Semiconductors and Semimetals: Defects, HgCd, Se, HgCdO and Te (Semiconductors and Semimetals, Vol. 16)

SEMICONDUCTORS AND SEMIMETALS VOLUME 16 Defects, (HgCd)Se, (HgCd)Te

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

ALBERT C . BEER BATTELLE COLUMBUS LABORATORIES COLUMBUS. OHIO

VOLUME 16 Defects, (HgCd)Se, (HgCd)Te

1981

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Toronto Sydney

San Francisco

COPYRIGHT @ 1981, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PWBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York,New York 10003

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

24/28 Oval Road, London NW1 7DX

Library of Coqress Cataloging i n Publication Data Main entry under title: Semiconductors and semimetals.

Includes bibliographical references. Contents: v. 1-2. Physics o f 111-V compounds-v. 3. Optical properties of 111-V compound$-v. 4. Physics of 111-V compounds--[etc.] v. 16. Defects, (HgCd)Se, (HgCdITe-1. Semiconductors-Collected works. 2. SemimetalsCollected works. I. Willardson, Robert K. 11. Beer. Albert C. ~ _ . . ~ QC610.9.!A7 537.6’22 01-7900 ISBN 0-12-752116-X ( v . 16) AACR2 ~

PRINTED IN THE UNITED STATES OF AMERICA 81 82 83 84

9 8 7 6 5 4 3 2 1

Contents LISTOF CONTRIBUTORS . .

PREFACE .

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vii ix

1 2 4 9 13 22 29

Chapter 1 The Effect of Crystal Defects on Optoelectronic Devices Henry Kressel List of Symbols . . . . . . . . Introduction . . . . . . . . . Photodetectors . . . . . . . . Light-Emitting Diodes and Laser Diodes . . . . Dislocations. . . . . . . . . OtherDefects . . . . . . . . VI. Surface and Hetero-Interfacial Recombination . . VII . Defects Contributing to LED and Laser Diode Degradation . . . . . . . . References .

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Chapter 2 Crystal Growth and Properties of Hg.,Cd. Se Alloys C . R . Whitsett. J . G . Broerman. and C . J . Summers I . Introduction . . . . . . . . . . . . 11. The Pseudobinary HgSe-CdSe System . . . . . . .

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54

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I. 11. I11. IV . V.

ID . Crystal Growth of Hg,, C&Se Alloys .

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

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Electron Energy-Band Structure . . . Dielectric Function . . . . . Magnetotransport . . . . . . Optical Properties . . . . . Electron Scattering Mechanisms and Transport summary

References

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Chapter 3 Magnetooptical Properties of Hg.,Cd.

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49

54 61 69 72 75 79 % 114 114

Te Alloys

M . H . Weiler I . Introduction . . . . . . II . Theory of Zinc-Blende Semiconductors. In . Intraband Experiments . . . . IV. Interband Experiments . . . . V

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119 127 145 161

vi

CONTENTS

V . Hg.,Cd.

Te Parameters References . . .

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Chapter 4 Nonlinear Optical Effects in Hg.,Cd. Paul W . Kruse and John F . Ready I . Introduction .

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INDEX . .

CONTENTS OF PREVIOUS VOLUMES.

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176 188

Te

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194 1% 207 213 218 230 242 246 247 251

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255

II. PhenomenologicalDescription . . . . III. Theoretical Development . . . . . IV. Relevant Roperties of Hg.. ,Cd=Te . . . V . Hg...Cd. Te Spin-Flip Ramon Lasers . . . VI. Observations of Multiphoton Mixing in Hg.,Cd. Te W . Optical phase Conjugation in Hg...Cd. Te . . Vm . Third-Order Resonant Nonlinear Susceptibility . IX. Applications. . . . . . . . References

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263

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

J. G . BROERMAN, McDonnell Douglas Research Laboratories, St. Louis, Missouri 63166 (53) HENRYKRESSEL, RCA Laboratories, Princeton, New Jersey 08540 ( 1 ) PAULW. KRUSE,Honeywell Corporate Technology Center, Bloomington, Minnesota 55420 ( 193) JOHN F. READY, Honeywell Corporate Technology Center, Bloomington, Minnesota 55420 (193) McDonnell Douglas Research Laboratories, St. Louis, C. J. SUMMERS, Missouri 63166 (53) M. H.WEILER,Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ( 1 19) C. R. WHITSEIT,McDonnell Douglas Research Laboratories, St. Louis, Missouri 63166 (53)

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Preface This volume contains four review articles that deal specifically with phenomena and materials. The content of the studies, however, provides specific information of direct importance for device applications and performance. The latter consideration is the basis of the first chapter, which examines the role of crystal defects in the performance of optoelectronic devices. Of most concern are questions of reliability and degradation. These considerations are of utmost importance in the design of optoelectronic systems, and the chapter discusses the types of materials defects that can most readily cause device degradation and lack of reliability. The second chapter is devoted to the ternary compound (HgCd)Se, which possesses properties of interest in the infrared sensor field. The article reviews in detail the existing information on crystal growth and electrical and physical properties. The companion ternary, namely, (HgCd)Te, is the subject of Chapters 3 and 4. This material is a foremost candidate for a variety of optoelectronic applications, including photoconductive and photovoltaic infrared detectors, infrared emitters, tunable lasers, and nonlinear optical devices. Chapter 3 concentrates on magnetooptical properties, including a review of the most accurate determinations of band parameters and transport properties. Nonlinear optical effects, mostly those arising from large values of the thirdorder electric susceptibility in (HgCd)Te, are the subject of Chapter 4. A great variety of useful devices results from phenomena such as resonant four-photon mixing, optical phase conjugation, and the spin-flip Raman laser, with its useful tuning capabilities. Incidentally, (HgCd)Te is now being used so extensively and is the constituent of so many devices in the planning stage that an entire future volume of Semiconductors and Semimeruls (namely, Volume 18) will be devoted to it, including preparation and crystal growth, electrical and physical characteristics, and applications. The editors are indebted to the many contributors and their employers who make this treatise possible. They wish to express their appreciation to Cominco American Incorporated and Battelle Memorial Institute for providing the facilities and environment necessary for such an endeavor. Special thanks are also due the editors’ wives for their patience and understanding. R. K . WILLARLXON ALBERT C. BEER

ix

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SEMICONDUCTORS AND SEMldETALS, VOL. 16

CHAPTER 1

The Effect of Crystal Defects on Optoelectronic Devices Henry Kressel LISTOF SYMBOLS. . . . . . . . . . . . . . . I. INTRODUCTION , . . . . . . . . '. . . . . . 11. PHOTODETECTORS . . . . . . . . . . . . . . 1. Solar Cells . . . . . . . . . . . . . . . 2. Photodiodes . . . . . . . . . . . . . . . 111. LIGHT-EMITTING DIODES AND LASERDIODES . . . . . . . IV. DISLOCATIONS . . . . . . . . . . . . . . . 3. Dislocations in the Crystal Bulk . . . . . . . . . 4. Dislocations at Hetero-Interfaces . . . . . . . . . V. OTHERDEFECTS . . . . . . . . . . . . . . . 5 . Stacking Faults . . . . . . . . . . . . . . 6. Precipitates . . . . . . . . . . . . . . . 1. Grain Boundaries . . . . . . . . . . . . . V1. SURFACE AND HETERO-INTERFACIAL RECOMBINATION. . . . . 8. Free Surfaces . . . . . . . . . . . . . . 9. Radiation Damqqe . . . . . . . . . . . . . 10. Recombination at Hetero-lnterfhces . . . . . . . . 11. /C&rrier LiJi.time and Uadiutice &i%ciuncy in Heterostructures . CONTRIBUTING TO LED AND LASERDIODE DEGRADATION V11. DEFECTS 12. Introduction . . . . . . . . . . . . . . . 13. The Effect of Dislocations . . . . . . . . . . . 14. The Effect of Damaged Diode Edges . . . . . . . . 15. Process-Induced Defects . . . . . . . . . . . 16. The Effect of Stress. . . . . . . . . . . . . REFERENCF~ . . . . . . . . . . . . . . . .

i

2 4 5 1

9 13 16 20 22 22 23 21 29 29 31 32 36 40 40 42 44 46 41 49

LIST OF SYMBOLS diode area depth of absorbing p-n junction band-to-band recombination coefficient minority-carrier diffusivity electron diffusivity hole diffusivity diode diameter width of the heterojunction diode active region

e f

I

Is, J J,

1

electron charge fill factor (which incorporates the effect of the solar cell's internal resistance) current short-circuit current current density current density through the bulk region of the diode

Copyright @ 1981 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-752116-X

HENRY KRESSEL

short-circuit current density of solar cell diode saturation current density laser diode threshold current density Boltzmann constant minoritycarrier diffusion length diffusion lengths for electrons and holes, respectively area density of interfacial centers recombination center density injected carrier pair density needed to reach threshold intrinsic carrier concentration initial (i.e., background) electron density incident solar-power density initial (i.e., background) hole density surface recombination velocity temperature voltage solar-cell opencircuit voltage

thermal electron velocity width of the diode space-charge region width of the diode spacesharge region at the surface misfit strain in heteroepitaxy internal quantum efficiency (spontaneous emission) quantum efficiency in the absence of dislocations solar-cell conversion efficiency radiation wavelength dislocation density carrier capture cross section of recombination centers carrier lifetime nonradiative lifetime including interfacial recombination ‘‘bulk’’ nonradiative lifetime radiative lifetime radiation dose

I. Introduction This chapter is concerned with the impact of major crystal defects on p-n junction devices designed for light emission and detection. These devices share a common need for carrier lifetime control and their performance can be severely degraded by crystal defects. However, light-emittingdevices are sensitive to defects which reduce the carrier lifetime by nonrudiutive recombination process whereas photodetectors are more generally affected by defects which shorten the minority-carrier diffusion length. Therefore, radiative or nonradiative recombination processes shorten the lifetime and thus reduce the diffusion length and hence the photodiode efficiency. We discuss, for the most part, devices made from single-crystal 111-V material, but there is growing interest in fabricating solar cells from polycrystalline materials in order to reduce their cost. In such devices, grain and twin boundaries may reduce the device quality. Certain crystal defects are most relevant to optoelectronic devices : (1) Recombination centers present at exposed surfaces and at heterojunction boundaries. (2) Dislocations and stacking faults incorporated in bulk-grown and epitaxial materials. (3) Dopant precipitates, which may be introduced in the course of doping. (4) Point defects (vacancies, interstitials, contaminants, and various

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

3

atomic complexes) which may be introduced during crystal growth or diffusion. These point defects constitute nonradiative recombination centers for the most part. (5) Grain boundaries in polycrystalline materials. They introduce recombination centers. The density and electrical activity of defects is influenced by the thermal history of the sample which explains the wide range of experimental observations. For example, the electrical activity of dislocations depends on the presence of contaminants because of atomic clustering within the strain field of dislocations. Because of the importance of p-n junctions, many studies have been made on the impact of defects on their current-voltage characteristics. Relevant major defects include recombination centers within the energy bandgap and precipitates within the space-charge region which produce local electric field increases. These “microplasma ” sites are regions of locally enhanced carrier avalanche multiplication or field emission. Microplasma sites degrade the diode breakdown voltage and result in high diode leakage currents. We consider first the impact of dispersed recombination centers on the current-voltage characteristics. Centers with ionization energies near midgap are particularly troublesome. According to well-known theory (Sah et al., 1957), the diode current density dependence on voltage includes an “ ideal junction ” component (injection current) and a contribution from space-charge region recombination. For a p + - n junction (where the spacecharge region extends mostly into the n side of the junction), J = - eD;l2 zl12

n: no

[

):;(

]+

exp - - 1

2

Here, Nt is the recombination center density, ot is the capture cross section of the centers, n, is the intrinsic carrier density, z is the carrier lifetime in the n side of the junction with electron density n o , D, is the hole diffusivity in the n side of the junction, W is the width of the space-charge region, and t),h is the thermal velocity of the carriers. In this chapter, we denote by “leakage current” all contributors to the J versus V characteristics except the first term of Eq. (1). In addition to the second term in Eq. (l), tunneling currents may be important in very highly doped junctions, but such junctions are not discussed in this chapter. Furthermore, recombination at the junction periphery gives a current contribution which we discuss in Section 8. Because of its exp(e1//2kT) dependence, the space-charge recombination current is generally negligible when compared to the injection current at high bias

4

HENRY KRESSEL

levels. In reverse bias, however, the junction current is typically dominated by the space-charge region recombination current.

II. Photodetectors In this section we review the basic parameters of solar cells and photodiodes with a view toward highhghting the impact of major lattice defects on device performance. An extensive review of solar cells can be found in the book by Hovel (1975). Photodiodes, particularly avalanche photodiodes, are reviewed by Stillman and Wolfe (1977) and Webb et ul. (1974). We distinguish between solar cells and photodiodes because their functions differ, although they share many similar requirements. Whereas solar cells are large area devices (a few square centimeters) operated in forward bias, photodiodes are generally small area devices (a few square millimeters) operated in reverse bias. Furthermore, avalanche photodiodes require near-ideal current-voltage characteristics and uniform properties in order to produce useful gain. Therefore, photodiodes require a considerably higher crystallographic quality than solar cells. Usefd solar cells and even simple photodiodes can be made from certain polycrystalline semiconductors, but such materials are unlikely to be used for avalanche photodiodes. The performance of photdetectors depends on the following key parameters, which are impacted by lattice defects: (1) The diffusion lengths of electrons (L,) and holes (L), which should be as high as possible for maximum minority-carrier collection from both sides of the p-n junction. This means that the minoritycarrier lifetime where De,his the carrier should be as long as possible since Le,h= diffusivity. (2) Surface recombination of photogenerated carriers, which reduces the collection efficiency. (3) The leakage current, which should be as low as possible. Therefore, defects in the junction spacecharge region or diode periphery should be minimized. Avalanche photodiodes should be microplasma-free. This means that local field enhancement, owing to inclusions or precipitates in the spacecharge region, must be avoided and surface fields must be minimized. (4) Degree of crystallinity. Although useful solar cells can be produced from polycrystalline material, the grain boundaries can degrade the junction current-voltage properties and may increase the lateral resistance of the device.

fi

1.

THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

5

1. SOLAR CELLS

Solar cells have been produced from several semiconductorsincluding Si, GaAs, and 11-VI compounds such as polycrystalline CdS. The most dramatic recent improvements in solar-cell efficiency have been made in GaAs cells by the addition of an AlGaAs heterojunction. Solar cells using GaAs are theoretically capable of yielding a higher conversion efficiency than Si cells: 24% versus -20% at air-mass-zero (i.e., above the atmosphere). They operate at higher temperatures than Si and require thinner active layers because of a higher absorption coefficient in the relevant spectral region. Early GaAs solar cells were fabricated by Zn diffusion to form a shallow (1-2 pm deep)p-n junction in an n-type substrate grown from the melt (Gobat et al., 1962). The efficiencies were about lo%,a value more easily achieved with Si, and the higher density of GaAs compared to Si meant a further disadvantage in the power/weight ratio in space vehicles. As a result, interest in GaAs solar cells essentially vanished only to be revived after 1968 upon the development of the AlAs-GaAs heterojunction technology which led to substantial performance improvements over Si cells by reducing the carrier loss due to surface recombination. Figure 1 shows schematics of the GaAs homojunction and AlGaAs/GaAs heterojunction solar cells. The parameters that control the solar-cell efficiency qscare the the opencircuit voltage V,, the short-circuit current density J,,, the fill factor F INCIDENT RADIATION p-TYPE REGION a j < L ,

(a)

------T n-TYPE

GaAs

Laj (b)

n-TYPE GaAs

FIG.1. (a) Simple GaAs homojunction and (b) AlGaAs/GaAs heterojunction solar-cell structures. Structure b uses a p-p heterojunction for surface passivation.

6

HENRY KRESSEL

(which incorporates the effect of the solar cell’s internal resistance), and the spectral distribution of the incident solar power density Pi : qsc

(2)

= FVocJsc/Pi*

The short-circuit current density Jsc results from the photogenerated carriers reaching the space-charge region, Jsc

W e+ &I,

(3)

where Cis a parameter which includes the photogeneration rate for electrons and holes. We assume that the diffusion lengths for electrons (L,) and holes (L,,) are much higher than the width of the space-charge region. Because the diffusion length varies as the square root of the carrier lifetime, any reduction in the lifetime (other factors remaining constant) decreases the cell efficiency. The open-circuit voltage is easily calculated assuming the “ideal * diode equation parameters [i.e., neglecting space-charge region recombination in Eq. (111: VOC

+

= (~~/e)lnC(Jsc/Js,) 11,

(4)

where [from the first term in Eq. (l)] the saturation current density J , is J,, = e D ~ / 2 n ~ / z ’ / 2 n , .

(5)

It is evident from Eq. (4) that V, is maximized by having a large shortcircuit current density and a low saturation current density. Carrier recombination at the cell surface (see Part VI) reduces the shortcircuit current density and is particularly detrimental in reducing the cell’s response to high-energy photons which are absorbed near that surface. Ellis and Moss (1970) have analyzed this effect in GaAs solar cells and have shown that surface recombination seriously reduces the cell efficiency if SLID 2 sinh(aj/L),

(6)

where D is the minoritycarrier diffusivity, aj is the depth of the absorbing p-n junction, and S is the surface recombination velocity. Figure 2 shows the calculated efficiency of a GaAs cell as a function of S for various values of the ratio aj/Le.It is clear that S seriously reduces the efficiencywhen it reaches lo5 cm/sec (unless the junction depth is extremely thin, which can increase the cell resistance too much). Since S can reach lo6 cm/sec for a GaAs surface (Section 8), the collection efficiency of the homojunction cell is typically limited by surface recombination. As discussed in Part IV, the addition of a p-p heterojunction, producing the structure shown in Fig. 1b, greatly reduces the surface recombination, thus making possible GaAs cells having a collection efficiency as high as -20%.

-

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

s.104

b. LL W

10

7

-

s.107

51

0

I

0.I

I

0.2

I

0.3

I

0.4

Oj/LC

FIG.2. The calculated efficiency of a p-n GaAs solar cell with a p-type junction depth uj as a function of the surface recombination velocity Sand of the ratio uj/Le.L, and L, are the electron and hole diffusion lengths in the p- and n-type sides of the junction, respectively. L, = 1.9 pm; L, = 4.8 pm; no = 5 x lo” ~ m - po ~= ; 5 x lo” ~ m - (After ~ . Ellis and Moss, 1970.)

2. PH~T~DIODES The photodiode is generally characterized by its quantum efficiency over a specified spectral region, its response to high-frequency signals, and its leakage current. For an avalanche photodiode, additional key parameters are the maximum useful gain and the voltage at which it occurs. The quantum efficiency depends on the design of the device and on the carrier diffusion length as discussed above for solar cells. The frequency response is affected by the capacitanceof the device and the response delay due to photogenerated minority carriers diffusing to the spacecharge region. Hence, the shorter the diffusion length, the easier it is to produce a fast detector (other factors remaining constant). Figure 3a shows a schematic cross section of a typical Si avalanche photodiode with an n+-pp+ structure. The space-charge region extends to the p-p+ boundary at the designed reverse-bias operating voltage. The thin n+ region is formed by diffusion. Note the deeper diffusion at the diode periphery. The purpose of this deeper “guard ring” diffusion is to reduce the electric field at the diode periphery which would otherwise be unacceptably high owing to the small radius of curvature associated with shallow diffused junctions. Figure 3b shows the current gain as a function of the reverse-bias voltage. The dark current should be low and the photogenerated carrier multiplication in the spacecharge region should be as uniform as possible

HENRY KRESSEL

8

I ELECTRIC FIELD

I MULTlPLlCATtON FACTOR

A

TEMPERATURE

VOLTAGE (b)

FIG.3. (a) Cross section of a n + - p - - p + avalanche photodiode of silicon showing the extent of the space charge region (dotted line) under full reverse bias. The electric field E peaks at the n-p interface.The carrier multiplication M in the space-chargeregion is shown (McIntyre, 1979). (b) Typical reverse-biascurrent gain as a function of voltage and temperature. (After Webb et nl., 1974.)

1. THE EFTECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

9

over the junction area (within a few percent). Typical multiplication values do not exceed 100 in practice. The major photodiode parameters which are impacted by lattice defects are the leakage current and the ability to obtain avalanche gain because of the detrimental effects of microplasma sites. There is a vast literature concerned with the effect of defects on the reverse-bias junction properties of silicon devices (Lawrence, 1973). Microplasma sites are-generally regions of local electric field enhancement in the space-charge region resulting from large-size foreign matter. Precipitates may consist of S O z , metals, or dopant clusters. Because of the locally increased field, avalanche carrier multiplication occurs in the vicinity of the defect site. Furthermore, if the field is sufficiently intense, the dark current may also increase locally owing to field emission. Goetzberger and Shockley (1960) first showed that metallic precipitates in the space-charge region of silicon p-n junctions result in leaky junctions. They also showed that junction properties could be improved by “gettering” treatments which removed the metal from the silicon. Dislocations, grain boundaries, and stacking faults in the device provide suitable sites for the formation of precipitates. Therefore, although excellent photodiodes can be reproduced in silicon, it is a more difficult task to produce avalanche detectors in lessdeveloped materials containing a significant density of dislocations or other defects. In Section 3 we will consider for illustration a heterostructure detector and the observed impact of dislocations on its leakage current.

In. Light-Emitting Diodes and Laser Diodes The quantum efficiency and the reliability (Part VII) of LEDs and laser diodes are strongly affected by lattice defects within the recombination region. The typical LED is ap-n junction structure formed by diffusion. The most widely used LEDs are of GaAs, GaAsP, and Gap; Zn is the most commonly used p-type dopant for devices which have diffused p-n junctions. Laser diodes are configured, as shown in Fig. 4, in the form of “broadarea” devices where the sides of the diode are sawn to prevent lasing in the lateral direction, or in the form of stripe-contact structureswhere the current is laterally constrained by resistive regions. Stripe-contact diodes are most widely used. The oxidedefined stripe device of Fig. 4b is one of several stripe-contact structures (for review, see Kressel and Butler, 1977). Devices which have the same bandgap semiconductor on both sides of the p-n junction are denoted as “ homojunction ” structures. Early laser diodes were homojunction structures. However, the most widely used laser diodes

HENRY KRESSEL PE WIDTH W

BROAD AREA (a 1

(b)

i

W' (C 1

f

(d)

FIG.4. Laser diode structures: (a) broad-area structure formed by cleaving two sides and sawingthe lateral sides;(b) stripecontact laser where the current is confined by openingacontact in an insulating layer of SiO, ; (c) single-heterojunction structure of AlGaAs/GaAs with recombination region width d; and (d) double-heterojunction structure of AlGaAs/GaAs. The bandgap energyprofile is shown next to each structure. The bandgap step is typically0.3 to 0.4 eV.

are now heterojunction structures with either one or two heterojunctions (Kressel and Butler, 1977). In a heterojunction structure, the recombination region is bracketed by either one or two heterojunctions in order to confine the injected camers and to produce a planar waveguide structure for transverse mode guiding. Figures 4c and 4d show cross sections of the single- and double-heterojunction laser. The width of the recombination region is dwhich takes values from 2 pm in the single-heterojunction laser to about 0.1 pm in the doubleheterojunction laser. An essential requirement for successful heterojunction laser operation is that the lattice match at the heterojunction boundaries be very close in order to minimize the density of nonradiative recombination centers introduced by misfit dislocations (Part VI). Heterojunction structures of AlGaAs/GaAs were the first devices developed, but since that time, heterostructures of InGaAsP/InP, InGaAs/GaAsP, and PbSnTe/PbTe have also been developed (to name but a few of the successfully realized heterostructures).

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

11

The quantum efficiency of an LED depends on the internal quantum efficiency and on the device structure. The latter is designed to maximize the external efficiency by minimizing the internal optical absorption. The key laser diode parameter is the threshold current density with the differential quantum efficiency as a secondary parameter. In general, it is desirable to produce devices with as low a threshold current density as possible. For example, devices designed for continuous wave operation at room temperature require a threshold current density under 3000 A/cm2. However, some lasers are designed for pulsed-power operation at high-power levels, and other factors such as resistance to optical damage must be considered. Such devices have wide (-2 pm) active regions. In general, the single-heterojunction laser is designed for high peak pulsed-power operation and its threshold current density is in the 10,000 A/cm2 range. Double-heterojunction lasers, on the other hand, are typically designed for cw operation and have threshold current densities below 2000 A/cm2. For AlGaAs/GaAs diodes -500 pm long and structures where 0.3 Id I2 pm (and similar heterostructures with bandgaps of about 1 ev) the threshold current density at room temperature varies as .Ilhz SOOOd, where d is in micrometers and Jlhis in amperes per square centimeter. However, devices having a high density of defects will have higher relative threshold current densities as discussed below. Nonradiative recombination centers reduce the carrier lifetime and thus decrease the internal quantum efficiency qi, where zo is the radiative lifetime, zn, is the nonradiative lifetime, and z-' = z; ' + .'z, The above-defined internal quantum efficiency is relevant to the efficiency of LEDs and to the threshold current density of laser diodes as well (see below). The carrier lifetime z in light-emitting diodes is deduced from the luminescence decay time following a current pulse. This measured lifetime includes the effect of both the radiative and nonradiative recombination processes. Unless the radiative lifetime is independently determined, the nonradiative lifetime cannot be independently measured. Therefore, in order to study the effect of a varying density of specific lattice defects on ,z, , it is necessary to evaluate their impact on the externally measured quantum efficiency. Assuming that the device structure is invariant with the lattice defect density under study, it is frequently reasonable to assume that external quantum efficiency changes follow changes in internal quantum efficiency qi . Owing to severe internal absorption in typical light-emitting structures, however, an absolute determination of the internal quantum efficiency is difficult.

12

HENRY KRESSEL

Changes in the internal quantum efficiencyarc frequently determined from the dependence of the photoluminescence efficiency on the defect densities. This permits a rapid and convenient assessment of the effects on a material by avoiding the need to fabricate a p-n junction. The impact of specific defects on laser diodes is more difficult to assess than in the case of LEDs owing to greater device complexity and higher operating current density. Specifically, quality lasers require planar active regions, careful control of the refractive index profile in the junction region, excellent ohmic contacts, and low thermal resistance to minimize junction heating. One may estimate the approximate impact of lattice defects on the laser threshold current density from their effect on the carrier lifetime and on the internal quantum efficiency for spontaneous emission. The lasing threshold is reached when the carrier pair density in the active region reaches a value sufficient to obtain an optical mode gain which just matches the optical losses characteristic to the laser cavity. Assuming unidirectional injection, the threshold current density is Jth

= (e&/z)d,

(8)

where d is the width of the active region, e is the electron charge, and N[h is the injected carrier pair density needed to reach threshold (typically 12 x loi8 cm-’ in GaAs at 300 K).Other factors being constant, a reduction in z will increase & . Equation (8) can be expressed in terms of the internal quantum efficiency for spontaneous recombination using Eq. (7) : Jth

=

(eNth/?izO)d.

(9)

On the basis of the above discussion, an approximation of the effect of lattice defects on the laser threshold current density may be ascertained from the same kind of studies which are useful for LEDs. However, caution is needed in interpreting the data owing to inherent differences between LEDs and laser diode operating principles:

(1) The laser diode reaches threshold at relatively high injection levels. Therefore, the carrier lifetime of interest is that appropriate at high carrier pair densities where radiative bimolecular recombination becomes important. As noted earlier, the typical carrier pair density is about 1-2 x 1Ol8 cm-’ in the threshold region of GaAs at 300 K, and the radiative lifetime is 2 to 3 nsec. Therefore, a low density of nonradiative centers will not impact the lasing threshold but will depress the quantum efficiency below threshold at low-injection levels where the radiative lifetime can easily exceed 10 nsec in the same device.

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

13

(2) At high forward bias, the p-n junction (or heterojunction) is predominantly operating in the regime of injection. Surface leakage currents (Section 8) and other nonradiative current components involving spacecharge region recombination or tunneling, which can be important at low bias values, will not significantly affect the laser threshold current density. (3) The internal quantum efficiency defined by Eq. (7) is not valid above lasing threshold, where the radiative carrier lifetime is shortened by stimulated recombination, if spontaneous lifetime values are used. The internal quantum efficiency should theoretically approach unity in the lasing region, as long as the junction current is channeled into a region of the device where stimulated emission occurs, the reason being the shortening of the stimulated recombination lifetime with increasing optical field intensity. IV. Dislocations The impact of dislocations on p-n junctions has been studied for many years, particularly in silicon (see review by Lawrence, 1973). Less information is available concerning the influence of dislocations on III-V compound devices although the general effects appear comparable to those reported for silicon diodes. Dislocations may, under certain conditions, reduce the carrier lifetime and mobility, degrade the quantum efficiency and the current-voltage characteristics of p-n junctions, and increase the LED and laser degradation rate (Section 13). As discussed below, it is well established that the radiative efficiency is reduced near a dislocation. However, the magnitude of the decrease varies, and conflicting experimental results have been reported concerning the effect of dislocations on carrier recombination. In fact, the theory of nonradiative recombination at dislocations is still uncertain. A comprehensive review of the impact of dislocations on the optical and electronic properties of semiconductors is beyond the scope of this chapter (see Schroter, 1979) and we limit ourselves to a review of the relevant highlights. The conflicts in the reported experimental work on dislocations and their impact on material and device properties arise largely from the uncertain effect of impurities. Impurities tend to segregate in the strain field surrounding dislocations, making it difficult to separate the role of the dislocations from that of the impurities clustered around them. Because some impurities (such as copper) are mobile at very low temperatures, even low-temperature plastic deformation does not ensure freedom from contaminant diffusion to dislocations. However, experiments show that not all dislocations are electrically active in a given sample. Figure 5 displays an electron-beaminduced current (EBIC) scan which shows that only two of the four edge

14

HENRY KRESSEL

FIG.5. Not all dislocationsin a given Si sample are electricallyactive. In (a), dark spots in the electron-beam-induced current scan coincide with two dislocation sites. In (b), we see that four dislocations are present in the micrograph as revealed by etching the Si sample. Thus, two dislocation sites have low electrical activity. These different behaviors of dislocations can be associated with the different degrees of their decoration or with the difference in their crystallographic structure. (After Jastrzebski, 1979.)

dislocations, revealed in a Si sample by the Dash etch, are electrically active. Local carrier recombination at the two other dislocation sites is minimal (Jastrzebski, 1979). All dislocationsare surrounded by strain fields, but edge dislocations have been postulated to have “dangling bonds” at the core atoms whereas such states are not expected at screw dislocations (Mueller, 1961). However, the

1.

THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

15

simple concept of dangling bonds, even for edge dislocations, is probably inadequate because the experimentally measured electronic properties of edge and screw dislocations appear to be quite similar. To explain the similarity, Heine (1968) suggested that the recombination centers associated with dislocations arise from their strain field which perturbs the lattice. Kamieniecki (1976) has postulated that the core of the dislocation can be treated in terms of an amorphous region of the crystal where the states formed are analogous to those encountered in amorphous materials. Assuming that the strain fields are important, we expect that effects which reduce them will also reduce the impact of dislocations on electronic properties. These include heat treatment which could lead to lattice reordering. The many studies of dislocations show the following major effects: (1) The carrier mobility is reduced when measured in the direction perpendicular to the dislocation axis. (2) Dislocations can (but need not) introduce donor and acceptor states. Kimerling and Pate1 (1979) concluded that two dominant states are associated with dislocations in Si: an electron trap at E, - 0.38 eV and a hole trap at E, 0.35 eV. In ultrapure Ge, Hubbard and Haller (1979) found levels around E, 0.08 eV for material grown in a hydrogen atmosphere in the [lo01 direction and levels at E, 0.2 eV for material grown in a nitrogen atmosphere. However, a change of the growth direction to [1 131 produced dislocations with no detectable levels. Based on the available data, we have to conclude that no definitive statements can be made concerning the electronic levels at dislocations. (3) Dislocations introduce mostly nonradiative recombination sites. (4) The average minority-carrier lifetime in the material is usually reduced with increasing dislocation density (at least for high densities). ( 5 ) Impurity precipitates at dislocations degrade the reverse-bias current-voltage characteristics of p-n junctions. The impact of “clean” dislocations may be minimal. (6) Dislocations accelerate the degradation rate of LEDs and laser diodes (Section 13).

+

+

+

The most troublesome effects of dislocations arise from items (3) to (6), but the specific correlation between dislocations and device properties depends on the type of dislocation, the method of introduction, the presence of impurities in the crystal, and the thermal history of the device. For example, Esquivel et al. (1976) reported studies of GaAs crystals bent at 700°C and oriented so as to introduce dislocations with either a Ga or As atom terminating the extra half-plane. The dislocations produced acceptors in n-type crystals, and the carrier mobility was reduced when measured in the direction perpendicular to the dislocation axis. This effect on the mobility

16

HENRY KRESSEL

was attributed to the scatteringof majority carriers by a space-charge region surrounding dislocations. 3. DISLOCATIONS IN THE CRYSTAL BULK a. Efect on LEDs and Lasers Dislocations introduce nonradiative centers as shown by numerous studies. However, there are questions concerning differences between “fresh and heat-treated dislocations, as well as the impact of impurities. Heinke and Queisser (1974) reported that “fresh” dislocations (introduced by plastic deformation at low temperatures) suppressed the radiative e5ciency by several orders of magnitude within a few micrometers of the dislocation core. However, a short heat treatment at temperatures as low as 600°C for 30 min recovered part of the radiative efficiency. Furthermore, they found that dislocations grown into the crystal have a relatively minor effect on the e5ciency compared to fresh dislocations. These conclusions have been questioned by Bohm and Fischer (1979) who confirmed that edge and screw dislocations reduce the nonradiative lifetime, but failed to discover a difference between fresh and heat-treated dislocations. These authors concluded that contaminants which migrated to the dislocations were responsible for the previously observed differences between dislocations introduced at low temperatures and those formed during crystal growth. Note that Bohm and Fischer (1979) also failed to detect specific recombination centers associated with the dislocations in their crystals, leaving open the reason for the enhanced nonradiative recombination at dislocations free from precipitates. Before reviewing the effect of dislocations on the quantum efficiency of LEDs, we consider their impact on the carrier lifetime. It is intuitively obvious that the lifetime will only be reduced when the average distance between dislocationsapproaches the minority-carrier diffusion length. Since the diffusion length is relatively short in direct bandgap materials (typically under 10 pm), the dislocation density must be rather large ( 2lo5 cm-’) for noticeable effects to be observed. The data generally show a reduction of the lifetime with increasing dislocation density, at least for substantial densities. Note, however, that a study in Si by Glaenzer and Jordan (1969) showed that dislocationsneed not affect the lifetime although recombination centers were found associated with the dislocations. These authors attributed their results to the presence of a potential barrier surrounding the dislocations which repels majority carriers. However, such a barrier would be absent if su5cient impurities accumulate in the strain field of the dislocation. Since contaminants and impurities tend to be present in most practical

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONICDEVICES

17

devices, it may be assumed that dislocations will impact the lifetime in the majority of cases. Data summarized by Vink et al. (1978) in Fig. 6 show lifetimeversus dislocation density data for Si, Ge, GaAs, and Gap. Note that the lifetime decreases inversely with the dislocation density in all materials. However, in Si and Ge the effect of the increasing dislocation density is felt at densities much below those in GaAs. The reason is the longer camer diffusion lengths in good quality indirect bandgap materials compared to those in GaAs. The effect of the nonradiative recombination at dislocations on the quantum efficiency follows from the reduction of the nonradiative carrier lifetime. However, assumptionsmust be made concerning the recombination rate at dislocations. Various models for quantitatively predicting the relationship between the dislocation density and quantum efficiency q have been discussed by Brantley et al. (1975). A simple expression derived by Roedel et al. (1979), which appears to be consistent with experiments, assumes that S = 00 at dislocations that are spaced on the average a distance Pd-” apart. Hence,

’

q / q O 2 (1

+ ~2L2Pd)-’.

(10)

Here L is the carrier diffusion length, pd is the dislocation density, and qo is the quantum efficiency in the absence of dislocations. Thus, the impact of 10

10

I

0

In 0

lo1

E

e

f

10’

2.

e

5

10’

E I 1

10

FIG.6. Minority-carrierlifetime as a function of dislocation density in Ge, Si, GaAs, and Gap. The origin of the data is given in Table I. (After Vink et al., 1978.)

HENRY KRESSEL

18

TABLE I

ORIGIN OF DATA SHOWN IN FIG.6

Swbol b 9

-

+ o

o

0 p

X

-k

+ A

4

Material

Resistivity (Q cm) Method of or dislocation carrier density ( ~ m - ~ )introduction

n-Ge n-Ge n-Ge n-Ge n-Ge p-Ge p-Ge p-Si p-Si p-Si n-Si n-Si n-GaP n-GaP n-GaP p-GaP p-GaAs p-GaAs

grown-in grown-in grown-in grown-in grown-in deformation grown-in grown-in grown-in deformation grown-in deformation grown-in grown-in grown-in grown-in grown-in grown-in

2.5 2 3-5

3040 8 2.2

800-2000 100 7-200

40 7-200

2-3 x 1017 1017

--

-

10'8 1019

-

Reference Okada (1955) Rosi (1958) Kurtz ef al. (1956) Kurtz et al. (1956) Wertheim and Pearson (1957) Wertheim and Pearson (1957) Wertheim and Pearson (1957) Lemke (1965) Bottger and Richter (1962) Glaenzer and Jordan (1969) Kurtz ef al. (1956) Glaenzer and Jordan (1969) Suzuki and Matsumoto (1975) Werhoven ef al. (1976) Blenkinsop er al. (1976) Werhoven et al. (1976) Ettenberg (1974) Titchmarsh et al. (1977)

dislocations on the efficiency of LEDs depends on the diffusion length. Devices with short diffusion lengths are less sensitive to a moderate dislocation density than devices with long diffusion lengths. Studies relating the dislocation density to the quantum efficiency of GaAs (Herzog et al., 1972),AlGaAs (Roedel er al., 1979),and GaP:N (Stringfellow er al., 1974) LEDs are summarized in Fig. 7 where the decrease in the efficiency with dislocation density is evident.

-. 0

.-0.8

1.6-

-

(3"0.c lo=

10'

DISLOCATION

10.

DENSITY (ern-')

FIG.7. The externally measured efficiencyof LEDs of GaAs, AlGaAs, and GaP as a function of the dislocationdensity: -.-a', GaAs (Henog et at., 1972); --- 0,AlGaAs (Roedel et aL, 1979); -A,Gap: N (Stringfellow et al., 1974). (After Vink et al., 1978.)

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

19

A study of the effect of the dislocation density on the threshold current density and differential quantum efficiency of AlGaAs/GaAs heterojunction lasers shows a significanteffectonly at dislocation densities in excess of about lo5 cm-2 (Druzhinina et al., 1974). However, the degradation rate of the lasers may be influenced by lower dislocation densities (see Section 13). In summary, we find that dislocations decrease the quantum efficiency of LEDs and increase the threshold of laser diodes. However, it is unclear from the experimental data whether the effects are solely due to dislocations or in combination with dopants and/or contaminants clustered in their vicinity. b. Eflect on Photodiodes

The influence of dislocations on the performance of photodiodes results mainly from the increase in the leakage current and the formation of microplasma sites which prevent useful avalanche diode performance. For illustration, we consider a detector designed for efficient collection at about 1.3 pm, shown in Fig. 8, which is produced by depositing a lattice-matchingepitaxial layer of InGaAsP on InP by vapor phase epitaxy (Olsen and Kressel, 1979). Close control of the lattice parameter is possible with this epitaxial technology, and the absence of misfit dislocations can be established by x-ray analysis. However, despite a close lattice match one fmds large differencesin the diode leakage current which have been traced to varying InP substrate quality. The dislocation density of the substrate, and consequent dislocation density in the epitaxial layer, causes differences in diode leakage current (Olsen, 1979). Figure 9 shows the leakage current as a function of reverse bias for two photodiodes. The leakage current of the diode with lo4 dislocations/cm2is much larger than in the case of the diode produced from a substrate with essentially no dislocations. However, the leakage current

I

1

FIG.8. Cross section of an avalanche photodiode of InGaAsP/InP designed for the 1.3-pm spectral region. The Zn-diffused region is shown below the p+-InP surface layer which is deposited in an area opened in the SiO, insulator. (After Olsen and Kressel, 1979.)

HENRY KRESSEL

FIG.9. Diode leakage current of InGaAsP/InP photodiode of the type shown in Fig. 8 (with a 1 0 0 - diameter) ~ as a function of the reverse-bias voltage applied to the junction. The figure shows data for two diodes. The diode with the lower leakage current was grown on a substrate having essentially no dislocations, whereas the other diode was grown on a substrate with lo4 dislocations/cm2.(After Olsen, 1979.)

density of the better diode is still much larger than the theoretically expected value indicating other defects which contribute to leakage in both devices. There is ample evidence in the literature (White, 1975; Milvidskii and Osvenskii, 1977) of many recombination centers in 111-V compounds owing to point defects which are either native in origin or associated with contaminants. It is not surprising, therefore, that the current-voltage characteristics, even in the best diodes, are generally far from ideal. However, continued progress in defect reduction can be expected to lead to improvements in photodiode properties. 4. DISLOCATIONS AT HETERO-INTERFACES

Misfit dislocations introduced in heteroepitaxy produce nonradiative centers. As shown in Fig. 10, the linear arrays of dislocations are clearly revealed by the dark regions in cathodoluminescence.These dislocationscan also be directly observed by x-ray topography (Olsen, 1975). The density of misfit dislocations depends on the lattice misfit strain Aa,/a, between the

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

21

FIG.10. (a) Optical photograph, (b) x-ray reflection topograph, and (c) scanning electron microscope cathodoluminescence photograph of the same region of an In,,,,Ga,,,,As/GaAs VPE sample. The optical photograph was taken on the original sample. The wafer was then angle-lapped and b and c were taken from the same region. The black lines indicate where a one-to-one correspondence can be made between the figures. (After Olsen, 1975.)

epitaxial layer and the substrate and on the layer thickness. The formation of misfit dislocations to relieve strain is prevented in very thin epitaxial layers where the energy needed to form dislocationsexceedsthe strain energy. Simple theory (Matthews, 1975) predicts that this occurs when the layer Aao . Thus, for typical epitaxial layers thickness does not exceed -a;/@ about 1 p m thick, the misfit strain should not exceed about if misfit dislocations are to be prevented. Even if misfit dislocations are formed, however, up to about 70% of the misfit strain can be accommodated elastically, i.e., without misfit dislocation formation (Olsen et al., 1978). The effect of this unrelieved strain on device properties is not firmly established, but there is evidence that elastic strain contributes to the degradation of LEDs and laser diodes (see Section 16) when the stress exceeds about lo8 dyn/cm2. If misfit dislocationswere simply confined to the heteroepitaxial interface, the mischief would be limited to an increase in the interfacial recombination velocity (Section 9). Unfortunately, the dislocations tend to bend into the

HENRY KRESSEL

22

epitaxial layer. These “inclined dislocations ” can then transmit the impact of the lattice misfit into the bulk of the epitaxial layer and thus contribute to a degradation of the layer properties. Therefore, it is highly desirable to minimize the lattice misfit between a substrate and the epitaxial layer. The most successful heteroepitaxial devices use lattice-matched materials (InGaAsP/InP or AlGaAs/GaAs, for example) where the lattice misfit (due to thermal expansion differences) does not exceed 5 x

V. OtherDefects 5. STACKING FAULTS

Stacking faults are commonly encountered in epitaxial layers. For example, the growth on substrates which are not sufficientlyclean can cause s t a c b g faults to form. Stacking fault studies in GaAs show that they introduce nonradiative recombination centers and thus reduce the minoritycarrier lifetime if their density is sufficiently high (Abrahams et al., 1971). Figure 11 shows the hole diffusion length in n-type GaAs being reduced with increasing stacking fault density (Young and Rowland, 1973). m e density is here defined as the stacking fault length per unit area of the sample.) However, as discussed earlier for simple dislocations, the role of contaminants, and impurities clustered at the stacking faults is not easily distinguished from the inherent impact of “clean” stacking faults. In Si crystals, for example, Marcus et al. (1977) found that stacking faults became electrically active with metallic precipitates segregated at the comers of the fault polyhedron. Dishman et al. (1979) found that only decorated stacking faults E

J w

g

I

lo2

I

Ill

lo3

I

I

Ill

I

I I l l

lo4

I!

lo5

STACKING F A U L T D E N S I T Y (m-’)

FIG.11. Hole diffusion length versus stacking fault density in n-type epitaxial GaAs layers grown on various underlying materials.., GaAs grown on GaAs; I, GaAs grown on AIGaAs; x , GaAs grown on Gap (Young and Rowland, 1973). The curve bounds the highest values of diffusion length measured for a given stacking fault density.

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONICDEVICES

23

crossing a Si p-n junction increase the diode leakage. Undecorated stacking faults appear harmless. Ravi et al. (1973) found that only a fraction of the stacking faults in Si diodes are electrically active and that their impact on the diode leakage current varies. Figure 12 shows the reverse-bias current-voltage characteristics for various diodes with varying numbers of stacking faults in the junction. The active number is usually smaller than the total stacking fault number. The smaller stacking faults, which Ravi et al. found to be more likely to be decorated, were also found to be more likely to be electrically active. 6.

,

PRECIPITATES

The precipitation of dopants when their concentration reaches their solubility limit is a common phenomenon in semiconducttm (Petroff, 1975). Etching studies as well as crystallographic examinations of such heavily doped crystals (Faktor and Stevenson, 1978) reveal many defects of which dopant precipitates are one class. The excess dopants may combine with native atoms in the crystal to form complex segregates. For example, the total Te concentration in GaAs :Te 10’ / 5 5

-

I06

a r

IOJ

I-

2 W

a

5

0

104

w v)

a

w > IOJ w

a

\02

ia

20

40

60

vr

ao

1

100

120

(V)

FIG.12. I-V characteristics of five Si diodes illustrating the effects of electrically active stacking faults on excess reverse-bias current. The total number of faults is shown at the ends of the curves and the total number of electrically active faults is indicated at the center. (After Ravi et al., 1973.)

HENRY KRESSEL

24

-1 2

10-1

W

+

f

r -

1

a

/*

'o-2 0

l -

0

Sn-DOPED} UNDOPED

Sn-DOPED

10-310'6

10"

10'8

1019

ELECTRON CONCENTRATION ( c ~ n - ~ )

FIG.13. Relative photoluminescence intensities for n-type GaAs and InP as a function of electron concentration (T = 297 K). The higher relative intensity of InP is due to its lower surface recombination velocity. (After Casey and Buehler, 1977.)

exceeds the freeelectron concentration for Te concentrations above about lo'* m-'. The excess Te precipitates in the form of Ga2Te3,which has been identified by electron diffraction (Kressel et af., 1968). Precipitation also occurs in Se-doped GaAs where Ga2Se3may form (Abrahams et af., 1967) and GaAs doped with the acceptor Zn (Abrahams et af., 1972) or the amphoteric dopant Si (Kressel et al., 1968). Excess dopants may precipitate at dislocations where they can be viewed by infrared transmission microscopy (Kressel et af., 1968). These precipitates form nonradiative regions in the crystal and, by shortening the nonradiative lifetime, reduce the quantum efficiency in n-type GaAs when the dopant concentration is high (Kressel et af., 1968). Figure 13 shows this effect for GaAs :Sn and InP :Sn (Casey and Buehler, 1977). A similar reduction in quantum efficiency occurs in highly Zn- and Ge-doped GaAs at doping levels near 10'' cm-' (Abrahams et al. 1972; Kressel and Ettenberg, 1973). In discussing crystals with high doping levels, it is important to distinguish basic limitations to the carrier lifetime owing to Auger recombination from lifetime limitations resulting from recombination at defect sites. The Augerlimited lifetime decreases as n2p in n-type material and p2n in p-type material. Therefore, it decreases steeply at high doping levels. The theory has been reviewed by Takeshima (1972). Figure 14 shows the electron lifetime in p-type Si as a function of acceptor concentration. The dashed curve is the theoretical Auger lifetime fit (Redfield, 1980) to the experimental data of Iles and Soclof (1975). As usual, there is substantial experimental scatter owing to difficulties in lifetime measurements, but the Auger curve provides

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

25

FIG.14. The electron lifetime in p-type silicon as a function of the acceptor concentration. The experimental data are from Iles and Soclof (1975). The curve is the theoretical value of the lifetime as limited by the Auger process. (After Redfield, 1979.)

a reasonable fit at high doping levels. Lower-measured lifetimes are due to recombination via defects. Jastrzebski et al. (1978) have calculated the Auger and radiative recombination lifetime in p-type GaAs as a function of temperature for various doping levels. There are substantial numerical uncertainties in the theoretical values, but Fig. 15 does show that the Auger lifetime becomes limiting in lightly doped crystals only at very high temperatures when the intrinsic carrier concentration becomes high. However, at room temperature the Auger lifetime is not believed to be limiting even at hole concentrations of 1.2 x 1019cmP3.For very highly Zn-doped GaAs crystals, Zschauer (1969) attributed the lifetime reduction to Auger processes even at room temperature. Although in heavily doped crystals lifetime limitations due to Auger processes are rarely encountered, dopant precipitates severely degrade the current-voltage characteristics of p-n junctions mainly because of local field enhancement in the space-charge region. Therefore, photodetectors are particularly sensitive because even a small density of precipitates can severely degrade the reverse-bias diode properties, making the fabrication of avalanche detectors impossible. In Si diodes, Busta and Waggener (1977) found that precipitates in the spacecharge region, rather than distributed point defects, produced “soft ” and leaky reverse-bias current-voltage characteristics. They fitted their data to the Fowler-Nordheim fieldemission model. The large leakage currents were explained by field emission at the precipitate sites where the local electric field reaches the values in the lo6 to lo7 V/cm range.

HENRY KRESSEL

26

TEMPERATURE

600 500 I

I

(K)

400

300

I

I

10-7

-P c w

f I-

10-8

‘

W

LL -I

I

10-9

I

I

2 3 RECIPROCAL TEMPERATURE, 1031 T ( K-?

FIG.15. Carrier lifetime as a function of temperature for p-type GaAs crystals with the hole concentrations indicated as measured at room temperature. The calculated curves are divided into two sections. At lower temperatures,the lifetime is calculated for radiativeprocesses. At high temperatures, the theoretical Auger lifetime is calculated because it becomes dominant as the free-carrier concentration increases.The data points are experimental values. (After Jastrzebski et al., 1979.)

Dopant precipitates are troublesome but contaminants are also common. Copper is well known to be a troublesome contaminant in Si devices. Mama (1979) studied single-crystal Si solar cells containing 10l6 copper atoms and found precipitates consisting of /?-CuSi. The impact of these precipitates on the solar-cell e5ciency varies with their location with respect to the p-n junction. Becuase of the formation of these precipitates in the course of heat treatment, the dispersed Cu is removed from the active region of the device. Therefore, the merage diffusion length in a single-crystalsolar cell may not be reduced. Hence, the short-circuitcurrent density is unaffected by the Cu as long as few precipitates form in the junction spacecharge region. Note that in polycrystalline solar cells, Cu precipitates in grain

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

27

boundaries crossing the junction can greatly increase the cell's dark current, thus reducing the opencircuit voltage. 7. GRAIN BOUNDARIES

Polycrystalline semiconductors are being explored for the fabrication of solar cells in order to reduce costs. In the case of silicon, which is receiving the most attention, the material may be prepared by casting or by the edge-defined growth (EFG) technique, which produces ribbons in a continuous process (Ravi et al., 1975). In either case, grain boundaries with varying degrees of electrical activity are present. A convenient technique for determining the electrical activity of boundaries and other large defects is by means of electron-bean-induced current (EBIC) measurements in p-n junction structures. The observed EBIC contrast depends on the beam-generated carriers which cross the p-n junction at the interface. Dark regions (i.e., regions of low-beam-induced current flow)result when relatively few camers are collected locally because of strong recombination owing to a low carrier lifetime. EBIC studies (Kressel et al., 1977a) have been made in polycrystalline Si solar cells epitaxially grown on Si ribbon substrates. These studies led to a determination of the electrical activity of grain and twin boundaries as well as inclusions of foreign matter such as Sic. Coherent twin boundaries are the least active boundaries in Si. (Twinning in the diamond cubic lattice consists of a 180" rotation about a <111) direction.) Low-angle grain boundaries (which are dislocation arrays) and large-angle boundaries are most active in promoting localcamer recombination (Schwuttke et al., 1978). In view of the likely presence of contaminants in these crystals, it is probable that precipitates at these boundaries are at least partly responsible for their electrical activity. Studiesof GaAs crystalshave shown that various deep levels are associated with grain boundaries. Spencer et al. (1979) found levels at 0.90, 0.41, and 0.29 eV attributed to native point defects because these ionization energies are similar to those found in electron-irradiated GaAs. In addition, other levels normally associated with Cu, 0,and various fast-diffusing contaminants were found, indicating that impurity segregation occurs, as expected, at grain boundaries. In addition to the impact of grain boundaries on the electrical properties of p-n junctions, grain boundaries can reduce the effective mobility of majority carriers. Resistivity measurements as a function of position in polycrystalline Si show that the large-angle grain boundaries are high resistivity regions (Kressel et ul., 1977a; Spencer et al., 1979). Therefore, the energy diagram of Fig. 16a is appropriate for n-type crystals. Because the

HENRY KRESSEL

(C)

FIG.16. Electronic description of a grain boundary in an n-type crystal. (a) Energy-band diagram for the grain boundary with the Fermi level indicated. A potential barrier results from the fact that deep levels are associated with the grain boundary. (b) Energy-band diagram for a grain boundary with an electric field applied to the sample across the boundary.(c) Equivalent circuit for the grain boundary indicating two back-to-back diodes. (After Spencer et a[.. 1979).

Fermi level is constant in the crystal without an applied field, there is a potential barrier associated with the grain boundary which repels majority carriers and thus impedes their transport across the boundary. As% result, the lateral conductivity of polycrystalline solar cells is lower than in a similarly doped singlecrystal cell. This effect decreases the fill factor compared to a similar structure singlecrystal cell. Furthermore, rectification is possible at grain boundaries, as shown in Fig. 16b with a potential applied, further adding to the cell resistance. The proposed grain boundary equivalent circuit, shown in Fig. 16c, consists of two back-to-back junctions. Note that the magnitude of the band bending at the grain boundary is a function of the excess carrier density in the crystal. Therefore, with increasing injection the potential barrier decreasesas analyzed by Card and Yang (1976). Increasing the injection level reduces the impact of the grain boundaries on the carrier transport but enhances their effect on carrier recombination (i.e., lower lifetime) becausethe recombination rate at the boundary increases with majority-carrier density as the potential barrier is reduced. Therefore, if we analyze a polycrystallinesolar cell, we expect that increased solar intensitywill result in a reduced cell resistance but also in a reduced carrier lifetime. There-

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

29

fore cell short-circuit current density depends on the incident light intensity, an effect analyzed by Card and Yang (1976). Is it possible to reduce the impact of grain boundaries on device properties? Of course, using materials free from contaminants is one approach. In addition, recent work on grain boundaries in Si suggests that their electrical activity may be reduced by a hydrogen treatment. Seager and Ginley (1979) found that monatomic hydrogen diffused into Si polycrystals decreased the potential barrier at the grain boundaries, suggesting a reduction in the density of states. The treatment produced stable boundaries over long periods of time at temperatures up to 320°C, indicating that the hydrogen remains in place at the boundaries.

VI. Surface and Hetero-Interfacial Recombination 8. FREESURFACES

Minority carriers recombine at semiconductor surfaces (or at interfaces with dissimilar materials) via recombination centers. The surface recombination velocity is given by

s = OtVthNss,

(11)

where 0, is the carrier capture cross section for the recombination centers, uth is the thermal electron velocity, and N,, is the area density of interfacial centers. For a free surface of n-type GaAs, S reaches about lo6 cm/sec, but it depends on the carrier concentration of the crystal as shown in Fig. 17 (Jastrzebski et al., 1976). Note that the surface recombination velocity varies with the semiconductor. For example, available data for n-type InP suggest that S is much smaller

,2-@

-

9-@-

/'8

I'i 105

1015

1016

loi7

1018

10~9

ELECTRON CONCENTRATION ( ~ r n - ~ )

FIG.17. Surface recombination velocity in n-type GaAs as a function of the electron concenRao-Sahib and Wittry (1969);& Lagowski et al. (1973). tration: 0,Jastrzebski et al. (1976):.,

30

HENRY KRESSEL

than in comparablydoped GaAs. This means that a higher radiativeefficiency is possible in photoluminescence of InP compared to GaAs as shown in Fig. 13 (Casey and Buehler, 1977). Furthermore, the surface recombination velocity of InP can be reversibly changed by the gas ambient, whereas irreversible changes are more likely to be observed on GaAs surfaces when exposed to various ambients because GaAs oxidizes much more readily than InP (Nagai and Noguchi, 1978). Surface states reduce the quantum efficiencybecause they are nonradiative recombination centers. The periphery of an LED or laser diode is such a surface in a typical mesa structure, and the effect is quite general as shown by Henry et al. (1978). The impact of surface recombination may be measured in LEDs where the efficiency changes with the area-to-perimeter ratio. It is experimentally found that the LED quantum efficiency (at low current densities) increases as the area is increased with respect to the perimeter. From an analysis of the current-voltage characteristics, it is possible to separate the fraction of the current flowing through the surface states at the junction periphery from the current flowing through the bulk of the diode. Leistiko and Bittmann (1973) showed that the current Zflowing as a function of voltage V in a diode can be expressed as = Jb

+ (nDj/Ad)Js

ws,

(12)

where J, = eniS[exp(eV/2RT) - 13.

(13) Here, A, is the diode area, Dj is the diode diameter, W, is the width of the spacecharge region at the surface,ni is the intrinsiccarrier concentration,and Jb is the current density through the bulk region of the diode. Because J , at high-injection levels increases as exp(eV/kT), whereas J , increases as exp(eV/2RT), the relative effect of the peripheral nonradiative current on depressing the efficiency decreases with increasing junction voltage. No simple ways to permanently reduce the surface recombination velocity of 111-V alloys by chemical treatments have been reported. Stringfellow (1976), who studied GaP LEDs, could not produce a permanent reduction of the surface recombination velocity, from its value of -2 x lo5 cm/sec, by covering the surface with silicon nitride. Fluorine treatments, however, did temporarily reduce the surface recombination velocity to about lo3 cm/sec. An effective and permanent passivation technique consists of covering the surface of GaAs (and probably other 111-V compounds) with closely lattice matching isotype heterojunctions. This produces a permanent reduction in the surface recombination velocity as discussed in Section 9.

1. THE EFFECT OF CRYSTAL DEFECTSON OPTOELECTRONIC DEVICES

31

For example, AlGaAs can be deposited on GaAs p-n junctions to reduce the surface leakage by several orders of magnitude (Casey et al., 1979). 9. RADIATION DAMAGE Irradiation with highenergy particles produces point defects which introduce nonradiative recombination centers in 111-V compounds, thereby decreasing the quantum efficiency of LEDs and increasing the threshold current density of laser diodes. A comprehensive device review was presented by Barnes (1977) and the properties of the centers formed were reviewed by Lang (1977). Proton bombardment is the most destructive treatment, and laser diodes exposed to substantial doses frequently cease to lase because of the large threshold current density increase. For example, AlGaAs/GaAs singleheterojunction lasers exposed to a dose of 25 pC/cm2(2-MeV protons) must be annealed at 500°C in order to at least partly recover some of their useful properties as shown by Minden (1976). The threshold current increase is not solely due to nonradiative centers, which depress the internal efficiency, but also to an increase in the absorptioncoefficient owing to the defects which are formed. The irradiation of devices with high-energy electrons or gamma rays produces relatively less damage than neutron irradiation and thus allows the study of the effect of native point defects on LED and laser properties. As reviewed by Barnes (1977), the effect of 2-MeV electron irradiation on the quantum efficiencydepends on the type of LED and on the operating current density. The carrier lifetime decreases with the dose a, owing to the introduction of nonradiative centers, following an expression of the form l/z = l/znr

+ l/zo + A@,

(14)

where A is a proportionality constant and ,z, is the initial nonradiative lifetime. Therefore, the internal quantum efficiency decreases with the dose @: qi = 7/70 = (1

+ 7,A@ +

70/7,,)-1.

