.20;" (0.47-0.50)"

(0.52)"

[O.6lg

[0.48]"' [0.075]" (0.285:h [0.21' [O. 141'

i From Ref. 82. 'From Ref. 83. From Ref. 83a. From Refs. 84, 84a. " From Ref. 84b. " From Ref. 84c. From Ref. 84d. p From Ref. 84e. 91n some cases of neutron irradiation a Idatively large gamma Rux may be present.

' '

6.

RADIATION EFFECTS

375

the substances listed. Furthermore, there is no assurance that any of the levels listed (except possibly for InSb at liquid nitrogen temperature) correspond to intrinsic defects. After a sufficiently large irradiation, the concentration of electrically active impurities is small compared to the concentration of structural defects introduced, and the position of the Fermi level should be determined solely by the distribution of defect levels in the forbidden This saturation value of the Fermi level, sometimes called the final Fermi level &;,should be independent of the amount of irradiation required to approach it. That is, ideally, the final Fermi level should be independent of the initial Fermi level. Similarly, the change in carrier density divided by the integrated flux that produces the carrier density change An/& when plotted as a function of the Fermi level position, should generate a curve that is characteristic only of the bombardment levels i n t r ~ d u c e d . ~Such - - ~ a plot is frequently used to estimate positions of energy levels. In real situations, however, plots of Anfq5 vs (, and the position 5, may depend on the initial properties if one or more of the following conditions are present : (a) macroscopic inhomogeneities, (b) long lifetime trapping effects, (c) impurity-defect complexes, and (d) annealing at the bombarding temperature. The final Fermi level 5, represents the Fermi level position that would result in the net bound charge of the radiation produced defects being zero, i.e., the ionized acceptors and donors cancel each other. Thus any conditions, J. S. Blakemore, "Semiconductor Statistics.'' Pergamon, London, 1962. F. H. Eisen. Bull. Am. Phys. Soc. 8, 235 (1963). " J. W. Cleland and J. H. Crawford. Jr., Phys. Rev. 95. 1177 (1954). '7aJ. W. Cleland and J. H. Crawford. Jr.. Phys. Rev. 93, 894 (1954). 7 8 H. J. Stein, Bull. Am. Phys. SOC. 7, 543 (1962). 7 9 L. W. Aukerman and R. D. Graft, P h p . Rev. 127, 1576 (1962). '98L. W. Aukerman, P. C. Peters, and R. D. Graft, Bull. Am. Phys. SOC.6, 177 (1961). L. W. Aukerman, P. W. Davis, R. D. Graft, and T. S. Shilliday. J . Appl. Phys. 34. 3540 (1963).

75

76

L. W. Aukerman, E. M. Baroody, R. D. Graft, and T. S. Shilliday, "Theoretical and Experimental Studies concerning Radiation Damage in Selected Compound Semiconductors," Rept. No. ARL 62-343, May 1962. J. W. Cleland and J. H. Crawford, Jr., Phys. Reo. 100, 1614 (1955). " M. A. Krivov, E. V. Malisova, and V. Malyanov, I Z D . Vysshikh. Uchebn. Zavedenii Fiz. No. 2, 1 1 4 118 (1963). ''=R. A. Laff and H. Y. Fan, Phys. Rev. 121, 53 (1961). 84 C. Georgopoulos. Thesis (unpublished), Grenoble Univ., March 1964. 84a C. Georgopoulos, M. Verdone, and D. Dautreppe, Compt. Rend. 257,2640 (1963). 84bR. Kaiser and H. Y. Fan, Phys. Rev. 138, A156 (1965). 84cN.A. Vitovskii, T. V. Mashovets, S. M. Ryvkin, and R. Yu. Khansevarov, Fiz. 'Tuerd. Tela 5, 3510 (1963) [English Transl.: Soviet Phys.-Solid State 5, 2575 (1964)J 84dF.H. Eisen, Phys. Rev. 148, 828 (1966). 84eT. V. Mashovets and R. Yu.Khansevarov, Fiz. Tuerd. Tela 7, 2229 (1965) [English Transl.: Soviet Phys.-Solid Srare 7 , 1796 (1966)l.

376

L. W. AUKERMAN

such as the type or energy of the bombarding particles or the temperature of irradiation, that result in a different distribution of energy levels will, in general, result in different [, values. 7. INDIUM ANTIMONIDE a. Gross Features The electrical properties of irradiated InSb are very sensitive to the temperature and type of irradiation. This fact is reflected by the lFvalues for different irradiation conditions. At liquid nitrogen temperature iFis close to or below85 E, (the top of the valence band) for 1-MeV electron irradia' , ~ ~is~closer t i ~ n , it* ~ is ~about E, + 0.03 eV for 4.5-MeV e l e ~ t r o n s ~and to the middle of the gap7*or above it85bfor fast neutron irradiation. It is as if 5, increases as the primary events become more severe. However, there are some conflicting reports concerning the effects of fast neutrons on InSb. These will be discussed later. The effect of irradiation temperature on l F ,for 4.5-MeV e l e ~ t r o n s , ~ ~ , ~ ~ ~ is as follows: at 80"K, 200"K, and room temperature, CF is at E, + 0.03 eV, E, 0.075 eV, and close to the conduction band, respectively. For room l F = E, - 0.015eV. Thus, intemperature creasing the temperature of irradiation also causes CF to increase. An increase corresponds to the creation of additional donors, or the shifting of in energy levels toward the conduction band. Since increasing the temperature of irradiation or the maximum energy imparted to the primary atom might tend to decrease the correlation between vacancies and interstitials, the above observation suggests that the degree of correlation has a profound effect on the energy distribution of the levels, shifting it away from the valence band as the degree of correlation is diminished. If the above observation seems speculative, perhaps it can still serve as a useful mnemonic device for recalling the various trends described above. Nevertheless, a tendency for strong interaction between defects in this material would not be surprising in view of the extremely large Bohr orbits for hydrogenic donors ( 500 A). Bombardment also introduces ionized scattering centers which affect the mobility7' and the magnetoresistance.85dThe magnitude of the changes in mobility indicate that some of the defects introduced are more than singly ionized.

+

c,

N

F. H . Eisen, Bull. Am. Phys. SOC. 7, 187 (1962). s5aThisfact is derived from Eisen'ss5 reported increase in carrier density resulting from I -MeV electron bombardment of fairly heavily doped p-type material ( lOI6/cm3). *"L. K. Vodopyanov and N. 1. Kurdiani, Fiz. Tverd. Tela 7, 2749 (1965) [English Trans!.: Soviet Phys.-Solid State 7, 2224 (1966)l. 85EReactor irradiation can also create an appreciable concentration of transmutations by virtue of the thermal and low-energy neutrons present. In the experiments cited here, transmutations were eliminated by shielding with cadmium and indium foils. 85dP.C. Euthymiou, Phys. Status Solidi 8. 131 (1965). 85

-

6.

371

RADIATION EFFECTS

Irradiation with Co-60 gamma ray^^^^^^^,^^+^^ (- 1.25 MeV) appears to produce effects similar to 1-MeV electron irradiation. Gamma irradiation at liquid nitrogen temperature decreases the n-type and increases the p-type carrier density. The latter is true, for example, if the hole concentration is - 5 x lOI5/cm3. In this case the trend is the same as observed for 1-MeV electrons, but opposite to that observed with 4.5-MeV electrons. Kurdiani”” investigated the optical absorption of n-type InSb irradiated at room temperature with reactor neutrons (8.16 x 10l6 neutrons/cm’). The irradiation decreased the absorption out to 17 p at 100°K and produced a broad absorption band at about 13.2 p (0.093 eV), which broadened considerably and shifted toward longer wavelengths on increasing the temperature. The decreased absorption is probably due to the decrease in carrier concentration that occurred as a result of the irradiation. Vacuum annealing to 350°C for 40 hours restored the initial properties. b. Energy Levels

The energy levels found in irradiated InSb were summarized in Table 111. Various methods were employed in obtaining the levels listed. The levels for 80°K irradiation were estimated from plots of An/4 vs Hall effect data,84,84aand from photoelectronic methods.84e The levels near the band edges for the 200°K irradiations were also obtained from ,444 vs [. The levels at 0.048 and 0.081 eV above the valence band were determined more accurately from the temperature dependence of the Hall coefficient. The 300°K levels were determined from temperature dependence of Hall coefficient ( E , - 0.02 eV) and PEM and PC measurements ( E , + 0.12 and E , + 0.05eV). Warming to 200°K after irradiation at 80°K with 4.5-MeV electrons has the same effect as irradiating at 200°K. In comparing energy levels determined by different methods one must keep in mind the fact that the activation energy obtained from Hall effect data usually corresponds to the value at zero degrees absolute, whereas the value obtained from An/4 vs [ plots is the value for the temperature at which the measurements are made. In the latter case there is the additional uncertainty in the “statistical weighting” factor75and the usual scatter of points. Focusing attention to the region 0.3-0.06 eV above the valence band one sees the following sequence : electron bombardment creates a level at about 0.03 eV (or 0.04 eV76*84d) at 80”K, one at 0.048 eV at 200”K, and one at 0.055 eV at room temperature. Fast neutron bombardment at liquid nitrogen [,70370a*78,84d

C. R.Whitsett, Bull. Am. Phys.Soc.3, 142(1958). K. D. Alexopoulos and R. B. Oswald. Jr., in “Radiation Damage in Solids.” Vol. 111. p. 49. Intern. Atomic Energy Agency, Vienna. 1963. 87aN.1. Kurdiani, Fiz. Tiierd. Tela 5, 2022 (1963) [English Transl.: Societ Ph):s.-Solid State 5. 87

1477 (196411.

378

L . W. AUKERMAN

temperature creates what might be the same level as room temperature electron bombardment, at about 0.056 eV. (The level at E, + 0.034 eV is probably at least in part due to the reported by Georgopoulos et large gamma flux present.) This sequence suggests the interesting speculation that these three levels may be related, possibly derived from vacancy-interstitial pairs or triplets having different degrees of correlation, as suggested above. Studies of lifetime performed by Laff and Fan83aindicate that the level at 0.055 eV was present before irradiation in rather small concentrations (approximately 1014/cm3)and is a donor. The level at 0.048eV in 200°K irradiated InSb also appear^^^,^'^ to be a donor. The charge on the 0.03-eV (or 0.04-eV) level has not been definitely established experimentally ; however, the increase in hole concentration reported by E i ~ e nimplies ~ ~ the existence of a shallow acceptor level. The 0.03-eV level, then, is in all likelihood an acceptor. The 0.055-eV level in unirradiated specimens is probably an isolated vacancy resulting from a slight degree of nonstoichiometry. The level at 0.12 eV is, according to the analysis of Laff and Fan, associated with the same defect in a different charge state. The 0.048-eV and 0.08-eV levels at 200°K may arise from the same vacancy, their energies being perturbed slightly by the presence of a nearby interstitial. The estimated introduction rate^^^*^^^,^^^ by 4.5-MeV electron irradiations are 3.5/cm, 1.5/cm, and 1.5/cm for the levels at 0.03 eV, 0.048 eV, and 0.55 eV, at 8O"K, 200"K, and 300"K, respectively. Since p-type InSb, heavily irradiated with 4.5-MeV electrons, has been observed to convert to n-type upon annealing to room t e m p e r a t ~ r e , ~ ~ " ~ ~ and since the donor observed at room temperature (-0.055 eV) lies in the lower half of the forbidden gap, it follows either that one or more additional defect is stable at room temperature or that the 0.055-eV level represents a multiply ionized state of the defect. The group of levels in the range 0.040.1 eV below E , in 200°K electron irradiated specimens present a net a c c e p t ~ r behavior ~ ~ , ~ ~when ~ one analyzes the mobility. Therefore, the energy level distribution goes through another rather drastic change between 200°K and 300°K. Complex formation might be important in that temperature range. The level at 0.3 eV below the conduction band produced by 1-MeV bombardment is also an acceptor.84d ~

1

.

~

~

7

~

~

~

c. Elongation

One of the most striking differences between III-V compounds and group IV semiconductors shows up in the response of lattice expansion to irradiation. Figure 2 shows the lattice expansion AL/L for InSb, GaAs, Ge, and Si as functions of the integrated 2-MeV electron flux at 77°K. This figure, taken from the work of V ~ o k , ' shows ~ , ~ ~that the specific lattice strain for

6.

0

379

RADIATION EFFECTS

4

4,

8 12 10" ELECTRON SIC^

16

20

FIG.2. Fractional lattice expansion at 77°K for various semiconductors. (From Vook.'*)

structural defects is much greater in InSb and GaAs than in Ge or Si. Vook shows that this is due to a greater lattice strain per defect, rather than a greater introduction rate. The lattice strain per Frenkel pair in lnSb is about one atomic volume, and in germanium only about 0.02 atomic volume. The physical reason for this result most probably involves the partially ionic character of the 111-V semiconductors. Vook suggests it arises from repulsion between the four charged nearest neighbors to a vacancy. However, no attempt was made to include the effects of misplacements. Vook performed irradiations at 2.0 and 1.4 MeV. The ratio of the rate of change of lattice strain with flux for these two energies was 4.5 k 1.5. This yielded a threshold energy of about 44eV, considerably larger than the value of -6 eV obtained by Eisen and Bicke16' with lower energy electrons. The possible causes of such discrepancies were discussed earlier in Section 4. Gonser and Okkerse88-88b irradiated GaSb and InSb crystals at 140°K with 12-MeV deuterons in a direction perpendicular to the (100)planes and studied x-ray reflections after 2 x 10" deuterons/cm2. Their observations indicated a radiation induced contraction and simultaneously an increase in lattice constant. These results were explained by proposing that displacement spikes are produced, i.e., regions where minute quantities of the specimen have melted and then solidified in a metastable amorphous form. Since the density of the amorphous (liquid) form is greater than that of the crystalline form, the lattice must expand in order to make up the difference in '

-

'' U. Gonser and B. Okkerse, Phys. Rev. 105,757 (1957). ssaU. Gonser and B. Okkerse, Phys. Rev. 109, 663 (1958). 88bU.Gonser and B. Okkerse, J . Phys. Chem. Solids 7, 55 (1958).

380

L. W. AUKERMAN

volume ; but the overall increase in average density requires a decrease in the macroscopic dimensions. The contraction disappears on warming to about 160°K; in fact, a slight expansion was observed above this temperature. The electrical resistivity increased suddenly with irradiation and then slowly decreased. The initial increase was probably due to the generation of point defects. The gradual decrease, it was suggested, was due to the amorphous regions, which should have a metallic-like conductivity. Kleitman and YearianX9irradiated GaSb, InSb, and Ge with 9-MeV deuterons and observed interferometrically an expansion of GaSb and InSb, but none of Ge, at 20°C. This is consistent with the results of Vook, since Kleitman and Yearian apparently did not have the required sensitivity for detecting ALIL in Ge. Vook found that considerable ALIL remained in InSb after annealing to room temperature. The nature of AL/L changes during annealing will be discussed in later paragraphs dealing with annealing behavior in InSb. d. Thermal Conductivity

The effects of 2-MeV electron irradiation at about 50°K on the thermal resistivity of high purity InSb have been studied by Vook.' 7,90,90a The added thermal resistance increased as the 1 power of the integrated flux, rather than the first power observed for GaAs. These results are illustrated in Fig. 3. Furthermore, the increase in thermal resistance was orders of magnitude greater for InSb than for GaAs, and was not proportional to the absolute temperature. Such large discrepancies between InSb and GaAs were not present in the elongation72experiment. The scattering of phonons by the strain field surrounding a defect or by the difference in mass between the defect and the host atoms has been treated by K l e r n e n ~ ~ and ' , ~ ~Ziman.92 ~ They predict that the defect thermal resistivity is proportional to the first power of the absolute temperature and to the first power of the concentration of defects. The failure of InSb to conform to this theory indicates that a different scattering mechanism must be present. Vook was able to account semiquantitatively for the behavior of InSb in terms of the electron-phonon scattering theory of K e y e ~ ,which ~ ~ ,is~based ~ ~ on the large effect of strain on the defect electronic wave functions when warped energy surfaces are present. This mechD. Kleitman and H. J. Yearian, Phys. Rev. 108, 901 (1957). F. L. Vook. Phys. Reo. 135. A1750 (1964). 90aF. L. Vook, Phys. Rev. 149,631 (1966). 9 1 P. G. Klemens, Proc. Phyb. SOC. (London) A@. 11 13 (1955). 91aP.G . Klemens, Solid Stute Phys. 7, 1 (1958). 92 J. M. Ziman, Can. J . Phys. 34, 1256 (1956). 9 3 R. W. Keyes, Phys. Rev. 122, 1171 (1961). y3nR. W. Keyes, IBM J . Rey. Develop. 5. 266 (1961). 8y

90

6.

RADIATION EFFECTS

1018

101’

1016

381

9 NUMBER OF ELECTRONS/cm2 FIG.3. Increase in additive thermal resistivity of InSb and GaAs upon low temperature 2-MeV electron irradiation. (From Vook.“)

anism is not effective for GaAs, which becomes nearly intrinsic rather than p-type upon irradiation. The scattering model treated by Keyes is sensitive to the occupation of electronic levels and their dependence on strain. A later calculation by E r d o actually ~ ~ ~ predicts ~ the 2 power dependence for point defects in the strong scattering limit. e. Annealing Experiments

The most thorough annealing experiments on InSb to date are those ~ ~isochronal ~ , ~ ~recovery , ~ ~ curves , ~ ~for , ~ ~ carried out by E i ~ e n . ~ ~ ,The typical p - and n-type samples irradiated with 1-MeV electrons at liquid nitrogen temperature are shown in Fig. 4. Five well-separated recovery stages are seen in the n-type specimens. Eisen designates these as Stages I through V, starting at the lowest temperature stage. Earlier70370a annealing studies on n-type specimens irradiated with 4.5-MeV electrons show the same general features, except that Stages 111 and IV apparently were not resolved. In both cases nearly all the observable damage in n-type InSb was removed after Stage V was completed. The monitoring property was carrier density (from Hall effect measurements). found the annealing of Co-60 gamma irradiated Georgopoulos, et InSb to be almost identical to that of electron-irradiated specimens. Reactor ~

1

.

~

~

9

~

~

~

93bP.Erdos, Phys Rev. 138, A1200 (1965). 94 F. H. Eisen. in “Radiation Damage in Semiconductors” (Proc 7th Intern. Conf.), p. 163. Dunod, Paris and Academic Press, New York, 1964.

382

L. W . AUKERMAN

TEMPERATURE, 'K

FIG. 4. Isochronal annealing of n-type (curve B) and p-type (curve A) InSb after I-MeV electron irradiation. (From Eisen.")

irradiation carried out under conditions of a rather large accompanying gamma flux produced essentially the same annealing characteristics, except that the stages 111, IV, and V were relatively more prominent. Georgopoulos observed the annealing of mobility to occur at a slightly lower temperature than the carrier density. Similarly, Eisen found for electron irradiated specimens that the annealing of mobility slightly precedes the annealing of carrier density. The intriguing consequences of these observations will be discussed later. The annealing behavior of p-type specimens is quite different for 4.5-MeV irradiation as compared to 1-MeV irradiation. One-MeV electron irradiation of p-type specimens with carrier density in the IOl6/cm3 range increases the carrier density and decreases the mobility, when performed at liquid nitrogen temperature. The first two annealing stages following this treatment decrease the carrier density and increase the mobility. The higher temperature annealing stages are quite complex, some of them temporarily reversing the annealing trend in carrier density. This behavior is illustrated in Fig. 4. Irradiation with 4.5-MeV electrons decreases the carrier density and mobility if the density of free holes is initially greater than 3 x lOl5/cm3, while the annealing stages between 80°K and 200°K continue this trend. That is, reverse annealing of both carrier density and mobility is observed. For more pure specimens ([ > CF) the trend is the same as for the 1-MeV experiments. In Fig. 5 the isochronal annealing of a typical p-type specimen (p = 1.2 x 1016/cm3)irradiated with 4.5-MeV electrons is shown. Annealing in the 200°K to 300°K temperature range tends to restore the initial electrical properties, but most of the apparent damage is still present at room temperature. In discussing the 4.5-MeV results, A ~ k e r m a n ~ ' , points ~ ' ~ ~ out ~ ~ that y5

L. W. Aukerman. J . Appl. Phys. 30, 1239 (1959).

6. 35 1

I

I

I

383

RADIATION EFFECTS I

I

I

I

I

I

I

a-, 0

- 1.0

7,

-0.8

s . k

0

c

-0.6 +I pL

- 0.4

I

0

FIG.5. Isochronal annealing of lattice expansion and carrier density in electron-irradiated p-type InSb. (After V ~ o k . ~ ~ )

annealing in the 78" to 200°K range always increases the net bound charge, as if the defects were becoming more donor-like. This observation holds for both p- and n-type InSb, and can be extended to the results of Eisen as well. The major difference between 1-MeV and 4.5-MeV irradiation seems to be the generation of additional donors by the latter, thus moving the final Fermi level farther into the forbidden energy region. Eisen's work with 1-MeV irradiated n-type InSb revealed48 properties of the various stages as shown in Table IV. For comparison some results of G e o r g o p o ~ l o (gamma s~~ irradiation) are included.

cF

TABLE IV SUMMARY OF

Annealing stage

I-MeV RECOVERY DATAFOR n-TYPE InSb

Center temperature Fractional recovery ( O K )

Activation energy (eV) Eisen4*

Georgop0ulos8~

I I1

90 150

0.232 0.615

0.34 rf: 0.01 0.60 k 0.02

111

175 210 275

0.034 0.027 0.092

0.70 k 0.02 0.79 k 0.02 0.96 k 0.03

IV V

0.62 0.74 0.79 0.86"

Assumed to be second order. The others were assumed to be first order.

384

L . W . AUKERMAN

Eisen was able to follow various annealing stages as a function of bombardment energy. Thus, the threshold for Stage 11 is about 270 keV and for Stage I about 400 keV. In studying the orientation dependence of the damage near these thresholds,67c968 Eisen finds that Stage 11 damage is produced at a faster direction than when it is directed rate when the beam is directed along a along a [ l l l ] direction, whereas the reverse is true for Stage I. When the direction, the impulse it gives bombarding electron is directed along a [i-li] to a n In atom is directed toward an interstitial position, i.e., an open space in the lattice, but the impulse given to an Sb atom is directed toward the next nearest In atom. Therefore, one expects easier formation of In displacements when the beam is in the [TTT] direction and easier formation of Sb displacements when the beam is in the [ l l I] direction ; in other words, Stage I1 is due to In displacements and Stage I is due to Sb displacements. Thus, the threshold displacement energy is about 6.4 eV for indium and about 9 eV for antimony. The different energy dependence of Stage I and Stage I1 damage provides a convenient and certain means of identifying the two stages. In this manner the annealing stage in p-type samples that occurs between 87" and 103"K, depending on the carrier density, has been identified as Stage 11. Otherwise, it could have been mistaken for Stage I annealing. No annealing was observed between 4°K and the region of Stage I annealing. The annealing of Stage I1 can be described as two independent almost first-order processes. The reaction constant, hence T,, the temperature at which the annealing is half completed in an isochronal experiment, is a function of carrier density. The failure of first-order kinetics to be strictly followed was attributed to the partial ionization of one of the defects involved. As pointed out in Section 3c, any reaction accompanied by the capture or release of a hole or electron may be described by nonlinear differential equations if the change in carrier density is appreciable during the reaction. Of two specimens with carrier density of l O I 4 and 10" electrons/cm2, respectively, the one with the lower carrier density passed the inflection point T, at a temperature about 7" lower than the other. Eisen showed that this is quantitatively consistent with the of Eq. ( 3 3 , which (assuming electronic equilibrium) predicts the reaction constant i to be inversely proportional to the carrier density [Eq. (44)]. In a more recent publication E i ~ e shows n ~ ~ that ~ there are actually two energy levels associated with Stage I1 annealing : one at E , + 0.04 eV and one at E , - 0.03 eV. Eisen attributes both Stage I and Stage I1 to close pair annihilation and indicates that essentially no formation of defect impurity complexes takes place in these stages. As mentioned above, the recovery of InSb irradiated with 1-MeV electrons appears to be considerably more complicated for p type than for n-type. However, the above model for Stage I1 annealing in n-type can be carried through to include Stage I1 in p-type also. Assuming

[m]

6.

RADIATION EFFECTS

385

that both acceptor levels associated with the annealing defects must be unoccupied in order for annihilation of the defects to occur, Eisen48,76*84d is able to account quite accurately for the different values of T, and apparent activation energy corresponding to different values of the Fermi level. Values of T, vary from 89°K for 1.4 x 1016/cm3 p-type to 203°K for 8 x 10i7/cm3n-type. The apparent activation energy lies in the range 0.250.71 eV. Much of this variation in activation energy results from the rather large activation energies for degenerate n-type specimens. For nondegenerate specimens the variation in activation energy is about equal to the band gap, as expected. Presumably Stage I annealing in p-type InSb occurs below liquid nitrogen temperature. Aukerman71 suggested that Stage I and I1 recovery after 4.5-MeV electron irradiation involves a separation of defects rather than (or in addition to) annihilation, the reason for this being the decrease in mobility of p-type samples during these stages of recovery. This effect is not reported for lower energy bombardments, where the degree of correlation between vacancies and interstitials is greater. It is as if the closest possible vacancy-interstitial pairs do not possess any donor character at all, but as they move apart they become donors and acceptors. Stein78 found that the annealing of fast neutron irradiated n- and p-type InSb differs considerably from that of electron- or gamma-irradiated specimens. In the temperature range 100" to 2 W K , a remarkable increase in the net donor concentration occurs in both p- and n-type samples. Near room temperature this trend is reversed, but complete recovery is not observed up to the highest temperature of anneal, 400°K. These observations appear to be in direct contradiction to those of G e o r g o p ~ u l o s , namely, ~ ~ ~ ~ ~that " pile irradiation produced effects very much like electron or gamma irradiation. The latter author performed irradiations in a reactor and consequently was exposing his specimens to a rather high flux of gamma rays. In addition, since cadmium was used to filter out the thermal neutrons, considerable high energy gamma flux must have resulted from the Cd (n, 7 ) reaction.95a Stein used a pulsed reactor ;consequently, fewer gamma-rays and no thermal neutrons were present. Furthermore, at 77°K Georgopoulos observed conversion from n- to p-type with pile irradiation, whereas Vodopyanov and KurdianiESb observed conversion in the opposite direction with a pulsed reactor. One possible explanation of the above apparent contradictions is that the neutron damage in the pile irradiations was largely masked by the effects of gamma irradiation. As mentioned earlier, the radiation-induced lattice strain, ALIL, is much greater for the 111-V compounds than for the group IV elements. Still the changes are very small, as Fig. 2 illustrates. Figure 5 shows the isochronal annealing curve72obtained for a specimen irradiated with 2 x 10192-MeV y5aJ. W. Cleland, R. F. Bass, and J. H. Crawford, Jr.. J . A p p l . Phys. 33. 2906 (1962).

386

L . W. AUKERMAN

electrons/cm2 at 50°K. The initially n-type specimen of InSb was converted to p-type by the irradiation. It can be seen that the isochronal annealing curve, including the reverse annealing in the earlier stage, bears a striking resemblance to the annealing of p-type specimens irradiated with 4.5-MeV e l e ~ t r o n s ,although ~ ~ , ~ ~the ~ latter received a factor of about 2000 less integrated flux. VookgO also studied the annealing of thermal conductivity changes resulting from 2-MeV electron irradiation. The isochronal annealing of both thermal and electrical conductivity follows the same trend, showing no evidence of reverse annealing near Stage 11. These specimens were also quickly converted to p-type by the irradiation. Analysis of the isochronal data [see Eq. (22)] for Stage I1 annealing of thermal conductivity indicated second-order kinetics with an activation energy of 0.26 eV. This activation energy agrees reasonably well with Eisen’s values, which range from 0.25 to 0.28 eV depending on the hole concentration. The second-order kinetics might result from the much greater defect density present in the thermal conductivity experiments. When the average separation between vacancies approaches the average vacancy-interstitial separation, correlation effects should start to diminish. In a later publication Vookgoa shows that the thermal conductivity change is greater in Stage I than in Stage 11, and that this difference is increased for larger integrated flux.

f: X - R a y Damage Arnold and Vook 7,96,96a report changes in the low temperature electrical and thermal resistivity of InSb resulting from exposure to X-rays. The specimen was converted by electron irradiation to high resistivity p-type. Exposure to x-rays at 77°K decreases the electrical resistivity and increases the thermal resistivity. These changes are in the same direction as those produced by electron irradiation and opposite to those resulting from annealing of low carrier density specimens. This effect was also observed in initially high resistivity p-type material not pre-irradiated with electrons. If, as Arnold and Vook suggest, these x-ray induced changes actually result from atomic displacements rather than, say, the trapping of electrons, then this is the first instance of such behavior having been observed in a semiconductor. Possibly the Varley m e c h a r ~ i s m ~or~ ~ some - ~ ~modification ~ of it is operating here. E i ~ e finds n ~ ~that these defects anneal at 100°K and can be separated from the defects that anneal in Stages I and 11. They are also produced by sub-threshold electrons at a slowly increasing rate with G. W. Arnold and F. L. Vook, Bull. Am. Phys. SOC.9, 290 (1964). 96aG.W. Arnold and F. L. Vook. Phys. Rev. 137, A1839 (1965). ’’ Comment by F. H. Eisen, in “Radiation Damage in Semiconductors” (Proc. 7th Intern. Conf.), pp. 254-255. Dunod, Paris and Academic Press, New York, 1964. 96

6.

RADIATION EFFECTS

387

energy from 80 to 500 keV. Eisen further points out that in electron irradiation, these defects rapidly saturate. He suggests they are due to impurity effects which do not show up in high carrier concentration material. g . Discussion

Consider the annealing which occurs in Stages I and 11. Since no annealing is observed in electron-irradiated samples at temperatures lower than that corresponding to Stage I (about 78°K for n-type specimens, slightly lower for p-type), we may assume that these two stages involve the least stable defects produced. Since Stage I is qualitatively similar to Stage 11, any model proposed for Stage I1 may also apply to Stage I. Eisen attributes Stage I1 annealing to close vacancy-interstitial recombination. This is very reasonable, especially in view of the essentially first-order kinetics for both p - and n-type specimens. It is also possible that interactions involving misplacements occur. The changes in carrier density brought about by Stages I and 11 annealing imply the removal of acceptors, since the electron concentration increases and the hole concentration decreases in n- and p-type, respectively. Consequently, the close vacancy-interstitial pairs (or whatever defects anneal out in Stages I and 11) behave as net acceptors, and, since an appreciable fraction of them must be ionized in 1016/cm3p-type InSb, their energy level must not lie too far above E , + 0.02 eV. Very probably, then, the level lying at about E, 0.03 eV (see Table 111) corresponds to these acceptors. Eisen proposes that Stage 11 annealing consists of the recombination of close indium vacancy-interstitial pairs and that two types of indium interstitials are created. This can account for the two first-order processes in Stage I1 and the fact that the mobility annealing in Stage I1 is shifted slightly (5-lOOC) to lower temperature relative to the carrier density annealing. To explain the latter effect it is assumed that one of the interstitial configurations has a higher scattering cross section than the other. It is furthermore indicated that the recombination probably occurs via vacancy migration rather than interstitial migration. The effect of 1-MeV electron irradiation is to create net acceptors, even when the hole concentration is of the order of 1016/cm3.However, 4.5-MeV electrons appear to create, in addition to the 0.03-eV acceptors, donors not produced by the 1-MeV irradiations. This accounts for the higher value of corresponding to the higher energy of bombarding electrons. These donors are presumably created by the high energy tail in the distribution of primary knock-ons. Although the energy distribution of the displaced atoms is heavily weighted toward low energies, a straightforward application of Eqs. (1i), (12), and (9) shows that 20% of the atoms displaced by 4.5-MeV electrons have an energy greater than the maximum recoil energy resulting from 1-MeV electrons. Presumably, the extra donors result from some of the knock-ons

+

cF

388

L . W. AUKERMAN

in this group. A rather small percentage increase in the number of donors can affect the final Fermi level quite appreciably. Fast neutron irradiation also produces a more donor-like distribution of energy levels. Presumably the net acceptors characteristic of low energy knock-ons anneal out preferentially during the lower temperature annealing stages. This accounts for the apparent reverse annealing of carrier density in 4.5-MeV irradiated p-type InSb when [ < iF.Another explanation is that the closest vacancy-interstitial pairs and triplets are acceptors, but, as new configurations are formed during annealing, the donor character is developed. The reverse annealing of AL/L (2-MeV irradiation) can also result from a preferential annealing mechanism. That is, a defect that produces a negative AL/L anneals out preferentially at low temperatures. No attempt has been made to understand the complex annealing behavior of p-type InSb at temperatures greater than 100°K. The remarkable similarity in the annealing of electrical and thermal conductivity seems almost fortuitous. One might not expect both properties to be affected the same way throughout the whole annealing range. In particular, near 105°K and 310”K, Vookgo observes reverse annealing in both properties. This would seem to imply an actual increase in the number of defects (scattering centers) during these two annealing stages. An alternative explanation, however, is provided by the Keyes model for phonon scattering. This model, which also accounts for the nonlinear flux dependence and the anomalous temperature dependence of thermal conductivityg0in InSb, implies that the thermal conductivity depends on the degree of occupation of certain strain-sensitive energy levels. The electrical conductivity is also a function of the occupation of energy levels, since the degree of occupation directly affects the carrier density and mobility. This model therefore may provide a link between electrical and thermal conductivity, whereby the latter responds to changes in the former. Although the comments of this section indicate that the experimental results fit into a fairly consistent pattern, the reader must not assume that all radiation effects in InSb are well understood. On the contrary, very little is actually understood, although InSb is probably somewhat further along in this respect than any other 111-V compound. Some unanswered questions concern which defects give rise to which energy levels, what processes are involved in the various annealing stages, and why there is no evidence of donor creation by 1-MeV electrons at low temperatures. The role of misplacements is another interesting question to ponder. 8 . INDIUM ARSENIDE a. Gross Features

Both electron’l and neutron”,98a bombardment convert p-type InAs to n-type and increase the carrier density of n-type samples, even those of

6.

RADIATION EFFECTS

389

fairly low resistivity (corresponding to carrier concentrations up to about 10' '/cm3). The electron irradiations were carried out at liquid nitrogen temperature and the neutron irradiations near room temperature. Both studies were performed on polycrystalline specimens. Peculiarities were observed for p-type specimens nearly converted to n-type." The behavior of the Hall effect and resistivity suggested the presence of a shunting n-type conductivity, possibly associated with grain boundaries, surfaces,99 or dislocation lines.'" Interpretation of the results is rendered difficult by the fact that only polycrystalline samples were available for the experiments. The rate of carrier removal by 4.5-MeV electrons was dp/d$ = 10 cm-' for nearly degenerate p-type InAs; and the carrier introduction rate for nearly degenerate n-type samples was dnldq5 = 6.1 cm- '. Partial annealing was observed between liquid nitrogen and room temperatures for electronirradiated specimens. Neutron-irradiated specimens were stable up to about 100°C. More thorough annealing studies were performed by Bauerlein.'"' The experiments were performed by irradiating p-n junctions with 0.4-MeV electrons to produce the damage and then observing the changes in the ratio of short-circuit current to beam current during subsequent irradiation with subthreshold electrons (200 keV). Under proper conditions, which in the present case are satisfied, the short-circuit current is proportional to the square root59-59b,'00of the number of recombination centers times their capture cross section. The diodes were fabricated by diffusing zinc into singlecrystal n-type InAs a distance of about 20 to 30 p. The isochronal annealing curve for a diode irradiated at 63°K is shown in Fig. 6. As in the case of p-type InSb, reverse annealing is observed at low temperatures. This probably indicates a redistribution or interaction of defects which increases their capture cross section. The last annealing stage, in the region of room temperature and above, obeyed first-order kinetics with an activation energy of 0.8 eV. The two stages of annealing between 60" and 150°K each obeyed first-order kinetics, but no consistent activation energy could be obtained from the variation of rate constant with temperature. However, from the temperature corresponding to the inflection points [see Eq. (29) or (30)], Bauerlein'" estimated activation energies to be 0.23 and 0.33 eV, respectively. He assumed the constant corresponding to A in Eqs. (29) and (30) to be the lattice frequency of InAs. - 7 x 10l2 sec-'. J. W. Cleland and J . H. Crawford. Jr.. Butt. Am. Phys. Soc. 3. 142 (1958). '"'R. K. Willardson, F. J . Reid, and E. M . Baroody, WADCTech. Rcpt. 57-593. Scpwiiiber 1957. " H. Rupprecht, Z. Nnturforsch. 13a. 1094 (1958). l o o J. R. Dixon. J . A p p l . Phys. 30, 1413 (1959). l o ' R. Bauerlein. Z. Natitrforsch. 16a. 1002 (1961).

390

L . W. AUKERMAN

TEMPERATURE, 'K

FIG.6. Isochronal annealing of minority carrier lifetime in InAs after 400-keV electron irradiation. (After Bauerlein.'o')

b. Discussion The increase in carrier density for nearly degenerate n-type samples indicates that lies essentially at the conduction band edge or above it. This behavior is rather unique, since for most semiconductors usually lies within the forbidden energy gap. The continual increase in n-type carrier density with irradiation indicates the introduction of very shallow donor states. N o direct evidence was encountered for compensating acceptors. Nevertheless, defect levels lying somewhere within the forbidden gap are probably introduced ; otherwise the hole removal rate would equal the electron introduction rate. The annealing of recombination centers at temperatures around 300°K and above, being first order, can be put in the form

rF

rF

where l / z is the rate constant and can be interpreted as the average jump frequency. Writing z as = A - leE/kT Bauerlein obtains for A only 1.5 x lo9 sec-', although for close vacancyinterstitial recombination a value of the same order of magnitude as the lattice frequency v,, ( r 7 x 10l2sec-') would be expected. The low value of the pre-exponential factor A suggests that many jumps are required

6.

