High Pressure in Semiconductor Physics II (Semiconductors and Semimetals, Vol. 55))

High Pressure in Semiconductor Physics II SEMICONDUCTORS AND SEMIMETALS Volume 55

Semiconductors and Semimetals A Treatise

Editedby R. K . Willardson CONSULTING PHYSICIST SPOKANE, WASHINGTON

EickeR. Weber DEPARTMENT OF MATERIALS SCIENCE AND MINERAL ENGINEERING UNIVERSITY OF CALIFORNIA AT BERKELEY

High Pressure in Semiconductor Physics II SEMICONDUCTORS AND SEMIMETALS Volume 55 VolumeEditors

TADEUSZ SUSKI UNIPRESS HIGH PRESSURE RESEARCH CENTER POLISH ACADEMY OF SCIENCES WARSAW, POLAND

WILLIAM PAUL PHYSICS DEPARTMENT AND DIVISION OF APPLIED SCIENCES HARVARD UNIVERSITY CAMBRIDGE. MASSACHUSETS

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Contents PREFACE . . . . LISTOF CONTRIBUTORS .

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Chapter 1 Parallel Transport in Low-Dimensional Semiconductor Structures D . K . Maude and J. C . Portal I. Introduction . . . . . . . . . . . I1. The Effect of Pressure . . . . . . . . . 1. Pressure Effects on 2D Electronic Properties of Semiconductor Structures . . . . . . . . . . . . 2. Tuning of the Land6 g-Factor by Hydrostatic Pressure . . . . . . . . I11. Integer Quantum Hall Effect 1. Overview . . . . . . . . . . . . . . . 2. Spin Texture Excitations (Skyrmions) . 3. Zero Hall Resistance in the Semimetallic GaSb/InAs System . IV . Fractional Quantum Hall Effect . . . . . . . 1. Introduction . . . . . . . . . . 2. Composite Fermions. . . . . . . . . V . Magnetophonon Resonance Effect Under Hydrostatic Pressure in GaAs/Alo.~~Ga~,7zAs, Ga, 471n0.s3A~IA1~.4~In0.5~r and in 1% .53As/ InP Heterojunctions . . . . . . G . . . 1. Pressure Dependence of the Effective Mass . 2. Amplitude of the Oscillations and y Damping Factor . . Acknowledgments. . . . . . . . . . References . . . . . . . . . . .

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Chapter 2 Tunneling Under Pressure: High-pressure Studies of Vertical Transport in Semiconductor Heterostructures P . C . Klipstein I. Introduction . . . . I1. Theory and Calculation. . 1. Ricco and Azbel Formulae 2. Transfer Matrix Method .

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3. Calculating the I-V Characteristic . . . . . . . . . . . . . . . 4. Space Charge . 5. Band Mixing . . . . . . . . . . 6. Strain Due to Pseudomorphic Growth . . . . . 111. Experimental Techniques . . . . . . . . IV . High Pressure Studies of Negative Differential Resistance . . 1. Early Days . . . . . . . . . . 2. Resonant Tunneling in Single-. Double-. and Multiple-Barrier . . . . . . . . . Heterostructures 3. Resonant Interband Tunneling. . . . . . . V. Concluding Remarks . . . . . . . . . Acknowledgments. . . . . . . . . . References . . . . . . . . . . .

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Chapter 3 Phonons. Strains. and Pressure in Semiconductors

Evangelos Anastassakis and ManuelCardona I . Introduction . . . . . . . . . . . . 1. Historical Review of Strain Effects . . . . . . . 2. Effects of Stress and Strain on Electrons and Phonons in Crystals . I1. Background . . . . . . . . . . . . 1. Phonons and Crystal Symmetry . . . . . . . 2. Strains. Stresses. and Crystal Symmetry . . . . . . 3. Experimental Techniques for Applying Stresses . . . . 111. Effects of Hydrostatic Pressure on Optical Phonons . . . . 1. Mode Gruneisen Parameters . . . . . . . . 2. Thermal Expansion: Quantum Effects at T = 0 . . . . 3. Phonon Linewidths and Lifetimes . . . . . . . 4. Phase Transitions . . . . . . . . . . IV . Effects of Strains on Optical Phonons . . . . . . . 1. Phonon Deformation Potentials . . . . . . . 2. Phonon Secular Equation . . . . . . . . . 3. Control Experiments . . . . . . . . . 4. Theoretical Models and Trends of Phonon Deformation Potentials 5. Other Uses of Phonon Deformation Potentials . . . . V . Strain Characterization of Heterojunctions and Superlattices . . 1. Elastic and Piezoelectric Considerations in Heterojunctions . . . . . . . . . and Superlattices . 2. Pressure and Temperature Dependence of Strains . . . . 3. Characterization of Strains through Raman Spectroscopy . . VI . Concluding Remarks . . . . . . . . . . Acknowledgments. . . . . . . . . . . Appendix . . . . . . . . . . . . References . . . . . . . . . . . .

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Chapter 4 Effects of External Uniaxial Stress on the Optical Properties of Semiconductors and Semiconductor Microstructures FredH . Pollak I . Introduction . . . . . . . . . . . . . . . . I1. Effects of Homogeneous Deformation on Electronic Energy Levels . 1. Critical Points at k = 0 . . . . . . . . . . . 2 . Bands at k # 0 . . . . . . . . . . . . . 111. Determination of Intervalley Electron-Phonon and Hole-Phonon Interactions . . . . . . . . . in Indirect Gap Semiconductors . IV. Piezo-Optical Response of Ge and GaAs in the Opaque Region . . . . . V . Intrinsic Piezobirefringence in the Transparent Region . . . . VI . Effects of External Stress on Quantum States . . . . . . . 1. Effects of X Parallel to [001] and [011] (Piezoelectric Effect) on an Ino.21Gao.7&/GaAs (100) Single-Quantum- Well Structure . . . . 2. Effects of X Parallel to [001] and [Oll] (Piezoelectric Effect) on a . . . . GaAs/GaAlAs (100) Single-Quantum-Well Structure . 3. Determination of the Symmetry of Excitons Associated with . Miniband Dispersion in InGaAslGaAs (100) Superlattices . . . 4. Asymmetrical GaAslGaAlAs (100) Double Quantum Wells . . . . 5. Effects of X Parallel to [001] on Bulk GaAs and GaAslGaAlAs Single . . . . Quantum Wells Grown on (100) Si Substrates . 6. Symmetry of Conduction States of GaAslAlAs Type I1 (001) Superlattices . . . . . . . . . . . 7. Determination of the Band Alignment in Si,-,Ge, /Si (100) . Quantum Wells. . . . . . . . . . V. Summary . . . . . . . . . . . . VI . Acknowledgments . . . . . . . . . . . References . . . . . . . . . . . .

236 238 238 254 264 266 270 271 272 277 278 278 283 288 290 295 296 296

Chapter 5 Semiconductor Optoelectronic Devices A . R . Adams.M . Silver. and J. Allam I . Introduction .

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111. Semiconductor Lasers . . . . . . . . . 1. Basic Laser Concepts . . . . . . . . 2. Laser Characteristics: Their Pressure and Temperature Dependence . . . . . . . . . . IV . Uniaxial Strain Effects: Strained-Layer Lasers . . . . V . Hydrostatic Pressure Measurements of Avalanche Photodiodes: The Band-Structure Dependence of Impact Ionization . . 1. Introduction . . . . . . . . . . 2. Physics of Impact Ionization in Semiconductors . . . 3. Pressure Results . . . . . . . . . 4. “Universal” Dependence of Avalanche Breakdown on Band Structure . . . . . . . . . . 5. Conclusions . . . . . . . . . .

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I1. Experimental Considerations

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VI . Summary . . Acknowledgments . References . .

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Chapter 6 The Application of High Nitrogen Pressure in the Physics and Technology of III-N Compounds S. Porowski and I. Grzegory I . Introduction . . . . . . . . . I1. Thermal Stability of AIN. GaN. and InN . . . I11. Solubility of N in Liquid Al Ga. and In . . . IV . Kinetic Limitations of Dissolution of Nitrogen in Liquid V . High Nz Pressure Solution Growth of GaN . . . . . . . . . . 1. Experimental . . . . . . . . . 2. Crystals . VI . Physical Properties of Pressure-Grown GaN Crystals . VII . Wet Etching and Surface Preparation . . . . VIII . Homoepitaxy . . . . . . . . . IX. Conclusions . . . . . . . . . . . . . . . . Acknowledgments . References . . . . . . . . .

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Chapter 7 Diamond Anvil Cells in High Pressure Studies of Semiconductors Mohammad Yousu f I . DAC: An Apparatus Par Excellence to Achieve Highest Static Pressure . . . . . . . . . . 1. Diamond as an Anvil Material . . . . . . . 2. Precision Fabrication of a Typical DAC . . . . . 3. Principle of the Alignment of Diamond Anvils . . . 4. Gasketing: A Turning Point in DAC Use . . . . 5. Preparation of the Sample in a Typical DAC Experiment . 6. Pressure Scale and Pressure Calibration . . . . . 7. Pressure-Transmitting Medium and the Limit of Hydrostaticity 8. Pressure Combined with Other Thermodynamic Fields . . . I1. Condensed Matter Physics Techniques Coupled to a DAC . 1. Optical Spectroscopy . . . . . . . . 2. X-Ray Diffraction . . . . . . . . . 3. Transport Properties . . . . . . . . . 111. High Pressure Studies of Semiconductors . . . . . 1. Conventional Semiconductors . . . . . . . . . . 2. Strongly Correlated Semiconductor Systems . IV . Concluding Remarks . . . . . . . . . Acknowledgments . . . . . . . . . . References . . . . . . . . . . . INDEX

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Preface In 1963 two review books were published which reported extensions to the of High monumental work contained in Bridgman’s treatise, The Physics (1931, 1949). The first of these, Solids underPressure, edited by Pressure Paul and Warschauer, included two chapters on semiconductors and the electronic properties of solids in addition to eleven chapters on atomic diffusion, equations of state, phase equilibria and phase transitions, magnetic properties, shock-wave techniques and geophysics. The second, Highin two volumes edited by Bradley, ranged Pressure Physics and Chemistry, more widely over topics in gases, liquids and solids, but also included seven chapters on solids, one of them on semiconductors. In the ensuing period the number of laboratories pursuing research in high pressures has much increased, in some part because of the easier availability of commercial piston-cylinder apparatus, such as that manufactured by the Unipress, in Warsaw, but most particularly because of the advent of the diamond anvil cell technique. This latter has permitted much higher hydrostatic pressures to be achieved in risk-free environments, and spectacular progress has been made in many areas of research. The work on solids has been reported at numerous conferences, sponsored, for example, by the AIRAPT (International Association for the Advancement of High Pressure Science and Technology), the EHPRG (European High Pressure Research Group), and by ad hoc groupings of research workers. There have been reviews and High-pressure Rearticles in journals such as Reviewof Modern Physics, and High Pressures - High Temperatures, and in serial books such search Physics. From the reports it is evident that the research on as SolidState semiconductors has been one of the most enduring endeavors. Thus it is that in this volume we have gathered together a number of review articles on subjects of current interest. In doing so we have drawn heavily on the Semiconducwork reviewed in the biennial conferences titled HighPressure held since 1984, which in 1988 became satellite conferences to torPhysics, the International Conferences on the Physics of Semiconductors (ICPS). We are grateful to the organizers of these conferences for their diligence in establishing a historical record of this subject, which continues to contribute significantly to the understanding of semiconductors under ambient conditions. The book chapters indicate unambiguously the vitality of an ongoing ix

X

PREFACE

investigation, and no doubt the details of the results reported in these chapters will need to be updated in a relatively short time. Nevertheless, we expect that these chapters of critical review, with their extensive bibliographies, will provide a source book for continued investigation for some time to come. With minor exceptions, this book follows that of Paul and Warschauer in presenting only enough of the details of technique to make the subject matter understandable. The exceptions obviously lie in those areas where new strides have been made possible by discontinuous, nonincremental, advances in available techniques. The editors wish to thank the contributors for their friendly collaboration: this is their book, and we have merely helped in assembling their work. The thanks of all of us are due to the staff of Academic Press for their patience and accommodation, and especially to Dr. Zvi Ruder. This volume is the second of two volumes on the subject.

List of Contributors

Numbersinparenthesis indicate thepageson whichtheauthorscontribution begins.

A. R. ADAMS (301), Department of Physics, University of Surrey, Guildford GU2 5XH, United Kingdom J. ALLAM (301), Hitachi Cambridge Laboratory, Hitachi Europe Ltd., Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, United Kingdom EVANGELOS ANASTASSAKIS (1 17), Physics Department, National Technical University, Athens 15780, Greece MANUELCARDONA(1 17), Max-Planck-Institut fur Festkorperforschung, Heisenbergstr 1, 70569 Stuttgart, Germany I . GRZEGORY (353), UNIPRESS, High Pressure Research Center, Polish Academy of Sciences, ul. Sokolowska 29, 01-142 Warsaw, Poland P. C. KLIPSTEIN ( 4 3 , Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom D. K . MAUDE(l), Grenoble High Magnetic Field Laboratory, MPI-FKF and CNRS BP 166, F-38042 Grenoble Cedex 9, France FREDH . POLLAK (235), Physics Department and New York State Center for Advanced Technology in Ultrafast Photonic Materials and Applications, Brooklyn College of the City University of New York, Brooklyn, N Y 11210 S. POROWSKI (353), UNIPRESS, High Pressure Research Center, Polish Academy of Sciences ul. Sokolowska 29, 01-142 Warsaw, Poland J . C. PORTAL(l), lnstitut National Des Sciences Appliqutes, Complexe Scientijique, F-31077 Toulouse Cedex, France M. SILVER(301), Department of Physics, University of Surrey, Guildford GU2 5XH, United Kingdom MOHAMMAD YOUSUF(381), Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, Tamil Nadu, India

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High Pressure in Semiconductor Physics II SEMICONDUCTORS AND SEMIMETALS Volume 55

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SEMICONDLJCTORS AND SEMIMETALS, VOL. 55

CHAPTER1

Parallel Transport in Low-Dimensional Semiconductor Structures D.K . Maude GRENOBLE HIGHMAGNETIC FIELDLABORATORY MPI-FKF AND CNRS GRENOBLE, FRANCE

J.C.Portal GRENOBLE HIGHMAGNETIC FIELDLABORATORY MPI-FKF

AND

CNRS

GRENOBLE. FRANCE

INSTITLIT NATIONAL DES SCIENCES APPI.IOU~-ES COMPLEXESCIENTIFIOUE TDULDUSE, FRANCE

I. INTRODUCTION ................................................................................................................. 11. THEEFFECT OF PRESSURE ..............................

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1. Pressure Effects on 2 0 Electronic Prope 2. Tuningof theLandkg-Factor by Hydrostatic Pressure ....................................... 111. INTEGERQUANTUM HALLEFFE ~r ............................................................................... 1. Overview ..................................................................................................................... 2. SpinTexture Excitations (Skyrmions) ................... 3. ZeroHaN Resistance intheSemimetallic GaSb/l ............................. IV. FRACTIONAL QUANTUM HALLEFFECT ........................ 1. Introduction .............. 2. Composite Fermions .................................................................................................. V. MAGNETOPHONON RESONANCE EFFECT UNDERHYDROSTATIC PRESSURE IN G a A s I A l o . ~ ~ G a o , 7 2 AG s, AND Gao.471no.53AsllnP HETEROJUNCTIONS ..... ...................................................... 1. Pressure Dependen ........................................................... 2.Amplitude oftheOscillations and y DampingFactor ......................................... Acknowledgments ...................................................... References ...................................................................... .........

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I. Introduction In this chapter we review the properties of two-dimensional (2D) electrons and holes in the presence of high magnetic fields applied perpendicular 1 VOl. 55

ISBN 0-12-752163-1

SEMICONDUCTORS AND SEMIMETALS Copynghl 0 1998 hy Academic Press All rightc of reproduction in any form reserved. W80/8784/98$25.00

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D. K. MAUDEAND J. C. PORTAL

to the 2D layer. Historically, two types of systems have been investigated: electrons on the surface of liquid helium (Williams et af., 1971; Grimes, 1978) and electrons (or holes) in semiconductor structures. For hydrostatic pressure investigations, only the semiconducting system is relevant. Early work concentrated on the metal-oxide semiconductor (MOS) structures based on the Si-Si02 system (for a review see Ando et al., 1982), which resulted in the discovery of the integer quantum Hall effect (IQHE) (von Klitzing etaf.,1980). The development of growth techniques such as molecular beam epitaxy allowed the realization of high-mobility ( p > lo6 cm2/V-’s-’) modulationdoped heterojunctions or quantum wells based on III-V semiconductors. The most commonly investigated system is the GaAs/AlGaAs heterojunction. Due to the different bandgaps of the two semiconducting materials, a two-dimensional electron gas (2DEG) is formed at the interface, usually due to charge transfer from a remote doping layer (Fig. 1).High mobility results from both the atomically flat interfaces and the greatly reduced ionized impurity scattering due to the inclusion of an undoped spacer layer (-100 nm) that spatially separates the electrons from the ionized donor impurities. It was in such a GaAdAIGaAs heterojunction system that the fractional quantum Hall effect (FQHE) was discovered (Tsui etal., 1982). High hydrostatic pressure has been recognized as a powerful tool in the study of semiconductor physics and, during the last decade, particularly in the field of low-dimensional systems studied in parallel tansport. The direct effect of applying hydrostatic pressure to semiconductors is to decrease the interatomic distance. Pioneering pressure studies by William Paul showed that even if the decrease of lattice constant is not large (generally of the order of 1% at 10 kbar). it is sufficient to produce the following effects: 1. Pronounced, significant shifts in the electronic states. Deep electronic levels are created by impurity dopant or any defects and consequently change the charge carriers transferred to the quantum well. 2. A change in the energy of the band structure of each semiconductor and therefore the band-structure offset at the interface between the two components of the heterostructure. The electronic transport effects are more pronounced in type I1 heterostructures (Beerens et al., 1987a,b) with both electrons and holes (InAs/GaSb heterostructures) in comparison with type I heterostructures with only electrons or holes (AlGaAs/GaAs heterostructures). High pressure induces a phase transition from a semimetal to a semiconductor. 3. A change in the quantum effect observed in the oscillatory behavior of the resistivity. such as the magnetophonon resonance (MPR), in

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

A l p 1JS n-type

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GaAs

P-tYPe

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FIG. 1. Schematic of the conduction and valence band structure in a typical GaAs/Al,Ga,~1 As heterojunction.

which the dependence of the effective mass, the scattering, the electron-phonon interaction, and 2D screening effects have been studied with hydrostatic pressure. 4. Reduction of the magnitude of the Land6 spin g-factor, change the relative strengths of the different fractions of the quantum Hall effect, and the energy gap in the first composite fermion hierarchy. Unlike other perturbations applied to study low-dimensional structures, such as electric and magnetic fields, alloying, or uniaxial stress, hydrostatic pressure preserves both the crystal symmetry and the atomic order. In heterojunctions the electrons (or holes) have quantized energy levels in one spatial dimension but are free to move in two spatial dimensions. The system is not strictly two-dimensional due to the finite spatial extent of the wave function and the penetration of the wave function into the

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D. K. MAUDEAND J. C. PORTAL

barrier. In addition, the applied magnetic field is not confined to the plane of the 2DEG, which has important consequences for Zeeman and orbital energies, which depend, respectively, on the total and the perpendicular component of the magnetic field. An applied magnetic field quantizes the electron or hole motion into Landau levels, and the induced gaps in the density of states lead to new effects and fascinating new physics, such as the integer and fractional quantum Hall effects. Hydrostatic pressure, which can be used to vary the bandgap of the semiconducting material and hence tune the Land6 g-factor through the spin-orbit interaction, is an extremely powerful tool for probing both the integer and fractional quantum Hall effects. The main advantages of high-pressure applications in complementary study of gated microstructures are the possibility of tuning the 2D carrier concentration without a change in the band-structure parameters and of tuning just the band structure of low-dimensional systems. Thus, the application of high pressure for systematic and controllable tuning of the electronic states seems ideally suited to fundamental studies of the electronic structure of low-dimensional systems and devices.

II. The Effect of Pressure EFFECTSON 2D ELECTRONIC PROPERTIES OF 1. PRESSURE SEMICONDUCTOR STRUCTURES Hydrostatic pressure coupled with a high magnetic field is commonly used in studies of magnetotransport effects such as Shubnikov-de Haas oscillations (Shubnikov and de Haas, 1930) and classical and quantum Hall effects. This technique was applied to the GaAslAlGaAs system, where it was found to cause a strong linear decrease in 2D carrier concentration. The effect of hydrostatic pressure on 2D systems is a reduction of 2D carrier concentration. Nevertheless, depending on the heterojunction components and the nature of the doping impurities, different pressure effects are predominant for different heterostructures under study (Dmowski and Portal, 1989; Grkgoris etal., 1987a). The observed decrease of 2D carrier concentration can be related either to the deeping of the donor levels in the doped layer of modulation-doped heterostructures and consequent electron transfer from the quantum well to the localized donor states (Beerens etal., 1988) or to the reduction of the band-structure discontinuity and intrinsic charge transfer between the two components of undoped heterostructures (Beerens etal., 1987a; Gauthier etal., 1987).

TRANSPORT I N LOW-DIMENSIONAL STRUCTURES 1 PARALLEL

5

of Donor Levels intheBarrier and Control Dependence a. Pressure Process of Carrier Concentration in theQuantum Well In GaAs/AIGaAs and GaInAs/AIInAs modulation-doped heterostructures in which a silicon impurity is used as a dopant, a deep electronic level in the barrier of the AlGaAs and AlInAs doped layers is created, which moves down into the gap relative to the r minimum when pressure is applied. Therefore the free-electron concentration in the doped layer, and consequently the charge transfer to the quantum well (GaAs and GaInAs), is reduced. Beerens etal.(1988) have developed a very useful model that enables one to account for the case in which both the quantum well and the doped layer conduct (e.g., high-temperature measurements). Such parallel conduction is particularly large in AlInAs doped layers and can strongly effect magnetotransport measurements and lead to mistaken characterization of the 2D electron gas by the classical Hall effect. Figure 2 shows as a function of pressure both the Hall concentration of a GaAs/AIGaAs heterostructure and the 2DEG concentration in a quantum well measured by Shubnikov-de Haas following a typical recording reported in Fig. 3. Pressure induces a decrease in both the 2DEG concentration and the 3D free-electron concentration in the doped layer. At sufficiently high pressures the parallel conduction through the doped layer becomes negligible compared with the contribution of the 2DEG. Thus, pressure allows one to eliminate parallel conduction and to verify whether the model used to discriminate the two components is correct (GrCgoris et al.1987b). This fact was used to study the 2DEG in a GaAdAlGaAs heterojunction at high temperatures, where it was shown that the 2DEG concentration increases with temperature and that the Si level in AlGaAs is temperature dependent (Beerens etal., 1988). This method was also used to deduce the electron concentrations in different layers of the GaInAdAlInAs heterojunction and the real mobility of the 2DEG. A drastic reduction in the parallel conduction at pressures higher than 7 kbar also allowed the first observation of the quantum Hall effect, which has never been observed in GaInAdAlInAs at ambient pressure (Fig. 4). This observation was only possible at high pressure, when the parallel conduction is strongly reduced. At low temperature, Fig. 5a shows typical Shubnikov-de Haas recordings at different pressures. Two series of oscillations are revealed, indicating that two subbands Eo and El are occupied. The electron concentrations No and Nl of these subbands, as deduced from the period of the oscillations, decrease with pressure (Fig. 5b). The total concentration No + N I agrees with the low-field Hall value NH = l / e R at H all pressures.

D. K. MAUDEAND J. C. PORTAL

6

0

5

10

15

PRESSURE (kbar) FIG. 2. Hall concentration as a function of pressure in a GaAs/Al,, Ga0.,,As heterostructure with drpncer = 300 A. The symbols are the experimental points. The full curves give the 2DEG concentration, and the broken curves represent the Hall concentration as calculated with the triangular-well model. (From Beerens ef al., 1988.)

1

PARALLEL TRANSPORT IN

LOW-DIMENSIONAL STRUCTURES

1200 I

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6

MAGNETIC FIELD (T) FIG. 3. Shubnikov-de Haas pTl( B )and quantum Hall p,,(B) effects measured for GaAsIAlGaAs at atmospheric pressure for two different cooling pressures p c :pCl= 1 bar (solid line) and pcz= 9 kbar (dashed line). The sheet electron concentrations are 3.90 X 10" cm-2 and 2.27 X 10" cm-' for pcl and p C z respectively. . (From Dmowski and Portal, 1989.)

In a GaInAdAlInAs heterojunction, the effect of pressure is to decrease the 2DEG concentration at a lesser rate than that of the 2DEG concentration measured in the GaAdAlGaAs heterojunction (Fig. 2). For the two cases, therefore, the decrease of the carrier concentration is stronger (a deepening of donor level in the barrier) than in the case in which the pressure affects only the band structure, as in GaInAs/InP heterojunctions (see the next section). Hydrostatic pressure has permitted a quantitative study of parallel-conduction effects in GaInAs/AlInAs heterostructures. The activation energy of the Si donor level in AlInAs is found to increase with pressure at a rate of 5 ? 1 meV kbar-I; this causes a strong reduction in the conductivity of the AlInAs layer (and therefore in parallel-conduction effects) as well as a decrease of the 2DEG concentration. At a pressure of about 7 kbar, the parallel conduction becomes negligible and Hall plateaux appear in pXy(B). This occurs while there are still two subbands populated in the well and therefore shows that the quantum Hall effect is not incompatible with the occupation of two subbands, but is quite sensitive to the presence of a parallel-conducting path. It has been shown that pressure can eliminate the parallel conduction in GaAs/Al,Ga, -,As heterostructures due to the deepening of the donor level

8

D. K. MAUDE AND J. C. PORTAL

5

10

15

MAGNETIC FIELD ( l ) FIG. 4. (a) Shubnikov-de Haas p,,(B)and (b) Hall p,,(B)recordings for a GaInAs/AIInAs heterojunction at 4.2 K for several 1987b.) pressures at high magnetic field. (From Grkgoris etal.,

in Al,Ga,-,As. This fact has been used to observe the properties of the 2DEG at high temperature, and a temperature variation of its concentration is found that is accounted for by a temperature dependence of the Si donor activation energy in A1,Gal-,As. As we can see, the r-band concentration in A10.28Ga,,72As,which is mainly responsible for the parallel conduction, drops rapidly with pressure. In the higher-pressure range, the X-band concentration increases because the X band and the donor level get slowly closer to each other due to slightly different pressure dependencies. The presence of the X-band electrons does not lead, however, to a significant parallel conduction because of their very low mobility.

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

9

h

N

6

2"

100

u 2

0

1 MAGNETIC FIELD (T)

0

5 10 15 PRESSURE (kbar)

FIG. 5. (a) Shubnikov-de Haas recordings for a GaInAslAlInAs heterojunctionfor several pressures at 4.2 K, given by curves (in kbar). (b) The carrier concentration measured from the Shubnikov-de Haas (triangles) and low-field Hall effect (squares) studies. No and N 1are the concentrations of the subbands E(,and E l .The line is the calculated 2D electron gas (From GrCgoris etal.. 1987b.) concentration (Ns).

The donor level in doped layers of AlGaAs increases at a rate of about 11meV kbar-' and is the same as that found in low-temperature conditions where there is no parallel conduction. However, application of pressure not only decreases the 2DEG concentration but also modifies the band structure of 2D systems. In some cases it can be desirable to change the carrier density while keeping the bandstructure parameters constant. Such a tuning is possible because in practically all the cases in which the decrease of the 2D carrier concentration is related to the shift of donor levels with pressure, it appears that these levels have a metastable (DX-like) character and the occupation of these levels at low temperatures can be frozen in an arbitrary state, different from that corresponding to thermodynamic equilibrium. Figure 3 presents the Shubnikov-de Haas and quantum Hall effects measured at 4.2 K and atmospheric pressure. The two different 2D carrier concentrations correspond to two different cooling pressures of 0 and 9 kbar, respectively.

10

D. K. MAUDEAND J. C. PORTAL

Dependence of theBand Structure and theConsequent b. Pressure Changein the2DEG Concentration of theQuantum Well

The pressure can change the band-structure offset at the interface, which leads to charge transfer between the two components of the heterostructure and consequently to a change in the 2DEG concentration in the quantum well. This was observed in pressure studies of a nonintentionally doped G a I n A s h P heterojunction with three electric subbands occupied (Gauthier etal., 1986, 1987). The application of pressure revealed a small decrease (1%per kbar) of the total 2DEG concentration. This result contrasts with GaAdAIGaAs modulation-doped heterostructures, in which this very important change of 2D electron density with pressure suppresses less important effect of pressure on other parameters that interfere in the description of quantum well and 2DEG behavior, such as conduction-band discontinuity, electron effective mass, and the positions and the distances between electric subbands. However, when donor levels in the doped layer are shallow or are placed high enough above the bottom of the conduction band, they do not contribute to the change in free-electron concentration with pressure, and the other pressure effects become more pronounced. This fact yields an opportunity to study these pressure effects in the GaInAs/InP heterojunction, which is just such a case. Of the three populated electric subbands in the sample that was studied, only two contributed to the quantum Hall effect (Razeghi etal., 1986). For a range of pressure up to 15 kbar, Shubnikov-de Haas and quantum Hall effect measurements, including how-field Hall effect, were performed at 4.2 K and are shown in Fig. 6 and Fig. 7. These values permit the deduction of the total electron concentration from high-field Shubnikov-de Haas measurements and from the position of the plateau ( n = 2) of the quantum Hall effect. The values are very close to those obtained by low-field Hall effect measurements, pointing out the fact that no parallel conduction has to be considered in this sample (Fig. 8). The magnitude of the observed decrease of the total electron population (1% per kbar) agrees well with the pressure change of the conduction-band discontinuity AEc in the tiangular well approximat ion. It is worth noting that the difference in the pressure coefficients of the gaps for a GaInAs/InP heterojunction is about three times greater than for the GaAs/AlGaAs system. No and N ,,the populations of the ground and first excited subbands, are deduced from the two sets of low-field Shubnikov-de Haas oscillations. Figure 9 shows the dependence of N , ,N o ,N , ,and N2 on pressure. N , ,the

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

0

5

10 15 MAGNETIC FIELD B (T)

11

20

FIG. 6. Shubnikov-de Haas recordings of G a I n A s h P heterojunction for five different pressures: A (15 kbar), B (12.1 kbar), C (8.8 kbar). D (4 kbar), and E (atmospheric pressure); T = 4.2 K. (From Gauthier cf aL, 1986.)

population of the third subband, is deduced from the difference between the total electron concentration N s and ( N o+Nl): hence the results for the evolution of this subband are not really accurate due to the addition of experimental errors. We assume the population of N2 to be fairly constant, with a value of 0.28 X 10" cm-'. A decrease of 0.95 X 10" cm-2 in total population leads to a decrease of 0.64 X 10" cm-l for the ground subband and 0.31 x 10" cm-' for the first excited subband, the third subband keeping roughly the same populations (Fig. 9). To confirm the population of the third subband even at high pressure, we applied a parallel magnetic field (0 = 900) (Fig. 10). At 15 kbar, we see a fall in resistance at about 1.5 T due to the suppression of intersubband scattering, indicating that the E2 subband is not entirely empty at zero magnetic field. The contribution of intersubband scattering decreases with increasing pressure. From low-field Hall effect experiments for each pressure applied, we deduced the Hall mobility of the 2DEG (Fig. 11). We noticed a decrease of about 30% in mobility for pressure up to 15 kbar. A fall in mobility of

D. K. MAUDEAND J. C .PORTAL

12

0

5

10 15 MAGNETIC FIELD 6 (T)

20

FIG.7. Quantum Hall effect recordings of GalnAdInP heterojunctionfor four different pressures: A (15 kbar), B (8.5 kbar), C (4.2 kbar), and D (atmospheric 1986.) pressure); T = 4.2 K. (From Gauthier et al.,

the same order of magnitude (28%) has been observed for pressure up to only 8.8 kbar (Sotomayor Torres etal., 1986) but in a sample displaying strong parallel conduction. The bandgap of GaInAs is pressure dependent, dEoldP= 10.7 meV kbar-' (Adachi, 1982); this gives rise to a change in the carrier effective mass (Hermann and Weisbuch, 1977). Magnetophonon resonance (MPR) experiments performed by Shantharama et al. (1985) in bulk GaInAs show an effective mass dependence on pressure of 1.9 t 0.15% kbar-'. Nevertheless, experiments performed in the heterojunction itself show a rate of increase of mass of 1 -+ 0.1% kbar-' (Gauthier etal., 1987), which is not sufficient to explain the observed fall of mobility where a strong dependence of mobility on effective mass is expected for GaInAs due to alloy scattering (Walukiewicz et al., 1984). Elsewhere, these values, which are lower than for bulk GaInAs alloy, have been tentatively explained by the difference in the compressibilities of GaInAs and InP inducing uniaxial stress and changing the nonparabolic contribution to the band structure and the electron-phonon interactions. Other scattering mechanisms, such as screening and alloy scattering, must be considered as being partly responsible for the observed mobility decrease with pressure. The most surprising effect for a GaInAs/InP heterojunction

1 PARALLEL TRANSPORT IN LOW-DIMENSIONAL STRUCTURES

V

5h

V

-

N

6

-

r

-0

4-

-

F

v)

z w

n

211

1

1

1

1

1

1

1

1

1

1

1

1

1

- *---a- - -?- -3

--I-I

$--AI

I

5

,

-_ ,

I

10 PRESSURE (kbar)

I

I

I

15

FIG. 9. Evolution of the population of the different subbands as well as total electron concentration versus pressure. 0 denotes N,;0, N O :0. N , ; A,N z . (From Gauthier e t d ,1986.)

13

D. K. MAUDEAND J. C. PORTAL

14

s 0

2

4

6

8

MAGNETIC FIELD B (T) FIG. 10. Resistivity as a function of parallel magnetic field for different pressures: A (15 kbar), B ( 6kbar), and C (atmospheric pressure). (From Gauthier etal.. 1986.)

was that althought the decrease of the total electron concentration at 15 kbar considerably surpassed the population of the third electric subband, this subband remained occupied with a fairly constant population (Fig. 9). This means that pressure does not depopulate the electric subbands successively. The electron effective mass for the sample studied was determined to be m = 0.0485 mofrom cyclotron resonance measurements at 0 kbar (Nicholas etal., 1985). We have calculated the Fermi energy for each Eisubband (in the parabolic approximation) as

for each pressure, taking into account the increase of mass. Hence we deduce the evolution of the gaps EI-Eo and E2-Eo(Fig. 12) with hydrostatic pressure, where Eo is the ground subband. These gaps decrease because the subbands do not depopulate with the same efficiency. The effect of pressure is thus twofold: It lowers the Fermi energy and decreases the gap between the electric subbands (Fig. 12). Although the Fermi level goes down relative to the ground electric

1 PARALLEL TRANSPORT IN LOW-DIMENSIONAL STRUCTURES

PRESSURE (kbar) FIG. 11. Hall mobility versus pressure in GaInAslInP heterojunction at 4.2 K up to a pressure of 15 kbar. (From Gauthier er al., 1986.)

30

h

G 0 u

I

l

25 -

r

r

b-

20L”--

-0-

0-

-

2E

v

,

15 L-

--A-

l

l

--

l

l

l

l

l

l

-

Q-,

-a--

l

l

l

l

l

l

l

l

l

l

,

l

l

,

l

-

”---e,

‘8

-

--4-

10 -

~

-I -Q---

P--

-

-Q---

‘k-

-

EF E2 E,

’ 0

--El

L - --A,--A

-

-

-

5-

-

-

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

PRESSURE (kbar) FIG. 12. Evolution of the positions in energy of the excited subbands El and E2 as well as the Fermi level E F .The origin is taken at the ground subband Eo. (From Gauthier et al., 1986.)

15

16

D. K. MAUDEAND J. C.PORTAL

subband, the second and third subbands also go down with pressure, the third subband keeping a fairly constant distance from the fermi level (Fig. 12). This indicates the strong opening of the quasi-triangular quantum well, which becomes more obtuse when the pressure increases. The considerable broadening of the quantum well (Gauthier etal., 1987) was confirmed by the difference in the spread of the wave functions of the ground and the next-higher-lying subband estimated for different pressures. The same pressure-dependent behavior of the carrier concentration of each electric subband is shown by the CdTe/InSb heterostructure system (Alikacem etal.. 1990). Investigating the electrical properties of single heterostructures consisting of a layer of nominally undoped CdTe grown on p-type InSb substrates, low-temperature Shubnikov-de Haas measurements indicate the presence of a two-dimensional electron gas at the interface, with at least six occupied electrical subbands and a total electron sheet density n = 6.7 X 10I2cm-2. The application of hydrostatic pressures up to 15 kbar reduces n to 2.6 X lo'* cm-2 and approximately doubles the electron mobility. This behavior is related to the existence of donorlike defect states at the interface that act to pin the Fermi level of the 2DEG. The electron sheet density in the ith subband is given by ni = 2epi/h, where P;' = [A(l/B)Ii is the frequency of the Shubnikov-de Haas oscillations. Spin splitting is not resolved. By Fourier transforming the magnetooscillations in 1/B, the pi value is determined for each subband contributing to the magnetoresistance. Such a transform is shown in Fig. 13 for 6 = 0". Six subbands are clearly resolved. The effect of hydrostatic pressure on the occupancies of the subbands is shown in Fig. 14. It can be seen that a pressure of 15 kbar reduces n by more than a factor of two. However, at least six subbands remain occupied up to pressures of at least 10 kbar. Although increasing pressure leads to a large drop in the total sheet density, the conductivity decreases by only 10%between 0 and 15 kbar. Hence, the mean mobility increases by a factor of two from its value at atmospheric pressure of p = 3200 cm2/V.s (T = 4.2 K). This indicates that the amount of charged-defect scattering, which limits the mobility, is markedly reduced by the application of pressure. The strong pressure dependence of n indicates that shallow donors arising from cross-doping are not the origin of the 2DEG, since, for low pressures, such donor levels would remain close to the conduction band edge with increasing pressure. A more plausible explanation is that the high density of conduction electrons arises from donorlike defect states at the interface. The interface states have a much weaker pressure dependence because they are highly localized and should not have the character of the r conduction band minimum. Therefore, when pressure is applied, conduction elec-

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

MAGNETIC FIELD P(T) FIG. 13. Fourier transform of the first derivative of the magnetoresistance for 0 = 0". The observation of six clear peaks indicates that at least six subbands are occupied. The (From occupancies are given by n, = 2ep,/h. Alikacem er aL, 1990.)

5 10 15 PRESSURE (kbar)

0

FIG. 14. Pressure dependence of the carrier concentrations ni in the six lowest subbands (i = 0-5). obtained from the Fourier transform spectra. (From Alikacem er al.. 1990.)

17

18

D. K. MAUDEAND J. C. PORTAL

trons trap out onto the interface states, which act to pin the Ferrni level (Hermann er al., 1988). A plausible band diagram for the CdTe/InSb heterostructure system is shown in Fig. 15. The high electron sheet density observed in a single InSb/ CdTe heterostructure arises from the presence of interface detect states of donorlike character. The pressure dependence of the Shubnokov-de Haas oscillations indicates that these states act to pin the Fermi level of the 2DEG and to limit the value of the low-temperature mobility. OF THE LAND^ g-FACTOR BY 2. TUNING HYDROSTATIC PRESSURE

Hydrostatic pressure reduces the lattice constant and hence increases the direct bandgap in 111-V semiconductors. For the quantum and fractional Hall effects in the GaAs/AIGaAs system, since the conduction and valence band offsets remain almost unchanged, the most important consequence is the change in the Land6 g-factor, which is modified via the spin-orbit interaction. In contrast, in the semimetallic GaSb/InAs system the conduction and valence band overlaps are strongly modified, which allows one to tune the electron and hole concentrations. n-CdTe

p-lnSb Increasing

FIG. 15. Schematic diagram of the band bending at the CdTelInSb interface. The donorlike defects are distributed across the interface and act to pin the Fermi level as pressure is applied. Ionized defect = +;defect following electron capture = 0.The energy of the lowest subband, i = 0. and the Fermi energy are shown. (From Alikacem er uf.. 1990.)

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

19

For the IQHE and the FQHE in GaAs/AIGaAs heterojunctions, the small value of the Land6 g-factor ( g = -0.44 in bulk GaAs) has important consequences. The Zeeman energy is generally small compared with the energy scale of the electron-electron interactions, leading to spin-unpolarized fractional states and to complex spin texture excitations (Skyrmions). The possibility of using hydrostatic pressure to tune the g-factor, and hence the Zeeman energy, through zero therefore makes hydrostatic pressure a powerful tool for investigating such effects. The pressure required to achieve the g = 0 limit in a GaAs/AlGaAs heterojunction is -17 kbars (Nicholas etal., 1996), slightly beyond the pressure of -15 kbars attainable with a conventional Cu-Be liquid clamp cell. For this reason it is advantageous to use a modulation-doped quantum well rather than a heterojunction. For quantum wells, the confinement energy, together with the increased penetration of the wave function into the barrier, acts to reduce the g-factor. The Land6 g-factor for an isolated quantum well can be calculated by allowing for the penetration of the wave function into the barrier following the method of Ivchenko and Kiselev (1992). The bulk band-edge g-factor was first calculated for the barrier (gs) and the well (gA) using the fiveband k.p approximation of Hermann and Weisbuch (1977):

The interband coupling and band-structure parameters used are defined in Table 1. The principal reason for using a five-band model is that it gives the correct value for the bulk GaAs (g = 0.44). The bulk value in each material g A , B is then corrected according to the relation AgAqB = h+BKi,B,where the coefficient hpiB (which depends on the interband matrix element, the bandgap and the spin-orbit splitting) is taken from Ivchenko and Kiselev (1992):

D. K. MAUDEAND J. C .PORTAL

20

TABLE I BANDSTRUCTURE PARAMETERS USEDTO CALCULATE THE LAND^ g-FACTOR IN AN ISOLATED~ . 3 5 G ~ . 6 s A s l G a A s I Alo.~Gaob~As QUANTUM WELL

1.519 0.341 4.659 4.488 0.07 28.9 6 0.02

1.94 0.32 -4.659" -4.488" 0.1 24.4 -6" 0.02

Nore. The

values are taken from Hermann and Weisbuch (1977) and Cohen and Marques (1990). " Assumed to be the same as for GaAs.

The wave vectors K A and K B are determined by the envelope of the wave function (Bastard, 1988) of the lowest electric subband in a well of width LA,

with

where L A is the well width, V is the conduction band offset and mA,B is the effective mass in the well and barrier. The constants A and B together with the wave vectors K A and KB are determined by matching +(z) and

1 d -$ ( z )at the interface ( z = I!,,&) and by normalizing the wave ~ A , dz B

function such that

I" -m

+(z)' dz = 1.

Finally, the effective g-factor is calculated by taking an average of the g-factor in the barrier and well weighted by the probability of finding the electron in each layer: LA/2

g = (gA

+ hf' K i )

1 $(Z)* dz

-Lu2

m

f

2(gB - h? K i )

1

LAO

$(Z)'

dz

(8)

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

0.50 . . . I

21

. . ,. . , . . . . . . ,. - . .

I

I

I

I

I

I

I

.-

_.____

6.8 nm QW

'

..... ... 10 nm QW

6, 9)

U t

m

-I

I . . . . I . . . . , . . . , I , . . . I . . . .

0

5

10

15

20 25 30

Pressure [kbar] FIG. 16. Pressure dependence of the Land6 g-factor for different quantum well widths calculated using the procedure described in the text and having the band-structure parameters indicated in Table I.

The calculated variation of the g-factor with hydrostatic pressure is shown in Fig. 16 for the GaAslAlGaAs system and for quantum well widths of 6.8, 10, and 15 nm. With decreasing well width, the pressure required to achieve the condition g = 0 decreases due to the increasing effective bandgap and the increased penetration of the wave function into the barrier. We note that, to a good approximation, the variation of the g-factor around g = 0 is linear and the slope is almost unchanged when compared with calculations that negelect the penetration of the wave function into the barrier. This is to be expected since the pressure changes the g-factor by modifying the spin-orbit interaction but has almost no influence on the penetration of the wave function into the barrier.

111. Integer Quantum Hall Effect 1. OVERVIEW There are a number of excellent reviews of the quantum Hall effect (see, for example, Chakraborty and Pietilainen, 1995) and thus we limit the introductory discussion here to a rather elementary hand-waving explanation of the basic physical principles involved. The integer quantum Hall effect observed in a two-dimensional electron gas can be explained within the framework of a single-electron picture

22

D. K. MAUDEAND J. C. PORTAL

and is a direct result of Landau quantization of the electron motion. The application of a magnetic field (B) adds a parabolic potential term to the Hamiltonian and for sufficiently high magnetic fields quantizes the orbital motion, so that the density of states consists of discrete, well-defined Landau levels of degeneracy 2eBlh(spin up and spin down) separated by an energy of hw, = heB/m*, where m* is the effective mass for carrier motion. Thus, there is a cyclotron gap in the density of states at the Fermi energy at even integer Landau filling factors Y = n,/(eB/h), which leads to exactly quantized plateaux in the Hall resistance R H = ( e 2 / h )When / v . the Zeeman term g p B B is included, the spin degeneracy is lifted and gaps in the density of states also occur at odd filling factors (spin splitting). Electrical resistance measurements are typically performed using a Hall bar geometry. At even integer filling factors, the Fermi level in the bulk lies in localized states between the highest filled and lowest empty spin degenerate Landau levels, and a,, = pxx = 0. Electrical current is carried by edge states: Electrons with their orbit center close to the edge of the sample execute skipping orbits, which renormalizes the Landau-level energies. Each filled Landau level in the bulk gives rise to a one-dimensional conducting channel at the edges of the sample. The conduction is dissipationless because backscattering would require an electron to be scattered from one edge of the sample to the other (Biittiker, 1988; Haug, 1993). 2. SPINTEXTURE EXCITATIONS (SKYRMIONS) While the IQHE described above can be completely understood in a single-particle picture, electron-electron interactions nevertheless can play a significant role in modifying the energetic size of the gaps in the density of states. It has long been known that at odd integer filling factors the (spin) gap is considerably enhanced when compared with the single-particle gap (Nicholas etaf.,1988; Usher etal., 1990). In particular, at filling factor v = 1, while the ground state is a ferromagnetic single-electron state, the excitation spectrum has been predicted (Bychkov etal., 1981; Kallin and Halperin, 1984; 1985) to consist of a many-body spin wave dispersion

where tn= is the magnetic length and lois a modified Bessel function. In accordance with Kohn's theorem (Kohn, 1961), optical measurements probe the neutral excitation at k = 0 and thus give a value for the bare

1 PARALLEL TRANSPORT. I N LOW-DIMENSIONAL STRUCTURES

23

gap E(0) = gpBB (Dobers et al., 1988). Transport measurements, on the other hand, are sensitive to the charged large wave vector limit E ( w )= g , u BB + (e’ .\/;;72/dB). This can be understood in the following way: The excitation flips a single spin, leaving a quasi-hole behind in the otherwise full lowest-spin Landau level. This quasi-electron-hole pair forms an “exciton”, which is a neutral particle and therefore cannot contribute to electrical transport. In order to contribute to the current, this exciton must be dissociated. The measured transport gap is thus enhanced by e 2 m / d B ,which corresponds to the Coulomb energy required to separate the quasi-electron-hole pair. The spin wave dispersion model successfully accounts for the many-body enhancement of the spin gap at v = 1 deduced from thermally activated transport, although the absolute value of the enhancement is somewhat overestimated. Theoretical work (Sondhi et al., 1993; Fertig etal., 1994) suggests that in the limit of weak Zeeman coupling, while the ground state at v = 1 is always ferromagnetic, the lowest-energy charged excitations of this state are a spin texture known as Skyrmions (Skyrme, 1961; Belavin and Polyakov, 1975). The expected experimental manifestations of Skyrmions are (1) a rapid spin depolarization around v = 1 and (2) a 50% reduction in the gap at v = 1 compared with the prediction for spin wave excitations. A considerable amount of experimental evidence now exists to support the theoretical picture of spin texture excitations: The spin polarization around v = 1 has been measured by nuclear magnetic resonance (Barrat et al., 1995) and by polarized optical absorption measurements (Aifer et al., 1996). In addition, transport measurements have been performed to investigate the collapse of the spin gap at low Zeeman energies (Schmeller etal., 1995; Maude etal., 1996). The size and energy of the Skyrmions depend on the ratio of the Zeeman and Coulomb energies, q = [ ( g p B B ) / ( e 2 / d B ) ]g P 2 cos 8, where 8 is the angle between that magnetic field and the normal to the plane of the 2DEG (B, = B cos 6).The correct regime to observe Skyrmions (q < 0.01) can thus be obtained in two ways: (1) working at low magnetic fields, 77 can be tuned (increased) by rotating the magnetic field away from the normal or (2) hydrostatic pressure can be applied to tune the g-factor, and hence 77, through zero. The first approach, successfully applied by Schmeller et al. (1995), has the disadvantage that at low magnetic fields it is not evident that Landau level mixing can be neglected (Kralik et al., 1995). Hydrostatic pressure has been used to tune the g-factor through zero in an AlGaAs/GaAs/AlGaAs modulation-doped quantum well with a well width of 6.8 nm (Maude et al., 1996). In this experiment the thermally activated transport gap at filling factor v = 1 was measured for a number of different pressures between 0 and 8 kbars. At each pressure the carrier

D. K. MAUDEAND J. C. PORTAL

24

concentration was carefully adjusted by illuminating the sample with pulses of light so that v = 1 occurred at the same magnetic field value of 11.6 T. For a 6.8-nm quantum well, the g-factor calculated using a five-band k.p model as described in Section I1 is zero for an applied pressure of 4.8 kbars. The dependence of the spin activation gap at v = 1 as a function of the g-factor is shown in Fig. 17. In contrast to the prediction of the spin wave approach (short dashed line), a deep minima is observed around g = 0. The expected variation for Skyrmion-type excitations is indicated by the solid line. The data reproduce well the expected 50% reduction in the spin gap, although the minima is significantly wider than predicted. Schmeller et al.(1995), using the derivative of the spin gap versus the Zeeman energy, estimated that s = 7 spins are flipped in the region 0.01 5 7) 5 0.02. Lines with slopes corresponding to s = 7 and s = 33 spin flips are shown in Fig. 17. While for lql 2 0.004 the data are consistent with s = 7, the slope around g = 0 implies a Skyrmion size of s = 33 spins. 1997a) on heterojunctions under pressure More recent work (Leadley elal.,

30'o.oo4

0

-0.002 \I

1

I

0.002 I

I

1

0.004

I

I

I/

P-----------L -----___________________________________---/:-----------=

lo-, -0.10

I

I

I

-0.05

I

I

,

I

,

-

, ,

0 g-factor

I

I

I

0.05

I

,

I

I -

0.10

Fit;. 17. The symbols indicate the measured gap at u = 1 (-11.6 T) as a function of the Land6 g-factor for a 6.8-nmquantum well (Maude etal.. 1996). The ratio of Zeeman and Coulomb energies, q = [ ( g p B B ) / ($/&la)] is indicated for reference. The solid line is the expected variation

of the gap with g-factor calculated for a Skymion-type excitation (Sondhi etnl.. 1993). while the short dashed line indicates the "bare" Zeeman dependence SlglpBB+ Es with s = 1 as predicted by the spin wave dispersion model. The long dashed and long-short dashed lines have slopes corresponding to s = 7 and s = 33 spin Aips, respectively.

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

25

shows a similar minima around 18 kbars corresponding to g = 0. The data are consistent with s = 35 spin flips, although the spin gap is reduced somewhat more than the 50% predicted by Skyrmion theory.

3. ZEROHALLRESISTANCE IN THE SEMIMETALLIC GaSblInAs SYSTEM The GaSblInAs heterostructure system can be grown to have an unusual crossed gap alignment: The conduction band of the InAs layers lies below the valence band of the GaSb layer. The resulting electron transfer creates spatially separated two-dimensional electron and hole gases in the InAs and GaSb layers, respectively. In addition to the intrinsic electron and hole populations, a surplus of electrons, presumably from interface donor states, can be present. Applying hydrostatic pressure decreases the band overlap, reducing the electron and hole concentrations, and can even induce a semimetal-semiconductor transition (Beerens etal., 1987a,b;Holmes et al., 1995~).In single-quantum-well structures the charge imbalance leads to compensated Hall plateaux, pxy = h/e2[ u, - vh], where u, and Vh are the integer electron and hole filling factors (Mendez et al., 1985). Nearly equal electron hole concentrations have been obtained in GaSb/InSa superlattice structures grown by metalorganic vapor phase epitaxy (MOVPE) (Daly etal., 1995). Using hydrostatic pressure to reduce the electron and hole concentrations, the extreme quantum limit (v, = v,, = 1) has been obtained, at which point the Hall resistivity disappears while the diagonal resistance diverges (Daly et al., 1996).

IV. Fractional Quantum Hall Effect 1. INTRODUCTION The fractional quantum Hall effect results in deep minima in the diagonal resistance, accompanied by exact quantization of the Hall plateaux at fractional filling factors (Tsui et al., 1982). Similar to the IQHE, this is the result of gaps in the density of states, unlike the IQHE, however, it is not possible to explain the presence of such gaps at fractional filling factors within the framework of a single-electron picture. The origin of the density of states is the interactions between electrons, the so-called many-body effects, for which quantitative theory is both complicated and computationally extremely time consuming.

26

D. K. MAUDEAND J. C. PORTAL

It was realized early on that the small electronic g-factor in the GaAs/ AlGaAs system further complicated the problem because the small Zeeman energy favors spin-unpolarized (or spin-reversed) fractional states at filling factors of v < 1 for which full polarization is otherwise expected (Halperin, 1983). The spin polarization of fractional states was measured experimentally by varying the Zeeman energy by rotating the magnetic field away from the normal (Clarke etal., 1989; Eisenstein etal., 1989) or by applying hydrostatic pressure (Morawicz et al., 1993). Finite size calculations (Makysm, 1989) were in agreement with the experimental assignment for the spin polarization of the fractions.

2. COMPOSITE FERMIONS Our understanding of the FQHE was considerably advanced by the composite fermion (CF) model of Jain (1989,1992), subsequently expanded on by Halperin etal. (1993). The C F model describes the transport properties in terms of noninteracting electrons bound to an even number, 2m, of magnetic flux quanta (ao= M e ) by the attachment of infinitessimal flux tubes of strength 2mao and introducing a Chern-Simons gauge field. The CFs obey Fermi-Dirac statistics when bound to an even number of flux quanta. At filling factor v = 1, the attached flux quanta exactly cancel the applied magnetic field and the system can be considered as a metallic system of noninteracting CFs in zero magnetic field and with a well-defined Fermi surface. On either side of v = 1, the CFs move in an effective magnetic field B* = B - B1/2.The fractional states can then be understood in an analogous way to the IQHE as arising from the gaps in the density of states due to the cyclotron motion of noninteracting CFs. This picture has received considerable experimental support (Du etal., 1993; Willet, 1993; Kang et al.. 1993). Within the CF model the spin polarization of the fractional states can be understood in a natural way by constructing two sets of CF Landau level (LL) fan charts for the lowest spin up and spin down electron Landau level. Figure 18 shows the CF LL fan chart calculated for an electron density of 0.44 and 1.2 X 10” cm-2 so the filling factor v = 4 occurs at 3.68 T and 10 T, respectively. The integer CF and fractional electron filling factors are indicated. The spin splitting of the two C F LL fan charts is in principle simply bare spin splitting gpBB given by the total magnetic field (to obtain a more quantitative agreement between experiment and the CF model, we are forced to enhance this bar spin splitting by a factor of -2). The CF LL separation fieB*lm * depends on the effective magnetic field. An additional

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

Bm 2.0

3.0

4.0

5.0

I

I

I

1

k )r

F

z

W

.

-1

-2.0

-2 -3 -1 .o

3 0.0

B'

2 1.o

2.0

12.5

15.0

m

Em 5.0

7.5

10.0

30

-

20

;

10

Y

)r

w

0 -1

-10

,

-5.0

-2 -3-4

4 3 2

1

1

-2.5

0.0

2.5

5.0

B" m FIG. 18. Composite fermion Landau level (CF LL) fan charts calculated for a GaAs/AI,Gal .,As heterojunction at ambient pressure (g = -0.44) using the empirical CF mass = 0.51 + 0.083 B* determined by Nicholas etal. (1996). The fractional electron and integer CF filling factors are indicated. The movement of the Fermi level through the LL structure is indicated by the heavy line. To obtain a quantitative agreement with theory we have assumed an enhanced spin gap of -2gwBB. (a) n = 4.4 X 10'" cm-'; (b) n = 1.2 X 10" cm-2.

27

28

D. K. MAUDEAND J. C .PORTAL

complication is that the CF LL does not evolve linearly due to the magnetic field dependence of the CF effective mass m* = (0.51 to 0.083 B*) me (Nicholas et al., 1996). For fractions with 1 < v < 2, the situation is further complicated due to the presence of an additional filled spin LL. For 1 < v < 2 the CF model requires a “composite Fermionization” of the holes (Du et al., 1995). This simple picture is able to at least qualitatively predict the evolution of the spin polarization of the fractional states with pressure and carrier concentration (magnetic field). The application of hydrostatic pressure decreases the Land6 g-factor and thus the separation of the two CF LL fans. Using pressure to approach the g = 0 limit, we expect to weaken odd numerator fractions while strengthening even numerator fractions. At g = 0, the system should behave as a doubly degenerate 2D layer with fractions occurring at twice the usual occupancies. The Zeeman energy that depends on the total magnetic field rotates the CF LL fan charts clockwise or anticlockwise around B* depending on the spin. This symmetry-breaking rotation gives rise to some interesting effects at certain carrier densities. In Fig. 19 we see that for a sample with a carrier density of 4.4 X 10” cm-2, the gap at 2/3 increases while the gap at 2/5 decreases with pressure (Holmes etal., 1994, 1995a,b) at a rate that is in reasonable agreement with the expected pressure dependence of the g-factor (Zeeman energy). This result, which is at first sight surprising because the 213 and 2/5 fractions both occur at a CF filling factor of 2 2 , can be explained quite naturally provided we assume an enhanced spin splitting of -2gcBB. As can be seen from Fig. 18a, at low densities 2/3 and 215 correspond to a spin flip transition. However, according to the LL scheme in Fig. 18a, reducing the Zeeman energy increases the gap at 213 while simultaneously decreasing the gap at 2/5 and is thus in agreement with the experimental observation. Finally, we turn our attention to the 113 fractional state, which corresponds to a CF filling factor of 1. Naively, one would expect that with increasing pressure the polarized 1/3 state, which corresponds to a CF cyclotron gap (at least for sufficiently high carrier densities), would be transformed into spin flip transition before disappearing altogether at g = 0. This is true provided the interaction between CFs is negligible, an assumption that is referred to as the mean field approximation. In reality, fluctuations in the electron (CF) density will give rise to a fluctuating magnetic field and hence lead to interactions between CFs. Thus, studying 1/3in the g = 0 limit can be used to probe the residual interaction between the CFs and probe the validity of the mean field approximation. The spin wave dispersion of CFs at 1/3 has been calculated by Nakajima and Aoki (1994), who estimated a gap enhancement = 0.06(e 2 / d BIn ) .addition, it has been

1 PARALLEL TRANSPORT IN LOW-DIMENSIONAL STRUCTURES

1.2,

O.8.0

.

,

2.5

.

I

5.0

.

I

7.5

.

29

1

10.0

Pressure [kbar] FIG. 19. Energy gap at filling factor 2/5 and 2/3 (CF filling factor 2 2 ) measured as a function of pressure for carrier densities of 4.4 X lo1"cm-3 (closed symbols) and 6.8 X 10" cm-3 (open symbols), taken from Holmes et al. (1994,1995b). The dashed lines are guides for the eye. The solid lines correspond to the expected pressure dependence of the Zeeman energy, -6.9 mK tesla-I kbar-'.

proposed that at CF filling factor 1 ( v = 1/3), spin texture excitations (Skyrmions) might be formed (analogously to v = 1 for electrons (Kamilla et aL, 1996). Measurements in heterojunction samples at a pressure of around 18 kbar would seem to support this picture (Leadley et af.,1997a,b).

30

D. K. MAUDEAND J. C. PORTAL

V. Magnetophonon Resonance Effect under Hydrostatic Pressure in GaAs/Ab.~Gao.7&, Gao.471n05JAs/A10.481noJZAS, and Ga,,.471~53As/InPHeterojunctions The hydrostatic pressure technique has been advantageously associated with MPR measurements to study not only the behavior of the effective mass of carriers under pressure, enabling a comparison with predictions of k.p theory, but also to provide some additional information about the electron-longitudinal optic (LO) phonon coupling in two-dimensional electronic systems. The MPR results are thus discussed in terms of nonparabolicity and 2D screening effects and permit studies of the dependence of MPR on the tuning of the 2D carrier concentration (see the review by Dmowski and Portal, 1989). A review of the magnetophonon effect can be found in Nicholas (1985). Several authors have reported MPR measurements in two-dimensional 1980; systems such as n-type GaAs/A1,Gal- ,As heterostructures (Tsui etal., Englert etal., 1982; Kido etal., 1982) or Gao.471no.s3As/InP (Portal etal., 1984; Brummel etal., 1984) and Gao.471no.53As/Alo.481no.s2As heterostructures (Portal et al.. 1984). Das Sarma and Madhukar (1 980) considered the influence of optic phonon coupling in two-dimensional systems, showing that it could lead to splitting of the cyclotron resonance close to the magnetophonon resonance condition. Das Sarma (1983, 1984) has considered polaron coupling in more detail, showing that the polaron interaction is enhanced in a twodimensional system due to the electronic confinement, but that effects such as screening and finite electric subband widths can reduce the strength to considerably less than the value in bulk material. Lassnig and Zawadskii (1983) have developed a theory for the transverse magnetophonon conductivity in heterostructures, including the effects of collision broadening. The first observations of MPR under hydrostatic pressure were reported and Gao.471no.s3A~/ for G ~ A s / A ~ ~ . ~ ~Gao.471no.s3As/Alo.481no.s2As, G~~.~~As, InP (GrCgoris elal., 1988; Gauthier et al., 1988). The magnetophonon resonance effect manifests itself as an oscillatory behavior of the resistivity as a function of the magnetic field B. These oscillations are due to electron-LO phonon interactions that become resonant when oLo= N o , = N e B N / m *

where oLoand w, are the phonon and cyclotron frequencies, rn" is the effective mass, N is the resonance index, and BN is the magnetic field at

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

31

the resonance. The series of oscillations are periodic in 1/Bwith a periodicity A(l/B) = e/wLom*. From the measurements of BN, the effective mass m* or the phonon frequency wLo can be deduced, provided that one of the two is known. The temperature must be high enough to have a sufficient phonon population, but not too high because thermal broadening of the Landau levels reduces the oscillation amplitude. Optimal temperatures are usually in the range of 100 to 250 K. The amplitude of MPR oscillations can give valuable information about the strength of the electron-phonon interaction. Stradling and Wood (1968, 1970) proposed the following empirical expression to describe the oscillatory part of the magnetoresistivity for bulk material:

-

Axx exp

-YOLO ~

WC

OLO

cos 237 WC

where y is the damping factor. 1.

PRESSURE

DEPENDENCE OF THE EFFECTIVE MASS

MPR measurements at constant pressure are shown (see Fig. 20) for both types of heterojunctions (see Table 11, adapted from GrCgoris etal., 1988). In each case, the resonance peaks shift toward higher fields with increasing pressure due to the increase of the effective mass. According to a formula proposed for bulk material [see Eq. (ll)], the amplitude of the oscillatory part of magnetoresistivity is proportional to exp(-yoLo/w,), where y is the damping factor. It appears that as the electron concentration in the heterojunction becomes very low under high pressure, the amplitude of the oscillation tends to reach a maximum value that is close to that of the bulk (Grkgoris et al., 1988). The resonance fields BN are determined as the values of magnetic fields corresponding to the tangent points between the resistivity peaks and the envelope curve of the oscillations. This method corrects directly for the apparent shift of the maxima due to the damping of the oscillations. The effective mass m * , as determined from Eq. (lo), is plotted in Figs. 21 and 22 as a function of pressure for two resonances ( N = 2 and 3 in Fig. 21; N = 1 and 2 in Fig. 22). In heterostructures, the choice of which phonons are to be considered in the MPR analysis in order to estimate the effective mass m * is, however, by no means obvious. Usually, the frequency of the phonon interacting with 2DEG is different from that observed in the bulk.

D. K. MAUDEAND J. C. PORTAL

32

0

5

10 MAGNETIC FIELD (T)

15

I

I

I

I

0

5

10 MAGNETIC FIELD (T)

15

20

20

FIG. 20. Typical magnetophonon resonance oscillations for representative pressures in (a) GaAs/Al,,BGao.7ZAsat 150 K and in (b) Gao.471n,53A~/A&,.4R In,,.szAsat 220 K. The dashed Line shows the expontential damping of the oscillations, A is the relative amplitude. and arrows point to the resonances. (From Grkgoris ef af.,1988.)

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

33

TABLE I1 TEMPERATURE AND PRESSURE DEPENDENCE OF THE CARRIER CONCENTRATION NsA N D THE MOBILITY ps OF THE 2D ELECTRON GAS IN BOTHSYSTEMS High pressure

Ambient pressure T (K) NS(10" cm-2)p (cm2V-' 4.2 150 290

2.3 2 2.6

) P (kbar)

SKI

Ns(P)(10" cm-2)p s ( P(cm2 ) V-'

GaAslAb.&a,,.72As 328,000 1.7 35,000 13.6 8.000 14.4

1.03 0.21 0.8

s-' )

138,200 21,Ooo

6,000

Gao.471n0.,AstA~.~nAs

4.2 150 220

5.15 6.02

72,000 29,000 16,500

14.5 13.6 13.6

4.9 5.1

50,000 24,000 13,000

For GaAs/AlGaAs and GaInAs :AlInAs, we have used the "GaAs-like" mode with wLo = 282 cm-' (Brummell etal., 1987) and the "InAs-like" mode with wLo = 222 cm-' (Nicholas et al., 1985). In GaAslAlGaAs we have assumed the same temperature and pressure dependence as found in bulk GaAs (Blakemore, 1982; Trommer et al., 1976), and for the InAs-like mode in GaInAdAlInAs we have assumed a similar temperature dependence and have taken the pressure dependence from the Griineisen formula (Mitra etal., 1969): wLo(GaAs) = 282 (1 - 4 x

T ) (1 + 1.5 x

P)

(12)

wLo(InAs) = 222 (1 - 4 X

T )(1 + 1.8 X

P)

(13)

where T is in K, P is in kbar, and wLo in cm-'. Note that these pressure and temperature shifts of wLo are small. When analyzing the MPR results in 2D systems, one has to consider the fact that the electric quantization provides an additional amount of kinetic energy that leads to an increased value of the effective mass compared with the band edge m* because of nonparabolicity effects. We now compare the MPR mass m* to the band-edge mass m?jmeasured in bulk GaAs and Gao.471no.53As (Shantharama etal., 1984, 1985). If the polaron correction to the effective mass is neglected, the mass enhancement due to nonparabolicity is then given by the difference between m* and m$ in Figs. 21 and 22. The mass enhancement has a relative magnitude of about 11%for GaAs/ AlGaAs and about 30% (smaller gap) for GaInAdAlInAs (GrCgoris etal., 1988). In both systems the nonparabolic mass enhancement decreases with

D. K. MAUDEAND J. C. PORTAL

34 0.085

i I

I

I

GaAs /Al,,,Ga,

I

,As

290K

0.080

0.075

0 0

8

I. 0

0.070

0.065 0 0 N=2

m O N=3

0.060 0

5

10

15

20

PRESSURE (kbar) FIG. 21. Pressure dependence of the effective mass in G ~ A s / A ~ ~ . ~ ~ G ~ . , ~ A s at 290 K. Solid symbols represent the MPR mass m*, and open symbols correspond to the band-edge mass m:. The error bar shown is the same at any pressure. The arrows show the nonparabolic enhancement to the mass. The solid line gives bulk results, after Shantharama etal. (1984). Note that both lines coincide at 290 K (from GrCgoris etal.. 1988.)

pressure (e.g., down to 18% at 15 kbar for GaInAdAlInAs) (GrCgoris ef al., 1988). This behavior is expected, since under pressure the electron concentration decreases and the gap increases, which brings a reduction in nonparabolicity. In the GaAs/AlGaAs system, however, the band-edge effective mass estimated for high pressure was enhanced by about 2% relative to that of bulk GaAs, which was accounted for by the polaron correction, not included in the calculation. Such an interpretation is consistent with the fact that electron-phonon coupling is increased at high pressure because the 2D carrier concentration becomes low and the screening of the electronphonon interaction is suppressed (GrCgoris etal., 1988).

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

0.050

E"

\

35

t HU

0.035

0

I

I

1

5

10

15

N=3

20

PRESSURE (kbar) FIG.22. Pressure dependence of the effective mass in Gao.,71m,s,As/ Alo.481no.52As at 220 K. The same remarks as in Fig. 21 apply here for the symbols and error bars. The solid line gives the slope in bulk G ~ . 4 7 1 q . s 3 A s at 210 K, after Shantharama etal.(1985);the dashed lines are from fiveband k.p theory (Hermann and Weisbuch, 1977). with P2 = 23.8 eV, (p’)’ = 2.9 eV, and C = -2 for the matrix elements. (From GrCgoris et al.. 1988.)

Interesting results were obtained for the GaInAdInP system [see Table 111, adapted from Gauthier etal., (1988)] where the phonon frequency oo interacting with the 2DEG was directly extracted using the effective mass obtained from cyclotron resonance experiments performed at the high temperatures at which the MPR effect is usually studied. The value of oo extracted for a variety of samples with different carrier concentrations and numbers of occupied subbands were found to decrease with increasing 2D carrier concentration. For the lowest concentration, the frequency was very close to the value of the GaAs-like LO mode in GaInAs (271 cm-'). For the highest concentration, the frequency fell to 240 cm-', which is between the InAs-like and GaAs-like phonon branches. This behavior is not yet

D. K. MAUDEAND J. C. PORTAL

36

TABLE I11 CHARACTERISTICS OF THE DIFFERENT SAMPLES STUDIEDOF G%.471&.53As/InP HETEROJUNCTIONS ASS~JMING A PRESSURE VARIATION OF THE DIRECT GAPE, OF 10.7 meV/kbar (Adachi, 1982) Number Phonon of energy dm$ldP dnH/dP m*/m, T nH electric w, (m, kbar-I) dm:ldEg (cm-' kbar-') at 4.2 Sample (K) (10'' cm-2) subbands (cm I ) (m.eV-') (lo9) K 52 92 82 83

120 150 150 15s

6 5.2 3.6 1.3

3 2 2 1

240 245 260 270

4.7s 6.03 7.1

0.044

0.056 0.066

-6 -8 -12.6

0.0485

0.0489 0.0490

Note. rn: and nH were attained with pressure dependence. (From Gauthier et al., 1988.)

understood and was suggested to be associated with a mixed bidimensional plasmon-phonon mode or intersubband plasmon-phonon mode. On the other hand, in GaInAs-based heterojunctions with AlInAs (GrCgoris et al., 1988) and InP (Gauthier etal., 1988) barrier layers, the bandedge effective mass was estimated to increase with pressure at a rate smaller than that estimated for bulk GaInAs (Shantharama etal., 1985). For GaInAslInP heterojunctions this deviation was found to be dependent on the 2D carrier concentration, being smaller for lower-concentration samples and more important for higher-concentration samples (see Fig.