(1 5)

It is evident from Eq. (15) that the impact of electron irradiation depends on the initial radiative and nonradiative lifetime of the device. Diodes with initially low radiative lifetimes are less sensitive to a given irradiation flux than those with initially high lifetimes. For example, GaAs:Si LEDs (grown by liquid-phase epitaxy) are more sensitive to irradiation because of their long carrier lifetimes (Barnes, 1977) compared to Zn-diffused LEDs. The study of electron or gamma ray irradiation effects on lasers and LEDs is complicated because defects may anneal in forward-bias operation even at room temperature. This effect results from the transfer of the bandgap

32

HENRY KRESSEL

energy, released by nonradiative electron-hole recombination, to localized lattice vibrational energy which in turn reduces the activation energy for point defect diffusion (Lang and Kimerling, 1974). The larger the bandgap energy, the more energy is released in the electron-hole recombination process and the more effective this enhanced displacement ‘process is expected to be. The effect has been demonstratedin GaAs (Langand Kimerling, 1974) and GaP (Lang and Kimerling, 1976) LEDs, where the density of recombinationcenterswas monitored by a capacitive technique as a function of time. The annealing process was greatly accelerated in forward bias compared to unbiased testing. Because of the above self-annealingduring high current density operation, it is difficult to study the impact of electron irradiation on the laser diode threshold current density and differential quantum efficiency. However, by operating the devices with short-current pulses, it is possible to deduce useful conclusions. Compton and Cesena (1967) studied homojunction GaAs lasers by this pulsed technique and concluded that the change in the internal quantum efficiency with irradiation accounted for the threshold current density increase. This result is qualitatively consistent with Eq. (9). 10. RECOMBINATION AT HETERO-INTERFACES The interfacial recombination at hetero-interfaces depends on the lattice misfit. In order to estimate the impact of lattice misfit on the interfacial recombination at heterojunctions, it is necessary to postulate an origin for the recombination centers. As noted earlier (Section 4), the misfit strain can be partly relieved by the formation of misfit dislocations but, in addition, point defectscan aqumulate at the interface. Thus, interfacialrecombination centers can be introduced by dislocations and/or point defects. It is also possible that nonradiative recombination processes are enhanced within the strain field at the interface (Johnston and Logan, 1976). Since there are insufficient data to allow a sorting of the various contributions, we shall use a simple model for estimating the interfacial recombination velocity as a function of lattice misfit. We assume the following: (1) misfit dislocations are formed to fully relieve the misfit strain and (2) the recombinationcenters are associated with the atoms which constitute the dislocation core. For a heterojunction on the (100) or (111) planes, the calculated interfacial recombination velocity S is (see, e.g., Kressel, 1974) =

KVth o t ( A a O / ~ O ) / a ~

7

(16)

where Aao/uo is the misfit strain, uth is the thermal velocity of the electrons ( x lo7 cm/sec at room temperature), 6,is the capture cross section for the recombination centers associated with the dislocation core. and K = 8 for the (100) plane and 8 / f i for the (1 11) plane.

1. THE EFFECT OF CRYSTAL DEFECTS OF OPTOELECTRONIC DEVICES

33

The value of S can be experimentally determined by several techniques. The two measurement methods of interest here, because of convenient application to suitable devices, consist of carrier lifetime and spectral response measurements. Consider first the dependence of the carrier lifetime on S. In a doubleheterojunction structure with a thin recombination region d and moderate S values, the carrier lifetime T is (Ettenberg and Kressel, 1976) l/z r l/zo

+ I/Z:~ + 2S/d,

(17)

where z is the “bulk” radiative lifetime and T : ~is the “bulk” nonradiative lifetime (i.e., excluding interfacial recombination). We assume equal S values at the two interfaces and SLID 4 1 ; L is the carrier diffusion length and D is the minority-carrier diffusion constant. For a single-heterojunctioninterface, the third term in Eq. (17) is S/d. From Eq. (17), it is possible to experimentally determine S from the dependence of z on d, assuming that z g is independent of d. The second method for determining S involves the spectral dependence of the short-circuit current I,, on incident light wavelength for heterojunction solar cells. An example of such a solar-cell structure of InGaP/GaAs is shown in Fig. 18 (Nuese, 1977). The photocurrent due to carriers generated near the surface by radiation strongly absorbed within the GaAs region is particularly sensitive to recombination at the InGaP/GaAs p-p interface. An estimate of S is obtained from curve-fitting I,, as a function of the incident radiation wavelength A by using known values of the GaAs absorption coefficient and minority-carrier diffusion length in GaAs. Detailed studies of the dependence of S on the lattice misfit were made using In,Gal - ,P/GaAs structures, where the lattice misfit was varied by changing the value of y during the vaporepitaxial deposition process, and misfit dislocations were known to be formed. Figure 19 shows S estimated from the dependence of z on din double-heterojunction structures (Ettenberg and Olsen, 1977) and from the spectral response of the single-heterojunction solar-cell structure of Fig. 18 (Nuese, 1977). The data show the expected increase of S with Aao/ao.The two solid lines correspond to Eq. (16) with o, = and lo-” cm’, respectively. Although substantial scatter is evident, the experimental values fall within the two lines : S = (3.8 f 1.2) x 107(Aao/ao).

(18)

Other interfaces studied include Pbo.B6Sno.l,Te/InP where doubleheterojunction lasers were analyzed. The lattice mismatch is about 0.3 %, and at 5 K, Kasemset and Fonstad (1979) found that the interfacial recombination velocity is about lo5 cm/sec. [Perhaps accidentally this value is consistent with Eq. (18).]

+-

PHOTOCURRENT

-y

1.0

~

5 J

g 0.8-

ILLUMINATION

I

~~

I-

ya 0.6 2

MISMATCH

n -GaAr

p-n JUNCTION

3

0

0.4

-

0.2

-

: gr a

01

0.60

0.65

0.70

0.75

0.80

0.85

b.

0.90

I

0.9;

WAVELENGTH ( p m )

a

b

FIG. 18. (a) Single-heterojunction solar-cell structure and (b) relative photodiode (short-circuit) current as a function of incident wavelength of the radiation for dimerent values of the lattice misfit. The values of S at the InGaP/GaAs interface deduced from the photodiode measurements are indicated. (After Nuese, 1977.)

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

I

35

n

/ \IT+

16'

= 16'" cmZ

lo-'

lo-'

LATTICE MISFIT STRAIN (hao/ ao)

FIG.19. Interfacial recombination velocity at an InGaP/GaAs interface as a function of the lattice misfit strain. The solid lines are from Eq. (16) with two values of the capture cross section a,.The experimental data were obtained by two different methods: D, solar-cell measurements using the dependence of the short-circuit current on incident light wavelength (Nuese, 1977); A, double-heterojunction diode measurements of the dependence of the carrier lifetime on heterojunction spacing. [S = (3.8 1.2) x lO'(Aa,/a,).] (After Ettenberg and Olsen, 1977.)

+

For very small misfit strains, misfit dislocations are usually not formed, thus leaving the crystal strained. But the absence of misfit dislocations does not mean the absence of interfacial recombination centers. Studies of AIGaAs/GaAs heterojunctions reveal substantial interfacial recombination without misfit dislocations. The origin of these recombination centers has not been established but could be due to the segregation of contaminants or vacancies at the interface. Therefore, the technology used to prepare the heterojunctions, the thermal history of the samples, as well as the contamination level of the materials, could influence the measured interfacial recombination velocity. The most comprehensive data concerning S were obtained from studies of T as a function of d in AlGaAs/GaAs double-heterojunction structures. The first studies (Ettenberg and Kressel, 1976) involved highly doped GaAs p-type recombination regions into which carriers were injected by a forward-biased junction. Later studies (Nelson and Sobers, 1978) also used double-heterojunction structures, but the minority carriers were produced by optical means (i.e., using incident radiation strongly absorbed within the

36

HENRY KRESSEL

HOLE DENSITY (cm-7

.,

FIG.20. Interfacial recombination velocity as a function of the hole concentration in the recombination region of double-heterojunction AI,,,Ga,,, As/GaAs diodes: Ettenberg and Kressel(l976); I, Nelson and Sobers (1978).

recombination region). Figure 20 summarizes the interfacial recombination velocity for Al,,Ga,.~As/GaAs heterojunctions as a function of the p-type doping in the GaAs recombination region. There appears to be a trend to increasing S with doping level but in all cases S < lo4 cmlsec-a low value compared to those of Fig. 19. Note that some of the available data (Ettenberg and Kressel, 1976) suggest that S in AlGaAs/GaAs double-heterojunction diodes does not remain constant with d. In fact, an apparent reduction in S with d 5 1 pm was noted. Other LED data are consistent with this result (Lee and Dentai, 1978). The data of Fig. 19 concern mostly heterojunctions where the epitaxial layer is in compression. However, when the layer is in tension, microcracks can easily be generated. When the heterojunction quality is thus grossly deteriorated, no meaningful correlation is possible between misfit strain and interfacial recombination velocity. One may assume that S is equal to the value found on a free surface. 11. CARRIER Lmnm AND RADIATIVE EFFICIENCY IN

HETEROSTRUCTURES The impact of a given interfacial recombination velocity on the carrier lifetime and radiative efficiencyin heterostructures depends on the density of injected carriers in the recombination regions and on its width d. Equation (17) provides an estimate of the Iifetime in a heterojunction device for given dand Svalues neglectingthe effect of the injected carrier density on either the radiative lifetime or on the interfacial recombination velocity. Hence, the internal quantum efficiency is computed from the ratio yli = z/zo.

1.

THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

37

Increasing the injected current density decreases the radiative lifetime. The radiative carrier lifetime as a function of the current density J is given by the general expression (Namizaki et al., 1974)

when no and p o are the initial (i.e., background) electron and hole densities, respectively, and B, is the band-to-band recombination coefficient. For a low carrier injection level relative to the background carrier concentration, Eq. (19) reduces to

+~0)I-l.

CBr(n0

TO

(20)

For high injection, where bimolecular recombination dominates,

z (ed/B,J)'/'.

(21) Because of a substantial density of nonradiative recombination centers in most materials, the carrier lifetime z is commonly shorter than zo . However, at sufficiently high injection levels, the lifetime in moderately doped direct bandgap semiconductors is expected to be limited by band-to-band radiative recombination. The carrier lifetime in an AlGaAs/GaAs :Ge double-heterojunction diode is shown in Fig. 21 as a function of the current density. The decrease in lifetime with increasing current is evident (Wittke et al., 1976). zo

J (A ,05000 I

loo I

1oO0500 200 I

1

I

50

20

I

x x

9-

x

I

x

X

8X

7-

- 6J!

z

X X

5X

4-

X

32-

lr 0' 0.02 0.040.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200.22 J-'/z ( ~ - ' / 2 cm)

-

FIG.21. Lifetime as a function of the inverse square root of the current density for a narrow active region DH LED with d = 0.1 pm and pa 10'' cm-3. The active region was GaAs and the surrounding material was A1,,,Ga0,,As. (After Wittke et al., 1976.)

38

HENRY KRESSEL

Given a material with few “bulk” nonradiative recombination centers 4 1) and neglecting Auger recombination, the internal quantum efficiency increases with J in the high-injection regime following an expression of the form (zo/z,,

Hence, qi = 0.5 is reached when J = 4eS2/dB,.

(23)

In the above simple model, we assume that S is independent of the freecarrier concentration in the recombination region, an assumption that may not hold at very high injection levels. However, the analysis does suggest that interfacial recombination substantially reduces the internal quantum efficiency as the heterojunction spacing is reduced even at high current densities. Consider, for example, a double-heterojunction diode where the carrier hole-electron pairs. (This is the injection level reaches 2 x 10” approximate density needed to reach threshold at room temperature in a typical GaAs laser.) Figure 22 shows the calculated internal quantum efficiency as a function of d for values of S = 0, lo3 cm/sec, and lo4 cm/sec. These curves show that even at the very high injection condition appropriate for laser operation, the value of S should be under lo4 cm/sec if very narrow

I . O . , , .

c *V

P0 LL W

sz 3

I

.

I

.

I

s=o

.

0.80.604

-

+

I-

J

a z

a

-

0.1-

0.2

-

W

I-

t

0

FIG.22. Calculated internal quantum efficiency [from Eq. (22)] as a function of d for three values of S in a double-heterojunction diode assuming an injected hole electron densityofrl x 1018cm-3andB,= 10-’0m-3/ see. It is assumed that S is independent of the free-carrier concentration.

‘

of2 ‘ d4

‘

d6

*

68

‘

I0

HETEROJUNCTION SPACING d ( p m )

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

39

PHOTON ENERGY ( e V f

FIG.23. Short-circuit current density/photon of an AlGaAs/GaAs solar cell initially and after the AlGaAs layer is removed (arbitrary units).The increased surface recombination degrades the cell performance. (After Woodall and Hovel, 1972.)

heterojunction spacings ( c 0.4 pm) are used. Typical laser diodes have heterojunction spacings of 0.1 to 0.3 pm in order to minimize the threshold current density. Excessive interfacial recombination will be reflected in an anomalously high threshold current density for a given d value because of the lowered internal quantum efficiency. Note that the above discussion concerns the internal quantum efficiency resulting from spontaneous carrier recombination, not stimulated carrier recombination. Therefore, the qi value under discussion particularly impacts the threshold current density, and not directly the differential quantum efficiency in the lasing region where the stimulated emission process (with its shortened carrier lifetime) dominates. With regard to heterojunction solar cells, the dependence of the efficiency on S varies with the structure and the light intensity (i.e., the nonequilibrium carrier concentration). The calculations of Ellis and Moss (1970) show that we need S 6 lo4 cm/sec in a GaAs solar cell in order to maintain the calculated efficiency close to the theoretical maximum of -25 % achieved when S = 0 (see Fig. 2). The addition of an Al,.,Ga,,,As heterojunction to the p-n homojunction, producing a p+-p-n structure as shown in Fig. l b (Woodall and Hovel, 1972), reduces the surface recombination velocity and hence improves the cell collection efficiency. Figure 23 shows the high short-circuit current density of an AlGaAs/GaAs cell with heterojunction initially present and its reduction after the AlGaAs removal because of the increased surface

40

HENRY KRESSEL

recombination of photogenerated carriers. The best AlGaAs/GaAs solarcell efficiencies are in the vicinity df 20% (see Woodall and Hovel, 1977, and references therein).

VII. Defects Contributing to LED and Laser Diode Degradation 12. INTRODUCTION Degradation in LEDs produces a quantum efficiency decrease with operating time. In laser diodes the threshold current density increases. As far as is presently known, the formation of defects in the recombination region during operation (which is mainly responsible for the degradation) is independent of whether the device is operating in the spontaneous or stimulated emission mode. The current density (i.e., the electron-hole recombination rate) is the major parameter affecting the degradation rate. Lasers may also degrade by a process related to the optical power density, whereby the crystal regions adjoining the facets become mechanically damaged. This degradation process, which will not be further discussed here, can be adequately controlled by limitation of the optical power density at the facets and the use of dielectric facet coatings (Ladany et al., 1977). The degradation of 111-V compound laser diodes and LEDs has been extensively studied since 1969 when it became evident that nonradiative defect centers were introduced within the GaAs laser recombination region during forward-bias operation (Kressel and Byer, 1969). It was also shown that the degradation rate was highest for devices which initially contained a high density of dislocations (Kressel et al., 1970). These observations contributed to the hypothesis that the energy released in nonradiative electronhole recombination contributed to the increase in the defect density within the recombination region by a process of enhanced atomic displacement. Earlier, Gold and Weisberg (1964), who studied the degradation of GaAs tunnel diodes, had suggested that Zn displacement was occurring in forwardbias operation by a “phonon kick” process whereby Zn displacement was eased by the energy locally released in nonradiative recombination at Zn sites. Although it is unlikely that accelerated dopant displacement is important in LED or laser diode degradation, it is now well established that the diffusion of some point deficts is accelerated by electron-hole recombination. Direct experimental evidence of a reduced diffusion activation energy with forward bias for point defects generated by the irradiation of GaAs diodes with 1-MeV electrons was reported by Lang and Kimerling (1974). Similar experimental data (Kimerling, 1978) have been obtained for other materials and a theoretical model has been presented by Weeks et al. (1975). In general,

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

41

the larger the bandgap of the material, the more energy is released in the electron-hole recombination process and, therefore, the more probable is the point defect diffusion enhancement. In fact, for large bandgap (- 2.2 ev) materials such as Gap, owing to the large amount of energy released, the diffusion of certain point defects in forward-biased diodes becomes an athermal process (Lang and Kimerling, 1976). In view of the importance of this recombination enhanced diffusion effect, it is important to examine all devices operating at high carrier injection levels for their susceptibility to property changes during operation. Susceptible devices include not only laser diodes and LEDs, but also microwave avalanche diodes as well as solar cells operated at high solar concentration levels where the current density can become quite large. In this section, we review the major experimental findings regarding the defects most relevant to the degradation of laser diodes and LEDs. The experimental work concerning the degradation of injection optoelectronic devices, resulting from internal defect increase, is consistent with the hypothesis of atomic displacement eased by nonradiative electron-hole recombination. The recombination enhanced diffusion of point defects has major consequences :(1) dislocation networks can grow by a process of climb resulting from the diffusion of native point defects to or from the dislocations ; and (2) the diffusion rate of impurity atoms could also be enhanced, particularly for species such as copper which may diffuse by an interstitial mechanism. In addition to the above phenomena which involve atomic diffusion, it is likely that dislocation motion and network growth by glide (a conservative process) is eased by nonradiative electron-hole recombination owing to the reduction in the energy otherwise needed for dislocation displacement. heteroEvidence for this process was found in Ino.,5Ga,,zsAso.,4Po~~~/InP structures by Mahajan et al. (1979). We limit our discussion to degradation phenomena which produce a decrease in the diode output as a result of defect density increase in the recombination region. For reference, it is useful to note that a 50 % reduction in quantum efficiency for a laser diode in the spontaneous emission mode typically corresponds to a 15-20% threshold current density increase (Ettenberg et al., 1974). This correlation permits the convenient testing of laser diodes below threshold and then relating the results to their lasing behavior (Kressel et al., 1978a). However, measurements in the spontaneous emission mode should be made at current densities comparable to those needed for lasing, i.e., at current densities usually in excess of 1000 A/cm2. The reason is that a low density of nonradiative centers, which may decrease the spontaneous emission efficiency at low current densities, will not noticeably affect the efficiency at high current densities.

42

HENRY KRESSEL

13. THEEFFECT OF DISLOCATIONS Studies of some degraded lasers and LEDs show large nonradiative regions introduced during operation. DeLoach et al. (1973) first observed these large nonradiative regions in cw AlGaAs/GaAs laser diodes and denoted them as “dark lines.” These regions were subsequently shown by Petroff and Hartman (1974) to consist of large dislocation networks which grew during operation by nonconservative climb, presumably by a process involving interstitial and/or vacancy diffusion to dislocations initially present in the device. Indeed, earlier work correlated the degradation rate of GaAs homojunction lasers and LEDs with the initial dislocation density in the device. Mettler and Pawlik (1972) found that a substrate dislocation density of 5 x lo3 cm-2 had little effect on GaAs:Si LED reliability but that dislocations in excess of 5 x lo4 cm-2 produced a noticeable reliability reduction. Kressel et al. (1970) found that an average dislocation density of 104 to lo5cm-’ resulted in lasers with high but varying degradation rates, as indeed expected because of the fact that the dislocation density in a given device of the wafer will vary. Since dislocations propagate from the substrate, it is clear that the dislocation density in the substrate should be minimized and that every effort should be made to grow structures as dislocation-free as possible. Misfit dislocations may be introduced in the growth of heterostructures with a lattice mismatch and these will impact reliability as illustrated below. An example of the detrimental effect of lattice mismatch on diode reliability is shown in Fig. 24 which compares the life of two sets of InGaAs/InGaP double-heterojunctiondiodes designed for emission at 1.06 pm (Olsen et al., 1979). One set of diodes had lattice-matched epitaxial layers (although the match to the GaAs substrate is obtained by severalgrading layers). The other set of diodes had 1% lattice-mismatched epitaxial layers, thus producing a high density of misfit dislocations in the recombination region. The latticemismatched diodes are seen to degrade much more rapidly than the latticematched ones. However, it is not possible to attribute these reliability differencessolely to different initial dislocation densities because the strain in the active region also differs. As discussed below, the strain difference could be relevant to the observed reliability difference. Despite the good lattice match, defects introduced during epitaxial growth of AlGaAs heterostructures may contribute to degradation (Ishida and Kamejima, 1979; Kawakami et al., 1978). In general, these defects consist of stacking faults and dislocationswhich thread through the epitaxial layers. It has also been suggested that devices produced from layers grown by liquid-phase epitaxy which exhibit strong “terracing ” can have above-

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

TIME

43

(h)

FIG.24. The effect of lattice mismatch on the reliability of InGaAs/InGaP double-heterojunction diodes emitting at 1.06 pm. The devices with a 1 % lattice mismatch degrade at a much higher rate than those without lattice mismatch. (After Olsen et al., 1979.)

average degradation rates (Kawakami et al., 1978), although the reason for this is unclear. The nonconservative climb of dislocations requires native point defects, and the question arises concerning their origin in diodes. In general, a supersaturation of vacancies and interstitials exists in crystals depending on the growth condition and cooling rate, and it would be interestingto correlate degradation processes with point defect densities. Unfortunately, direct measurements of point defect densities are difficult and information can only be inferred indirectly. In this category ofexperiments fall comparative studies of diodes with GaAs :Ge in the recombination region which were grown at various rates by liquid-phase epitaxy (Ettenberg and Kressel, 1975). The growth conditions which were believed to produce the lowest initial arsenic vacancy concentration in the crystal also produced the LEDs with the longest operating life. However, there is also a possibility that point defect diffusion from the substrate into the epitaxial layer is a factor in the degradation rate dependence on growth rate. Presumably, crystals grown at high growth rates are less likely to be contaminated by this out-diffusion process than more slowly grown crystals (Jastrzebski et al., 1979).

44

HENRY KRESSEL

Other experimental work pointing to the effect of vacancies on degradation concerns GaAsP LEDs. It was found by Haemmerling and Huber (1977) that exposing the material to a high arsenic overpressure during Zn diffusion can reduce the degradation rate. This effect was attributed to the reduction of the arsenic vacancy concentration in the material. In GaAsP/InGaP double-heterojunction diodes, Kressel et aZ. (1978b) found vacancy loops resulting from the condensation of excess vacancies in the crystal. These diodes degraded relatively rapidly. 14. THEEFFECT OF DAMAGED DIODEEDGJLS The preparation of laser diodes by cleaving two facets (to form the mirrors of the Fabry-Perot cavity) and sawing the other two sides (in order to prevent cross-lasing) is an old and familiar technique (Fig. 4a). However, such structures are liable to relatively rapid degradation owing to the dislocations introduced by the sawing (Kressel et al. 1977b; Ladany and Kressel, 1974). The generally used method of making reliable laser diodes (and LEDs for edge emission) is by the use of stripe contacts whereby the current is channeled into the device by a contact with a width less than that of the diode chip. Thus, the two lateral active diode sides are effectively buried within the bulk, and the current flow is away from two exposed edges. There are various techniques for forming stripe-contact structures. Two widely used ones consist of a metal stripe making contact in an area opened within an insulating surface (oxide-defined stripe) (Fig. 4b). Another technique consists of proton bombardment of the lateral regions of the device in order to increase the resistivity outside the ohmic contact region (for review, see Kressel and Butler, 1977). The dramatic effect of damaged edges on reliability is seen in Fig. 25 which compares the life of laser diodes produced by sawing the sidewalls and by the use of a stripe contact (Henshall and Hinton, 1978). Whereas the stripe-contact lasers exhibit no degradation, the sawn-side devices can be seen to fail very rapidly. A study of AlGaAs LEDs shows a direct correlation between the number of exposed edges and the degradation rate. The diode structures studied are shown in Fig. 26. Figure 27 shows that the diodes with no exposed edges (Fig. 26a) has the most stable output despite having a high operating current density, whereas the diode with four exposed edges (Fig. 26) has the worst reliability (Kressel et al., 1977b). A visual examination of the degraded devices showed nonradiative regions extending inward from the sawn edges. Transmission electron microscopy showed that these nonradiative regions consist of dislocation networks which extend from the work-damaged sides onZy into the active region but

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

a

\, 0

100

200

1

300

I

400

45

I 500

TIME (h)

FIG.25. Experiment comparing single-heterojunction AlGaAs/GaAs lasers fabricated using different processes in order to test the impact on reliability. The sawn-cavity lasers had exposed work-damaged edges and degraded rapidly. The stripe-contact lasers, formed using proton bombardment in order to restrict the current flow away from the edges, d o not degrade. The lasers were operated pulsed with pulse lengths of 200 ns at 15 KHz. (After Henshall and Hinton, 1978.)

not into the passive bounding layers of the devices (Henshall and Hinton, 1978). This supports the conclusion that electron-hole recombination is necessary for the growth of the dislocation networks. It is unknown whether these dislocation networks expand by a nonconservative climb mechanism, by conservative glide, or by a combined process.

(b) (C 1 FIG.26. Various AlGaAs LED structures used to study the effect of exposed edges, where electron-hole recombination occurs, on reliability. (a) All four sides are sawn; (b) one exposed edge is cleaved; (c) no exposed edges. (After Kressel et al., 1977b.)

46

HENRY KRESSEL

I .2 1.1 I-

n 3

5

t

I.O.--a

0.9-

0

t- 0.80

2 0.7W

2

I-

0.6-

a a

0.50.4

-

0.30.2

-

0.1I

t

I

I

I 1 1 1 1

I

I

I I I1111

I00

1000 HOURS

I

I

1

1 1 1 1 1

10.000

1 100.000

FIG.27. Relative light output as a function of operating time for three sets of AI,,,,Gao.,,As: [Si( 100)]double-heterojunction LEDs.Thestructures studied are shown in Fig. 26and thediodes were operated at various current densities. - x ,four exposed edges (operated at 1000 A/cm2); one exposed edge (operated at 4000 A/cm2); -0,no exposed edges (operated at 3500 A/cm2).The devices with no exposed edges have the best reliability, with the reliability decreasing as the number of exposed edges increases. (After Kressel et a/., 1977b.)

--+.

15. PROCESS-INDUCEDDEFECTS In addition to defects which propagate into the active region of the device from the substrate or are introduced during epitaxial growth, various defects may also be introduced during device fabrication and handling. For example, scratching the surface above the active region introduces dislocations into the material. A more subtle effect is caused by selective Zn diffusion through an oxide mask under conditions where the surface concentration is very high and the Zn diffusion front reaches the recombination region (Ladany and Kressel, 1974). Kawakami et al. (1978) found, by transmission electron microscopy, that many small dislocation loops of a n interstitial character were sometimes formed in selective Zn diffusion near the edge of the oxide mask defining the stripe area. This could result from the local stress concentration at these apertures in the oxide mask. These dislocations then constitute the seed defects for degradation.

1.

THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

47

16. THE EFFECT OF STRESS Externally applied stress to LEDs and laser diodes during operation may rapidly accelerate their degradation. Eliseev and Khaidarov (1975) studied single- and double-heterojunction AlGaAs/GaAs lasers operated under uniaxial compression perpendicular to the junction plane. The degradation increased with stress and resulted from the growth of dislocation networks by a process of climb and probably slip as well. It is supposed that initially present dislocations constitute the starting nuclei for the multiplication process. The stress provides a driving force for the dislocation expansion which could be eased by the energy released in nonradiative recombination. Direct evidence for the detrimental effect of external stress on reliability of GaAs devices was also obtained in studies where the solder used to attach the laser to a diamond heat sink was changed from indium (a soft solder) to a Sn solder which placed considerable stress on the device (Hartman and Hartman, 1973). However, it is unclear whether these stress-related degradation phenomena are always important. For example, it is not known whether such effects readily occur in diodesmade from low bandgap materials where the energy available from nonradiative electron-hole pair recombination is relatively low compared to the atomic displacement energy. Stresses can also be produced by the isolating oxide in stripe-contact lasers. Goodwin et af. (1979) found that the magnitude of the stress depends on the shape of the oxide profile and its thickness. A stress value below 10' dyn/cm2 in the active region was found necessary for reliable devices. We now turn our attention to stresses in diodes resulting from lattice misfit strain. Here, the stress is determined on the basis of lattice parameter measurements which give strain values. Consider the AlGaAs/GaAs structure. Although the lattice parameters of GaAs and AlGaAs are equal at the growth temperature of about 800°C, their thermal coefficients of expansion differ, which leads to strain in the device at room temperature (Ettenberg and Paff, 1972). The AlGaAs layers are always in compression because they have a larger lattice parameter than GaAs. Therefore, a GaAs layer sandwiched between two AlGaAs layers is in tension. However, the strain can be reduced with the addition of A1 to the middle layer, as shown in Fig. 28, reaching a condition of no strain at an A1 concentration of about 5%. With the further addition of Al, however, the epitaxial layers are increasingly in tension and quite high stress values can be reached. The question arises as to the effect of the above stress on device reliability. It is known (Ettenberg et al., 1974) that the reliability of AlGaAs/GaAs diodes improves as small amounts of A1 are added to the recombination region (thus forming Al,Ga, - y As/Al,Ga, -,As structures) at least until x = 0.1. Several factors may account for this improvement of which the

48

HENRY KRESSEL

R

10

lo6 N

1

5

\

-

c %

0

v) v)

-10'

W

a I-

v)

-10

-10

i 0.25

0.0

0.05

0.10

0.15 X-Al

0.20

FIG.28. Stressin the active region of AI,Ga, -,As/A1,Ga1 -,As heterostructuresas a function of the Al concentration x.-, y = 0.25; y - x = 0.25; ---, y = 0.5. (After Olsen and Ettenberg, 1978.)

---.

reduced stress and the change from tension to compression is one. However, other factors for the improved reliability could include the elimination of oxygen from the recombination region and a lower point defect density in AlGaAs compared to GaAs. More direct evidence for the detrimental effects of built-in stress on degradation is suggested by the experiments of Ladany and Kressel(l978) who found that Al,Ga,-~s/Al,Ga,-,As lasers where x > 0.1 tend to degrade with increasingAlcontent. As notedearlier, anincreasein the Alcontentofthe epitaxial layer increases the lattice misfit stress between the layers and GaAs substrate. However, by increasing the thickness of the first Al,Ga, -,As layer of the heterostructure, it is possible to reduce the stress in the recombination region because the layers become more nearly comparable in thickness to the GaAs substrate which now bends to partly relieve the stress in the

1. THE EFFECT OF CRYSTAL DEFECTS ON OPTOELECTRONIC DEVICES

49

epitaxial layers. Ladany et al. (1979) found that a reduction in the stress from 2 x lo8 dyn/cm2 to 5 x lo7 dyn/cm2, by the use of thick epitaxial layers, resulted in a substantial improvement in the reliability of lasers. This stress reduction for reliable operation is consistent with that obtained by Goodwin et al. (1979) who, as noted above, concluded that the stress should not exceed 10' dyn/cm2.

REFERENCES Abrahams, M . S., Blanc. J., and Buiocchi, C. J. (1972). fhilos. M q . 23, 795. Abrahams, M. S., Buiocchi, C. J.. and Tietjen, J. J. (1967). J. Appl. f h y s . 38, 760. Abrahams. M. S., Buiocchi. C. J.. and Williams. B. F. (1971). Appl. f h j s . Lett. 18, 220. Barnes, C. E. (1977). Rep. SAND76-0726. (Available from NTIS, U.S. Dep. Commer., Springfield, Virginia.) Blenkinsop. I. D., Harding, W., and Wright, D. R. (1976). Electron. Lett. 12, 503. Bohm. K.. and Fischer, B. (1979). J. Appl. f h y s . 50, 5453. Bottger. 0..and Richter, E. (1962). Z. Naturforsch.. Teil A 17, 529. Brantley, W. A., Lorimer, 0.G., Dapkus, P. D., Hazko. S. E., and Saul, R. H. (1975). J . Appl. f h y s . 46. 2629. Busta, H. H., and Waggener, H. A. (1977). J. Electrochem. SOC.124, 1424. Card. H. C.. and Yang, E. S. (1976). IEEE Int. Electron. Devices Meet. Tech. Dig. p. 474. Casey, H. C., Jr., and Buehler, E. (1977). Appl. f h y s . Lett. 30, 247. Casey, H. C., Jr., Cho. A. Y.,and Foy, P. W. (1979). Appl. f h y s . Lett. 34, 594. Compton. D. M. J., and Cesena, R. A . (1967). IEEE Trans. Nucl. Sci. 14, 15. DeLoach, B. C., Hakki, B. W., Hartman, R. L., and DAsaro, L. A. (1973). Proc. IEEE61, 1042. Dishman. J. M., Haszko. S . E., Marcus, R. B., Muraka, S. P., and Sheng, T. T. (1979). J . Appl. P h j s SO. 2689. Druzhinina. L. V., Bublik. V. T., Dolginov, L. M., Eliseev, P. G.. Kerbelev, M. P., Osvenskii, V. B.. Pinsker, 1. Z.. and Shumskii, M. G. (1974). Zh. Tekh. Fi;. 44,No. 7, 1499 [Engl. transl., Sor. f h y s - T e c h . f h y s . 19, 935 (1975)l. Eliseev, P. G., and Khaidarov, A. V. (1975). Kruntoqyu Elektron. (Moscow) 2. No. I , 127 [Engl. transl.. Sor. J . Quantum Elec,tron. 5. 73 (1975)l. Ellis. B., and Moss, T. S. (1970). Solid-State Electron. 13, 1. Esquivel. A. L., Sen, S.. and Lin, W. N . (1976). J. Appl. Phys. 47, 2588. Ettenberg. M. (1974). J . Appl. f h w . 45, 901. Ettenberg, M.. and Kressel, H. (1975). Appl. f h y s . Lett. 26, 478. Ettenberg. M., and Kressel. H. (1976). J . Appl. fh-vs. 47, 1538. Ettenberg, M., and Olsen. G . H. (1977). J . Appl. f h y s . 48, 4275. Ettenberg, M., and Paff, R. J. (1972). J. Appl. f h y s . 43, 3760. Ettenberg, M.. Kressel, H., and Lockwood. H. F. (1974). Appl. f h y s . Letr. 25, 82. Faktor. M. M., and Stevenson. J. L. (1978). J . Electrochem. Soc. 125, 621. Glaenzer. R. H., and Jordan, A. G . (1969). Solid-State Electron. 12, 247. Gobat, A. R.,Lamorte. M. F., and Mclver, G . W. (1962). IRE Trans. Mil.Electron. 6 . 20. Goetzberger, A..'and Shockley. W. (1960). J. Appl. fhj,s. 31, 1821. Gold, R. D., and Weisberg, L. R.(1964). Solid-State Ekectron. 7 , 81 I . Goodwin. A. R., Kirkby. P. A.. Davies, I. G . A.. and Baulcomb. R. S. (1979). Appl. f h y s . Lett. 34, 647.

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Haemmerling, H., and Huber, D. (1977). J. Electron. Muter. 6 , 581. Hartman, R. L., and Hartman, A. R.(1973). Appl. Phys. Lett. 23, 147. Heine, V. (1968). Phys. Rev. 146, 568. Heinke, W., and Queisser, H. J. (1974). Phys. Rev. Lett. 33, 1082. Henry, C. H., Logan, R. A., and Merritt, F. R. (1978). J. Appl. Phys. 49, 3530. Henshall, G. D., and Hinton, R.E. P. (1978). IEEJ. Solid-State Electron Devices 2, 31. Herzog, A. H., Keune, D. L., and Craford, M.G. (1972). J . Appl. Phys. 43,600. Hovel, H. J. (1975). “Solar Cells.” Academic Press, New York. Hubbard, G. S., and Haller, E. E. (1979). Electron. Muter. Conf., Boulder, Colo. Iles, P. A., and Soclof, S. I. (1975). Proc. IEEE Photovoltaic Specialists Con(, Ilth, New York, p. 25. Ishida, K., and Kawakami, T. (1979). J. Electron. Muter. 8, 57. Jastrzebski, L. L. (1979). Unpublished observations. Jastrzebski, L. L., Lagowski, J., and Gatos, H. C. (1976). Appl. Phys. Lett. 27, 537. Jastrzebski, L. L., Lagowski, J., Gatos, H. C., and Welukiewicz, W. (1978). GaAs Relat. Compd. Conf. Proc. No. 45. Symp., Phys. Soc., London, 1978 p. 437444. Jastrzebski, L. L., Lagowski, J., and Gatos, H. C. (1979). J. Electrochem. Soc. 126, No. 12,22312234. Johnston, W. D., Jr., and Logan, R.A. (1976). Appl. Phys. Lett. 28, 140. Kamieniecki, E. (1976). J. Phys. C 9, 1211. Kasemset, D., and Fonstad, C. G. (1979). Appl. Phys. Lett. 34,432. Kawakami, T., Saito, H., and Shinoda, Y . (1978). Rev. Electr. Commun. Lab. 26, 1152. Kimerling, L. C. (1978). Solid-State Electron. 21, 1391. Kimerling, L. C., and Patel, J. R. (1979). Appl. Phys. Lett. 34, 73. Kressel, H. (1974). J. Electron. Muter. 3, 747. Kressel, H.. and Butler, J. K. (1977). “Semiconductor Lasers and Heterojunction LEDs.” Academic Press, New York. Kressel, H., and Byer, N. E. (1969). Proc. IEEE 57, 25. Kressel, H., and Ettenberg, M. (1973). Appl. Phys. Lett. 23, 51 1. Kressel, H., Abrahams, M. S., Hawrylo, F. Z., and Buiocchi, C. J. (1968). J. Appl. Phys. 39, 5139. Kressel, H., Byer, N. E., Lockwood, H. F., Hawrylo, F. Z., Nelson, H., Abrahams. M. S., and McFarlane, S . H. (1970). Metall. Trans. 1, 635. Kressel, H., D’Aiello, R. V.,Levin, E. R.,Robinson, P. H., and McFarlane, S. H. (1977a). J. Cryst. Growth 39, 23. Kressel, H., Ettenberg, M., and Lockwood, H. F. (1977b). J. Electron. Muter. 6, 467. Kressel, H., Ettenberg, M., and Ladany, I. (1978a). Appl. Phys. Lett. 32, 305. Kressel, H., Nuese, C. J., and Olsen, G. H. (1978b). J. Appl. Phys. 49, 3140. Kurtz, A. D., Kulin, S. A., and Averbach, B. V . (1956). Phys. Rev. 101, 1285. Ladany, I., and Kressel, H. (1974). Appl. Phys. Lett. 25, 708. Ladany, I., and Kressel, H. (1978). Electron. Lett. 14, 407. Ladany, I., Ettenberg, M., Lockwood, H. F., and Kressel, H. (1977). Appl. Phys. Lett. 30,87. Ladany, I., Furman, T. R.,and Marinelli, D. P. (1979). Electron. Lett. 15, 342. Lagowski, J., Baltov, I., and Gatos, H. C. (1973). Surf.Sci. 40,216. Lang, D. V. (1977). Inst. of Phys. Conf. Ser. No. 31, Review of Radiation Induced Defect in I I I - V Compounds, p. 70. Lang, D. V., and Kimerling, L. C. (1974). Phys. Reu. Lett. 33,489. Lang, D. V., and Kimerling, L. C. (1976). Appl. Phys. Left. 28,248. Lawrence, J. E. (1973). In “Semiconductor Silicon” (H. R. Huff and R. R. Burgess, eds.), p. 17. Electrochem. SOC.,Princeton, New Jersey.

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Lee, T. P., and Dentai, A. G. (1978). IEEE J. Quantum Etectron. 14, 150. Leistiko, O., Jr., and Bittmann, C. A. (1973). Solid-State Electron. 16, 1321. Lemke, H. (1965). fhys. Status Solidi 12, 125. Mclntyre, R. J. (1979). Unpublished observations. Mahajan, S., Johnston, W. D., Jr., Pollack, M. A,, and Nahory, R. E. (1979). Appl. fhys. Lett. 34, 717. Marcus, R. B., Robinson, M., Sheng, T. T., Haszko, S. E., Muraka, S. P., and Katz, L. E. (1977). J. Electrochem. Sor. 124, 425. Matthews, J. W. (1975). In “Epitaxial Growth,” Part B (J. W. Matthews, ed.), p. 562. Academic Press, New York. Mettler, K., and Pawlik, D. (1972). Siemens Forsch.-Entwicklungsber. 1,274. Milvidskii, M. G., and Osvenskii, V. B. (1977). Sou. fhys. Crystallogr. 22, 246. Minden, H . T. (1976). J. Appl. fhys. 47, 1090. Mueller, R. K. (1961). J. Appl. fhys. 32, 640. Nagai, H., and Noguchi, Y. (1978). Appl. fhys. Lett. 33, 312. Namizaki, H., Kan, H., Ishi, M., and Itoh, R. (1974). Appl. Phys. Lett. 24, 486. Nelson, R. J., and Sobers, R. G. (1978). Appl. fhys. Lett. 32, 761. Nuese, C. J. (1977). J. Electron. Muter. 6, 253. Okada, J. (1955). J. Phys. Soc. Jpn. 10, 1110. Olsen, G. H. (1975). J. Cryst. Growth 31, 223. Olsen, G. H. (1979). Unpublished observations. Olsen, G. H., and Ettenberg, M. (1978). In “Crystal Growth”(C. H. L. Goodman, ed.), Vol. 2, Chap. I, pp. 1-56. Plenum, New York. Olsen, G. H., and Kressel, H. (1979). Electron. Lett. 15, 141. Olsen, G. H., Nuese, C. J., and Smith, R. T. (1978). J . Appl. fhys. 49, 5523. Olsen, G. H., Nuese, C. J., and Ettenberg, M. (1979). IEEEJ. Quantum Electron. 15, 688. Petroff, P. (1975). Lattice Defects Semicond., Inst. Phys., London, 1974, Conf: Ser. No. 23, p. 73. Petroff, P., and Hartman, R. L. (1974). J . Appl. fhys. 45, 3899. Rao-Sahib, J. S., and Wittry, D. B. (1969). J. Appl. fhys. 40,3745. Ravi, K . V., Varker, C. J., and Volk, C. E. (1973). J. Electrochem. Soc. 120, 533. Ravi, K. V., Serreze, H. B., Bates, H. E., Morrison, A. D., Jewett, D. N., and Ho, J. (1975). Proc. IEEE Photovoltaic Specialists Conf., Ilth, New York p. 280. Redfield, D. (1980). IEEE Trans. ED-27,766. Roedel, R. J., Van Neida, A. R., Caruso, R., and Dawson, L. R. (1979). J. Electrochem. SOC.126, 637. Rosi, F. D. (1958). R C A Rev. 19, 349. Sah, C. T., Noyce, R. N., and Shockley, W. (1957). froc. IRE45, 1228. Salama, A. M. (1979). J. Electrochem. Soc. 126, 114. Schroter, W. (1979). Defects Radiat. E$ Semiconduct., Inst. fhys.,’ London, Con$ Ser. No. 46, p. 114-127. Schwuttke, G. H., Ciszek, T. F., Yang, K. H., and Kran, A. (1978). I B M J . Res. Dev. 22, 335. Seager, C. H., and Ginley, D. S. (1979). Appl. fhys. Lett. 34, 337. Spencer, M., Stall, R., Eastman, L. F., and Wood, C. (1979). J . Appl. fhys. 50, 8006-8009. Stillman, G. E., and Wolfe, C. M. (1977). In “Semiconductors and Semimetals”(R. K. Willardson and A. C. Beer, eds.), Vol. 12, p. 291. Academic Press, New York. Stringfellow, G. B. (1976). J. Vac. Sci. Technol. 13, 908. Stringfellow, G. B., Liridquist, P. F., Cass, T. R., and Burmeister, R. A. (1974). J. Electron. Muter. 3, 497. Suzuki, T., and Matsumoto, Y. (1975). Appl. fhys. Lett. 26, 431. Takeshima, M. (1972). J. Appl. fhys. 43, 4114.

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Titchmarsh, J. M., Booker, G. R., Allenson, M., and Rowland, M. C. (1977). As quoted in Wight, D. R. (1977). J. Phys. D 10,431. Vink, A. T., Werkhoven, C. J., and Opdorp, C. V. (1978). “Semiconductor Characterization Techniques,” p. 259. Electrochem. Soc.,Princeton, New Jersey. Webb, P. P., McIntyre, R.J., and Conradi, J. (1974). RCA Rev. 35,234. Weeks, J. D., Tully, J. C., and Kimerling, L. C. (1975). Phys. Rev. B 12, 3286. Werhoven, C. J., van Opdorp, C., and Vink, A. T. (1976). Gallium Arsenide Relat. Compd., Inst. Phys., London, Conf. Ser. No.33a, p. 317. Wertheim, G. K., and Pearson, G. L. (1957). Phys. Rev. 107,694. White, A. M. (1975). J . Muter. Sci. 10, 714. Wittke, J. P., Ettenberg, M.,and Kressel, H. (1976). RCA Rev. 37, 159. Woodall, J. M., and Hovel, H. J. (1972). Appl. Phys. Lett. 21, 379. Woodall, J. M., and Hovel, H. J. (1977). J. Crysr. Growth 39, 108. Young, M. L., and Rowland, M. C. (1973). Phys. Status Solidi A 16,603. Zschauer, K. H. (1969). Solid State Commun. 7, 1709.

SEMICONDUCTORS AND SEMIMETALS. VOL . 16

CHAPTER 2

Crystal Growth and Properties of Hg. -.Cd. Se Alloys C. R . Whitsett. J . G . Broerman. and C. J . Summers

. . . . . . . . . . . . . . . . . . . 2 . Constitutional Phase Diagram . . . . . . . 3 . Free Energy of Mixing . . . . . . . . . 4 . Vapor Pressure . . . . . . . . . . . CRYSTAL GROWTH OF Hg, ..Cd. Se ALWYS . . . . 5 . Bridgman Crystal-Growth Method . . . . . . 6 . Ingot Preparation . . . . . . . . . . 7. Crystal-Growth Kinetics . . . . . . . . 8. Thin Film and Epitaxial Growth . . . . . . 9. Annealing and Chemical Doping . . . . . . 10. Defect Structure . . . . . . . . . . . ELECTRON ENERGY-BAND STRUCTURE . . . . . . 1 1. Perfect-Semimetal to Semiconductor Transition . . 12. Band Structure Calculational Model . . . . . DIELECTRIC FUNCTION. . . . . . . . . . 13. Electronic Part of the Dielectric Function . . . . 14. Lattice Part of the Dielectric Function . . . . MAGNETOTRANSPORT. . . . . . . . . . 15. Shubnikoo-de Haas Effect . . . . . . . . 16. Hall Effect . . . . . . . . . . . . . OPTICAL PROPERTIES . . . . . . . . . . 17. Fundamental Absorption Edge . . . . . . . 18. Optical Reflectivity . . . . . . . . . . 19. Free-Carrier Reflectivity . . . . . . . . 20. Lattice Reflectivity . . . . . . . . . . 21 . Photoconductivity . . . . . . . . . . ELECTRON SCATTERING MECHANISM AND TRANSPORT. 22. Electron Scattering Mechanisms . . . . . . 23. Polar Longitudinal Optical-Phonon Scattering . . 24. Nonpolar Optical-Phonon Scattering . . . . . 25. Acoustic-Phonon Scattering . . . . . . . 26. Electron-Hole Scattering . . . . . . . . 27 . Compositional-Disorder Scattering . . . . . 28. Defect Scattering . . . . . . . . . .

I . INTRODUCTION .

HgSe-CdSe SYSTEM. 11. THEPSEUDOBINARY 1. Crystallographic Phases . . . .

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C. R. WHITSETT, J. G . BROERMAN, AND C. J. SUMMWS 29. Experimental and Theoretical Electron Mobilities 30. Thermal Transport . . . . . . . . .

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I. Introduction The Hg, -,Cd,Se and Hg, -,Cd,Te alloys are quite similar. (See Volume 18 of this treatise.) Each system is a solid solution of a symmetry-induced zero-gap semiconductor and a wide-gap semiconductor, and the energyband gap increases monotonically with increasing x. The selenides have higher melting temperatures than the tellurides, which makes their preparation in sealed quartz capsules more difficult, and they are n-type as grown in contrast with the tellurides which tend to be p-type. The Hg, -,Cd,Se alloys have potential for infrared detection applications as they are stable n-type conductors. HgSe and CdSe crystallize into two different structures, but the compositions of Hg, -,Cd,Se alloys that are of importance for infrared detector applications crystallize in the cubic, zinc-blende structure, and the existence of a second crystal structure for highcadmium compositions is of no practical consequence. The electron energy-band structuresof the Hg, -,Cd,Te and Hg, -,Cd,Se alloys are nearly identical, and their intrinsic electrical and optical characteristics are similar. The most significant difference between the two alloy systems is that the telluride alloys form with a Hg deficiency, which tends to make them p-type, and the selenide alloys form with a stoichiometricexcess of Hg, which makes them n-type. The annealing procedures and chemical dopants that must be employed to control the electrical characteristics are therefore different for the two alloy systems. Wide-gap 11-VI semiconductors have been reviewed by Reynolds et al. (1965), Thomas (1967), and Aven and Prener (1967), and the emphasis of this article is therefore on HgSe and compositions of Hg, -,Cd,Se alloys (x < 0.70) that have possible application as thermoelectric coolers and infrared sensors.

II. The Pseudobinary HgSe-CdSe System 1. CRYSTALLOGRAPHIC PHASES

The compound HgSe crystallizes in the cubic, zinc-blende structure. In this structure, the Hg atoms form a face-centeredcubicsublattice, and the Se atoms form another face-centeredcubicsublattice displaced one-quarter of a body diagonal from the Hg lattice. The lattice constant a. of HgSe at 20°C has been measured as 0.6088 nm by Cruceanu et al. (1966), 0.6085 nm by

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg, -,Cd,Se

55

ALLOYS

Swanson et al. (1957a), 0.6086 nm by Singh and Dayal(1970), and 0.6086 nm by Kalb and Leute (1971). The compound CdSe crystallizes in the hexagonal, wurtzite structure. The 20°C lattice constants, a and c, have been measured as 0.4299 nm and 0.7010 nrn by Swanson et al. (1957b), 0.4300 nm and 0.7020 nm by Kot and Mshenskii (1964), 0.4309 nm and 0.7024 nrn by Stuckes and Farell (1964), 0.4298 nm and 0.7009 nrn by Cruceanu and Niculescu (1965), and 0.4301 nm and 0.7014 nm by Kalb and Leute (1971). The zinc-blende and wurtzite lattices are both formed by tetrahedral bonding between anions and cations. The closest packing of atoms occurs in the (111) planes of the zinc-blende lattice, and the (111) planes of each atom species have an ABCABC . ..stacking sequence. The packing is identical in the wurtzite (OOO1) planes, but the stacking sequenceis ABAB ... The Se-Se interatomic spacing is nearly equal in HgSe (0.4304 nm) and CdSe (0.4301 nm). For comparison, the Te-Te spacing is 0.4569 nm in HgTe and 0.4583 nm in CdTe, both of which have the zinc-blende structure. The variation with x of the lattice constant for Hg,-,Cd,Se alloys as determined by Kalb and Leute (1971) is compared in Fig. 1 with the lattice +

0.6485 ~

CdSe Mole fraction of CdSe. x

FIG.1. Variations of cubic unit-cell lattice constant with x in tke Hg,-,Cd,Se and Hg, -,rCd,Te alloy systems: for the Hg,-,Cd,Se alloys with x > 0.81, J2a is plotted, where a is the hexagonal unit-cell edge. [Hg,-,Cd,Se data from Kalb and Leute (1971); Hg,-,Cd,Te data from Woolley and Ray (1960).]

56

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R. WHITSETT, J. G . B R O W N , AND C. J. SUMMERS

constant variation for the Hg, -,Cd,Te alloys determined by Woolley and Ray (1960). The plot for the wurtzite phase of Hg,-,Cd,Se in Fig. 1 is of because this quantity is directly comparable with the cubic lattice constant a,. Between x = 0.77 and x = 0.81, at 20°C the zinc-blende structure transforms to the wurtzite structure. Iwanowski (1975), using samples whose homogeneity was established by electron microprobe analyses, determined that the mass density of Hg, -,Cd,Se decreases linearly with x through both the zinc-blende and wurtzite phases as shown in Fig. 2. Precision mass density measurements can thus be used to measure alloy compositions,and with care the composition of a sample approximately 4 x 4 x 1 mm can be determined to an accuracy of k0.002 mole fraction CdSe.

PHASE DIAGRAM 2. CONSTITUTIONAL Nelson et al. (1977) determined the phase diagram of the pseudobinary HgSe-CdSe alloy system by measuring the thermal-arrest temperatures 8.5

1

1

1

1

1

1

1

1

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8.0

7.5

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I

E

-f.-2:

7.c

6.1 (Wurtzi I

61

i

(Zinc ~ e n d e ~ I

-

I

5.:

1

1 0.2

1

1 0.4

1

1

1

0.6

1 0.8

1 1 .o

cdse Mole fraction of CdSe. x

FIG.2. Variation of density with x for the Hg, -,Cd,Se system. (From Iwanowski, 1975.)

2.

CRYSTAL GROWTH AND PROPERTIES OF

1200

Hg, -,Cd,Se

ALLOYS

57

-

Zinc Men&

a

0.2

0.4

0.6

0.8

bSe

1.o Cdse

Mole fraction of CdSe in melt, X,

FIG.3. Experimental data points and calculated liquidus and solidus curves for the HgSe-CdSe pseudobinaryalloy system;,).( data points obtained from thermal arrest measurements; (0),data points from first-to-freezetip of Bridgman grown ingots. (From Nelson et d., 1977.) - - -,ideal solution theory; -, nonideal solution theory.

during heating and cooling of different alloy compositions and the first-tofreeze composition of slowly frozen ingots grown by the Bridgman method (see also Whitsett et al., 1976). Their results, shown in Fig. 3, were corrected for the loss of mercury from the melt to the vapor phase by using vapor pressures calculated on the basis of Raoult's law. Values reported for the melting temperature of CdSe are 1239 f 3°C by Reisman et al. (1962), 1250 f 2°C by Kulwicki (1963), 1252 f 0.5"C by Strauss and Steininger (1970), 1258 f 3°C by Mason and OKane (1961), 1259 f 4°C by Cook (1968), and 1264 It 10°C by Sysoev et al. (1967). The average of these values is 1254"C, which is plotted in Fig. 3. The melting temperature of HgSe was reported by Strauss and Farrell (1962) to be 799 f 1"C, and this value also was measured by Whitsett and Nelson (1969). For x values between 0.33 and 0.55, the Hg, -,Cd,Se alloys solidify with the wurtzite crystal structure but transform peritectically to the zinc-blende crystal structure at 947.5 f 4.0"C. Nelson et al. (1977) observed no thermal arrest from a peritectic reaction for x = 0.60, and the peritectic isotherm must therefore terminate between 0.55 and 0.60. The two-phase, wurtzite and zinc-blende, solid region (see insert in Fig. 3) has a small width at the peritectic temperature because no influence on the shape of the solidus curve near the peritectic temperature was discerned. The phase diagram curves in Fig. 3 do not represent phase equilibria at a constant pressure but,

58

C. R. WHITSETT, J. G. BROERMAN, A N D C. J. SUMMWS

rather, at different pressures determined by the solid-liquid-vapor equilibrium for each alloy composition. The pressure coefficient of the melting temperature of CdSe has been reported to be -0.0015 K/MPa by Jayoraman et al. (1963). No value has been reported for the pressure coefficient of the melting temperature of HgSe, but the value of 0.045 K/MPa for HgTe reported by Jayoraman et al. (1963) can be used as a guide. Using these values, Nelson et al. (1977) showed that the effect of the vapor pressure on the liquidus and solidus temperatures is small (<0.5”C) and can be neglected. 3. FREE ENERGY OF MIXING From fits of the phase diagram data by a thermodynamic, nonidealsolution theory for completely miscible solutions, interaction parameters for the solid and liquid phases were obtained by Nelson et al. (1977). This approach is justified for the HgCdSe alloy system because the peritectic transition has little effect on the slope of the solidus line. The changes in the chemical potentials that result from mixing x moles of component A and (1 - x) moles of component B in the liquid state are

+ BL(xL)’

Api = RT ln(1 - xL)

(1)

and

A& = R T In xL + BL(l - xL)’, where R is the molar gas constant, T is the temperature, and L denotes the liquid state. The coefficient B L is the free energy of mixing that results from the reforming of chemical bonds and van der Waals forces between neighboring atoms; for an ideal solution, BL = 0. The changes in the chemical potentials in the solid state upon mixing A and B are Ap; = R T ln(1 - xs)

+ BS(xs)* + LA(T/TA- 1)

(3)

and

A& = RT ln(xS) + p ( 1

+ LB(T/TB- l),

(4) where L A and LBare the molar enthalpies of fusion, T A and T’ are the melting temperatures of A and B, respectively, and S denotes the solid state. The coefficient p is the free energy of mixing in the solid state and is usually larger than BL because it has contributions, such as from lattice distortion, that are in addition to the energy of mixing in the liquid. Each of the free energy-of-mixing coefficients, @ and BL,can be expressed as the sum of an enthalpy and an entropy, i.e., BL = B:

- xS)’

+ BkRT

(5)

2.

CRYSTAL GROWTH AND PROPERTIES OF

and BS = BY

Hg, -,Cd,Se

ALLOYS

59

+ BS,RT.

Using a technique developed by Foster and Woods (1971), Nelson et al. (1977) found the values for the coefficients BL and P that give the leastsquares fit to the HgSe-CdSe phase diagram data. In their calculations, they used the values of 30.7 kJ/mol for the heat of fusion of HgSe (Steininger, 1970) and 43.9 kJ/mol for the heat of fusion of CdSe (Strauss and Steininger, 1970). The least-squares fit was obtained for BL = 27.57 kJ/mol - 3.09RT

(7)

P = 30.93 kJ/mol

(8)

and

-

2.93RT.

The liquidus and solidus curves determined by Eqs. (7) and (8) are plotted in Fig. 3, where the ideal-solution curves (BL = BS = 0) are also plotted for comparison. The relative magnitudes of the enthalpy and entropy contributions in Eqs. (7) and (8) are probably not significant because Nelson et al. (1977) obtained almost identical rms deviations between the theoretical and experimental liquidus and solidus curves by assuming either zero enthalpy of mixing or zero entropy of mixing. Figure 3 shows that both the liquidus and solidus points fall significantly below the ideal-solution curves. The departure from ideal behavior is greater than the experimental error and possible errors resulting from inaccuracies in the vapor pressure corrections to the phase diagram data. The fact that the liquidus and solidus lines fall below the ideal curves suggests that there is a strong degree of association in both the liquid and solid phases of the HgCdSe alloy system which lowers the free energy below the ideal-solution free energy. This suggestion is supported by analyses of the phase diagram of Cd-Se solutions reported by Jordan (1970), which show that a strong bonding occurs between Cd and Se mixed in stoichiometric proportions. No comparable analysis has been reported for the Hg-Se system. 4. VAPORPRESSURE

Nelson et al. (1977) calculated the vapor pressure in equilibrium with Hg, -,Cd,Se at the liquidus temperature in a sealed capsule as a function of x on the assumption that the vapor species were Hg, Cd, and Se,. They assumed Raoult's Law to be valid for the Hg,-,Cd,Se alloy system, in which case the elemental partial pressures are given by the relations

60

C. R. WHITSETT, J. G. BROERMAN, AND C. J. SUMMERS

where P!: is the partial pressure of element Y in equilibrium with the alloy or compound X and x is the melt composition at the liquidus temperature. If the reaction between HgSe and its vapor is (HgSe)liquid

L

(Hg)gas + gseZ)gas,

(12)

then at equilibrium,

or

where KH@' is the thermodynamic equilibrium constant for the reaction. The value for K H e ewas calculated from the following expressions for the Gibbs free energy for the reaction given by Eq. (12): AGHgSe =

- R T In KHgSe

(15)

AGHgse =

AHHgSe - T

(16)

and

In these expressions,AH,,, is the enthalpy change and ASH,, is the entropy change upon forming liquid HgSe from the vapor phases of the constituent elements. The Gibbs free energy of dissociation of solid HgSe (Brebrick, 1965) is L\G[(HgSe),,,,

-

(Hg),,,

+ gSe2)gas]= (175.4 - 0.1774T) kJ/mol.

(17) If the heat of fusion of HgSe, 30.7 kJ/mol (Steininger, 1970), is subtracted from Eq. (17), the result is AGHgSe =

(144.7 - 0.1774T) kJ/mol,

(18)

which can be substituted into Eq. (15) to determine pgSe at a given temperature. The calculation for 6;" is similar to the HgSe calculation. The Gibbs free energy of dissociation of solid CdSe (Wosten, 1961) is AG[(CdSe)so,id

-

(Cd)gas + &se)2 gas] =

(329.45 - 0.1997T) kJ/mol,

(19)

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg1 -,Cd,Se

ALLOYS

61

Liquidur temperature (OC)

900

799

loo0

1100

12001254

m.

-200

e,

1 E a

L

0 >

15

175

- 150

.

-

125

= c

5

5u: -

10 .