RADIATION EFFECTS

391

before the defect can be annihilated. Possibly the defects migrate to sinks (i.e., dislocations or other imperfections) where they are rendered ineffective. This would be consistent with the first-order kinetics. Nevertheless, since the energy of the bombarding electron was so low, Bauerlein'" prefers a model in which close vacancy-interstitial pairs recombine. To explain the low value of A, Bauerlein proposes a mechanism involving many defect interchanges before recombination can take place. 9. GALLIUM ARSENIDE

a. Gross Features and Electrical Properties Both n- and p-type GaAs are decreased in carrier density by irradiation with e l e ~ t r o n sor~ fast ~ * neutrons.80 ~~~ The final Fermi level resulting from large doses of fast neutrons appears to lie near the middle of the forbidden gap.809102 Figure 7 shows the conductivity of an n-type sample as a function of the integrated fast neutron flux. Shields of boron and cadmium eliminated most of the low energy neutrons so that the transmutation rate was negligible. The intrinsic conductivity of GaAs at 1IOT, the temperature of irradiation, is estimated to be approximately ohm-' cm-'. The curve shown for AlSb will be discussed in subsequent paragraphs.

FIG.7. Change of conductivity with fast neutron bombardment of GaAs and AISb. lo'

L. W. Aukerman. Proc. Intern. Con1 Semicond. Phys., Prague, I960 p. 946. Czech. Acad. Sci., Prague, and Academic Press, New York, 1961.

392

L . W. AUKERMAN

The free carrier density of n-type GaAs is decreased by irradiation a t rates of approximately 8/cm and 0.5/cm by fast neutron and I-MeV electron bombardment, respectively. The known energy levels in irradiated GaAs are listed in Table 111. No levels within 0.1eV of the conduction band were detected; however, levels very close to the conduction band, if present, would probably be difficult to detect. The levels were determined by temperature dependence of the Hall e f f e ~ t ~ ~and - by ~ ~photoelectronic , ~ ~ * ~ ~techniques.84c ~ Specimens of GaAs irradiated with fast neutrons to a very low carrier density exhibited peculiar behavior" in some of their electrical properties. For example, photoconductivity and photo-Hall effect having extremely long relaxation times (hours) were observed. In addition, the temperature and dependence of the Hall mobility pH was anomalously steep80~'02,103 could not be accounted for by scattering from ionized centers. Slopes of In pHvs In Twere observed to be greater than 10 in some instances. Slow surface states l o 4 might be responsible for the above-mentioned photoeffects in GaAs. Longo and Wang105report similar effects in deuteronirradiated and gold-doped silicon. Anomalously steep mobility curves for neutron-irradiated Ge,lo6 Si,'07 and A1Sblo8have been reported. Irradiation with co-60 y rays carried out by Cleland et ~ 1 . , " ~ at room temperature decreased the carrier density of n-type GaAs at a rate of approxicm-'. This is much larger than the corresponding mately 1.4 x removal rates for Gel1' and Si,'" about 10-3cm-1 for each. However, Krivov and Malyanov" l a calculated a cross section for forming Frenkel defects by CO-6Oy rays. Their cross section predicts an introduction rate cm-2, in good agreement with Cleland et al. They assumed a of 1.6 x threshold of about 24 eV, which is somewhat larger than Bauerlein's value. The level quoted by Krivov et a1.83 (see Table 111) at E, + 0.05 eV appears to have been introduced at a rate of about 2 orders of magnitude greater than this. In the author's opinion, because of the very small dose used, this level may have been due to causes other than bulk damage. R. K. Willardson. J . A p p l . Phys. 30. 1 1 58 (1959). T. B. Watkins. Pri~jir.S e n i i r i ~ r ~ 5, d . 3-51 (1960). ' 0 5 T. A. Longo and E. Y. Wang, Proc. Second Corzj. N ucl. Rudiurioli Effects 011 S e m i c o d . Drrices, M m e r i a l , a d Cirruits, New York, 1959, p. 14. Cowan, New York, 1960. l o 6 J. W. Cleland. J. H. Crawford, Jr.. and J. C. Pigg, Phys. Reo. 98. 1742 (1955). lo' G. K. Wertheim, Phjs. R m . 111.1500(1958). I o 8 F. J. Reid, A. C. Beer, and R . K. Willardson, Bull. Am.Phy.7. SOC.2,356(1957). J. W. Cleland. R. F. Bass. and J. H. Crawford. Jr.. in "Radiation Damage in Semiconductors" (Proc. 7th Intern. Conf.), p. 401. Dunod, Paris and Academic Press, New York, 1964; and O a k Ridge National Laboratories Solid State Div. Annual Progress Report for Period Ending Aug. 31, 1961, ORNL-3213, p. 71. ' l o J. W. Cleland, J. H . Crawford. Jr., and D. K. Holmes. Phys. Rec. 102. 722 (1956). I E. Sonder and L. C . Templeton. J . A p p l . Phys. 31. 1279 ( I 960). ' l I d M .A . Krivov and S. V. Malyanov,Izr. Vysshikh. Uchehii. ZavedeniiFiz. No. 4, 147-151 (1963). '03 lo'

'

6.

RADIATION EFFECTS

393

The mobility change for a given change in carrier density"

corresponding to lightly or moderately electron-bombarded n-type GaAs is consistent with the Brooks-Herring1 l 2 model for ionized scattering centers, provided the introduced acceptors are assumed to be no more than doubly ionized. The behavior of this ratio also indicates that the level at E , - 0.13 eV is a donor. In Fig. 8 the quantities An and A(l/p) are, respectively, the total changes in carrier density and the reciprocal of Hall mobility produced by electron or gamma irradiation, and c is the Fermi energy after irradiation. The data (1-MeV electron were taken from the measurements of Aukerman et d S 0 irradiation), and Cleland et al. O 9 (gamma irradiation). The quantity A(l/p) is controlled by the total number of scattering centers weighted by the square of their charge, whereas I -An[ is controlled by the net acceptor concentration, or by the number of centers weighted by the first power of their charge. The points marked with diamonds and squares are values calculated by the Brooks-Herring model assuming singly ( 0 )and doubly (n) ionized acceptors, and no ionized donors, when c is near E , . The rise in the calculated curves for higher E , - [, or lower carrier density, results from the effect of conduction electrons on polar scattering and the scattering by chemical impurities. The greater rise of the experimental points in the region 0.1 eV and greater is obviously connected with the partial ionization of the level at 0.13 eV. The indication is that this level becomes positively charged as c approaches it from above, thus decreasing the net negative bound charge. In other words, the level at E, - 0.13 eV is a donor.

Ec-L eV FIG.8. Fermi level dependence of the change in reciprocal mobility per change in carrier density for electron (0) and gamma (A) irradiated GaAs. Calculated points (11) and ( 0 )are and Aukerman et al.") explained in the text. (Constructed from the data of Cleland et '12

P. P. Debye and E. M. Conweli, Phys. Reu. 93,693 (1954)

394

L. W . AUKERMAN

The values of A(l/p)/lAn\ obtained from samples lightly irradiated with fast neutrons are about twice those obtained from electron-irradiated specimens with equivalent doping. This suggests that the neutron-produced defects are either more highly ionized, or are more compensated than those produced by electrons. Magnetoresistance r n e a s ~ r e m e n t s ’also ~ ~ suggest the presence of multiply charged defects in neutron-irradiated GaAs. Grimshaw,”’” by carefully analyzing the Hall mobility of n-type GaAs, showed that assuming the defects to be singly ionized could not account for the decrease in mobility upon bombardment with 0.4-MeV electrons. However, by assuming the bombardment centers to be doubly ionized, he obtained good agreement. The effects of radiation damage on the electroluminescence of GaAs diodes will be discussed in Section 13, which deals with devices. Cathodoluminescence studies will also be taken up in that section. b. Elongation and Thermal Conductivity

As in similar experiments with elongation and thermal conductivity experiments usually employ considerably greater amounts of irradiation than do experiments involving various electrical properties. The increase in specific length expansion AL/L with integrated flux7’ for GaAs is almost identical to that of InSb, as shown in Fig. 4. Much of the discussion for InSb in that section applies also to GaAs. Irradiation with 2-MeV and 1.6-MeV electrons at 95°K yielded AL/L@ = 1.0 x cm’ and 6.8 x cm’, respectively. Vook7’ used these values with the Seitz-Koehler theory to estimate the threshold energy for GaAs. The value obtained for Tdwas about 45 eV. This is considerably higher than the 10-eV value obtained by B a ~ e r l e i n utilizing ~ ~ - ~ ~minority ~ carrier lifetime, a very sensitive indicator of radiation damage. The changes in thermal conductivity (1/K) with integrated flux obtained by V ~ o k ” * for ’ ~ GaAs ~ and InSb are shown in Fig. 3. Unlike InSb, GaAs bombarded with 2-MeV electrons exhibits a linear increase in 1/K with integrated flux. This difference was discussed in Section 7c. The behavior of 1/K with integrated flux and with absolute temperature is, for GaAs, consistent with the point-defect scattering theories of K l e m e n ~’”~and ~ , Ziman?’ ~ c. Annealing Experiments

Annealing experiments on electrical property changes in GaAs have to date been carried out only at room temperature and above. Figure 9 illustrates803’02the difference in the annealing properties of 1-MeV electron and ‘IZa

‘13

J. A. Grimshaw, in ”Radiation Damage in Semiconductors” (Proc. 7th Intern. Conf.), p. 377. Dunod, Paris and Academic Press, New York, 1964. F. L. Vook, Phys. Rev. 135, A1742(1964).

6. 1

1.3

I

395

RADIATION EFFECTS

I

I

I

I

I

t-

1

IP o

1.1

L dJ

0.9

TEMPERATURE OF ANNEAL:C

FIG.9. Isochronal annealing of neutron-irradiated (curve A) and electron-irradiated (curve B) n-type GaAs.

fast neutron bombarded n-type GaAs. The damage monitoring property was electrical conductivity. The markers labeled p,, and represent the conductivities before irradiation with electrons and neutrons, respectively. These results suggest that fast neutrons produce two kinds of damage: one kind that anneals at about 220°C and a more stable component that disappears in one or more higher temperature stages. Part of the less stable component may result from the rather high gamma flux present. The neutronirradiated specimen was shielded with cadmium to prevent transmutations by thermal neutrons ;consequently, the specimen was exposed not only to the gamma flux normally present in a reactor, but also the more energetic gamma rays95aresulting from the Cd(n, y) reaction. Although the electron-irradiated samples recovered most of their initial conductivity after annealing for moderate times in the 200 to 250°C range, recovery of the fast neutron irradiated specimens did not go to completion at temperatures under 600°C. No overshoot or reverse annealing was observed in any of the electrical property annealing studies. Attempts to analyze the higher temperature annealing stages of neutronirradiated specimens lead to inconsistent results, possibly due to complications resulting from contamination by rapidly diffusing impurities such as copper."4 The stage in the vicinity of 220°C was studied for electron- and neutron-irradiated n-type specimens. The results of isothermal annealing curves indicate that in both cases the unrecovered fraction cp can be decomposed into two first-order processes as follows cp = uexp(-i,t) + (1 - a)exp(-&t). 'I4

C. S. Fuller and J. M. Whelan, J . Phys. Chem. Solrds 6,173 (1958).

396

L . W. AUKERMAN

Adopting the convention that 1, > 2, (1, corresponding to the quicker process) it was found that for electron-irradiated specimens Al could be expressed as ,Iol exp(E,/kT) with E , = (1.10 k 0.05) eV. O n the other hand, 1, was correlated with the carrier density : 2,N,/n = ,Io2exp( - E 2 / k T ) , with E, = (1.55 k 0.05)eV. In accordance with the discussion in Section 3b, it appears that the ,I, process is Fermi-level dependent. But, interestingly enough, the dependence of 1, on n is just the opposite of the dependence which would be predicted employing in a simple manner the mass action law [see Eqs. (35) and (4411. Thus a somewhat more complicated model than that suggested by (35) is required. A similar analysis of the 220°C annealing stage in neutron-irradiated samples failed to show any Fermi-level dependence. The results for both electron and neutron irradiation are summarized in Table V.