PRESSURE (kbar) FIG. 23. Change of the band-edge effective mass rnd with pressure for the three samples studied compared with the bulk increase. We clearly see that the slope gets closer to the bulk value as the carrier concentration decreases. (From Gauthier etal., 1988.)

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

37

23). Taking into account the increasing phonon frequency with decreasing carrier concentration revealed in GaInAs/InP, the pressure dependence of the effective mass will be even lower and thus differ more from the bulk results. It is worth noting that for the series of samples studied (Gauthier etal., 1988) the increase of oowith decreasing carrier concentration was equivalent to a pressure coefficient dwo/dp= 0.4 cm-' kbar-', which is of the same order of magnitude as the pressure coefficient resulting from the Griineisen formula. It may thus be said that the scattering of a 2DEG in GaInAs by LO phonons may be controlled so that the dominant interaction is with one of the two phonon modes, InAs or GaAs. These two cases correspond to InAs made dominance in GaInAs/AlInAs heterojunctions. GaAs made dominance in GaInAs/InP heterojunctions. The conclusion must therefore be that the best estimates of the pressure dependence of the mass in GaInAs/InP heterojunctions are the values deduced from the higher-concentration structures with less pressure dependence of the carrier concentration. These give values substantially lower than for the bulk GaInAs alloy but closer to the values found in binary 111-V compounds (Shantharama etaf., 1985). Possible causes for this could include differences in the compressibilities of GaInAs and InP leading to the appearance of uniaxial stress, changes in the nonparabolic contribution to the band structure, or changes in the electron-optic phonon interactions. The results taken with the lower carrier-concentration samples indicate a progressive increase in the pressure dependence of the mass toward the bulk value. Because the subband size increases, this would suggest that structural factors play a lesser role. 2. AMPLITUDE OF THE OSCILLATIONS AND y DAMPING FACTOR The relative amplitudes ApIpo of all the resistivity peaks are plotted as a function of pressure. The amplitude is defined as the half-separation between the envelope curves of the oscillations at the positions of the of the resistivity. The damping of the maxima (A,,,) and minima oscillations appears not to be exponential at low pressure, but it is useful nonetheless to define the damping factor as

D. K. MAUDEAND J. C. PORTAL

38

0

5

10

15

20

PRESSURE (kbar) FIG. 24. Pressure dependence of the damping factor in G a A s / A h 8 G%,2Asfor two resonances at (a) 150 K, (h) 215 K, and (c) 290 K. The arrows indicate the extreme values reported in the literature for hulk GaAs (Stradling and Wood, 1970: Senda er al., 1979; Kid0 and Miura, 1983). The lines are guides for the eye. (From Grkgoris etal., 1988.)

Results are presented versus pressure in Fig. 24 for a heterojunction of G~AS/A~~.~~G~~.~~AS. In GaAs/Alo,zsGao.72Asheterostructures, the amplitude of all peaks increases with pressure and seems to reach a maximum value at pressures (Fig. 20) that are higher €or larger values of N [see Eq. (lo)]. It appears that for low electron concentrations, the amplitude of the oscillations tends to reach a maximum value that is close to that of bulk GaAs. In this situation (high pressure) the damping parameter is about 1, as in bulk GaAs (Fig. 24). At lower pressure, we have situations in which certain peaks have reached their maximum amplitude and others have not, and this results in a nonexponential decrease of the amplitude with B- . In other words, the damping factor depends on magnetic field in that case

1 PARALLEL TRANSPORT I N LOW-DIMENSIONAL STRUCTURES

39

3

2

Ir’

1 c

GaAs IAl,,,Ga,,,~s 215 K 0 N=2 N=3

0 0

5

10

15

20

PRESSURE (kbar) Fro. 24. (continued)

(Fig. 24), and this dependence changes with pressure, that is, with the electron concentration. These results indicate that the nonexponential increase in the amplitude is linked to the 2D character and the degeneracy of the system. For very low values of carrier concentration N s ,the system tends to behave as in the 3D case, as can be expected, since the electron wave functions then have large spreadings.

Acknowledgments

It has been a pleasure to work with the excellent scientists who have come to Grenoble High Magnetic Field Laboratory or those who have collaborated efficiently in this area of high-pressure semiconductor physics and particularly in transport properties of low-dimensional semiconductor structures.

D. K. MAUDEAND J. C .PORTAL

40

.

3

1 I

2 c

Ir’ c.

I

1

GaAs /AID28Ga,, ,As 290 K 0

N=2

rn N=3

0 0

5

10

15

20

PRESSURE (kbar) FIG. 24. (continued)

Several important contributions to this paper come from exceptional PhD students: G. GrCgoris, J. Beerens, S. Ben Amor, D. Gauthier, N. G. Morawicz, M. L. Williams, M. S. Daly; inspirational collaboratorsL. Dmowski, P. Wisniewski, T. Suski, R. J. Nicholas, D. R. Leadley, D. M. Symons, S. Holmes, A. N. Utjuzh, L. Eaves, M. Potemski, M. Lakrimi, K. W. J. Barnham, A. Briggs, E. E. Mendez, L. Esaki, L. L. Chang, and J. L. Robert; and A. Y. Cho, D. L. Sivco, L. L. Chang, M. Razeghi, F. Alexandre, J. J. Harris, C. T. Foxon, M. Henini, N. J. Mason, P. J. Walker, D. Ashenford, B. Lunn, G. Hill, and M. A. Pate, who have supplied many exquisite crystals. REFERENCES Adachi, S. (1982). J .Appl.Phys.S3 (12), 8775. Aifer, E. H.. Goldberg, B. B., and Broido, D. A. (1996). Phys.Rev.Lett. 76,680. Alikacem, M., Leadbeater, M. L.. Maude, D. K., Davies, M., Eaves, L., Heath, M., Dmowski, L., Portal, J. C.. Ashenford, D., and Lunn. B. (1990). Surf Sci.229,428. Ando, T., Fowler. A. 8..and Stern, F. (1982). Rev.Mod. Phys.54,437.

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41

Barrat, S. E., Dabbagh, G., Pfeiffer, L. N., West, K. W., and Tycko, R. (1995). Phys.Rev. Lett. 74, 5112. Bastard, G. (1988). Wave Mechanics AppliedtoSemiconductor Heterostructures p. 74. Les editions de physique, Les Ulis (France). Beerens, J., Grtgoris, G., Ben Amor, S.. Portal, J. C., Mendez, E. E., Chang, L. L., and Esaki, L. (1987a). Phys.Rev.B 35,3039. Beerens, J., Grtgoris, G., Portal, J. C., Mendez, E. E., Chang, L. L., and Esaki, L. (1987b). Phys.Rev.B 36,4742. Beerens, J., GrLgoris, G., Portal, J. C., Robert, J. L., Mercy, J. M., and Alexandre F. (1988). Semicond. Sci.Technol. 3, 577. Belavin, A. A., and Polyakov, A. M. (1975). JETP Lett. 22,245. Blakemore, J. S. (1982). J.Appl.Phys.53, R123. Brummell, M. A., Nicholas, R. J., Brunel. L. C., Huant, S., Baj, M., Portal, J. C., Razeghi, M., di Forte-Poisson, M. A., Cheng, K. Y., and Cho, A. Y. (1984). Surf Sci.142, 380. Brummell, M. A., Nicholas, R. J., Hopkins, M. A., Hams, J. J.. and Foxon, C. T. (1987). Phys.Rev.Lett, 58, 77. Buttiker, M. (1988). Phys.Rev.B 38,9375. Bychkov, Y. A., Iordanskii, S. V., and Eliashberg, G. M. (1981). JETP Lett. 33, 152. Chakraborty, T., and Pietilainen, P. (1995). Springer Series inSolidState Science 85. Clarke, R. G., Haynes, S. R., Suckling, A. M., Mallet, J. R., Wright, P. A,, Harris, J. J., and Foxon, C. T. (1989). Phys.Rev.Lett. 62,1536. Cohen, A. M., and Marques, G. E. (1990). Phys. Rev.B 41,10608. Daly, M. S., Lubczynski, W., Wareburton, R. J., Symons, D. M., Lakrirni, M., Dalton, K. S. H., Van der Burgt, M., Nicholas, R. J., Mason, N. J., and Walker, P. J. (1995). J.Phys.Chem. Sol. 56, 453. Daly, M. S., Dalton, K. S. H., Lakrimi, M., Mason, N. J., Nicholas, R. J., Van der Burgt, M., Walker, P. J., Maude, D. K., and Portal, J. C. (1996). Phys.Rev.B 53, R10524. Das Sarma, S., and Madhukar, S. (1980). Phys.Rev.B22,2833. Das Sarma, S. (1983). Phys.Rev.B27,2590. Das Sarma, S. (1984). Phys.Rev.B29, 2334. Dmowski, L., and Portal, J. C. (1989). Semicond. Sci.Technol. 4, 211. Dobers, M., von Klitzing, K., and Weimann, G. (1988). Phys.Rev.B 38,5453. Du, R. R., Stromer, H. L., Tsui, D. C., Pfeiffer, L. N., and West, K. W. (1993). Phys.Rev. Len.70,2944. Du, R. R., Yeh, A. S. Stormer, H. L., Tsui, D. C., Pfeiffer, L. N., and West, K. W. (1995). Phys.Rev.Len.75,3926. Einstein, J. P., Stormer, H. L., Pfeiffer, L., and West, K. W. (1989). Phys.Rev.Lett. 62, 1540. Englert, T., Tsui, D. C., Portal, J. C., Beerens, J., and Gossard, A. C. (1982). SolidState Commun. 44, 1301. Fertig, H. A., Brey, L., Cote, R., and MacDonald, A. H. (1994). Phys.Rev.B 50, 11018. Gauthier, D., Dmowski, L., Ben Amor, S., Blondel, R., Portal, J. C., Razeghi, M., Maurel. P., Omnes, F., and Laviron, M. (1986). Semicond. Sci.Technol. 1, 105. Gauthier, D., Dmowski, L., Ben Amor, S., Blondel, R., Portal, J. C., Razeghi, M., Maurel, (Stockholm), P., and Omnes, F. (1987). Proc.18thInt.Conf Phys.Semiconductors p. 629. World Scientific, Singapore. Gauthier, D., Dmowski, L., Portal, J. C., Leadley, D., Hopkins, M. A,, Brummell, M. A., Nicholas, R. J., Razeghi, M., and Maurel, P. (1988). Superla@. and Microstr. 4,201. Gregoris, G.,Beerens, J., Dmowski, L., Ben Amor, S., and Portal, J. C. (1987a). Optical Properties ofNarrowGap Low Dimensional Structures (NATO AS1 Series B, vol. 152). p. 337.

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Grbgoris, G., Beerens. J., Ben Amor, S., Dmowski, L., Portal, J. C., Sivco, D., and Cho, A. V. (1987b). J. Phys.C:SolidStarePhys.20,425. Gregoris, G.. Beerens, J., Ben Amor, S.. Dmowski, L., Portal, J. C., Alexandre, F., Sivco, D. L.. and Cho. A. Y. (1988). Phys.Rev.B 37, 1262. Grimes, C. G. (1978). Surf Sci.73, 379. Halperin, B. I. (1983). Helv.Phys.Acfa. 56, 75. Halperin, B. I., Lee, P. A., and Read, N. (1993). Phys.Rev.B 47, 7312. Haug, R. J. (1993). Semicond. Sci.Technol. 8, 131. Hermann, C., and Weisbuch, C. (1977). Phys. Rev.B 15,823. Hermann, R., Kraak, W., Handschack, S., Shuring, T., Kusnick, D., and Schnackenburg, B. (1988). Phys.Status Solid(b)145, 157. Holmes, S.. Maude. D. K.. Williams, M. L., Harris, J. J., Portal, J. C., Barnham, K. W. J., and Foxon, C. T. (1994). Semicond. Sci.Technol. 9, 1549. Holmes. S.. Maude. D. K., Williams, M. L.,Harris, J. J., Portal, J. C., Barnham, K. W. J., and Foxon, C. T. (1995a). J. Phys.Chem.56,459. Holmes, S., Harris, J. J., Maude, D. K., Portal, J. C., Williams, M. L., Barnham, K. W. J., and 22nd Int. Conf on thePhysics ofSemiconductors, Vancouver Foxon, C. T. (1995b). Proc. 1994 (Lockwood. D. J.. ed.). p. 1015 World Scientific, Singapore. Holmes, S., Yuen, W. T., Malik. T., Chung, J., Norman, A. G., Stradljng, A. R., Harris, J. J., Maude, D. K., and Portal, J. C. (1995~).J. Phys.Chem.56,445. Ivchenko. E. L., and Kiselev, A. A. (1992). Sov.Phys. Semicond. 26,827. Jain, J. K. (1989). Phys.Rev.Lett. 63, 199. Jain, J. K. (1992). A d v .Phys.41, 105. Kallin, C., and Halperin, B. I. (1984). Phys.Rev.B 30, 5655. Kallin, C., and Halperin. B. I. (1985). Phys.Rev.B 31,3635. Kamilla, R. K., Wu, X. G.. and Jain, J. K. (1996). SolidStareCommun. 99, 289. Kang, W., Stormer. H. L.. Pfeiffer. L. N.. Baldwin. K. W.. and West, K. W. (1993). Phys.Rev. Lett. 71, 3850. Kido. G., Miura, N.. Ohno, H.. and Sakaki. H. (1982). J. Phys.SOC.Jpn.51,2168. Kido. G.. and Miura, N. (1983). J. Phys.Soc. Jpn.52, 1734. Kohn, W. (1961). Phys.Rev.123, 1242. Kralik. B., Rappe, A. M., and Louie, A. G. (1995). Phys.Rev.B 52, R11626. Lassnig, R.. and Zawadskii, W. (1983). J. Phys.C16, 5435. Leadley. D. R., Nicholas, R. J.. Maude. D. K., Utjuzh. A. N., Portal, J. C., Harris, J. J., and 79,4246. Foxon, C. T. (1997a). Phys.Rev.Lett., Leadley, D. R., Nicholas, R. J.. Maude, D. K., Utjuzh. A. N., Portal, J. C., Harris, J. J., and Foxon, C. T. (1997b). Proc. 12thConference on theElectronic Properties of TwoDimensional Systems, Tokyo, September 1997. Special issue of Physica, 1998. Maksym. P. A. (1989). J. Phys.Condens. MutterI, 6299. Maude, D. K., Potemski, M.. Portal. J. C.. Henini, M.. Eaves, L. E., Hill, G., and Pate, M. A. (1996). Phys.Rev.Lett. 77,4604. Mendez, E. E., Esaki, L., and Chang. L. (1985). Phys.Rev.Lett. 55,2216. Mitra, S .S., Brafman, 0.. Daniels, W. B.. and Crawford. R. K. (1969). Phys.Rev.B 186,942. Morawicz, N. G., Barnham. K. W. J.. Briggs. A,, Foxon, C. T., Hams, J. J.. Najda, S. P., Portal, J. C., and Williams. M. L. (1993). Semicond. Sci.Technol. 8, 333. Nakajima, T.. and Aoki, H. (1994). Phys.Rev.Lett. 73, 3568. Nicholas, R. J. (1985). Prog.Quantum.Electron. 10, 1, and references therein. Nicholas. R. J., Brunel. L. C., Huant. S., Karrai. K., Portal, J. C. Brummell, M. A,, Razeghi, M., Cheng. K. J., and Cho. A. Y. (1985). Phys.Rev.Lett. 55,883. Nicholas. R. J.. Haug, R. J.. von Klitzing, K.. and Weimann, G. (1988). Phys.Rev.B 37,1294.

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Nicholas, R. J., Leadley, D. R., Daly, M. S.. van der Burgt, M., Gee, P., Singleton, J.. Maude, D. K., Portal, J. C., Harris, J. J., and Foxon. C. T. (1996). Semicond. Sci. Techno!. 11,1477. Portal, J. C., GrBgoris, G., Brummell, M. A., Nicholas, R. J., di Forte-Poisson, M. A., Cheng, K. Y., and Cho, A. Y. (1984). Surf Sci. 142,368. Razeghi, M., Duchemin, J. P., Portal, J. C.. Dmowski, L., Rerneni, G.. Nicholas, R. J.. and Briggs, A. (1986). Appl. Phys. Left. 48,712. 75,4290. Schmeller, A., Eisenstein, J. P.. Pfeiffer. L. N.. and West. K. E. (1995). Phys. Rev.Lett. Senda, K., Shimomae, K.,Kasai, K., and Hamaguchi. C. (1979). J.Phys. SOC.Jpn. 47,551. Shantharama, L. G., Adams, A. R., Ahamd. C. N., and Nicholas, R. J. (1984). J.Phys. C. 17, 4429. Shantharama, L. G., Nicholas, R. J., Adams. A. R., and Sarkar, C. K. (lY85)J. Phys. C 18, L443. Shubnikov, L., and de Haas, W. J. (1930). LeidenCommun. 207a. 207c. 207d, 210a. Skyrme, T. (1961). Proc.R. SOC.Lond.262, 237. Sondhi, S. L.. Karlhede, A., Kivelson, S. A., and Rezayi, E. H. (1993). Phys. Rev.8 47,16419. Sotomayor Torres, C. M., Claxton, P. A., Roberts, J. S., Stradling, R. A,, and Wasilewski, Z. 170, 464. (1986). Surf Sci. Stradling, R. A., and Wood, R. A. (1968). J. Phys.Cr SolidStare Phys. 2, 1711. Phys. 3,L949. Stradling, R. A., and Wood, R. A. (1970). J . Phys. C: SolidState Trommer, R., Anastassakis, E., and Cardona, M. (1976). In LightScattering in Solids (Balkanski, M. etal., eds.), p. 396. Flamarion, Paris. Tsui, D. C., Englert, T., Cho, A. Y., and Gossard, A. C. (1980). Phys. Rev.Lett. 44, 341. 48,1559. Tsui, D. C., Stormer, H. L., and Gossard, A. C. (1982). Phys. Rev.Lett. Usher, A., Nicholas, R. J., Hams, J. J.. and Foxon. C. T. (1990). Phys. Rev.B 41, 1129. von Klitzing, K., Dorda, G., and Pepper, M. (1980). Phys. Rev.Lett. 45, 494. Walukiewicz, W., Ruda, H. E., Lagowski. J.. and Gatos, H. C. (1984). Phys. Rev.B 30,4571. Willet, R. L., Ruel, R. R., West, K. W., and Pfeiffer, L. N. (1993). Phys. Rev.Lett. 71, 3846. Williams, R., Crandall, R. S., and Willis, A. H. (1971). Phys. Rev.Lett. 26, 7.

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SEMICONDUCTORS AND SEMIMETALS. VOL. 55

CHAPTER2

Tunneling under Pressure: High-pressure Studies of Vertical Transport in Semiconductor Heterostructures P. C.Klipstein CLARENDON LABORATORY DEPARTMENT OF PHYSICS UNIVERSITY OF OXFORD OXFORD, UK

I. INTRODUC~ION ................................................................................................................. 11. THEORY AND CALCULATION. 1. Riccoand AzbelFormula 2. Transfer MatrixMethod..... 3.Calculating theI-V Characteristic ........................................................................... 4.SpaceCharge..................... 5. Band Mixing............................................................................................................... 6.Strain Due toPseudomorphic Growth................................................................... 111. EXPERIMENTAL TECHNIQUES ..... IV. HIGH-PRESSURE STUDIESOF IFF ...................... 1. EarlyDays.................................................................................................................. 2.ResonantTunneling inSingle-, Douhle-, and Multiple-Barrier

................................................................................... g ............................................

................................................................................... Acknowledgments ...................................................................... References ........................................................................................................................

45

41 47

48 51 54 57 63 64 66 66

68 101 110 111 112

I. Introduction Since the early days of quantum mechanics, the possibility of an electron or other particle tunneling through a barrier forbidden by classical physics has led to new understanding and to applications in many areas. The behavior of alpha decay in nuclear physics, Zener breakdown in semiconductors, and the Nobel prize-winning Esaki tunnel diode and scanning tunneling microscope are but a few examples. When Esaki and Tsu went on to propose the idea of a semiconductor superlattice in 1970 [l],the new physics of 45 VOl. 55 ISBN 0-12-752163-1

SEMICONDLICTORS AND SEMIMETALS Copyrlght D 1998 by Academic Press All rights of reproduction in any form reserved. o o x n i x 7 x ~ i ~$25 x 00

46

P. C. KLIPSTEIN

such structures rapidly came to be viewed in terms of resonant tunneling of coherent electrons through the material. Studies of resonant tunneling, even in devices containing only a few layers, can thus provide essential insight into the physics of more complex heterostructure materials. Most recently, this has been beautifully exemplified in the quantum cascade laser [2].which is a semiconductor device based directly on several interdependent tunneling processes. In a typical resonant tunneling structure, an external voltage controls the energy difference of quasi-bound or resonant states that are in different regions of the sample but whose wave functions overlap, giving rise to a current with a nonlinear dependence on voltage. The device frequently exhibits negative differential resistance (NDR). However, it is often the case that several competing processes may contribute to the current so that the application of an external perturbation, such as a variation in temperature, pressure, or magnetic field, can be necessary to obtain a full understanding of the tunneling mechanism. Of these, pressure has the additional advantage of giving access to states that are inaccessible at ambient pressure, and can thus open the way to new physics and understanding. In this chapter, I focus o n the contributions of high-pressure experimentation to the development of understanding of some important NDR systems. The first semiconductor NDR systems were developed in the 1950s and 1960s and included the transferred electron device, or Gunn diode, and the tunnel diode. From the late 1970s new techniques for heterostructure growth generated great interest in the double-barrier resonant tunneling diode and related multiple-barrier structures. Although the Gunn diode is not strictly a tunneling device, it drew attention to the usefulness of pressure for the study of NDR. With the emergence of the InAs/(AlSb)/GaSb broken-bandgap material combination, the Esaki or tunnel diode has recently undergone a revival in which the interband tunneling may usefully be studied at high pressures. Such studies have yielded unexpected results related to the nature of the InAs-GaSb interface. Since the early days of resonant tunneling in double-barrier structures (DBSs), high pressure has been used to probe the tunneling mechanism, with many and varied interpretations. High pressure can change the effective mass of the tunneling states and can increase the role played by conduction band states of other symmetries than the usual r-symmetry state. In the case of GaAs/AIAs and GaSb/AISb, novel types of resonant tunneling have recently been created at high pressure when a new symmetry becomes dominant: the X band for GaAs/AIAs and the L band for GaSb/AlSb. These new results will be discussed in some detail. This chapter is arranged as follows. In Section 11, theoretical principles useful for the analysis of resonant tunneling are discussed. These include

2 TUNNELING UNDER

PRESSURE

47

the transfer matrix method for calculating energy levels, wave functions, and transmission coefficients through a general potential profile. Next, selfconsistent calculation methods for determining the potential profile for a quantum well or a 2D accumulation layer containing charge are treated. The theory is then described for dealing with situations in which bands of more than one symmetry are involved, notably T-X mixing. Section I11 contains a brief discussion of the apparatus used for achieving high pressure in resonant tunneling experiments. In Section IV, the results and conclusions of a number of experimental studies will be reviewed, dealing with each of the subject areas outlined above. Finally, Section V draws general conclusions of what may be learned from tunneling under pressure and finishes with some speculations for the future.

11. Theory and Calculation

1. RICCOAND AZBELFORMULAE As monolayer control of the composition of heterostructures was becoming a real possibility, Ricco and Azbel [3] proposed a simple theory to describe the resonant tunneling properties of a DBS. They treated the DBS as the quantum mechanical analogue of an optical Fabry-Perot resonator, in which the coherent interference of monochromatic waves leads to resonances in the transmission coefficient at certain energies. If the two barriers have transmission coefficients TI and T2 (TI 5 T2), then on resonance T l / T 2whereas , off resonance, the the transmission coefficient is Tp transmission coefficient is Tv TIT2.Unfortunately, it very soon became clear that such an analysis was unable to describe the real peak to valley ratio, Zp/Zv,in the I-V characteristic of a typical resonant tunneling device (e.g., see Fig. 10a, curve i). The reason for this lay in the fact that energy and momentum conservation could prevent all of the electrons in the emitter contact from taking part in the resonant current, while in the valley current, nonresonant inelastic scattering by phonons [4] or plasmons [5] could provide a significant additional contribution. Luryi [6] showed that for a highly doped emitter contact with no (or negligible) spacer layer to separate it from the barriers, only an equatorial slab of the 3D electron Fermi sphere could take part, containing a fraction d(SE/E,)of the total emitter electron concentration, where SE is the width of the quasi-bound resonant state in the well, given approximately by A / T , and T is its lifetime. When there is an undoped spacer layer of appreciable thickness (comparable to the extent of the lowest-energy wave function confined in it), 2D electrons can be confined in the “triangular” potential well of an accumula-

-

-

48

P. C. KLIPSTE~N

tion layer adjacent to the first barrier, but again only those in the lowest 2D level will contribute to the peak current [7]. To calculate the expected peak and valley currents more accurately, the value of the transmission coefficient over the full range of energies of electrons in the emitter contact is required. To obtain this, the transfer matrix method has become a widely used tool.

2. TRANSFER MATRIXMETHOD The wave function of an electron in a crystal is described by Bloch's theorem, 9 = u,.*(r) * eik.’, where u J r ) is crystal periodic and is different for each of the bands in the crystal (n = band index). In an extension of the bulk k p perturbation theory to describe heterostructures, the wave function can often be approximated near a particular band extremum at k = k,as W = En Q,n(r)un,k,l(r), where all the bands in the crystal are included in the summation, and a, are the envelope functions. It is assumed that un,ko(r) is the same for each band in each material, so that all Q,,(r) are continuous across a heterojunction interface. This is reasonable for materials with similar structures and lattice constants, such as GaAs and AlAs. If all other bands are far away in energy, so that the amplitude of a, is much greater than that of any of the other envelope functions, it may be shown that Qo obeys the Ben-Daniel and Duke Hamiltonian for an electron with effective mass m* and potential energy V&), the energy of the bulk band extremum [8, 91. The effect of the remote bands is simply to change the effective mass from the free-electron value [9]. V,(r) is usually a slowly varying function of r,such that the material may be divided up into thin layers in which V,(r) = V j= constant in the ith layer. Due to reflections at the interfaces, the envelope function (dropping the subscript 0) in any given layer i contains forward and backward traveling = Qj(z)eikii.',where z is the growth waves: Q, = (a.e;P + p . e-ik~,.r)eikil.r direction, kZ,;= 2m7(EZ- V ; ) / h EZ, is the energy of the particle after substraction of kinetic energy due to in-plane motion, and kllis the in-plane wave vector. From the explicit form of Q,, it is easily shown that for a layer i extending from zl to z?, ~ o s ( k , . L) ~

m7 - sin(k,.;. L ) kz.;

-

X 4 . i

--.

m7

sin(k,,, - L )

cos(k,,;- L)

-vQ,j

2 TUNNELING UNDER PRESSURE

49

where z2 - z1 = L and the 2 X 2 square matrix is known as the transfer Mi for the layer i. For EZ < Vi the trigonometric functions become matrix, -

hyperbolic functions. Integrating the Ben-Daniel and Duke Hamiltonian across an interface at coordinate zintbetween layers i and j gives the boundary condition for the derivative of the envelope function, so that the following conditions hold:

which are also consistent with conservation of probability flux. For a region between coordinates zL and zR, divided into n layers, it follows from Eqs. (1) and (2) that the wave function and its derivative at the edges of the region are related by

Mg is the transfer matrix for the complete system.

-

For a potential with bound states, we may write the condition for a bound state as

where zR and zLare chosen to be in regions where the wave function only contains a term that decays exponentially with a displacement away from the region of space occupied by the bound state. Dividing the resulting pair of simultaneous equations gives

50

P. C. KLIPSTEIN

where Ma,are the elements of M;. The energies of the bound states are those which satisfy Eq. (5). Thefunction on the left-hand side of Eq. ( 5 ) can vary very rapidly with E Z ,so great care must be taken to locate all the bound states correctly. The unnormalized wave function is easily calculated by choosing values for CD and l/rn*-V@ at zL in the ratio 1 :kL/rnEand using the transfer matrix M $ as in Eq. (3) to calculate the wave function at z, 5 zR, where the transfer matrix is constructed from s thin layers, the last of which contains zs.(If zL is at the base of an infinite barrier, the ratio 0: 1 should be used.) Once obtained, the wave function is easily normalized. For a superlattice, MI; is calculated for one period of length D and set equal to eiqD I , w h e r e q i s the superlattice wave vector. This then becomes a simple eigenvalue problem, where the dispersion, q ( E , ) is , determined from the eigenvalue, and the wave function from the eigenvector. For energies where the states are not bound, the transfer matrix may be used to calculate the transmission coefficient. The wave functions at the left- and right-hand side of the region of interest may be written

Using Eq. (3), the transmission coefficient is easily shown to be

can be derived in an A formula for the reflection coefficient R = (B12/IA\2 analogous way. For 2D + 2D tunneling, considered in detail in Section IV.2.e the resonant current can be estimated, in principle, from the energy splitting SE between symmetric, as,and antisymmetric, aA,combinations of degenerate emitter and collector wave functions involved in the tunneling. An electron localized in one well can be considered to be a linear combination of as and CDA whose probability density will shift from one well to the other in a time T = nh/&. This is an example of quantumbeats. For symmetric samples of the type considered below with 60- or 70-A-wide AlAs X-band wells separated by a 40 A GaAs barrier, the splittings for

2 TUNNELING UNDER PRESSURE

51

the ground states of each well (effective mass 0.25 mO)are -0.037 and -0.030 meV, respectively.' They correspond to T values of 56 and 69 pS. Note that the price of localizing the electron in one of the wells is that energy is no longer well defined. This is no different from the use of a wave packet to estimate the time for tunneling through a single [lo] or double barrier [ll].The use of the Bardeen transfer Hamiltonian [12], which has been so successful for 3D + 3D and 3D 2D tunneling [13], is problematic for 2D -+ 2D tunneling [14] because the Fermi golden rule only applies when there is a finite density of final states (for z-motion), and a suitable treatment has not yet been formulated. Further discussion of 2D -+ 2D tunneling is delayed until Section IV.2.e.

THE I-V CHARACTERISTIC 3. CALCULATING

Using the formula for the transmission coefficient T(EZ) in Eq. (7), it is possible to calculate the current in a 3D -+ 3D or 3D + 2D tunneling structure comprising an emitter layer, a barrier or barrier and well structure, and a collector layer by summing the transmission over all the carriers in the 3D emitter. Their kinetic energy, E = EZ + E X , Y ,is divided into contributions for motion parallel or perpendicular to the z-direction. If the effective mass in the barriers is different from the effective mass in the emitterlcollector and well, then it is easy to show that conservation of both momentum parallel to the layers and energy leads to the transmission coefficient for the structure becoming a function of the emitter energies Ez and E X , Yinstead , of just EZ.The current density is

where fE is the Fermi-Dirac function in the emitter, assumed to be in thermal equilibrium at temperature T, and V is the applied bias. For a uniform isotropic mass throughout the structure, it can easily be shown that this gives

[ (

J = S / ;In exp 'Calculation carried out by H. Im

-

(EZ - EF)) + 11 T(Ez, V )dEZ kT

(9)

P. C. KLIPSTEIN

52

0.0

0.5

Bias (V)

1.0

1.5

FIG. 1. Theoretical resonant tunneling characteGstic at 1 bar and 77 K for a DBS with 30-A AlAs barriers and a 50-A GaAs well. The emitter is taken to have a 3D electron distribution, and the effective mass throughout is taken as 0.067 mo,that of the r mass in GaAs.

By performing this calculation for a range of applied biases, the current voltage characteristic may be derived. Note that the result is inaccurate near the origin since Eq. (8) only considers the current flowing in one direction. However, most resonant tunneling occurs at a large enough bias for this to be unimportant. Results at 77 K for the r conduction band of a DBS with 30 AlAs barriers and a 50 GaAs well are shown in Fig. 1, where the barrier height is 1.03 eV and the effective mass of 0.067 rno for GaAs has been used throughout [15].2 Figure 1exhibits the usual triangular form of resonant tunneling expected for 3D 2D tunneling. The smoothness of the turn on at around 0.5 V is due to the finite temperature used. The peak current calculated in this way is usually fairly accurate, but it is clear from Fig. 1 that the valley current is grossly underestimated. An important reason for this is that all inelastic processes have been neglected from this purely coherent model of the tunneling current. Of these, phonon scattering is known to be important, because a phonon replica of the main resonance is often observed when the emitter electrons are biased one longitudinal optic (LO) phonon energy above the resonant level in the well [4]. Another important contribution

A

?Since the mass in AlAs (m* mass difference are small.

A

- 0.15 m,,) is fairly similar, corrections to Fig. 1 due to the

2 TUNNELING U N D E R PRESSURE

53

for AlAs barriers is that of the conduction band with X symmetry, which will be discussed in detail in Section IV.2.b. The foregoing argument has been based on a coherent picture of resonant tunneling. There has been much discussion of the validity of this picture as opposed to a sequential model, in which the electrons tunnel through the emitter barrier into the resonant level in the well, scatter to other states in the well, and then tunnel out through the collector barrier. Clearly the sequential model is the correct picture for resonant tunneling through an excited subband of the well of a DBS when the scattering time from excited to ground subbands is shorter than that for coherent tunneling out of the excited subband through the collector barrier, a situation that frequently occurs for typical DBS dimensions (see Section IV.2.d). However, for resonant tunneling through the ground subband, the successful application of the Bardeen transfer Hamiltonian to DBSs [13] demonstrates that phase coherence may be lost between the electron’s entry into the well and its departure. Thus, intrasubband scattering that results in lifetime broadening of the resonant level that is less than the energy range of the 3D electron distribution in the emitter does not affect the current [16]. Therefore, in most cases of DBSs of typical dimensions with a true 3D emitter, the coherent or sequential tunneling pictures give identical results for the peak current. It is often observed that the peak and valley currents are sensitive to pressure, and this has been related to changes of effective mass in the barriers or changes of the barrier height. In principle, a calculation such as that of Fig. 1 can be used to fit these changes. However, the following simpler approach is often used to obtain qualitative agreement [17]. For 3D + 2D tunneling, the resonance peak occurs when the well state, E,,,, is biased to Ez = 0 in Eq. (9). For kT 4 E F ,and approximating the transmission to be constant at the Ricco-Azbel value, T l / T 2 ,for EZ over the width of the resonant state, SE,and zero outside it, Eq. (9) gives J p rn T l / T 2SE for the peak current density. SE is proportional to the rate at which electrons tunnel out of the well, so it is proportional to T 2 .The coherent contribution to the valley current, Jv, will obey the Ricco and Azbel formula, T I . T 2 ,but this is always much smaller than the actual value because the inelastic contribution is much greater. However, for phonon scattering at the emitter barrier the current will depend on the (squared) overlap of an emitter state with the resonant state in the well, which will be proportional to TI, and at the collector barrier, on the overlap of the resonant state in the well with a collector state, which will be proportional to T 2 ,so the product Tl . T2may still be proportional to the valley current. Thus, we have Jp cc Tl and will assume Jv rn T I T2. Taking Ti cc

e-2kldl

54

P. C .KLIPSTEIN

and k,= -/ti, where A; is the effective barrier height approximately equal to (Vj) - Ere,and (VJ is the average potential of the barrier (most accurately estimated at low applied bias), we find

Note that the peak current exhibits a pressure dependence similar to that for a single bamer. Similar results may easily be obtained using the sequential picture. If the occupation of the well state is fw, then Jp Tl(l - fw) T2fw T1T2/(T, + T 2 ) Tl (for T2 Tl), and Jv T I -T2. Cury et al. have used a WKB calculation for the barrier transmission, Tia e-&i, and a;= ( m / h sk ) d(Vj(z) - Ere,) dz, which is more precise [18]. If the bamer heights do not change with pressure, this gives the same result as Eq. (lo), with kidj replaced by a;.

-

-

-

-

-

4. SPACECHARGE The preceding analysis has assumed that no charge has been stored in the well. A simple Fabry-Perot analysis of light in an etalon shows that the wave amplitude inside the etalon is a maximum on resonance. Thus, a similar analysis for electrons in a DBS shows that there can be a substantial amount of charge stored in the well. This can create significant modifications to the I- V characteristic, delaying the resonant condition until higher bias and causing intrinsic bisrability, in which the resonant bias on increasing the voltage is higher than on decreasing [19]. To deal with such effects, it is necessary to adopt a self-consistent approach, in which the wave function and hence the charge distribution are calculated iteratively from the potential, for example, using the transfer matrix method of Section 11.2. At each stage, the charge distribution is used with Poisson’s equation to recalculate the potential. The potential for the next stage is then constructed by combining a small fraction of the new potential with the previous potential. By repeating the calculation until the potential, and therefore the bias across the sample, becomes stable, the self-consistent solution is obtained. The self-consistent calculation is fairly straightforward for a “square” quantum well or superlattice potential containing charge, but the more difficult situation often arises of a “triangular” potential whose shape is

2 TUNNELING UNDER PRESSURE

55

determined entirely by the distribution of charge that it contains. Examples of the latter situation are an accumulation layer of electrons (or holes) adjacent to the emitter barrier of an n-type (or p-type) DBS, and the charge layers adjacent to the interface (or AlSb barrier) in a semimetallic InAs/ (AlSb)/GaSb heterostructure. In Section IV.3.b, the NDR properties of an n-InAs/p-GaSb heterojunction are discussed in detail. The method used there to calculate the charge distribution is described in the remainder of this section. The results are easily adapted for the case of a single n-type or p-type accumulation layer in a DBS. The potential, cp, for a 2D electron gas in the region z I0 or a 2D hole gas in the region z 2 0 is related to the charge by Poisson’s equation:

and p i , @ h , i ( ~ ) are the 2D populations and normalized where ni,@e.i(z) wave functions of the occupied electron and hole subbands, respectively. The potential energy of an electron is Z I O

V ( z )= {;::;(z)

z20

where A is the energy difference between the valence and conduction bands at the interface (z = 0) of the InAs/GaSb heterojunction. nA and nD represent ionized acceptor and donor concentrations in the n-InAs and p-GaSb materials, respectively, which it is assumed are slightly compensated (in real materials this should always be true). At low temperatures the majority InAs donor levels and GaSb acceptor levels in the flat-band region will not be fully occupied and the Fermi level will be pinned very close to their respective energies, ED and EA. In the band-bending region there will also be negatively charged ionized acceptors in addition to the 2D electron charge in the InAs and positively charged donors in addition to the 2D hole charge in the GaSb. This is necessary sothat the Fermi level in the flat-band region can be adjusted to lie at the correct energy in the bandgap. To obtain the first wave functions, the Fang-Howard potential is a good initial approximation for the potential [20,21]. When a magnetic field is applied in the z direction, ionized donors (concentration, ND) and acceptors (NA) may also appear in the n- and p-type

56

P. C. KLIPSTEIN

band-bending regions, respectively, to achieve continuity of the Fermi level between the magnetic donor or acceptor levels and a nearly full confined Landau level. Their appearance will depend on the values of N D and nA in the n-type material and the values of NA and nD in the p-type material [22]. For a field ( B x B, Y ,0) perpendicular to the z direction, the dispersion in ( k x ky) , depends on the solution of a one dimensional Schroedinger equation with an effective potential Veff(z) (in the Landau gauge),

whose wave functions, @ ( k x k, y,z), are used together with the background compensation charge to calculate the self-consistent potential. For zero magnetic field eq. (11) can be integrated by parts to give, with Eq. (12), the following [23]: V(Z)

=

The boundary conditions are

u

[A

z=oz=O+

(15)

where n and p are the total 2D charge concentrations and Zn and Z, are the widths of the band-bending regions in the n- and p-type materials, respectively. The values of Z, and 2, are adjusted so that for a given n and p the Fermi level has the correct energy in the flat-band region. The electric displacement is continuous at the interface, so n + nAZn = p + nDZp,which simply expresses charge balance. For simplicity, charge penetration into the vertical confining barrier has been neglected, but can easily be taken into account if necessary.

2 TUNNELING UNDER PRESSURE

57

Usually, nA and rtD are not well known, but it is found that the confinement energies and subband occupations are hardly affected by the choice of these parameters. For example, in an n-InAs/p-GaSb heteroj~nction~, changing both from 1 X 10'5cm-3 to 4 X 10'5cm-3 changes n from 4.56 X 10"cm-2 to 4.37 X 10"cm-2 a n d p from 4.47 X 10*1cm-2to 4.24 X 10"cm-2. With nA = nD = 1 X 10'5cm-3, a change of donor binding energy from 2 to 20 meV only changes the lowest electron and hole confinement energies by less than 0.2 meV and their 2D concentrations by less than 2%.

5. BANDMIXING a.

Introduction

In Section 11.2, the envelope function expansion based on a single band lying far in energy from all others was considered. In this section, we consider the situation in which several bands lie close in energy. It will be useful to allow crystal periodic functions u,,k(r) from more than one symmetry point, k r,in the Brillouin zone to be included in the envelope-function expansion of the wave function. For a superlattice,

where q is the superlattice wave vector and U , is a superlattice periodic function. Different k g are allowed as long as the product @;(z) * ~ , , ~ : ( r ) still satisfies the superlattice Bloch condition. Similar to the one-band case, it is often possible to define a new Hamiltonian acting on the envelope functions alone. Important examples for only one value of k,"are reviewed in Ref. [9]. In the following two sections, however, T-X mixing that involves states with different kf will be discussed. Note that although it is always possible to construct the wave function by summing many terms containing crystal periodic functions from a single symmetry point, in this case the use of more than one symmetry point results in more slowly varying envelope functions and fewer significant terms leading to greater physical insight. 3The following parameters were used: A = 0.155 eV; = 0.1 mu;mh,, = 0.3 tqI;me.i,= 0.023 mo;m , , ( k ,=)0.023 rno[l + (0.86fLZ/E,2mo)(k:)]. No light-hole subband was occupied [24].

58

P. C. KLIPSTEIN

GaAs AlAs

GaAs

\'

j

conduction band

I

1

valence band

growth direction

- z a) b) FIG. 2. (a) Schematic energy dispersion of bulk GaAs and AlAs in the first Brillouin zone, along the [Ool] axis (minimum direct and indirect superlattice bandgaps are shown by solid and broken arrows, respectively). (b) Bandgap profile for the holes and the r and X electrons in a superlattice.

b. Allowedr - X ,Mixing

In the conduction band of a GaAs/AlAs superlattice, it is possible to have significant mixing between the zone center r states, Ur, at k: = 0 (with s-like orbital symmetry), and the zone edge Xz states, ux,, at k", = [0, 0, 11 * (a is the cubic lattice constant). Figure 2 shows the energy dispersion of bulk GaAs and AlAs along [Ool] and the band profiles experienced by the r electrons, X electrons and r holes. It should be clear that in each material there are six minima of the X band located near the equivalent (100) faces of the bulk Brillouin zone. Figure 3 shows their constant energy surface. In AIAs, the two longitudinal minima with a long wave vector in the z direction (X, minima) have a large effective mass of m: = 1.0 2 0.1 mo,whereas the four transverse, or Xx, Xy,minima have a smaller mass in this direction of m& = 0.25 ? 0.01 mo [25].3aThere are two types of X state separated by an energy gap: X I , which is p-like on group I11 sites and d i k e on group V sites, and X3, which is p-like on group V sites and s-like on group 111 sites [%I. XI is predicted to lie below X3 in GaAs and AlAs, a fact supported by the observation of strong coupling to the XI symmetry LO and transverse acoustical (TA) phonons at the indirect absorption edge of AlAs [27]. However, recent Raman [28] and high-resolution photoluminescence (PL) measurements in type I1 super'"The symbols rn: and m%.ufor the heavy and light (or longitudinal and transverse) effective masses will be used throughout this chapter.

2 TUNNELING UNDER PRESSURE

59

FIG. 3. Constant energy surface for X-band e l m trons. (From Ref. 15.)

lattices [29] have also shown strong coupling to the superlattice mode with LA(X) character, indicating that the lowest state has some X3character. Let it be assumed that the lowest X I state can mix with a r state that is near in energy. Treatments of the mixing by expanding bulk r and X crystal periodic functions in one material as a linear combination of those in the other are unphysical because they lead to functions with different amplitudes in alternate monolayers [30,31]. A preferred approach is to use the following two-band Hamiltonian, which combines independent Ben-Daniel and Duke "single-band'' Hamiltonians Hr and H x , but allows mixing to take place at the interfaces, at coordinates zi[32, 331:

As shown in Section 11.2, the Hamiltonians Hr and H x , describe the properties of the dominant envelope function within the envelope-function expansion. These envelope functions are denoted @r and aXz in Eq. (17) and are continuous at the interfaces. Nonzero aileads to mixing between the wave functions Qrur and @xzuxz that changes the boundary condition in Eq. (2) for the derivatives of the envelope functions, as may be seen by

60

P. C. KLIPSTEIN

integrating Eq. (17) across an interface at zi. The new boundary conditions, 4, become with a,= (h2/2rnoa)

-

1 0 0 0 -

0 1 2 0 U

0 0 1 0 t? - 0 0 1 ll

where MIFis known as the interface matrix.

-

There have been many treatments of T-X mixing based on tight-binding or pseudo-potential calculations (see references 3-11 of Ref. [34]) or effective-mass models similar to Eq. (17) [32-341. While it is clear from all of these that the sudden change of potential at the interface is the cause of the mixing, there has been no clear explanation for the microscopic origin of the mixing. An explanation may lie in the asymmetric component, vint(r), of the crystal potential in the region of the interfacial plane of As atoms. This crystal potential may be written /3(vcaAs+ v A I A ~ )+ vint(r),where VGaAs(r) and V A I A ~ ( ~are ) the bulk crystal periodic potentials and /3 = 4. The asymmetry is due to the presence of a plane of Ga atoms on one side of the interface and A1 atoms on the other. The mixing is given approximately by a; (UrIVintlUX,) [35]. The potential, vint(r),has even parity in the x and y directions, but approximately odd panty in the z direction: vint(x, y , zAS - z) = -Vint(yI X , z - Z A ~ ) ,where zASis the coordinate of the interfacial plane of As atoms. Parity considerations show that the mixing is much stronger for X3 than XI, while the dependence of vint(r)on the number of Al-As-Ga bonds shows that the mixing is proportional to x for a GaAs/Gal-,A1,As interface, both results that are in agreement with the tight-binding calculation of Ando and Akera [36]. It also follows that oimay be written = = ah-x~ .e-i,(2r/o).z, for mixing between r and XIz, and &X,. p,.e-"(2"/")'zi for mixing between r and X3z, where Pi is +1 for a normal interface (AIAs on GaAs) and -1 for an inverted interface (GaAs on AlAs), and and a;-x, are constants. For both types of mixing, mi alternates in sign as a given interface is moved by successive monolayers because of the sign alternation of the overlap of the r and X crystal periodic functions near the interfacial As atoms. This sign alternation was first pointed out by Aleiner and Ivchenko to achieve agreement between effec-

-

2 TUNNELING UNDER PRESSURE

61

tive-mass models and microscopic tight-binding or pseudo-potential models, in which only the latter were sensitive to layer number [34]. Equation (17) may be generalized to a 3 X 3 Hamiltonian matrix containing diagonal terms Hr, H x I z ,and Hxgzand off-diagonal Sfunction terms with (YY-'~, ( Y T - ~ ~ and , ( ~ 7 1 - ~=3 ( ~ ~ , ~= ~c 1 $ ~ 3u* iPi.~The~ interface ~ ~ ~ matrix ~ ~ is ) obtained as before, by integration across an interface. Ando and Akera [36] showed that the transfer matrix method may be used in a way similar to the method for a single band, intercalating an interface matrix between the transfer matrices for each layer. For example, it follows from Eq. (18) that

= M,, X

-

X

X

where h& is the transfer matrix for the whole system. Xia notes that the Ando and Akera interface matrix for T-X mixing calculated by tightbinding methods leads to a non-Hermitian Hamiltonian because the offdiagonal elements in Eq. (18) must be complex conjugates and hence must have the same sign if real [32]. This requirement is also necessary for conservation of probability flux through the interface. Xia compares the T-Xz interface matrix approach with a pseudo-potential calculation for a GaAslz/AIAslzsuperlattice and obtains very good agreement for = 0.338 eV A (or t = 0.5) [32], whereas Liu uses a value of 0.1 eV A [33] based on the pseudo-potential model of Marsh [37]. Ando and Akera [36] point out that the interface matrix can be useful for other systems, such as an InAslGaSb interface. In this case the interface matrix contributes mixing in addition to the kap mixing of the standard bulk Kane Hamiltonian [38]. Such an interface can be prepared GaAs-like or InSb-like [39], and its properties are discussed further in Section IV.3.b.

62

P. C.KLIPSTEIN

It is likely that a different interface matrix will be appropriate for each interface type. 4 Several workers [34, 361 draw attention to the fact that since crf;-'~, the k - pmixing between XIz and X3z (responsible for the camel's back band structure near the bulk X point) is important if X1 is below X3. In this case, the k - pinteraction between ux,,and ux,, must also be included as off-diagonal terms in the 3 X 3 Hamiltonian (in addition to the axl-',) [9,34]. T-X mixing in the vicinity of the XlZstates then takes place indirectly because they contain a small admixture of X3z; that is, a term @x,,ux,z must appear in the envelope expansion of the wave function. is small (-0.33 eV or essenUnfortunately, it is unclear whether (~Tg-~l tially zero. Both results have been proposed, based on pseudo-potential and tight-binding calculations, respectively [32, 361. Considering mixing between r and the lower Xz band, the complete Xz wave function l Z f ) ,becomes a linear combination of the ’I and Xz wave functions of appropriate symmetry (e.g., same superlattice wave vector) in the absence of r-X mixing:

A)

17

For simplicity only the r state nearest in energy has been included in Eq. (20) because its coefficient has the smallest denominator (r states further in energy and mixing with states in the upper Xz band can easily be included as necessary). From a 3 X 3 generalization of Eq. (17), it follows that the mixing potential is given by

where both envelope-function components of the lower Xz wave function have been included. In a typical heterostructure, with layer widths of -50 to 100 A, V , 1 meV [40]. It is often assumed, incorrectly, that V , is proportional to the overlap integral of the r and Xz envelope functions [30, 41, 42, 431 rather than the product of their values at the interface.

-

c.

Forbidden r - X x ,and y Xz-Xx.y Mixing

In a perfect heterostructure, there can be no mixing between the lowest Xz and Xx or Xy states. However, disorder will lead to mixing [105], as

63

2 TUNNELING UNDER PRESSURE

shown by the following simple model, which assumes interface steps of height equal to one bulk unit cell (it may easily be generalized to steps of different height). Thus, if z, is the position of the unit cells within the disordered layer of a single interface, Z, < ziare the cell coordinates of GaAs, and z, > ziare the cell coordinates of AlAs, the interaction Hamiltonian due to disorder may be written as

where f(x,y ) is a function with a value of 1 for GaAs and 0 for AlAs that maps out the interface step distribution at the interface coordinate z = z;, ~ y , z - z,) and VAIAs(r) = VAlAs(X, y , z and VGaAs(r) = Z Z ,VGaAs(X,

cz

z,) are the crystal potential functions for the two materials. Calculating matrix elements from the unmixed wave functions, the disorder-induced analogous to the allowed potential for mixing between Xz and Xxor X y , mixing potential Va in Eq. (21), is

where the sum is now over all interfaces with coordinates z;;y, is a constant that depends on an integral containing the crystal potentials, VGaAsand VAIAsr and the XIBloch functions of the Xz and Xxstates; and F j ( k xk,y ) is the Fourier transform of J ( x , y ) with K 2r/a (it is assumed that F i ( ~0),= Fi(O,K ) ) . The envelope functions are for X I , although the small X3contribution should not be ignored in a more detailed treatment. Similarly, disorder-induced mixing can take place between the r and Xx or Xy states, with mixing potential VL = y XiF;(K,0) @xx(zi) . @,-(zJ = y * El F;:(K,0) . @X,,(Z;) . Qr(zi),where y is a constant.

-

-

-

6. STRAIN DUETO PSEUDOMORPHIC GROWTH The details of the correct application of the Hamiltonian for anisotropic strain to states of different symmetry inside the bulk Brillouin zone are beyond the scope of this chapter, and are covered elsewhere in this book. However, an important result to which reference will be made later is the effect of strain on the X minima. For an in-plane biaxial compressive

64

P. C. KLIPSTEIN

strain, such as occurs when AlAs is grown pseudomorphically on GaAs, the energies of the bulk Xx.y states are reduced relative to the Xz states. In the case of AlAs in a heterostructure grown on a GaAs substrate that therefore defines the lattice parameter, the splitting is -23 meV [44]. This results in Xz(l) as the lowest confined state for AlAs widths thinner than -50 A, and Xx,y(l) for thicker AlAs widths [44,451. For a Si/Sil-,Ge, heterostructure grown on a relaxed Si,-xGexvirtual substrate, the opposite is true, and the Xx,y states in the Si lie above the Xz states. For x 0.35, the splitting is -230 meV [46]. Stress can also be applied externally. Compressive stress in the growth direction lowers the energy (relative to the valence band edge) of the XZ states, while the Xx.y states are slightly up-shifted due to the competing effects of the hydrostatic component of the stress and the biaxial in-plane tensile stress component [47]. An in-plane stress applied along the [ l l O ] or the [ i l O ] directions causes a piezoelectric field in the [001] direction whose sign reverses on changing from one direction to the other. Such effects have been detected in I7-profile tunneling in GaAs/AlAs DBSs, in which the piezoelectric field can act parallel or antiparallel to the applied electric field [48].

-

111. Experimental Techniques

The experimental techniques necessary for electrical measurements at high pressures, with or without the presence of a magnetic field, have recently been reviewed in detail in Ref. [49]. Most of the experiments described in the next section have been carried out in miniature piston and cylinder cells, such as that shown in Fig. 4 (designed by M. Eremets and co-workers [49]). A force of up to several tons is applied by a hydraulic press onto a removable tungsten carbide push rod, and the pressure is then maintained by tightening the top screw of the cell. For magnetic measurements, the cylinder body is usually made from beryllium copper (pressure range -14 kbar) or a NiCrAl alloy [50] (pressure range -30 kbar). Similar materials may be used for the pistons, although a low-magnetic grade of tungsten carbide is also suitable and less prone to damage. Maraging steel has been used successfully up to 20 kbar for nonmagnetic measurements [17]. Brass gaskets coated in indium form the seals, while the petroleum pressure-transmitting medium (100-120°C fraction) is conveniently contained inside a Teflon capsule, which provides the seal at low pressures. Electrical feedthroughs are most conveniently made through a

2 TUNNELING UNDER PRESSURE

65

Pressure medium

Glued electrical

FIG. 4. A schematic diagram of a miniature clamp piston and cylinderpressure cell. Typical dimensions: ID (inner) -4 mm, OD (outer) -27 mm, length (outer) -80 mm. (Based on a design by M. I. Eremets; from Ref. 15.)

small hole in one piston and held in place with Stycast 2850FT resin, cured with 3.65% by weight of Catalyst 9, pumped to remove small air bubbles. Pressure inside the cell is measured with a resistive manometer, usually manganin wire (previously pressure cycled) or n+-InSb, the calibrations for which are discussed in Ref. [49]. Prior to the popularization of the miniature piston and cylinder cell, NDR measurements were often carried out in a Bridgeman anvil apparatus [51]. Such systems have the advantage of very high pressures (-100 kbar) and large working volumes, but it is usually difficult to work below 77 K or in a large magnetic field.

66

P. C .KLIPSTEIN

IV. High-pressure Studies of Negative Differential Resistance 1. EARLYDAYS One of the first important contributions of high-pressure measurement to semiconductor physics was to establish that the energies of the different conduction band minima in Si, Ge, and 111-V semiconductors exhibited different pressure coefficients with respect to the valence band maximum [52]. Typically the direct r bandgap and the indirect L bandgap (minima along the (111) directions) increase at about 10 and 5 meV kbar-', respectively, while the indirect X bandgap (minima along the (100) directions) decreases at about 1 meV kbar-'. This fact assisted the identification of different minima responsible for the conduction band edges of Si (X band) and Ge (L band), while a crossover from r to X was observed in GaAs at about 40 kbar [53]. At about the same time, new semiconductor devices and applications were being developed, such as the tunnel diode [54] and the Gunn diode [55]. Studies on tunnel diodes contributed to the study of band structure, since the interband tunnel current is very sensitive to the band edges nearest to the tunneling energy, whose energies often change in a characteristic way with hydrostatic or uniaxial stress. Different uniaxial stress orientations are sensitive to the orientations of the different conduction band minima. Sharp phonon-related features can also be seen in the current, whose energies were studied as a function of pressure. Some of these results are reviewed in Ref. [56]. High pressure played a crucial role in understanding the mechanism for NDR in the Gunn diode. In a Gunn diode, a large electric field applied across a region of n-type GaAs causes an instability in the DC electrical properties. Using Bridgeman anvil techniques, Hutson etal. observed that the instability threshold voltage decreased above -10 kbar, so that it is reduced by -40% at 25.7 kbar [57]. Several mechanisms were under consideration at the time, including hot electron interactions with phonons or plasmons, trapping of hot electrons at impurities, and energy-dependent scattering of hot electrons [55]. The Ridley-Watkins mechanism of electron transfer from the high-mobility r minimum to the lower-mobility X minima was originally rejected by Gunn because the electron temperature did not appear sufficiently high [55]. However, Hilsum showed that the large density of states of the X minima could reduce the required electron temperature [58].The high-pressure results supported the Ridley-Watkins mechanism, because pressure reduced the separation between the r and X minima by about 60%,increasing the efficiency of electron transfer. Electron transfer turned out to be the correct mechanism. Similar results were also obtained for GaAsl_,P, alloys [59].

67

2 TUNNELING UNDER PRESSURE

A more detailed Bridgeman anvil study by Pickering etal.[60] showed that in some samples the threshold field and voltage initially increased due to the increase of the r electron effective mass. The reason advanced for why the voltage in some other samples did not increase is that high fields may already be present in the contacts due to Schottky barrier formation, which can nucleate the charge domains responsible for the electrical instabilities. This is consistent with the behavior observed more recently by Pritchard in an n+-n-n+GaAs sample in which the 1-pm-long n-region ( n = 1 X 10I6 cm-’) contained at its midpoint a 500-A graded Gal-,Al,As ramp up to x = 0.3 [61]. Pulsed I-V measurements were used to determine the voltage threshold for the onset of NDR, V , , which showed a strong decrease with pressure (Fig. 5 ) . Electrons leaving the ramp are already hot, so the behavior should be similar to that in samples in which high electric fields already exist in the contacts, as observed. This result contrasted with a control sample, in which the n-region ( n = 2 X 10l6 cm-’) was 1.7 p m long and did not contain the ramp. In this case, Vt increased with pressure up to 8 kbar, the largest value applied, consistent with an increase of the r electron effective mass.

W 1P)-Vt(lbar) W(1bar)

-0.10

t

-013

I

I

2

4

1

1

I

6 8 10 PRESSURE(Kbarl

I

i1

12

14

FIG. 5. Fractional change in the NDR threshold voltage with pressure at 77 K for the graded bandgap Gunn diode described in the text. (From Ref. 61.)

68

P.C. KLIPSTEIN

With the advent of heterostructures, high-pressure studies have focused principally on tunneling mechanisms in double- (and higher-multiple) barrier structures, including superlattices, and on novel types of broken-gap interband tunnel diodes. The progress of these studies is described in the following two sections. 2. RESONANT TUNNELING IN SINGLE-, DOUBLEAND MULTIPLE-BARRIER HETEROSTRUCIWRES

a. Introduction In the mid-l980s, it became possible to grow double-barrier resonant tunneling structures with clear resonant features in their I-V characteristics and substantial peak to valley ratios (PVRs). There began an intensive study into the dependence of the resonant properties on well and barrier widths, barrier composition, etc., and an exploration of ways to increase the PVR and sustain it up to room temperature. While the general properties were easy to understand in terms of, for example, the Ricco-Azbel theory, the real resonant properties were not well reproduced by simple coherent tunneling models that did not take into account inelastic scattering and charge buildup in the well. High pressure was often used as a means of gaining a better understanding of the tunneling mechanism and hence of finding ways to achieve better agreement between model and experiment and to enhance the resonant properties. Important contributions included establishing the role of the X band in destroying the r-like tunneling in GaAs/AIAs DBSs [62] and a demonstration of the existence of intrinsic bistability by using pressure to control the efficiency of charge storage in the well [63, 641. Subsequently, it was observed that at sufficiently high pressure the X band could be populated [65-671; this observation generated a new area of study into the novel resonant properties of the X band and 2D -+ 2D tunneling [68-721. Somewhat analogous L-band resonant properties have been observed in GaSb/AlSb DBSs [73, 741. b.

r and X

Related Tunneling

Figure 6 shows the relative alignment of the r and X conduction band profiles in a GaAs/AlAs heterojunction, based on the parameters in Ref. [75] and assuming that 64% of the r bandgap difference is accommodated in the conduction band. Since the offset between r,(GaAs) and &(AIAs) is very important to much of the subsequent discussion, the dependence on this percentage is plotted in Fig. 7. For 64%, the offset is 130 meV.

2 TUNNELING UNDER PRESSURE

AlAs

GaAs L

-I - ;-

A

I

X,-

69

-480meV --- -

rc

I

103OmeV

-------130meV

V

I

1 I

Z

I

(36%(

I

i

rv

I580meV

I

FIG.6. The electronic band alignments at a GaAs/AlAs interface with a 64 :36 band offset ratio. based on the parameters in Ref. 75. (From Ref. 15.)

For GaAs/Al,Ga,-,As, the T,(GaAs)-T,(AlAs) and T,(GaAs)-&(AlAs) barriers scale roughly linearly with x , although small nonlinear corrections should not be ignored for precision studies, estimates for which may be found in Refs. [75] and [76]. When pressure is applied, the r,(GaAs)X,(AlAs) barrier reduces at approximately 13 meV kbar-' [77,78]. The L minima of both GaAs and AlAs are about 300 meV above T,(GaAs), providing only a small step in the L profile [79]. There has been little evidence for the involvement of the L minima in any of the studies described here for the GaAs/A1,Gal-,As system. This is almost certainly because 2004

60

62 AEJAE,

64

66

8

(percent)

FIG. 7. The dependence of the absolute offset between Tc(GaAs) and X,(AIAs) on the band offset ratio. (From Ref. 15.)

70

P. C .KLIPSTEIN

they are indirect in k-space, have a fairly high mass for motion in the z direction: and lie significantly higher in energy than the low-mass indirect XXTy minima in AIAs, which therefore provide a more transparent barrier for indirect tunneling. One of the first high-pressure studies of a DBS was by Mendez etal. [SO] on two GaAs/Ga,-,AI,As samples; one with 100-A barriers ( x = 0.4) and a 40-A well, the other 50-Abarriers ( x = 1) and a 50-Awell. The first sample lost NDR at only -10 kbar, perhaps because the well was narrow and therefore needed a large electric field at resonance that reduced the effective barrier heights, and also because the emitter barrier was wide. In a subsequent paper discussed later, it was shown that the role played by the X minima increases with emitter barrier width [62]. The valley current at high bias also changed over to a higher pressure coefficient beyond 4 kbar and was related to indirect tunneling via the transverse X minima by fitting to Eq. (10) and ignoring the kzdzterms, as for a single barrier. For the second sample, with pure AlAs barriers, the NDR was lost by 2 kbar. The strong pressure dependence of NDR was another clear indication that the lighter, transverse X minima are involved. In a subsequent paper by the same authors, GaAs/AlAs DBSs with wells of 50 or 60 A and barriers with widths of 23, 30 and 40 A were studied [62]. It was found that the PVR became larger for thinner barriers and was less easily suppressed with pressure (Fig. 8). This behavior was modeled in terms of competing contributions at the emitter: a direct resonant r-like contribution and an indirect Xx.y nonresonant contribution. The latter involves scattering through a large in-plane wave vector of - 2 d aby phonon emission/absorption and interface roughness, tunneling through the Xx,y (AlAs) barrier, and scattering back by --27r/a to the r subband in the well. It cannot contribute significantly at biases below the peak bias, V,, because only beyond this bias does the second scattering process involve a real final state. Above V,, the nonresonant contribution increases rapidly with bias due to lowering of the effective emitter barrier height. Thus the peak current is relatively insensitive to pressure because the nonresonant channel is effectively closed, but the valley current increases rapidly with pressure because at this bias the channel is open and becomes more efficient as the -130-meV T,(GaAs)-Xx.y(AIAs) barrier decreases. The changeover from r-to X-mediated tunneling with increasing barrier thickness or pressure can be modeled quite well by adding r and X FowlerNordheim terms with the appropriate masses and barrier heights but with 4The masses parallel and perpendicular to the [lll]axis of a given L valley are listed by S. Adachi [J.Appl. Phys. (1985) 58, Rll], while the mass in the z direction is derived by F. Stern and W. E. Howard [Phys. Rev. E (1967) 163, 8191 as Ilm, = (1/3) [(2/m,) + (l/miJ] = 2/3m,, giving values greater than 0.1 for m, in both materials.

2

TUNNELING UNDER PRESSURE

71

FIG. 8. Comparison of the I-V characteristics at 77 K at representative p!essures (in kbar) for a 23-60-23-A and a 40-50-40-A GaAs/AIAs DBS. (From Ref. 62.)

-

a prefactor for the Xx,u channel of B M to represent the scattering efficiency. It is interesting to note that the effect of the X minima in the collector barrier is much smaller. Due to the potential difference across the DBS on resonance, the X minima in the collector barrier of, for example, the 23-60-23 A sample studied in Ref. [62] cross the energy of the resonant

level in the well at a pressure of only a few kbar. However, the resonance retains a strong PVR beyond 9 kbar. The reason is that transport via Xz is suppressed due to the small value of V, 1 meV and the difference in the in-plane dispersions for the resonant r state and the collector-barrier Xz state. Therefore, particularly for a thin collector barrier, the current can remain strongly r mediated for all states except those at some in-plane wave vector for which there is an almost direct coincidence in energy [33]. For a 52-A barrier, Carboneau etal. calculated that the r-and Xz-mediated currents were approximately equal using a I'-XIz effective-mass model similar to Eq. (17) [81]. Thus, for a 30-A barrier, the r contribution should be considerably larger than the Xz contribution. Transport through the collector Xx.y states of a DBS will not destroy the resonance because even for when they lie below the resonant state, the prefactor of BM transfer to Xx,y will contribute a broadening SE,,,IE,,, where Ere, is the confinement energy of the resonant level in the well. In single-barrier structures, weak NDR due to direct T-X-T tunneling has been observed and studied as a function of pressure and magnetic field by a number of workers [81-851. The prefactor, BM,used by Mendez, Calleja, and Wang for the Xx,ymediated current in the emitter barrier [62] was chosen to be consistent with thermionic emission studies, for example, by Solomon etal. [86] or Pritchard etal.[79] on single GaAs/AI,Gal-,As barriers with x = 0.8 and x = 1, respectively. A r,(GaAs)-&(AIAs) barrier was found in the region of -150 meV, and a steady reduction in the effective Richardson coefficient for x = 1. The reduction was observed for x > 0.4, by a factor of could be explained by the efficiency of scattering from r to X. Thermionic emission studies of the T,(GaAs)-&(AlAs) barrier height and its pressure coefficient are a little problematic, since shifts as low as -7 meV kbar-' and as high as -17 meV kbar-' have been reported [87, 881. The reason for these large discrepancies from the expected value of --13 meV kbar-I is presently unresolved. Diniz etal. [89] studied a 28-50-28-A GaAs/AlAs sample at temperatures down to 4.2 K. The peak current decreased with pressure up to 4 kbar at 4.2 K, but increased with pressure at room temperature. It was suggested that Xx.y-mediated tunneling is enhanced at higher temperatures due to an increased phonon population that provides the required in-plane momentum, and enhances the prefactor BM . Apparently, higher temperatures can also open the Xx,y channel at biases below V , ,in contrast to the case at low temperatures. At 4.2 K a large pressure coefficient of increase in the valley current was fitted to Eq. (10) for a single barrier, implying that a large effective mass of 1.1mo was required, and leading to the conclusion that the longitudinal and not the transverse minima were involved at this

-

-

-

-

73

2 TUNNELING UNDER PRESSURE

temperature. This would require a much smaller value for the prefactor BMthan that adopted in Ref. 62, for the current due to Xz to be greater than that due to Xx,y perhaps due to the lower temperature. c. Pressure Dependence of the Resonant Current Unrelated to the X Band

For a single GaAs/A1,Gal-,As barrier system, Eaves et al. observed that the tunnel current decreased slowly with pressure up to -10 kbar [90] and were able to fit the variation to Eq. (lo), in which the main contribution was due to the increase in the barrier effective mass. A larger decrease beyond 10 kbar was initially attributed to the possible involvement of the X minima in the barrier, but later was shown to be due to pressure-induced freeze-out of the carriers in the highly doped emitter layer [91]. Similar freeze-out effects have been observed in other studies [63]. There have been several further studies of this type on double-barrier and related resonant tunneling structures [63, 641, in which the X minima are sufficiently high in energy not to contribute significantly to the tunnel current, for example, in GaAs/A1,Gal_,As DBSs with x < 0.4 and thicknesses below 100 A. The pressure dependence of the current can then be estimated by Eq. (lo), often with the reasonable assumption that the band offsets and hence the tunnel barrier height are only weakly pressure dependent. For example, assuming the band offset ratio in G ~ A S / A I ~ . ~ ~ G ~ ~ . ~ ~ does not vary with pressure [78], -d(ln A l ) / d P 0.1% kbar-' 0.25 d(ln rn~,,,)/dP, so the mass terms have a larger effect than the barrier-height terms [17]. A comparison between this approach and a full coherent tunneling calculation similar to Eq. (8) (allowing for different in-plane masses in the well and barrier) has given fairly good agreement for the pressure dependence of the peak current [17]. Measurement of the pressure dependence of the effective mass in the well can be carried out by tunneling into Landau levels created by a magnetic field applied perpendicular to the layers. This has been done both for DBSs [92] and 2D + 2D tunnel structures in which inversion and accumulation layers are separated by a single barrier [93]. Equation (10) has been used on GaAs/AlAs DBSs of varying barrier thickness to investigate the relative strengths of r- and X-mediated tunneling, since the X channel will have a much smaller barrier height than the r [89]. For 14-50-14-A and 17-50-17-A samples, the r,(GaAs)-r,(AlAs) barrier dominates between 300 and 4.2 K consistent with Ref. [62], which showed that the r channel dominates for thin barriers. Therefore, a reduction in current can be related to the increase in the T,(AlAs) electron mass.