\\

0

-0

100 -75

;

,8

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t.0-

&I.%

cd&

Mole fraction of CdSe in capsule. x

FIG.4. Partial vapor pressures of Hg, Cd, and Se, and the total equilibrium vapor pressure for Hg, -,Cd,Se alloys at their liquidus temperatures. (From Nelson et al., 1977.)

and the heat of fusion of CdSe is 43.9 kJ/mol (Strauss and Steininger, 1970), giving AGCdSe= (285.55 - 0.1997T) kJ/mol.

(20)

The Hg, Cd, and Se, vapor pressures calculated by Nelson et al. (1977) from these relations are plotted as functions of x in Fig. 4. Their calculations yielded a maximum vapor pressure (21 MPa) for a Hg, -,Cd,Se melt with x = 0.65 at its liquidus temperature of 1100°C. The neglect of higher polymers of Se in the vapor is undoubtedly a source of inaccuracy of the calculation. Munir et al. (1972) showed that the pressure of Se, in equilibrium with HgSe is large compared with that of Se, and that the vapor pressures of other Se polymers cannot be neglected. However, the major error source is the assumption of Raoult's law, which probably leads to overestimating the Hg vapor pressure for all values of x. 111. Crystal Growth of Hg, -,Cd,Se Alloys

5. BRIDGMAN CRYSTAL-GROWTH METHOD The preparation of Hg, -,Cd,Se alloys by the Bridgman crystal-growth method has been reported by several investigators (Nelson et al., 1977; Iwanowski et al., 1978;Iwanowski, 1975; Kumazaki et al., 1977; Stankiewicz et al., 1974; Borisov et al., 1971). However, apart from the work of Nelson

62

C. R. WHITSETT, I. G . BROERMAN, AND C. 1. SUMMERS

et al. (1977), few details have been given of the experimental procedures and results of this method.

6. INGOTPREPARATION Hg, -,Cd,Se alloy ingots are prepared by reacting the elemental constituents in sealed, evacpated, quartz capsules. The capsules are made from water-free, fused-quartz tubing, which should have concentric inner and outer diameters and be free of visible striae or other flaws that can reduce the fracture resistance of the quartz. Alloys with x values up to 0.6 can be prepared with small risk in 5-mm i.d. x 15-mm 0.d. capsules and with success most of the time in 10-mm i.d. x 20-mm 0.d. capsules, provided that adequate precautions are taken. Alloys with x values up to 0.4 have been prepared in 12-mm id. x 18-mm 0.d. capsules, but this size capsule frequently ruptures. To prepare a capsule, a length of quartz tubing is first sealed at one end, cleaned, and annealed with a hydrogen torch or in a furnace at 1200°C until no strains are visible at room temperature in polarized light. The elemental constituents, which must be pure and oxide-free, are inserted into the tube next; the cadmium should be added last to minimize the possibility of its reacting with Hg before the capsule is evacuated. The sealed end of the loaded capsule is inserted into an ice bath to minimize Pa, and sealed evaporation of the elements, evacuated to a pressure < at the open end. The last-sealed end should be annealed with a hydrogen torch until no strains are visible in polarized light. The encapsulated elements are reacted by heating the capsule to a temperature above the liquidus temperature of the composition being prepared. This can be done safely by supporting the capsule inside a nickel-alloy pipe with threaded pipecaps on each end and heating this assembly in a tubular rocking furnace. Nickel-alloy thermocouple wells brazed into one of the pipe-caps can be designed to support the capsule within the pipe. Capsule explosions, when they occur, usually occur at a temperature near 700°C during the first heating of a capsule and rarely after the alloy has once been successfully melted. Capsule failures apparently are caused by minute, random flaws in the quartz tubing and are not correlatable with the rate of heating the capsule during the initial reaction of the elements, although the explosions invariably follow a sudden increase in the temperature of the reacting elements caused by the heat of reaction. The capped nickel-alloy pipes contain the debris when an explosion occurs. Successful alloy preparation depends upon the concentricity of the inner and outer diameters of the quartz capsule, the absence of flaws in the quartz, no thinning of the quartz wall at the sealed ends, and the absence of strains in the quartz. After the alloy has been melted and rocked, the rocking furnace is rotated to a vertical position and cooled. The resultant polycrystalline alloy ingot

2.

CRYSTAL GROWTH AND PROPERTIESOF Hg, -,Cd,Se

ALLOYS

63

can be recrystallized by the Bridgman crystal-growth method or by graingrowth annealing. The ingots can be removed from the quartz capsule by cutting longitudinal slots on opposite sides of the capsule with a precision diamond-rim saw. The two halves of the slotted capsule fall away from the enclosed ingot. Alternatively, the quartz capsule can be dissolved in HF. 7. CRYSTAL-GROWTH KINETICS Ingots grown by directionally freezing an alloy solution under conditions of near equilibrium and complete mixing in the melt exhibit along the growth axis a varying compositional gradient, whose theoretical profile (Pfann, 1952) is given by CS(1) = kCo(1 - l/L)k?

(21)

In Eq. (21), C,(I) is the solute (CdSe) concentration in the solid at a distance 1 along an ingot of length L, C, is the initial solute concentration in the melt, and k is the equilibrium distribution coefficient at the temperature of the liquid-solid interface. The distribution coefficient, k is the ratio of :he concentrations of CdSe in the liquid and solid at the crystal-melt interface and is assumed to be independent of composition in the derivation of Eq. (21). Equation 21 shows that the ingot composition decreases uniformity from an initial value of kCo at 1 = 0, to zero for 1 = L. An alternative method of directional freezing is under conditions of elemental diffusion in the melt and negligible mixing. The theoretical compositional profile then depends not only on the initial melt composition and equilibrium distribution coefficient but also on the relative values of the solidification growth rate R and the solute (CdSe) diffusion rate D in the liquid solution (Smith et al., 1955). For k > 1 (or k c l), the compositional profile typically has an initial transient of progressively decreasing (or increasing) solute concentration, a region of constant composition equal to the initial melt composition, and a final region of rapidly decreasing (or increasing) solute concentration. The length of the center steady-state region can be increased at the expense of the initial and final transients by increasing the rate of solidification, R. For R >> D, the widths of the initial and final transients are small, and Cs(l) is essentially independent of 1 and equal to C,. However, to prevent constitutional supercooling and to obtain crystals of good quality, it is necessary that G, the temperature gradient in the liquid ahead of the growth interface, obey the relationship (Tiller et al., 1953)

where rn is the slope of the liquidus line and k > 1.

64

C.

R. WHITSE", J. G . B R O W N , AND C. J. SUMMERS

C, = 0.418 k - 1.88 0 . 5 i tI

D = 5 x 10-5cm2Isec R = 1.O pmlsec L = l5Acm

'

C. = 0.321

0.3

R = 1.Ojhnlsec L - 15.5cm

0.2 i 0

I

I

I

5

10

15

Distance dong ingot, Q (cml

FIG.5. Theoretical and experimentalcompositional profilesof Hg, -,Cd,Se ingots grown by the fast Bridgman method. (From Summers and Nelson, 1980.) -, theory; ----, experiment.

Diffusion-limited crystal growth has been investigated by Summers and Nelson (1980). Figure 5 shows the theoretical and measured compositional profiles of two crystals grown at 1 pm/sec by the Bridgman method. For both crystals the ratio G/R exceeded that required for diffusion-limited crystal growth. To eliminate microscopic inhomogeneities and strains, the crystals were annealed for 48 h at temperatures 50°C below their solidus temperatures before they were slowly cooled to room temperature following growth. The theoretical compositional profiles agree reasonably well with the experimental profiles. The difference between the experimental and theoretical profiles at the start of growth is possibly due to a different rate of freezing at the bottom of a capsule because of its taper. The slow increase in Cd composition along the central section of large-x ingots may be associated with end effects that are difficult to include in the theoretical calculations; a second factor is that the distribution coefficient for Cd may depend on melt concentration and growth rate, whereas it is assumed to depend only on temperature in the calculations. 8. THINFILM AND EPITAXIAL GROWTH

Kot et al. (1962) prepared thin films of Hg,-,Cd,Se by serially evaporating HgSe and CdSe onto mica substrates at 80-120°C and subsequently

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg, -.$d,Se

ALLOYS

65

Layer thickness (pm)

FIG.6. Cadmium distribution in epitaxial Hg, -,Cd,Se layers grown by the isothermal close-spaced method. (From Savitskii et al., 1975.)

annealing the films. They determined the energy gap as a function of x from electrical conductivity, photoconductivity, and optical absorption measurements and observed that the energy gap had a sublinear dependence on x, decreasing from 1.62 eV for CdSe to ~ 0 . eV 4 for pure HgSe. These results, however, are not confirmed by subsequent investigations performed on Bridgman grown samples. Epitaxial layers of Hg, -,Cd,Se have been prepared on singlecrystal CdSe substrates by Savitskii et al. (1975) using the isothermal close-spaced method first used by Cohen-Solal et al. (1966) for Hg,-,Cd,Te. Figure 6 shows the CdSe distribution measured by Savitskii et al. (1975) for layers grown at 650°C for periods between 1 and 50 h. Two mechanisms are identified in the growth process. For times less than 5 h, the principal growth mechanism is the vapor deposition of CdSe which is released into the capsule by the dissociation of the substrate during the initial capsule warmup and the first stages of growth. This source of CdSe ceases after the deposition of a continuous layer of HgSe and the capture of the CdSe remaining in the vapor. For longer growth times the concentration of Cd in the epitaxial layer is determined by its diffusion rate out of the substrate, and consequently, the CdSe depth profiles become flatter as shown in Fig. 6. From the dependence of the epitaxial layer composition on layer thickness in this time regime, Savitskii et al. (1975) estimated the Cd diffusion coefficient and activation energy to range between 1.15-8.0 x lo-" cm2/sec and 0.7-1.0 eV, respectively, in Hg, -,Cd,Se alloys with x values between 0.25 and 0.5. AND CHEMICAL DOPING 9. ANNEALING For all values of x, the electron concentration of as-grown HgSe and Hg, -,Cd,Se crystals is reduced by annealing at 200-400°C in vacuum and

66

C . R. WHITSETT, J. G . BROWMAN, AND C. J. SUMMERS

increased by annealing in Hg vapor. Whether the lower limit of the electron concentration achievable by vacuum annealing is set by impurities or by Hg interstitials has not been established. By studying the change in the electrical properties of Bridgman grown HgSe ingots after annealing in Hg and Se vapor, Kumazaki et al. (1976) deduced the pressure-temperature-compositionphase diagram for HgSe-Se and the existence region of HgSe. Electron concentrations between 5 x 10'' and 1 x 1019 ~ m were - ~obtained by varying the Hg pressure and temperature. For 0.1 Pa < Pip < lo3 Pa and 200°C < T < 500"C, the electron concentration n obeys the relation n = 2.13 x 10"exp

ry3

. (P!p)1/3 ~ m - ~ .

~

Several other investigators, Wright et al. (1962), Whitsett (1965), and Volkov et al. (1971), have reported an increase of the electron concentration in HgSe by annealing in Hg vapor from the usual as-grown value of 3 x 10" cm-3 to nearly 10'' ~ m - As ~ .shown in Fig. 7, Kumazaki et al. (1976) found that annealing HgSe between 500 and 600°C in Se vapor produces a large increase in the electron concentration. However, Lehoczky et al. (1974) obtained electron concentrations as low as 3.6 x 10l6 cm-3 in HgSe by a combination of vacuum and Se-vapor annealing. Iwanowski (1975) investigated the effects of annealing Hg, --x Cd,Se crystals with x < 0.20 in equilibrium with Se at 200-240°C and with Hg

F 1017

0.0015

0.0020

dc

0.0025

Inverse temperature ( ~ - 1 )

FIG.7. Electron concentration in HgSe as a function of annealing temperature in Se vapor. (From Kumazaki et al., 1976.)

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg,

-,Cd,Se

ALLOYS

67

at 240°C. The electron concentration decreases to one-third its value in as-grown crystals during the first 100-200 h of Se-vapor treatment and changes little thereafter. Accompanying the decrease of the electron concentration is a threefold increase of electron mobility. Annealing in Hg vapor doubled the electron concentration in Hg0.83Cd,.,, Se and slightly decreased the electron mobility. Nelson and Summers (1977) have shown that the electron concentration in Bridgman ingots grown at rates c0.2 pm/sec not only decreases during vacuum annealing at 300-400°C but continues to decrease after annealing if the crystal is in air or oxygen. Electron concentrations between 9.4 x 1OI2 and 7.0 x 1014 cm-3 were achieved in Hg, -,Cd,Se samples with x values between 0.49 and 0.55 by annealing in a dynamic vacuum of Pa at 295°C for 137 h and subsequently aging in air at room temperature for 1300 h. The electron concentrations in the aged samples were independent of temperature, and the electron mobilities were low, w 100 cm2/V sec. These results can be rationalized on the basis that at the relatively high annealing temperature of 295"C, large concentrations of vacancies are formed, which are filled during aging by in-diffusing oxygen atoms that act as acceptors. Subsequent work established that the rate of change of electron concentration during aging in air or 0, increases with the vacuum-annealing temperature and that aging in vacuum or Ar has no effect. Summers and Nelson (1980) showed also that vacuum annealing has little effect and post-annealing aging has no effect on Hg,-,Cd,Se crystals grown at a faster rate ( x 1 pm sec-I). No effectivep-type dopants have been found for HgSe, although the elements Li and Na have been shown to be substitutional, p-type dopants in CdSe (Nassau and Shiever, 1972). Li is an acceptor dopant (Johnson and Schmit, 1977) in Hg,-,Cd,Te. Li, Na, and the other group IA elements of the periodic table are logical p-type dopant choices because they have one valence electron per atom and will thus act as acceptors if they substitute in the Hg,-,Cd,Se lattice for Hg or Cd, which have two valence electrons per atom. Nelson (1977) prepared Hg,.,Cd,,Se ingots doped with 5 x 10" atoms/cm3 of Li, Na, and K and 5 x l O I 9 atoms/cm3 of Li and K. The electron concentrations and mobilities in these ingots as-grown are listed in Table I. Their results show that K is not an effective dopant in Hg,., Cdo.4 Se alloys. For Na-doped ingots, they obtained ambiguous results; the addition of 5 x 10l8 Na atoms/cm3 reduced the electron concentration to w 1.4 x l O I 5 cmL3,but the higher-doping level of 5 x 10'' Na atoms/cm3increased the electron concentration by three orders of magnitude to 2 x 1OI8 ~ m - Li ~ was . the most predictable dopant, and electron concentrations of 6.2 x 10'' and 4.7 x loi4 cmP3 were obtained for ingots doped with 5 x 1OI8 and 5 x l O I 9 Li atoms/cm3, respectively. Thus, Li appears to function as an acceptor in Hg,,,Cd,.,Se and partially

68

C. R. WHITSETT, J. G. BROERMAN, AND C. J. SUMMERS

TABLE I ELECTRICAL CHARACTERISTICS OF As-Grown H G ~ . ~ CSEDSAMPLES ~ . ~ UNDOPED AND DOPeD WITH

Dopant Ingot concentration no. Dopant (atoms/cm3)

40H 401 403 40K 40L 40M

None Li Na K Li Na

0 5 5 5 5 5

x io1* x 1OI8 x 1OI8 x 1019

x

lOI9

LI, NA, AND K

Electron concentration at 4.2 K (~rn-~)

3.3 x 6.2 x 1.4 x 2.0 x 4.7 x 2.0 x

Electrical resistivity (Q cm)

4.2 K 1017 0.00354 1015 2.20 loi5 4.68 lot7 0.00372 1014 45.8 10" 0.000887

300 K 0.0101 3.02 9.55 0.0164 49.6 0.00172

Electron mobility (cmZ/vsec) 4.2 K 300 K 1870 5350 333 457 467 952 1900 8400 268 290 3520 1810

compensates the excess-Hg donors. For low doping levels, Na similarly acts as an acceptor, but at high doping levels, Na is a donor. Analysis of the electron mobility in the K-doped Hgo.6 Cdo.4Se indicated that the material was not compensated; the K atoms either remain neutral, become positively ionized, and/or precipitate from the alloy lattice. The electron-mobility analysis for the Li-doped samples indicated that the acceptor concentrations are approximately equal to the electron concentrations that exist in undoped, as-grown, Hgo.6 Cdo.4 Se; this suggests that Li atoms enter the crystal lattice as acceptors until their concentration is sufficient to compensate the excess-Hg donors, and that higher concentrations of Li are ineffective in converting the alloy to p-type. In low concentrations Na behaves similarly to Li in compensating the excess-Hg donors, but in high concentrations, Na is a donor. Vacuum annealing of the K-, Li-, and Nadoped crystals for 64 h at 258°C changed theelectron concentrations in all samplesto x 4 x 10l6 and the electron mobilities of the vacuum-annealed doped samples were the same as for undoped, vacuum-annealed Hgo.6 Cdo.4 Se. For example, the electron concentration in the ingot doped with 5 x 10l8 Na atoms/cm3 increased from 1.4 x 1015cm-3 to 4.0 x 10l6 cm-3 during annealing, and the electron mobility increased from 470 cmZ/v sec to 1470 cm2/v sec. Apparently, Li and Na readily diffuse from HgO.6Cdo.4Se when it is heated in vacuum. The electrical resistivity of Li- and Nadoped Hgo.6Cdo.4Se changes slowly with time in air at room temperature because of the loss of Li and Na from the Hgo.,Cd0.,Se lattice. Thus, neither Li nor Na are practical as dopants in Hg0.,Cd,,Se because the doped alloy must be maintained at cryogenic temperatures to preserve its properties.

2.

CRYSTAL GROWTH AND PROPERTIESOF Hg,

-,Cd,Se

ALLOYS

69

10. D ~ CSTRUCTURE T The defect chemistry of Hg, - ,Cd, Se alloys can only be inferred from that known for HgSe. The simplest defect model assumes a concentration of interstitial Hg atoms, vacant Hg lattice sites, interstitial Se atoms, and vacant Se lattice sites. From an analysis of the electron concentrations of HgSe crystals quenched from different temperatures after being annealed in Hg or Se vapor, Kumazaki et al. (1976) calculated the formation enthalpy to be 44 kJ/mol (0.46 eV per pair) for Hg interstitial-vacancy Frenkel pairs and 60 kJ/mol(O.62 eV per pair) for Hg-Se Schottky vacancies. They also obtained -14 kJ/mol (-0.14 eV per atom) as the heat of sublimation of interstitial Hg. Kumazaki et al. (1977) also studied the defect properties of Hg, -,Cd,Se alloys by ultrasonic attenuation.

IV. Electron Energy-Band Structure 1 1.

TO

PERFECT-SEMIMETAL

SEMICONDUCTOR TRANSITION

In 1963, Groves and Paul (1963, 1964) proposed the perfect-semimetal band structure for a-Sn. It was soon realized by them and by Harman et al. (1964) that the mercury chalcogenides, HgSe and HgTe, also belong to this class of materials. This unusual band structure is shown in Fig. 8. In a normal InSb-type semiconductor, the s-like r6 level is the edge of the lowest-lying conduction band, and the p-like Ts level is the edge of the highest valence band. The next valence band is the spin-orbit split-off r,

\

/

r6

I

--

/

HgSe

Crossover

Zero gap

Hg,-,CdxSe

CdSe

Positive gap

FIG.8. Energy-band model for Hg, -,Cd,Se alloys.

70

C. R. WHITSETT, J. G. BROERMAN, AND C. J. SUMMWS

band. In the perfect semimetal structure, the r6 level has fallen to a position between r8and r7.Because r6 and the light-hole part of Ts are k p coupled, they invert, the light-hole part of Ts becoming a small-mass conduction and r6 becoming a valence band. The fundamental energy gap is therefore identically zero because the conduction and valence parts of Ts are degenerate by symmetry. CdSe is a wide-gap semiconductor, and a simplified energy-band model of the CdSe-HgSe pseudobinary alloy system is shown in Fig. 8. As the mole fraction x of CdSe is increased, the virtual-crystal r6 level rises with respect to the Ts level, crossing it at about x = 0.1. Above this critical value of x, the alloy is a normal InSb-type semiconductor. The transition from zinc-blende to wurtzite crystal structure at x x 0.8 is accompanied by a small splitting of T8.

-

12. BANDSTRUCTURE CALCULATIONAL MODEL The calculational model used most often to describe the band structure near the zone center for zinc-blende-structure materials was developed by Kane (1957) for InSb. The cell periodic part of the wave function can be expanded in a complete set of zone-center wave functions,

If the expansion is truncated to include only T6, r7,and Ts, one obtains for the energies of the light hole, conduction, and split-off bands En = E:,

+ h2k2/2m,,

(25)

where the E: are the solutions of the secular equation El3

+ ( A - E,)E"

- (&A

+ P 2 k 2 ) E - ($)AP2k2 = 0.

(26)

The doubly degenerate wave functions of the conduction, light-hole, and split-off bands are Ik, n, f) = e""[a,IiSa,)

f b,l(X T i Y ) a , )

+ c,IZaT)],

(27)

where X, Y, and Z are the basis set of rI5chosen in a coordinate system with the z axis along k,S is the rl wave function, a* are Pauli spin functions for spin parallel (+) and antiparallel (-) to k, P is the momentum matrix element between S and X , E , is the T6-Ts energy gap, A is the rI5spinorbit splitting (Er8 - E,-J and a, b, and c are functions of energy. For the conduction band near the zone center, b, and c, are much larger than a, in the semimetallic range of x where EG < 0, and the wave function is nearly pure p-type. In the semiconductor range of x where E , > 0, b, and c, are

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg,

-,Cd,Se

ALLOYS

71

much smaller than a,, and the wave function is nearly pure s-type. The heavy-hole energy is

4"= E,,

fi2k2/2PU,mo,

(28) where p" is the effectivemass of the heavy holes and the heavy-hole wave functions are -

Ik,u, k) = e i k ' r ( l / f i ) l ( Xk iY)a,)

(29) in the approximation where the warping produced by higher bands has been spherically averaged and the nonzero slope at the origin leading to a small overlap with the conduction band away from k = 0 has been approximated by a small overlap E,, at the zone center. If higher bands are included in the interaction with the conduction band, the energy has an additional term which is not isotropic. When spherically averaged, it is of the form (Seiler et al., 1971)

(h2k2/2m,J[a2A' + b2M' + c'L'

+ &(b2 - 2c2)(L' - M' - N ' ) ] , (30)

where A', M', L', and N' are matrix elements with higher bands. These terms are small in comparison with the terms arising from the interaction between r6,I-,, and rs. This formalism provides a convenient description in terms of a few parameters of not only the energies but also most other kinematic quantities entering the theory of the transport, optical, and dielectric properties of Hg,-,Cd,Se alloys. Moreover, since X , Y, and Z transform like the D(') representation of the rotation group and a* transform like D("2),the overlap matrix elements which appear in the theories can be evaluated purely geometrically in terms of these same few parameters. The momentum and density of states as functions of energy are derivable directly from the secular equation. Because the conduction- and valence-band states near k = 0 are the lowest-energy states for electrons and holes, respectively, carriers generated by impurities or an external field rapidly thermalize to these levels. Knowledge of the band structure near k = 0 is therefore essential to understanding the electrical and optical properties of this alloy system. In a magnetic field both the conduction and valence bands are shifted and quantized into an infinite series of spin-split Landau levels as shown in Fig. 9 (see, e.g., Smith, 1967). The separations between conduction-band Landau states hw, and spin states ho,are related to the electron effective mass m* and g factor g by the equations and

72

C. R. WHITSETT, J. G . BROERMAN, AND C. J. SUMMERS

I} n=0

FIG.9. Energy-band structure for a semiconductor in a magnetic field.

k=O

where B is the magnetic field, B is the Bohr magneton, and M , is the spin quantum number, which has values of k$ for conduction electrons. Both m* and g are related to the Kane band parameters by the equations 1

(33)

m* and

[ (

g=21+

(34) I - - :*)(,: 2A)]. These equations show that as EG decreases, m* decreases, g increases, and both hw, and hw, therefore increase. Because of the valence-band degeneracy at k = 0, the energy level scheme is considerably more complicated than shown in Fig. 9 (Smith, 1967).

V. Dielectric Function The dielectric response determines the optical and, to a large extent, the electronic transport properties of a material. In the Hg, -,Cd,Se alloy system, the dielectric function changes radically between the semimetallic and semiconductor composition regimes. The reason for this change is the presence in the semimetal regime of the zero-energy TScc-, TSv excitation,

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg,

-,Cd,Se

ALLOYS

73

which leads to a response function of a unique type for the perfect semimetal band structure. The dielectric function can be divided into two parts, one arising from the polarizability of the ionic lattice and the other arising from intra- and interband electronic transitions :

do, 4 ) = &lat(W, 4 ) + EeI(W, 4).

(35)

13. ELECTRONIC PART OF THE DIELECTRIC FUNCTION For the electronic part of the dielectric function, the random phase approximation (Ehrenreich and Cohen, 1959) provides a relatively simple computational framework of reasonable accuracy for the Hg, -,Cd, Se alloys. It yields the following expression for the dielectric function:

is the Fermi occupation function. Equation (36) can be divided where into intra- (n = n') and interband (n # n') parts. First consider the semimetallic range of composition. It is convenient to divide the interband part into a part E=, arising from TScc,TSvexcitations and a background part &b arising from all other interband excitations. Because of the large energy gaps involved, &b is regarded as independent of o and 4 for small values of these variables. The zero-frequency case (static dielectric function) was analyzed by Liu and Brust (1968a,b), Sherrington and Kohn (1968), and Halperin and Rice (1968) at absolute zero with no impurity carriers. They showed that because of the zero-energy excitation, era has the form EF:V,

4 ) = ko/q,

(37)

where ko depends on conduction- and valence-band effective masses. This represents a new type of singular behavior, intermediate between a metal ( E a 1/q2) and an insulator ( E = constant). The introduction of impurity carriers removes the zero-energy transition and hence the l/q interband singularity but introduces an intraband or free-carrier singularity. Liu and Tosatti (1969, 1970) find for this case under degenerate conditions the interband part is given by EPF(0,4 ) = (8C2m*mo/7th2k~)[1 - *q/kF)'],

(38)

where k, is the Fermi momentum of the degenerate electron gas. Broerman et al. (1971) find for the intraband part E y ( m

4) = ( k W ) f ( 4 > ,

(39)

14

C. R. WHITSETT, 3. G. BROERMAN, AND C. I. SUMMERS

I

I

I

1.o

f(q)

0.5

-

-

c P3,2 electron

0

0

0.5

1.o qIKF

1.5

2.0

FIG. 10. Intraband screening functions for S,,, electrons (upper curve) and P3,, electrons (lower curve). (From Broerman et al., 1971.)

where k,, is the Fermi-Thomas momentum andf(4) is a function, shown in Fig. 10 (lower curve), which greatly lowers the screening at highmomentum transfer from that of a free or s-like electron (upper curve in Fig. 10). This factor arises as an effect of the p-like symmetry of the conduction-band wave function on the wave function overlap integral in Eq.(36). The interband part, even though no longer singular when impurity carriers are present, can be extremely large in a low electron-concentration sample. Broerman (1970a) analyzed the temperature dependence of E?: (0, 0) and found a T-'/' behavior for intrinsic samples. Samples with extrinsic conduction electrons smoothly approach the degenerate case as the temperature decreases, while p-type samples display peaking behavior as the Fermi energy passes through Tswith decreasing temperature. Sherrington and Kohn (1968) examined the frequency dependence of Er8(m,0) and found an m- ' I 2 singularity at absolute zero in an intrinsic sample. Broerman (1972a) examined the frequency dependence in the presence of degenerate impurity carriers with infinite lifetime and found that the real part of the dielectric function has a logarithmic singularity and the imaginary part has a finite discontinuity at a frequency corresponding to excitation to the Fermi level. The singular behavior is removed at finite temperatures or finite lifetimes, but strong peaks remain at low temperatures. The preceding derivations are based on the parabolic-band approximation. Broerman (1970b) showed that this is reasonably accurate for the static case. However, a more elaborate theory incorporating the band nonparabolicity must be employed for optical calculations for those materials with small IE,, - Er, I, at frequencies extending far into the conduction band. The unusual behavior of the perfect-semimetal dielectric function has a number of observable effects on the optical and transport properties of the alloy systems. The effects have been seen experimentally in HgSe, HgTe,

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg, -,Cd,Se

ALLOYS

75

and a-Sn. A review, concerned primarily with low-temperature phenomena, has been given by Broerman (1972b). The effects in the Hg,-,Cd,Se alloy system will be seen in the following sections on optical and transport properties. Above the semimetal-semiconductor transition composition, the electronic dielectric function is that of a normal semiconductor. An analysis of the dielectric function in the neighborhood of the fundamental gap will be shown in the section on optical properties. PART OF THE DIELECTRIC FUNCTION 14. LATTICE

The lattice part of the polarizability in the Hg,-,Cd,Se alloy system exhibits two-mode behavior (Summers et al., 1981). Transverse optical(TO) phonon absorption for both HgSe and CdSe appear in the infrared spectrum of the materials. The lattice dielectric function for small q is thus of the form

where wTi is the TO-phonon frequency, e& is the total transverse effective charge, Mi is the reduced mass, N iis the number of unit cells per unit volume, and Ti is a broadening parameter of the ith type unit cell. The values of these parameters over a large range of compositions have been determined and will be described in the section on optical properties.

VI. Magnetotransport Magnetotransport studies, although less direct than optical studies, have proved indispensable in the determination of the band structure of the Hg, -,Cd, Se alloys, especially in the semimetallic and narrow-bandgap semiconductor regimes where the r6-r8absorption is difficult to observe. 15. SHUBNIKOV-DE HAMEFFECT

In a magnetic field B, the electron states are condensed into a series of Landau cylinders in momentum space parallel to the direction of the magnetic field. As the field increases, the cylinders expand and are, one by one, excluded from the volume of k space enclosed by the Fermi surface. If the energy separation of the cylinders is much larger than kT (but still smaller than EF)and the scattering rate is much smaller than the cyclotron frequency, then the passage of each cylinder through the Fermi surface results in a peak of the electricalconductivity of the electron gas. The resulting oscillatory magnetoconductivity, known as the Shubnikov-de Haas effect, has 1/B frequency proportional to the extremal cross-sectional area of the

76

C. R. WHITSETT, J. G . BROERMAN, A N D C. J. SUMMERS

Fermi surface normal to the magnetic field. The temperature dependence of the oscillations is a sensitive function of the carrier effective mass at the Fermi surface. A quantitative determination of the band structure parameters of HgSe was made by Whitsett (1965) using the Shubnikov-de Haas technique. He measured the periods at 4.2 and 1.2 K of the Shubnikov-de Haas oscillations in n-type samples with conduction-electron concentrations between 2 x 10’’ and 5 x loi8cm-3 and determined the dependence of the electron effective mass at the Fermi surface on electron concentration. He analyzed the effective mass results using a simplified Kane model (only r8 and r 6 interaction)and determined the parameters P and EG to be 7.1 x eV cm and -0.25 eV, respectively. He also observed anistropies in the Shubnikovde Haas period and beats in the oscillations for the magnetic field in the (1 11) and (100) directions and suggested that these were caused by small bulges on the Fermi surface in the (1 11) directions. The lack of inversion symmetry of the zinc-blende lattice allows splitting of the conduction band at a general point in the Brillouin zone into its two “spin” states, and this, in principle, can produce two Fermi surfaces of slightly different extremal areas which would produce the beating observed by Whitsett. Roth et al. (1967) and Roth (1968) showed that these splittings produce two orbits in the HgSe conduction band and lead to the orientationdependent beating observed by Whitsett. Galazka et al. (1971) showed that the size of the conduction-band warping is consistent with T8 symmetry but not r6.In a later investigation, Seiler et al. (1971) performed an accurate measurement of the warping and beating effects and analyzed the data with the full Kane theory (k p interaction of r6,r,, and re) with higher-band corrections. They thus determined the higher-band parameters and the spin-orbit splitting (0.45 eV) and revised Whitsett’s determination of P and EG to P = 7.2 x eV cm and EG = -0.22 eV. Bliek and Landwehr (1969) measured both the Shubnikov-de Haas and the de Haas-van Alphan effects in HgSe in pulsed fields up to 21T. At these very large fields, spin splittings in the oscillation are easily observable. Their results indicate that inversion-asymmetry splittings, rather than warping of the Fermi surface, are responsible for the beating at lower fields seen by Whitsett. The Shubnikov-de Haas effect has also been used to probe the band structure of Hg, -,Cd,Se alloys with small x. Stankiewiecz et al. (1974) performed such measurements on alloys with 0.05 < x < 0.19, which range spans the semimetal-semiconductor transition. They found P to be nearly constant in this range and E , to vary approximately linearly with x, with the assumption that the spin-orbit splitting and higher-band couplings were the same as in HgSe. Giriat et al. (1975) measured the conduction-electron g factor for a sample with x = 0.072 and found it to agree with theory. The value of g

-

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg, -,Cd,Se

ALLOYS

77

at the bottom of the conduction band is predicted for a sample of this x to have the extremely high value of - 116. Dobrowolski and Diet1 (1976) examined the spin-split oscillation in the same composition range as Stankiewicz et al. (1974) for samples with electron concentrations between 10'7-10'8 ~ m - The ~ . measurements were performed at 2 K in magnetic fields up to 7 T and showed that the g factor decreased with increasing electron concentration (i-e., increasing Fermi energy) as predicted by theory. At high temperatures the thermal broadening of the Fermi surface and decreased electron lifetime reduces the magnitude of the Shubnikov-de Haas oscillations. To overcome this loss of resolution, Byszewski et al. (1976) used pulsed magnetic fields up to 32 T to measure Shubnikov-de Haas oscillations in HgSe at 77 K. They used a modification of the Pidgeon and Brown (1966) theory and obtained values for E , and Pin good agreement with those determined by Lehoczky et al. (1974) from Hall effect measurements. 16.

HALL

EFFECT

The conduction-electron population depends sensitively on the band parameters if the impuritycarrier concentration is much less than the intrinsiccarrier concentration. Thus, in principle, the Hall effect can be used to determine the temperature dependence of the band structure of a material. In practice, this is difficult because a realistic model of the band structure (and hence much computational time) and measurements on a large number of samples with differing impurity-carrier concentrations are required. Such a determination of the temperature dependence of the band parameters of HgSe was performed by Lehoczky et al. (1974), who measured the Hall coefficient at temperatures from 4.2 to 300 K of nine samples of HgSe with electron concentrations from 3.6 x 10l6 to 5 x lo" ~ m - ~ . For the scattering mechanisms operative in these samples, the Hall effect scattering factor is approximately unity, and n = l/R,e, where n is the electron concentration and R, is the Hall coefficient. They derived a theoretical expression for n using the Kane three-band model with higher-band corrections. A simultaneous, nonlinear, least-squares fit to the data of all samples was then performed at 23 temperatures between 4.2 and 300 K. The spin-orbit splitting and higher-band couplings were regarded as fixed at the values determined by Seiler et al. (1971) at 4.2 K. The hole mass and valence-band overlap were treated as adjustable parameters and assumed' to be the same for all samples at all temperatures. The r8-r6 bandgap and momentum matrix element P were treated as parameters having the same values for all samples but varying with 'T; and the ionizeddonor concentration was assumed to be temperature independent and equal to the 4.2-K

78

C. R. WHITSETT, J. G . B R O W N , AND C. J. SUMMERS

/Intrinsic

k loo 200 300 50

150

250

Temperature (K)

FIG.11. hast-squares fit to the temperature dependence of the electron concentration measured for HgSe samples having a range of 4.2 K electron concentrations.Also shown is the temperature dependence of the intrinsic electron concentration. (From Lehoczky et al., 1974.)

electron concentration for each sample. An example of the fits obtained for four samples is shown in Fig. 11. The values of EG and P used to obtain these fits are shown in Fig. 12. The analysis yielded a valence-band mass of 0.78mo and a valence-band overlap of Egi = 5.04 meV. This value for the overlap seems large and may be due to the presence of quasistationary states predicted by Gelmont and Dyakonov (1971), which would be impossible to distinguish from valence-band states for samples in this range of Fermi energy. As can be seen from Fig. 12, P is nearly independent of temperature, as would be expected. E G , on the other hand, decreases rapidly as the temperature increases, i.e., Ts rises very rapidly with respect to Ts, and this is in contradiction to the Brooks-Yu theory (Yu, 1964). However, Pidgeon and Groves (1967) report a similar behavior for HgTe over a restricted low-temperature range, and thus this anomalous behavior appears to be characteristic of the Hg chalcogenides. As will be seen in the section on optical properties, the anomalous behavior of the gap occurs also in the semiconductor range of the alloy system. No similar determination of the temperature dependence of the gap in the semimetal regime for x # 0 has been reported.

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg1 -,Cd,Se

ALLOYS

79

0 0

0 0

0 0 0

0

*....

8.0

-

7.5

3

0

0

0 0

........ 0

E 0

(0

200

100

300

7.0

z

Temperature (K)

FIG.12. Temperature dependence of EG and P obtained from a simultaneous least-squares fit to Hall coefficient data measured for nine HgSe samples. (From Lehoczky et a[., 1974.) ~

VII. Optical Properties 17. FUNDAMENTAL ABSORPTION EDGE Because of the large electron concentrations normally found in HgSe, the band structure parameters of this compound have been determined largely from galvanomagnetic measurements, and optical determinations of the band structure parameters for HgSe have only recently been reported. Guldner et al. (1977) reported near-bandgap infrared magnetotransmission measurementsat 4.2 K for unoriented crystals of HgSe and Hg0.985 Cdo.ol Se for magnetic field strengths up to 6 T. The Ts-+ r8 magnetooptical transitions were investigated for the spectral region 220-270 meV in the Voigt geometry with linearly polarized radiation parallel to the field direction and in the Faraday configuration with right ( 0 ' ) and left ( 0 - ) circularly polarized radiation. Transition energy versus magnetic field-strength fan charts such as shown in Fig. 13 were obtained and analyzed using the k * p model of Pidgeon and Brown (1966). Values of E G = -0.274 f 0.002 eV and A = 0.39 f 0.01 eV were obtained for HgSe and E G = -0.230 f 0.002 eV and A = 0.42 f 0.01 eV for Hg,.g8,Cdo.,,,Se. For both samples the value of P was determined to be 7.20 & 0.26 x lo-' eV cm. Szuszkiewicz (1977) reported the optical transmission results shown in Fig. 14 for 5-pm thick samples of n-type HgSe between 4.2 and 300 K and

80

C. R. WHITSETT, J. G. BROERMAN, A N D C. J. SUMMERS

310

300

290

-> -E

280

270 YI C

280

250

240

230 0

1

2 3 4 Magnetic field (T)

5

6

FIG.13. Dependence of interband transition energies on magnetic field strength in Hgo,,,,Cdo,o,,Se. Theory and experimental data for EllB (Voigt geometry) and u+ and ucircularlypolarized radiation (Faraday geometry) are shown. q are interband transitionsdefined in the reference. (From Guldner et al., 1977.) 20 I

r

%-

15

c

::

r

10

8

.-6

i5

a

0

0

0.1

0.2

0.3

0.4

0.5

Energy (eV)

FIG.14. Absorption coefficient spectra for HgSe at 10 K. Details of theoretical curves are given in text. (From Szuszkiewicz, 1977.)

2. CRYSTAL GROWTH AND PROPERTIES OF Hg, -,Cd,Se ALLOYS

81

attributed the absorption edges at x0.08 and w0.34 eV to the onset of the optical transitions A and B, shown in the insert of Fig. 14, which occur between the heavy-hole band and the Fermi level, and the light-hole band and the Fermi level, respectively. The data were analyzed using an absorption coefficienttheory based on the Kane energy-band model, The best fit to the data is shown as a solid line in Fig. 14 and was obtained using the parameters EG = -0.26 eV, P = 7.2 x lo-* eV cm, and A = 0.40 eV. The dashed line in Fig. 14 was calculated using the parameters obtained by Seiler et al. (1971). Although the calculated curves fall considerably below the data, they describe well the positions of the absorption edges. The optically determined 4.2-K values for EG differ significantly from the value of -0.22 eV determined for HgSe from galvanomagnetic data. The magneto-optical measurements of Guldner et al. (1977) show that the conduction-band g factor is negative with a value of - 110 for HgSe and - 25 for Hg,.,,, Cdo.015 Se. Interband Faraday rotation measurements performed by Kireev et al. (1972) on Hg,.,Cd,.zSe and Hg,.,Cd,.,Se show that for both alloy compositions the rotation is negative near the fundamental bandgap and that these alloys have negative g factors. The sign of the g factor is, therefore, expected to remain negative for high-x alloy compositionsbecause the g factor for CdSe, although positive, is small. For CdSe, Piper (1967) has measured a g-factor value of 0.68 from a 4.2-K electron-spin-resonance experiment, which is in excellent agreement with values of 0.7 and 0.53 f 0.03 obtained by Hopfield (1961) from analysis of exciton spectra and by Walker et al. (1972) from analysis of spin-flip Ramanscattering experiments, respectively. The band parameters of CdSe have been determined from optical studies of the absorption edge (Parsons et al., 1961; Grynberg, 1964; Kofdk et al., 1967) and most accurately by Wheeler and Dimmock (1962) from a detailed experimental and theoretical study of the exciton structure at 1.8 K in magnetic field strengths up to 3.5 T. The optical absorption measurements show that the absorption edge for light polarized parallel to the c axis is 0.0173 eV higher than the edge for light polarized perpendicular to the c axis and that between 80 and 413 K both absorption edges move to higher energies with decreasing temperature. Values of 1.669 and 1.694 eV are reported by KoiiAlc et al. (1967) and Parsons et al. (1961), respectively, for the lowest-energy gap at 300 K. A 5-K va!ue of 1.877 eV is obtained from an extrapolation of the available low-temperature absorption-edge data. Wheeler and Dimmock (1962) found that the lowest-energy gap at 1.8 K occurs at 1.8412 eV and that the crystal-field splitting of the valence band at k = 0 is 0.0248 k 0.0019 eV. A hole effective mass of 0.45 0.09rnOwas obtained, and it was determined that near k = 0 the conduction band is approximately spherical with an electron effective mass 0.13 f 0.003mo.

82

C . R. WHITSETT, J. G . BROERMAN, AND C . I. SUMMERS

Studies of the valence-band anisotropy have been reported by Gutsche and Lange (1964). The first attempt to establish the compositional dependence of the energy gap for the Hg,-,Cd,Se system was made by Slodowy and Giriat (1971), who measured the fundamental absorption spectra at 295 and 100 K for alloys with nominal x values between 0.2 and 0.6. Bridgman grown crystals with electron concentrations between lo" and loz8cm-' were used, and thus the optical absorption edges were shifted to higher energies as a result energy gap of the Moss-Burstein effect. Values for the fundamental r8--r6 were obtained by subtracting estimates of the conduction-band Fermi levels from the experimental determinations of the optical absorption edge. Slodowy and Giriat (1971) deduced that the 295-K energy gap of the alloys increases linearly with x and that the temperature coefficient of the energy gap, dE,/dT, decreases linearly with increasing x values as shown in Fig. 15. Measurements of the fundamental absorption spectrum have also been reported by Kireev and Volkov (1974) for Hg,-,Cd,Se samples with x values between 0.1 and 0.5 at temperaturesof 300 and 100 K. They observed broad absorption edges, which made impossible an analysis of the data to extract energy-band parameters. Kirik et al. (1974) measured the temperature dependence of the absorption edge in Hgo.85 Se and showed that its

-

b

-Y 2

t

-z

-5*

6A

\

4-

-

\

-

2-

0 -

-2

-

Ly 0

-4-6-8

-

-107 0

Hose

I

'

0.2

I

I

0.4

I

I

0.6

k b fraction WS.,x

'

0.8

I

1.0

cdse

FIG.15. Dependence of the energy-band gap temperature coefficient on Hg, -,Cd,Se alloy composition. (From Slodowy and Giriat, 1971.)

2.

CRYSTAL GROWTH AND PROPERTLES OF Hgl -,Cd,Se

ALLOYS

83

spectral dependence conforms to Urbach's rule and indicates the presence of a strong electron-phonon interaction. Recently, Summers and Broerman (1980) have made an extensive study of the near-bandgap optical properties of Hg, -,Cd,Se alloys for eleven x values between 0.15 and 0.68 over the 5-300 K temperature range. This study used well-characterized samples, which were grown by the Bridgman method and annealed to obtain electron concentrations of 1 x 9 x 10l6 ~ m - The ~ . homogeneity and average compositions of the samples were established by infrared scanning and precision massdensity measurements, respectively. From optical transmission measurements between 1 and 25 pm, similar to those shown in Fig. 16 (Whitsett et al., 1977), the compositional and temperature dependence of the refractive index and absorption spectra were obtained by analyzing the long wavelength channel spectra and the near-bandgap transmission data, respectively. The absorption spectra were least-squaresfit by an optical absorption theory formulated in the random phase approximation and expressed as a function of the following principal Kane energy-band parameters ; fundamental energy gap, momentum-matrix element, and spin-orbit splitting. The presence of extrinsic and thermally excited electrons, the nonparabolicity of the band structure, and the energy dependence of the light-hole and heavy-hole valence-band to conduction-band optical transition probabilities were fully considered. An example of the best fit to the data obtained by Summers and Broerman (1980) for a sample with an x value of 0.253 at temperatures of 30, 125,and

Wave number (crn-1)

FIG.16. Infrared transmittance spectra measured at different temperatures for a Hg, - .Cd,Se sample with x = 0.194 and n = 4.6 x 1016cm-3.(From Whitsett et al., 1977.)

234

C . R. WHITSETT, J. G . BROERMAN, AND C. J.

SUMMERS

200 K is shown in Fig. 17. From similar analyses of all samples, the compositional and temperature dependences of P and E, were obtained. They eV cm, independently of sample temfound that P = 8.0 f 0.2 x perature and composition. For alloys with x < 0.45, E , increases linearly with temperature, and the magnitude of the Moss-Burstein shift of the optical absorption edge progressively increases as x and the temperature decrease. For x values greater than 0.45, the temperature dependence of the energy gap is nonlinear above 70 K, and the nonlinearity for higher-x samples increases such that for x > 0.555 the energy gap has a negative temperature coefficient above 70 K, as shown in Fig. 18 for a sample with x = 0.684. For high-x samples, the energy-gap temperature dependence is thus similar to that for most III-V compounds. The compositional dependence of the energy-gap temperature coefficient obtained by Summers and Broerman (1980) lies above the curve obtained by Slodowy and Giriat (1971)andisgivenbyaE,/aT = 8.3 x 10-4eVK-'forx = OanddE,/aT= 0 for x = 0.60. The compositional dependence of the fundamental energy gap obtained by Summers and Broerman (1980) is shown in Fig. 19 for temperatures of 10,100,200, and 300 K. The solid lines in Fig. 19 are given by the expression E , = -0.209(1 - 7.172~- 2 . 1 7 4 ~ ~ )

+ 7.37 x + 2.00

lOw4(l- 1.277~- 0.151x')T

10-~(1 + 2 3 ~ 5 599.4x2)~2, ~ (41) which was obtained from a least-squares fit to all the data. In the above equation E G is in eV, T is in Kelvins, and x is the mole fraction of CdSe. At present the temperature dependence of the energy gap of this alloy system and those of the HgCdTe and PbSnTe systems, which exhibit similar temperature dependences, have not been adequately explained (Guenzer and Bienenstock, 1973). At all temperatures the compositional dependence of the energy gap is found to bow below a linear interpolation between the energy gaps of HgSe and CdSe. The largest departure from linearity occurs at 10 K (approximately 0.15 eV), and the nonlinearity is small at 300 K. Equation (41) gives a value of -0.201 eV for HgSe, but if only the 10-K data are fitted, a value of -0.216 eV is projected for HgSe. The weight of all available data, optical and electrical, supports the followingvalues for the low-temperature (< 10-K) band parameters of HgSe: E, = -0.22 eV, A = 0.45 eV, and P in the range 7.2-8.0 x eVcm. 18. OPTICAL REFLECTIVITY Borisov et al. (1971) measured the room-temperature reflection spectra of zinc-blende structure Hg,-,Cd,Se alloys with 0 c x c 0.79 over the

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg,-,Cd,Se

ALLOYS

85

4-

100 0.19

1

1 0.21

1

1

1

0.23

1

1

0.25

1

1

0.27

0.29

c

51

9 100 0.23

-

I

I

I

I

0.25

0.27

0.29

0.31

0.30

0.32

0.33

1 o4

c

I

E

103

4-

.-01 ._0 r % 102

8 C

._

K 51 9

101

100 0.26

0.28

0.34

0.36

Photon energy (eV)

FIG.17. Comparison of theoretical (solid curves) and experimental (dashed curves) absorp~ , 125 K, tion coefficient spectra for Hgo.,,,Cdo,253 Se at (a) 30 K with n = 3.06 x 10l6 ~ r n - (b) with n = 3.40 x 10Ib ~ m - and ~ , (c) 200 K, with n = 3.64 x loi6 C I I - ~ . (From Summers and Broerman. 1980.)

86

C. R. WHITSETT, J. G . B R O W N , AND C. J. SUMMERS 1.02

-2

1.01

Q

w C 1

.oo

0.99 100 200 Temperature (KJ

0

300

FIG.18. Dependence of the fundamental energy gap on temperature for Hg,,,,,Cd,,,,,Se with n = 2.9 x lot6cm-'. (From Summers, 1980.)

1 20

1 00

5-K

-

k

liiiedr

appi oxim.ilion

080

U w

6

060

m c D)

040 c m

3

L" L "

0 20 0 0 20 040

L 03

0

01

02

04

05

06

07

Mole fraction CdSe. x

FIG.19. Compositional dependence of the fundamental energy gap of Hg,-,Cd,Se for temperaturesof lo(.), 100(4),200(.),and 3 0 0 K (A).(FromSummersand Broerman, 1980.)

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg, -,Cd,Se

-[llll

k=O

ALLOYS

87

[lOOl-

Crvstal momentum

FIG.20. Schematic of energy-band model expected for Hg,-,Cd,Se

alloys.

energy range 2.0-5.6 eV. They observed a broad reflection peak whose position moves with composition. This peak is assigned to the El optical transition energy, which as shown in Fig. 20 occurs between the valence and conduction bands along the A direction in the Brillouin zone. The main side-band observed on the high-energy side of the peak is attributed to the El + A1 transition energy. Also observed in the spectra are features attributable to el + A, transitions between the conduction and valence bands at the L point, and to E , transitions between the valence band and a higher conduction band near k = 0. The origin of the A-type transitions is unclear. The A and el + A, transitions are observed only in alloys with x < 0.30, and the type E , transitions are observed only in alloys with x > 0.4. The compositional dependences of these energy gaps and the L-point spinorbit splitting energy are depicted in Fig. 21. The El and El + A, energies have little dependence on x below x = 0.35 but increase linearly with x for higher x values. The values obtained by Borisov et al. (1971) for the El and El A, energies in HgSe are in reasonable agreement with the values 2.85 and 3.15 eV, respectively, obtained by Cardona et al. (1967) from electroreflectance measurements. An extrapolation of the curves in Fig. 21 to x = 1 gives transition energies in excellent agreement with values obtained by Ludeke and Paul (1967) from transmission measurements on zincblende structure CdSe thin films formed by evaporation on BaF, substrates. Borisov et al. (1971) suggested that the change in the compositional dependence of the El-type transitions near x = 0.35 could result from higherband structure changes associated with the semimetal-to-semiconductor transition (although this occurs for x = 0.12) or radical band-shape changes in solid Hg, -,Cd,Se solutions. At present such conjectures are premature,

+

88

C . R. WHITSEl’lT, J. G . BROERMAN, AND C. J. SUMMERS

2.0

1 0

1

1

0.2

1 0.4

1

1 0.6

1

1 I 0.8

HgQ

1.0 CdSe

Mole fraction CdSe, x

FIG.21. Compositional dependence of the positions of reflection peaks and the A-point spin-orbit splitting energy Al for Hg, -,Cd,Se alloys with 0 < x < 0.79. (From Borisov et al., 1971.)

and further experimental data are required. Electroreflectance studies on etched surfaces would elucidate both the assignment and the exact compositional energy dependencesof optical transitions in this alloy composition range, and data for higher photon energies could help establish the band structure at other symmetry points in the Brillouin zone. 19. FREE-CARRIERREFLECTIVITY

Room-temperature reflectivity and magnetoreflectivity measurements have been reported for HgSe in the region of the freecarrier plasmon reflection edge by Wright et al. (1962) and Volkov et al. (1971) for samples with electron concentrations between 5 x lo” and 3 x lo1’ ~ r n - ~ The . electron effective mass for each sample was determined from the position of the minimum in the reflectivity spectrum and its shift with magnetic field strength, and the effective mass as a function of electron concentration is plotted in Fig. 22 and compared with effective mass ratios obtained by Aliev et al. (1965) from thermoelectric power measurements and a theoretical curve computed by Lehoczky et al. (1974) using band parameters

2.

CRYSTAL GROWTH AND PROPWTIES OF

Hg, -,Cd,Se

ALLOYS

89

Electron concentration ( c m 3 )

FIG.22. Dependence of electron effective mass on electron concentration for HgSe. [From Lehoczky et al. (1974); A,from Wright et al. (1962); 0 ,from Aliev et al. (1965).]

obtained from electrical measurements. The increase in effective mass with electron concentration is related to the decrease in conduction-band curvature with increasing energy, as has been well documented for 111-V compounds. Figure 22 shows that the optical determinations of m* are greater than those determined by thermoelectric-power and electrical measurements, probably because the values are obtained from a classical analysis of the data in which the high-frequency dielectric constant is considered to be constant. Manabe and Mitsuishi (1975), who extended the reflection studies on HgSe to approximately 100 pm (and to lower temperatures x100 K), also experienced difficulty in analyzing their data using classical theory. From a Kramers-Kronig analysis of their data, they obtained the frequencies of the modes of oscillation and fit them by a coupled plasmon-phonon theory. They obtained 132 & 2 cm-' for the transverse optical-mode frequency, but to fit the electron concentration dependence of the high-frequency plasmonlike mode (Fig. 23) they had to vary E,. Values between 17 and 12 are inferred for E, for samples with 300-K electron concentrations of 5.9 x 1017 and 2.0 x lo1' cm-j. A similar dependence of E , on electron concentration was observed by Volkov et al. (1971), who obtained values between 10.2 and 8.2 as the electron concentration increased from 7 x 1017 to 5 x 10'' ~ m - It~ is. now realized that the assumption that E , is independent of electron concentration (and frequency) is invalid for semimetallic and small-bandgap semiconductors. For these systems both interband and intraband dielectric functions vary considerably in the same frequency range and a distinct separation of their effects by classical theory is thus impossible. The contribution to the high-frequency dielectric constant from transitions between the Tsand r8 bands at k = 0 is dependent

90

C. R. WHITSETT, J. G . BROERMAN, AND C. J. SUMMERS

t

3

4

6

6

7

(Nlm'l'12 x 1 8

FIG.23. Dependence of coupled plasmon-phonon modes on electron concentration for HgSe. (From Manabe and Mitsuishi, 1975.)

on the electron concentration and temperature as a consequence of the Moss-Burstein effect, and the small separation between these bands results in a strong frequency dependence of E, in the mid-to-far infrared. These effects have been studied theoretically by Broerman (1972a) and experimentally by Grynberg el al. (1974) in HgTe, where the correct treatment is shown to resolve discrepancies of classical analyses. Because of the high electron concentrations in HgSe and low-x Hg, -,Cd,Se samples, the far-infrared absorption of these materials is large and prevents the use of transmission measurements for determining band parameters. This difficulty has, in part, been overcome by the development of the strip-line technique in which the power transmitted by a special waveguide structure, one side of which is formed by the sample, is monitored. By adjusting the strip-line width to wavelength ratio and the strip-line orientation relative to the magnetic field, different traveling wave modes and, therefore, components of the dielectric tensor can be investigated (von Ortenberg, 1974). Using this experimental technique and the water-vapor laser as an intense source of highly monochromatic 118- and 337-pm radiation, Schwarzbeck et al. (1976) and Schwarzbeck and von Ortenberg (1978) observed cyclotron, spin-fiip, and combined cyclotron-spin-flip resonances

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg, -,Cd,Se

ALLOYS

91

in HgSe. Their measurements were performed below 4.2 K in magnetic field strengths up to 8T and analyzed by a k * p theory similar to that developed by Pidgeon and Brown (1966) to obtain band parameters of EG = -0.22 eV, P = 7.2 x eV cm, and A = 0.45 eV, which agree with values obtained from galvanomagnetic studies. However, in a later study Schwarzbeck et al. (1978) obtained values for the anisotropic conductionband parameters that were in strong disagreement with the values obtained by Seiler et al. (1971) from Shubnikov-de Haas measurements on oriented HgSe samples. The reason for this disagreement is not known and requires a better understanding of the strip-line technique. Additional structure observed in the magnetic field transmission spectrum of strip-line HgSe structures at wavelengths of 118,195, and 337 pm has been attributed to donor-state to conduction-band transitions by Schwarzbeck et al. (1978). The impurity transition energy is observed to be below the conduction-band edge in the magnetic field induced energy gap and to increase linearly with field to a maximum x 10.5 meV at 7.8 T; the data are well fitted by a high-field hydrogenic impurity model. At radio- and microwave frequencies the opacity of high-conductivity samples can be dramatically altered by a strong magnetic field. The magnetic field renders the sample transparent to helicon waves, which are circularly polarized in the same sense as the cyclotron motion of the mobile carriers and propagate parallel to the applied magnetic field. Helicon wave propagation is basically a high-frequency transport phenomenon, which provide a contactless method for measuring electrical transport parameters. This phenomenon has been observed in Hg,-,Cd,Se alloys with x values below 0.25 and electron mobilities greater than lo5 cm2/v sec by Furdyna (1979), who measured the variation of the helicon phase with magnetic field and obtained values for the electron concentration and electron mobility by fitting helicon interference curves. In contrast with HgSe, the wide bandgap of CdSe permits a clear separation to be made between inter- and intraband effects. Measurements of the 300-K free-carrier absorption and reflectivity of unoriented samples (Kubo and Onuki, 1965) show that there is little dependence of the effective mass on electron concentration and that the conduction-band curvature is parabolic, with an effective mass of 0.15 f 0.1 m, at k = 0. This value is considerably higher than the low-temperature value of 0.13 f 0.01 mo reported by Wheeler and Dimmock (1962), and the 40-K value of 0.118 f 0.002 m, obtained by Chamberlain et al. (1972) from a cyclotron resonance measurement. An increase of 10% is expected in the effective mass value between liquid-helium and room temperature because of the temperature dependence of the energy gap, and this factor makes the various effective mass values agree within experimental error.

92

C. R. WHITSETT, J. G. BROERMAN,AND C. J. SUMMERS

20. LATTICEREFLECTIVITY The far-infrared reflectivity of singlecrystalline, wurtzite CdSe has been studied in the reststrahlen region by several authors. In all these studies the radiation was linearly polarized parallel and perpendicular to the c axis of the crystal, and the fundamental lattice vibration frequencies and dielectric constants were determined from a classical-oscillatorfit to the data. Results of these studies are given in Table 11. An anisotropy of approximately 4 cm-', or 3 %, is observed in the transverse optical mode frequency. These results are in good agreement with lattice parameter measurements, which show that for the wurtzite form of CdSe the c-axis lattice parameter is z 1 % greater than the lattice parameter perpendicular to the c axis. The infrared reflectivity studies measure the transverse optical-mode frequency and the high- and low-frequency dielectric constants, from which the longitudinal mode frequency can be calculated using the LyddaneSachs-Teller relationship. Band-edge emission studies (Gnatenko and Kurik, 1970; Bosacchi and Franzosi, 1975) and application of Urbach's rule to the temperature dependence of the fundamental absorption edge (Nitecki and Gaj, 1974) independently give a value of 218 cm-' for the longitudinal optical mode frequency, approximately 7 cm- higher than the value obtained from infrared reflectivity measurements. A study of the room-temperature far-infrared reflection spectra of mixed crystal Hg,-,Cd,Se alloys has recently been reported by Summers et al. (1981) who showed that all samples have a plasma reflectivity edge at low wave numbers and two restrahlen bands at 140-160 cm-' and w 180 cm-',

'

TABLE I1 OPTICAL-MODE FREQUENCIES AND DIELECTRIC CONSTANTS FOR CdSe MEASURED Transverse optical-mode frequency (cm- ') Ellc E l c 167" 171 166b 172 168' 171

Longitudinal optical-mode frequency (cm- l )

Ellc 209 211 211

E l c 211.5 210 211

High-frequency dielectric constant EollC

9.95 10.16 9.6 l0.2od

Static dielectric constant

-501 C

E,IlC

9.1 9.29 9.3 9.53

6.05 6.3 6.1

1C 5.95 6.2 6.1 E,

6.05e 5.95

' From LeToullec (1967). From Geick et al. (1966). From Verleur and Barker (1967). From Berlincourt et al. (1963). From Shiozawa (1967).