TABLE V

ANNEALING PARAMETERS

FOR THE

2k0C

STAGE IN tZ-TYPE

GaAs

1, process

~

~~~~

~

Electron irradiation Neutron irradiation

2.9 x 10' 1.1 x lo6

1.1 0.9

1.1 x 1013n/N, 1.6 x lo7

1.55 0.94

0.154.30 0.44.5

A , and A , are the pre-exponential terms, as in it = A,e-Er'rT.

a is the fraction of this stage which anneals via the A l rate (A, > A*).

Although the gross behavior of the 220°C stage in neutron-irradiated GaAs is similar to that in electron-irradiated samples, nevertheless there are several differences in the detailed characteristics. These consist of the different magnitude of the 1, rate constant, its different dependence on carrier density, and the different values of a corresponding to the two cases. No correlation between any of the rate constants and etch pit counts or degrees of compensation was observed. Silicon was the major dopant for all but one specimen. This specimen, doped with tellurium, exhibited E., and ,I2 values consistent with the others. P-type GaAs anneals somewhat differently than n-type, showing three resolvable stages above room t e m p e r a t ~ r e . ~About ~ . ~ ~ "20 % of the damage anneals in a stage centered at 155"C, about 70% at 240"C, and the rest at 350°C. Detailed isothermal studies were not carried out on p-type specimens ;

6.

397

RADIATION EFFECTS

however, the isochronal stage at 240°C fits a first-order process with an activation energy of about 1.3 eV.l15 The annealing of p-type and n-type samples need not have precisely the same appearance since the carrier density tends to be controlled in n-type by acceptors and in p-type by donors. Furthermore the different Fermi level can affect the rate processes involved. The annealing of lattice strain" and thermal conductivity" has been carried out by Vook over the temperature range of 50" to 600°K. Figure 10 shows the isochronal annealing of the thermal resistivity change brought about by a 50°K irradiation of n-type GaAs with an integrated flux of about lo" 2-MeV electrons/cm2. The annealing occurs over a broad temperature range with no apparent evidence of discrete stages. The annealing of lattice strain is rather similar to that of thermal conductivity (Fig. lo), i.e., the isochronal curve is very broad. In this case the annealing was extended to 600"K, and a jog centered at 500°K somewhat similar to but broader than that observed by Aukerman and Graft for electrical conductivity was observed. After this stage, there is a residual contraction, i.e., there is an overshoot of the annealing. The resulting negative strain is not well understood. Vook suggests it could be caused by the presence of divacancies. Such large amounts of irradiation might tend to favor the creation of divacancies.

50

100

W

200 250 300 TEMPERATURE, O K

350

4M)

450

FIG. 10. Isochronal anneal of thermal resistivity of GaAs after 2-MeV electron irradiation. (From Vook."3)

"*

L. W. Aukerman, E. M. Baroody, and R. D. Graft, "Theoretical and Experimental Studies Concerning Radiation Damage in Selected Compound Semiconductors," Second Interim Rept. Contract No. AF-33(616)-8064, August 1961.

398

L . W . AUKERMAN

Annealing of lattice strain and thermal conductivity in the vicinity of room temperature has no counterpart in the annealing of electrical conductivity in n-type samples. However, the conductivity type of Vook's specimen after irradiation is not known, since the final Fermi level for electron-irradiated GaAs has not been determined. The annealing of p-type specimens above room temperature has been r e p ~ r t e d , but ~ ~ whether . ~ ~ ~ or not annealing occurs at room temperature was not discussed. Several important conclusions to be drawn from these experiments are : low temperature annealing spread over a large temperature interval definitely does occur ; a stage is observed at about 500°K in agreement with the conductivity annealing; and a net contraction remains after this stage is completed. Annealing spread over a wide temperature range has been observed from time to time after fast neutron irradiation and is frequently analyzed' ' 6 * 1 1 6 a in terms of processes having a spectrum of activation energies. This type of annealing is frequently associated with more complex type of damage such as would be likely to occur if the defects were in some manner interacting with one another. Vook finds from the temperature dependence of the added thermal resistivity that during annealing the defects change their phonon frequency scattering dependencies and can no longer be considered as point defects. The direction of the changes in temperature dependence suggests that the point defects are clusteringinto colloids or precipitates to provide a scattering behavior more like boundary scattering. This result is not surprising in view of the relatively large value of integrated flux (- 1019/cm2). d. Discussion of Annealing Results

It is seen that the 220°K annealing stage of electrical conductivity in both electron- and neutron-irradiated specimens follows first-order kinetics. This fact should provide a valuable clue to the underlying mechanism. Recombination of close vacancy-interstitial pairs (or vacancy-misplacement-interstitial triplets) as well as migration to sinks (possibly modified b y trapping effects) could be considered as possible mechanisms since both can lead to first-order (or nearly first-order) kinetics. The latter mechanism would require the rate constants A l and 2, to be proportional to the concentration of sinks,49 whereas the experimental results indicate that were this the case the sink concentration would be nearly constant from specimen to specimen. Etch pit counts, on the other hand, seemed to indicate a variation in the dislocation densities (dislocations being the most likely candidates for sinks) by as much as two orders of magnitude from specimen to specimen. On this basis Aukerman et a1.79,79aconsidered diffusion to sinks as less likely than close pair recombination for the 1, and A, processes. W. Primak, Plzys. Rev. 100, 1677 (1955). 116aV.Vand, Proc. Phys. Soc. (London) A 5 5 222 (1943).

6.

399

RADIATION EFFECTS

Although Vook observed annealing of thermal conductivity’ l 3 and lattice expansion72at quite low temperatures, this annealing cannot be close pair recombination because it is too broad. Possibly the annealing observed by Vook is associated with the presence of a relatively large number of divacancies. Such large integrated doses might tend to emphasize the creation of divacancies which, if more mobile than single vacancies or interstitials, could give rise to low temperature annealing processes. Grimshaw”2a reports no apparent annealing in the mobility between liquid nitrogen and room temperature, provided the total dose is not too great. Returning to the A,, A2 processes, there is some additional information that needs to be considered, namely, the pre-exponential factors A of Table V. For close vacancy-interstitial recombination, A NN vo = 10I3/sec. Thus, for the ;Lz process in the electron case A is consistent with closest pair recombination, provided the model suggested by Aukerman and Graft is valid (i.e., no recombination unless a postulated electronic state lying above the Fermi level is occupied). O n the other hand, the other A values are much too low for close pair recombination and suggest instead some sort of diffusive mechanism requiring many jumps. For annihilation at a fixed number of sinks the factor A is approximately v , / N j where N , is the average number of jumps made by a defect before it is annihilated.’’6b This gives N j values lying in the range lo5to lo’, which means that the average defect migrates a distance of cm before annihilation. This separation is too large the order of 3 x to allow correlated recombination. (It is slightly larger than the estimated average separation of radiation-induced defects.) The magnitudes of the pre-exponential terms suggest, therefore, migration over rather large distances, possibly to sinks or very deep traps, for the ;L1 process in neutronand electron-irradiated specimens and for the & process in neutron-irradiated specimens. The magnitudes of A are consistent with sink densities of the order of 1013/cm3if they have spherical symmetry, or 10s/cm2 if they have cylindrical symmetry (e.g., dislocation). If the annealing involves diffusion to sinks, it would appear that the sink concentration is nearly constant. A concentration of lo8 dislocations/cm2 seems unduly large, especially in view of the fact that only lo4 to lo6 etch pits/cm2 were counted on a sampling of the specimens irradiated. An interesting speculation arises: if divacancy formation is fairly probable in GaAs (Vook’s results would tend to suggest that it is), the sinks may be vacancy clusters formed by precipitation of divacancies during irradiation. The different behavior of the ,I2 process in the case of neutron irradiation does not contradict the close pair hypothesis for electron irradiations, since the degree of vacancy-interstitial correlation is expected to be much lower for

-

L’6bActuallyA % (d2/rS2)u,where d is the jump distance and rs the average distance between sinks.’ A simple random-walk calculation makes N , x rsZ/d2.

400

L . W. AUKERMAN

the neutron case. It should be kept in mind that many more close-pair configurations are possible in compounds than in elements, 2nd some of these might be quite stable at room temperature. e. Optical Properties

Optical properties of irradiated semiconductors can frequently be a useful tool for studying the positions of energy levels. However, in the case of fastneutron irradiated GaAs any discrete levels which would show up in an optical absorption spectrum appear to be completely masked by a very 3’~~ far beyond the absorpstrong absorption of unknown 0 r i g i n ~ ~ extending tion edge. In Fig. 11 the apparent room temperature absorption coefficient CI (or attenuation coefficient, if light scattering is predominant) is presented as a function of photon energy for a number of irradiated GaAs specimens. The absorption coefficient fits a I-’ dependence quite well for wavelengths beyond about 12,000A( 1 eV). Near the band edge the curves turn upward slightly as if the band edge were made “fuzzy” by irradiation. The dashed curve, representing the band edge for an undamaged specimen of low carrier

-

0.I

I

PHOTON

10

ENERGY, eV

FIG. 11. Optical absorption in GaAs after fast-neutron and 1-MeV electron irradiation,

6 . RADIATION

EFFECTS

401

density, was taken from the data of Spitzer and Whelan.’’7 The strong absorption near 0.07 eV is a lattice absorption band. It is seen that increases in absorption (or attenuation) by orders of magnitude can result from rather moderate neutron bombardment. A heavy 1-MeV electron bombardment, on the other hand, did not produce the A - 2 dependence but produced a less drastic spread in the band edge. Similar behavior was observed in neutron-irradiated CdTe and CdS. Although the nature of this absorption is not understood, light scattering regions produced by transient high temperature and pressure (e.g., thermal spikes or displacement spikes) has been suggested.” McNichols and Ginell”’ interpret the A - 2 dependence in terms of the model for spikes suggested by Gonser and Okkerse.88-88b They calculate the infrared absorption for a distribution of small metallic inhomogeneities in an otherwise uniform nonmetallic crystal and obtain excellent agreement with the A-2 region of neutron-irradiated GaAs.

10. GALLIUM ANTIMONIDE Dimensional changes in GaSb resulting from deuteron irradiation were discussed in Section 7 along with similar effects in InSb. Some effects on electrical properties resulting from fast neutron irradiation were studied by Cleland and Crawford,82 who show evidence for two levels at approximately 0.14 and 0.2 eV above the valence band (see Table 111) and no levels close to the conduction band. The irradiations were carried out in polycrystalline material at either room temperature or about 150°K. Fast neutron irradiation at either temperature decreases the carrier density and mobility of both p- and n-type specimens of rather high initial carrier density, and an n-type specimen was converted to p-type after a rather large integrated flux. Thus, the final Fermi level is in the lower half of the forbidden gap. The effect of annealing is to increase the net acceptor concentration or decrease the net donor concentration in both 11- and p-type specimens. This results in reverse annealing for the n-type samples and “overshoot” in the p-type (i.e., during annealing, the p-type carrier density increases beyond the pre-irradiation value). Experiments with control samples seem to indicate that these effects are not artifacts due to thermally created defects or accidental contamination with impurities. The levels at E , - (0.12 to 0.2) eV, E, - (0.47 to 0.50)eV, E , -t0.48 eV, and E, + 0.075eV in Table 111 were determined84b by optical absorption and photoconductivity experiments on n- and p-type GaSb samples irradiated with 4.5-MeV electrons at 0°C. The latter level was found to be present before irradiation and was presumed to be a structural defect resulting from ‘I8

W. G. Spitrer and J. M. Whelan. Phys. Rec. 114, 59 (1959). J. I . McNichols and W. S. Ginell, J . Appl. Phys. 38. 656 (1967).

402

L. W. AUKERMAN

nonstoichiometry. The above irradiation conditions resulted in iF= E , + 0.05 eV. All evidence of electron damage is removed by heating for one hour at 500°C. Neutron damage,82 on the other hand, is not annealed out at temperatures up to 550°C. Gonser and Okkerse88-8'b suggest that deuteron irradiation of GaSb and InSb at low temperatures creates small regions of an amorphous phase which result in a slight shrinkage. Since this effect is attributed to displacement spikes, it should be present also in the case of fast neutron irradiation. The observed shrinkage, however, anneals out below room temperature, and in fact there remains at room temperature a slight expansion. Presumably the amorphous regions recrystallize at a temperature below 300"K, although the recrystallization may not be perfect, i.e., there may still remain a considerable amount of structural defects in the form of lattice misfits, dislocation loops, and vacancy clusters, which may be expected to be very difficult to anneal and may account for some of the neutron damage present a t relatively high temperatures. K. Thommen has investigated Hall effect and resistivity of p-type GaSb in the temperature range 15°K to 300°K after irradiation with I-MeV electrons and various annealing treatments.' 8a No recovery was observed between 15" and 100°K.The first annealing stage at 123°K obeyed first-order statistics with an activation energy of 0.31 eV. Other recovery stages occurred at 163, 203, and 365°K. The dominant radiation induced acceptor lies at E , + 0.023 eV. 1 1. ALUMINUM ANTIMONIDE

Radiation damage studies on AlSb are seriously hampered by the fact that this material is quite hygroscopic. Irradiation with fast neutrons"' decreases the carrier concentration of both n- and p-type samples. Some irradiated p-type samples showed anomalous mobility effects similar to those observed for neutron-irradiated germaniumlo6and The shallow minimum in the conductivity vs integrated flux curve for p-type AlSb irradiated with fast neutrons (Fig. 7) was first interpreted as a conversion from p-type to n-type. However, subsequent studies95 revealed that this behavior is caused by a thin surface layer, which masks the bulk conductivity whenever it becomes lower than the order of to ohm- ' cm- Attempts to remove this thin film by etching were unsuccessful, but lightly sandblasting the surface either removed the film or sufficiently disrupted it to decrease the conductivity of the AlSb specimen of Fig. 7 after irradiation to a value lower than ohm-' cm-l. Sandblasting has a negligible effect on unirradiated samples. Thus, it appears that the final Fermi level of AlSb lies near the center of the gap, as in the case of GaAs and silicon. 118aK. Thommen, Phys. Reu. 161,769 (1967).

6.

RADIATION EFFECTS

403

12. INDIUMPHOSPHIDE Irradiation" at room temperature with an integrated flux of about 1.2 x 1017/cm2fast neutrons decreases the conductivity of an n-type InP specimen from 56 ohm-' cm-' to about 0.04 ohm-' cm-'. The specimen was shielded from thermal neutrons with cadmium and boron. The temperature dependence of the Hall coefficient after irradiation indicates the presence of an energy level 0.285 eV below the conduction band (see Table 111). The Hall mobility was decreased from 3300 to 1000cm2/volt-sec by the irradiation and was neither abnormally low nor showed any abnormal temperature dependence. The isochronal annealing of a rather lightly irradiated n-type specimen shows a broad annealing range between 100" and 275"K, followed by a relatively sharp stage at about 340°K. Neither of the abovementioned InP samples was completely recovered after annealing up to 550°C. Thus some of the neutron created defects show a high degree of stability as in the case of other IiI-V compounds.

IV. Radiation Damage in Devices 13. GENERAL REMARKS Solid state devices that are sensitive to minority carrier lifetime can be expected to be sensitive to radiation damage also. It has long been known,"' for example, that junction transistors and diodes are quite sensitive to radiation effects. The minority carrier lifetimes of presently available 111-V compounds are so small that junction transistors made of these materials are still quite experimental. However, other device applications may become feasible in the near future. This section will be concerned only with certain diode applicatims, namely, solar cells, electroluminescent diodes, and tunnel diodes. The first two are quite sensitive to minority carrier lifetime. Tunnel diodes are affected by means of a different mechanism. The change in minority carrier lifetime z can be expressed as follows'2o : l/t

=

1/70

+K4

(63)

where zo is the initial lifetime, 4 is the integrated bombarding flux, and K is a constant. This equation expresses the fact that the total recombination rate is the sum of the recombination rates through different centers, the rate of recombination through bombardment centers being proportional to 4. The quantity K , of course, depends on the type of bombardment as well as the F. J. Reid, in "Effects of Radiation o n Materials and Components" (J. F. Kircher and R. E. Bowman, eds). p. 473. Reinhold, New York, 1964. l Z o J. J. Loferski, J . Appl. Phys. 29, 35 (1958). l9

404

L . W. AUKERMAN

properties of the centers introduced. Equation (63) has been amply verified 2 2~ for e ~ p e r i m e n t a l l y ~ ~ 1- *~1 ~ . ' ~germanium and silicon.

a. Solar Cells The type ofsolar ell'^^-^^^^ usually employed (e.g., for space applications) has a p-n junction close to the illuminated surface. The purpose of the p-n junction is to separate hole-electron pairs created by absorption of photons. Thus, only photons of energy greater than the band gap of the material are effective.The ideal band gap for the solar spectrum in space is about 1.5 eV.124 Present-day commercial solar cells are made of silicon ( E , = 1.08 eV), while higher band gap materials such as GaAs, InP, and CdS are being developed. In order to obtain maximum ionization near the junction,'the junction depth is made quite small, 0.5 to 2 p. The junction is illuminated through this thin diffused layer or "skin." Typical commercial solar-cell efficiencies lie in the range of 10 to 15%. The best experimental GaAs solar cells are now comparable in efficiency to commercial silicon cells. An interesting comparison of silicon and GaAs solar cells, with respect to their spectral response and radiation damage tolerance, has been carried out by W y s o ~ k i . ' ~ ~The * ' ~ photoresponse ~" of silicon under sunlight results primarily from the longer wavelength photons which penetrate beyond the junction into the base region. The photoresponse of GaAs results from photons absorbed in the diffused layer close to the surface. The main reason for this difference is the more abrupt absorption edge of GaAs. Figure 12 illustrates the greater tolerance of GaAs to 17.6-MeV protons. For very low energy particles, which create damage close to the surface, the GaAs cells are of course inferior to silicon. b. Luminescence and Light-Emitting Diodes The discovery of rather efficient recombination radiation of energy close to the band gap from forward biased GaAs diodes'26-'26b created wide0. L. Curtiss, Jr., J . W. Cleland. J. H. Crawford. Jr., and J. C. Pigg, J . A p p l . Phys. 28, 1161 (1957). G. C. Messenger and J. P. Spratt. Proc. I.R.E. 46. 1038 (1958). 1 2 3 M. B. Prince, J . A p p l . Plrys. 26, 534 (1955). IZSaR.L. Cummerow, Phys. Rev. 95, 16 (1954). 121bR.L. Cummerow, Phys. Rev. 95, 561 (1954). P. Rappaport. R C A Rev. 20, 373 (1959). 1 2 5 J. J. Wysocki, J . Appl. Phys. 34, 2915 (1963). 125aJ.J. Wysocki. IEEE Trans. Nucl. Sci. NS-10, 60 (1963). I z 6 J. I. Pankove and M. Massoulie, Bull. Am. Phys. SOC.7. 88 (1962). lZoaD.N. Nasledov, A. A. Rogachev, S. M. Ryvkin, and B. V. Tsarenkov. Fiz. Tvrrd. Trln 4 1063 (1962) [English Transl.: Soviet Phys.-Solid State 4, 782 (1962)l. 126bR.J. Keyes and T. M. Quist, Proc. I.R.E. 50, 1822 (1962).

6.

405

RADIATION EFFECTS

14 I3 0.

2 120

6 10'0

I

'

I "

to"

I

'

' I '

' 47HC-9.2% '

10'2

' I '

I013

I

' ' I -

1014

(PROTONS/cm2)

FIG. 12. Power output of GaAs and Si n / p solar cells vs the integrated proton flux. (After Wy~ocki.'~'")

spread interest. Shortly afterwards, the first semiconductor laser was ann o ~ n c e d . ' ~ ' - ' ~The ' ~ principal incoherent emission band (edge emission) from a typical forward biased GaAs diode at 77°K peaks slightly lower than the band gap at about 1.47 eV. This band is usually assumed to be generated by the direct radiative recombination of electrons and holes or by recombination of electrons with holes trapped at shallow acceptors (the p-type dopant), and is usually accompanied by two minor broad bands at 1.28 and 0.95 eV, respectively. 2 8 An excellent review of radiative recombination has been given by Gershenzon.I28a Since radiation damage creates levels within the forbidden gap, one might expect irradiated GaAs diodes to show additional bands. However, this does not appear to be the case. Both fast neutron'29 and electron damage'30-'30b decrease all three bands approximately equally and neither adds new bands. Thus the radiation damage appears to create only nonradiative recombination centers, which compete with the radiative processes and make them less efficient. M. 1. Nathan, W. P. Dumke, G. Burns, F. H. Dill, Jr., and G. J. Lasher, Appl. P h p . Letters 1, 62 (1962).

"'=R. N. Hall, G . E. Fenner, J. D. Kingsley, T. J. Soltys, and R. 0. Carlson, Phys. Rev. Letters 9, 366 (1962).

1Z7bT. M. Quist, R. H. Rediker, R. J. Keyes, W. E. Krag, B. Lax. A. L. McWhorter, and H. Zeigler, Appl. Phys. Letters 1, 91 (1962).

*''G. Burns and M. I. Nathan, Proc. I E E E 52,770(1964). 128aM.Gershenzon, in "Semiconductors and Semimetals" (R. K. Willardson and Albert C. Beer, eds.), Vol. 2, p. 289. Academic Press, New York, 1966. l Z 9 M. C. Petree, Appl. Phys. Letfers 3, 67 (1963). M. F. Millea and L. W. Aukerman, Appl. Phys. Letters 5, 168 (1964). 130nM.F. Millea and L. W. Aukerman, Bull. Am. Phys. SOC.9,646 (1964). I 3 O b M . F. Millea and L. W. Aukerman. J . Appl. Phys. 37. 1788 (1966).

406

L. W. AUKERMAN

However, the cathodoluminescence spectra, according to Loferski and Wu,I3l seem to indicate that in n-type GaAs additional bands are created

by 400-keV electron bombardment. Photoluminescence of zinc-diffused GaAs also shows additional structure on a band near 1.4eV at 4.2"K. A r n ~ l d ' ~ ~describes , ' ~ ~ " a series of three sharp lines repeated at intervals of 0.01 1 eV, the transverse acoustical phonon energy. These lines were brought out by first irradiating very heavily (1.3 x lo'* e/cm2) with 0.6-MeV electrons and then annealing up to 220°C. Before annealing all luminescence was completely eliminated by the irradiation. In an earlier work' 3 3 concerning orientation effects it was suggested that the degradation of luminescence was brought about chiefly by arsenic defects. The sharp spectra could not be formed when the incident energy was above a value lying between 1.0 and 1.5 MeV. Arnold suggests therefore that the real threshold in GaAs is near 50 eV, and that the damage produced at lower energies results from interaction with structural defects, such as gallium and arsenic vacancy pairs. A number of radiation damage experiments have been conducted by Millea and A ~ k e r m a n ' ~ ~in , ' ~order ~ " to reveal some of the mechanisms responsible for the electroluminescence and current-voltage characteristics of zinc-diffused GaAs diodes. If it is assumed that the principal recombination band results from electrons injected into the p-region and that the radiative recombination probability is proportional to the acceptor or hole concentration, it can be shown'34 that I,, the electroluminescent intensity at a given voltage, would be proportional to the minority carrier lifetime. E ~ p e r i m e n t a l l y ,however, '~~ it was found (see Fig. 13) that I , was proportional to the square root of minority carrier lifetime ;assuming Eq. (63)to be correct that is

I, cc (1 + Z~K$)-''*.

(64)

This is the result expected if the recombination probability is independent of the acceptor concentration. Furthermore, utilizing short circuit current measurements during b ~ m b ard rn en t,'~these authors were able to demonstrate an inverse correlation between electron diffusion length (proportional to and the donor concentration, such that

l/z,

=

10-9ND+ 6 x 10-64.

(65)

J. J. Loferski and M. H. Wu, in "Radiation Damage in Semiconductors" (Proc. 7th Intern. Conf.) p. 213. Dunod, Paris and Academic Press, New York, 1964. 1 3 2 G. W. Arnold, Phys. Rev. 149.679 (1966). 13ZaG. W. Arnold and G. W. Gobeli, Proc. Santa Fe Conj Radiation Effects in Semiconductors, Plenum Press, New York, 1968. 1 3 3 G . W . Gobeli and G. W. Arnold, Bull. Am. Phys. Soc. 10, 321 (1965). 1 3 4 L. W. Aukerman, M. F. Millea, and M. McColl, IEEE Trans. Nttcl. Sci. NS-13, 174(1966). 1 3 ' L. W. Aukerman, M. F. Millea, and M. McColl, J . A p p l . Pliys. 38, 685 (1967). 13'

6.

RADIATION EFFECTS

407

INTEGRATED FLUX ( c m - 2 ) FIG. 13. Degradation of electroluminescent intensity I , and short-circuit current J,, with bombardment by 2-MeV electrons on a zinc-diffused GaAs p-n junction. The subscript “zero” refers to initial values, and J , refers to the electron beam current.

These two facts very strongly suggest135that the radiative transition starts from the donor level rather than from the valence band as had previously been assumed. do not behave according to Heavily doped diodes ( N , 2 2 x Eq. (64). In this case the intensity at a given bias has a tendency to saturate with The results seem to indicate that much of the electroluminescence comes from the depletion region. It is proposed’36 that, at liquid nitrogen temperature, recombination occurs via “tail states”128ain the depletion region. The increase in forward current A J , resulting from electron bombardment, follows the empirical relation’ 34

AJ cc q3 exp SV,

(66)

where V is the applied bias and S < q/kT This behavior is true for either lightly doped or heavily doped diodes and is consistent with the hypothesis that the added current results from nonradiative recombination via tail states in the depletion region. 136

L. W. Aukerman and M. F. Millea, J . Appl. Phys. 36,2585 (1965)

408

L . W . AUKERMAN

Radiation damage was also used by Logan et al.13' to show that the electroluminescence of G a P diodes occurs outside the depletion region. A direct correlation between quantum efficiency and minority carrier lifetime was established. Using this correlation and Eq. (63), the authors were able to fit their curves of efficiency vs gamma dosage. Since the minority carrier lifetime was derived from measurement of diffusion length (photoconductive response during y-irradiation) the authors concluded that the bands aregenerated outside the space-charge region by injected current ;otherwise, variation of diffusion length would not affect the quantum efficiency. Saji and Inuishi' 3 8 reported some interesting observations concerning the effects of Co-60 gamma irradiation of GaAs laser diodes. Not only did the threshold current increase, but also the lasing wavelength shifted slightly to shorter wavelength with increasing y-ray irradiation. During annealing at 200°C the change in threshold current tended to recover, but the lasing wavelength shifted to still shorter wavelength. c. Esaki Diodes Considerable interest in the effects of energetic radiation on Esaki or tunnel diodes'39 has arisen from the expectation that these devices should be relatively radiation resistant, and from the desire to understand more thoroughly the nature of the excess or valley current. Although it is true that the peak current is relatively little affected by irradiation, the excess current is quite strongly increased by radiation damage.140*'40aThis fact is illustrated in Fig. 14 for a GaAs diode irradiated with various amounts of 2-MeV electrons. It is generally agreed that the excess current results from some tunneling mechanism involving electronic levels in the forbidden band.'41,'42 Thu s, as irradiation increases the density of localized states in the forbidden band, the excess current increases. At a given voltage the excess current is proportional to the integrated f l ~ ~ . ~ ~ At low temperatures the excess current sometimes (but not always) shows some structure; that is, humps are present, as in Fig. 14. The presence of these humps has been interpreted in terms of tunneling between a band and a discrete energy l e ~ e l . ' ~ ~ - ' ~ ~ R. A. Logan, H. G. White. and R. M. Mikulyak, A p p l . Phys. Letters5.41 (1964). M. Saji and Y. Inuishi, Japan J . Appl. Phys. 4, 830 (1965). 139 L. Esaki. Phys. Rev. 109, 603 (1958). 14' T. A. Longo, Bull. Am. Phys. SOC.5, 160(1960). 14'aJ. W. Easley and R. R. Blair, J . Apgl. Phys. 31, 1722 (1960). 1 4 1 T. Yajima and L. Esaki, J . Phys. SOC.Japan 13, 1281 (1958). A. G. Chynoweth, W. L. Feldmann,and R. A. Logan, Phys. Rev. 121,684(1961), 1 4 3 R. S. Claassen, J . Appl. Phys. 32, 2372 (1961). 144 C. B. Pierce, H. H. Sander, and A. D. Kantz, J. Appl. Phys. 33,3108 (1962). C. B. Pierce and A. D. Kantz. J . Appl. Phys. 34, 1496(1963). 13'

13'

~

~

6.

.. 0.01

I

I

I

409

RADIATION EFFECTS

I

'

-

'

I

I

I

I

I

I

I

I

VOLTAGE, V

FIG.14. Current-voltage curves for a GaAs tunnel diode at 78°K. Curves 0, 1,3, and 10 were taken following 2-MeV electron irradiation of, respectively, 0,0.25,0.74,and 3.2 x 10" electrons/ cm2. (After C l a a ~ s e n . ' ~ ~ )

From Fig. 14 it is seen that electron irradiation of a GaAs tunnel diode produces humps centered at approximately 0.4 and at 0.8V. Pierce et aZ.,144,145 who irradiated a number of GaAs Esaki diodes at 77"K, estimate the two humps to be centered at 0.4 and 0.9 V. There is an ~ n c e r t a i n t y ' ~ ~ - ' ~ ~ involved in trying to assign a definite energy level to a given hump. The difficulty arises as a result of the uncertainty in the Fermi level positions on the two sides of the junction and the broadness of the humps. C l a a ~ s e n ' ~ ~ assumes that the first hump is due to the level at E , - 0.13 eV. The second hump then predicts the level at E , - 0.52 eV (see Table 111). An isochronal annealing experiment between - 100°C and 275°C indicated a slight change in the magnitude of the hump near room temperature and a large change near 220"C, the latter being similar to the annealing of electrical conductivity. An apparent reverse annealing of the 0.4-eV hump occurred at about 125°C.

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Solid Solutions and Impurity Effects

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CHAPTER 7

Phenomena in Solid Solutions N . A . Goryunova I;. P . Kesamanly D . N . Nasledov I.

INTRODUCTION

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 413

11. SUBSTITUTIONAL SOLIDSOLUTIONS I N SYSTEMS INVOLVING 111-V COM-

THEIREQUILIBRIUM PHASEDIAGRAMS . . . . . . 1 . Solid Solutions of 111- V Compounds . . . . . . . . . 2. Solid Solutions between 111- V and Other Compounds . . . . . 111. PREPARATION AND CHARACTERIZATION OF EQUILIBRIUM ALLOYS . . I V . PHENOMENA I N SOLIDSOLUTIONS WITH ISOVALENT SUBSTITUTION . . 3. Cationic Substitution . . . . . . . . . . . . . 4. Anionic Substitution . . . . . . . . . . . . . . V. PHENOMENA IN SOLID SOLUTIONS OF HETEROVALENT SUBSTITUTION.. 5. Solid Solutions with Normal Tetrahedral Structure . . . . . 6 . Solid Solutions with Defect Tetrahedral Structure . . . . . VI. CONCLUSIONS . . . . . . . . . . . . . . . . . POUNDS, AND

415 416 417 424 426 426 432 450 45 1 455 451

I. Introduction Mixed crystals or, as they are more precisely designated, substitutional solid solutions are very common in the group of 111-V compounds. Such solid solutions may be produced by combining 111-V compounds either one with another or with other substances, especially with those belonging to the same crystallochemical group of diamond-type semiconductors, i.e., those that are similar in structure and in type of chemical bonding. This frequency of occurrence of solid solutions in the group 111-V compounds did not become evident immediately. At the early stage of investigations it did not seem likely that 111-V compounds would always form mixed crystals, even one with another. However it has since been found that in all cases studied in detail the difficulties were associated with the kinetics of formation, and that solid solutions could be produced in a homogeneous and equilibrium condition using improved techniques. It has also been found that the distribution of atoms of two or more kinds in the diamond-type lattice of zinc blende can take place both statistically at random and in an 413

414

N. A . GORYUNOVA, F. P. KESAMANLY, AND D. N. NASLEDOV

ordered form. The fact that, in the course of an ordinary preparation of 111-V solid solutions involving a rapid cooling from the melt, nonequilibrium alloys are produced may be explained by the directional character of the covalent bonding which hampers diffusion. The rapid developments involving the preparation of solid solutions of HI-V compounds which started in 1953 were due to the requirements of semiconductor applications. On studying the properties of 111-V compounds, the investigators concerned with their applications soon arrived at the conclusion that in order to ensure adequate progress in semiconductor engineering one should, besides the 111-V compounds, have materials with properties intermediate to those of the known 111-V compounds. One might expect to attain these by alloying 111-V compounds one with another. It was feared however that the electron scattering due to the statistically disordered arrangement of atoms of the third or fifth group in the periodic system would significantly reduce the mobility of current carriers. This would deprive the alloys of their similarity with 111-V compounds and preclude their application in the fields in which 111-V semiconductors are used. We shall see later that these fears were groundless. A second objection was that substitutional solid solutions formed with 111-V compounds, as well as those of another kind might differ in composition from pseudobinary sections of ternary systems or, in other words, differ from the stoichiometric ratios corresponding to the starting compounds proper. These deviations might result in a large number of impurity levels as well as in making impossible the subsequent purification of the alloys to the extent approaching intrinsic conduction. It will be shown later that the possibility of such a deviation from the stoichiometric proportions is different for various kinds of solid solutions, Now for 111-Vcompounds in solid solution with one another, these deviations could not be detected experimentally ; hence one might conclude that in the solid solutions of 111-V compounds they do not exceed the deviations from the stoichiometric ratios existing in the starting binary compounds. However at the beginning of the investigations into the physical properties of solid solutions, this was not evident, and likewise other problems, some of which remain unsolved up to now, were not clear. We shall give later a chemical classification of solid solutions, and we will arrange the material in accordance with this classification. There is no doubt that physical phenomena in the solid solutions of 111-V compounds depend directly on the chemical nature of the starting components as well as on the mechanism of formation of the solid solution. All these factors determine the character of the energy spectrum with which the physical properties of the corresponding solid solution are closely associated. However because of lack of experimental data, this dependence cannot always be established with sufficient confidence.

7. PHENOMENA

IN SOLID SOLUTIONS

415

One may hope that the suggested chemical classification may be used as a starting point in the interpretation of future experimental data.

II. Substitutional Solid Solutions in Systems Involving III-V Compounds, and Their Equilibrium Phase Diagrams In the family of diamond-type semiconductors one distinguishes normal tetrahedral phases derived from semiconductors of the IV group and defect tetrahedral compounds differing in the existence of vacant sites in the diamond-type structure. Correspondingly the solid solutions considered below may be classed either as normal or defect diamond-type (tetrahedral) semiconducting phases. The formation of substitutional solid solutions from starting components, one of which represents a III-V compound, and another-a compound belonging likewise to diamond-type semiconductors-may take place in different ways. In cases where both interacting compounds are of the III-V type, “anionic” or “cationic” isovalent substitution occurs. Also, substitution of an “anion” and “cation” may take place at the same time. Actually the terms “anionic” and “cationic” are a matter of convention since in such covalent compounds as the III-V groups there cannot exist cations and anions as in the case of salts. If the second compound in the solid solution is of a normal tetrahedral (diamond-type)phase but ofa type other than of the III-V system (for instance, of the II-VI, the II-IV-V,, the 11I,Vl3, or other types), then heterovalent substitution (in the case of defect phases) will take place involving a simultaneous formation of vacant sites proportional to this substitution. It may be supposed from the beginning that, in the case of isovalent substitution, the substituting atom will not be an active impurity affecting the type of conduction. And conversely, it is quite natural to consider the case of heterovalent substitution from the standpoint of formation of donor or acceptor levels by the substitution atom. The introduction of vacant sites in the course of forming solutions of defect compounds (heterovalent substitution) represents an additional complicating factor. The formation of substitutional solid solutions is characterized by certain phase diagrams. They are of different form depending on the chemical nature of the constituents undergoing substitution. The investigation of solid solutions has advanced fairly far. However there is a considerable difference between the large number of systems in which the existence of substitutional solid solutions has been established by physicochemical analysis and a much smaller number of systems where physical properties of these solid solutions have been investigated. The following section lists the systems based on III-V compounds where the