-

-

P. C .KLIPSTEIN

74

d. Intrinsic Bistability

Double-barrier structures often show hysteresis in the NDR region of the I-V characteristic. This can be due to several causes, including the load line of a series resistance greater than the magnitude of the NDR, spontaneous current oscillation driven by the negative resistance [94], and intrinsic bistability [19]. Intrinsic bistability is due to charge storage in the well that can give rise to several very different current values in certain bias ranges, so that the current flowing depends on the previous bias history. Due to the other possible causes of hysteresis, however, the observation of intrinsic bistability was initially contentious. A high-pressure study provided elegant confirmation of its existence [63, 641. An asymmetric AlolG%.&/GaAs DBS was studied which comprised 111-A and 83-A Alo.4Gao.6As barriers and a 58-A well. The charge storage is greatest when the 83-A barrier is closest to the emitter because of the thicker restraining collector barrier. Figure 9a shows how the peak bias decreases and the peak current increases for the first resonance as pressure

30

- 15

2c

-

-

1c

4

4

- 105

3 0

-

-

1c

-5 5 I / /

C

FIG. 9. Pressure dependence of the forward-bias (V > 0) I-V characteristics at 4.2 K for a 111-58-83-A Alo,4Ga,l,,As/GaAs DBS. (a) and (b): First resonance: (c) second resonance. Pressures are as follows: i, 1 bar: ii, 7.4 kbar: iii, 12.7 kbar; iv, 14 kbar; v, 16.8 kbar; vi, 19.3 kbar. In (a) the current scale refers to curve iv. Curve v has been reduced by a factor of 4 (i.e., peak current = 125 PA), and curve vi by a factor of 450. (From Refs. 63. 64.)

75

2 TUNNELING UNDER PRESSURE

is increased above 14 kbar. By 16.8 kbar, the resonance has become sharp and symmetric instead of asymmetric, the current has increased from 15 p A at 1 bar to 125 PA, and the bias has dropped to -0.25 V from its original 0.6 V. Most of the bistability has been lost by 14 kbar, when the X band edge in the collector barrier is still about 60 meV above the resonant state in the well. The sheet density ( n , ,determined from magnetotunneling oscillations) and the dwell time ( T = n,e/J) in the well have been reduced from 2 X 10" to 1 X lo1' cm-* and from 0.7 to 0.2 pS, respectively. The current increases in spite of a reduction in n, because of the increased transparency of the collector barrier caused by the opening of the T-X channel. The phonon satellite stays nearly the same size (note the different scales used for each curve in Fig. 9) because it is a result of scattering after transmission from the accumulation layer through the first barrier (this transmission is more weakly pressure dependent). The second resonance also shows bistability arising from electrons tunneling through the second confined state in the well but rapidly scattering down into the lower state, where they are stored for a significant dwell time. In reverse bias (Fig. lo), the peak bias and current density remain stable

-160

-

-120

= l

a

Y

-60

-40

v

(V)

FIG. 10. Pressure dependence of the-reverse-bias ( V < 0) I-V characteristics at 4.2 K for a 111-58-83-A Alo.4Gh.6As/GaAsDBS. (a) First resonance; (b) second resonance. Pressures are as follows: i. 1 bar; ii, 5 kbar; iii, 10.8 kbar; iv, 14 kbar; v, 19.3 kbar. In (a) the current for curve v is reduced by 1000. In (b) the current for curve iii has been reduced by a factor of 4. The peak at --0.5V in curve i of (a) is due to LO phonon-assisted tunneling via the first confined well state. (From Refs. 63, 64.)

76

P. C .KLIPSTEIN

with pressure and there are no bistability effects. In this case, the first barrier is thicker and dominates the pressure dependence, so the current actually falls due to the d(ln m*)ldPterm in Eq. (10). The first and second resonances disappear at a lower pressure compared with forward bias. This can be explained by the mechanism of Ref. [62], in which nonresonant conduction through transverse X states relative to resonant conduction through the r profile increases with emitter barrier thickness. The main resonance in forward bias is lost by 19.3 kbar. At this point, scattering into the collector X state is said to be about 2 Ttransit 20 fs. This figure is considerably shorter than the scattering rate determined by time-resolved optical studies in type I1 superlattices, in which the scattering rate from r to X states is -10 pS [95]. It is probably incorrect, because, as discussed in Section IV.2.b, the NDR is not as sensitive to T-X mixing and scattering at the collector barrier as it is to scattering into transverse X states in the emitter barrier [62]. At 19 kbar, Xx,y in the emitter will be about 120 meV above r,(GaAs), so the loss of NDR is consistent with the fact that NDR is also absent in a high-quality 70-40-70-AGaAs/AIAs DBS at 1 bar (when Xx.y is also about 120 meV higher in energy), as discussed later and shown in Fig. 17 [96]. The loss of the second resonance at 12 kbar is used to estimate a scattering time into the collector X states of about 2 Ttransit 10 fs. This may be an underestimate similar to that for the first resonance. The larger bias across the emitter barrier could lower the effective Xx,y energy in the emitter barrier and cause even more rapid quenching of the r resonance. Indeed, for reverse bias the smaller electric fields at resonance and the larger X confinement energy in the thinner collector barrier should lead to higher pressure thresholds for loss of NDR if due to the T-X scattering channel in the second barrier, not lower thresholds as observed.

-

-

e. BeyondtheTypeZI Transition At zeromagnetic field

As discussed in Section IV.2.b, the initial effect of pressure in GaAsl AlAs DBSs is to enhance the valley current and to destroy resonant tunneling through the r profile. However, around 1990 it was observed that by continuing to apply pressure new resonant behavior starts to appear close to the type I1 transition, defined as the pressure at which the lowest states in the TCand X, profiles cross each other [65,66]. Figure 11 shows the I-V behavior over a wide range of bias for four DBS samples with 40-AGaAs layers and equal AlAs layer thicknesses of 40,50, 60, and 70 A. Figure 12 shows both the I-V and the 1 -Vbehavior near the origin on an expanded

2 TUNNELING UNDER PRESSURE

1

t

77

"

-1000

-1000

Bias (V)

t FIG. 11. I-Vcharacteristics of four LAIAs-40-LALAs GaAsl AlAs DBSs, where L A I A s = 40, 50, 60, or 70 A, measured at the temperatures and pressures indicated (20-pm mesa diameter). The resonances due to elastic tunneling processes are indicated by arrows.Note the higher temperaturesneeded to excite the Xx.y(l) emitter state for the two smallest values of LAIAs. (From Ref. 96.)

scale for the thinnest and thickest barriers. Full sample details can be found in Ref. [96]. It is clear that the effect of pressure is to make the X profile into the ground state of the conduction band, populating the AlAs wells with electrons transferred from the doped GaAs emitter and collector layers. This has been demonstrated most directly by a sharp increase in capacitance with pressure above the type I1 transition [71]. Initially the I-V behavior near the origin was ascribed to tunneling between confined longitudinal

P.C. KLIPSTEIN

78

FIG. 12. I-V and differential conductance characteristics at 4.2 K for low bias and above the type I to type I1 transition. (a) The 70-A sample, which displays characteristics that are nearly symmetric about zero bias. (h) The 40-A sample, which displays asymmetric characteristics.

states [65, 661 because only these were populated in the samples, which, in each of the studies, had -40-A AlAs layers. The confinement energies of the Xx.y states in the AlAs layers are greater and so it was thought that the population of these states would be negligible. However, the studies were both carried out at 77 K, and when measurements were performed on similar samples down to 4.2 K, it was found that the first two resonances nearly vanished below 77 K. Figure 13 shows an example of this behavior for a 30-40-30 sample at 14.5 kbar at temperatures below 100 K, which led Austing et al.to ascribe the first two resonances of this sample to processes involving an initial XX,Jl) state [69]. From an Arhennius plot, Xx.y( 1) was found to be about 20 meV higher in energy than the populated Xz(l) state. The first resonance was ascribed to Xx,y(l) + Xx.y(l) and the second, broader resonance to Xx,y(l) + Xx.y(l) + P ,where P is a phonon. Clearly, the lower mass of the transverse states strongly promotes the tunneling of these states. A simple estimate of the ratio of the currents for the Xx.y(1) -+Xx,y(1) and X,( 1) -+ X,( 1) processes in a 30-50-30 A sample at 4.2 K.5 The third gave a value of -10’ at 77 K,-1 at 11 K and

A

’Using more accurate values for the masses [mx,y(GaAs) = 0.23 mo.mx,y(AIAs) = 0.25 mo,mZ(GaAs) = 1.3 m,,,rnZ(A1As) = 1.1 rno] and a well depth of 0.35 eV, from which E x . y ( l ) - &(I) = 31 meV is easily calculated for 30-A wells, Eq. (1) of Ref. 6.9 gives a current ratio of -5 X lO’at 77 K. 1at -20 K, and -3.5 X 10.” at 4.2 K for a 30-40-30-A sample.

2 TUNNELING UNDER PRESSURE

'O1 60

J

79

103K

FIG. 13, Temperature dependence of the I-V characteristic for a 30-40-30-A GaAslAIAs DBS below 103 K (15-pm mesa diameter), showing suppressionwith decreasing temperature of the first two resonances, which have an Xx,y(l) emitter state, but not the third and fourth resonances, which therefore have an Xz(l) emitter state. (From Ref. 69.)

and fourth resonances in Fig. 13 were ascribed to processes involving Xz(l) as the initial state because these resonances did not freeze out and because they appear steplike, consistent with tunneling between sub-bands that each have a different in-plane momentum. Additional information about the alignment of levels in the two X wells could be obtained from the bias values of the resonances, which were attributed to Xz(l) -+ Xx,y(l) + Pi, where Piis a different momentum-conserving phonon for each resonance. In the light of more recent work on the Xx,y(l) -+ Xx.y(l) + P resonance

80

P.C .KLIPSTEIN

that observed two resonances in which P is TOGaAs and TOAIAs 1961, it is likely that the third and fourth resonances in Fig. 13 also involve GaAs and AlAs variants of the same symmetry phonon. In one early study [65],it was observed that after the appearance of the new low-bias X-related resonances, a resonance similar to the original r-mediated resonance reappeared. It was thought to be the original r resonance, but subsequent low-temperature measurements on samples with 30- and 40-A AlAs layers showed that this resonance also froze out at low temperatures, demonstrating that the initial state is Xx,y(l) [69]. It is the same resonance as in Fig. 11 for the 40-A sample at 10.5 kbar and 80 K. Comparing this resonance with the resonances for the other samples with thicker AlAs layers in Fig. 11 shows that the bias reduces with increasing AlAs thickness and that an additional resonance appears for the 60- and 70-A samples. These resonances are due to the Xx,y(l) + Xx,y(2) process for the 40- and 50-A samples, where they are thermally activated (since Xz(l) is the emitter ground state), and to XX.,(l) + Xx,y(2) and Xx,y(l) + Xx,y(3) processes for the 60- and 70-A samples, where they are not thermally activated (since Xx,y(l) is the emitter ground state). The bias positions of the resonances as a function of pressure agree well with a selfconsistent Schrodinger-Poisson calculation [96]. The increase with pressure of the Xx.y(l) -+Xx.y(2) bias position is found to be quite sensitive to the ratio r of stored charge in the emitter and collector AlAs wells (the smaller the ratio the greater the bias). Figure 14 shows the X !nd r profiles calculated for the Xx.y(l) + XX,,(2) resonance in the 70-A sample. The basic calculation procedure (assuming no T-X mixing) follows the transfer matrix and Poisson solution techniques of Sections 11.2 and 11.4. Modelling is facilitated by the assumption that the Fermi energy is continuous across the doped emitter contact and spacer, and the emitter AlAs layer. The high density of states in the Xx,y(l) level results in Fermi level pinning to within 1 meV above Xx.y(l) (at zero temperature), allowing the potential profile of the entire accumulation region to be calculated self consistently. The electric field at the emitter spacer/AlAs interface is then used with a second self consistent model in which charge is added to the emitter and collector wells in the ratio r : 1 until the collector and emitter states are aligned as desired. Finally, the depletion region in the collector contact is easily calculated from the resulting electric field at the collector AIAs/ spacer interface. The sum of the potentials across these three distinct regions within the device gives the total applied bias. It should be clear from Fig. 14 that as the pressure increases, charge transfers from the r accumulation layer to the first X well. This also explains why with increasing pressure the nonresonant current usually increases because of nonresonant transport through the emitter Xx.y states, but then

2 TUNNELING UNDER PRESSURE

81

z (Angstroms)

FIG. 14. Self-consistent calculation of the r and X conduction band profiles at 9 kbar for the 70-40-70-&. GaAslAlAs sample of Fig. 11. Note that the Fermi level of the occupied bound state, rACC(l). in the GaAs emitter accumulation layer is aligned with the lowest confined state in the emitter AlAs layer. Xx,y(l). (Calculation by J. M. Smith.)

usually starts to decrease at the point where the X-related resonances begin to appear [69]. The decrease is due to a reduction in the amount of r accumulation charge, which reduces the current due to r-mediating tunneling. At the same time the X-resonant current is enhanced due to an increase in the amount of emitter X charge. Inspection of the Z’-V plot reveals additional resonant features not immediately apparent in Fig. 11. Such a plot for the 70-A AlAs sample is shown in Fig. 15, where additional phonon-mediated resonances due to Xx,y(l) 4 Xx,Y(l) + P G ~ X AX~. ,Y ( ~+ ) X X , Y ( ~+) PAIA~, and X X , Y ( ~+ ) Xx,y(2) PAIAs are indicated by broken arrows. For a given phonon energy, the alignment between levels in the two AlAs wells is defined, and the selfconsistent calculation of the bias can be performed as a function of pressure and rvalue, as for the zero-phonon processes. A comparison betyeen such calculations and the observed bias values for the 60- and the 70-A samples is shown in Fig. 16, with the fitted values for the layer thicknesses, the effective mass m&, the rvalues, and the phonon energies shown in Table I. It turns out that all these parameters can be fitted unambiguously because of the different variation of the pressure dependence with each parameter [96]. It is also possible to obtain a value (to within 510meV) for the T,(GaAs)-&(AlAs) energy difference of 130 meV for the Xz profile and 106 meV for the Xx.y, assuming a pressure dependence of -13 meVlkbar

+

82

P. C. KLIPSTEIN

FIG. 15. Conductance data for the 70-40-70-A GaAs/AIAs sample of Fig. 11 at 9.8 kbar and 4.2 K. The solid arrows indicate elastic processes: the dotted arrows indicate their phonon satellites. (From Ref. 96.)

[96]. This corresponds to an average energy of 114 meV for the unsplit X band edge and thus to a band offset ratio of 63%in Fig. 7. The mass values in Table I are accurate to within 20.03 mo,the main degree of uncertainty being in the doping of the emitter and collector, determined from electrochemical profiling [96]. These mass values provide clear evidence for the involvement of the transverse X x . ystates in the tunneling and provide a new determination for the transverse mass. The phonon energies are accurate to within 2 2 meV and provide clear evidence for the involvement of TO phonons and not the more usual LO phonons observed in r-profile resonant tunneling. The scattering is probably intravalley rather than intervalley [96]. It should also be noted from Table I that quite a substantial charge is to be found in the collector AIAs, and that it is greatest for the zerophonon processes, as expected. Figure 16 shows that the forward-bias value of a given resonance is less than the reverse. This is due to differences in the thicknesses of the two wells. The fitted values in Table I show that the well closest to the substrate (this well is the emitter in reverse bias) is wider by about 1-1.5 monolayer (ml) than the other well (1 ml = 2.83 Similar behavior for the 40- and 50-A samples can be seen qualitatively in Figure 11, where the Xx.y(l) + Xx,y(2) resonance has a larger reverse bias. The difference in layer widths is due to details of the molecular beam epitaxy (MBE) growth [96], in which it is difficult to compensate exactly for the fact that the growth rates of the first and second AlAs layers are different. The difference in layer

A).

2 TUNNELING UNDER PRESSURE

11f

I

h

2: 4 Y

v

83

'

I \ \

Pp

P:,

FIG. 16. Best fits to the bias position pressure dependencies of the 70-40-70-A (upper) and 60-40-60-A (lower) GaAslAlAs DBSs, up to the type I to type I1 transition, using the Schrodinger-Poisson model. Open symbols correspond to theory, closed to experiment, squares to forward bias, and circles to reverse bias. Each resonance is identified in the figure. The fitted parameters are given in Table I. (From Ref. 96.)

widths can lead to a large asymmetry in the lowest Xx.y(l) + Xx,y(l) resonance for the samples with Xz(1) as the emitter ground state because at zero bias the Xz(l) levels are almost exactly aligned after the type I1 transition. This alignment is due to a large density of states that leads to a very small Fermi energy with respect to the bottom of each occupied subband. Therefore the Xx,y(l) levels are misaligned and can only become aligned at a small reverse bias. The consequence is a much larger resonant I-V peak in reverse bias than in forward bias and an asymmetric 1’-V peak, for example in the sample with 30 A AlAs layers in Fig. 13. In

P. C. KLIPSTEIN

84

TABLE I

PARAMETERS USEDFOR THE THEORETICAL CURVES OF FIG.16 Sample Parameter

60-40-56 A

70-40-67 A

0.24 0.25 2 1.4 1.2 2.5 32 43

0.24 0.25 5 1.5 1.5 3 33 42

Note. m: is for the Xx.y( 1) resonance,rn? for the Xx,y(2) resonance. r,. . . r4 correspond to the ratio of charge densities in the emitter and collector AlAs layers for the four resonant processes in order of ascending bias. (From Ref. 96.)

contrast, the lowest level in each AlAs layer of the 70-A sample is Xx,y(l), and so these are nearly aligned at zero bias for pressures above the type I1 transition, leading to nearly symmetric I-V and Z’-V curves in Fig. 12a. The accurate fitting of the difference in well widths disproves an earlier contention that the asymmetry in samples with layer widths of about 30 A could be due to differences in scattering at the normal and inverted interfaces, which are known to have different degrees of interface roughness [97]. Scattering is required to conserve in-plane momentum at finite bias. The argument was based on a semiclassical model of the tunneling efficiency if scattering occurred before or after tunneling [98]. That this argument is incorrect will be seen from the quantum mechanical argument presented in the section on 2D + 2D tunneling (p. 92). It is interesting to note that the sample with 70-A AlAs layers shows excellent resonances at high pressure (e.g., Fig. ll),and it is clearly a very high-quality structure. Yet at ambient pressure, the resonant properties appear very disappointing (Fig. 17). This is because the emitter barrier is very thick and transport via the transverse Xx,yminima dominates the electrical properties at 1 bar [62]. Although the preceding discussion has been centered on the use of have used compreshydrostatic pressure to populate the X minima, Lu etal. sive uniaxial pressure in the growth direction to lower the energy of the XZ(AIAs) minima substantially below the energies of the Tc(GaAs) and Xx,y(AIAs) minima in a 23-50-23-A GaAs/AIAs structure [67]. There is a

85

2 TUNNELING UNDER PRESSURE

0

. I " ' "

n

0.5

'

~

'

00

'

I

0.5

~

"

'

I

1.o Bias 01)

'

"

'

1

1.5

-1 -

-2

-

strong similarity between the uniaxial results (at 77 K) in Fig. 18 and the hydrostatic results of Fig. 12. At low bias, resonances start to appear above a uniaxial stress of about 16 kbar, and by 24 kbar a peak is observed at ?40 mV with a broader one at approximately 5140 mV. Comparing these results with those for sample A of Ref. [69], which has nearly similar well

Voltage [mv] FIG. 18. I-V characteristic at 77 K of a 23-50-23-A GaAslAlAs DBS under the [OOl] uniaxial stress values indicated (in kbar). (From Ref. 67.)

86

P. C. KLIPSTEIN

and barrier dimensions, the resonances for sample A under hydrostatic pressure have current densities approximately 50 times greater than in the uniaxial case. For sample A the concentration of Xzelectrons is -10" ~ m - ~ , whereas for the uniaxial case the concentration is hard to estimate since it is supplied by a depletion layer in the GaAs in which the r and Xz minima are close in energy, but cannot exceed -loi2 cm-3 (calculated assuming r-like donors). The observed current densities are perhaps consistent with tunneling between a small number of thermally activated Xx,y electrons for the hydrostatic case and a larger number of Xz electrons for the uniaxial case, and are discussed further in a later section (p. 94). Ambient pressure studies of tunneling have been reported in a Si/Si0_65 Geo.3s/Si/Si0.65Geo.3S/Si DBS grown on a Si0.65Geo.3svirtual substrate, where at low temperatures tunneling takes place through the Xz profile, but where there is a more weakly modulated Xx,y profile close to the top of the relaxed Si0.65Ge0.35barriers. It was reported that the NDR showed an unusually strong temperature dependence and that one current resonance was only observed above 50 K, for which models were proposed involving phonon absorption or emitter quantization [46]. In the light of the hydrostatic pressure results discussed here for GaAs/AlAs DBSs with AlAs thicknesses of less than 50 A, an alternative explanation is suggested. In the GaAs/AIAs DBSs, only Xx.y current is detected, although Xz(l) is the emitter ground state. Similarly, the resonant behavior in SilSb.65Ge0.35 at higher temperatures may be related to thermal excitation to the XxSylevels, through which transport and tunneling can take place much more easily. The results of this section have shown how strong 2D + 2D resonant tunneling can be observed in the GaAs/AIAs system at high pressure. The conductance peak for the 70-40-70-A sample in Fig. 15 bears a strong similarity to the conductance peaks on separately contacted GaAs quantum wells on either side of an A1,Gal,As barrier that were observed by several workers [99,100]. It is extremely difficult to make separate contacts directly to individual wells, and necessarily large well and barrier widths have been used to separate the wells. In the present system, high pressure can be used to make separate contacts to the wells, and much thinner wells and barriers can be used. Coupled-well systems have exhibited novel Coulomb gap effects due to interaction between the tunneling electron and the remaining hole when a magnetic field is applied perpendicular to the layers [101, 1021. The present system has the potential to exhibit analogous properties, although higher magnetic fields must be applied because of the larger effective masses of the X-band states. Although such Coulomb gap effects have not yet been observed definitively for X-band 2D + 2D resonant tunneling in static magnetic fields up to 15 T, other useful results have been derived and are discussed in the following two sections.

2 TUNNELING UNDER PRESSURE

87

Magnetic field studies

At pressures just beyond the type I1 transiB parallel to thez direction. tion, all the 2D electrons at finite bias may reside in the emitter AlAs layer, as shown by the band profile of Fig. 19. In the 70-40-70-A sample of Fig. 15, the process Xx,y(l) Xx,y(l) + TOAIAsis particularly suitable for studying Landau levels formed in the collector AIAs, because it is stronger than the process involving TOGaAs and the phonon provides the necessary in-plane momentum change for tunneling between the first Landau level in the emitter and higher Landau levels in the collector [103]. Figure 20 shows the appearance of oscillations in the I"-V plot at 10.2 kbar that correspond to the inter-Landau-level tunneling associated with this resonance. From the bias values of the minima, the Schrodinger-Poisson calculation can be used to make an accurate determination of the corresponding potential difference (u value) between Landau levels in the adjacent wells. The r value (slightly changed from pressures below the type I1 transition) is accurately calibrated for this pressure by ensuring that at zero field the u value (defined in Fig. 19) corresponds to the TOAIAs energy of 42 meV [96]. The resulting fan diagram of u versus B is shown in Fig. 21, together with the best fit to the expected dependence: u = ETo + n.(fieBlrn&,). Here mieo=drn$.y+rng is the mass appropriate to the Xx,y states, and the fitted value is mge0= (0.56 t 0.04)rn".

GaAs

AlAs

GaAs

AlAs

GaAs

-+

Ionized donor states

FIG. 19. Schematic band profile for a GaAsiAIAs DBS pressurized beyond the type I1 transition and biased to the Xx.y(l) + Xx,y(l) + T O A I A s resonance. The potential difference, u, is defined in the diagram. (From Ref. 103.)

88

P. C. KLIPSTEIN

p = 102kbar

T = 4.2K

-160 -180 -200 -220 -240 -260 -280 -300 -320 Bias (mV) FIG. 20. I"-Vcharacteristic for the 70-40-70-A GaAs/AlAs sample of Fig. 11 at magnetic fields of 0 and 15 T applied parallel to the z direction, for 10.2 kbar and 4.2 K. Alignment of the Xx,y(l) + X x . ~ ( l+ ) T O A I A s resonance is indicated by the minimum at around - 185 mV, and three further minima are clearly visible at high field. (From Ref. 103.)

52

I

H

.

4

'

,

'

I

'

,

.

,

.

,

.

Calibrated u values

50 -

40

5 0

2

4

10

12

14

6B (Tf

FIG. 21. Comparison of the measured u values (squares) and the field dependence of u predicted by the Landau level equation, using the best-fit mass of rn& = 0.56 rno(solid lines). (From Ref. 103.)

89

2 TUNNELING UNDER PRESSURE

Magnetotunneling has been used to study the T-X-T tunneling through a single AlAs barrier at ambient pressure, from which a value m k Y = (0.28 2 0.03)mohas been determined [85].This is in reasonable agreement with the cyclotron resonance value in Ref. 25 and the value mk,y= (0.25 ? 0.03)rnodetermined from the zero-field pressure-dependence results of Fig. 16. B perpendicular tothez direction. Figure 22 compares the Z -V plots of the 70-40-70-A sample pressurized to about 9 kbar, just beyond the type

0 I

-200

-100

,

v (LV)

I

100

L

200

FIG. 22. Conductance curves for the 70-40-70-A GaAslAlAs sample of Fig. 11 in three different E-field orientations, at 4.2 K and 9 kbar. (From Ref. 104.)

90

P. C.KLIPSTEIN

I1 transition, for fields oriented parallel and perpendicular to the z direction [104]. For the parallel case, only the conductance very close to the origin is suppressed. This may be a signature for the onset of the Coulomb gap, but further work must be done to prove or disprove it. However, for fields oriented perpendicular to the z direction, the conductance peak at the origin is much more strongly affected, in contrast with the higher-bias phonon-assisted tunneling peaks, which are only weakly perturbed. The zero-bias conductance peak is split, falling at the origin and rising into two symmetrical peaks on either side. The splitting is greater for fields oriented along [110] than [loo]. Such splitting has been observed before in 2D + 2D tunneling systems, and is explained by the change in wave vector, A k y = -(eBx/h) Az, between the edges of the emitter and collector subbands, where Az = (@ clzl@c) ( @ E ~ Z ~ @ E )= 110 A (991. In the present case at Bx = 15T, Aky > kF.y, where kF.Yis the Fermi wave vector. This is true if the y direction is in either the heavy or the light mass directions. Figure 23 shows how the shift destroys the conductance at zero bias by misaligning the emitter states from collector states at the same energy. The value of the zero-bias conductance is therefore directly related to the interface-roughness Fourier components F , ( q for ) the normal interface and F2(q)for the inverted, with q A k y . It decreases steadily with increasing field as expected, and is almost identical for B 11 [loo] and B 11 [110]. The effect of interface roughness on 2D 2D tunneling is discussed further in the next section.5a The conductance peaks at small finite bias occur when the occupied emitter electrons again overlap the empty collector states, as depicted at the top and bottom of Fig. 23. This almost doubles the Z-V peak to valley ratio compared with the zero-field plot shown in Fig. 12, because now the tunneling at finite bias can approximately conserve in-plane momentum. It is not possible to apply the Schrodinger-Poisson model with confidence at such small biases, but the larger splitting for B (1 [110] is consistent with a smaller mass that determines the dispersion. Therefore, the splitting for B 11 [loo] appears to be determined by mk,while for B 11 [110] the dispersion is expected to be consistent with

-

SaFouriercomponents of one dimensional interface disorder, f(y), are used here, because such simplified notation is used in the next section where a parallel magnetic field will be kF.Y.F l ( 9 and ) F 2 ( 9 )may be replaced by two dimensional discussed further. When Aky 9 )and F2(0. 9 )of the disorder, f ( x y). . Fourier components F,(O.

*

2 TUNNELING UNDER PRESSURE

Reverse Bias

\

lpkY, I I I I I I

I I

I I

I

I I

I I I

FIG. 23. A magnetic field along the x direction causes a shift in k ybetween the emitter and collector electron subbands. Therefore, tunneling that approximately conserves parallel wave vector can take place at finite bias in both directions, but at zero bias a finite wave vector must be supplied (by impurities, defects, and interface roughness) in order for tunneling to proceed. (From Ref. 104.)

91

92

P. C. KLIPSTEIN

For B 11 [lo01 there is no evidence of another peak at higher bias due to mg,y, perhaps because for the bias at which the condition for such zerophonon tunneling is fulfilled, the current may already be dominated by phonon-assisted tunneling. Diamagnetic depopulation of the Xx minima is too small an effect to be the cause. It is possible to apply the Schrodinger-Poisson model to the shift of the second resonance, Xx.y(l) +Xx.y(2), with B (1 [lo01 and B (1 [110] [104]. For B 11 [loo], a value for the longitudinal mass is obtained of (0.93 2 O.ll)rno, consistent with the literature value md = (1.0 2 O.l)mo [25]. For B 11 [110], a value of m ~ l = o l(0.81 2 0.14)rno is obtained, which is about twice the value expected from Eq. (24). The reason for the large discrepancy is not clear at present, but may be related to the camel’s back structure of the X band edge, which has not yet been taken into account. Discussion of 2 0 + 2 0 tunneling Quantumbeats.In Section 11.2, the splitting SE between symmetric and antisymmetric states on resonance was related to the quantum beat time T,which in the case of the well ground states of the 70-40-70-A sample is 69 pS. If the electron is scattered out of the collector well - for example, to lower-energy r states (in the GaAs collector layer) -in a time faster than this (typically T~ 10 pS [95])6and no scattering can take place while the electron is in the emitter well, then the T value might be expected to give a true measure of the time for tunneling. It then follows that the current density is J nEe/T, where nE is the electron concentration in the emitter AlAs layer. A more accurate value is obtained by considering that scattering to the lower-energy r collector states can take place as soon as the electron begins to appear in the collector well. The amplitude of the wave function in the+collectorwell increases with time as sin2(?rf/27). Thus J = nEe/Twhere , sin2 (nt127)dt = T ~ which , for T G Tgives a tunneling time T = ‘?&. In practice such a procedure appears to overestimate the current density by several orders of magnitude. For example, for the Xx.y(l) -+ Xx,y(2) process in the 70-40-70-A sample, 6E 0.095 meV, 17 pS. At 9 kbar, very close to which corresponds to T 22 pS, or T the type I1 transition, a self-consistent calculation similar to that of Fig. 14 gives nE 4 X 10 cm-’. These values predict a current density of 3.8 X lb A c m 2 , which is about 400 times greater than observed at 9 kbar

-

-

so

-

-

-

-

6Feldman et al.[95] measured the time for scattering from a higher confined T state to lower confined Xz and Xx,y states. Here, we are considering the reverse process of scattering from Xx,yto T, so the result will be modified by h a 1 density of states effects and the absence of zone center phonon contributions due to T-X mixing. The order of magnitude is not expected to change, and agrees with that estimated by Teissier etal.[43].

2 TUNNELING UNDER PRESSURE

93

(30 pA in a 20-pm diameter device). Since 6E is much less than the typical inhomogeneous broadening due to impurities in the AlAs layer and to interface steps, this is not surprising. Such broadening should be similar in size to the photoluminescence line-width of a few meV observed in MBEgrown type I1 superlattices with comparable AlAs layer widths (e.g., -5 meV for superlattices with 23-A AlAs layers [lOS]). Therefore, only a small fraction of the area of the sample will be undergoing quantum beats at any one bias, even if nominally at resonance. For the first resonance, Xx,y(l) -+ Xx,y(l), the conductance peak at the origin (Fig. 15) has a finite width. When integrated with respect to bias, this can give strong NDR (Fig. 12a). Conduction at finite bias when subbands are slightly misaligned can be explained either by inhomogeneous broadening, or by Eq. (22), which allows the appropriate Fourier component of the , supply the necessary in-plane momeninterface step distribution, f ( xy,) to tum for elastic scattering between the emitter and collector subbands. Although any model for the current depends sensitively on the definition of the interface disorder, some important aspects may be demonstrated by the following example based on Eq. (22), for a single Xx or Xy valley in which the disorder is restricted to one dimension, f(y). For a given wave vector in the x direction, k,, and assuming wells of equal thickness, the four degenerate envelope functions for the emitter and collector states in the absence of interface roughness are = QE(z). eikXx.ezikEy and +3,4 = @-(z).e'Q . e z i k ~Matrix y. elements of the roughness potential may be constructed as Mij=

AEX, * (+il(fi S(z- z a ) + f ~ S ( z- zd))l

+ fZS(z

- z b )+

fi S(z- G )

(25)

+j)

where the four interfaces, in order from the left to the right of the DBS, are located at za, . . . , z d , and AEx, = (uXIxlVGaAs - V A I A ~ ~ is U ~the ,~) X-band offset. It has been assumed that the roughness of the normal interface is different from that of the inverted [97]. The contribution, SJ,to the current density, J , is obtained by diagonalizing the 4 X 4 Hamiltonian constructed from M i j where , the roughness at the ith interface has Fourier components F i ( qand ) F j ( - q= ) F,*(q), with q = kc - k E . For simplicity, it has also been assumed that Fourier components at other wave vectors (e.g., kc + kE) are absent. The new eigenstates are symmetric and antisymmetric combinations of and +3 or J4! and @4 and somay undergo quantum beats between emitter and collector. Their energy splitting in forward bias (emitter on left-hand side) is

6E = 2 . AEx,. d R ( q ) . R*(q)

94

P. C. KLIPSTEIN

with

The phase factors y1,. . . ,y4allow for the phases of the Fourier components to be different at each interface. Assuming that these phases vary randomly with position across the sample area, and that the current is proportional to the RMS value of SE, the current, HL+R, is given by

is obtained by interchanging F l ( qand ) In reverse bias, the current, HJR+L, Fz(q) in Eq. (27). It was originally proposed that differences in the roughness of the normal and inverted interfaces could account for asymmetries of as much as 6 :1 in the peak current of the first Xx,y(l) + X X , ~ ( resonance l) [98]. However, since the bias for the first resonance is very small, the envelope functions are very nearly symmetrical within each well, so that @E(za) = @E(z~,), @E(&) = @ E ( Z ~ ) . @c(Za) = @ c ( Z b ) and @c(Zc) = @ c ( Z d ) is a very good approximation. Thus the currents J L - R and J R - L must be very nearly equal, and differences in interface roughness do not appear to contribute significantly to the asymmetry. Convincing evidence that the asymmetry arises from differences in AlAs thickness of as little as one monolayer was presented in an earlier section (p. 82). For magnetic fields applied parallel to the interfaces, it was proposed in the previous section that the conductance at zero bias is related to the Fourier components of the roughness. According to Eq. (27), it will be proportional to {FI(q)+ F2(q) }0.5, with 4 A k y ,the wave vector change induced by the field. Since the current is the sum of forward and reverse currents, and near the origin will vary linearly with bias as the number of electrons in the emitter well increases and the number of electrons in the collector well decreases, the conductance has the same functional form as Eq. (27), with each of the four products @E(zi)2@c(zi)2equal to one another. Effect of T-X mixing. The occurrence of T-X mixing leads to some interesting effects in the case of Xz(l) + X,(n)tunneling (n 2 1). This should normally be very difficult to observe due to the large Xz longitudinal

-

95

2 TUNNELING U N D E R PRESSURE

effective mass, but has been reported in uniaxial stress experiments by Lu er al.[67]. In their 23-50-23-Asample, the splitting between symmetric and antisymmetric wave functions for the Xz(l) -+ Xz(l) resonance is only SE ii: 7 neV. However, the following argument shows that it is possible for mixing with all the higher T(m) subbands and continuum states to increase the splitting (rn 2 1). For simplicity, only T-XlZ mixing is assumed initially (i.e., the mixing potential has the form of Eq. (21) with only the first term in the square bracket) and the double-well system is taken to have an infinite potential at z = za and z = z d , sothat all r and X wave functions are zero at these coordinates. The purpose is to illustrate some important consequences of the mixing. For an even number of GaAs monolayers, second-order perturbation theory on the unmixed r and Xz states gives a splitting 6 E = SE SErnix with 6Ernix= Em (-1 ) " 1 4 a ~ @ ) l . ( m ) ( Z b ) 2 @ X , z , ~ z b ) 2 / ( E r ( , ) - EX,,(l))? where z b is the coordinate of one of the GaAs interfaces. For odd m , r ( m )repels only the symmetric Xz,,,(l) state ( a = S), while for even m it repels only the antisymmetric ( a = A). The reverse is true for an The size odd number of GaAs monolayers, so that 6 E = SE + SErnix. of the repulsion of both symmetric and antisymmetric states is estimated quite well by considering only the first term in the sum (repulsion by r(l) only). For the alignment of r and X profiles at the high uniaxial pressures employed by Lu el al.[67] and assuming a n 0.33 eV A [32], the repulsion is -100 peV. The exact splitting, 6E, may be calculated by performing the complete summation or by determining the energies of the symmetric and antisymmetric states using the transfer matrix technique in the presence of band mixing (Eq. (19)), and is -5 peV. This is much greater than SE, leading to an enhancement of the quantum beat frequency and hence the zero-bias conductance (in a perfect sample) by several orders of magnitude. The effect remains if the mixing is between r and X3z, although the size of 6Emixchanges. a0 is then due only to the X3z component of the Xz state (second term in square brackets in Eq. (21)), as discussed in Section II.5.b, and the @ X 3 Z , n envelope in the above summation, must be obtained function, which replaces ax,,,,, from a 2 X 2 Hamiltonian with HXlzand HX,, diagonal terms and k-p off-diagonal terms. When the emitter and collector subbands are slightly misaligned, the enhancement of the tunneling current caused by the allowed T-X mixing may vanish. As shown in the 1D example of the previous section, symmetric and J!,I~ or and J!,I~ are formed by and antisymmetric combinations of the appropriate Fourier component of the roughness potential, F,(4).Mix* e+ikry or ing with the states @,.(m)(z)* eikxx. etik8." and Qrcm,(z) * @ r ( m ) ( .~eikxx ) . e-ik8y and @r,,,(z) . erkx.r. e-jkcY respectively of the T ( m )sub7

-

+,

96

P. C .KLIPSTEIN

band or continuum state now produces equal repulsions for the symmetric and antisymmetric Xz states, each with a total energy shift:

@E(Zc)@r(m)(Zc) + @E(Zb)@r(m)(Zb) and @C(zb) + @ c ( Z ~ ) @ ~ ( ~ ) ( Zhave ~) been neglected, Ex, is the energy of the

where the small terms @r(m)(Zb)

degenerate emitter and collector states, and corrections of the denominators due to the in-plane energy dispersions of the Xz and T(m) subbands are considered negligible. Equation (28) holds whether it is T-XIz or T-X3z mixing that dominates, provided the appropriate value is used for a. and QE and aCare the appropriate envelope functions for XI=or X3z.Because each T(m)subband or continuum state now couples separately to the emitter and collector parts of the symmetric and antisymmetric eigenfunctions of the interface roughness potential, this result is independent of the number of monolayers in the central GaAs layer.' Since there are now equal repulsions for the symmetric and antisymmetric states, their energy splitting remains the same as in the absence of r-X mixing, so the quantum beat frequency and tunnel current due to the disorder, fCy), when emitter and collector states are misaligned, should be unaffected by the mixing, in contrast to the aligned case. This creates the interesting possibility of a barrier that is much more transparent when on resonance than when off. Although the earlier model of disorder was introduced to illustrate some possible consequences of interface roughness and T-X mixing, it is important to emphasize that the assumptions used may limit a direct comparison with real systems. In particular, roughness was only considered in one dimension and its Fourier components which couple states within the same AlAs layer were not included. These components will lead to inhomogeneous broadening of the emitter and collector states, and even, possibly, to localization in the x-y plane. In the presence of significant inhomogeneous broadening it is probably better to think, as earlier, of different areas of the sample resonating at slightly different bias values. T-X mixing will affect the quantum beat frequency between symmetric and antisymmetric 71t can also be shown, using Eq. (22). that interface-roughness-induced mixing between T ( m ) and XZ states with different in-plane wave vectors is strongly suppressed for n monolayer steps ( n even) due to the opposite z parities of ur and ux,, at a group 111 site. Effects due to steps with odd n, or other forms of disorder cannot be completely discounted, although they are not considered further here.

2 TUNNELING UNDER PRESSURE

97

states, but the roughness (i.e., localization and dependence of T-X mixing on AlAs monolayer number) will allow both of them to interact with each r state, although with different strengths. The precise details of the disorder may thus have a profound effect on the strength of Xz + Xz tunneling. This may be the reason why Austing etal. observe a small, possibly Xz(l) + Xz(l), resonance near the origin at close to 4.2 K (when Xx,u states are not populated), in a 30-50-30-A sample but not a 30-40-30-A sample [69]. For the 40-40-40-A sample (grown in a different reactor) in Fig. 12(b), a clear low bias resonance is observed at 4.2 K which could also be attributed to Xz(l) + Xz(l). In this case, the splitting without T-X mixing between 90 neV in a perfect symmetric and antisymmetric X,(l) states is SE sample. For an emitter electron concentration, nE 10l1ern-', this corresponds to a current density, J nEe/r, of 0.69 A/cm-2 or a resonant current in a 20 p m diameter sample (the diameter of the sample in Fig. 12(b)) of -2 pA. With mixing, the simple model introduced earlier shows that the current is enhanced approximately one hundredfold. If inhomogeneous broadening results in only a small fraction of the area of the sample contributing to the resonance, the value of the current enhanced by T-X mixing is more consistent with the observed resonant current of a few microamps, and it is not unreasonable to attribute this resonance to Xz(l) + Xz(l). Similar considerations could also be consistent with the value of 14 pA for the resonant current observed by Lu etal.[67] in their uniaxial stress experiments (see Fig. 18) for which SE 7 neV, corresponding to a predicted current of 43 p A with nE 10l2cm-2. For this case it was shown earlier that T-X mixing can lead to a nearly one thousandfold increase in the energy splitting and the current, which, again, could reasonably compensate for inhomogeneous broadening effects that limit the active area of the sample. Finally, it is interesting to consider the effect of T-X mixing on the transfer of charge out of the GaAs emitter accumulation layer and into the 2D + 2D tunneling system. For DBSs with AlAs thicknesses below 50 A, pressurized below the type I1 transition, the r(1) subband in the GaAs accumulation layer formed under bias (similar to that depicted in Fig. 14) will pin close to the X,(l) emitter ground state, sothat r(1) and Xz(l) states are degenerate for some small in-plane wave vector. The T-X mixing due to the interface potential forms symmetric and antisymmetric combinations of these states, with energy splitting SE&, that undergo rapid quantum beats transferring charge in and out of the emitter AlAs layer. In this way charge may be scattered into nearby Xz(l) states at slightly different in-plane wave vector. This remains true even if an X,(n) collector state is also brought into alignment with the Xz(l) state by adjusting the

-

-

-

-

P.C .KLIPSTEIN

98

I

e

b,

-

-------

r e L

e

1 i-y#5

PI10kB8r

GBsbAtsbcasb~Gasb

FIG. 24. Band diagram of the GaSblAlSb DBS at (a) P = 0 kbar and (b) P = 10 kbar. The thin solid and dashed lines denote the conduction band edges of the r and L valleys, respectively. The thick solid and dashed lines denote the confinement states in the quantum well resulting from the F and L valleys, respectively. (From Ref. 73.)

bias, since S&& sk8When scattered out of the beating r(1)and Xz(l) combination, beats may take place between symmetric and antisymmetric combinations of Xz(1) and X,(n), as already considered. -An analogous situation holds for DBSs with AlAs layers thicker than 50 A. However, in this case the T-X mixing is due to interface roughness potential, V i ,discussed in Section 11.5.~. f

L-BandTunneling

Jimenez etal.[73] have pointed out that in a GaSb-AlSb-GaSb-AlSbGaSb DBS, the L minima in the GaSb conduction band lie close in energy to the r minimum. With the application of pressure up to 10 kbar, the ordering of the minima can be made to reverse (Fig. 24). The pressures at which the crossing takes place between the lowest L and r states in the well and in the emitter accumulation layer can be different because of the different confinement energies in these two regions. For a 34-60-34-A DBS, the resonance through the r profile is suppressed at -7 kbar and two new XItis then necessary to diagonalize the 3 X 3 Hamiltonian for the r(1)emitter state and the symmetric and antisymmetric Xz states at energies 5 &/2 with respect to the r(1)state [only T-X mixing with the r(l)emitter state is considered]. The quantum beats between Xz states [in the absence of r(l)]are broken and replaced by quantum beats between symmetric and antisymmetric combinations of r(1)and Xz( 1) emitter states.

2 TUNNELING UNDER PRESSURE

*l'o[J, -0.5

8

.. ,?,

.,&:.-.-. I

0.

- 0.0

0.0

jj

I?!

#.I

$'.

99

*!

,,ip" i '.

I l ; l !

-0.2

-0.5 2 , ' j

:

Reverse Voltage (V) FIG. 25. Measured pressure dependence of the I-Vcharacteristics of a (100)-grown 34-60-34-A GaSb/AlSb DBS at 77 K. The inset shows an amplified view of the lower region of the curve. (From Ref. 73.)

resonances appear, as shown in Fig. 25. At pressures significantly above the crossover, both emitter and well have L-symmetry ground states, and the two new resonances are attributed to the processes L-L-L(1)-L-L and L-L-L(2)-L-L, where L(1) and L(2) are the first and second confined L-symmetry states in the well. The second resonance is seen to shift up in bias with increasing pressure. This is because the peak current increases with pressure while the contacts become more resistive due to the heavier mass and hence lower mobility of the L-symmetry electrons. Growth on (111) orientation substrates shows the same r-profile resonance but no resonance through the L profile. It is suggested that the contributions of the two heavy mass valleys [oriented along (111) and (?Ti)] should be weak because they will not penetrate the barrier significantly and also because the confined states are too close together to be resolved. Contributions from the six lighter valleys are absent because only the heavy electron states in the emitter accumulation layer are populated. Perhaps temperatures higher than 77 K might be necessary to populate the lighter emitter states and reveal these features.

100

P. C .KLIPSTEIN

For the (100) orientation at 7 kbar, the bulk r and L minima are very close. Therefore, in the well, L(l) is lower than I'(l),whereas in the accumulation layer both r and L confined subbands are populated. The r-r-r(l)-r-l-resonance can therefore lose electrons to L(l) that cannot then tunnel out of the well as easily; a significant charge can thus build up in the well. This results in a large hysteresis in the I-V characteristic due to intrinsic bistability. A similar situation occurs for L-L-L(2)-L-L, but coherent tunneling out of the well is easier in this case and consequently there is less scattering to L( 1).Therefore a large hysteresis is observed €or the r ( l ) process but not for the L(2) process (see Fig. 26). A perpendicular magnetic field of -12 T reduces the range of allowed energies and wave vectors for phonon scattering, and it is observed that the hysteresis decreases substantially [74]. Resonant tunneling in the complementary DBS structure, InAs-A1SbInAs-A1Sb-InAs, has not been studied as a function of pressure, although the proximity of the valence band of AlSb to the energy of the tunneling electrons should make the transmission coefficient of the barrier very sensi15

-1

10

-

-Voltage sweep from -1.5 V to 1.5 V -Voltage sweep from 1.5 V to -1.5 V

0-

1 , . , .,

, .

,, .

,, ,,, ,, ,. , ,,, , . 1 -1.0 -0.5 0.0 0.5 1.0 1.5 Voltage (V) FIG. 26. Measured I-V characteristic of a (100)-grown 34-60-34-A GaSb/AISb DBS at 4 K and 7 kbar. (From Ref. 74.) -15

-1.5

I

2 TUNNELING U N D E R PRESSURE

101

tive to pressure [106]. However, this system has attracted attention in other ways, because very high current densities can be achieved while maintaining a respectable PVR. Therefore, microwave power has been detected up to very high frequencies in microwave oscillator circuits [107-1091. Other antimonide-based heterostructures have also attracted considerable attention over the past decade due to the variety of NDR devices that can be grown and the impressive PVRs that can be achieved in interband tunneling structures. Some devices of this type are discussed in the next section.

3. RESONANT INTERBAND TUNNELING a. Structures Containing AlSbBarriers The extreme type I1 band alignment in InAslGaSb heterostructures [110] makes a new type of tunnel diode possible in which interband tunneling takes place between electrons in InAs and holes in GaSb. Such devices attracted a great deal of attention when it was discovered that large PVRs could be obtained in devices containing AlSb barriers separating electrons in InAs from holes in GaSb [ill-118). Studies in devices containing no AlSb barriers showed that resonant interband tunneling could also be observed, but with a much smaller PVR [119,120]. There was some debate over whether the interband tunneling involved heavy or light holes, since for zero in-plane wave vector there should be negligible electron-heavy hole mixing, whereas mixing with light holes is allowed [121, 1221. For increasing in-plane wave vector, the k-pmixing with heavy holes increases [123]. High pressure increases the direct bandgaps of InAs (14 meV kbar-') and GaSb (10 meV kbar-'), thereby reducing the energy overlap between the InAs conduction band and the GaSb valence band. High-pressure studies have shown that the peak current and bias decrease with pressure as expected for a decreasing overlap [124,125]. For a GaSb-AISb-InAs-AlSbGaSb device, Mendez et al. have reported oscillations in the zero-bias conductance as a function of magnetic field, related to Landau levels from the two-dimensional electron system tunneling through an essentially threedimensional hole distribution [126]. At very high fields the Land6 g-factor has been observed to depend on the filling factor of the electron Landau levels (1271, while at very low temperatures additional features appear in the zero-bias conductance oscillations, for which Coulomb gap _ _ effects and tunneling via impurity- or interface-related states have been proposed [128].

102

P. C. KLIPSTEIN

b. N D R ofn-InAs/p-GaSb Diodes Reverse bias. When no AlSb barrier is present, large electric fields exist at zero bias in the InAs and GaSb layers and the electronic structure comprises two-dimensional electron (2DEG) and hole (2DHG) gases located on either side of the interface. The band profile in an n-InAs/p-GaSb diode with a band overlap of 155 meV is shown in Fig. 27. This was calculated using the self-consistent effective-mass calculation described in Section 11.4 for noninteracting electrons and heavy holes. The 2DEG and 2DHG concentrations are each close to 4.5 X 10" cm-2. There are two occupied electron subbands and two occupied heavy-hole subbands, but no occupied light-hole subbands. A special feature of such a device is that the ideal interface comprises a monolayer of either InSb or GaAs [39]. Interface growth has been optimized to the point at which such monolayer control is possible (129, 1301. High pressure and high magnetic field can be used to provide new understanding of the two interface types [131]. Full details of the n-InAs/p-GaSb diode samples discussed here are to be found in Ref. [131]. The band overlap, A, is expected to be a function of the interface type [132-1341. The relative sizes of AGaAsand AlnSbmay be deduced from the variation with pressure of the current due to Zener-like tunneling at a fixed reverse bias (InAs positive with respect to GaSb), shown in Fig. 28 [131].

140-

20

-

-3000

InAs -2000

-1000

GaSb 0

1000

2000

3000

Z / A

FIG. 27. Self-consistent effective-mass calculation of the band-edge profile and of the wave functions of occupied electron (E,, and El) and heavy-hole (HH,, and HHI) subbands, for an n-InAsip-GaSb diode with a band overlap of 155 meV at zero bias and temperature, in the vicinity of the Fermi energy, EF. There are no occupied light-hole subbands. n = 4.56 X 10" and p = 4.47 x 10" cm-'. (From Ref. 24.)

2

c P, 0.8

,.

" m

103

TUNNELING UNDER PRESSURE

.............. GaAs, pressure

upshifted 3 kbar

\

0.61

b = N 0.41 9

0.040

T

5

7

10 15 20 Pressure (kbar)

25

FIG. 28. Pressure dependence of normalized current at 77 K and -0.1 V bias in an n-InAslp-GaSb diode for each interface type. (From Refs. 130, 131.)

The corresponding band alignment is shown in Fig. 29. Since the current depends principally on the overlap of full bulk GaSb valence states with empty bulk InAs conduction states, it should be proportional to a universal function of A. In Fig. 28, the curve of I versus P for the device with the GaAs-like interface lies almost exactly on top of that for the InSb-like interface if it is first shifted up by -3 kbar. This shows that A has a very similar pressure variation for both interface types and that P y s b -

a)

b) P>P,

P = 1 bar

Ids V

0-

*

,d.\\d-43

FIG.29. Schematic band diagram of an n-InAs/p-GaSb diode at a fixed reverse bias, V, for a pressure of (a) P = 1 bar and (b) P > PT. Two types of conduction process are shown at 1 bar: Those indicated by solid arrows are expected to be dominant over those indicated by dashed arrows, which involve the electrons tunneling through a barrier. For P > PT, only the tunneling processes can contribute to the current.

104

P.

c. KLIPSTEIN

P p A s= 3 kbar, where PT is the threshold pressure at which A + 0. Parallel transport measurements on superlattices grown in the same reactor show that dAldP = -9.5 meV kbar-' for both interface types, AlnSb = 155 meV and ACaAs = 125 meV [135]. Comparing these results with Fig. 28 shows that PT is the pressure close to which the Zener-like current appears to exhibit a point of inflection (see arrows). The band overlap used in the calculation in Fig. 27 is therefore for an InSb-like interface.

Forward bias.For InAs negative with respect to GaSb, NDR is observed. The general features may be explained by an initial increase in the overlap of the 2DEG and 2DHG as their confinement energies increase with bias. However, at a critical bias a semimetal-to-semiconductor transition (SMSCT) occurs that is associated with the region of NDR because electron and hole states n o longer overlap in energy. NDR strengthens with decreasing temperature, particularly for the InSb-like interface (Fig. 30). When pressure is applied, the decrease of band overlap reduces the 2DEG and 2DHG concentrations, so that these go to zero at the pressure, P p S bof , the SMSCT. Figure 31 shows the I-V characteristics as a function of pressure for the InSb-like interface at 77 K. The decrease of band overlap and 2D concentrations is reflected by the reduction in the bias position of the NDR peak and the value of the peak current. Figure 31b shows very clearly that the NDR vanishes at P p S bthat lies between 12.7 and 15.8 kbar, so that at higher pressure there is no resonant component to the current. This is in good agreement with the parallel-transport and reverse-bias Zener tunneling measurements, which predict PFSbclose to 16 kbar (131, 1351. The calculated subband energies at ambient pressure are plotted as a function of bias in Fig. 32. The lowest confined hole subband (HHO) crosses the lowest confined electron subband (EO) at bias V,, while it lies 30 meV (approximately the LO phonon energy) above EO at V30. These calculations have been repeated at pressures up to -12 kbar, beyond which the accuracy becomes significantly limited by the assumed values for n ~nA , , and the donor and acceptor binding energies (because the fixed and mobile charge concentrations become comparable). Figure 33 shows a comparison between the observed peak and valley voltages and the calculated values' of 9Althoughelectron and hole interactionshave been neglected in this self-consistent effectivemass model, this should not result in significant errors in the estimates of V , and V , because at these biases there are no degenerate electrons and holes, sointeractions will be weak for all in-plane wave vectors. For comparison, a self-consistent calculation in which the k-p interaction between electrons and holes (and between heavy and light holes) is included using Rev. B 53,4630) with the appropriate (unadjusted) the Hamiltonian of Los er nl.(19%) (Phys. Luttinger parameters is in progress.

2 TUNNELING UNDER PRESSURE

105

10.0 r

Voltage / V FIG. 30. Forward-bias I- V characteristics of n-InAs/p-GaSb diodes with (a) InSb-like and (b) GaAs-like interfaces at the temperatures indicated. (Measured by U. M. Khan-Cheema.)

Vc and V30. There does not appear to be very good correspondence because the slopes of the observed and calculated voltage curves are very different. However, if it is assumed that a contact resistance Rc is in series with the device, the peak and valley voltages can be replotted as V;E'= Vp - Zp.Rc and V $ = V , - Zv - Rc. This has been done in Fig. 33 for Rc values of 10 fl and 20 R. The slopes of the curves of V , * ( Pand ) V : ( P )for Rc = 20 R are now very similar to those of V c ( P ) and V3,,(P). A resistance Rc of 20 R is quite consistent with the resistance of the GaSb mesa adjacent to the junction, which has a diameter of 10 pm and a depth of about 0.4 pm, if

P. C. KLIPSTEIN

106

Y

0.0

0.1

0.2

0.4

0.3

0.5

Voltage (V)

0.31

0.00.00

9kbar/

0.05

'

5

0.10

.

s

13/

.

0.15

16/ 1.-

I

0.20

'

1

0.25

Voltage (V) 0) FIG. 31. Forward-bias I-V characteristics at 77 K of an n-InAslp-GaSb diode with an InSb-like interface at pressures up to 19.8 kbar, plotted (a) over a wide bias current range and (b) near the origin. (From Ref. 24.)

-

the mobility of the -1 X 10l6 cm-3 holes is 1600 cm2 V-' s-' [136] and the total resistance of the contact itself and wires into the pressure cell is 1-2 0.If the agreement in slopes is taken as good support for this value of R c ,Fig. 33 shows that the NDR peak occurs close to V , when electron and hole bands no longer overlap, whereas the NDR valley occurs close to V3,when electrons cannot scatter to hole states by a single phonon. This behavior has been reproduced €or several different mesas fabricated from

2 TUNNELING UNDER PRESSURE

107

FIG. 32. Calculated energies of the 2D electron and hole subband edges versus bias, at 1 bar and 0 K, for an n-InAslp-GaSb diode with an InSb-like interface. EF and E: are the electron and hole quasi-Fermi levels. (From Ref. 24.)

the same wafer. It provides a clear understanding of the mechanism for NDR and fairly conclusive proof that the light-hole states do not make a significant contribution to the resonant current. The resonant current must therefore be due primarily to mixed electron and heavy-hole states at finite in-plane wave vector that have a large amplitude on both sides of the interface. The pressure dependence of NDR for the GaAs-like interface is shown in Fig. 34. The NDR is suppressed in a way similar to that for the InSblike interface, but it does not vanish until almost 20 kbar, a pressure much higher than P p A sdeduced from the Zener tunneling measurements or parallel-transport measurements, which both indicate that PpAs lies below PFSb.A similar analysis of the pressure dependence of the peak and valley voltages, as performed for the InSb-like interface, is shown in Fig. 35. Again a value of Rc 20 0 gives best agreement between the slopes of V , * ( P ) and VG(P)and the value, V c ( P )calculated . for A = 125 meV. However, V c ( P )is much too small to be related directly to the NDR voltages, and it must be concluded that a different mechanism is responsible for NDR ), corresponds fairly at the GaAs interface. The bias value, V , * ( Pactually well to the situation in which EO is aligned -140 meV above HHO. Also, the threshold pressure in Fig. 34 of -22 kbar for loss of NDR is approximately 9 and corresponds to an energy separation of bulk band kbar above PyaAs edges, Tc(InAs) - Tv(GaSb) 9 X 9.5 = 85.5 meV. The bias at which the NDR vanishes in Fig. 34 is -0.1 eV, in contrast to the negligible voltage

-

-

0

2

4

6

8

10

12

Pressure / kbar FIG. 33. Peak and valley voltages, V , and V,, at 77 K plotted against pressure for an n-InAs/p-GaSb diode with an InSb-like interface. The error bars indicate a 20.5 kbar uncertainty in the manometer pressure calibration. The dashed lines indicate the calculated threshold voltages, V , and V,, defined in the text. The dotted lines show how allowing for the existence of a series resistance of 10 or 20 R can affect the experimental data. (From Ref. 24.)

at which the NDR vanishes in Fig. 31 for the InSb-like interface. All of these values indicate that the resonance is not directly related to the band overlap but takes place through some inelastic mechanism when a bandgap of roughly 140 meV exists between the electron and hole systems. The origin of this large inelastic energy, which is too great to be related to a single phonon, is unclear and must be the subject of further investigation. Experiments have also been carried out in very high (pulsed) magnetic fields, where the electrons and holes each condense into a single Landau

2 TUNNELING UNDER PRESSURE

109

10

8

E 6 CI

K

g 4

5 2 0 ..

0.0 0:1 0:2 013 014 015 016 Voltage 01) (a)

12

15

ia 21

0.0

0.1

0.2

0.3

Voltage (V) (b)

FIG. 34. Forward-biasI-V characteristicsat 77 K of an n-InAsfp-GaSb diode with a GaAs-like interface at pressures up to 21.2 kbar, plotted (a) over a wide bias and current range and (b) near the origin. (From Ref. 24.)

level for fields above -20 T and small applied bias [131]. The energies of the Landau levels increase with field and cause the electron and hole concentrations to decrease in a way that is analogous to the effect of pressure. The results of the magnetic field studies provide further support for the conclusions of the high-pressure experiments, namely, that the resonant conduction mechanisms for the two interface types are quite differ-

P. C. KLIPSTEIN

110

500

xi

400

>

------

300

\

> 100 0

0

5

10

15

20

Pressure / kbar FIG. 35. Peak and valley voltages, Vp and V,,at 77 K plotted against pressure for an n-InAs/p-GaSb diode with a GaAs-like interface. The error bars indicate a 20.5 kbar uncertainty in the manometer pressure calibration. The dashed line indicates the calculated threshold voltage, V c ,defined in the text. The dotted linesshow how allowing for the existence of a series resistance of 10 or 20 fl can affect the experimental data. (From Ref. 24.)

ent and that the peak of the resonant current through the GaAs interface occurs after the electron and hole states have uncrossed.

V. Concluding Remarks Advances in heterostructure growth over the past 10 years have allowed the physicist to engineer new band structures in semiconductors by imposing

2 TUNNELING IJNDER PRESSURE

111

an artificial periodicity on the material in the growth direction. The most straightforward result for a single band is the creation of new subband structure leading to mini-gaps and modified densities of states. Quantum confinement in other directions is beginning to lead to further modifications of electronic states, for example, in quantum wires and dots, and even to single-electron behavior. However, some of the most interesting vertical transport results of the past 10 years have been related to mixing, or changing the order, of states of different bulk crystal periodic symmetries, for example, the r valence states and the r, X, and L conduction states. In the heterostructure, novel band alignments and quantum confinement can be used to position the states in question close in energy so that mixing effects become pronounced. However, it is not always possible to control this energy separation as much as desired by heterostructure design alone, or to vary it continuously within the same sample. Both of these restrictions have been overcome using high pressure, which is therefore unique in the insight it has provided into the fundamental quantum mechanical nature of semiconductors and their heterostructures. In addition, pressure has been used to vary bandgaps and tune effective masses or g values in order to probe their role in tunneling or other processes. When combined with a large magnetic field that modifies the density of states or dispersion in a characteristic way, a very powerful set of probes has been achieved. It has been the purpose of this chapter to present a number of such studies and to discuss their contributions to the current understanding of semiconductor heterostructures. With the development of new heterostructure material combinations, improved nanostructure fabrication, and pulsed magnets leading to evergreater magnetic fields, it can be said with some confidence that high pressure will continue to make a useful contribution for at least the next 10 years.

Acknowledgments

First and foremost, this review would not have been possible without the help and stimulation of several generations of research students at Oxford. They have contributed substantially to the development of many of the ideas and techniques presented here, in particular D. G. Austing, J. M. Smith, U. M. Khan-Cheema, H. Im, W. Chaudhry, and G. Rau. I am grateful to Dr. J. M. Smith and Dr. U. M. Khan Cheema for assistance with the preparation of many of the diagrams. I am also very much indebted to Dr. M. I. Eremets, whose ingenuity, technical knowledge, and willing

112

P. C. KLIP~TEIN

assistance have advanced the application of high-pressure methodology at Oxford, and to Prof. R. H. Friend, who first introduced me to the excitement of high-pressure research and, with J. Simmons, also provided technical assistance in the early stages. The author acknowledges financial support from the Engineering and Physical Sciences Research Council and the Royal Society of the UK, and the Human Capital and Mobility Network program of the EU, without which this chapter could not have been written.

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[65] Mendez, E. E., and Chang, L. L. (1990). Surf Sci. 229,173. [66] Austing, D. G., Klipstein, P. C., and Roberts, J. S. (1990). Solid State Commun. 75,697. Phys.69,8241. [67] Lu, S. S., Meng, C. C., Williamson, F., and Nathan, M. I. (1991). J. Appl. [68]Otthaman. Z., Peck, A. J., Bending, S. J., Saunders, G. A,, Rossmanith, M., Ploog, K., and von Klitzing, K. (1991). 1. Appl.Phys.70, 1548. [69] Austing, D. G., Klipstein, P. C., Higgs, A. W., Hutchinson, H. J., Smith, G. W., Roberts, J. S., and Hill, G. (1993). Phys.Rev.B 47, 1419. (701 Austing, D. G., Klipstein, P. C., Roberts, J. S., Button, C. B., and Hill, G. (1993). Phys. Rev.B 48 11905. [71] Austing, D. G., Klipstein, P. C., Roberts, J. S., Mistry, C. P., and Hill, G. (1993). J. Appl.Phys.74,7340. [72] Carnahan, R. E., Martin, K. P., Higgins, R. J., Park, B. G., Wolak, E., Lear, K. L., and Harris. J. S. (1994). Semicond. Sci.Technol. 9, 500. [73] Jimenez, J. L., Mendez, E. E., Li, X.,and Wang, W. I. (1995). Phys.Rev.B 51, 7938. [74] Jimenez. J. L., Mendez, E. E., Li, X., and Wang, W. I. (1995). Phys.Rev.B 52, R5495. [7S] Adachi, S. (1985). J.Appl.Phys.58,R1. [76] Mizuta, H., and Tanoue, T. (19%). ThePhysics andApplications ofResonant Tunneling Diodes, Cambridge University Press. [77] Skolnick, M. S., Smith, G. W., Spain, 1. L., Whitehouse, C. R., Herbert, D. C., Whittaker, D. M., and Reed, L. J. (1989). Phys.Rev.B 39, 11191. [78] Wolford, D. J., Kuech. T. F., Bradley, J. A,. Cell, M. A,, Ninno, D., and Jaros, M. B4, 1043. (1986). J. Vac.Sci. Technol. [79] Pritchard, R., Austing, D. G., Klipstein, P. C.. Roberts, J. S., Higgs, A. W., and Smith, G. W. (1990). J. Appl.Phys.6, 205. [SO] Mendez, E. E., Calleja, E.. and Wang, W. I. (1986). Phys.Rev.B 34,6026. [81] Carbonneau, Y., Beerens, J., Cury, L. A., Liu, H. C., and Buchanan, M. (1993). Appl. Phys.Len.62, 1955. [82] Beresford, R., Luo, L. F., Wang, W. I., and Mendez, E. E. (1989). Appl.Phys.Len. 55, 1555. [83] Ohno, H., Mendez, E. E., and Wang. W. 1. (1990). Appl.Phys.Lett. 56,1793. [84] Otthaman. Z., Geim, A. K., Bending, S. J., Syme, R. T., and Smith, R. S. (1993). Semicond. Sci.Technol. 8,1483. [85] Finley, J. J., Teissier, R. J., Skolnick, M. S., Cockburn, J. W.. Grey, R., Hill, G., and Pate, M. A. (1996). Phys.Rev B 54, R5251. [86] Solomon, P. M., Wright, S. L., and Lanza, C. (1986) Superlatt. Microstruct. 2,521. Sci. [87] Austing, D. G., Klipstein, P. C., Roberts, J. S., and Hill, G. (1995). Semicond. Technol. 10, 616. [88] Rossmanith, M., Syassen, K., Bockenhoff. E., Ploog, K., and von Klitzing, K. (1991). Phys.Rev.B 44,3168. [89] Diniz, R., Smoliner. J.. Gornik. E., Meiners, U., Brugger, H., Wisniewski, P., and Suski, T. (1993). Semicond. Sci.Technol. 8, 1352. (901 Taylor, D. C., Guimaraes, P. S. S., Snell, B. R., Eaves, L., Sheard, F. W., Toombs, G. A,, Portal, J. C., Dmowski, L., Singer, K. E. Hill, G., and Pate, M. A. (1986). Surf Sci. 174, 472. [91] Maude. D. K.. Portal, J. C., Dmowski, L., Foster, L., Eaves, L., Nathan, M., Heiblum, 59, 815. M., Harris. J. J.. and Beall, R. B. (1987). Phys.Rev.Lett. [92] Cury, L. A., Pritchard, R., Celeste, A., Portal, J. C.. Eaves, L., Davies, M., Health, M., Middleton. J. R.. Dmowski, L., Sivco. D. L., and Cho, A. Y. (1990). Proc.Int. Conf Phys.Semicond. 20, 1333.

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[93] Suski, T., Gschlossl, C., Demmerle. W., Smoliner, J., Gornik. E., Bohm, G., and Wei59, 2436-2438. mann, G. (1991) Appl. Phys.Lett. [94] Sollner, T. C. L. G. (1987). Phys.Rev.Len.59, 1622. 1951 Feldman, J., Nunnenkamp, J., Peter, G.,Gobel, E., Kuhl, J., Ploog, K., Dawson, P., and Foxon, C. T. (1990) Phys.Rev.B 42, 5809. (961 Smith, J. M., Klipstein, P. C., Grey, R., and Hill, G. (1998). Phys.Rev.B. 57,1740. [97] Foxon, C. T., and Joyce, B. A. (1990). In Growthand Characterisation of Semiconductors (R. A. Stradling and P. C. Klipstein, eds.), pp. 35-64, Adam Hilger, Bristol and New York. [98] Smith, J. M., Austing, D. G.,Grey, R., Roberts, J. S., Hill, G., and Klipstein, P. C. (1995). J.Phys.Chem.Solids 56, 475. [99] Smoliner, J., Demmerle, W., Berthold, G., Gornik, E., and Weimann, G. (1989). Phys. Rev.Let.63, 2116. [loo] Eisenstein, J. P., Pfeiffer, L. N., and West, K. W. (1991). Appl. Phys.Lett. 58, 1497. [loll Eisenstein, J. P., Pfeiffer, L. N., and West, K. W. (1995). Phys.Rev.Lett. 74, 1419. [102] Turner, N., Nicholls, J. T., Linfield, E. H., Brown, K. M., Jones, G. A. C., and Ritchie, D. A. (1996). Phys.Rev.B 54, 10614. [lo31 Smith, J. M., Klipstein, P. C., Grey, R., and Hill, G. (1998). Phys.Rev.B. 57, 1746. [lo41 Smith, J. M., Klipstein, P. C., Grey, R., and Hill, G. (1998). [I051 Tribe, W. R., Klipstein, P. C., and Smith, G .W. (1994). In 22ndInt. C o n jon thePhysics of Semiconductors (Lockwood. D. J., ed.), pp. 759-762, World Press. [lo61 Chow, D. H., McGill, T. C., Sou, 1. K., Faurie. J. P. and Nieh, C. W. (1988) Appl. Phys. Lett. 52, 54. [lo71 Soderstrom, J. R., Chow, D. H., and McGill, T. C. (1989). J. Appl. Phys.66,5106. [I081 Soderstrom, J. R.. Chow, D. H., and McGill, T. C. (1990). IEEE Electron Dev.Lett. 11, 27. [I091 Brown, E. R., Soderstrom, J. R., Parker, C. D., Mahoney, L. J., Molvar, K. M., and McGill, T. C. (1991). Appl. Phys.Len.58, 2291. [110] Sai-Halasz, G. A,, Tsu, R., and Esaki, L. (1977). Appl. Phys.Letr. 30, 651. [ l l l ] Chow, D. H.. Soderstrom, J. R., Collins, D. A,, Ting, D. 2.-Y., Yu, E. T., and McGill, T. C. (1990). SPIE 1283 (Quantum Well and Superlattice Physics 111), 2. [112] Soderstrom, J. R., Chow, D. H.. and McGill, T. C. (1989). Appl. Phys.Lett. 55, 1094. [113] Luo, L. F., Beresford, R., and Wang, W. 1. (1989). Appl. Phys.Lett. 55,2023. [114] Beresford, R., Luo, L. F., Logenbach. K. F., and Wang, W. I. (1990). Appl. Phys.Lett. 56, 551. [I151 Chow, D. H., Yu, E. T., Soderstrom, J. R. Ting, D. Z.-Y., and McGill, T. C. (1990). J. Appl. Phys.68,3744. [I161 Beresford, R., Luo, L. F., Logenbach, K. F., and Wang, W. I. (1990). Appl. Phys.Lett. 56, 952. [117] Logenbach, K. F., Luo, L. F., and Wang, W. L. (1990). Appl. Phys.Lett. 57, 1554. [118] Mendez, E. E., Nocera, J., and Wang, W. 1. (1992). Phys.Rev.B 45,3910. [119] Collins, D. A., Yu, E. T.. Rajakarunanayake, Y., Soderstrom, J. R., Ting, D. 2 - Y . , 57, 683. Chow, D. H., and McGill, T. C. (1990). Appl. Phys.Lett. [120] Luo, L. F., Beresford, R., Logenbach, K. F., and Wang, W. I. (1990). J.Appl. Phys. 68,2854. [I211 Yu, E. T., Collins, D. A., Ting, D. 2 . - Y . , Chow, D. H.. and McGill, T. C. (1990). Appl. Phys.Lett. 57, 2675. [122] Yang, L., Wu, M. C., Chen, J. F., Chen. Y. K., Snider. G. L., and Cho, A. Y. (1990). J.Appl. Phys.68,4286. [I231 Ting, D. Z.-Y., Yu. E. T., and McGill. T. C. (1991). Appl. Phys.Lett. 58, 292.