2.

CRYSTAL GROWTH AND PROPERTIES OF Hg,

-,Cd,Se

I

100

0

300

200

ALLOYS

93

500

400

Wave number (cm-1)

FIG.24. Comparison of experimental and theoretical reflectancespectra for a Hg, -,Cd,Se alloy with x = 0.347 and n = 7.84 x 1 0 L 6 ~ m -(From 3. Summers et al., 1978.)

which are assigned to the HgSe and CdSe sublattices,respectively. Parameters for the frequency, strength, and damping of the lattice and electronic vibrational modes in the Hg,-,Cd,Se system were obtained from fits to each experimental reflectance spectrum by a synthesized reflectance spectrum constructed from a superposition of damped harmonic oscillators and a Drude-Zener term to represent the lattice and the electronic contributions to the complex dielectric constant, respectively. An example of the fits obtained is shown in Fig. 24. The compositional dependences of the transverse optical lattice vibrational frequencies for the HgSe and CdSe sublattices obtained from these analyses are shown in Fig. 25. The data are in

-

-

-

3

75

2

50-

-

25-

-

'=

8

0

I

1

I

1

I

I

94

C . R. WHITSETT, J. G . BROERMAN, AND C . J. SUMMERS

excellent agreement with measurements for HgSe and CdSe and show that o,(HgSe) increases with x while oTo (CdSe) decreases slightly. The magnitudes and compositional dependences of the high-frequency dielectric constant obtained from these measurements are in good agreement with refractive index values obtained from transmission measurements. The longitudinal optical-mode frequency of HgSe has been determined from the 4.2-K current-voltage characteristic of HgSe-A12 0,-Pb junctions to be approximately 169.4 cm- (Niewodniczanska-Zawadzka and Rauluszkiewicz, 1976). This value agrees reasonably with a value of 180.6 cm- calculated from the 300-K infrared reflectivity data of Summers et al. (1981) on low-x Hg,-xCd,Se samples.

'

21. PHOTOCONDUCTIVITY Photoconductivity was fist observed in the Hg, -,Cd,Se alloy system by Ziborov et al. (1973) for x values between 0.24 and 0.92. They observed, as shown in Fig. 26, that the wavelength of peak photoconductive response moves to longer wavelengths from 0.7 to 5.0 p,with decreasing x value. In some alloys with x < 0.24 a bolometric response with a spectral sensitivity to beyond 10 pm was observed. The measurements were made at 295 and

-

1.1

1

x = 0.24 1 .o

I

1

0.34

I

I

0.46 0.54 0.63

I

0.92

I

0.9

0.8

;

'H

.-

E

0.7

0.6

g 0.5

c

a

3.--

0.4

a

z

0.3 0.2 0.1

0 Energy (eW

FIG.26. Photoconductivity spectra of Hg, -,Cd,Se alloys with x values between 0.24 and 0.92, ---, 100 K, 77 K. (From Ziborov et a/., 1973.) ~

2.

CRYSTAL GROWTH AND PROPERTIESOF

Hg, -,Cd,Se

ALLOYS

95

Temperature, T (K)

--

lo6,[

105

g

lo4r

4d

a

-

--

---

-

---

ld,

--

0

c

a

2 0

102

=-

--

--

-

10'

-

-

x = 0.55

--

.-t

a

I

-

.-2> .-

=8

70

I l l

r --

-s> e

I

I

-

n

10090 80

300 200 150

l

l

t

"

0.005

l

l

t

l

l

0.010

l

'

t

-

0.015

Inverse temperature (K1)

FIG.27. Peak photoconductiveresponsivityas a function of inverse temperature for vacuumannealed and aged Hg,-,Cd,Se samples. (From Summers, 1980.)

100 K and showed that the temperature and compositional dependences of the spectral response are in agreement with optical absorption data. Similar studies have been performed by Summers (1980), who measured the photoconductive responsivity for x values between 0.256 and 0.54 at temperatures between 5 and 300 K. Vacuum annealing and doping with Li was found to reduce the electron concentration and improve the lowtemperature photoconductive performance of alloys with x > 0.4. The peak responsivities of photoconductive elements fabricated from two samples that were annealed in vacuum at 295°C for 137 h and aged for 1300 h are shown in Fig. 27 as functions of the reciprocal temperature. For both elements the responsivity increased by two orders of magnitude, to values > lo5 V/W, as the sample temperature was lowered from 300 K to below 100 K. This large increase is attributed to minority-carrier trapping or surface trapping effects, which increase the effective lifetime of the photonexcited electrons and thus the magnitude of the photoconductive response. The effect of trapping was also observed in the photoconductive decay signal and contributes a long tail to the signal as shown in Fig. 28 for a sample

96

C. R. WMTSETT, J. G. BROERMAN, AND C. J. SUMMERS

FIG.28. Effect of minority-carrier trapping on photoconductive decay for an element with x = 0.325 at (a) 35 K and (b) 14 K. (From Summers, 1980.)

with x = 0.328, which has a photoconductive spectral response out to ~ 4 . pm. 0 For low-x alloys, the large electron concentrations, >2 x 1OI6 cm- 3, and Auger recombination make the magnitude of the photoconductive signal very small.

W I . Electron Scattering Mechanisms and Transport There are no reports of p-type specimens of Hg,-,Cd,Se having been prepared, and therefore only electron transport will be considered here. The emphasis will be on small-x alloys, and work on CdSe will be discussed only as necessary while a somewhat more elaborate discussion of HgSe will be presented. 22. ELECTRON SCATTERING MECHANISMS As with most other properties of the alloy system, the electron scattering mechanisms are profoundly affected at the semimetal-semiconductor

2 . CRYSTAL GROWTH AND PROPERTIES OF

Hg, -,Cd,Se

ALLOYS

97

transition. The reasons for these changes are as follows: (1) At the semimetal-semiconductor transition, the wave function at the conduction-band edge abruptly changes from p-type (r,)to s-type (r,)symmetry. This change of symmetry produces changes in the magnitudes of the electron scattering amplitudes as well as the type of lattice modes with which the electron interacts. It also produces changes in the intraband screening of those interactions characterized by charged and dipolar couplings (longitudinal optical-phonon, electron-hole, and charged-defect interactions). (2) Below the semimetal-semiconductor transition composition, interband scattering by optical phonons is allowed, while above this composition, the process is weakly forbidden by symmetry and strongly forbidden by conservation of energy for gaps greater than the optical-phonon energy. (3) Below the semimetal-semiconductor transition, the interband dielectric function contains an anomalous term arising from the zeroenergy rsV-rscexcitations. This anomalous term is large and strongly electron-concentration and temperature dependent. It thus strongly affects charged- and dipolar-coupled scattering mechanisms. Above the semiconductor-semimetal transition, the interband screening is that of a normal insulator. The scattering mechanisms which one would expect a priori to be of importance in the Hg, -,Cd, Se alloy system are polar longitudinal opticalphonon scattering, nonpolar optical-phonon scattering, acoustic-phonon scattering, electron-hole scattering, compositionaldisorder scattering, and charged- and neutraldefect scattering. A brief description of these mechanisms follows, and differences between the semimetal and semiconductor compositions are indicated. 23.

POLAR LONGITUDINAL OPTICAL-PHONON SCATTERING

Because both HgSe and CdSe are highly ionic materials, polar longitudinal optical-phonon scattering dominates other scattering mechanisms at room temperature over the entire range of compositions. In this interaction, the electrons are coupled to the longitudinal optical-phonon dipole moment, as screened by the high-frequency dielectric function of the material, and are scattered by the emission or absorption of an LO phonon. The semimetal and semiconductor ranges of alloy composition differ both in the allowed transitions and in the screening of this interaction. In the semimetal range of composition, both intraband and interband scattering processes are allowed. In low electron-concentration samples at intermediate temperatures, the interband process can be of considerable importance because of the high density of final states available in the valence

98

C. R. WJ.-IITSJ3TT, J. G . BROERMAN, AND C. J. SUMMERS

band (Lehoczky et al., 1974). Furthermore, the magnitude and the angular dependence of the scattering amplitudes in the semimetal regime are grossly different from those for a free electron (or normal semiconductor) because of the plike character of the conduction- and valence-band wave functions (Lehoczky et al., 1974; Broerman, 1969; Ehrenreich, 1957). The screening of this interaction in the semimetal range of composition is different from that in a normal semiconductor in both its interband and intraband parts. The interband part contains a term arising from the r8:t)TSvexcitation which has a complex dependence on frequency, (Sherrington and Kohn, 1968; Broerman, 1972a), impurity concentration (Broerman, 1972a; Liu and Tosatti, 1970), and temperature (Broerman, 1970a). The intraband part, because of the p-like character of the wave function, contains a factor which greatly lowers its magnitude at high-momentum transfer from that of a free electron (Broerman et al., 1971). Suitable approximations for the screening in HgSe, which are applicable to the semimetal range of the alloy system, have been derived by Lehoczky et al. (1974), who also derived expressions for the scattering matrix elements and the contribution to the Boltzmann equation for this scattering mechanism. In the semiconductor regime of alloy composition, the possibility of single-phonon interband scattering is energetically forbidden for gaps greater than the LO-phonon energy. For gaps smaller than the LO-phonon energy, the process is energetically allowed but only weakly allowed by wave function symmetry. The screening of the interaction is that for a normal semiconductor. At small gaps, the conduction-band wave function has an appreciable admixture of p-like component at the Fermi energy for all but the lowest impurity concentrations. The effect of the p admixture, as well as the extreme nonparabolicity of the conduction-band dispersion relation, must be considered in any realistic treatment of scattering in the small-gap compositions. The Hg, -,Cd,Se system has another property that complicates the description of optical-phonon scattering. The HgSe and CdSe TO modes appear separately in the infrared spectrum for all alloy compositions (Summers et al., 1981). A self-consistent treatment of scattering by a twomode disordered system has not been developed. However, the LO modes calculated from the experimentally determined TO modes have only a small separation (Nelson et al., 1978). Furthermore, the TO modes are quite broadened (Summers et al., 1981). Thus a single-mode approximation should produce small error in scattering calculations,and this approximation is therefore made. In low and medium electronconcentration samples (n 10l8 cmP3), the optical-phonon energy is an appreciable fraction of the Fermi energy, and thus an elastic approximation for the scattering amplitude is inappro-

-=

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg, -,Cd,Se

ALLOYS

99

priate. Although this causes no difficulty in the formulation of the Boltzmann equation for the perturbed distribution function, the Boltzmann equation in this case becomes a linear finitedifference equation, and a simple relaxation time solution is not possible. The mobility and other transport coefficients must, therefore, be calculated using variational or iterative techniques (Ehrenreich, 1957). OPTICAL-PHONON 24. NONPOLAR

SCAZ 'TERING

Electrons can interact with both longitudinal and transverse-optical phonons oia a deformation-potential interaction. This interaction was first investigated by Ehrenreich and Overhauser (1956) in the context of hole scattering in the valence band of germanium and has since been developed for zero and small-gap semiconductors by Boguslawski (1975) and Szymanska and Dietl (1978). The interaction is not possible in a pure s-type as in a normal semiconductor). The process is, however, strongly band (r6, allowed in the semimetal regime of composition where the wave function at the band edge is p-type, and it can also occur in the semiconductor regime at energies where there is an appreciable p-type admixture. Because of its weak coupling, it should be relatively unimportant in comparison with polar LO-phonon scattering, although Dietl and Szymanska (1978) pointed out that it can explain discrepancies between theory and experiment noted by Lehoczky et al. (1974) for high electron-concentration samples of HgSe. 25. ACOUSTIC-PHONON SCATTERING In acoustic-phonon scattering, electrons interact with acoustic phonons through a deformation-potential coupling. For a normal wide-gap semiconductor with a pure s-type conduction band, the electrons interact only with the longitudinal branch. However, if the conduction band is p-type or contains p-type contributions, the electrons can interact with the transverse modes. Thus, below the semimetal-semiconductor transition, the transverse and longitudinal modes interact with the electrons with comparable strength. Above the transition, the interaction is mainly with the longitudinal mode but with the transverse interaction increasing with increasing energy (or pfunction admixture) and thus becoming relatively important at high ionizedimpurity concentrations or high temperatures. A formalism for handling this situation has been developed by Zawadski and Szymanska(1971) and applied to the alloy system by Lehoczky et al. (1974) and by Nelson et a). (1978). 26. ELECTRON-HOLE SCATTERING An electron interacts with the gas of high-mass valence electrons via the Coulomb interaction screened by the electronicpart of the dielectricfunction. If the valence-band gas is nondegenerate, the scattering probability is

100

C. R. WHITSETT, J. G. BROERMAN, AND C. J. SUMMERS

identical to that for an interaction with a distribution of fixed scattering centers of density equal to the hole concentration (Lehoczky et al., 1974). For a degenerate hole distribution, the scattering probability has an additional dependence on momentum transfer and valence-band symmetry type. However, since the q and symmetry dependence does not exceed a factor of 0.5 for the most important scattering processes and because the hole distribution is far from degenerate, the fixedcenter description is generally adequate for electron-hole scattering. This interaction is screened by the electronic part of the dielectric function, it being assumed that the lattice cannot respond rapidly enough to follow the electron-hole relative motion. Below the semimetal-semiconductor transition, the anomalous TEct,TEvcontribution must be included in the screening as well as the reduction in intraband screening resulting from the p-type symmetry of the wave function. 27. COMPOSITIONAL-DISORDER SCATTERING The usual description of an alloy assumes a virtual crystal model with an “average” periodic potential, which yields as its eigenvalues a dehable, unbroadened, band structure. However, the random potential associated with the compositional disorder of the alloy remains, and this potential produces scattering between the eigenstates of the virtual crystal Hamiltonian. The treatments of this scattering process which are usually adopted for numerical calculations of the mobility are equivalent to a method attributed originally to Brooks (Makowski and Glicksman, 1973). The compositionaldisorder potential is represented as a nondilute random distribution of square-well scattering centers, with depths equal to the differences between energy gaps of the end-point compounds and with the dimensions the Ts-rE of the unit cell. This is a crude representation of the actual situation, but it allows computation of scattering amplitudes and a scattering term for the Boltzmann equation that probably is in error by only a small factor. It does give good agreement with experiment in the Hg,-,Cd,Se system (Nelson et al., 1978). 28. DEFECT SCATTERING An electron is scattered by ionized defects via the Coulomb interaction screened by the static dielectric function. Again, the interaction changes at the semimetal-semiconductor transition. Below the transition, the screening must include the anomalous TEc-rEyinterband term (Broerman, 1970a, 1970b), which is strongly dependent on electron concentration and temperature, as well as the modification to the intraband term produced by p-like wave function symmetry (Broerman et al., 1971). Above the transition, the screening is that of a normal semiconductor.

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg, -,Cd,Se

ALU)YS

101

A realistic representation of the potential of a neutral defect requires a detailed knowledge of the defect and a large amount of computation. The qualitative features of a short-range scatterer can be retained with the use of a &function or square-well potential, in which case the calculation is equivalent to that for compositional-disorder scattering. This approximation should be adequate provided the dimensions of the scatterer are small in comparison with 1/qmax(where qmaxis the maximum momentum transfer), which is the case for scatterers with unitcell dimensions. If band nonparabolicity is neglected, this scattering mechanism leads to a mean free path that is independent of electron energy (Lehoczky et al., 1974; Nelson et al., 1978; Broerman, 1971), and this will be discussed with regard to experimental data in the next section.

29. EXPERIMENTAL AND THEORETICAL ELECTRON MOBILITIES

Blum and Regel (1951) reported that HgSe is a high-electron-mobility semimetal and that the electron mobility decreases with a nearly T - 5 / 2 dependence. They also reported a discontinuity in the conductivity and temperature coefficient of the conductivity at the melting temperature, which has not been subsequently investigated. Tsidil'kovskii (1957) reported a T - 5 / 2behavior of the mobility from 12 to 270 K and a T - 3 behavior from 300 to 430 K. For a high electron-concentration sample, Rodot and Rodot (1960) reported a T - mobility temperature dependence while Gobrecht et al. (1961) found a T - 3 . 5dependence for a low-concentration sample and concluded that the dominant scattering mechanism depends on electron concentration. These widely varying results, along with the assumptions of a parabolic conduction-band structure and the relaxation time approximation in the early analyses, created confusion that was partially resolved by Harman (1960), who used the Kane nonparabolic model for the conductionband structure. Harman concluded that optical-phonon scattering is the dominant mechanism. However, as the previous discussion of scattering mechanisms makes clear, the situation is far more complicated than this. Broerman (1969, 1970~)performed a calculation of the low-temperature, ionized-impurity-limited mobility which specifically included the wave function symmetry dependence of the scattering amplitude and extended this to include the effects of the anomalous r8"-r8= excitation on the screening of the interaction (Broerman, 1970b,d). The results of this calculation are shown in Fig. 29. The highest observed mobilities (Whitsett, 1965; Galazka et al., 1971) are clearly in agreement with the calculation for a p-like band with screening by the anomalous dielectric function (top curve). The middle curve in Fig. 29 is a calculation for a p-like band with no anomalous screening, and the lower curve is for a normal s-like band. This figure strikingly illustrates the differences for this scattering mechanism between

102

C. R. WHITSETT, J. G . BROERMAN, AND C. J. SUMMERS

-

0 101’

I

I

I

r l

I

1018

I

I

1

1019

Electron Concentration (cm31

FIG.29. The 4.2-K Hall mobility of HgSe (Broerman, 197Od). [A and A,from Whitsett (1965); 0,from Galazka et al. (1971).] The solid curve is calculated with the perfect semimetal

dielectric function and the effect of rewave function symmetry on the scatteringamplitude. The dashed curve includes the effect of T8 wave function symmetry but uses a concentration-independent dielectric constant measured on a high-concentrationsample. The dotted curve is for a reband with the same effective mass and dielectric constant.

the semimetal and semiconductor regimes. The differences are larger for lower impuritycarrier concentrations, which have seldom been achieved in HgSe, but which are obtained in HgTe (Broerman, 197Od). Lehoczky et al. (1974) reported a comprehensivestudy of the temperaturedependent transport properties of HgSe. They measured the electron concentrations and mobilities as functions of temperature for a large number of samples with known annealing histories and determined the valenceband mass and the temperature dependence of Er8-Er6 from an analysis of Hall coefficient data using the full Kane nonparabolic band structure with higher-band corrections. A theory of electron scattering in the semimetal regime was then constructed which included interband and intraband polar LO-phonon scattering, longitudinal and transverse acoustic-phonon scattering, electron-hole scattering, and scattering by charged and neutral defects. The theory included the anomalous interband and intraband screening as a function of temperature, frequency, and chargecarrier concentration, and employed the full Kane model with higher-band corrections for the band structure. The mobilities were calculated using a variational technique (Ehrenreich, 1957). A particularly interesting pair of samples from this study is shown in Fig. 30. The lowconcentration sample

2.

CRYSTAL GROWTHAND PROPERTIES OF

Hg, -,Cd,Se

ALLOYS

103

0 . ND = 3.6 x 10l6 ma

4 = 2.0 x 1 0 4 c m

A - ND = 3.70 x

lo1'

cm'

4=1.0xio4cm

"0

I 50 200 300 100

150

250

Temperature (K)

FIG.30. Temperature dependence of the electron mobility of an HgSe sample in two annealing states (Lehoczky et aL, 1974). The low-concentration sample (upper curve) shows the rapid decrease characteristic of interband LO-phonon scattering at low temperature and the change in slope at about 70 K where intraband LO-phonon scattering begins to dominate. The highconcentration sample (lower curve) does not exhibit this effect because its Fermi energy is larger than the LO-phonon energy. The theoretical curves include scattering by donors and neutral defects.

(upper curve) has a low-temperature Fermi energy that is comparable with the LO-phonon energy, and the mobility drops rapidly from the lowtemperature peak, as is expected from interband LO-phonon scattering, and suddenly changes slope at about 70-K where ordinary intraband LO-phonon scattering begins to dominate. The high electron-concentration sample (lower curve) has a low-temperature Fermi energy that is much higher than the LO-phonon energy and exhibits a normal temperature dependence. The theoretical curves for both samples include scattering by neutral centers with mean free paths of about 2 pm, which were included in the analysis to account for the low-temperature mobilities being lower than that calculated for scattering by singly ionized donors only, which is true for most HgSe samples. The density of neutral scatterers required to account for this mobility deficit was nearly constant within each sample series (i.e., the same physical sample subjected to a succession of annealings to vary the electron concentration), although it differed between sample series. This fact led Lehoczky et al. (1974) to conclude that HgSe probably has a stable neutral defect that is frozen in during crystal growth, a

104

C. R. WHITSETT, J. G . BROFXMAN, A N D C. J. SUMMERS

conjecture which is supported by thermal-conductivity studies (Whitsett et al., 1973). The agreement with experiment for this theory, both for temperature dependence and charge-carrier concentration dependence, is excellent for impurity-carrier concentrations below 2 x 10" cm-j. Above this carrier concentration, the room-temperature mobilities are lower than calculated from the theory. A possible reason for this disagreement is a breakdown in the screening approximation. However, Dietl and Szymanska (1978) have pointed out that, with reasonable values for the deformation potentials, the nonpolar optical-phonon scattering mechanism (Ehrenreich and Overhauser, 1956; Boguslawski, 1975) can remove this disagreement at high carrier concentrations, where it becomes more comparable to polar LO-phonon scattering because it is not screened by the free carriers. On the basis of an analysis of Nernst-Ettingshausen data, Dietl and Szymanska (1978) disagree with the conjecture of neutral defects by Lehoczky et al. (1974) and conclude that acceptors or doubly ionized donors are responsible for the mobility deficit at low temperatures. The calculation of the longitudinal Nernst-Ettingshausen coefficient (Dietl and Szymanska, 1978) along with experimental data at 30 K is shown in Fig. 31. The dashed curve is calculated for donors and neutral defects, while the solid curve is for donors and acceptors. However, Iwanowski et al. (1978) conclude that low-x Hg, -,Cd,Se alloys probably do contain neutral defects in sufficient density to provide the limiting mechanism for the low-temperature mobility. The relative concentrations of the various charged and neutral defects may depend on the conditions of growth of each crystal. The preceding description of electron scattering and mobility in HgSe is applicable to the Hg,-,Cd,Se alloys in the semimetal regime with the addition of compositional-disorder scattering, which is extremely small in

Electron conmtration. n ( ~ r n - ~ l

FIG.31. The longitudinal Nernst-Ettingshausen coefficientof HgSe as a function of electron concentrationat 30 K (Dietl and Szymanska,1978). The dashed curve is calculatedfor scattering by donors and neutral defects and the solid curve for donors and acceptors. 10,from Dietl and Szymanska (1978); 0, from Shalyt and Aliev (1964).]

2.

CRYSTAL GROWTH AND PROPERTIES OF

Hg, -,Cd,Se

ALLOYS

105

the composition range below the semimetal-semiconductor transition. In the following discussion of the electron mobility in the alloy, the emphasis is therefore on those compositions above the transition. Relatively few studies of the electron mobility in Hg, -,Cd,Se alloys have been made. A qualitative description was given by Cruceanu and IonescuBujor (1969). Iwanowski and Diet1 (1976) investigated the low-temperature mobility of alloys with x = 0.05, 0.1, and 0.2 and reported values which were consistently lower than those calculated for singly ionized donors and compositional-disorder scattering only. These earlier studies were limited to relatively small ranges of temperature and alloy composition. Recently Nelson et al. (1978) reported a study of the electron mobility between 4.2 and 300 K for semiconductor compositions below the wurtzite transition. The data were analyzed using a theory which included polar LO-phonon scattering, acoustic-phonon scattering, electron-hole scattering, compositional-disorder scattering, and charged- and neutraldefect scattering. The band structure parameters used in the calculation were those obtained from optical measurements by Summers and Broerman (1980). The lattice effective charges, optical-phonon frequencies, and dielectric constants used were those obtained by Summers et al. (1981) from infrared reflectivity measurements. The only consequential adjustable parameter of the analysis is the compositionaldisorder potential. In Fig. 32 is shown a composite of the room-temperature data (for samples with conduction-electron concentrations less than 10' cmP3) from the study by Nelson et al. (1978). The theoretical curves are shown for three values of the disorder potential &is. One would expect Edisfor this alloy system to be approximately 2 eV, and the highest measured mobilities are within about 10 % of the values calculated for Edis= 1.5 eV. The agreement between theory and experiment at 200 and 100 K is shown in Figs. 33 and 34, respectively. As the temperature decreases, the mobilities depend increasingly on impurity concentrations, and extrinsic scattering processes become important at about 100 K. The temperature dependence of the mobility of a mid-x (x = 0.35) sample of the study is shown in Fig. 35. Also shown in this figure are the theoretical mobility pH and the contributions to the mobility of the more important scattering processes (i.e., the mobilities which would be found if only the indicated mechanism were present). The 4.2-K mobility of this sample was less than that calculated from singly ionized donors only, and acceptors were added to match the calculation to experiment at 4.2 K. The lowertemperature scattering is dominated by ionized-defect scattering. As the temperature increases, higher-lying states in the conduction band become populated, and these more energetic electrons are not as strongly scattered by the long-range screened-Coulomb potential. The mobility thus increases

106

C. R. WHITSETT, J. G . B R O W N , AND C. J. SUMMERS 3 x lo4

si

1o4

>

1

N

-E I

-

j

103

E

-

I"

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mole fraction CdSe. x

FIG.32. 300-K Hall mobility of semiconducting Hg, -,CdxSe samples as a function of alloy composition (Nelson ei al., 1978). The theoretical curves are for various values of the disorder potential Edisassuming 10l6donors/cm3,no acceptors, and no neutral defects.

Mole fraction CdSe, x

FIG.33. 200-K Hall mobility of semiconducting Hg, -,Cd,Se samples as a function of alloy composition (Nelson et al., 1978). The experimental points are for samples with n, (4.2 K) between loi6 and lo' ~ m - The ~ . theoretical curve is calculated with a value of 1.5 eV for the disorder potential, no acceptors, and no neutral defects.

2.

CRYSTAL GROWTH AND PROPERTIESOF

Hg, -,Cd,Se

ALLOYS

107

P

> mi

-6

1 o3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mole fraetion CdSe, x

FIG.34. 100-K Hall mobilitiesof semiconducting Hg, -,Cd,Se samples as a function of alloy composition (Nelson et al., 1978). The experimental joints are for samples with n, (4.2 K) between loi6 and lOI7 cm-j. The theoretical curves are calculated for donor concentration of 10l6and l o L 7cm-3, a disorder potential of 1.5 eV, no acceptors and no neutral defects. The data points are for specimens with n 2 loJ7cm-3 at 4.2 K.

with temperature until it is limited by the short-range compositionaldisorder potential and the LO phonons. Above 150 K, the dominant scattering is by LO phonons. The mobility maximum for intermediate-x alloys gives information on the disorder potential. If Edisis raised to 2 eV (with a corresponding decrease in donor compensation to match the 4.2-K data), then no maximum occurs in the calculated mobility because the decrease in ionizeddefect scattering is counterbalanced by an increase in disorder scattering, leading to a lowtemperature mobility plateau. Conversely, a calculation with no disorder scattering results in a peak which is much higher than that experimentally observed. A similar consideration eliminates neutral defects as important scatterers in the sample shown in Fig. 35. An attempt to match the 4.2-K mobilities of most mid-x samples by adding neutral scatterers removes the mobility peak because scattering from any short-range potential increases with energy. These arguments do not rule out highly ionized donors, rather than

108

C . R. WHITSETT, J. G . BROERMAN, AND C. J. SUMMERS

1 t1

lo30

50

100

150

Temperature

200

250

300

(K)

FIG.35. Hall mobility as a function of temperature of a mid-x (z0.35) semiconducting The theoretical Hg, -,Cd,Se sample with a 4.2-K electron concentration of 1.02 x 1OI6 curve for pH was calculated with a value of 1.5 eV for the disorder potential, no neutral defects, and sufficient acceptors to match the observed 4.2-K mobility. Also shown are the contributions to the mobility from the individual scattering mechanisms.

acceptors, as the source of the low-temperature mobility deficit. However, the sample shown in Fig. 35 would require doubly and triply ionized donors to match the 4.2-K calculation to experiment. As one moves to higher-x values, as shown in Fig. 36 (x N 0.55), even higher ionization states are required. This is unlikely and leads to the conclusion that the ionized defects in the mid- and high-x alloy compositions are probably singly ionized donors and acceptors. Results for a low-x sample (x = 0.2) are shown in Fig. 37. The calculation assumes a concentration of singly ionized donors equal to the 4.2-K electron concentration and no acceptors or neutral defects. The data lie a nearly constant 12 % below the theory over the entire range of temperature, which is considered to be good agreement. At low temperatures, the mobility is

2. CRYSTAL GROWTH AND

PROPWTIES OF Hg,

-,Cd,Se ALLOYS

I"

0

50

100

150

Temperature

200

250

109

300

(K)

FIG.36. Hall mobility as a function of temperature of a high-x (z0.55) semiconducting sample with a 4.2-K electron concentration of 5.16 x 10l6 cm-3 (Nelson et al., 1978). The theoretical curve for pH was calculated with a disorder potential of 1.5 eV, no neutral defects, N , = 1.216 x 10'' ~ m - and ~ , NA= 7.00 x 10l6 cm-'. Also shown are the contributions to the mobility from the individual scattering mechanisms.

dominated by ionized-donor scattering with a smooth transition to LOphonon scattering. The low-temperature peak does not appear for the sample shown in Fig. 37 because the higher-momentum states of the low effective mass band do not populate as rapidly with increasing temperature as the high mass bands of the mid- and high-x compositions. An inspection of Fig. 32 reveals that most samples with x below 0.3 have mobilities as much as 40% below those of the high-mobility samples, even at room temperature. These low mobilities cannot be explained by compensation or highly ionized donors because the contribution to the mobility at room temperature from charged-defect scattering is negligible. In Fig. 38 are shown the mobilities of one of the low-mobility, low-x samples (x = 0.27) for its entire annealing history. The sample as grown had a 4.2-K electron

110

C. R. WHITSmT, J. G . B R O W N , A N D C. J. SUMMERS 5 x 106

106

105

104

0

50

100

150

200

250

300

Temperature IK)

FIG.37. Hall mobility as a function of temperature of a low-x (z0.19) semiconducting Hg,-,Cd,Se sample with a 4.2-K electron concentrations of 4.29 x loi6cm-3 (Nelson et al., 1978). The theoretical curve for pH was calculated with a disorder potential of 1.5 eV, no acceptors, no neutral defects, and a donor concentration equal to the 4.2-K electron concentration. Also shown are the contributions to the mobility of the individual scattering mechanisms.

concentration of 1.19 x 1017~ m - It~ was . then annealed in vacuum to an electron concentration of 5.89 x 10l6 ~ m - and ~ , finally annealed in Hg ~ . theoretical curves were vapor to a concentration of 2.84 x 1017~ m - The calculated by assuming a concentration of neutral defects sufficient to match the 4.2-K mobility for the as-grown sample to experiment. This concentration corresponds to about l O I 9 square-well-potential scatterers of depth 2.5 eV and extending over a sphere of radius equal to one lattice constant. No changes were made in the neutral scattering for the other annealing states, for which only the donor densities were adjusted, being set equal to the 4.2-K electron concentrations. The agreement with experiment, especially at low temperature, is good. As for the case of pure HgSe, the analysis suggests the existence of a stable neutral defect in low- and mid-x Hg,-,Cd,Se alloys. Iwanowski et al. (1978) have reported a study of the temperaturedependent mobility for 0.05 x < 0.25, a range including the semimetalsemiconductor transition. They report generally good agreement between experimental measurements and a theoretical analysis similar to that of Nelson et al. (1978). In summary, the picture which emerges of the intrinsic scattering mechanisms in Hg,-,Cd,Se is simple. Over the entire composition range, the dominant scattering mechanism at room temperature is polar LO-phonon

-=

2.

CRYSTAL GROWTH AND PROPWTIES OF

Hg, -,Cd,Se ALLOYS

111

105

ND = n (4.2 K) = 5.89 x 10l6crn-3

P

t

1 4

lo30

50

100

150

200

250

300

Temperature (K)

FIG.38. Hall mobility of a low-x (z0.27), low-mobility semiconducting Hg, -,Cd,Se ] vacuum anneal [n, (4.2 K) = 5.89 x sample: as-grown [n, (4.2 K) = 1.19 x 1017~ r n - ~after 10l6 ~ m - after ~ ] Hg anneal [n, (4.2 K) = 2.84 x 1017c m - 7 (Nelson et al., 1978). The thebretical curves were calculated with a value of 1.5 eV for the disorder potential, no acceptors, donor densities equal to the 4.2-K electron concentrations, and a density of 1019 cm-' neutral defects which are unaffected by annealing.

scattering. At very high impurity concentrations ( > 5 x 10l8 ~ m - non~ ) polar optical-phonon scattering can be of importance in the semimetal range of compositions. At room temperature, compositional-disorder scattering is about 5 %of the total at the semimetal-semiconductor transition and increases to about 20% at the wurtzite transition. At room temperature and below, electron-hole and acoustical-phonon scattering are small. The defect scattering, however, is complex, and there is evidence for four kinds of defects. Both HgSe and CdSe are n-type, and thus there are donors associated with native defects of both compounds in the alloy system. CdSe is typically highly compensated (Samorjai, 1963), and there is evidence for increasing densities of an acceptor with increasing x. There is

112

C. R. WHITSEWT, J. G. B R O W N , AND C. J. SUMMERS

evidence for a stable neutral defect in HgSe and low- and mid-x Hg, -,Cd,Se alloys, a conclusion which is supported by thermal conductivity studies in HgSe (Whitsett et al., 1973). Because annealing in Hg vapor always increases the electron concentration, the HgSe donor is believed to be associated with a stoichiometric excess of Hg in the lattice and may be a Hg interstitial. The decrease in electron concentration and low-temperature mobility upon prolonged exposure to air suggests that the acceptor associated with CdSe may be oxygen. Little else is known about the defects, and a detailed study of the relation between the electrical transport properties and the growth and annealing conditions of HgSe, CdSe, and Hg, -,Cd,Se alloys possibly could reveal more of the microscopic nature of the defects. 30. THERMALTRANSPORT The lattice thermal conductivity of HgSe was first measured by Aliev et al. (1966) and subsequently by Nelson et al. (1969) and Whitsett et al. (1973). Aliev et al. (1966) measured the total thermal conductivity in the temperature range 30-220 K. They determined the change in thermal conductivityupon the application of a strong magnetic field, and by assuming that the entire change was due to the quenching of the electronic part were able to separate the total conductivity into its lattice and electronic parts. Nelson et al. (1969) measured the thermal conductivity of HgSe samples in the temperature range 1.2-300 K. Their data are not only higher at low temperatures than those of Aliev et al. (1966), but also exhibit a pronounced dip in the range 4-30 K, which they attributed to a third-order resonant scattering from localized defect modes (Wagner, 1963). This resonance process proceeds as follows. A defect in a lattice introduces a localized part in the phonon wave functions. For most frequencies, the plane wave part dominates, but there may occur an interval around a resonant frequency for which the localized part dominates, and this will introduce the possibility of scattering on the defect by processes more complex than simple Rayleigh scattering. If the defect has no internal modes, then the secondorder process in which a phonon annihilates and excites a quantum of the localized mode is forbidden. However, the third-order process involving a two-phonon annihilation and excitation of the resonant mode is allowed. This process leads to a scattering correction which is sharply cut off on the low-temperatureside. More complete data and analyses in the range 1.2-300 K were reported by Whitsett et al. (1973). They analyzed the data assuming that the dominant scattering mechanisms were Rayleigh scattering, boundary scattering, phonon-electron scattering, normal and umklapp phonon-phonon scattering, and third-order resonant scattering. Examples of their data and theoretical fits are shown in Fig. 39. The data points and solid curves are the lattice thermal conductivities, and the dashed curves are the calculated

2.

CRYSTAL GROWTH AND PROPWTIESOF Hg,

-,Cd,Se

ALLOYS

113

Temperature (K)

FIG.39. Lattice thermal conductivity as a function of temperature for HgSe samples with ~ . dashed curves are calculated electron concentrations of 2.1 x lo” and 2.2 x lo’* ~ r n - The electronic thermal conductivities.(From Whitsett et al., 1973.)

electronic thermal conductivities. At temperature below about 3 K, the dominant scattering mechanisms are boundary, electron-phonon, and Rayleigh point-defect scattering. At about 3 K, the resonant localized mode process begins to turn on as higher-energy phonons appear, and the conductivity rapidly drops. Above 30 K, normal and umklapp phonon-phonon processes begin to dominate the resonant mode, which for HgSe lies within the acoustic branch. The identity of the defect responsible for the resonant mode remains unclear. Whitsett et al. (1973) varied the sulfur and oxygen concentration (the two most likely contaminants), but the results were inconclusive. Whitsett et al. (1973) also correlated the ionizeddefect concentrations in a large number of samples with the strengths of the Rayleigh scattering, which should be proportional to the number of point defects. This analysis indicates that the samples contained large numbers of neutral defects, a result which is consistent with the analysis by Lehoczky et al. (1974) of electron transport data. The identity of the neutral defect is not known.

114

C. R. WHITSETT, J. G. BROERMAN, AND C. J. SUMMERS

IX. Summary Despite the large body of published work on HgSe and Hg,-,Cd,Se alloys, there is still little understanding of the effect of crystal-growth processes on the electronic properties of these materials and their defect chemistry. This knowledge is required to grow crystals with specific properties and will possibly enable p-type samples to be grown. The difficulty in obtaining charge-carrier conversion is presently a serious obstacle to the technological development of this alloy system. Apart from this, much has been learned about the properties of the Hg, - ,Cd, Se system by the application of sophisticated measurement techniques and appropriate theoretical analyses. The basic properties of the HgSe-CdSe pseudobinary phase diagram have been measured and analyzed to enable quantitative crystal-growth studies to be performed. The principal features of the energy-band structure and its dependence on alloy composition and temperature have been established, and a similar understanding has been obtained of the mechanisms controlling the transport properties of these alloys. The intrinsic properties of the system are similar to those of the Hg, -,Cd,Te system, and studies of one system complement the other. This is particularly true of crystal-growth and chemical-defect studies because the properties of both alloys are strongly affected by small changes in crystal stoichiometry. The application of such measurement techniques as electron-spin resonance and deep-level transient spectroscopy to crystals prepared under a range of well-defined, reproducible, conditions is expected to add significantly to the understanding of the defect chemistry of both Hg, -,Cd,Se and Hg, -,Cd,Te alloys. Magnetooptical studies of the Hg, -,Cd,Se system will result in more accurate band parameters for this system and a better understanding of impurity energy levels. REFERENCES Aliev, S. A., Korenblut, L. L., and Shalyt, S. S. (1965). Fiz. Tverd. Tela 7 , 1673 [Engl. transl., SOP.Phys.-Solid State 7 , 13571. Aliev, S. A,, Korenblut, L. L., and Shalyt, S. S. (1966). Fiz. Tuerd. Telo 8, 705 [Engl. transl.. Sor. Phys.-Solid Slate 8, 5651. Aven, M., and Prener, J. S. (1967). “Physics and Chemistry of 11-VI Compounds.” Wiley, New York. Berlincourt, D., Jaffe, H., and Shiozawa, L. R. (1963). Pbys. Rev. 129, 1009. Bliek, L. M., and Landwehr, G. (1969). Pbys. Status Solidi31, 115. Blum, A. I., and Regel, A. R. (1951). Zb. Tekb. Fiz. 21, 316. Boguslawski, P. (1975). Phys. Starus Solidi B 70, 53. Borisov, I. N., Kireev, P. S., Mikhailen, V. M., and Bezborodova, V. M. (1971). Fiz. Tekb. Poluprovodn. 5, 829 [Engl. transl., SOP.Phys.-Semicond. 5, 7341. Bosacchi, B., and Franzosi, P. (1975). Phys. Srarus Solidi B 70, K139.

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Brebrick, R. F. (1965). J. Phys. Chem. Solids43, 3846. Broerman, J. G. (1969). Phys. Rev. 183, 754. Broerman, J. G. (1970a). Phys. Rev. Lett. 25, 1658. Broerman, J. G. (1970b). Phys. Rev. B 1,4568. Broerman, J. G. (1970~).Phys. Rev. Lett. 24,450. Broerman, J. G. (1970d). Phys. Rev. B2, 1818. Broerman, J. G. (1971). J. Phys. Chem. Solids 32, 1263. Broerman, J. G. (1972a). Phys. Rev. 8 5 , 397. Broerman, J. G. (1972b). Phys. Semiconduct., Int. Conf., 11th 2,917. Pol. Sci. Publ., Warsaw. Broerman, J. G., Liu, L., and Pathak, K. N. (1971). Phys. Rev. B4,664. Byszewski, P., Kowalski, J., and Galazka, R. R. (1976). Phys. Status Solidi B 78, 477. Cardona, M., Shaklee, K. L.,and Pollak, F. H. (1967). Phys. Rev. 154,696. Chamberlain, J. M., Simmonds, P. E., and Stradling, R. A. (1972). Phys. Semicond., Int. Conf., 11th 2, 1016. Pol. Sci. Publ., Warsaw. Cohen-Solal, G., Marfaing, Y., and Bailly, F. (1966). Rev. Phys. Appl. 1, 11. Cook, W. R. (1968). J. Am. Ceram. SOC.51, 518. Cruseanu, M. E., and Ionescu-Bujor, S. (1969). J. Muter. Sci. 4, 570. Cruceanu, M. E., and Niculescu, D. (1965). C. R. Acud. Sci. 261, 935. Cruceanu, M. E., Nistor, N., and Niculescu, D. (1966). Kristullogrujiyu 11, 305. Dietl, T., and Szymanska, W. (1978). J . Pbys. Chem. Solids 39, 1041. Dobrowolski, W., and Dietl, T. (1976). Phys. Semicond., Int. Conf., 13th p. 447. North-Holland Publ., Amsterdam. Ehrenreich, H. (1957). J. Phys. Chem. Solids 2, 131. Ehrenreich, H., and Cohen, M. H. (1959). Phys. Rev. 115, 786. Ehrenreich, H.,and Overhauser, A. W. (1956). Phys. Rev. 104,331. Foster, L. M., and Woods, J. F. (1971). J . Electrochem. SOC.118, 1175. Furdyna, J. K. (1979). Unpublished observations. Galazka, R. R., Seiler, D. G., and Becker, W. M. (1971). In “Physics ofSemimetals and Narrow Gap Semiconductors” (D. L.Carter and R. T. Bate, eds.), p. 481. Pergamon, New York. Geick, R., Perry, C. H., and Mitra, S. S. (1966). J. Appl. Phys. 37, 1994. Gelmont, B. L., and Dyakonov, M. I. (1971). Fiz. Tekh. Poluprovodn. 5, 2191 [Engl. transl., Sou. Phys.-Semicond. 5, 1905 (1972)l. Giriat, W., Gluzman, N. G., Domanskaya, L. I., and Tsidil’kovskii, I. M. (1975). Fiz. Tekh. Poluprovodn. 9, 1024 [Engl. transl., Sou. Phys.-Semicond. 9,6751. Gnatenko, Y . P., and Kurik, M. V. (1970). Opt. Spektrosk. 29, 339 [Engl. transl., Sou. Phys. Opt. Spectrosc. 29, 1791. Gobrecht, H., Gerhardt, V.,Peinemann, B., and Tausend, A. (1961). J. Appl. Phys. 32,2246. Groves, S., and Paul, W. (1963). Phys. Rev. Lett. 11, 194. . Dunod, Paris; Academic Groves, S.,and Paul,W. (1964). Phys. Semicond., Znt. Conf., 7 t h ~41. Press, New York. Grynberg, M. (1964). Phys. Semicond., Int. Conf., 7th p. 136. Dunod, Paris; Academic Press, New York. Grynberg, M., LeToullec, R., and Balkonski, M. (1974). Phys. Rev. B 9, 517. Guenzer, C. S., and Bienenstock, A. (1973). Phys. Rev. B 8,4655. Guldner, Y., Rigoux, C., Dobrowolska, M., Mycielski, A., and Dobrowolski, W. (1977). In “Physics of Narrow Gap Semiconductors” (J. Rauruszkiewicz, M. Gorska, and E. Kaczmarek, eds.), p. 87. Pol. Sci. Publ., Warsaw. Gutsche, E., and Lange, H. (1964). Phys. Semicond., Int. Conf., 7th p. 129. Dunod, Paris; Academic Press, New York. Halperin, B. I., and Rice, T. M.(1968). Rev. Mod. Phys. 40,755.

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Harman, T. C. (1960). Phys. Rev. 118, 1541. Harman, T. C., Kleiner, W. H., Strauss, A. J., Wright, G. B., Mavroides, J. G., Honig, J. M., and Dickey, D. H. (1964). Solid State Commun. 2, 305. Hopfield, J. J. (1961). J. Appl. Phys. 32,2277. Iwanowski, R. J. (1975). Acta Phys. Pol. A 47, 583. Iwanowski, R. J., and Dietl, T. (1976). Phys. Status Solidi B 75, K83. Iwanowski, R.J., Dietl, T., and Szymanska, W. (1978). J. Phys. Chem. Solids 39, 1059. Jayoraman, A., Kement, W., Jr., and Kennedy, G. C. (1963). Phys. Rev. 130,2277. Johnson, E. S., and Schmit, J. L. (1977). J. Electron. Mater. 6,25. Jordan, A. S . (1970). Met. Trans. 1,239. Kalb, A., and Leute, V. (1971). Phys. Status Solidi 5, K199. Kane, E. 0. (1957). J. Phys. Chem. S o l a 1,249. Kireev, P. S., and Volkov, V. V. (1974). Fiz. Tekh. Poluprouodn. 7, 1419 [Engl. transl., Sou. Phys.-Semicond. 7,9491. Kireev, P. S., Volkova, L. V., and Volkov, V. V. (1972). Fiz. Tekh. Poluprouodn. 5,2085 [Engl. transl., Sou. Phys.-Semicond. 5, 1816.1 Kirik, M.V., Marchik, I. V., and Pan’kiv, P. M. (1974). Ukv. Fiz. Zh. (U.S.S.R.) 19, 1563. Koxihk, C., Dillinger, J., and Prosser, V. (1967). In “11-VI Semiconducting Compounds” (D. G. Thomas, ed.),p. 850. Benjamin, New York. Kot, M. V., and Mshenskii, V. A. (1964). Izv. Akad. Nauk SSSR 28, 1067. Kot, M. V., Tyrziu, V. G., Simashkevich, A. V., Maronchuk, Y.E., and Mshenskii, V. A. (1962). Fir. Tverd. Tela 4, 1535 [Engl.transl.: Sou. Phys.-Solidstate 4, 11283. Kubo, S., and Onuki, M. (1965). J. Phys. SOC.Jpn. 20, 1280. Kulwicki, B. M. (1963). Ph.D. Thesis, Univ. of Michigan, Ann Arbor. Kumazaki, K., Matsushima, E., and Odajima, A. (1976). Phys. Status Solidi A 37, 579. Kumazaki, K., Yagawa, F., and Adajima, A. (1977). In “Internal Friction and Ultrasonic Attenuation in Solids” (Ryukiti R. Hasiguti and Nubuo Mikoshiba, eds.), p. 61 1. Univ. of Tokyo Press, Tokyo. Lehoczky, S. L., Broeiman, J. G., Nelson, D. A., and Whitsett, C. R. (1974). Phys. Reu. B 9, 1598. LeToullec, R. (1967). Thesis, Univ. Paris, Paris. Liu, L., and Brust, D. (lW8a). Phys. Rev. 173,777. Liu, L., and Brust, D. (1968b). Phys. Reu. Lett. 20,651. Liu, L., and Tosatti, E. (1969). Phys. Rev. Lett. 23, 772. Liu, L., and Tosatti, E. (1970). Phys. Reu. B 2,1926. Ludeke, R.,and Paul, W. (1967). Phys. Status Solidi 23,413. Makowski, L., and Glicksman, M. (1973). J. Phys. Chem. Solids 34,487. Manabe, A,, and Mitsuishi, A. (1975). Solid State Commun. 16,743. Mason, D. R., and OKane, D. F. (1961). Phys. Semicond., Znt. Con$, 5th p. 1026. Academic Press, New York. Munir, Z. A., Meshi, D. J., and Pound, G. M. (1972). J. Cryst. Growth 15, 263. Nassau, K., and Shiever, J. W. (1972). J. Cryst. Growth 13/14, 375. Nelson, D. A. (1977). Unpublished. Nelson, D. A., and Summers, C. J. (1977). Unpublished observations. Nelson, D. A., Broerman, J. G., Paxhia, E. C., and Whitsett, C. R. (1969). Phys. Rev. Lett. 22, 884. Nelson, D. A,, Summers, C. J., and Whitsett, C. R.(1977). J. Electron. Muter. 6, 507. Nelson, D. A., Broerman, J. G., Summers, C. J., and Whitsett, C. R. (1978). Phys. Rev. B 18, 1658. Niewodniczanska-Zawadzka, J., and Rauluszkiewicz, J. (1976). Phys. Semicond., Znt. Con$, 13th p. 443. North-Holland Publ., Amsterdam.

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Nitecki;R., and Gaj, J. A. (1974). Phys. Status Solidi B62, K17. Ortenberg, M.(1974). ZEEE Trans. Microwave Theory Tech. 22, 1081. Parsons, R. B., Wardzynski, W., and Yoffe, A. D. (1961). Proc. R . Soc. London, Ser. A 262,120. Pfann, W. G. (1952). Trans. AZME 194, 747. Pidgeon, C. R., and Brown, R. N. (1966). Phys. Rev. 146, 575. Pidgeon, C. R., and Groves, S. H. (1967). In “11-VI Semiconducting Compounds” (D. G. Thomas, ed.), p. 1080. Benjamin, New York. Piper, W. W. (1967). In “11-VI Semiconducting Compounds” (D. G. Thomas, ed.), p. 839. Benjamin, New York. Reisman, A., Bekenblit, M., and Witzer, M. (1962). J. Phys. Chem. 66,2210. Reynolds, D. C., Litton, C. W., and Collins, T. C. (1965). Phys. Status Solidi 9,645. Rodot, M., and Rodot, H. (1960). C. R. Acud. Sci. 250, 1447. Roth, L. M. (1968). Phys. Rev. 173, 755. Roth, L. M., Groves, S. H., and Wyatt, P. W. (1967). Phys. Rev. Lett. 19, 576. Samorjai, G. A. (1963). J. Phys. Chem. Solids24, 175. Savitskii, V. G., Alekseenko, L. I., Filatova, A. K., and Slavkovskii, I. S. (1975). Zzu. Akud. Nuuk SSSR, Neorg. Mater. 11, 1746 [Engl. transl., Znorg. Muter. (USSR) 11, 14961. Schwarzbeck, K., and von Ortenberg, M. (1978). Phys. Semicond., Znt. Conf., 14th p. 1125. Inst. Phys., London. Schwarzbeck, K., von Ortenberg, M., and Landwehr, G., and Galazka, R. R. (1976). Phys. Semicond., Znt. Conf., 13th p. 435. North-Holland Publ., Amsterdam. Schwarzbeck, K., Bangert, E., and von Ortenberg, M. (1979). J. Mugn. Mug.Mar. 11, 119. Seiler, D. G., Galazka, R. R., and Becker, W. M. (1971). Phys. Reo. B 3,4274. Shalyt, S. S., and Aliev, S. A. (1964). Fiz. Tuerd. Telu 6. 1979 [Engl. transl., Sou. Phys.-Solid State 6, 1563 (1964)l. Sherrington, D., and Kohn, W. (1968). Reu. Mod. Phys. 40,767. Shiozawa, L. R. (1967). Sixth Quarterly Rep., Contract N. AF 33(657)-7399. U.S. Air Force Aeronaut. Res. Lab., Wright-Patterson Air Force Base, Ohio. Singh, H. P. and Dayal, B. (1970). Actu Crystallogr., Sect. A 26,363. Slodowy, P. A., and Giriat, W. (1971). Phys. Status Solidi B 48,463. Smith, S . D. (1967). In “Handbuch der Physik” (S. Flugge, ed.), Vol. XXV/2a, p. 234. SpringerVerlag, Berlin and New York. Smith, V. G., Tiller, W. A., and Rutter, J. W. (1955). Can.J. Phys. 33, 723. Stankiewicz, J., Giriat, W., and Dobrowolski, W. (1974). Phys. Status Solidi B 61, 267. Steininger, J. (1970). J. Appl. Phys. 41, 2173. Strauss, A. J., and Farell, L. B. (1962). J. Znorg. Nucl. Chem. 24, 1211. Strauss, A. J., and Steininger, J. (1970). J. Electrochem. SOC. 117, 1420. Stuckes, A. D., and Farell, G. (1964). J. Phys. Chem. Soliak 25,477. ‘ Summers, C. J. (1980). Unpublished observations. Summers, C. J., and Broerman, J. G. (1980). Phys. Reu. B 21, 559. Summers, C. J., and Nelson, D. A. (1980) Unpublished observations. Summers, C. J., Broerman, J. G., Nelson, D. A., and Whitsett, C. R. (1978). Phys. Semicond., Znt. Conf., 14th p. 165. Inst. Phys., London. Summers, C. J., Bhatnager, A. K., and Nelson, D. A. (198 1). To be published. Swanson, H.E., Gilfrich, N. T., and Cook, M. I. (1957a). Natl. Bur. Std. (US.), Circ. 7 , 35. Swanson, H. E., Gilfrich, N. T., and Cook, M. I. (1957b). Narl. Bur. Stand. (US.), Circ. 7,539. Sysoev, L. A., Raiskin, E. K., and Gur’ev, V. R. (1967). Znorg. Muter. (USSR) 3, 341. Szuszkiewicz,W. (1977). In “Physics of Narrow Gap Semiconductors” (J. Rauluszkiewicz, M. Gorska, and E. Kaczmarek, eds.), p. 93. Pol. Sci. Publ., Warsaw. Szymanska, W., and Dietl, T. (1978). J. Phys. Chem. Solids 39, 1025. Thomas, D. G., ed. (1967). “11-VI Semiconducting Compounds.” Benjamin, New York.

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Tiller, W. A., Jackson, K. A., Rutter, J. W., and Chambers, B. (1953). Actu Metull. 1,428. Tsidil'kovskii, I. M. (1957). Zh. Tekh. Fiz. 27,9 [Engl. transl., Sou. Phys. Tech. Phys. 2,9]. Verleur, H. W., and Barker, A, S.,Jr. (1967). Phys. Rev. 155, 750. Volkov, V. V.,Volkova, L. V., and Kireev, P. S.(1971). Fiz. Tekh. Poluprouodn. 4,1437 [Engl. transl., Sou. Phys.-Semicond. 4, 12301. Wagner, M. (1%3). Phys. Rev. 131, 1443. Walker, T.W.,Litton, C. W., Reynolds, D. C., Collins, T. C., Wallace, W. A., Gorrell, J. H., and Jungling, K. C. (1972). Phys. Semicond.,Znt. Conf., 11th p. 376. Pol. Sci. Publ.,Warsaw. Wheeler, R. G., and Dimmock, J. 0.(1962). Phys. Rev. 125,1805. Whitsett, C. R. (1965). Phys. Rev. 138, A829. Whitsett, C. R.,and Nelson, D. A. (1969). J. Cryst. Growth 6,26. Whitsett, C. R., Nelson, D. A., Broerman, J. G., and Paxhia, E. C. (1973). Phys. Rev. B7,4625. Whitsett, C. R.,Broennan, J. G., Nelson, D. A., and Summers, C. J. (1976). Final Rep., Off. Nav. Res., Contract No. NOOO14-74-C-0318. Washington, D.C. Whitsett, C. R.,Summers, C. J., Nelson, D. A., and Broerman, J. G. (1977). Final Rep., Air Force Mater. Lab., Contract No. F336174-C-5167. Dayton, Ohio. Wosten, W. J. (1961). J. Phys. Chem. 65, 1949. Woolley, J. C., and Ray, B. (1960). J. Phys. Chem. Solids. 13, 151. Wright, G. B., Strams, A. J., and Harman, T. C. (1962). Phys. Rev. 125,1534. Yu,S. C. (1964). Ph.D. Thesis, Harvard Univ., Cambridge, Massachusetts. Zawatiski, W., and Szymanska W. (1971). Phys. Status Solidi45,415. Ziborov, A. I., Bezborodova, V. M., and Kireev, P. S . (1973). Fiz. Tekh. Poluprovodn. 6,2045 [Engl. transl., Sov. Phys.-Semicond. 6, 17401.

SEMICONDUCTORS A N D SEMIMETALS, VOL. 16

CHAPTER 3

Magnetooptical Properties of Hg, - Cd,Te Alloys M . H. Weiler I. INTRODUCTION . . . . . . . . . . . . . 1. Magnetooptics of Semiconductors . . . . . . 2. Other Techniques . . . . . . . . . . . 11. THEORY FOR ZINC-BLENDE SEMICONDUCTORS . . . . 3. Quasi-Germanium Model. . . . . . . . . 4. Selection Rules . . . . . . . . . . . 5. Approximate Expressions. . . . . ., . . . 111. INTRABAND EXPERIMENTS . . . . . . . . . . 6. Cyclotron, Combined, and Phonon-Assisted Resonances 7. Spin-Flip Transitions . . . . . . . . . . 8. Transport Measurements . . . . . . . . . IV. INTWBAND EXPERIMENTS . . . . . . . . . . 9. Exciton Corrections . . . . . . . . . . 10. Magnetooptical Results . . . . . . . . . 11. Parameter-Fitting Techniques . . . . , . . . V. Hg, -,Cd,Te PARAMETERS. . . . . . . . . 12. Summary of Results for Hg, -,Cd,Te . . . . . 13. Conclusion. . . . . . . . . . . . . REFERENCES

. . . . . .

.

. . . .

.