416

N. A. GORYUNOVA. F. P. KESAMANLY, AND D . N . NASLEDOV

existence of substitutional solid solutions has been established. This list becomes enlarged very rapidly. The corresponding references may be found in the monograph of one of the authors‘ and in the book2 edited by R. Willardson and H. Goering. References to publications not listed in the latter are given.

1 . SOLIDSOLUTIONS OF 111-V COMPOUNDS a. Normal Tetrahedral Phases, Isovalent Substitution

(i) Boron compounds have been inadequately investigated from this point of view, but there are indications of a high solubility of BP and BAS in each other,3 and of the formation of solid solutions in the GaAs system.3a (ii) Aluminum compounds probably form solid solutions with one another, although the corresponding systems have not yet been investigated. Little study has been given also to systems in which the nitride and phosphide of aluminum interact with other 111-V compounds. The existence of substitutional solid solutions throughout the whole concentration range has been established for the following systems except the last one : AlAs-GaAs

AlSb-GaSb

AlAs-InAs

AlSb-lnSb AlP-GaP3b

(iii) Gallium compounds, excepting the nitride whose behavior in such systems has not been investigated, form solid solutions with one another throughout the whole concentration range : Gap-GaAs GaAs-GaSb, with other 111-V compounds

Gap-InP GaAs-In As GaSb-InSb, as well as with aluminum compounds (as mentioned previously). N . A. Goryunova, “Khimiya almazopochobnykh poluprovodnikov.” Leningrad. Izd-vo Leningradskogo Universiteta, 1963 [English Tmnsl.: “The Chemistry of Diamond-like Semiconductors’’ (translated by Scripta Technica : J . C. Anderson, ed.). Chapman & Hall. London, 19651. J . C. Woolley in “Compound Semiconductors” (R. K. Willardson and H . L. Goering, eds.). p. 3. Reinhold, New York. and Chapman & Hall. London, 1962. C. C. Wang, M. Cardona, and A. Fischer, R C A Rev. 25, 159 (1964). 3aS.M.Ku, J Electrochem. Soc. 113, 813 (1966).

7.

PHENOMENA IN SOLID SOLUTIONS

417

(iv) Indium compounds, except for the nitride, form substitutional solid solutions with one another throughout the whole concentration range : InP-InAs InAs-InSb as well as with other 111-V compounds (see above). One may note that in this list of systems of substitutional solid solutions there are no examples of substitution of “cations” or “anions” which are located far from one another in the periodic table (the only exception being the system AlSb-InSb). Such systems await study. The systems that have been studied are those formed by a simultaneous substitution of a “cation” and “anion”. InP-GaSb3‘

Ga-In-As-Sb3‘

I ~ P - G ~ A s ~Ga-In-P-Sb3‘ ~ Ga-In-P-As3‘ Phase diagrams of the systems listed represent, in the majority of cases, diagrams of the first type by Roozeboom3‘ which correspond to typical solid solutions of substitution. An example is the phase diagram of AlSbInSb (Fig. 1).

2. SOLD SOLUTIONS BETWEEN 111-V

AND

OTHERCOMPOUNDS

a. Normal Tetrahedral Phases, Heteroiialent Substitution ( 1 ) ZZ1-V-ZV Compounds. The elements of group IV dissolve in 111-V compounds only at concentrations of about 1 at. %. However through rapid cooling one may produce metastable solid solutions of germanium and gallium antimonide throughout the whole concentration range. 3bG.A. Kalyuzhnaya, D. N. Tretiakov, A. S. Borshchevskii, and A. A. Vaipolin, Issledocaniya PO Poluprovodnikam-Novye Poluprovodnikovye materialy. Gosudarstvennoe Izdatel’stvo “Kartya Moldovenyaske” (D. N. Nasledov, N. A. Goryunova, D. V. Gitsu. V. N. Lang. and S. I. Radautsan. board of editors), p. 123, Kishivev. I964 [English Transl.: Soviet Research in New Semiconductor Materials (D. N. Nasledov and N. A. Goryunova, eds.). p. 80. Consultant Bureau. New York. 19651. 3cN. A. Goryunova and V. I. Sokolova. Izv. Modavsk. Filiala Akad. Nauk SSSR Nr. 3(69). 31 ( 1 960). 3dN.N. Sirota and L. A. Makovetskaya, Dokl. Akad. Nauk Belorussk S S R 7,230 (1963) [English Transl.: “Period of Identity and Microhardness of Semiconductor Solid Soh tions of InPGaAs”. FTD-TT-65-2000 (1966)l. 3eE.K. Muller and J. L. Richards, J . Appl. Phys. 35,1233 (1964). “See, for example, H. W. Bakhuis Roozeboom, “Die heterogenen Gleichgewichte vom Standpunkte der Phasenlehre.” Vieweg & Sohn, Braunschweig (In several volumes and parts, published between 1901 and 1918).

418

--

N. A. GORYUNOVA, F. P. KESAMANLY, AND D. N.

0 In Sb

25

50 mole ' 7

7s

FIG. 1 . Phase diagram for AISb-InSb system. 0 : heating:

( 2 ) ZZZ-V-ZZ-VZCompounds i. Alp-ZnS v. GaAs-ZnSe ii. AlSb-CdTe vi. GaSb-ZnSe3h iii. GaP-ZnS3b,3g vii. GaSb-CdTe3' iv. GaP-ZnSe3b viii. InP-CdS

-7.-*----1

NA>LCUUV

400 AL sb x : cooling. (After Goryunova.')

ix. x. xi. xii. xiii.

InP-CdSe InAs-ZnTe InAs-CdTe InAs-HgTe InSb-CdTe

In some of the above systems (ii, ix, xii), solid solution was obtained throughout the whole concentration range. In case xi, however, the limit of solubility has been found to be as low as 30 mole % ;in case xiii, 5 mole % (on the 111-V end), and in case i, 1 mole %. (3) ZZZ-V--ZZ-ZV- V, Compounds i. GaAs-ZnGeAs, v. InAs-CdGeAs, vi. InAs-CdSnAs, ii. InP-CdGeP, 3c iii. InAs-ZnGeAs, vii. InSb-ZnSnSb, iv. InAs-ZnSnAs, viii. InSb-CdSnSb, The phase diagrams for some of the above systems are presented4 in Figs. 2 and 3. Whereas in the foregoing systems both binary components had the same zinc-blende structure, in cases of interaction between 111-V compounds and ternary compounds of various types, solid solutions form with substances of a similar but not the same structure. As a rule, in ternary compounds of the diamond group a certain distortion of the zinc-blende structure and ordering of the cationic lattice sites take place. This is perhaps the reason why ternary compounds with a strongly pronounced tetragonality, for 3gM.Harsy and I . Bertbti, Phys. Stat. Solidi 11, K135 (1965). 3hI. I. Burdiyan and B. P. Korolevskii. Uch. Zap. Tiraspol'sk.Gos. Ped. Inst. p. 127 (1966). 3'I. I. Burdiyan and A. I. Mokeitshik, Uch. Zap. Tiraspol'sk. Gos. Ped. Inst. p. 125 (1966).

7.

PHENOMENA IN SOLID SOLUTIONS

419

600-

’3 2 I

PInAs

mole %

FIG.2. Phase diagram for 2lnAs-ZnSnAs, system. and

Zn Sn As,

0: cooling;

x : heating. (After Borchers

example, CdGeAs,, essentially d o not form solid solutions with 111-V compounds, as compared with others in which the range of existence of solid solutions is usually fairly large. The ternary components of systems vii and viii represent the so-called hypothetical compounds which cannot be obtained separately. However they exist as solid solutions with the corresponding 111-V compounds within a fairly broad concentration range. Such a system represents a base of a more complex system involving an element of group IV, Gap-Si-ZnSiP,, which has been studied by Loebner et ul.’ The authors have revealed the existence of a region of solid solutions containing amphoteric silicon. Independently from these authors and almost at the H. Borchers and R. G. Maier, Metall 17, 1006 (1963). E. E. Loebner. I. J. Hegyi. and E. W. Poor, in “Metallurgy of Elemental and Compound Semiconductors” (Metallurgical Society Conferences, Vol. 12. R. Grubel. ed.), p. 341. Wiley (Interscience), New York, 1961.

420

N. A. GORYUNOVA, F . P. KESAMANLY,ANDD . N. NASLEDOV

1

qi I

I

I

0

1

1

100 PInAs

I

80

I

,

I

60

I

i__i

40

mole OA

20

0

Cd GOAS#

FIG.3. Phase diagram for 2lnAs-CdGeAs, system. (After Borchers and Maier.4)

same time, this phenomenon of dissolution of a group IV element in normal tetrahedral phases has been discovered by one of us and our co-workers.6*7 Loebner and his co-workers have suggested that the hypothetical ordered phase ZnGa6Si,,P6 serves as a base for solid solutions of the quaternary system Zn-Ga-Si-P. In Fig. 4 is shown the region of existence of solid solutions in this quaternary system. (4)

III-v-z-w*- v, InSb-AgSn,Sb,

( 5 ) III-v-I-rII-v2 InSb-CuInTe, N. A. Goryunova, A. A. Vaipolin, and Tsing Ping-hsi. Physics and Chemistry. Reports. XIX Scientific Conference [in Russian], Leningrad (1961), p. 27. ' N , A. Goryunova, V. I. Sokolova, and Tsing Ping-hsi, Dokl. Akad. Nauk SSSR 152,363 (1963) [English Transl.: Proc. Acad. Sci. U S S R . Phys. Chem. Sect. 152,808 (196311.

7.

421

PHENOMENA IN SOLID SOLUTIONS

Zn

SL R C

A

B

PGaP

4sc

FIG 4 Range of solid solutions in Z G ~ P - ~ S I - Z ~ Ssystem IP, (After Loebner rt a/ 5 ,

(6) I I I - I/--I2 - I V-VI

, InP-Cu,GeSe,

3E

GaAs-Cu,GeSe, InSb-Ag,SnTe,

(7) III-V-I,-I/-VI, InAs-Cu,AsSedb InSb-Ag,SbTe, After one of the authors had for the first time produced solid solutions within a broad concentration range for the hypothetical compound CdSnSb, in InSb' [see item (3) above] and had shown that this is possible for other hypothetical compounds as well,9 the phenomenon was observed also for such compositions as shown below. '"N.A. Goryunova. G. K. Averkieva, and A. A. Vaipolin, Fizika. Dokl. K XXIII Nauchn Konferentsii Leningr. 1nzh.-Stroit. In-ta. Leningrad. 1965. p. 52. N. A. Goryunova, A. A. Abdurakhrnanova, and M. 1. Aliev, Dokl. Akud. Nuuk A z e r b . S S R 21. 13 (1965). N. A. Goryunova and V. D. Prochukhan. Fiz. T w r d . Telu 2, 176 (1960) [English Trans/.; Sooiet Phys.-Solid State 2. 161 (196O)l. N. A. Goryunova. A. V. Vojtsekhovskii, and V. D. Prochukhan, Vcstn. Leniiigr. U n i c . S er. Fiz. i Khim. 10. 156 (1961).

76

422

N. A. GORYUNOVA, F. P. KESAMANLY,AND D. N. NASLEDOV

(8) ZZI-V-Ill:-ZV-Vl

InSb-In,GeTe InSb-In,SnSe InSb-In,SnTe. Here the substitutional solid solutions were also found to exist within the concentration range 5 to 30 mole % although compounds of the type 111,-IV-VI do not exist individually.” b. Defect Tetrahedral Phases, Heterovalent Substitution (1) i. ii. iii. iv. v. vi.

111-7/-111,-v1, BP-B,Se, vii. GaAs-Ga,Se, AlSb-Al,Te, viii. GaAs-Ga,Te, Gap-Ga,S, ix. GaSb-Ga,Se, Gap-Ga,Se, x. GaSb-Ga,Te, Gap-Ga,Te, xi. InP-In$, GaAs-Ga,S, xii. InP-In,Se,

xiii. InP-In,Te, xiv. InAs-In,S, xv. InAs-In,Se, xvi. InAs-In,Te, xvii. InSb-In,S, xviii. InSb-In,Se, xix. InSb-In,Te, In systems of this type, the second binary component is of the defect zincblende structure. The formation of solid solutions in these cases is accompanied by both “anionic” substitution and introduction of vacant sites. Phase diagrams for some of these systems’ are given in Figs. 5 and 6. For the majority of the systems investigated, the solubility is fairly high (10 to 100 mole %), the broadest region of solid solution bordering the 111-V compound. A more complicated interaction, which has not yet been interpreted in any detail, has been found to exist in systems ix, xi, xiii, xvii. In systems of the type 111-V-111,-VI, , investigations revealed substantial deviations from the stoichiometric ratios in the corresponding pseudobinary sections. In addition, one of the authors suggested that the composition regions in the concentration triangle of the systems 111-V-VI at the 111-V rich end would represent more or less large areas corresponding to solid solutions with the diamond structure. The existence of homogeneous alloys of such a structure is made possible due to different valence states of the group 111 element, as well as to structural defects in the cation or anion parts of the lattice or to the segregation of anions. The presence of such a “lobe” in systems Ga-Sb-Te and In-Sb-Te has been Not only solid J . C. Woolleyand E. W. Wi1liams.J. Electrochem. Soc. 111.210(1964). loaN. P. Luzhnaya, G. K. Slavnova. Z. S. Medvedeva, and A. A. Eliseev, Z h Neorgun. Khim. 9. 1174 (1964) [English Transl.: Russ. J . Inorg. Chem. 9,642 (1964)l. I ‘ I . P. Molodyan and S. I. Radautsan, IzL.. Akad. Nauk SSSR Ser-. Fiz. 28. 1017 (1964) [English Trunsl.; Bull. Acad. Sci. U S S R 28. 918 (1964)l.

lo

7.

PHENOMENAIN SOLD SOLUTIONS

423

solutions but compounds as well are possible in these systems. The recent investigations revealed the range about 5-10 % of the solid solutions in the system InSb-AuIn, . ‘ l a We suppose the existence of very many systems is on the basis of 111-V with the various solid solutions and the new chemical compounds. T”C

4

700-

4 I

690.

I

50

25

0

I n2So,

,

75

In As

mole %

FIG.5. Phase diagram for InAs-In,Se, system. (After Goryunova.’)

0 IT)^ T s

20

40

60 mole %

80

(00

In As

FIG.6. Phase diagram for InAs-In,Te, system. (After Goryunova. I ) ‘IaV. K. Nikitina. A. A. Babitsyna, and Yu. K. Lobanova, Izc. Akad. Nauk SSSR, Ncorgan. Materialy 3, 311 (1967) [English Transl.: Inorganic Materials 3,275 (1967)l.

424

N. A . GORYUNOVA, F. P. KESAMANLY,AND D. N. NASLEDOV

In other analogous systems the formation of diamond-type phases was established only along the sections 111-V-111-V and III-V-VI.'2-14 Alloys of intermediate composition were not studied. 111. Preparation and Characterization of Equilibrium Alloys

Difficulties experienced in the preparation of equilibrium alloys precluded for a long time both the utilization of solid solutions and the investigation of equilibrium diagrams. However as early as 1939, Stohr and Klemm produced equilibrium alloys of silicon and germanium by grinding samples, annealing the powder thus obtained, and subjecting the material to a repeated grinding and annealing.I5 In 1956 Petrov published a paper on the processes of nonequilibrium crystallization of substances with covalent bonding. l 6 The first systems of solid solutions 111-V-111-V for which a transition to the equilibrium condition during anneal has been established were the systems 111-V-111-V studied by Goryunova and F e d ~ r o v a . ' ~ It has been revealed that annealing in the powder form reduces greatly the time required for the transition to the equilibrium condition. However powdered materials are inconvenient for the investigation of electrical properties and for semiconductor applications where 111-V compounds are used. Therefore in all subsequent work on the preparation of solid solutions of 111-V compounds annealing in the powder form was used only to find a principal answer to the question on the possibility of existence of solid solutibns in these systems. In order to accelerate the transition to the equilibrium condition while at the same time obtaining large crystals, and preferably single-crystal samples, attempts were made to start annealing at temperatures approaching the solidus line in the given system of solid solutions. A preliminary determination of the solidus line may be accomplished by the x-ray method proposed by Woolley.' The methods ordinarily used to prepare solid solutions in the form of polycrystalline ingots are the methods of zone recrystallization and directional freezing which are described by Woolley and others in the book' mentioned earlier. I'

l3

l6

B. P. Kotrubenko. V. N. Lange, and T. I. Lange, IzL.. A k a d . Nauk S S S R Ser. Fiz. 28. 1007 (1964) [English Transl.: Bull. Arad. Sci. U S S R 28. 909 (1964jl. H. Hahn and D. Thiele. Z . A m r g Alleg. Chem. 303. 147 (1960). M. S. Mirgalovskaya and E. V. Skudnoba, Zh. Neorg. Khhn. 5, 1113 (1959) [English 7r.an.d. Russ. J . Inorg. Chen~.4, 506 (1959)]. H. Stohr and W. Klemm. Z. Anorg. Alleg. Chem. 241. 305 (1939). D. A. Petrov, Zh. Fiz. Khim. 21, 1449 (1947). N. A. Goryunova and N. N. Fedorova, Zh. Tekhn. Fiz. 25. 1339 (1955).

7.

PHENOMENA IN SOLID SOLUTIONS

425

In the field of preparation of materials having a high melting point and high pressure of dissociation a new method has become widespread. It is based on the growth of mixed 111-V crystals from their solution in metallic components’ (“solute buildup”). This method consists in a continuous introduction of the substance being dissolved into the solvent until the compound of interest begins to form as crystals at a rate approaching the equilibrium rate. The required temperature gradient over the section of the metal melt is produced by means of a two-temperature furnace, and the vapor pressure of the volatile component is controlled by changing the temperature of the condensed vapor phase (a single-zone furnace). This method permits one to obtain dense single-phase polycrystalline ingots at temperatures of about 250°C below the melting point. During the last few years, wide use has been made of the method of preparation and simultaneous doping of solid solutions in the form of single-crystal layers using chemical transport reactions, as for example with GaAs-GaP solid solution^'^ and other materials. The method consists in the transport of a substance from a high-temperature zone to a low-temperature one using the reaction with iodine. Homogeneous single-crystal layers are obtained at a comparatively low temperature. The principal advantage of the method is that it permits one to produce homogeneous layers of solid solutions of a desired concentration. It is much more difficult to attain homogeneity using other methods. It is interesting to note that this technique has been used to construct the first laser employing the solid solution of a 111-V compound.20 A check of the quality of the obtained solid solutions should first of all include an x-ray analysis. With good homogeneity the lines obtained with x-ray powder analysis will be narrow and not blurred. Photometric measurements permit an accurate determination of the point at which the lines corresponding to a solid solution become exactly as broad as those of the initial starting compounds. Other important means of characterization are investigations of microstructure and microhardness, which can also provide information on the degree of homogeneity of the solid solution. The homogeneity of an ingot is checked frequently by measuring the electrical conductivity over the length

l9

*’

E. P. Stambaugh. J. F. Miller, and R. C. Himes. in “Metallurgy of Elemental and Compound Semiconductors” (Metallurgical Society Conferences, Vol. 12. R. Grubel. ed.). p. 317. Wiley (Interscience), New York, 1961. San-Mei Ku. J. Electrochem. SOC. 110. 991 (1963). N. Holonyak, Jr.. D. C. Jillson, and S. F. Bevacqua, in ”Metallurgy of Semiconductor Materials“ (Metallurgical Society Conferences, Vol. 15, J . B. Schroeder. ed.). p. 49. Wiley (Interscience). New York. 1962.

426

N. A. GORYUNOVA, F. P. KESAMANLY,AND D. N . NASLEDOV

of the ingot. There is no doubt that such physical properties as optical properties may likewise serve as a means of detection of inhomogeneities in a sample of an alloy.

IV. Phenomena in Solid Solutions with Isovalent Substitution 3. CATIONIC SUBSTITUTION a. Indium Antimonide-Gallium Aiitimonide

The investigation of physical properties of solid solutions of In,Ga, -,Sb is considered in a number of publication^.^'-^^ Figure 7 illustrates the temperature dependence of the Hall constant in samples of In,Ga,-,Sb of different composition with hole conduction. As the temperature increases from 78"K, the Hall constant at first decreases slowly, which indicates the increase in the concentration of holes. after which it reverses its sign. The temperature at which the sign of the Hall constant reverses shifts to higher temperatures with an increase in gallium content in the alloy. The values of the energy gap E , determined by the temperature dependence of the Hall constant and the electrical conductivity in the intrinsic region increase linearly with gallium content up to a composition of 70% gallium antimonide, after which a sharp violation of the linearity is exhibited. As for the dependence of E , on composition which was determined by optical means, it is nonlinear over most of the composition range. The value of the effective electron mass increases monotonically, and the mobility of electrons decreases with the increase of gallium content in the alloy. Despite the fact that the effective hole mass also increases monotonically with the increase of gallium content in solid solution, the mobility of holes does not depend on composition and varies within the range 600750 cm2/volt-sec. The latter indicates that the ratio of electron-to-hole J. S. Blakemore, Can. J . Phys. 35, 91 (1957). V. I. Ivanov-Omskii and B. T. Kolomiets. Dokl. Akad. Nuuk S S S R 127. 135 (1959) [ E q l i s h Trans/.: Pvor. Acad. Sri. U S S R Phys. Chern. Sect. 127.553 (1959)l. 2 3 V. I. Ivanov-Omskii and B. T. Kolomiets. Fiz. Tvcvd. Tela 1. 913 (1959) [English Transl.: Soviet Phys.-Solid State 1. 834 (1959)l. 2 4 J . C. Woolley, J. A. Evans, and C. M. Gillet. Proc. Phys. SOC.(Loridoti)74.244 (1959). 2 5 V. I . Ivanov-Omskii and B. T. Kolomiets. Fiz. Tvevd. Tela 2. 388 (1960) [English Trans.: Socirt Phys.-Solid State 2. 363 (1960)l. *' J . C. Woolley and C. M. Gillett. J . P h j s . Chmm. Solids 17.34 (1960). " V. I. Ivanov-Ornskii and B. T. Kolorniets. Fiz. Tuerd. Tela 3. 3553 (1961) [Euglish Transl.: Soviet Phys.-Solid State 3, 2581 (1962)l. '* V. I. Ivanov-Omskii and B. T. Kolomiets. Fiz. Tvrrd. Tela 4. 299 (1962) [English Transl.: Soviet Phys.-Solid State 4,216 (1962)l. 2 9 I. S. Baukin, V. I. Ivanov-Omskii, and B. T. Kolomiets. Izu. Akad. Nauk SSSR. Ser. Fiz. 28. 1000 (1964) [English Transl.: Bull. Acad. Sci. U S S R 28. 902 (196411. 21

22

7.

PHENOMENA IN SOLID SOLUTIONS

427

3 2.6 -0

6

2.2

0 L

”(8

’2 p

4.4

X

a

p, 0

i.0

4

0.6

02 Log T FIG.7. Temperature dependence of Hall constant R , for samples ofGaSb-InSb. + : 24 mole :b‘ GaSb. 76 mole % InSb : 0 : 76 mole GaSb, 24 mole ”i; InSb; 0: GaSb. (After Woolley and Gillett.26)

:;

mobilities depends on alloy composition, namely, it decreases with the increase in gallium antimonide content. This suggests the explanation that the substitution of gallium for indium does not change the mobility of holes and is reflected mainly in a change of the electron mobility. Similar facts are known for other systems.30 The small deviation of the concentration dependence of electron mobility from a linear law, together with the high absolute value of the mobility, suggests that the contribution to the total electron scattering of the disordered structure of the alloy is small. If this mechanism were predominant, then on the curve relating electron mobility with composition a minimum would have been observed at the point corresponding to a 1 : 1 component ratio. Thus i t can be concluded that electron scattering by the disordered structure of an alloy in the solid solution of In,Ga,-,Sb is not of major importance. The thermal gradient effects of Nernst-Ettingshausen in the alloys of In,Ga, -,Sb of various compositions with electron conduction in the temperature range 100 to 500°K were investigated by Ivanov-Omskii and K o l o m i e t ~ . ~ ~Figure - ~ ’ 8 shows the results oftheir study on a sample ofequimolecular composition, the concentration of electrons being 2 x 10’’ ~ m - ~ . 30

B. A. Efimoba. T. S. Stavitskaya, L. S. Stil’bans, and L. M. Sysoeva. Fiz. E e r d . Trla 1, 1325 Tratsl. . Societ Phys.-Solid Statr 1. I2 17 ( 1 960)l.

( 1959) [English

428

N . A. GORYUNOVA, F. P. KESAMANLY,AND D. N . NASLEDOV

4.0 r

Ol

-1.0

FIG.8. Curves for conductivity (u),Hall coefficient (R). thermal emf (a). and coefficients of longitudinal ( Q " ) and transverse (Q') Nernst-Ettingshausen effects for GaSb-InSb. (After Ivanov-Omskii and Kolomiets.25)

At low temperatures the Nernst-Ettingshausen coefficients are negative, which may indicate a significant scattering of electrons on ionized impurities. The increase in the negative value of the Nernst-Ettingshausen coefficients near 200°K is caused by the contribution due to current carriers of opposite sign (holes). The authors note that the Hall constant in this sample does not depend on magnetic field strength in the range 100Ck7000 Oe. The electrical and optical properties of the alloy In,Ga, -,Sb of an equimolecular composition are close to those of indium arsenide although the melting point of the alloy is lower. Presented below are values of some parameters for this alloy of equimolecular composition : Energy gap at WK Electron mobility at 300°K Hole mobility at 300°K BE,/AT Effective electron mass Effective hole mass

0.42 5 0.02 eV 30,000 cm2/volt-sec 600 cm2/volt-sec -(4.0 5 0.5) x 10-4eV/deg (0.04 O.O1)mo (0.25 k 0.05)m0

The Nernst-Ettingshausen coefficients in gallium antimonide and alloys with cpntents of indium antimonide of up to 10% are essentially positive. This may indicate a similarity between the band structures of gallium antimonide and of its alloys with a small content of indium antimonide. With an increase of indium antimonide content in the alloy, the energy gap between the bands (000)and (1 11) increases. The signs of the Nernst-Ettingshausen coefficients in alloys containing up to 10 % of indium antimonide are not determined by the scattering mechanism.

7.

PHENOMENA IN SOLID SOLUTIONS

429

b. Indium Arsenide-Gallium Arsenide A second solid solution of cationic substitution in which the starting compounds possess the same band structure is the alloy In,Ga, -,As. The properties of this alloy are sparsely studied, and data given in the two publicat i o n 1~, 3~2 dealing with the investigation of transport effects and optical properties are limited and in some respects even contradictory. In the earlier p ~ b l i c a t i o nI, ~for example, the dependence of the energy gap on composition has been found to be linear whereas in a subsequent paper32 it is stated that this dependence is linear up to 80 % of gallium arsenide in the alloy. The Hall constant in alloys of In,Ga,-,As with high indium arsenide content does not depend on temperatures within the temperature range 100400°K. In alloys that contain more than 50% gallium arsenide the Hall constant increases with increase in temperature. The thermal conductivity of the lattice for the alloy In,Ga, -,As at room temperatures decreases significantly as compared with the starting compounds and attains a minimum value of 0.05 W/cm-deg at a 1 : 1 component ratio.3 In Fig. 9 are given all the presently available data on the composition dependence of the thermal conductivity of isovalent solid solutions.33 As seen from Fig. 9, thermal conductivity changes with alloy composition in a similar way for GaAs-GaP and InP-InAs. According to the data obtained by Abrahams et aL3’ the density-of-states effective mass of the electrons as determined by thermal emf and Hall coefficient measurements does not depend on alloy compositim. It is difficult with such a conclusion, particularly since the dependence of the energy gap on composition is close to a linear law, that is, the band structures of the starting compounds and alloys are identical and the effective masses of electrons in pure components a t electron concentrations specified in the article in question differ almost by a factor of 3. The effective electron masses in indium arsenide and gallium arsenide are 0.026 and 0.07 that of the mass of a free electron, respectively. The effective electron mass in the alloys GaAsInAs increases with increase in the content of GaAs and depends on the concentration. 3a The Hall mobility in the alloy In,Ga, -,As decreases monotonically as the content of gallium arsenide increases up to 70%.

’

M. S . Abrahams, R. Braunstein. and F. D Rosi. J . Phys. Chem. Solids 10. 204 (1959). J . C . Woolley, C. M. Gillet, and J. A. Evans. Proc. P ~ J J Soc. S . (London)77.700 (1961). 3 3 J . R. Drabble and H. G. Goldsmid, “Thermal Conduction in Semiconductors.” Pergamon, Oxford. 1961. ”*E. F. Hockings. I. Kudman, T. E. Seidel. C. M. Schmelz. and E. F. Steigmeier. J . A p p l . Phys. 37. 2879 (1966). 31

32

430

N. A. GORYUNOVA, F. P. KESAMANLY, AND D. N . NASLEDOV

Q

25x1750 2

mole% GaAs in GaP

5

~

0 a5075m ~ 0

mole% I nAs mole% in

GQAS

Ln

InP

InAs

FIG. 9. Thermal conductivity of alloys between 111-V compounds at room temperature. (After Drabble and G o l d ~ m i dcurve ~ ~ ; I data from we is^^^ : 2 from Abrahams et ~ 1 . :~3 ' from Bowers et d 4 ' )

c. Alurniiium A7Jti~lOlJid~-ri7diUm Antimonide In the solid solution of cationic substitution Al,In, -,Sb the conduction bands of the starting components differ in structure, and the bend on the curve illustrating the dependence of the energy gap on composition is one more proof of this difference. One of the authors of this review has studied the temperature dependence of transport phenomena in this alloy ~ y s t e m . ~ ~ , ~ ' Figure 10 shows the temperature dependence of the electric conductivity and the Hall constant36 for samples of different composition. viz. ( I ) 9InSbAISb, (2) 7.5 InSb-2.5 AISb. (3) InSb-A1Sb. All the crystals exhibit hole conduction at low temperatures, their electric conductivity depending only weakly on temperature. The transition to intrinsic conduction shifts to higher temperatures with the increase of aluminum content in the alloy. In the low-temperature region the Hall coefficient varies little with temperature, and the Hall constant does not depend on the magnetic field strength. The same crystals were used to study the temperature dependence of magnetoresistance at 5000 Oe. In the low-temperature region the magnetoresistance is very small. which may be attributed to a low mobility of holes. 34

35

36

Ya. A p e \ and D. N. Nasledov. 131..Akud. N u u k Tirrkm. S S R 3. 3 ( I 959). Ya. Agaev. 0. V. Emel'yanenko, and D. N. Nasledov. F i z . Trrrtf. T d u 3. 194 (1961) [ E ~ g l i d i Trattai.: S0cic.t Phys.-Solid Stutr 3. 141 (1961)l. Ya Agaev and D. N. Nasledov. Fiz. Twrd. Tela 2. 826 (1960) [English Trurtsl.' Sorier PJI~sSolid State 2. 758 (1960)l.

7.

PHENOMENA IN SOLID SOLUTIONS

log 6

og

3.0

-10

2.5

-2.5

2.0

- 20

45

- 15

10

- 1.0

0.5

431

I

,0.5

I

FIG. 10. Temperature dependence of Hall constant and electric conductivity in A1Sb-lnSb samples. 1 : 9 InSb-AISb: 2: 7.5 InSb-2.5 AISb: 3 : InSb-AISb. (After Agaev and N a ~ l e d o v . ~ ' )

However it begins to rise with an increase in temperature and attains a maximum in the same region as in the case of the Hall coefficient. The Hall mobilities of holes for samples (1 1, (2), and (3). were. respectively, 600. 400. and 80 cm2/volt-sec, which indicates that hole mobility decreases with the increase of aluminum content in the alloy. Studied in this alloy were also the temperature dependences of the transverse and longitudinal thermomagnetic effects of Nernst-Ettingshausen. A regular variation. from one sample to another, of the temperature dependence curves for the Nernst-Ettingshausen coefficients was observed. which indicates that the electronic processes that determine thermomagnetic effects in solid solutions of Al,In, _,Sb are generally of.the same nature as those in the starting components of this alloy. The negative sign of the Nernst-Ettingshausen coefficient in the extrinsic region evidently indicates the absence of significant scattering of current carriers on the alloy lattice. Scattering associated with distortions of the ideal structure in the alloy. which originate from a random distribution of atoms of the components involved in the sites of the common lattice, results in the same relationship between the mean free path and energy of current carriers as in the case of scattering on acoustical vibrations of the lattice,25 and hence in a positive sign of the Nernst-Ettingshausen effect. In the alloy AIJn, -,Sb. this effect

432

N. A. GORYUNOVA, F. P. KESAMANLY,AND D. N. NASLEDOV

however does not become positive at any temperature. This permits one to conclude that scattering on the disordered structure of the alloy in the Al,In,-,Sb system is small. This means essentially that the mobility of current carriers in the samples investigated is limited by foreign impurities and can be increased by purification of material.

d. Aluminum Antimonide-Gallium Antimonide In crystals of solid solutions of AI,Ga,-,Sb with hole conduction, the temperature dependencies of electrical conductivity, Hall coefficients, and thermal emf were s t ~ d i e d . ~ ~The - ~ ' temperature dependence curves for these effects are similar to those obtained for the starting components, the only difference being that with the increase of aluminum content in the alloy the temperature of the transition to intrinsic conduction in samples with the concentration of impurities of about 10" cm-3 shifts gradually from 700 to 900°K The mobility of current carriers increases with increase in gallium content of the alloy. The mobility of holes a t room temperature varies within the range 75 to 250 cm2/volt-sec. The differential thermal emf and the effective mass of holes both increase with an increase in aluminum content of the alloy. The dependence of the energy gap on composition is nonlinear. In one p ~ b l i c a t i o na ~linear ~ relationship was found between the lattice parameter and the energy gap. The band structure for the alloy Al,Ga,-,Sb has also been i n v e ~ t i g a t e d . ~ ~ . ~ ~ 4. ANIONIC SUBSTITUTION a. Indium Arsenide-Indium Phosphide (1) Electrical Properties. The temperature dependences of electric conductivity a ( T )and Hall coefficient R ( T ) in mixed crystals of InAs,P -, of different compositions were studied by we is^^^ in the temperature range 20 to 500°C. Figure 11 shows values obtained by him for the Hall constant in samples with composition corresponding to x = 0.8. Doping mixed crystals I. I. Burdiyan and B. T. Kolomiets. Fiz. Tterd. Tela 1. 1165 (1959) [English TramZ.: Soviet Phys.-Solid Scare 1. 1067 (1960)l. 3 R I. I. Burdiyan and B. T. Kolomiets, Voprosy metallurgii i fiziki poluprovodnikov: poluprovodnikovye soedineniya i tverdye splavy. Trudy 4-go soveshchaneya. Moskva. Izd-vo Akad. Nauk SSSR. 1961. p. 127. 39 I. I. Burdiyan. Ya. A. Rozneritsa, and G. I. Stepanov. Fiz. Tuerd. Tela 3, 1879 (1961) [ E n g h h Trunsl.: Soviet Phys.-Solid State 3, 1368 (196111. 40 J . F. Miller, H. L. Goering, and R. C. Hirnes. J . Electrochem. Soc. 107. 527 (1960). 4 1 H. Ehrenreich, J . Appl. Phys. 32, 2155 (1961). 41 C. A. Mead and W. G. Spitzer. Phys. Reu. Letters 11. 358 (1963). 4 3 H. Weis's, 2. Naturforsch. l l a . 430 (1956). 37

7.

PHENOMENA IN SOLID SOLUTIONS

FIG.11. Temperature dependence of Hall constant in InAs,,lnP,, samples, C-G : n-type samples. (After we is^.^^)

433

crystals. I and 2: p-type

of InAs,P, -x with various impurities can be used to produce both n- and ptype material just as in the case of the starting compounds. Curves 1 and 2 in Fig. 1 1 correspond to p-type samples obtained by zinc doping; other samples are of n-type, the most “impure” of them being produced by adding a donor impurity (sulphur) to crystals with lower electronic concentrations. Curves for crystals of other compositions are similar to those presented in Fig. 11. Analysis of the data of Weiss shows that the ratio of electron and hole mobilities, b, is high both in mixed crystals of InAs,P,-, as well as in InP and InAs (see Table I). The values of b determined from the relation R,,,IR,,

=

(b - 1I2f4b,

where R,,, is the highest value between its zero value and the value corresponding to the intrinsic region, and R,, is the value in the purely extrinsic region, are equal to 60 and 100. The higher the content of phosphorus in the InAs,P,-, alloy, the higher is the temperature at which the intrinsic conduction becomes evident.

434

N . A. GORYUNOVA, F. P. KESAMANLY,AND D . N. NASLEDOV

The temperature dependence of Hall mobility for crystals with x = 0.8 is presented in Fig. 12. from which it is seen that the behavior resembles very much that of the mobility in indium arsenide : in fact, when the temperature is raised above room temperature the mobility decreases-at first following a T - ’ law, and at still higher temperatures corresponding to a T - 3 ’ 2law. Data on the dependence of electron ( U , ) and hole (Up)mobilities, as well as of energy gap, on composition (x)are given in Table I.43 TABLE I DEPENDENCE OF ENERGY GAPAND MOBILITIES ON COMPOSITION I N InAs,P,

- li

X

E , (eV) U , (cm’/volt-sec). 20°C U , (cm2/volt-sec),20°C

I .34 3400 50

-.

4600 -

0.83 0.8 7000 40

0.63 11.100 60

0.58 11.300 -

0.45 23.000 240

I

240’ iI I / T’.K FIG. 12. Temperature dependence of electron Hall mobility in InAs, *Po crystals. (After we is^.^^)

7.

PHENOMENA IN SOLID SOLUTIONS

435

With an increase in phosphorus content in the alloy, the energy gap increases whereas the mobilities of electrons and holes decrease monotonically. The presence of a linear relationship between the energy gap and composition implies that the band structures of the starting compounds and of the mixed crystals are essentially the same, and the effective mass ofcurrent carriers increases with an increase in phosphide concentration in the alloy. In order to investigate the prospects for possible application of mixed crystals of indium arsenide-indium phosphide in Hall transducers used in the measurement of strong magnetic fields, the dependences of the Hall constant and electron mobility on magnetic field strength were m e a ~ u r e d " ~ in the range of 0 to 170 kG in crystals with x = 0.8. The Hall field was found to increase linearly with increase in magnetic field strength up to 170 kG, so that the Hall coefficient did not depend on magnetic field strength. As for the electric conductivity and mobility, they become several times less with the magnetic field strength increasing to 170kG. Figure 13 shows on a log-log scale the relationship of a(O)/a(B) - 1 with field strength. From this it is seen that the expression for electric conductivity as a function of magnetic field strength may be presented in the form

Despite the fact that electron mobility decreases strongly with increase in magnetic field strength, the dependence of the Hall angle tan 0 = :RUB does not exhibit a maximum in the range of magnetic fields applied, as is present in the case of indium antimonide and arsenide (Fig. 14). Ehrenreich carried out theoretical calculation^^^ of the mobility of electrons as a function of composition, taking into account electron scattering on both polar lattice vibrations and impurity ions. The total mobility was determined using the expression 1 1 1 ---

u

+

-

Upolar

~-

Uirnpurity

.

The values of mobility thus obtained agree well with the experimental data of we is^.^^ In addition, Ehrenreich evaluated the possible contribution due to electron scattering by the alloy and succeeded in showing that the maximum value of this contribution to the total mechanism of scattering cannot exceed 10%. It is seen from Fig. 15 that if electron scattering on the alloy were prevalent, then the curve relating electron mobility with composition would have a minimum at x = 0.5. in other words, electron scattering on the alloy in the solid solution of InAs,P, - x is not predominant at low temperatures. 44

45

E. Rraunersreuther. F. Kuhrt. and H . Lippmann, Z . NatwforscP. 15a. 795 (1960). H . Ehrcnrcich. J . Phys. Chrrn. Solids 12. 97 (1959).

436

N. A. GORYUNOVA, F. P. KESAMANLY,AND D. N. NASLEDOV

"': 4

6 4t

2l

aiid 2

4 08W6G

0 FIG. 13. Dependence of u(O)/a(B) on magnetic field strength in InAs,,,P,,, ? ; 2 . InAs; 3 : InSb. (After Braunersreuther cr 0 1 . ~ " )

crystals. 1 :

InAs,,P,

B FIG. 14. Dependence of Hall-angle tangent on magnetic field strength. 0 : InSb : A : InAs : 8Po? . (After Braunersreuther et ~ 1 . ~ 9

x : InAs,

7.

PHENOMENA IN SOLID SOLUTIONS

InP

moLe fractions

437

In As

FIG. 15. Estimated electron mobility due to alloy scattering in InAs,P,_, mixed crystals. Curue 1 : theory assuming linear interpolation of static and dynamic dielectric constants. longitudinal optical frequency. reduced atomic mass. and volume per unit cell. Curtie 2 : theory assuming linear interpolation of effective ionic charge. (After E h r e n r e i ~ h . ~ ~ )

(2) Thermal Properties of the Mixed Crystals. Thermal properties of are treated in article^^^-^^ dealing with thermal mixed crystals of InAs,P, -, emf, thermal conductivity, and transverse Nernst-Ettingshausen effect. The results of measurements of the temperature dependence of thermal emf in crystals with x = 0.8 are given in Fig. 16. As seen from the figure, the behavior of the corresponding curves for all samples (including those not given here) are in qualitative agreement with theoretical predictions. The fact that the curves pertaining to p-type crystals lie, at high temperatures in the part before the bend in the intrinsic conduction region, above the curves corresponding to n-type crystals can be attributed to a large ratio of electronto-hole mobilities. This effect supports the conclusions made from electrical 46 47