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[124] Mendez, E. E. (1992). SurfSci. 267,370. [I251 Nogaret, A., Maldonado, M. A., Carnahan, R. E., Martin, K. P., Higgins, R. J., Aristone, F., Maude, D. K.,Portal, J. C., Chen, J. F., and Cho, A. Y. (1993). Phys. Rev.B 47,13872. [126] Mendez, E. E., Ohno, H., Esaki, L., and Wang, W. I. (1991). Phys.Rev.B 43,5196. 37, 779; [127] Mendez, E. E., Nocera. J., and Wang, W. I. (1994). SolidStateElectronics (1993). Phys.Rev.B 47, 13937. [128] van Son, P. C., Milliken, F. P., Mendez, E. E., and Wang, W. I. (1994). Surj Sci. 305,516. and (1291 Lyapin, S. G., Klipstein, P. C., Mason. N. J., and Walker, P. J. (1994). Superlattices 15, 499. Microstructures (1301 Klipstein, P. C., Lyapin, S. G., Trojan, I., Mason, N. J., Khan-Cheema, U. M., Symons, D. M., Daly, M. S., Lakrimi, M., Nicholas, R. J., Murgatroyd, I. J., Seong, T. Y., Booker, G. R., Hill, G., and Walker, P. J. (19%). In Inr. Conf on Semiconductor Hereroepitmy (B. Gil, ed.), pp, 512-522, World Scientific. (1311 Khan-Cheema, U. M., Klipstein, P. C., Mason, N. J., Walker, P. J., Eremets, M. I., and Hill, G. (1996). In 23rdlnt. Conf on thePhysics of Semiconductors (Scheffler and Zimmerman, eds.), pp. 2271-2274, World Press. [132] Foulon, Y..and Priester, C. (1992). Phys. Rev.E 45,6259. B11,1693. [133] Hemstreet, L. A., Fong, C. Y.,and Nelson, J. S. (1993). J. Vac.Sci.Technol. 810, 1 7 4 . [134] Dandrea, R. G., Duke, C. B.. and Zunger, A. (1992). J. Vuc.Sci.Techno!. [135] Daly, M. S., Symons, D. M., Lakrimi, M., Nicholas, R. J., Mason, N. J., and WaIker, 362,205. P. J. (1995). Surf Sci., [136] Metzler, R. A., and Becker, W. M. (1973). Phys. Rev.B 8,5604.

SEMICONDUCTORS AND SEMIMETALS. VOL. 55

CHAPTER3

Phonons, Strains, and Pressure in Semiconductors Evangelos Anastassakis NATIONAL TECHNICAL UNIVERSITY PHYSICS DEPARTMENT AIHENS. GREECE EUROPEAN UNION

Manuel Cardona MM-PLANCK-INSTIIUT FUR F E S T K O R P E R C ~ R S C ~ I ~ , N ~ STUITGART, FEDERAL REPUBLIC OF GERMANY EUROPEAN UNION

......................................... 1. Historical Review of Strain Effectr ..........................................................................

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

118 118 121 127 127 132 138 144 148 152 155 156 163 164 167 171

4. Phase Transitions ...................... IV. EFFECTS OF STRAINS ON ORICAL NS ................................................................ I. Phonon Deformation Potentia ....................... 2. Phonon Secular Equation ............................................................. ............................. 3. Control Experiments ............................._ ................................. 4. Theoretical Models and Trends of Phonon Deformation Potentials __........_....._ 191 5. Other Uses of Phonon Deformation Potentials ..................................................... 194 V . STRAINCHARACTERIZATION OF HETEROJUNCTIONS AND SUPERLATTICES __.. 199 1. Elastic and Piezoelectric Considerations in Heterojunctions and Superlattices ......,.....................................,...,............................................................... 199 .............. 2. Pressure and Temperature Dependence of St 205 3. Characterization of Strains through Rnnian Sp 213 .................................. 220 VI. CONCLUDING REMARKS .................................................................................................. .............. ........ Acknowledgments 222

117 Vol. 55 ISBN 0-12-75216.1-1

SEMICONDUCTORS AND SEMIMETALS Copyright 0 1998 hy Academic Press All rights of reproduction in any form reserved. WSO/X~WYX 125.00

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Appendix ..........................................................................................................................

222

References ........................................................................................................................

224

The effect of strain on the electronic states of semiconductors was intensively investigated in the 1950s and 1960s. The results obtained were highly instrumental in reaching our present understanding of the band structures of these materials. The study of the effects of strains, both uniaxial and hydrostatic, on the vibrational eigenstates (phonons) is more recent and has helped in the understanding of the symmetry of such states and their anharmonic properties. Raman spectroscopy is probably the most powerful technique for such studies. Once these effects are calibrated versus the magnitude of the stresses, the Raman spectra of phonons in stressed materials can be used to estimate the magnitude of the various components of semiconductor nanostructures that have built-in stress due to lattice mismatch and differences in the expansion coefficients of their components. Raman spectroscopy has brought about a renaissance of interest in the effects of stress on vibrational states. In this chapter we review such effects and their macroscopic origin. We also discuss the application of the measurements of effects of stress on phonons to the determination of built-in stress in semiconductors and semiconductor nanostructures.

I. Introduction Application of external stresses to a crystal changes the elemental translation vectors and can also change, in general, the coordinates of the atoms in the primitive cell [ 11.This results in considerable changes in the electronic states of the crystal and concomitantly in the properties of the lattice vibrations, observable even for moderate strains. At higher strains phase transitions may occur as a result of a crossover of the total energy of the crystal with that of another phase [2]. For general stresses such phase transitions may never be reached, as a result of crystal fracture. If the applied stress is isotropic (i-e., hydrostatic), fracture does not occur and rather high pressures can be applied provided adequate equipment is available: Phase transitions can then be observed. 1. HISTORICAL REVIEW OF STRAJN EFFECTS The investigation of the structural and electrical properties of solids under stress was pioneered by P. W. Bridgman (Nobel Prize Laureate,

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119

1946) at Harvard University. He developed equipment for electrical measurements and was able to see a sharp decrease in the mobility of germanium versus quasi-hydrostatic pressure, which we know today is due to the crossing of the [lll]with the [lo01 minima upon increasing the pressure [ 3 ] . The usefulness of optical measurements under hydrostatic (or quasihydrostatic) pressure was early realized by Drickamer and co-workers, who designed and built windows for operation under hydrostatic [4] and under very high quasi-hydrostatic pressure (up to 14 GPa) [ 5 ] , and also by W. Paul and co-workers [6,7]. The possible use of glass windows to perform optical measurements at pressures up to 3 GPa appeared in the literature as early as 1930 [8]. Investigations of lattice vibrations (i.e., phonons) under stress are more recent. Early work involved long-wavelength acoustic phonons measured by ultrasonic techniques [9], acoustic phonons at the Brillouin zone (BZ) boundary detected by inelastic neutron scattering [lo], and measurements of optic phonons in alkali halides by means of infrared (IR) spectroscopy under quasi-hydrostatic pressures, using diamond windows [ll].A big leap forward took place with the introduction of laser Raman spectroscopy. Raman measurements were performed at room temperature under pressures up to approximately 1 GPa using a transparent oil as a pressuretransmitting fluid. Samples transparent to the exciting laser frequency [ 121, as well as opaque samples 1131,were investigated. The shifts of the observed phonon peaks with volume were represented by a mode Gruneisen parameter y,: d In wi yi=-- d l n V It was found that the yi’s for optical phonons at the center of the Brillouin zone lie between +1 and +2. For polar materials, the longitudinal opticaltransverse optical (LO-TO) splitting was found to decrease with increasing pressure [12], a fact that has been repeatedly confirmed for zinc blende-type materials [14], with the exception of S i c [15]. This fact has been attributed to an increase of the ionicity with increasing bond length [14]. The change of the LO-TO splittings with pressure is an important factor in the pressure dependence of the static dielectric constant of ionic crystals [16]. Raman measurements were not limited to first-order spectra, that is, to phonons at the center (r point) of the BZ. Phonons at the edge of the BZ were also measured by recording second-order Raman spectra. The transverse acoustical (TA) phonons were found in many cases to have an anomalous negative Gruneisen parameter [17, 181.

l20

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

The next leap forward was made with the adaptation of the diamond anvil cell (DAC) to optical measurements [19], in particular Raman spectroscopy [20]. Ever since, the DAC has been used profusely to determine Gruneisen parameters of phonons in semiconductors, not only at the r point (onephonon scattering) but also at the edge of the BZ (using two-phonon Raman spectra) [18]. In this manner, a large database concerning the dependence of phonons on hydrostatic pressure has been obtained, which can be used to check developments in ab inirio calculation methods [21] and also to characterize built-in stresses in bulk semiconductors and semiconductor nanostructures. In such structures strain has been shown to be instrumental in determining interface roughness [22]. The strain due to lattice mismatch has also been proven to be the driving mechanism for the production of self-organized quantum dots (e.g., of InAs on GaAs) [23]. Interest in the subject has also been triggered by the improved laser performance found for semiconductor quantum wells under strain [24]. Moreover, a number of optoelectronic and microwave devices that make use of the effects of tailormade strain fields in semiconductor microstructures have been proposed [25]. Electrical measurements under uniaxial stress have been performed in semiconductors since the mid-1950s. They were instrumental in disentangling the complex and diverse nature of the electronic band edges of germanium and silicon (see the piezoresistance measurements, performed up to 0.01 GPa, discussed in Ref. [26]). Impurity band conduction under uniaxial stress was thoroughly investigated by Fritzsche and his group using stresses up to approximately 0.6 GPa [27]. Most of the work on phonons reported here was performed using the technique of Ref. [28], which enables one to reach up to about 2.5 GPa in Si, 1.5 GPa in Ge, and 1 GPa in GaAs. Very detailed information on parameters of the electronic band edges has been obtained by measuring cyclotron resonance under uniaxial stress [29]. Uniaxial stress techniques are particularly suitable for optical measurements because they do not require pressure-supported windows. They have been used profusely to investigate the band structure of semiconductors since the late 1950s [30-321. Modulation spectroscopy, in particular electroreflectance [33], was found to be a very powerful spectroscopic method when combined with uniaxial stress techniques. More recently, experiments under uniaxial stress have led to valuable information on the relaxation of spin polarization and the elusive spin splittings of the electron energy bands in noncentrosymmetric semiconductors (e.g., GaAs) [34, 351. Measurements of phonon properties under uniaxial stress go back to the work of Anastassakis er af. [36] for silicon. These authors discovered that the phonon splittings induced by a (1001 stress had the opposite sign than

3

PHONONS, STRAINS, AND

PRESSURE I N SEMICONDUCTORS 121

those induced by a [111] stress, a fact that later has been shown to hold for most zinc blende and diamond-type semiconductors (exceptions: diamond and BN; see Section IV.4.b). In 1987 the effects of uniaxial stresses on the IR effective charges (i.e., on the LO-TO splittings) were thoroughly investigated for GaAs [37]. The uniaxial stress work just mentioned refers to phonons at or close to the r point of the BZ. While a considerable database is available for the effect of hydrostatic pressure on phonons at other points of the BZ (in particular around its boundary [18, 20, 38, 39]), little is known about the effects of a pure shear stress (i.e., a traceless uniaxial stress) [40]). We mention here the work performed on silicon by inelastic neutron spectrosCOPY P11.

OF STRESS AND STRAIN ON ELECTRONS AND 2. EFFECTS PHONONS IN CRYSTALS

In the previous section we considered either the effects of strain or those of stress on electrons and phonons in crystals. The reason is that sometimes (e.g., when the effects are produced directly by external forces) the primary cause lies in the applied stresses. Other times (e.g., in the cases of mismatch strains) the primary cause lies in internal constraints that directly change the lattice parameters. We shall assume in most of this work that the strains are linear in the stresses and vice versa. The stresses and strains are secondrank symmetric tensors that, if linearity holds, are related by a fourth-rank tensor whose components are either called elastic stiffness (strain + stress) or elastic compliance (stress -+ strain) constants. These parameters are discussed in detail in Section 11.2. We shall here consider briefly the symmetry-based theoretical background needed to understand the effects of strains on electron and phonon states in crystals. The corresponding effects of stress can be calculated from the effects of strain by means of the elastic compliance tensor, provided linearity holds. The observed effects of strains are indeed linear in most cases since the magnitude of the strain is limited by the yield strain of the samples. Much larger strains, however, can be applied with hydrostatic techniques. In this case nonlinear behaviors are often observed (the strain is only limited by either the equipment or by phase transitions). Figure 1, as an example, shows the dependence on hydrostatic pressure of the two lowest electronic gaps of GaAs [42]. The direct gap, rY5+ I'f, displays a clearly sublinear behavior above 4 GPa. For comparison, Fig. 2 shows the corresponding shifts observed for the LO

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

GaAs T=300K I I

/

Phase Trans1tion I

I

I

5

x>

15

PRESSURE ( GPa 1 FIG. 1. Direct absorption edge r;, 4 r; (open circles) and indirect gap Tis --* Xi’ (solid circles) of GaAs as a function of pressure. The solid lines represent the result of least squares fits to the experimental data using a quadratic or linear function, respectively. The dashed-dotted line represents the linear component of the quadratic fit to the ris + Ti gap. (From Ref. [42].)

and TO phonons of GaAs, which are also sublinear versus P above 4 GPa [43]. These sublineanties are due, in part, to the usual stiffening of the bulk modulus B (B = -dP/dIn V) with increasing pressure. Consequently, if an equation of state instead of the bulk modulus is used to convert pressure into strain, the nonlinearities of Figs. 1and 2 become considerably weaker (see the upper scale in Fig. 2, which represents the strain). In the case of electronic states (with or without spin effects), it is customary to discuss the effects of strain by the method of invariants [44,45].The same method can be used for phonons. Its aim is to reduce the expectation values of a perturbation (in our case, strain) to the lowest number of adjustable parameters possible, compatible with the symmetry properties of the state under consideration. The secular equation [Eq. (45)], which in Section IV is derived by inspection upon consideration of the wave functions in Table I, can also be readily derived by the method of invariants.

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123

-Aa I a,

260b

I

I

2

I

I

I

4

I

6

I

8

Pressure (GPa) FIG. 2. TO and LO frequencies of GaAs measured by Raman spectroscopy versus pressure (lower. linear scale) and versus lattice constant a (upper, nonlinear scale). The dashed lines represent the linear behavior found at low pressures: the full, nonlinear behavior has been fitted with quadratic functions (solid lines). (From Ref. [431.)

The basis of the method is the construction of a combination of products of “angular momentum” operators J , with the strain components:

C a ! i : +( ~ ~ , JJA)EA,,

(2)

AP

(JJ,, + J,JA)~,,, has at least one component that transforms like the fully symmetric irreducible representation (IRR) rl of the relevant point group

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EVANGELOS ANASTASSAKIS AND MANUELCARDONA

of the crystal. We choose the coefficients in Eq. (2) in such a way that Eq. (2) belongs solely to rl.The superscript (1) denotes the number of rl IRRs generated by the products ( J J , + J,JA)EA,.Each one of them corresponds to a free adjustable parameter in the effective strain Hamiltonian. In this manner, for the valence bands of Ge, Si, or GaAs at the r point of the BZ, the so-called Pikus-Bir Hamiltonian becomes

+ 2%5d{[J,Jy + JYJX]€,+ C.P.} where c.p. represents the three circular permutations of the coordinates x , y, and z that are referred to the crystal axes. cxin Eq. (3) depends on three adjustable parameters a, b, and d. Likewise, the strain Hamiltonian for the optical phonons of a diamond-type crystal at r will be shown in Section IV to depend on three parameters labeled KII,K12,and K4, which, by analogy to the electronic case, are called phonon deformation potentials (PDP). (KI1+ 2KI2corresponds to a, Kll - K12to b, and KU to the d of Eq. (3), except for numerical factors.) The operators J,,J y, and J ,of Eq. (3) are not, strictly speaking, angular momentum operators but operators that change like angular momentum components upon the operations of the point group under consideration. To calculate the effect of a given strain tensor on a degenerate set of states one must evaluate the matrix elements of .Kbetween these states. If we know the behavior of these states upon point group operations in terms of corresponding powers of x , y, and z (such as given in Table I), we can easily calculate matrix elements of JA and JAJ, using the expressions for angular momentum operators found in any standard quantum mechanics book. Most of the experimental data concerning the dependence of phonons on strain have been obtained for the optical phonons of diamond- and zinc blende-type semiconductors. In the diamond structure these phonons are threefold degenerate. The combination of PDPs KI1 + 2KI2 represents the effect of a hydrostatic strain (an irreducible component of a general strain): It shifts the phonons without splitting them. Kll - K12 represents the splitting due to a traceless uniaxial strain along [lo01(another irreducible component of a strain), while KU represents the splitting for a [lll]strain (the remaining irreducible component of a symmetric second-rank tensor in a cubic crystal). In a zinc blende-type crystal, and for wave vectors k larger than 2nlh (A is the reststrahlen wavelength), the optical phonons are split by the electrostatic field that accompanies them into a singlet (LO),

3 PHONONS, STRAINS, A N D PRESSURE IN SEMICONDUCTORS

125

which vibrates along k,and a doublet (TO) vibrating perpendicular to k. The LO-TO splitting is much larger than the splittings that can be achieved with uniaxial strain and therefore has priority in determining the vibrational eigenvectors: LO modes near r in a polar cubic crystal always vibrate along k. The strain may, however, be instrumental in determining the TO eigenvectors perpendicular to k [35-371. Although this chapter is mainly concerned with Raman and IR-active phonons with k = 0, we shall next discuss the effects of a strain on a generic phonon at an arbitrary point of the BZ ( k# 0). The first consideration is the existence of often large degeneracies associated with the so-called star of k,that is, all k vectors into which the original k transforms upon application of all operators of the crystal point group. Except for the hydrostatic component of the strain, which leaves the star of k invariant, a strain usually splits states belonging to the star of k.It is customary to treat the degeneracies of states with a fixed, given k and their strain splitting first and then tackle the lifting by strain of the degeneracies among different k sof the star. We consider the point-group operations that leave invariant a given k , not located at the edge of the BZ. This subgroup of the point group is called the group of k (Gk). We figure out all the invariants corresponding to those of Eq. (2) but this time for the group Gk. In this manner we can obtain the shifts and splittings of the subset of degenerate states “belonging to the wave vector k. These splittings are sometimes referred to as intravalbecause of their relationship with electron valleys of the elecleysplittings tronic band structure. Once these effective Hamiltonians have been worked out by the method of invariants, as discussed previously, one must consider the corresponding intervafley spfitrings. A simple way of doing this is by constructing the invariant

R - E .k- - 1-3 tr E

(4)

where d is the unit vector along k and the trace eliminates the hydrostatic component of E that does not produce any intervalley splittings. Equation (4) must be multiplied by a free parameter (deformation potential) for every irreducible component of the strain. The preceding discussion applies to the interior of the Brillouin zone. At the zone boundary, Gk may have to be expanded to include operations that transform k into another boundary point that differs from k by a reciprocal lattice vector. This procedure is straightforward. The situation may (but need not) become rather complicated for zone boundary k sin nonsymmorphic space groups, that is, groups that include screw axes, glide

126

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

planes, or both. In this case Gkmay have to be expanded to include some glide or screw operations. Such is the case for the X and the W points (see Ref. [46] for a discussion of the W point of face-centered cubic crystals, including the effects of strain) of the BZ in the diamond structure that contains glide planes. A full discussion of the effect of strain on electronic states (including spin and several important points of the BZ) and phonons can be found in a superb article by Kane [47]. As an example of the splitting patterns to be expected at a generic point of the BZ, Fig. 3 shows the splittings produced on the L3 phonons of zinc blende (TA or LA phonons, [111] direction, edge of the Brillouin zone)

[IOO]strain

[I 1I ] strain

intervalley splitting

k II [I 1I ] FIG. 3. Intervalley and intravalley splitting of transverse phonons along the 1111) direction obtained upon application of either a [111] or a [I001 pure shear strain.

3

PHONONS, STRAINS, A N D PRESSURE IN SEMlCONDUCrORS

127

by a traceless shear strain along either the [loo] or the [ l l l ] directions. In Kane’s notation the intravalley splittings for a [lo01 strain are determined by the deformation potential d: , while the corresponding splitting for a [ l l l ] strain is determined by d : .The intervalley splitting for a [ l l l ] strain is determined by d : . As already mentioned, a few measurements of the behavior of phonons along high-symmetry directions of k ( k# 0) upon application of uniaxial stress are available but rather incomplete [48]. The most reliable data have been obtained by inelastic neutron scattering. Raman measurements are also possible for critical points at the edge of the BZ, using the secondorder spectra [49]. Another possibility that has hardly been explored is to change the phonon wave vector by means of phonon confinement in a short-period quantum well that is then subjected to strain, a somewhat difficult undertaking (see Section V).

11. Background

In this section we review phonons in relation to crystal symmetry, based on group theoretical considerations. Detailed introductions to these subjects can be found in Refs. [50] and [51]. The terminology and matrix representation of strains, stresses, and other tensor quantities in a particular crystalline environment are also discussed: their relation in various reference systems is demonstrated through specific examples of heterojunctions (HJs) and superlattices (SLs).

1. PHONONS AND CRYSTAL SYMMETRY A crystalline compound with N atoms per primitive cell (PC) sustains an enormous number of vibrational normal modes, each characterized by a wave vector q that is taken to lie within the (first) Brillouin zone (BZ). The corresponding frequencies are given by 3N functions of q (the socalled dispersion relations) defined within the BZ. Three of these functions, the acoustical branches, exhibit zero frequencies (o= 0) at the zone center (phonon wave vector q = 0); Brillouin spectroscopy is one way to study the dispersion of acoustical phonons for small magnitudes of q. The remaining 3N - 3 branches, the optical branches, sustain phonons with nonzero frequencies throughout the BZ. Sometimes one of these frequencies may approach zero. It then signals a phase transition, and one speaks of a soft mode.

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Optical phonons with q 0 can usually be studied through first-order infrared (IR) or Raman spectroscopy. Some q = 0 phonons may be forbidphonons. den to both kinds of spectroscopies; they are then called silent Optical phonons with 4 # 0 can be investigated through second- or higherorder IR, Raman spectroscopy, or both. The effects of external perturbations, such as strains, on phonons are best studied through first-order optical spectroscopies. There are good reasons for this: The spectra correspond to interactions between electromagnetic radiation and a single q = 0 phonon. Hence they are, in general, stronger than second- or higher-order interactions involving several phonons. A second reason is that the 4 = 0 optical phonons, called thephononsfrom now on unless otherwise specified, are described by quasi-macroscopicparameters, the normal coordinates, involving the same atomic displacements within each primitive cell. The vibrations of a PC are thus similar to those of a molecule and appear spectroscopically as sharp, quasi-discrete features. They offer an accurate, convenient, and elegant way to formulate the problem of phonons at q = 0 and external perturbations in accordance with the point-group symmetry of the crystal.

a. GroupTheoretical Considerations

The collective character of the normal coordinates associated with a particular phonon in a crystal is reflected by the fact that their basis is an array containing the displacements of each and every individual atom participating in the phonon. The normal coordinates transform under the symmetry operations of the crystal group in the same way as the basis functions of one of the irreducible representations (IRRs) of the point group. Therefore, the number of different phonon symmetry types in a particular crystal is restricted by the IRRs of the point group; the phonon types are labeled accordingly. Even the three 4 = 0 acoustical phonons, representing basically parallel displacements of the entire PC, belong to one or more IRRs, which, in this the case, always coincide with those occurring in the reduction of T(r), three-dimensional representation of the point group to which a polar vector rbelongs. These IRRs can be easily identified by inspection of the character table of the point group. Another important point concerning the T(r) IRRs is that only phonons belonging to one of them are IR active in the first order; that is, they are capable of absorbing IR radiation of their own frequency. From now on we shall refer to IR-active phonons as polar phonons.For a consistent definition of basis functions for all 32 point groups, the reader is referred to Ref. [52]. For convenience we include in Table I the tables of representations and the associated basis functions for

3 PHONONS, STRAINS,A N D PRESSUREIN SEMICONDUCTORS

129

TABLE I IRREDUCIBLE REPRESENTATIONS AND BASISFUNCTIONS OF THE POINTGROUPS Td (GaAs, ZINCBLENDESTRUCTURE)AND C,, (GaN, WURTZITE STRUCTURE)

Note.The degeneracies of the (A, B ) - ,E-, T (or F)-type IRRs are 1, 2, 3, respectively. X, 7,7, are the corresponding components of an axial vector (or, equivalently, those of a second-

rank antisymmetric polar tensor). The table for the cubic point group 0 is the same as T d , except that the functions [x, y, z] belong to T I ,not to T 2 .The cubic point group for diamondtype crystals, Oh,is simply the product of 0 with the (E, I) group, where / is the point inversion operation.

the point groups Td and C,,, which correspond to the zinc blende and the wurtzite structures, respectively. For later use, we also introduce at this point the symmetrized double product T(x) = (T(r) X r(r))s; it represents the six-dimensional reducible representation (RR) of any second-rank symmetric tensor, for example, the electric susceptibility of a crystal x.' The reduction of T(X) into IRRs is easily obtained by inspection of the character table and the corresponding basis functions: T(x) includes only the IRRs that have second-order functions of x , y ,and z as basis functions [52]. Each IRR is included as many times as the number of such functions divided by the dimensionality of the IRR; the latter is read from the character table (left-most character column, not shown in Table I; see Ref. [52])and, for the crystallographic point groups, can be 1,2, or 3. Only phonons belonging to the reduction of r(X) are Raman active in the first order; that is, they can scatter monochromatic radiation of frequency (mi)into the frequencies wi % oj, where wj is the frequency of the jth-type phonon (- for Stokes, + for anti-Stokes). 'We consider only the symmetric components of y , as well as of the derivative of ,y with respect to the phonon displacements, called the Raman tensor, provided the laser and scattered frequencies are far away from electronic resonances. Close to resonance, antisymmetric components can appear. See F. Cerdeira, E. Anastassakis, W. Kauschke, and M. Cardona, Phys. Rev. Lett. 57, 3209 (1986).

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The assignment of IRRs to the 3 N phonons follows from a straightforward group-theoretical procedure, based on information about the lattice space group and each atom's exact position in the unit cell [53].For this purpose we designate a general atomic dispIacement u as belonging to the 3Ndimensional reducible representation I’(u). Its reduction reveals all possible types of optical phonons plus the three acoustical q = 0 phonons sustained by this particular lattice. b. More Definitionsthrough Specific Examples

We consider the following examples: SrTi03,perovskite structure, N = 5, point group

o h ,space

group 0;

The reductions of T(r) and r(x) follow from the character table of the point group o h .That of T(u), yielding the possible phonon symmetries, follows from the tables given in Refs. [53].The IRRs of T(u) inside parentheses designate the acoustical phonon part of T(u); those outside parentheses designate the optical phonon part. Optical phonons with the same symmetry as r(r) are IR active (three sets of triply degenerate polar phonons of symmetry Flu,each set corresponding to a different frequency). Optical phonons with the symmetry of one of the IRRs of T(x) are Raman active (there are no Raman-active phonons in the perovskite structure). Finally, phonons belonging to the F2,, IRR, which occurs in r(u) but not in r(r) or T(X), are both IR and Raman inactive (i.e., silent phonons; note that these phonons can often be observed in hyper-Raman scattering [54]). Both Flu and F,, are three-dimensional IRRs; hence, each counts as three units (threefold degeneracy) toward the total dimensionality 3N = 15 of T(u). Each type of phonon has a degeneracy given by the dimensionality of the IRR to which it belongs. This degeneracy can only be split by lowering the symmetry of the point group, for example, through the application of a nonhydrostatic stress. CaF2, fluorite structure, N

=

3,point group Oh,space group 0;

3 PHONONS,STRAINS, A N D PRESSURE IN SEMICONDUCTORS

131

In this case there is one set of Raman-active optical phonons (FZgIRR) and one set of IR-active phonons (FluIRR), both triply degenerate.

Si, diamond structure, N

=

2, point group O h ,space group 0:

There is one set of Raman-active phonons (&,, triply degenerate) and no IR-active phonons. GaAs, zinc blende structure, N group T:

=

2, point group Td (Table I), space

Here, there is one set of triply degenerate optical phonons that is both Raman and IR active ( F 2 )A. threefold degeneracy is found for those modes at q + 0, as required by the cubic symmetry. With slightly increasing q, however, they split into a singlet and a doublet, the former corresponding to atomic displacements parallel to q and the latter to displacements perpendicular to q. Such phenomena are common to all polar phonons and result from the electric dipole moment that accompanies the atomic displacements. Displacements parallel to q generate an electric field that enhances the vibrational restoring force. The corresponding phonons are called longi(LO). This electric field is absent for displacements perpentudinal optical optical (TO) phonons; their frequency dicular to 4:One speaks of transverse lies below that of the LO phonons. For IR-active (polar) modes to appear in the Raman spectra (and for Raman-active modes to appear in the IR spectra), the point group must be noncentrosymmetric. If there is a center of inversion, the phonons are either odd or even with respect to the inversion. IR phonons must then be odd because they have to couple to the odd electric field of the photon, whereas Raman phonons must be even because they couple simultaneously to both the incident and the scattered electric field, that is, to the product of two odd excitations. An LO-TO splitting is thus expected to exist in GaAs and to manifest itself through a split peak in the corresponding Raman spectrum, one at the LO frequency oL and the other at the TO frequency w. Such splittings do not exist in the Raman spectra of Si and

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EVANGELOS ANASTASSAKIS AND MANUELCARDONA

CaF2 because of the complementarity principle of IR and Raman activity previously enunciated: An IR-active mode Flucannot be Raman active (F2g),and vice versa. Near resonance, however, it is possible to violate this complementarity and to observe Raman activity from nominally Ramaninactive modes, for example, scattering by the Flumode in the antifluorites Mg2Siand Mg2Ge[%I.Similar situations occur in the presence of symmetrymodifying perturbations, such as an applied electric field: In this case, Raman or IR activity can be induced in nominally Raman- or IR-inactive modes, respectively. These are examples of the so-called morphic effects, reviewed in Ref. [56] (e.g., scattering by the Flumode in SrTi03 [57] and IR absorption by FZgmode in diamond [58], both under an external electric field). GaN, wurtzite structure, N

=

4, point group C,,(Table I), space group

G”

The phonons A l and El are both IR and Raman active, while the E2 phonons are Raman active only and the B1 phonon is silent. Both IRactive phonons exhibit LO-TO splittings that manifest themselves in their Raman spectra. It should be noted again that in polar crystals the LO-TO splitting vanishes for q + 0. It appears only for magnitudes of q that correspond to wavelengths (A = 2 d q ) much smaller than that of the electromagnetic waves of the same frequency. This condition is fulfilled in Raman backscattering with visible or near-IR lasers. In the intermediate q-region, accessible in forward-scattering experiments, the TO dispersion relations apbecome complex as mixed photon-phonon particles, called polaritom, pear [59].

2. STRAINS, STRESSES, AND CRYSTAL SYMMETRY Strains (E) and stresses (a)are second-rank tensor fields [60]. For the present purposes we shall treat them as symmetric (note that the “antisymmetric” component of a strain tensor corresponds to a rigid rotation). Their components are related through Hooke’s law in its generalized form.

3 PHONONS, STRAINS, A N D PRESSURE I N SEMICONDUCTORS

133

where SApvp and CAP,are the components of the fourth-rank compliance and stiffness tensors, respectively. In Eq. (10) summation over repeated indices is assumed, and all Greek indices run from 1 to 3. Symmetry relations reduce the total number of independent components of either S or C to a maximum of 21. T o eliminate redundancy in the large number of components of these fourth-rank tensors, one often uses the 6 X 6 symmetric matrix notation:

E, =

sijuj

(12)

where i,j = 1, . . . , 6 and uI = u l lu4 , = ~ 2 3 el , = e l l , and e4 = €23 + E~~ = 2eZ3,plus all circular permutations of the indices. The compliance constants S i j= Sjj are related to the corresponding tensor components through Sj, = S,,,

for i, j

S i j= 2S,,,

for i 5 3, j > 3, v # p

S i j= 4S,,,,,

for i ,j > 3, A # p; v # p

53

(14)

Note that the 6 X 1 and 6 X 6 matrices in Eqs. (12) and (13) do not transform liketensors under the operations of the point group. The 6 X 6 matrices representing S and C relative to the reference system S, corresponding to the crystallographic axis xcv ( v = 1-3) are tabulated in Ref. [60] for each of the 32 crystal classes and for isotropic media (e.g., glasses). Relations between individual components Cjland S j j for the cubic crystal classes are included in the appendix. To establish a consistent terminology, we present below all possible forms that a stress, strain, or other second-rank symmetric tensor b can have relative to the orthonormal system S, : x,, ( v = 1-3) in which it is diagonal. The system S, may not coincide with that of the crystallographic axes, S,.

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EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

The terms bisorropic and pure bisotropic were introduced in Ref. [61], the remaining terms are from Ref. [60]:

buniaxial . .

=

(O

0 b)

where dots stand for zero. We give below two examples of practical interest to show how these terms are applied to strained cubic HJs (and SLs) grown along high-symmetry directions. A third example pertains to hydrostatic pressure.

3 PHONONS, STRAINS, AND PRESSURE I N SEMICONDUCTORS

135

a. GrowthAlong[OOl]

In the cubic crystal classes, xc111 [loo], xc211 [OlO], and xc311 [Ool]. Latticemismatched cubic epilayers (bulk lattice constant a )grown along xc3on a cubic substrate (bulk lattice constant a,)develop a coherent, homogeneous strain that, relative to S,, has the form (15b):

€ =

where €11 is the in-plane component of the strain. For so-called pseudomorphicgrowththe lattice constant along the interface plane is continuous across that interface, a fact that generates an interface strain that is determined by the lattice mismatch or misfit:

e l ,the strain normal to the plane, can be expensed in terms of ell. The 6 X 6 matrix for C. relative to the S, of cubic classes, has the form [60]

C=

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

l36

Applying Eq. (13), together with (16), (18),and the equilibrium boundary condition u3 = 0 (meaning that no external forces are applied along the axis of growth), yields the following results:

Hence,

The parameter B in Eq. (19c) is the bulk modulus of the cubic epilayer; it is defined from

P represents a hydrostatic pressure that causes a relative volume change d In V. Based on the established terminology [(15a) through (15g)], the strain, in this example, is bisotropic and the stress is pure bisotropic. We regard the commonly used designation of 6iaxial as misleading, if not incorrect.

6.

Growth Along [110]

As a second example, we consider a lattice-mismatched cubic epilayer grown along x; 11 [110]. Relative to the new system S' : x i 11 [Ool], x i 1) [lTO], x i 1) [110], the strain has the same form given in Eqs. (16) and (17). It is convenient to apply Eq. (13) in the primed system S'. The

3 PHONONS, STRAINS, AND PRESSURE IN S E M I C O N D U ~ O R S 137

conversion of C into C' is best approached through the compact algorithm described in Ref. [62] and included in the appendix. Equation (13), combined with the results of the appendix and the boundary condition u; = 0, yields

Hence,

In this example the strain is bisotropic and the stress is biaxial. c. Hydrostatic Pressure

A hydrostatic pressure, or a pressure for short, represents the simplest case of stress, namely, an isotropic stress of the form (15a). Being usually compressive, the pressure is treated as a negative scalar stress, - P (
-P(S11+ 2S12) = - P / 3 B= ( dln V ) / = 3 dV/3V(
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EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

Similarly, a pressure applied to a bulk uniaxial (noncubic) crystal causes an anisotropic contraction. Consider, for instance, a crystal with wurtzite structure such as GaN or CdS. According to Eq. (12) and to the form of S for the hexagonal crystallographic system [60], the strain produced by a pressure is bisotropic, similar to that which exists in a lattice-mismatched cubic epilayer [i.e., Eq. (16)] except that its origin is different,

where h-1 and K I are the linear compressibilities of the hexagonal crystal parallel and normal to the xlx2plane, respectively [60]. The same expressions hold for trigonal and tetragonal crystals, the relevant part of the S matrix being isomorphic in all these cases. Finally, in orthorhombic crystals the strain produced is of the general form (15g), that is,

where K " ( v = 1-3) is the linear compressibility along the crystallographic axis x , .In crystals with even lower symmetry the situation is more complex because the crystal axes are not determined by symmetry and even change upon the application of pressure.

3. EXPERIMENTAL TECHNIQUES FOR APPLYING STRESSES a. Uniaxial Compressive Stress

A traditional and versatile apparatus for applying uniaxial stress on solids for optical and electrical experiments is described in Ref. [28] and depicted in Fig. 4. The force is produced by the elongation of a spring and applied to the sample (typical cross section -2 mm2) via a 10/1 lever arm. The force is measured by either the calibrated elongation of the spring [28] or by means of a force gauge integrated into the system [63]. In more recent versions, the spring has been replaced by other devices such as a pneumatic hydraulic cylinder-piston system [a]. The samples are cut in the form of rectangular rods of nearly square cross section, with the ratio of length to cross-sectional dimension equal

3 PHONONS, STRAINS, A N D PRESSURE IN SEMICONDUC~ORS 139 ,-LEVER

ARM

EPOXY’

FIG. 4. Typical stress apparatus used for optical measurements [ZS]. The applied force is obtained from the elongation of the spring, as measured with a linear variable differential transformer [28, 631.

to -7 for maximum endurance. Usually, the rod is glued with epoxy resin into holes drilled in a pair of cylindric brass cups into which the rod cross section fits snuggly. In this way, premature breaking due to thermal expansion differences between the sample and brass while cooling or heating is avoided. In a slightly different approach, the samples are not glued but simply held on flat stainless steel pistons [65]. Either fine pieces of fused silica or soft metal shims are placed between the polished surfaces of the sample and the pistons to ensure homogeneous contact. The cups are mounted into a frame (Fig. 4); the force is applied to the lower cup through a pin mounted on a counterframe connected to the lever arm by means of a pull rod. The sample frame can be fitted into a liquid nitrogen cryostat for operation at 80 K as well as at room temperature. Operation with the frame immersed in liquid helium is also possible. In this case formation of bubbles is prevented by pumping below the A point of 4He. Alignment of all parts along the long axis of the sample is very important for reaching high stresses before the sample breaks. Stresses as high as 2.5 GPa have been reached with Si samples using the improved versions of this setup.

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EVANCELOS ANASTASSAKIS AND MANUELCARDONA

The advantages of this apparatus are precise stress calibration, application of low-to-high stresses (depending on material) along preselected crystaliographic directions, bulk homogeneity of the stress, and operation at low temperatures. Its main disadvantages are that it requires bulk pieces of material, which are not always available, and special preparation of the samples. Some (520%)relaxation of the stress very near the sample surface, as revealed by Raman experiments with opaque semiconductors [66, 671 and by far-infrared absorption measurements in InAs quantum wells [68, 691, may introduce errors in the pressure dependence of measured parameters.

b. PureBisotropic Stress Pure bisotropic tensile stresses of the form (1%) have been applied to (111) Si plates using two different mechanical devices. In the first device [70], the plate (-40 mm in diameter, -1 mm thick) was squeezed between a circular ring (18 mm in diameter) and a concentric spring-loaded circular die (6 mm in diameter); the central sag did not exceed the thickness, and the resulting stress was calculated, within 15% error, using complicated equations from elasticity theory [71]. In the second device [72], much thinner (111)plates (-30 mm in diameter, -0.2 mm thick) were squeezed between the same bearing and a concentric spring-loaded steel sphere (12 mm in diameter); the stress at the center was calculated from the strain components, which were measured through independent X-ray diffraction. A similar apparatus using (001) Si wafers (0.28 mm thick) or (001) GaAs/ AlGaAs quantum well membranes (20-30 pm) was used in Refs. [73], in which the stress was generated either by a sphere or by a hydrostatic pressure P;it was shown that at the center of the wafer or the membrane the stress could be related in a simple manner to the geometric characteristics of the sample. In a different version, the wafer was replaced by a stripe (long dimension a) supported at two short ends, and the central force was exerted at the center of the stripe by a sharp metallic edge normal to a; the tensile stresses produced were uniaxial of the form (lse), along the long dimension of the stripe [74]. The advantage of these techniques is that rather high stresses can be attained. There are serious disadvantages, however, since the area of homogeneous stresses is limited and not well defined and the calibration procedure, whatever it is, may be hampered by reproducibility and geometric problems. Moreover, these devices are not appropriate for bulk measurements under homogeneous stresses and can easily introduce nonlinearities into the data because of the strong inhomogeneous deformations.

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS

141

Another way of applying bisotropic and pure bisotropic stresses of the forms (15b) and (15c), respectively, is through epitaxial growth. The origin and description of the stresses in SLs and HJs were presented in Section II.2.a, Eqs. (20) and (21), respectively. Homogeneous stresses up to 1 GPa are easily obtained in epilayers with thicknesses less than the critical value required for the generation of dislocations (undercritical layers). Usually the strain can be measured independently (e.g., by X-ray diffraction, transmission electron microscopy, or low-energy electron diffraction), and Rutherford backscattering [75] can be employed to assess the degree of coherence and homogeneity of the layers. In purely undercritical epilayers, where the strain fully reflects the lattice mismatch, the elastic state is homogeneous and well described by Eqs. (20) or (23). One speaks of pseudomorphic For overcritical epilayers, there is at least partial strain relaxation growth. due to the formation of dislocations. Some residual strains often remain, their origin being partly the lattice mismatch and partly the difference in the thermal expansion coefficients (thermal mismatch). As a result of partial strain relaxation, the elastic state is not homogeneous: it depends on position, thickness, and growth conditions, and cannot be described by a unique model. The disadvantage of epitaxy for the production of strains is that each epilayer is strained to a fixed nominal value: Careful selection of substrate and growth technique is required to achieve a specific required strain. Tuning of the strain in the epilayer is possible through the use of a mixed crystal (alloy) epilayer or substrate [75]. After the strained epilayer has been grown, additional fine tuning can be obtained by varying its temperature if the thermal expansion coefficients of epilayer and substrate are different [76] or by placing it under hydrostatic pressure [77-801, a subject to be further discussed in Sections IV.3.f and V.2. An alternative procedure is to glue the sample tightly and uniformly on a bulk holder of different thermal expansion coefficient and vary the temperature of the composite system. Depending on the holder, it is possible to apply either tensile or compressive pure bisotropic stresses that result in bisotropic strains [81].

c. Diamond AnvilCell

In the last two decades enormous progress has been made in the physics of semiconductors under high pressure, owing to the availability of diamond anvil cells (DAC) [19, 20, 821. The principle of operation and technical details of the various versions of these fist-size rigs have been reviewed in several articles [83] and will only be summarized here. The advantages of the DAC technique are low cost, small size, the need of only minute samples

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EVANGELOS ANASTASSAKIS AND MANUELCARDONA

(-100 pm), relatively accurate calibration (kO.1 GPa), extensive range of pure hydrostatic pressures (210 GPa), operation at low and high temperatures, the possibility to perform electrical transport measurements and also Fourier transform IR (FTIR) measurements, and, last but not least, safety. The main disadvantages are the rather elaborate loading procedure and possible changes of the isotropic character of the pressure during the operation. As a rule of thumb, pressures are hydrostatic and homogeneous up to -10 GPa. To minimize uniaxial components and inhomogeneities above 10 GPa, it is highly advisable to use helium as the pressure-transmitting medium. Even at room temperature helium freezes at about 10 GPa, but it seems to remain sufficiently soft to transmit hydrostatic pressures, provided the samples are not too thin [84]. The DAC may also be used for applying uniaxial stresses on thin films deposited on the surface of one of the diamond anvils [85]. The pressuretransmitting liquid used in this case is alcohol, which becomes very viscous at low temperatures and virtually freezes at 200 K. It then acts as a piston pressing the film uniaxially. Stress generated in this manner has been shown to produce a splitting of the lowest edge exciton (Z1-Z2) of a (111) CuBr film; this splitting is observed in the optical absorption spectra. Uniaxial stresses as high as 3.5 GPa were reached before plastic deformation occurred in the film. The uniaxial stress can be calculated from the measured hydrostatic pressure by considering the actual boundary conditions on the film in a way similar to that followed for pseudomorphic epilayers. In a similar study, the sample, a semiconductor laser diode, was placed between the diamond anvils with conducting metallized films between their surfaces [86]. The emission wavelength was observed to undergo a linear red shift at a rate of -25 nm/GPa, up to 0.6 GPa. The calibration of the stress was based on established information relating the wavelength to the applied tensile strains. These observations are consistent with boundary conditions corresponding to a nearly frictionless contact between the laser and the anvils.

d. OtherPossibilities

The uniaxial stress apparatus described in Section II.3.a has been used for stressing multiple quantum wells (MQWs) along the direction of growth [87]. Careful sample preparation is required in this case; the details are depicted in Fig. 5. The sample used in this study [87] consisted of 300 GaAslAIAs periods (18/16 monolayers, total thickness 2.8 pm), with Ga,AI,-,As (x 0.3) cladding and a GaAs substrate. This structure, glued

-

3 PHONONS, STRAINS,

AND

PRESSURE IN SEMICONDUCTORS

143

DETAIL OF THE SAMPLE #---.

/

MQW’S

/

SILICON ROD

\

/

/

suBST R AT E ‘,

\

I /

/

y- /

SILICON ROD FIG. 5. Sample detail for uniaxially stressing a multiple quantum well (MQW). A modified version of the stress rig shown in Fig. 4 is used for applying the force. (From Ref. [87].)

between two Si bars, can sustain compressive stresses as high as 0.6 GPa. Such an arrangement was used in the study of transmission and photoluminescence spectra of MQWs. The stress results in remarkable changes in the optical response of the MQW. In an earlier report the technique was shown to be very informative in the study of the electronic band structure of GaAs/AlAs superlattices with the stress applied both parallel and perpendicular to the growth axis [88]. Ultrasonic waves (2 MHz, s 10 W/cm2) have been used to generate time-dependent deformations in crystals. These deformations have been probed by measuring the corresponding shift and splittings of degenerate phonon frequencies in CaF2 and ZnSe, which can amount to as much as 1.4% and 0.7%, respectively [89]. These effects have been qualitatively interpreted as the result of a reduction of the elastic constants in the presence of the ultrasonic wave. Since the strains attained are of considerable magnitude, the technique can only become quantitatively useful provided a reliable procedure to measure the relevant components of the strain tensor is developed. Shock compression experiments have also been used to produce large uniaxial strains. For diamond and a-quartz crystals of various orientations, the strains can be revealed by their effects on the phonon frequencies as

144

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

determined by time-resolved Raman spectroscopy [90]. The shock waves are produced by bombarding the samples with projectiles accelerated to speeds of 0.34 to 0.37 km/s with a compressed-gas gun. The longitudinal and lateral stresses can be determined from nonlinear elasticity theory. In this manner, peak longitudinal stresses ranging from 10 to 50 GPa have been achieved with lateral stresses one to two orders of magnitude smaller. (It is interesting that such bisotropic stresses result in uniaxial strains in the direction of propagation, that is, we have a situation opposite to that of a uniaxially stressed bar.) The time-dependent spectra show phonon frequency shifts and degeneracy liftings that agree with theoretical predictions (see Section IV.3.a and Figs. 17 and 18). Systematic studies involving tensile stresses have been carried out in fibers in which, by relatively simple means (e.g., hanging weights, mechanical stretching), homogeneous, well-calibrated tensile stresses of the order of 510 GPa can be obtained. For instance, up to 10 GPa have been applied to silica fibers [91]. The effects of such stress have been demonstrated by measuring shifts of vibrational frequencies proportional to the applied stress; forward Raman scattering through the fiber was used. SimiIarly, up to 0.7 GPa were applied to graphite fibers [92] and probed through backscattering from the cylindric surface of the fiber (see also Section IV.3.h).

111. Effects of Hydrostatic Pressure on Optical Phonons

As already mentioned, rather high hydrostatic pressures (up to -30 GPa) can be applied nowadays to a solid inside an optical diamond anvil cell. The highest pressure that can be reached is determined either by the strength of the diamond-gasket system or by the occurrence of a phase transition in the material being investigated. The strongest, most stable material in this respect is diamond itself. Recent ab initio theoretical calculations [93] indicate that under uniaxial conditions, such as those prevailing in the diamond windows of the DAC, the diamonds should be stable up to -500 GPa. This explains the power of the DAC for experimenting under ultrahigh pressures. Under pure hydrostatic pressure the abinitio calculations indicate that the diamond structure should persist up to the colossal pressure of 3 TPa (3000 GPa = 30 Mbar). At this pressure the volume of diamond is reduced by a factor of two (see Fig. 6). These facts are particularly striking when one considers that under normal pressure and temperature the diamond structure is actually metastable: The graphite structure has a lower free enthalpy.

3 PHONONS, STRAINS, AND PRESSURE I N SEMICONDUCTORS 145

z

k a , L 3 v) v)

a,

E

1

Atomic Volume

(A3)

FIG. 6 . Pressure vs volume equation of state, as obtained in an ab inifio molecular dynamical calculation [93]. The parallel lines at pressures above 1.7 TPa signal an incipient phase transition to the SC4 (simple cubic with four atoms per unit ce1l)phase. The dots indicate a fit with Eq. (28) leading to the parameters V , = 5.615 A31atom, B = 4.42 TPa, and B’ = 2.9.

The equation of state shown in Fig. 6 predicts a saturation of the volume with increasing pressure, a rather general phenomenon that is responsible, at least in part, for the nonlinearities shown in Figs. 1 and 2. The equation of state P ( V )shown in Fig. 6 has been obtained numerically, point by point, through elaborate ab initio calculations [93]. Several analytic expressions have been proposed to represent this equation [94], the simplest and most popular being the Murnaghan equation [95], which follows immediately from the assumption that the bulk modulus B increases linearly with pressure according to the expression dP - B ( P )= B dlnV

+ B’P

146

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

The dimensionless parameter B‘ of Eq. (27) takes, for most solids, values that lie between 3 and 5. Integration of Eq. (27) leads to the Murnaghan equation:

where Vois the volume at P = 0. The dots in Fig. 6 represent the predictions of Eq. (28) for B = 4.42 (the experimental value) and B‘ = 2.9. They show that Murnaghan’s equation [Eq. (28)] reproduces rather well the results of the ab initio calculation of the equation of state. Before we discuss in detail the effects of a hydrostatic pressure on the frequencies of phonons in crystals, we should keep in mind that the corresponding experimental data are obtained as a function of applied pressure, whereas for calculations (e.g., of the ab initio variety) the natural independent variable is either the volume or the lattice constant. Hence, in order to compare experimental results with theoretical predictions we must use an equation of state [e.g., Eq. (28)]. As an example we display in Fig. 7 the results of ab initio calculations of the Raman frequency of diamond plotted versus pressure [96]. Experimental data have been obtained up to pressures of 38 GPa; they agree with the calculations of Fig. 7. In order to display the theoretical results as a function of pressure (instead of volume, the natural “theoretical” variable), the authors of [96] made use of the theoretical equation of state obtained numerically in the same ab initio total energy calculation. As is often the case (see Fig. 2), the frequency of the Raman phonons is a sublinear function of P at high pressures. This fact reflects, at least in part, the sublinear behavior of the volume versus P displayed in Fig. 6. However, in the case of diamond, and for the wide pressure range accessible to ab initio calculations, a plot of the Raman frequency versus volume shows a supralinear behavior displayed also by the o ( V )curve of Fig. 7. In view of the good agreement found between the Raman frequencies measured for diamond and the calculated ones for pressures up to 38 GPa, the use of calculations such as those of Fig. 7 has been suggested for calibrating pressure scales [96]. Most of the investigated phonon frequencies increase with increasing P. The rare cases of frequencies that decrease with Pare believed to be loosely related to an incipient instability that signals a phase transition. The best known of such cases involves the transverse acoustical (TA) phonons at the edge of the Brillouin zone in Ge, Si, and zinc blende-type semiconductors. These TA frequencies can be obtained from the second-order Raman

3 PHONONS, STRAINS, A N D PRESSURE IN SEMICONDUCTORS

Atomic Volume

147

(A3)

1.7

1.6

1.5

I

I

I

2100 n 7

€

0

W

> g

1900

a, 3

0

!

LL

c 1700 0 c

0 I=

a

1500

1300

0

150

300

450

600

Pressure (GPa) FIG. 7. Calculated frequency o(P)of the zone-center optical phonon of diamond (dots and solid line through them) plotted against calculated pressure in the range 0-600 GPa. o(V)is the corresponding plot versus volume. The upper horizontal scale is linear in volume. (From Ref. [96].)

spectra as measured in a DAC. Figure 8 shows the dependence of the frequencies of TA-related second-order Raman features versus P measured for GaP [97].Notice that the behavior of features that decrease with increasing P is supralinear in P, a fact that is probably related to the instability mentioned earlier (see Section 111.4). It is worth mentioning that for diamond, and contrary to the case of most other tetrahedrally bonded materials, the TA frequencies at the edge of the zone increase (although less than those of higher modes) with increas-

148

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

0

TO-TA (Xi L)

o 2TA (K; 2)

I

0

5

10

Pressure (GPa) FIG. 8. Excitation frequencies extracted from the two-phonon Raman spectrum of GaP (300 K) in the 2TA and TO-TA regions as a function of pressure. (From Ref. [97].)

ing P [98], a fact that corresponds to the stability of the strongly covalent structure: Pressure-induced phase transitions do not occur up to about 3 TPa (see Fig. 6; keep in mind, however, that diamond is metastable with respect to graphite).

1. MODE GRONEISEN PARAMETERS We have discussed in the previous subsection the nonlinear behavior of phonon frequencies versus P at ultrahigh pressures. At low pressures, and within the experimental accuracy, this behavior is often assumed to be linear and is represented by the mode Gruneisen parameter yi introduced in Eq. (1). In the standard case in which wiincreases with increasing P, yi is positive (typical values of yi lie between 1 and 2); in the opposite case (e.g., for TA phonons at the BZ edge), it is negative. We should point out that yi is sometimes obtained through a linear fit of frequencies measured at low pressures, while sometimes a quadratic fit is made over a larger

3 PHONONS, STRAINS,AND PRESSURE I N SEMICONDUCTORS 149

pressure range. If the behavior versus P (or V )is sublinear, a quadratic fit usually leads to a larger linear term (i.e., a larger mode Griineisen parameter) than the simple linear fit. This fact accounts for some of the discrepancies among the measured yi found s in the literature. Perhaps the most complete measurements of mode Gruneisen parameters throughout the whole Brillouin zone have been performed by inelastic neutron scattering for germanium at room temperature, using a hydrostatic pressure cell with metallic lead as the pressure-transmitting medium [39]. Figure 9 shows the yi obtained in Ref. [39] for TA phonons along the r -+

2.0 1.5

1 .o

-+-

0.5

+Q

0

-0.5 -1 .o -1.5

-2.0 FIG. 9. Mode Griineisen parameters in the [loo] and [ I l l ] directions of the Brillouin zone as determined from the frequency changes between 0 and -5 GPa (dots) and 0 and 2.3 GPa (circles). The squares, diamonds, and triangles refer to ultrasonic, tunneling, and Raman measurements, respectively. The line correcalculations. (From Ref. [39].) sponds to results of ab inifio

150

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

X [loo] and K + L [ l l l ] directions of the BZ of germanium. Note that over most of the BZ yi(TA) is negative, as mentioned in the previous subsection. However, when approaching r, the sign of ’y; becomes positive, a fact that signals the standard behavior of the elastic constants: The stiffness constants Cij increase (i.e., the material becomes more stiff) with increasing P. We recall that for diamond the yi(TA) are positive throughout the whole BZ. The case of the zinc blende-type variety of silicon carbide (3C-Sic) is of particular interest because it is intermediate between diamond [yi(TA) > 01 and silicon [yi(TA) < 01. For 3C-Sic (silicon carbide with [99]) to be zinc blende structure), yi(TA) has been calculated (ab initio positive near X and negative near L. The magnitude of yi(TA) in both cases is, however, rather small ((yi(TA)\ O.l), as expected for a material intermediate between Si and diamond. The negative values of yi(TA) discussed above are responsible for the anomalies in the expansion coefficients found at low temperatures for most tetrahedral semiconductors (see Section 111.2). An extensive compilation of 7;’s available up to 1988, obtained either experimentally or theoretically for diamond and zinc blende-type crystals, has been given in Ref. [18]. Since then, considerable progress has been made, especially because of the development of ab initio theoretical techniques. Figure 9 illustrates the excellent agreement between the theoretical calculations (solid line) and the experimental points for germanium [39]. Although a check of the theoretical results against experimental data is always highly desirable, whenever the latter are not available one can use ab initio calculated values of yi with little risk if required. (The results of sticks-and-balls lattice dynamical calculations with adjustable harmonic and anharmonic force constants may be useful but are a priori less reliable than the ab initio results: see, for instance, Ref. [99] for 3C-Sic and [lo01 for Si and Ge.) Particularly satisfying are the calculated negative values of yi shown in Fig. 9, especially in view of the difficulties encountered in developing a simple, physically intuitive model of such a phenomenon. As discussed in the previous subsection, the only simple interpretation is that it signals an instability, possibly leading to a phase transition with increasing P. We cannot, within the length of this review, list all experimentally or theoretically determined Griineisen parameters. Instead we have confined ourselves to listing some of the most recent data for the isoelectronic series Ge, GaAs, ZnSe, and CuBr in Table 11. In the tables for all four materials one can see the negative values of yi for the TA phonons at the edge of the zone. When approaching the r point, the 7;’sreverse sign for germanium and maybe also for GaAs (with the possible exception of the TA(1) phonon, which may keep very small negative values at y;). More strongly negative

-

3

151

PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS

TABLE I1 THEORETICAL VALUESOF THE GRUNEISEN PARAMETERS OF Ge, GaAs. ZnSe, AND CuBr AT VARIOUS POIN.I.S OF THE BRILLOUIN ZONE

X LO TO(2) TO(1) LA TA(2) TA(1) GaAs LO TO(2) TO(1) LA TA(2) TA(1) ZnSe LO TO(2) TO(1) LA TA(2) TA(1) CuBr LO TO(2) TO(1) LA TA(2) TA(1) Ge

1.17 1.49 1.49 1.17 -1.36 -1.36 1.01 1.62 1.62 1.22 -1.15 -1.15 1.04 1.88 1.88 1.24 -1.43 -1.43

r

K 1.05 1.05 1.05

.

. (-1.31) (-1.31) (1.73) (1.73) (-1.62) (-1.62) (0.9) (1.6) (1.6) (1.1) (-1.3) (-1.3)

. .

1.24 1.50 1.58 1.23 0.15 -0.74 1.25 1.71 1.84 1.18 -0.24 -1.19

(-1.53) (-1.53)

.

(-0.4) (-1.62)

. .

(-0.4) (-1.3)

0.62 0.32 1.09 1.18 1.18 1.34 0.57 0.01 1.18 1.47 1.47 1.23 -0.07 -0.60

. . . .

(-0.4) (-0.4)

.

.

(-0.4)

.

L

(1.12) (1.12) (1.12) (1.31) (0.61) (0.37) (1.23) (1.39) (1.39) (1.3) (0.53) (0.05) (0.9) (1.4) (1.4) (1.45) (0.16) (-0.60) (1.9) (2.4) (2.4)

1.67 1.28 1.28 0.55 -1.18 -1.18 1.62 1.40 1.40 0.56 -1.13 -1.13 1.63 1.70 1.70 0.51 -1.43 -1.43

.

. . . . (-1.51) (-1.51) .

(1.5) (1.5)

. (-1.7) (-1.7) (0.9) (1.6) (1.6) (1.1) (-1.5) (-1.5)

. . . .

(-0.4) (-0.4)

Note.Experimental data are given in parentheses. (From Refs. [18] and [loll.)

values are seen in the table for the T A phonons of ZnSe and CuBr. These negative values are reflected in the temperature dependence of the thermal expansion coefficients, to be discussed in Section 111.2. Table I1 suggests that the optical phonons at the zone edge have larger Griineisen parameters than those at the zone center. This was already noticed in 1971 [13] while measuring Gruneisen parameters of Ge,-Sil_, mixed crystals. In such mixed crystals (alloys) the wave vector conservation selection rule is relaxed as a result of the violation of translational invariance, and the first-order Raman peaks correspond, at least in part, to phonons in regions of the BZ where the phonon density of state has a maximum, that is, at the edge of the BZ. Correspondingly, an increase in yi is observed when increasing x from 0 to 0.5, and a decrease is again observed from 0.5 to 1. The explanation given for this behavior is based

152

EVANGELOS ANASTASSAKIS

AND

MANUELCARDONA

on the so-called Blackman-Brout [lo21 sum rule, which can be derived with some restrictive assumptions concerning the dynamical matrix:

C o f ( q=)constant 1

where the summation is carried out over all phonon bands and the “constant” depends on the volume. Differentiation of Eq. (29) with respect to In V leads to

2y i w f (=q )constant I

The negative values of yifor TA phonons at the BZ edge lead to an increase of yi(TO) or yi(LO), or both, with increasing q, in agreement with the experimental observations. 2. THERMAL EXPANSION: QUANTUM EFFECTS AT T = 0 The Gruneisen parameter y was initially introduced to quantify the overall anharmonicity that is required to interpret the thermal expansion of a solid [103]. The volume expansion coefficient (Y can be written in terms of Y as (Y

=

(y)

= yC,/B

P

where B is the isothermal bulk modulus and C, the specific heat per unit volume at constant pressure. Within the Debye model of the lattice vibrations (which assumes only acoustic modes with linear dispersion relations) the Gruneisen parameter y can be written as dlnO

y=

-m

where 0 represents an average Debye frequency. The lattice dynamics of the crystals discussed throughout this chapter are very far from the Debye model: The dispersion relations of the acoustic modes are nonlinear and a number of often flat optical branches appear. As seen in Section 111.1, one can define for each mode a mode Gruneisen parameter y similar to that in Eq. (32) but with 0 replaced by the corre-

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS

153

sponding mode frequency w;. It is obvious that to obtain the thermal expansion due to the anharmonic behavior of the lattice vibrations, Eq. (31) must be replaced by an average of the contributions of all phonon frequencies and mode Griineisen parameters yi . The corresponding expression at zero pressure can be obtained from the free energy F( V ,T ) ,which includes the lattice vibrations, by calculating the equilibrium volume through the sohtion of [dF(V, T)/aV] = 0. In this manner one finds [104,105] 1 v(7 )= V ( 0 )+ 2B

?iwiyi coth

(5) (33)

where kB is the Boltzmann constant. Through differentiation of Eq. (33) one can obtain the volume thermal expansion coefficient:

a ( T )=

1 6; 2BTV(O)? fLwiYicosh ti- 1

(34)

Comparison of Eq. (34) with (31) indicates that indeed y should be replaced in Eq. (31) by an average of the Gruneisen parameters y; of all vibrational modes, the acoustic modes being dominant at low Ts (because of the statistical factor &[cosh6,- 11 - I ) whereas the optical modes become important with increasing T. Equation (34) indicates rather clearly that a( T ) can be a nonmonotonic function of T whenever yi of s different signs contribute to it. This is indeed the case for most of the tetrahedrally bonded materials discussed here (but not for diamond; see Section 111.1). For Ge and GaAs (see Table 11) the yi of s the acoustic modes near the center of the BZ are positive. This leads to the standard behavior for T < 10 K, a positive a( T ) (see Fig. 10 for GaAs). As the temperature increases, TA frequencies close to the edge of the BZ, with yi< 0, begin to play a role in Eq. (34). Consequently, a( T) reverses sign and the crystal anomalously contracts with increasing temperature. At T > 50 Kin GaAs, LA and optical modes play an increasingly important role and a( T ) becomes positive again, as illustrated in Fig. 10. In this figure the points are experimental [106], whereas the solid line was obtained with Eq. (34) using ab initio calculated wi sand yi [105]. s As shown in Table 11, the Gruneisen parameters of phonons near the center of the BZ are also negative in the case of ZnSe (and probably also CuBr). In this case, a( T ) is expected to remain negative even at the lowest

154

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

8 x 1 O4

-

6x10-6

4.10-6

1 1

GaAs I

L

2x10" 0x10" I

I

I

I

100

200

I

300

Temperature (K) FIG. 10. Linear thermal-expansion coefficient a of GaAs versus ternperature [ 1051. Diamonds and squares indicate experimental data from Refs. (1061. Note the double sign reversal of a measured and calculated at low temperatures.

temperatures. Upon increasing T, it becomes positive (at about 50 K for ZnSe) [l05]. Let us consider next Eq. (33) in the limit of T -+ 0: 1 V (T + 0 ) = V ( 0 ) +2B

2h ~ i y i

(35)

The second term in the right-hand side of Eq. (35) represents the zeropoint quantumcorrecrion (it vanishes for h -+ 0) to the equilibrium volume of the crystal. In a monoatomic crystal (e.g., diamond, germanium, silicon), the quantum correction to the crystal volume (also to the lattice constant ao) is proportional to M-"* according to Eq. (35), a typical result for such zero-point quantum effects. Ab inirio calculations have been performed for such monoatomic crystals as a function of the isotopic mass M [104]. For was predicted between diamonds diamond a change Aaolaoof 1.5 X with M = 12 and M = 13 amu. This prediction is in excellent agreement with experimental results for diamonds grown with different isotopic compositions [1071. Measurements in agreement with theoretical predictions have also been reported for Ge [lo81 and Si [104, 1091. We point out in closing this section that the isotope effects just mentioned are only observed

3 PHONONS,STRAINS, A N D PRESSURE I N SEMICONDUCTORS 155

at low temperatures (more precisely, at temperatures well below the average Debye temperature 0).This follows directly from Eq. (33): For T + 0, coth( 5;:/2) = 2kT/tiwi and, accordingly, V (T ) becomes independent of the isotopic mass.

AND LIFETIMES 3. PHONONLINEWIDTHS

Within the harmonic approximation in a perfect crystal, phonons of a given q have an infinite lifetime. The corresponding first-order Raman spectra should then be composed of &function-like peaks with a full width at half maximum (FWHM) r determined by the spectrometer resolution, which can be made rather high (50.1 cm-I). However, crystals are neither perfect nor exactly harmonic. Among the most ubiquitous “imperfections” are the mass fluctuations associated with the natural isotopic abundances of the constituent atoms [110] or with artificially fabricated crystals with tailor-made isotopic compositions [111).Another rather general broadening mechanism is related to the anharmonic decay of a given phonon into two phonons or more (although two-phonon processes seem to dominate, except at the highest temperatures). These processes require energy and wave vector conservation. For a (Raman) phonon at q = 0, with frequency w o , the energy conservation equation becomes

and the corresponding linewidth can be written

where r is an angular frequency (in radlsec), (Ha)the matrix element for the anharmonic decay, and N , ( w l+ w 2 ) the density of two-phonon states with a total angular frequency w l + w2 = w . Equation (37) can be evaluated ab initio using electronic band-structure total-energy techniques. The calculations performed for diamond, silicon, and germanium [112] lead to values of the width and its temperature dependence that agree reasonably well with the experimental data [113]. When dealing with Eq. (37), one has to keep in mind that Nd(wl+ 02) is, in some range of w , a smooth function of w = w1 + w2 but can also exhibit sharp structures related to van Hove singularities (usually at the edge of the BZ). State-of-the-art ab initio calculations have errors that are difficult to assess (the largest are probably associated with the use of the local density approximation [LDA]). The errors may be different, both in

156

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

sign and in magnitude, for the different phonons under consideration, so it can happen that the calculated wo falls on a sharp van Hove singularity of o = o1 + y for the calculated N d ( o l+ y),while experimentally it does not. The opposite can also occur. These possibilities should be carefully checked and corrected by hand through appropriate frequency shifts before comparing calculated anharmonic linewidths with the experimental ones. For this endeavor, hydrostatic pressure may be a very important experimental variable. If either wI or y (or both) belongs to a TA branch, the application of pressure will shift oowith respect to w1 + y . Sometimes wo will cross a van Hove singularity of w1 + ~2 in the course of applying pressure, and large changes in r may result. Such changes were predicted to occur for Si starting at about 7 GPa [112], but experiments performed in a DAC did not reveal them [114]. After correcting the calculated frequencies in the manner discussed above, a smooth increase in r with pressure was found. The maximum pressure attained (10 GPa) was not sufficient for wo to reach the van Hove singularity (1141. This is illustrated in Fig. 11. While the existing database for such phenomena is rather meager, a few additional interesting results are found in the literature. Perhaps the most spectacular one refers to the TO Raman phonon of GaP [115]. The Raman frequency qoof GaP falls nearly at the top of a van Hove singularity involving TA + LA phonons around the X point. Consequently, the wo peak is rather broad (r = 15 cm-' and asymmetric. This coincidence has been detuned at a pressure of -9 GPa because wo moves up in frequency while TA + LA stays nearly stationary. Another system that should be investigated in a DAC is CuCI. Even at P = 0, spectacular effects are seen in this material involving resonance of the TO phonons with strong twophonon density of states [116]. Measurements that have been performed at pressures below 0.7 GPa already suggest that interesting effects should appear at higher pressures [117].

4.