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

. . 119 . . 121 . . 126 . . 127 . . 131 . . 136 . . 139 . . 145 . . 145 . . 155 . . 159 . . 161 . . 161 . . 163 . . 171 . . 176 . . 117 . . 184 . . 188

I. Introduction Since the discovery of cyclotron resonance in germanium just over 25 years ago (Dresselhaus et al., 1953; Lax et al., 1954), magnetooptical studies of semiconductors have evolved into a set of powerful tools for determining the electronic band structure of semiconductors. By carrying out a variety of optical measurements in high magnetic fields, it is possible to identify the symmetry properties of the fundamental extrema and, for example, to make accurate measurements of the bandgap and the conduction- and valence-band effective masses and g factors. These are the parameters of a group-theoretical model for the Hamiltonian of the electrons and holes in the material. With this model, and a knowledge of the parameters, one then can predict and understand the behavior of the electrons and holes in other experimental configurations and in device applications. Magnetooptical techniques have reached perhaps their most advanced state of sophistication when applied to semiconductors which crystallize 119 Copyright @ 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-752116-X

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in the zinc-blende structure. These are the 111-V compounds, such as InSb and GaAs, and the 11-VI compounds, such as HgTe, CdTe, and CdS. The most successful applications have been the narrow-gap zinc-blende semiconductors, of which the prototype is InSb because of the high-quality samples available. We have recently shown (Weiler, 1978) that the quasigermanium model originally proposed by Pidgeon and Brown (1966) provides an excellent description of the behavior of electrons in both the conduction and the valence bands of InSb. A very successful application of magnetooptical techniques has been the determination of the electronic band structure of the alloy system Hg,-,Cd,Te. It was a magnetooptical study of the semimetal HgTe by Groves et al. (1967) which established that the quasi-germanium bandstructure model could be used to describe the energy bands in HgTe as well as in gray tin (Groves and Paul, 1963), but with a negative, or inverted, energy gap. This model has now been used successfully (Weiler et al., 1977; Guldner et al., 1977a,b) to describe the electronic energy bands of the alloy system Hg, -,Cd,Te for up to 30% CdTe (x = 0.3). Over this wide range of alloy composition x, the model parameters are essentially constant except for the energy gap which changes from -0.3 eV (inverted or semimetallic structure) to +0.2 eV (normal or semiconducting structure). These bands are with E , 2 0. illustrated in Fig. 1 for HgTe, CdTe, and Hg,.,,Cd,.,,Te In this chapter we will show from our recent work that the quasigermanium model gives a consistent description of the behavior of both the electrons and holes in Hg,-,Cd,Te and will give the dependence of the model parameters on alloy composition and temperature. After reviewing the techniques of magnetooptics and comparing them with other techniques,

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we will describe in Part I1 the quasi-germanium model for Hg, -,Cd,Te, including approximate analytic expressions which are often useful. In Part I11 we will review the results of measurements on transitions within the conduction band (intraband measurements). In Part IV we will review the results of measurements on transitions from the valence band to the conduction band (interband measurements). Finally, in Part V we describe the results of these measurements, which are a set of parameters, depending on alloy composition and temperature, for the quasi-germanium model for the electrons and holes in Hg,-,Cd,Te. We will derive fairly simple expressions for these parameters and for other quantities of interest in device applications. An extensive review of the literature for Hg, -,Cd,Te was given,by Dornhaus and Nimitz (1976). In this chapter we will concentrate on the magnetooptical results, particularly for alloys in the small-gap semiconductor region. 1. MAGNETOOPTICS OF SEMICONDUCTORS Magnetooptical experiments are, as the name implies, optical measurements on samples in an applied magnetic field. The first such experiments were carried out by Zeeman (1897) on atoms. Optical experiments are known to yield extremely accurate determinations of sharp energy-level features. The application of high magnetic fields to semiconductors introduces such sharp features (called Landau levels) into the energy-level structure, which then can be studied by optical techniques. Usually the accuracy of the results is limited by the intrinsic width of the levels, due to sample impurity or inhomogeneity, rather than by any limitation of the apparatus. In fact, magnetooptical techniques are most useful when samples are available of sufficient quality so that reasonably sharp spectral features are observed. These techniques, which include absorption and reflection spectroscopy and polarization measurements such as Faraday rotation, have been reviewed extensively; in particular, by Lax (1963), Lax and Mavroides (1960, 1967), Dresselhaus and Dresselhaus (1966), Palik and Wright (1967), and Aggarwal (1972), among many others. In this short introduction we will simply review some of the most important techniques and results. a. Landau Levels An electron in a simple, parabolic, isotropic conduction band at the center of the Brillouin zone (k = 0) obeys the energy-momentum relation E = E,

+ h2k2/2m,,

where E is the energy measured relative to the top of the valence band, E , is the energy gap, hk is the cystal momentum, and m, is the (conductionband) effective mass. In the presence of an applied magnetic field H, the

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electronic motion in the plane perpendicular to H becomes quantized and the motion parallel to H is unaffected. The energy-momentum relation becomes, for electron spin up or spin down, E$. = E,

+ (n -t- &)hwcf *ggpBH + h2k&/2mc,

(2)

where n is the Landau quantum number, w, = eH/rncc is the cyclotron frequency, gc is the effective (conduction-band) g factor, pB 3 heH/rnc is the Bohr magneton, and hk, is the crystal momentum parallel to H. The density of states per unit volume and per unit energy interval dE, corresponding to the case where the magnetic field H is zero, is, from Eq.(l), P(E) = (2.2)- y2rnc/h2)3/2(~ -~,)1/2

(3)

(see, for example, Callaway, 1964, pp. 27ff.). The density of states in the presence of the magnetic field is (see, for example, Dingle, 1952) p(E) = (47~~)‘(2mc/h2)”2(eH/hc) X [ E - E , - (n *)hac & + g c p ~ W - 1 / 2 . n = o s= +, -

1

+

(4)

Whereas the H = 0 density of states in Eq.(3) shows only a smooth edge as E 4 E,, especially when broadening is included, the density of states in Eq. (4) shows a series of sharp peaks whenever E approaches one of the energy levels in Eq. (2) for k, 40. This is shown schematically in Fig. 2.

Eg ENERGY

FIG.2. Density of states near the energy gap E , (solid curve); density of states near the first Landau level in an applied magnetic field (dashed curve).

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b. Intraband Transitions

The peaks in the density of states in Eq. (4) become sharp optical features corresponding to transitions among the Landau levels (intraband transitions). A sharp transition from level n to level n + 1 is observed at an optical frequency w = w, and is called cyclotron resonance; a spin-flip transition takes place at the spin-flip frequency w = w, = JgclpBHand is called spin resonance; in the zinc-blende semiconductors a transition is allowed from level n, s = + to level n + 1, s = - at a frequency w = w, w, and is called combined resonance. Other transitions, for example, n to n + 2, are also allowed under certain circumstances; these are called harmonics of cyclotron resonance. One may also observe cyclotron transitions from n to n + 1 accompanied by the emission of a longitudinal optical phonon this and similar cyclotron-harmonic transitions at frequency w E w, -toL0; are called phonon-assisted transitions and are useful for measuring the optical-phonon frequency and the electron-phonon coupling. All of these transitions, and similar ones among valence-band states, are called intraband transitions because they take place within one band. Studies of intraband transitions are extremely valuable for measuring the effective masses and g factors for a given band. These results are limited by the fact that the band must be occupied; that is, electrons or holes must be present, This requires either that the sample be at a fairly high temperature, or that impurities (donors or acceptors) be present. Both of these introduce scattering or line broadening. The requirement for sharp cyclotron resonance lines is o,z %- 1, where z is the lifetime. This is discussed in the review by Lax (1963). In addition to line broadening, carriers introduce a plasma frequency we = (4.nne2/m,~o)1/Z, where n is the carrier.density and E~ is the low-frequency dielectric constant. The cyclotron resonance shifts to wh = (w: + The sample is opaque below the plasma frequency. These effects are discussed by Palik and Wright (1967).

+

c. Interband Transitions

An electron having a simple valence band in a magnetic field has energy E:, * = - ( n

+ $)hwV f +g,,PnH - h2k&/2m,,

(5)

where w, = eH/m,c is the valence-band cyclotron frequency, gv is the hole g factor, and m, is the valence-band effective mass. Transitions from states in the valence band to states in the conduction band are called interband transitions. They also produce sharp optical features for wz %- 1 where Aw = E , is the photon energy. Interband transitions among levels of different Landau quantum number n and different spin states, particularly for

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more complex bands, provide independent measurements of the conduction- and valence-band parameters such as m, and m,. Furthermore, all the transitions extrapolate for H + 0 to the energy gap E,. This is a more reliable way of measuring E , than attempting to locate the position of the gap from the absorption @e at H = 0. Since sharp transitions are observed for or 9 1 with ho x E,, and thus, usually, o % o,,interband transitions are generally sharper than intraband transitions. This distinction is less important for narrow-gap materials. Furthermore, interband transitions are observed in pure samples at low temperatures, further reducing the.scattering. d. Measurement Techniques

The most straightforward measurement techniques for both interband and intraband transitions are optical absorption and reflection. Interband transitions are usually, but not always, observed in reflection since absorption coeacients are high. Intraband transitions, conversely, are usually observed in transmission since they are weaker. In either case, a line-shape analysis as discussed by Lax (1963) or Aggarwal(1972), for example, yields the position of the transition, usually near the peak of the absorption or reflection. A few typical line shapes for interband reflection and intraband absorption are shown in Figs. 3 and 4. For interband transitions the resonant frequency is usually in the visible or near-infrared region accessible to standard spectroscopic techniques, and, for typical magnetic fields, the intraband frequencies are in the microwave or far-infrared region and studied using far-infrared lasers or Fourier transform spectrometers. For narrow-gap materials both interband and intraband transitions can lie in the infraredregion (2 to 20 pm) and can be studied using grating spectrometers. The optical polarization, in relation to the magnetic field, determines the transitions observed. In the Faraday configuration (left or right circular polarization, aL or a,, respectively, with light propagation parallel to the 0.10 I

I

I

I

2

-

%a08-

a I a 0.06

-

1

-

>.

r, 0.040 W

0.02-

a QOW -0.1

I

0 0.I MAGNETIC FIELD ( H - H o I / H o

0.2

FIG. 3. Calculated interband magnetoreflection curves for wz = 2000(a), 1000(b), SOO(c), 2OO(d), and lOO(e), versus magnetic field H near the resonant field H , for a particular interband transition, where the reflectivity is equal to R,.

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MAGNETOOPTICAL PROPERTIES OF Hg,

-,Cd,Te

ALLOYS

125

W,/W

FIG.4. Calculated intraband magnetoabsorption for uL polarization for different values O f W7.

magnetic field) one typically observes intra-conduction-band transitions n + n & 1 (cyclotron resonance). In the Voigt configuration (light propagation transverse to the field, polarization transverse or parallel, d or IC, respectively) one observes cyclotron resonance or combined resonance. For interband transitions the selection rules for simple bands are n + n (An = 0) for Q polarizations and An = 0 with a spin flip for IC.For the zinc-blende semiconductors the selection rules are more complicated and will be discussed later in this chapter. In addition to absorption and reflection measurements, other optical techniques are useful for probing Landau levels in semiconductors. One such technique is Raman scattering, first proposed for semiconductors in a magnetic field by Wolff (1966) and observed for n-type InSb by Slusher et al. (1967). The strongest and narrowest such transition is spin-flip scattering, where the Stokes frequency is equal to the spin-flip frequency 0,. This, of course, provides a direct measure of the conduction-band g factor. Spin resonance has also been detected optically by Weisbuch and Hermann (1977) by observing changes in the polarization of the interband recombination radiation induced by microwave radiation at the spin-flip frequency. In only a few semiconductors has the spin resonance been observed using microwave EPR techniques (Goldstein, 1966). Another class of magnetooptical techniques involves measurements of the changes in polarization of light reflected or transmitted by the

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semiconductor. One such effect is Faraday rotation, that is, the rotation ofthe plane of polarization of light as it propagates parallel to the magnetic field. It occurs because the indices of refraction are different for the left and right circularly polarized components (oL and oR)of the linearly polarized beam. The rotation has sharp structure at frequencies corresponding to the transition allowed for either oLor uR,hence it can be used to measure these transition frequencies. Faraday rotation studies of semiconductors have been reviewed by Pillar (1972). The sensitivity of all of the above optical techniques can be enhanced in many cases using modulation techniques. By modulating an applied electric field, stress, magnetic field, or the sample temperature, or the wavelength or intensity of the incident light, one can use lock-in amplification to improve the signal-to-noise ratio. Such techniques are most useful for fairly weak transitions with a large background. This subject has been reviewed by Aggarwal(l972). 2. OTHERTECHNIQUES

Another major category of experimental techniques for studying semiconductor band structure are measurements of oscillationsin either galvanomagnetic properties such as electrical resistivity in a magnetic field, or magnetic properties such as the susceptibility. The oscillations in the resistivity (Shubnikoff-de Haas effect) or in the magnetic susceptibility (de Haas-van Alphen effect) occur when at a given magnetic field, the H = 0 Fermi level coincides with the peak in the density of states of one of the Landau levels (see Kittel, 1963, p. 220). These techniques are useful for studying materials with large numbers of carriers. These techniques are particularly useful for mapping out complicated Fermi surfaces of metals, semimetals, and doped semiconductors. They also have been applied with success to the narrow-gap semiconductors, and we will discuss the results for Hg,-xCdxTe in Section 8 of this chapter. Another technique which has proved useful for studying semiconductors is the magnetophonon effect. Peaks occur in the resistivity transverse to an applied magnetic field whenever the longitudinal optical-phonon frequency wLo is equal to an integral multiple of the cyclotron frequency a,. This effect does not require high carrier concentrations. Magnetophonon effects are particularly useful for studying hot electrons. The subject has been reviewed recently by Peterson (1975). The technique has been used to study Hg, -,Cd,Te and the results will be discussed in Section 8 below. In general, transport studies are particularly valuable for studying electron scattering and conduction mechanisms in various temperature and electric field regimes. Hall measurements, besides determining the mobility and carrier concentration of a sample, can be used to study impurity levels when

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127

freeze-out occurs (Putley, 1966). Unfortunately, the donor impurity levels in Hg, -,Cd,Te are so shallow that freeze-out has never been achieved. Many other techniques for studying semiconductors have been reviewed in earlier volumes of this series (see especially Volumes 1, 8, and 10). Where such techniques have been used to study Hg, -,Cd,Te, we will discuss the results later in this chapter. In general, the techniques of magnetooptics, especially measurements of interband and intraband transitions, provide the most direct and accurate determination of electronic properties in pure materials near the fundamental gap.

11. Theory for Zinc-Blende Semiconductors The energy bands near the fundamental gap of a zinc-blende semiconductor are illustrated on the left-hand side of Fig. 5. The gap occurs at the zone center, or r point, where the states transform according to representations of the 5 crystal symmetry group. Without spin, the S-like conduction band transforms as r,, and the triply degenerate P-like valence band as r, . Including spin, the valence band splits into bands transforming as Ts(J = $)and r, (J = 4) with spin-orbit splitting energy A. The conduction band transforms as r6(J = 4). In an applied magnetic field H, the bands are quantized into Landau levels as illustrated on the right-hand side of Fig. 5,

-3 2

3 -2 l-

I-0

-0

b set

a set

-

-I

0-

I23-

I2-

-I

-

0-

I-

2-

3-

I2-

3-

3-

-I

0I-2

-

-l

23 -

3

FIG.5. (a) Zinc-blende semiconductor energy bands (H = 0) and (b) in an applied magnetic field H.Note that vectors are indicated by over arrows in figure and by bold letters in text.

M. H.WEILER

128

where we show the density-of-states peak (kH = 0) for each level. In this part, we will develop a model Hamiltonian for calculating the energies of these Landau levels. With certain approximations, these levels are coupled in groups of four only, so that calculating the energies involves diagonalizing 4 x 4 matrices. The a-set levels are the spin-up state in the conduction band, with Landau quantum number n, the states m, = 5 (with n - 1) and -3 (with n 1) in the valence band, and the m, = 3 state in the split-off band (with n + 1). A similar coupling occurs among the states in the b set. The set of coupled levels are all labeled with the conduction-band quantum number in Fig. 5. Thus the valence-band states labeled - 1 are not coupled to any conduction-band level. In order to obtain the complete Hamiltonian for the set of coupled bands illustrated in Fig. 5, there have been two different approaches based on grouptheoretical techniques. In the first approach (Kane, 1957) one finds interband matrix elements paBof the k p perturbation Hamiltonian hk * pa,& and also of the spin-orbit Hamiltonian. For the case of InSb, Kane enumerated all possible combinations of these matrix elements to second order in k among the single-group basis states rl (conduction band S) and r4(valence bands X, E: and 2).The first-order matrix elements pas and second-order combinations involving intermediate states belonging to different representations of the single group were adjustable parameters of the perturbation Hamiltonian. This Hamiltonian was expressed in terms of linear combinations of the functions X, I: and 2 and of the spin functions t and 1, which diagonalize the spin-orbit interaction. The Hamiltonian involves matrix elements coupling the various basis states including adjustable parameters multiplying functions to second order in k. A second method for obtaining this Hamiltonian was that of Luttinger (1956), who used a grouptheoretical analysis to find all allowed matrix elements of k and k x k among the valence-band states transforming as the Ts representation of the double group. His result involved adjustable parameters which were linear combinations of those of Kane, but included an additional parameter q which is nonzero only in the presence of spin-orbit splittings of the intermediate states. Luttinger’s results were extended by Roth et al. (1959) to include the r7 split-off band. Pidgeon and Brown (1966) included the Ts conduction band in their analysis, combining the results of Kane and of Roth et al. They called their model the quasi-germanium model since they neglected the effects of inversion symmetry so that the model also applies to semiconductors such as Ge and Si. The use of this model in the analysis of magnetooptical experiments was reviewed in detail by Aggarwal(l972). We have used the second approach (Weiler, 1977; Weiler et al., 1978) to obtain a complete set of parameters for the coupled r6,r,, and Ts bands, and found three new parameters in addition to those of Kane and Roth et al. which have the same origin as Luttinger’s parameter q. Our

+

-

3.

MAGNETOOPTICAL PROPERTIES OF Hg - ,Cd,Te ALLOYS

129

group-theoretical treatment makes use of the Koster et al. (1966) (KDWS) tables of coupling coefficients. The appropriate set of basis functions is given below, in terms of the basis functions used by Kane (1957): a set:

st,

11) = J1;/2,1/2

=

13)

= -(~/J~xx

= $!/2,3/2

15) = &,

+ iy)t,

-112

= (i/&C(X

- iY)t

+ 2213,

(6)

= - ( i / A c ( x - int - zii; 17) = G,-l/2 b set:

12) = &,

-112

16) = $&2,

112

14) = +!/2,-3/2

IS>

=

&,1/2

= Sl,

=

-(i/fi)C(X

+ in1 - 2

= -
m

WJ,

(7)

+ i v t + z11.

These basis states are not identical to those of any of the important earlier papers on the zinc-blende Hamiltonian. However, this previous work involves several inconsistent basis sets. The considerations involved in choosing our set were (1) to make the 4 x 4 matrices for the a and b sets real, (2) to agree with the KDWS tables, and (3) to agree with the most extensive previous work: (a) Luttinger (1956), (b) Roth et al. (1959), (c) Pidgeon and Brown (1966), and (d) Aggarwal (1972). Our set satisfies (l), (2), and (3a); it differs slightly from (3b) to satisfy (2), from (3c) to satisfy (l), and from (3d) to satisfy (3a). Using the KDWS tables of coupling coefficients for the & group, we found the matrices involving terms to second order in k among these basis functions. For example, two terms in k x k are F i = 2kf - k i - k,Z and F: = d ( k ; - k:) which belong to the twofold T3 representation. The matrix elements of these functions among the r6, r7,and l-8 states are proportional to the complex conjugates of the table entries on p. 91 of KDWS. The resulting matrices must be Hermitian and be invariant under time reversal. These conditions require certain parameters to be either zero or else purely real or imaginary. All the real, independent parameters found in this manner are listed in Table I. The l-8 x l-8 parameters yl, y z , y3, K , and q were shown by Roth et al. (1959) also to involve the split-off band r7when one starts with singlegroup representations. In the full double-group picture the r7 x r7 and r7 x r8 parameters are independent of the r8 x r8 ones and are denoted by y;, 7; ,y; , K', and K". The conduction-band (r6)effective mass parameter

M. H. WEILER

130

TABLE I PARAMETERS OF THE k

.p HAMILTONIAN AMONGTHE rs,r7,AND rs B m - E m e STATES

F, the "linear-k" parameter C for Ta, and the r6 x Ts parameters P and G were defined by Dresselhaus (1955), Kane (1957), and Dresselhaus et al. (1955) in terms of single-group basis states; the r6 x r7 and r7 x Ta parameters are denoted by C , P', and G . We also obtain three new parameters: N , , N z , and N , . N , contributes to the conduction-band g factor, similar to K in the valence bands; N 2and N, represent additional couplings between the r6 conduction band and the Ta valence bands. These new parameters, like q, arise from the spin-orbit splitting of higher bands. The parameters q and p = (7, - y2)/2 are the so-called Ge warping parameters because, if these were zero, in a material like Ge with inversion symmetry, the energy bands would be isotropic or independent of the direction of k. The parameters C, G, N z , and N 3 are called the inversion asymmetry parameters because they are zero in materials like Ge with inversion symmetry but nonzero in the zinc-blende materials. The complete 8 x 8 matrix &' for the parameters listed in Table I is given in Table 11, which also includes the band-edge energies E , and A relative to the Ta valence band, and the parts of the free-electron terms not included in the definition of y1 and IC (see Luttinger, 1956). Other terms used in Table I1 are

kZ E k?

+ k; + k;,

= k, k ik,, F i = d ( k ; - k;),

kf

F:

= 2k:

F:

= {k,, kf} E (k,k* + k'k,),

-

k; - k;,

H, = i[k,, k,],

Hf

=

f[k',

F:

E

{k,, k,},

k,].

For simplicity, Table I1 is given in atomic units h

=m =

1.

(8)

3.

MAGNETOOPTICAL PROPERTIESOF Hg,

-,Cd,Te ALLOYS

131

3. QUASI-GERMANIUM MODEL For a magnetic field H in an arbitrary direction defined by the spherical polar angles 8 and 4, we perform the following coordinate transformation illustrated in Fig. 6:

k, = cos 8 cos 4 k ,

k,

=

cos 8 sin 4 k ,

k,

=

-sin 8 k ,

-

sin 4 k ,

+ cos 4 k ,

+ cos 4 sin 8 k,, + sin 4 sin 8 k,,

(9)

+ cos 8 k,.

A similar transformation was given by Luttinger (1956) for the case 4 = 45" which confines the magnetic field to the (110) crystal plane. The new coordinate axes (1,2, 3) are shown in Fig. 6 where the magnetic field is along the 3-axis. The corresponding rotation of the basis states results in a transformation of the k - p Hamiltonian according to X(8, 4) = U t X U (see Weiler, 1977). We then set k,

=

(l/Afi)(a

+ a+),

k,

k , = (i/M)(a - a'),

= k,,

(10)

where A = (hc/eH)"' is the Landau radius, hk, is the component of the momentum along the direction of the applied magnetic field, and a+, a are raising and lowering operators for harmonic oscillator functions 4,,:

a+4, = J.+1-4n+

1,

a4n = & 4 n -

1-

(11)

N is the number operator

N&,, = a+a$,, = n4,, and [a, a']

= aa+ - a+a = 1.

The resulting X ( 8 ) separates into two 4 x 4 matrices for the a and b sets

if one neglects terms proportional to k,, q, C , G, N,, and N, and most terms proportional to the warping parameter p 3 &(y3 - y,). One can include some warping by way of two valence-band effective mass parameters

where

TABLE I1 k * p HAMILTONIAN FOR A ZINC-BLENDE SEMICONDUCTOR'

1 -Pk-

Jz

+ -GF;

3

iJ5

+Y3F; 2

1

1

-P k '

J5

----yiF$

9

1

I

-A

I I

--w: II 9 I 1

I

I I I

-&J'1k2

I I

I

I

1

+ -C'k,

Jz

+ + $)H, (K'

I I I I

I

I ------- -- 1-------1

Js i-y3F;

I

2

I

+&C'k+

I

I

I I

-(K'

+ f)H-

The terms used are defined in Eqs. (8). The upper triangle is the Hermitian conjugate of the lower triangle.

i I I I I I

I

I

+t~2F:

I I I

I

M. H. WEILER

134

Y

I

FIG.6. Coordinate transformationto place HI13 axis in new (1,2,3) system; (X,Y,2)are the cubic crystal axes.

These parameters were defined by Luttinger (1956) only for the case 4 = 45" = n/4,corresponding to the magnetic field H in the (110) plane, which makes the second term in Eq.(16) equal to zero. The average of this second term over all angles 6 and 4 is 3, which is equal to that of the first term, so that f(6, 4) = 5. The solutions to the Schriidinger equations s a l a > = Eala>,

Sblb)

= Eblb)

(17)

are of the form

w i t h n 2 - 1 andwitha;'=b;'=a;'=a~=b,' = b 06 -- b -8' = b o 8- = 0. For n 2 1 there are eight independent solutions [ la(n)), I&))] for each n, which are denoted, in order of decreasing energy, conduction band Cla'(n)>, lb"(n)>l, heavy hole Cla-(n)>, Ib-(n)>l, light hole Cla+(n)), Ib+(n))], and split-off band [I as@)), Ib"(n))]. These states are illustrated in Fig. 5. The energies of these levels are calculated numerically, for a given set of parameters and for a given value of magnetic field, by solving Eq. (17), that is, by diagonalizing the 4 x 4 matrices given in Table 111. Here the terms involving y l , y', and y" in the fourth rows and columns are included in the single-group approximation P' N P, y; N yl, etc. Also in Table 111, P is included using the interband energy E , = 2mP2/h2, and s = heH/rnc is the free-electron cyclotron energy. Table I11 is now in ordinary

TABLE I11 QUASI-G~ MODELHAMlLToNlANS IN .@?I

pg+ sC(2n

f

1)F

+ N , + n + 11

(l/J2)s[(2n

1

(1 7)

+ 3))'

-

K

- 11

136

M. H.WEILER

energy units since E,, A, E,, and s have the dimensions of energy and F, IC, N,,and the 7’s are dimensionless. The Hamiltonians of Table I11 are equivalent to those of Pidgeon and Brown (1966) and of Aggarwal (1972), except for corrected sign errors and when account is taken of the different basis sets. Similarly, Table I11 is also equivalent to Eqs. (B-9) and (B-10) of Roth et al. (1959) for the Ts and r7 band energies. Approximate expressions for solution to E!q. (17) will be given in Section 5 below. First we find the selection rules for optical transitions among these states. 4. SELECTION RULES

The strongest allowed optical transitions among the states of Eq. (18) are those proportional to the interband matrix element P; that is, we find the optical perturbation X6, by replacing k by k (eA/hc) in the terms proportional to P in the Hamiltonian in Table 11. Here A is the light vector potential in the radiation gauge. The resulting transition matrix elements are

+

where E is the optical electric field, L+ and 8- are the unit polarization vectors (& k i & ) / f i for, respectively, right and left circular polarization 0, oL transverse to the magnetic field H, and L3 is a unit polarization vector parallel to H referred to as the x or EllH polarization. Thus the selection rules for both interband and intraband transitions (before renumbering of the valence bands) are

The selection rules for interband transitions given by Roth et al. (1959) and by Pidgeon and Brown (1966) were, for oL and o,, An = 0 and -2 rather

3. MAGNETOOPTICAL PROPERTIESOF Hg, -,Cd,Te

ALLOYS

137

4 , 4

i

I

3

1

I3

FIG. 7. Intra-conduction-band transitions allowed in the quasi-germanium model (a, and w, + w,) and induced by warping (30,). Levels are labeled as in Fig. 5.

I I I

I

b

2

II

than + 1 and - 1, respectively. This is because they renumbered all valenceband states n --+ n + 1so that the state numbers n correspond to the quantum number of one of the larger terms [the coefficient of state IS) or 14) in Eqs. (18)]. We do not renumber the valence-band states, so that each set of states n are coupled in the quasi-germanium model (see, for example, Groves et al., 1970). The allowed intra-conduction-band transitions are illustrated in Fig. 7. The aL transition occurs at the cyclotron frequency o N o,,and the x transition at the “combined resonance” frequency o N o,+ o,,where o,is the spin-flip frequency given by ho, = g c p B H . Other transitions are allowed because of the approximate nature of the quasi-germanium model. Terms have been neglected, proportional to the warping and inversion asymmetry parameters which have little effect on the energies but do induce extra transitions. For example, for certain orientations of the magnetic field with respect to the cubic crystal axes, a term proportional to the warping parameter p couples the n = 4 level to the n = 0 level. This, combined with the a, transition allowed in Eq. (20), provides a mechanism for an n = 0 to n = 3 transition which occurs approximately at the third harmonic o x 3w,. This transition is illustrated in Fig. 7. We have analyzed these cyclotron harmonic transitions (Weiler et al., 1978) and have obtained the selection rules given in Table IV. The transitions are labeled following Bell and Rogers (1966): a transition from a, to a,,+,,, is denoted by m 0,; from a, to b,,, by m o,+ 0,; and from b, to a,,, by m o,- 0,. In fact, the conduction-band Landau levels are not equally spaced, so that the transitions are not precisely harmonics. Some of these selection rules have also been obtained by Bell and Rogers (1966) and by Zawadzki and Wlasak (1976) for the case HllCOOl].

M. H. WEILER

138

TABLE IV k, = 0 AND EXTRATRANSITIONS FOR kH # 0,FOR ALLOWED (A) AND INDUCED BY WARPING (w) AND INVERSION ASYMMETRY (I)

INTRA-CONDUCTION-BAND TRANSITIONS FOR OPTICAL POLARIZATION UL, UR, AND IT. BOTH

kH = 0 OL

All orientations A

*R

‘uc + 0%

0,

3 0 , - w,,

50,

+ 0,

kH # 0 IT

w, - w,, 3wc 0,

+

20,

+

IT

*R

*L

0,

ws

3wc

Some of these transitions have been observed in InSb by us (Weiler et al., 1974)and by Favrot et al. (1976) and Lee (1976). They follow the selection rules in Table IV except for a strong 20, transition observed for E I Hll[OOl] which may be an impurity induced transition (Mycielski et al., 1977; Bastard et al., 1979). These transitions can be used to obtain useful

information about the conduction-band parameters. They have not yet been observed in Hg, -,Cd,Te. Some interband transitions allowed by the selection rules in Eq. (20) are illustrated in Fig. 8, where we show the first few transition allowed among the a-set levels for oL and o, polarization. A similar set of transitions is allowed among the b-set levels for these polarizations. For the a polarization, the transitions are from the a set (or 6 set) in the valence band to the b set (or a set) in the conduction band. A rich spectrum of transitions is observed, which can be analyzed, as shown below, to obtain the parameters of the quasi-germanium model Hamiltonian for a given material.

3.

MAGNETOOPTICAL PROPERTIES OF Hg,

-,Cd,Te

139

AUOYS

3-

3

2I-

-0

0 FIG.8. Some interband transitions allowed in the quasi-germanium model. (a) and (b) Levels are labeled as in Fig. 5.

-I

r,

-I-

I

+

2

1

3-

I-

2-

3-

0I -

23-

Additional interband transitions, induced by the warping and inversion asymetry perturbations, are also allowed. These have been observed in InSb by Pidgeon and Groves (1969) and analyzed to obtain estimates for the warping and inversion asymmetry parameters. While there is evidence for such transitions in Hg,-,Cd,Te (Weiler, 1977), the lines were too broad to permit such analysis for these materials.

5. APPROXIMATE EXPRF,SSIONS Aggarwal (1972) has given approximate solutions for the conductionand valence-band energies from Eqs. (1 7) which are useful for understanding the effects of the various parameters. We have corrected and extended these to include some terms proportional to N , and also to q. The conductionband energies are, to first order in H , E[a"(n)] N E , E[bc(n)] N E ,

where mc = eH/m,c and

m C

( n 4- $))hot

*gcpBff,

+ (n + ))bC - 4gg,pBH,

(21)

M. H. WEILER

140

The higher-band contributions F and N, were discussed by Johnson and Dickey (197Q and Hermann and Weisbuch (1977). The valence-band energies are, for n 2 1,

-s((n 4)~: - Y ’ ~ 4.“ (5 f)4 k {[yf - (n 4)y’L - K L - 38 - f ) q ] ’ 3n(n l)(y”L)2}”’), (234 E[b*(n)] N -s((n + 4)yf + Y ’ ~- 4 ~ ’- (3 + fk f { [ y k (n 4)y’L - K L - 3%- f)cl]’ 3n(n l)(y””)’)”’), (23b) where s E heH/mc is the free-electron cyclotron energy, f = f(0, 4) and yf, Y‘~,etc., are the parameters originally defined by Luttinger (1956): E[u*(n)]

+

31

+

+

+

+

+ + +

= ~ E , J E P+> yl, yirL = ~ E , J E B+) yii, yf

+

+ +

+ ~ E J E , )+ K .

yiL 3 ~ E J E , ) yi, I C 3 ~

(24)

For the special cases n = -1 and 0, there are no heavy-hole states. The approximate light-hole energies are

E[u’(- l)]

N

-+s[Y~

- y’L - K ~ ] ,

E [ b + ( - l)]

N

-b[y‘;

+ Y ’ ~- 3 ~ ~ 1 ,

E[u+(O)]

N

-+~[3y4 - 3yfL- ICL],

E[b+(O)]

11

-$[3y?

(25)

+ 3yfL - 381.

Notice that the E J E , terms cancel in the n = -1 energies so that these energies have the magnetic field dependence of heavy-hole states. For large n, the energy differences in Eq. (23) are approximately equal to sm/mi, where the effective masses for the light- and heavy-hole bands are given by

For E d E , % y’, y” this gives

-

m 1 + 37”). (27) 2 m+ mFor this case the light-hole band is nearly isotropic and is nearly the mirror image of the conduction band (equal but opposite curvature or mass); the heavy-hole band has a large effective mass (- y ; l ) which is sensitive to the warping effects contained in y’ and y”. For the case of narrow-gap materials, when the energy gap E , is small compared to the intraband energies such as hw,, the expressions in Eqs. (21) m

2E 3 4

- - -2 N

1 + 71 + T(Y’ + 37’9,

- - 71 - -(y’

3.

MAGNETOOPTICAL PROPERTIES OF

Hg - ,Cd,Te

ALLOYS

141

and (23) are then no longer accurate, except for the expressions for the heavyhole levels (u-, b - ) which are accurate at higher magnetic fields until the heavy-hole cyclotron energy becomes comparable to the energy gap. For this case we can still obtain approximate expressions for the conductionelectron and light-hole energies if the spin-orbit splitting A is much larger than E , and the intraband energies. Such expressions have been given by Yafet (1966) and Kacman and Zawadzki (1971). We will give similar expressions appropriate to our basis set [Eqs. (6), (7)]. The energies of the conduction-band levels, including the momentum k, along H, are given approximately by

(28) where o,= eH/m, c and m, and gc are defined in Eqs. (22). The corresponding eigenvectors [Eqs. (18) extended to the full 8 x 8 matrix] are, for n 2 0 and with E, = E[ac(n)], Eb = E[b”(n)], and D, = [E4(2E, - EB)I1/’,Db = CEb(2Eb - E B ) ] ~ ’ ~ , E4 4 n

I b‘(n))

1

2:

Db

M. H. WEILER

142

Similar expressions for the light-hole energies [with a minus sign in front of the radical in Eq. (28) and for which Eqs. (29) are also valid] are not very accurate since the approximation y1 x yz x y3 x K x q x 0 has also been made. The heavy-hole states which are decoupled from the conduction band, and which will correspond to the energies given in Eq. (23) with

(n +

0 1 + 2C2)bn- 1 4 n+ 1

Jm~

la-(n))

2:

[(n

0 0

+ 1 + 2f)(4n + 1 + 2C2)]-1/2

,

~J6nA 0 0

-

0 - c J 6 ( n r )0+ n -

Ib-(n))

2:

[(n

+ 2C2X4n + 3 + 2C2)1-1’2

0

The states for n = - 1 corresponding to the energies in Eq. (25) have components a5 N 1,

(3 1 4

or

b,

1:

1

with all other terms zero. These are the states at the top of the valence band, along with the n = 1 heavy-hole states. The energies in Eq. (28) are often referred to as the two-band or Bowers and Yafet (1959) or Lax (Lax et al., 1961) model. The model has been used extensively to analyze intra-conduction-band experiments in small-gap materials (Weiler et al., 1967). It has proved invaluable in studying electronic behavior in combined electric and magnetic fields. The energy levels given by Eq. (28) are not evenly spaced as they would be for a simple band; this effect is referred to as nonparabolicity because it reflects the fact that the

3.

MAGNETOOPTICAL PROPERTIES OF

Hg, -,Cd,Te

A L ~ Y S

143

energy-momentum relation for H = 0 is not parabolic. If the spacing of successive levels is used to define a cyclotron effective mass m,(H) from E[u'(n

+ l)] - E[u"(n)] = heH/m,(H)c,

(32)

this mass increases with increasing n and with increasing magnetic field. Another feature of the zinc-blende energy levels are the so-called quantum effects: the light- and heavy-hole energies given in Eqs. (23) are also not evenly spaced, particularly for low values of n. While the most accurate energies result from diagonalizing (for kR = 0) the 4 x 4 matrices in Table 111, we find for Hg,-,Cd,Te with x z 0.2 that the error in Eq. (28) is less than 3 % (1 or 2 meV) for magnetic fields less than 10 T. Eliminating E, from Eqs. (22) gives a generalized relation similar to that given by Roth et ul. (1959),

Better approximations for the materials with larger gaps, especially for the light-hole energies, are given by Aggarwal(l972). From his expressions for the energies expanded to second order in H , one can obtain, for example, the contribution of the valence-band parameters to the conduction-band energies which can be generalized to include warping effects. The Luttinger parameters (y,, y 2 , y3, K , and 4 ) can be evaluated approximately (Suzuki and Hensel, 1974) in terms of the single-group parameters F , G, H I , and H , defined by Dresselhaus (1955) [or A, B, C,and D defined by Kane (1957)l: 71 N -1 72 N

y3 N

-gF' + 2G + 2H1 + 2H2) +$q,

+ 2G - HI - H2) - $4, -3F'- G + HI - H2) + 44, -&F

(34)

K E - ~ - ~ F ' - G - H1 + H2) - :49 4 -BCA/~(~mf,, where E(T4) is the energy, relative to the top of the valence band, of the higher r4 conduction band when its spin-orbit splitting A is zero. The expression for 4 was first given by Hensel and Suzuki (1969). The parameter H , results from interactions with higher Tsbands; since these are normally far removed, one can usually set H , N 0 which gives (Vrehen, 1968) 71 - 27, - 373

+ 3 K -k yq + 2

0.

(35) This and the other approximate expressions given in this section are appropriate for pure samples. These can be modified to take into account 2:

144

M. H.WEILER

the effects of carrier concentration. Because of the Pauli principle, the conduction-electron concentration, for example, must equal the integrated density of states up to the Fermi energy E , :

where the density of states in a magnetic field is given in Eq. (4) for the case of a simple conduction band. When only the lowest Landau level is occupied, this gives E f (for simple bands) to be E: = ihw, - $lg,lpfiH

+ 2n4n2h2A4/m,,

(37)

which gives the Fermi level wave vector kH = k, as k, = 2n2nA2, (38) where, as above, A2 = hc/eH. The latter expression, for k,, will be valid for more complex bands as long as only one level is occupied. With this, we can calculate the energies of electrons at the Fermi level, at least for narrow-gap materials with large spin-orbit coupling, using Eq. (28):

J-w,

E, = -+E, + (39) again, for the case where only one Landau level is occupied, and with E; given in Eq.(37). The expressions given here [Eqs. (28), (3911 unfortunately cannot be simplified further to give, for example, expressions for the effective mass or g factor as a function of magnetic field and carrier concentration, as long as one determines these quantities from energy-level differences (optical measurements). The so-called density-of-states effective mass found from aE,/akH gives, from Eqs. (28) and (39), md = mCJl

h2kHfmd

(40)

+ 4E:/E,,

(41)

which is equivalent to expressions given before, for example by Lax et al. (1961). The cyclotron resonance effective mass and the inverse of the spinresonance g factor will have similar, but more complex dependences on magnetic field and carrier concentration. The expressions in this section have all been given for the case where E , > 0, that is, for a semiconductor. Expressions similar to Eqs. (28) and (29) were given by Kacman and Zawadzki (1971) for the case E , < 0 appropriate to Hg,-,Cd,Te with x 5 0.16. The light-hole level for E , > 0 becomes the conduction band for E , < 0 (see Fig. 1). Since this band also depends on the Luttinger parameters yl, y 2 , y3 and K which Kacman and

3.

MAGNETOOPTICALPROPERTIES OF Hg

- ,Cd,Te

ALLOYS

145

Zawadzki neglect, their expressions for E , < 0 are less accurate for the conduction band than those for E , > 0. For relatively large 1E , I, for example, for HgTe, the expressions in Eqs. (23) may be more accurate at modest magnetic fields.

III. Intraband Experiments In this section we will review the results of intraband experiments on Hg, -,Cd,Te, including recent unpublished results of our own on samples of various alloy composition, temperature, and carrrier concentration. In Part V, we will use these results, combined with those of the interband experiments reviewed in Part IV, to obtain values for the parameters discussed in Part I1 as a function of alloy composition and temperature. 6. CYCLOTRON, COMBINED, AND PHONON-ASSISTED RESONANCES The first intraband magnetooptical experiments on the Hg, -,Cd,Te alloy system were carried out by Ellis and Moss (1971), then by Kinch and Buss (1971), McCombe et al. (1970a,b), and Poehler and Ape1 (1970). The results of Kinch and Buss (1971), using Fourier transform #spectroscopy, are reproduced in Figs. 9 and 10, for a sample with x = 0.204 k 0.004 in the Faraday configuration at a sample temperature T = 2.1 K and with a carrier concentration n 1: 7 x 1014 cmF3. The complex line shape near 160 cm- in Fig. 9 and the break in the curve of the transmission minimum versus magnetic field in Fig. 10 are caused by the interaction between the electron and the HgTe-like longitudinal optical phonon at 138 cm(17 meV). Interaction with a CdTe-like phonon at 157.5 cm-' was observed only in the Voigt configuration with samples of much higher carrier con). and Buss analyzed their data to obtain a centration (- 10l6 ~ m - ~Kinch

'

'

c (

0.6 o,5

"!

1

1

0.41

3.96kG 5.28 7.04 I

40

I

80

1

1

120

10.12 12.76 1

I

I

160

i

19.38

'

200

240

FREOUENCY (cm-')

FIG.9. Normalized transmission in the Faraday configuration at various H , for a sample at T = 2.1 K with x = 0.204, n 7 x LOl4 ~ r n - thickness ~, -4.3 p (after Kinch and Buss, 1971).

-

146

M. H. WEILER

r-'

200

1

I

.*

'TO

I

ov 0

I

I

0-

/

/'

''

2

I

4

I

6

I

I

8 I0 H (kG)

I

I

12

14

!

1 6

I

18

FIG. 10. Cyclotron frequency versus magnetic field from the curves in Fig. 9 for (HgCd)Te at T = 2.1 K (after Kinch and Buss, 1971).

value for the electron-phonon coupling constant (Frohlich, 1954) a = 0.037 k 0.008. Because of the low carrier concentration, the transmission minima in Fig. 9 occur at the cyclotron resonance frequency, with no distortion due to plasma effects, as indicated in Fig. 10. The dashed line in Fig. 10 was calculated using the Lax model [Lax et al., 1961; see also Eq. (2811 for the energy of a transition from the a set, n = 0 level to the a set, n = 1 level. The best-fit parameters were E , = 61.7 f 4 meV, rn, = 4.66 f 0.07 x m assuming A = 0.96 eV, giving [see Eqs. (2211 E , = 19.8 eV and gc = -200 assuming F = N , = 0. Because of the low effective mass and high g factor, the conduction-band levels are spaced far apart even at magnetic fields of a few kilogauss so that only the a-set (spin-up), n = 0 level is occupied with I lOI5 cm-3 carrier concentrations. From the absorption edge at 4.2 K with no magnetic field (see Fig. 2) Kinch and Buss found E, = 59 & 3 meV consistent with the determination from the cyclotron resonance data. McCombe et al. (1970b) observed cyclotron resonance, and also combined resonance and LO phonon-assisted cyclotron resonance, using a farinfrared pulsed gas laser, in a sample with x = 0.203, n N 9 x l O I 4 ~ m - ~ , at T = 4.3 K, in the Voigt configuration where the calculated resonant frequencies were corrected for the plasma shift using op= 25 cm-'. Their experimental results are given in Fig. 11. They fit their data to the model of Bowers and Yafet (1959) [Eq. (28)] for the cyclotron transition for E IH[ac(0) + ac(l)], the combined resonance transition for EIIH[a'(O) + bc(l)], and phonon-assisted cyclotron resonance o = o,+ oLousing E, = 64 It 3 meV and E , = 18.5 f 1 eV, assuming A = 0.96 eV and

3.

MAGNETOOPTICALPROPERTIES OF Hg,

0

10

30

20

40

-,Cd,Te

50

H (kOe)

60

ALLOYS

70

147

.,

FIG.1 1. Transitionfrequencies versus magnetic field for a sample with x = 0.203at T = 4.3 K : 0,spin-up, n = 0 cyclotron resonance [ v , ( t ) ] ; combined resonance [v,,.]; and phononassisted resonance [ v , ( t ) + vm] (After McCombe et al., 1970b). Data fitted to the model of Bowers and Yafet. E, = 0.064 eV; A = 0.96 eV; E, = 18.5 eV.

+,

F = N, = 0. This gives the band-edge effective mass and g factor as rn, = 5.0 x m and gc = - 172 for this alloy composition and temperature. McCombe and his co-workers (1967) had been the first to observe combined resonance in InSb. McCombe (1969) demonstrated that this transition was allowed by the small energy gap, since terms proportional to hw,/E, in Eqs. (19) give nonzero matrix elements, for EIIH, for the transition ac(n) + b"(n + 1). The matrix elements were calculated by Sheka (1964) from the Bowers and Yafet (1959) model. Using the approximate expressions for the conduction-band states in Eq. (29), the transition matrix element in Eq.(19) becomes, for the optical electric field EllH and for D, x Db x E , ,

1:

eEE (n + 1)s + 3 w E , [ 2m ] ~

{E[b"(n

+ l)] - E[ac(n)]}. (42)

Since the energy difference in curly brackets is approximately proportional to H,the transition intensity is approximately proportional to H 3 at low fields. From the phonon-assisted transitions, McCombe et al. (1970b) found the LO-phonon frequency to be 140 cm-' which is consistent with the results of Kinch and Buss (1971), and also with the infrared reflection measurements by Baars and Sorger (1972), Carter et al. (1971), and Mooradian and Harman (1971). Similar phonon-assisted transitions have been observed in InSb

M. H. WEILER

148

(McCombe et al., 1967;Johnson and Dickey, 1970;Weiler et al., 1974) and explained by Enck et al. (1969)and Bass and Levinson (1965). Poehler and Apel (1970) also observed cyclotron resonance, with an infrared pulsed gas laser at 48 pm, on samples with x = 0.15,0.19, and 0.245 at T = 4.2 K.The samples had carrier concentrations at 77 K of 70,4, and 7 x 1014 ~ m - respectively. ~ , The observed cyclotron effective mass mt = e H / o , c was 0.004,0.006,and 0.013 m, respectively. Without information on the value of E , or of the energy gap at 4.2 K,or data at several wavelengths and magnetic fields, these mass values cannot be used to determine the band-edge effective mass m,. The most important correction is that due to the fact that ho, is larger than or comparable to E , for these measurements [see Eq. (2811,the effect of carrier concentration, using Eq. (28) for kf instead of the H = 0 expression used by Poehler and Apel (1970), is negligible except for the x = 0.15 sample where, in fact, we expect more than one Landau level to be occupied at the experimental magnetic field of 0.9 T. Cyclotron resonance has also been observed using a 337-pm HCN laser by Knowles and Schneider (1978),on samples with x = 0.20 and 0.215,with 77-K carrier concentration of 7 and 5 x 1014 ~ m - respectively, ~ , at temperatures from 77 to 150 K. They found the band-edge effective mass using a simplified expansion of Eq.(28)to second order in h,; at 77 K the effective and 9.7 0.3 x for mass ratios were m,/m = 8.9 & 0.3 x x = 0.20 and 0.215, respectively. These mass ratios increased approximately linearly with temperature. Narita et al. (1971)carried out cyclotron resonance experiments in reflection at wavelengths from 45 to 150 pm, at T = 4.3 K, on samples with x = 0.176 and 10-K carrier concentrations of 8.5 x 1014 ~ m - From ~ . a linear fit of the a,versus H curve at fields from 0.4 to 0.8 T, they obtained a m. As we shall discuss below, even value for the effective mass of 6.2 x at these low magnetic fields such a linear fit gives a cyclotron effective mass larger than the band-edge mass m,. More recently, Kim and Narita (1976) have studied cyclotron resonance and also interband transitions and fit their data using the quasi-germanium model (Table 111). Their samples were all on the semimetallic side (E, < 0), with alloy composition x = 0, 0.07,O.126,and 0.148.They used values for E , calculated from the expression of Wiley and Dexter (1969)intended to fit HgTe and CdTe: E , = 18

+ 3x

(43)

eV

and values of E , from the helicon-propagation results of Wiley and Dexter (1969)combined with the results of Guldner et al. (1973)for HgTe: E , = -304

+ 0.5T(1 - 2x) + 1914x

meV

(44)

with Tin degrees Kelvin. Making the approximation of isotropic bands YZ

= Y3 = 7

(45)

they found, for the Luttinger parameters,

which gives a heavy-hole effective mass of 0.38 m. We have recently carried out unpublished cyclotron and combined resonance measurements, in transmission, on three of the samples which we studied earlier (Weiler et al., 1977) in interband magnetoreflection. The experiments were performed using an infrared grating monochromator, using wavelengths from 12 to 30 pm, with the sample held in the Voigt configuration for transmission, on the cold finger of a liquid He Dewar in the bore of a 15-T Bitter magnet as illustrated in Fig. 12, which shows the various samples and polarization configurations. Some typical transmission curves are given in Fig. 13. The sample temperature was measured once using a thermocouple to be T N 24 K. The experimental curves show a progression from high to low carrier concentration for the three samples. The samples have alloy compositions of x = 0.196, 0.220, and 0.265 with 77-K carrier concentrations of 8.7, 2.4, and 3.2 x loi4 ~ r n respectively, -~ with Hall mobilities of 2.1, 1.1, and 0.8 x lo5cm2/v sec, respectively.

FIG. 12. Experimental configuration for magnetooptical measurements. (a) Voigt transmission: EIIH, E IH. (b) Faraday reflection; uL,0., (c) Voigt reflection; E(IH,E IH.

I

0 wc

I

2

4

6

;li,

8

-

10

12

1I4

t 0

wc+ ws I

I

I

I

I

v) L

c 3

-

D

I.

I

i

1

Wc + WS

z

0

-

ul

0.5-

ul

- 0.

I ul

z

a a

1

+

wc

2

4

6

8

10

I

12

14

0

2

4

8

6

10

12

I(

MAGNETIC F I E L D ( T )

(b)

I.

0.

2

4 6 8 10 M A G N E T I C FIELD (0)

I

I

12

14

(TI

.

FIG.13. Transmissionversus magnetic field in the Voigt configurationfor three Hg, -,Cd,Te 2.1-8.7 x 10'4cm-3. (a) For E I H the intra-conduction-band samplesat T = 24K for n : transitions are cyclotron resonance (0,) and phonon-assisted resonance (0, + wLo);(b) for EllH transitions are combined (w, + w s )and phonon-assisted (me + w, + wLo) resonances. For (a): top curve, x = 0.196; hw = 67.9 meV; middle curve, x = 0.220, hw = 67.9 meV; bottom curve, x = 0.265, hw = 65.7 meV. For (b): top curve, x = 0.196, hw = 75.5 meV; bottom curve, x = 0.220, hw = 81.0 meV.

3. MAGNETOOPTICAL PROPERTIESOF Hg, -,Cd,Te

ALLOYS

151

For the x = 0.196 sample we were able to observe cyclotron resonance (ac),combined resonance (0, us),and the corresponding phonon-

+

assisted transitions (0, + oLo,o,+ o,+ oLo). From these we determined ho,, to be 16.4 f 0.7 meV or 132 f 6 cm-', consistent with the results of McCombe et al. (1970b), Baars and Sorger (1972), Carter et al. (1971), and Mooradian and Harman (1971). For this sample at cyclotron resonance, the transmission is nearly zero (the samples are all - 1 mm thick). For the x = 0.220 sample we observed cyclotron resonance and very weak combined resonance, and no phonon-assisted transitions. For the x = 0.265 sample we could observe only a relatively weak, cyclotron resonance line. The curves of transition energy versus magnetic field are given in Figs. 14-16 for these samples. In each case the carrier concentration was low enough, and the transition frequencies high enough, so that the plasma shift was negligible. From the data for the cyclotron frequency o,we compute the cyclotron effective mass mC(H)= eH/w,c; these results are given in Fig. 17 for the three samples. The solid curves are calculated from the quasi-Ge model

0

4 MAGNETIC

8

12 FIELD ( T I

16

FIG.14. Transition energy versus magnetic field for a sample with x = 0.196 ( T = 24 K). Transitions are identified as in Fig. 13. Curves are calculated from the quasi-germanium model. 0 ,E l H; A, EIIH.

MAGNETIC

FIELD ( T I

FIG.15. Transition energy versus magnetic field for a sample with x = 0.220 (T = 24 K). Transitions are identified as in Fig. 13. Curves are calculated from the quasi-germaniummodel. 0,E IH; A, EIIH.

0

4 8 12 MAGNETIC F I E L D ( T I

16

FIG.16. Transition energy versus magnetic field for a sample with x = 0.265 (T = 24 K). The transition is identified as cyclotron resonance. The curve is calculated from the quasigermanium model. 0,E IH.

3. MAGNETOOPTICAL PROPERTIES OF Hg, -,Cd,Te 0.025

I

I

I

I

I

I

I

I

I

ALLOYS

153

al

- 0.020 \

-

1

E"

mm

2 0.01 5 W

1 IV W IA

LA

z 0 a

0.0 I C

I-

0

1

V

>.

V

0.005

0

I I

I

4

I

MAGNETIC

I

8

I

12

FIELD ( T I

FIG.17. Cyclotron effective mass versus magnetic field (open circles) from the data in Figs. 13-16, with the data of Ellis and Moss (1971) for x = 0.22 (+)and 0.25 ( x ) a1 T = 24 K. Solid curves: calculated from quasi-germaniummodel; dashed lines: linear extrapolations to H = 0.

[Table 1111 for the transition ac(0)+ ac(l), using parameters deduced from the interband measurements on these samples, with a value E , = 19 eV which was found to give a good fit to both interband and intraband data. The other parameters will be discussed in Part IV. For these samples, the band-edge effective masses are m, = 6.2, 8.9, and 13.8 x m for x = 0.196, 0.220, and 0.265, respectively. The corresponding effective masses deduced from linear fits (dashed lines) to the data would be 8.5, 11.9, and 14.8 x lop3 m which are significantly larger especially for the lower-x (smaller-gap) samples. The curve of m,(H) versus H has significant curvature; in order to obtain the band-edge mass mc, data must be taken over a wide range of fields in order to determine this curvature, or else one must make an independent determination of E , or E,. The early cyclotron resonance results of Ellis and Moss (1971) for samples with x N 0.22 and 0.25 at temperature near 4.2 K are included in Fig. 17.

M. H.WEILER

154

Although they made a linear extrapolation to H = 0 to obtain m, N 10.4 f 0.3 x m and 13.2 k 0.3 x lo-' m, respectively, their data are consistent with ours for x = 0.220 and 0.265 for which we obtain band-edge m and 13.8 x m, respectively. effective masses 8.9 x For our x = 0.196 and 0.220 samples we also observed combined resonance w, + w,, which is the d ( 0 )to b'( 1) transition. The difference between the combined resonance frequency and the cyclotron frequency is then ws(l), the n = 1 spin-flip frequency. From this we obtain the field-dependent, n = 1 g factor g,(l, H ) using hw,(l) = g,(l, H ) p B H .These results are given in Fig. 18. Again the solid 'and dashed lines are calculated from the quasigermanium model using parameters deduced from the interband measurements. The band-edge g factors are g, = - 143 and -94 for x = 0.196 and 0.220, respectively. As for the effective mass data, the curvature, or else E , or E,, must be measured in order to deduce band-edge parameters from the high-field data. Also, since the magnitude of the n = 1 g factor decreases much more rapidly with field than that of the n = 0 g factor, the bandedge g factor may be deduced more easily from experiments involving spin flip of the electrons in the n = 0 Landau level which will be discussed in the next section. -1501

0 0

I

I

4

I

I

8

I

I

MAGNETIC FIELD (TI

FIG.18. Effective g factor for the n = 1 level versus magnetic field for x = 0.196 (circles) and 0.220(triangles) at T = 24 K from the data in Figs. 13-15. Solid and dashed curves calculated from the quasi-germaniummodel.

3.

MAGNETOOPTICAL PROPERTIES OF

Hg, -,Cd,Te

ALLOYS

155

7. SPIN-FLIP TRANSITIONS The most direct way to measure the conduction-electron g factor is to observe spin resonance as a magnetic-dipole transition using microwaves. Since this is a weak transition, these measurements have not been carried out on many semiconductors, including Hg, - ,Cd,Te. For Hg, -,Cd,Te in the narrow-gap semiconductor range, and for InSb, spin-flip transitions have been observed directly using optical techniques (McCombe et al., 1970a; McCombe and Wagner, 1971), and by spinflip Raman scattering (Sattler et al., 1974). Both processes occur most strongly in the narrow-gap materials. In this section, we will summarize the results of both measurements. The observation of electric-dipole-excited spin resonance in Hg,., Cd,,,,,Te by McCombe et al. (1970a) had been predicted by Rashba and Sheka (1961) to be allowed by inversion asymmetry, and by Yafet (1963) and Sheka (1964) to be allowed by spin-orbit coupling which has been found to be the dominant effect (see Table IV). Using an extension of Eqs. (19) to include the full eight-component states in Eqs. (29), we obtain the transition matrix element for the 0, polarization,

(bc(n)lXwla’(n)>=

eEE - 8 4hk,(E[b‘(n)] - E[a‘(n)]}, 6mwE,

(47)

where we have neglected terms of order l/A2. The transition takes place only for k, # 0, so that the intensity has a greater-than-linear dependence on the carrier concentration n. The integrated absorption for the case where only the lowest Landau level is occupied is (Sheka, 1964; McCombe et al., 1970a)

where we have recalculated the equation of McCombe et al. (1970a) in cgs units. Their experimental results obeyed approximately this 1/H dependence on magnetic field. Transmission curves from the experiments of McCombe et al. (1970a) for a sample of alloy composition x = 0.193 with a carrier concentration of 3 x 10’’ cm-3 at T N 4.2 K, are given in Fig. 19. The labels CRA and CRI denote. circular polarization in the cyclotron resonance active and inactive sense (oL and oR), respectively. The transmission minima labeled CRI are the electric-dipole-excited spin-flip transitions. These transmission minima, as well as the observations of cyclotron and combined resonance in this sample (McCombe, 1972),are plotted versus magnetic field in Fig. 20.

156

M. H. WEILER

ti.lLuJ 8

10

12

14

16

18

B (kG)

B (kG)

(a)

(b)

FIG.19. Transmission versus magnetic field in the Faraday configurationfor a sample with two different wavelengths for (a) CRA and (b) CRI circular polarizations (after McCombe, 1973). For a, I = 118.6 pm and for b, L = 171.7 pm. x = 0.193 at

The n = 0 g factor from the spin-flip data, and the n = 1 g factor from the difference between combined resonance and cyclotron resonance, are given in Fig. 21. In both Figs. 20 and 21 the solid lines were calculated using the model of Bowers and Yafet (1959) with the parameters E , = 58 meV, E , = 19.1 eV, A = 0.96 eV, and F x N, x 0. The band-edge g factor is approximately -200. The parameter L in Fig. 21 is identical to our Landau quantum number n. The deviation from the curve of the L = 0 points near 10 k G is probably due to the shift of the spin-flip transition away from kEI= 0 which is largest at low fields where the density of states is broader. The n = 0 data had earlier been fit (McCombe et al., 1970a) using EB = 56 meV and E , = 18.5 eV which gives approximately the same g factor [see Eq.(22)]. Spin-flip Raman scattering was first observed in Hg,-,Cd,Te by Sattler et al. (1974). Since this subject is being reviewed elsewhere in this volume, we will confine ourselves to a simple discussion of the results for the g factor in Hg, -,Cd,Te. As for the optically excited spin-flip and combined resonance transitions discussed above, conduction-electron spin-flip Raman scattering is allowed only in the presence of spin-orbit coupling in the valence band. Wolff (1966)

-

:

3.

:

MAGNETOOPTICAL PROPERTIES OF

Hg, -,Cd,Te

ALLOYS

157

400

E

0 300

21

I00

0

30 4 0

10 20

50

70 80 90

60

100

B(kG)

FIG.20. Transition frequencies versus magnetic field for x = 0.193: spin-flip resonance (triangles), cyclotron resonance (closed circles), and combined resonance (open circles) (after McCombe, 1972). The positions of the longitudinal and transverse optical-phonon frequencies are indicated by LO and TO, respectively. --, theory of Bowers and Yafet; E, = 0.058 eV; E, = 19.1 eV.

r

-2001 0

I

I

,

I

I

,

I

10

20

30

40

50

60

70

, 80

B(kG)

FIG. 21. Effective g factors for x = 0.193 from the data in Fig. 20; for the L = 0 (closed theory of Bowers circles) and L = 1 (open circles) Landau levels (after McCombe, 1972). -, and Yafet; Eg = 0.058 eV; Ep = 19.1 eV.

M. H.WEILER

158

and Yafet (1966, 1973) have shown that the matrix element for spin-flip Raman scattering with pump frequency wo is proportional to

which is zero if A -,0 and is resonant when hwo x E, or E, + A. Thus for pumping with a CO, laser at 10 p m , the spin-flip Raman scattering will be resonant for Hg,-,Cd,Te samples with E, 2 120 meV. Sattler et al. (1974) used a sample at T N 22 K with x = 0.234 and n 1: 5.4 x lOI4 cmP3with E, x 132 meV (based on infrared transmission at 16 K). A linear extrapolation of data from magnetic fields 0.4 to 2.0 T gave a band-edge g factor -79 f 3. At these low magnetic fields such an extrapolation is justified (see Fig. 18). Later measurements (Weber et al., 1975) on the same sample using a chopped-cw CO, laser instead of a TEA laser, with the sample at T N 12 K, gave gc = -82 f 3. Kruse (1976) observed spin-llip Raman scattering using a Q-switched CO, laser from a sample with x = 0.23, n x 1.1 x 10’’ cm-3 at magnetic fields from 0.2 to 1.4 T. Kruse reported an extrapolated g value of gc x -86 for the Stokes and second-Stokes signals. For a similar sample at T 5 K, Walukiewicz et al. (1978) analyzed the spin-flip scattering data using the quasi-germanium model with the parameters of Weiler et al. (1977) with E, = 130 meV, A = 1 eV, F = 0.7, and n, = 0, which gives a band-edge g factor gc = -80. Their data for the n = 0 spin-flip energy h a , = Ig,(H)lpsH and the effective g factor Igc(H)I

-

-

100

- 80

-60 0

-40 -

0

B rn

20 0

4

8 12 16 20 24 MAGNETIC FIELD ( k O e )

28

FIG.22. Spin-flip energy and effective g-factor versus magnetic field from spin-flip Raman scattering from a sample with x = 0.23 (after Walukiewicz et al., 1978). ---, theory.