48 49

H. Weiss. Ann. Physik 4, 121 (1959). R. Bowers. J . E. Bauerle, and A. J . Cornish. Bull. Am. Phys. SOC.4, 134 (1959). R. Bowers. R. W. Ure. J . E. Bauerle. and A. J. Cornish, J . Appl. Phys. 30, 930 (1959). Ya. Agaev and 0.Ismailov, Izv. Akad. Nauk Turkin. SSR. Ser. Fiz.-Tekhn., Khim. i Geol. Nauk No. 5, 9 (1962).

438

N. A . GORYUNOVA, F. P. KESAMANLY,AND D. N. NASLEDOV

, 0

I

Y

l

j

4.0

Q

)

U

15

I

L

~

.

2.5

2.0

~

_

!1

3.5d

SO

YTOK FIG.16. Temperature dependence of differential thermal emf in InAs,,,P,

crystals (after

we is^^^):

Sample R,. cm3/C

1

2

A

B

C

D

E

+5

f1.l

-400

-180

-95

-40

-5

measurements. From thermal emf values measured at high temperatures effective masses of electrons and holes were calculated for mixed crystals of various compositions on the assumption that scattering of current carriers takes place on thermal vibrations of the lattice. These calculations have shown that the effective mass of electrons and holes increases monotonically with the increase in phosphorus content of the alloy. This could be expected since, for crystals with the same band structure, theory predicts a monotonic increase of the effective mass of current carriers with increase in energy gap of the crystal. However, large values obtained for the effective electron mass, such as, for instance, in* = 0.06rn0 in the case of indium arsenide (and from thermal emf measurements in a strong magnetic field in a sample with the same electron concentration m* = O . O 3 5 n ~ , ~imply ~) a conclusion that scattering of current carriers on thermal vibrations of the lattice at high temperatures cannot be considered as the only mechanism involved. The dependence of thermal conductivity of a crystal lattice at room temperature on alloy composition is shown in Fig. 9. In binary compounds, L. L. Korenblit, D. V. Mashovets, and S. S. Shalyt. Fiz. Tuerd. Tela 6, 559 (1964) [English Trans/.: Souiet Phys.-Solid State 6. 438 (196431.

7.

PHENOMENA IN SOLID SOLUTIONS

439

thermal conductivity increases from antimonide to arsenide and phosphide just as the energy gap increases. However, alloys have a considerably lower thermal conductivity than do indium arsenide and phosphide. The minimum value of heat conductivity observed in the alloy InAs,P,-, at x = 0.5 is about 40 % of the value for indium arsenide. Figure 9 shows also experimental which lie below those data4' for thermal conductivity in the alloy InAs,P, -, obtained by Weiss. This fact, as well as some other considerations, is regarded as casting doubt on the quantitative accuracy of the data of Weiss, although it is emphasized that qualitatively they reflect correctly the character of the effect.33 The absence of a monotonic behavior in the variation of thermal conductivity which was observed for mobility indicates a difference between the scattering mechanisms of electrons and phonons. On the basis of temperature dependence measurements of the transverse Nernst-Ettingshausen coefficient, a conclusion is made49 that at low temperatures the scattering of electrons in the alloy with x = 0.8 occurs mainly on impurity ions, whereas at high temperatures the contribution due to the scattering on lattice vibrations becomes important. The positive sign alone cannot be considered as a proof of the scattering taking place predominantly on acoustical lattice vibrations since, as was shown by calculations of K o l o d ~ i e j c z a kthe ~ ~ Nernst-Ettingshausen effect can be positive also when scattering occurs on optical vibrations of the lattice provided the conduction band is nonparabolic in shape. The mobility at high temperatures follows the T - law, which likewise does not support the suggestion of the scattering on acoustical vibrations. Apparently electron scattering takes place on optical vibrations, and the fact that mobility follows a law steeper than T-'I2 may be explained by an increase of effective electron mass with temperature which was observed in pure indium phosphide in long-wavelength Faraday effect measurements. 5 2 (3) Optical and Photoelectric Measurements. Infrared transmission and reflection in samples of n-type mixed crystals of InAs,P, - and of the starting compounds having electron concentrations from 1 to 4 x 10'6cm-3 d ~the ~ wavelength region I to 35 p, were measured by O ~ w a I in The absorption edge shifts monotonically to shorter wavelengths with increase of phosphorus content in the alloy.54 The temperature dependence of the absorption edge in alloys in the range 90 to 470°K was also investigated.

51

52

53

J . Kolodziejczak and L. Sosnowski. A r t a Phys. Polon. 21, 399 (1962). F. P. Kesamanly, E. E. Klotyn'sh. Yu. V. Mal'tsev, D. N. Nasledov. and Yu. I. Ukhanov, Fiz. Tuerd. Tela 6. 134 (1964) [English Transl.: Souiet Phys.-Solid State 6. 109 (1964)l. F. Oswald, Z . Naturforsch. 14a. 374 (1959). G . B. Dubrovskii. Fiz. Tuerd. Tela 5,954 (1963) [English Trans/.: Soviet Phys.-Solid State 5, 699 (1963)l.

440

N. A.

GORYUNOVA, F. P .

KESAMANLY,AND

D.

N.

NASLEDOV

On the basis of these measurements Oswald obtained the following relation for the energy gap InAs,P, --x as a function of composition and temperature :

E , = 1.42 - 0 . 9 8 ~- (4.6 - 1 . 1 ~ x) 10-4T (ev), where x is the fractional arsenic content. 0 d x d 1 : 100 d T Q 500°K. Also discussed by O ~ w a l dis ~the ~ dependence of refraction coefficient on wavelength in the normal dispersion region. Data on the absorption coefficient were used to calculate, by the formulas of Drude and Frohlich, the values of electron effective mass, which were found to vary regularly from 0.02m0 for indium arsenide to 0.lorn, for indium phosphide. These values are in qualitative agreement with those obtained by Weiss. In Oswald’s papers3 conclusions are also made concerning the existence of a polar nature of bonding in starting compounds and alloys. An investigation of photoconductivity and photomagnetic effect on polycrystalline samples of alloys with x = 0.9 and x = 0.8 was carried out” in order to determine regularities in variation of spectral response and to find the lifetimes of electrons and holes and their temperature dependencies. Figure 17 shows the spectral response of photosensitivity of the alloys investigated. The maxima ofphotoconductivity at 80°K for alloys with x = 0.9 and x = 0.8 lie, respectively, at iL = 2.5 and 2.0 microns. The values of the energy gap determined by the Moss rule are, respectively, 0.45 and 0.57 eV. The temperature coefficient of energy gap variation for the alloy with x = 0.8 is, according to these measurements, 2.7 x eV/deg.

t

0’

I

I

,

2

,

,

,

,

3

>.r FIG. 17. Spectral response of photoconductivity in alloys of InAs,P,-, system. l a : InAso.9Po.,, T = 80°K; 2a: InAso.8Po.,, T = 296°K; l b : same as l a but with white-light background quenching; 2b: same as 2a but T = 80°K. (After Agaev and S l o b o d c h i k ~ v . ~ ~ ) 55

Ya. Agaev and S. V. Slobodchikov, Izv. Akad. Nauk Turkm. SSR, Ser. Fiz.-Tekhn., Khim. i Geol. Nauk No. 1, 14 (1965).

7.

PHENOMENA IN SOLID SOLUTIONS

441

Figure 18 illustrates the temperature dependence of photoconductivity for an alloy with x = 0.8, curves for other compositions being similar. The lifetimes of electrons and holes determined from these data at room temperature in alloys with x = 0.9 and x = 0.8 are, respectively, about l o p 7 and 10-6sec. For the alloy with x = 0.9 a decrease in temperature brings about an increase in the lifetime of electrons, the lifetime of holes at first decreasing (down to 220°K) and then exhibiting a rise. For another sample with x = 0.8, the general pattern oflifetime variation is the same with the exception that at low temperatures the lifetimes of electrons and holes differ from each other by one to two orders of magnitude.

tI Y 3

I

*\

/

100

200

300

T”K FIG.18. Temperature dependence of photoconductivity in InAso,8Po,,. I : without quenching; 2 : with quenching. (After Agaev and S l o b o d ~ h i k o v . ~ ~ )

b. Gallium Arsenide-Gallium Phosphide (1) Mobility of Electrons. In crystals of mixed composition of the type GaAs,P,-, with 0.5 < x < 1.0 the electron mobility was investigated by Ku.I9 He obtained an interesting relationship between electron mobility and composition for samples with the same concentration of current carriers, as is presented in Fig. 19. It is seen that the electron mobility remains approximately constant (at the level of mobility in gallium arsenide) for compositions from pure gallium arsenide up to about 15% of gallium phosphide content. but falls off sharply at higher gallium phosphide contents. The ~~” that electron results of investigations by Tietjen and W e i ~ b e r g confirm 55aJ. J. Tietjen and L. R. Weisberg. Appl. Phys. Letters 7. 261 (1965).

442

N . A. GORYUNOVA, F. P. KESAMANLY,AND D. N . NASLEDOV

GaAs

MOLE

FRACTIONS

GaP

FIG. 19. Dependence of electron mobility on composition ( 1 - x) in GaAs,P, _ x crystals. 0: electron concentration 6 x l o i 7~ 1 3 1 (After Ku.I9) x : electron concentration 1.5 x l o L 7 cm-3 ;

scattering on the alloy in the solid solutions is not predominant. The mobility relationship discussed above may indirectly indicate a change in the conduction band structure caused by changes in composition. A decrease of mobility is apparently due to a transfer of electrons to the minimum where their effective mass is considerably larger. The temperature dependence of electron mobility in mixed crystals of GaAs,P, -, has not yet been studied. However investigation of volt-ampere characteristics of p-n junctions in crystals of mixed composition yielded indirect i n f ~ r m a t i o non ~ ~the temperature dependence of electron mobility. These diodes were prepared by diffusing zinc into a tellurium-doped crystal. Thus the series resistance of the diode was determined mainly by the resistance of the n-region. At room temperature the series resistance of the diode calculated from the slope of the volt-ampere characteristic is 1 ohm. As the temperature decreases, it increases up to about 2 ohms and then drops sharply to approximately 0.5 ohm. Unexpected in the behavior of these diodes is a rapid increase of resistance with cooling followed by a stepwise decrease at a further decrease of temperature. This fact may be satisfactorily 56

N. Holonyak. S. F. Bevacqua, and C. V. Bielan, Appl. Phys. Letters 3,47 (1963).

~ ~ .

7.

443

PHENOMENA IN SOLID SOLUTIONS

explained by a model of the conduction band in which the (O00)minimum with a small effective mass at room temperature lies somewhat higher than the absolute minimum, where the effective mass is larger and electron mobility is small. As the temperature decreases, the minima shift with respect to the top of the valence band a t different rates, the absolute minimum which is located not at the center of the Brillouin zone moving away from the valence band faster than does the (000)minimum. The observed increase of resistance is associated with the decrease of electron concentration, at the absolute minimum, with the decrease of temperature, whereas the stepwise drop of resistance is due to a sharp increase of the mobility of electrons because of their transfer to the (OOO) minimum which becomes absolute at this temperature. Fenner studied5' the dependence of resistance, of n-type mixed crystals of GaAs,P, -,for 0.36 < x < 0.39, on pressure up to 15,000atm in the temperature range 190 to 363°K and observed a strong pressure dependence of the resistance (see Fig. 20). The temperature dependence of the Hall constant for the crystals investigated by him is illustrated by Fig. 21. The results obtained are explained by the existence of an additional minimum too BOF

80. 402010

Pro, 8

0

6

4 2

I

2

4

6

8-40

12

44

PRESSURE IIO'atrn I FIG. 20. Dependence of resistivity on pressure in GaAs,P,-, crystals with x 363°K; 0 : 300°K: 0 : 192°K: 0 : 193°K. (Alter F e r ~ n e r . ~ ~ ) 57

G. E. Fenner, Phys. Ren 134, A1 113 (1964).

=

0.359. A :

444

N . A . GORYUNOVA, F. P. KESAMANLY, A N D 9. N. NASLEDOV

H)(l°F

t

FIG.21. Temperature dependence of Hall coefficient in GaAs,P, - I crystals at atmospheric pressure. 0 : Y = 0.359: : Y = 0.372 :V : x = 0.367: 0: s = 0.24. (After F e n ~ ~ e r . ~ ? )

in the conduction band of mixed crystals where the mobility of electrons is small [apparently at k = (100)l.As the pressure is raised, the gap between the (000) absolute minimum and the (100) minimum decreases and electrons move to the (100) minimum, with the result that the resistance of the crystal increases. The results obtained are compared with theoretical calculations based on the Ehrenreich m ~ d e l . ~The * , ~agreement ~ between experiment and theory is good if the parameters are varied in a systematic way. In the purest sample the ratio v of light and heavy holes is large and falls off rapidly with increase in temperature. In other crystals v does not significantly depend on temperature. On the basis of experimental evidence, Fenner comes to the conclusion that the (000) and (100) minima in mixed crystals intersect at x = 0.45. The mobility of holes in crystals of mixed composition was not studied. ( 2 ) Optical Absorption Spectrum. Optical properties of mixed crystals near the intrinsic absorption edge were studied from the standpoint of determination of the transmission edge. Measurements were carried out mainly on polycrystalline samples of solid solutions of gallium arsenidegallium phosphide of various thicknesses and degrees of perfection. Data 5R

H . Ehrenreich. Phys. Rrr. 120, 1951 (1960)

7.

PHENOMENA IN SOLID SOLUTIONS

GaAs

mole fractions

FIG.22. Dependence ofenergygap on composition ( 1

- x) in

445

Go P GaAs,P, -.crystals. (After Ku.19)

on the dependence of the energy gap on composition obtained by different authors are given in Fig. 22. The majority of the authors represent this dependence for crystals of GaAs,P, --x in the form of two straight lines intersecting near x = 0.5.59*60However, because of an insufficiently well-grounded determination of E , (by the transmission edge), inhomogeneity of fine-grain samples, and a number of other factors which resulted in considerable scatter of experimental points, it is only with care that one could try to interpret these data. More reliable data on the behavior of the absorption spectrum in the region of the intrinsic absorption edge were obtained recently in the investigations of the barrier photoeffect at the metal-to-semiconductor contact.6 Qualitative information was obtained on the absorption spectrum near the intrinsic absorption edge which permitted a more accurate determination of the energy gap to be carried out for crystals of different compositions. An example of a spectral response of the surface photo-emf is shown in Fig. 23 for compositions with x = 0.8 and x = 0.20. Such a dependence for photoresponse is interpreted as resulting from indirect transitions to the (100)minimum, and of direct ones to the (000) minimum. The energy locations of the minima were determined by representing the spectral response of the photo-emf as a sum of three components associated respectively with absorption before the edge, that due to indirect transitions, and that caused

'' San-Mei Ku and J. F. Black, Solid State Electron. 6. 505 (1963). 6o

T. A. Fulton, D. B. Fitchen, and G. E. Fenner. A p p l . Phys. Letters 4.9 (1964) W. G. Spitzer and C . A. Mead. Pliys. Rev. 133, A872 (1964).

446

N. A. GORYUNOVA, F. P. KESAMANLY,AND D. N. NASLEDOV

1.5

20

6

5

h v , eV

FIG.23. Spectral response of surface photo-emf in GaAs,P, right curve: x = 0.20. (After Spitzer and Mead.61)

-x

crystals. Left curve : x = 0.78 :

by onset of direct transitions. In this way data were obtained on the dependence of the threshold for indirect and direct transitions on composition (see Fig. 24) which reflect more accurately the composition dependence of the energy gap than any earlier information. Data given by Spitzer and Mead" permit one to draw the conclusion that the (100) minimum is absolute for compositions from x = 0 to x = 0.65 (Fig. 24). and not up to x = 0.5 as was believed previously. The absorption spectrum for crystals of mixed composition was studied in detail in the near infrared region.62It was found for n-type crystals that the band of additional infrared absorption observed in n-type G a P exists in mixed crystals of GaAs,P,-. only for x ,< 0.5. From an analysis of the extent and shape of this band, which are only slightly affected by a decrease of temperature, a suggestion was made on the connection of the infrared absorption band with electron transitions from hydrogen-like donor levels lying close to the bottom of the conduction band to a higher minimum of the same band which is split from the absolutc minimum because of the absence of an inversion center in these crystals. The p-type crystals also exhibit an infrared absorption band,63 which is attributed to electron transitions from the valence band which is split off 19,41*56959

'* 63

J. W. Allen and J. W. Hodby, Proc. Phys. Soc. (London) 82. 315 (1963) J . W. Hodby. Proc. Phys. SOC. (London) 82. 324 (1963).

7.

447

PHENOMENA IN SOLID SOLUTIONS

" I

Ga As

mote fractions

GaP

FIG.24. Dependence of thresholds for direct and indirect transitions on composition (1 - s) in GaAs,P, _ x crystals. A, A : direct transitions; 0, 0 : indirect transitions. A, : composition determined by x-ray fluorescence. (After Spitzer and Mead.6 ')

due to spin-orbit interaction, to the band of heavy holes. It was shown that the position of the band edge determined by the magnitude of spin-orbit splitting of the valence band at the (000) minimum changes linearly from 0.127 0.08 eV in GaP to 0.33 eV in GaAs. Determinations of the indices of refraction of GaAs,P, - x crystals of different compositions were done by measurement of the transmission maxima for plane-parallel samples.64 It was found that the refractive index changes linearly with composition from 3.01 for gallium phosphide to 3.31 for gallium arsenide. Recently, Abagyan et al? have studied the intrinsic absorption edge for mixed crystals of gallium arsenide-gallium phosphide grown from a vapor phase. On the basis of the data obtained, the auth01-8~come to the conclusion that it is necessary substantially to correct and supplement present ideas on the position of the conduction band minima, both in crystals of mixed composition and in gallium arsenide. The corresponding data on the position of the conduction band minima as a function of composition are given in Fig. 25. The dashed line was drawn through the points corresponding to the onset of a sharp increase in the absorption coefficient. 64

65

S. A. Abagyan. S . M. Gorodetskii. T. B. Zhukova, A. 1. Zaslavskii, A. V. Lishina. and V. K. Subashiev:-Fiz. Tverd. Tela 7. 200 (1965) [English Transl.: Soviet Phys.-Solid State 7, 153 ( 1 965)]. S. A. Abagyan. A. V. Lishina. and V. K. Subashiev, Fir. Tverd. Tela 6. 2852 (1964) [English Transl.: Soviet Phys-Solid State 6, 2266 (196511.

448

N . A. GORYUNOVA, F. P. KESAMANLY,AND D. N . NASLEDOV

3,

0

GaP

1

0.2

0.4 06 0.8 1.0 mde feactions GaAs

FIG.25. Energy positions of conduction band minima as function of composition in GaAs,P, _ I crystals. Dashed line corresponds to onsets of sharp increases in absorption coefficients. (After Abagyan et ~ 1 . ~ ~ )

In gallium arsenide at room temperature the (OOO) minimum lies at approximately 0.06 eV above the energy marking the onset of the sharp increase of the absorption coefficient.66Assuming that in crystals of mixed composition the energy interval between the beginning of the sharp increase of the absorption coefficient and the position of the (OOO) minimum is the same as for gallium arsenide, the authors65 have drawn in Fig. 25 a straight line corresponding to the (OOO) minimum 0.06 eV above and parallel to the straight line corresponding to the beginning of the absorption coefficient increase. It follows from this estimate that the (OOO) minimum in gallium phosphide lies near 2.77 eV. Extrapolating the lines to gallium arsenide yields new information on the position of the conduction band minima for gallium arsenide. The (100) minimum is found to lie at 1.62 eV, which is 0.1 eV below the estimate made b e f ~ r e . ~ The ~ . ~ extrapolation ' procedure also yielded the result that in gallium arsenide there exists a (1 11) minimum lying 0.05 eV above the absolute minimum.65 The conclusion on the location of the (1 11) minimum in the conduction band of gallium arsenide should be considered with care since published 66

M. D. Sturge, Phys. Reu. 127, 768 (1962).

7.

PHENOMENA IN SOLID SOLUTIONS

449

data on the independence of the effective mass of the density of states for electrons in gallium arsenide on the concentration of current carriers up . ~ ~ experito about 10l8cm-3 are in contradiction with this c o n c l ~ s i o nNew ments are apparently required to confirm the location of this minimum in the conduction band of gallium arsenide. These data help explain the dependence of electron mobility on composition illustrated by Fig. 19. As seen from Fig. 25, for compositions where the mole fraction of GaAs is less than 0.75, the (100) minimum is absolute. Since in this minimum the effective mass of electrons is considerably larger, their mobility is substantially less than at the (OOO) minimum. ( 3 ) Reflectivity Spectra. The spectra of ultraviolet reflectivity for gallium arsenide and gallium phosphide crystals, as well as for mixed crystals in this system, have been investigated by Abagyan et a1.68 Their data on reflection for the GaAs and GaP crystals agree with other published values6' In gallium arsenide, the first maximum is of a doublet nature, whereas in gallium phosphide no doublet is observed. The doublet is not revealed also in crystals of mixed composition, although in crystals with x = 0.56 the first maximum is somewhat broadened. The dependence of the position of the reflection maximum on composition is shown in Fig. 26. The energy separation between the peaks of the doublet in the first maximum in gallium arsenide is 0.21 eV. The presence of this doublet is attributed to spin-orbit splitting of the valence band at L and X points. Indeed, the magnitude of the energy separation is close to the theoretical estimate at the center of the Brillouin zone, viz., 0.35 eV.66 From the fact that the doublet structure of the first reflection maximum is not revealed in gallium phosphide and crystals of mixed composition, it is concluded that in gallium phosphide as well as in mixed crystals of a close composition the first maximum is of a different nature than in the case of gallium arsenide. A comparison between the slopes in the dependence of the position of reflection maxima on composition (Fig. 25) and in that of the minima of the conduction band on composition (Fig. 26) shows that the slope pertaining to the first reflection maximum is close to that for the (111) minimum, whereas the position of the second reflection maximum changes in a manner similar to the behavior of the (100) minimum of the conduction band. This enabled the authors68 to associate the first reflection maximum in gallium 67

68

Yu. M. Burdukov, 0. V. Emel'yanenko, N. V. Zotova, F. P. Kesamanly, E. E. Klotyn'sh, T. S. Lagunova, D. N . Nasledov, V. G. Sidarov, G. N. Talalakin, and V. E. Shcherbatov, Izv. Akad. Nauk SSSR, Ser. Fiz. 28,951 (1964) [English Transl.: Bull. Acad. Sci. U S S R 28,855 (1964)l. S. A. Abagyan, V. K. Subashiev. and S. P. Singkhal, Fiz. Tuerd. Tela 6. 3186 (1964) [EngZish Transl.: Sooiet Phys.-Solid State 6, 2546 (1965)l.

450

N. A. GORYUNOVA, F. P. KESAMANLY, AND D. N. NASLEDOV

3-

%

41

%

\

,

FIG.26. Positions of reflectivity spectra maxima as functions of composition for GaAs,P, - x crystals. Upper curve: first maximum (transitions of A3 - A , type); lower curve : second maximum (transitions of X, - X ,type). (After Abagyan et

phosphide and in mixed crystals with transitions at the points of the band structure lying in (111) directions ; and the second maximum of reflection, with electron transitions to the (100)minimum. V. Phenomena in Solid Solutions of Heterovalent Substitution

Beside the solid solutions where isovalent substitution of elements takes place, interest exists in the properties of solid solutions of 111-V compounds produced by heterovalent substitution. All solutions of 111-V compounds formed through heterovalent substitution, which have been studied, may be divided into two groups, differing in crystal structure : (i) Solid solutions with normal tetrahedral structure in which up to now only several systems have been studied, for instance 111-V with 11-VI and 111-Vwith 11-IV-V,. (ii) Solid solutions with defect tetrahedral structure, such as the systems of 111-V with 111,-VI,.

7.

451

PHENOMENA IN SOLID SOLUTIONS

5 . SOLIDSOLUTIONS WITH NORMAL TETRAHEDRAL STRUCTURE It may be said that the investigation of crystals in this group of solid solutions is only beginning, and data published up to now do not provide a complete characterization of properties even for the materials which have been studied. We shall discuss here in detail the results obtained by the authors in recent years on InAs-CdTe and InAs-CdSnAs, solid solutions.

a. Indium Arsenide-Cadmium Telluride In InAs-CdTe we have studied the temperature dependencies of electric conductivity, Hall coefficient and transverse Nernst-Ettingshausen coefficient. In addition, differential thermal emf and transmission spectra at room temperature have been studied.69 In Fig. 27 are presented experimental data on the dependence of the energy gap and electron mobility on composition. As seen from this figure, the energy gap depends linearly on composition and the mobility of electrons decreases monotonically from the value for indium arsenide to that for cadmium telluride. It is worthwhile to note that some solid solutions of 111-V and 11-VI compounds exist in a limited range, in this case up to 30 % cadmium telluride content. The dashed part of the lines in Fig. 27 corresponds to extrapolation done by the authors.

3 4

1.5

3000

e.

-

2000

1.0

.a

3

5

h

Po

I3

LIJ

3 ,n

1000 E

0.5

L 20 40 60 80

0

InAs

mole %

I3 Cd Te

FIG.27. Dependence of energy gap and mobility on composition in InAs-CdTe system. Electron concentrations for most specimens were above ~ r n - ~(After . Vojtsekhovskii et 69

A. V. Vojtsekhovskii. F. P. Kesamanly. B. K. Mityurov. and Yu. V. Rud. Ukr. Fiz Zh. 10. I349 (1965).

452

N. A . GORYUNOVA, F. P. KESAMANLY,AND D. N. NASLEDOV

By analogy with the InAs-InP solid solution where a similar dependence of these parameters on composition was observed, one may draw the conclusion that the band structures of the starting compounds and of solid solutions are the same. It can be seen from Fig. 28 illustrating the temperature dependence of transport phenomena for a sample of 95 InAs-5 CdTe that the NernstEttingshausen coefficient is positive and increases with increase in temperature. As for the electric conductivity and Hall coefficient, they are essentially independent of temperature. The same behavior of the two latter quantities is exhibited in crystals of the alloy InAs-ZnTe.

100

200

300 400 T'K

600

FIG. 28. Temperature dependence of transport phenomena for 95 InAs-S CdTe sample. (After Vojtsekhovskii et

As was mentioned earlier, the positive sign of the Nernst-Ettingshausen effect may occur only if current carriers are scattered on acoustical lattice vibrations or on the disordered alloy structure. It is true however that, if the conduction band is nonparabolic in shape, scattering on polar lattice vibrations may also result in a positive Nernst-Ettingshausen effect. Since the curve relating electron mobility to composition does not exhibit any minimum, one may suggest that in this solid solution the role of scattering on the alloy is also small. The value of the scattering parameter r (in the expression relating mean free path I with current carrier energy E, 1 E') is determined by the formula

-

k 7T2 Q' = -8- r)-UH, e 3P where Q' is the transverse Nernst-Ettingshausen coefficient, and U , is the Hall mobility. It is found that r varies from 0.5 a t 90°K to 0.1 at 550°K.

7.

PHENOMENA IN SOLID SOLUTIONS

453

These values of the scattering parameter were used to calculate the values of the reduced Fermi level ,ii by the formula

From this, the density-of-states effective mass for electrons was calculated, which was found to be the same for pure indium arsenide and its alloys, namely, about 0.05rn0. It was shown” that the thermal conductivity of InAs-CdTe solid solutions decreases with increase in telluride content at room temperature. Transport phenomena in solid solutions of InSb-CdTe and InAs-HgTe have been investigated by Sharavskii and c o - ~ o r k e r s . ~

b. JnAs-CdSnAs, Temperature dependences of transport phenomena in alloys of InAsCdSnAs, were s t ~ d i e d ~from ~ - ~the ~ standpoint of comparison of the properties of CdSnAs crystals. It has been mentioned already that these alloys can be obtained essentially in any composition. The curves relating electric conductivity 0 of alloys and starting compounds with temperature are similar. The temperature dependence curve of the Hall coefficient R ( T )for alloys of any composition irrespective of conduction type exhibits a clearly pronounced maximum before the sharp decrease caused by intrinsic conduction. In some crystals, the Hall coefficient increases by a factor of 2 to 8. This suggests that the increase of R at high temperatures is not due to a change in the scattering mechanism of the current carriers with temperature. The increase of R before intrinsic conduction could be explained as due to transition of current carriers from one band to another ; however this interpretation cannot be considered as well-grounded because of a lack of published data on the band structure, even for the starting ternary compound. It should be noted that crystals of alloys with hole conduction doped with selenium change the type of conduction. However the behavior of the temperature dependence of and R , as well as the order of magnitude of the Hall (r

70

7‘

’’ 73

74

75

A. V. Vojtsekhovskii, LJkr. Fiz. Zh. 8, 1027 (1963). E. N. Khabarov and P. V. Sharavskii. Dokl. Akad. Nauk SSSR 155,542 (1964) [English Transl.: Soviet Phys. “Doklady” 9, 225 (196411. A. Inyutkin, E. Kolosov, L. Osnach, V. Khabarova, E. Khabarov. and P. Sharavskii, Zzv. Akad. Nauk SSSR, Ser. Fiz. 28, 1010 (1964) [English Transl.; Bull. Acad. Sci. U S S R 28, 91 1 (1 96411. S. Mamaev. D. N. Nasledov, and V. V. Galavanov, Fiz. Tuerd. Tela 3, 3405 (1961) [English Transl.: Societ Phys.-Solid State 3, 2473 (1962)l. D. N. Nasledov. S. Mamaev, and 0. V. Emel’yanenko, Fiz. Tuerd. Tela 5, 147 (1963) [English Transl.: Soviet Phys.-Solid State 5, 104 (1963)l. P. Leroux-Hugon, Compt. Rend. 255, 662 (1962).

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N. A. GORYUNOVA, F. P. KESAMANLY, AND D. N. NASLEDOV

coefficient, do not change after doping. This indicates that the mobilities of electrons and holes in InAs-CdSnAs, alloys with a high content of the second component d o not differ much from each other. This conclusion is supported by the fact that the temperature dependence of the Hall coefficient for crystals with hole conduction does not exhibit a reversal of sign. Contrary to the case of starting compounds where the Nernst-Ettingshausen effects change sign from negative to positive at room temperature, in the alloys they retain the positive sign within the whole temperature region studied (100-600°K). We think that this fact is due to a predominant contribution of current carrier scattering on the disordered alloy structure in crystals of InAs-CdSnAs, . The results of the measurement of the differential thermal emf and Hall coefficient were used to calculate effective electron masses in alloys with electron conduction, which were found to lie in the range 0.035-0.060m0. Despite some peculiarities of alloys associated with the difference in crystal structure of the starting compounds, the effective mass of electrons in them is small and does not significantly depend on composition. The most probable value of m* for the whole system may be taken as 0.045m0,which is in agreement with published data for the starting components at electron concentrations of about 10l8cm-3.50376 An interesting feature is a bend which is observed on the curve relating the energy gap E , with alloy composition (Fig. 29). Analogous dependence

0.51

y

m

X

0 2 0 4 0 6 0 8 0 ~ 0 0

Cd Sn As,

mole %

2 InAs

FIG.29. Dependence of energy gap on composition in InAs-CdSnAs, system. 1 : as obtained from a(T ) :2:as obtained from R ( T ) .(After Mamaev ~ t a l . ’ ~ ) 76

W. G. Spitzer and J. N. Wernick, Solid State Electron. 2. 96 (1961).

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PHENOMENA IN SOLID SOLUTIONS

455

was revealed in the system Insb-InA~.’~“ In the course of investigation of a number of semiconducting alloys it has been found that if two substances dissolve in one another in any ratio, then the energy gap will change regularly or even linearly provided the band structures of these substances are the same : if, on the other hand, their band structures are different, the curve may exhibit bends.” The bend observed in Fig. 29 might perhaps indicate that some change takes place in the band structure of solid solutions. On the other hand, examples of minima d o exist where there is no evidence of essential change in band s t r ~ c t u r e . ” ~

6. SOLID SOLUTIONS WITH DEFECT TETRAHEDRAL STRUCTURE Compounds of the III,VI, type (for instance, In,Te,) a s well as 111-V compounds have a zinc-blende crystal structure : however they are defective with respect to the sublattice of element A (one-third of the sites in the sublattice A are vacant). We will consider here the systems InSb-In,Te,, InAsIn,Te,, InAs-In,Se,, and GaAs-Ga,Se,. In all these systems a small admixture of the second component (a fraction of a percent) acts as a donor impurity resulting in a strong increase of electron concentration and hence of electric conductivity of the alloy in question. It was that the annealing of InSb-In,Te, samples brings about an ordering of the vacancy sublattice. Electron mobility decreases monotonically with the increase of In,Te, content, whereas the thermal emf at first increases, and after that remains practically constant. The dependence of the thermal conductivity of alloys on the content of In,Te, is shown in Fig. 30, where a sharp decrease of conductivity is seen caused by introduction of small amounts (up to 1 %) of In,Te, , after which it does not change significantly. The thermal resistance of InSb-In,Te, samples with a small content of the second component, increases linearly with temperature within the range 80 to 5 W K , whereas in samples with In,Te, content above 1 % a deviation is observed above room temperature, the beginning of the deviation shifting to lower temperatures with the increase of In,Te, content. These results are explained79 as due to a strong effect of the defect in the structure of In,Te,. An introduction of In,Te, results in the formation of point defects in the lattice of indium antimonide. The number of these defects depends upon the actual amount of InzTe, introduced. Of predominant 76qJ. C. Woolley and J . Warner. Can. J . Phys. 42. 1879 (1964). D. Long. J . Appl. Phys. 33, 1682 (1962).

”

71a

’’ 79

See for example, C. Hjlsum in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.). Vol. 1. p. 3. Academic Press, New York, 1966. M. I. Aliev and A. Y u . Dzhangirov. Fiz. T v u d . Tela 6, 2415 (1964) [English Transl.: Soviet Phys.-Solid State 6. 1916 (1965)l. M. I. Aliev and A. Yu. Dzhangirov, Fiz. Tuevd. Tela 5, 3338 (1963) [English Transl.: Soviet Phys.-Solid State 5. 2447 (1964)l.

456

N . A . GORYUNOVA, F. P. KESAMANLY, A N D D . N . NASLEDOV

b

3

InSb

4

1

I

1

5 mole o/o

1%Tea

FIG.30. Dependence of thermal conductivity of InSb-In,Te, on In,Te, content (after Aliev and D~hangirov'~): Curve

T"K

1

2

3

4

110 213 300 460

role in the transfer of heat in indium antimonide is phonon thermal conductivity. Phonons experience scattering on defects, this scattering being more pronounced at lower temperatures. This is confirmed by a larger deviation in the thermal conductivity curves observed with a decrease in temperature. Reflection was studied" in the range 1.0-60eV for pure indium antimonide and arsenide, as well as for their alloys of different compositions. Peaks observed on the reflection curve are attributed to transitions at the center of the Brillouin zone and at (1 11) and (100)points. Transport effects in InAs-In,Te, do not differ in behavior from those observed in InSb-In,Te,.81.82 In alloys rich in indium arsenide, the electron D. L. Greenaway and M. Cardona, Proc. Intern. Con$ Phys. Semicond.. Exeter, I962 p. 666. Inst. of Phys. and Phys. SOC., London, 1962. *' J . C. Woolley, B. R. Pamplin. and J. A. Evans, J . Phys. Chem. Solids 19, 147 (1961). 8 2 D. B. Gasson, 1. C. Jennings, J. E. Parrott, and A. W. Penn, Proc. Intern. Con$ Phys. Semicond.. Exeter, 1962 p. 68 I . Inst. of Phys. and Phys. SOC.,London, 1962.

7.

PHENOMENA IN SOLID SOLUTIONS

451

gas is strongly degenerate. This is explained by a high solubility of tellurium in indium arsenide. Because of the strong degeneracy of the electron gas, the values of the energy gap determined by the absorption band edge will be too high due to the Burnstein effect. These values may be corrected through determination of the depth of the Fermi level by thermal emf measurements. The effective mass of the density of states for electrons in samples with 3 % content is 0.084.09 of the free electron mass. This value is close to the value of the effective electron mass in crystals of indium arsenide with a strongly degenerate electron gas ( n = t O l 9 cm--,). Reflection curves resemble those for InSb-In,Te,.80 In InAs-In,Se, alloys the electron mobility falls off sharply when going over from indium arsenide to 9 : 1, I : 1 compositions, after which it changes only slightly with the increase of indium selenide concentration. The thermal conductivity of samples of different composition decreases with temperature according to the T - law. As one goes over from indium arsenide to alloys, the thermal conductivity decreases sharply as the concentration of In,Se, increases up to 50%, and for a further increase in concentration it rises slowly following a nearly linear law. The thermal conductivity of samples is completely due to p h ~ n o n s . ~ ~ . * ~ In this group of solid solutions we have studied solid solutions of GaAsGa,Se,.85,86 Their thermal emf first increases with increase in selenide content, reaching a maximum value a t 30 % of Ga,Se, , and then reverses sign becoming positive (at 6 5 % of Ga,Se,). The energy gap dependence found by the measurements of the electric conductivity and Hall constant in the intrinsic region is nonlinear. VI. Conclusions Mixed crystals or solid solutions of substitution between 111-V compounds represent semiconducting tetrahedral phases. Isovalent substitution taking place in these phases does not result in a formation of impurity (donor or acceptor) levels. Deviations from stoichiometric proportions in solid solutions of this kind do not exceed those typical oithe starting 111-V compounds. Many physical properties of these substances are intermediate with respect to those of the starting 111-V compounds. 83