PHASETRANSITIONS

As already mentioned, tetrahedrally bonded crystals (with the exception of diamond) undergo phase transitions at pressures within reach of the DAC. In fact, the explosive development of the field of high-pressure phases in the past decade is due largely to the DAC and to the increasing availability of X-ray synchrotron sources for structural studies under pressure inside a DAC [118]. Raman investigations have also been helpful in sorting out the different phases that appear under pressure [119, 1201. Ab initio total energy calculations have also reached a degree of maturity (thanks to the

STRAINS, A N D PRESSURE I N SEMICONDUCTORS 157 3 FHONONS,

t O'A

'

2I

'

6 I 8I Pressure (GPa) 4I

I

'

'

10

FIG. 11. Pressure dependence of the full width at half maximum of the Raman lines in Si and Ge. Filled and open squares refer to increasing and decreasing pressure, respectively. The solid lines represent linear fits. The dashed lines were calculated in Ref. [112]. The dash-dotted line represents the results of the calculation for Si (after Ref. 11141.)

increasing availability of faster and more powerful computers) that enables the reliable prediction of high-pressure structures and the corresponding transition pressures [121]. The calculation of the crystal structure, phonon spectra, and electronphonon interaction of a high-pressure phase led to the prediction (and experimental observation) of superconductivity at moderate T,’s(e.g., -7 K) in a number of such phases [122].

158

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

a. PhaseTransitions: Ah lnitio Calculations

In this subsection we give a few examples of high-pressure phases of tetrahedral semiconductors and discuss the treatment of the total energies of the various phases in order to obtain the transition pressures. For more detail and data on other materials, the reader should consult Ref. [118] and other papers that have appeared in Phys.Stat. Sol( b ) 198 (1996). Let us consider Si, Ge, and Sn at low temperatures (little is known about the temperature dependence of the phase transitions under consideration). In the case of Sn, the low-temperature, low-pressure phase is the semimetallic, tetrahedral one (a-tin or gray tin), which, for P = 0, converts into the metallic @tin structure above 13°C. Semimetallic a-tin transforms at low temperatures for P 3 0.8 GPa into p-tin, a metal that becomes a superconductor below 3.7 K. It is therefore reasonable to assume that Ge and Si will also transform into the p-Sn tetragonal structure at high pressures. The calculated transition pressures are 11 GPa for Si and 10 GPa for Ge, in reasonable agreement with the experimental results. The 0-Sn phases of Ge and Si are superconducting ( T ,= 5.35 and 6.3 K, respectively). These critical temperatures are higher than that of 0-Sn (3.7 K), a fact that follows from the BCS theory on account of the higher Debye temperature. The power of the a6 inifio techniques for calculating the electronic structure, the phonon spectra, and even the electron-phonon interaction has been used to estimate superconducting critical temperatures T, in the p-Sn phases of Ge and Si [122]. Such estimates are rather difficult in general, but in the case of the p-Sn phases they are made easier by adjusting inaccurately known parameters (such as the Coulomb pseudopotential) so as to reproduce the well-known T,of p-Sn at P = 0. In the case of lesserknown high-pressure phases (without superconducting analogue at P = 0), a precise estimate of T, is not possible. One can, nevertheless, estimate McMillan’s electron-phonon coupling parameter A and, in this manner, obtain a qualitative feeling for the possibility of superconductivity and the magnitude of T,. In Ref. I1231 a value A = 0.38 was estimated for the p-Sn manner, A = 3 was calculated for its phase of silicon. In the same ab initio simple hexagonal phase (sh), which exists above -15 GPa. This value of A signals the possibility of superconductivity in the sh phase of Si with T, = 6.3 K (the T,of the j3-Sn phase). Superconductivity has indeed been observed in the sh phase of Si. We show the corresponding T,versus P in Fig. 12 [123]. The decrease from 8 to 3.5 K with increasing pressure correlates with the calculated decrease in A. The experimental determination of high-pressure phases, usually performed by X-ray diffraction, is often not trivial. Most of the transitions are

STRAINS, AND PRESSURE IN SEMICONDUCTORS 3 PHONONS,

9

8

7

sz W

p

6

5

‘.

4

3

1

15 20 Pressure (GPa)

25

FIG. 12. T, as a function of pressure for high-pressure phases of silicon (indicated by horizontal dashed lines). The points represent the measured values with experimental error bars; the two types of points denote different samples. The dashed line is used as a guide for the eye. (From Ref. [123].)

of the first order and, correspondingly, may require adequate nucleation. Hysteresis phenomena, involving supercooling and superheating and also other kinetic barriers to the transformation, are often observed (remember again that diamond is rather “stable” under normal conditions, in spite of being thermodynamically metastable). Therefore, theoretical guidance, in the form of total energy calculations of the various possible structures versus volume, is needed. Typical computational results (obtained for T = 0) are illustrated in Fig. 13 for germanium [124]. The lowest minimum

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

160

-7.74

n

-7.79

E 0 w a \

2

Y

>.

P

a

C

Lu

-7.84

-7.89

1 0.45 0.55

0.65

0.75

0.85

0.95

1.05

Normalized volume FIG. 13. Energies of germanium crystals versus volume. The volumes are normalized to the calculated equilibrium volume of 22.5616 A’. Inset: detailed structure of the total-energy curves near the transitions. The arrows indicate the range of transition volumes for the 0-Sn to simple hexagonal (sh) and the calculated sh to hcp transitions. (From Ref. [124].)

corresponds to the equilibrium volume V,of the diamond structure at P = 0. The volume in the abscissa has been normalized to V,. A lower volume is obtained at the pressure P = dE/dV(where the appropriate units must be used); P is thus given by the slope of the corresponding energy versus volume curve. A first-order phase transition to a higher energy phase is obtained at the value of P at which the tangent to the lowest energy curve becomes tangent to the curve for the p-Sn structure (dashed line in Fig. 13).

3 PHONONS, STRAINS, AND PRESSLJRE I N SEMICONDUCTORS

161

Using the same construction, transitions from the p-Sn phase to other phases are found as shown in the inset of Fig. 13. According to the ab initio calculations these phases are simple hexagonal (sh) and hexagonal closepacked (hcp). The p-Sn phase of germanium is calculated from Fig. 13 to appear at -10 GPa, in agreement with experimental observations [124, 1251. The p-Sn to simple hexagonal transition occurs for Ge at 80 GPa. A transition to an hcp structure is predicted on the basis of Fig. 13 at P = 80 GPa; it has indeed been observed to occur at this pressure. At P = 100 GPa Fig. 13 predicts a transition to an hcp structure, which has not been observed experimentally. Instead, X-ray diffraction in a DAC has revealed a transition to a double hcp (dhcp) structure at P = 100 GPa. More recently [118], a transition from p-Sn to an orthorhombic structure with space group Imma has been found in a small pressure range around 75 GPa. This orthorhombic structure had been predicted by Lewis and Cohen [126]. The orthorhombic structure has a free internal structure parameter and can be considered as intermediate between p-Sn and sh. Similar low-symmetry structures, with hitherto unknown internal structure parameters, seem to be dominant at high pressures for most zinc blende-type materials. Contrary to previous belief, these materials do not exhibit a &%-like phase [118]. The high-pressure phases of Si are roughly similar to those of Ge. Some of the differences that have been observed can be attributed to the presence of 3d electrons in the Ge core that soften the d-wave component of the atomic pseudo-potential [121]. We close this subsection by mentioning that several metastable structures are obtained for Ge and Si at atmospheric pressure after pressurizing them to obtain the p phase and then releasing the pressure. They were reported in 1964 by Kasper and Richards [127] and are referred to as SiIII (bodycentered cubic, 8 atoms per primitive cell), GeIII (simple tetragonal, 12 atoms per primitive cell), and GeIV (body-centered cubic, Ia3 structure). The SiIII phase of Si has been investigated by ab initio theoretical methods and found to be stable with respect to the p-Sn phase. The path through which the SiIII and GeIII form upon releasing the pressure required to keep the p-Sn stable remains unclear [128]: The equivalent phase of C (labeled BC8) has been calculated to be stable between 1 and 3 TPa [93]. Failure to appear in the molecular dynamical simulations of Fig. 6 indicates that it is probably hindered by enthalpy barriers (4 evlatom) [93]. An additional metastable phase has been found experimentally for Si. It is similar to the wurtzite structure, with two interpenetrating hcp lattices (2H; 4 atoms per primitive cell). This structure is obtained by annealing SiIII (obtained as described above) for 24 hours at 250°C [129].

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6. PhononsinHighPressure Phases: Raman Spectra

A b initio total energy techniques can be used to calculate the frequencies and the eigenvectors of high-pressure phases. This has been done by Lewis and Cohen for the high pressure phase (P-Sn, hcp) of Si, Ge, and Sn [120]. Knowledge of the phonon frequencies and the electron-phonon interaction constants is essential in attempts to predict superconductivity and the corresponding T,in the metallic high-pressure phases of these materials [130, 1231. The p-Sn structure is composed of two body-centered tetragonal (bct) sublattices (see inset in Fig. 14). There are thus three sets of optical phonons vibrating along z or along x, y (the latter are twofold degenerate each, as corresponds to the equivalence of the x and y directions in a tetragonal crystal). The vibrations of one sublattice against the other, either along z

!-

l(bl1 , OO

-I

l

20

,

lLo, , 40

I

60

I

i

j

80

Pressure (GPa) FIG. 14. Frequencies of the TO and LO Raman-active phonons of Ge in the B-Sn structure. The dots are calculated results, while the crosses represent experimental data 11201. The inset sketches the tetragonal unit cell of p-Sn. (From Ref. [IZO].)

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS

163

(singlet, labeled LO here) or along x and y (doublet, TO), are Raman allowed. The ratio cla is about 0.55. Consequently the LO phonons can be regarded as mainly bond bending, while the TO ones correspond to a stretching of the bond. Hence the TO frequencies should be higher than their LO counterparts. As shown in Fig. 14, the frequency of the LO modes is nearly independent of pressure, while that of the TO modes increases by nearly a factor of two in the stability range of the p-Sn phase of germanium. The calculated Raman frequencies and their dependence on pressure are in excellent agreement with experimental data, not only for Ge but also for Si and Sn in the p-Sn phase [119, 1201. In the case of Si, the behavior of the Raman phonons under pressure is somewhat different from that shown in Fig. 14 for Ge. The LO modes decrease superlinearly with increasing pressure, a fact that signals the phase transition to the sh or hcp phases (15 and 30 GPa, respectively). In Ge this transition occurs at much higher pressures (-80 GPa). These differences have been attributed to the presence of 3d electrons in the core of Ge. The Raman spectrum of wurtzite-type Si has also been observed on samples obtained after annealing a SiIII specimen at 250°C for 24 hours. An asymmetric peak is found in the measured Raman spectrum with the maximum of 520 cm-’ (the same position as in diamond-type Si), sloping down slowly toward lower frequencies [131]. It has been interpreted as consisting of three components: the highest ones of r,+ and rt symmetry (nearly degenerate) and a lower one, at 502 cm-’, of rq symmetry.

IV. Effects of Strains on Optical Phonons

The effects of strains on the frequencies and the polarization directions of q = 0 optical phonons have been employed as a tool for strain characterization experiments in crystalline systems ever since the first observation of such effects in Si [36]. The method involves Raman (and sometimes IR) measurements of phonon frequency shifts with respect to unstressed samples, and their conversion into strain or stress components via the phonon deformation potentials (PDPs). Of key importance in such studies is the knowledge of precise values for the PDPs. These are obtained from control experiments, in which the frequencies of the phonons under consideration are “calibrated” against well-defined applied strains (or stresses). Both types of experiments can be quantified by using the phonon secular equation. In control experiments the frequency shifts of a particular phonon are measured as a function of applied strain (or stress). These results serve as input in a secular equation that leads to values for the PDPs of the

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EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

phonon under consideration. In characterization experiments the PDPs are taken from the literature and the frequency shifts are measured using the Raman (or IR) spectra; with this input, the secular equation leads to magnitudes and signs for the unknown strain (or stress) components in the scattering volume. If the strains are inhomogeneous, the shifts assume a range of values. As a result, the observed average Raman spectra become, in general, broader and asymmetric, thus leading to information not only about the average strains but also about their spatial distribution. A full microscopic treatment of the problem is rather complex. However, the following general considerations have proved themselves sufficient and have been used for most applications in the last three decades [56, 132, 1331. The basic definitions of PDPs and details for handling the secular equation are discussed next at some length because of the practical importance of the subject. We note that according to the state of the art, PDPs can be calculated ab initio on the basis of the total energies or restoring forces obtained from first-principles electronic band structure calculations [94].

1. PHONON DEFORMATION POTENTIALS We introduce, in a semiphenomenological way, the normal coordinate upassociated with a q = 0 optical phonon of degeneracy p = 1-3. Within the

harmonic approximation and with no strains present, the p-fold degenerate phonon frequency wp is associated with an effective force constant kg that is a second-rank tensor, defined in the subspace of dimensionality p, according to

where S , is the Kronecker delta and (b an appropriate crystal potential energy. The index p has been dropped from k;,w i , and ug for simplicity. Thus, with no strains present, we can write ko= w21,where I is the p X p unit matrix and w 2 the pfold-degenerate eigenvalue corresponding to ko, with eigenvectors along the p unit vectors of any orthonormal basis in the p-dimensional subspace. In the presence of a strain, kochanges to a new force constant k = ko+ Ak, which is no longer proportional to the p X p unit matrix and can have nondiagonal as well as diagonal elements. Accordingly, the p-folddegenerate eigenvalue w2 of ko changes to p new eigenvalues R2 of k, whereby the degeneracy can (but need not) be partially or totally lifted.

3 PHONONS,STRAINS, AND PRESSURE IN SEMICONDUCTORS

165

Finding the new eigenvalues and eigenvectors (strain-modified frequencies and polarizations) will be discussed in detail in Section IV.2. The second-rank tensor Ak can be expanded to terms linear in the strain components as

or, in suppressed index notation, 1

1

Aki= Ki,ej j = 1-6, i = 1,2,6 f o r p = 2 3 1-6

(39b)

K can thus be represented by an i X j matrix (not a tensor!); in general, the square part of the latter is not symmetric upon interchanging the two symmetric pairs afi and KP (see the discussion later). The components KapKp or K , are defined as the strain phonon deformation potentials, or PDPs. The number of nonzero components of Kaaxp depends on the crystal point group and on the dimensionality p of r,, where r,,is the IRR of the phonons under consideration (see the examples in Section 1I.l.b). According to group theory, the criterion for KaaK,, to have at least one nonzero component is that the reduction

includes rl,the fully symmetric IRR of the point group [132]. r(e) is the RR for the strain tensor, identical to T(x)introduced in Section I1.l.a (note that near resonance, antisymmetric components of rpX T,, may also have to be considered; see Section 1I.l.a). It is easy to show that in all cases we have n1# 0 (both factors [r, X T,Is and T(x)include rl,the former being the square of a representation, the latter including the representation of a unit matrix). Thus, the matrix K has nonzero components for all IRRs of all crystal classes, of which as many as n I are independent components. For instance, in the case of triply degenerate phonons belonging to any of the IRRs F,,, Fig, FZu,F2g ( F , , F2) of Oh ( Td)of cubic structures (Section II.l.a), the reduction (40) yields nl = 3, hence there are three independent components of K,which we designate by Kll, K I 2 ,K M in suppressed notation, relative to the crystallographic axes S,.

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EVANGELOS ANASTASSAKIS AND MANUELCARDONA

For threefold-degenerate phonons, K is isomorphic to the elastooptical (The same applies to nondegenerate tensor P [60] since, in general, Kij # Kji. and twofold degenerate phonons of noncubic crystals, except that in these cases the nonzero components of K are fewer than those of P because of the lower dimensionality of the phonons.) However, for the three cubic classes Oh, 0,and T d ,the tensor P has the same symmetry as C.Since we are mainly interested in crystals of Oh and Td symmetry, from now on we shall treat K symmetrywise in the same way as C.Therefore, the matrix form (18) and the transformation algorithm of the appendix apply also to K (no factors of 2 or 4, [see Eq. (14)] are used when converting K ,Ak from 4 to 2 and from 2 to 1 index notation, respectively). Notice that according to Eq. (39b), the strain PDPs Kij are obtained from the expansion of Ak versus strain. We can also define stress PDPs L, from the stress expansion of Ak,by analogy to Eq. (39b). The corresponding sets of components are interrelated through L = K * S. Here we use the definition of Eq. (39b) since it is the strain PDPs that are usually found in the literature. For historical reasons 11331, the three components K l l , K 1 2 ,and K4 in the cubic classes Oh, 0,and Tdare often designated by p .q,and r,respectively. These parameters have the dimensions of 0’. In practice, it is more appropriate to use the contracted tensor notation K,i and keep the PDPs normalized to w2. Such dimensionless parameters Rij usually take values from 0 to 5 for the various threefold-degenerate phonons of different materials and are easier to use. They have a simple physical meaning in terms of a fictitious applied unit strain. We give the definitions of the dimensionless Kijbelow for the triple-degenerate phonons, since we will follow this notation:

Finally, it should be stressed that Kij and K , take different values for the TO and LO versions of polar phonons at values of q sufficiently large to display the full LO-TO splitting. For nonpolar phonons - for example, the F2gphonon in diamond- and fluorite-type crystals - this complication does not exist and we can confine ourselves to q = 0. We will use the labels T and L , for example, K $ = K i / w $ and so forth, whenever it becomes necessary to distinguish between transverse and longitudinal PDPs. This distinction will be further discussed in Section IV.5.a.

3 PHONONS,STRAINS, AND PRESSURE IN SEMICONDUCTORS

167

2. PHONON SECULAR EQUATION Formally, the eigenvalues and eigenvectors of the second-rank tensors koand k can be found by diagonalizing the corresponding p X p matrices, that is, by solving the phonon secular equations

Iko- w211 = 0 and

Jk- R21(= 0

(42)

20 AR

(44)

or, equivalently,

where A

=

R2 - m2

Equation (44) yields the eigenfrequencies of the initially p-fold-degenerate phonons after application of the strain. Specifically, for p = 1 Eq. (44) gives a shift from w to 0,for p = 2 it gives the two perturbed frequencies, and for p = 3 it gives either three nondegenerate perturbed frequencies (complete splitting) or two perturbed frequencies of degeneracy one and two, usually termed the singlet and the doubletphonon components (partial splitting) or, for a hydrostatic strain, a shifted degenerate triplet. In each case the corresponding eigenvectors are obtained from Eq. (43). It is essential that all nonzero strain components be specified before diagonalizing Eq. (39b). Two examples are given next to illustrate the use of the secular equation in cubic and noncubic crystals.

a. CubicCrystals (Oh, Td)

The first example concerns triply degenerate phonons of the cubic classes Oh and T d ,for example, the FZgphonon of Si ( o h ) the , F2 phonon of GaAs (Td), the Fluand FZgphonons of CaF2 ( o h ) ,or the F,, and F2uphonons of SrTi03 (oh).All these phonons will be affected by the same strain in a similar way (except for different numerical values of the PDPs and for LO-TO split phonons). Suppose that E; # 0 (i = 1-6) in the system S,, where K has the simplest possible matrix form (18). Equation (43) can be explicitly written for a general strain by keeping in mind that each of the

168

Kiici

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

+ Kiz(6z + ~ KUE6

K44%

3 )

K44%

K44%

Kiiez + K d e 3 + € 1 ) K44~

-

A

K44E4

Klle3

=O

+ K d E i + €2) - A

The eigenvectors obtained from (45) correspond to the polarizations of the perturbed phonons given in terms of the basis vectors of the S, system. From an experimental point of view, the new phonon components can be most conveniently observed in Raman scattering experiments, since by making appropriate use of polarization selection rules it is possible to observe each component separately. This, of course, is possible only for Raman-active phonons, for example, FZg and T 2 .The components of T2 can also be seen in IR experiments (they are Raman and IR active), those of Flu only in IR experiments (they are Raman inactive; the different eigenvectors can also be separated, at least in part, by using polarized light). The components of F,, (silent mode) cannot be seen in either Raman or IR experiments; they require a different type of experimentation, such as electric-field-induced Raman scattering [133,1341,hyper-Raman scattering [54], or inelastic neutron scattering. Diagonalization of Eq. (45) leads to general expressions for the new . the case eigenvalues and the corresponding eigenvectors in terms of K i jIn of polar phonons, Eq. (45) is only strictly valid for q exactly equal to zero. As mentioned earlier, for values of q sufficiently large that the LO-TO splitting is fully developed, one must use two sets of K i j one : for the transverse phonons, KC, and the other for their longitudinal version, K f ; . The difference between Kf;and KC is related to the dependence of the LO-TO splitting on the strain, which can be translated into a dependence of an effective charge on strain [37]. For polar phonons in cubic materials, the stress splittings that can be achieved before the sample breaks are small compared with LO-TO splittings. Hence, the latter determine the LO eigenvectors, not Eq. (45): The LO displacements take place along the q direction. The problem simplifies considerably if q lies along a highsymmetry direction and so does the strain. In this case Eq. (45) also yields the eigenvectors of (LO-TO)-split phonons under stress [37, 1351. Depending on E, Eq. (45) may not be easily diagonalizable. The secular equation is, in general, a cubic equation whose analytic solution is always possible but cumbersome (numerical solution is often to be recommended). It sometimes happens, however, that the strain is much simpler (fewer

3 PHONONS, STRAINS, AND PRESSURE I N SEMICONDUCTORS

169

nonzero components) if written as Q ’ relative to a rotated system S’ :xl (v = 1-3), for example, one in which the axes x : are parallel to the edges of the sample if it is a parallelepiped with the strain applied to its base. There are then two ways to proceed [136]: (i) From E’ we obtain E in S, by rotation, and then proceed to diagonalization of Eq. (45), as described above. (ii) Equation (43) can be written and solved in any system of axes S’, that is,

where all quantities are referred to S’. In Eq. (46) there are fewer nonzero components E; than in (45), but on the other hand one has to know the necessary components K:ireferred to S ‘; this requires a straightforward computation, which is described in the appendix. In short, it may be simpler to diagonalize Eq. (46) instead of (45), or vice versa, depending on the specifics of each particular problem (and the background of the reader). An inherent advantage in choosing S ,however, is that the eigenvectors are referred to S’ and may turn out to be in a simple geometrical relation to the strain or stress configuration. If, for instance, the strain is produced by a uniaxial stress u and one of the eigenvectors happens to be along the axis a, the corresponding phonon is justifiably defined as the singlet component along a, or simply thesinglet; the other two eigenvectors are either singlets perpendicular to u (e.g., for u 1) [llo]) or a doublet perpendicular to u (e.g., u 1) [Ool]or u 1) [ill]). The possibility of choosing S’ rather than S, was pointed out in Ref. [136] and applied recently by De Wolf etaf. in handling the secular equation for a particular Si device [137]. Using S’, the eigenfrequencies appear as solutions of a quadratic equation, whereas in S, the solution of the cubic equation mentioned above is required. Let us add one final remark about strain characterization based on triply phonon degenerate phonons: According to Eq. (45), written for a nonpolar (e.g., the FZgphonon of Si), there are at most six unknown strain components involved in the shifts and at most three shifts AR that can be obtained from the experiment. Thus, a complete determination of all six strain components from the shifts of the triply degenerate nonpolar phonon is not possible in diamond-type crystals. It has been suggested that the missing information can be obtained from the intensity of the Raman spectra [138]. On the other hand, if the “triply degenerate” phonon is polar (e.g., the Fl phonon

170

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

of GaAs) for q in the region of the full LO-TO splitting, there are six unknown strain components and six shifts. Thus, in principle the strain characterization can be complete, provided all six shifts have been measured from Raman experiments and all six PDPs K ; and K $ are known from control experiments. b. NoncubicCrystals As a second example we consider GaN, a wurtzite-type crystal of current interest. Using Eqs. (9a) and (9b), the criterion (40) gives

Equation (47a) implies that two independent PDPs are required to describe the effect of a general strain on an A l phonon. For one set of E l phonons, however, three PDPs are required according to Eq. (47b). The same reductions hold for BI and E 2 , which have the same degree of degeneracy as Al and E l , respectively [132]. We labeled the two PDPs of a nondegenerate , Equation (43) for either A, or B1 is trivial: phonon A l or B, as ( K 3 1K33).

and Eq. (44) corresponds to a shift

where K X 2 = K j l . Since A l is polar, two sets of (K31rK33)are required: one for the TO and one for its LO counterpart. The values K $ , Ki3were determined recently from low-temperature Raman measurements in strained GaN epilayers grown on sapphire substrates [139]. Only one set of PDPs is needed for the nonpolar phonon BI. The three PDPs of the twofold-degenerate phonons E l (at q = 0) and EZconsist of three nonzero independent components each, ( K I 1 K, 1 2 KI3). , Equations (43) and (44) for either E l or E2 thus become

fll,= 0 + h"/20

ZJ

= 1,2

(49b)

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS 171

where Kll = K Z 2 K , 1 2= K Z 1 K , I 3= K23 # K32rand K66 = ( K I 1 K12)/2. We recall that q, = 2eI2. The new eigenvectors still lie on the x I x 2plane but are rotated relative to x1 and x2 because of the K66Eg term. Two sets of ( K I 1K,I 2 ,K13)are needed for the polar E l , and only one for each set of E2 s(note that in wurtzite-like GaN there are two such sets; one is Galike, the other N-like). Values for K I 1+ K I 2and K13 of the N-like Ez were obtained recently from Raman experiments [139]. The characterization of strains can be more complete in crystals with more than one type of phonons, for example, GaN, than in diamond-type crystals with only one triply degenerate nonpolar phonon. According to Eq. (48a), at most three unknown strain components are involved in the shift of A ,; the same holds for B1. (The components appearing in the shift of B1,however, are the same as those appearing in A , , though their effects are numerically different.) Furthermore, according to Eq. (49a), at most four unknown strain components are involved in the two pairs of shifts of the El phonon (LO and TO); the same unknown strain components are also involved in the two pairs of shifts (N-like and Ga-like) of the two E2 phonons. Presumably, enough of those shifts can be measured experimentally, for example, from appropriate Raman scattering configurations (excluding B1,which is silent). Thus, by combining all the available shifts it should be possible to calculate the values of all four strain components and even to perform consistency checks. This, of course, presupposes that the necessary PDPs are available from control experiments. In situations like the one just described, where there is a surplus of shifts, the information from the strain characterization can be fed back into the expressions for the measured shifts of any phonons with unknown PDPs, thus leading to a set of values for those PDPs.

3. CONTROL EXPERIMENTS

From the previous two examples it becomes clear that strain characterization of a particular crystal by Raman spectroscopy requires separate control experiments for a number of different phonons of that crystal. In diamond-type crystals there is only one choice, the triply degenerate nonpolar phonon at q = 0; in zinc blende-type crystals there are two choices, the TO and LO polar phonons for q # 0. In all cases, the shifts are measured as a function of an externally applied stress u,and the PDPs are calculated using Eq. (45) or (46) adapted to the applied strain. Such control experiments have been performed for a number of materials and involve Raman scattering experiments in various configurations (or sometimes IR reflection

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EVANGELOS ANASTASSAKIS AND MANUELCARDONA

spectroscopy [140]). Crystal bars with several of the following orientations are used:

where u is always along the axis 3, the long dimension of the bar. In the following sections we classify the various standard Raman procedures followed according to the transparency of the material to the laser wavelengths in use. a. Centrosymmetric CubicCrystals: Regionof Transparency We use samples a, b, and c. Using procedure (i) mentioned in Section IV.2.a shortly before Eq. (46), we find the following for the strain in Sc:

With any one of these t w o strains, Eq. (45) leads to partial removal of the triple degeneracy. The new eigenvalues are a singlet R3 = Rs(eigenvector parallel to a)and a doublet R, = R2 = Rd (eigenvectors along any two mutually perpendicular directions normal to a).Part of the shift of each component (AC&,, only observable in combination with a, and ad) is due to the hydrostatic content of u.The total (observed) shifts of the singlet and doublet components can be expressed in terms of ARh and the relative splitting A 0 between singlet and doublet as follows [66, 361:

3 PHONONS, STRAINS, A N D PRESSURE I N SEMICONDUCTORS

173

where AR,, = (ARs + 2ARd)/3 = (a0/6)

(Rll + 2R12)(SII+ 2S12) (52a)

The slopes ( A Q / u ) and (ARdla) for the two directions of u are determined from two Raman spectra, that of the singlet band centered at Rsand that of the doublet band centered at Rd. The values of I?,, - K12and are then obtained from Eq. (52b), while those of I?,, and R12are obtained from R,,- K12,and the value of I?,, + 2R12from Eq. (52a). Furthermore, the corresponding mode Griineisen parameter can be calculated from

where u/3,the hydrostatic content of the uniaxial stress, is negative for compressive stress and positive for tensile stress. (We recall that compressive stress is considered negative.) Since I?,, + K12is an invariant of K in all cubic classes, the Griineisen parameter can be obtained from Eq. (53) for any direction of a;thus, we can obtain extra information that may be used for a consistency check. The necessary Raman configuration for observing the Rsand Rd bands are chosen among the following sample and scattering configurations (given in Porto's notation, for definition see next page):

174

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

The meaning of, for instance, (54a) is as follows (Porto's notation): The incident (scattered) light wave vector k' (k") is along xal (?&) with polarization along xa2 (xa3); the scattering is produced by the phonon at Rd(l)x a l )that , is, the doublet component of the phonon with displacements alongxal. For later use we included the forward-scattering geometries (0"). For cr 11 [Ool], Rd and R, can be obtained from either a (90") or b (180"). For u 11 [ill], R,, and R, can be obtained from c in either 90" or 180" geometry. Clearly, a and c in 90" geometry suffice to yield the necessary slopes from control experiments of transparent cubic crystals. Whatever geometries are chosen, the results should be the same since Eqs. (50) through (53) are tied to the directions of o,not to the Raman configuration used. Complete removal of the triple degeneracy is obtained by a uniaxial stress u 11 [110]. that is, from sample d. The corresponding strain tensor in S, is

Diagonalization of Eq. (45) leads to three singlets at frequencies 0,, R2, and f13 [66,137] such that

where A f l h , and ARcl")are given by Eqs. (52a) and (52b). Instead of Eq. (52a) we must use

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS

175

while Eq. (53) remains unchanged. The three eigenvectors are along X d l, necessary Raman configurations for observand a,are chosen among the following: ing i l l , a;?,

x d 2 ,and x d 3 ,respectively. The

d: 180"

x d l ( x d 2x-d 2 ) x d l :a l ( lxld l )

:a1(11 xdl)

x d l ( x d 3x,d 3 ) y d I

(59a)

and a3can be obtained from the 90" Since all three frequencies .R1,a,, geometry, this sample orientation is unique in that all three PDPs can be obtained from only one sample and only one geometry (90") by proper use of the polarization selection rules (59c). Otherwise, if the PDPs are already available from samples a and c, the information from d can be used for a consistency check. Materials of this type for which control experiments have been performed are diamond [141], Si [142], and the fluorides BaF2. CaFz [143], and SrF2 [144]. Since the line at 1064 nm (1.165 eV) of the Nd:YAG CW laser coincides with the indirect gap of Si at 110 K, complete experiments on Si were performed using that line at 110 K, that is, in the region where it is still transparent (90" scattering geometry). Typical spectra and data are shown in Figs. 15 and 16. An alternative experimental procedure has been applied in diamond (and a-quartz), whereby shock waves along [110] and [OOl] were combined with time-resolved Raman spectroscopy [90]. The time-dependent uniaxial strains applied to diamond along [110] or [OOl] resulted in splittings into three singlets or a singlet and a doublet, in accordance with the results for the d and a sample, respectively. The effects were seen very clearly in near backscattering geometry. Figure 17 shows a sequence of spectra taken at 10-ns intervals, as the pulse crosses the diamond plate along [110]. The splitting into three singlets is clearly visible. Figure 18 shows a similar spectrum corresponding to a longitudinal compressive stress of 15.2 GPa along [110]. All three PDPs can be determined from such data, provided the selection rules (54a) and (59a) are properly applied and the strain profile is precisely known. On the other hand, the high strains present may introduce nonlinearities in both the C and K coefficients.

176

EVANGELOS ANASTASSAKIS

AND

MANUELCARDONA

Si

535

525

ZII [ l l l ]

515

Wavenumber (cm-') FIG. 15. Raman spectrum of Si under 2 GPa along [lll] at 110 K. excited by a Nd :YAG laser. A Hg lamp was used for calibration of the frequency scale. (From Ref. [142].)

b. Centrosyrnmetric CubicCrystals: Regionof Opacity

In such situations 90" and 0" geometries are inapplicable and the measurements can be performed only in the backscattering geometry. Samples b and c can yield all slopes required, according to rules (55a) and (56a); sample d yields only two of the three slopes needed, (fl,/cr and flJa), according to rules (59a) and (59b), thus leading to incomplete information.

Uniaxial Stress (GPa) FIG. 16. Singlet and doublet shifts An,,and had, respectively, of Si versus X 11 [ I l l ] at 110 K obtained from spectra of the type illustrated in Fig. 15. HP: hydrostatic shift calculated from Eq. (52a).

3 PHONONS, STRAINS, AND PRESSURE I N SEMICONDUCTORS

1200

1400 Wavenumber (cm")

1300

177

1500

FIG. 17. Shock waves polarized along [110] in a diamond plate result in a splitting of the triple-degenerate (0") phonon Raman band into three singlets ( w , , w 2 . and w 3 ) . Time-resolved Raman spectroscopy in near backscattering geometry is used [90].The spectra were taken at 10-ns intervals as the pulse traversed the plate. The splittingsbecome more clear when the scattering volume is under maximum deformation.

The first such control experiment was performed on Si in the opaque region (He-Ne laser, 632.8 nm [36]). Later it was repeated with the 647.1-nm line of a Kr+ laser (penetration depth 2.5 X lo3 nm) [145]. In both cases, the possible relaxation of the applied stress near the surface might have affected the value of PDPs because of the small penetration depth. For this reason the experiment was repeated in the region of transparency, where no such relaxation is expected 11421. Germanium [66] and the antifluorites MgSiz and MgSn,?[146] also belong to this category and have been studied using lasers in the visible spectrum. c. Noncentrosymmetric Cubic Crystals: Regionof Transparency

The determination of the six PDPs, KC and Kh, requires a more incisive use of the polarization selection rules (54a) to (54c) through (56a) to (56c) and (59a) to (59e). It is the orientation of the sublattice displacement relative to the scattering wave vector q = k’ - k" that determines whether the phonon is TO or LO [or of mixed character (TO + LO) for large strains]. Accordingly, Eqs. (51a) and (51b) through (53) and (58a) and

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

1320

1340

1360

1380

1400

Wavcnurntwr (cm ' 1

18. Similar spectrum as in Fig. 17 correspondingto a longitudinal compressive stress of 15.2 GPa along [llO]. Splittings as high as 36 cm-' are induced. (From Ref. [90].) FIG.

(58b) can be applied as before, provided one adds the labels T or L as appropriate. For instance, (54a) suggests that the displacement of the phonon &(((xa1) is parallel to q, hence the scattering involves the LO version of the doublet component; its frequency is designated by a,",and Eqs. (51b), (52a), and (52b) are labeled by the superscript L accordingly. The forward-scattering geometry is needed here to measure the frequency flk from b and c samples, which cannot be obtained otherwise. This problem has been approached as follows [147, 371: In forward scattering, the q vector is very small, lying partly in the polariton dispersion region of the TO component [51, 1481. However, if we choose slightly oblique incidence, a small component qo parallel to u will be present. According to (5512) and (56c) the corresponding phonon is polarized along a (it is therefore a singlet) and along qo (a fact that makes it LO); therefore its frequency is In short, the geometries selected for obtaining KC and K $ .are

3 PHONONS,STRAINS, AND PRESSURE IN SEMICONDUCTORS 179

Combining values for these data with Eqs. (51a) to (53) and (58a) to (58b) leads, as before, to (KT,- KT’,Y T ) and ( K f l- K h ,y ~ )that , is, to KT1, K f i K, Z ,K k .

c:180" xcl(xcz, X&c1:

xr1(xC2, x c 2 ) ~:Qfi , 1 + fig

(61a)

Equations (51a) to (53) and (58a) to (58b) then lead to values for ( K L , ( K L ?YL). Using the above approach, control experiments have been performed in GaP [147], GaAs [37], indium-hardened GaAs [149], AlSb [150], and InP [67]. Data for additional cubic transparent materials have appeared in the literature, namely, ZnSe [66] (only for the PDP of TO phonons) and Bi12Ge020[143]. In the latter crystal (class T ) there are more PDP components because of its lower cubic symmetry.

Y T ) and

CubicCrystals: Regionof Opacity d. Noncentrosymmetric

In this case the unique features of forward scattering under slightly oblique incidence (previous case) cannot be exploited; in principle, one might attempt grazing-angle incidence within the 180" geometry in b and c samples in a way that will produce a sufficient component qw that would cause Raman scattering by an LO singlet phonon. However, because of the large index of refraction in the region of opacity, it is difficult to have a sufficiently large q,;for this reason no data have been reported for this geometry. On the other hand, the missing information required to determine 0,"can be taken from the Griineisen parameter y L ,provided its value is obtained from other sources, such as Raman experiments under hydrostatic pressure. Indeed, we can express in terms of yL ;the resulting expression becomes, using Eqs. (53) and (52a) [151]

thus bypassing Eqs. (60c) and (61c). Otherwise, the computation proceeds as before.

180

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

The present approach was followed in GaAs [151] before the data in the transparency region had become available [37], and more recently in InSb [152], where the small energy gap leaves little hope for experiments in the transparency region. The first PDP data on GaAs, GaSb, and InAs were obtained in the region of opacity [66].They were, however, incomplete since the LO components were not measured and the y L value was not invoked.' It should be noted that the counterpart of Eq. (62a) for the TO mode is

In principle, Eq. (62b) can be used if either one of R$ or fl,$ is not directly available or not sufficiently reliable. The same applies also to d samples; from Eqs. (53) and (58b) we find, for either TO or LO,

namely, y can be used to obtain one (and only one) of the three singlet slopes. Such procedures, however, often suffer from error propagation problems. e. Strained Superlattices and Heterojunctions

A different concept of control experiments involves measuring, via Raman spectroscopy, the frequency shifts of variably strained layers of SLs and HJs. By growth, undercritical layers are coherently and homogeneously strained (pseudomorphic growth) to specific levels of strain (see Section V.l) that, in certain cases at least, can be calculated; a collection of such SLs or HJs provides information similar to a macroscopicbar under variable uniaxial stress. This procedure has been followed in Refs. [153] and [154], where (001)ZnS/ZnSe and (001)ZnTe/ZnSe SLs, respectively, were used in an attempt to extract values for Z?,, - K12 in ZnS, ZnTe, and ZnSe without making any distinction between TO and LO PDPs. The analysis for undercritical SLs, and by extension HJs, is described next. The strain or stress components developed in the vth type layer (v = 1, 2) of SLs depend on the relative thickness of that layer ( p y= h,/d, where d = hl + h2 is the width of one period and p 1 + p 2 = 1). This follows 'The same applies to cubic S i c (p-Sic): Using a diamond anvil cell for hydrostaticpressure and an individual whisker microtension apparatus for applying stress along [lll],DiGregorio etal.succeeded in calibrating the shift and splitting of the TO phonon with stress. From the geometries (61a. first) and (61b) combined with Eqs. (51) to (53), they obtained information that can lead to values for -yT and K&. See J. F. DiGregorio, T. E. Furtak, and J. J. Petrovic, J. Appl. Phys. 71, 3524 (1092).

3 PHONONS, STRAINS, AND PRESSURE I N SEMICONDUCTORS 181

directly from the definition of the in-plane isotropic strain E!, in the vth layer of an undercritical SL grown pseudomorphically along an arbitrary direction

This is analogous to Eq. (17) for HJs, except that the substrate lattice constant is replaced by a common in-plane lattice constant of the SL, given by

G ,is the shear modulus of the vth material, a well-defined function of the stiffness constants Ci;. of the material for each direction of growth [155]. Setting p1 = 0 and p 2 = 1 yields results for epilayers (h) grown on much thicker substrates (h, % h). By analogy to the a-, c-, and d-oriented samples (rods) described at the beginning of Section IV.3, we have a-SLs: growth along [OOl] c-SLs: growth along [lll] d-SLs: growth along [110]

We examine these standard orientations separately: (A) a-SL’s: For growth along [OOl], the shear modulus is [155]

The normal strain E: and the pure bisotropic stress a,developed in each layer are given by Eqs. (19b) and (19c,d), respectively. (All carry the index v.) Equations (63a), (63b), and quantities, except all, (19c) provide explicit expressions for E!, and u,in terms of p y Unless . necessary, the index v shall be dropped from now on. The secular equation for a triply degenerate phonon in the presence of a pure bisotropic stress in the (001) plane leads to a singlet and a doublet, similar to a uniaxial stress u 11 [OOl] (sample a). In the uniaxial stress experiments, it was the applied stress u that produced the strain; hence, all results were expressed in terms of u.In the present case, it is the lattice misfit strain that produces the stress; we therefore express the results in terms of €11, which is as-

182

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

sumed to be a known quantity for either layer of a given a-SL, according to Eqs. (63a), (63b) and (64).With a strain of the form (16) and (19b), the secular equation (45) gives

with eigenvectors parallel and perpendicular to xa3, respectively. The hydrostatic shifts and the shear splittings are now expressed in terms of €11:

The hydrostatic shift Ailh is twice that of Eq. (52a), and the relative shift AC! is opposite in sign to the AC!(O0') given in Eq. (52b). (The same results hold for an overcritical SL, but in this case €11 is smaller than given in Eq. (63a).) The slopes l l ( ~ l l ) / ~ l l or R ( p ) /can p be obtained from Raman measurements on a series of samples of different mismatch strains €11 for either layer. For the standard geometry used (backscattering):

a,"

only the phonon of either layer can be measured. According to Eqs. (65a) and (53),the slope of ilk (a3) combined with yL ( y ) taken from other sources can, in principle, lead to the values of K;,and Kf2for polar phonons and of K l l and KI2for nonpolar phonons of either layer. (B) c-SLs: For growth along [lll],Eqs. (63a) and (63b) are valid, with

Furthermore,

3 PHONONS, STRAINS, AND PRESSLJRE I N SEMICONDUCTORS 183

The paragraph preceeding Eq. (65a) holds also in the present case. The secular equation gives

with eigenvectors parallel and perpendicular to xC3,respectively. Again, the hydrostatic shift Aflh is twice that of Eq. (52a) and the relative shift A f l is opposite in sign to Afl(ll1) given in Eq. (52b) found for a bar configuration. The following Raman configurations are allowed in the backscattering geometry:

Both 0: and C4,' of either layer can be measured. Their slopes, together with yT and y L obtained from other sources, can lead to the PDPs K & and K$,. If the phonon is nonpolar, all values of K , can be obtained by combining the results of an a-SL and a c-SL. If it is polar, the values KTl and KTz cannot be obtained from the combined results of an a-SL and c-SL. Additional measurements from d-SLs are necessary. ( C )d-SLs: For growth along [110], we have

while vl, az(# ul), are given by Eqs. (22c,d) and (28b), respectively [155]. In these SLs, the triply degenerate phonon of either layer splits into three singlet components, under the bisotropic strain €11 of Eqs. (63a,b). The secular equation leads to AO1 = WEI~

( c l l + c12 + 2C44)Kll + 2(-clZ -t 2c44)K12 2(C11 + c 1 2 + 2C44)

(724

184

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

with eigenvectors x d l, xd2,and xd3,respectively. The hydrostatic shift, according to Eqs. (58b) and (72a-c), is

The following Raman backscattering geometries are allowed only for scattering by TO phonons:

Both and a,' of either layer can be measured; the slope of fly together with y Tfrom other sources can lead to values for KTland KT2only. Alternatively, both slopes together with yT can lead to all values KT, KT2,and KL. In short, if the phonon is polar, all values K Z L ,KTjL,and K 2 L can be obtained only by combining the results of a-, c-, and d-SLs. The problem with this method is that for each orientation, a set of SLs is required for better statistics and to ensure that the growth is pseudomorphic. Moreover, growing c- and d-SLs is not always possible. On the other hand, the method is uniquely appropriate in situations in which the uniaxial stress method cannot be applied, either because large-size bars of good quality cannot be grown or kept stable with time (e.g., AlAs) or because the bulk material is too soft to withstand the necessary uniaxial stresses (e.g., ZnS, ZnSe). It is noted that both materials of the SL can be investigated simultaneously, and the results can be transcribed to HJs simply by setting h, + h. $ Heterojunctions underHydrostatic Pressure The strains in HJs and SLs can be tuned by applying a variable hydrostatic pressure, provided their bulk moduli differ. Thus, a single sample under pressure can yield the same information as several samples of different nominal strains at zero pressure. For undercritical systems the strains, and correspondingly the frequency shifts, vary linearly with the pressure P. The complete analysis for HJs of the three orientations ([Ool], [lll], [Oll]) has been presented in Ref. [156], following the first experimental observations of Ref. [78] on a (Ool)ZnSe/GaAs heterojunction. The main results are as follows.

3 PHONONS, STRAINS,AND PRESSURE IN SEMICONDUCTORS

185

Let a, and B,be the zero-pressure lattice constant and bulk modulus of the substrate, and AB = B, - B.The in-plane strain in the epilayer of the HJ is independent of orientation and equal to the lattice misfit, that is, €11 = ( a s / a ) - 1. For convenience, we define

= Rll + For example, (100) = C1,, (124) = Cll + 2C12 + 4C4, [122] 2K12 - 2Ku,and soforth. Indices T and L should be added to [a/3y] when treating polar phonons. The corresponding P-dependent relative frequency shifts are listed next.

(A) a-HJs

For P = 0, these expressions coincide with Eqs. (65a,b). The selection rule of (67) continues to hold. Therefore, measuring An:as a function of P and fitting the data to Eq. (75a) leads to two equations from which the two unknown parameters, I??, and K f 2 ,can be determined. (Remember that -6yL = l?;, + 2Rf2.) (B) C-HJs

For P = 0, these expressions coincide with Eqs. (69a,b). The polarization selection rules (70) continue to hold. Measuring AClkas a function of P and fitting the data to Eq. (76a) leads to two equations from which the two unknown parameters, K& and y L ,can be determined. Similarly, measuring AOZ as a function of P and fitting the data to Eq. (76b) leads to K14and y T .

186

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

(C) d-HJs

(e)3 =

-3f [ y

+

2(112)I112l]+g[Y-zT/rzj AB

,ll2]]

(77c)

For P = 0, these expressions coincide with Eqs. (72a,b,c). Since the polarization selection rules (73) continue to hold, measuring An[ as a function of P and fitting the data to Eq. (77a) leads to two equations from which the two unknown parameters Kf,, KT2 can be determined. Similarly, measuring AOT as a function of P and fitting the data to Eq. (77b) leads to two equations with three unknown parameters, Kf,,KT2,and K&. Combining the two measurements leads to values for all three of them. (Remember that -6y, = KT1 + 2Kf*.) In conclusion, all six PDPs of a polar phonon, K&,,, can be determined from combined measurements of Raman backscattering under pressure on the three standard HJs. For nonpolar phonons, only the a- and c-HJs are necessary for obtaining Z?,,,,2+,. In all cases the substrates are assumed to be unstrained by the lattice mismatch, and their phonon frequencies are affected only by the pressure. The method requires strict linearity of the elastic deformation and the piezo-Raman effects. Cui ef al. have obtained data at room temperature under pressure from an undercritical (001)ZnSe/GaAS heterojunction (f = -2.53 X [78]. Figure 19 shows Raman spectra from the HJ (solid line) and from a bulk sample of ZnSe (dashed lines). The spectra are taken at ambient pressure (a), at the critical pressure P,, = 21 kbar (b)-where the unit cells of epilayer and substrate are cubic and equal in size-and above P, (c). Notice that the difference of the phonon frequencies is zero at P,, and nonzero and opposite in sign above and below P,. Figure 20 shows the frequency differences of Fig. 19 as a function of P. The P-dependent frequencies measured from the strained epilayer and from a chip of bulk ZnSe are w ( P )and w,,(P). respectively. The inset shows the observed differences

3 PHONONS, STRAINS, A N D PRESSURE IN SEMICONDUCTORS 187

-

: epilayex

I

I

270

28 0

I

I

260

270

I

I

250 260 PHONON FREQUENCY ( c n i ' ) FIG. 19. Raman spectrum of the LO phonons of ZnSe obtained from an undercritical (001)ZnSe/GaAs HJ (solid lines) and from a bulk ZnSe sample (dashed lines). At the critical pressure of 21 kbar (b), the unit cells of the epilayer and the substrate are cubic and equal in size, and the corresponding frequencies coincide (tetragonal distortions inside each layer are expected to be small). (From Ref. [78].)

o(P) - w(0) for epilayer and bulk ZnSe. The data for the epilayer are fitted by the linear function o ( P ) - o(0)

= 3.3P

with the frequencies in cm-' and P in GPa. From this we obtain

(784

188

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

0

4

Pressure (GPa)

a

FIG.20. The frequency differences of Fig. 19 as a function of P. - w ( P )- w b ( P ) where , o(P)and wb(P)correspond to the

Aoc-b

strained epilayer and bulk ZnSe. Inset: o(P)- w(0) for epilayer and bulk ZnSe. (From Ref. [78].)

where wb(0)= 252 cm-'. The fitting value for the measured difference w(0) - wb(0) is 1.1 cm-' (Fig. 20). From Eqs. (78b) and (75a), the values of Kfl and Kfz listed in Table I11 were obtained [156]. The listed value of y Lwas obtained from the data for bulk ZnSe shown in the inset of Fig. 20. Hydrostatic pressure data [157] have given y T = 1.7 and y~ = 1.15 (note that for most zinc blende-type materials y T > y L ;see Section 111). The values of KT,, KT2,K&, and y Thave been obtained from measurements under uniaxial stress [66]. No data are available from c- or d-HJs; thus K& is not available. The same applies for the (001)ZnTe/InAs heterojuncwhich was studied in a similar manner under tion (f = -4.9 X pressure ( P , 5.5GPa) at 13 K [158]. The analysis leads to the values of Kf,.K f z and , yT listed in Table 111.

-

g.

Far-Infrared Spectroscopy: PolarPhononsinCubicMaterials

Fourier transform IR spectroscopy has been used in the past to measure frequency shifts of TO and LO phonons under uniaxial stress in a and c samples of cubic crystals like GaAs [65,140] and InP [67]. The polarization selection rules are easier to obtain than for Raman spectroscopy: Each

189

3 PHONONS,STRAINS, AND PRESSURE I N SEMICONDUCTORS TABLE 111

NORMALIZED DEFORMATION POTENTIALS AND MODE G R ~ N E I S PARAMETERS EN FOR THE RAMANPHONONS OF DIAMONDA N D ZINCBLENDE-TYPE CRYSTALS (SEMICONDUCTORS) Material

0

Si Ge Diamond GaAs

521 300 1332 269 (292) 268 (291) 304 (351) 317 (338) 227 224 (233) 218 220 (241) 204 (251) 179 (208) 179 (190) 366 (403) 1063 (1305)

GaAs :In InP AlSb GaSb InAs ZnSe ZnTe InSb GaP

BN

R,I

R,z

K44

Y

- 1.83 -1.47 -2.81 -2.40 (-1.7) -2.35 (-1.55) -2.5 (-1.6) -2.1 (-1.6) -1.9

-2.33 - 1.93 -1.77 -2.70 (-2.4) -2.65 (-2.40) -3.2 (-2.8) -2.6 (-2.6) -2.35

-0.71 -1.11 -1.90 -0.90 (-0.55) -0.55 (-0.15) -0.5 (-0.2) -0.7 (-0.3) -1.1

1.08 0.89 1.06 1.29 (1.09) 1.27 (1.05) 1.48 (1.19) 1.21 (1.15) 1.10 (1.21) 0.85 1.21 (1.06) 1.90 (0.92) 1.60 (0.99) 1.44 (1.17) 0.87 (1.07) 1.13 ( 1.20)

-

-

-

-0.95

-2.10 -4.00 (-2.28)

-0.76

-2.75 (-0.94) (-2.30) -2.45 (-1.72) -1.35 (-1.45) -2.90 -

-

(-1.70) -3.04 (-2.65) -1.95 (-2.5) -1.95 -

-

-0.43 -

-0.54 (-0.22) -0.6 (-0.5) -2.45 -

Ref.

Note.Values in parentheses refer to LO phonons. Frequencies in cm-'. Temperature 300 K, except for Si (110 K) and ZnTe (13 K).

reststrahlen spectrum, properly analyzed, yields both TO and LO frequencies. (Note that under normal incidence, photons couple to TO but not to LO phonons. The LO frequencies can, however, be obtained through fitting of the reflectance spectra, a delicate procedure for the degree of accuracy required.) If the IR radiation is polarized parallel to a,the measured frequencies are a,' and a,"; if polarized perpendicular, they are and a,".Since the photons do not couple to the LO phonons, the q vector corresponding to the LO frequency extracted from the reflection spectrum bears no relationship to the q of the photon. It is simply parallel to the IR electric field. Thus, all four slopes are obtained for each sample from one set of spectra as a function of u.Combining these data with Eqs. (51a,b) through (53) leads, as before, to all six PDPs K?L and the Gruneisen parameters yT,L[65, 671. With a d sample, incidence along x d lleads to

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

190

03" and a,'.",and incidence along x d z leads to a,'." and a,'.".Combining these data with Eq. (58a) yields all six PDPs K$" and the Gruneisen . parameters Although the technique is simple to apply and the volume probed extends deeper than in Raman backscattering, it has not been employed as extensively as Raman spectroscopy. Recently, far-infrared absorption was used to study the strain dependence of the TO phonon frequency in InAs quantum wells (QWs) [68]. Similarly,(CdTe),(ZnTe), SLs of various thicknesses (n = 2-5) have been measured by far-infrared reflectivity in order to determine the effect of thickness on the strains (through the measured frequency shift) [69]. The measured spectra are shown in Fig. 21; the TO frequencies of both constituents appear shifted by different amounts, thus demonstrating the presence of thickness-dependent strains in both layers. h. Fibers underTensile Stresses

Stretched fibers of different materials have been studied through Raman spectroscopy. In the study of Ref. [92], the doubly degenerate phonon E2g of graphite at 1580 cm-' was observed to shift linearly with strain and to split into two components; numerical values of the PDPs were obtained from the observed slopes. A tabulation of all PDPs available up to 1977,including noncubic crystals, can be found in Ref. [56]. Updated values, mainly for semiconductors, available up to the end of 1989 were included in Ref. [136]. We further 1.3

1.2 1.1 1 b '5 0.9 0.8 E 2 0.7 0.6 0.5

3

0.4

0.3

120

130

140

150 160

170 180

190 200

210

Frequency (cm-') Fro. 21. Far-infrared spectra (lines exhibiting noise, T = 10 K) of CdTetZnTe SLs for n = 2-S monolayers at near-normal incidence. The arrows show the TO phonon frequencies obtained from the fitted spectra (smooth lines). (From Ref. [69].)

3 PHONONS, STRAINS. AND PRESSURE I N SEMICONDUCTORS

191

update this last tabulation by incorporating whatever data were available to us up to the end of 1996 (Table 111).

4. THEORETICAL MODELSA N D TRENDS OF PHONON DEFORMATION POTENTIALS The phonon deformation potentials, like the third-order elastic constants, can be regarded as anharmonic parameters. A number of theoretical models have been used to quantitatively describe and calculate most harmonic (second-order elastic constants, zone-center phonon frequenparameters cies, and the complete phonon dispersion relations): Keating’s twoharmonic-parameter model [162,163], based on a valence force field (VFF) model [164] that includes bond-bending and bond-stretching forces; its extended version [165]; and a number of other semiempirical models (e.g., the band charge model [166] and Cochran’s shell model mode [167]) have all been reasonably successful in reproducing a large number of harmonic properties. Among these harmonic parameters, we mention the internal stress parameter 5 [168], which determines the relative displacement of the two sublattices upon application of a uniaxial stress along [ l l l ] . Rather little reproducible experimental information is available for 4‘ [169], so we must often rely on theoretical estimates [142, 170-1731. The number of semiempirical parameters increases when trying to account for anharmonic properties [142, 1741. In this case recently developed ab initio techniques, based on calculations of the total energy using the electronic band structure obtained with the local density approximation for exchange and correlation, have been considerably successful [168,173-1811. Berry phase techniques that enable the determination of dynamical charges (responsible for the LO-TO splittings) have been recently developed and have already achieved considerable success [182]. a. Theoretical ModelsforthePhonon Deformation Potentials

An early model [133] based on quasi-harmonic approximations with only nearest-neighbor bond-stretching forces and the corresponding anharmonicities generated by a Morse potential failed to describe the measured PDPs of Si. An early generalization of the harmonic VFF model that included anharmonic parameters provided considerable insight into the possible origin of the signs of K,,- K I zand K, but failed to reach quantitative agreement with experiments using a reasonable number of anharmonic parameters [66]. Early anharmonic VFF models were used to fit the third-

192

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

order elastic constants [178]. They turned out to be insufficient to represent the PDPs, which became known later. A simultaneous fitting of 22 experimental quantities with harmonic and anharmonic constants using a generalization of Keating’s model with 11 fitting parameters [174] did not lead to acceptable values for the PDPs of Si. A modified Keating model has been used in Ref. [179] (and a bond charge model in Ref. [loo]) to find analytical expressions for the PDPs of both nonpolar (Si, Ge) and polar (GaAs) crystals. The model includes strain-dependent force constants, and the necessary strain-modifier parameters are obtained using the available experimental values of PDPs as input parameters. Explicit expressions for p, q,and r as a function of the anharmonic VFF parameters are given in Ref. [179]. A procedure that makes the best use of available experimental information and theoretical models was introduced in [142] and applied to Si and diamond in two steps. In the first step, the expressions of Keating’s twoharmonic-parameter model [162] for the two harmonic force constants a (bond stretching), /3 (bond bending) and the internal strain parameter 5, were simultaneously fitted to accepted values for the second-order elastic constant C, and the zone-center phonon frequency. The values of [obtained in this way compare well with the experimental values although, as is well known, the frequencies of the TA phonons at the X point do not. The three parameters (Y, /3, and 5 were thereafter used as fixed constants. In the second step, Keating’s three-anharmonic-parameter model (7 for bond stretching, 8 for bond bending, and Z for change in bond bending produced by bond stretching) was extended to include two additional VFF anharmonic force constants (7 and 3) and used to express the six third-order elastic constants cijk (known from experiments) and the three PDPs in terms of the five anharmonic constants 7,F,E, 3. Through fitting to the cijk values it was possible to obtain values for these five anharmonic constants and, subsequently, for the PDPs of Si that were in quite good agreement with the experiment. For diamond, the second step was modified because a complete set of C,jkwas not available. The experimental values of three hydrostatic pressure coefficients were used instead, together with the experimental PDPs, to obtain values for 7,F, E, 7,3 and Cijk. As already mentioned in Section IV.4, ab initio techniques have been recently developed that allow the determination of the total energy of a crystal on the basis of the calculated electronic band structure. The only delicate point of these first-principles methods is the treatment of manybody effects, such as exchange and correlation among electrons, by means of the local density approximation (LDA). This approximation treats the exchange and correlation potential seen by any given electron as a local function of the total electronic density. The atomic coordinates enter the

s,

3 PHONONS,STRAINS, AND PRESSURE IN SEMICONDUCTORS

193

calculations as free parameters, and therefore interatomic force constants can be calculated toanyorderofanharmonicity basically by taking derivatives of the total energy with respect to atomic positions. Rucker and Methfessel have recently performed ab initio calculations for Keating model parameters and investigated their scaling with the lattice parameter for Si, Ge, and C [183]. These results are very useful for estimating vibrational frequencies of crystalline Si-Ge-C alloys. A6 initio calculations of the splittings that correspond to R,,- KI2 and karas well as y, have been reported in Refs. [175] and [176] for Si and in [94] for diamond. These ab initio results are in reasonable agreement with the experimental data. No a6 initio work appears to exist for the PDPs of polar crystals, with the exception of a report concerning BN, where an ab initio pseudopotential linear-combination-of-atomic-orbitals method was used to compute strain parameters [180]. When properly interpreted, these strain parameters can yield reasonable values for the TO-phonon PDPs of BN [160], which are in accordance with the general trends of PDPs discussed in the next subsection (no experimental values for the PDPs of BN exist at present).

b. Trendsof thePhononDeformation Potentials Some general trends can be established for the PDPs from the expressions obtained in Ref. [142]; in simplified form, they can be written as

where p is the reduced mass of the primitive cell, and the anharmonic parameters ?I and 3 have been ignored as having no influence on these qualitative arguments [160]. A comparison of the diamond and Si data [141] yields the scaling behavior 171 a,52.9 and IZl a$.5, while from Ref. [66] m2p ao3.Equations (79a,b) become

-

-

-

where R and Q are positive factors, common to all members of the diamond-zinc blende family; they represent the anharmonicity of the va-

194

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

lence forces. The first terms in the right-hand sides of Eqs. (80a,b), equal to one, are geometric contributions related to the changes in the bond angles induced by the strain. The balance between these geometric terms and the anharmonic contributions (proportional to Q and R, respectively) and BN [181] determines the sign of El, - K,,and K4. In diamond [141] 5 0.12, whereas for all other group IV and III-V materials that do not This contain C, B, or N (first row in the periodic table) l 0.55 [172]. difference arises from the strong covalency of the bond between first-row elements, which implies that a and fl are of the same order, while for the other materials (Y + fl (strong covalency implies strong rigidity of the tetrahedron formed by the bonds). On this basis, the R term of Eq. (80b) varies little from material to material [changes in ( 1 - l )compensate the changes in a;’]. Because the K, term is known to be negative in diamond, we conclude that the anharmonic term prevails over the geometric term and gives rise to negative values of Ku of a larger magnitude for diamond and BN than for materials not involving first-row elements. In the case of El, - K12,because of the strong dependence on a. the anharmonic term may or may not dominate over the geometric term and the sign can be either positive or negative. In diamond and BN the anharmonic term dominates and the sign is negative. For the other materials, with much larger a,,, the opposite situation occurs and the sign is positive. It is worth noticing that, with the exception of ZnTe, all entries of Table 111 follow these qualitative rules. The PDPs of ZnTe given in Table I11 should thus be remeasured.

-

-

POTENTIALS 5. OTHERUSESOF PHONONDEFORMATION The phonon deformation potentials are anharmonic microscopic parameters that can be linked to other microscopic or macroscopic properties of the crystal, as demonstrated in the three examples that follow. a.

Effective Charge: Photoelastic Coefficients

This subject concerns only IR-active (polar) phonons of cubic crystals that, as discussed in length in previous sections, exhibit independent sets of PDPs for TO and LO phonons. The connection of these PDPs with other microscopic parameters can be established through the well-known relation

3 PHONONS,STRAINS, AND PRESSURE IN SEMICONDUCTORS

195

where e* is the TO phonon effective charge, V, the unit cell volume, and E , the optical dielectric constant at high frequencies (as opposed to the static dielectric constant E, at zero or low frequencies). Differentiating Eq. (81) with respect to a strain yields the following relations [37, 1841

where the -1 term in Eq. (82a) is absent for ij = 44,55, and 66, and SE E, - E,. The new coefficients introduced here are

kfl= ( ~ E , J ~ E ) (photoelastic ~ ~ tensors)

k ;=

&Iij = ( 8 In e*/ae)ij (effective charge strain tensor)

=

(83a) (f33b)

all being fourth-rank tensors, similar in symmetry to the elastooptical tensor

P.Physically speaking, the difference K b - K : reflects the strain dependence of E,, E ~ and , e*.By analogy to the mode Griineisen parameter y given in Eq. (53), a similar parameter y* can be defined for the volume dependence of e*:

a In e*

=

-(fill + 2fil2)/3,

(844

where P is the magnitude of the compressive hydrostatic pressure. Thus, experimental knowledge of all six PDPs can lead to the values of &Iij from Eq. (82a) and of kfjfrom Eq. (82b) [67, 371. The only problem is that k ; are not always accurately known. On the contrary, y* can be obtained from Eq. (84a) in terms of &I or ,, from Eq. (84b) provided k ;are available. No ab initio calculations for the shear components of f i iexist. j Ample experimental evidence suggests that y* is negative for most zinc blende-type semiconductors [67, 37, 161, 1851; that is, e* decreases with increasing P. One can make the reasonable assumption that e* depends on the bond length L , which decreases with increasing P. Under this assumption one can conclude that e* increases with increasing L, except for Sic (see Table IV). Extensive ah inifio calculations of the dependence of e* on volume have been published in Ref. [186]. The corresponding Griineisen parameters y* are indeed negative for a number of 111-V compounds but positive for Sic

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

1%

TABLE IV CALCULATED AND MEASURED VALUES OF THE GRONEISEN PARAMETER FOR THE TRANSVERSE EFFECTIVE CHARGE OF A NUMBER OF ZINCBLENDE-TYPE SEMICONDUCTORS

YT.k

Y .*.p e&

GaAs

AlAs

AlSb

GaP

InP

GaN

Sic

SiGe

ZnSe”

-0.82 -0.73 2.18

-0.56 -0.94 1.91

-0.69 -0.49

-0.55 -0.59

2.04

2.55

-0.14 2.70

+0.26 +0.67 2.70

+0.43

-0.88’ 2.206

-0.04 -0.26 2.03

0.08’

Note. Data from Refs. [ l a ] and [191] unless otherwise specified. Obtained from yLo = 1.18 and y~,-, = 1.47 calculated ab initio in Ref. [lo51 From Ref. [190]. From Ref. [189].

a

[187,188] and the hypothetical ordered SiGe compound (in the zinc blende structure): see Table IV.

b. Electrostriction Coefficients An interesting connection has been shown to exist between the PDPs of triply degenerate phonons in polar or nonpolar crystals and the electrostriction tensor y [192, 1931. The three independent components of y for crystals of Oh,0, and Td symmetry have been shown to be y11 = yllll =

[Sll(kL - E,) + 2S12(kh + E,)]E012

Y M = 4Yuz3 = S M ( ~ &- Ed%

(854

(85c)

where kfjare related to K $ L through Eq. (82b), and e0 is the permittivity of vacuum. Contrary to the cases of S and y, the 112 and 114 factors are not necessary for the conversion of the ksindices. The ksterms in the brackets of Eqs. (Ma-c) represent the dielectric response of the material, whereas the remaining terms originate from Maxwell’s stress tensor [193]. As an example we consider GaAs: Using E, ( e m ) = 12.9 (10.9), kyl,12,44= 17.5, 8.7, 8.5, and the K T , Lvalues from Table 111, we find from Eq. (82b) that k l l . ki2,k& = 29.7, 14.2, 14.6 and from Eqs. (85a-c) that yI1,y12, yM = -(0.18, 6.8, 2.5) X lo-’’ m2/V2,respectively. c. PDPs for Polycrystaltine Materials

We consider the PDPs for triply degenerate phonons of randomly distributed polycrystallites of cubic materials, polar as well as nonpolar. The

3 PHONONS, STRAINS, A N D PRESSURE I N SEMICONDUCTORS

197

computation for relating their PDPs to those of the corresponding bulk materials proceeds in two steps [194]: 1. Through an appropriate averaging procedure, the elastic constants C$ of the polymaterial are expressed in terms of those of the bulk material. The so-called Voigt-Reuss-Hill (VRH) average is adopted here. According to this computation,

where CF(R)are the Voigt (Reuss) averages given by

c,",= CII

-

cp, = c;

- 2C215C'

2c/5

+ c/5

c y 2 = c1*

c;",= cy, + C2/5C'

C& = (cp,- c 9 / 2 = C& - 3C2/10C'

(87b)

with

2. The PDPs for the polymaterial, K$, are expressed in terms of the bulk values, Kij, using the same averaging procedure. Since both C* and K* are fourth-rank tensors of the same symmetry, we have

where KC, K, and K' are defined by analogy to Eqs. (87a) and (88). The Reuss average is obtained through a more complicated procedure, which finally yields

198

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

Lf is the Reuss average of the stress PDPs L,, Section IV.1:

introduced in

where L = L , , - L I 2- LM,and

Thus, starting from Eq. (92) backward, the PDPs of the polymaterial can be calculated in terms of the bulk values Kijand C,.Based on the values for silicon given in Table I11 and C,,= 167.7, CI2= 65.0, and C , = 80.4 GPa at 80 K [195], the results for and R$ of poly-Si are

eij

Ljj= (-4.0, -8.9, -8.8)

e$= (-10.0,

-5.9, -4.1)

X

X

GPa-' GPa-'

K $ = (-2.5, -2.0, -0.3) Similarly for poly-diamond with C,,

=

(934 (93b) (93c)

1076, 125, 575.8 GPa at 80 K [196],

t,j = (-2.3,

- 1.2, -3.3)

x

GPa-'

(944

e$ = (-3.2,

-0.8, -2.5)

X

GPa-'

(94b)

K$

(-3.9, -1.2, -1.3)

(94c)

Application to strained polysystems by solving the appropriate secular equation is straightforward. In a bisotropically strained polylayer with €11 known from independent sources, the normal strain and the stresses are the same as in a crystalline layer grown along [Ool]. The results are identical to those of Eqs. (19b,c) except that all parameters should be replaced by their starred counterparts. The secular equation (45) is valid with only diagonal terms, and with K,,replaced by K $ . The phonon frequency splittings are those in Eqs. (65a,b) and (66a-c). If a strain is applied externally on a bulk polysystem, the general formalism based on Eq. (45) and K $ should be followed. Here the definition of axes is arbitrary.

3

PHONONS, STRAINS, A N D P R E S S U R E IN SEMJCONDUCTORS

199

V. Strain Characterization of Heterojunctions and Superlattices The number of compounds used in the fabrication of SLs and HJs has increased in recent years and includes combinations and alloys of group IV, and III-V and II-VI diamond- and zinc blende-type compounds. Many optoelectronic applications rely on the properties of such tailor-made systems. Depending on the conditions of growth and the thickness of the layers, such a system may be substantially strained, owing to lattice or thermal mismatch or both (thermal mismatch results from differences in the thermal expansion coefficients; see Section II.3.b). The strains depend on the direction of growth of the layers, represented by the unit vector N , and can have profound effects on the lattice dynamics and the electronic band structure of the system, with concomitant consequences (positive as well as negative) on its performance. In addition, the strains may induce strong piezoelectric (PZ) fields that always point along (or opposite to) N . Both strains and PZ fields can be tuned through a hydrostatic pressure or variable temperature. Whereas the vast majority of systems investigated correspond to [001]grown layers, advancements in the technologies of epitaxial growth have made it possible to fabricate systems grown along [111], [110], or even lower-symmetry directions (e.g., [113], [120], [211]). In short, all five parameters-strains, PZ fields, pressure, temperature, and direction of growthcan influence the performance of a system and be the object of fundarnental studies. 1. ELASTIC AND PIEZOELECTRIC CONSIDERATIONS IN HETEROJUNCTIONS AND SUPERLATTICES In this section we briefly review the elastic and PZ aspects of SLs and HJs, including nonzero pressures and variable temperatures. For detailed discussions on these subjects the reader is referred to Refs. [61,76,77,155, 156,197,1981, where the elastic and PZ properties of undercritical HJs and SLs grown along arbitrary directions are examined. The approach is based on the use of Hooke’s law for a force-free elastic continuum with an expression for the in-plane strain, consistent with pseudomorphic growth. An alternative procedure [19Y] makes use of a constraint equation for the unit transition vectors under minimal energy, reaching identical results. a.

Definitions

We consider a general direction of NII[13m3n33. Both layers ( v = 1 , 2) are of cubic materials of the diamond (0,) or zinc blende ( T dPZ) , family.