3.

MAGNETOOPTICAL PROPERTIES OF Hg,

-,Cd,Te

ALLOYS

159

are reproduced in Fig. 22, where the dashed curve is the calculated spinflip energy. From the field dependence of the spin-flip scattering intensity, they concluded that acceptor levels with binding energy E , N 8 meV, as well as the valence-band levels, acted as intermediate states in the spinflip Raman process. This value for the energy of an acceptor level above the valence band is consistent with other data. (See, for example, Nimtz and Tyssen, 1976, and references therein; Dornhaus and Nimtz, 1976; Bichard et al., 1978.)

8. TRANSP~RT MEASUREMENTS For completeness in discussing the intraband magnetooptical results for Hg, -,Cd,Te, we review in this section the results of other measurements of the conduction-band-primarily transport measurements. These results have been reviewed extensively by Dornhaus and Nimtz (1976). The experimental curves of transverse and longitudinal magnetoresistance measured by Suizu and Narita (1973) for a sample with x = 0.178, n = 3.79 x 1OI6 ~ m - and ~ , T = 1.52 K are reproduced in Fig. 23. Although

H (kG)

FIG.23. Transverse (a) and longitudinal (b) magnetoresistance curves for a sample with x = 0.178 and n = 4 x 1Ol6 cm-3 (after Suizu and Narita, 1973). The two curves in (a) have different voltage scales. The arrows indicate the calculated peak positions. T = 1.52 K. I = 10 rnA. (a) H II ; Vo = 0.2861 mV; (b) H I ( I ; Vo = 0.2946 mV.

M. H.WEILER

160

the detailed interpretation of such data involves a determination of the scattering mechanisms in the material, the Shubnikov-de Haas peaks in the magnetoresistance occur approximately at magnetic fields where each Landau level passes through the H = 0 Fermi level. Narita and Suizu (1975) interpreted these data in terms of a parabolic (large-gap) model which is not accurate for samples with x = 0.178. The large spin splitting of the Landau levels is clearly observed in Fig. 23. Dornhaus and Nimtz (1977) studied the transverse magnetoresistance in the extreme quantum limit (hC 9 kT) for a number of Hgl-xCd,Te samples at T = 4.2 K. Their results are summarized in Table V, for the energy gap and bandedge effective mass and g factor. Antcliffe (1970)carried out Shubnikov-de Haas measurements on samples with x = 0.204 and n > 10’’ cm-’ at T N 4.2 K, and interpreted the results using the nonparabolic model of Kane (1966) [Eq. (39)]. They found E , = 63 +_ 8 meV, E , = 17 +_ 1.4 eV, giving rn, = 5.6 +_ 0.25 x m, with an extrapolated gc = - 164. Kozacki et al. (1976) studied a semimetallic sample (x = 0.155) with a very small, negative gap (Eg = -7.1 meV). They found a g factor at the Fermi level of - 780, which is the largest value ever reported for Hg, - ,Cd,Te. For samples with low carrier concentration, and therefore a small Fermi energy, resistivity peaks can be observed when the Landau levels pass through the optical-phonon energy (the magnetophonon effect). These peaks were studied by Kahlert and Bauer (1973) for a sample with x = 0.212 and n = 1.6 x 10’’ cm-’ at 4.2 K. Their results are reproduced in Fig. 24. They interpreted their results using the model of Lax et al. (1961) with parameters A = 0.96 eV, E , = 93 meV, rn, = 5 +_ 0.3 x 1O-j m, and gc = - 172 f 10. These results in Eqs. (22) imply E , = 26.6 eV. They also found the longitudinal optical-phonon energy to be ha,, = 17.1 meV. From the temperature dependence of the magnetophonon peaks from 65 to 125 K, they found a value for the temperature derivative of the energy

-

TABLE V RESULTS OF DORNHAUS AND NIMTZ (1977) FOR

Hg,-,Cd,Te SAMPLES AT T = 4.2 K

0.165 0.17 0.185 0.2 0.216

20 15 8.9 3.4 4.5

14 24.5 47 70 96

1.2 2.0 4.2 6.5 8.8

800 500 240 156 105

3.

MAGNETOOPTICAL PROPERTIES OF I.6

I

I

I

I

Hg, -,Cd,Te I

ALLOYS

161

I

B(kG)

FIG.24. Magnetophonon oscillations in the longitudinal magnetoresistance for a sample at T = 4.21 K (after Kahlert and Bauer, 1973). with x = 0.212 and n = 1.6 x 10’’

gap dE$dT = 7.6 x eV/K, neglecting the contribution of levels higher than n = 0 which become populated at higher temperatures. Kahlert (1978) later estimated that a value of (dE,)/dT obtained in this way for InSb, was a factor of 2 too large.

IV. Interband Experiments In this section we will review the results of interband experiments on Hg, -,Cd,Te, with emphasis on the results for semiconducting samples (EB > 0). Because an interband transition involves the creation of an electron and a hole which experience a mutual Coulomb attraction, bound electronhole pairs called excitons can be created. In Section 9 we discuss the corrections which one makes to interband magnetooptical data for exciton effects. In Section 10 we will review the magnetooptical results, including our recent magnetoreflection results, and in Section 11 we will discuss the techniques used for obtaining the quasi-germanium model parameters from our interband data. These parameters and the results of other techniques including intraband measurements will be discussed in Part V. 9. EXCITON CORRECTIONS Elliott and Loudon (1960) calculated the effect on the magnetoabsorption spectrum of the Coulomb attraction between electrons and holes in simple parabolic bands [Eqs. (1) and (511. They showed that the absorption spectrum has a peak corresponding to the lowest N = 0 hydrogenlike bound state of the electron-hole pair, which occurs below the free interband transition by the exciton binding energy E,. The only other feature of the spectrum is a weak edge near the N = 1 exciton level (see Johnson, 1967a).

M. H.WEILFiR

162

It was noted by Johnson (1967b) that for high magnetic fields the N = 0 exciton peak lies below the lowest interband transition by an approximate binding energy

E,

N

1.6Ry1l3,

(50)

where R is the effective Rydberg (R, = 13.7 ev) R = R,p/mlcf

(51)

and y is the reduced magnetic field y = ms/2pR,

(52)

where s was defined after Eq. (23), p is the reduced effective mass for the transition p-l = m,- 1 mV-', (53)

+

and IC,is the static dielectric constant. Equation (50) is a reasonable approximation for y % 1. For the zinc-blende semiconductors the results of Elliott and Loudon (1960) are inadequate because of the complicated valence-band structure, and because of nonparabolicity for the narrow-gap materials. Altarelli and Lipari (1974) calculated the effects of exciton binding for InSb including the complex valence band but not nonparabolicity (the quasi-germanium model in the limit hw, 4 E,). Their calculation was made for y 2 5. As was suggested by Elliott and Loudon (1960) and by Vrehen (1968), the results tabulated by Altarelli and Lipari (1974) for transitions to the conductionband Landau level n can be found within a few percent from E,(n)

N

1.6R[y/(2n

+ 1)]1'3

(54)

for n = 0, 1, and 2. Since the exciton effects have not been calculated in the quasi-germanium model, one must take nonparabolicity into account in an approximate way by introducing a field-dependenteffective mass. A reasonable expression for this [considering the approximate nature of Eq. (5411 is to use in Eqs. (51) and (52) p(n, H)N p[1

+ 2(2n + l)ms/pEJ'/'

(55)

following Eq. (41). For a typical interband transition in semiconducting Hg, -,Cd,Te withlc, = 16(BaarsandSorger, 1972),m, N 10-3m,m, N 0.4, E, N 100meV, the exciton binding energies range from about 7 to 9 meV for transitions to the n = 0 to 2 conduction-band levels, with y ranging from about 50 to 9.

3.

MAGNETOOPTICAL PROPERTIES OF Hg,

-,Cd,Te

ALLOYS

163

-

Compared to the interband energies being measured, and the energy differences of 50 meV between adjacent transitions, these exciton corrections are not insignificant; however, the corrections are small enough so that errors of a few percent in EB are not important. On the other hand, since the change in p in Eq. (55) is very large (from 0.001 to -0.04 m), the change in EBdue to nonparabolicity is also quite large (from 2 to 8 meV). An extension of the calculation of Altarelli and Lipari (1974) to include nonparabolicity using the full quasi-germaniummodel would be very useful in making these exciton corrections. Because the exciton corrections discussed in this section are relatively small, which is true in general for the small-gap materials, it was not previously considered necessary to make these corrections. Recently we have found for InSb that these corrections are necessary to have a consistent description of the interband and intraband data using the quasi-germanium model (Weiler, 1978). In Section 11 below we show that the same is true for Hg,-,Cd,Te.

-

10. MAGNETOOPTICAL RESULTS

The first interband magnetooptical experiments on Hg, -,Cd,Te were carried out by Harman et al. (1961) and extended by Strauss et al. (1962). They measured magnetoreflection as well as the Hall coefficient and resistivity, on a number of samples with x = 0.136 to 0.20, and carrier concentrations of 2 to 3 x 1015 cm-3. An approximate fit was made to interband and cyclotron resonance reflection peaks observed for a sample with x = 0.17 at 4.2 K. The model of Bowers and Yafet (1959) was used without exciton eV cm corrections, with parameters E, = 6 meV and P = 8 x (E, = 2mPz/h2 = 17 ev), giving m, N 4 x m. The fit to the interband data was not very good because the complexity of the valence-band structure had been neglected. The first study of interband transitions accompanied by an analysis using the quasi-germanium model, again without exciton corrections, was carried out by Groves et al. (1971) on a sample with x = 0.161 0.003 with carrier concentration n LX 9.4 x 1014 cm-3 at liquid helium temperatures. Interband magnetoreflection data were taken with the sample at -25 and 95 K. The energy gap E , was approximately - 10 meV (semimetallic configuration) at the lower temperature, and + 10 meV at the higher temperature. The other parameters determined were Ep -N 18.5 eV and heavy-hole mass mh 1: 0.28 f 0.1 m. Kim and Narita (1976) studied far-infrared magnetoabsorption on samples of Hg,-,Cd,Te with x = 0 (HgTe), 0.07, 0.126, and 0.148, all of which had E , < 0. They also used the quasi-germanium model without exciton corrections to analyze their data, which included cyclotron resonance

164

M. H. WEILER TABLE VI

RFSULTSOF GULDNER et al. (1977a,b)FOR Hg,-,Cd,Te SAMPLESAT T = 4.2K

0.01 0.025 0.05 0.105 0.115 0.15 0.185 0.215 0.25 0.28

-285 -261 -207 -110 -90 -30 35 86 161 208

18.3 18.5 18.8 18.8 18.8 18.8 18.9 19.0 19.0 19.0

and a number of interband transitions. We have discussed their results in Section 6 above. The most extensive magnetooptical studies of Hg, -,Cd,Te were the transmission experiments carried out by Guldner et al. (1977a,b) following their earlier work (Guldner et al., 1973; Tuchendler et al., 1973) on HgTe. They studied samples of Hg,-,Cd,Te with x = 0.01 to x N 0.3, spanning the semimetallic (EgN -300 mev) to semiconducting ( E g N +200 mev) region. They studied interband transitions for all their samples, and intraband transitions for the semimetallic samples, in both the Faraday and Voigt configurations. The transition energies were fit to the model of the Pidgeon and Brown (1966) with F = N, = 0 and no exciton corrections.

X

FIG.25. Energy gap Eo versus alloy composition x (after Guldner et al., 1977b).

3.

MAGNETOOPTICAL PROPERTIES OF

Hg, -,Cd,Te

ALLOYS

165

,pi-5 / p '

P

wn

-300

-200

0 E, ( m e V )

-100

100

200

FIG.26. Interband coupling parameter E, versus energy gap Eo (after Guldner et al., 1977b).

Their results for E, and E , for their samples at T = 4.2 are given in Table V1 and in Figs. 25 and 26. Since their data were taken on unoriented samples, in their analysis they made the approximation y z N y3 = f. They fit their data using A = 1 eV (Camassel et al., 1974) and y1 = 4.5

k 1.5,

K =

-1 f 0.5,

71

- 27 = 2.5

& 0.5,

(56)

where the last equation implies a heavy-hole effective mass [from Eq.(26)J m = 0.4 k 0.1 m for all x. Guldner et al. (1977a) also saw in the semimetallic samples evidence of acceptor states within the conduction-band continuum. Transitions were observed to these states from the r6light-hole band. The binding energy of one acceptor level (A,) was -0.8 meV, and that of the other level (A,) varied from 2.5 meV at x < 0.06 to 5 meV at x N 0.1. Transport measurements (Mauger et al., 1974) have indicated acceptor binding energies ranging from 5 meV at x N 0.1 to 19 meV for a sample with E, N 0. In their magnetoabsorption data for a semiconducting sample with x = 0.28, Guldner et al. (1977b) observed structure due to polaron effects, that is, pinning and splitting of the interband transition energies when the cyclotron frequency was near the LO phonon frequency oLo. Pinning was observed in the Faraday configuration at the CdTe-like mode haLo = 19.5 meV, and in the Voigt configuration at this mode and also the HgTe-like mode at ho,, = 17.0 meV. This behavior has been given a detailed analysis by Swierkowski et al. (1978) as shown in Fig. 27, which compares the data of Guldner et al. (1977b) for the energy of the ac(l) level (the transition energy ho less the energy gap and the calculated valence-band energy) using the parameters A = 1 eV, E , = 208 meV, E , = 19 eV, y , = 5,f = 1.5, K = -0.5, F = N, = 0, and electron-phonon coupling constants a = 0.037 and 0.046 to the 17.0- and 19.5-meV phonon, respectively. These results are comparable to those of Kinch and Buss (1971).

M. H.WEILER

166

I

I

30

25

20

I

35

I

40

_-_-I 45

H (kG)

FIG.27. Adjusted transition energies (circles and triangles) versus magnetic field from the data of Guldner et al. (1977b) compared to calculated energies (solid lines) including electronphonon coupling (after Swierkowski et al., 1978) for Hgo.,,Cd,,2,Te. A,Voigt, bhh(2)4 4(1); 0 ,Faraday, a,dO) -,d l ) .

Recently, Dobrowolska et al. (1978) carried out magnetotransmission measurements on HgTe (x = 0) at temperatures from 8 to 92 K.The data were fit using the quasi-germanium model without exciton corrections, with A = 1.08 eV, Ep = 18.2 k 0.2 eV, and the parameters of Groves et al. (1967) 71 = 3.0,

72

= -0.5,

73 = 0.67,

K =

-1.3,

F = 0. (57)

They indicated that they would have obtained a better fit with a small, positive value for F; however, based on a single-group analysis (Weiler, 1977), F is expected to be negative since it results from interaction between the conduction band and r, bands at higher energies. Dobrowolska et al. also obtained values for the energy gap as a function of temperature which were fit to E,=

[

-302+-

31+T

with T in degrees Kelvin. We carried out magnetoreflection experiments on ten samples of Hg, -,Cd,Te in the narrow-gap semiconductor region 0.175 I x I 0.269 (Weiler, 1977; Weiler et al., 1977). Our samples were purchased from Cominco, Inc. The sample characteristics are given in Table VII, which lists the alloy composition x as determined either by microprobe measurements or from the transmission edge observed with an infrared spectrometer, using a curve supplied to us by Cominco. Also given are the 77-K carrier concentration and Hall mobility, and the crystal axis and/or the orientation (0,4)of the reflecting face normal.

TABLE VII PHYSICAL AND ELECTRICAL CHARACTERISTICS OF Hg,- .Cd,Te SAMPLES

NML Alloy composition x sample Spectrophono. Nominal !Microprobe tometer 796 798 800 801 802 804 805 806 809 810

0.20 0.20 0.26 0.22 0.20 0.30 0.30 0.26 0.23 0.23

0.213 k 0.004 0.182 k 0.015 0.248 k 0.014 0.196 k 0.013 0.175 _+ 0.013 0.269 k 0.017

0.265 k 0.005 0.247 k 0.005 0.220 f 0.005 0.221 f 0.005

77 K

Carrier concentration 1.3 x 1014~ 4.5 3.2 8.7 1.1 2.1 2.1 1.1 2.4 2.8

Hall mobility

3 1 18.9 ~ x~

1.4 0.8 2.1 2.2 0.5 0.5 0.7 1.1 1.1

Orientation Axis 6, 4

lo5 cm2/V sec [lll]

[Ool] [Ool] [Ool] [Ool] [Ool] [111]

70" 30" 54.7 45 39 42 82 32 0 45 0 45 0 45 0 45 0 45 54.7 45

M. H. WEILER

168

The samples were mounted on the cold finger of a liquid He Dewar in the Faraday configuration as shown in Fig. 12. We measured the sample temperature in such circumstances to be 24 K with the Dewar filled with liquid helium, and 91 K using liquid nitrogen. The optical apparatus consisted of a grating spectrometer equipped with gratings, filters, and polarizers to span wavelengths from -20 to 1 ,urn. The Bitter magnet had a maximum field of 15 T'. The typical interband magnetoreflection data are shown in Figs. 28 and 29. Figure 28 gives magnetoreflection spectra, taken at a fixed photon energy 261.6 meV as a function of magnetic field, for both oL and oR polarizations on a sample with x = 0.213 (sample no. 796) at 24 K. For comparison, in Fig. 29, we show spectra taken with the same apparatus (Weiler, 1977) on a sample of InSb, which also shows, for comparison, the magnetoabsorption peaks observed by Pidgeon and Brown (1966) for E I H. The Hg,-,Cd,Te peaks are significantly broader than the InSb peaks. Comparison with the theoretical lineshapes in Fig. 3 gives oz = 10oO or I

I

I

U

Z

ul

LL W

a

I

60

I

80 100 MAGNETIC FIELD ( h G )

FIG.28. Magnetoreflection curves in the Faraday configuration (cR and uL),. for a Hgo.7R7Cdo.21 ,Te sample with x = 0.213 at 24 K. Peaks are labeled as transitions according to Table VIII. hw = 261.6 meV.

3.

MAGNETOOPTICAL PROPERTIES OF I

I

I

Hg, - .Cd,Te I

ALLOYS

169

I

6

I

40

I

I

1

60 80 100 MAGNETIC FIELD ( k G )

I

120

FIG.29. Magnetoreflection curves similar to Fig. 28 for InSb, compared to the magnetoabsorption results of Pidgeon and Brown (1966) for InSb [Ool] with E IH. ho = 387.1 meV; T = 24 K.

z w 2 x lo-” sec for the InSb sample, but oz w 100 or z w 2 x 10- sec for the Hg,-,Cd,Te sample. Figure 30 gives a similar spectrum for the same Hg,-,Cd,Te sample at 91 K. The lines are only slightly broader, indicating that much of the line width is due to the inhomogeneity of the alloy composition. In Fig. 31 we plot, for this same sample, the photon energy versus magnetic field for each reflectivity peak. The solid and dashed curves are those calculated, using the quasi-germanium model with exciton corrections as described above, using parameters obtained from a fitting procedure to be described in the next section. The lines were identified by computing transition intensities using Eqs. (19) and approximate sets of parameters to find the strongest lines. These transitions are listed, for both cL and oR polarizations, in Table VIII. We found, in contrast to the observation of Guldner et al. (1977b), that the strongest transitions at high photon energy or quantum number, were pairs of transitions from the a- or b-set heavy-hole levels to the conduction band.

uR

9.10

f

I

I

60

80

I

I

100 120 MAGNETIC FIELD (kG)

FIG.30. Magnetoreflection curves for the sample as in Fig. 28 (Hg,,,,,Cd,.,,,Te) T = 91 K and ho = 291.7 meV.

but at

as in Fig. 28

.the transitions

3.

MAGNETOOPTICALPROPERTIES OF

Hg, -,Cd,Te

ALLOYS

171

TABLE VIII INTERBANDTRANSITIONSIDENTIFIEIJ FOR Hg, -,Cd,Te

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1s 16 17 18

a+(-l)-+cf(O) b+( - 1) -+ b'(0) a - ( l ) + d(2) b-(l) -+ M(2) b+(O) b"(1) a-(2) + d(3), b-(2) + M(3) b'(4) a-(3) 6(4), b-(3) b'(S) a-(4) d(S), b-(4) b+(l) M(2) a-(5) -+ d(6), b-(5) -+ b'(6) a-(6) d(7), b-(6) b'(7) a-(7) -+ d(8), b-(7) -+ b'(8) b+(2) -+ b'(3) a-(8) -+ d(9), b-(8) -+ b'(9) a-(9) -+ &(lo), b-(9) -+ b'(10) a-(10) -+ aC(ll), b-(10) + b'(l1) b+(3) M(4) a-(11) -+ d(12), b-(11) -+ W(12) -+

-+

-+

-+

-+

-+

-+

-+

-+

a-(l) d(0) b-(l) + b"(0) a-(2) -+ d(1) a+(l) d(0) b-(2) -+ bc(1) a-(3) -+ d(2) b-(3) -+ W(2) a-(4) -+ d(3), b-(4) -+ b"(3) a+(2) d(1) a-(S) -+ d(4), b-(S) -+ b'(4) a-(6) -+ d(5), b-(6) -+ bc(S) a-(7) -+ d(6), b-(7) -+ b'(6) a+(3) -+ d(2) a-(8) -+ d(7), b-(8) -+ b'(7) a-(9) -+ d(8), b-(9) -+ b'(8) a-(10) -+ 6(9), b-(10) -+ W(9) a+(4) + d(3) a-(11) + d(lO), b-(11) -+ b'(10) -+

-+

-+

11. PARAMETEX-FITTING TECHNIQUES The process of identifying each transition such as those in Fig. 31 and finding the band parameters for a best fit between theory and experiment involved several steps making use of the M.I.T. time-sharing computer system. First, approximate parameters were chosen for each set of data. Second, these were used to calculate the expected interband transition energies as a function of magnetic field, by diagonalizing the Hamiltonians in Table 111, and these energies were plotted on photocopies of the data. The plots which gave a reasonable fit to the data were used to identify most data points with one or, at most, two particular interband transitions. These were stored in an on-line disc dataset and used as input to the leastsquares fitting routines described below. These routines provided new parameter sets with which to make new plots and complete the line identifications. The data points were fit to the transitions listed in Table VIII, which were calculated to be significantly stronger than the other transitions allowed by the selection rules in Eqs. (20). The least-squares fit calculation was based on a method described by Reine (1970). Given a set of N energy and magnetic field pairs ( E i , Hi) corresponding to each observed reflectivity peak, and a function E(H,) for

M. H.WEILER

172

calculating a theoretical transition energy for a given magnetic field H i , we wish to minimize the root-mean-square deviation

6=

I"'

l N C [ E ( H , ) - Ei]2

-

"i=I

,

(59)

where the function E(H) depends on a set of parameters aj, j = 1, . .., M. Reine showed that to minimize 6 to first order in the corrections 6aj to the parameters aj -,aj

+ 6aj

(60)

one must solve the M x M system of equations

Successive corrections are made to the parameters aj until satisfactory convergence to a minimum is obtained. Reine calculated the derivatives [dE(Hi)]/daj in Eq.(61) using approximate expressions for the transition energies to first order in H . This was sufficiently accurate for large-gap materials such as GaAs and GaSb. Since the transition energies in small-gap materials are strongly nonparabolic, i.e., depend on higher-order terms in H,we used exact expressions for these derivatives, obtained by differentiating the fourth-order determinantal equation for the energy eigenvalues with respect to the parameters aj. The details of this calculation are given in Weiler (1977). The parameters of the quasi-germanium model Hamiltonian in Table 111 are the energy gap E,, the spin-orbit splitting A, the interband coupling energy E,, the higher-band effective mass parameters F, y l , y2, and y3, and the g factor contributions N,,K, and q. In the absence of a direct experimental determination of the spin-orbit splitting, we use the estimated value A N 1 eV (Moritani et al., 1973). Since q and N, are expected to be small for Hgl-xCd,Te, we set these equal to zero. Thus we have a possibility of determining seven parameters: E,, E,, F, y,, y 2 , y3, and K. However we found that the data were not sensitive enough to fit all these parameters simultaneously. We found as shown below a method for obtaining constraints on the Luttinger parameters yl, y 2 , y3, and K . With these constraints we have been able to make an approximate determination of the best values for F and E,, and then for the Luttinger parameters. We then found, for each set of data, the energy gap

4-

The transitions listed in Table VIII include two, aJ3) and 0,(6), from the heavy-hole levels a- (1) and a-(3), to the conduction-band level a"(2). We

3.

MAGNETOOPTICAL PROPERTIES OF

Hg, -,Cd,Te

ALLOYS

173

used the energy difference between these two to obtain constraints on the Luttinger parameters y,, y, ,y 3 , and K. The expressions in Eqs. (23) give E[cR(6)1 - E[eL(3)1

2syH,

(62)

where, to lowest order in H , YH N

$[71y,

-

34($

+ 3y") - 6K - 3 9 - 2f)q],

(63)

which is of the same order of magnitude as the high-quantum-number inverse heavy-hole effective mass m/m- in Eq. (26):

m/m- N y1 - +(y'

+ 3y"),

(64)

where both results are valid only if EJE, %- y l , y', y", and K. The above result for Y,, is much less sensitive to the small parameter q than a similar energy difference E[aL(2)] - E[O&)] For q

N

E

Q[ -71

+ 2(7' + 37") - 6~ - &9 - 2f)q-J.

(65)

0 Eq. (63) can be rewritten

Y d e , 6)= YH[(loo)l + %(Y3 - YZ)[f(e, 6)- l1-

(66)

We could measure the energy differences in Eq. (62) only for those samples for which we could resolve the [cL(3), aL(4)] and [aR(6), cR(7)] doublets. Representative spectra for these transitions are shown in Fig. 28. The doublets are clearly resolved, but for other samples the [aR(6),aR(7)] doublet was only barely resolved. For each sample where the doublets were resolved, we plotted the magnetic field and photon energy positions of the aL(3)and cR(6) peaks as shown in Fig. 32 for the same sample whose spectrum is given in Fig. 28. From this plot the energy differences, and hence yH, were determined; the average yH found for each sample are given in Table IX. The first part of the Table gives yH for the [Ool] oriented samples, giving an average over all these samples of yH[(ool)]

%

2.9

0.2,

(67)

where 0.2 is the probable error. Then the second part of Table IX gives yH(8,#) for the other samples. Equations (66) and (67) were used to give the value for (y3 - yz) to yield an average value ( ~ 3 -

72) z 0.8 f 0.3.

(68)

This was the first determination of the warping for Hg,-,Cd,Te. Since the Dresselhaus parameter H , (or the Kane parameter D ) results from the interaction between the valence band and higher Tsbands, in the

174

M. H. WEILER

MAGNETIC FIELD (kG) FIG.32. Detail of transition energies versus magnetic field for the same sample as in Fig. 31

(Hg,~,,,Cd,,,,,Te at T = 24 K), for the determination of the valence-band energy difference f l a - ( l l l - ECa-WI. TABLE IX &SULTS FOR THE HEAVY-HOLE MASSPARAMETER YH POR SAMPLES WITH Hl/[OOl], AND POR OTHW ORIENTATIONS

802 804 805 809

Sample

f(@ 4)

796 798 810

0.202 O.OO0 O.OO0

8, 4

3.46 f 0.09 2.42 f 0.09 3.05 f 0.10 2.77 f 0.04 YH(@

4)

1.82 f 0.04 2.13 f 0.08 2.04 f 0.08

Yo-Y2

1.12 f 0.21 0.64 f 0.17 0.71 f 0.17

3. MAGNETOOPTICALPROPERTIES OF Hg, -,Cd,Te ALLOYS

175

single-group approximation, and since these bands are very remote in HgTe and CdTe, we set H , = 0 which gives another constraining equation: y , - 2y, - 3y3

+ 3K + yq + 2

N

0.

(69)

In our earlier work (Weiler, 1977; Weiler et al., 1977) we also set the Dresselhaus parameter F' (or Kane parameter A) equal to zero and had four equations [with Eqs. (67)-(6911 which we solved immediately for y,, y,, y 3 , and K. Since the latter approximation, that the higher rl binds are remote, is much less justified, we have used only the three equations [Eqs. (67)(69)] as constraints to the parameter-fitting equations [Eqs. (61)] using the method of Lagrange multipliers. We had recently (Weiler, 1978) developed this technique in order to fit our data for InSb. With these constraints and correcting for exciton effects as described in Section 9, we found the best fit to both our interband and intraband data for samples at both 24 and 91 K, was obtained using E , = 19.0

0.5

and F = -0.8 f 0.3

with no systematic dependence on alloy composition or temperature. The value of E, is larger than our earlier result (EP= 17.9 eV) and agrees with the results of Guldner et al. (1977a,b) for semiconducting samples, although they used F = 0 and made no exciton corrections. The results for the Luttinger parameters are, with q = 0, y , = 3.3 f 0.2, 73 = 0.9 f 0.1,

y, = 0.1 K =

f 0.1,

-0.8 f 0.1.

(72)

The change from our earlier parameters (Weiler, 1977; Weiler et al., 1977) which were y1 = 3.0, y z = -0.3, y3 = 0.5, and K = -1.2 comes from the exciton corrections, from the new value for E, and from allowing F ' Z 0. [Here F' = -2.4 f 0.3, G = -1.1 f 0.2, H , = -4.1 f 0.3 with H z = 0.1 The new parameters result in nearly unchanged values for the heavy-hole mass [Eq. (26)]

m- [Ool] = 0.40 f 0.07 m, m- [llO] = 0.49 f 0.11 m, m- [lll] = 0.53 f 0.12 m.

(73)

With these parameters fixed, the best-fit values for E, are given for our samples in Tables X and XI. The uncertainty in E, is approximately f 3 meV, m, and that that in m, and m, [from Eqs. (2611 is approximately f 2 x

176

M. H. WEILER TABLE X RFsULTS

FOR

Hg,-,Cd,Te SAMPLES AT 24 K

Sample

x

E8 (mev)

802 798 801 796

0.175 0.182 0.196 0.213 0.220 0.221 0.247 0.248 0.265 0.269

29 36 81 109 118 122 170 164 187 185

809 810 806 800 805 804

m,

m)

m, [001](10-~ m)

-gc

2.3 2.8 6.2 8.3 9.0 9.2 12.7 12.3 13.9 13.8

418 343 143 103 94 91 62 64 55

2.3 2.8 6.2 8.2 8.9 9.2 12.6 12.2 13.8 13.7

56

TABLE XI RESULTSFOR Hg, -,Cd,Te SAMPLES AT 91 K Sample

x

E,(meV)

802 796 809 810 805

0.175 0.213 0.220 0.221 0.265

51 128 139 139 206

m , ( i ~ ~ m )m, [001](10-~m) 4.0 9.6 10.4 10.4 15.1

4.0 9.7 10.5 10.5 15.2

-gc

234 86 78 78 49

in gc is approximately +3%. These parameters give excellent agreement to both our intraband and interband data, as indicated in Figs. 14-18 and 31. In the next section we will summarize our results and those of others for the parameters of Hg,-,Cd,Te as a function of alloy composition and temperature.

V. Hg,-,Cd,Te Parameters In this concluding part we will arrive at a set of parameters for the quasigermanium model for Hg, -,Cd,Te which give excellent descriptions of the properties of the conduction and valence electrons near the fundamental gap. First, we will mention some theoretical estimates which have been made of these parameters. Lawaetz (1971) extrapolated from the known parameters of Si and Ge, using measured quantities wherever possible, to obtain estimates of the parameters of a number of zinc-blende semiconductors at T = 0 K. For HgTe (x = 0) he estimated E , = 18.0 eV, y1 = 1.12, y2 = -0.29, y 3 = 0.34,

-0.95, and q = 0.06. For CdTe (x = 1) he estimated E, = 20.7 eV, y , = 0.98,yz = -0.27, y 3 = 0.30,uZ = -3.43, and q = 0.05. More recently IC =

Hermann and Weisbuch (1977) reexamined existing data for the conduction-band effective mass and g factor in CdTe and concluded the E, = 18.5 f 1 eV rather than 20.7 eV, with E, = 1.606 eV (Segall and Marple, 1967) and A = 0.927 eV (Camassel et al., 1974). Values of E , calculated from pseudopotential electronic wave functions have been 9.1 eV for CdTe (Bowers and Mahan, 1969) which is a factor of 2 too small, and E, = 2((S(~,lX)(~/m= 18 eV for Hgo.8,4Cdo.,46Te (Katsuki and Kunimune, 1971). The spin-orbit splitting A is approximately 1 eV for the entire range of alloy compositions (Chadi and Cohen, 1973). Chadi and Cohen (1973) calculated the energy gap of Hg,-,Cd,Te at T = 0 K to be approximately linear from 0.3 eV for x = 0 to 1.6 eV at x = 1, with a zero energy gap at x = 0.165. Katsuki and Kunimune (1971)found the zero gap crossing to be at x = 0.146 at 77 K. We will now discuss the magnetooptical results for the parameters of the quasi-germanium model for Hg, - ,Cd, Te. These are summarized in Table XII. Except for the energy gap E,, the parameters are approximately independent of alloy composition and temperature. 12. SUMMARY OF RESULTS FOR Hg, -,Cd,Te

The results of our magnetoreflection experiments on samples of different orientation provide the most accurate determination of the higher-band parameters yl, y2, y,, and IC [Eq. (72)] and F [Eq. (71)] and particularly the anisotropy y, - yz [Eq. (6811.These,in turn,determine theheavy-holeeffective mass [Eq. (73)] which has not been measured directly. These parameters, TABLE XI1 PARAMETERS OF THE Q U A S I - G e

MODELFOR Hg, -,Cd,Te" EP

A Y1 Y2 Y3

K

F 4 Nl

19 eV 1 eV

3.3 0.1 0.9

- 0.8 -0.8 0.0 0.0

* E , = [-304 + (0.63T2/ + TXl - 2 ~ +) 1 8 5 8 ~+ 54x2] mev. 11

M. H.WEILER

178

which result from the interactionwith remote bands in the k papproximation, seem to be independent of alloy composition. This is not surprising since these remote bands have nearly the same energies, especially with respect to the Tsband, in both HgTe and CdTe (Chadi et aL, 1972). One would expect that F,the contribution to the r6 band effective mass due to higher r4bands, would increase for Hg, -,Cd,Te samples with gaps approaching 1 eV since then the nearest band is only about 5 eV away. A bit more surprising is the result that the r6-rs band coupling energy Ep is probably nearly the same (18 to 19 eV) for HgTe, CdTe, and the Hg,-,Cd,Te alloys, We found Ep = 19 eV to give the best fit to both interband transitions including exciton corrections and to intraband transitions for semiconducting samples (x 2 0.17); the same exciton corrections applied to the interband data for HgTe (Groves et al., 1967;Guldner et al., 1973; Tuchendler et aL, 1973) and the semimetallic samples (Guldner et al., 1977a,b) may result in better fits for E, = 19 eV, instead of 18 to 18.5 eV, for this alloy composition range as well. Saleh and Fan (1972) found E , = 19 eV from magnetoabsorption measurements on HgTe. Consequently, we choose E , = 19 eV for all alloy compositions 0 5 x I 1. There seems to be no variation of E, with temperature, at least below 90 K.

r4

0.8

0.7 0.53

-x-x-x-x-x-:

0.6

->, 0.5

0.46

-

/xcxcx-x-x-x-x-x->

0.405

a

a

W

>-

0.4

o 3.85xx -

W

0.3

0.35

~ x ~ ~ ~ x - x - - x - x cx-x-x---x-x-x-x

[r

W

-x-x-x-x-x-x-x

:x

x z x - x

-/X

7 x - x -

x-x- - - - - - . x

0.2 0.23

/x-

0.1

0 -

0

1

1

40

1

1

80

1

!

120

1

1

160

1

1

200

1

1

240

1

/

280

FIG.33. Energy gap of Hg,-,Cd,Te versus temperature, deduced from optical absorption data for samples of different alloy composition x (after Kruse et al., 1971).

3.

MAGNETOOPTICALPROPERTIES OF Hg, -,Cd,Te

A L ~ Y S

179

The only parameter of the quasi-germanium model that changes significantly with alloy composition and temperature is the energy gap. In principle the energy gap can be determined from the location of the absorption edge in zero magnetic field as is shown in Fig. 3. This method is limited by the lack of exciton structure, and by the difficulty of locating the position corresponding to the energy gap on a broadened and smoothly varying edge. Kruse et al. (1971) and Scott (1969) made such measurements on Hg,-,Cd,Te over a wide range of temperatures and alloy compositions. Representative data of Kruse et al. (1971) is reproduced in Fig. 33. Scott (1969) fit his data with an expression linear in x and T as did Wiley and Dexter (1969) from data on helicon propagation. Schmit and Stelzer (1969) also made a linear fit, with a small term cubic in x, to their data on the photoconductive response of Hg - ,Cd,Te. These expressions, similar to that given in Eq.(44),are discussed by Dornhaus and Nimtz (1976) in their review article on Hg, -,Cd,Te. More accurate results, at least for T 5 100 K and for x 6 0.3, have come from magnetooptical measurements. Dobrowolska et al. (1978) have measured the energy gap of HgTe at temperaturesfrom 8 to 92 K using interband magnetoabsorption as discussed in Section 9 [see Eq. (5711, with E , N 18.2 eV. Their data, along with earlier data from various measurements by other authors, are reproduced in Fig. 34. They fit their results with the expression given in Eq. (58). A better fit to all the data in Fig. 34 is made using the expression

,

E,=

[

-304+- 0*63T2] mev, 11+T

(74)

which is the solid curve in Fig. 34. The dashed curve is the best fit to a linear T dependence E,

=

[-307

+ 0.61T1

meV,

(75)

which is not a very good approximation to the data at T --* 0. We obtained an expression for the energy gap of Hg, -,Cd,Te from our magnetoreflection data by fitting our results for E , in Tables X and XI to the expression E,=

[

-304+- 0.63T2 (1 - 2x) 11+T

+ 1912x - Cx(1 - x)

1

meV, (76)

which is chosen to reproduce the HgTe data [Eq. (74)], the change in sign of the temperature dependence at x 1: 0.5 (see, for example, Scott, 1969), the energy gap of CdTe at T --* 0 [ E , = 1.608 eV correcting for exciton effects (Zanio, 1978)], and includes a term Cx(1 - x) which represents a bowing of the gap away from a linear dependence on x. The temperature dependence in Eq. (76) from Eq. (74) gives a better fit to our data at 24 and

180

M. H. WEILER

TEMPERATURE (K)

FIG.34. Energy gap of HgTe versus temperature. Data: A, Groves et al. (1967, 1970); A, Szuszkiewicz(1977); 0,Raymond and Verie (1970); 0,Guldner et al, (1973); 0 ,Verie and Decamps (1965); x, Otmezguine et a/. (1970); $, Dobrowolska et al. (1978).

91 K than the expression of Dobrowolska et al. (1978) [Eq. (58)]. We find a value for the bowing parameter C = 54 & 11 meV which gives E,=

[

-304+-

0.63T2 (1 - 2x) 11+T

+ 1858x + 54x2

This expression agrees very well with the data of Guldner et al. (1977a,b) and Dornhaus and Nimtz (1977) from Tables V and VI taken at T = 4.2 K as shown in Fig. 35. The fit is somewhat better than the dashed curved calculated from the expression of Kim and Narita (1976) [Eq. (44)]. In Fig. 36 we compare curves calculated from Eq. (77) for T = 24 and 91 K with our data and with data of other authors converted to T = 24 K using the temperature dependence in Eq. (77). Considering the uncertainty in x of our samples (the uncertainty as indicated by error bars in the T = 24 K data is about the same for the T = 91 K data), with similar or only somewhat smaller uncertainty stated by the other investigators, Eq. (77) gives very good agreement with the data.

3.

MAGNETOOPTICAL PROPERTLESOF Hg,

7

-

7

I

1

ALLOYS

-,Cd,Te I

181

/

200

too

> al

E

O

a

a

c3

>-

0

a W

=

W

-100

- 200

L

I

0.05

0

I 0.10

I 0.15

I

I

0.20

0.25

0.30

A L L O Y COMPOSITION, x

FIG.35. Energy gap of Hg,-,Cd,Te for 0 Ix I 0.3 at T = 4.2 K, compared to the data of Guldner et al. (1977a,b) ( 0 )and Dornhaus and Nimtz (1977) (A).Solid curve: calculated from Eq. (77); dashed curve: from Kim and Narita (1976) [Eq. (44)].

Using the parameters from Table XI1 in the expressions in Eqs. (22) we obtain the following expressions for the band-edge effective mass m, and g factor ge in the conduction band as a function of x and T:

m, - -Om6 g, =

+ E, ( T ) ( 1 0 0 0 + E.). 19,000

m

--

-[(?)( +

667

)

lo00 - 21, 1000 E ,

+

(78)

where E , is given, in meV, in Eq. (77). The light-hole mass m, is nearly equal to m,. These expressions are compared with our results from Tables

182

M. H. WEILER

0.16

0.2 0 0.24 0.28 A L L O Y COMPOSITION. x

FIG.36. Energy gap of Hg,-,Cd,Te versus alloy composition at T = 24 and 91 K: I$, this work T = 24 K; 0 this work, T = 91 K ; and data of other authors adjusted to T = 24 K ; A.Guldner et al. (1977a,b); H:a, Groves et al. (1971); b, Strauss et al. (1962); c, McCombe et al. (1970a); d, Kinch and Buss (1971); e, McCombe et al. (1970b); f, Antcliffe (1970); g, Kahlert and Bauer (1973); h, Weber et al. (1975); i, Poehler and Ape1 (1970); j, Swierkowski et al. (1978). Lines calculated from Eq. (77) for T = 24 K (solid line) and 91 K (dashed line).

X and XI, and with those of other authors, in Figs. 37 and 38. The agreement is reasonable with all the experimental data. Figure 38 illustrates the large increase in the g factor of Hg, -,Cd,Te near the zero-gap crossing. In the presence of an applied magnetic field as indicated in Figs. 17 and 18, or with substantial carrier concentration as indicated in Eq. (41), the effective mass increases and the g factor decreases from the band-edge values. Values of E, calculated from Eq.(77) for several temperatures are given as a function of alloy composition in Fig. 39. In Figs. 40 and 41 values of rn, and gc calculated from Eqs. (78) are given as a function of E,. For use in calculating the TrCband energies in Eq. (23), the Luttinger parameters are 7: = 3.3 + 6333/E,, 7: = 0.1 + 3167/E,, (79) K~ = -0.8 3167/E,, yf; = 0.9 + 3167/E,,

+

3.

MAGNETOOPTICALPROPERTIES OF

I

I

I

I

Hg, -,Cd,Te

ALLOYS

183

I

0.20 0.24 0.28 A L L O Y COMPOSITION, x

FIG.37. Conductionband-edge effective mass m, versus alloy composition.Curves calculated from Eq. (78) using Eq. (77) at T = 24 K (solid line) and 91 K (dashed line) compared to data as identified in Fig. 36, A:data of Dornhaus and Nimtz (1977) at T = 4.2 K and k, Knowles and Schneider (1978) adjusted to T = 91 K.

where again E , in meV is given in Eq. (77). These expressions may be used in Eq. (26) to find the light- and heavy-hole masses. Taking the average of j j r = 7" = 7 in Eq. (15) with7 = 3 gives

For HgTe at 4.2 K these equations give yk = -17.6, y'; = -10.3, y i = - 9.5, K~ = - 11.2, and yL = -9.8 which are somewhat larger in magnitude than the parameters of Guldner et al. (1973) and Uchida and Tanaka (1976) but agree with those of Groves et al. (1967) and with the results of Saleh and Fan (1972) from magnetooptical studies of HgTe. Our expression, Eq. (77), for E , versus x and T agrees fairly well with that of Scott (1969) for our range of x and T The expression of Schmit and Stelzer (1969) does not fit our data, nor do the more recent expressions of Finkman and Nemirovsky (1979) derived from studies of the Urbach tail of the absorption edge for absorption coefficients a 5 lo00 cm- and for 0.205 5 x 5 0.220 with 150 < T < 320 K. The equation of Finkman and Nemirovsky gives the photon energy E(a) for a given value of the absorption

184

M. H. W L E R I

1

- 50C

0

-400

IV

2 c3

w -300

c3

n W

I

n 2

a

m

2

-200

0

IV 3

0

z

8

-100

0

FIG.38. Conduction bandedge g factor gcversus alloy composition.Curves calculatedfrom JZq. (78) using Eq. (77) at T = 24 K (solid line) and 91 K (dashed line) compared to data identified as in Fig. 37 and L, Walukiewicz et al. (1977) adjusted to T = 24 K.

coefficient, and requires a N 25,000 cm- to approximate our data for the energy gap E , at T = 91 K with, however, a significantly different slope dE/dx. 13. CONCLUSION In this chapter we have presented evidence, from magnetooptical and other experiments, that the quasi-germanium model, with the parameters listed in Table XII, provides an excellent description of the electronic energy levels in the alloy system Hg,-,Cd,Te over the entire range of alloy composition. The parameters for CdTe (x --+ 1) are, of course, deduced from magnetooptical experiments on Hg,-,Cd,Te with x 5 0.3 and are only approximate. These parameters are difficult to measure directly as x + 1

3.

MAGNETOOPTICALPROPERTIES OF Hg,

-,Cd,Te

ALLOYS

t 100

185

1 I

I 0.16

I

I

I

I

t

I

I

0.20

I 0.24

I

I

I

I 0.28

I

'

1

0.32

X

FIG.39. Energy gap of Hg, -,Cd,Te calculated from Eq. (77) for several values of temperature T (K).

because of the relatively large exciton effects and the small magnetic effects due to the large effective masses. We have presented the most extensive review of the magnetooptical data for Hg, -,Cd,Te alloys in the narrow-gap semiconductor region 0.2 6 x 6 0.3 since these materials are the most useful technologically as described in a later volume in this series. The electronic energy levels for these alloys may be calculated to a high degree of accuracy using the quasigermanium model Hamiltonians in Table I11 with the parameters in Table XI1 and the energy gap from Eq. (77). For interband transitions in a magnetic field, the transition energy is reduced by the exciton binding energy given, approximately, by Eq. (54). The Hamiltonians in Table I11 are in a form suitable for eigenanalysisby computer (Weiler, 1977). In Section 5 we have given a number of approximate analytical expressions for the energy levels. The conduction-band energy levels can be calculated to within a few percent using Eq. (28) (the Lax or Bowers and Yafet model) with m, and gc from Eqs. (78) and E , from Eq. (77). If the energies are needed

186

M. H.W L E R

FIG. 40. Conduction band-edge effective mass m, of

Hg,-,Cd,Te calculated from Eq. (78).

at substantial carrier concentrations n, the expressions in Eqs. (38)-(39) may be helpful. Transition probabilities may be calculated using Eqs. (19) with the approximateeigensolutions in Eqs. (29) and (30). For the case where the magnetic field is sufficiently small, and E, sufficiently large, so that the intraband energies ho,,etc. are small compared to E,, then the expressions in Eqs. (21), (23), and (25), with the parameters in Eqs. (77)-(79),may be used to calculate the Landau level energies in the conduction and valence bands separately. The expressions in Eqs. (23) and (25) are accurate for the heavyhole levels [Eq.(23) for n > 11 and the n = - 1 light-hole levels [Eqs. (2511 as long as the heavy-hole cyclotron energy -heH/m- c is small compared to E,; this is true even at high magnetic fields in most cases. The success of the quasi-germanium model in describing the electronic properties of Hg, -,Cd,Te is remarkable considering the fact that these materials contain a high degree of disorder. The band structure can be calculated quite accurately assuming the potential due to the Hg and Cd ionic cores is given by (1 - x) times the Hg potential plus x times the Cd potential (the virtual crystal approximation).The difference between the real potential and this mean potential gives a random, fluctuatingfield whose only effect

3. MAGNETOOPTICALPROPERTIESOF Hg, -,Cd,Te ALLOYS

187

FIG.41. Hg,-,Cd,Te conduction band-edge g factor gc and its reciprocal calculated from Eq. (78).

observable seems to be an additional scattering (alloy scattering). No effects have been attributed to clustering of Hg- or Cd-rich regions, which is statistically probable and has been observed by Vanier et al. (1977), other than the occurrence of rather large linewidths such as we found in magnetoreflection (Figs. 28 and 30). Perhaps these large linewidthsmask any effects of clustering which might occur. An interesting extension of the work on Hg,-,Cd,Te has been made recently in the study of Hg, -,Mn,Te and Cd, - x Mn,Te alloy systems. These materials crystallize in the zinc-blende structure and their behavior is remarkably similar to Hg,-,Cd,Te except that in an applied magnetic field the large magnetic moments of the Mn++ ions interact with the conduction and valence electrons to produce anomalously large spin splittings (Bastard et al., 1977, 1978), hence very large g factors and Faraday rotation angles (Gaj et al., 1978) and interesting helicon propagation effects (Holm and Furdyna, 1977). The results of magnetooptical and transport measurements have been successfully analyzed using the quasi-germanium model extended to include the exchange interaction with the M n + + (Bastard et al., 1978;

188

M. H. WEILER

Kossut, 1976). As has been the case for the Hg, -,Cd,Te system, magnetooptical techniques have played a key role in developing successful models for understanding the behavior of these materials.

ACKNOWLEDGMENTS We are grateful to Professor Benjamin Lax and Dr. Roshan L.Aggarwal for their advice and support during the course of our studies of Hg, -.Cd,Te, which were carried out at the Francis Bitter National Magnet Laboratory, M.I.T. We would like to thank the following individuals for permission to reproduce portions of their work, and for sending corrections to the manuscript: Drs. G. Bastard, G. Bauer, D. Buss, M. Dobrowolska, Y.Guldner, P. Kruse, B. McCombe, S. Narita, C. Rigaux, and C. Vtrit. We would also like to thank Drs. M. Balkanski, H. Y. Fan, S. Tanaka, and W. Zawadski for sending preprints of their recent work.

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M. H. WEILER

h

McCombe, B. D. (1973). Presented at Narrow-Gap Semiconductor Conference, 2nd, Nice (unpublished). McCombe, B. D., Bishop, S. G., and Kaplan, R.(1967). Phys. Rev. Lett. 18,748. McCombe, B. D., Wagner, R. J., and Prim, G. A. (1970a). Phys. Rev. Lett. 25,87. McCombe, B. D., Wagner, R.J., and Prim, G. A. (197Ob). SolidState Commun. 8,1687. Mauger, A., Otmezguine, S., Verie, C., and Friedel, J. (1974). Proc. Int. Conf. Phys. Semicond., 12th, 1974 p. 1166. Mooradian, A., and Harman, T. C. (1971). J. Phys. Chem. SoliciS32, Suppl. 1,297. Moritani, A., Taniguchi, K., Hamaguchi, C., and Nanai, J. (1973). J. Phys. SOC.Jpn. 34,79. Mycielski, J., Bastard, G., and Rigaux, C. (1977). Phys. Rev. B 16, 1975. Narita, S., and Suizu, K. (1975). Prog. Theor. Phys., Suppl. 57, 187. Narita, S., Kim, R. S., Ohtsuki, O., and Ueda, R. (1971). Phys. Lett. A 35,203. Nimtz, G., and Tyssen, E.(1976). Phys. Status Solidi A 38, K171. Otmezguine, S., Raymond, F., Weill, G., and Vent, C. (1970). Proc. Int. Conf. Phys. Semicond., IOth, 1970, p. 536. See also Otmezguine, S. (1975). Ph.D. Thesis, University of Paris. Palik, E.D., and Wright, G. B. (1967). In “Semiconductorsand Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 3, p. 421. Academic Press, New York. Peterson, R. L. (1975). In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 10, p. 221. Academic Press, New York. Pidgeon, C. R., and Brown, R. N. (1%6). Phys. Rev. 146,575. Pidgeon, C. R.,and Groves, S. H. (1969). Phys. Rev.186,824. Piller, H. C. (1972). In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 8, p. 103. Academic Press, New York. Poehler, T. O., and Apel, J. R.(1970). Phys. Lett. A 32, 268. Putley, E.H. (1966). In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 1, p. 289. Academic Press, New York. Rashba, E.I., and Sheka, V. I. (1961). Fir. Tverd. Tela 3, 1863. Sou. Phys.-Solid State (Engl. Transl.)3, 1357 (1961). Raymond, F., and Vtrit, C. (1973). Surface Sci. 37, 515. See also Raymond, F. (1975) Ph.D. Thesis, University of Paris. Reine, M. B. (1970). Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge (unpublished). Roth, L.M., Lax, B., and Zwerdling, S. (1959). Phys. Rev. 114,90. Saleh, A. S., and Fan, H . Y. (1972). Phys. Status Solidi B 53, 163. Sattler, J. P., Weber, B. A., and Nemarich, J. (1974). Appl. Phys. Left.25,491. Schmit, J. L., and Stelzer, E. L. (1969). J. Appl. Phys. 40,4865. Scott, M. W. (1969). J. Appl. Phys. 40,4077. Segall, B., and Maple, D. (1967). In “Physics and Chemistry of 11-VI Compounds” (M. Aven and J. S. Prener, eds.), North-Holland Publ., Amsterdam. Sheka, V. I. (1964). Fiz. Tverd. Tela (Leningrad)6,3099; Sou. Phys.-Solid State (Engl. Transl.) 6,2470 (1965). Slusher, R. E., Patel, C. K. N., and Fleury, P. A. (1967). Phys. Rev. Lett. 18,77. Strauss, A. J., Harman, T. C., Mavroides, J. G., Dickey, D. H., and Dresselhaus, M. S. (1962). Proc. Int. Conf. Phys. Semicond. [6th], 1962 p. 703. Suzuki, K., and Hensel, J. C. (1974). Phys. Rev. B9,4184. Suizu, K., and Narita, S. (1973). Phys. Lett. A 43, 353. Swierkowski, L.,Zawadski, W., Guldner, Y.,and Rigaux, C. (1978). Solid State Commun. 27, 1245. Szuszkiewicz, W. (1977). Phys. Stafus Solidi E 81, K119. Tuchendler, J., Grynberg, M., Couder, Y., Thome, H., and Le Toullec, R. (1973). Phys. Rev. E 8, 3884.

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MAGNETOOPTICALPROPERTIES OF Hg, -$d,Te

ALLOYS

191

Uchida, S., and Tanaka, S. (1976). J. Phys. SOC.Jpn. 40, 118. Vanier, P., Pollak, F. H., and Raccah, P. M. (1977). Appl. Opt. 16,2858. Verie, C., and Decamps, E. (1965). Phys. Status Solidi 9,797. Vrehen, Q. H. F. (1968). J . Phys. Chem. Solids 29, 129. Walukiewicz, W., Stoelinga, J. H. M., Aggarwal, R. L., Lax, B., and Kruse, P. W. (1978). In “Physics of Narrow-Gap Semiconductors” (J. Rauluszkiewicz, M. Gorska, and E. Kaczmarek, eds.), p. 81. Elsevier, Amsterdam. Weber, B. A,, Sattler, J. P., and Nemarich, J. (1975). Appl. Phys. Lett. 27, 93. Weiler, M. H. (1977). Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge (unpublished). Weiler, M. H. (1979). J. Mugn. Magn. Muter. 11, 131. Weiler, M. H., Zawadzki, W., and Lax, B. (1967). Phys. Rev. 163, 733. Weiler, M. H., Aggarwal, R. L., and Lax, B. (1974). SolidState’Comrnun. 14,299. Weiler, M. H., Aggarwal, R. L., and Lax, B. (1977). Phys. Rev. B 16, 3603. Weiler, M. H., Aggarwal, R. L., and Lax, B. (1978). Phys. Rev. B 17, 3269. Weisbuch, C., and Hermann, C. (1977). Phys. Rev. B 15, 816. Wiley, J. D., and Dexter, R. N. (1969). Phys. Rev. 181, 1181. Wolff, P. A. (1966). Phys. Rev. Lett. 16,225. Yafet, Y. (1963). Solid Stare Phys. 14, I . Yafet, Y. (1966). Phys. Rev. 152, 858. Yafet, Y. (1973). In “New Developments in Semiconductors” (P. R. Wallace, R. Harris, and M. J. Zuckerman, eds.), p. 469. Noordhoff, Leyden. Zanio, K. (1978). In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 13. Academic Press, New York. Zawadzki, W., and Wlasak, J. (1976). J. Phys. C9, L663. Zeeman, P. (1879). Versl. Kon. Akad. Wet. Amsterdam 5, 181 and 242. Zeeman, P. Philos. Mag. 43, 226.

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SEMICONDUCTORS AND SEMIMETALS. VOL . 16

CHAPTER 4

Nonlinear Optical Effects in Hg. -.Cd. Te* Paul W . Kruse and John F. Ready I . INTRODUCTION . . . . . . . . . . . . . . . I1. PHENOMENOLOGICAL DESCRIPTION . . . . . . . . . . 1. Effect of a Magnetic Field upon the Conduction Band of a Semiconductor . . . . . . . . . . . . . . 2. Spontaneous andStimulated Raman Scattering by Free Electrons: Spin-Fl@Raman Laser . . . . . . . . . . . . 3. Resonant Four-Photon Mixing . . . . . . . . . . 4. Optical Phase Conjugation . . . . . . . . . . . I11. THEORETICAL DEVELOPMENT. . . . . . . . . . . 5 . Spin-Flip Raman Laser . . . . . . . . . . . . 6. Resonant Four-Photon Mixing . . . . . . . . . . 7. Optical Phase Conjugation . . . . . . . . . . . IV. RELEVANT PRO PERT^ OF Hg, - .Cd, Te . . . . . . . . 8. Dependence of Forbidden Energy Gap upon Composition and Purity . . . . . . . . . . . . . . . . 9. Dependence of Electron Effective Mass upon Composition . . 10. Dependence of g Value at Conduction-Band Edge upon Composition . . . . . . . . . . . . . . . 11. Optical Properties . . . . . . . . . . . . . V. Hg,-,Cd, Te SPIN-FLIP RAMAN LASERS . . . . . . . . 12. Experimental Arrangement . . . . . . . . . . . 13. Threshold . . . . . . . . . . . . . . . 14. Tuning . . . . . . . . . . . . . . . . 15. Eficiency . . . . . . . . . . . . . . . 16. Temporal and Spatial Characteristics . . . . . . . . OF MULTWHOTON MIXING IN Hg, -,Cd, Te . . . VI . OBSWVATIONS 17. Experimental Arrangement . . . . . . . . . . . 18. Properties of Four-Photon Mixing . . . . . . . . . 19. Properties of Six-Photon Mixing . . . . . . . . . VII . OPTICAL PHASE CONJUGATION IN Hg, -,Cd, Te . . . . . . 20. Experimental Arrangement . . . . . . . . . . . 21 . Experimental Results . . . . . . . . . . . .

194 196 196 199 203 205 207 207 211 213 213 213 214 215 217 218 219 220 220 223 226 230 230 231 238 242 242 243

* Research supported by the Air Force Office of Scientific Research (AFSC). United States Air Force. under contract F49620.77.C.028 . The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. 193

Copyright @ 1981 by Academic F'ress. Inc. All rights of reproduction in any form reserved. ISBN 0-12-752116-X

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PAUL W. =USE

AND JOHN F. READY

VIII. THIRD-ORDER RESONANT NONLINEAR SUSCEPTIBILITY. . . . 22. Estimate of Resonant Nonlinear Susceptibility from Raman Laser Threshold . . . . . . . . . . . . . . . 23. Estimate of Resonant Nonlinear Susceptibility from Eficiency of Four-Photon Mixing . . . . . . . . . . . . IX. APPLICATIONS . . . . . . . . . . . . . . . 24. Applications for Tunable Infrared Laser Sources . . . . . 25. Comparison to Indium Antimonide Spin-Flip Raman Lasers. . 26. Comparison of Spin-Flip Raman Lasers to Other Tunable Infrared Sources. . . . . . . . . . . . . . REFERENCES

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

246 246 246 241 241 249 250 25 1

I. Introduction The alloy semiconductor Hg, -xCd,Te has established itself as the most useful of all materials for infrared detection. Among the properties which give rise to that utility are the ability during growth to adjust its energy gap from a value of 0 eV (x = 0.17) to 1.4 eV (x = 1.0) and a direct energy gap over all values of composition (Long and Schmit, 1970). Those same properties make it extremely useful as a nonlinear optical material, in which the effects of interest depend upon the interaction of incident photons with the free electrons (or holes) in Hg, - ,CdxTe. The term “nonlinear optics” in general refers to those effects in which the polarization of a material depends in a nonlinear manner upon the electric field of an incident electromagneticwave. In conventional optics the polarization induced by an electric field is proportional to the field, the proportionality constant being the electric susceptibility,which in the general case is a tensor. In nonlinear optics, the polarization depends upon a power series in the electric field such as p = q,(x(’)E + x(’)E’

+ xt3)E3+ x ( ~ ) +E .. ~ .),

(1)

where P and E are the magnitude of the polarization and electric field vectors, respectively, and E~ is the permittivity of free space. Here for simplicity we neglect the tensor properties of the susceptibility. Then fl) is known as the first-order susceptibility, x(’) is the second-order, x(3) is the third-order, x(4) is the fourth-order, etc. Since the wave equation is expressed in terms of the polarization of the medium, an intense electromagnetic wave (e.g., from a laser) propagating through a nonlinear optical medium can give rise to diverse optical effects arising from the higher-order susceptibility terms. It can be shown (Bloembergen, 1965; Baldwin, 1969; Yariv, 1967; Harper and Wherret, 1977) that x(’) is responsible for second harmonic generation, sum and differencefrequency generation, parametric oscillation, optical rectification, and the linear electrooptic effect (Pockels effect). The third-order susceptibility (Bloembergen, 1965; Baldwin, 1969; Yariv, 1967; Harper and Wherret, 1977) x(3) gives rise to third harmonic generation, the quadratic

4.