L. I. Berger and S. 1. Radautsan, Voprosy metallurgii i fiziki poluprovodnikov; poluprovodnikovye soedineniya i tverdye splavy. Trudy 4-go soveshchaniya. Moskva. Izd-vo Akad. Nauk SSSR, 1961. p. 129. 84 S. I. Radautsan and B. E.-Sh Malkovich. F'iz Tuerd. Tela 3. 3324 (1961) [English Transl.: Souiet Phys.-Solid State 3. 241 3 (196211. 8 5 D. N. Nasledov and I. A. Feltin'sh. Fiz. Tiierd. Tela 1. 565 (1959) [English Trailsl.: Societ Phys.-Solid State 1. 510 (1959)l. 86 D. N. Nasledov and I. A. Feltin'sh. Fiz. Tverd. Tvla 2. 823 (1960) [English Trans/.: Societ Phys.-Solid State 2, 755 (1960)l.

458

N. A. GORYUNOVA, F. P . KESAMANLY, AND D. N . NASLEDOV

Thus solid solutions can fill in the total range of the energy gap and charge carrier mobility range from the highest melting-point III-V compounds with an energy gap of about 6 eV and electron mobility of about 1000 cm’/ volt-sec to the lowest melting-point compounds having an energy gap of a few tenths of electron volt and a mobility of tens and hundreds of thousands cm2/volt-sec. This undoubtedly is of a great importance for semiconductor applications which might require both alloys with definite properties and ingots or layers with gradually changing properties. A stepwise change of properties observed in solid solutions between III-V compounds with different band structures may be also used in some areas. A convincing example has been the preparation of a laser on the basis of a solid solution of gallium phosphide in gallium arsenide. In semiconductor applications involving the use of thermoelectric properties, solid solutions will have definite advantages. In solid solutions an additional scattering of phonons takes place, which is associated with the increase in the number of atoms of different kinds in the lattice of the substance, since the wavelength of phonons is commensurate with interatomic distances. Solid solutions between III-V compounds and other substances represent systems in which the interaction is complicated by the fact that substituting atoms will act as donors or acceptors according to their valency. Typical for such compositions is an extrinsic type of conduction determined by the most active substituting atom. Of considerable theoretical interest are the processes of mutual compensation of atoms in heterovalent solid solutions. The high concentration of impurities which may be attained in III-V compound systems of heterovalent substitution may likewise become interesting in various applications.

CHAPTER 8

Electrical Properties of Nonuniform Crystals* R . T . Bate I . INTRODUCTION . . . . . , . . , . , . . . . . 459 11. ORIGIN AND NATURE OF INHOMOGENEITIES USUALLY ENCOUNTERED IN CRYSTALS GROWN FROM THE MELT . . . . . . . . . . 460 1. Normal Segregation . . . . . . . . . . . . . . 460 2 . Inhomogeneities Produced by Fluctuations in Growth Rate (Striations) . . . . . . . . . . . . . . . . . . 463 3. Statistical Inhomogeneity . . . . . . . . . . . . 463 111. TYPICAL EFFECTSOF INHOMOGENEITY ON ELECTRICAL PROPERTIES . 464 4. Electrical Conductivity . . , . . . . . . . . . . 464 5. Magnetoresistance . . . . . . . . . . . . . . 465 6. Hall Effect . . . . . , . . . . . . . . . . 467 I . Other Effects. . . . . , . . . . . . . . . . 468 8. Helicon Damping . . . , . . . . . . . . . . 469 1V. CALCULATION OF ISOTHERMAL TRANSPORT EFFECTSI N INHOMOGENEOUS CONDUCTORS . . . . . . , . . . . . . . . . . 47 1 9. Microscopic Case . . . , . . . . . . . . . . 47 1 10. Intermediate Case . . . , . . . . . . . . . . 472 11. Macroscopic Case . . . , . . . . . . . . . . 473 v. DETECTION OF INHOMOGENEITIES , . . . . . . . . . . 474 12. High-Resolution Potential Probing . . . . . . . . . . 474 475 13. Anisotropy and Symmetry Relations . . . . . . . 14. Photovoltaic Effects . . . , . . . . . . . . . . 475

I. Introduction When measurements of various intensive properties of solids are carried out, it is often assumed that the property being measured either does not vary from point to point within the sample or if it does, that the measurement yields the average value of the parameter in question. Although this assumption may well be true in some cases, there are many exceptions to be found in the study of transport properties of semiconductors. The variation of electron or hole concentrations from point to point, which results from an inhomogeneous distribution of donors or acceptors within the semiconductor, can produce startling anomalies in the Hall mobility, magnetoresistance, thermal conductivity, and the magnetic-field dependence of the Hall coefficient. Moreover, it appears likely that the proper conditions for growth of perfectly homogeneous crystals of 111-Vcompounds from the melt may never

* This article was written while the author was at the Battelle Columbus Laboratories. 459

460

R. T. BATE

be achieved and that all existing melt-grown samples must be considered inhomogeneous to some extent. In view of this, one must inquire which of the transport effects can be measured and interpreted with confidence for existing materials, and conversely, which measurements should be viewed with suspicion because of the probable influence of inhomogeneity. Among the various inhomogeneity effects to be discussed in this chapter, the case in which both n- and p-type regions are present in the same sample will be specifically excluded. The electrical behavior of such samples is often very complex, exhibiting gross deviations from Ohm’s law, multiple changes in sign of the Hall coefficient or thermoelectric power as a function of temperature, etc. Although these effects are qualitatively understood, a precise description of them necessarily involves all the complexities of p-n junction theory and is beyond the scope of this discussion. A number of papers dealing with various aspects of this problem are given in the literature. We shall further restrict our considerations to sufficiently low electric fields for Ohm’s law to be obeyed. The redistribution of carriers in an inhomogeneous semiconductor upon application of an electric field and various thermal effects can cause deviations from the linear dependence of current density on electrochemical potential gradient at relatively low currents. Several of these effects have been investigated by Baranskii and co-workers. References to Baranskii’s extensive contributions in this area are given by Beer.3 11. Origin and Nature of Inhomogeneities Usually Encountered in Crystals Grown from the Melt

1. NORMAL SEGREGATION a. Impurity Gradients

The growth of crystals from the melt is usually a relatively slow process in which the liquid and solid are nearly in equilibrium. Because the presence of impurities affects the freezing temperature, there is a tendency for a continuous impurity gradient to form in crystals grown from the melt.4 These gradients are not usually very steep except in crystals grown from a small volume of melt, or in the portion of a large melt which freezes last. Thus the influence of gradients of this type is negligible in many cases. B. R. Gossick, J . Appl. Phys. 30, 1214 (1959). 0. Madelung, Z . Naturforsch. 14a, 951 (1959). *‘J. R. Dixon, J . Appl. Phys. 30,1412 (1959). A. C. Beer, “Galvanomagnetic Effects in Semiconductors,” p. 326. Academic Press, New York, 1963. W. G. Pfann, Solid State Phys. 4,429 (1957).

8.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

461

b. “Cores” and the “Facet Effect” It has been known for several yearmsthat a radial gradient in carrier concentration often occurs in crystals grown by the Czochralski technique. Although this effect was originally ascribed to insufficient stirring of the melt near the center of the rotating crucible, the principal effect is actually associated with the formation of planar facets (usually on (1 11) planes) at the solid-liquid i n t e r f a ~ e . Nucleation ~,~ of growth steps on these surfaces apparently requires an inordinately large degree of supercooling, and the segregation of impurities at the facet is quite different from elsewhere on the interface. The result is usually one or more well-defined “cores” of higher impurity concentration, which trace out the positions of the facets as growth proceeds. An example of such a core revealed by autoradiography is shown in Fig. 1. The impurity concentration in a core may exceed that in the rest of

FIG.1. Autoradiograph of a slice from a pulled crystal of InSb doped with Se75.The crystal was pulled in a [ l l l ] direction and was cut perpendicular to the growth axis. (After Allred and Bate.6)

the crystal by as much as a factor of 10. This phenomenon has been studied in detail for InSb,5p6 Ge,’ and GaSb, and facets have been observed on decanted interfaces of GaAs crystals.

’ J. B. Mullin, in “Compound Semiconductors” (R. K . Willardson and H. L. Goering, eds.), ’

p. 365. Reinhold, New York, 1962. W. P. Allred and R. T. Bate, J . Etectrochetn. Soc. 108, 258 (1961). J. A. M. Dikhoff, Solid State Electron. 1,202 (1960).

462

R. T. BATE

c. Cellular Growth When crystals are grown from melts containing large concentrations of impurities, a phenomenon known as constitutional supercooling may occur.8 As solidification proceeds, impurities which lower the melting point are rejected into the liquid. If the liquid is not adequately stirred, regions of very high impurity concentration will build up immediately in front of the advancing interface. The temperature below which further freezing can occur at the

FIG.2. Cellular growth pattern in Bi-Sb alloy (100 x ) revealed by etching in concentrated HNO,.

interface may then be much lower than the freezing point in the liquid farther from the interface. Lowering the temperature sufficiently for freezing to continue at the interface may then cause supercooling of a layer of liquid further from the interface. When this happens, freezing occurs ahead of the normal planar interface and the interface assumes a “scalloped” shape. This phenomenon, which is commonly observed in mixed crystals grown from the melt, is called cellular growth, and results in the occurrence of “cells” within the crystal which are highly enriched in one constituent. Such cell structure observed in a Bi-Sb alloy is shown in Fig. 2.

’ J. W. Rutter and B. Chalmers, Can. J . Phys. 31, 15 (1953).

8.

2.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

INHOMOGENEITIES PRODUCED BY

463

FLUCTUATIONS IN GROWTH RATE

(STRIATIONS) It is well known that the segregation of impurities during solidification depends strongly on the growth rate." In most of the crystal-growing techniques commonly employed, the growth rate is uniform if averaged over a period of I s 3 0 sec or more, but rather sharp, short-time fluctuations can occur. These short-term growth rate fluctuations result from several influences, all of which produce rapid temperature fluctuations at the growing interface. The final result is a periodic variation in the impurity concentration along the direction of crystal growth. Such variations in a crystal are commonly called striations. Some of the influences which tend to produce striated crystals are discussed below.

a. Rotation of Crucible and Seed When crystals are grown by the Czochralski technique, both the crucible and the seed may be rotated, usually in opposite directions, to provide stirring of the melt and to compensate for thermal asymmetry. If a transverse temperature gradient exists, the temperature at any point of the interface not on the axis of rotation will then fluctuate with a period (or periods) corresponding to these periods of rotation. This is the most common cause of striations, and the fluctuations that result usually have periods in the range 0.01 to 0.1 mm for common rotation and growth rates5 An example of striations caused by seed and crucible rotation is shown in Fig. 3.

b. Temperature Control Striations have been observed in crystals grown with no rotation of Although they may have a variety of causes, one which crucible or has been confirmed is the fluctuation of temperature due to the temperature control cycle." If the heat capacity of the crystal-growing system is not made large, the temperature will fluctuate as the heater power is switched on and off by the temperature controller. If the control cycle is rapid, the time rate of change of temperature can be quite large even though the total temperature excursion is small. The resulting variations in growth rate may produce pronounced striations. 3. STATISTICAL INHOMOGENEITY

Herring'' has pointed out that even crystals prepared under ideal conditions will have inhomogeneous distributions of impurities which can H.C. Gatos, A. J. Strauss, M. C. Lavine, and T. C . Harman, J . Appl. Phys. 32,2057 (1961) lo

N. Albon, J . Appl. Phys. 33, 2912 (1960). C. Herring, J . Appl. Phys. 31, 1939 (1960).

464

R. T. BATE

FIG.3. Striations in Se-doped InSb crystal revealed by CP4 etch. These striations resulted from seed and crucible rotation during pull. (After Allred and Bate.6)

strongly influence their electrical properties. If we divide the pure crystal up into “boxes” of equal volume, and then distribute donor atoms randomly over these boxes, we will naturally find that not all boxes contain the same number of donor atoms. Actually, the donor population of the boxes will follow a Poisson distribution. In order that charge neutrality be preserved, the conduction electron concentration must also fluctuate from one box to another as long as the dimensions of the boxes are greater than a Debye (or Fermi-Thomas) screening length. These fluctuations can be quite large, particularly at low temperatures or in compensated material. 111. Typical Effects of Inhomogeneity on Electrical Properties

4. ELECTRICAL CONDUCTIVITY

Although the influence of inhomogeneity on the conductivity of semiconductors has apparently not been studied in detail experimentally, general physical considerations suggest a sizable effect. The results of Herring‘ predict that the measured conductivity will be less than the average value for the case of intermediate scale fluctuations, while for the macroscopic

8.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

465

case, there are instances in which conductivities larger or smaller than the average will be found. If both n- and p-type regions are present, the conductivity can be orders of magnitude smaller than the average, and pronounced deviations from Ohm’s law may be observed. A discussion of this latter case is beyond the scope of this chapter. Errors in conductivity measurements due to inhomogeneity can lead to serious discrepancies in the determination of Hall mobilities. A discussion of this problem will be found in Section 6c below.

5. MAGNETORESISTANCE a. Magnetic-Field Dependence

Carrier-concentration inhomogeneities probably have a more profound effect on the high-field transverse magnetoresistance than on any other transport property. The occurrence of negative magnetoresistance, and the nonsaturation of magnetoresistance in strong magnetic fields-two puzzling phenomena observed in many semiconductors-can be easily explained if inhomogeneity is taken into account. Bate and Beer12 and Bate et al.13 have shown that either of these effects can occur in macroscopically inhomog e n e o u ~ samples, l~~ while Herring” indicates that nonsaturation of magnetoresistance will occur in semiconductors possessing inhomogeneities of intermediate scale. In the absence of quantum effects, classical transport theory predicts that the resistance of an extrinsic semiconductor should become independent of magnetic field in sufficiently strong magnetic fields. Specifically,the conditions wzo $ 1 and hw < kT should be satisfied for all bands contributing appreciably to conduction. Here w is the cyclotron resonance frequency, zo is the momentum relaxation time, and the other symbols have their usual meanings. The first relationship characterizes the strong-field region, while the second insures that quantum effects are negligible. In place of saturation of magnetoresistance, however, one often observes instead that the resistance increases nearly linearly with increasing magnetic field.14 The effect is particularly pronounced in semiconductors such as PbTe’ which are characterized by wide ranges of compositional stability, and occurs even when quantum effects should be completely negligible. The work of Herring’ gives strong evidence that such a near-linear increase in resistance is characteristic ofthree-dimensionally inhomogeneous semiconductors. Furthermore, R. T. Bate and A. C. Beer, J . A p p l . Phys. 32, 800 (1961). R. T. Bate, J. C. Bell, and A. C. Beer, J . A p p l . Phys. 32, 806 (1961). 13”For the meanings of the terms “microscopically” and “macroscopically” inhomogeneous, and the “intermediate case,” see Part IV. l4 R. T. Bate, R. K. Willardson, and A. C. Beer, J . Phys. Chem. Solids 9,119 (1959). R. S. Allgaier, Phys. Reo. 112, 828 (1958).

l2 l3

466

R. T. BATE

when the product cozo (-10-8poH, where p,, is the carrier mobility in cm2/volt-sec and H is the magnetic-field intensity in oersteds) is large compared to unity, very slight deviations from homogeneity can produce the effect. Figure 4 illustrates the typical behavior for PbTe. Similar effects have been noted, for example, in germanium15=and in InSb.'5b

Magnetic Field, gauss

FIG.4. Plot illustrating behavior of magnetoresistance in inhomogeneous PbTe. Note in particular the curve labeled "PbTe (18761-70)(77"K)."The linear dependence on H is typical of inhomogeneous material. The absence of a linear range for PbTe sample 18761-47 (which had a higher carrier concentration) indicates a more homogeneous carrier distribution.

b. Anisotropy

The anisotropy of magnetoresistance in striated samples of n-type InSb ~ most striking feature of their has been studied by Rupprecht, et ~ 1 . 'The results is the marked deviation in strong magnetic fields from the symmetry '"S. M. Puri and T. H. Geballe, Semiconductors and Semimetals 1, 203 (1966). 15%ee, for example, H. Weiss, Semiconductors and Semimetals 1, 315 (1966). l 6 H. Rupprecht, R. Weber, and H. Weiss, Z. Naturforsch. 15a, 783 (1960).

8.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

467

required by spherical constant-energy surfaces. For example, they found that the transverse magnetoresistance at 78°K in a field of 10 kG varied by more than a factor of ten, depending on the direction of current and magnetic field with respect to the original direction of pull of the crystal. According to Weiss,' 5 b , 1 7 the anomalies which they observe are associated with striations lying in { 111f planes, the crystal having been pulled in a [ 1111direction. These observations emphasize an important point : inhomogeneities associated with impurity segregation and striations will in general possess a symmetry lower than that of the crystal. Because of this, the anisotropy of magnetoresistance can be used to detect inhomogeneities of this type in cubic semiconductor^.'^^ This point will be discussed further in Section 13. 6 . HALLEFFECT

a. Magnetic-Field Dependence Although the Hall effect is much less sensitive to inhomogeneity than is the magnetoresistance, pronounced anomalies can be observed when gross inhomogeneities occur. An example is the anomalous magnetic-field dependence of the Hall coefficient observed by Bate et ~ 1 . 'in~ n-type InSb containing a step-function discontinuity in carrier concentration (n-n+ junction). Junctions of this type resulting from the facet effect are very common in pulled crystals of InSb. If the junction is near the Hall probes, the Hall voltage will be an average of the Hall voltages in the two regions weighted by their conductivities. Since the conductivity of n-type InSb decreases rapidly with increasing magnetic field in a way which is sensitive to carrier concentration, the weighting factors in the average will have different magnetic-field dependences. Thus the measured Hall coefficient can be a strong function of magnetic field. An example of such behavior is shown in Fig. 5.

b. Symmetry Relations In a homogeneous cubic semiconductor, the Hall effect is isotropic in both the weak field and strong field limits.18 However, if the material is inhomogeneous, the measured Hall coefficient will in many cases be a function of the off-diagonal and diagonal components of the magnetoresistivity tensor and will become anisotropic, particularly in strong magnetic fields. The observation of anisotropy of the Hall coefficient in the strong-field limit should therefore be considered a sign of inhomogeneity-provided other extraneous influences, such as contact effects, are eliminated. " H.

Weiss, J . Appl. Phys. 32, 2064 (1961).

'*See, for example, Ref. 3, pp. 44 and 250.

468

R . T. BATE 1.9 I.e

1.7 I.6 6

m

-a Q)

-&

1.5 1.4

v)

0

,“

1.3

1.3

1.2

1.2

LL

I.I

1.1

I.o

1.0

-

E

-cn8 0

IL

0.90 10

0.90

lo3

10‘

lo4

lo5

H in Gauss FIG. 5. Magnetic-field dependence of the normalized Hall coefficient of two high-purity n-type InSb samples. Sample A is homogeneous with a carrier concentration of 4.2 x l O I 4 cm313. Sample B contains two regions of differing carrier concentration. The boundary between these two regions is roughly perpendicular to the direction of the current and is near the center of the sample, where the Hall probes are located. The carrier concentrations in these two regions are about 5 x l O I 4 cm-3 and 5 x lOI5 cm-3. (After Beer.”)

c. Hall Mobility

Generally speaking, the Hall mobility of an inhomogeneous semiconductor will be less than the average of the Hall mobility over the sample, principally because the resistivity is higher than the average value.” However, there are exceptions to this rule for certain types of macroscopic inhomogeneity. There is good reason to believe that microscopic inhomogeneities can also decrease the carrier mobility directly by scattering current carriers, since random fluctuations in composition constitute a deviation from strict lattice periodicity. The so-called “alloy scattering” is an example of this mechanism.

7. OTHEREFFECTS a. Thermal EfSects Inhomogeneity is known to have an influence on other electrical and transport properties, such as the Seebeck effect, and thermomagnetic effects. l9

M. Glicksman, Phys. Rev. 111, 125 (1958).

8.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

469

However, the influence of inhomogeneity is not very strong in these cases. Puri and Geballe'sa~20have in fact pointed out that because the Nernst field is small in large transverse magnetic fields, measurements of the magneto-Seebeck effect are much less affected by inhomogeneity than are magnetoresistance measurements and can thus provide unambiguous evidence for quantum transport in strong magnetic fields. The enhancement of the electronic part of the thermal conductivity by inhomogeneity is an important phenomenon in semiconductors possessing high thermoelectric figures of merit. When the Seebeck coefficient a varies from point to point within a material, circulating electrical currents exist in the presence of a temperature gradient, resulting in a net transfer of Peltier heat from hot to cold regions.'l This effect will become more important as the figure of merit a Z T / K pincreases ( K here is the thermal conductivity of homogeneous material, and p is the electrical resistivity). It is interesting to note in this connection that, although thermodynamics places no upper limit on the figure of merit of an homogeneous thermoelectric material, a calculation using a method similar to Herring's' shows that such an upper limit, of the order of ( ( a - ( a ) ) 2 / ( a ) 2 ) - ' , does exist for a randomly inhomogeneous material.' l a This parameter might well be as small as unity in polyphase materials or alloys in which cellular growth occurs.

8. HELICONDAMPING The propagation of low frequency circularly polarized waves in metals and semiconductors in the presence of a magnetic field is a phenomenon of considerable interest.22*22a These waves, called helicon waves,2zb can be thought of as a dynamic manifestation of the Hall effect.23The low frequency dispersion relation for helicon waves with wave vector k, of frequency o, propagating parallel to an applied magnetic field H , in an isotropic conductor isz3 k' = 4nio/(p, + i R H , ) , (11 where p T is the resistivity for current flow perpendicular to H , . Depending on the sign of the Hall coefficient R, either a right or left circularly polarized S. M. Puri and T . H . Geballe, Phys. Rev. Letters 9, 378 (1962).

'*C. V. Airapetiants, Zh. Tekhn. Fiz. 27, 478 (1957) [English Transl.: Soviet Phys.-Tech.

Phys. 2, 429 (1957)l. 21"Foran explanation of the symbols, see Section 10, especially footnote 24a. 2 2 R. Bowers, C. Legendy, and F. Rose, Phys. Rev. Letters 7,339 (1961). ""See, for example, B. Ancker-Johnson, Semiconductors and Semimetals 1,379 (1966). 22bP.Aigrain, Proc. Intern. Con$ Semicond. Phys., Prague 1960 p. 224. Czech. Acad. Sci., Prague, and Academic Press, New York, 1961. 23 R. G. Chambers and B. K. Jones, Proc. Roy. SOC.(London) A270,417 (1962).

470

R. T. BATE

wave will propagate with little attenuation in strong magnetic fields, and the spatial variation of the least attenuated mode of circular polarization goes as e+ikz,where k = k, + iki and

Since the spatial dependence goes as e-kizeikrz, k; 2nkr- is the wavelength. If RHO 9 pT, we get

is the skin depth and

and

6,

=

k;

’

--*

(IRHo(/pT)([RHol/no)”2.

(5)

The subscript c is to remind us that we are dealing with only one circularly polarized mode. Note that the skin depth for this mode increases as the 5 power of the magnetic field if the resistivity saturates. However, as we have seen, inhomogeneity often prevents saturation of the magnetoresistance. We can assess the effect of random inhomogeneities on a scale small compared to the helicon wavelength by means of Herring’s formula.’‘ For a material with no bulk magnetoresistance effect (for example a degenerate semiconductor with spherical constant-energy surfaces), the transverse resistivity due to random inhomogeneities is (see Section 10)

It is noted that if ( ( n - (n))’> << ’, the Hall angle (= R H , / p , ) will be large when R H O 9 p o . In view of Eq. (6), the expression for the skin depth can be written

IRHOI dc

= Po

\RHO[112

+ ( n , 4 ) l R H o , * ~ ( ~ )’

(7)

where A2 = ( ( n - (n))’)/(n)’. Thus as the field is increased beyond a critical field H , the dependence of skin depth on H , changes from H;‘’ to HA/’. The critical field is given by the relation

8.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

471

or

H

4 x lo8 npoA’ ’

=--

(9)

where H , is in oersteds and p, is the carrier mobility in cm2/volt-sec. For po = lo6 cm’/volt-sec and A = 0.1, the transition occurs at about 13 kOe. Since the damping time constant for standing helicon waves is inversely proportional to the resistivity,’ it will also be influenced by inhomogeneity through the magnetoresistance effect. In sufficiently strong magnetic fields, the time constant should be inversely proportional to H , . At sufficiently high frequencies or low magnetic fields, the helicon wavelength may be reduced to distances characteristic of the carrier density fluctuations, so that Herring’s theory no longer applies. The damping effect of inhomogeneities might be less pronounced under these conditions.

IV. Calculation of Isothermal Transport Effects in Inhomogeneous Conductors Herring has pointed out’ that inhomogeneities can be separated into three classes depending on whether the scale of fluctuation is (1) comparable to the lengths associated with individual electrons or scattering centers (Debye length, mean free path, etc.), (2) large compared to these but small compared to sample dimensions, or (3) comparable to sample dimensions. We will call situation ( 1 ) the Microscopic Case, (2) the Intermediate Case, and (3) the Macroscopic Case.

9. MICROSCOPIC CASE This is the most difficult of the three cases to handle because it requires solution of the Boltzmann equation in a form more general than that usually considered. Progress on this case has been made recently by Frisch and M~rrison.’~ a. The Boltzmann Equation in an Inhomogeneous Semiconductor

The general form of the Boltzmann equation isz4

where (4n3)-’f(k,r)dk is the number of electrons per unit volume at r whose wave vectors lie in the interval dk = dk,dk,dk,. In the usual calculation of the isothermal transport coefficients, f is taken to be independent 24

H. L. Frisch and J. A. Morrison, Ann. Phys. (N.L) 26, 181 (1964).

412

R . T. BATE

of position so that the second term in (10)vanishes. However, this term will be appreciable when microscopic fluctuations in carrier concentration occur, and in addition [df/dt],,,, will be position dependent. Frisch and Morrison have solved Eq. (10) in the relaxation time approximation and have shown that it predicts nonsaturation of the magnetoresistance in strong magnetic fields. Their result for a microscopically stratified or striated medium is very similar to that obtained by Herring for the intermediate case. 10. INTERMEDIATECASE a. Fourier Analysis (Herring)

The intermediate case, as defined above, has received much more attention from theorists than have either of the other cases, culminating with Herring's definitive paper.' By an elegant argument involving Fourier analysis, Herring showed that a relation can be found between the average current density and electrochemical potential gradient in an inhomogeneous conductor and that this relation will, under certain conditions, be independent of the detailed form of the inhomogeneity. Specifically, it is required that the fluctuations in conductivity be small in magnitude and random, yet on a large-enough scale that a conductivity tensor can be uniquely defined at each point of the medium. Although these restrictions must be applied in his derivation, Herring, and also Frisch and Morrison, have pointed out that his results are approximately valid in many cases where one or more of the above restrictions is violated. Several of Herring's equations for isotropic conductors and isotropic inhomogeneities are reproduced here for easy :

(i) Effective conductivity 0eff

(r

= (0)[1 -

+(((a - (o>)2)/(0)2)1.

(11)

$ ( ( ( P - (P>)2>/(P>2)l.

(12)

(ii) Effective resistivity p Perf

=

(P)[1

-

Note that, although ( p ) # (e)-l we should have perf = 0,:. This latter relation is satisfied by the above formulas only for small fluctuations. (iii) Weak field Hall coefficient R(0)

24"Thebrackets indicate spatial averages, i.e., ( b )

=

'

V - J b dV

8.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

473

(iv) Strong field Hall coefficient

Ref,!

=

( R - ')

(subject to certain restrictions)

(14)

(v) Thermoelectric power a

(vi) Strong field transverse magnetoresistance (fluctuations in carrier concentration only)

provided the fluctuations in n are isotropic.

11. MACROSCOPIC CASE The mathematical formulation of the problem in this case does not differ in principle from that used by Herring for the intermediate case. However, it is now no longer possible to define an effective conductivity tensor which is independent of sample shape and size. Thus the boundary conditions at various sample surfaces must be carefully considered.

a. The Equation of Continuity When steady-state conditions are attained after a voltage is applied to a conductor, the total current entering any volume element of the material must be equal to that leaving it if there are no sources or sinks within it. This is the equation of continuity, which can be stated mathematically as follows :

V.j=O,

(17)

where j is the current density vector. At all surfaces of the sample not covered by contacts, this equation leads to the boundary condition : j n == 0,

(18)

where jn is the component of current density normal to the surface. In an isotropic conductor situated in a z-directed magnetic field, the current density is given by j

=

oE,

(19)

where E = - VV is the negative gradient of the electrochemical potential V,

474

R. T. BATE

at a = -pa,

0

Bat

0

a,

0 ,

0

01

where p = R H / p , is the tangent of the Hall angle. Substitution of (19) and (20)into (17) yields the partial differential equation

The solution of this equation for certain special cases has been discussed by Bate and Beer,I2 by H l a ~ n i k and , ~ ~by ~ Beer.25 The most interesting phenomena occur when the inhomogeneity and boundary conditions are such that the Hall field is a function of position, even far away from the end contacts. This will happen, for example, when a gradient in carrier concentration exists along the direction of the current. The application of a transverse magnetic field in this case distorts the current lines and tends to concentrate the current along one side of the sample. Thus, depending on probe placement and magnetic field direction, the resistance may appear to decrease or increase when the magnetic field is applied, even if there is no bulk magnetoresistance. V. Detection of Inhomogeneities Techniques such as metallographic examination, x-ray diffraction, and electron beam microanalysis are often used to detect inhomogeneity in polyphase materials or single-phase materials in which the concentrations of the constituents are of like magnitude. However, in semiconductors, the impurities which determine the electrical properties may be present in concentrations of only a few parts per billion, in which case they will not even be detectable by the above methods in their present state of development. One is thus usually forced to resort to indirect means of detecting inhomogeneity. Some of these methods are discussed in the following sections. 12. HIGH-RESOLUTION POTENTIAL PROBING The most direct way to detect small scale variations in resistivity is of course to measure the resistivity point by point. An apparatus to make such 24bl. Hlasnik, Solid-State Electron. 8, 461 (1965) See Ref. 3, p. 312.

25

8.

ELECTRICAL PROPERTIES OF NONUNIFORM CRYSTALS

475

measurements has been developed at Battelle by Mengali.26 It is capable of mapping variations in resistivity with a resolution approaching 1 mil (0.025mm). However, such resolution, as good as it is, may still not be sufficient to resolve striations which result from seed and crucible rotation. Further improvements in this technique would be difficult because of problems in probe fabrication.

13. ANISOTROPY AND SYMMETRY RELATIONS As was mentioned previously, the gross inhomogeneities often encountered in pulled crystals usually possess a symmetry lower than that of the crystal. One can often make use of this fact to detect their presence in cubic crystals. Consider, for example, a sample of a cubic semiconductor in the form of rectangular parallelepiped having its longest dimension parallel to an axis possessing fourfold rotational symmetry. If the cross section perpendicular to this symmetry axis is square,26aand if identical resistivity probes are placed on all four equivalent sides of the sample, then the perturbations due to contact effects will also possess fourfold rotational symmetry. Thus the transverse magnetoresistance measured with the current along the fourfold axis should possess fourfold symmetry in the absence of any inhomogeneity of lower symmetry. If striations are present in planes not perpendicular to the fourfold axis, the transverse magnetoresistance will not exhibit fourfold symmetry, and the presence of the striations is revealed. If contact perturbations can be eliminated, then the cross section perpendicular to the fourfold axis may be of arbitrary shape, and only one set of resistivity probes is required. By measuring directional magnetoresistance in two different specimens cut so that the current directions were at two different angles with respect to the direction of pull of the crystal, Weiss was able to determine the direction of the normal to the layers of equal doping in the specimen.15bThese data also permitted him to separate the magnetoresistance effect due to the inhomogeneities from that indicative of the material itself. 14. PHOTOVOLTAIC EFFECTS

A powerful technique for detection of inhomogeneity in semiconductors, which has been somewhat overlooked, is the so-called “bulk photovoltaic effect.” This effect, which was discussed by is the appearance of a voltage across an inhomogeneous semiconductor when a ray of light is a description of the apparatus, see 0. J. Mengali and M. R. Seiler, Aduan. Energy Con2. 59 (1962). 26rA circular cross section or any shape possessing fourfold or higher rotational symmetry is equally suitable. Z‘bJ. Tauc, Czech. J . Phys. 5, 178 (1955). ” For

cersion

476

R. T. BATE

incident on the surface at a point far from any contacts. The voltage is approximately proportional to the gradient of the resistivity at the point where the light is incident. The effect is easily observable in pure germanium and silicon, and in high-resistivity GaAs, but is more difficult to observe in low resistivity materials. An automatic recording technique which uses the bulk photovoltaic effect to map resistivity variations has been described by Oroshnik and Many.27 They employed a motor-driven scanning mechanism to move a light beam along the sample and recorded the photovoltage vs distance along the sample on an X-Y recorder. The extension of this technique to less photosensitive material requires an intense light source, and the improvement of the resolution of smallscale inhomogeneities such as striations requires a very sharply focused beam. These requirements suggest the investigation of lasers for use as light sources in the photovoltaic technique. 27

J. Oroshnik and A. Many, J . Electrochem. SOC.106,360 (1959).

'' A. C. Beer, J. Appl. Phys. 32,2107 (1961).

Author Index Numbers in parentheses are footnote numbers and are inserted t o enable the reader to locate those cross references where the author’s name does not appear at the point of reference in the text. A Abagyan, S. A., 447,448,449,450 Abbasov, A. S., 92, 93(177), 95, 96(180), 100. 149, 157 Abdurakhmanova, A. A,, 421 Ablova, M. S., 9, 12, 13, 14, 15 Abrahams, M. S., 202,429,430 Abrikosov, N. Kh., 27 Agaev, Ya., 430,431,437,439(49), 440,441 Ageev, N. V., 106, 110 Ageeva, D. L., 106, 1 lO(207, 208) Aigrain, P., 299,469 Airapetiants, C. V., 469 Albon, N., 463 Alexander, B. H., 10, 14,22 Alexopoulos, K. D., 377 Aliev, M. I., 421, 455, 456 Allen, C. M., 322 Allen, J. W., 14, 17(28), 185, 205, 206,216,446 Allgaier, R. S., 465 Allison, H. W., 196 Allred, W. P., 60, 461, 464 Ancker-Johnson, B., 469 Anderko, K., 41. 50(17), 55, 56(17), 57. 58, 60,61.64,65(17. 92). 127 Anderson. R. L., 219 Andreatch, P., Jr., 342 Anosov, V. Ya., 48, 65(51), 88, 96(51), 104(51) Antonov-Romanovsky, V. V., 360 Aoki, M., 233 Arnold, G. W., 386,406 Arseni, K. A,, 237 Attard, A. E., 110 Augustyniak, W. M., 365, 368, 371(60, 60a) Aukerrnan, L. W., 373,374(70,70a, 79,79a, 80, 81), 375, 376(70, 70a), 377(70, 70a), 378(70, 70a), 381(70, 70a). 382, 385, 386(70, 70a), 388(71), 389(71), 391, 392(79, 79a, 80, 102). 393, 394(80, 102). 396(79, 79a), 397, 398, 400(95, 102), 401(80), 402(80, 9 9 , 403(81), 405,406.407

Austin, I. G., 138, 339, 341(32) Averkieva, G. K., 26,421 Azaroff, L. V.. I10

B Babitsyna, A. A,, 423 Bagchi, S. N., 110, 11l(214) Balluffi, R. W., 346 Banbury, P. C., 372 Banus, M. D., 38, 141(11). 339, 340(50), 341 Baraff, G. A., 272,277.278,281,2X2.2X3.284, 285, 286,294,299,301 Baranov, B. V., 23 Bardeen, J., 181 Baroody, E. M., 352, 355(33), 369(33), 374(81). 375, 388(98a), 389, 397, 403(81) Barrer, R. M., 182, 189(39) Bartelink, D. J., 272 Bartning, A. M., 217 Bass, R. F., 385, 392, 393(109), 395(95a) Batdorf, R. L., 288, 291(16), 292(16), 294(16), 297(16), 301(16), 315 Bate, R. T., 235,461,464,465,467.474 Bateman, T. B., 342 Bates, C. H., 339 Batsanov, S. S., 124 Bauerle, J. E., 430(47), 437,439(47) Bauerlein, R., 370, 371(64,64a, 64b), 389, 390, 391, 394 Baukin, I. S., 426 Baxter, R. D., 235 Bazhenova, L. N., 12, 16(25a) Becke, H., 216, 239(113) Becker, J. H., 58,64(99) Becker, R., 69 Becker, W. M., 235 Bednar. J., 61, 64 Beeler, J. R., Jr., 353 Beer, A. C., 58, 392, 402(108), 460, 465, 467(3, 13), 468, 474, 476