200

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

For HJs no index is used for the layer, and v = s for the substrate ( v will often be omitted for simplicity),while a ,and h,are the bulk lattice constants and the thicknesses of the two layers, respectively. The lattice misfit is defined asf = (a2/u1)- 1.The SLs are in a free-standing state. The volume of the layers is free of misfit dislocations, and the uniform straidstress component arrays are those prescribed by the elastic continuum theory. The system is subcritical, that is, the individual layer thicknesses are smaller than the corresponding critical thicknesses h,,(h,, is the critical thickness beyond which misfit dislocations start to appear in the corresponding layer). Primed and unprimed components refer to the systems S' and S,, respectively. The definitions of Eqs. (17), (21), and (63a) and those in the appendix continue to hold. Several parameters needed throughout this presentation are introduced here in the order of appearance in the applications under discussion. The parameter

A = CIIC&+ (CC4/2)(110)(1

-

T33) + C2(121)(13m3n3)*

(95)

characterizes each layer grown along a specific N , and is easily computed in terms of Cij and N. The factors C and T33are defined in the appendix, and the notation (121) etc. is explained in Eq. (74). Within each layer, the in-plane strain €11 is isotropic, given by Eq. (63a). (Slight anisotropies in low-symmetry directions N have been shown to have negligible effects [200].) The retrugonal distortion of the layer is defined as [61]:

Computing A€ requires prior knowledge of €11. However, the reduced tetrugonu1distortion AC is readily computed from N a n d Cijof the SL or HJ layer and is thus independent of €11. The so-called shearmodufusof the layer is a positive-definite quantity defined as [1551 G = 3B(3 - AS)

(97)

G enters the expression for the common in-plane lattice constant of the layers

3 PHONONS, STRAINS, A N D PRESSURE I N SEMICONDUCTORS 201

which is equivalent to Eq. (63a). Equation (98) applies to the free-standing state only and can be derived from the requirement of minimal elastic freeenergy density. If the SL is grown on a buffer (with parameters hb, G b , and ab), the following three possibilities exist: The S L is grown incoherently on an unstrained buffer. In this case the SL remains in its own free-standing state and Eq. (98) continues to hold. The SL is grown coherently on the buffer and hb 2 h,. The buffer is also strained as if it were a third constituent of the SL; equation (98) is valid with the terms hbGbaband hbCbadded to the numerator and denominator, respectively. The SL is grown coherently on the buffer (as in case 2 ) but with hb 9 h,. The buffer is practically unstrained and imposes its lattice constant to the SL, that is, all = a b . For simplicity, we shall consider only case 1 from now on. Analogous definitions apply to HJs. b. Strains and Stresses in Superlattices The results of the elastic continuum theory for the strain and stress components in layer v = 1 are as follows [61]:

202

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

where T,j are defined in the appendix. The same expressions hold for the layer v = 2 provided the numerator G2h2in Eq. (99a) is replaced by - G l h l . The lattice constant normal to the plane is different for each type of layer and can be expressed as [155]

It should be emphasized that the strain components in Eqs. (99a-d) refer to the system S’ of a superlattice grown along NJlxj.For completeness we give below the strain components expressed in the system S, [197]:

3B~11 --

A

(102)

Applications of these results have already been given in Section IV.3.e for SLs grown along [Ool], [lll], and [llo]. We give below the main results for some less usual directions of growth N [61, 155, 1971:

where the definition (112) etc. is givcn in Eq. (74). (B) N = [ 120]/%'3, with x ; (1 [Ool] and x i (1 [2iO]lG: (104a)

3 PHONONS. STRAINS. A N D PRESSURE IN SEMICONDUCTORS 203

(C) N

=

[113]/V%, with x; )I [332]/%%,

~ $ 1 1[ i l O ] / f i :

Ar = (3B/A)(C& + 38CC4/112 + 27C2/113)

(105a) (105b)

This analysis is readily applicable to strained HJs (thickness h )grown on unstrained substrates (thickness h,)in any direction N . For the case h,9 h the results are €11

=f

E'

=f- AE = f ( l

-

AC)

( 106a)

where Aa = a, - a is the absolute lattice misfit. Notice that the in-plane lattice constant and strain are completely independent of N . Numerical applications of these results can be found in Ref. [155].

c.

Piezoelectric Fields

Strained piezoelectric layers in SLs and single- or multi-quantum wells with polar constituent materials can sustain piezoelectric fields [201]: This effect vanishes in centrosymmetric crystals. A number of theoretical [2012031 and experimental [204-2111 investigations dealing with this problem have appeared in recent years. Depending on the magnitude of the strains and the PZ constants, the fields may reach significant values, exceeding los V/cm, provided the concentration of mobile carriers is low enough to keep these fields unscreened. In the presence of PZ fields the overall behavior of the layered system is modified. Changes in the electronic band structure are among the most important consequences, as demonstrated by linear and nonlinear optical techniques [204-2091. Furthermore, the PZ fields are expected to affect the degeneracy and frequency of the long-wavelength optical phonons in a way similar to the effects induced by misfit strains [156] (note that the PZ fields must belong to the same IRR as the strains that cause them).

204

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

According to the definition of the PZ effect as a polarization proportional to an applied straidstress [60], this strain-induced polarization can be written in terms of the strains as (in the S' system) [198]

(108c) where a polar cubic crystal of either the Td or T class has been assumed; in such a crystal, e represents the only independent component of the PZ tensor. The factors TA,," are defined in the appendix. For ideal heterointerfaces, only P; gives rise to a PZ field E; and only Pll, the in-plane component of P,gives rise to a PZ displacement Dll. The electrostatic equations applied to each layer yield [201]

where K~ is the static dielectric constant of the layer (relative to vacuum). Equations (109a,b) combined with Eq. (108c) directly provide the nonzero components of the PZ field and displacement for an arbitrary direction of growth. The following general rule becomes clear from Eq. (108c): PZ fields in ideal heterointerfaces are induced only for those directions of growth in which all three direction cosines, I,, m3,and n3,are nonzero. Such directions are defined as polar axes. In real SLs, however, the heterointerface roughness or the steplike faces in low-symmetry SLs may induce in-plane PZ field components due to PI1 [212]. Photoluminescence and photoreflectance spectroscopy are the usual techniques for observing PZ effects in SLs, and MQWs. A few distinct Raman experiments have been reported, however, where the PZ effects manifest themselves through the appearance of forbidden Raman scattering. Specifically, the Frohlich interaction of LO phonons with electrons in the

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS

205

presence of a PZ field induces scattering by the LO phonons in configurations where such scattering is normally forbidden [206, 209, 2131. The situation is illustrated in Fig. 22: The Raman spectra are obtained from a [211]-orientedGal-,In,As/GaAs MQW ( x = 0.15) far from resonance (top) and near resonance (bottom) [209], under intense (solid line) and weak (dashed line) illumination. For the scattering configuration used, only the TO phonon peak should appear; this is indeed the case away from resonance, regardless of laser intensity (top, spectra in Fig. 22). Because of the presence of a PZ field, scattering by the LO phonon appears near resonance [213] but only under weak laser intensity (bottom, dashed line). Under strong intensity, the photoexcited carriers screen the PZ field, thus weakening the electric-field-induced scattering by phonons.

2. PRESSURE AND TEMPERATURE DEPENDENCE OF STRAINS Superlattices and heterojunctions show interesting changes of their elastic state when placed under variable pressure and temperature (T). It is useful, therefore, to have some general criteria for predicting the elastic behavior of a particular combination of layers under variable P or T. The problem

-3 x 103 W/crn2

--- 3 x 102 W/crn2

250

260

270

290 300 RAMAN SHIFT (cm-1) 280

310

320

FIG. 22. Raman spectra of a [211]-oriented Gal-,In,As/GaAs MQW ( x = 0.15) far from resonance (top) and near resonance (bottom), under

intense (solid line) and weak (dashed line) illumination. For the geometry used, only TO scattering is observed away from resonance, regardless of intensity (top). LO scattering appears near resonance only under weak intensity (bottom, dashed line). (From Ref. [209].)

206

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

is treated here in its general form for SLs and HJs grown along N [76]. Pressure and temperature effects are examined separately.

a. Hydrostatic Pressure Effects on theElastic State ofSLs and HJs

Pressure-induced effects have been observed in the phonon spectra through Raman spectroscopy [78-80,214,2151, and in the electronic band structure through photoluminescence [80, 2161 or other optical spectroscopic techniques [go, 2171. In particular, pressure has been shown to affect the conversion of electronic type I to type I1 quantum wells [218]. Of key importance is the P dependence of d ( P )and ~ ' ( p )and , AE(P)= d ( P )E I ( P )As . before, compressive hydrostatic pressure is defined as negative and set equal to -P, so that P > 0. In the presence of a sufficiently small P,the common lattice constant of a pseudomorphic layered system, and the strain components, are linear functions of P (Hooke's law). Under this assumption, which implies that P 4 3B,and that the shear moduli G , are independent of P , from the requirement of minimum free-energy density we find [77]:

The in-plane strains relative to the P become

=

0 state (netP-induced strains)

with

is the linear By definition, Y’ = 2 , l when Y = 1,2, respectively; K , = 1/3B, compressibility of layer v and AK = - K~ ; is a generalized lattice misfit pressure, to be discussed later. and AB = B2 - Bl; P,nis a critical fp

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS

207

The corresponding tetragonal distortion [defined in Eq. (96)] takes the form AeY(P)= Aev(1 -

E)

(113a)

For nearly matched constituents (A€”= 0) it can be shown that hYIGv’ AeY(P)= PAB A C v ( 3B1B2 hlGl + h2G2 ~

)

(113b)

Thus, a P-dependent tetragonal distortion begins to develop even in latticematched systems, provided Bl # B2. Furthermore,

It is emphasized that all strains here have been defined relative to the = 0 state; therefore, Eqs. (1 11) and (114) include the anisotropic part as well as the isotropic part - P/3B,. The latter does not appear in the expression for the tetragonal distortion. Notice that according to Eq. (113a), AE(P) of both layers becomes zero P, (see also Section IV.3.f), i.e., ALE,(P,) = 0. The at the criticalpressure condition P, > 0 requires that BYfv- AB and f vhave the same sign for either value of v. Unless this condition is satisfied, the tetragonal distortion cannot be tuned to zero. At P,,, the unit cells of both layers recover their cubic shape, and the net (hydrostatic) strains can be written as

P

At P, the common lattice constant of both layers becomes

It is also clear that AeY(P)reverses sign at P,. Like a”(P),the critical pressure P, is a parameter characterizing the system as a whole and not the layers individually. For most combinations of constituents, P, turns out to be in the range under 80 GPa. Technically, such pressures are attainable; on the other hand, they are often higher than the critical pressures Pg where phase transitions for either or both layers may take place

208

EVANGELOS ANASTASSAKIS AND MANUELCARDONA

(see Section 111). Examples of SLs where this is not true, that is, P,,, < Pg, are ZnSeIGaAs, InAsIZnTe, and GaSb/ZnTe [78]. For the vast majority of known SLs the relation B,fv-e AB is valid and the term Bufv can be ignored in the presence of AB.A detailed analysis of these cases can be found in Ref. [77]. The preceding results are easily adapted to HJs in which h, P h and €11 = f. For most cases we have Bf 4 AB and the results are

33

(

a"(P) = as(P) = us 1 - P 3Bs

€Il(P) = f - -

AE(P)= A (1

+

1

( p )=

&f$

1

(117a)

-

( B + AZAB) 3BB,

(117b)

(117c)

(117d)

It is possible to deduce the linear compressibilities of the SL layers for directions parallel and perpendicular to that of growth. Such information may be particularly useful in analyzing experimental data involving X-ray diffraction under pressure. The P dependence of the in-plane and normalto-the-plane lattice constants is expressed by

where K ' and K : are the linear compressibilities governing the in-plane and normal-to-the-plane linear lattice contraction, respectively, under P; they are expected to depend on the individual linear compressibilities K , and to exhibit a weak dependence on P through the P dependence of K , (a nonlinear effect!). The expressions of all and d ( P )combined with the P-dependent strains, relative to the P = 0 state, yield [76]

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS 209

K\L(P)refers to the SL as a whole and is isotropic within the plane. Likewise,

where A K ( P )= K ~ ( P-)K ~ ( Pand ) u' = 2, 1 when u = 1, 2, respectively. Clearly, K : ( P )refers to each layer individually; the normal-to-the-plane compressibility of the entire SL is obtained from (119c) We can now define the effective bulk compressibility KsL and effective bulk modulus BSLof the entire SL as

For most common HJs (Bf < A B ) the corresponding results are all = a,

a"(P)= all(1 - I&)

= a,(P)

= K ~ ( P-)A K ( P ) AC KR,(P) = K ~ J (= P K"(P) ) = K , ~ ( P ) K’(P)

(121a)

(121~)

where h, %- h, AB = B, - B, and AK = K, - K. The critical pressure P,,, at which the tetragonal distortion becomes zero was measured from Raman experiments under P for a (001)ZnSe/GaAs HJ, as described in Section IV.3.f [78]. In fact, the entire range of measurements of Refs. [78, 1581 shows that the linear approach to the problem is sufficient to describe the pressure effects on the strain components. Additional information concerning effects of electric fields on polar phonon frequencies can be obtained from such data [156].

210

EVANCELOS ANASTASSAKIS AND MANUEL CARDONA

b. Variable Temperature Effects on theElastic State of SLs and HJs

The temperature dependence of strains and stresses in SLs and HJs can be treated by complete analogy to P. Only terms linear in the T will be considered here. Most, if not all, physical properties of crystals exhibit a dependence on T that may be particularly large when a phase transition is approached. Therefore, the full treatment of the problem is rather complicated and one can only approach it under certain simplifying assumptions that allow only the dominant macroscopic temperature effects to be considered. The key parameter here is p, the linear thermal expansion coefficient (TEC). (Note, however, that the thermal expansion is often not linear at low T [see Section 1111: Even sign reversals in the thermal expansion coefficient can be found [105].) The elastic stiffness components C,j are considered here to be T independent. We define as AT = T - Toany temperature interval relative to an arbitrary reference value To (e.g., room temperature so as to be in the linear expansion region). As in the case of P ,we only consider undercritical systems ( h5 hc), where the strains are due to lattice misfit; such strains are uniform over the entire volume of the layer and do not normally depend on the growth temperature Tg.In overcritical systems ( h> h,)the strains are of mixed type, misfit (only partial) and thermal; they depend on TRand h;and are not uniform over the volume of the layer. In excessively thick systems the strains are purely thermal. Overcritical systems are more complicated in this regard and will not be treated here. A close look into the P and T effects allows one to transcribe the corresponding relations between the two types of phenomena through the following substitutions;

where Sp = pz - P I .It should be remembered that P stands for the absolute value of the pressure (>O), whereas AT = T - To can take negative as well as positive values. We do not present here the detailed results but rather emphasize the main conclusions and refer the reader to Ref. [76). At TRthe two layers grow on each other coherently; at any lower temperature T they remain in registry with each other, parallel to the plane, following a common in-plane lattice constant all(T) that is determined on thermodynamical grounds and depends on the direction of growth. This generates an in-plane bisotropic elastic strain on each layer that depends on all( T ) and can be computed after all(T) becomes known. The value a"(T ) at T # To is given by Eq. (110) transcribed according to Eq. (122), with all parameters

3 PHONONS, STRAINS, AND PRESSURE IN SEMICONDUCTORS 211

substituted by their corresponding value at T. The counterparts of Eqs. (111) and (112) become

where

The physical meaning of the critical temperature is that at T, the tetragonal distortion of both layers becomes zero and both unit cells recover their cubic shape. The critical temperature exists for all material combinations, in principle. According to Eq. (124), it can be lower or higher than To,but in all cases it must be positive (T, > 0). If, for a particular material combination, Eq. (124) yields T, < 0, this means that there is no real temperature at which the unit cell of these materials, combined in an undercritical SL, recover their cubic shape. We should keep in mind, however, that even if the unit cells of the bulk constituents remain “cubic” (i.e., all = uJ, the unit cell of the SL has a number of internal structure parameters that, like the parameter 5 mentioned in Section IV.4 and Ref. [168], cannot be determined from the macroscopic elastic constants. Very little quantitative information is available about these parameters [219]. In a similar manner we can transcribe all P-related equations of Section V.2.a, including all types of TEC, that is, the in-plane linear TEC for the SL as a whole, &,(T); the normal-to-the-plane TEC for either layer, P:(T); the normal-to-the-plane TEC of the entire SL, piL;and the volume TEC of the entire SL, bSL. They are obtained from equations equivalent to (119a-c) and (120). The results for HJs are easily obtained by transcription of Eqs. (117a-d) and (121a-e). Only undercritical structures have been considered so far in connection with P and T effects. There is ample experimental evidence for either type of effects; in Refs. [78-80,158,2201 the effects of P have been demonstrated very clearly (see Section IV.3.f). In Refs. [221] and [222] typical examples of undercritical systems are reported that reveal, through photomodulated spectroscopy and X-ray diffraction, effects due to temperature. Figure 23 shows data for .d(T) and & ( T ) measured through X-ray diffraction for the undercritical HJ (001)ZnSe/GaAs [222]. The dashed and solid lines represent calculation, based on the analysis of Ref. [76], for a T-dependent and T-independent TEC, respectively. Improvement of the agreement with

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

212

.-a

O

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

ZnS e / ZnTe

L

vl

-4

b

-121

0

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

ZnSe/GaS b -12

-

C T e m p e r a t u r e (T)

FIG. 23. In-plane and normal-to-the-planestrains as a function of Tfor three undercritical heterojunctions [76]. Dashed lines: results computed taking into account the T dependence of the thermal expansion coefficient. Solid lines: results computed without taking into account this Tdependence. The data points were obtained by X-ray diffraction [222].

the experimental data can be obtained by using thermal expansion coefficients averaged throughout the corresponding temperature region. Similar computed results are also shown in Fig. 23 for ZnSe/ZnTe and ZnSe/GaSb HJs, for which no experimental data are available at present. The situation in overcritical systems is more involved since the strains are of mixed character (lattice and thermal misfit), they depend on thickness, and they may not even be homogeneous within each layer. Similar effects are expected to occur for P and for T; temperature effects have in fact been observed in numerous cases of overcritical systems, for example,

3 PHONONS, STRAINS, AND PRESSURE I N SEMICONDUCTORS 213

Refs. [223,224]. To our knowledge, no systematic theoretical treatment of such effects has been presented sofar. Raman spectroscopy is certainly a very direct technique for observing T- and P-tuned strains in SLs and HJs, as demonstrated in Refs. [78, 791. Other techniques are also appropriate, for example, photoluminescence [224-2271, photoreflectance [80, 1581, and piezoreflectance and electroreflectance [158]. In these situations the effects of T and P on strains manifest themselves through the corresponding changes in the electronic band structure of such systems.

3. CHARACTERIZATION OF STRAINS THROUGH RAMAN SPECTROSCOPY The phenomenological approach reviewed here has been used routinely for strain characterization of numerous material combinations in SL and HJ systems, through Raman spectroscopy. A classical example is the Sion-sapphire (SOS) film [228], which exhibits a bisotropic strain due to a strong lattice mismatch between Si and sapphire and the difference of their thermal expansion coefficients. Using Eqs. (65a) and (19a-d) and the PDPs of Si (Table 111), we find, for backscattering along [Ool],

a(GPa)

=

-0.22 Afi,7(cm-')

(125b)

with €11 = (S,, + SI2) u [see Eq. (19d)l. ignoring minor corrections arising from the noncubic structure of the substrate. Equation (125a) is equivalent to (65a). Accordingly, the observed shift of +3.5 cm-' in Ref. [228] is interpreted as a (compressive) stress u = -0.77 GPa. One should keep in mind that Eq. (125b) is valid only for backscattering in Si along [OOl]. The corresponding results for the c and d HJs or SLs coincide with Eqs. (69a,b) and (68b), and (72a-c) and (19c,d), respectively. In all cases, the strain can be obtained in sign and magnitude but nothing can be said about its exact origin, that is, the relative contributions of lattice and thermal mismatch. Selected references to work published in the 1984-1990 period were presented in Ref. [56]. Work published prior to 1984 has been reviewed in Ref. [229]. An extensive review on the lattice dynamical aspects of strained and unstrained SLs, including literature related to Raman spectroscopy, can be found in Ref. [230].

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EVANGELOS ANASTASSAK~S AND MANUELCARDONA

Because of the large number of references available, it is beyond the scope of the present review to include anything but some general remarks on the way the method is being applied and to refer briefly to a number of characteristic reports on families of materials in order to show the potential behind this technique. a. General Remarks Availability of P D Ps. In characterization problems, the analysis presented in Sections IV.2 and IV.3 is directly applicable with the roles of PDPs and u reversed: The known parameters now are the PDPs, while the stress or strain components are the parameters to be found (see the example of SOS earlier). Difficulties in such experiments arise when the PDPs are not available, or, if available, are incomplete and unreliable. Such situations occur frequently with polar phonons, where the distinction between E $ and K $ is not always being realized. Very often the values of E$ are used for K$,if the latter happen to be unknown, or vice versa [231] (see also Section V.3.f). In other cases, when the PDPs are not available, arbitrary values are guessed [232]. Often, the PDPs of a material of the same family are used [69, 2331 or, if the material is an alloy, the PDPs are obtained through interpolation between those of the end members [234]; we point out, however, that this practice has been criticized in Ref. [235]. Angulardispersion. Most strained SLs and HJs involve simple strain patterns (like those analyzed in Section IV.3e); as a result, the phonon eigenvectors of the strained material are parallel and normal to the growth axis. In this case, the phonon wave vector q involved in a Raman characterization experiment performed under exact backscattering conditions is along one of the strain-modified eigenvectors, and the observed frequencies coincide with the strain-modified eigenfrequencies; the analysis is then straightforward. On the other hand, there are situations in which q is not along an eigenvector of the strain secular equation; in that case, for polar phonons the observed frequencies do not necessarily coincide with the eigenfrequencies of the strain secular equation. Such deviations result in miredfrequencies and the interpretation of the observed shifts is more complicated, especially when strain relaxation and asymmetric Raman line shapes are involved. This happens in four cases at least:

1. The incidence or scattered light, or both, travel at grazing or Brewster angle incidence on a bisotropically strained polar HJ grown in highsymmetry direction, for example, (001) or (111) GaAs HJs on various substrates [236, 2371.

3 PHONONS, STRAINS, AND PRESSURE I N SEMICONDUCTORS

215

2. The incident or scattered light, or both, travel along the growth axis or at grazing angle on a bisotropically strained HJ grown in lowsymmetry directions, for example, (112) GaAs HJs on Si or CaFz substrates [238]; now the eigenvectors are neither parallel nor perpendicular to the growth axis. These are the cases involving nondiagonal strain components in S,. 3. Cases involving IR-active phonons in polar materials. The LO eigenvectors are determined by the direction of q, not by the strain secular equation, if the LO-TO splitting is much larger than the effects of strain. For large strains one must diagonalize the sum of the strain plus the electrostatic dynamical matrix. The latter is proportional to the dyadic 4 :4. The directions of the phonon eigenvectors thus depend on the magnitude of the strain. 4. The strain configuration in high-symmetry HJ systems is more complicated than those preceding, leading to a situation similar to case 2. These cases are encountered in discontinuous patterns [233, 2391, LOCOS (local oxidation of silicon) structures and strained stripes [137,233,239,240], trenches [241,242], and deformed wafers [138]. It has been argued that measurements of frequency shifts and scattering intensities at grazing incidence can, in principle, lead to information about all strain components [138, 236, 237, 243, 2441. In short, it is necessary to handle frequency shifts and selection rules properly when investigating strained crystals in nontrivial geometries. These subjects were treated recently in Refs. [135, 2451. It was shown that for phonon propagation in arbitrary directions in strained ZB crystals, the angular dispersion of the mixed frequencies and phonon polarizations can be expressed in terms of the eigenfrequencies and eigenvectors of the strained crystal [135]. Furthermore, the selection rules are shown to bear a close relation to those of unstrained crystals [245]. (For selection rules far from resonance in oblique backscattering geometries of unstrained crystals see Refs. [246], in which the standard semiconductors Si, Ge, and GaAs are discussed.)

b. Strain Mapping The availability of micro-Raman spectroscopy with spatial resolution of the order of 5 1 p m has generated endless possibilities of strain mapping across microareas of material systems and microelectric devices. The main difficulty encountered in these cases is the complexity of the strain patterns,

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

216

which may or may not be known from independent elasticity theory models. Knowledge of the nonzero strain components is necessary before proceeding to the diagonalization of the secular equation. The latter is the next point of difficulty but, as discussed in Section IV.2.a, can be handled by an appropriate choice of axes. This was demonstrated in Ref. [137] for the Si system shown in Fig. 24; the figure depicts a Si substrate surface perpendicular to x i 11 [Ool], carrying a stripe of Si3N4/poly-Siwith its long axis along x i (1 and its width along x i (1 [110]. The stress tensor was expressed in S’ (nonzero components g!,,ui3,mi3). In the S’ axis the eigenfrequencies can be determined from a quadratic equation, whereas in S, the solution of a cubic equation is required a priori. The solid line is the result of theoretical predictions for the frequency shift based on the secular equation and a strain pattern calculated independently using an edge-force model [247]. According to the Raman data, such a model is adequate for this particular structure [137]. Strains and structural variations present in narrow (1 pm across) In,Gal-,As ( x = 0.25) stripes grown on a patterned GaAs substrate have been mapped via Raman microprobe spectroscopy; the system exhibited substantial frequency variations due to variable defect density across the pattern [248].

FlO]

200 100 d

d

0

2

-u

w

-100

5

10

15

20

Y

25

Position (pm) Fio. 24. Profile of the Raman shift and uniaxial stress along x i 11 [110] obtained through micro-Ramanmeasurements.The scan covers the region near and beneath a 9.4-pm-wide Si3N4/poly-Siline. Solid curve: theoretical fit for an edge-force (f) model. [From Ref. [137]. Reprinted with permission from J. Appl. Phys. 79,7148 (19%).0 1996, American Institute of Physics].

3 PHONONS,STRAINS,A N D PRESSURE IN SEMICONDUCTORS

217

Several reports have appeared in the last decade describing the mapping of strains/stresses in various structures (e.g., silicides [249] and metal structure [240]) or in even more complicated systems like trenches [241,242] and microscopic islands via second-order Raman spectroscopy [250]. Extensive reviews have appeared dealing with the variety of strain-mapping applications through micro-Raman spectroscopy [137,251].

c. Si/GeStructures

Comprehensive reviews on theoretical and experimental work concerning strained layers of Ge,Sil-, in HJs and SLs with extensive literature, including Raman work, can be found in Refs. [252-2541. Si/Ge superlattices have received considerable attention due to their optical and electronic properties and their potential role in the fabrication of quasi-direct energy-gap semiconductors. There is extensive literature related to short-period strained SLs [253] and their Raman spectra [255], in particular. Figure 25 shows a classic example of the Raman Si-Si band from two short-period (110)Si3Ge9SLs, one undercritical (thickness 350 A, €11 = 4%) and the other overcritical (2400 A, €11 = 3%, somewhat relaxed) [256]. In each case there are two spectra, corresponding to the two scattering configurations of Eq. (73) for d-SLs. Both phonons are transverse, with frequencies rR1 and rR2 [see Eqs. (72a-c)] differing from the bulk value because of strain effects. Although it is certain that other mechanisms (such as interdiffusion, interface roughness, and confinement) participate in the frequency softening, the observation of sharp polarization selection rules suggests the presence of (tensile) strain effects. In a similar work [257], (001)Si,Ge, short-period strained superlattices with different nominal strain values were used to separate the confinement from the strain effects on the phonon frequencies, both for Si-Si (under tensile strain) and Ge-Ge (under compressive strain).

d. GaAdOtherCompounds GaAs is one of the most common constituents of SLs and HJs. Ignoring the GaAs/AlAs family, which is nearly strain free, hence practically uninteresting for the present discussion,we briefly refer to distinct strained material combinations probed by Raman spectroscopy.' 'For optical probing of such negligible strains in HJs of AI,Ga,-,As/GaAs, see S. Logothotides, M. Cardona, L. Tapfer, and E. Bauser, J.Appl.Phys.66,2108 (1989).

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

218

X,=501.7nm

=T,

Si,Ge,

Si -Si

&LO

L60

77K

L80

500

Energy Shift (cm-') FIG. 25. Raman spectra of the Si-Si vibrations measured for tyo [ 1101 Si3GeySLs. one undercritical (thickness 350 A ) and one overcritical (2400 A). Solid and dotted lines correspond to spectra of the two frequencies a, and n, determined through Eqs. (72a-c) for d-SLs, respectively. Reprinted from Ref. [256] with permission from Elsevier Science.

In,Gal -,As/GaAs structures are interesting in nonlinear optoelectronic systems because of their built-in strains and the concomitant PZ fields [258]. Free-standing, moderately strained (001) and (lll)In,Gal-,As/GaAs SLs were studied, and the strains obtained from the LO phonon shifts were mapped. Combined with X-ray depth-profiling measurement, these results yielded a three-dimensional picture of growth-related strain characteristics [259]. Strong PZ fields are developed in the (111) samples, which can be probed best through photoluminescence [260]. Strained HJs of (001)In,Ga, ,As produced evidence of a stepwise mechanism of strain release beyond the critical thickness [261], of nonuniform strain relaxation ~

3 PHONONS,STRAINS, A N D PRESSURE IN SEMICONDUCTORS

219

toward the surface [262], and of composition-dependent strains in both layers of the corresponding SLs [263]. Highly strained SLs are obtained for x = 1 (lattice misfit 7.16%); combined X-ray and Raman studies on a series of thin (001)InAs/GaAs SLs revealed information on the various competing mechanisms of strain relaxation [264];from independent Raman work, information has been obtained on the frequency softening [265] and the LO phonon localization in the GaAs layers [266]. The (001)GaAdGaP strained SL (lattice misfit -3.6%) has been used as a model system for theoretical calculations [267] and experimental studies [268] of the LO phonon frequency shifts induced by the combined effects of strains and confinement [269]. GaAs/InP strained SLs (lattice misfit 3.8%) have been studied extensively; they have revealed strong misfit and thermal strains in accordance with the PDP theory [237, 2701. Finally, strained HJs between GaAs and insulators can also be investigated through Raman spectroscopy. In the case of a GaAs epilayer on CaF2 substrates [244], and a Si02 insulating epilayer on GaAs substrates [271], the strains have been attributed to thermal mismatch between epilayers and substrates.

e. OtherIII- V/III-V Systems The list of combinations of III-V compounds, besides GaAs, appearing in strained HJs and SLs is long (see Ref. [59], Fig. 9.2). In many cases the components of the strain have been determined through Raman measurements using PDP values, subject to the general reservations given in Section V.3.a. The usual objective of all these investigations is to correlate lattice quality, strain profile, and PZ fields in connection with optoelectric applications. The epilayer materials are mostly Gap, GaSb, InAs, InP, InSb, AlSb, and their alloys. A few examples are GaP/GaAs and Inl-,A1,Sb/InSb [272] (where the strain profile was investigated in relation to the epilayer thickness); GaP/Si [273] (lattice misfit strains versus thermal strains were investigated), InAs/InP (strained islands [274] and QWs and SLs [275] were investigated); InGaAdAlGaAs [276] (photoluminescence and Raman strain characterization is discussed). Likewise the following SLs have been investigated: GaPlInP [277] (effects of strain on the dispersion curves); InAs/In,-,Ga,Sb [278] (investigation of the threading dislocations and growth optimization); InAsIInAsSb [279] (Raman strain characterization); InAs/AlAs [280] (Raman, X-ray, and TEM characterization); InAs/InP 12811 (effects of strains on confined models): InAs/AlSb [282] (an investigation of interface bonds): and GaSb/AISb [22] (a study of alloy regions at the interface).

220

f

EVANGELOS ANASTASSAKIS AND MANUEL CARDONA

II-VI/IIIV and II-VI/II-VI Structures

Heterojunctions and superlattices based on various combinations of 11-VI compounds are of interest because of broad applications to tunable optoelectronic devices in the visible region (blue) not covered by 111-V structures [283]. As already mentioned, the strains are device parameters of key importance: therefore, strain characterization through Raman or other spectroscopic techniques is commonplace in the current literature. In this connection one could refer to the major systems ZnSe/GaAs [284, 285],ZnSe/ZnS [286], ZnSe/alloys [287], Zn/Te [288], and various combinations [289]. Unfortunately, no complete sets of PDP values are available for the 11-VI compounds, and a usual practice is to make guesses {69] or to use the K f values, wherever available, for Kk or vice versa [153,154, 231,232, 285,2901. An entire generation of reports is based on such practices, often leading to disagreement between experiment and theory.

VI. Concluding Remarks The main purpose of this chapter has been to discuss the effects of strain, both hydrostatic and uniaxial, on the vibrational properties of solids, with special emphasis on technologically important tetrahedral semiconductors. A brief review of other related effects of strain, such as those that appear on the electronic band structure, has been given. Effects related to the dependence of phonon frequencies and eigenvectors on pressure, such as the thermal expansion coefficient and the effect of isotropic masses on the lattice parameters, have also been discussed. The approach used has been three-pronged: 1. We have discussed experimental data and cast them in a quantitative phenomenological framework involving phonon deformation potentials (PDPs). 2. Various semiempirical representation of the observed strain effects (i.e., based on springs-and-balls lattice dynamical models with adjustable parameters) have been introduced and used to obtain an intuitive, qualitative physical feeling of the observed effects (e.g., why the splittings induced by strains along either [lo01 or [111] usually have opposite signs). We have also briefly discussed various attempts at calculatthat is, without adjustable ing the observed effects ab inifio, parameters. These state-of-the-art methods are based on the total

3 PHONONS, STRAINS, A N D PRESSURE I N SEMICONDUCTORS

221

energies calculated from the one-electron band structures with manybody effects (i.e., exchange and correlation) treated within the local density approximation (LDA). 3. Once the effects of strain on phonons are quantified in terms of PDPs, measurements by means of Raman or IR spectroscopy can be used to determine the state of strain in a sample and, in conjunction with micro-Raman techniques, its topography. Unfortunately, our present knowledge of the effects of strain on the structrual and vibrational properties of crystals is rather incomplete and a lot of work is waiting to be performed. Structural parameters, such as the internal strain parameter 3 of tetrahedral materials and also the position of single layers of the components in SLs and MWQs, are rather poorly known. In many cases, the lack of experimental data forces us to use theoretical estimates obtained with methods that have not been sufficiently tested against experiment. This is an area in which more work should be done. Concerning vibrational properties, while a fair amount of information is available for group IV and 111-V semiconductors about the phonons at the center of the Brillouin zone, little information is available for the technologically important 11-VI compounds, let alone their I-VII counterparts (e.g., CuCl [116]). The database concerning phase transitions under hydrostatic pressure is becoming extensive and reliable and so is their theoretical understanding. However, virtually nothing is known about phase transitions induced by uniaxial strain: The samples often break before undergoing a phase transition; thus, theoretical investigations would be desirable. The effect of uniaxial stress on diamond has been studied by ab inirio techniques in Ref. [93]. The results, relevant to the diamonds of the diamond anvil cell, show that mechanical instability only sets in at the colossal pressure of 500 GPa. The investigation of fracture under a uniaxial theoretical methods is just starting. load by ab inirio Anharmonic vibrational properties, such as the pressure dependence of phonon lifetimes and linewidths, are just beginning to be investigated [112, 1141; in this case a rich phenomenology is expected [115]. Little information is available concerning nonlinear effects of strain that appear under hydrostatic pressure if phase transitions do not prevent it [96]; see Figs. 2 and 6. The database extant for the effects of strain on phonons at general points of the BZ is also very meager: Measurements by inelastic neutron scattering in uniaxial stressing rigs, such as those described in Ref. [41], and measurements under hydrostatic pressure [39] are badly needed. Likewise, measurements of the combined effects of confinement (or zone folding for acoustic phonons) and strain on the frequencies and phonon linewidths of SLs and MQWs (and also of quantum wires and dots) are

222

EVANCELOSANASTASSAKIS AND MANUELCARDONA

highly desirable. After theoretical analysis they should provide a rather detailed understanding of the effects of strain on the lattice dynamics of semiconductors. We repeat again that we make no pretense of completeness in the long list of references: A recent literature survey, covering back to 1978, has yielded more than lo00 papers related to effects of strains on phonons.

Acknowledgments E. Anastassakis gratefully acknowledges partial support by the General Secretariat for Research and Technology, Greece. M. Cardona acknowledges support by the Fonds der Chemischen Industrie. Thanks are due to S. Birtel for superb word processing of a difficult manuscript that contains contributions of two geographically separated coauthors.

Appendix The system S ’ :x;x;xX; is defined by the orthogonal axes x i (1 [Ilmlnl], x i 11 [12m2n2], and x i 11 [13m3n3] in terms of the direction cosines. The transformation matrix a that takes the system S, to S’ is written

For the cubic crystal only, the relationship between the 6 matrices Ch and C , is [62]

X

6 stiffness

where C = CI1- CI2- 2C- and T: = 1for i = j 5 3 and zero otherwise. Cij is defined in Eq. (18). The fully symmetric fourth-rank tensor is defined as

3 PHONONS, STRAINS, A N D PRESSURE I N SEMICONDUCTORS

223

where i, j express, in contracted notation, the symmetric pairs of indices h p and vp, respectively. Latin indices run from 1 to 6; Greek indices run are easily computed by inspection from 1to 3. The various components of Tij of Eq. (A.1). No factors of 2 or 4 [see Eqs. (14)] are required in converting Tijto TAPUP or vice versa [60]. An expression similar to (A.2) holds for S;,except that S = Sl1 - S12- S-12. and factors of 2 or 4 are required when converting Tij to TAcYp, by analogy to Sij[62]; see also Eq. (14). Consider the example treated in Section II.2.b. We have

According to Eqs. (A.3) and (A.4), the only nonzero components of T are TI, = 1 and T22= T2, = T32= T33= Tda= 1/2. By using Eqs. (A.2) and IS), we find

A third-rank tensor TAPK was introduced in Section V.l.c, in connection with PZ effects in SLs of polar cubic crystals. It is defined as T A= ~([Am, ~ + l,mA)n, + c.p. = 2(l,m,n,

+ c.P.) - cApn

(A.6)

c.p. means cyclic permutation over I, m, and n, and cApKis the antisymmetric unit tensor. For all cubic classes the relations between C and S in S, are [60]

The equations obtained by permuting C and S are also valid. In reference systems other than S,,these relations are more complicated (see appendix in Ref. [61]).

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EVANGELOS ANASTASSAKIS AND MANUELCARDONA

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SEMICONDUCTORS AND SEMIMETALS. VOL. 55

CHAPTER4

Effects of External Uniaxial Stress on the Optical Properties of Semiconductors and Semiconductor Microstructures FredH. Pollak PHYSICS DEPARTMENT T~(.HNO OGY I NEWYORKSTATECENTERFOR AUVANCEU AND

IN

ULTRAFAST PHOTON~C MATERIALS

APPLICA~ONS

OFTHE C i m U N I V I : R S ~ oiI Y N i w YURK BROOKLYN COLLEGE

NY BROOKLYN.

I. INTRODUCTION....................................... ............................. 11. EFFECTS OF HOMOGENEOUS DEFORMA ERGY LEVELS ..... 1. Critical Points at k = 0 ..................................... ............................ 2. Bands at k # 0 ..................................................................................... 111. DETERMINATION OF INTERVALLEY ELECTRON-PHONON AND HOLE-PHONON INTERACTIONS IN INDIRECT GAPSEMICONDIJCTORS ......................... Iv. PIEZO-OPTICAL RESPONSE OF Ge AND GaAs IN THE O P A Q U E RE v. INTRINSIC PIEZOBIREFRINGENCE IN THE TRANSPARENT REGION ... VI. EFFECTSOF EXTERNAL STRESS ON QUANTUM STATES .............................................. 1. Effects of X Parallel to [OOl]and [Oll] (Piezoelectric Effect) on an Ino.21Gao.,OAs/GaAs (100) Single-Quantum- Well Structure ..... 2. Effects of X Parallel to [OOl] and [ O l l ](Piezoelectric Effect) on a GaAdGaAlAs (100) Single-Quantum- Well Structure ............................ 3. Determination of the Symmetry of Excitons Associated with Miniband ............................ Dispersion in InGaAdGaAs ( I 00) Superlattic 4. Asymmetrical GaAdGaAlAs (100) Double Q ............................ 5. Effects of X Parallel to [OOl] on Bulk GaAs and GaAs/GaAIAs Single Quantum Wells Grown on (100) Si Substrates .................................................... 6. Symmetry of Conduction States for GaAs/AlAs Type I1 (001) Superfattires. 1. Determination of the Band Alignment in Sil.,Ge,/Si (100) Quantum Wells.. VII. Summary ......................................................................................................................... Acknowledgments .......................................................................................................... References ....... ...................................

236 238 238 254 264 266 270 27 1 272 277 278 278 283 288 290 295 296 296

This chapter reviews the effects of external uniaxial stress and biaxial strain on the optical properties associated with the intrinsic electronic states of diamond-, zinc blende-, and wurtzite-type semiconductors. The effects 235 Vol. 5s ISBN 0-12-752163-1

S E M I C O N D I I ~ I ' O R SA N D SEMIMETALS Copyright D IYYX hy Academic Press All rights uf reproduction in any form re\erved. OOXO/X784/YX $ZS.W

236

FREDH. POLLAK

of homogeneous deformations at critical points at k = 0 (direct bandgap) and k # 0 have been analyzed. For the latter situation, conduction band indirect and (111) direct gap minima at A1 and L, as well as r25,(r15)-A1 transitions are considered. The influence of the spin-exchange interaction in conjunction with the applied stress and strain also will be discussed for both cubic and hexagonal materials. The polarization effects created by the reduction in symmetry of the cubic materials are analyzed. Experiments on bulk and thin-film materials give important information about deformation potentials, the symmetry of interband transitions, spin-exchange parameters, intervalley electron-phonon and hole-phonon interactions, and piezo-optical coefficients for both the transparent and opaque regions. External stress investigations of quantum transitions in GaAs/GaAIAs, internally strained InGaAs/GaAs, and SiGe/Si heterostructures also will be considered. These studies have made it possible to determine the symmetry of excitons associated with miniband dispersion, the properties of symmetrical GaAs/GaAIAs double quantum wells (QWs) and the band alignment in SiGe/Si QWs. Large [110] stresses have produced quantum-confined Stark effect redshifts and intensity variations due to the stress-induced piezoelectric field along the (100) growth direction. Stress along [001] on internally strained bulk GaAs and GaAs/GaAlAs QWs fabricated on Si (100) has revealed a very unusual polarizationdependent effect.

I. Introduction The application of a homogeneous strain in a solid produces changes in the lattice parameter and, in some cases, in the symmetry of the material. These in turn produce significant changes in the electronic band structure [l-81 and vibrational modes [5-8, 10, 111. All configurations of homogeneous strain can be divided into two contributions: the isotropic or hydrostatic components, which give rise to a volume change without disturbing the crystal symmetry, and the anisotropic component, which in general reduces the symmetry present in the strain-free lattice. For the electronic states, energy gaps are altered, and in some cases degeneracies are removed. Effective masses are affected by the variations in energy gaps as well as by changes in interband matrix elements. The strain dependence of electronic levels can be characterized by deformation potentials, that is, the energy shift per unit strain, which are typically in the range of from 1 to 10 eV. For vibrational states, there is a shift of phonon frequencies and also, in

4 EXTERNAL UNIAXIAL STRESS/~EMICONDUCTOR OPTICAL PROPERTIES 237

some cases, a destruction of symmetry. The “deformation potentials” of the lattice vibrations, that is, the relative change in phonon frequency per unit strain, are of order 1. The electronic and vibrational deformation potentials have been measured experimentally for a large number of diamond-, zinc blende- and wurtzite-type semiconductors. In addition, there have been several theoretical calculations. During the past several years, there also has been considerable interest in strained-layer quantum wells (QWs) and superlattices (SLs) from both fundamental and applied points of view. Strained-layer heterostructures allow the use of lattice-mismatched materials without the generation of misfit dislocations. This freedom from the need for precise lattice matching widens the choice of compatible materials and greatly increases the ability to control the electronic and optical properties of such structures. To fully describe the electronic energy levels of such strained heterostructures, it is important to have information about the effects of strain on the properties of the host materials [5,7, 12-17]. These include strain-induced changes in energy gaps, splittings due to the lowering of symmetry, and variations in effective masses. The application of an external stress can also alter the electronic levels of such strained-layer systems in interesting ways. The changes in vibrational modes are extremely useful for the characterization of the distribution of strain in the components of the heterostructure. This chapter reviews the effects of external homogeneous stresses on the optical properties associated with the electronic states of the highest valence bands and lowest conduction bands of diamond-, zinc blende- and wurtzitetype bulk materials. Special emphasis is placed on the band extrema at the center of the Brillouin zone (BZ). Electronic deformation potentials for a number of relevant bands will be summarized. In addition, the effects of an external stress on the quantum levels of QWs and SLs will be discussed. The properties of strained-layer QWs, SLs as well as epilayers with builtin strain, have been considered in Refs. [5], [7], and [ll-141 and will not be repeated here, with the exception of epilayers of GaN. For this material there is only one reported article on the effects of an external strain. Information about the nature of the direct gap excitons, including deformation potentials, has been obtained by varying the strain in epilayers fabricated on various substrates and under different growth conditions. The effects of strain on vibrational levels, impurity centers, and free carriers (cyclotron resonance and free-carrier absorption) will not be considered in this chapter. These phenomena are discussed in Refs. [5-81, [lo], and [ l l ] (phonons), Refs. [l] and [2] (free carriers), and Refs. [l] and [6] (impurity centers).

238

FREDH. POLLAK

11. Effects of Homogeneous Deformation on Electronic Energy Levels

The influence of a homogeneous strain on the electronic energy levels of critical points in diamond- and zinc blende-type semiconductors has been discussed by a number of authors [l-5, 18-23]. In the following sections, the strain-induced changes in the conduction and valence band extrema at k = 0, conduction band minima at A(X) (Si, Gap, AISb, AlAs) and L (Ge), as well as &-Al interband transitions will be described. For purposes of discussion, we present the band structures of Si and GaAs along the (111) and (100) directions of the BZ as calculated by the pseudo-potential method [24] in Figs. 1 and 2, respectively.

1. CRITICAL POINTSAT k a.

=

0

Diamond-and ZincBlendeTypeSemiconductors

In the absence of strain or spin-orbit splitting, the valence-band edge at is a sixfold-degenerate multiplet with orbital symmetry Tzs. (diamond) or TI5 (zinc blende). The

k

= 0 in diamond- and zinc blende-type materials

6 4 cz

>

2

Q)

XI

4

W

r25'

x4

L

A

r

A

FIG. 1. Band structure of Si along (001) and (111). (From Ref. [24].)

X

4 EXTERNAL UNIAXIAL

STRESSISEMICONDUCTOR OPTICAL PROPERTIES

239

6 4 n

>

X?

Q)

U

)r

2

x6

F a,

5 0 -2

x7 x6

L

A

r

A

X

FIG. 2. Band structure of GaAs along (001) and (111). (From Ref. [24].)

spin-orbit interaction lifts this degeneracy into a fourfold-degenerate (including spin) P3/*multiplet ( J= 3/2, M j = ?3/2, 21/2 in spherical notation) and a multiplet (J = 1/2, M j = +-1/2),as shown in detail in Fig. 3. The spin-orbit Hamiltonian is

where L and c a r e the angular momentum and Pauli spin operators, respectively. Also given in Fig. 3 is the double group notation for the spin-orbit split in diamond- and bands as well as the lowest conduction bands [I'2f(rl)] zinc blende-type semiconductors. Because of the J = 3/2 degeneracy, the valence bands have warped energy surfaces. The fundamental direct gap, that is, the energy difference between the r;(r6)conduction and &(T8) valence bands, is denoted as Eo. The spinorbit splitting is Ao, and Eo + A0 is the transition energy between the conduction band and the spin-orbit split r;(r7)bands. A strain with a uniaxial component splits the J = 3 / 2multiple1 into a pair of degenerate Kramers doublets [l-61. The three valence bands for the case of a compressive uniaxial strain are shown schematically in Fig. 3,

240

FREDH. POLLAK

Stress Axis FIG.3. Valence bands ( J = 312,MJ = 5312, 2112 and J = 112. M, = 2112 in spherical in most diamond- and zinc blende-type notation) and the lowest conduction band [r2,(r,)] semiconductors for unstrained (left) and strained (right) crystals. Also indicated is the double group notation. (From Ref. [25].)

where the bands are labeled v l , v2, and v3. In addition, the hydrostatic component of the strain will shift the energy gap between the valence bands and the lowest-lying conduction band. The transition between the conduction band and the vivalence band is denoted Eo(i),where i = 1, 2, 3. The removal of the J = 3/2 degeneracy produces energy surfaces that are ellipsoids of revolution around the uniaxial strain axis. It has been shown [l-61 that the strain Hamiltonian f i r for)a p-like multiplet can be expressed as

where tr(E) is the trace of the strain matrix E, ejj denotes the components of the strain tensor, and cp denotes cyclic permutation with respect to the indices x, y. and z .The quantity a, represents the intraband (absolute) shift of the orbital valence bands due to the hydrostatic component of the stress

4 EXTERNAL UNIAXIAL STRESS~SEMICONDUCTOR OFTICAL PROPERTIES 241

(intraband or absolute hydrostatic deformation potential), while b and d are uniaxial deformation potentials appropriate to strains of tetragonal and rhombohedra1 symmetries, respectively. At k = 0 the conduction band minima for the diamond- and zinc blendetype solids (except for Si and "zero-bandgap" materials such as a-Sn and HgTe) are an antibonding s-state with symmetry, r2,(r,). The effect of a strain is to produce a hydrostatic shift given by

H:’)= a,tr(E)

(3)

where a, is the intraband (absolute) hydrostatic deformation potential of the T,(T,) conduction band [l-61. We take our wave functions in the IJ,M,) representation that makes H,, diagonal. For the s-like r2*(r1) conduction band [3-51, p/2, 1/2), 11/2, -1/2),

=

1s t )

(44

=

1s .1)

(4b)

ri(r8)and T;(T7) valence bands can be written as 13/2,3/2) = (l/z/z)(X + iY)t (4c)

whereas the p-like

13/2,1/2) = (l/&)(X

+ iY).1 - (

(3/2, -1/2)

=

-(l/VZ)(X

-

iY)7

13/2, -3/2)

=

-(l/fi)(X

-

iY).1

-

m ) Z

7

( a ) Z .1

(44

(4e) (40

11/2,1/2) = ( l / v 3 ) ( X+ iY)1 4- ( 1 / v 5 ) Z 7

(4g)

iY)t - (l/v5)2.1

(4h)

p/2, -112)

= ( l / l h ) ( X-

At the BZ center, the matrix elements connecting the T T ( r l )conduction band with the rA(rx)and T:(r7)valence bands are zero. The strain-dependent Hamiltonian matrix of the valence band (neglecting the small strain-dependent spin-orbit splitting terms) is 13/2, +3/2)"

13/2, -t1/2)"

\1/2, -t1/2)"

-a,tr(e)- - SE:,

0

0

0

0

-a,tr(E)-

+ a~;,

z/z6E:,

v’?

SET,

-a,tr(E)- - A,,

(5)

242

FREDH. POLLAK

where rn denotes either the [Ool] or [ l l l ]direction [l-51. Since the strain does not remove the Kramers degeneracy, there is a similar expression for the (S .1 ), ( J ,-A#,)” manifold. Solving the Hamiltonian in Eq. ( 5 ) yields the following for the stress dependence of the v2, v l , and v3 bands:

SEvl =

+ (A0

-

6EFV)/2- (1/2)[A$+ 2Ao 6EFv + 9(6EFv)2]“’2’(6b)

SEv3 = SEZ,V+ (A” - SEFV)/2+ (1/2)[Ac+ 2A0 6EZv + 9(6EFv)2](112) (6~) The band v2 corresponds to 13/2, 23/2)“, whereas v l and v3 are straininduced linear combinations of 1312, ?1/2)“ and 1112, 21/2)“ [3-51. Combining Eqs. (3), (4a), (4b), (6a), (6b), and (6c), one can write the following for the stress dependence of the E0(2),Eo(l), and Eo(3)bandgaps:

-

(1/2)[A:+ 2Ao 6EFv + 9(6E~v)z]‘”z’

(7b)

where Eo is the zero-strain bandgap. It can be shown that the transition E o ( l ) is allowed only for the polarization of the light perpendicular to the uniaxial strain axis, whereas E0(2) and E0(3) can be observed for both perpendicular and parallel configurations [3-51. In this treatment we have neglected the small strain-induced coupling between the rl conduction and rlSvalence bands [3,51. For an external uniaxial stress X along either [OOl] or [ l l l ] ,

UNIAXIAL STRESS/SEMICONDUCTOR OPTICAL PROPERTIES 243 4 EXTERNAL

where S, are the elastic compliance constants and a is the interband hydrostatic pressure coefficient [3, 51. Figure 4 depicts the energies of Eo(l), E0(2), and E0(3)for GaAs as functions of external stress X 1) [OOl] for light polarized parallel (11) and perpendicular (I)to the stress axis [25]. The measurements were made

I ,87

I ,86

I .85 h

2

v

1,84

> (3

1,83

z W

, I ,82

I ,51

1,5c

I ,49

I 2

STRESS

1 4

1 6

8

10

( 10gdyn cm-2)

FIG. 4. The energies of the Eo(l), E,,(2), and E 4 3 )features of GaAs at 77 K for X (1 [Ool] for light E 11 X and E I X. The dashed lines represent the linearized portion of the curves. (From Ref. [25].)

244

FREDH.POLLAK

using the electroreflectance technique. The solid lines are least squares fits to Eq. (7), making it possible to evaluate the deformation potentials a and b. Similar studies for X 11 [111] have yielded values for d. For a two-dimensional in-plane (001) or (111) strain, which has a uniaxial component along [Ool] or [l 1 11, respectively, we have

C ~ E "=, a[2 ~ - PIE

where the Cij are the elastic stiffness constants [3, 51. Table I lists the experimental values of the interband hydrostatic pressure coefficient (a) and shear deformation potentials (b and d) for the k = 0 bands of a number of diamond- and zinc blende-type semiconductors. These parameters have been measured by a variety of optical and transport experiments. Theoretical calculations of these parameters, including absolute hydrostatic pressure coefficients, for various materials have been performed by a number of authors [26-321. The determination of the intraband (absolute) hydrostatic pressure coefficients is more difficult in relation to the interband deformation potentials, a topic also discussed by Refs. [26] and [27]. Nolte etal. [33] have measured the absolute band-edge hydrostatic deformation potential (a,) for GaAs and InP. The situation for [I101 uniaxial or (110) biaxial strain is more complicated, since for this low-symmetry direction the 13/2, 3/2)('") band is coupled to both the 13/2, -1/2)('") and 11/2, -1/2)("l) states by terms proportional to (C"'m" S.i - 6Ek!f"), where i = X or E [34]. The stress-dependent wave functions for the k = 0 valence bands also are given in Ref. [34].

h. WurtziteType Materials

Because of the lower symmetry of the wurtzite-type semiconductors in relation to diamond (Oh)and zinc blende (T,)-type materials, the

4

EXTERNAL UNIAXIAL STRESWSEMICONDUCTOR OITICALPROPERTIES 245

degeneracy of the p-multiplet is split not only by the crystal field (cf) but also by the spin-orbit (so) interaction. The combined Hamiltonian for this effect can be expressed as [l,4, 91

where L , = ( l / f i ) 2 ( LiL,) x and a, = ( l / d ) ( a 2 X icy). The fundamental bandgap of unstrained wurtzite materials exhibits three excitonic features, namely, A, B, and C , which correspond to r?-r;,r5-r: (upper), and r5-r; (lower) interband transitions, respectively. The operator H Y w ) that describes the effect of the strain on the valence band in the wurtzite materials is given by [l,4, 91

where el = E, + eyy,E, = E ~ , - eyV ? 2ieXy,and ez2 = E, ? kyz. The operator H!".") that represents the effects of the strain on the rl(I'7c) conduction band is written as [l,4, 91

Using the Hamiltonians of Eqs. (10) to (12) and the wave functions of Eq. (4) (quasi-cubic model), expressions for the energies for the A, B, and C excitons in strained wurtzite materials can be obtained. In the quasicubic model Ab = Ac.Considering only terms up to second order in the strain, these expressions are as follows (neglecting the spin-exchange interaction):

FREDH. POLLAK

246

TABLE I THEINTERBAND HYDROSTATIC DEFORMATION POTENTIALS OF THE DIRECTGAPEo AND SHEARDEFORMATION POTENTIALS OF THE ra.(rls) VALENCE BANDS OF A NUMBER OF DIAMONDAND ZINCBLENDE-TYPE SEMICONDU~ORS

Material

Hydrostatic Deformation Potentials a (ev)

Si

-5.1"

Ge

- 12.7 -9.56b -9.95h -11.7 -8.80 0.2' -9.77 - 6.70 -8.46 -9.43 -6.36 -10.7 -t 0.2' -9.11 ? 0.2"

GaAs

G%78Ah22As

Ga0.73Ab.nAs InAs

GaSb

AlSb InP GaP

*

-8.6 ? 0.2' -6 -5.7 -5.9 -6.9 -8.3 -8.28 -8.2 -7.9 -8.3 -5.9 -2.0 -0.9 -6.35 -6.6 -9.3 - 9.9 -8.2 -7.9 -8.3

Shear Deformation Potentials b (ev) -2.27 -1.92 -2.10 -2.14 -2.2 -2.4 -2.6 - 2.86 -2.21

-2.0 ? 0.2' -1.7 -2.0 -2.0 -2.08 2 0.2' - 1.88' -1.86' -2.1' -1.90 5 0.2' -1.8

d (ev) -5.1 -4.84 -4.85 -5.3 -5.1 -3.5 -4.7 -5.3 -6.6 -4.43 ? 0.6' -4.55 -5.4 -5.3

-5.47 t 0.6'

-3.6

-1.8 -2.0

-4.6 -4.8

-1.35

-4.3

-2.0

-5.0 -4.2 -4.6 -4.8

- 1.55 -1.8 -2.0

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OPTICAL PROPERTIES 247 TABLE I continued Hydrostatic Deformation Potentials a (eV)

Material ~

~~

Shear Deformation Potentials b (ev)

~

InSb

~

d (ev) ~

~

-2.05

-5.0

CdTe

-7.7 -7.6 -7.5 -8.0 -5.1

-1.2

ZnTe ZnSe ZnS

-5.8’ -5.4k -4.0k

-1.8’ -1.2k -0.538 -0.7k

-5.49 -4.8h -4.6’ -3.8k -3.79

Note.All values taken from Landolt-Bornstein - Numerical Dataand Vols. 17a and 17b Functional Relationships inScience and Technology, (0. Madelung, M. Schultz, and H. Weiss, eds.), Springer, New York, 1982, unless otherwise indicated. Ref. 1261. Evaluated from a = -(d&ldP)[(C,1 + 2C12)/3]. Ref. [70]. Ref. [71]. Ref. [72]. ’Ref. [61]. Ref. 1731. Ref. [74]. ‘Ref. [75]. ’Ref. [76]. Ref. 141.

where D1,2 = dl,2 + D;,2, and

Ei

-E

i

=

(1/2)(3Ab

+ A,)

-

(1/2)[(Au

-

Ah)’

+ 8A26]’/2

(14a)

FREDH. POLLAK

248

{

a!?= (1/2) 1 2

[(A,

- Ab) - Ab)’ 8 A~]”’

+

l1-W Wurtzite materials. The effects of an external stress for X (1 [ O O O ~ ] (c-axis), X 11 [ll30] (a-axis), and X 11 [1132] on the A, B, and C excitons of bulk CdS [4, 351, CdSe [4], and ZnO [4, 361 have been reported. These studies have made it possible to evaluate the various coefficients A,, Ab, 4, D2,D3,Dd.Ds, and 1061. Figure 5 shows the experimental data for X 1) [OOOl] on the A, B, and C excitons of CdS as measured by wavelength-modulated reflectivity at 77 K 1351.For this stress direction E+ = E?, = 0 and hence all terms in Eqs. (13) containing D5and D6 are zero. Thus EA will have a linear stress dependence whereas EBand Ec will contain both linear and quadratic terms, the latter being due to their mutual interaction. Thus it is possible to identify the character of the various features, that is, the lowest-lying transition is indeed r:-r;, and so forth.

CdS 77°K

h

2.60 v

x

> (3 -

II [ O O O I ]

t

2.58

z

STRESS (lo9d y n e -crn’* 1 11 [Oool] (c-axis). (From

FIG.5. The exciton energies of CdS at 77 K as a function of X Ref. [35].)

UNIAXIAL STRESSISEMICONDUCTOROPTICAL PROPERTIES 249 4 EXTERNAL

Wurtzite GUN. To date only one experiment involving external uniaxial stress has been reported on the wurtzite group I11 nitrides [37]. However, it is possible to vary the in-plane stress in layers of such materials by appropriate growth conditions [38], including the use of different substrates [38, 391 and orientations [40]. For example, Shikana et al. [38] have fabricated GaN (0001) on sapphire using either GaN or AIN buffer layers to control both the compressive and tensile in-plane biaxial strain. Buyanova et al. [39] have reported results of GaN (0001) fabricated on S i c substrates with different built-in tensional biaxial strain. The work of Ref. [40] involved GaN grown on a-plane sapphire. However, in all these cases there are certain limitations since the uniaxial component of the biaxial strain was parallel to either the c-or a-axis. In the experiment of Yamaguchi et al. [37], the external uniaxial stress was applied along the a-axis. The experimental results of Ref. [38] are plotted in Fig. 6 as a function of the strain along the c-axis ( E ~ ~ The ) . energies of EA, EB,EC,and En=2 (first excited state of the A exciton) all increase with increasing biaxial compressive strain. EA is a linear function of e Z Z ,and the ground state binding energy of the A exciton, Eex,A, is not affected by the strain. From the difference between EA and En=2, Eex,A is estimated to be 26 meV. From an analysis of the data, these authors have obtained values of Acf = 22 meV and As,, = 15 meV, which are close to those reported previously. The value of Aso agrees well with that of cubic GaN (17 ? 1 meV). This = analysis yielded the following physical parameters: [ D l- (C33/C13)D2] 38.9 eV, [ D 3- (C33/C13)D4]= 23.6 eV, and the energy gap in unstrained crystal (EgA)equals 3.504 eV. Note that in Fig. 6 the A and B lines cross at E,, = -0.01 and that there is an anticrossing of the B and C exciton lines similar to that displayed in Fig. 5 for the case of external stress parallel to the c-axis. From an analysis of their external uniaxial stress experiment, Yamaguchi et al. [37] found D4= -3.4 eV and ID3/= 3.3 eV. The work of Ref. [40] yielded D5= -2.4 eV. Values for the deformation potentials that describe the hydrostatic component of the strain can be obtained independently by hydrostatic pressure experiments, as discussed in Ref. [9]. c. Spin-Exchange Effects at k

=

0

In the previous sections the effects of strain on the band structure at k = 0 have been discussed. However, these bands still possess spin degeneracy that is not removed within the one-electron framework described above. Therefore, in the case of excitons there are two particles involved. The

FREDH. POLLAK

250

3.6

% 3.55 -

-+

I

'

"

I

'

"

,

/

' I

wurtzite-GaN T=lOK

/ f /

/

/ /

Exp. Theor.

n

EA

-

/

i /

/

-

-

0

B W

9 z

3.5

0

2 z

Eu 3.45 X W

3.4 STRAIN

E,

FIG. 6. Exciton resonance energies of GaN as a function of eZZ.Theoretical fit lines also are shown.(From Ref. 1381.)

lowering of the symmetry of the system by the strain (uniaxial or biaxial) and the spin-exchange interaction results in a spin splitting of the spin degeneracy [3-51. Diamond-andzinc blende-type semiconductors. We shall now consider as a specific example the effects of a [Ool] uniaxial strain on the exciton spectrum of the lowest direct gap in diamond- and zinc blende-type materials. The results for [111] strain are qualitatively similar.

UNIAXIAL 4 EXTERNAL

STRESSISEMICONDUCTOR OPTICAL PROPERTIES

251

The exciton part of the Hamiltonian can be written as

where BE is the exciton binding energy and the second term describes the crystalline exchange interaction [3-51. The operators f f h and a, operate on the valence-hole and conduction-electron wave functions, respectively. Both BE and the spin-exchange constant ( j ) are assumed to be independent of stress. The total Hamiltonian is then described by Eqs. (l),(3), (5), and (15). The basis functions can be written as [3-51

where the valence-hole wave functions ( J ,I+_ M,)(OO') are given by Eqs. (7) and the conduction-electron functions transform as IS 1) and IS 1). In the following discussion, energies that equally affect all the states of Eq. (29), that is, the spin-orbit interaction and hydrostatic pressure term, will be dropped and the interaction with the spin-orbit levels will be neglected. The Hamiltonian is then quasi-diagonal in the basis of Eq. (16):

FREDH. POLLAK

252

where i = X or E. From Eqs. (17), the exciton energy levels are (including the hydrostatic term)

El = E2 = 6 E P 1 )+ &EL!’) + (1/2)j E3 = S E f f )’ )6E&Y1) + (1/2)j E4 = S E P ’ )- 6E&Y’)- (5/6)j

X

{ [ 6 E $ 7 )-I(8/3)jSE$Y) 2 + (16/9)j2}"*

It can be shown that, of the above levels, only 3, 5, and 7 (which belong to the representation F2)are optically active at zero stress, with level 3 being allowed for parallel configurations, while 5 and 7 are allowed for perpendicular configurations. Application of a uniaxial strain splits the 3 level from 5 and 7 and causes the 6 or 8 level to become optically active for perpendicular configurations. Figure 7 shows the stress-induced splittings of the exciton lines of ZnTe, as observed in the wavelength-modulated reflectivity spectra for X 11 [ l l l ] for E 1) Xand E IX [41]. These results are in agreement with the previously mentioned considerations. Note that the lowest energy line is observed only at finite stresses for E I X. The exchange-splitting parameter has been determined from the zero-stress extrapolation. Similar experiments have been reported for ZnS [4] (cubic) and ZnSe [4]. ZZ-VZ Wurtzite materials. Langer etal.[4] also observed a polarizationdependent splitting of the A and B exciton lines for X perpendicular to the c-axis with the light incident along the c-axis for wurtzite CdS, CdSe, and ZnO. These results for CdS are shown in Fig. 8.The highest and lowest energy features are observed only for E IX , whereas the two intermediate energy transitions are seen only for E 11 X.The origin of these polarizationdependent splittings is the spin-exchange interaction of Eq. (15). The solid lines are fits to theoretical expressions that make it possible to evaluate the spin-exchange parameter j .

UNIAXIAL 4 EXTERNAL

STRESSISEMICONDUCTOR OPTICAL PROPERTIES

253

I .5

->

I .o

Q)

- 0.5 E

W

o o o

” 4

- 0 0 -

0

0

-

-0.5i

a

0 I

I

1

I

I

II

1

I

FIG. 7. Splitting of the exciton lines for ZnTe for X )I [ l l l ] for light E I( X and E I X. The lowest-energy line is seen only at finite stresses for the perpendicular configuration. (From Ref. (411.)

Rohner [42] has calculated theoretical values of this parameter for CdS, CdSe, ZnO, ZnS, and ZnSe, all of which are in good agreement with experiment. Wardzynski and Suffczynski [41] have observed that the exchange splitting depends exponentially on the interatomic distance of the cation and anion of the compound.

FREDH. POLLAK

254

2.59

2.58 h

>

Q)

2.57

6 z 256

-

I

W

2.55

2.54

lTLZ

I

PRESSURE ( kbad FIG. 8. The effe' of X perpendicular to the c-axis with the light incident along the axis on the A and B exciton lines of wurtzite CdS for E _L X and E 11 X. (From Ref. [4].)

2. BANDSAT k # 0

For band extrema or interband critical points at k # 0, the shear component of the applied uniaxial stress can cause three effects [3, 51: (1) band states of different k , which are degenerate because of the symmetry of the crystal, may have their degeneracy reduced depending on the projections of their k onto the stress direction (interband splitting), (2) a splitting of degenerate orbital bands whose k are not parallel to the stress, and (3) a stress-induced coupling between neighboring bands [e.g., the A1 conduction

4 EXTERNAL U N I A X I A L STRESSISEMICONDUCTOR OPTICAL PROPERTIES 255

band minima and nearby A2, band in silicon (see Fig. l)]. The second and third effects mentioned above are denoted intraband splittings. Deformation-potential theory for many-valley cubic semiconductors was first considered in Refs. [18] and [19] as well as Ref. [23]. In the notation of Brooks [MI, the stress-induced hydrostatic shift and interband splitting of a band-extrema interband critical point due to effect (1) is given by

AE = ii.{g1tr(e)l - + &[E

-

(1/3)tr(~)l]).A -

(19)

where ii is a unit vector in the direction of the band extrema or critical point in k-space, 1 is the unit dyadic, and kl and cF2 are the hydrostatic (absolute) and shear (absolute) deformation potentials, respectively. The intraband effects will depend on the specific bands under consideration. We shall now consider two specific examples of major interest, namely, the conduction band minimum along the equivalent (100) directions of the BZ, which occurs in such indirect gap materials as Si, Sil-.Ge, (x < O M ) , Gap, and AlAs, and the Ll conduction band minima in Ge (equivalent (111) directions). Table I1 lists the band-edge (absolute) deformation potentials cKl and d2 for Si(Al), Ge(LI), GaP(A1), AISb(Al), and AIAs(Al). For the sake of completeness, values for the indirect gap interband hydrostatic pressure deformation potential (Jl + a , )are also listed. Band Minima atA, a. Conduction

In addition to the linear shifts and splittings given by Eq. (19) there is also a nonlinear shift due to the stress-induced coupling between neighboring A1 and A2,(A1)conduction bands (see Figs. 1 and 2). For the [loo] direction of the BZ this interaction is given by

where cC2is a shear deformation potential [3,5]. Components for the other equivalent (100) directions of the BZ are obtained by cyclical permutations of x, y ,and z . Combining Eqs. (19) and (20), we can write [3, 51

FREDH. POLLAK

256

TABLE I1 BAND-EDGE DEFORMATION POTENTIALS 81, C,, AND 1 @[ AND THE INTERBAND HYDROSTATIC DEFORMATION POTENTIAL ( B1 + a,) FOR THE CONDUCTION BANDMINIMA OF SEVERAL INDIRECT GAPDIAMONDAND ZINC BLENDE-TYPE SEMICONDUCTORS

Material

Cl

(4+ a,)

62

1G1

(eV)

(ev)

(eV)

(ev)

si (A1)

-3" -7.4h

Ge(L1)

-5.6" -4.3 -6.1 -12.6 (b) 13.0

Gap ( A d AlSb (AI) AlAs (A I )

4.6

3.8' 3.1' 1.6' 1.2' 1.5' -5.7d

1.6'.' 2.2'

-8.77 -9.0 -8.1 -8.6 -9.2 - 18.7 - 19.3 - 16.3 - 15.9 -6.5 -6.3 -5.4 -5.1 ? 0.71

8.0

Note.All values taken from Landolt-Bomstein -NumericalData and Functional Relationships in Science and Technology, Vols. 17a and 17b (0. Madelung, M. Schultz, and H. Weiss. eds.), Springer, New York, 1982, unless otherwise indicated. Ref. [77]. Ref. (781. rrs(ris)-Alindirect gap. r2s.(rls)-Llindirect gap. ' Evaluated from (8,+ a,)= - ( d E / d P ) [ (+C l2CI2)/3]. l /Ref. (661.

with

where ,?:(A,) and E:[Azf(Al)] are the zero-strain energies of the A, and Az,(A,) bands, and AEc(Al) and AEc[Azf(Al)] are given by Eq. (19) with the appropriate deformation potentials. In Si, E:(Al) - E:(A2.) = 0.8 eV (see Fig. 1). Table I1 lists 8; for Si. For these cases there is a Uniaxial [001] stress and biaxial (001) strain. uniaxial component of the strain along [Ool] and an in-plane symmetry,

4 EXTERNAL UNIAXIAL STREWSEMICONDUCTOR OPTICAL PROPERTIES 257

thus the quantities eXy = ex, - zYz = 0 and therefore there is no A1/A2,(A1) coupling. For the [Ool] singlet band the shift is given by

AELml] = S E E ! - (2/3)SE&y,:' i = Xor E whereas for the [lo01 and [OlO] doublet valleys

AE~ool,'olO1 = 8EK.j + (1/3)SE&!,:) i = Xor E where for X

11 [OOl]

&p') H,c,X- 4(S11 + 2 S l 2 ) X

SELy.2= 82(Sll - S12)X

while for the biaxial situation

SEF:? = 81 [2 - A'Oo"]~

8EkT.i)

-

d2[1

+ Atrn1)]~

Since for (100)direct gap materials like Si the quantity 4 < 0, a compressive stress or strain along [OOl] moves the singlet [OOl] band below the doublet bands. [ I l l Uniaxial ] stress and (111)biaxial strain. For these configurations there is no removal of the interband degeneracy of the equivalent (100) conduction band minima. However, there is an intraband A1/A2-(A1) coupling, as discussed earlier. Thus, to second order in the strain, the shift of all the A1 conduction band minima is given by

where

and (1/6)S44X Exy

=

-(1/3)[1

+ A("')]E

uniaxial stress biaxial strain

b. Conduction Band Minima atL l

Another situation of considerable interest is the degenerate conduction band minima along (lll), such as occurs for Ge(L,). Since there are no

258

FREDH. POLLAK

other bands near L, , we neglect intraband-mixing terms analogous to Eq. (20). Thus, for the L1 conduction band minima, only Eq. (16) applies.