NONLINEAR OPTICAL EFFECTS IN Hg,

-,Cd,Te

195

electrooptical effect (Kerr effect), stimulated Brillouin and Rayleigh scattering, stimulated Raman scattering, four-photon mixing, and optical phase conjugation from four-photon interactions. The three nonlinear optical effects which concern us, stimulated Raman scattering, four-photon mixing, and optical phase conjugation, arise from the third-order susceptibility f 3 ) . In general, there are three mechanisms? which contribute most to f 3 ) . First, for those materials whose crystalline structure lacks a center of inversion symmetry, a contribution arises due to the valence electrons. Hg, -.Cd,Te, having the zinc-blende structure (434, is such a material. Second, for those materials for which the energy gap is sufficiently small that the conduction band is nonparabolic, i.e., the Kane model (Kane, 1966) is obeyed, a contribution arises from the interaction of the free electrons with the incident photons. Hg, -,Cd,Te, for which 0.17 I x I 0.30, is such a material. Third, in the presence of a magnetic field which quantizes the free electrons into their Landau states (see Section 1) there is also a contribution arising from a resonant interaction between the free electrons and the incident photons. Of the above contributions to in Hg,-,Cd,Te for which 0.17 I x I 0.30, by far the largest arises from the resonant interaction between the free electrons and the incident photons when the sample is in a magnetic field. Thus our discussion of stimulated Raman scattering is concerned with spin-flip processes in which an electron in a lower spin state undergoes a transition to an upper spin state as a result of an interaction with a photon. So too, our discussion of four-photon mixing is concerned with resonant processes in which the scattering of incident photons by free electrons is greatly enhanced at a critical magnetic field. However, our discussion of optical phase conjugation is in terms of a nonresonant contribution arising from conduction-band nonparabolicity. Although a resonant contribution is expected if the sample is in a magnetic field, no experimental data are available as of early 1980. This review is concerned only with those three nonlinear optical effects which have been demonstrated experimentally in Hg, - ,Cd,Te, i.e., the spin-flip Raman laser (Sattler et al., 1974; Weber et al., 1975; Kruse, 1976; Norton and Kruse, 1977; Walukiewicz et al., 1977; Kruse et al., 1978), resonant four-photon mixing (Khan ef al., 1979; Bridges et al., 1979; D. J. Muehlner, K. Kash, M. A. Khan, P. A. WoltT, and R. A. Wood, unpublished data, 1979), and optical phase conjugation (Khan et al., 1980a).That is not to say, however, that experimental verification of other nonlinear effects will not soon appear. One which is of considerable interest is far-infrared generation,

t N.B. Jain and Steele (1980) have identified a fourth mechanism based upon a photoexcited electron-hole plasma.

1%

PAUL W. KRUSE AND JOHN F. READY

either by differencefrequency mixing or as an output of the spin-flip Raman laser. Experimental evidence should appear within the near future. The large body of literature on nonlinear optical effects in InSb is worth noting. As of 1979 it totaled nearly 100 publications, most of which are on the spin-flip Raman laser (Colles and Pidgeon, 1975; Smith, 1976). Other topics includefour-photon mixing (Pate1et al., 1966; Brueck and Mooradian, 1973; Nguyen et al., 1976) and far-infrared generation (Brignall et al., 1974; Bridges and Nguyen, 1973; Shaw, 1976). Because Hg, _,Cd,Te is “InSb-like” in a band-structuresense, soon after the initial experimentaldata on spin-flip Raman laser effects were obtained it became evident (Wolff, 1971) that similar results might be found in Hg, -,Cd,Te. Four reasons can be cited for studying nonlinear optical effects in Hg, -,Cd,Te. First, the effects reveal basic properties of Hg, -,Cd,Te, e.g., effective mass, which are valuable in understanding the material and in exploiting it in infrared detectors and focal plane arrays. Second, there is a well-established materials technology, so that Hg, - ,Cd,Te crystals of controlled composition and purity can be prepared. This capability allows matching of the bandgap to the photon energy of a pump laser so as to provide resonant enhancement of the nonlinear effects. Third, the nonlinear optical effects of interest occur very efficiently in small-gap Hg,-,Cd,Te, more so than in any other material. Fourth, these nonlinear optical effects will eventually prove useful in themselves, and Hg, -,Cd,Te devices and systems exploiting them will be marketed. In the following parts we will describe in general terms the nonlinear optical effects, identify the reasons why the effects occur so well in Hg, -,Cd,Te, build the theoretical foundations for the effects, report upon the experimentalconfirmationand details, and discuss potential applications.

II. Phenomenological Description 1. EFFECT OF A MAGNETIC FIELD UPON THE CONDUCTION BAND OF A

SEMICONDUCTOR

The nonlinear optical effects in Hg, -,Cd,Te of greatest interest require that the sample be within a magnetic field. Although they occur both in the conduction and valence bands, the conduction-band effects are much more pronounced and therefore are of primary interest. The discussions which follow are limited to the conduction band, but also apply to the valence band by the same reasoning. In the absence of a magnetic field the density of states in the conduction band exhibits a parabolic dependence upon energy for wide-gap ( 20.25 eV) semiconductors. For narrow-gap semiconductorsthe k .p interaction (Kane, 1966) causes a deviation from the parabolic dependence, the magnitude of

4. NONLINEAR OPTICALEFFECTSINHg, -,Cd,Te

197

FIG.1. Density of states dN/de as a function of normalized energy e/hmCRin the presence of a magnetic field. The zero case field dN,/ds is also shown. Here m,, is the cyclotron resonance frequency. [From Roth and Argyres (1966).]

which depends inversely upon the energy gap. In the presence of a magnetic field in either case the conduction-band states coalesce into the so-called Landau states (see Fig. 1). Each Landau level is characterized by an orbital quantum number L and a spin quantum number S. Every orbital level has associated with it two spin states: spin up (i.e., spin vector parallel to the magnetic field vector) and spin down (i.e., antiparallel). A representation of what is commonly referred to as the “Landau ladder” is illustrated in Fig. 2.

FIG.2. Schematic energy-leveldiagram illustrating the splitting of the energy levels into the Landau ladder in the presence of a magnetic field. Here Land S are, respectively,the orbital and spin quantum numbers, mCRis the cyclotron resonance frequency, and wsF is the spin-flip frequency.

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PAUL W. KRUSE AND JOHN F. READY

The energy of the states in the Landau ladder is given by (Roth and Argyres, 1966)

EL,s = E ,

+ (h2k,2/2m,) + (heH/m,c)(L + i)+ SgBH.

(2) The factor eH/mec is known as the cyclotron resonance frequency mCR.Here H is the magnetic field, assumed to be in the z direction, k, is the wave vector in the z direction, E , is the energy at the bottom of the conduction band at k, = 0 and H = 0, h is Planck’s constant h divided by 2n, e is the electronic charge, c is the speed of light, and fl is the Bohr magneton. The values which L and S can take on are

s = +’2, -La2 L = O , 1 , 2,..., Because the nonlinear optical effects are most pronounced in small-gap semiconductors in which the conduction band is nonparabolic, we shall confine our interest to small-gap Hg, -,Cd,Te. In general this means that the range of interest is 0.17 I x I 0.30. The parameter me of Eq. (2) is the electron effectivemass. For a small-gap semiconductor it is given by the Kane model (Kane, 1966) as

where m, is the mass of an electron, Eg is the forbidden energy gap, A is the valence-band spin-orbit splitting energy, which is about 1.0 eV in Hg, -,Cd,Te, and E , is a parameter introduced by Kane whose value is 18 eV in Hg, -,Cd,Te and several other semiconductors. The effectivemass value given by Eq. (3) is that at the bottom of the conduction band. At higher energies the effective mass increases due to conduction-band nonparabolicity. The remaining parameter of Eq. (2), g. is the spin-level splitting factor, commonly referred to as the “g value.” The value of g is of paramount interest in ascertaining the magnitude of the nonlinear optical effects of interest to us. Equation (2) shows that g characterizes the magnetic field dependence of the splitting of an orbital level into the spin-up and spindown states. Because of nonparabolicity, g decreases with increasing energy within the conduction band. The g value at the bottom of the band is given by (Zawadzki, 1963)

In the nonlinear optical effects of interest to us, the free-electron concentration is such that the Fermi level lies within the conduction band, near the bottom of the band. To a good approximation the observed effective mass

4. NONLINEAR

OPTICAL EFFECTS IN

Hg, -,Cd,Te

199

and g values are those at the bottom of the band. Thus, in this chapter we shall neglect any corrections to g and me arising from nonparabolicity. It can be seen from Eq. (4)that if the electron effective mass were m,, then g would have its classical value of 2. As the effective mass decreases, i.e., as the (direct) energy gap becomes small, g becomes large and negative. It is customary to refer to Ig I as g without referring to the negative value in that case. When me 4 m, and 2A b 3E,, conditions which are valid for small-gap Hg, -,Cd,Te, then N

(5)

mohe-

Because me is of the order of O.OO5m0 in Hg,.2Cdo.sTe, g is approximately 200. It is the extremely large g values (i.e., extremely small electron effective masses) in small-gap Hg, -,Cd,Te which make the nonlinear optical effects so large in it.

2. SPONTANEOUS AND STIMULATED RAMAN SCATTERING BY FREE ELECTRONS : SPIN-FLIPRAMAN LASER Consider now the interaction of an incident photon with the free electrons of a semiconductor sample located within a magnetic field. We shall assume that the sample is n-type with a free-electron concentration n, at the operating temperature such that the Fermi level lies within the conduction band near the bottom of it. In this case the sample is somewhat degenerate. An example isHg,.7,Cdo.,,Tewithne = 1 x 10,’ cm-3at 4 K.Inpracticetheoperating temperature is near 4 K, so the sample is mounted on the cold finger, or else immersed, within a liquid helium optical Dewar. The Dewar in turn is mounted between the pole faces of an electromagnet. The sample is prepared in the form of a resonant cavity such that the surfaces of incidence (front surface) and of exit (back surface) are extremely flat and parallel. In the absence of a magnetic field, the Fermi level lies at some value within the conduction band depending upon the free-electron concentration and density of states. As the magnetic field is increased from zero, the conduction band coalesces into the Landau ladder. As the field continues to increase, the separation between the orbital levels increases and the spin-level splitting within a given level also increases [see Eq. (211. At any field the electrons fill the Landau levels from the bottom as far up in energy as there are electrons at that energy to fill the available states. As the field continues to increase, a value will be reached such that all the electrons reside in the lower spin state (spin up) of the lowest orbital level (see Fig. 3). In Hgo.77Cd0.2,Te this is approximately 10 k G at 4 K. Consider now that a stream of monochromatic photons of energy hv, E, is incident upon the front surface of the cavity which is in a field and at a temperature such that all the electrons are in the lowest state. Because the

-=

PAUL W. KRUSE AND JOHN F. READY

-

I 2 3

*a a

a

t m

a 4, t

Q

a W

z

W

-

0 0

1

2

WAVE VECTOR

3

4

5

6

(PARALLEL T O FIELD)

(ARBITRARY U N I T S )

FIG.3. Schematic energy level diagram of a degenerate semiconductor in a magnetic field, illustratingthe relation of Fermi levels to levels with orbital quantumnumber L and spin quantum number S. The levels are denoted (L, S). (a) Low magnetic field; (b) high magnetic field.

photon energy is low, no free electron-hole pairs will be generated. However, scattering will occur in which the photons, interacting with the electrons, will lose energy to them. Some of the electrons will undergo a “spin-flip’’ transition involving a virtual intermediate state such that after the interaction they will be found in the upper spin state of the lowest orbital level (see Fig. 4) from which they relax back to the lower level. Such processes, in which the incident photons change energy as a result of inelastic collisions without being annihilated, are known as Raman scattering. Exiting the sample

4. NONLINEAR OPTICAL EFFECTS IN Hg, -,Cd,Te

201

FIG.4. Interaction of an incident photon with an electron in the lower spin state, causing a spin-flip transition.

therefore will be not only those photons which have not interacted, but also photons of energy hv, given by hv, = hv, - gBH,

(6) where gBH is the energy difference between the spin states of the lowest orbital level. Expressed in terms of wave numbers J, and f,,, Eq. (6) is

Js = 0, - gBH/hc.

(7)

The incident radiation is referred to as the “pump” radiation. If it is coherent, monochromatic, and sufficiently intense, and if the semiconductor sample is in the form of a resonant cavity, then the radiation emerging from the sample has all the properties of laser radiation, including coherence, monochromaticity, and a threshold. In this event we have a spin-flip Raman laser. The AL = 0, AS = + 1 transition, in which the scattered photon emerges with an energy decrease of gBH, is known as first Stokes (I Stokes). The first-Stokes photons can also scatter from the free electrons, losing energy such that exiting the samples are photons whose energy is 2gbH less than those incident. This is known as second-Stokes (11-Stokes) radiation. Higher-order processes (I11 Stokes, IV Stokes, etc.) are also possible, as well as anti-Stokes (AS) processes. In the latter case a photon scatters from an electron already in an upper spin state, causing the emission of a photon whose energy has increased by gBH. Higher-order anti-Stokes (I1 antiStokes, etc.) are also possible. It is apparent that the radiation associated with the higher-order Stokes and anti-Stokes processes is less and less intense as

202

PAUL W. KRUSE AND JOHN F. READY

the order increases. Figure 5 illustrates the spectrum of the radiation exiting the sample. The theoretical tuning curves are shown in Fig. 6. Spin-flip Raman laser action was first described theoretically by Yafet (1966), following an analysis by Wolff (1966) predicting a large cross section for Raman scattering from Landau levels. Most of the studies to date have been concerned with InSb (Colles and Pidgeon, 1975; Smith, 1976), where the pump laser is either a CO, (Pate1 and Shaw, 1971) or a CO one (Mooradian et al., 1970). In the former case the cross section is relatively low due to a lack of resonant interaction, whereas CO laser emission near 5 pm is in resonance with the 5-pm absorption edge of InSb at 4 K. It will be shown in Part I11 that the cross section for the scattering of photons from free electrons increases by orders of magnitude as the energy gap is decreased from a value much larger than the incident photon energy to one almost equal to it. Thus the second advantage of Hg,-,Cd,Te becomes apparent. Not only is the g value very high when x is low, but it is possible to adjust the x value during growth so that the energy gap matches that of the pump laser, i.e., the absorption edge lies at the laser wavelength. Because of the great interest in the utility of CO, lasers, the most interesting x value is 0.23. That is to say, the absorption edge of Hg,.77Cdo.2,Teat 4 K is near 10 pm.For Hg0.77Cd0.23Te at 4 K the g value is 80. All of the Hg,-,Cd,Te spin-flip Raman laser studies at the time of this writing have been concerned with x = 0.23 material.

IS

IIS Y-

P;)MP

Ins

V.

FIG.5. Schematic of first-Stokes, second-Stokes, and anti-Stokes emission from a spin-flip Raman laser. As magnetic field increases, the separations of the various lines from the position of the pump increase. (a) H = H 1 ;(b) H = H 2> H1.

4.

NONLINEAR OPTICAL EFFECTSIN

MAGNETIC FIELD

Hg, -,Cd,Te

203

-

FIG.6. Schematic tuning curve for a spin-flip Raman laser, with spin-level splitting factor g.

Thus it can be seen that a tunable infrared laser can be constructed in which a CO, laser is used to pump a resonant cavity of Hg,.,,Cd,.z3Te mounted in an optical Dewar at 4 K within an electromagnet. As the magnetic field is increased, first-Stokes radiation is emitted from the sample. It will be seen that this tunes at 3.8 cm-'/kG. Second-Stokes and anti-Stokes signals are also seen. A full description of the experimental observations can be found in Part V.

3.

RESONANT

FOUR-PHOTON

MIXING

The second nonlinear optical effect of interest is resonant four-photon mixing. The term applies to the interactions between a semiconductor and the electromagneticfield involving four photons, two which are incident and two which are emitted. As pointed out in Part I, four-photon mixing is a third-order process, i.e., involving x(~),which can arise from valence-electron interaction, conduction-band nonparabolicity, or effects arising from quantizing the electronic states in a magnetic field. We are concerned here with four-photon mixing arising from the third interaction listed above involving the free-electron spin states, the effect of which is much greater than either of the other two in small-gap Hg,-,Cd,Te. As in the spin-flip Raman laser, let there be a sample of an n-type semiconductor in an optical Dewar at, say, 4 K mounted between the pole faces of an electromagnet.It is not necessary, however, that the samplebe a resonant

204

PAUL W. KRUSE AND JOHN F. READY

cavity, since four-photon mixing does not involve the generation of laser radiation. Let there be incident upon the sample two collinear beams of laser radiation, perpendicular to the field, whose optical frequencies are w1 and 0,, where o1> 0,.From the sample will emerge not only 0,and 0,but also two new frequencies o3and o4given by 0 3

= 20,

-0

2

and o4 = 20,

- 0'.

If the magnetic field is absent, in small-gap semiconductors w3 and o4 arise from conduction-band nonparabolicity. The signals are detectable, but small. However, in the presence of a magnetic field whose magnitude is adjusted to meet the condition gBHlh

=0 1

-0 2 ,

(10)

a resonant interaction occurs involving the spin levels, the effect of which is very large. This is known as resonant four-photon mixing. Because resonant four-photon mixing involves scattering photons from free electrons, the forbidden energy gap of the sample at the operating temperature must exceed the energy of the incident photons to prevent the incident radiation from being absorbed by intrinsic processes. The effect is the largest when the energy gap E, is adjusted to match the incident photon energy. In practice, this means E , N ho,.For example, the absorption edge of Hgo.77Cd0.23Te at 4 K is at 10 pm. If o,is the R(20) line of a CO, laser at 975.94 cm- (10.247 pm) and o2is the P(20) line at 944.21 cm- (10.591 pm), then o1- 0,= 31.73 cm-'. Because g = 80 in Hgo.77Cdo.23Te,the resonant condition, Eq. (lo), is met for H = 8.50 kG. In this case,

'

o3= 1007.67 cm-' (9.923 pm),

o4 = 912.48 cm-' (10.959 pm).

Note that o4lies past the absorption edge and is detectable, but o3is selfabsorbed. We shall confine our attention to w4. It is useful and correct to envision resonant four-photon mixing in the following way (Nguyen et al., 1976). The magnetic field aligns the electron spins. The incident photons at w1 and 0,drive the spins so as to precess about the field at the difference frequency o1- 0,.This is especially pronounced when the resonance condition is met. The frequency u4 is generated when o2then interacts with the precessing spins to generate the difference frequency 0 4

=0

2

- (0'- 02) = 2 0 2 - U'.

(11)

4. NONLINEAR OPTICAL EFFECTS IN Hg, -,Cd,Te SPIN

- FLIP

RAMAN LASER PUMP AT WI

LASING EDGE

205

ACTION AT Wl-gBH/%,W,

wI + g#Hc/h

- 2g8H/h

II

POLARIZATION II II l . II WITH RESPECT TO H

RESONANT

FOUR-PHOTON

PUMP

MIXINO

WI AND W e , SET SO W I - y ’ 98H/h

AT

ABSORPTION EDGE

W3= 2 W,- W2 = WI+ 9 8 H/h

w

II II

POLARIZATION WITH RESPECT TO n

~

~ = ~w i 9w8 n ~~t i w

I II

~

9-w 2 II

-W

FIG.7. Comparison of spin-flip Raman laser and four-photon mixing signals. Polarization selection rules are also shown.

It is instructive to note that under the resonance condition o4lies at the first-Stokes frequency of o2(second Stokes of a,), whereas w3 is at the first anti-Stokes of o,(second anti-Stokes of wz). Thus the energy separation between w3 and w,, o1and 02, and w2 and oqis in each case gBH (see Fig. 7). Also shown in the figure are the polarization selection rules for the two effects and for comparison the relevant frequencies for the spin-flip Raman laser. Four-photon mixing has been studied in various media, the o3process sometimes being known as CARS (coherent anti-Stokes Raman scattering) and the w4 process as CSRS (coherent Stokes Raman scattering),pronounced “scissors.” However, our concern is with the resonant process in small-gap Hg, -,Cd,Te. The first studies of four-photon mixing due to conductionband nonparabolicity effects in small-gap semiconductors were reported by Pate1 et al. (1966). The first studies of resonant four-photon mixing in the small-gap semiconductor InSb were first reported by Brueck and Mooradian (1973). Recently, Nguyen et al. (1976) employed four-photon mixing in InSb to determine the transverse and longitudinal spin-relaxation times. Resonant four-photon mixing in Hg, -,Cd,Te will be discussed in Part VI.

4. OPTICALPHASECONJUGATION The third effect of interest is optical phase conjugation. The term “phase conjugation” refers to the generation of an outgoing electromagnetic wave which has a phase distribution in space that is the complex conjugate of an

206

PAUL W. KRUSE AND JOHN F. READY

incoming electromagnetic wave. The medium which performs this phase conjugation is known as a “phase conjugate mirror.” Thus a wave front of arbitrary spatial distribution incident upon a phase conjugate mirror will be reflected with the identical spatial distribution of the wave front, so that the two waves will identically overlap. Consider an information bearing beam, e.g., one generated by passing laser radiation through a photographic transparency of a scene, which passes through a scattering medium such as frosted glass, thereby scrambling the information. If reflected from a phase conjugate mirror, every ray will retrace its originalpath through the scattering medium, so upon emerging the beam will have the original image content restored. Methods for mechanically deforming the surface of a conventional mirror so as to generate a phase conjugate return are known as “adaptive optics.” This discussion centers upon methods for generating phase conjugate returns in nonlinear optical materials. Various methods have been cited (Yariv, 1978). That of interest here is four-wave mixing in Hg,-,Cd,Te. The experimental configuration is similar to that of four-photon mixing,? but with an important difference. The output from one laser is split into two beams which are directed at opposite sides of the sample; these are known as the counterpropagating pump beams. Radiation from a second laser is incident upon one side of the sample at a small angle with respect to one of the counterpropagatingpump beams. Interference between this pump beam and the signal beam establishes, by a third-order nonlinearity, a phase grating within the sample. The other pump beam, diffracted back up the signal beam, is a phase conjugate of the signal. Note that it is not necessary for the sample faces to be normal to any of the beams. Dispersion of the refractive index limits the sample thickness needed to maintain phase coherence if the two lasers operate at different frequencies. The most simple means to overcome this is to use a single laser for all three beams (signal, counterpropagating pumps). This is known as degenerate four-wave mixing. The experimental data presented in Part VII on optical phase conjugation in Hg, - ,Cd, Te have been obtained by degenerate fourwave mixing (Khan et al., 1980a). On the other hand, if two lasers are employed, if the sample is at about 4 K in a magnetic field, and if the field is adjusted to the same resonant condition as in resonant four-photon mixing, i.e., gBWh = 0 1 - 0 2 , (10) then a much larger phase conjugate return signal should be seen. This is known as optical phase conjugation by resonant four-wave mixing. As of

t Although convention dictates the expressions “four-photon mixing” and “four-wave mixing’’ as used herein, in reality they are interchangeable terms.

4.

NONLINEAR OPTICAL EFFECTSIN

Hg, -,Cd,Te

207

early 1980, a resonant four-wave mixing experiment has not been reported in Hg,-,Cd,Te.

III. Theoretical Development In order to establish the foundation for interpreting the data concerning spin-flip Raman laser action and four-photon mixing to be presented in Parts V and VI, here we shall present the theoretical basis for these effects. As stated in Part I, they arise from the third-order nonlinear susceptibility f 3 ) . Although in the general case the susceptibility contains both real and imaginary parts, in a practical sense for Hg,-,Cd,Te only the imaginary part is of consequence. Therefore x‘3’

= x (3)‘ -

$3’’’

N

-p”

(12)

and the third-order nonlinear polarization is given by

P = - ~ E f3’” O IE, I’E,,

(13)

where E, is the pump electric field vector and E, is the scattered electric field vector. In the case of stimulated Raman scattering, where there is only one pump laser, E, represents it. For four-photon mixing, JE,J2 is replaced by IEp1l2,and E, by EP2,where E,, and E,, are the electric fields associated with the two pumps. 5. SPIN-FLIP RAMANLASER

Consider now the spin-flip Raman laser. Our analysis follows that of Dennis (Dennis et al., 1972). The wave equation including the source term arising from the nonlinear polarization is given by

V

X

V

X

E d2E, E,+-’-= c2 dt2

471d’P c2 dt2’

where E, is the Stokes dielectrictensor and c is the speed of light. For simplicity we shall consider only the one-dimensionalcase. Then the solution to Eq. (14) is

where n, is the refractive index and cr, is the optical absorption coefficient, both measured at angular frequency 0,. In addition, E,(x) is the magnitude of the Stokes electric field at x and E,(O) is the value at x = 0. For simplicity we have dropped the double-prime notation on x(~).We define the Raman laser gain per unit length G as G = (471I

x‘~’Io&, C) 1 E , 12,

(16)

208

PAUL W. KRUSE AND JOHN F. READY

where lx(3)[represents the absolute value of x(~).Inspection of Eq.(15) shows that if the gain G exceeds the absorption loss a,, then the wave will grow. By forming the Hg, -,Cd,Te sample as a resonant cavity in the x direction, i.e., with the faces perpendicular to x made flat and parallel, oscillations will build up and laser action will occur. As with all lasers, the threshold for laser action is given by p,exp(G - a,)/ 2 1,

(17)

where 1 is the cavity length and p, is the reflectivity of the end faces at angular i.e., frequency us, Ps = C(n, - 1)/(ns

+ IN2.

(18)

In order to make practical use of Eq. (16), it is necessary to relate the Raman gain G to the Raman scattering cross section and the spin-level splitting factor g. The derivation is beyond the scope of this review; see Dennis (Dennis et al., 1972) for details. It can be shown that the peak intensity gain per unit length G , is given by

Here I, is the intensity of the pump radiation, i.e., I, = !‘,/Ap

=

n,~lE,1~/8n,

np is the refractive index at the pump frequency, P , is the pump power, A, is the cross-sectionalarea of the pump beam on the sample, r is the linewidth (full width at half maximum) expressed in radians/second of the spontaneous scattered radiation, (ii 1) is a Boltzmann factor relating to the modes of radiation in the cavity, and (do/dR) is the scattering efficiency given by

+

Here 1 $) I is the peak value of I I. But the scattering efficiency can also be expressed in terms of a scattering cross section 0 as (Pate1 and Shaw, 1971) do/dQ

=

on,f(E,, H).

(22)

Here n, is the number of free carriers (electrons in n-type Hg, -,Cd,Te) per unit volume andf(E,, H) is a factor ranging from zero to one which is expressed in terms of the Fermi level E, and magnetic field H. We shall have more to say about it later.

4.

NONLINEAR OPTICAL EFFECTSIN Hg,

-,Cd,Te

209

Yafet (1966) derived the relationship between the spin-flip scattering cross section (r and the spin-level splitting factor 9. The exact expression can be approximated with little loss in accuracy by

(23) Here e is the electronic charge, m, is the free-electron spin mass, oois the pump angular frequency, o,is the spin-flip angular frequency, and E , is the forbidden energy gap. The spin mass is defined in terms of the g value as (Wolff et al., 1976) A m,

[see Eq. (4)]. Consider now the factors in Eq. (23). The first is the classical (Thomson) scattering cross section of a photon by a free electron, the magnitude of which is 8.9 x cm2.The second factor accounts for the free-electron effective mass in semiconductors being less than that of the normal mass of an electron. We have already noted that the small effective electron mass in small-gap Hg,-,Cd,Te gives rise to a large g value. Thus the scattering cross section in small-gap Hg,-,Cd,Te is much larger than in semiconductors with large energy gaps or with indirect gaps. The third factor in Eq. (23) describes resonance enhancement. The pump energy h o p must exceed that of the energy gap in order that the energy not be absorbed by formation of free electron-hole pairs. However, as the pump energy is reduced toward that of the gap, the denominator becomes small and the cross section increases rapidly. Thus it is desirable to pump a spin-flip Raman laser with photons whose energy is very close to, but just less than, that of the gap. The fourth and fifth factors in Eq. (23) are both slightly less than unity. They have little effecton the value of the scattering cross section. Consider now the factorf(E,, H) in Eq. (22), the value of which ranges from zero to unity. It describes the approach to the quantum limit. Spin-flip Raman laser action occurs in degenerate semiconductors, that is, those in which the Fermi level lies in the conduction band. As the magnetic field is increased above zero, the electron energies coalesce into the Landau ladder, which fills from the bottom up with the available electrons. As the field increases, the Fermi level intersects lower and lower levels (see Fig. 3). If there remain electrons in the upper level of a spin-state pair within which spin-fliptransitions are occurring,the efficiency will be reduced. The quantum limit occurs at the magnetic field for which all the carriers lie in the lowest spin level, i.e., the Fermi level lies between the upper and lower spin states.

210

PAUL W. KRUSE AM) JOHN F. READY

In that casef(E,, H)is unity. To a reasonable approximation this condition is met when

Thus when values of x(3) are to be determined experimentally the magnetic field is adjusted to a value such thatf(Er, H)is unity. The remaining factor that deserves mention is the Boltzmann factor found in Eq. (19). This is a small correction, frequently neglected, whose value is given by ii+l=

1 exp@BH/kT) - 1

+

’*

In practice, gj?H/kT % 1, so that the Boltzmann factor is taken as unity. It is sometimes desirable to express the third-order susceptibility in terms of the cross section, freeelectron concentration, and linewidth. Combining Eqs. (21) and (22) we find

I $) I

= 2c4an, f(Er ,H)nJho:rn,,

(27)

which reduces to under the usual conditions in whichf(E,, H) N 1 and n, 1: n,. Therefore, combining Eqs. (19), (21), and (28) under these assumptions we find

G, =

16n2Z,w, IxL3’I - 32n2Z,c20n, n2c2 n2ho,Jr *

(29)

Thus the above equations trace the interrelationships between the semiconductor parameters me, n,, E8, g , r, n,,, n,, and a, the sample length I, the laser pump parameters o,,up,and I,, the magnetic field H,the scattering parameters G,, do/dLl, and D, and the third-order nonlinear susceptibility A large nonlinear susceptibility x(3) gives rise to a large Raman gain G [Eq. (1611 and therefore a large peak gain G, [Eq.(2911. In terms of material parameters, large values of G, arise from small spontaneous linewidths r and large scattering efficiencies do/dCI [Eq. (19)]. Large scattering efficiencies arise from large values of cross section 0, large values of free electron concentration n,, and sufficient field H to achieve the quantum limit [Eq. (2211. Large values of cross section D arise from small effective mass me, i.e., large g values, and resonant enhancement [Eq. (2311. Thus small-gap Hg,-,Cd,Te, with its small effective mass, has a large g value, a large scattering cross section, and a large third-order nonlinear susceptibility.

4. NONLINEAR OPTICAL EFFECTSIN Hg, -,Cd,Te

211

Of course, a large g value also gives rise to a large change in spin-flip output frequency per unit change in field [see Eq. (6)]. Although Eq. (22) suggests that heavy doping is desirable, this is not necessarily true for two reasons. First, for laser action it is necessary that the Raman gain G exceed the absorption losses as [Eq. (1711. Heavy doping can cause free-carrier absorption to be the dominant absorption mechanism. The value of the free-carrier absorption coefficient aFC,given by the classical Drude-Zener theory, is

aFC= e3n,/o~cns~om,2pL,,

(30)

where p, is the electron mobility. Thus free-carrier absorption will increase with increased n,. Second, the spontaneous Raman linewidth depends in a complicated manner upon n,. As n, is increased, the linewidth also increases, which reduces the scattering cross section. The optimum free-electron concentration for Hg,-,Cd,Te spin-flip Raman lasers is in the 1015range.

6. RESONANTFOUR-PHOTON MIXING

As pointed out in Part I, in the general case four-photon mixing can arise from nonlinearities due to bound electrons, due to free electrons in the absence of a magnetic field (conduction-band nonparabolicity), and due to free electrons in the presence of a magnetic field under the resonance condition in which the difference in energy of the two pump photons, Am, - rim2, equals the spin-level splitting energy, gBH. Resonant fourphoton mixing due to spin nonlinearities is much stronger than the others in small-gap Hg, - ,Cd,Te. The derivation of the appropriate equations describing resonant fourphoton mixing also begins with the wave equation, Eq. (14). However, in this case the electron system is pumped not by a single laser but by two, at frequencies w1 and 02,where 0, > o2by convention. Scattered photons emerge from the sample at frequencies w 3 and w4, given by o3= 2 0 , - 0 2 , 0 4 =

20, - 0 1 .

It turns out (Wynne and Boyd, 1968; Yablonovitch et al., 1972; Wynne, 1969) that the peak gain per unit length at w4 under nonresonant conditions is a slight modification of Eq. (29), i.e.,

212

PAUL W.KRUSE AND JOHN F. READY

where the approximation has been made that the refractive index does not change with frequency. Here I 2 is the intensity of the laser emitted at 0 2 , i.e., I , = P2/A,where A is the beam area. In contrast to the spin-flip Raman laser, four-photon mixing does not give rise to stimulated emission. Therefore, it is not the threshold for laser action, where the gain just exceeds the loss, that is of interest. Instead, we are interested in the power output at 0,. This can be determined from the gain expression, Eq.(31). Note that this refers to the gain of the electric field and must be squared to represent a power gain. Since the power gain is the ratio of output power at 0,to input we find power at o,,

P4 = (G,,J2P1L2 = 256n4~2P$P, I(x‘”)

12C/n4c4A2.

(32)

Here P , and P2 are the power inputs at o1and 0 2 ,P4 is the power output at 04,A is the cross-sectional area of each of the pump and output beams, assumed to be equal, and L is an effective length in the direction of the beam over which the frequencies are mixed, taking into account coherence and absorption effects. L is given by

where 1 is the length of the sample in the direction of the incident radiation, i=&-l,and

Ak

=

2k2 - kl - k4,

Aa

= a2

(344

+ (a,/2) -

(34W

where the ki and ai values are the magnitudes of the wave vectors and absorption coefficients at the appropriate frequencies. The expression for x(3) for four-photon mixing is similar to that for the spin-flip Raman laser, namely

1x(3)1

= 2c4on,/hw,w3

r,

(35)

which becomes Eq. (28) when o1= o2= 0,. Note that this applies to the resonant condition, where x(3) is a maximum. In examining Eq.(32) we note that the output power at w4 depends upon the square of the input power at o2 and linearly upon the input power at 0,. This expression is therefore valid only below a threshold, above which saturation occurs. By measuring P,, P , , and P4 it is possible to obtain a direct measurement of x ‘ ~ )for thin samples whose length is less than the coherence length under conditions in which self-absorption can be neglected.

4.

NONLINEAR OPTICAL EFFECTS IN

Hg, -,Cd,Te

213

7. OPTICALPHASECONJUGATION As pointed out in Part 11, optical phase conjugation by degenerate fourwave mixing bears a close resemblance to four-photon mixing. However, only one laser is employed and the sample is not in a magnetic field. The phase conjugate return signal is a small modification to Eq. (32): P4/P, = 256n4w2P2P2Ix(3)12L2/n4c4A2.

(324

Here P4 is the power in the phase conjugate return, PI is the power in the incident signal beam, and P, and P,are the powers in the counterpropagating beams. The effective overlap length L within the sample in which the interaction occurs is given by Eq. (33), taking into account that Ak = 0 and Aa = a, i.e., L~ = [(I - e-0')/a]2e-a'. (334 Here 1 is the length of the sample in the direction of the incident radiation and a is the optical absorption coefficient at the laser operating frequency. The principal contribution to the third-order nonlinear susceptibility x(3) in small-gap Hg,-,Cd,Te when the sample is not in the magnetic field resonant condition has been found to be conduction-band nonparabolicity. Wolff and Pearson (1966) have shown x(3) due to nonparabolicity to be given by

Thus a large value of x(3) is expected for semiconductors with small effective mass which are pumped by lasers whose wavelength lies at the absorption edge of the semiconductors. This is the same material requirement for the spin-flip Raman laser and resonant four-photon mixing.

IV. Relevant Properties of Hg, - CdxTe The previous parts have determined the dependence of the spin-flip Raman laser and resonant four-photon mixing effects upon selected parameters describing the properties of Hg,-,Cd,Te. In this section we shall present the values of those parameters. 8. DEPENDENCE OF FORBIDDEN ENERGY GAPUPON COMPOSITION AND PURITY

Schmit and Stelzer (1969) have shown that the forbidden energy gap E, of Hg,-,Cd,Te can be expressed in terms of composition x and temperature T in the following way: E, = 1 . 5 9 ~- 0.25

+ 5.2333 x

T(l - 2.08~)+ 0 . 3 2 7 ~ ~ .(36)

214

PAUL W. KRUSE AND JOHN F. READY 0.50 0.45

0.4 0

0.3 5 0.30 0.25 0

W

0.20 0.15

0.10 0.05

0 0

50

loo

150

200

250

300

f IK)

FIG.8. Temperature dependence of the energy gap E, and the corresponding wavelength A, of Hg, -,Cd,Te as a function of temperature T and composition variable x. [From Schmit and Stelzer (1969).]

This function is plotted in Fig. 8. From this it can be seen that for those applications in which a C 0 2 laser emitting in the 9.6-10.6 pm region is the pump, and in which the operating temperature is 4 K or thereabouts, the composition for which resonant enhancement occurs is x 1: 0.23. Similarly for resonant enhancement with a CO laser operating between 5 and 5.5 pm, with the sample temperature at 4 K,the appropriate composition is x N 0.30. These statements apply both for spin-flip Raman laser operation and resonant four-photon mixing. In the latter case, both pump lasers operate near the absorption edge.

9. DEPENDENCE OF ELECTRON EFFECTIVE MASSUPON COMP~~SITION The second parameter of interest, the free-electron effective mass, also depends upon composition. As the x value is increased from 0.17, ie., the composition at which the conduction and valence bands meet, the energy

4. NONLINEAR OPTICAL EFFECTS IN Hg, -,Cd,Te

215

gap increases from 0 eV. As the gap increases,the conduction-band curvature, that is, the dependence of energy upon momentum, decreases. Since the electron effective mass depends inversely upon the second derivative of energy with respect to momentum, it increases as the energy gap increases. For small direct-gap semiconductors, including Hg, -,Cd,Te for which 0.17 5 x 5 0.30, the conduction-band energy deviates from the simple parabolic dependence upon momentum, following instead the Kane (1966) model. Thus the effective mass for a given x value is no longer independent of energy within the conduction band but increases with increasing energy above the bottom of the band. However, it is a sufficiently good approximation for the purities of interest here to employ the effective mass at the bottom of the conduction band. Thus we shall calculate that value and employ it in the g-value determination to follow. The effective mass me at the bottom of the conduction band is given by Eq-(3)9 m0_ _

(3)

me

where m,, is the mass of an electron, E, is the forbidden energy gap, A is the spin-orbit splitting energy, which is about 1.0 eV in Hg, -,Cd,Te, and E, is a parameter introduced by Kane whose value is 18 eV in Hg, -,Cd,Te and several other semiconductors. Introducing the values for A and E , in Eq. (3) and simplifying, we find

me/mO1: 0.078Eg,

(37)

where E, is expressed in eV. This approximation is valid in Hg,-,Cd,Te over the composition range from x = 0.17 to x = 1.0, i.e., the range in which Hg,-,Cd,Te is a semiconductor, and over the temperature range from 0 to about 100 K. Figure 9 illustrates (Long, 1968) the theoretical dependence of me/mo upon x based upon the above expression for the range 0.17 I x I 1.00. Also shown for completeness is the dependence for 0 I x 50.17, although it is not applicable to the nonlinear optical effects of interest to us. Some experimentally determined values are also shown. 10. DEPENDENCE OF g VALUE AT CONDUCTION-BAND EDGEUPON

COMPOSITION From the dependence of effective mass upon x, the dependence of g upon x can be calculated from Eq. (4).The value so determined will be that at the bottom of the conduction band. Substituting Eqs. (36) and (37) and A = 1.0 eV into (4), we find the dependence of g upon x over the range from 0.17 I x I 0.30 to be that illustrated in Fig. 10 (Long, 1972).

216

PAUL W. KRUSE AND JOHN F. READY

r Oo30A

me

0 0 15

m. 0.0 I 0 0.005

r

0 0

0 0 5 0.10 0.15 0.20 0.25 0.30 0 3 5 0.40 X

FIG.9. Dependence of conduction band-edge effective mass me upon composition variable (E, = 18 eV). Experimental data points are. shown. [From Long (1968). Reprinted by permission of John Wiley & Sons, Inc.]

x at 0 K in Hg, -,Cd,Te

0

ANTGLIFFE

8\/

K-

/

SATTLER ET AL.

50 40

r , , ,

30 2 0 0.15

0.20

0.25

0.30

X

FIG. 10. Dependence of .(I value at conduction band edge upon composition variable x at 0 Kin Hg,-,Cd,Te (E, = 18 eV). [After Long (1972); Experimental data from KNee (1976), Sattler et al. (1974), and Antclifk (1970).]

4.

NONLINEAR OPTICAL EFFECTSIN Hg, -,Cd,Te

217

The most useful Hg,-,Cd,Te composition, that which is resonant with a COz laser, is x = 0.23. Assuming T = 4 K and A = 1.0 eV, we find the following values from these equations for Hg0.77 Cd0.Z3Te: Eg = 0.1208 eV,

1g I =

rn,/mo = 0.00946,

86.6.

Note that g is negative. It is the absolute value which is of interest. In practice, it will be seen that a better fit to the data is g = 80 at x = 0.23 and T = 4 K. This is shown in Part VI.

11. OPTICAL PROPERTIES The reflectivity and absorptivity of Hg0.7, Cd0,z3Te have been measured as a function of wavelength near the band edge. The measurements were performed using a Fourier transform spectrometer.These measurements are important because they define the absorption of the material for the various wavelengths involved in both spin-flip laser action and in four-photon mixing. The material must be reasonably transparent at both the wavelength of the pump laser and the wavelength of the output. At the same time resonance enhancement makes it desirable to work as close as possible to the band edge. Thus detailed knowledge of the absorption coefficient is needed in order to balance these requirements. Figure 11 shows the optical absorption coefficient for Hg0.77Cd,.z3Te as a function of wave number near the band edge. The positions of some important COz laser pump lines are indicated for comparison. Measurements of the reflectivity p are needed to determine the dispersion of the index of refraction n, according to the relation p = [(n - 1)’

+ k2]/[(n +

+ kz],

(38)

with k the extinction coefficient. Over the wavelength region of interest, at wavelengths slightly longer than the band edge, the refractive index decreases linearly with decreasing wave number, with a value of dnldL

N

2.5 x

cm-’.

(39)

Knowledge of the dispersion is needed to evaluate the distance over which the different beams interact coherently. When two beams of slightly different wavelength interact in a material, dispersion causes the waves to get out of step after they traverse some path length in the material. The coherence length, the distance in which the beams can be regarded as interacting coherently, is given by the factor L defined in Eq. (33). In Part VIII we will use the experimental results to derive a value for f 3 ) , using the absorption coefficient and dispersion data appropriate to Hg0.77Cd0.Z3Te*

218

PAUL W. KRUSE AND JOHN F. READY 45

40

35

-5

7

30

c

2

w0

25

5W 0 0

L

20

0

ta

I5

m

a

10

5

0

WAVE

NUMBER (cm-')

FIG.11. Optical absorption coefficientof Hg,,,,Cd,,,,Te as a function of wave number and temperature. The dashed curve is interpolated. [From Kruse (1976).]

V. Hgl -,Cd,Te Spin-Flip Raman Lasers In this part we describe some of the typical properties of spin-flip Raman lasers based on Hg, -xCd,Te. Such lasers have been operated at a number of laboratories (Sattler et d., 1974; Weber et d., 1975; Kruse, 1976; Norton and Kruse, 1977;Walukiewicz et d.,1977; Kruse et al., 1978; D. J. Muehlner, unpublished data, 1978).

4.

NONLINEAR OPTICAL EFFECTS IN

Hg, -,Cd,Te

219

12. EXPERIMENTAL ARRANGEMENT

A typical experimental configuration is shown in Fig. 12. Details may differ but the essential features can be understood with reference to the figure. The COz laser employs a rotating mirror to Q switch repetitively. Other experimental configurations have used pulsed COz TEA lasers (Sattler et al., 1974; Walukiewicz et al., 1977) and chopped continuous CO, lasers (Weber et al., 1975; Muehlner, unpublished data, 1978). The pump laser is line tuned with a diffraction grating which serves as the high reflectivity mirror for it. The polarization of the pump laser beam is parallel to the applied magnetic field, the orientation which produces maximum efficiency of the spin-flip interaction. The pump laser beam is focused on the sample of Hg,.,, Cdo.23Te which is contained within a liquid helium optical Dewar. The sample is positioned between the pole faces of an electromagnet. At liquid helium temperature, Hg,.,, Cd0.23Te is not highly transparent at a pump laser wavelength of 9.6 pm (see Fig. 11). It is reasonably transparent to 10.28-pm radiation, and more transparent at 10.6 pm.Resonance enhancement will be larger at 10.28 than at 10.6 pm; thus the optimum wavelength for the pump laser is 10.28 pm. The sample is in the form of a right circular cylinder with diameter 12.7 mm and length between 0.75 and 5 mm. The end faces are polished flat and parallel to 40 arcsec. The uncoated faces of the sample, which have about 30% reflectivity near 10 pm, form the end mirrors for the spin-flip laser resonant cavity. In the collinear configuration shown in Fig. 12, the transmitted pump laser radiation and the spin-flip signal emerge in the same direction and must be separated by the frequency selecting element, usually a monochromator. The polarization of the spin-flip laser radiation is perpendicular to the polarization of the pump laser. Thus a wire grid polarizer is useful for separating the spin-flip signal from the transmitted pump radiation. An

~I//~~;CAL (HG,CD) T E

G

R

I

t GO2 LASER

tq +

n

m

S

-

SAMPLE

DEWAR ( HG ,CDl TE DETECTOR

PLlFlER

x -Y RECORDER

FIG.12. Diagram of experimental apparatus for spin-flip Raman laser. o,= o1- g/?H/h. G, grating; R , rotating mirror; L, lens; E , electromagnet;P , electromagnet power supply; M, monochromator;S, magnetic field sensor.

220

PAUL W. KRUSE AND JOHN F. READY

infrared detector sensitive in the 9-12 pm wavelength interval is employed to monitor the intensity of the radiation passing through the monochromator. Detectors that have been employed include Hgo.8Cdo.2Teones operating at 77 K, with a long wavelength limit near 14 pm. Cudoped Ge detectors operating at 4 K have also been used. The output of the detector is amplified, and fed into a boxcar integrator, which averages over a number of laser pulses. The integrated signal is fed to the y axis of an x-y recorder, whereas the output of a magnetic field probe is fed to the x axis. In this manner a plot of the signal at a fixed wavelength is obtained as a function of magnetic field, and the field at which the signal at the wavelength peaks is determined. By repeating the experiment for other monochromator settings,and plotting the peak signals as a function of magnetic field, data of the type illustrated in Fig. 6 are obtained (see also Fig. 14). 13. THRESHOLD The Hg,.,, Cd0.,,Te spin-flip Raman laser operation is characterized by a sharp threshold in pump laser power. The value of the threshold is below lo00 W / m 2 for typical conditions, and values as low as 400 W / m 2 have been reported (Weber et d., 1975). Figure 13 shows the spin-flip laser output as a function of peak power density for a 10.28-pm pump. The threshold near 1000 W/cm2 is clearly defined. Above threshold the output increases rapidly with increasing power density, up to 3000 W/cm2. Above 3000 W/cm2 the output saturates, probably because of sample heating. 14. TUNING The frequency of the detected spin-flip laser radiation as a function of applied magnetic field up to 13.5 kG is shown in Fig. 14. This shows the output for three different pump lines: 9.6, 10.28, and 10.6 pm.The largest signals are obtained with the 10.28-pm pump. It is possible to observe several different signals, including first-Stokes, second-Stokes, and antiStokes radiation. The first-Stokes signal is the primary spin-flip radiation at frequency w, = coo - gBH/h. When the intensity of the Stokes radiation builds up, it can serve as the pump for a second spin-fliptransition, generating second Stokes radiation at frequency w,, - 2gBH. The second-Stokes radiation tunes with a slope twice as great as the first-Stokes radiation. When the laser is operating, electrons are present in the upper spin state of a given Landau level. These may interact with pump laser photons in a process which drives the electrons down to the lower level, producing anti-Stokes radiation at frequency coo + gBH. The anti-Stokes radiation tunes in the opposite direction to first Stokes with frequency increasing with increasing magnetic field. All three of these lines are observed with the 10.28-pm pump, which produces the largest conversion efficiency. With the 10.6-pm pump,

4.

NONLINEAROPTICAL EFFECTS IN

I

0

Hg, -,Cd,Te

lo00 2000 3000 4000 5 0 0 0 PEAK PUMP POWER DENSITY (W/cm2

221

6000

FIG.13. First-Stokes signal at 10.67 pm as a function of 10.28-ympump peak power density for a magnetic field of 9.3 kG.[From Kruse (1976).]

because of the smaller resonance enhancement, only the first-Stokes line is observed. With the 9.6-pm pump, the first- and second-Stokes lines are observed, but because the anti-Stokes frequency lies within the absorption band of the material, it is not observed. As Fig. 14 indicates, with a magnetic field from 0 to 13.5 kG, continuous tuning from 9.6 to 11.2 pm is possible by making suitable choices of pump laser lines and processes. We emphasize that the first-Stokes line is much more intense than either the second-Stokes line or the anti-Stokes line for a given pump laser line. The tuning for Hgo.,,Cd,,,3Te is found to be 3.8 cm-'/kG, which corresponds to g = 80. Observations extending up to 28 kG (Walukiewicz et al., 1977) indicate that g decreases with increasing magnetic field, reaching a value around 65 at 28 kG.

222

PAUL W. KRUSE AND JOHN F. READY

MAGNETIC FIELD (kG)

FIG.14. Tuning curves for a Hgo.,,Cdo.23Tespin-llip Raman laser, showing first-Stokes, second-Stokes, and anti-Stokes lines for pump wavelengths of 9.6, 10.28, and 10.6 pm. -, 9.6-pm pump; ----, 10.28-pm pump; --.-, 10.6-pm pump; S, Stokes line; 2S, 2ndStokes line; AS, Anti-Stokes line.

4.

NONLINEAR OPTICAL EFFECTS IN Hg,

-,Cd,Te

223

Fine tuning characteristics have been investigated by inserting a FabryPerot interferometer between the spin-fip laser and the monochromator. The coarse tuning characteristics illustrated in Fig. 14 can be influenced by the presence of the discrete modes in the laser cavity. In principle, the cavity modes can cause pulling of the laser frequency toward that of a cavity mode. As the magnetic field is varied, the laser frequency can shift discontinuously between adjacent cavity modes. This mode hopping would lead to a fine structure in the tuning curves, with breaks in the curve at positions of the mode hops. Such behavior was in fact not observed in the Fabry-Perot investigations of a sample with carrier concentration of 1 x 10'' cm-3 (Ready and Kruse, unpublished data, 1979). No mode hopping was found, even when the pump laser power was reduced to near threshold. The spin-flip laser instead tuned continuously. The low finesse of the uncoated laser cavity reduced the effect of the cavity modes on the tuning characteristics. Continuous tuning without mode hopping has been also observed in InSb spin-flip Raman lasers under some conditions (Pidgeon and Smith, 1977; Patel, 1973), particularly when the reflectivity of the ends of the cavity is low. The linewidth of the Hg,.77 Cd,.,,Te spin-flip Raman laser has also been measured using a Fabry-Perot interferometer (J. F. Ready and P. W. Kruse, unpublished data, 1979). The linewidth was found to be an increasing function of magnetic field (see Fig. 15). The extremely narrow linewidths of the continuous InSb spin-flip Raman laser (Patel, 1972) were not achieved.

15. EFFICIENCY High conversion efficiencies have been observed, up to approximately 9 % (Norton and Kruse, 1977). Figure 16 shows the peak external conversion efficiency when the pump wavelength is 10.26 pm. The maximum conversion efficiency occurred near 7 kG. At lower fields the signal decreased to near zero around 4 kG and then increased again, reaching 6 % near 2.5 kG.The behavior is due to different levels in the Landau ladder crossing the Fermi level as the magnetic field is changed. The maximum output occurs when the Fermi level lies between the upper and lower spin states of the lowest Landau level. For a field near 7 kG, this situation applies, and the output is large. If the field decreases to 4 kG, the upper spin state of the L = 0 level falls below the Fermi level, and there is no emission involving the L = 0 state. However, as the field decreases further, higher lying Landau levels cross the Fermi level and contribute to the stimulated emission. This accounts for the increase in output as the field is reduced below 4 kG. At still lower fields, there should be a series of maxima and minima as higher lying Landau levels cross the Fermi level.

224

PAUL W. =USE

0.6

-

'g

t + e

S w

z

I

0.5

.

0.4

-

L

AND JOHN F. READY

0.3

-I

0.2

.

0.1

.

0

0

2

4

MAGNETIC

6

8

10

FIELD ( k 6 )

FIG.15. Spin-flip Raman laser linewidth as a function of magnetic &Id. The solid line is a least-squaresfit to the experimental points. (Ready and Kruse, unpublished data.)

Other workers have observed similar maxima whose exact magnetic field value varies with the pump laser wavelength (Walukiewicz et al., 1977). The spin-flip laser output falls to zero as the magnetic field is increased above 28 kG. In addition to the main features of the variation of output power with magnetic field, which have been shown in Fig. 16, smaller features have been observed. These are shown in Fig. 17 in the region from 1.4-2.4 kG. An oscillatory behavior with a period of 71 G is observed because of the presence of the Fabry-Perot modes of the laser cavity. The observation of the cavity modes indicates that the spontaneous linewidth is less than 70 G (0.27 cm- ') at values of the magnetic field near 2 kG. The modes disappear at fields greater than 4 kG, indicating that the linewidth has become greater than the mode separation. Figure 17 also shows a set of sharp narrow peaks between 1.4 and 1.5 kG. Similar sets of narrow spikes are observed at 360, 710, and 1080 G. A de-

4.

NONLINEAR OPTICAL EFFECTSIN Hg, -,Cd,Te

225

0.I

8cn

0 W

>

z 0 0

0.01

X W Y

a W

a

0.001

I

0

I

I

I

I

I

1

2

4

6

8

10

12

H (kG)

FIG.16. Peak external conversion efficiency for Hg,.,,Cd,,,,Te spin-flip Raman laser pumped at 10.26 pm as a function of magnetic field If. [From Norton and Kruse (1977).]

tailed view of the structure of the lines in the 710-G manifold is shown in Fig. 18. The lines are separated by 12 G. These lines are produced when the spin-flip transition overlaps a weak satellite line in the COz pump laser, a condition corresponding to four-photon mixing. In other words, several lines present in the pump laser are mixed through the nonlinearity of the Hg,.,, Te. If one input frequency is the most intense pump frequency, the R(20) line of the C O , laser, combinations of lines which become resonant near 710 G are

+ R(14) - R(18), R(20) + R(16) - R(20), R(20) + R(18) - R(22), R(20)

etc. As the magnetic field is tuned thiough the value corresponding to the frequency of one of these combinations, resonant four-photon mixing can

PAUL W. KRUSE AND JOHN F. READY

226 1432 C

I

1.4

I

I

1.6

I

I

I

I

2.0

1.8

I

I

2.2

I

1

2.4

H (LG)

FIG.17. Structure in the laser output due to cavity modes. [From Norton and Kruse (1977).]

occur. For the particular series above, the lines will be spaced by about 13 G, a value which corresponds well to the 12-G spacing shown in Fig. 18. We may also obtain the spontaneouslinewidth from Fig. 18. The observed value is about 8 Gat a field of 700-720 G. Sincethe tuning rate is 3.8 cm- '/kG, this corresponds to 0.0304 cm- l . This is much smaller than the cavity mode spacing of 71 G (0.27 cm-') mentioned earlier. 16. T E M P O W AND SPATIAL CHARACTERISTICS

The temporal behavior of the spin-flip signal for a Q-switched laser pump is shown in Fig. 19. This shows data taken at a field of 8.1 kG, near the maximum efficiency. The transmitted pump signals, reduced by narrowing the monochromator slitwidth, are also shown for comparison, with the magnetic field off and magnetic field on. The spin-flip signal rises more rapidly than the pump laser pulse, reaching its peak before the peak of the pump laser pulse. The spin-flip laser pulse duration of 50 nsec (FWHM) is narrower than that of the 100-nsecduration pump laser pulse. The double peak behavior shown in this figure is not a universal feature of the spinflip laser emission. Pump depletion is clearly shown when the magnetic field is on. The transmitted pump laser intensity is greatly reduced, because of conversion of pump laser energy to spin-flip laser output. Similar behavior

4.

NONLINEAR OPTICAL EFFECTSIN

Hg, -,Cd,Te

227

715 G

0.I

l l r l l r l

600

700 720 H

740

(GI

FIG. 18. Structure of peaks observed in laser output in the 7 1 0 4 region. [From Norton and Kruse (1977).]

due to pump laser depletion has also been observed with C 0 2 TEA laser pumps (Walukiewicz et al., 1977). Attempts to operate Hg, - .Cd,Te spin-flip Raman lasers continuously have so far been unsuccessful. Operation of a Hgo~7,Cdo.z,Tespin-flip Raman laser with a chopped cw pump laser indicates that a Raman laser pulse occurs early in the chopped pump pulse but then dies out after a few milliseconds, probably because of sample heating (Muehlner, unpublished data, 1978). The emission from the spin-flip-Ramanlaser also shows a marked spatial variation. The spatial profile of the laser emission is relatively complicated with the emission predominately arising from small areas on the sample (Kruse et al., 1977). Figure 20 shows one example of the spatial dependence of the spin-flip laser output as the pump laser beam was scanned in a horisample. The inset zontal direction across the face of the Hg,.,,Cd,.,,Te

228

PAUL W. KRUSE AND JOHN F. READY

T

it

,

SPIN

50

- FLIP,

8.1 kC

100

150

w-.

200

4

TIME (nsec)

FIG. 19. Laser signals and transmitted pump signals, as a function of time. The transmitted pump signals were obtained with the spectrometer slits adjusted to reduce the signal. [From Norton and Kruse (1977).]

shows the track of the scan which extended for 0.89 cm across the crystal, which was roughly circular with two flattened edges. The laser emission was strong over a 3-mm portion of this scan, and was either zero or relatively weak outside that region. Other scans at different vertical positions could delineate the regions of maximum emission, which typically had dimensions of a few millimeters. The results shown in Fig. 20 were obtained with the sample at 12 K. When the temperature is lowered to 2 K, the irregular structure is less pronounced. A larger fraction of the sample contributes to the laser emission, but most of the laser output still arises from the small regions of high efficiency (“sweet spots”). ’ Figure 20 includes a scan of the relative transmission of the sample for the pump laser radiation across the same horizontal line. As the figure indicates, regions of high transmission are correlated, although not perfectly, with regions of high laser efficiency. It is tempting to ascribe the

J- (

0 . ! 9

I

lc-- 1.27 cm -1

TRANSMISSION

\r

I !

HORIZONTAL POSITION (cm)

FIG.20. Spatial structure of spin-flip Raman laser signal and spatial variation of transmitted pump radiation. The inset shows the position of the spatial scan on the crystal face.

230

PAUL W. KRUSE AND JOHN F. READY

variations in transmission to compositional variations, which would shift the band edge slightly. However, the same pattern of transmission variability persists when the sample is at 300 K,a temperature at which the band edge has moved to near 7 pm. Small changes in composition could not account for the spatial variation of the 1O-pm transmission at 300 K. The cause of the nonuniformity of the transmission and its relation to the efficiency of the spin-flip laser operation is not understood.