477

478

AUTHOR INDEX

Belasco, M., 227 Bell, J. C., 465,467(13) Belov, N.V., 41(41), 45,46(41),47 Bendik, M.A,, 226,227 Bensch, H., 110 Berg, L. G., 134 Berger, L. I., 457 Bergmann, A., 60 Bering, B. P., 124 Berkovich, E. S., 5,6,21 Bernard, W., 181 Bert6ti. I., 418 Besco, D.G., 353 Bevacqua, S. F., 425.442,446(56) Bever, M.B., 65,68,69,71,72(125), 86,87. 89(165), 94,99,102,123,134,135,136,138,

149

Brewer. L.. 41(28), 43,87 Brill, R.,106, 1 lO(202,203) Brinkman, J. A., 359,365 Brouwer. G., 175 Brown, F.. 353 Brown, R. D., 323,324(57) Brown, W . L., 360,361,365,368,371(60,60a) Bube, R.H., 170,172(16a) Buckle, H., 22 Buehler, M.G.,236 Bugrova, G. I., 25 Buiocchi, C. J., 202 Bullis, W. M., 193 Burdiyan, I. I., 24,418,432 Burdukov, Yu. M., 449 Burns, G., 405 Burstein, E., 323 Bushey, A. H., 51 Byron, E. V., 16

Bickel, P. W., 368,369,371(61),379 Bielan, C. V., 442,446(56) Bierbaum, C., 4,1 i(4) Billington, D. S., 344, 347(3) C Biltz, W., 126 Birchenall, C. E., 182,189(42), 191,192,235. Cahn, J. H., 349 371,372(67a) Caldwell, J. P., 219 Black, J. F., 214,445,446(59) Campbell, 1. E.,62,64,75(122),85 Blair, R.R.,408 Cardona, M., 416,456,457(80) Blakemore, J. S., 374(75),375,377(75),426 Carlson, R.O., 405 Blanc, .I.,170,172,215 Carman, J. N., 132,141 Blewitt, T. H., 353 Carpenter, G. B., 110 Blount, E. I., 373 Casey, H.C. Jr., 178,181. 207,208,211,212, Blunt, R.F., 58,61,64 231,236 Bochvar, A. A., 41 Catterall, J. A,, 86 Boltaks, B. I., 182,189(43), 191. 192,222,233, Chaikovskii, E.F., 190 235.237.238,251.252,253,254.255 Chakraverty, B. K., 170 Bommel, H . E., 340(54,59). 341 Chalmers, B., 462 Bond, W. L., 132,141 Chamberlain, L. L., 139 Borchers, H., 418(4), 419,420 Chambers, R.G., 469,471(23) Borshchevskii. A. S.. 10. 1 l(18). 12. 14.lX(18). Champlin, K. S., 314,315(39) 19,20, 24,28,29(43), 31(43), 53,416(3b), Chang, L. L., 181, 196, 197, 207,208(37a),

417,418(3b). Bougnot, J., 235 Bowers, R.,430,437,439(47),469 Bradley, C. C., 331 Bragg, W. H.. 41(40),45,106 Bragg, W.L.. 41(40).45 Brammer, A. J.. 342 Brandhorst, H., 340(47), 341 Braunersreuther, E., 435,436 Braunstein, R.,429,430(31) Breger, A., Kh., 110.111(212)

211(37a), 212(37a),213,217,226(114),236 Chen, W.S., 236,255 Chrenko. R.M., 365 Christensen, A. U., 87, 88(168), 149(168) Chynoweth, A . G., 293,297,298. 299,300 (18.20),302,303(26),304(26),305(22), 308,

309(29), 313(28), 314,315,338,408 Chzhen’-Yuen’, Lyu, 10, 68,69,136,149 Claassen, R.S., 408,409 Clark, J. C., 5,9(8), 12(8), 14(8), 19(8),20(8) Cleland, J. W., 346, 374(77,77a, 82), 375,

479

AUTHOR INDEX

376(77, 77a), 385, 388(98), 389, 392, 393, 395(95a), 401, 402(106), 404 Clement, N. J., 38, 134, 135, 136, 137(10),138, 141, 149 Closer, W. H., 346 Clusius, K., 103, 149 Cochran, W., 110, 121 Cohen, B. G., 302, 303(26), 304(26), 314(26) Conradt, R.. 314, 315(45) Conwell, E. M.. 181,264. 393 Corbett, J. W., 344, 364, 365 Coriell, A. S., 338 Cornish, A. J., 430(47), 437,439(47) Cottrell, T. L., 41(27), 42. 86 Crawford, J . H. Jr., 344, 346, 347(3), 374(77, 77a, 82), 375, 376(77, 77a), 385, 388(98), 389, 392, 393(109), 395(95a), 401,402(106),

Dixon, J. R., 389, 460 Dorfman, Ya. G., 106 ]>rabble, J. R., 342, 429, 430, 439(33) Dreyfus, R. W., 170 Drickamer, H. G., 328, 329, 330, 331, 335, 339, 340, 341(23,46) Drougard, M., 228 Drowart, J., 77, 78, 79, 80, 81, 82, 85. 91, 98, 149 Duane, W., 106 Dubinskii, S. A,, 61,62,63,64 Dubrovskii, G. B., 439 Dumke. W. P., 405 Durup, J., 349, 386(24e) Dworkin, A. S., 86, 149 Dzhafarov, T. D., 222 Dzhangirov, A. Yu., 455, 456

404

Cronin, G. R.. 177, 202 Cummerow, R. L., 404 Cunnell, F. A., 61, 64, 205, 206, 213, 215, 217(99), 219,225 Curtiss, 0. L., Jr, 404

D Dacey, G. C., 315 Damask, A. C., 364,365,398(49) Danilov, V. I., 69 Darken, L. S., 69 Darnell. A. J., 339. 340(54. 59). 341 Dautreppe, D.. 374(84a), 375. 377(84a), 378 (84a). 381(84a). 385(84a) Davies, J . A., 353 Davis, P. W., 374(80), 375, 391(80), 392(80), 393(80). 394(80), 401(80), 402(80) Debye, P. P., 181. 393 Dederichs, P. H., 353 Dehlinger, V., 69 Derick, L., 51, 64, 76, 85 Deryabina, V. I., 28 Descamps, J., 22 DeSorbo, W., 103,149 Detweiler, R. M., 371 Deubner, D. C., 255 Dienes, G. J., 344, 347(2), 349, 350(2), 351, 364, 365,398(49) Dikhoff, J. A. M., 461 Dill, F. H.. Jr., 405

E Easiey, J. W., 408 Edel’man, F. L., 15 Edmond, J. T., 61,64(114) Edwards, A. L., 328, 329(1), 330, 331, 339(9) Effer, D., 235 Efimova, B. A,, 427(30), 429 Ehrenreich, H., 328,432,435,437,444,446(41), 448(41, 58) Eisen, F. H., 191, 192, 235, 364, 368, 369, 371, 372, 374(76, 84d), 375, 376, 377(76, 84d), 378,379,381,382,383,384,385,386 Elie, G., 21 9 Eliseev. A. A,, 422 Emel’yanenko, 0. V.. 430,449,453 Emerson, W. B., 6 Englert, A,, 353 Epstein, A. S., 219 Erdos, P., 381 Erginsoy, C., 353 Ermolenko, E. N., 91, 96, 98(175), 149 Esaki, L., 408 Esposito, R. M., 372 Etter, P. J., 235 Euthymiou, P. C., 376 Evans, D. A,. 325 Evans, E. L.,41,42.86,89,123,125(20), 126(20), 149 Evans, J. A,, 178,202(25,26),426,429, 456 Evans, R. C., 46, 150(47)

480

AUTHOR INDEX

F Fairfield, J. M., 187, 371 Fan, H. Y., 121, 193, 330, 344, 374(5, 5a, 83a, 84b), 375, 378,401(84b) Fane, R. W., 202,203,204 Farrell, L. B., 341 Fedorenko, A. I., 353 Fedorova, N. N., 25,53,60,424 Fein, A. E., 352, 369(32) Feinleib, J., 331,332(13) Feldmann, W. L., 299, 338,408 Feltin’sh, I. A,, 178, 202, 457 Fenner, G. E., 331,334,405,443,444,445 Feoktistova, N. N.,9, 12, 13, 15(16) Ferguson, W. S., 93 Fern, R., 228 Feshbach, H., 349 Field, N. I., 5,9(8), 12(8), 14(8), 19(8), 20(8) Figielski, T., 313 Firth, E. M., 110 Fischer, A,, 416 Fitchen, D. B., 445 Flatley, D., 216,239(113) Fletcher, R.C., 360, 361 Flicker, H., 370,371(63), 372 Folberth. 0. G., 41(25), 42, 51, 64, 76, 77, 79,81,86(70). 90, 125, 126. 158(173) Foy, P. W., 315 Frank, F. C., 184, 185(47), 188, 214 Frankevich, E. L., 41(22), 42 Frederikse, H. P. R.,58,61, 64(99) Frenkel, J., 166, 171 Frenkel, Ya. I., 145 Frieser, R. G., 198, 199 Frisch, H. L., 471 Frohlich, H., 270 Frosch, C. J., 51,64,76,85 Fugita, F. E., 346 Fuller, C. S., 173, 182, 185(44), 186, 189(44), 2M), 202, 203, 208(17), 226(44), 229, 230, 234,237,395 Fulton, T. A., 445

G

Gadzhiev, S. N., 97,99, 105, 136, 149 Galavanov, V. V., 453,454(73) Gal’chenko, G. L., 86,149 Garber, R. I., 353

Garibov, 1. M., 99, 149(184) Garland, C. W., 131, 342 Gasson. D. B.. 456 Gates, H. C., 38, 141(11), 155(301), 158, 339, 342,463 Geballe, T. H., 466, 469 Gebbie, H. A., 138, 331, 339, 341(32,45) Geller, S., 340(55), 341 Georgopoulos, C., 374(84, 84a), 375, 377(84, 84a), 378, 381, 383, 385 Gerasimov, Ya. I., 95, 96(180), 100, 103, 104 (191), 149(191) Gerlich, D., 131 Gershenzon, M., 188, 197, 308, 312, 405, 407( 128a) Gertsfel’d, K. F., 145 Geyderikh, V. A., 103, 104(191), 149(191) Gibson, J. B., 353 Gielessen, J., 335 Giesecke, G.,41(37), 45, 51(37), 53, 55, 56, 58, 60(37), 61(37) Gillett, C. M., 178, 202(25), 426, 427, 429 Gilvarry, J. J., 139, 142 Ginell, W. S., 401 Glagoleva, N. N., 22 Glasford, G . M., 219 Glazov, V. M., 5, 9(9), 10, 22, 57, 58, 59, 60, 61(95),62,63, 64.65,68,69, 136, 149 Glicksman, M., 320,321, 322,468 Glicksman, R., 219 Gobeli, G . W., 406 Goering, H.L., 432 Goetzberger, A,, 314, 315 Goikhman, A. Ya., 219, 226(123) Goland, A. N., 353 Gold, R. D., 219 Goldfinger, P., 77, 78, 79, 80, 81, 82, 83, 85, 86,91, 94, 95(154), 98, 100, 149 Gol’dshmidt. V. M., 4, 15, 36, 41(36). 44. 45 51, 54(36). 55, 149 Goldsmid, H. G., 429,430,439(33) Goldstein, B., 190,191,198, 199,200,201,203, 205,219, 225,235, 239, 252, 371, 372(67b) Golodushko, V. Z., 77, 78, 81, 82, 83 Gololobov, E. M., 58, 60, 61, 110, 111, 115, 119, 120(230), 126, 131, 142 Golutvin, Yu. M., 150 Gombas, P., 122 Gonser, B. W., 62, 64(122), 75(122), 85(122) Gonser, U., 346,379.401,402

AUTHOR INDEX

Gooch, C. H., 206, 213, 215, 217(99),219, 225 Goodman, C. H. L., 157 Gordy, W., 41(23), 42, 124, 125, 126 Gorodetskii, S. M., 447 Gorton, H. C., 196 Goryunova, N. A., 10, 11, 12, 16(25), 18(18), 19(18, 25), 23, 24, 25, 26, 27, 28, 29(22), 31(22), 49, 50(54), 53, 60,64, 89(54), 416, 417,418,420,421.422(1), 423, 424 Goss, A. J., 202, 203, 204 Gossick, B. R., 346,460 Gottlicher, S., 110 Graft, R. D., 374(79, 79a, 80, 81), 375, 391(79, 79a, 80), 392(79, 79a, 80), 393(80), 394(80), 39q79, 79a), 397, 398(79, 79a), 401(80), 402(80), 403(81) Greenaway, D. L., 456,457(80) Greene, R. F., 173, 174(19), 177(19) Greenfield, 1. G., 61,62,64 Gregory, E., 339, 340(57), 341 Grigorieva, V. S., 26 Grigorovich, V. K., 5, 38, 65 Grimm, H. G., 106, 1lO(202) Grimmeiss, H. G., 51 Grimshaw, J. A., 372,394,399 Groves, S., 331,332(13) Guertles, W., 60 Gul’tyaev, P. V., 18, 103, 104(192) Gummel, H. K., 308, 309(29) Gunn, J. B., 338 Gupta, D. C., 227 Gurney, R. W., 169, 170(15), 193 Curry, R. W., 69 Gurvich, L. V., 41(22), 42 Gusev, I. A., 249,251,252,253,255 Guseva, L. N., 106, llO(196, 205, 206) Gutbier, H. B., 91, 99, 149 Gutorov, Yu. A., 191,235

H Haebler, H., 86 Hahn, H., 41(34), 44,45, 50, 90,97, 149, 424 Haisty, R. W., 202, 227, 228(131), 232 Haitz, R. H., 314, 315, 317 Hall, J. J., 342 Hall, R. N., 52, 53, 54, 56, 57, 58, 61, 62, 63, 73, 74, 188, 215, 219(56), 230, 405 Hall, W. F., 75,76,85,86(144), 88, 149 Halperin, B. I., 181,207

481

Ham, F. S., 364 Hanneman, R. E., 38, 141, 339, 341(43) Hansen, M., 41. 50(17), 55, 56(17). 57, 58, 60, 61,64,65(17,92), 127 Harman, T. C., 463 Harper. J. A,, 190 Harrap, V., 193 Harrity, J. W., 373 Hirsy, M., 418 Harvey, W. W., 180 Hasiguti, R. R., 365 Hass, M., 121 Hassel, O., 46 Haucke, M., 38. 146(6),147(6) Havighurst, R. J., 106 Hayes, W., 234 Hazelby, D., 196 Heasell, E. L., 248, 251 Hegyi, I. J., 419,421(5) Heinen, K. G., 177, 201, 203(90) Henkel, H. J., 219 Henneke, H., 251 Henvis, B. W., 121 Herbstein, F. H., 131 Hermann, C., 106, 110(203,204) Herring, C., 336, 463, 464, 465, 468(11), 469, 470, 471. 472 Ileymer, G ., 4 1, 132(18). I4 I Heynert, G., 69 Hicinbothem, W. A., 320, 321, 322 Hildenbrand, D. L., 75,76, 85, 86(144), 88, 149 Hill, D., 12, 49 Hilsum, C., 17, 37, 44, 455 Himes, R. C., 425,432 Hinge, K. S., 79,89, 149 Hllsnik, I., 474 Hobstetter, J. N., 188,21 l(57) Hoch, M., 79,89, 149 Hochman, R. F., 15 Hockings, E. F., 68,69,133, 149,429 Hodby, J. W., 446 Holmes, D. K., 347, 348(18), 349. 353, 392 Holonyak, N . Jr., 219,425,442,446(56) Horiye, H., 373 Hosemann, R., 110, 11l(214) Hosler, W. R., 58, 61, 64(99) Howard, N. R., 289 Howard. R. E., 349, 386124~) Hrostowski, H. J., 180 Huggins, R. A., 185, 187(49),192, 193

482

AUTHOR INDEX

Hull, G . W., Jr., 340(55, 57), 341 Hulme, K. F., 239 Hultgren, R., 138, 141 Hume-Rothery, W., 109, 152 Hutson, A. R., 338 1

Iandelli, A,, 53, 56, 64 Ingles, T. A.. 49 Inuishi, Y., 408 Inyutkin, A,, 453 Iofa, B. Z., 83 Ioffe, A. F., 155 Ishinio, S., 365 Ismailov, O., 437,439(49) Ivan’ko, A. A., 12, 16(25a) Ivanova, R. A,, 28.29 Ivanov-Omskii, V. I., 22, 29, 426, 427, 428, 43 l(25) Izeika, T., 314

J James, H. M., 370, 373 James, R. W., 110, 139,144 Jamieson, J. C., 339, 340 Jan, J.-P., 338 Jayaraman, A., 137, 138,141, 338,339,342 Jennings, I. C., 456 Jeunehomme, M., 81, 82, 83, 85, 94, 95(154), 100, 149 Jillson, D. C., 425 Johnston, W. D., 76, 85 Jones, B. K., 469,471(23) Jones, H., 155 Jones, M. E., 197, 205, 206, 207, 215, 217(97), 239, 240(82) Jones, P., 196 Joshi, S . K., 131 Jost, W., 182, 189(40) Jungbluth, E. D., 214 Junker, H. J., 132, 141(265) Juza, R., 41(%44,45, 50, 90,97, 149

K Kafalas, J . A,, 339 Kaiser, R., 374(84b), 375,401(84b) Kalyuzhnaya, K. A,, l4,416(3b), 417, 418(3b)

Kamenetskaya, D. S., 69 Kanai, Y., 320, 322 Kantz, A. D., 408, 409(144, 145) Kanz, J. A., 187, 188(55), 190,203(63), 207(63), 209(55), 217(63), 222(55), 223, 227, 243, 245(55), 253 Kapustinskii, A. F., 36, 147, 150 Kasper, J. S., 340(47), 341 Kauer, E., 41(30), 43, 44, 50(30) Keating, P. N., 178,202(27) Kefeli, L. M., 21 Keil, G., 315, 317(46) Keller, H., 203 Kelley, K. K., 87, 88, 149 Kelley, R. H., 371 Kemp, J. E., 239 Kendall, D. L., 169, 185, 187, 188(14, 55), 190,192, 193, 197, 198,199,200(14,83), 205, 206, 207, 208, 209, 211, 215, 217, 222, 223, 225, 226, 227, 228, 231, 232, 239, 240(82), 243, 245(55), 254(14) Kennedy, G . C . , 137, 138, 141(270, 272), 339 Kern, W., 216,239(113) Kesamanly, F. P., 439, 449,451.452(69) Keyes, R. J., 404,405 Keyes, R. W., 332, 333, 334, 335(16), 336, 337, 342, 380 Khabarov, E. N., 26,453 Khabarova, V., 453 Khandelwal, G. R., 349 Khansevarov, R. Yu., 374(84c, 84e), 375, 377(84e), 392(84c) Kholodnyi, L. P., 15 Khrushchov, M. M., 5 Kikuchi, M., 314,315(40) Kinchin, G. H., 350, 351 King, E. G., 87, 88(168), 149(168) King, J. H., 138, 339, 341(32,45) Kingsley, J. D., 405 Kiosse, G. A., 27 Kischio, W., 51 Kittel, C., 41 Kleimack, J . J., 288, 291(16), 292(16), 294(16), 297(16), 301(16), 315(16) Kleitman, D., 380 Klemens, P. G., 380,394 Klement, W., Jr., 138, 141(272),339 Klemm, W., 132,141,424 Kleppa, 0. I., 102, 149 Kleshchinskii, L. I., 26

AUTHOR INDEX

483

Klontz, E. E., 345, 368, 372 Laitinen, H. A., 93 Klotyn’sh, E. E., 439,449 Landsberg, P. T., 325 Klotz, H., 105, 149 Lange, T. I., 9, 14(17), 15, 424 Knoop, F., 6 Lange, V. N.,9, 14(17), 15,424 Kochetkova, N. M., 96, lOl(181) LaPaca, S., 41(38), 45, 49(38) Koehler, J. S., 343, 344(1), 346, 347(1), 351(1), Lark-Horovitz, K., 344, 368, 372, 373, 374(5, 352, 365(1), 368(1) 5a, 70a). 375(5, 5a), 376(70a), 377(70a), Koelsch, H., 53 378(70a), 381(70a), 382(70a), 386(70a) Koenig, S. H., 323, 324(57) Larrabee, G . B., 177, 204, 227, 228, 229, 230, Kogan, L. M., 219,226 232, 234 Kohn, W., 368 Lasher, G . J., 405 Kolodziejczak, J., 439 Lavine, M. C., 340(50), 341,463 Kolomiets, B. T., 22, 426, 427, 428, 431(25), Lax, B., 405 432 Lax, M., 181,207 Kolosov, E., 453 Lazarus, D., 182, 189(46), 254(46) Kondrat’ev, V. N., 41(22), 42 Lee, C. A,, 288, 291, 292(16), 294, 297, 299, Koren, N. N., 51, 53(68) 301, 315(16) Korenblit, L. L.. 438,454(50) Legendy, C., 469 Korolevskii, B. P..418 Lehmann, C., 353 Kornilov, A. N., 86, 149(160) Leibfried, G., 347, 348(19), 353 Korol’kov, G . A., 22 LeMay, C. Z., 21 5 Koster, W., 53, 54, 55, 56, 57, 60(84), 61, 62, Lenie, C., 50, 64 67,73 Leroux-Hugon, P., 453 Kotrubenko, B. P., 424 Letaw, H. Jr., 371 Kozhina, I. I., 28 Levy, P. W.. 346 Kozlovskaya, V. M., 80. 81, 83, 84, 85, 92, Libby, W. F., 339, 340(54,59), 341 105, 149 Lindemann, F. A,, 139 Krag, W. E., 405 Lindenberg. W., 132, 141(265) Krebs, H., 38, 146, 147 Lindhard, J., 352 Krivov, M. A.. 374(83), 375, 392 Liner, L. V., 8 Kroger, C., 86 Lipmann, H., 435,436(44) Kroger, F. A.. 164, 165, 166, 172, 173, 177(2), Lipson, H., 65 1 78, 179. 182(2). 185(2), 36 1 Lishina, A. V., 447.448(65) Ku, San-Mei, 416, 425, 441, 442, 445, 446 Liu, C. V., 93 (19, 59) Liu, T. S., 4, 56, 58, 61, 62, 63, 64, 73 Kubaschewski, 0..41,42, 86. 89, 123, 125(20, Llinares, C., 235 163), 126(20). 128, 149 Lobanova, Yu. K.. 423 Kuczynski, G. C., 15 Loebner, E. E., 419,421 Kudman, I., 178, 202(29), 429 Loferski, J. J., 368, 370, 371(59, 59a, 59b, 63). Kuhrt, F., 435, 436(44) 372,389(59,59a, 59b), 403,404(59.59a, 59b). Kulikov, G . S., 192, 238 406 Kulp, B. A,, 371 Logan, R. A., 132, 141, 288, 291(16), 292(16). Kurdiani, N. I., 376,377,385 294(16), 297(16), 298, 299, 301(16), 302, 303, Kurnakov, N. S., 21, 48, 62, 65, 66 304, 305, 306, 308(22), 314(26), 315(16), 408 Kuznetsov, G. M., 41 Long, D., 332, 335(15), 455 Kuznetsov, V. D., 20 Longini, R. L., 173, 174(19), 177(19), 209. 214, 219(104) L Longo, T. A., 392,408 Laff, R. A,, 193,374(83a), 375,378 Lorimor, 0. G., 234 Lagunova, T. S., 449 Lovelace, K., 253

484

AUTHOR INDEX

Lumsden, J., 69, 144 Luzhnaya, N. P., 422 Lyons, V. J.. 81, 82

M MacDonald, R . S., 365 MacKay, J. W., 345 Macres, V. G., 191, 239(65) Madan, I. A., 27, 28, 29 Madelung, O., 460 Maeda, K., 314, 315(44), 317(44) Mah, A. D., 87, 88(168), 149 Maier, R. G.,418(4), 419,420 Makarov, E. S., 41(43), 45 Makovetskaya, L. A., 25, 53,417 Malisova, E. V., 374(83), 375, 392(83) Malkovich, B. E.-Sh., 457 Malkovich, R. Sh., 213 Mal’tsev, Yu. V., 439 Malyanov. V., 374(83), 375, 392 Malysh, G. K., 213 Mamaev, S., 453,454 Manca, P., 155 Mansuri, Q. A., 55 Many, A,, 476 Marezio, M., 341 Margrave, J. L., 87, 149 Mariano, A. N., 339 Marina, L. I., 51, 64, 76, 77, 78, 85, 86(75), 91. 149 Marinace, J . C., 219, 225(115), 226(115), 308. 309(3 I), 3 12(31) Markovskii, L. Ya., 41(29), 43, 44(29), 48(29). 49(29), 64(29), 75(29), 85(29) Marsh, S. P., 339 Martin, J. E., 339, 341 Mashovets, D. V., 438,454(50) Mashovets, T. V., 374(84c, 84e), 375, 377(84e), 392(84c) Massing, G., 132 Massoulie, M., 404 Masters, B. J., 371 Matkovich, V. I., 49 McAfee, K. B., 293 McCargo, M., 353 McColl, M., 406, 407(134, 135) McDonald, B., 31 5 McDonald, T. R. R.,339, 340(57). 341

Mclntyre, R. J., 314, 315(41) McKay, K. G., 287, 288, 289(14), 292, 293, 297, 299, 300(18). 308, 313(28), 314, 315(14) McKinley, W. A,, 349 McLachlan, D., 139 McNichols, J. L., 401 McQueen, R. G., 339 McSkimin, H. J., 342 McWhan, D. B., 340(55, 57), 341 McWhorter, A. L., 405 Mead, C. A,, 432,445,446,447,449(61) Medvedev, V. A., 41(22), 42 Medvedeva, Z. S., 422 Meechan, C. J., 359 Mefferd, W. L.. 60 Mehta, R.,211, 221, 222 Mengali, 0. J., 475 Menger, H., 49 Merzbacher, E., 349 Meskin, S. S., 219, 226(123) Messenger, G . C., 404 Meyer, N., 272, 276 Michel, A. E., 308, 309(31), 312 Middleton, A. E., 58 Mikulyak, R. M., 188, 197, 308, 312,408 Milgram, M., 353 Millea, M. F., 192, 405, 406, 407 Miller, D. P., 346 Miller, J. F., 425, 432 Miller, R. C., 190, 337, 338(27) Miller, S. E., 235 Miller, S. L.. 288, 292, 293 Mil’vidskii. M. G., 8, 15 Minamoto, M. T.. 322 Minomura. S.. 335. 339(23), 340. 341 Mirgalovskaya, M. S., 424 Mitra, S. S., 131 Mityurov, B. K., 451,452(69) Miyauchi, T., 157 Mlodzeevskii, A. B., 69 Moissen, N. N., 49 Mokeitshik, A. I., 418 Mokrovskii, N. P., 38,65(9), 132, 133, 141 Moll, J . L., 173, 174(20), 179, 272, 273, 276. 278,293,299, 302 Molodyan, I. P., 27, 28,422 Monteil, E.. 235 Moore, R.G . , Jr., 227 Mooser, E., 121 Morin, F. J., 173, 208(17)

AUTHOR INDEX

Morrison, J. A., 471 Mott, B. W., 4, 8, 9(6) Mott, N. F., 106, 149, 155, 169, 170(15), 193 Miiller, E. K., 417 Muller, R. L., 20 Mulliken, R. S., 41(24), 42, 125(24) Mullin, J. B., 461,463(5) Murin, A. N., 249,251,252(155a), 253, 255

N Naber, J. A,, 370 Nachtrieb, N. H., 38, 134, 135, 136, 137(10), 138, 141, 149 Nagasaki, H., 341 Nakazawa, F., 365 Nashel’skii. A. Ya.. 11. 12. 25, 51, 64(74. 75), 7 W 4 , 75). 77(74), 78(74, 75), 85(74, 75), 86(75), 91(74, 75). 149(74, 75) Nasledov, D. N., 60, 178, 202, 404, 430, 431, 439,449,453,454(73), 457 Nathan, M. I., 308, 309(31), 312(31), 405 Natta, G., 54 Nedumov, N. A,, 38, 65 Negreskul, V. V., 27, 28 Nelson, R. S., 346, 350, 353 Nesmeyanov, A. N., 41(26), 42, 43, 83, 123. 141 Neufeld, J., 350, 351, 352, 369(31) Neugebauer, C. A., 87, 149 Neuhaus, A,, 49 Neumann, B., 86 Neuringer, L. J., 330 Newman, R., 308 Newton, R. C., 137, 141(270), 339 Nichols, D. K., 345 Nikitina, V. K., 423 Nikolaenko. G . N., 60 Nikol’skaya, A. V., 95. 96(180), 100, 103, 104, 149 Nowicki, D. H., 62,64(122), 75(122), 85(122) Nyguist, H. L., 8

0

Oelsen, 0..134 Oelsen, W., 134 Oen, 0. S., 349. 353

485

Okai, B., 341 Okkerse, B., 379.401,402 Olekhnovich, N. M., 41(33), 43, 45, 54, 110, 111, 113, 117 Oliver, D. J., 324, 325 &en, W., 69 Ormont, B. F., 20, 157 Oroshnik, J., 476 Orth, R. W.. 251.252 Osborne, J. F., 201, 203, 204, 227, 228, 229, 230, 234 Osnach, L. A,, 26,453 Ostrovskaya, V. Z., 25 Osvenskii, V. B., I 5 Oswald, F., 121,439,440 Oswald, R. B. Jr., 377 Owen, E. A., 58

P Palms, J. M., 352 Pamplin, B. R., 178, 202(26), 456 Panish, M. B., 181,207,208(37, 37a), 211(37a), 212(37a), 216, 236 Pankove, J. I., 404 .Paranjape, B. V., 270 Park, K. C., 131, 342 Parrott, J. E., 456 Parsons, J. R., 346 l’ashintsev, Yu. I., 18, 41(39), 45, 54. 55(39, 86), 131, 142 Passerini, L., 51, 54 Patel, J. R., 14 Patrick. L., 308 Patterson, W., 372 Paul, W.. 328, 329. 330, 331, 332(10, 13), 333, 334(2) Pauling, L., 123, 165 Paulus, R., 41(32), 43, 45(32) Pavlova, N. G., 27 Pearson, G. L., 181, 191, 192, 196, 197, 207, 211,213,216,231,239(65), 299,315 Pearson, W. B., 41(21), 42, 121, 127 P’eart. R. T.. 228 Pease, R. S., 41(31), 43, 45. 350. 351 Peet, C . S.. 196 Pell, E. M., 21 1 Penn, A. W., 456 Peresada. G. I., 137, 141

486

AUTHOR INDEX

Peretti. E. A., 4, 56, 58, 61, 62, 63, 64, 73 Perri, J. A., 41(38), 45, 49(38) Peters, C. G., 6 Peters, C. L., 106, 1lO(202. 203) Peters, P. C., 374(79a), 375,391(79a), 392(79a),

Q Quarrington, J. E., 158, 159 Quist, T. M., 404, 405

R

396(79a), 398(79a)

Petree, M. C., 405 Petrov, A. V., 18, 103, 104(192) Petrov, D. A., 57,58,59,60(95), 61(95), 62,63, 64. 65,424

Petrusevich, R. L., 226(127), 227 Pfann, W. G., 460,463(4) Pfister, H., 41(37), 45, 51(37), 53, 55, 56, 58, 60(37), 61(37)

Philipchuk, B. I., 5 Phillips, J. C., 131 Pierce, C. B., 408,409 Piercy, G. R., 353 Piesbergen, U., 89, 96, 98, 101, 104(172), 129, 130, 131, 149 Pigg, J. C., 392, 402(106), 404 Pilkuhn, M. H., 216, 219 Pilor, A., 219 Pines, B. Ya., 69. 190 Pines, D., 267 Platzman, R. L., 349, 386(24e) Pogodin, S. A., 21, 48, 61, 62, 63, 64, 65(51), 88, 96(51), 104(51) Pokrovskii, N. L., 124 Pollak, M., 336, 337 Polyakov, A. S., 83 Ponyatovskii, E. G., 137, 141 Poor, E. W., 419, 421(5) Popper, P., 49 Poretskaya, L. V., 27 Portnoy, W. M., 371 Post, B., 41(38), 45, 49(38) Potter, R. F., 131 Potts, H. R., 191, 239(65) Powell, C. F., 62,64(122), 75(122), 85(122) Presnov, V. A,, 158 Preston, G. D., 58 Preuss, H., 125 Primak, W., 398 Prince. M. B.. 404 Prochukhan, V. D., 2628,421 Prostoserdova, 1. V., 249 Pugh, E. N., 191 Pumper, E. Ya., 249 Puri, S . M., 466, 469

Rabenau, A., 41(30), 43,44, 50(30), 51 Racette, J. H., 188, 215, 219(56), 230 Radautsan, S. I., 27, 28, 29,422, 457 Rappaport, P., 368, 371(59, 59a, 59b), 372, 389(59, 59a, 59b), 404 Raynor, G. V., 109, 152 Rebinder, P. A,, 20 Rediker, R. H., 405 Reed, B. S., 187, 188(55), 209(55), 222(55), 223, 243, 245(55)

Regel’, A. R., 15, 38, 65, 132, 133, 141 Reid, F. J., 235, 388(98a), 389, 392, 402(108), 403

Reiss, H., 173, 182, 185(44), 186, 189(44), 200, 208,226(44)

Rembeza, S . I., 237 Renner,T., 50,64,75,79,85,86,89,90,95(151), 96(173), 98(151), 149, 158(173)

Rezukhina. T. N.. 96. 101(181) Rhys-Roberts, C., 325 Richards, J. L., 61, 64(114), 417 Richman, D., 51, 52, 53, 56, 68, 69, 77, 78. 79, 80, 81, 82, 84, 85, 133, 149

Rieskarnp, K. H., 134 Riser, H., 201, 203(90) Roberts, F. E. R., 224,228 Roberts, V., 158, 159 Robinson, M. T., 353

Rogachev, A. A.. 404 Rohmann, A,, 125 Romanenko, V. N., 29 Roozeboom, H . W. Bakhuis, 417 Rose, A., 307 Rose, D. J., 314 Rose, F., 469 Rose. F. W. G., 179 Rose-Innes, A. C., 17, 37, 44 Rosenberg, A. J., 155(301), 158, 179 Rosi, F. D., 429,430(31) Rosov, V. V., 24 Roth, H., 181 Roth, W. A,, 86 Roy, R., 339 Rozneritsa, Ya. A., 432

AUTHOR INDEX

Rozov, V. V., 131, 144 Rubenstein, M., 52, 53, 73 Rubinshteiin, R. N., 80, 81,92* 105(152), 149 Rud, Yu. V., 451,452(69) Ruehrwein, R. A., 49,75(57), 79(57), 85 Ruge, I., 314, 315, 317(46) Rundquist, S., 41(35), 44,45,49(35), 75(35) Rupprecht, H.,214,215,216,219, 389,466 Rutter, J. W., 462 Rybka, V.. 233 Ryvkin, S . M., 374(84c), 375. 392(84c). 404

S Saffrendi, C. S., 345 Sagar. A,. 334, 337. 338( !8,27) Saidov, M. S., 158 Saji, M., 408 Samoylov, 0. Ya., 147 Samsonov, G. V., 12, 16(25a), 41(29), 43, 44, 48(29), 49(29), 64, 75(29). 85 Samuels. L. E., 191 Sand, R., 339 Sander, H. H., 408,409(144) Sangster, R. C., 132, 141 Sarkisov, E. S., 157 Sarkisyan, D. A., 5 Sasmar, D. J., 86, 149(159) Sato, S., 87. 149 Sattler, A. R., 352 Scarlett, R. M., 314, 315 Schachinger, L., 103, 149 Schade, R., 121 Schillinger, W., 323, 324(57) Schillman. E., 236, 237 Schissel, P. O., 87, 149 Schmelz, C. M., 429 Schmid, A. P., 181 Schmitt, R. A., 345 Schneider. A., 41, 105, 132(18). 141. 149 Schnorrenberg, E., 41(32), 43, 4.5(32) Schol, K., 48, 53, 54, 56, 64, 73, 77, 79, 80, 81, 82, 83, 85 Schottky, G., 178 Schottky, W. F., 65, 68, 71, 72(125), 89, 94, 99, 102, 123, 134, 135, 136, 138, 149, 165, 166, 170 Schreiber, J. Jr., 372

487

Schiirrnann, E., 69 Schwuttke, G. H., 214 Sclar, N., 323 Scott-Monk, J., 370, 371(63) Seidel, T. E., 429 Seiler, M. R., 475 Seitz, F., 41, 42, 266, 343, 344(1), 347(1), 349, 351(1), 352, 365(1), 368(1) Seltzer, M., 227 Semenchenko, V. K., 124 Semenkovich, S. A,, 157 Seregin, P. P., 251, 252(155a) Shafer. M.. 53. 54 Shalyt, S. S.. 438,454(50) Sharavskii. P. V., 26, 453 Sharifov, K. A,, 97, 99, 105, 136, 149, 157 Shaw, D., 196 Shcherbatov, V. E., 449 Shchukarev, S. A,, 16 Sheleg,A. U., 109, 110, 111, 131, 142 Shepherd, M. L., 330 Shewman, P. G., 182, 189(41), 207(41) Shibata, A,, 219 Shih, K. K., 216 Shilliday, T. S., 374(80, 81), 37.5, 391(80). 392(80), 393(80), 394(80), 401(80), 402(XO), 403(81)

Shishiyanu, F. S., 222, 233,234 Shockley, W., 173, 174(20), 179, 266, 272, 276, 314, 315, 317, 319 Shortes, S. R., 190, 203(63), 207, 217, 227 Shreder, I. F., 69, 73, 136 Sidarov, V. G., 449 Siedel, T., 177 Silsbee, R. H., 3.50, 353 Silvestri, V. J., 81, 82 Singkhal, S. P., 449, 450(68) Sinke, G. C., 41, 89(19), 96(19), 101. 104(19), 129 Sirota, N. N., 18,24, 25, 36, 38, 41(33, 39, 42), 43. 45, 51. 53, 54. 55(39, 86). 58, 60, 61. 65, 77, 78, 81, 82, 83, 91, 93, 96. 98(175). 104, 109, 110, 111, 113, 115, 117, 119, 120(230), 126, 130, 131, 142, 144, 149, 150, 154, 155(13), 157, 417 Skudnova. E. V., 27,424 Skuratov, S. M., 86, 149(160) Sladek, R. J., 335 Slavnova. G. K.. 422 Slifkin, 192, 371

488

AUTHOR INDEX

Slobodchikov, S. V., 60,440,441 Slykhouse, T. E., 328, 329(1), 330(1), 331 Smirnov, L. S., 366 Smirnova, A. D., 14 Smirous, K., 61, 64 Smith, C. S., 336 Smith, P. L., 138, 339,341 Smith, R. L., 6, 61, 62,64 Smits, F. M., 182, 185(45), 189(45), 190, 208 (45), 254(45) Smoluchowski, R., 349, 386(24c, 24d) Snow, A. J., 110 Snyder, W. S., 350, 351, 352, 369(31) Sokolov, V. I., 222,234,251,252,253,254,255 Sokolova, V. I., 24, 53, 60(82), 417, 420, 421(3c) Sollertinskaya, E. S., 226(127), 227 Soltys, T. J., 405 Sommerfeld, A., 106 Sonder, E., 392 Sondland, G. F., 6 Sosin, A,, 364, 365, 366(50) Sosnowski, L., 439 Spies, K., 41(32), 43, 45(32) Spiridonov, V. P., 124 Spitsyn, A. V., 366 Spitzer, W. G., 121, 132, 141(265), 234, 330, 401,432,445,446,447, 449(61),454 Spratt, J. P., 404 Sprokel, G. J., 187 Stackelberg, M. V., 41(32), 43,45 Stambaugh. E. P., 425 Stavitskaya, T. S., 427(30), 429 Steigmeier, E. F., 429 Stein, H. J., 346, 374(78), 375, 376(78), 377(78), 385 Stepanov, G. I., 432 Stepanov, N. I., 69 Stil’bans, L. S., 427(30), 429 Stocker, H. J., 192,226(72), 253 Stohr, H., 424 Stolnitz, D., 216, 239(113) Stolyarov, 0. G., 15 Stone, B., 12,49 Strack. H., 202,227 Stratton, R., 267, 268(3), 269, 270, 322, 325 Strauss, A. J., 202, 341, 463 Stromberg, L. F., 340(56), 341 Stull. D. R.,41,89(19),96(19),101, 104(19),129 Sturge. M. D., 186, 448, 449(66)

Subashiev, V. K., 447,448(65), 449,450(68) Suchet, J. P., 158 Suzuki, K., 314, 315(44), 317(44) Swenson, C . A., 340(56), 341 Syrkin, Ya. K., 11, 12, 17(21), 124 Sysoeva, L. M., 427(30), 429 Sze, S. M., 239

T Tachikawa, K., 314, 315(40) Takhtareva, N. K., 10, 11, 12(18), 14, 18(18), 19(18), 20,23(22), 25, 29(22,43), 31(22,43), 53 Talalakin, G. N., 449 Tammann. G., 57, 59, 125 Tanuma, S., 341 Tararin, V., 66 Tatevskii, V. M., 124 Tauc, J., 475 Taylor, H. L., 146 Taylor, J. H., 331, 335 Taylor, K. M., 50,64 Taylor, W. H.. 110 Templeton, L. C?,392 Terman, L. M., 295 Terpilowskii, J., 103, 104, 149 Teutonico, L. J., 346 Thibault, N. W., 8 Thiel, A., 53 Thiele, D., 424 Thoma, B., 54, 55, 56, 57,61,62,67,73 Thomas, W. J . O., 124, 125(246), 126 Thommen, K., 402 Thompson, M. W., 346,350, 353 Thomsen, D. O., 353 Thornsen, P. V., 352 Thurmond, C. D., 168, 216(13) Tietjen, J. J., 202, 441 Tittman, B. R., 340(54, 59), 341 Tokuyama, T., 314 Tolkachev, S. S., 28 Tolpygo, K. B., 121 Toman, L., 5,9(8), 12(8), 14(8), 19(8), 20(8) Tomizuka, C . T., 192 Torun, A,, 313 Tovpentsev, Yu. K., 26 Tretiakov, D. N., 11, 12, 14, 20, 23(22), 29(22, 43). 31(22, 43), 416(3b), 417, 418(3b)

AUTHOR INDEX

Treuting, R. G., 192 Truell, R.,346 Trzebiatowskii, W., 103, 104, 149 Tsarenkov. B. V.. 404 Tsing, Ping-hsi, 420 Turnbull. D.. 184. 185(47), 188, 214 Tweet, A. G.. 193

U Ukhanov, Yu.I., 439 Ulrich, W., 53, 54, 60(84) Urasov, G. G., 57, 58, 59,60,64 Ure, R. W., 437

V Vaipolin, A. A,, 416(3b), 417, 418(3b), 420,421 Valyashko, M. G., 41(29), 43, 44(29), 48(29), 49(29), 64(29), 75(29), 85(29) Van Artsdalen, E. R., 86, 149(159) Van Bueren, H. G., 254 Vand, V., 398 van den Boomgaard, J., 48, 53, 54, 56, 64, 73, 77,79,80,81, 82, 83, 85 van der Meulen, Y. J., 235 van Laar, J. J., 66,69(126), 73, 136 van Lint, V. A. J., 345 van Maaren, M. H., 235 van Overstraeten, R., 272, 273, 278. 293, 299 van Reijen, L. L., 110, 11l(215) van Roosbroeck, W., 320 Van Wazer, J. R., 76 Varley, J. H. O., 349, 386 Vasil’ev, V. P., 95, 96(180), 100, 149(186) Vavilov, V. S., 344,374(6), 375(6) Vedeneev, V. I., 41(22), 42 Vekilov, Yu. Kh., 15 Verdone, M.. 374(84a), 375,377(84a), 378(84a), 381(84a), 385(84a) Vernon, S. N., 341 Veszelka, J., 59, 60, 64 Vieland, L. J., 69, 74, 84(143), 177, 178, 198, 199,200,202(29), 204(85), 227 Vigdorovich, V. N., 5, 9(9), 11, 12, 22, 25, 51, 64, 76, 78, 85, 86(75), 91, 149 Vineyard, G. H., 344, 347(2), 349, 350(2), 351, 353 Vink, H. J., 164 Vitovskii, N. A., 374(84c), 375, 392(84c)

489

Vittorf, N. M., 69 Vodopyanov, L. K., 376, 385 Vogel, R., 69 Voitsekhovskii, A. V., 26-28, 421, 451, 452, 453 Vol, A. E., 61,66 von Klitzing, K. H., 335 Vook, F. L., 346, 352, 373, 378, 379, 380, 381, 383, 385(72), 386, 388, 394, 397, 399 Vosko, S., 349, 386(24c) W Wackerle, J., 339 Wagner, C., 69, 70, 94, 149, 165, 166(7, 8, 9), 170(7, 8, 9) Waite, T. R., 360, 365 Waller, I., 139, 144 Wang, C. C., 10,22,416 Wang, E. Y., 392 Warekois, E. P., 339 Warner, J., 455 Warschauer, D. M., 328, 329(3), 330 Wartenberg, H., 132, 141 Watkins, G. D., 364, 365 Watkins, T. B., 392 Watt, L. A. K., 251,252,255 Weber, R., 466 Wegener, H., 353 Wei, L. Y., 239 Weibke, F., 86, 125(163), 128 Weisberg, L. R., 170, 172(16a), 199, 200, 215, 219, 441 Weiser, K., 51, 52, 53, 54, 64, 77, 78, 79, 80, 84(72), 85,86,92,98, 149, 215, 228 Weiss, H., 158, 159, 165, 430, 432, 433, 434, 435,437,438,466,467,475 Welker, H., 60, 61,64, 125, 154, 158, 159, 165

Weller, W. W., 87,88(168), 149(168) Wentorf, R. H., 49 Wernick, J. N., 454 Wertheirn, G. K., 392 Weyland, H., 38, 146(6), 147(6) Whelan, J . M., 177, 229, 342, 395.401 White, H. G., 305, 306, 408 White, W. B., 339 White, W. E., 51 Whitsett, C. R.,377 Wieber, R. H., 196 Wiegand, D. A,, 349, 386(24d) Wiegmann, W., 288, 291(16), 292(16), 294(16),

490

AUTHOR INDEX

297(16), 301(16), 315(16) Wikner, E. G., 373 Willardson, R. K., 58, 60, 388(98a), 389, 392, 394(103) 402( 103. 108). 465 Williams, E. W., 178, 202(25a), 422 Williams, F. V., 49, 75, 79(57), 85, 149, 202 Williams, G. P. Jr., 192 Williams, R. M., 219 Williams, W. S., 87, 149 Wilson, A. J. C., 65 Wilson, R. B., 248,251 Witte, H., 109, 110 Wittig, J., 340(60), 341 Wolfel, E., 109, 110 Wolff, G. A,, 5, 9(8), 12, 14, 19, 20 Wolff, P. A., 272,278,293,309,310,312 Wolfstirn, K. B., 202, 203, 230, 234, 237 Woodall, J., 228 Woolley, J. C., 178, 202(25, 25a, 26, 27), 416, 422,424,426,427,429,455,456 Wu, M. H., 372,406 Wurst, E. C . Jr., 190, 203(63), 207(63), 217(63) Wyatt, M. E., 345 Wysocki, J. J., 404, 405

Y Yajima, T.. 408 Yakobson, S. U., 51, 64(74), 76(74), 77(74). 78(74), 85(74), 91(74), 149(74)

Yamashita, J., 325 Yanovich, V. D.: 144 Yearian, H. J., 380 Yeh, T. H., 198,203 Yep, T. O., 235 Yoseli, N., 233 Yoshida, M. J., 345 Yurkov, B. Ya., 370 Yushkevich, N. N., 93,96,104,149

2

Zakharova, N . Ya., 51,64(75), 76(75), 78(75), 85(75), 86(75), 91(75), 149(75) Zallen, R., 329, 331, 332(10, 13) Zaslavskii, A. I., 447 Zeigler, H., 405 Zeldes, P., 181 Zhdanov, G. S., 110, 11 l(212) Zhdanov, H. S., 50 Zhemchuzhni, S. F., 66 Zhigach, A. F., 41(29), 43, 44(29), 48(29), 49(29), 64(29), 75(29), 85(29) Zhukova, T. B., 447 Zhuze, V. P., 158 Ziman, J. M., 380 Zirman, J. V., 50 Zotova, N. V., 449

Subject Index A Acoustical phonons, see also Collisions, Scattering momentum change, 265 velocity randomizing, 265 Activity, 167, 176, 177, 179-181, 208, 21 1 coefficient, 179-181, 212 Allotropic transformations, 46 Alloys, see Mixed crystals, Solid solutions Aluminum, electron density distribution, 108, 109 Aluminum antimonide absorption edge, pressure dependence, 33 1 corrosion resistance, 60 covalent radii, 45 Debye temperature, 130, 131, 142 dynamic ionic displacements, 143 effective ionic charge, 120 electron density distribution, 114, 115, 118, 119 electronegativity, 126 entropy, 89, 90, 149 of atomization, 90, 149 of formation, 90, 129, 148 of fusion, 136 free energy of atomization, 90, 148 of formation, 90, 148 heat of atomization, 90, 123, 148 of formation, 89, 90, 123, 128, 141, 148 of fusion, 68, 136, 141 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19

irradiated material, 402 Knoop hardness, 20 lattice constant, 45, 58 melting point, 60, 64, 136 pressure dependence, 140 microhardness, 12, 23 molar volume, 127 phase transition parameters (pressure), 340 self-diffusion, 189, 190 system phase diagram, 57-59 volume change (melting), 140 x-ray density, 58, 127 Aluminum antimonide-gallium antimonide band structure, 432 conductivity, 432 effective mass, 432 energy gap, 432 Hall coefficient, 432 microhardness, 24 mobility, 432 thermoelectric power, 432 Aluminum antimonide-indium antimonide alloy scattering, 43 1, 432 conductivity, 430, 431 energy gap, 430 Hall coefficient, 431 microhardness, 23, 29, 31 mobility, 43 1 Nernst-Ettingshausen coefficient, 43 1 Aluminum arsenide covalent radii, 45 Debye temperature, 131 effective ionic charge, 120

491

492

SUBJECT INDEX

Aluminum arsenide (Cont.) electron density distribution, 112, 113, 116, 117 electronegativity, 126 heat of atomization, 89, 123, 148 of formation, 89, 123, 128, 148 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19 lattice constant, 45, 54 melting point, 55, 64 microhardness, 12 molar volume, 127 system phase diagram, 55 vapor pressure, 79 x-ray density, 127 Aluminum arsenide-indium arsenide, 416 microhardness, 24 Aluminum nitride energy gap, 50 entropy, 88, 149 of atomization, 88, 149 of formation, 148 free energy of atomization, 88, 148 of formation, 88, 148 heat capacity, 88 heat of atomization, 88, 148 of formation, 87, 88, 128, 148 ionic radii, 45 interatomic distances, 45, 127 lattice constant, 45, 50 melting point, 50, 64 molar volume, 127 vapor pressure, 75, 76, 85 x-ray density, 50 Aluminum phosphide effective ionic charge, 120 electronegativity, 126 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19 lattice constant, 45, 51 melting point, 51, 64 molar volume, 127 x-ray density, 51, 127 Amorphous regions, irradiated materials, 379, 380, 402 Amphoteric impurity, 173, 174, 177, 178 in GaAs, 177, 178, 234

Analog series, see also Chemical bonding Debye temperature, 18 melting point GaSb, 18 GaAs, 18 Gap, 18 InSb, 18 InAs, 18 InP, 18 reduced hardness, 18 Anisotropy, hardness Gap, 14 InSb, 14, I5 microhardness, 9-1 1 Annealing kinetics, 357-368 atomic rate constant, 363 electronic rate constant, 363 first order, 359, 360, 362-364, 366, 384, 387, 389-391, 398, 402 migration effects, 364 trapping effects, 364 second order 360, 364, 366, 383 divacancy formation, 364 trapping effects, 365, 367 Annealing, radiation damage, 345, 352, 355, 356 activation energy, 357-359, 364 determination of, 358, 359 Fermi level dependence, 364 migration, 366 trapping, 366 clustering, 365 complex formation, 365, 367 defect annihilation, 365 defect-impurity complex formation, 365 diffusion-controlled reactions, 360 frequency factor, 357 GaAs, 394-400 InSb, 381-388 electrical conductivity, 388 thermal conductivity, 388 InAs, 389-391 InP, 403 isochronal (pulsed), 356-360, 381386, 389 isothermal, 356-360 radiation induced, 345 recovery, InSb, 383

SUBJECT INDEX

reverse, 388 p-type InSb, 388 stages, 356 vacancy-interstitial pair annihilation, 360-362, 384 Antidefects, see Antistructure defects Anti-Schottky defects, 170, see also Interstitial defects Antistructure defects, 165, 170, 172, 355, see also Defects, Misplacements excited, 172 pair, 171 self-diffusion, 182 Atomic scattering factor, 107-1 11 Atomic spacing, see Interatomic distances Atomic volume, 38, 39, 42, 125-129 Atomization energy, 20, see also Heats of atomization, Free energy of atomization, specific listing of materials versus interatomic distance, I5 1 Atomization entropy, see Entropy of atomization Atomization heats, see Heats of atomization Auger process, 267, 323-325 Avalanche breakdown, 286-293 charge multiplication factor, 287 current buildup, 286-289 nonuniform fields, 289-293 p-n junctions, 286-289 Ge, 292 Si, 292 positive feedback, 286, 288

B Band blending, 207, see also Band tailing Band structure large defect concentrations, 181 pressure effect, 328, 329 strain, 336, 337 Band tailing, 181, 207, 211, 213, 407 Baraff curves, see Ionization rate (universal plots) Bergsman testing machine, 4, see also Microhardness Berthollide phases, 63-66 Bierbaum testing machine, 4, see also Microhardness

493

Bismuth-thallium system berthollide phases, 66 physical properties, 66 Boiling point, elements, 40, 42 Holtzmann equation, 268 solution, 272-274, 277-285 Boltzmann-Matano analysis, 207, 209, 212, 235, 241 Bonding, 106-132, see also Chemical bonding Boron antimonide, 49 Boron arsenide interatomic distances, 45, 127 ionic radii, 45 lattice constant, 45, 49 molar volume, 127 vapor pressure, 19 x-ray density, 49, 127 Boron nitride heat of formation, 86, 128, 148 interatomic distances, 45, 127 lattice constants, 45, 49 melting point, 64 molar volume, 127 vapor pressure, 75, 76, 85 x-ray density, 49, 127 Boron phosphide heat of formation, 148 interatomic distances, 45, 127 ionic radii, 45 lattice constant, 45, 49 microhardness, 12 molar volume, 127 vapor pressure, 75, 85 x-ray density, 49, 127 Breakdown, 269ff, see also Avalanche breakdown bulk semiconductors, 320-322 InSb, 320-322 collective breakdown field, 271 impact ionization, 264, 316, 323-325 junctions, tunneling, 299 low-electron densities, 272-286 microplasma, 294, 298, 313-319 thermal ionization, 323 Breakdown condition, 288, 289 Brine11 tests, 7, see also Hardness Brittleness elastic recovery, 10

494

SUBJECT INDEX

Brittleness (Cont.) hardness measurements, 4-7, 30 standards, 10, 11 zero standard, 11 Brouwer’s approximation, 175 C Cadmium diffusion GaAs, 194, 198, 225, 226 built-in electric field, 232 effectof dislocations, 226 uphill diffusion, 233 InSb, 195, 238, 251-253 copper interstitials, 251 InAs, 195, 226, 237 Cadmium sulfide predicted ionization rate, 307 threshold displacement energies, 371 Cadmium telluride energy gap, 18 hardness, 18 Cadmium telluride-indium antimonide microhardness, 26 transport phenomena, 453 Cadmium telluride-indium arsenide alloy scattering, 452 conductivity, 451, 452 effective mass, 453 energy gap, 451 Hall coefficient, 451, 452 Nernst-Ettingshausen coefficient, 451, 452 thermal conductivity, 453 Cadmium tin antimonide-indium antimonide, 4 18 microhardness, 26, 27, 31 Cadmium tin arsenide-indium arsenide alloy scattering, 454 conductivity, 453 effective mass, 454 energy gap, 454 Hall coefficient, 453 microhardness, 27, 29, 31 Carrier density critical, collisions, 268 pressure effect, 334 Carrier lifetime, determination, 298 Carrier removal rate, see also Damage rates