[OOI] Uniaxial stress and (001) biaxial strain. These types of stress and strain do not remove the degeneracy of the equivalent L1 minima and hence there is only a hydrostatic pressure shift given by AEc(L1) = SEF;!

i= X o r E

(27)

where S E E ) is given in Eqs. (24a) and (24b) for the uniaxial stress and biaxial strain situations, respectively. [ I l lUniaxial ] stress and (111) biaxial strain. These configurations make the [lll] conduction band minima (singlet) inequivalent to the remaining L, states (triplet). The energy shifts are thus from Eq. (19):

where

c.

SE#,:-; is given in Eq. (26a) for the uniaxial and biaxial cases, and

~zs.(rls)-Al Indirect Transitions

Such transitions occur in indirect bandgap materials such as Si, Gap, AIAs, and AlSb and alloys such as GaAlAs (in a certain composition range). The application of a stress or strain along any crystallographic direction removes the degeneracy of the r2s,(r15) valence band. A configuration that has a uniaxial strain along either [Ool] or [110] also destroys the degeneracy of the equivalent A1 minima. Schematic representations of the effects of a compressive uniaxial stress X ]I [lll],XI[[Ool], and

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OFTICALPROPERTIES 259

J

J=3/2

v2

A2

XI1 [ I l l ]

XI1 [OOI]

XI1 [IIO]

FIG. 9. Schematic representation of the stress-induced splittings of the rzS.(rl5) valence and A , conduction bands for the case of a compressive stress or strain along [lll], [OOl], and [110]. The spin-orbit split band v3 is not shown. (From Ref. [34].)

X 11 [110] are shown in Fig. 9 [3,5,34]. For compressive stress the top of the valence band is split into the vI and v2states, as discussed in Section 1I.l.a. (1111 Uniaxial stress and (111) biaxial strain. For these cases the degeneracy of the equivalent Al minima is not removed and there is only a shift due to the hydrostatic component of the stress. Therefore, two transitions (Al and A 2 )would be observed, where from Eqs. (6) and (25) the stressdependent energies are given by

E*I

+

= Eind(0)

-

aV)(Sl1 + S12)X- (A-&f:,?)/2 (30b)

where Eind(0) is the zero-stress value of the indirect bandgap, and &EL::,%is given in Eq. (8). It can be shown that both transitions are allowed for both E I( X and E I X . Corresponding expressions can be obtained for biaxial (111) strain.

260

FREDH. POLLAK

[001], (1101 Uniaxial stress and (001) (110) biaxial strain. For these two configurations the (100) minima are no longer degenerate and have the alignments illustrated in Fig. 9 for the case of compressive uniaxial stress. For the case of X (1 [001], expressions for the stress dependence of the B,-B4 features can be obtained from Eqs. (6), (23), and (24). It can be shown that B , ,B3,and B4 are allowed for both polarizations. B2 is seen only for E 1) X [29]. The situation for X 11 [ 1101 is more complicated due to the stress-induced coupling of (3/2,+3/2) to 13/2, +1/2) and 11/2, ?1/2) as well as the nonlinear term of Eq. (17). Complete expressions for the stress and polarization dependence of C,-C4 are given in Ref. [34]. Biaxial strains that have compressive uniaxial components along the [Ool] and [110] directions would produce results qualitatively similar to those illustrated in Fig. 9.

d. Interband Transitions along(112) Another important set of bands are the A1 and A3 valence bands, which occur along the equivalent (111) directions of the BZ zone. The orbital degeneracy of A3 is removed by the spin-orbit splitting. The spin-orbit split transitions between A3 and A, are denoted as El and El + A1, where A1 is the spin-orbit splitting of A3 (see Fig. 2). A I0011uniaxial strain does not remove the k-space degeneracy of these bands (no interband splitting) but does cause an intraband effect, that is, a strain dependence of the separation between the A3 orbital bands. On the other hand, a [ l l l ] uniaxial strain produces both interband and intraband splitting; the [lll] band is split off from the remaining three bands ( [ l n ] , [TIT], and [nl]), and there is an intraband effect on this latter group whose k does not lie along the uniaxial strain direction [3, 5, 251.

[00l] Uniaxial stress and (001)biaxial strain.For a [001] uniaxial stress or (001) biaxial strain, the energies of the El and El + A, transitions are given by the following (neglecting exciton effects, which are considered in the next section) [3, 5, 251: AE, = (A1/2) + S E E A(El + A,)

=

(A1/2)+ S E E

- (1/2)[A7

+ (8/3)(SELy))2]1’2(31a)

+ (1/2)[A7 + (8/3)(SE~y))2]1’2(31b)

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCI-OR OPTICAL PROPERTIES 261

where z

where

=

X or E, with

&?band

is . now an interband hydrostatic deformation potential and

The quantity 0: is an interband deformation potential for the A3 valence band for a [OOl] uniaxial stress or (001) biaxial strain. [ I l lUniaxial ] stress and (111)biaxial strain. A [ l l l ] uniaxial stress or (111) biaxial strain that has components along the [ l l l ] direction preferentially selects out the [ l l l ] direction (singlet) while making equal angles with the other six (111) directions (triplet). This gives rise to the interband splitting between the singlet and triplet states. In addition, there is an intraband effect for the triplet states. The strain-dependent energy eigenvalues for the El and El + A1 singlet states are thus from Eq. (19):

where

and

where

&pband

is now an interband shear deformation potential.

262

FREDH. POLLAK

By using the wave functions with the proper transformation properties, it can be shown that for the triplet state, the energy shifts caused by the strain are given by

and

where 0:is an intraband deformation potential appropriate to a [ l l l ] uniaxial stress or (111) biaxial strain [25].

e. Exciton Effects at k # 0 In the above section on the El and El + A1 transitions, we have neglected spin-exchange effects, that is, Eq. (15). For example, a [001] uniaxial stress or (001) biaxial strain does not remove the equivalence of the (111) bands, and hence only two polarization-independent transitions given by Eq. (31) are predicted. That is, the Kramers degeneracy of the A, and A3 bands is not removed in the one-electron picture. However, inclusion of the spinexchange effect does destroy this degeneracy, as in the case of the bands at k = 0 [3, 5, 251. For the case in which there is a uniaxial component along [OOl], it can be shown that the strain-dependent El and El + Al energies, including the spin-exchange interaction, are given by AEi = (A1/2)+ S E E )- (l/2)[A: + (8/3)(6ELy1’2 Sj)2]1’2(37a) A(El + A1)

=

(A1/2)+ SEE1)+ (1/2)[A; + (8/3)(6E$y1)? S,)2]1/2(37b)

UNIAXIAL STRESWSEMICONDUCTOR OPTICAL PROPERTIES 263 4 EXTERNAL

and

is the probability that the electron and the hole are in the same where p ( 0 ) lattice site, and j is the exchange interaction between Wannier functions [3,5,251. Figure 10 shows the effects of X 11 [Ool] on the El and El + A, features of GaAs for E 11 X and E IX at 300 K measured using electroreflectance [25]. The applied stress causes a shift in the “center of gravity” of the El and El + Al features, an increase in the energy separation between these

STRESS

(I 09dyn cm-2)

FIG. 10. The effects of X 11 [OOl] on the E l and El + A , features of GaAs at 300 K for light E 1) X and E I X. (From Ref. [25].)

264

FREDH.POLLAK

two peaks, and a small polarization-dependent splitting for each optical structure. The polarization-dependent splitting, due to the exchange interaction, is such that for the El transition, the component for E 11 X occurs at a higher energy than the E I X component, whereas the ordering is reversed for the El + A, transition. Similar results have been obtained for the El and El + A1 features of GaAs at 77 K and of Ge at both 300 K and 77 K [25].

111. Determination of Intervalley Electron-Phonon and Hole-Phonon Interactions in Indirect Gap Semiconductors For indirect gap semiconductors (e.g., Si, Gap, Ge) the application of an external stress can be employed not only to determine the nature of the transition (i.e., T-A(X) or T-L) and the associated deformation potentials but also to gain information about the electron-phonon (EP) and hole-phonon (HP) intervalley scattering matrix elements [43-481. In multivalley indirect semiconductors the fundamental absorption process is phonon assisted; it proceeds by two mechanisms involving EP as well as HP scattering matrix elements in such a way that for a given phonon these two processes can interfere either constructively or destructively. Therefore, the nature of this interference phenomenon inhibits the evaluation of these matrix elements separately in cubic semiconductors by measuring only one relevant parameter, for example, absorption coefficient or photoluminescence (PL). This difficulty has been overcome by the application of a uniaxial stress along appropriate crystallographic axes that reduces the symmetry of the conduction or valence bands. By measuring the stress and polarization dependence of the amplitude of phonon-assisted indirect transitions, combined with previously measured absolute values of the relevant absorption coefficient, it has been possible to evaluate the EP and HP scattering matrix elements for the transverse optical (TO) phonon in Si [43-481 as well as the longitudinal acoustic (LA) and transverse acoustic (TA) phonons in GaP [44-481. For Ge the situation was more complicated, but valuable information about the LA-phonon-assisted process was obtained [46, 481. The absorption coefficient a/of an indirect gap semiconductor, resulting from an electronic transition between states differing in energy E, and accompanied by the creation or annihilation of a phonon or the Zth branch, is given by

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR O ~ I C APROPERTIES L 265

where E is the photon energy, Q is the wave vector of a phonon of frequency wl(Q), the upper and lower signs of T and f refer to phonon emission and absorption, respectively, and nQ is the phonon occupation number [43,44, 481. L [ E - Eg ifLwl(Q)] is a line-shape function. The frequency independent (over a small photon energy range) term, Afiof Eq. (39), is related to the “strength” of the transition. The parameter A is a constant involving certain material quantities such as index or refraction and electron and hole effective masses. The term fr is the “oscillator strength” of the indirect transition between valence state *v,k and conduction state * c , p proceeding via an intermediate state ‘Pj,k(electrons) or ‘Pj,p (holes) and is given by

where Hf(Q) is the effective electronic perturbation due to the creation or annihilation of a phonon of the Ith branch having polarization t and wave vector Q, C is the unit polarization vector of the incident photon, and p is the linear momentum of the electron. Equation (40) contains the matrix elements of the EP (SL-,,,) and HP (SL-ph) interaction given by the following: SL-ph =

(‘Pi.kIHXQ)I*c,p)

(414

For diamond- and zinc blende-type materials the intermediate state for is either l?l,c(r2,,c) or r15,c, or both, depending on which phonon is involved. In Si, for example, the intermediate conduction state for the TO, TA, and LA phonons is rI5,, (the lowest-lying conduction state), whereas The intermediate state for $-ph is the top of for the LO phonon it is r2s,c. the valence band at k = 0 [48]. The solid line in Fig. 11 depicts the wavelength-modulated absorption spectrum of the TO-phonon-assisted indirect exciton in Si at 77 K for stress X = 0. Also plotted is the spectrum for X = 3.5 X lo9 dyn/cm2along [OOl] for E )IX (dashed line) and E IX (dotted line) [43,48]. The four observed peaks, B ,to B4,correspond to the transitions noted in Fig. 9. Similar results have been obtained for X )I [ l l l ] , in which case only two peaks, A l and A 2 ,are seen (see Fig. 9). From a line-shape fit to the stress data it was Se-ph

266

FREDH. POLLAK

SILICON 77OK

-> d

7

211[OOl]

X.3.51 x

-4

lo9 dyn cm-2

4-

I

-E

r

?

X

X

;I:

2-0

<;*-$.--scale ;

I 1.00

Wavelength (pm)

FIG. 11. The solid line is the wavelength-modulated absorption spectrum of the TOphonon-assisted indirect exciton in Si at 77 K for stress X = O. Also plotted is the spectrum for X = 3.51 X loy dyn/cm2 along [Ool] for E 11 X (dashed line) and E I X (dotted line). (From Ref. [43].)

possible to evaluate the amplitudes of the various transitions [i.e., B k , B h (m = 1-4) and A ! ,A t (n = 1, 2)]. Based on theoretical expressions for these amplitudes it was possible to obtain the ratio S;f_o,h/S;f_o,h. Furthermore, from the absolute absorption coefficient it was then possible to determine both S T-:, and S;f_O,,. These values were found to be in good agreement with a theoretical calculation based on the rigid ion model [45, 481. Similar results were obtained for the electron-phonon and hole-phonon matrix elements of the LA and TA phonons in Gap. For Ge the situation was more complicated because the dominant phonon-assisted transition, corresponding to the LA phonon, proceeds via both rT,c and rI5,= intermediate conduction band states [48].

IV. Piezo-optical Response of Ge and GaAs in the Opaque Region Using a rotating-analyzer ellipsometer, Etchegoin and co-workers have measured the three complex components Pll(o),P I 2 ( o and ) , PM(o)of the linear piezo-optical tensor Pijkl(w) of Ge [8, 491 and GaAs [8, 501 in the photon energy range of 1.6 to 5.6eV. The measured components of the piezo-optical tensor are in good agreement with prior Kramers-Kronig (KK) analysis of piezoreflectance data. Each component of P j j k lwas ( o )also

4 EXTERNAL UNIAXIAL STREWSEMICONDUCTOR OFTICAL PROPERTIES267

compared with a band structure calculation performed using the empirical pseudo-potential method. Improved deformation potentials d yterband, ?interband , D:, and D:,as well as the spin-exchange parameter S,, were r2 obtained for the El and El + A, transitions (see Table 111). w ) can be related to the change in The piezo-optical components P,,kl( each component of the tensor, A€,], and to the stress tensor by

Measurements were made for X along the [ill] and [OOl] directions. For each direction and each stress, two measurements were performed to find the components of ~ ( w in) the direction parallel and perpendicular to the stress. In principle only three independent measurements are needed to obtain all three components of the piezo-optical tensor. The authors have measured four configurations, only three of which are independent. The additional set of results was used to check the consistency of the data since it must lead to a linear combination of the three independent components of Pijkl(w). Another consistency criterion of the data is that the real and TABLE I11 INTERBAND( 6 t l e r b a n d , &Yband) AND ~ N T R A B A N D(0: AND 0:) DEFORMATION POTENTIALS AND SPIN-EXCHANGE PARAMETERS S, FOR THE As-A1 TRANSITIONS I N SEVERAL DIAMONDAND ZINC-BLENDE-TYPE SEMICONDUCTORS

Si

-4.6"

8.7"

5u.h

4m.h

Ge

-5.0

9.8 10.0 2 0.4' 7.5 2 0.7' 9.80 2 0.6' 6.4h 6.4h

5.89

6.1b -3.1 2 0.9' 0 2 0.5".' -6.4 2 1.5g

-5.0 2 0.3'

GaAs InSb GaSb

-4.3 2 0.3' -4.85 +- 0.59 -3.25h -4.0h

-5.6 2 0.4' 3.4 ? 0.3h.' -4.85 2 0.8#

3.9 2 0.4'.'/1.0 2 0.3'.' 10 ? I'J"'4.6 f 0.6'.' 8.0 Z 1.5X."/6.1 2 1.2'.'

Note. All values taken from Landolt-Bomstein - Numerical Dataand Functional Relationships inScience and Technology, Vols. 17a and 17b (0.Madelung, M. Schultz, and H. Weiss. eds.). Springer, New York. 1982, unless otherwise indicated. a Ref. [79]. Absolute value. Ref. 1251. El feature. ’ E land A, feature. 'Ref. [44]. a Ref. [45]. Ref. [go].

268

FREDH. POLLAK

imaginary components of AE(w) (and hence Pijkl)be related by a KK inversion. Figure 12plots the real (solid lines) and imaginary (dashed lines) compoP I 2 ( w and ) , PU(w)for Ge [49]. It was found that indeed nents of Pll(o), the real and imaginary parts of these three quantities were KK transforms of each other. Similar results were obtained for GaAs [50]. The piezo-optical coefficients PI1(a),P I 2 ( oand ) , Pa(o) for Ge [49] and GaAs [50] were calculated using the empirical pseudo-potential approach, not taking into account the spin-orbit splitting. However, the authors point out that it was not possible, for numerical reasons, to use in the calculation

4.1

-0.9

-

-5.9

F

'Q

L 0)

b Y Y

2.5

3

-.-

CI

a

-2.:

-5J

--

, I

I

-25.1

---

I

R O ~ part I Imaginary part I

-45.1

2

3

I t I t I t

-

,I

4

5

Photon energy (eV) FIG. 12. The experimentally determined real (solid lines) and imaginary (dashed lines) components of Pll(o),P,,(w), and Per(w) for Ge. (From Ref. [49].)

UNIAXIAL 4 EXTERNAL

STRESS/SEMICONDUCI'OR OPTICAL PROPERTIES

269

the small values of stress used in the experiment. Therefore for the computation of these parameters they took a value of 2000 MPa. The various calculated imaginary components (dashed lines) of P I (,w ) , P I 2 ( w ) and , P U ( w ) for Ge are compared with the measured ones (solid lines) in Fig. 13 [49]. In order to display all the results for different critical points and different components at the same time, the plots are given in arbitrary units (which are the same for each pair of theoretical and experimental points) and vertically shifted in some cases. There is overall good agreement between experiment and theory. References [49] (Ge) and [50] (GaAs) also reported results similar to Ref. [25] for the stress dependence of the energies of the El and El + Al transitions for both X )I [001] and X 11 [lll]. For the former case a polarization-dependent splitting of both the El and El + Al features were observed, similar to Fig. 10. These results were analyzed using Eqs. (37a) and (37b). These results have made it possible to evaluate the deformation potentials &?band 'interband , &, and 0:as well as the spin-exchange 6 2 parameter aj for these materials, as listed in Table 111. 7

Photon eneruy (eV)

FIG. 13. Comparison of the calculated (dashed lines) imaginary components of P l l ( w ) , Plz(o),and PU(o)for Ge with the measured ones (solid lines). (From Ref. [49].)

270

FREDH. POLLAK

V. Intrinsic Piezobirefringence in the Transparent Region The piezobirefringence (PB) of Ge [51, 521, Si [51, 521, GaAs [51, 521, GaSb [52, 531, InAs [47, 481, and InSb [52, 531 and its dispersion in the region below the fundamental absorption edge have been measured at room temperature for X 11 [Ool] and X 11 [lll]. The carrier concentration of the samples was low enough to make free-carrier PB negligible in the spectral region of the measurements. The data were analyzed using an interband one-electron model for the optical constants. For Ge and GaAs the effects of the dispersion of the Eo,Eo + Ao, E l ,El + A , , and E2 were included, whereas for GaSb, InAs, and InSb only the influence of Eo and Eo + A. was considered. The data for Si were fit with a one-electron model that includes dispersion centers at 4.4 eV ( E 2 )and 3.2 eV (Eh and El). In general these models provide a good description of the experimental data. Figure 14 displays the experimentally determined linear PB (a)for GaSb for X 11 [Ool] (open circles) and X I [ l l l ] (crosses) at room temperature as a function of reduced frequency [53]. The solid lines are least squares fits

2.0

-2tI -3

0.5

I

I

I 0.8

0.6

I 0.7

up,

(hw, = 0.74eV)

I

1

I

I 0.9

I

FIG. 14. Experimentally determined linear piezobirefringence coefficient (a)for GaSb for X 11 [OOI] (open circles) and X 11 [ill] (crosses) at room temperature as a function of reduced frequency. The solid l i e s are least squares fits to a theoretical expression that explicitly takes into account contributions from Eo and Eo + Ao. (From Ref. [53].)

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OPTICAL PROPERTIES 271

to a theoretical expression that explicitly takes into account contributions from Eo and Eo + Ao:

where hwoand noos are the energies of the Eo and Eo + A. gaps, respectively, and the functions g ( x ) and f(x) are given by the following: g(x) = ( 1 / x 2 )[ 2(1 - X ) ’ Q

-

(1 + 4 1 / 2 1

f(x= ) ( 1 / x 2 ) [2 ( 1 - x)-1/2 - (1 + x)-1/2]

(43b)

The parameter D includes the nondispersive contribution from all the higher-lying gaps to the linear PB coefficient below the fundamental bandgap. The parameter C" is given by

where P is a matrix element of the momentum between the top valence band and lowest conduction band, and p$ is a joint density of state mass. From the fit it was possible to evaluate the deformation potentials b and d.

VI. Effects of External Stress on Quantum States For all directions of external X, the shifts and polarization dependence of the quantum states originating in heavy-hole-like or light-hole-like (outplane) valence subband states will vary with the stress qualitatively similar to Eo(l) and E0(2),respectively, as discussed earlier, except for any changes related to a piezoelectric field [5,7,16,17]. The heavy-hole (HH) transitions have a small stress dependence because of the cancellation of the hydrostatic and shear components of the strain; light-hole (LH) transitions have a large stress dependence because of the addition of these components. Although LH features have a linear dependence on the stress, the HH resonances have a nonlinear component because of the strain-induced coupling to the spin-orbit split band. The HH feature is allowed for both E 11 X and E I X,whereas LH signals are seen only for E I X.

212

FREDH. POLLAK

The situation for XI1 [110] is somewhat more complicated. For this lowersymmetry direction, there is not only an induced piezoelectric field along the (001) growth direction but also a nonlinear stress dependence exhibited by the LH features due to the coupling with the spin-orbit split band as well as between the HH and LH subbands. A number of authors have reported the effects of an external uniaxial stress on the optical properties associated with the quantized states in GaAs/GaAlAs QWs and SLs [16, 54-58]. These experiments have been discussed in Ref. [5] and will not be repeated here.

1. EFFECTSOF X PARALLEL TO [001] AND [ O l l ] (PIEZOELECTRIC EFFECT)ON AN Ino.2,Gao,79As/GaAs (100) SINGLE-QUANTUM-WELL STRUCTURE The effects of large, external uniaxial stress (X) along [001] and [011] on the quantum states of a strained-layer Ino.21G~.79As/GaAs (100) single quantum well (SQW) have been investigated using photoreflectance (PR) at 300 K [17, 59, 601. For X 11 [Ool] (-8 kbar) the stress-induced shifts of the various quantum transitions are in good agreement with a theoretical calculation based on only deformation potential theory, including the straininduced coupling between LH and spin-orbit split valence bands. For this stress direction there were no significant changes in the intensities of the various PR features. The situation for X )I [110] (-18 kbar) was quite different. For this lower symmetry stress direction there was a dramatic increase in the intensities of the 12H and 13H “symmetry-forbidden” features relative to the “symmetry-allowed’’ 11H peak. There was also a significant redshift of 11H, 12H, and 13H at high stresses. The notation rnnH(L)denotes a transition from the rnth conduction to n th valence subband of heavy ( H ) or light (L )-hole (out-plane) character. This phenomenon has been interpreted in terms of an electric field generated along the [loo]growth direction induced by the piezoelectric coupling for this stress direction. The experimental energies of the various transitions are in good agreement with a theoretical calculation based not only on deformation potential effects but also a quantum-confined Stark effect (QCSE) redshift due to the stressinduced piezoelectric field. The agreement between experiment and theory (including the piezoelectric effect) shows that the full piezoelectric field is being generated across the SQW. Because of the different stress and polarization dependence of HH and LH transitions, a uniaxial stress can be used to discriminate between these two types of transitions.

4 EXTERNAL UNIAXIAL

sTRESS/sEMICONDUCrOR OPTICAL PROPERTIES

273

a

A

1.15

1.20

1.25

1.30

1.35

1.40

Energy (eV) FIG. 15. (a) Photoreflectance spectrum (solid line) of an Ino.zlG~7pAs/GaAs (001) SQW at 300 K for X = 0. The dashed line is a line-shape fit that yields the energies of the mnH(L) transitions indicated by arrows at the top of the figure. (b) The solid lines are the PR spectra for a stress of 7.78 kbar along [OOl] for E I X and E 11 X.The dashed lines are line-shape fits yielding the energies indicated by the arrows. (From Ref. [59].)

Figure 15a shows the PR spectrum (solid line) of the sample at 300 K for X = 0. The dashed line is a line-shape fit that yields the energies of the rnnH(L) transitions indicated by arrows at the top of the figure. The solid lines in Fig. 15b are the PR spectra at 300 K for a stress of 7.78 kbar along [OOl] for E I X and E 11 X .The dashed lines are line-shape fits. The obtained energies are denoted by arrows together with their identification. A comparison of Fig. 15a and 15b shows that the energies of the heavyhole transitions (11H, 12H, 13H) are relatively insensitive to the external stress (cancellation of hydrostatic and shear terms) and are seen for both E I( X and E _L X,whereas 11L has shifted to higher energies (addition of hydrostatic and shear terms) and is seen only for E I X .The stress and polarization dependence of the various transitions are similar to those observed for GaAs/GaAlAs (100) QWs and SLs for X 11 [OOl], that is, stress perpendicular to the growth direction [5]. The solid lines in Figs. 16 to 18 represent the experimental data for X 11 [Oll] for 8.52, 15.8, and 17.6 kbars, respectively. The dashed lines are line-shape fits. In Fig. 16 the polarization dependence of 11L is clearly

FREDH. POLLAK

274

1 n

- Expt.

TI1

,-shape fit

Y,

X

c,

.A

=

[0111

8.52 kbm

d 3 0 11L x 5

1

22H

M I

1

23H

,11H 12H , 13H

1.15

1.20

1.25

1.30

1.35

1.40

Energy (eV) FIG. 16. The solid lines are the experimental PR spectra of an Ino.21G%,79As/GaAs (001) SQW at 300 K for X 11 [Oll] of 8.52 kbar. The dashed lines are line-shape fits yielding the energies indicated by arrows. (From Ref. [59].)

evident. What is most significant about these spectra is the large change in the amplitude of the symmetry-forbidden 12H and 13H features in relation to the symmetry-allowed 11H (and 11L) transition. This effect is not observed for X 11 [Ool] (see Fig. 15). For the highest stress there is also a considerable redshift of the heavy-hole peaks due to the QCSE related

1

- Expt.

0

0

11H

-1 ..15

12H

1.20

13H

1.25

1.30

1.35

1.40

Energy (eV) FIG.17. The solid lines are the experimental PR spectra of an I%.21G%,79As/GaAs (100) SQW at 300 K for X (1 [Oll] of 15.8 kbar. The dashed lines are line-shape fits yielding the energies indicated by arrows. (From Ref. [59].)

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OP~ICAL PROPERTIES 275

1.15

1.20

1.25

1.30

1.35

1.40

Energy (eV) FIG. 18. The solid lines are the experimental PR spectra of an Ino.zlG~,79As/GaAs (100) SQW at 300 K for X 11 [Oll] of 17.6 kbar. The dashed lines are line-shape fits yielding the energies indicated by arrows. (From Ref. [59].)

to the piezoelectric field. This phenomenon was not observed in previous or subsequent X 11 [Oll] piezo-optical studies of QWs because of the generally low stresses (-5 kbar) that were employed [16]. The dots in Fig. 19 plot the stress dependence of the energies of the

0

I

11H

12

0

1

2

.

3

0

4

0

0

5

6

.

7

.

8

9

stress (kbar) FIG. 19. The dots plot the experimental stress dependence of the energies of the various intersubband transitions of an I~.z1Gao,79As/GaAs (100) SQW for X // [OOl]. The solid lines are a theoretical deformation potential calculation. (From Ref. [60].)

276

FREDH. POLLAK

various intersubband transitions for X 11 [Ool].The solid lines are a theoretical deformation potential calculation of the effects of the external stress using a Hamiltonian that explicitly includes the coupling with the spin-orbit split (SOS) band. The stress dependence of the energies of the transitions for X 11 [ O l l ] is displayed in Fig. 20. Note that there is a large redshift for heavy-hole-like features such as 11H, 12H, and 13H. This effect cannot be explained on the basis of the coupling with the SOS band. This large redshift, in addition to the changes in intensities shown in Figs. 16 to 18, has been explained on the basis of an electric field, F,, generated along the [ 1001 growth direction by the piezoelectric coupling 1.5

sill

1

[0111

Expt. - Calc. w i t h F 0

._

Calc. without F

1.4 n

3 Q) W

h

hD

5

1.3

c:

w

1.2

0

2

4

6

8

10

12

14

16

18

stress (kbar) FIG.20. The dots plot the experimental stress dependence of the energies of the various intersubband transitions of an In,,21Gao.79As/GaAs(100) SQW for X 11 (0111.The solid lines are a calculation based on both deformation potential theory and the QCSE induced by the piezoelectric field F,. The dashed line is the calculated energy of the 11H transition without F,.(From Ref. 1601.)

4 EXTERNAL UNIAXIAL

STRESS/SEMICONDUCTOR OPTICAL PROPERTIES

277

caused by stress along the low-symmetry [Oll] direction. If there is no interface charge this field is given by

where el4 is the piezoelectric constant, S4 is the elastic compliance conis the permittivity of free stant, K is the dielectric constant, and space. An external [Oll] stress X = 17.6 kbar generates an electric field F, = 1.85 X lo5 V/cm along the [loo] growth direction. The solid lines in Fig. 20 are a theoretical calculation that includes not only deformation potential effects, similar to Fig. 19, but also the QCSE related to F,.The dashed line is a calculation of the stress dependence of the 11H feature based on only deformation potential theory, that is, without the QCSE induced by the piezoelectric field. This strain-induced QCSE is responsible for the large redshift of 11H, 12H, and 13H. The calculation with the field also has yielded the intensities (overlap integrals) of these features. For X = 0 the intensities of the three “heavy-hole” transitions are in the ratio 100:0:2; the nonzero value for the “parityallowed” 13H feature is due to the finite depth of the QW. The straininduced electric field distorts the wave functions sothat for X = 17.6 kbar (1.85 X lo5 Vlcm) this ratio was changed to 7 : 15 :20, that is, 13H is now the most intense. This general trend is in agreement with the data of Figs. 16 to 18. TO [OOl] AND [Oll] 2. EFFECTSOF X PARALLEL (PIEZOELECTRIC EFFECT)ON A GaAs/GaAlAs (100) SINGLE-QUANTUM-WELL STRUCTURE

The effects of large X 11 [OOl] (=lo kbar) and X 11 [Oll] (=16 kbar) on a G ~ A S / G ~ ~ . ~(100) ~ A ~SQW ~ . ~structure ~ A S (lw = 100 A) at 300 K have been investigated using PR [17,61]. Measurements were made for E I X and E 11 X . For X 11 [Oll], the authors have observed a redshift of several intersubband features of the SQW, an increase in the intensities of several “symmetry-forbidden” SQW transitions, and Franz-Keldysh oscillations (FKOs) in the signal from the direct gap of the GaAlAs barrier layer. These effects were not seen for X 11 [OOl]. However, in contrast to the InGaAs/GaAs SQW discussed earlier, the magnitude of the electric field associated with the redshift (i.e., the QCSE) and FKOs is considerably smaller than the piezoelectric field deduced from the piezoelectric coefficients. This difference has been interpreted in terms of trapped charge at surfaces or interfaces.

278

FREDH. POLLAK

From the stress-induced splittings and shifts of the direct gap of the Ga,,73Alo.z7Asfor X ( 1 [Ool], it was also possible to evaluate the hydrostatic [a = -(8.6 I+_ 0.2) eV] and shear [ b= -(1.90 I+_ 0.2) eV] deformation potentials for this A1 composition. 3. DETERMINATION OF THE SYMMETRY OF EXCITONS ASSOCIATED WITH MINIBAND DISPERSION IN InGaAdGaAs (100) SUPERLATTICES Boring etal. [16, 621 have reported both experimental and theoretical demonstrations of the effects of X 11 [Oll] on an InGaAdGaAs (100) SL. This work, in which the applied stress is perpendicular to the growth axis, has revealed a mixing between the first light-hole (confined) state and second heavy-hole (unconfined) state caused by the application of the stress in the crossed configuration with regard to the growth axis. The stresses were low enough so that piezoelectric effects were not important. The fundamental sequence of the SL was 2.7 nm ( L , )of Ino.llGa,,s9As and 11 nm (LB) of GaAs repeated 20 times. Figure 21 shows the energies of a number of quantized electron (e) to heavy-hole-like (hh) and light-hole-like (Ih) features at both the minizone center (r)and edge (Z) [e.g., elhhl (r),elhhl (Z), and so forth] as a function of X 11 [Oll]. The measurements were made using reflectivity at 2 K. Note the dramatic anticrossing at about 2 kbar between the ellhl (r)and elhhz (r)transitions. This observation suggests that the Ihl (r)and hhz (r)states have the identical parities with respect to the macroscopic symmetry elements of the SL. The dashed and solid lines in Fig. 21 display theoretical calculations based on the envelope-function and tight-binding models, respectively, that include the interaction with the spin-orbit split band but neglect any piezoelectric effects. 4. ASYMMETRICAL GaAs/GaAlAs (100) DOUBLE

QUANTUM WELLS Gill etal. [63] have performed an investigation of XI1[Oll] on the excitonic levels of GaAslGaAlAs (100) double asymmetric quantum wells (DAQWs) [%I. The stresses were low enough (X 5 5 kbar) so that stress-induced piezoelectric effects were not important. The results, when compared with what is obtained from SQWs, exhibit some peculiar behavior that is typical of the asymmetric structures. Asymmetry makes it possible to obtain novel couplings between excitonic states, which cannot be accounted for without

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OPTICAL PROPERTIES 279 1

1

1

1

1

1

1

= L,

Ga

In

As-GaAs

1

1

2.7 nm

Lb = 1 1 n m T = 2 K

1530

1520 I

2 E 1

1510

P a, C

w

500

490

e,,hhl(,r1

1480 1470 0

1

2

3

4

5

6

7

8

9 1 0

Stress ( kbar 1 FIG. 21. The solid rectangles are the experimental energies of a number of quantized electron (e) to heavy-hole-like (hh) and light-hole-like (Ih) features at both the minizone center (r)and edge (2)as a function of X 11 [ O l l ] for an Ino.llGao.89As/GaAsSL. The dashed and solid lines are theoretical calculations based on the envelope-function and tight-binding models, respectively. (From Ref. [62].)

an accurate theoretical treatment of the Coulomb interaction, including intersubband mixing of the valence band wave functions. Gil and co-workers have demonstrated that under the applied strain the excitons can change their character from oblique or interwell (electron and hole wave functions peaking in different wells) to vertical or intrawell (electron and hole wave functions peaking in the same well). The samples investigated were two GaAs/Gal-,A1,As ( x 0.3) DAQWs in which a wide GaAs QW (Lw 100 A) is coupled to a narrow one (Lw 40 A) across a thin barrier ( L B 20 A or 40 A).

-

-

-

-

FREDH. POLLAK

I 0 kbar

0.4 kbar

0.E kbar 1 . 4 kbar

1.9 kbar

2.5 kbar 3.0 kbar

1.55

1.57

1.59

1. 61

ENERGY (eV> FIG. 22. Evolution of the low-energy part of the 2 K reflectance spectrum for a GaAs/ Gal-.AI,As ( x 0.3) DAQW sample with LB 20 A for several values of the applied [ O l l ] stress. Two structures, denoted (a) and (b), are observed, corresponding to the ey-hy and e7-I: excitonic transitions. (From Ref. [63].)

-

-

Figure 22 shows the evolution of the low-ener y part of the 2 K reflectance spectrum for the sample with LB 20 for several values of the applied [110] stress. Two structures, denoted (a) and (b), are observed, corresponding to the eT-hr and er-lp excitonic transitions. Here the notation is as follows: e, h, and 1 refer to electronic, heavy-hole-like and lighthole-like states, respectively, while the subscripts indicate the quantum number of the electronic states and a superscript w or n indicates whether a given level originates from the wider or narrower well, respectively. For example, the first heavy-hole level of the narrow well might give rise to the third heavy-hole level of the entire DAQW; it is thus denoted h;. The

- 1

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OPTICAL PROPERTIES 281

identification of the structures depicted in Fig. 22 is confirmed by their stress behavior. The heavy-hole exciton shifts to higher energies with a smaller slope in relation to the light-hole feature, as discussed earlier. What is shown in Fig. 22 does not seem to be very different from what has been seen for the stress dependence of SQWs (see, for example, Fig. 15). However, a careful examination shows that the (b) feature has a lower stress coefficient for X L 2.6 kbar in relation to lower stresses. The open rectangles in Fig. 23 depict the experimental values of the stress dependence of the various observed features, including higher energy resonances not shown in Fig. 22. Note that there is a change in slope at 1730 17 10 1690

1670 CI

> 0)

E 1650

v

>

$ w

-- II -

1630

-

1610

-

Z

W

I 1

I

1

1

a

1

1

!

* '

1 I

:

1

c;hz

FIG. 23. The open rectangles are the experimental values of the stress dependence of the 0.3) DAQW sample with various observed exciton features of a GaAslGal-,AI,As ( x LB 20 A for applied [ O l l ] stress. The solid lines are a theoretical calculation. (From Ref. [63].)

-

-

282

FREDH. POLLAK

X = 2.6 kbar for the light-hole features eTl7 and ezlt. The open rectangles in Fig. 24 depict the corresponding data for the sample with the wider barrier. In this case the change in slope of e7lr and ell; occurs at X = 1.5 kbar. The solid lines in Figs. 23 and 24 represent theoretical calculations of the stress dependence of the various excitonic features, taking into account not only the shift of the intersubband energies but also the variations in exciton binding energies. For example, this computation shows that e7-17 and eZ-12" exchange their slopes with e;Y-hY and eJ-hY for stresses higher than about 2.6 and 1.5 kbar for the samples with the narrow (Fig. 23) and wide (Fig. 24) barriers, respectively. Thus, these authors have observed

r

1730 1710

1690 1670 CI

> u

E

Y

1650

z 1630 W

z

W

1610 1590 1570

1550 0

1

2

3

STRESS

(kbar)

FIG. 24. The open rectangles are the experimental values of the stress dependence of the various 2bserved exciton features of a GaAs/Ga,-,Al,As ( x 0.3) DAQW sample with LB 40 A for applied [011]stress.The solid lines are a theoretical calculation.(FromRef. [63].)

-

-

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OP~ICAL PROPERTIES 283

an anticrossing of certain heavy- and light-hole features that is not seen in SQWs and is due to the asymmetry of the DAQW.

TO [OOl] ON BULKGaAs AND 5. EFFECTSOF X PARALLEL GaAslGaAlAs SINGLE QUANTLJM WELLSGROWN ON (100)

Si SUBSTRATES Microstructures of GaAs/Si (100) are under an in-plane biaxial tensile strain arising during postgrowth cooling due to the differences in their expansion coefficients. At 300 K this strain is 1.5 kbar, while at 77 K it is 2.5 kbar [17,64,64]. For bulk GaAs/Si the biaxial strain causes a shift and splitting of Eo such that Eo(l)occurs at a lower energy in relation to Eo(2). In QWs the ordering of the out-plane HH and LH subbands can be altered by varying the width of the well. In a sufficiently wide QW the ground state is a LH subband, whereas for a narrow configuration the reverse is true. The effects of large external uniaxial stress along [OOl] (-10-20 kbar) on the optical features associated with biaxially strained bulk GaAs and two GaAsfGaAlAs SQWs, with Lw of 52 A and 200 A, grown on (100) Si have been studied using PR at 77 K and 300 K [17,64,65]. Measurements were made with incident light polarized parallel and perpendicular to X . This stress configuration makes it possible to externally alter the LH and HH splitting in both the bulk material and the SQWs. In the SQW of width 200 A , the ground state valence band was continually tuned from LH to HH. In the bulk material, a stress-induced anticrossing of the Eo(l) and Eo(2) features of the fundamental gap was observed, with an interesting polarization effect. The solid curves in Figs. 25 and 26 are the PR spectra at 300 K for the Lw = 52 A sample for X ( 1 [OOl] with E I X and E 11 X , respectively. In the X = 0 traces, the features around 1.4 eV correspond to the internal strain split Eo of bulk GaAs, and the doublet in the vicinity of 1.53 eV is due to the 11H and 11L transitions in the SQW. The dotted lines are least squares fits of the data to an appropriate line-shape function that makes it possible to determine the energies of the transitions accurately. For clarity, only the fits for X = 0 and 10.4 kbar are shown. The obtained energies are denoted by arrows. At zero external stress, E0(2)lies below Eo(l) because of the built-in biaxial tensile strain. However, for this narrow SQW the large quantum confinement results in 11H being the ground state. From the Eo(2)-E0(1) splitting, a built-in tensile strain of 1.5 kbar at 300 K has been deduced. The effect of the external stress is to cause a blueshift

FREDH. POLLAK

284

I

zp

LZ=52A Calc.

, I

E&l)

,

300K

SQW

IllH I

Energy (eV) FIG. 25. The solid curves are the PR spectra at 300 K for GaAs and a GaAs/GaAIAs SQW (Lw = 52 A) fabricated on Si (100) for various X 11 [001]with E 1) X.The dashed lines are a theoretical calculation. (From Ref. [651.)

of the E0(2) and 11L features while Eo(l) and 11H remain relatively stationary. At the highest stresses, E0(2)lies above Eo(l)and the HH-LH splitting has been substantially increased. The dashed lines in Figs. 25 and 26 are a theoretical calculation of the effects of both the internal and external strains on the energy positions of Eo and the quantum transitions of the SQW, including the coupling to the SOS band. This calculation also has yielded the polarization-dependent intensities of the various features, as will be discussed later. At the highest stresses, for E I X both Eo(l), E0(2) and 11H, 11L are seen, whereas for E 1) X only Eo(l) and 11H are observed, in agreement with the previous discussion of polarization effects.

UNIAXIAL STRESS/SEMICONDUCTOR O P ~ I C APROPERTIES L 285 4 EXTERNAL

1.40

1.45

1.50

1.55

1.60

Energy (eV) FIG. 26. The solid curves are the PR spectra at 300 K for GaAs and a GaAsIGaAlAs SQW (Lw= 52 A) fabricated on Si (100) for various X (1 [OOI] with E IX.The dashed lines are a theoretical calculation. (From Ref. [65].)

For the Eo transitions there is actually an anticrossing of E0(2)and Eo(l) at about 2 kbar due to the fact that Xis not collinear with the growthhuiltin strain axis. This anticrossing also produces a very interesting polarization effect. For E IX (see Fig. 25) the intensity of the feature that started out as Eo(2) is zero and that of Eo(l)is maximum. It should be noted that in this anticrossing regime the designations Eo(l) and E0(2) are no longer valid because of the strong stress-induced coupling. The intensities of the Eo transitions as a function of X (including the built-in strain) have been calculated for both polarization directions. At X = 0 the intensities of Eo(2)and Eo(l)are in the ratio of 1:3, in agreement

286

FREDH. POLLAK

with the experimental data of Figs. 25 and 26. At a value of X = 2.4 kbar for E I X , the intensities of the features that started out as E0(2) and Eo(l) are zero and maximum, respectively. This is a consequence of the anticrossing. Note that in Fig. 25, the intensity of the lowest-lying feature is zero. At the highest stresses (10.4 kbar), E0(2) for E (1 X is essentially zero, whereas for E 1 X the ratio of Eo(l)to E0(2) is approaching 1:3. Furthermore, at the highest stress similar intensity effects should apply for 11H and 11L. These results are in good agreement with the data shown in Figs. 25 and 26. The solid lines in Figs. 27 and 28 are the PR spectra at 77 K of the Lw = 200A sample for different values of X 11 [OOl] with E I X and E 11 X,

,

’.

,

1.9

‘.

3.7

’

11H

1.48

1.50

1.52

1

1.45

1.50

1.55

1.60

1.65

1.70

Energy ( e V ) FIG. 27. The solid curves are the PR spectra at 77 K for the quantum transitions of a GaAslGaAlAs SQW ( L w = 200 A) fabricated on Si (100) for various X [I [00l] with E (1 X. The dashed lines are a theoretical calculation. (From Ref. [65].)

4 EXTERNAL UNIAXIAL STRESSISEMICONDUCTOR OPTICAL PROPERTIES 287

77K

1.9

Calc.

,

11H

b

1.48

I

1.52

1.50

kbar

33H

1

33L

-

11H

1.45

1.50

22H

1.55

11L

1.60

0

22L

1.65

1.70

Energy ( e V ) FIG. 28. The solid curves are the PR spectra at 300 K for the quantum transitions of a GaAslGaAlAs SQW (Lw = 200 A ) fabricated on Si (100) for various XI1 [001] with E I X. The dashed lines are a theoretical calculation. (From Ref. [65].)

respectively. The observed features are due to rnnH(L) quantum transitions in the SQW. For this sample the signal from Eo of the GaAs was very weak. The dotted lines are least squares fits of the data to an appropriate line-shape function. The obtained energies are indicated by the arrows. For clarity, only the fits for X = 0 and 19.6 kbar are shown. Note that at X = 0, 11L lies below 11H. The influence of the external stress is to cause a blueshift of the LH features while the HH peaks remain relatively stationary. Note that at the highest stress 11L lies above both 11H and 22H. These figures also show that at the highest stresses for E 1) X only HH features are observed, whereas for E IX both HH and LH resonances are allowed. This is in agreement

288

FREDH. POLLAK

with our previous discussion concerning the stress and polarization dependence of these various features. A theoretical calculation of the effects of both the internal and external strains and quantum confinement on the energy positions of the quantum transitions has been performed, including the coupling to the spin-orbit split band. These results are shown by the dashed curves in Figs. 27 and 28. As shown in the insets of Figs. 27 and 28, there is actually an anticrossing for 11H-11L at about 2 kbar because Xis not collinear with the quantization axis of the SQWs. The origin of this effect is thus similar to the anticrossing of E0(2)-E0(l) in Figs. 25 and 26. It should be noted that at the anticrossing the designation HH or LH is no longer valid because of the strong mixing. This anticrossing also produces a related polarization-dependent effect similar to that discussed earlier. The intensity of the lowest-lying SQW feature, which started as 11L for X = 0, is very weak for E IX and maximum for E II x. OF CONDUCTION STATES FOR GaAs/AlAs (001) 6. SYMMETRY TYPEI1 SUPERLAITICES

In the following discussion, we have changed notation to be consistent with the reported work. Thus, in this section the SL is (001). While most uniaxial stress experiments on 111-V QWs or SLs have dealt with quantum transitions associated with the direct gap of the bulk constituent material, Lefebvre et al.[17, 66, 671 have determined the symmetry of the conduction band states related to the X point in the BZ (see Fig. 2 ) for type I1 GaAs/AlAs (001) SLs using both on-axis (X11 [Ool]) and inplane (X /( [llO]) stresses. From their stress dependence the symmetry of the conduction ground state is labeled in terms of X, or X,,y conduction states. In some structures, X, is the conduction band ground state and the ordering between X, and X,,y is reversed by in-plane stress. In others, the situation is reversed and an explanation of the results needs to take into account the lattice matching of AlAs slabs to GaAs. The shear tetragonal deformation potential (6;) of the X states in AlAs has been obtained. In contrast to the more classical GaAs/Gal-,A1,As ( x < 0.3) system, the “barrier” material in a GaAs/AlAs SL has an indirect gap, with the lowest conduction band of Xz symmetry (see Fig. 2). The relative ordering and spacing of each pair of extrema (rg and X;) in the bulk materials, together with the magnitude of the valence-band offset between the GaAs and AlAs, act in such a way that both type I and type I1 SLs can be fabricated, depending on the relative thicknesses of the GaAs and AlAs layers. However, in type I1 SLs the GaAs is the barrier for electrons and

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OITICALPROPERTIES 289

the well material for the holes. In this latter case the electron wave functions are composed of bulk states of Xg symmetry. The question that was addressed by this experiment was the nature of the electron ground state: X, (band along the growth direction) or (bands perpendicular to the growth direction). The perturbation of the XC,bulk conduction band states depends on the orientation of the stress, as discussed in Section II.2.c and shown schematically in Fig. 29, assuming X, to be the ground state. For compressive X 11 [loo], that is, along the growth direction, the shear component of the stress causes the X, level to move down in energy at a rate of -(2/3)6 while the X,,ylevels increase in energy at a rate of (1/3)6, where S = c&(Sl1- Slz).On the other hand, for X 1) [011] (i.e., perpendicular to the growth axis) the motion of the levels is reversed, with the rates shown becomes in the figure, so that for a given stress there is a crossing and the ground state. Figure 30 shows the conduction band shifts for the two stress orientations for a sample with a 4.34-nm period in which the AlAs constituted 60% of the period. The measurements were made using PL at 2K. Identifying these results with the theoretical sketch of Fig. 29, the authors deduced that (1)the ground state was Xz-like, (2) S 8.2 2 0.1 meV/kbar, so that gZ = -5.1 1- 0.7 eV for AlAs, and (3) a zero-stress splitting occurred between the X,-and Xx,y-likestates of -6.5 meV. The origins of the weak nonlinearities are discussed in Ref. [66].

-

I 1

I I

L I [ 110 1 [ 001 1

UNIAXIAL STRESS FIG. 29. Schematic behavior of the three zone-edge-relaxed XS conduction bands under [lo01 (on-axis) and [Oll] (in-plane) applied uniaxial stress. (From Ref. (661.)

FREDH.POLLAK

290

5

++

- 10

*+

n101

\

P d . 3 4 nm g=0.6 T=2 K

to 011

2

1 STRESS

3

(kbaf)

FIG. 30. Contribution of the shear part of the strain field on the conduction band shifts for [00l] (pluses) and [110] (crosses) stress orientations for a GaAslAlAs (001) SL sample with a 4.34-nm period in which the AlAs constituted 60% of the period. (From Ref. [MI.)

For a sample with a 1-nm period and the AlAs again being 60%of the period, a similar experiment revealed that the situation was reversed from that of the material just discussed and that the ground state had X,,?-like character. OF THE BAND ALIGNMENT IN 7. DETERMINATION Si,-,Ge,/Si (100) QUANTUM WELLS

Houghton etal.[68] have deduced a type I band alignment from a lowtemperature PL experiment in which [lo01 and [110] compressive and tensile uniaxial stresses were applied to Sil-,Ge,/Si QWs (0.15 5 x 5 0.38) using a beam-bending technique. Signals were observed from both the

4 EXTERNAL UNIAXIAL STREWSEMICONDUCTOR OFTICALPROPERTIES 291

QWs and the Si layers in the samples. To interpret their experiment they calculated the effects of both the internal strain and external stress on the valence and conduction bands of the system. The conduction band (CB) configurations for type I (electron confinement in the SiGe QWs) and type I1 (electron confinement in the Si barriers) are illustrated in the inset of Fig. 31. Spatially direct type I transitions are shown as solid (Si) and dotted (SiGe) arrows. For a type I1 CB, the transition is spatially indirect (dot-dashed arrows). For the QWs the effects of the coherent internal strain and quantum confinement are such that the highest valence band is the 1312, t 3/2) band, which in this chapter is denoted as the v; state, while the lowest conduction states are the degenerate Aolo and A l W levels. As discussed earlier for the valence bands of bulk Si, the effect of an external [loo] or [110] stress is to make v113/2, 2 1/2) or v213/2, ? 3/2) the ground state for compression or tension, respectively. However, for the A conduction bands compressive [loo] stress makes Alw the minimum energy state and tensile [loo] stress makes Aolo and Awl the minimum energy states, whereas for [110] external stress the opposite is true. With [loo] stress the type I and type I1 configurations are hard to distinguish since under either tension or compression the lowest Si and SiGe are

-0.4

0.0

-0.2

X

0.2

0.4

(GPa) Compression Tension FIG. 31. The solid lines show the calculated shifts of the lowest transition energy for bulk Si for both compressive (vl-AllXI) and tensile ( V ~ - A , , Awl) ~ ~ . external [lo01 stress. The dotted lines show the lowest calculated SiGe type I transitions for compression (v;-Alo0) and tension (v;-Aao), ignoring confinement. The dashed line shows the lowest SiGe transition for a 2.5-nm MQW under tension. The solid circles are the experimental data from the bulk Si portions of the sample. The open symbols are PL shifts from 2.5-nm (triangle) and 3.7-nm (circles) QWs with a Ge composition of 15%. (From Ref. [68].)

292

FREDH. POLLAK

of the same type and shift downward at almost the same rate. The solid lines in Fig. 31 show the calculated shifts of the lowest transition energy for bulk Si for both compressive (vl-Aloo) and tensile (v2-Aol0, Awl) external [loo] stress. The dotted lines show the lowest calculated SiGe type I transitions for compression (v&A and tension (vi-Aolo), ignoring confinement. The dashed line shows the lowest SiGe transition for a 2.5-nm multiple quantum well (MQW) under tension. The solid circles are the experimental data from the bulk Si portions of the sample. The open symbols are PL shifts from 2.5-nm (triangle) and 3.7-nm (circles) QWs with a Ge composition of 15%. It was found that the results were not very sensitive to Ge composition. The experimental results are in agreement with the theory. For [110] stress the situation is different. The Aolo, Aloe degeneracy is not lifted so that, for low tensile stress, the SiGe PL peak shifts upward as the SiGe Aolo, Aloe and the SiGe v; valence band (W) move apart. If the CB were of type I1 character, the lowest-energy transition would be Si Awl to SiGe v;, as illustrated in the inset of Fig. 31, giving a negative shift under low [110] tensile stress (dot-dashed line in Fig. 32). The solid lines in Fig. 32 are the calculated shifts for bulk Si for the vl-Aolo, A l and ~ v2-Am1 transitions. For compression the former is the lowest energy state; for tension, the latter. The solid circles are the experimental data from the bulk Si part of a sample. The dotted, dashed, and dot-dashed lines in Fig. 32 show the lowest calculated transitions for SiGe type I ignoring confinement, SiGe type I for a 1-nm QW under tension, and Si-SiGe type I1 under tension, respectively. Figure 32 displays the experimental PL shifts from a number of SiGe QW samples. Sample A ( x = 0.15) consists of 2.5-nm (open circles), 3.7-nm (open triangle base down), and 6.0-nm (open diamond) QWs. Sample B ( x = 0.15) consists of 2-nm (open triangle base up) and 4-nm (open squares) QWs. The plus symbols are from sample C (6.0-nm QW, x = 0.20), and the crosses are from sample D (1.2-nm QW, x = 0.38). Note that for tensile stress all the PL features from the MQWs shift to higher energies while the bulk Si data exhibit a redshift. The authors have deduced a type I CB offset from the above experiment. However, Thewalt ef al.[69] have called this interpretation into question and presented an experiment that indicates a type I1 CB alignment in coherently strained Si,-,Ge,/Si (100) QWs. They studied the PL energy shifts under both (100) biaxial and [Oll] uniaxial external stresses. They conclude that the CB offset is type 11, even though the PL observed in all previously published literature, such as Ref. [63], is type I. Their results on a 3-nm wide Si0.7Ge0.3/Si (001) single QW are summarized in Fig. 33, which shows the energy shifts of the Sil-,Ge, no-phonon (NP)

4

EXTERNAL UNIAXIAL

-0.3 -0.2

STRESSISEMICONDUCTOR OPTICAL PROPERTIES

-0.1

0.0

0.1

02

293

0.3

X

Tension (GPa) Compression FIG. 32. The solid lines show the calculated shifts for bulk Si for the v , - A , ~ ~ AIW ~ , , and VZ-AWI transitions. For compression the former is the lowest energy state; for tension, the latter. The solid circles are the experimental data from the bulk Si part of a sample. The dotted, dashed, and dot-dashed lines show the lowest calculated transitions for SiGe type I ignoring confinement, SiGe type I for a 1-nm QW under tension, and Si-SiGe type I1 under tension, respectively. Also displayed are the experimental PL shifts from a number of SiGe QW samples. Sample A (x = 0.15): 2.5 nm (open circles), 3.7 nm (open triangle base down), 6.0 nm (open diamond) QWs; Sample B (x = 0.15): 2 nm (open triangle base up), 4 nm (open squares) QWs. The plus symbols are from sample C (6.0 nm QW, x = 0.20) while the crosses are from sample D (1.2 nm QW, x = 0.38) (Ref. 68).

line versus stress for a wide range of excitation intensities, as measured relative to the position of the NP line at zero external stress and the lowest excitation level used. The inset shows schematically, for zero CB offset, the type I and I1 transitions (solid arrows) for zero external strain, in which case the transition energies are equal, barring small excitonic corrections that will lower the type I energy. Tensile [ O l l ] uniaxial or (100) biaxial strain reduces the band splittings in the QW while removing the band degeneracies in the Si (dashed horizontal lines), thus raising the energy of type I transitions while lowering those for type I1 (dashed arrows). Note that for type I transitions the stress effects on both the CB and VB add to produce a large blueshift, whereas for the type I1 transition the redshift is due to the difference between the large decrease in the Si CB edge and the smaller increase in the VB energy in the QW. The results of their detailed calculations of the strain dependence of the type I and I1 transition energies are shown in the body of Fig. 33 as solid and dashed lines, respectively. Note that only the slopes of the lines are significant; the type I and type I1 curves were shifted vertically to match

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50

0

50 100 Tensile Strain (UPa)

150

200

FIG. 33. Energy shifts of the NP lines over a wide variety range of excitation energies versus applied tensile stress. Energies are relative to the NP line at zero external strain and illumination with lO-’I,, (I, = 10Wlcm’). The solid (dotted) lines are theoretical predictions for a SiO.,Ge,,,JSi(001) QW. The inset illustrates the effects of external strain on type 1/11 transitions (Ref. 69).

the high and low excitation data, respectively. For both types of stress, the high-excitation data show good agreement with the blueshift calculated for type I transitions, which also is in agreement with previously reported results and is consistent with the relatively high excitation levels used in those earlier studies. However, as the excitation level, and thus the charge accumulation, is lowered, we observe a strong deviation from type I behavior above some characteristic stress that decreases with decreasing excitation. This signals the changeover from type I to type I1 PL. The 10-5Z0 data show a type I blueshift only at very low stress, and then a shift down in energy as predicted for type I1 PL. At lo-’ Zo the PL is redshifted for all nonzero stresses, which strongly suggests that the actual band alignment is type 11. The excitation-level dependence of the transition energy at fixed strain seen in Fig. 33 is exactly what would be expected for a band-bending induced changeover from type I1 to type I PL in a system having a type I1 CB offset that increases with increasing stress. Considering the behavior at high (001)

4 EXTERNAL UNIAXIAL STRESS/SEMICONDUCTOR OFTICAL PROPERTIES 295

stress, at low excitation the energy is not strongly dependent on the excitation level, since the electric fields are still small. Then, as the type I1 to type I changeover is approached, the energy rises rapidly for small increases in excitation and levels off again when in the type I regime. This is because of the much smaller band bending once extra electrons are allowed to enter the QW, together with much shorter lifetimes for the type I situation. Thewalt and associates have verified that when we observe type I PL (with or without strain), the PL lifetime is in the ps to sub-ps region, as is typical for Si bound exciton and normal Si,_,Ge, QW PL. However, when type I1 PL dominates, the observed PL decay time was in the ms region, up to 10 ms at the lowest excitation. These very long experimental PL decay times emphasize the need for extremely low excitation levels in order to avoid the effects of band bending-with 10 ms lifetime, and assuming 100% carrier collection into the QW, only 4 pW/cm2 are required to produce an electron-hole pair density of 10" cm-*, which can cause substantial energy shifts.

VII. Summary This chapter has reviewed the effects of external uniaxial stress and biaxial strain on the optical properties associated with the intrinsic electronic states of diamond-, zinc blende-, and wurtzite-type semiconductors. Using deformation potential theory, the effects of homogeneous deformations at critical points at k = 0 (direct bandgap) and k # 0 have been analyzed. For the latter situation conduction band minima at A 1 and L1 as well as rz5,(rlS)-Al indirect and (111) direct gap transitions were considered. The influence of the spin-exchange interaction in conjunction with the applied stress or strain was considered for both cubic and hexagonal materials. The polarization effects created by the reduction in symmetry of the cubic materials also were analyzed. Uniaxial stress (or built-in strain in the case of GaN) experiments on bulk and thin-film materials, based on the considerations just mentioned, have yielded important information about deformation potentials, the symmetry of interband transitions, spin-exchange parameters, intervalley E P and H P interactions, and piezo-optical coefficients for both the transparent and opaque regions. Many of these experimental studies have been compared with theoretical calculations of the relevant quantities. External stress investigations of quantum transitions in QWs and SLs have also been fruitful. Work has been done on a number of systems, including GaAs/GaAlAs, strained-layer InGaAsIGaAs, and SiGe/Si.

2%

FREDH. POLLAK

These studies have made it possible to determine the symmetry of excitons associated with miniband dispersion, the properties of asymmetrical GaAsIGaAlAs double quantum wells, and the band alignment in SiGe/Si QWs (although there is still some controversy about the interpretation of these investigations). The application of large stresses along the [Ol 11 direction has produced QCSE redshifts and intensity variations due to the stress-induced piezoelectric field along the growth direction of the microstructures. The study of stress along [Ool]on internally strained bulk GaAs and GaAslGaAlAs QWs fabricated on Si (100) has revealed a very unusual polarization-dependent effect. While there has been considerable work done on the quantum states of the systems just mentioned, to date there have been no reports concerning the influence of external stress or strain in other heterostructures, such as II-VIs (either wide- or narrow-gap), Sb-based III-Vs such as GaSb/GaAlSb and InGaSb/GaSb, or mixed II-VI/III-V configurations such as ZnSe/GaAs. To date only one experiment dealing with external stress on wurtzite GaN has been published. Therefore, there is still much work to be done in this field.

Acknowledgments The support of National Science Foundation grant #DMR-9414209, PSClBHE grant #666424, and the New York State Science and Technology Foundation through its Centers for Advanced Technology program is gratefully acknowledged.

REFERENCES [l] G. E. Bir and G .L. Pikus, in SymmetryandStrain Effects inSemiconductors, John Wiley, New York. 1974. [2] I. Balslev, in Semiconductors and Semimetals, Vol. 9 (R. K. Williamson and A . C. Beer, eds.), Academic Press. Boston, 1972, p. 403. [3] F. H. Pollak, SurrSci. 37, 863 (1973). [4] D. W. Langer, R. N. Euwema, K. Era, and T. Koda, Phys.Rev.B 2,4005 (1970). [5] F. H. Pollak, in Semiconductors and Semimetoh, Vol. 32 (T. P. Pearsall, ed.), Academic Press, Boston, 1990, p. 17. [6] A. K. Ramdas and S. Rodriquez, in Semiconductors and Semimetals, Vol. 36 (D. G. Seiler and C. L. Littler, eds.), Academic Press, Boston, 1992, p. 137. [7] F. H. Pollak, Proc.Mat.Res.SOC.Symp.405, 3 (19%). Solidi (b)D8, 5 (1996). [8] M. Cardona, Phys.Stat. andSemimetals, vol. 52, ed. by J. I. Pankove and T. D. Moustakas [9] B. Gil, in Semiconductors (Academic Press. Boston).

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[lo] E. Anastassakis, in Dynamical Properties of Solids (G. K. Horton and A. A. Maradudin, eds.), North Holland, New York, 1980, p. 161. [ l l ] E. Anastassakis and M. Cardona, this volume. [12] I. J. Fritz, R. M. Biefeld, L. R. Dawson, and T. E. Zipperian, in Semiconductors and Semimetals, Vol. 24 (R. Dingle, ed.), Academic Press, Boston, 1987, p. 459. [I31 J. Y. Marzin, J. M. Gerard, P. Voisin, and J. A. Brum, in Semiconductors and Semimetals, Vol. 32 (T. P. Pearsall, ed.), Academic Press, Boston, 1990, p. 56. [14] R. People and S. A. Jackson, in Semiconductors and Sernimetals, Vol. 32 (T. P. Pearsall, ed.), Academic Press, Boston, 1990, p. 119. [15] H. Temkin, D. Gershoni, and M. Panish, in Semiconductors and Sernimetals, Vol. 40 (A. C. Gossard, ed.), Academic Press, Boston, 1994, p. 338. [16] B. Gil and P. Boring, Jpn.J. Appl.Phys.32, Suppl. 32-1, 93 (1993). [17] F. H. Pollak and H. Qiang, Jpn.J. Appl. Phys.32, Suppl. 32-1, 101 (1993). and Electron Physics, Vol. 8 (L. Marton, ed.), [18] H. Brooks, in Advancesin Electronics Academic Press, New York, 1955. [19] C. Herring and E. Vogt, Phys.Rev.101, 933 (1956). [20] W. H. Kleiner and L. M. Roth, Phys.Rev.Lett. 2, 334 (1959). [21] J. C. Hensel and G. Feher, Phys.Rev.129, 1041 (1963). [22] H. Hasegawa, Phys.Rev.129, 1929 (1963). [23] E. 0. Kane, Phys.Rev.178, 1368 (1970). [24] J. R. Chelikowsky and M. L. Cohen, Phys.Rev.B 14,556 (1976). [25] M. Chandrasekhar and F. H. Pollak. Phys.Rev.B 15,2127 (1977). [26] A. Blacha, H. Presting, and M. Cardona, Phys.Stat. Sol. (b)126, 11 (1984). [27] J. A. Verges, D. Glotzel, M. Cardona, and 0. K. Anderson, Phys.Stat. Sol. (b)113, 519 (1982). [28] 0. H. Nelson and R. M. Martin, Phys.Rev.B 32, 3792 (1985). [29] N. E. Christensen and 0. B. Christensen, Phys.Rev.B 33,4739 (1986). [30] M. Cardona and N. E. Christensen, Phys.Rev.B 35, 6182 (1987). [31] C. G. Van de Walle, Phys.Rev.B 39, 1871 (1989). [32] A. Qteish and R. J. Needs, Phys.Rev.B 45, 1317 (1992). 59, 501 (1987). [33] D. D. Nolte, W. Walukiewicz, and E. E. Haller, Phys.Rev.Lett. [34] L. Laude, F. H. Pollak, and M. Cardona, Phys.Rev.B 3, 2623 (1971). Conference on the [35] J. E. Rowe, M. Cardona, and F. H. Pollak, in 1967International II-VISemiconducting Compounds(D. G. Thomas, ed.), W. A. Benjamin, New York, 1967, p.112. [36] J. E. Rowe, M. Cardona, and F. H. Pollak, SolidStareComm. 6, 239 (1968). [37] A. A. Yamaguchi, Y. Mochizuki, C. Sasaoka, A. Kimura, M. Nido, and A. Usui. Appl. Phys.Lett. 71, 374 (1997). [38] A. Shikana, T. Azuhata, T. Sota. S. Chichibu, A. Kuramata, K. Horino, and S. Nakamura, J. Appl. Phys.81,417 (1997). [39] I. A. Buyanova, J. P. Bergman, B. Monemar, H. Amano, and I. Akasaki, AppL Phys. Lett. 69, 1255 (1996). [40] A. Alemu, B. Gil, M. Julier, and S. Nakamura, Phys.Rev., 857, 3761 (1998). Comm. 10,417 (1972). [41] W. Wardzynski and M. Suffczynski, SolidState [42] P. G. Rohner, Phys.Rev.B 3, 433 (1971). [43] 0. J. Glembocki and F. H. Pollak, Phys.Rev.B 25,1193 (1982). [44] 0. J. Glembocki and F. H. Pollak, Phys.Rev.B 25,1179 (1982). 48,413 (1982). [45] 0. J. Glembocki and F. H. Pollak, Phys.Rev.Letts. [46] 0. J. Glembocki and F. H. Pollak, Phys.Rev.B 25,7863 (1982).