VI. Observations of Multipboton Mixing in Hg, -,Cd,Te In this part we describe observations of four-photon mixing (Khan et al., 1978, 1979; Bridges et al., 1979; Muehlner, Kash, Khan, Wolff, and Wood, unpublished data, 1979) and six-photon mixing (Khan et al., 1979) in Hg,-,Cd,Te, with x near 0.23. The radiation from two pump lasers at angular frequencies o1 and 0, (where o1> 0,) is combined to produce radiation at frequency w3 = 20, - 0, and o4 = 20, - o1via a resonant four-photon mixing process involving the spin nonlinearity. 17. EXPERIMENTAL ARRANGEMENT A typical experimental arrangement is shown in Fig. 21. The sample, of diameter 12.7 mm and thickness between 0.75 and 5 mm, mounted in a liquid helium immersion optical Dewar, is at a temperature near 2 K. The Dewar is mounted between the pole faces of an electromagnet. The sample is pumped by two Q-switched CO, lasers, which emit pulses of 2Wnsec duration and peak power of 1kW. The relative orientation of the sample and the electric field vectors Eland E, of the two pumps is sketched in Fig. 22. This arrangement of the pump laser electric field vectors reduces the potential contribution to the signal arising from conduction-band nonparabolicity (Nguyen et al., 1976). The two pump laser beams are combined with a Ge beam mixer and focused on the sample collinearly by a lens of 25-rnm focal IHG. CD) TE SAMPLE

- .

PUFIER

FIELD SENSOR

FIG.21. Experimental arrangement for multiphoton mixing. [From Khan et al. (1979).] G, grating; M,stationary mirror; B, beam mixer; R, rotating mirror; L, lens; E, electromagnet; P , electromagnet power supply.

4.

NONLINEAR OPTICALEFFECTS IN Hg, -,Cd,Te

231

E,

m 5 mm

FIG.22. Relative orientations of the polarization vectors Ei, the wave vectors ki, and the magnetic field H in four-photon mixing.

length. The peak laser power on the sample is kept below 400 W/cm2 in order to avoid spin-flip Raman laser emission. The pulses from the two pump lasers are synchronized by Q switching both lasers with the same rotating two-faced mirror. Both pump lasers are separately line tunable with individual diffraction gratings. 18. PROPERTIES OF FOUR-PHOTON MIXING The outputs from the nonlinear process at frequencies o3= 20, - 0, and 0, = 20, - o1are polarized as shown in Fig. 22. The output signals, separated from the transmitted pump laser radiation by a monochromator, are monitored by a Hg0.80Cdo.20Teinfrared detector. The output of the detector is summed by a gated integrator, the output of which is fed into the y axis of an x-y recorder. The output at O, is much stronger than the because the sample is less transparent at frequency 03. output at 03, Typical recorder traces with magnetic field used as the input to the x axis are shown in Fig. 23, for two different areas on the sample. The magnetic field was scanned by the signal from a function generator which was input to the electromagnet power supply. A magnetic field sensor drove the x axis of the recorder. Signals from two different positions on the sample are shown. A strong resonance is observed at o3and 0,when the magnetic field satisfies the condition Io1- w2 I = gBH/h. Because of compositional nonuniformity in the sample, the peaks appear at different values of magnetic field, characterized by different values of g. A map of the variation of g with position on a sample is shown in Fig. 24. The areas labeled “sweet spot” are areas from which spin-flip laser action was strong. (See Section 16.) The linewidth of the four-photon peaks shown in Fig. 23 is approximately 70 G. The observed linewidth varies with experimentalconditions, including carrier concentration, magnetic field, pump laser power, and location on

232

PAUL W. KRUSE AND JOHN F. READY

I

I

z8

7.9

I

ao

I

8.1

I

I

I

I

9.0

9.1

9.2

I

8.2

8.3

I 9.3

I

1

9.4

9.5

H (kG)

FIG.23. Four-photon output as a function of magnetic field H for two different areas with different values of g and x, as indicated. Curve (a), g = 84.22; x = 0.2277; curve (b), g = 73.60, x = 0.2341. [From Khan et al. (1979).]

8035

(0.230)

0620

0660

0.700

a740

HORIZONTAL MICROMETER

0.780

o . 8 ~ 0 0860

osw

READING (INCHES)

FIG.24. Four-photon mixing scan of Hg,.77Cdo.2,Te sample indicating measured g values with corresponding x values in parentheses. The positions are indicated by the readings of the micrometers used to move the sample. The "sweet spots" are areas of relatively high Raman 1aseroutput.T x 10 K.w, = 975.94cm-'(10.247 pn),R2Oline,w2 = 944.21 cm-'(10.591 pm), P20 line. wg = 2w2 - w , = 912.48 cm-' (10.959 pm).

4. NONLINEAR OPTICAL EFFECTS IN

MAGNETIC

Hg,-,Cd,Te

233

FIELD ( G I

FIG.25. A Lorentzian line shape with a full width at half maximum of 15.8 G. [From Muehlner, Kash, Khan, Wolff, and Wood, unpublished data (1979). Figure courtesy of M. A. Khan.]

the sample. In some cases (Muehlner, Kash, Khan, Wolff, and Wood, unpublished data, 1979) linewidth as narrow as 6.6 G (FWHM) has been observed. Figure 25 shows an example of a peak with a width of 15.8 G (FWHM), which is very accurately Lorentzian to at least 150 G from its center. Power broadening is illustrated in Fig. 26. The peaks can become nonLorentzian and grow very broad as the input power density is increased. The development of the satellite peak with increasing laser power is not understood. The minimum expected linewidth varies with carrier concentration and with magnetic field. Four-photon production via a spin-flip transition can occur only when the Fermi level lies between the upper and lower spin states. Because of conduction-band nonparabolicity, the spread in energy difference between upper and lower spin states varies between the positions in momentum space where the upper spin state crosses the Fermi level and the position where the lower spin state crosses the Fermi level. This spread in energy gives a contribution to linewidth. This contribution to linewidth of the four-photon transition is a function of carrier concentration and magnetic field. For constant carrier concentration, the linewidth first increases with increasing magnetic field to a maximum value and then decreases (Bridges et al., 1979).

234

PAUL W. KRUSE AND JOHN F. READY

4500

5000

6000

5500

MAGNETIC

FIELD

6500

(0)

FIG.26. Four-photon line shapesfor various values of pump laser power density,as indicated. [From D. J. Muehlner, Kash, Khan, Wolff, and Wood,unpublished data (1979). Figure courtesy of M. A. Khan.]

Under many conditions, the four-photon mixing peak is observed to be split into discrete components (Khan et al., 1978; Bridges et al., 1979; Muehlner, Kash, Khan, Wolff, and Wood, unpublished data, 1979). The behavior is illustrated in Fig. 27. In some cases the peak may be split into three or four discrete components. This behavior is possibly due to the presence of regions of discretely different g values in the area illuminated by the two pump beams. The nature of the regions and of the boundaries separating them is not understood. In one investigation (Bridges et al., 1979) the split peak behavior was suggested to be caused by regions of different g value left on the edges of samples, when the pump beams were perpendicular to the growth axis of the sample. However, similar behavior has been observed in samples in which the pump beams were parallel to the growth axis (Khan et al., 1979).

4.

NONLINEAR OPTICAL EFFECTS IN

235

Hg, -,Cd,Te

*.

a a a c

-

m

a 4

6.0

6.5

7.0 7.5 0.0 MAGNETIC FIELD (kG)

8.5

9.0

9.5

FIG.27. Four-photon signal showing split peak. Pump frequencies 4 2 0 ) and P(20). Am = 31.73 an-’.T = 12 K.

The linewidth thus is a complicated function of the parameters of the experiment. It appears that in many cases the sample condition dominates the line shape, leading to broadening, irregular line shapes and splitting of the line into discrete peaks. In selected regions of the samples narrow Lorentzian lines may be observed. Tuning of the output signal as a function of frequency w1 for fixed w2 and of w2 for fixed w1 is shown in Figs. 28 and 29. The solid lines represent the calculated values for the four-photonfrequency o4 = 2w2 - olas a function of the indicated values of w1 and w 2 . Curves are shown corresponding to selected values of the constant pump frequency. Scaling of the output signal at w4 with input power is indicated in Fig. 30. For the data representing scaling with o1power, the power at w2 was constant, and vice versa. The scaling with respect to the power in the wiinput is linear, and the scaling with the power in the w2 input is quadratic. The solid lines represent the expected functional dependencies, i.e., quadratic in w2 power and linear in w1 power. These experimentalresults follow the expected results [see Eq.(3211 closely. For larger values of pump laser intensity, above a few hundred W/cm2, saturation of the four-photon output has been observed (Muehlner, Kash, Khan, Wolff, and Wood, unpublished data, 1979). The saturation is demonstrated in Fig. 31, which shows the square root of the efficiency (? = P4/P2) as a function of pump laser power density [(P1P2)”2].These data can be

236

PAUL W. KRmE AND JOHN F. READY

918

913

-

912

-

917 916 915 914

-

c I

sE 3*

-

910 909

911

908 907

-

906!4~

: 4 1

&?

~2

544

cm-V

645

26

641

FIG.28. Variation ofw, with o2for fixed ol, as indicated. solid curyes are calculated values of 20, ol. Points are experimental results.

-

used to derive values for zl,the spin-lattice relaxation time, and z2, the transverse spin relaxation time (Nguyen et al., 1976). Analysis of the results of Fig. 31 yields z1 N 4 x lo-' sec and z2 N 2 x lo-'* sec. It is also possible to observe a four-photon mixing signal when the sample is at 77 K, as indicated in Fig. 32. This shows the power in the o4line as a function of magnetic field, The linewidth has broadened to 2500 G. The g

4. NONLINEAR OPTICAL EFFECTS r~ Hg, -,Cd,Te 918

-

915

-

237

-

,

-5 912

3

-

911

W2- P ( 2 0 )

908

-

907

-

974

w2 *

975

976

977

978

979

944.21 cm’l

P ( 22) = 942.40 cm-l

980

981

wI (cm-9 FIG.29. Variation of w, with w, for fixed w 2 ,as indicated. Solid curves are calculated values of 20, - wl.Points are experimental results.

value has shifted by a large amount from its value at lower temperature, i.e., from 80 to 58. The value at 77 K is in reasonable agreement with the value of 60 at 91 K, interpolated from Weiler’s data for x = 0.23 (Weiler, unpublished data, 1978). The efficiency of the four-photon mixing process at 2 K was determined by calibrating the detector against a power meter. The value of the ratio P,/P2 was found to be 1.47 x under conditions such that PI = 69‘W,

238

PAUL W. KRUSE AND JOHN F. READY

loor

so 80

-

-

40

u,

k

z 3

&

20-

4

a

t m

a d

w

a w 1

x

c a a

I0

-

8 -

6 4 -

c a

0

I

2

4

6

810

INPUT POWER (ARBITRARY UNITS)

FIG.30. Scaling of power in w, signal with power in the w1 pump and in the w2 pump. 0 ,o1 input; 0,w2 input.

P , = 180 W, and the area over which the beams overlapped was about 0 25 cm2. This ratio will be used later to estimate the value of f 3 ) (see Part VIII). The efficiency of four-photon mixing may be increased by an order of magnitude by phase matching in a noncollinear configuration (Khan, et al., 1980b). It should be noted explicitly that spatial regions on the sample for which four-photon mixing was relatively efficient did not necessarily correspond to regions of the largest spin-flip Raman laser output. Note also that whereas spin-flip Raman laser emission occurs only in “sweet spots,” four-photon mixing signals are found everywhere in the sample.

19.

PROPERTJES

OF SIX-PHOTON

MIXING

A resonant six-photon mixing signal at frequency 3 0 , - 2wl, smaller than the four-photonsignal, has also been observed (Khan et al., 1979).Six-photon

c

a

% % % %

\

.

- 0

I

0

200

400

600

800

1000

1200

I

I

1400

1600

I 1800

POWER DENSITY (W/crn2)

FIG.31. Saturation of the efficiencyoffour-photon mixing as input power density I, increased. The square root of the efficiency q is plotted. Here tj is the power at m4 divided by the power at w 2 . [From Muehlner, Kash, Khan, Wolff, and Wood, unpublished data (1979). Figure courtesy of M. A. Khan.]

9.0

9.5

10.0

10.5 11.0 11.5 MAGNETIC FIELD (kG)

12.0

12.5

FIG.32. Four-photon signal as a function of magnetic field at 77 K. Pump frequencies are R(20) and P(20); Am = 31.73 cm- '. [From Khan et al. (1979)J

239

240

PAUL W. KRUSE AND JOHN F. READY

mixing is also a resonant process, displaying a sharp maximum when the magnetic field H satisfies the condition I o2- o1I = g/?H/h. The o6signal is polarized perpendicular to the magnetic field. Figure 33 shows the relative output signal from the six-photon process, as compared to the output signal from the four-photon process. These peaks are observed simultaneously. In Fig. 33 the two traces are displaced vertically. The o, signal was obtained under conditions where the input to the sample was attenuated by a factor

I

U J

c 2

3

* a a

a

t m

a 4, a 3

W

0

n

5 a c

3

0

I 7.9

I 8.0

1

I

I

I

8.1 8.2 8.3 8.4 MAGNETIC FIELD (kG)

I 8.5

I 8.6

FIG.33. Output in the four-photon signal w4 and the six-photon signal w6 (vertically displaced) as a function of magnetic field at 12 K. The pump power was attenuated when the w, result was obtained. Pump frequencies are P(18) and R(20); Am = 29.95 cm-'.

4.

NONLINEAR OPTICAL EFFECTS IN Hg,

-xCdxTe

241

of about 10. According to the scaling law, this would imply that the output power from the sample under similar conditions should be larger by a factor of about 10oO in the o4as compared to the o6line. The substructureprobably arises from compositional nonuniformity in the irradiated region of the sample. Figure 34 shows the tuning of the six-photon signal as a function of wl, for selected lines from one of the COz pump lasers, with 0, fixed (0, = P(18) = 945.99 cm- '). The solid curve shows 3w, - 2wl, for the given w1 and ozshown.

-7 f

-

890

-

889

-

888

-

887

-

886

-

3" 885 884

-

883

-

882

-

881

-

880

I 972

I 973

I

I

974

975

I

I 976 0,

977

I 978

979

(cm-')

FIG.34. Frequency of o6signal as a function of wl.for fixed 02, as indicated. The solid line is the calculated 30, - 2w, for w2 = P( 18) = 945.99 cm- '. The points are experimental data.

242

PAUL W. KRUSE AND JOHN F. READY

A number of factors demonstrate that the observed signal is due to sixphoton mixing. First, the presence of both pump laser beams is required; when either pump laser is blocked separately the signal is zero. Second, the signal tunes resonantly with magnetic field, with the proper tuning for the signal at 30, - 20,. Finally the scaling with the input power has been verified. The scaling is proportional to the cube of the power at frequency 0, and to the square of the power at frequency ol, the correct scaling for a sixphoton mixing process (Chabay et al., 1976). The observed scaling with pump power is sufficient to rule out a sequential process in which 0, is produced by four-photon mixing and then mixes in a second four-photon process according to 0,

+ w4 - 0,= 20,

- 0,= 3w, - 20,.

The scaling for this sequential process should occur as the fifth power of the 0, pump power and the square of the w1pump power, which is inconsistent with the observed results. However, a sequential process is possible in which o4is produced and then mixes with 0, and o1according to 0 4

-k 0

2

-0

1

= 30,

- 201.

The scaling for this process would also depend upon the third power of the 0, pump power and the square of the 0, pump power, the same as for the direct six-wave mixing process. It appears likely that this sequential process may be the cause of the o6signal.

VII. Optical Phase Conjugation in Hg, -,Cd,Te In this part we describe optical phase conjugation experiments in Hg, -,Cd,Te by degenerate four-wave mixing (Khan, Kruse, and Ready, 1980a; Jain and Steele, 1980). 20. EXPWIMENTAL ARRANGEMENT The experimental arrangement is illustrated in Fig. 35. Samples of n-type Hg, -,Cd,Te (x = 0.216,0.225,0.231, and 0.232) were mounted in an optical Dewar. Counterpropagating pump beams were obtained by beam splitting the output of a CO, TEA laser operating on the P(22) line at 10.62 ,urn with a 200-nsec pulse width. The probe beam, also obtained by beam splitting the CO, laser output, was incident upon the samples at a 6" angle to the pump beam. The overlap between pump and probe was 4-6 mm in diameter. The samples were not oriented normal to either pump or probe beams. The pulse amplitude of the probe beam was determined by replacing the samples with a gold mirror normal to the beam and detecting the return by a Ge:Cu photoconductor. The signal amplitude was displayed on an oscilloscope. The

4.

NONLINEAR OPTICAL ~

C T INS Hg, -,Cd,Te

243

FIG.35. Experimental arrangement for degenerate four-wave mixing in Hg, -,Cd,Te. [From Khan, Kruse, and Ready (1980a).]

pulse amplitude of the phase conjugate return was determined in the same way by removing the gold mirror. The power reflection coefficient was determined by dividing the peak amplitude of the phase conjugate return by that when the gold mirror was substituted. The phase conjugate return disappeared when any of the input beams was blocked, as it should. When an aberrator of roughened KC1 was placed in the probe beam, the diameter of the return from the gold mirror doubled, whereas that from the sample remained the same, again as it should. 2 1. EXPERIMENTAL RESULTS Figures 36 and 37 illustrate typical data relating to the dependence of the power reflection coefficient P4/P1 upon incident pump intensity Here I, = P,/A and I ; = P’/A. As expected from Eq. (32a), a slope of two is found at low intensities. The reason for saturation at high intensities has not been determined. Table I lists the parameters of the four samples employed in the experiment, the operating temperatures, the measured values of f 3 ) , and the theoretical values of predicted from the theory of Wolffarid Pearson (1966), Eq. (35a). The measured values are obtained from Eq. (32a) using the measured ratios P4/P1in the relatively low power region where the power reflection coefficient has not saturated. There is agreement between the measured and calculated values to within an order of magnitude, which may be considered reasonable in this type of experiment. The values of f 3 ) , arising from conduction-band nonparabolicity in the absence of a magnetic field, are of the order of lO-’esu. As we shall discuss in Part VIII, values for the third-order resonant nonlinear susceptibility, arising from the spin nonlinearity in the presence of a magnetic field, are larger by 3 orders of magnitude. Thus, much

m.

lo

r-

FIG.36. Power reflection coefticient as a function of counterpropagating pump intensities I, and I, in H&.,6,Cdo.za2Te at 12,77, and 295 K. NDis the extrinsic electron concentration. ~ . Khan, Kruse, and Ready (198Oa).] Sample 1: thickness = 0.05 cm; ND = 1 x 10" ~ m - [From

4

-pi

z

0

; I-

A

10-1

LL

w a K

-

P

-

s

10-2

T.77 K T.295 K

T.295 K

1 I

I

l

l

I

105

I

I l l

I

I06

-& (W/cmz)

I

I

II

Id

FIG.37. Power reflection coefficient as a function of counterpropagating pump intensities I, and P, in Hgo.,s4Cdo.2,6Te at 77 and 295 K. N, is the extrinsic electron concentration. Sample 4: thickness = 0.05cm; ND = 8 x 10'5cm-3; x = 0.216. [From Khan, Kruse, and Ready (1980a).]

244

TABLE I COMPARISON OF EXPERIMENTALLY DETERMINED VALUES OF x(3)WITH TIBORETICAL VALUES BASEDON CONDUCTION-BAND NONPARABOLIC~TY Extrinsic Absorption edge electron concentration f 3 ) (295 K), (esu) f 3 ) (77 K), (esu) x(’) (12 K), (esu) X (w4 Sample value 12 K 77K 295 K (cm-’) Theoretical Experimental Theoretical Experimental Theoretical Experimental

1 2 3 4

0.232 0.231 0.225 0.216

9.8 9.9 10.8 12.4

8.6 8.7 9.3 10.4

6.1 6.1 6.4 6.8

1 x 1015 8 x lOI5 8.4 x 1 0 1 5 7 x 1015

1.2 x 1.5 x 3.4 x 2.1 x

10-8 10-8 10-8

9.6 10-8 1.1 x 1.3 1 0 - 7 3.8 10-7

3.6 10-9 3.3 x 3.9 x 10-8 8.9 x 1 0 - 8

1.0 6.8 8.2 4.0

x 10-7 x lo-’ x 10-7

x 10-7

3.2 x 1 0 - 8 3.7 x -

2.0 x lo-’ 7.0 x -

246

PAUL W. KRUSE AND JOHN F. READY

larger phase conjugate signals should be observed from phase conjugation using Hg, -xCdxTe in the presence of a magnetic field. As of early 1980,this experiment had not yet been reported. We note from the results shown in Figs. 36 and 37 that at cryogenictemperatures, phase conjugate reflections up to 9% have been observed, due to conduction-band nonparabolicity. Even at room temperature, phase conjugate reflections greater than 1 % are attainable.

VIII. Third-Order Resonant Nonlinear Susceptibility The results described in Parts V and VI may be used to estimate the thirdorder resonant nonlinear susceptibility x(3)of Hg,.,, Cdo.23Te.The estimates derived here are the peak values, obtained under resonant conditions.

22. ESTIMATE OF RESONANT NONLINEAR SUSCEPTIBILITY FROM RAMANLASERTHRESHOLD According to Eq. (29), the gain G for buildup of Raman intensity in the presence of pump laser power density I , is given by

G

= ~~A~Z,W,~X(~)(/~~C~.

(29)

The threshold for laser action is defined by Eq. (17)as p,exp(G - a,)l 2 1.

(17)

The experimental observations indicate that the threshold for laser operation for a pump laser wavelength of 10.28 pm, a reflectivity ps of 0.309 and a loss coefficient a, of 5 cm- in a 5-mm thick sample is approximately lo00 W/cm2.Substitutingin Eq.(17)leads to a gain of 7.35 cm- at threshold. Then substitution into Eq. (29) leads to an estimate of 1x(3)1 = 2.8 x esu.

23. ESTIMATE OF RESONANT NONLINEAR SUSCEPTIBILITY FROM EFFICIENCY OF FOUR-PHOTON MIXING Equation (32) gives the output power at frequency 0, in a four-photon mixing process P , = 2 5 6 ~ ~ (X‘~’I’P; 02 P1L2/n4c4A2,

(32)

with the coherence factor L2given by

L2

= I(ei(Ak)le-(A@)l -

l)/(i

Ak - L\dl)(2e-E4l.

(33) under conditions The efficiency P4/P2 was measured to be 1.46 x such that P, = 69 W,P2 = 180 W, A = 0.25 cm2. In this measurement, 0 ,

4.

NONLINEAR OPTICAL EFFEC'LS IN Hg, -,Cd,Te

247

was the R(20) line and o2was the P(20) line of the pump lasers. From the values of the dispersion and absorption in Section 11, the parameters in the coherence factor are found to be a4 = 5.0 cm-', Aa = 7.5 cm-', and Ak = 2.8 cm-'. This leads to

L' = 9.97 x 1 0 - ~cm2 for a 5-mm thick sample. Thus the coherence length L is about 3.16 x cm, i.e., the sample was much thicker than the coherence length. Substitution of these values into Eq. (32) then yields

lx(3)l = 1.0 x 1 0 - ~esu. We emphasize that these estimated values are the peak values obtained under resonant conditions. Since f 3 ) is almost entirely imaginary and negative (Uennis et al., 1972), we have The estimates from the two different sets of measurements differ by a factor around 3. The uncertainties in measurement of power density mean that the estimates are each uncertain to about f 50%. Taken together, these two estimates indicate that f3) N

- i(1 - 3)

x

esu

in Hg0.77 CdO.2 3 Note that this resonant contribution is about 103-104 times that of the nonresonant value arising from conduction-band nonparabolicity (see Part VII).

M.Applications 24.

APPLICATIONS FOR TUNABLE INFRARED LASERSOURCES

There are a number of developing applications for tunable infrared laser sources (Schlossberg and Kelley, 1976). The Hg, -xCd,Te spin-flip Raman laser must be considered as a possible candidate for some of these applications; however, it must be compared with other types of laser sources which compete for the same applications. The most obvious application is that of infrared spectroscopy (Smith, 1976). Laser sources offer extremely high brightness and extremely narrow linewidth. Thus one can perform infrared spectrometry with a resolution much higher than is available from conventional spectrometers. The development of such laser sources represents a qualitatively different method of spectroscopy.

248

PAUL W. KRUSE AND JOHN F. READY

A second application for tunable laser sources involves the monitoring of air pollution. The narrow-linewidth laser source may be tuned to an absorption line characteristic of the pollutant. Systems involving single-ended monitoring of pollutants in volumes of air at a distance have been demonstrated. Tunable laser sources also could be useful as optical heterodyne local oscillators. In conventional radio-frequency heterodyne receivers, the local oscillator is tuned as the radio-frequency stage is tuned, so that the intermediate frequency is constant. In most previously demonstrated optical heterodyne systems, the local oscillator (a laser) has not been tuned and the detector had to be fast enough to respond to the differencefrequency. Use of a tunable laser as a local oscillator would allow conventional operation, similar to the operation in the radio-frequency portion of the spectrum. A number of applications in photochemistry and photochemical process control would be possible. These applications are still only poorly dehed, but offer promise for the future. Applications involving laser-assisted isotope separation are under investigation. This specifically includes separation of uranium isotopes by irradiation of UF, molecules with a wavelength near 16 pm. The linewidth must be very narrow, and must match the wavelength of the isotopically selected species exactly. Laser-assisted isotope separation is a topic of considerable current research interest. Tunable infrared lasers are required for many of the isotope separation schemes that are being studied. Tunable infrared lasers could also influence a variety of applications in the submillimeter region of the spectrum. Difference frequency generation using tunable infrared lasers would yield sources variable over a broad range in the submillimeter region. Applications such as submillimeter spectroscopy, laser downconversion for millimeter wave oscillators, and submillimeter imaging radar systems to penetrate inclement weather are possibilities. For military applications, frequency-agile lasers for optical communications, optical radar, and target designation could be valuable in advanced systems. The applications mentioned above are ones for tunable infrared lasers. In addition, four-photon mixing also provides for applications such as very high are resolution spectroscopy. If two input beams, at frequencies oland 02, tuned so that mi - w2 is close to a Raman-active resonance of the material being studied, the four-photon signal at frequency o3= 2 0 , - o2may be observed, and can be used to characterize the material. The use of wave mixing spectroscopy has mainly been carried out in the visible portion of the spectrum. It has the capability for measurements in systems like flames, plasmas, and luminescent crystals, for which conventional methods are not well adapted (Levenson, 1977). Under the name CARS (coherent anti-Stokes

4.

249

NONLINEAR OPTICAL EFFECTSIN Hg, -,Cd,Te

Raman spectroscopy), four-photon mixing has been used for diagnostic probing in hostile environments, such as flames (Hall et al., 1979; Eckbreth, 1978). 25. CoMpARISON To INDIUM ANTIMONIDE SPIN-FLIP RAMAN

LASER

A number of spectroscopic applications have been demonstrated using the indium antimonide spin-flip Raman laser as a source. For example, the absorption spectrum of ammonia was recorded as a function of wavelength by varying the magnetic field (Shaw and Patel, 1971). Applications in air pollution monitoring were demonstrated by measurement of nitric oxide in a sample of air obtained along a highway (Kreuzer and Patel, 1971). Nitric oxide concentrations as low as 0.01 parts per million could be obtained with an integration time of the order of 1 sec and a spin-flip Raman laser power around 10 mW. Spectroscopy of the molecule OCS was performed with a resolution around 0.02 cm- and for the molecule SbH3with a resolution of 0.01 cm-' (Smith, 1976). In addition, optoacoustic spectroscopy with an InSb spin-flip Raman laser has been demonstrated. This technique can be useful when mode hopping of the laser causes problems with conventional absorption spectroscopic techniques. Good quality spectra of the OCS molecule were obtained (Smith, 1976). A continuous spin-flip Raman laser was used as an absorption spectrometer with an intracavityabsorption cell. Spectra of nitric oxide and carbonyl sulfide were obtained (Dutta et al., 1977). Because of the high brightness and short pulse duration of pulsed spin-flip Raman lasers, one could carry out timedependent spectroscopy. One could study the variation of populations of molecular species as a function of time by infrared spectroscopy. Because conventional light sources have relatively low intensity, such measurements were not previously possible. Chemical reactions of transient species of molecules can be monitored by measuring their infrared absorption as a function of time. T h e Hg, -,Cd,Te spin-flip Raman laser can be regarded as a competitor for similar types of applications. Table I1 compares properties of the two types 'of spin-flip Raman lasers. The status of development of the indium antimonide spin-flip Raman laser reflects the fact that it was developed earlier. Conver.sion efficiencies and maximum power outputs are higher. However, the greater tunability and the possibility of covering a broader wavelength range with a lower magnetic field may be regarded as advantages for Hg,-,Cd,Te devices. In addition, the ability to change the combination of pump laser and material composition will allow development of Hg, -,Cd,Te spin-flip Raman lasers for particular wavelength regimes. Such devices could cover specific wavelengths with relatively small magnetic fields. For example,

250

PAUL W. KRU.93 AND JOHN F. READY

TABLE I1 PROPERTIES OF SPIN-FLIP RAMAN LASERS

InSb Maximum wavelength for first-Stokes output for 10.6-pn pump Tunability Maximum reported optical conversion efficiency Maximum reported power output (CO, pump)

Minimum observed linewidth Maximum reported operating temperature a

12.8 pm @ 100 kG 2.0 cm-'/kG

11.2 p n @ 13 kG 3.8 cm-'/kG

z 50 % (CO pump) 1-kW peak T = 0.25 p s e ~ R = 4 pps" tl = 10-3 I1 kHz (cw) 45 K

10 % (CO, Pump) 100-W peak T = 0.25 psec R = 500 p p ~ " tj = lo-' 3 x 109 HZ (pulsed) 15 K

-

pps = pulses per second.

a spin-flip Raman laser consisting of a CF, laser pump operating at 15.3 pm and a Hgo.,9, Cdo.zosTecavity at 4 K would allow resonant operation near 16 pm. Such a laser could readily be tuned with a reasonably small magnetic field through the 16-pm region which is of interest for experiments in the photochemistry of UF6. 26. COMPARISON OF SPIN-FLIP RAMANLASERS TO OTHW TUNABLE INFRARED SOURCES

There are other types of infrared sources of high brightness operating in the same general wavelength region as spin-flip Raman lasers. Parametric oscillators and semiconductor laser diodes are strong competitors. Parametric oscillators (Byer and Herbst, 1977) offer tunable infrared sources covering the wavelength region out to 25 pm. They offer the advantages of not requiring a large magnetic field and of operating at ambient temperature. However, they are tuned either through temperature variation or through mechanical motion (rotation of the sample). Therefore the tuning will not be at a high rate, as is possible with the electronic tuning of spin-flip Raman lasers. In addition, the linewidth of spin-flip Raman lasers can be narrower. For applications requiring simplicity of construction, parametric oscillators may offer an advantage. For applications involving rapidity of tuning or high resolution, spin-flip Raman lasers have an advantage. Semiconductor lasers have been tuned over relatively narrow regions by variation of current, temperature, magnetic field, and pressure. As compared to spin-flip Raman lasers, tunable semiconductor lasers offer smaller size

4.

NONLINEAR OPTICAL EFFECTS IN

Hg, -,Cd,Te

251

and simpler construction. However, such lasers generally share the disadvantage of requiring cryogenic operation. In addition, their power output is less. For a given device, the semiconductorlaser is not tunable over as large a range as the spin-flipRaman laser. The usefulness of tunable semiconductor diode lasers for remote monitoring of concentrations of gases in the atmosphere and for measuring constituents of automobile exhaust has been well established (Hinkley and Kelley, 1971; Hinkley, 1972; McNamara, 1972; Antcliffe and Wrobel, 1972; Hinkley et af., 1976). The ultimate usefulness of the Hg,-,Cd,Te spin-flip Raman laser will be determined by future developments which reduce the complexity of the device. Construction of a laser with epitaxial layers of Hg, -,Cd,Te could considerably reduce the size of the device, allowing the active material to be positioned within the pole faces of a small permanent magnet. The magnetic field would define the center of the tuning range. Tuning would be accomplished with a solenoid. This would restrict the tuning range, but would allow construction of a smaller, more practical device. Such a laser could be cooled by a liquid helium cryostat. The total package would be of reasonable size (- 1 ft3) and could be a self-contained unit suitable for use outside of the laboratory. In order that Hg, -,Cd,Te spin-flip Raman lasers leave the laboratory, engineeringdevelopments such as these will have to be achieved.

ACKNOWLEDGMENTS We wish to express our appreciation to the management of the Honeywell Corporate Technology Center, and to Mr. Max Swerdlow of the Air Force Office of ScientificResearch for their continuing support of the research upon which this chapter is based. Our investigations were done in collaboration with our Honeywell colleagues Dr. Paul Norton and Dr. M. Asif Khan, upon whose work much of this is based. Mr. Donne11 Woodson and Mr. Richard Brinda were of much assistance to us in the laboratory. We have benefitted greatly from many discussions and collaborativeinvestigationswith Professor Peter Wolff, Professor Dirk Muehlner, Professor Roshan Agganval, and Professor Benjamin Lax, all of the Massachusetts Institute of Technology. We are also indebted to Dr. M. Asif Khan for his critical reading of the manuscript and to Mr. Robert Lancaster of the Honeywell Electro-Optics Center, who supplied the Hg,.77Cd,.23Tecrystals for use in our experiments. We thank Profs. D. J. Muehlner, K. Kash, M. A. Khan, P. A. WOE, and R. A. Wood for allowing us to use some of their unpublished results on four-photon mixing work performed at the Massachusetts Institute of Technology. We also appreciate the efforts of Darlene Rue in typing several drafts of this manuscript.

REFERENCES Antcliffe, G. A. (1970). Phys. Rev. B 2, 345. Antcliffe, G. A., and Wrobel, J. F. (1972). Appl. Opt. 11, 1548. Baldwin, G. C. (1969). “An Introduction to Nonlinear Optics.” Plenum, New York. Bloembergen, N. (1965). “Nonlinear Optics.” Benjamin, New York.

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Bridges, T.J., and Nguyen, V. T., (1973).Appl. Phys. Lett. 23, 107. Bridges, T.J., Burkhardt, E. G., and Nguyen, V. T.(1979).Opt. Commun.30,66. Bngnall, N., Wood, R. A., Pidgeon, C. R., and Wherrett, B. S. (1974). Opt. Commun. 12, 17. Bmeck, S. R. J., and Mooradian, A. (1973). Opt. Commun. 8,263. Byer, R. L.,and Herbst, R. L. (1977).In “Nonlinear Infrared Generation” (Y. R. Shen, ed.), p. 81.Springer-Verlag, Berlin and New York. Chabay, I., Klauminzer, G. K., and Hudson, B. S.(1976).Appl. Phys. Lett. 28,27. Colles, M.J., and Pidgeon, C. R. (1975). Rep. h o g . Phys. 38,329. Dennis, R. B., Pidgeon, C. R., Smith, S. D., Wherrett, B. S.,and Wood, R. A. (1972).Proc. R. SOC.London, Ser. A 331,203-236. Dutta, N., Warner, R. T.,and Wolga, G. J. (1977). Opt. Lett. 1, 155. Eckbreth, A. C. (1978). Appl. Phys. Lett. 32,421. Hall, R. J., Shirley, J. A., and Eckbreth, A. C. (1979). Opt. Lett. 4,87. Harper, P.G.,and Wherrett, B. S.,eds. (1977). “Nonlinear Optics.” Academic Press, New York. Hinkley, E. D. (1972).Opto-electronics 4,69. Hinkley, E. D.,and Kelley, P. L. (1971).Science 171,635. Hinkley, E.D.,Ku, R. T.,Nill, K. W., and Butter, J. F. (1976). Appl. Opt. 15, 1653. Jain, R. K.,and Steele, D. G. (1980). Appl. Phys. Lett. 37, 1. Kane, E.0.(1966). In “Semiconductors and Semimetals” (R. K.Willardson and A. C. Beer, eds.), Vol. 1, p. 75.Academic Press, New York. Khan, M. A., Kruse, P. W., and Ready, J. F. (1978).J. Opt. Soc. Am. 68,1378. Khan, M.A., K N ~ P., W., and Ready, J. F. (1979).Opt. Comrmm. 28,374. Khan, M.A.,Kruse, P. W.,and Ready, J. F., (198Oa).Opt. Lett. 5,261. Khan, M. A., Bogart, T. J., Kruse, P. W., and Ready, J. F. (1980b).Opt. Lett. 5,469. b u r , L. B., and Patel, C. K. N. (1971).Science 173,45. Kruse, P.W. (1976).Appl. Phys. Lett. 28,90. Kruse, P.W., Norton, P., and Ready, J. F. (1977).J. Opt. SOC.Am. 67,1424. Kruse, P.W., Norton, P., and Ready, J. F. (1978). Proc. E - 0 Laser, 1978 p. 523. Levenson, M. D.(1977).Phys. Today May, p. 44. Long, D. (1968).“Energy Bands in Semiconductors.” Wiley, New York. Long, D. (1972). Unpublished data. Long, D., and Schmit, J. L. (1970).In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 5, p. 175.Academic Press, New York. McNamara, F. L. (1972).Opt. Eng. If, 9. Mooradian, A., Brueck, S.R. J., and Blum, F. A. (1970). Appl. Phys. Lett. 17,481. Muehlner, D.J., (1978),unpublished data. Muehlner, D. J., Kash, K., Khan, M. A., Wolff, P. A. and Wood, R. A, (1979). Unpublished data. Norton, P., and Kruse, P. W. (1977).Opt. Commun.22, 147. Nguyen, V. T.,Burkhardt, E. G., and WON,P. A. (1976).Opt. Commun. 16,145. Patel, C. K. N. (1972). Phys. Rev. Lett. 28, 649. Patel, C. K.N. (1973). I n “Fundamental and Applied Laser Physics” (M. S. Feld, A. Jovan, and N. A. Kurnit, eds.), p. 689.Wiley, New York. Patel, C. K. N., and Shaw, E. D. (1971).Phys. Rev. B 3, 1279. Patel, C. K. N., Slusher, R. E., and Fleury, P. A. (1966).Phys. Rev. Lett. 17,101 1. Pidgeon, C.R., and Smith, S.D. (1977).Infru Phys. 17,515. Ready, J. F., and Kruse, P. W. (1979).Unpublished data. Roth, L. M., and Argyres, P. N. (1966).In “Semiconductorsand Semimetals”(R. K. Willardson and A. C. Beer, eds.), Vol. 1, p. 159.Academic Press, New York. Sattler, J. P., Weber, B. A., and Nemarich, J. (1974).Appl. Phys. Lett. 25,491.

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Index C

CdSe band parameters, 81, 91 band structure, 69 dielectric constant, 92 dopants, 67 effective mass, 91 heat of fusion, 59 lattice constant, 55 lattice reflectivity, 92-94 LO-mode frequency, 92 melting temperature, 57 optical absorption edge, 81,91 polar LO-phonon scattering, 98 TO-mode frequency, 92, 93 CdTe, LO phonon, 145, 165 Crystal defects, see also Defects effect on optoelectronic devices, 1-49 general discussion, 2-4 laser diodes, 9-13, 15-18, 30-32, 3638, 40-49, see also Laser diodes light-emitting diodes, 9- 12, 15- 18, 30-32, 36-38, 40-49, see also Light-emitting diodes photodetectors, 4-9, see also Photodetectors; Photodiodes; Solar cells Current-voltage characteristics, 3, 13, 20, 23, 25, 30 effect of precipitates, 25 effect of stacking faults, 23

point defects, 2, 9, 12, 31, 32, 35, 44, see also Point defects precipitates, 2, 9, 15, 23-27, see also Precipitates recombination centers, 2-7, 15, 22, see also Recombination; Recombination centers stacking faults, 2, 9, 22, 23, see also Stacking faults de Haas-van Alphen effect, 126 Dislocations, 2, 9, 13-22, 24, 32-35, 4043, 46 acceleration of degradation, 15 carrier-lifetime reduction, 15- 17 dangling bonds, 14, 15 dark regions, 20, 21, 42 effect on efficiency of LEDs, 15- 19,4049 hetero-interface, 20-22, 30-36 impurity segregation, 13, 15 misfit, 10, 20-22, 32-35, 42 nonradiative recombination sites, 15- 17 recombination centers, 10, 13, 15- 17

E Effective mass, 71, 72, 198, 214-216, see also specific materials density of states, 144 field-dependent, 162 F

D Defects, see also Crystal defects dislocations, 2, 10, 13-22, see also Dislocations grain boundaries, 3, 9, 27-29, see also Grain boundaries microplasma sites, 3, 4, 9, 19

Faraday rotation, 125 G g

255

factor, 71, 72, 81, 122, 144, 158, 198, 199, 201, 208, 209, 215-217, 221 spin mass, 209

256

INDEX

Grain boundaries, 3, 9, 27-29 deep levels, 27 impurity segregation, 27 potential barrier, 28 band bending, 28

H Hall measurements, 77-79, 126 (HgCd)Se, 53- 114 annealing, 65-68 electron concentrations, 67, 68 electron mobility, 67 band parameters, 76, 77, 79, 81, 84-88 band structure, 69-72, 87 Landau levels, 71,72 semimetal-to-semiconductor transition, 69,70

Cd diffusion coefficient, 65 comparisons with (HgCd)Te, 54, 55 crystal growth, 61-68 annealing, 65-68 Bridgman method, 61 composition profile, 63, 64 constitutional supercooling, 63 diffusion-limited, 64 directional freezing, 63 epitaxial layers, 65 kinetics, 63,64 reaction of elements, 62, 63 defect properties, 69 density, 56 dielectric function, 72-75 electronic part, 73-75 lattice contribution, 75 dopants, 67, 68 effective mass, 71,72,78 electrical characteristics, 68 electron mobility, 104- 112 composition dependence, 106, 107 temperature dependence, 108- 110 electron scattering mechanisms, 96112

acoustic phonon, 99 compositional disorder, 100 defect, 100, 101, 104 dominant mechanisms, 110- 112 electron-hole, 99, I00 nonpolar optical phonon, 99 polar LO phonon, 97-99

energy gap, see also band parameters dependence on composition and temperature, 82-86 g factors, 71, 72, 76, 77, 81 helicon waves, 91 lattice constant, 55 magnetotransport effects, 75-79 Hall effect, 77 Landau levels, 71, 72 Shubnikov-de Haas effect, 75-77 optical absorption edge, 79-85 optical reflectivity, 84, 87, 88 lattice effects, 92 peritectic transition, 57, 58 phase diagram, 57 photoconductivity, 9 6 % effect of trapping, 95, 96 thermodynamic relationships, 58-61 TO-mode frequency, 93 vapor pressure, 61 (HgCd)Te band parameters, 146-148, 151, 153158, 160, 161, 164-166, 174-187

band structure, 120 bandgap, see also band parameters composition dependence, 148, 164, 177-180, 213, 214

temperature dependence, 148, 177180, 213, 214

effective mass, 1 6 1 4 8 , 153, 154, 160, 163, 165, 174-177, 181-183, 186, 214, 216 electron-phonon coupling constant, 146, 165 exciton binding energies, 162 g factor, 146, 147, 154-158, 160, 176, 181, 184, 187, 199, 209, 215-217, 221

importance for spin-flip Raman laser, 209-211, see also spin-flip Raman laser importance of band nonparabolicity, 213

optimum electron concentration, 211 interband magnetooptical studies, see Interband magnetooptical studies intraband magnetooptical studies, see Intraband magnetooptical studies LO-phonon frequency, 147, 151, 160 magnetooptical properties, 119- 188, see also Interband magnetooptical

257

INDEX

studies; Intraband magnetooptical studies; Magnetooptical phenomena magnetoreflection studies, 163, 166- 171 nonlinear optical effects, 193-251, see also Nonlinear optical effects optical phase conjugation, 195, 206, 242-

HgTe, see also (HgCd)Te energy gap, 180 LO phonon, 145, 165

I

246

optical properties, 217, 218 parameters relevant for spin-flip laser and four-photon mixing, 213-218 effective mass dependence on composition, 214-216 gap dependence on composition and temperature, 213, 214 g-factor dependence on composition, 2 15-2 17

optical properties, 217-218 polaron effects, 165 resonant four-photon mixing, 195, 203, 211, 230-238, 246, 247, 249

spin-flip Raman laser, 195, 199, 210, 211, 218-230, 246-251 applications, 247, 248

comparison with InSb laser, 249, 250 transport measurements, 159- 161 HgSe, 54- 114, see also (HgCd)Se annealing, 66 electron concentration, 66 band parameters, 76-79, 81, 84, 87, 91 cyclotron resonance, 90-91 dopants, 67 effective mass, 78, 88, 89 dependence on electron concentration,

lnSb magnetoreflection data, 168, 169 spin-flip Raman laser, 202, 249, 250 applications, 249 comparison with (HgCd)Te laser, 249, 250

Interband magnetooptical studies, 161- 176 exciton correction, 161-163 (HgCd)Te band parameters, 164-166, 174- 187

magnetoreflection studies, 163, 166- 171 parameter-fitting techniques, 171- 176 Intraband magnetooptical studies, 79-81, 145- 161

combined resonances, 145- 151 cyclotron resonance, 145- 154 (HgCd)Se, 79-81 (HgCd)Te band parameters, 146- 148, 151, 153-158, 160, 161

phonon-assisted resonance, 145- 147, 151 (HgCd)Te electron-phonon coupling constant, 146, 165 (HgCd)Te LO-phonon frequency, 147, 151, 160

spin resonance, 155- 159

88, 89

electron mobility, 101- 104 electron scattering mechanisms, 96- 112 defect, 100, 101, 104 nonpolar optical phonon, 99 g factor, 81 heat of fusion, 59 lattice constant, 54 LD-mode frequency, 94 magnetotransport effects, 75-79 Hall effect, 77-79 Shubnikov-de Haas effect, 75-77 melting temperature, 57 optical absorption edge, 79-81 optical reflectivity, 87-94 phase diagram, 66 spin-flip resonance, 90,91 thermal conductivity, 112, 113

L Landau levels, 71, 72, 121, 122, 127, 128, 197-200

Laser diodes, 9- 13, 15- 18, 30-36, 40-49 broad area, 9, 44-46 damaged edge, 44-46 degradation, 15- 19, 40-49 from dislocations, 15- 19, 40-46 from stress, 47, 48 effect of strain, 21, 42 effect of stress external, 47 from lattice misfit, 47, 48 heterojunction, 10, 11, 20 importance of lattice match, 10, 2022, 42, 43, 47, 48

258

INDEX

homojunction, 9, 19 misfit dislocations, 10, 20-22, 42, 47, 48 nonradiative recombination centers, 10, 11, 15, 16

optical damage, 11,40 quantum efficiency, 9, 11-13, 24, 30,

Landau levels, 71, 72, 121, 122, 127, 128, 197-199

measurement techniques, 124- 126, 149 Faraday configuration, 124, 149 Voigt configuration, 125, 149 theory, zinc-blende semiconductors, 127- 145

38-41

effect of nonradiative recombination, 38-40, see also Recombination stripe contact, 9,4446 threshold current density, 11- 13, 19, 31,

approximate solutions for band energies, 139-144 density of states effective mass, 144 quasi-germanium model, 128, 131-

32, 39-41

136

Light-emitting diodes (LEDs), 9- 13, 1518, 30-32, 36-38, 40-49

carrier lifetime, 11 degradation, 15- 1 9 , 6 4 9 from dislocations, 15- 19, 40-43 effect of strain, 21,42 effect of vacancies, 43, 44 importance of lattice match, 10, 20-22, 42, 43

quantum efficiency, 9, 11, 12, 18, 24, 30, 38-40

selection rules, 136- 139 Magnetophonon effect, 126, 160, 161 (HgCd)Te, 160, 161 Magnetoreflection studies, 163, 166171

Microplasmas, 3, 4,9, 19 Multiphoton mixing, 195, 203-205, 211, 212, 230-242, 246, 248, 249

determination of third-order resonant nonlinear susceptibility, 246, 247 experimental arrangement, 230, 231 output of four-photon mixing, 231240

M Magnetooptical phenomena, 71,72, 79-81, 121- 188, see also Interband magnetooptical studies; Intraband magnetooptical studies density of states in magnetic field, 122, see also Landau levels g factor, 71, 72, 81, 122, 144 (HgCd)Se, 81 (HgCd)Te, 146, 147, 154-158, 160, 176, 181, 184, 187

HgSe, 81 (HgCd)Se, 79-81 interband transitions, 123, 124, 138, 139, 166, 170, 171, 174, see also Interband magnetooptical studies parameter-fitting techniques, 171- 176 intraband transitions, 123, 125, 137, 138, 147, 150- 152, see also Intraband magnetooptical studies combined resonances, 123, 145- 151 cyclotron resonance, 123, 145- 159 spin resonance, 123, 155- 159

efficiency, 237-239 linewidth, 231, 233-235 magnetic-field dependence, 23 1, 232 polarization, 231 power broadening, 233, 234 saturation, 235, 239 tuning, 235-237 output of six-photon mixing, 238, 240242

relevant properties of (HgCd)Te, 213218

effective mass dependence on composition, 214-216 gap dependence on composition and temperature, 213, 214 g-factor dependence on composition, 215-217

optical properties, 217, 218 resonant four-photon mixing, 203-205, 211, 212,230-238, 248, 249

applications, 248, 249 comparison with spin-flip Raman laser, 205 peak gain, 211 power output, 212

INDEX

N Nonlinear optical effects, 193-251 in (HgCd)Te, 193-251, see also (HgCd)Te polarization, 194 susceptibility, second order, 194 optical rectification, 194 parametric oscillation, 194 Pockels effect, 194 second harmonic generation, 194 sum and difference frequency generation, 194 susceptibility, third order, 194- 1%, 246, see also Third-order resonant nonlinear susceptibility Brillouin scattering, 195 contributing mechanisms, 195 four-photon mixing, 195, see also Multiphoton mixing importance in (HgCd)Te, 195, 196, see also (HgCd)Te importance in InSb, 196, see also InSb Kerr effect, 195 optical phase conjugation, 195, see also Optical phase conjugation Raman scattering, 195, see also Raman scattering Rayleigh scattering, 195 third harmonic generation, 195 0

Optical phase conjugation, 195, 205-207, 213, 242-246

adaptive optics, 206 experimental arrangement, 242, 243 importance of band nonparabolicity, 213

typical results, 243-246 determination of third-order susceptibility, 245 power reflection coefficient, 244 Optoelectronic devices, see also (HgCd)Te; Laser diodes; Nonlinear optical effects; Photodetectors; Photodiodes; Solar cells

current-voltage characteristics, 3, 20, see also Current-voltage characteristics effect of crystal defects, 1-49, see also Crystal defects junction current, 3 leakage, 3, 9, 13 recombination, 3, 10- 13 tunneling, 3, 13 multiphoton mixing, 195, 203-205, 211, 212, 230-242, 248, 249, see also Multiphoton mixing optical phase conjugation, 195, 205-207, 213, 242-246, see also Optical phase conjugation photodetectors, 4-9, see also Photodetectors photodiodes, 7-9, 19, 20, see also Photodiodes spin-flip Raman laser, 195, 199-203, 205, 207-211, 218-230, 246-251, see also Spin-flip Raman laser

P Photodetectors, 4-9, see also Photodiodes; Solar cells factors affecting performance, 4 leakage current, 4, 9 material crystallinity, 4 minority-carrier lifetime, 4 surface recombination, 4, 6 Photodiodes, 7-9, 19, 20, see also Photodetectors avalanche type, 7-9, 19 carrier multiplication, 7-9 current- voltage characteristics, 3, 20, 23, 25, 30, see also Current-voltage characteristics dislocations, 19, 20 frequency response, 7 leakage current, 9, 19, 20 microplasma sites, 9, 19, see also Microplasmas at precipitates, 9 quantum efficiency, 7 reverse bias, 7, 8 Point defects, 2, 9, 12, 31, 32, 35, 44 from radiation damage, 3 1, 32

260

INDEX

Precipitates, 2, 9, 15, 23-27 complex segregates, 23

R Raman scattering, 125, 158, 195, 199-203, 208, see also Spin-flip Raman laser cross section, 208, 209 Recombination, see also Recombination centers at dislocations, 15- 17 diffUsion enhancement, 40,41 hetero-interface, 10, 30-36 effect on carrier lifetime, 36, 37 effect on quantum efficiency, 30, 3840 misfit dislocations, 10,22, 32-35 point defects, 32 reduction, 30, 31 lattice matching, 30, 31 surface, 2-4,6, 7, 29-31 Recombination centers, 2-7, 10, 11, 13, 15-17, 22, 32, see also Recombination nonradiative, 2, 3, 10, 11, 13, 15-17, 22, 24-26, 31-36

Auger process, 24-26 radiation-damaged regions, 31, 32

s Shubnikov-de Haas effect, 75-77, 126 (HgCd)Te, 160 Solar cells, 5-7, 26, 27, 28, 34, 39, 40,see also Photodetectors effect of precipitates, 26,27 efficiency, 5-7 effect of nonradiative recombination, 38-40 fill factor, 5, 6 GaAs compared to silicon, 5 heterojunctions, 5 open-circuit voltage, 5, 6 short-circuit current density, 5, 6 surface recombination, 6 Spin-flip Raman laser, 195, 199-203, 205, 207-211, 218-230, 246-251 anti-Stokes radiation, 201, 202, 205, 220-222

applications, 247-249 IR spectroscopy, 247 isotope separation, 248 monitoring air pollution, 247 optical heterodyne local oscillator, 247

tunable submillimeter wave source, 248

cavity modes, 224-226 compared with other tunable IR sources, 250, 251

comparison with resonant four-photon

mixing,205 conversion efficiency, 223- 226 as function of magnetic field, 225 determination of third-order resonant nonlinear susceptibility, 246 experimental arrangement, 219,220 (HgCd)Te laser, 195, 199, 210, 211, 218230, 24&251

compared to InSb laser, 249, 250 importance of (HgCd)Te, 209-211 optimum electron concentration, 211

InSb laser, 202, 249, 250 applications, 249 compared to (HgCd)Te laser, 249, 250

linewidth, 223, 224, 226, 227 as a function of magnetic field, 224 output vs. time in nsec, 226-228 pump depletion, 226 relevant properties of (HgCd)Te, 213218

effective mass dependence on composition, 214-216 gap dependence on composition and temperature, 213, 214 g-factor dependence on composition, 215-217

optical properties, 217, 218 sample spatial variation of emission, 228-230

Stokes radiation, 201, 202, 205, 220222

structure, 224-227 theoretical developments, 207-21 1 Raman laser gain, 207-210 spin mass, 209 threshold. 208

261

INDEX

threshold, 220 tuning, 220-223 Stacking faults, 2, 9, 22, 23 carrier-lifetime reduction, 22 nonradiative recombinationcenters, 22 precipitate segregation, 22

T Third-order resonant nonlinear susceptibility, 246, 241 from efficiency of four-photon mixing, 246, 241 from Raman laser threshold, 246

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Contents of Previous Volumes Volume 1 Physics of III-V Compounds C . Hilsum, Some Key Features of 111-V Compounds Franc0 Bussuni, Methods of Band Calculations Applicable to 111-V Compounds E. 0. Knne, The k . p Method V. L. Bonch-Bruevich, Effect of Heavy Doping on the Semiconductor Band Structure Donuld Long, Energy Band Structures of Mixed Crystals of 111-V Compounds Laura M . Roth and Perros N . Argyres. Magnetic Quantum Effects S. M . Puri and T . H . Gebulle, 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. Novkova, Thermal Expansion U. Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J. R . Drabble, Elastic Properties A . U. Mac Rae and G . W . Gobeli, Low Energy Electron Diffraction Studies Robert Lee Mieher, Nuclear Magnetic Resonance Bernurd GoMsfein, Electron Paramagnetic Resonance T.S. Moss, photoconduction in 111-V Compounds E. Anrodik and J . Tauc, Quantum Efficiency of the Internal Photoelectric Effect in InSb G. W . Gobeli and F . G . Allen, Photoelectric Threshold and Work Function P . S. Pershun, Nonlinear Optics in 111-V Compounds M . Gershenzon, Radiative Recombination in the III- 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. Stienvalt and R . F . Potter, Emittance Studies H . R. Philipp and H . Ehrenreich, Ultraviolet Optical Properties Manuel Curdonu, 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. Lux and J . G . Mavroides, Interband Magnetooptical Effects

263

264

CONTENTS OF PREVIOUS VOLUMES

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

Volume 4 Physics of HI-V Compounds N. A. Goryunova, A. S. Borschevskii, and D. N. Treriakov, Hardness N. N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds AmBv Don L. Kendall, Diffusion A. G. Chynoweth, Charge Multiplication Phenomena Robert W.Keyes, The Effects of Hydrostatic Ressure on the Propdes of III-V Semiconductors L. W.Aukerman, Radiation Effects N. A. Goryunova, F. P. K e s d y , and D. N. Nasledov. Phenomena in Solid Solutions R. T. Bate, Electrical Properties of Nonuniform Crystals

Volume 5

InfraredDetectors

Henry Levinstein, Characterization of Infrared Detectors Paul W.Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. B. Prince, Narrowband Self-Filtering Detectors Ivars Melngailis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides D o W Long and Joseph L. Schmir, Mercury-cadmium Telluride and Closely Related Alloys E. H. Purley. The pyroelectric Detector Norman B. Stevens, Radiation Thennopiles R. J. Keyes and T. M. Quisr, Low Level Coherent and Incoherent Detection in the Infrared M. C. Teich, Coherent Detection in the Infrared F. R. A r m , E. W. Sard, B. J . Feyton, and F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers, Jr., Microwave-Based Photoconductive Detector Robert Sehr and Rainer Zuleeg, Imaging and Display

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

Volume 7 Application and Devices: Part A John A. Copeland and Stephen Knight, Applications Utilizing Bulk Negative Resistance F. A. Padovani, The Voltage-Current Characteristics of Metal-Semiconductor Contacts P. L. Hower, W. W. Hooper, B. R. Cairns, R. D. Fairman, and D. A. Tremere, The GaAs Field-Effect Transistor Marvin H. White, MOS Transistors G. R. Antell, Gallium Arsenide Transistors T. L. Tansley, Heternjunction Properties

CONTENTS OF PREVIOUS VOLUMES

265

Volume 7 Application and Devices: Part B T. Misawa, IMPATT Diodes H. C. O k a n , Tunnel Diodes Robert B. Campbell and Hung-Chi Chang, Silicon Carbide Junction Devices R. E. Enstrom, H. Kressel, and L. Krassner, High-Temperature Power Rectifiers of GaAs,,P,

Volume 8 Transport and Optical Phenomena Richard J . Srirn, Band Structure and Galvanomagnetic Effects in 111-V Compounds with Indirect Band Gaps Roland W. Ure, Jr., Thermoelectric Effects in 111-V Compounds Herbert Piller, Faraday Rotation H.Barry Bebb and E. W. Williams, Photoluminescence I: Theory E. W. Williams and H. Barry Bebb, Photoluminescence II: Gallium Arsenide

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

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

Volume 11 Solar Cells Harold J . Hovel, Introduction; Carrier Collection, Spectral Response, and Photocurrent; Solar Cell Electrical Characteristics; Efficiency; Thickness; Other Solar Cell Devices; Radiation Effects; Temperature and Intensity; Solar Cell Technology

Volume 12 InfraredDetectors(II) W . L. Eiseman, J . D.Merriam. and R. F . Porrer, Operational Characteristics of Infrared Photodetectors Peter R. Brarr. Impurity Germanium and Silicon Infrared Detectors E. H. Purley, InSb Submillimeter Photoconductive Detectors G. E. Stillman, C . M. Wove, and J . 0.Dimmock, Far-Infrared Photoconductivity in High Purity GaAs G. E. Stillman and C . M. Wolfe, Avalanche Photodiodes P. L. Richards. The Josephson Junction as a Detector of Microwave and Far-Infrared Radiation E. H. Purley, The Pyroelectric Detector-An Update

266

CONTENTS OF PREVIOUS VOLUMES

Volume 13 Cadmium Telluride Kennerh Zonio, Materials Preparation; Physics; Defects; Applications

Volume 14 Lasers, Junctions, Transport N . Holonyak, Jr. and M. H . Lee, Photopumped III-V Semiconductor Lasers Henry Kressel and Jerome K . Burler, Heternjunction Laser Diodes A . Van &r Ziele, Space-Charge-Limited Solid-state Diodes Peter J . Price, Monte Car10 Calculation of Electron Transport in Solids

Volume 15 Contacts, Junctions, Emitters B . L. Sharma, Ohmic Contacts to 111-V Compound Semiconductors Allen Nussbaum, The Theory of Semiconductor Junctions John S. Escher, NEA Semiconductor Photoemitters