GaAs, 393 InSb, 377 InAs, 389 radiation damage, 369-373, 375 Carrier transfer effect, 336-3 38 pressure dependence, 333, 338 Cascade process, 352, 365, see also Displacement cascade channeling, 354 Cell, electrolytic, see Electrolytic cell Cellular growth, 462 Channeling, 350, 353, 354, 360 Characteristic temperature, see Debye temperature Charge multiplication factor, 287 measurement techniques, 293-300 light injection, 294-298 microplasma, 3 15 threshold field, 292 Charge multiplication phenomena, 263325, see aiso Pair production Charge neutrality condition, 173, 175 Chemical bonding, 36, 37, 106-132 analog series, 15-17 isoelectronic series, 17-19 energy gap, 18 hardness, 18 metallization, 16, 19 microhardness, 15-21 Closest packing, see Tightest packing Clusters, see also Defects, Complexes radiation induced, 346, 364, 365 Cobalt diffusion, InSb, 195, 238, 255 Collision density, 279 Collisions, 263, see also Scattering acoustical phonon, 264 critical carrier density, 268 electron-electron, 267 ionizing, 264 density, 280 momentum changing, 265 optical phonon, 264 mean free path, 266 scattering matrix, 266 pair production, 264 threshold energy, 266 polar modes, 322 velocity randomizing, 265, 267

495

SUBJECT INDEX

Collisions, elastic, 345, 349 radiation induced, 345 Collisions, inelastic, 352, see also Ionization (energy losses) radiation induced, 345, see also Ionization Common-ion effect, 173 Complexes, 169, 170, 178, see also Defect association, Clusters radiation induced, 346, 364, 378, 387 Compton effect, 349 Concentration gradients, uphill diffusion, 212, 223, 224, 232, 258 Conduction band structure, pressure effect, 328, 329, 336, 337 Conductivity hopping mechanism, 324 inhomogeneity, 464, 465, 472 irradiated material, 377-402 pressure dependence, 332-335 Zn-diffused layers (GaAs), 220, 221 Constitutional supercooling, 462 Coordination number, 36, 37, 41, 46, 147 change after melting, 147 Copper bromide energy gap, 18 hardness, 18 Copper diffusion AlSb, 194, 196 GaAs, 188, 198, 229-231 complex formation, 230 solubility, 230 uphill diffusion, 23 1 Ge, 185, 188 InSb, 195, 238, 253, 254 solubility, 253 InAs, 195, 237 Si, 188 Copper indium telluride-indium arsenide microhardness, 27, 28, 31 Copper iodide energy gap, 18 hardness, 18 Covalent bond, 36, 106-119 Covalent radii, 45, 46 Critical carrier density, electron-electron collisions, 268 Crowdion, 353

Crystal lattice influence, radiation damage, 353 Crystal structure, 41-47 Crystallochemical properties, 36-47

D Daltonide phases, 48, 62-66 Damage rates, see also Carrier removal rates radiation induced, 347-355, 369, 373 Damaged regions “electrical” size, 346 mobility measurements, 346 Debye temperature, 43, 130-132, 142, see also specific listing of materials Defect equilibria, 164-181 concentration, 176, 177 Defects, 165-1 8 1, see also Disorder antistructure, 165 association, 165 atomic, 165-178 clusters, radiation induced, 346, 364 divacancies, 168, 169 interstitial, 165, 360-362 intrinsic, 360 metastable, 169 microplasma inducing, 294 regions, radiation damage, 346 stoichiometric, 169 vacancy, 165, 167, 168, 360-362 Defect association, 165, see also Complexes antistructure pairs, 171 complexes, 169, 170, 178, 179 donor-vacancy association, 173, 178, 179 Frenkel disorder, 171, 172 impurities, 173 interstitial-divacancy, 204, 225 interstitial pairs, 170 vacancy interstitial, 171, 172 Defects, large concentrations, 179-181 band structure variation, 181 Debye-Huckel effects, 180 Fermi-Dirac degeneracy, 179, 180 ionization energy variation, 179, 180 Defects, rearranged, 169, 256 Defect structure, diffusion, 164

496

SUBJECT INDEX

Defect zinc-blende structure, 422 Degenerated eutectic, 67, 69 specific compounds, 52, 53, 55, 56, 60, 61 Density x-ray, tabulation, 127 Diamond, molar volume, 127 Dielectric permeability, versus energy gap, 156 Diffusion, 163ff accumulative process, 187 activation energy, 189 Boltzmann-Matano analysis, 207, 209, 212, 235, 241 coefficient, 189 dissociation, 185, 187 impurity, 164, 183 111-V compounds, 193-256 vacancy exchange, 183 interstitial, 182-185 interstitial-substitution, 185-188, 209, 214-216, 223-226, 247 isoconcentratioQ technique, 197, 207, 213 parallel mode, 184 pre-exponential factor, 189 self, 164, 182, 183 direct exchange, 182 impurity effects, 183 ring exchange, 182 111-Vcompounds, 189-193 sheet resistance techniques, 220-222, 228 substitutional, 182-185 uphill, 187, 225, 226, 231, 233 Diffusion activation energy, 189 Diffusion coefficient, 189 Dislocations decoration, 188 diffusion, 186-188 divacancy generation, 188, 204, 211 effect, p-n junction location, 226 Lomer-Cottrell reaction, 214 loops (InSb), 254 microplasma production, 3 15 radiation-induced loops, 365 vacancy generation, 186 varying density, Zn-diffusion in InSb, 244

Disorder, see also Defects antisymmetric Frenkel (vacancy-interstitial), 166, 171, 172, 182 interstitial-antistructure, 166, 172, 173 vacancy-antistructure, 166, 172 impure compounds, 173-18 1 amphoteric impurity, 173 symmetric, 165 antistructure, 165 interstitial, 165 Schottky (vacancy), 165-170, 174, 182 Disordered state, 38 Displacements crystal lattice effects, 353, 354 definition, 350, 351 focusing effect, 353, 354 ionization induced, 349 primary, 347, 350 radiation induced, 347 secondary, 349-353 threshold energy, 347, 350, 354, 355, 368, see also Threshold displacement energy diatomic solids, 355 Displacement cascade, 350, 352, 354, 355 channeling, 354 Displacement spike, 365, 379, 402 Dissociation pressure, 75-85, see also Vapor pressure Distribution function, 268, 324, 325 approximately symmetric, 272 displaced Maxwellian, 268, 269-272 highly anisotropic, 284, 285 spiked, 277 Divacancies, 168, 169, 174, 188, 225,256 generation, 188 Mn-diffusion (GaAs), 228 mobility, 188 radiation induced, 365, 399 self-diffusion, 182, 183 vibrational entropy, 169, 193 Donor-vacancy association, 173, 178, 179 Dynamic ionic displacements, 139-147, see also specific listing of materials

SUBJECT INDEX

E Effective electronic charge, 267 Effective ionic charge, 106, 120-122, 267, 372, see also Ionicity AISb, 120 AIAs, 120 AIP, 120 GaSb, 120 GaAs, 120 Gap, 120 InSb, 120 InAs, 120 InP, 120 Effective mass, pressure dependence, 332, 333 Effusion method, heat of formation, 89 Elastic constants, pressure dependence GaAs, 342 Ge, 342 Elastic properties, pressure effects, 341, 342 Electric field built-in Cd gradient, 232, 258 Zn gradient, 212, 223, 224, 258 critical, 266, 269 distortion, low-temperature breakdown, 323, 324 nonuniform, avalanching, 289-293 threshold (breakdown), 266, 292 threshold (charge multiplication), 292 Electrochemical potential, 473 Electroluminescence, see Luminescence spectra, Light emission, Injection lasers, Luminescent diodes Electrolytic cell, thermodynamic properties determination, 92, 93, 95, 100, 103, 104 Electromotive force method, see Electrolytic cell Electron density distribution, 36, 106-123 Al, 108, 109 AlSb, 114, 115, 118, 119 AIAs, 112, 113, 116, 117 GaSb, 114, 115, 118, 119 GaAs, 112, 113, 116, 117 Ge, 108, 109

497

InSb, 114, 115, 118, 119 InAs, 112, 113, 116, 117 NaCI, 108, 109 Electronegativity, 43, 123- 126 AISb, 126 AIAs, 126 AIP, 126 GaSb, 126 GaAs, 126 Gap, 126 InSb, 126 InAs, 126 InP, 126 Electron-electron collision, 267, see also Collisions momentum change, 267, 268 Electronic excitation, 352, 353, see also Collisions, inelastic Electron multiplication, see Charge multiplication phenomena, Pair production Electron orbital radius, 180 Electron-phonon collisions, see Collisions, Scattering Electron temperature, effective, 269 Electron transfer effect, see Carrier transfer effect Elongation, irradiated materials, 385 GaAs, 378, 379 Ge, 378, 379 InSb, 378, 379 Si, 378, 379 Energy bands, see Band structure Energy gap, see also Forbidden zone, specific listing of materials pressure effect, 328, 329 temperature derivative vs. heat capacity, 161 versus atomization energy, 155 versus enthalpy GaSb, 159 InSb, 160 versus interatomic distance, 151 versus temperature GaSb, 159 InSb, 160 Energy levels, radiation induced, 373-403, 405-409

498

SUBJECT INDEX

Energy levels (Conr.) determination, 374 effect of complexes, 375 effect of inhomogeneities, 375 effect of special annealing temperatures, 375 effect of trapping, 375 via occupational statistics, 374 via optical excitations, 374 GaSb, 374, 401, 402 GaAs, 374, 392, 409 InSb, 374, 377, 378 InAs, 390 InP, 374, 403 Energy-loss mechanisms, 264-268 optical phonons, 265 polar modes, 267 Entropy of atomization, 88-105, see also specific listing of materials Entropy of formation, 86-105, see also specific listing of materials Entropy of fusion, 42, 71, 136, 140 Entropy of solution, 74 Esaki diodes, see Tunnel diodes

F Facet effect, 461 Fermi-Dirac statistics, 179, 180, 240, see also Defects, large concentrations Fermi level, irradiated material AISb, 402 carrier removal rate dependence, 374 GaSb, 401 GaAs, 391 InSb, 376, 383 saturation value or final Fermi level, 375 Field, nonuniform electric effective width, 292 field distribution, 292 linear gradient junction, 291 step junction, 290 built-in bias, 290 space-charge width, 290 Figure test, 187 Focusing effect, 353, 354, see also Radiation damage, Channeling

Forbidden zone, 36, 155-160, see also Energy gap Free energy of atomization, 43, 90-105, 148 Free energy of formation, 71, 86-105, 148, see also specific listing of materials Frenkel defects, 346, 347, 360, see also Frenkel disorder, Vacancyinterstitial defects annealing, 361 lattice strain, 385 InSb, 379 Frenkel disorder, 166, 171, 172, 182, see also Frenkel defects paired defects, 171, 172 self-diffusion, 182, 183 Fusion heats. see Heats of fusion

G Gallium antimonide absorption edge, pressure dependence, 33 1 conductivity, pressure effect, 334 covalent radii, 45 Debye temperature, 18, 130, 131, 142 dynamic ionic displacements, 143, 146 effective ionic charge, 120 electron density distribution, 114, 115, 118, 119 electronegativity, 126 energy gap, 18 entropy, 149 of atomization, 97, 149 of formation, 96, 129, 148 of fusion, 136, 140 free energy of atomization, 97, 148 of formation, 96, 148 heat of atomization, 97, 123, 148 of formation, 71, 94-96, 123, 128, 148 of fusion, 68, 135, 136, 140, 149 impurity diffusion, 235, 257 divacancy process, 257 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19 irradiated material annealing, 401, 402

SUBJECT INDEX

carrier density, 401 dimensional changes, 401 Hall effect, 402 mobility, 401 resistivity, 402 Knoop hardness, 20 lattice constant, 45, 60 melting point, 18, 61, 64, 74, 136 pressure dependence, 138, 140 microhardness, 12, 23, 24, 32 molar volume, 127 phase transition parameters (pressure), 340 piezoresistance, 336, 337 reduced hardness, 18 self-diffusion, 189, 191, 235 system phase diagram, 61, 62 volume change (melting), 133, 140 x-ray density, 127 Gallium antimonide-indium antimonide effective mass, 426, 428 energy gap, 428 Hall coefficient, 426-428 microhardness, 22, 23, 29, 3 1 mobility, 426, 428 Nernst-Ettingshausen coefficient, 428 Gallium arsenide absorption edge, pressure dependence, 33 1 acceptor solubility enhancement, 178 conductivity, pressure effect, 334, 335 covalent radii, 45 Cu diffusion, 188 Debye temperature, 18, 130, 131, 142 dynamic ionic displacements, 143, 146 effective ionic charge, 120 elastic constants, pressure dependence, 342 electron density distribution, 112, 113, 116, 117 electronegativity, 126 electron mobility, pressure dependence, 333 emission edge, pressure effect, 33 1, 332 energy gap, 18 entropy, 149 of atomization, 94, 149 of formation, 93, 129, 148 of fusion, 136

499

free energy of atomization, 94, 148 of formation, 93, 148 heat of atomization, 94, 123, 148 of formation, 91-93, 123, 128, 148 of fusion, 68, 134, 136, 141, 148 hopping processes, 324 impurity impact ionization, 324 impurity-vacancy association, 178 injection lasers, 331, 405, 408 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19 ionization rates, 302-305, 307 irradiated specimens annealing studies, 394-400 carrier removal, 391, 393 conductivity, 391 elongation, 394 Hall effect, 392 magnetoresistance, 394 mobility, 392-394 photoconductivity, 392 photoelectronic techniques, 392 thermal conductivity, 394 Knoop hardness, 55 lattice constant, 45, 55 light emission (avalanche breakdown), 308-3 12 low-temperature breakdown, 324 luminescent diodes, 404-408 melting point, 18, 56, 64, 74, 136 pressure dependence, 138, 140 microhardness, 12 microplasma noise, 3 14 molar volume, 127 phase transition parameters (pressure), 340 reduced hardness, 18 self-diffusion, 189, 190, 235, 371 solar cells, 404 system phase diagram, 56, 57 threshold displacement energies, 371 trapping centers, pressure effect, 335 tunnel diodes, 408, 409 vapor pressure, 79-82, 85 volume change (melting), 140 x-ray density, 55, 127 zinc diffusion, 185

500

SUBJECT INDEX

Gallium arsenide-gallium phosphide alloy scattering, 442 band minima, 443-450 direct gap, 446-448 energy gap, 445 Hall coefficient, 443, 444 indirect gap, 446-448 mobility, 441-444 optical absorption, 444-448 photo emf, 445, 446 reflectivity spectra, 449, 450 refractive index, 447 resistivity, pressure dependence, 443 spin-orbit splitting, 447 thermal conductivity, 430 Gallium arsenide-gallium selenide, 457 Gallium arsenide-gallium sulfide, 422 microhardness, 28. 29 Gallium arsenide-indium arsenide effective mass, 429 Hall coefficient, 429 mobility, 429 thermal conductivity, 429, 430 Gallium arsenide-indium phosphide, 416 microhardness, 25 Gallium nitride entropy of formation, 148 heat of atomization, 90 of formation, 90, 128, 148 interatomic distances, 45, 127 ionic radii, 45 lattice constants, 45, 50 melting point, 50, 64 molar volume, 127 x-ray density, 50, 127 Gallium phosphide absorption edge, pressure dependence, 33 1 Dehye temperature, 131, 142 dynamic ionic displacements, 143 effective ionic charge, 120 electronegativity, 126 emission edge, pressure effect, 332 entropy of atomization, 91, 149 of formation, 91, 148 of fusion, 136 free energy of atomization, 91, 148 of formation, 148

heat of atomization, 91, 148 of formation, 91, 128, 141, 148 of fusion, 68, 136, 148 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19 ionization rates, 305, 306 Knoop hardness, 20 lattice constant, 45, 51 light emission (avalanche breakdown), 308, 312 melting point, 18, 51, 53, 64, 74, 136 microhardness, 12, 24 microplasma noise, 3 14 molar volume, 127 reduced hardness, 18 system eutectic, 50 system phase diagram, 52, 53 vapor pressure, 76-78, 85 x-ray density, 127 Gallium phosphide-gallium selenide, 432 microhardness, 29 Gallium phosphide-gallium sulfide, 422 microhardness, 28 Gallium phosphide-indium phosphide, 416 microhardness, 24 Gamma ray irradiation, 346, 347, 360 Compton effect, 349 pair production, 349 photoelectric effect, 349 Germanium amphoteric impurity (in GaAs), 177 Cu diffusion, 185, 188

Debye temperature, 131, 142 dynamic ionic displacements, 143, 146 elastic constants, pressure dependence, 342 electron and hole ionization rates, 292, 293 threshold, 300-302 electron density distribution, 108, 109 energy gap, 18 entropy, 129 of fusion, 141 hardness, 18 heat of fusion, 141 impurity impact ionization, 323, 324 junctions, charge multiplication, 293

SUBJECT INDEX

light emission (avalanche breakdown), 308-3 12 low-temperature breakdown, 323, 324 melting point, pressure dependence, 138, 140 microplasma noise, 3 14 molar volume, 127 piezoresistance, 336 predicted ionization rate, 307 self-diffusion energy, 37 1 threshold displacement energies, 37 1 volume change (melting), 133, 140 Germanium-silicon alloys microhardness, 10, 22, 29 Gold diffusion GaAs, 194, 198, 234 InSb, 195, 238, 255 Gray tin Debye temperature, 131, 142 dynamic ionic displacements, 143, 146 phase transition, pressure effect, 339, 341 Gunn effect, 338 threshold field, pressure dependence, 338

H Hall coefficient irradiated material GaSb, 402 GaAs, 392 InSb, 377 InP, 403 Magnetic field dependence, inhomogeneous specimens, 467, 468 pressure dependence, 335, 337, 338 strong field, statistical inhomogeneities, 473 symmetry relations, inhomogeneous specimens, 467 weak field, statistical inhomogeneities, 472 Hall constant, see Hall coefficient Hardness, 3-34, see also Microhardness AlSb, 12 AlAs, 12 anisotropy, 9-11, 14 Brine11 tests, 4-7 chemical bonding effect, 15-2 1

501

correlations, 16-22 definition, 3-7 GaSb, 12 GaAs, 12 Gap, 12 InAs, 12 Kurnakov rules, 10, 21. 22, 29, 30 Mohs scale, 11-13 scratch test, 4 InSb, 14, IS surface hardening, 8, 9 surface preparation, 7-9 tests, 3-7 Heat capacity, 43, see also specific listing of materials Heats of atomization, 36, 43, 152, see also Atomization energy, specific listing of materials elements, 40 versus atomic number. 153 Heats of formation, 36, 86-105, 140, see also specific listing of materials effusion method, 89 from phase diagram data, 69-71 versus atomic number, 150, 153 versus energy gap, 156 versus heat of atomization, 154 versus interatomic distance, 151 Heats of fusion, 36, 42, 133-136 AISb, 68, 136 GaSb, 68, 135, 136 GaAs, 68, 134, 136 Gap, 68 InSb, 68, 134-136 InAs, 68, 134, 136 InP, 68, 134, 136 Heats of mixing, 70 Helicon waves, 469-47 1 damping, 469, 471 dispersion relation, 469, 470 inhomogeneity effect, 470, 471 Heterovalent substitution, 417-424 mutual compensation, 458 Hot-electron effects, 264 Hot electrons, see ulso Distribution function, Charge multiplication, Carrier transfer effect Gunn effect, 338 Hydrostatic pressure, 327-335, 338-342

502

SUBJECT INDEX

I Impact ionization, 264 impurity levels, 264, 316, 323-325 Impurity diffusion, 183, 256-258 AlSb, 194, 196 concentration gradient, 208 GaSb, 194,235 GaAs, 194, 197-235 Gap, 194, 196, 197 InSb, 195, 237-256 InAs, 195, 236 vacancy exchange, 183 Impurity effects electron irradiation, 387 self-diffusion, 183 Impurity gradients, 460 Impurity incorporation, 174 Impurity levels impact ionization, 264 pressure effect, 332 Indentation test dynamic, 3-7 static, 3-7 Indenter Knoop, 6, 7, 11, 12 Vickers, 6, 7 Indium antimonide absorption edge, pressure dependence, 33 1 acceptor solubility enhancement, 178 bulk breakdown, 320-322 complex formation, 378 conductivity, pressure effect, 334, 335 covalent radii, 45 Cu solubility, 253 Debye temperature, 8, 130, 131, 142 dislocation loops, 254 divacancies, 254, 257 dynamic ionic displacements, 143, 146 effective ionic charge, 120 electron density distribution, 114, 115, 118, 119 electronegativity, 126 electron mobility irradiated specimens, 376 pressure dependence, 333 energy gap, 18 enthalpy of formation, 71, 254

entropy, 149 of atomization, 105, 149 of formation, 103-105, 129, 148 of fusion, 136, 141 equilibrium vacancy concentration, 254 free energy of atomization, 105, 148 of formation, 102-105. 148 gamma irradiation, 377 Hall effect (breakdown region), 32 1 heat of atomization, 105, 123, 148 of formation, 102-105, 123, 128, 148 of fusion, 68, 134-136, 141, 148 impurity diffusion, 237-257 activation energies, 256 composition independence, 257 divacancy process, 257 interstitial-substitution~lprocess, 256 impurity-vacancy association, 178 inhomogeneities, 461, 464, 468 instabilities, 321, 322 interatomic distances, 45, 127 interstitial Cu solubility, 253 ionic radii, 45 ionicity, 19 irradiated materials annealing experiments, 38 1-388 elongation, 378-380 Hall effect, 377 lifetime measurements, 378 magnetoresistance, 376 photoconductivity, 377 photoelectromagnetic effect, 377 photoelectronic measurements, 377 thermal conductivity, 380, 38 1 x-ray damage, 386, 387 Knoop hardness, 20 lattice constant, 45, 61 melting point, 18, 62, 64, 74, 136 pressure dependence, 138, 140 microhardness, 12-15, 17, 23, 26, 28 molar volume, 127 phase transition parameters (pressure), 340, 341 piezoresistance, 336 pinch effect, 321, 322 predicted ionization rate, 307 reduced hardness, 18 reverse annealing, 388

SUBJEC'T INDEX

self-diffusion, 189, 190, 238, 371 substitutional solubility, 253 system phase diagram, 61-63 threshold displacement energies, 371 vapor pressure, 83-85 volume change (melting), 133, 140 x-ray density, 127 Indium antimonide-indium arsenide, 4 17 microhardness, 25 Indium antimonide-indium telluride defect scattering, 456 mobility, 455 thermal conductivity, 455, 456 Indium antimonide-silver indium telluride microhardness, 27, 28, 31 Indium antimonide-zinc tin antimonide, 418 microhardness, 27 Indium arsenide absorption edge, pressure dependence, 331 acceptor solubility enhancement, 178 conductivity, pressure effect, 335 covalent radii, 45 Debye temperature, 18, 130, 131, 142 dynamic ionic displacement, 143 effective ionic charge, 120 electron density distribution, 112, 113, 116, 117 electronegativity, 126 electron mobility, pressure dependence, 333 entropy, 149 of atomization, 102, 149 of formation, 101, 129, 148 of fusion, 136 free energy of atomization, 102, 148 of formation, 101, 148 heat of atomization, 102, 123, 148 of formation, 99-101, 123, 128, 148 of fusion, 68, 134, 136, 141, 149 impurity-vacancy association, 178 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19 irradiated specimens annealing studies, 389 carrier removal rate, 389

503

Knoop hardness, 20 lattice constant, 45, 56 melting point, 18, 56. 57, 64, 74, 136 pressure dependence, 138, 140 microhardness, 11, 12, 25, 27, 28, 32 molar volume, 127 phase transition parameters (pressure), 340 predicted ionization rate, 307 reduced hardness, 18 system phase diagram, 56, 58 threshold displacement energies, 37 1 vapor pressure, 81-83, 85 volume change (melting), 140 x-ray density, 127 Indium arsenide-indium phosphide absorption edge, 439, 440 alloy scattering, 435, 437 conductivity, 432 effective mass, 438, 440 energy gap, 434, 440 Hall coefficient, 432-436 microhardness, 25 mobility, 434, 435 Nernst-Ettingshausen coefficient, 439 photoconductivity, 440, 441 polar scattering, 435 thermal conductivity, 430, 437-439 thermoelectric power, 437, 438 Indium arsenide-indium selenide, 422, 457 microhardness, 27 Indium arsenide-indium telluride, 422, 456 microhardness, 27 Indium arsenide-mercury telluride microhardness, 26 transport phenomena, 453 Indium arsenide-zinc germanium arsenide, 418 microhardness, 27, 29 Indium diffusion GaSb, 194, 235 GaAs, 198, 232 Indium nitride entropy of formation, 148 heat of atomization, 97 of formation, 97, 128, 148 interatomic distances, 45, 127

504

SUBJECT INDEX

Indium nitride (Cont.) ionic radii, 45 lattice constants, 45, 50 melting point, 64 molar volume, 127 x-ray density, 50, 127 Indium phosphide absorption edge, pressure dependence, 330, 331 Debye temperature, 130, 131, 142 dynamic ionic displacement, 143 effective ionic charge, 120 electronegativity, 126 electron mobility, pressure dependence, 333 entropy, 149 of atomization, 99, 149 of formation, 98, 129, 148 of fusion, 136 free energy of atomization, 99, 148 of formation, 98, 148 heat of atomization, 98, 99, 148 of formation, 97, 98, 128, 148, 150 of fusion, 68, 134, 136, 141, 149 interatomic distances, 45, 127 ionic radii, 45 ionicity, 19 irradiated specimens, 403 Knoop hardness, 20 lattice constant, 45, 53 light emission (avalanche breakdown), 308 melting point, 18, 53, 64, 74, 136 pressure dependence, 138, 140 microhardness, 12, 25 molar volume, 127 phase transition parameters (pressure), 340 predicted ionization rate, 307 reduced hardness, 18 resistivity, pressure effect, 335 self-diffusion, 189, 191, 371 system eutectic, 53 system phase diagram, 53, 54 threshold displacement energies, 37 1 vapor pressure, 77-80, 85 volume change (melting), I40 x-ray density, 53, 127

Indium phosphide-indium selenide, 422 microhardness, 28, 29 Inhomogeneity, see also Nonuniform crystals conductivity, 464, 465 detection, 425, 426 anisotropy, 475 photovoltaic effects, 475 potential probing, 474 growth-induced cellular growth, 462 constitutional supercooling, 462 cores, 461 facet effect, 461 impurity gradients, 460 striations, 462, 463 Hall mobility, 465, 468 hardness measurements, 10, 29-3 1 helicon damping, 469-47 1 macroscopic, 473, 474 magnetoresistance anisotropy, 466, 467 magnetic-field dependence, 465, 466 metallic inclusions, 401 microscopic, 471, 472 radiation damage, 352 Seebeck effect, 469 statistical, 462, 463, 472, 473 thermal conductivity, 469 Injection electron (at cathode), 288 electron-hole (via light), multiplication measurements, 294-298 Injection lasers, 331, 332, 405 mode jumping, 332 pressure effect, 331, 332 Interatomic bonding, 106-132, see also Chemical bonding Interatomic distances, 45, 127 Interionic distances, 41, 45 tabulation, 45, 127 versus atomic number, 152 Internal friction techniques, complex defect studies, 170 Interstitial defects, 165, 170, 373, see also Defects anti-Schottky defects, 170 Cu in InSb, 251 solubility, 253

505

SUBJECT INDEX

diffusion, 182-186 interstitial-antistructure, 172 ionization state, 171 pairs, self-diffusion, 182, 183 radiation-induced, 345 Interstitial-substitutional diffusion, 185188, 209, 214-216, 223-226 Mn diffusion (GaAs), 228 retrograde diffusion, 23 1 Zn diffusion (InSb), 247 Ionic bond, 36, 106-125, see also Ionicity Ionic displacements, see Dynamic ionic displacements Ionic radii, 41, 45, 46 Ionic volumes, 39 Ionicity, 17, 379, see also Effective ionic charge 'effect on hardness, 17, 18, 19 Ionization collisions, 264 energy losses, 352, 353 impact, 264 mean free path, 267, 272 pair production, 264 probability, 276, 279 radiation-induced, 345, 352 strong field, 264 threshold, 273, 298-300 Ionization potentials, elements, 40, 42 Ionization rate, 268, 269, 272-286 determination (microplasma breakdown), 298 exponential factor, 275 field dependence, 268, 269 GaAs, 302-305 Gap, 305, 306 high fields, 272-275

low fields, 275-277 predicted (various materials), 307 Si, 300-302 unequal, 288, 289, 292 universal plots, 281-283, 301-307 Ionization state interstitial, 171 vacancy, 167 Zn in GaAs, 205-207 Ionization threshold determination, 298300 end corrections, 299

Iron diffusion InSb, 195, 238, 255 Isoelectronic series, 46, see also Chemical bonding hardness CdTe, 18 CuBr, 18 CuI, 18 GaSb, 18 GaAs, 18 Ge, 18 InSb, 18 AgI, 18 ZnSe, 18 ZnTe, 18 Isovalent substitution, 4 15-417

J Junctions, 289-300 breakdown (tunneling), 299 dislocations, 3 I5 effective width, 299 low-field ionization rate, 293 photoresponse (reverse bias), 297, 298 reverse bias, threshold (multiplication), 300 reverse current, 293 uniform, 315

K Knock-on atom, 347-351, see also Radiation damage Knoop hardness, 19, 20, see also Microhardness AlSb, 20 GaSb, 20 GaAs, 20 Gap, 20

InSb, 20 InAs, 20 InP, 20 Kurnakov classification, 48, 62-66 Kurnakov rules, 10, 21, 22, 29, 30 1

Lattice constants, 41-45, 49-60 Lattice dynamics, 37, 139-147 Light emission, 307-313, see also Luminescent diodes, Luminescence spectra

506

SUBJECT INDEX

Light emission (Cont.) avalanche breakdown, 308 bremsstrahlung, 308, 3 13 efficiency, 3 13 radiative transitions, 307, 308 impurity levels, 309 interband, 309 intraband, 309-3 11 spectra, 308, 3 11-3 13 Liquidus curves, 48, 67, 68, 72-74, see also Phase diagrams, specific listing of materials Lithium diffusion GaSb, 235 complex, 235 double-acceptor nature, 235 GaAs, 194, 198, 234, 258 amphoteric impurity, 234 Li-Zn complexes, 234 Localized damage, 345, 346, see also Radiation damage Lomer-Cottrell reaction, 214 Luminescence spectra, pressure effect, 331 Luminescent diodes, radiation damage, 404-408

M Magnesium diffusion GaAs, 194, 198, 227 InAs, 195, 226 Magnetoconductivity tensor, 474 Magnetoresistance anisotropy, inhomogeneous specimens, 466, 467 magnetic field dependence, inhomogeneous specimens, 465, 466 strong field, inhomogeneous specimens, 473 Manganese diffusion amphoteric behavior, 228 divacancy model, 228 GaAs, 194, 198, 227, 228, 258 interstitial-substitutional model, 228 neutral interstitial, 228 sheet-resistance technique, 228 Mass action relation, 166, 167, 173, 179 Mean free path ionization, 267 optical phonon, 266

Melting, 132-147 heats of fusion, 133-136 pressure dependence, 138, 140 volume change, 42, 132, 133 Melting point, 36, 40, 42, 54, 68, see also Melting temperature, specific listing of materials Melting temperature, 36, 40, 42, 54, 68, see also specific listing of materials elements, 40, 42 versus atomic number, 153 versus energy gap, 156 versus interatomic distance, 151 Mercury diffusion GaAs, 198, 227 InSb, 195, 238, 253 Metallic bond, 106-119 Metallization, see Chemical bonding Metastable defects, 169, 172 Microhardness, 4, 9-34, see also Hardness, specific listing of materials AISb, 12 AMs, 12 anisotropy, 9-1 1 Gap, 14 Ge, 9 InSb, 13-15 Si, 9 Bergsman testing machine, 4 Bierbaum testing machine, 4 BP, 12 brittleness, 10, 11, 30 crystal growth conditions, 12 crystal strain, 10 definition, 4-7 GaSb, 12 GaAs, 12 Gap, 12 InSb, 12-15, 17 InAs, 11, 12 InP, 12 inhomogeneity, 10, 29-3 1 Knoop indenter, 6, 7, 11, 12, 19, 20 PMT-3 testing machine, 4-6 reduced, 17 substructure effect, 10 tests, 3-7 Vickers indenter, 6, 7

507

SUBJECT INDEX

Microplasmas, 294, 298, 3 13-3 19 breakdown, 298 defect inducing, 294 dislocation effect, 3 15 extinction, 3 15 impurity fluctuation, 315, 317-319 initiation, 3 15 noise, 313, 314 optical excitation, 3 17 thermal excitation, 3 17 traps, 315, 316 Misplacements, 355, 360-362, 388, see also Antistructure defects Mixed crystals, 413, see also Solid SO~Utions hardness, 21-32 heterovalent substitution defect tetrahedral structure, 455457 normal tetrahedral structure, 451454 111-V-IV compounds, 417 111-V-11-VI compounds, 418 111-V-11-IV-V, compounds, 4 18 111-V-I-IV,-V, compounds, 420 111-V-I-111-V, compounds, 420 111-V-I,-IV-VI, compounds, 42 1 111-V-I,-V-VI, compounds, 42 1 111-V-111,-IV-VI compounds, 422 111-V-111,-VI, compounds, 422 isovalent substitution, 111-V compounds, 416, 417 properties anionic substitution, 432-450 cationic substitution, 426-432 Mobility damaged regions, 346 field dependence, 264, 270-272 inhomogeneities, 465, 468 irradiated materials GaSb, 401 GaAs, 392-394 InSb, 376 InP, 403 pressure dependence, 333 carrier transfer effect, 333 interband scattering, 334 Mohs scale, 11-13 Molar volumes tabulation, 127

Monomorphic transformations, 46 Multivalley semiconductors shear piezoresistance, 336, 337

N Neutron irradiation, see also Radiation damage inhomogeneous damage, 346 Rutherford scattering, 348, 349 scattering cross section, 348 ultrasonic attenuation, 346 Noise, microplasma, 3 13, 314 Nonstoichiometry, 48, 164, 166, 256 solid solutions, 414, 422 Nonuniform crystals, 459-476, see also Inhomogeneity 0 Octahedral sites, 46, 47 Optical absorption intrinsic edge, pressure effect, 329-33 1 irradiated InSb, 377 Optical phonons, see also Collisions, Scattering energy loss, 265 velocity randomizing, 265

P Pairing, see Defects, association Pair production, see also Charge multiplication phenomena avalanche breakdown, 286-293 Auger process, 267 energetic particles, 3 19, 320 gamma irradiation, 349 mean energy, 3 19 threshold energy, 266 Phase diagram, 37, 47-74, see also specific listing of materials A1Sb-InSb system, 418 2GaP-4Si-ZnSiP2 system, 421 21nAs-CdGeAs2 system, 420 InAs-In, Se, system, 423 InAs-In, Te, system, 423 21nAs-ZnSnAs2system, 419 Zn in GaAs, 216 Phase transitions, pressure, 338-341 Phonon distribution function, 265 Photoelectric effect, 349

508

SUBJECT INDEX

Physicochemical properties, 3, 36 hardness, 3-34 Piezoresistance, 336, 337 electron transfer effect, 336-338 multivalley semiconductors, 336 p-type Ge, 336 p-type InSb, 336 Pinch effect, InSb, 321, 322 PMT-3 testing machine, 4-6, see ako Microhardness Polymorphic transformations, 46 Precipitation, 214, 364 Zn in GaAs, 214 Pre-exponential factor, 189, 193, 257 vibrational entropy effect, 193 Pressure effects, see also Hydrostatic pressure, Piezoresistance band structure, 328, 329 conductivity, 332-335 effective mass, 332 elastic properties, 341, 342 electroluminescence, 33 1, 332 Hall effect, 335, 337, 338 optical absorption, 329-33 1 phase transitions, 338-341 Primary knock-on, 347, 387, 388, see also Radiation damage

R Radiation annealing, 345 Radiation damage, 343-409, see also Displacements, specific listing of materials anisotropy, 384 channeling, 350 annealing, 345, 355-368, see atso Annealing, radiation damage cascade process, 352 crystal lattice influence, 353, 354 damaged regions, “electrical” size, 346 devices, 403-409 luminescent diodes, 404-408 solar cells, 404 tunnel diodes, 408, 409 dislocation loops, 365 displacement threshold, 347, 350, 354, 355 electron bombardment, 344, 369-402 focusing, 353, 354

impurity effects, 387 incipient damage, 369 investigation techniques mobility measurement, 346 sputtering experiments, 346 ultrasonic attenuation, 346 localized, 345, 346 neutrons, fast, 343, 344, 385, 388-394, 396, 399-403 orientation dependence, 384 primary knock-on, 347, 387, 388 rates, 347-355 recovery, 345, 355, 356, see also Annealing, radiation damage InSb, 383 replacement collisions, 351 replacement threshold, 351 subthreshold, 369, 386 Radiation effects, 343-409, see also Radiation damage Recombination nonradiative, 405, 407 radiative, 307-3 13, 405 Recovery, see Annealing, radiation damage Reduced microhardness, 17, 20, 21, see also Microhardness GaSb, 18 GaAs, 18 Gap, 18 InSb, 18 InAs, 18 InP, 18 Replacement collisions, 351, 354, 355, see also Radiation damage replacement-vacancy ratio, 35 1, 352 threshold, 351 Resistivity inhomogeneities, 472 pressure dependence, 333-335 Retrograde diffusion, 23 1 interstitial-substitutional model, 23 1 Reverse bias p-n junction, charge multiplication, 295-298 threshold, 300 Runaway critical field, 269 critical velocity, 322

SUBJECT INDEX

Rutherford cross section, 349

S Scattering, see also Collisions acoustical phonons, 264 alloy, 414 phonons, 458 interband, pressure effect, 334 optical phonons, 264 Scattering cross section, neutron irradiation, 348 Schottky-antistructure disorder, 172 Schottky disorder, 166-170, 174, 182, see also Disorder divacancies, 174 Schottky-antistructure disorder, 172 self-diffusion, 182, 183 Scleroscopic tests, 3, see Indentation test (dynamic) Scratch test, 4, see also Hardness InSb, 14, 15 Seebeck coefficient, see also Thermoelectric power inhomogeneous specimens, 469, 473 Selenium diffusion GaAs, 194, 198, 202, 203 InAs, 195, 236 polyatomic molecules, 203 Self-diffusion, 164, 182, 183, see also Diffusion activation energy, 371, 372 AlSb, 189, 190 antistructure defects, 182 direct exchange, 182 Frenkel disorder, 182 GaSb, 189, 191, 235 GaAs, 189, 190, 198, 235 impurity effects, 183 InSb, 189, 193, 238 InP, 189, 191 ring exchange, 182 Schottky disorder, 182 stoichiometric effects, 182 Sheet-resistance techniques, diffusion, 220-222, 228 Short-range order, 37 Silicon amphoteric impurity (GaAs), 177 Cu diffusion, 188

509

Debye temperature, 131, 142 dynamic ionic displacement, 143, 146 eiectron and hole ionization rates, 292, 293 entropy of fusion, 141 heat of fusion, 141 impurity impact ionization, 323, 324 light emission (avalanche breakdown), 308, 311 low-temperature breakdown, 323-324 melting point, pressure dependence, 138, 140 microplasma noise, 3 14, 3 15 molar volume, 127 pair production particle energy, 3 19 predicted ionization rate, 307 self-diffusion, 371 solar cells, 404 threshold displacement energy, 37 1 volume change (melting), 140 Silicon carbide light emission (avalanche breakdown), 308 predicted ionization rate, 307 Silicon germanium alloys, see Germanium-silicon alloys Silver diffusion GaAs, 194, 198, 233 InSb, 238 Silver iodide energy gap, 18 hardness, 18 Skin depth, helicon waves, 470 critical field, 470 Slip plane Ge, 14 InSb, 14 Sodium chloride electron density distribution, 108, 109 Solid solutions, 413-458, see also Mixed crystals alloy scattering, 414 anion replacement, hardness, 25 cation replacement, hardness, 22-25 characterization, 424-426 defect tetrahedral phases, 422-424 hardness, 21-32 Kurnakov rules, 10, 21, 22 heterovalent substitution, 4 17-424

510

SUBJECT INDEX

Solid solutions (Cont.) isovalent substitution, 416, 417 nonequilibrium effects, 414 normal tetrahedral phases, 416-422 preparation, 424-426 Solidus curves, 48, see also Phase diagrams, specific listing of materials Solute buildup, 425 Sphalerite structure, 37, 38, 46-48 Specific heat, see Heat capacity Specific volume, 38, 125-129 Spin resonance, diffusion studies, 164 Statistical inhomogeneities, 464, 468, see also Scattering (alloy) Stoichiometric defects, 160, 182, see also Nonstoichiometry solid solutions, 414 Strain, see Pressure effects, Piezoresistance Striations, 463, 464 Structure, 36-47 Substitution, heterovalent, see Heterovalent substitution Substitution, isovalent, see Isovalent substitution Substitutional solid solutions, 413, see also Solid solutions, Mixed crystals Sulfur diffusion GaAs, 194, 197-202 InAs, 195, 236 Surface hardening, 8, 9 Surface preparation, hardness test, 7-9 Symmetry relations, inhomogeneous specimens Hall effect, 467 magnetoresistance, 466, 467

T Tellurium diffusion GaSb, 195, 235 GaAs, 198, 203 InSb, 195, 237, 238 I d s , 195, 236 polyatomic molecules, 203 Temperature of fusion, 35ff, see also Melting temperatures Tetrahedral phases defect, 415, 422-424 normal, 415-422

Tetrahedral sites, 46, 47 Thermal conductivity inhomogeneous specimens, 469 irradiated material GaAs, 380, 394 InSb, 380, 394 Thermal spike, 365 Thermally stimulated currents, complex defect studies, 170 Thermodynamic properties, 36, 40, 42, 43, see also listings of specific properties, specific listing of materials from phase diagram data, 69-74 tabulation, 42, 43, 148, 149 Thermoelectric power, see also Seebeck coefficient pressure effect, 337, 338 Threshold displacement energy, 347, 350, 354, 355, 368-373, 394 experimental determination, 368-372 cathodoluminescence measurements, 372 subthreshold tail, 369, 386 Thulium diffusion, GaAs, 194, 198, 231 retrograde diffusion, 23 1 Tightest packing, 41, 45, 46 Tin, amphoteric impurity (GaAs), 177 Tin diffusion GaSb, 195, 235 GaAs, 194, 198, 203-205 InSb, 195, 237-239 InAs, 195, 236 Townsend coefficient, see Ionization rate Trapping centers binding energy, pressure effect, 335 during annealing, 364-367, 398 from irradiation, 344 ionization effects, 345 Tunnel diodes radiation effects, 408, 409 Zn-doped, degradation, 219 Cu interstitial contamination, 219 electron injection dissociation, 2 19 Zn interstitial diffusion, 219 Zn vacancy association, 219 Tunneling diffusion measurements, 202 junction breakdown, 299 Two-stream instabilities, 322

SUBJECT INDEX

U Ultrasonic attenuation, neutron-irradiated Si, 346 Uphill diffusion, 187, 225, 226, 258 accumulative process, 187 complex formation, 233 interstitial-substitutional model, 226, 231

V Vacancies, 165, see also Defects, Schottky disorder, Disorder, Divacancies concentration variation, 168 ionization state, 167 neutral, 167, 168, 174 radiation induced, 345, 360 vacancy-exchange diffusion, 183-185 Vacancy-interstitial defects, 171, 172, 346, 347, 360-362, 387, 391, see also Frenkel defects correlation, 360, 376, 378, 386, 387, 399 Valence band, 328 pressure effect, 328, 329, 336 Vapor pressure, 75-85, see also specific listing of materials Velocity distribution, see also Distribution function spiked, 276, 277 Velocity randomizing, see Collisions Volume change, fusion, 42, 132-140

W Wurtzite structure, 37, 38, 46, 47

X X-ray density, 127, see also specific listing of materials

Z Zinc diffusion, 185 AlSb, 194, 196

511

band tailing, 258 GaAs, 185, 194, 198, 205-225 alloy source, 217, 218 cooling cycle, 225 electromigration, 222 interstitial-substitutional, 209, 214-216, 223-225 isoconcentration condition, 210 p-n junction, 217, 218 profile conditions, 210 sheet-resistance measurements, 220-222, 228 state of ionization, 205-207 ternary phase diagram, 217 two-step experiments, 222 ZnAs, source, 215, 216 Gap, 196, 197 InSb, 195, 238-250 interstitial-substitutional model, 247 p-n junction depth, 248 rate-limitation process, 248 sheet-resistance technique, 249, 250 time dependence, 244, 248 two-step experiment, 245, 246 InAs, 195, 236, 237 strong concentration dependence, 205-225, 239-250, 257 Zinc incorporation, 174 alloy source, 217, 218 precipitation, GaAs, 214 surface concentration, GaAs, 206, 217 ZnAs, source, 215, 216 Zinc oxide, predicted ionization rate, 307 Zinc selenide energy gap, 18 hardness, 18 threshold displacement energies, 371 Zinc telluride energy gap, 18 hardness, 18 Zinc-blende structure see Sphalerite structure

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