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[47] 0. J. Glembocki and F. H. Pollak, Proc.16th Int.Con$ Physics of Semiconductors, Montipellier, 1982 (M. Averous. ed.). North Holland, Amsterdam, 1983, p. 546. on [48] F. H. Pollak and 0 . J. Glembocki. Proc.of theNATO AdvancedStudyZnstitute Elecrronic Structure, Dynamicsand Quantum Structural Properties of CondensedMatter (J. T. Devreese and P. Van Camp, eds.), Plenum Press, New York, 1985, p. 451. [49] P. Etchegoin, J. Kircher, M. Cardona, and C. Grein, Phys.Rev.B 45,11721 (1992). [SO] P. Etchegoin, J. Kircher, M. Cardona, C. Grein, and E. Bustarret, fhys. Rev.B 46, 15139 (1992). [51] C. W. Higginbotham, M. Cardona, and F. H. Pollak, Phys.Rev.184,821 (1969). of Opfo-Electronic Materials (W. A. Albers, [52] M. Cardona and F. H. Pollak, in The Physics Jr., ed.), Plenum, New York, 1971, p. 81. (531 P. Yu. M. Cardona, and F. H. Pollak, fhys. Rev. B 3, 340 (1971). [54] C. Jagannath, E. S. Koteles, J. Lee, Y. J. Chen, B. S. Elman, and J. Y. Chi, Phys.Rev. B 34,7027 (1986). [S5] A. Sooyakumar, A. C. Pinczuk, A. C. Gossard, D. S. Chemla, and L. J. Sham, Phys. Rev.Lett. 58, 1150 (1987). [56] J. Lee, C. Jagannath, M. 0.Vassell, and E. S. Koteles, Phys.Rev. B 37,4164 (1988). [57] B. Gil, P. Lefebvre. H. Mathieu, G. Platero, M. Alterelli, T. Fukunaga, and H. Nakashima, Phys.Rev.B 38,1215 (1988). [58] R. T. Collins, L. Vina, W. I. Wang, C. Maihiot, and D. L. Smith, Proceedings ofthe Society of Photo-optical Instrumenfation Engineers, Vol. 792. SPIE, Bellingham, 1987, p. 2. [S9] H. Qiang. F. H. Pollak, C. Mailhiot, G. D. Pettit, and J. M. Woodall, Phys.Rev.B 44, 9126 (1991). [60] H. Qiang, F. H. Pollak, C. Mailhiot. G. D. Pettit, and J. M. Woodall, SurjSci.267, 103 (1992). Comm. 84,51 (1992). [61] H. Qiang, F. H. Pollak, and R. N. Sacks, SolidState [62] P. Boring, J. M. Jancu, B. Gil, D. Bertho. C. Jouanin, and K. J. Moores, Phys.Rev.B 46,4764 (1992). [63] B. Gil, P. Lefebvre, P. Bonnel, H. Mathieu. C. Deparis, J. Massies, G. Nu, and Y. Chen, Phys.Rev.B 47, 1954 (1993). [64] H. Qiang, F. H. Pollak, K. Shum. Y. Takiguchi, R. R. Alfano, S. F. Fang, and H. MorkoG, Appl.fhys. Lett. 60, 2651 (1992). [65] H. Qiang, F. H. Pollak, K. Shum, Y.Takiguchi, R. R. Alfano, S. F. Fang, and H. Morkoq, Phil. Mag. 870, 381 (1994). (661 P. Lefebvre, B. Gil, H. Mathieu, and R. Planel, fhys. Rev.B 39, 5550(1989). [67] P. Lefebvre, B. Gil, H. Mathieu, and R. Planel, fhys. Rev.B 40, 7802 (1989). [68] D. C. Houghton, G. C. Aers, S. T. Eric Young, E. Wang, and N. L. Rowell, Phys.Rev. Lett. 75,866 (1995). [69] M. L. W. Thewalt, D. A. Harrison, C. F. Reihart, J. A. Wolk, and H. Lafontaine, Phys. Rev.Lett. 79,269 (1997). [70] H. Qiang, F. H. Pollak, and G. Hickman, SolidState Comm. 76, 1087 (1990). (711 N. Lipshitz, A. Jayaraman, R. A. Logan, and R. Maines, Phys.Rev.B 20,2398 (1979). [72] S. Logothetidis, M. Cardona, L. Tapfer, and E. Bauser, J. Appl.Phys.66,2108 (1989). Rev.B 1,672 (1970). [73] A. Gavini and M. Cardona, fhys. 174) F. Meseguer. J . C. Merle, and M. Cardona, SoLidState. Comm. 50,709 (1984). Rev.B 24,4652 (1981). [75] B. A. Weinstein, R. Zallen. M. C. Slade. and A. DeLozanne, fhys. fhys. SolidState 7, 1881 (1966). [76] A. A. Kaplianski and L. G. Suslina, Soviet [77] K. Murase. K. Enjouji, and E. Otsuka, Jpn.J. Appl.Phys.29, 1255 (1970).

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SEMICONDUmORS AND SEMIMETALS, VOL. 55

CHAPTER5

Semiconductor Optoelectronic Devices A.R.Adams and M. Silver DEPARTMENT OF Pmwcs UNIVERSITY OF SURREY GUILDFORD. UK

J.Allam HITACHI CAMBRIDGE LABORATORY HITACHI EUROPE LTD CAVENDISH LABORATORY CAMBRIDGE. UK

I. INTRODUCTION.................

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

11. EXPERIMENTAL CONSIDE 111. SEMICONDUCTOR LASERS .. ........................................................ I . BasicLaserConcepts .................... ependence ................ 2. LaserCharacteristics: TheirPressure and ................................. IV. UNIAXIAL STRAINEFFECTS: STRAINED-LAYE V. HYDROSTATIC PRESSUREMEASUREMENTS OF OTODIODES: THEB A N D - S T R U ~ JDEPENDENCE RE OF IMPACT IONIZATION ................................. 1. Introduction ........................................................ 2.Physics of ImpactIonization inSemiconductors .................................................. 3. Pressure Results ..................................................

5.Conclusions .................................... VI. SUMMARY ......................................................................................................................... Acknowledgments References ................................................................

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

301 302 305 305 308 326 327 327 329 332 344 347 348 349 349

I. Introduction Semiconductor optoelectronic devices now play an essential role in many aspects of our daily lives, such as communications and information technology. The efficient operation of such devices depends critically on their exact electronic band structure, which, as described in earlier chapters, can be changed continuously and in a most convenient manner by the application 301 Vol 55 ISBN 0.12-752163-1

SEMICONDUCTORS AND SEMIMETALS Copyright 0 1998 by Academic P ~ c r s All rights of reproduction in any form reserved WX01X7X4198 $25 00

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of hydrostatic pressure. This technique enables a better understanding of the basic physical processes to be obtained and allows the design of optimum structures. In this chapter we will first consider semiconductor lasers that make use of the efficient radiative recombination provided by direct bandgap III-V semiconductors and their alloys. We give a very brief introduction to basic laser characteristics and then describe how pressure measurements have been used to study trends in device characteristics. These measurements show that there are three important regimes of bandgap. At small bandgaps, corresponding to wavelengths longer than about 1.3 pm, which are important for optical fiber communication, pressure measurements show that the threshold current is dominated by Auger recombination. This process decreases with increasing bandgap so that in GaAs-based lasers operating around 860 nm (as used, for example, in standard optical pickup systems such as CDs), we find that the threshold current increases linearily with pressure as expected when only spontaneous emission occurs. At still larger bandgaps we consider visible red lasers operating around 650 nm, which have been developed for such applications as high-density data storage on video discs or for communications using plastic fibers. Their pressure dependence shows that, because the direct bandgap is almost as large as the indirect bandgap, efficient operation becomes limited by thermally activated leakage currents via the conduction band satellite minima. While hydrostatic pressure influences the direct bandgap and the structure of the conduction band, it has very little influence on the valence band. In contrast, in-built strain in quantum well lasers primarily affects the valence band. This has been used so effectively to improve the operation of almost all types of semiconductor lasers that its effects are described briefly for completeness. Avalanche photodiodes are one of the most successful forms of semiconductor optical detectors. The avalanche process gives rise to internal gain when the photogenerated carriers are drawn into a region of very high electric field where they are able to multiply by impact ionization. Highpressure studies of such diodes have made it possible to investigate their dependence on band structures and thus to propose a “universal model” to explain and predict the voltage at which impact ionization occurs. This result is not only significant for optoelectronic devices but also for all electronic devices using high electric fields for maximum power generation.

11.

Experimental Considerations

In this chapter we primarily describe the large influence on device performance caused by the changes with pressure of the direct bandgap, the r

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valley to satellite valley separations in the conduction band, and the influence of the uniaxial strain on the heavy-hole to light-hole splitting. These have already been described in detail in earlier chapters; however, there are certain techniques and other aspects of high-pressure measurement that should be kept in mind when dealing with optoelectronic devices, and these are described briefly here. Modern semiconductor optoelectronic devices have complex internal structures, so that, for example, a separate-confinement, multiple-quantumwell laser may consist of four or more different alloy compositions in a complicated sequence of quantum wells and heterojunctions to provide carrier confinement and optical wave guiding. However, 111-V materials with the same lattice spacing have almost the same compressibility [l], so hydrostatic pressure induces very little uniaxial strain in lattice-matched structures and the movements of the band extrema in each layer may be considered to be the same as if it were bulk material. Early work by Kirchoefer etal. [2] on AIGaAs/GaAs quantum well lasers bonded with In to TO-18 headers sometimes showed kinks in the observed rate of change of photon energy with pressure, which they associated with the effects of shear strain generated by the n-type and p-type confining layers as hydrostatic pressure was applied. In strained-layer structures the hydrostatic pressure might be expected to reduce the shear strain, since materials with larger lattice constants are more compressible. However, this effect will be very small and has not been identified as having any effect up to the maximum pressures of about 30 kbar used here. The change with pressure of the band offsets also can be a concern. In GaAs/AI,Ga,-,As the valence band offset has been measured to vary linearly with x and to increase by (1.1 2 0.2) meV.kbarr' [3]. In the InP/InGaAs and GaAdGaAsSb heterojunctions there is no measurable change in valence band offset within experimental accuracy. These three results are in good agreement with the changes predicted by the theory of Van de Walle and Martin [4]. Finally, there will be changes in the quantum confinement energy of electrons in quantum wells due to the change in well depth caused by the different pressure coefficients of the well and barrier materials and due to the increasing effective mass with bandgap in the well. In all the results presented here, the effects listed above proved much smaller than the influence on the performance of Fabry-Perot lasers caused by the changes with pressure of the direct bandgap and operating wavelength, the r valley to satellite valley separations in the conduction band, and the effects of uniaxial strain on the heavy-hole to light-hole splitting. Electrical contact to the device is normally made through gentle pressure contacts to unbonded laser chips or with thin gold bond wires to avalanche photodiodes. For lasers this means that current pulses of less than one

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microsecond must be used at a low repetition rate if device heating is to be avoided. Great care must be taken if devices soldered to heat sinks or standard device packages are used, since the sample can be subjected to unwanted stresses caused by the different compressibilities of the very different materials involved. Nevertheless, the authors have found that if softer solders such as indium are used, measurements on devices bonded to heat sinks can be made that are reliable and give the same results as unbonded chips. In exceedingly skillful measurements, Patel etal. have succeeded in making measurements on electrically pumped lasers within the confines of a diamond anvil high-pressure cell [5]. This required bringing leads through the gasket and using a 45-degree mirror to deflect the beam. A short laser with a thinned substrate was used, but the pressure could be measured using ruby luminescence techniques. However, almost all the results presented graphically here were obtained using a liquid pressure-transmitting medium in a piston and cylinder apparatus. The pressure was determined from the resistance of a calibrated manganin gauge, and the electrical leadthroughs were insulated using Vespel seals. Immersion in the liquid presented no problems for measurements on avalanche photodiodes; however, it causes a change in the reflectivity at the end facets of the laser and can produce a significant change in the threshold current and related characteristics. In lasers with uncoated facets, an increase in threshold current is observed due to a decrease in reflectivity caused by the refractive index of the liquid. However, in devices with antireflection coatings, a decrease in threshold current may be observed. Furthermore, since the refractive index of the liquid increases with increasing pressure, this will also influence the pressure dependence of the device characteristics. The magnitude of these effects depends on the length of the laser and the exact design of the optical cavity [see Eq. (3)]. As an example, for normal devices we see an increase of threshold current of between 5 and 10% when immersed in pentane. This corresponds to a change in refractive index of the external medium from 1to 1.35.A pressure of 10 kbar increases this further to about 1.44 [6], which will therefore be expected to produce an increase in threshold current of less than 3% provided the laser is not operating close to gain saturation. These changes are small compared with those brought about by the effect of pressure on the electronic band structure of concern here. The change in refractive index of the III-V material itself can be seen as a decrease in the lasing wavelength of distributed-feedback lasers [7] and in vertical cavity surface emitting lasers [8] since in these devices the lasing wavelength is determined by the optical cavity. The change in refractive index is an order of magnitude less than that of the liquid and normally can be ignored in Fabry-Perot lasers.

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Since temperature fluctuations of a few degrees had a relatively large effect on the breakdown voltage of the photodiodes and on the threshold current of some lasers, the temperature of the system was monitored with a thermocouple and a long settling time (up to about 1hour for photodiodes) allowed between changing the pressure and making a measurement. The temperature dependence of the devices at atmospheric pressure was determined in a separate experiment, and the results from the pressure run were corrected for temperature fluctuations. The measurements were made for increasing and decreasing pressures, and no hysteresis was observed. Optical fibers, which have been designed to be integrated with the types of semiconductor optoelectronic devices described here, make ideal highpressure optical windows [9]. One way that this can be achieved is to remove the plastic cladding from the end of the fiber and glue it with epoxy resin into a close-fitting hole through the center of a steel plug that is then sealed in the usual manner using aluminum foil and the appropriate amount of unsupported area. It is then possible to use commercially available couplings to plug the other end of the fiber directly into an analytical apparatus, such as an optical spectrum analyzer, which can be several meters from the highpressure cell. This provides excellent and relatively trouble-free optics. At Surrey the fiber is taken, together with the electrical leads, through the top, moving piston of the piston and cylinder apparatus. The devices are also mounted directly onto the same piston and can therefore be tested prior to insertion into the high-pressure cell. Because of the flexibility of the fibers the sample can be taken freely in and out of the cell without any change in the optical arrangement. Although the pressure within the cell can be measured by standard instruments such as a manganin wire gauge, for many laser experiments the pressure is a dummy variable and really we are interested, for example, in the variation of threshold current with lasing wavelength. Indeed, since a laser emits strongly in a narrow line that closely follows the bandgap energy, which itself has a large and normally well-known pressure coefficient, the change in laser emission wavelength can often provide a convenient and accurate measure of the pressure, provided the temperature is kept constant.

In. Semiconductor Lasers 1. BASICLASERCONCEPTS

To obtain lasing action from a forward-biased p-n junction, it must be able to perform two essential functions. First, it must confine the injected

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electrons and holes within the active region so that effective population inversion can be achieved. Second, it must provide an optical cavity to contain the photons stimulating the lasing emission. Figure 1 shows schematically how this is achieved in a single-quantumwell, separate-confinement heterostructure (SQW-SCH) laser. Figure l a shows the band alignments assuming an undoped, unbiased structure. The lowest bandgap (E,) material is the quantum-well active region that holds the high density of electrons and holes and produces optical gain. In order

LA?!f

I

T

FIG. 1. The band alignments in an undoped single-quantum-well, separate-confinement heterostructure laser. (a) An undoped structure. (b) The refractive index profile and the resulting confined optical mode (dotted line). (c) With n- and p-doped cladding regions forward biased to produce population inversion in the quantum well.

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to produce the beneficial quantum confinement effects, the well normally has a thickness of less than 10 nm. This is not enough to achieve sufficient optical guiding, and so on each side of the well there is a region of intermediate bandgap whose width is an appreciable fraction of the lasing wavelength within the refractive semiconducting material. The outer cladding regions of the structure normally consist of the material with the widest bandgap (Egc)available within the alloy system that can be grown with good electrical, optical, and structural quality. The resulting refractive index profile is shown in Fig. l b together with the profile of the guided optical wave, indicated by the dotted line. It should be noted that the optical confinement factor (t), the fraction of the light overlapping the active quantum well region, is only about 1%and increases as the wavelength of the light decreases and the guided mode becomes more strongly confined. A practical example of an SQW-SCH laser is one operating at 1.5 pm with a well of InGaAs, an SCH region of InGaAsP, and outer cladding regions of InP. These outer regions are heavily doped so that the structure forms a p-n junction. This doped structure, forward biased for lasing action, is illustrated in Fig. lc. Assuming the electron-electron and electron-phonon scattering is very much faster than electron-hole recombination, we can extrapolate the Fermi level for electrons in the heavily n-doped cladding region as a quasiFermi level (F,) across the SCH region and into the well. A similar argument applies to the quasi-Fermi level for holes (F"). The work of Bernard and Duraffourg [lo] showed that transparency for photons of energy Ephoccurs when F, - F, = E p h

Practically, this means that the junction must be forward biased until F, - F, > E,

where E, is the lowest allowed transition energy within the well, that is, E , plus the sum of the electron and hole quantum confinement energies. At energies greater than F, - F, the photons experience net absorption, but in the energy range F, - F, > Eph> E, the photons experience a net gain due to stimulated emission. This gain is sufficient to cause lasing action when it is equal to gth =

a

+ (1/L) ln(1lR)

(3)

308

A. R. ADAMS, M. SILVER,AND J. ALLAM

a is the loss along the optical cavity caused by intervalence band absorption [ll]and other free-carrier absorption that does not regenerate an electronhole pair, and by scattering out of the guide by imperfections. The second term is the photon loss due to emission out of the end facets of the laser. L is the length of the laser and R the facet reflectivity. At a particular temperature, g t h is reached at a particular threshold current density ( J t h ) and carrier density (nth). As the current is increased above J t h , the photon density in the cavity quickly increases until the extra rate of stimulated emission equals the extra rate of carrier injection. Thus, above threshold, the carrier concentration and gain remained pinned at n t h and g t h , respectively, and all additional current goes into stimulated emission. This is why lasers are more efficient than light-emitting diodes. Because almost none of the current below Jth goes into stimulated emission, it is a major objective of all laser design to make J t h as low as possible; the value of Jth thus dominates the rest of the discussion on lasers. We will see from the results of the hydrostatic pressure measurements described later that the behavior of J t h depends critically on the value of E, and on the value of Egc- (F, - Fv).

2. LASERCHARACTERISTICS: THEIRPRESSURE AND TEMPERATURE DEPENDENCE Dependence of theLasingWavelength a. Pressure The measured variations of the lasing wavelength with pressure for a variety of Fabry-Perot lasers are shown in Fig. 2. In normal edge-emitting lasers, as described earlier, the length of the optical cavity ( L )is between about 200 pm and 1 mm. This means that the energy separation of the Fabry-Perot modes is of the order of a few meV. Thus when a plot of lasing wavelength against pressure over many kbar is made, as shown in Fig. 2, it appears to vary quite smoothly as the peak of the gain spectrum moves from one longitudinal mode to the next. Although some laser parameters such as intervalence band absorption [ll]will change with pressure, calculations taking into account the rate of gain (&) and increase in optical confinement factor ( 6 )with bandgap show that 12th and P t h are almost independent of pressure. As a result, the peak of the optical gain, and thus the photon energy, follows very closely the variation of the bandgap with pressure. To illustrate this point, Fig. 2 shows the results obtained for a large variety of different 1.5-pm-wavelength laser structures. As can be seen, they all have the same pressure coefficient within experimental accuracy. Also shown are the results for two representative 1.3-pm devices and,

309

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

GaInP +1% 685nm

140

+ GaInP +1% 672nm 0 InGaAsP

Bulk 1.3pm o InGaAsP +1% 1.3pm o InGaAsP Bulk 1S p m A InGaAs unstrained 1.5pm v InGaAs +0.6% 1S p m a InGaAs +0.8% 1S p m DInGaAs -1.12% 1.5pm rn GaAsP -0.82% 760nm

120 100

9 4 80 W

M

.-

3’

60 40 20

0

0

2

4

6

8

10

12

14

16

Pressure (kbar) FIG. 2. The change with hydrostatic pressure of the photon energy from a selection of laser structures.

for contrast, results on a 980-nm InGaAslGaAs laser and two visible red lasers that have distinctly different bandgap pressure coefficients. b. Pressure and Temperature Dependence of theThreshold Current

Semiconductor lasers are operated in a wide range of environments and at extremely high power densities. Therefore, their temperature characteristics are often the limiting factor in their range of application and so are of considerable technical and commercial importance. Unfortunately, it is often found that performance degrades swiftly with increasing temperature above room temperature. The reasons for this are complex, and to unravel

310

A. R. ADAMS, M. SILVER, AND J. ALLAM

the causes it is extremely useful to consider their temperature and pressure dependencies together. As we will see, the causes for the temperature variation of the threshold current depends on the operating wavelength. We will therefore divide the discussion into three sections that consider devices of increasing complexity. First we will consider lasers operating in the 860 to 980 nm range, in which the threshold current density J t h is controlled by spontaneous radiative recombination alone. Second, we will consider visible red lasers, in which the temperature dependence of J t his strongly influenced at room temperature by additional thermally assisted electron leakage currents. Finally, we will study infrared lasers (1.3-1.5 pm), in which the threshold current is dominated by Auger recombination at room temperature. Lasersoperating intheWavelength rangeof 860to980 nm. The close lattice matching afforded by the GaAs/AlGaAs system facilitated the first fabrication of III-V direct bandgap heterostructure semiconductor lasers. These devices emit close to 860 nm and have been extensively investigated both experimentally [ 121 and theoretically [13]. Hydrostatic pressure measurements have proved a useful tool for demonstrating that these GaAs-based devices have a performance that is very close to ideal. Pressure dependence:For an ideal laser the internal optical loss is effectively zero; therefore, the device is working close to optical transparency. Furthermore, for an ideal laser the threshold current may be assumed to be due purely to spontaneous emission. The threshold current density can then be approximated in the Boltzmann statistics regime to

where nthis the threshold carrier density, B is the bimolecular recombination coefficient, and e the electron charge [14]. For bulk devices,

where m, and mh are the effective masses for electrons and holes, respectively, and T is the absolute temperature. The application of hydrostatic pressure increases the bulk bandgap of the laser ( E g )by about 10 meV. kbar-’. Both B and the electron effective mass are proportional to E,,whereas the hole mass is largely insensitive to the change in band-

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

311

gap [15]. As a result, the bandgap dependence of the radiative current density is

where Bo and no are independent of the bandgap. A similar calculation can be done for a quantum well to give

where L, is the quantum well width. The resulting bandgap dependence of the current density is therefore

where Boow and nOQWare independent of the bandgap as before. In reality, the bandgap dependence of the radiative current in a quantum well may be a weaker power than 2, because quantum wells require relatively high material gain to reach threshold compared with bulk devices. This results in the carrier density being less sensitive to bandgap, as we shall see later. From this simple analysis, we expect the radiative recombination current for bulk and quantum well laser diodes to increase with pressure. This has been confirmed by experimental results such as those shown in Fig. 3, which presents the variation of the threshold current normalized to the atmospheric pressure values for a bulk GaAs laser up to 8 kbar pressure. The theoretical variation of J t h from Eq. (6)shows good agreement with experiment. The agreement between theory and experiment indicates that the threshold currents of GaAs lasers are dominated by spontaneous emission and that they are close to ideal. From this analysis it is possible to optimize the design of the laser for low threshold current (for example, see Ref. [16]). As we will see in Section IV, one way to improve the lasing characteristics is to use strained quantum wells as the active region. Compressive strain can be most conveniently achieved by the introduction of indium to form InGaAs quantum wells. This has the added advantage of decreasing the bandgap and allowing lasing to be achieved at 980 nm. Commercially this is very important, since such diodes can be used for pumping erbium-doped optical fiber amplifiers.

A. R. ADAMS, M. SILVER, AND J. ALLAM

312

1.15 r

1

---

1.10

0 Bulk GaAs laser experiment Theory for lossless laser ( E F )

c,

.-t

0

a

0

0 t

20

,,"0

,, ,' ,' 0

0

0

1.05

0.95

0

2

4

6

a

Pressure (kbar) FIG. 3. Variation of the normalized threshold current with pressure for a bulk GaAs laser. Also shown is the predicted variation for an ideal lossless device.

Temperaturedependence: The increase of threshold current density with temperature can usually be fitted over a useful range of temperatures by an exponential of the form [17]

where T is the absolute temperature and Tois a characteristic temperature often quoted in the literature that can be found from

As a result, low values of Tocorrespond to a large variation of the threshold current with temperature. It is now possible to deduce an upper value for To when the threshold current is dominated by radiative recombination. For an ideal quantum well in the Boltzmann regime [14],

313

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

BO Jth= B(T)n(T)* = -(noT) T

= BonzT

where Bo and no are constant with temperature. This gives

This equation gives the maximum To at 300 K for a laser dominated by radiative transitions as

To = 300 K

(13)

which is close to the maximum that is measured in experiments [18, 191. Visible lasers.Temperature and pressure dependence: Figure 4 shows the measured variation of J[h with temperature for two quantum well lasers [20]. One contained quantum wells with 1%compressive strain and operated at a wavelength of 670 nm at 293 K; the other laser had wells with 0.7% tensile strain and operated at 635 nm. Both had (Alo.~G~.6)0.51no.5P separate-confinement wave guide regions between (A10.7G%.3)o.51no.5Pcladding regions. As can be seen from Eq. (ll),when f t his dominated by radiative recombination we would expect that Jth T.This appears to be obeyed well by both lasers at temperatures below about 250 K. However, above this temperature Jth begins to increase quite swiftly, indicating the onset of a new current path. What is more, we see that the effect is much larger in the 635-nm than in the 670-nm wavelength device, indicating that the

1

~

0

635nm theory 670nm expt 670nm theory

1

4

i

i

i

r’

0

100

200

300

400

Temperature (K) FIG. 4. Experimental and calculated temperature dependence of the threshold current density in a 635-nm QW laser with 0.7% tensile strain and a 670-nm QW laser with 1 % compressive strain.

A. R. ADAMS, M. SILVER, AND J. ALLAM

314

process is strongly dependent on the bandgap of the active region and therefore that it is a clear candidate for high-pressure studies. Figure 5 shows the measured pressure variation of the threshold current density as the temperature is increased from 308 to 338 K in a laser of bulk G%.51no.5Pbetween (Alo.7G~.3)0.51no.5P cladding regions operating at 655 nm [20]. As can be seen, Jth becomes more temperature sensitive as the bandgap is increased with increasing pressure, in agreement with the results of Fig. 4. What is more, Fig. 5 provides information about how the absolute value of Jth increases with increasing pressure at a constant temperature. To understand the results shown in Figs. 4 and 5, it is useful to consider the band-structure profile at lasing threshold illustrated in Fig. lc. The device essentially retains many of the characteristics of a simple p-n junction in which the current has the form Jd = Jo[exp(eV/KT)

-

11

(14)

where V is the applied forward bias, which is F, - F, as shown in Fig. lc. The sue of the forward-bias diffusion current Joexp(eV/KT) is controlled by the potential barrier presented to carriers by the cladding regions. From Fig. l c and using the sign convention shown there, this is &e = E , - AEp - (F,- F,) for electrons and Ebh = Egc+ BE, - (F,- F,) for holes. We therefore have a leakage current that will increase almost exponentially as the bandgap of the active region, and hence (F,- F,), increases. Furthermore, the (Alo.7G~.3)o.sI~.5P cladding regions have an indirect bandgap with the X minima lowest, therefore, Egcdecreases with

0

1

2

3

4

5

6

Pressure (kbar) FIG. 5. Experimental and calculated temperature and pressure variation of the threshold current density of a visible bulk GaInP/ AlGaInP laser.

5 SEMICONDUCTOR OPTOELEC~RONIC DEVICES

315

pressure as F, - F, increases. This leads to a swift decrease in the potential barriers Ebeand Ebh. The full experimentally and theoretically determined model used to fit to the results shown in Figs. 4 and 5 is far too complex to be dealt with here; the reader should turn to Ref. [20] for the details. Briefly, the direct and indirect bandgaps and their pressure coefficients were first determined by photoluminescence measurements in a diamond anvil cell as described in Chapter 7 of this volume. The conduction and valence band offsets were also determined by the same method so that, for example, one could take account of the thermal escape of electrons from the well that meant that n # p in the well region and band bending occurred. The temperature and pressure dependencies of the refractive indices of the structure, and hence of c, were calculated and thus the magnitude and variation of the threshold gain determined. The r conduction band and the quantum-confined valence subband states were calculated using k.p theory, and bandgap renormalization effects were also taken into account. Finally, having calculated the distribution of electrons and holes throughout the structure, the thermal leakage current was calculated using a simple diffusion-type model of the form

eDn,l J ,= In tanh(x,ll,,) for electrons, where nC1is the electron density at the edge of the p-cladding region that was determined assuming Boltzmann statistics, xpis the distance to the metal contact, and 1, is the effective diffusion length of electrons in the p-cladding region. The calculated threshold currents are shown in Figs. 4 and 5 and, as can be seen, good agreement with the temperature and the pressure dependencies of all three types of device is obtained. The work just described shows how hydrostatic pressure measurements can be used t o determine the magnitude of carrier leakage currents that limit the wavelength and temperature range possible in a given materials system. Earlier work had shown how the thermal excitation of carriers to satellite minima already begins to affect the threshold current when F, - F, is increased in narrow GaAdAlGaAs quantum wells just due to the increased confinement energy [21]. More recent work has determined the effect of tensile strain, temperature, and hydrostatic pressure on GaAsP/ AlGaAs single-quantum-well lasers and again showed the crucial role played by the X minima [22].

Long-wavelength devices. Lasers operating in the 1.3- to 1 . 5 - ~ mwavelength range have great technological importance because they are used

316

A. R. ADAMS, M. SILVER, AND J. ALLAM

in long-haul optical communications networks [23]. These networks use silica-based optical fibers that have zero dispersion at 1.3 pm and their minimum optical loss at 1.55 pm. Laser devices that operate in this longwavelength regime are fabricated in the InP-based materials system using either ternary InGaAs material or InGaAsP quaternary material as their active layer. It was observed early in their development that they performed in a very different manner to the GaAs-based lasers operating in the 980to 860-nm wavelength range [17]. In particular, the threshold current density increases much more quickly with temperature and at room temperature is nearly an order of magnitude larger than that predicted for radiative recombination alone. This poor performance has serious implications for the application of these lasers in communications modules because they must be thermoelectrically cooled, thus greatly adding to the cost and maintenance problems of the system. In addition, relatively heavy-duty drive circuitry is required to generate the large currents required for threshold. There has therefore been intense activity over more than a decade or so to try and overcome this problem. The introduction in the late 1980s of lasers with strained-layer quantum wells did produce impressive reductions in the threshold current of these lasers compared with bulk devices, resulting in submilliamp threshold currents. However, the temperature sensitivity remained, with To no greater than 100 K in 1.5 pm devices [24], whereas for 980- to 860-nm GaAs-based devices, as we have seen, values of To up to 300 K have been reported. Temperature andpressure dependence:The large temperature sensitivity of the threshold current of all long-wavelengthInP-based devices implies that their characteristics are not dominated by radiative recombination. Instead, there must be another intrinsic process that is strongly bandgap dependent, since it does not appear in the shorter-wavelength devices. Thus, hydrostatic pressure measurements again provide a very useful means to study the bandgap dependence of the dominant mechanisms responsible for the threshold current and its temperature sensitivity. Figure 6 shows the variation of the threshold current with pressure of a 1.48-pm unstrained InGaAsP quantum well laser. The pressure range up to 27 kbar corresponds to a wavelength shift from 1.48 to 1.18 pm. As pressure is applied, there is at first a rapid decrease in the threshold current, It,,, but the decrease is slower at higher pressures, where I,,is less than 50% of its value at atmospheric pressure. This is very different from the performance of all the GaAs-based devices described earlier, which showed an increase of the threshold current with pressure. This behavior reinforces the view that the threshold current of long-wavelength devices is dominated by neither radiative recombination nor carrier leakage. Furthermore, it is clear from Fig. 6 that the mechanism decreases very strongly with increasing bandgap,

5 SEMICONDUCTOR OFTOELECTRONIC DEVICES

317

wavelength (pm) 1.48

1.40

1.30

1.20

Pressure (kbar) FIG. 6 . Variation of the threshold current with hydrostatic pressure up to 27 kbar for a long-wavelength quantum well laser.

immediately explaining why it is not significant in the shorter-wavelength devices. There has been much controversy about the mechanisms controlling the performance of long-wavelength devices. For example, proposed explanations include a possible large contribution to the threshold current from stimulated emission [25], from intervalence band absorption (IVBA) [ll], from differential gain effects [26, 271, and from leakage currents [28]. Indeed, Patel etal.[29], studying 1.3-pm compressively strained QW lasers in a diamond anvil cell, observed a decrease in Jthup to about 12 kbar followed by an increase to nearly the atmospheric pressure value above 20 kbar. This was explained in terms of the change in differential gain, optical losses, and barrier leakage currents. However, although some of these effects may be important in some circumstances, high-pressure measurements indicate that generally the threshold current is dominated by nonradiative Auger recombination [30-341. Augerrecombination: Auger recombination is best illustrated by looking at the band structure of a typical semiconductor in reciprocal k-space

A. R. ADAMS, M.

318

~ L V E R AND ,

J. ALLAM

(a) band-to-band

S

(c) phonon-assisted

(d) phonon-assisted

" cv I

FIG. 7. Main Auger processes in long-wavelength material. An electron-hole pair recombines across the bandgap to excite either an electron high into the conduction band, or a hole deep into the valence spin-orbit band. In the phonon-assisted processes the electron-hole pair recombine across the bandgap via an intermediate state (I) by emission or absorption of a phonon.

as shown in Fig. 7. It can be thought of as a three-particle process in which a conduction electron and hole recombine across the bandgap and excite either (a) a conduction electron high into the conduction band (CHCC), or (b) a hole deep into the spin-orbit valence band (CHSH). This is a nonradiative process since an electron and hole have recombined across the bandgap without producing a photon. Furthermore, the process must conserve energy and momentum, resulting in a large change in wave vector

5

SEMICONDUCTOR OPTOELEmRONIC

DEVICES

319

between the initial and final states for the process, as shown in the figure. However, it is also possible to aid momentum conservation by interaction with a phonon via an intermediate state (I), as shown in Fig. 7 for (c) the phonon-assisted CHCC process and (d) the phonon-assisted CHSH process. It is interesting to note that Auger recombination is the reverse of impact ionization, in which a very energetic electron or hole gives up its energy to excite an electron across the bandgap, thereby generating an electronhole pair. It is this process and its variation with bandgap that is the main concern of the second half of this chapter about photodetectors. Several attempts have been made to calculate Auger recombination rates in long-wavelength devices, but there are many problems still outstanding. However, by using Boltzmann statistics it is straightforward to derive a simple expression for the three-particle Auger recombination current. For the two-electron, one-hole Auger process, such as the CHCC process, this leads to [30]

while for the two-hole, one-electron Auger process, such as the CHSH process, the current is of the form

where CCHccand CcHsHare the Auger recombination coefficients in units of cm6ss1. In general, therefore, the Auger current varies as the cubed power of the carrier density. The carrier density dependence of the threshold current has been identified using spontaneous emission measurements [33, 341. The integrated spontaneous emission ( L ) from buried heterostructure devices was measured as a function of current below threshold, as illustrated in Fig. 8a. As one would expect, the integrated spontaneous emission pins at threshold as the carrier density pins, and no change in the spontaneous emission spectrum is seen above 19 mA for the device shown. Since L is proportional to the radiative current, we can rearrange Eq. (4) to obtain

We can therefore use this as a probe to discover the dependence of the threshold current on n. We assume some general form,

320

A. R. ADAMS,M. SILVER, AND J. ALLAM

1.20

1.30 1.40 1.50 wavelength (prn)

1.60

14.0 3.5

-

3.0

-

2.5

-

.Strained

1.5krn device

threshold

A

v e

z?

d

2.0

-

1.5 I 1.oo

0

threshold

(22.95)

1.20

1.40 Log (L’”)

1.60

1

FIG. 8. Integrated spontaneous emission (15) as a function of current ( I )below threshold for a long-wavelength buried-heterostructure laser. Note that when the carrier density pins at threshold, so does the spontaneous emission.

where z identifies the dominant contribution. For example, if radiative recombination dominated, z = 2, whereas if Auger recombination dominated the current near threshold, z = 3. By plotting log(1) against log(L1lz) it is possible to find z from the gradient of the line. Figure 8b shows a plot of log(Z) against log(L”’) for a 1.5-pm wavelength quantum well device. The slope of the line is close to 3, showing that I 0~ n3,which, as described above, is indicative of Auger recombination. This appears to be true for currents down to I&.

5 SEMICONDUCTOR OFTOELECTRONIC DEVICES

321

Pressure measurements, such as those shown in Fig. 6, can be interpreted further to help determine which particular Auger process is active because the different mechanisms are expected to have characteristic variations with bandgap. Figure 9 shows the variation of the normalized threshold current density as a function of bandgap deduced from the pressure measurements in Fig. 6. Also shown are a number of theoretical calculations. The solid line is the calculated variation of the threshold radiative current density with bandgap. It only accounts for about 10% of the threshold current density at zero pressure and increases marginally with pressure in a manner similar to that seen for the 860- to 980-nm GaAs-based devices. The measured decrease of the threshold current density with pressure is therefore dominated by the decrease of Auger recombination with increasing bandgap. From Eqs. (16) and (17), this could be due to a decrease of either the cubed carrier density or of the Auger coefficient itself. We Wavelength (prn) 1.0

-

?$- t;\ *.

.-c 0.8 rn

C

a E 0.6 ? L

*(* '.'.',.

---_-

r,.

;,

*.

-0

.Experiment Radiative (InAD) .._ ..... CHCC-dr ,+, I CHCC-ph + ICHSH-dr + I,, - - - - - - CHSH-ph + I,

I **.

\.*

a;,

\. *> '\.*

3

0 .c

O 0.4

C 0 .40

s 0.2

'**

-

-

0.0 800

*+. '. .'-*'.-

'.. -.-..

-.-.. *---..._...

850

900 950 1000 Emission energy (mev)

2 . .

--1050

FIG. 9. Variation of the normalized threshold current density for the laser in Fig. 6, along with a number of theoretical curves. The theoretically calculated radiative current is shown as a solid line and increases with bandgap. Also shown is the calculated threshold current assuming that the remainder of the current not attributed to radiative recombination (about 90%) is associated with one of the four main Auger processes.

322

A. R. ADAMS, M. SILVER, AND J. ALLAM

first consider the cubed threshold carrier density for both a two-electron, one-hole process (CHCC) and a two-hole, one-electron process (CHSH). Over a bandgap range corresponding to pressures of 0 to 27 kbar, it is found that the cubed carrier density is virtually constant. At first this may seem surprising since we have seen, for example in Eq. (7), that ?zth for an ideal laser with a threshold close to transparency increases with m, and hence with bandgap. In addition, quantum wells require large material gains to reach threshold due to their small optical confinement factor. This would lead to a further increase in the pressure dependence of the threshold carrier density because the material gain decreases with increasing bandgap [35].However, in the structure considered here, these effects are almost entirely cancelled out by the increase in the optical confinement factor (t), which increases with the square power of the bandgap [36]. As a result the cubed threshold carrier density remains almost constant with pressure. The relative insensitivity of the cubed carrier density to pressure implies that the reduction in the nonradiative current density must be due to a decrease with bandgap of the Auger coefficient itself. Figure 9 includes theoretical calculations that predict the variation of the threshold current density with pressure, assuming that the remaining 90% of the current can be attributed to one of the four dominant Auger processes. Simple analytical expression can help to understand the variation of these processes with bandgap. The band-to-band Auger process can be expressed as a thermally activated process, given by [28, 37, 381

where Ct is independent of temperature, but includes the terms due to the Coulomb interaction matrix between the initial and final Auger states. A E is an activation energy which, for parabolic bands, is of the form

where m,, mh,and m, are the electron, hole, and spin-orbit masses, respectively, and & is the spin-orbit splitting energy. These simple expressions show that the Auger coefficient will decrease with bandgap. In addition, the exponential term suggests that the Auger coefficient could be very

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

323

sensitive to the band structure and temperature, and hence bandgap, especially for large AE. For the phonon-assisted Auger processes, cph( T ) also depends on the band structure, varying as [39]

where El is the energy associated with the forbidden intermediate state, x = hwo/kT, tho is the longitudinal optical (LO) phonon energy, and

C$, is a constant that is also proportional to the Coulomb interaction matrix between the initial and final Auger states, El is given by

for the CHCC and CHSH phonon processes, respectively. Since El is much larger than the phonon energy in long-wavelength devices, it can also be seen that the phonon-assisted Auger coefficients will decrease with increasing bandgap. However, since the phonon processes do not contain an exponential term as in the band-to-band case, they are far less sensitive to the band structure and temperature. The results of a more sophisticated calculation for the Auger recombination coefficient [40], and hence threshold current, are shown in Fig. 9. The threshold current density derived from the CHCC band-to-band process decreases very rapidly with bandgap compared with experiment. As a result, the predicted threshold current would be almost entirely due to radiative recombination above a bandgap energy E, = 950 meV. This is caused by the large activation energy associated with the CHCC process. However, the CHSH band-to-band process has a weaker bandgap variation, in better agreement with experiment. This is because, at the k vectors involved in the Auger process, mh > m, = m,. Therefore the activation energy for the CHCC process, as given in Eq. (21), is approximately proportional to E,,whereas that for the CHSH process is approximately proportional to (E, - A0)/2.As a result, the CHSH process has a smaller activation energy, giving it a weaker bandgap dependence. The phonon-assisted processes also have relatively weak bandgap variations. The relaxation of the k-selection rule mediated by momentum from phonons makes these pro-

324

A. R. ADAMS, M. SILVER, AND J. ALLAM

cesses generally far less band-structure dependent, and hence they have a considerably weaker bandgap dependence. The evidence from the pressure experiments suggests that we can rule out the band-to-band CHCC Auger process as the dominant process at long wavelength. The number of candidates can be reduced further by eliminating the phonon-assisted CHSH process. This is made possible by considering the absolute values of the Auger coefficients, which depend both on the magnitudes of the band structure terms and on the matrix element terms. For the band-to-band processes it has been generally believed that the CHSH process dominated over the CHCC process in the long-wavelength range [30]. This was despite theoretical results that showed that the overlap integral for a CHCC process was two orders of magnitude larger than for a CHSH process [41]. However, it could be argued that despite the larger CHCC overlap integral, its exponential term would greatly reduce the magnitude of the resultant Auger coefficient compared with the CHSH process. For the same reason, the CHCC process decreases more rapidly with bandgap than the CHSH process, as seen in pressure measurements such as those shown in Fig. 9. For the phonon-assisted processes this is not the case because the two processes have very similar variations with bandgap, and hence the band structure terms in the Auger coefficient are of the same magnitude. However, the matrix elements of the two phonon processes still differ by two orders of magnitude. As a result, we can use the relative magnitudes of the overlap integrals to infer that the phonon-assisted CHCC process will dominate over the phononassisted CHSH. This analysis leaves the two possible dominant processes in long-wavelength quantum well devices as the band-to-band CHSH process and the phonon-assisted CHCC process. Further work is required to identify which, if either, is the dominant of these two processes. Despite this final problem, the high-pressure measurements and their theoretical analysis provide strong evidence that Auger recombination is the dominant contribution to the threshold current density of long-wavelength lasers. This has important implications for the temperature sensitivity of these devices. Tofornear-infrared lasers:It is now possible to deduce an upper value for To when the threshold current is dominated by Auger recombination. The threshold current is now proportional to nth( T)3where it can be shown from Eq. (4)that for an ideal laser nth( T) is proportional to T. In addition, we use the generalized exponential form of the Auger coefficient in Eq. (15) to derive an analytical expression for To of the form [42]

T To = AE 3+-

kT

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

325

In the very best possible case, the Auger coefficient would be temperature insensitive and hence A E = 0. This gives the maximum To for an Augerdominated laser as 300 To =- 3 = lOOK

which is close to the maximum measured by experiment. More realistically, the Auger coefficient has a temperature dependence that for both the phonon-assisted processes and for low-activation-energy band-to-band [43] processes gives AE a value not far from kT,and hence we find T o = -300 - -75K 3+1

which is close to a typical value for a good long-wavelength device [24]. A more detailed analysis involving a nonideal temperature dependence of the quantum well carrier density shows that this can cause further reductions in To.

C. Summary of thePressure and Temperature Dependencies of Semiconductor Lasers

Hydrostatic pressure measurements on semiconductor lasers reveal clear trends in the intrinsic characteristics associated with their electronic band structure. In long-wavelength (15 1 . 3 pm) lasers, the bandgap is sufficiently small that Auger recombination provides the dominant current path, leading to a characteristic temperature, To 5 100 K at 300 K. The Auger recombination current decreases swiftly as the bandgap is increased by increasing pressure in a manner that suggests that the particular Auger process in operation is either the band-to-band CHSH process or the phonon-assisted CHCC process. As a result, when we investigate lasers operating in the 980- to 860-nm range, no evidence of Auger recombination can be found. Their characteristics are close to ideal, with a threshold current that increases with pressure and To 300 K at 300 K, as is expected when only radiative recombination is occurring. When we go to yet wider bandgap devices, for example, the visible red lasers considered in detail earlier, a new problem arises because the direct bandgap is approaching the indirect bandgap to the X minima. It then becomes impossible to provide a cladding region with a bandgap that is sufficientlylarge to prevent the thermal leakage of electrons out of the active region and into the p-type cladding region. This causes To to again decrease and become similar to

326

A. R. ADAMS, M. SILVER,AND J. ALLAM

that for the longer-wavelength lasers. However, now the threshold current is found to increase quite quickly with hydrostatic pressure. These results show that it will be difficult to obtain higher-temperature operation in InGaAIP-based devices operating at wavelengths shorter than about 630 nm. This fact forms one of the driving forces for developing new materials systems such as GaInN, with its much higher indirect band minima.

IV. Uniagial Strain Effects: Strained-Layer Lasers In Section I11 we have often referred to measurements made on strainedlayer lasers. This is because it is possible to improve the characteristics of semiconductor lasers considerably by deliberately growing the quantum well in a state of strain [44,451. The effects, which occur primarily in the valence band, are complicated and have been reviewed elsewhere [46]. Basically, strain has two major effects on lasers. 1. The strain splits the degeneracy of the light- and heavy-hole bands. This reduces the density of states at the top of the valence band and therefore reduces the carrier concentration required to achieve population inversion. 2. The uniaxial strain distorts the cubic symmetry of the electronic bonds between the atoms and helps match the symmetry of the wave functions at the top of the valence band to the one-dimensional symmetry of the laser beam. Thus, tensile strain raises the energy of the lighthole band with respect to the heavy-hole band. This results in increased gain for the transverse magnetic (TM) mode and suppresses the spontaneous emission of photons in other directions or polarizations [47]. Conversely, compressive strain raises the heavy-hole band above the light-hole band, producing increased gain for the transverse electric (TE) mode. Since both effects cause injected holes to be channeled more efficiently into those electronic states in which they can interact more effectively with the laser emission, they (a) reduce the threshold current, (b) increase the efficiency, (c) increase the operating frequency, (d) reduce the linewidth, and (e) reduce the chirp [48]. Figure 10 shows a compilation by Peter Thijs [24] of the measured strain dependence of threshold current density per well in long-wavelength (1.5 pm) InGaAs(P) quantum well lasers. The reduction in threshold current density with both tensile (left) and compressive strain (right) can be clearly seen. The maximum threshold current density occurs in layers with a small tensile mismatch in which the effects

327

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES 6

0’ -3

-2

Tension

-1

Strain(%)

0

1

2

Compression

FIG. 10. Variation of threshold current density with strain for a variety of 1.5-pm lasers. (Compiled by P. J. A. Thijs and reproduced with his permission.)

of strain and quantum confinement cancel and the heavy- and light-hole bands are degenerate. Interestingly, in optical fiber applications in which the polarization of the signals becomes scrambled, it is often important to have amplifiers [49] or optical modulators [50] that are polarization insensitive. In these cases the requirements are the opposite to those in lasers, and strain is used to carefully tailor the wave functions so that the interaction is exactly the same for both the TE and TM polarizations.

V. Hydrostatic Pressure Measurements of Avalanche Photodiodes: The Band-Structure Dependence of Impact Ionization 1. INTRODUCTION

In this section, we describe hydrostatic pressure measurements of avalanche photodiodes (APDs), which have yielded valuable information on the band-structure dependence of impact ionization. Impact ionization is an important process affecting the performance of semiconductor electronic devices (see, for example, Sze [51]). Ionization creates secondary electron-hole pairs that lead to photocurrent multiplica-

328

A. R. ADAMS, M. SILVER, AND J. ALLAM

tion in APDs (for a comprehensive review, see Capasso [52]) and to microwave generation in certain transit-time (IMPA’IT) diodes. Ultimately, current multiplication leads to avalanche breakdown, which imposes a fundamental limit on the applied voltage (the breakdown voltage Vb) for all semiconductor junctions and hence, for example, limits the power output of transistors. It is therefore important to have an understanding of the dependence of impact ionization on the material properties in order to design structures with desirable avalanche characteristics. However, impact ionization presents a challenging test of our understanding of the physics of hot carriers in semiconductors because it is determined by the band structure and the electron-electron and electron-phonon scattering rates at high electron energies away from symmetry points of the Brillouin zone. The process of Auger recombination in long-wavelengthlasers described in Section 111.2 is the reverse process to impact ionization. However, although Auger recombination in lasers generates hot carriers high in the bands, it always starts with two carriers near the band edge at the center of the Brillouin zone. As we will see, impact ionization involves carriers with kinetic energies much greater than the direct bandgap and so it is even more complex. In early experimental work [53]on the effects of crystal orientation, alloy composition, and temperature on avalanche photodiodes, it was observed that the impact ionization coefficients were strongly dependent on details of the band structure. This led to considerable interest in the possibility of “band-structure engineering” [52] to control impact ionization, for example, using semiconductor alloys, heterostructures, or multiple quantum wells. However, some subsequent experimental [54] and theoretical [55] work has not supported such a strong dependence, at least for steady-state transport in bulk devices, and the band-structure dependence of impact ionization has remained an open issue. More recently, we have studied impact ionization in several materials (Si, Ge, GaAs, Al,Gal-,As, (A1,Gal-,)o.521no.48P, and Al,Gal-,As/GaAs superlattices and multiple quantum wells, and GaAs/In,Gal -,As strainedlayer superlattices) using band-structure variation by the application of hydrostatic pressure, compositional variation, and layer width variation [56-611. Here we describe the pressure measurements that have revealed the important role of the distribution of final electron states in the Brillouin zone. In the following section, we describe the processes determining the probability of impact ionization, and indicate how hydrostatic pressure measurements might be expected to throw light on the complex physics involved. We then discuss earlier measurements in InAs and InAsl-xPx,which, with increasing pressure or composition x , exhibit a transition between avalanche

5 SEMICONDUCTOR OPTOELECIXONIC DEVICES

329

breakdown and intervalley transfer. We present our measurement of breakdown in GaAs, Ge, and Si p-n photodiodes. These materials show quite different behavior, and the results are interpreted in terms of the different band structures. Extending these conclusions to other wide-gap semiconductors, we discover a new “universal” relationship [62, 631 between the avalanche breakdown voltage and the band structure in wide-bandgap semiconductors. This has important implications for our understanding of impact ionization and for the design of avalanche devices employing semiconductor alloys and multilayers. 2. PHYSICS OF IMPACT

IONIZATION IN

SEMICONDUCTORS

Before presenting the results of hydrostatic pressure experiments, we consider the physical processes that govern impact ionization in a semiconductor device: the electron kinematics and scattering (electron-electron and electron-phonon) dynamics. Impact ionization refers to pair generation by electron-electron scattering. Figure 11 shows a high-energy electron in state 1 (wave vector kl, energy E l )interacting with a valence electron 2, yielding final states 1’and 2’ subjects to (E, k) conservation. The existence of an energy gap ( E , )

FIG. 11. Pair creation by impact ionization, initiated by an electron in state (El, kl).

330

A. R. ADAMS, M.

SILVER, AND

J. ALLAM

between the valence and conduction bands imposes a minimum energy or threshold (&) for the impact ionization process, which in general exceeds Eg due to the finite dispersion of the energy bands. Ethis determined from the energy band structure by numerical minimization of the initial state energy [64, 651 and is an important parameter for simple theories of ionization. Numerical calculation of the transition rate for impact ionization, ~ i(k, / ), can be performed using semiclassical first-order perturbation theory, with pseudo-potential electron band structures and wave functions [66-681. The first such calculation was performed by Kane, for the case of Si [66]. The rate increases slowly with energy above threshold and only becomes significant at energies well above Eth. It has been found [67] that ~ i(k, , ' ) can be approximated by a simple energy-dependent expression 7 i / ( E=)P ( E - Eth)awhere E = E ( k l ) T;/ , is zero for E < Eth,and a is typically in the range 3 to 5. Analytical solution for direct-gap parabolic bands leads to the above expression with a = 2, the well-known Keldysh formula [68]. In tetrahedral semiconductors, the conduction band is characterized by minima at the r, L, and X points of the Brillouin zone, separated by an energy range AT-L-X. The consequences for optical absorption and lowfield transport are well known. We investigate here the consequences at high fields, where breakdown occurs. Figure 12 shows the band structures along principal symmetry axes for the materials studied: InAs, GaAs, Ge, and Si, representing wide- and narrow-gap, and direct- and indirect-gap semiconductors. The thick lines represent states that can initiate impact ionization. Note that the lowest energy for impact ionization depends on the band and the direction through the crystal that we are considering. (For simplicity, only processes with parallel initial and final states are calculated, that is, a subset of the possible transitions [65]). Significant differences in the ionization properties can be expected as a result of the different band structures. Impact ionization is the inverse of the Auger effect discussed earlier; however, there is an important difference associated with the population of initial states. The Auger effect can be observed in the absence of an electric field, with thermally populated initial states; hence only the lowest conduction band valley is involved. However, initial stater; for impact ionization may have energies significantly greater than E, abo*,ethe band edge. If the excess energy is larger than A r - L - x , final electron states may be distributed between the r, L, or X conduction band minima. We can also make comparisons with optical absorption. The ionization thresholds can be viewed as a generalization of the van Hove singularities associated with critical points in the electronic joint density of states, which give rise to

331

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES 6

4

2-2 v

x m $ 0 B

-2 -4

6

6

4

4

-

2 5

2-2

v

x

F

U x l

E O

0:

a,

-2

-2 -4

L-T-Xt

k ( l l 1 ) k(100)

~

k(110)

r

L+T-X

k ( l l 1 ) k(100)

-rk(110)

-4

FIG. 12. Pseudo-potential band structures of (a) InAs, (b) GaAs, (c) Ge, and (d) Si. The lines represent the calculated bands. Thick lines represent states capable of initiating impact ionization (processes with wave vectors along principle symmetry axes).

features in the optical response [69]. Hydrostatic pressure has proved useful in identifying regions of the Brillouin zone associated with particular spectral features [70]. Hence we can expect that pressure will be useful in studying band structure dependence of impact ionization rates, in particular the final-state location in k-space. depends also on the distribution The ionization rate 7~’(kl)f(kl)d3kl function f(k,),obtained from the semiclassical Boltzmann transport equation with appropriate electron-phonon scattering rates. In direct gap materials with E, 4 Ar-L-x (represented by narrow-gap polar materials such as InAs), electrons are confined to the r valley for all values of the applied electric field. The dominant scattering is with polar phonons. Small-angle scattering results in electron motion predominantly along the field direction, leading to a rapid increase in energy with electric field. On the other hand, for direct semiconductors with AI.-L-x < E, (e.g., GaAs) and for indirect semiconductors (Ge, Si) intervalley scattering via the deformation-potential interaction dominates. The isotropic scattering results in efficient momentum and energy relaxation, and electron transport is in a drift mode. Hence, the distribution function is also strongly dependent on the band structure. In general, accurate determination of f ( k) ,for wide-gap semiconductors

332

A.

R. ADAMS,M.SILVER,AND J. ALLAM

at high fields requires numerical solution, for example, by a Monte Carlo method [71]. However, in the following discussion of the pressure variation of impact ionization, we employ much simpler analytical models to reveal the dependence on the principle variables.

3. PRESSURE RESULTS

The electron (a)and hole ( p ) ionization coefficients were not measured directly in the experiments described next, due to the difficulty of selectively injecting electrons or holes into the device within the high-pressure cell. Instead, the avalanche breakdown voltage V , was measured. The devices studied were p-n junction photodiodes, whose electrical characteristics were measured using a programmable current source and electrometer. Measurements of the photocurrent multiplication were also made to ensure that the devices were uniform and that breakdown was not related to edge effects or microplasmas.

a. InAsand InAsl-,Px Our first investigations of avalanche breakdown in semiconductors using hydrostatic pressure were performed on InAs [72] and on InAsl-,PX [73]. InAs is a narrow direct-bandgap material in which the intervalley separation (-1 eV) is significantly greater than the bandgap (-0.4 eV). Hence, impact ionization limits the carrier energy in the r valley and inhibits intervalley transfer [74]. If the bandgap is increased as a result of varying the composition or applying hydrostatic pressure, Ethcan be increased to the point at which it exceeds the intervalley separation, allowing intervalley transfer to occur. Figure 13 shows the conduction band energy at r, X, and L points for InAs as a function of hydrostatic pressure, and also for InAsl-,PX as a function of composition x . The estimated threshold energy (Eth - Er) becomes comparable to the intervalley separation (EL- Er)at a pressure* p 30 kbar in InAs or composition x = 0.35 in the alloy. The threshold field FT for current instability in InAs was measured using ohmic-contacted, bulk-grown crystals, and pressures up to 50 kbar applied with the sample potted in epoxy resin using a Bridgman opposed anvil system. At pressures p < 33 kbar, the current increased rapidly at FT 1 kV.cm-', consistent with avalanche breakdown. For p > 33 kbar, the field was limited instead by Gunn oscillations associated with the onset

-

*Henceforth p will refer to pressure, not to the hole density.

333

5 SEMICONDUCTOR OFTOELECTRONIC DEVICES

hydrostatic pressure p (kbar)

0.4

,/

.-..........---

...-- .......*I

(x)

]

0.0 0.0 0.2 0.4 0.6 0.8 1.0 phosphorus composition x FIG. 13. Conduction band energy in InAsl-,P, at r, X, and L points: variation with composition x (solid tines) and hydrostatic pressurep (dashed lines). The variation of the ionization threshold energy is also shown (chained line).

of negative differential conductance (NDC). Detailed measurements on InAsl-,P, were performed using epitaxially grown samples and a piston and cylinder apparatus to reduce the possibility of nonhydrostatic pressure [73].Figure 14a shows the composition dependence of FT. Although dFT/dxdoes not vary significantly over the full range of x ,the detailed behavior of the current breakdown again indicated avalanche breakdown at x < 0.35, and the Gunn effect for x > 0.35.The pressure dependence up to 15 kbar was also measured for samples with a range of compositions (Fig. 14b). An abrupt change in the pressure dependence was observed at x = 0.35. The results were compared with Stratton’s prediction [75]for the critical field for energy run-away (polar-phonon breakdown) in parabolic bands

A.

334

R. ADAMS,M.SILVER,AND J. ALLAM

n F

'E c?

>

s.

0’

2 0.2 0.4 0.6 0.8 1 phosphorus composition x

0

0.2 0.4 0.6 0.8 1 phosphorus composition x

FIG. 14. (a) Composition dependence of threshold field FT for InAsl..P,. (b) Pressure coefficient of FT as a function of composition x for In Asl-,P,.

5 SEMICONDUC~OR OFTOELECTRONIC DEVICES

335

where Ep is the phonon energy, m* is the r valley effective mass and c0 and E, are the static and high-frequency dielectric constants. The predicted pressure coefficient is (l/F,)(dF,/dp) = 2.5 eV-'(dE,/dp), which is half the = 0.06 kbar-' = measured value in the avalanche regime (l/FT)(dFT/dp) 6 eV-'(dE,/dp). This implies that Ethrwhich scales with E,, should enter explicitly the expression for the breakdown field, as it does in more realistic models with nonparabolic bands, in which the energy increases rapidly but does not run away [76, 771. The behavior in the regime where current oscillations were observed is more complex. Electron transfer from r to L causes NDC in materials such as GaAs or InP. The threshold energy for intervalley transfer is EL - E r , which has a different dependence on pressure than E,. Hence, the pressure coefficient changes abruptly in InAs,-,P, at x = 0.35. Avalanche breakdown and intervalley transfer are strongly interdependent. In InAs, energy relaxation by impact ionization suppresses intervalley transfer. In InP, intervalley scattering reduces electron heating and hence a much larger field is required for impact ionization. The high carrier densities present as a result of majority carrier injection from ohmic contacts allow charge and field redistribution and the creation of high-field domains in the Gunn regime. This prevents the study of breakdown under conditions of uniform field. Hence, in subsequent studies we have used reverse-biased p-n junction diodes with low dark current so that impact ionization can be studied in materials where intervalley scattering dominates.

b. GaAs GaAs is a direct-gap semiconductor in which the intervalley separation is small compared with the bandgap. Hence, intervalley transfer will occur at lower fields than those required for impact ionization. A full description of this work can be found in Ref. [61]. GaAs p+-i-n' structures with 1 pm i-regions were grown by metalorganic vapor phase epitaxy and fabricated into mesa photodiodes. Devices with low dark current and sharp breakdown were selected for the high-pressure measurements. Figure 15 shows the dark current as a function of reverse bias at atmospheric pressure and under hydrostatic pressure up to 10 kbar. Avalanche breakdown at -32 V is indicated by an order-of-magnitude increase in the current within a 0.02 V increment. The inset of Fig. 15 shows the breakdown in more detail. Figure 16a shows V ,at pressures up to 14 kbar for three devices. Variation of Vboat atmospheric pressure between devices is the result of small varia-

336

A. R. ADAMS, M. SILVER,AND J. ALLAM

reverse bias (V) FIG. 15. Reverse-bias I-V characteristic in GaAs at nominal pressures of 0 , 2 , 4 , 6 , 8. and 10 kbar.

tions in the electric field profiles or the series resistance. However, the normalized pressure coefficients (1/VW)(dVb(p)ldp) of the three devices kbar-’. are in good agreement, with an average value of -(3.3 2 0.4) X The pressure coefficient is small and opposite in sign to that measured for InAs, in spite of the fact that they are both direct-bandgap materials with similar pressure coefficients of the bandgap and phonon energies. To model the experimental data, we must account for the effect of pressure on both the electronic and vibrational properties. We have used Ridley’s lucky-drift model [78,79] with the ionization probability modeled by the Keldysh equation, which has proved successful in fitting experimental data for ionization coefficients in wide-gap semiconductors with a small number of adjustable parameters (EthrE,, P, and A, a mean free path for phonon scattering). The values of A obtained have been found to follow a “chemical trend” [79]

5

SEMICONDUCTOR OPTOELECTRONIC DEVICES

337

32.1

32.0

31.9

>" 31.8

31.7 FIG. 16. (a) V , for GaAs devices 1 to 3 ,determined from dark current breakdown for increasing (solid symbols) or decreasing (hollow symbols) pressure. (b) Comparison of experimentally measured pressure dependence of breakdown voltage (symbols) with the calculated dependence for a threshold energy independent of pressure (solid line) or scaling with Er,E x ,EL,or (E)(dashed lines).

where n(E,)is the phonon occupation number, p is the mass density, and the appearance of Ethreflects the energy range of electrons. E ,was obtained by suitable averaging of the longitudinal optical (LO) and longitudinal acoustical (LA) zone-edge phonons (common phonon model [79]).Expression (28)is consistent with intervalley phonon scattering with isotropic deformation potentials and phonon energies, and nonparabolic bands. Hence the chemical trend supports the validity of the model in estimating the effects of pressure on ionization coefficients due to both variation in phonon scattering rate and ionization probability. The ionization coefficients in GaAs [80] were fitted using values of Eth= 1.7 eV for electrons and 1.4 eV for holes and E, = 0.029 eV, and values of A and P were determined. We can obtain the pressure dependence of vbby suitable differentiation of the lucky-drift equations, and obtain (to lower order of approximation) --=--+ 1 dVb vb

dP

1 dEth 1 dE, 1 d h 2Eth dP 2Ep dP AdP

-

1 dEt, 1 dE, d p (29) Eth dp 2Ep dP PdP

A. R. ADAMS, M. SILVER, AND J. ALLAM

338

n

0

W

>"

A

a

Y

>"

Ftc. 16. continued

(dpldp) is determined from the Murnaghan equation of state, with the value of bulk modulus BT obtained from the literature. The variation of phonon energy with pressure is determined from a mean Gruneisen parameter obtained by averaging over the modes in a manner consistent with the common phonon model. The pressure dependence of the Keldysh prefactor P is not known, and we assume that Ethdominates the variation of the ionization rate. Furthermore, the effect of pressure-induced changes in Ar-L-x on the hot-electron distribution function is not considered. The density and phonon energies increase with pressure, causing vb to decrease with a pressure coefficient (-1.92 +- 0.08) X kbar-' (solid line in Fig. 16b). The overall behavior of (dVb/dp) is then determined by the sign and magnitude of (d&/dp). Figure 16b shows v&)/vb(o), calculated when the pressure coefficient of Ethis equal to that of the r (10.7 meV.kbar-'), L (3.8 mevekbar-'), and X (-1.3 meV-kbarr') conduction band minima "1. The best fit to the experimental data is obtained for d&,/dp = (2.7 f. 0.1) meV . kbar-'. This value is close to the coefficient (2.8 meV .kbar-' ) of a Brillouin-zone-averaged indirect energy gap (E), which can be approximated as [62] ( E ) =1- ( E , - + 4 E , + 3 E x ) 8

(30)

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

339

The existence of an effective threshold energy that scales with ( E ) implies that final electron states of the pair-production process are distributed between conduction band valleys according to the density of states. We show later that this conclusion holds for many wide-gap semiconductors, leading to a simple means for determining the ionization coefficients and breakdown voltage. c.

Ge

Ge is isoelectronic with GaAs and has a similar lattice constant. However, the absence of ionic contributions to the bonding results in smaller energy gaps, and Ge is an indirect-gap semiconductor with the minimum conduction band energy occurring at the L point. The intervalley separations are small, and the scattering environment similar to that for GaAs at high energies, hence we might expect that the materials will exhibit a similar dependence of avalanche breakdown on pressure. The breakdown voltage was measured in G e p-n photodiodes under hydrostatic pressure up to 15 kbar. Figure 17a shows the current-voltage characteristics at pressures of 0, 6, and 15 kbar. The curvature of the I-V characteristic at breakdown is due to the series resistance of the contacts and electrical feedthroughs. When biased below breakdown, the dark current decreases with pressure. The breakdown voltage, defined as the reverse bias for a given current in the range 5 to 50 PA, is shown in Fig. 17b. The with pressure up to =8 kbar, but then decreases breakdown voltage increases as the pressure is further increased. This behavior is unlike the linearly increasing or decreasing variation with pressure observed in InAs, GaAs, and Si. The change in the value of dvh/dp suggests that different ionization processes may be dominant at different pressures. If we assume that the process that dominates at atmospheric pressure is gradually shut off with increasing pressure, then we might expect the total pressure coefficient to be described by

where (dVh/dp)l is the coefficient for process 1 that is shut off at pressure pO, and (dVb/dp);!is the coefficient for process 2. The dashed lines in Fig. 17b are a fit to this expression; excellent agreement with the experiment is obtained with the parameters (1/VhO)(dvb/dp)l = +2.1 X kbar-', (1/Vho)(dVb/dp);! = -4.4 X 10-j kbar-', and p o = 8.5 kbar. Hence the

A. R. ADAMS, M.SILVER, AND J. ALLAM

340

n

a

1o

-~

W

CI

c

2

L

3 0

1 o-6

29.8

30.0 30.2 reverse bias (V)

FIG. 17. Pressure dependence of breakdown in a Ge avalanche photodiode. (a) Current-voltage characteristic at 0, 6. and IS kbar. (b) Breakdown voltage (measured for dark currents of 5, 10, 20, and SO PA) as a function of pressure. Solid symbols represent experimentalpoints. Dashed lines are a theoretical fit as described in the text. After [57], by permission of Gordon and Breach Science Publishers.

behavior at pressure >8.5 kbar (process 2) is similar to GaAs. At lower pressures, an additional process (1) not present in GaAs occurs, for which the pressure coefficient of the threshold energy is estimated as +4 meV . kbar-I, assuming the same pressure dependence of phonon scattering as for GaAs. We note that this value is equal to the pressure coefficient of the indirect bandgap. A possible explanation for this behavior can be found by examining the lowest-energy electron- and hole-initiated ionization processes, as shown in Fig. 18. Due to the small dispersion of the lowest conduction band and highest valence band at the L point, Ethis almost equal to the indirect bandgap E, for both electron- and hole-initiated processes. If the bandgap is increased by applying hydrostatic pressure, a point is reached at which the bandgap exceeds the width of the band containing the initial state, so that the process is no longer allowed by energy conservation. A similar effect is observed [58] in Gel-,Six alloys for x > 0.3. In GaAs, electronand hole-initiated processes with initial and final states along the (111) axis are forbidden for the same reason. Note that we only describe processes

341

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

30.3

->

n

a>

30.2

0

a 0

.cI

> 30.1

I

29.9 -i

1

1

1

1

~

1

1

1

1

1

1

1

1

1

1

FIG. 17. continued

involving parallel wave vectors oriented along high-symmetry axes. Processes can occur off-axis or for nonparallel wave vectors [65], but at higher energies due to the low transverse mass. In any case, the increased bandgap will lead to a reduced contribution to the total ionization rate by carriers with final states in the L valley. Detailed numerical calculations are required to confirm the above interpretation. If this picture is correct, these data represent clear evidence for multiple ionization processes and for so-called antithresholds [64]. Different ionization processes can be turned on or off by varying the band structure using hydrostatic pressure or compositional variation. d.

Si

Si is an indirect-bandgap material with the conduction band minima close to the X points. The intervalley separation AI.-L-x is large: the X-L separation is comparable to E,, and the r point is 2 to 3 eV above the band edge and hence probably plays no role in impact ionization. Si is of interest as a rare example of a material in which a and 0 are significantly different - a desirable property for avalanche photodiodes, improving the

A. R. ADAMS, M. SILVER,AND J. ALLAM

342

3

2 n

'a,1 W

%

g 0 c

a,

-1

-2

L

r

L

r

L

r

L

r

k(l11)

3

2 n

% 1

W

%

P O c

a,

-1

-2

k(l11) FIG. 18. Ionizationprocesses in Ge along (111): (a) electron-initiated, EU,= 0.8 eV; (b) hole-initiated, Eth= 0.9 eV. After [57], by permission of Gordon and Breach Science Publishers.

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

343

noise figure, bandwidth, and stability [52]. Although it has been noted that eV the lowest-energy thresholds for electrons and holes are different (1.2 for electrons and 1.7 eV for holes) [58],the detailed origin of the difference between CY and p has not been definitively shown. The current-voltage characteristics in the region of avalanche breakdown of a Si p’-n junction photodiode at pressures from 0 to 10.3kbar are shown in Fig. 19a. Breakdown was abrupt, and v h was determined at a current of 1 PA. The normalized pressure dependence of v h for three different devices is shown in Fig. 19b. The solid symbols are for increasing pressure, and hollow symbols for decreasing pressure. Devices 1 and 2 were bonded to transistor headers, whereas device 3 was suspended from fine gold wires to eliminate uniaxial stress effects. The normalized pressure coefficients ( l/Vho)(dVb/dp) of the three devices are in excellent agreement, being -2.60X kbar-’, -2.54X kbar-I, and -2.60X kbar-’, respectively. The negative pressure coefficient of v h in Si is almost an order of magnitude larger than in GaAs - despite the pressure coefficient of the bandgap The results were interpreted using a being an order of magnitude smaller. similar lucky-drift model as for GaAs, but with an electric field distribution appropriate for a p+-n junction. Numerical variation of the parameters revealed the pressure dependence

Figure 19b (broken line) shows the computed variation assuming (dEth/dp) = 0. It can be seen that variation of vibrational parameters (p, E p )dominates the pressure dependence, with the largest term arising from (dpldp). Figure 19b indicates that the pressure coefficient of the effective threshold = (dE,/dp) = (dE,/dp) = energy is negative. Assuming (dE,,/dp) -1.3meV.kbar-’, we obtain the dashed line in Fig. 19b, which overestimates the negative pressure coefficient. The best fit to experiment is obtained with (dEth/dp) = -0.4meV .kbar-I. The value of (dElh/dp) has been separately estimated by calculating Ethfor various ionization processes from pseudo-potential band structures under hydrostatic pressure. For the lowest-energy process, the values of (d&,/dp) are -1.2meV. kbar-’ for electrons and +0.3 meV kbar-’ for holes. We can account to some extent for multiple ionization processes and “soft” thresholds by estimating an average (dEth/dp) for processes that have Ethwithin =I eV for the lowestenergy process. These average values are -0.6 meV .kbar-’ for electrons dp) and 0.0 meV. kbar-’ for holes, which predicts ( l / V h , ) ( d V hin/ excellent

-

A. R. ADAMS, M. SILVER, AND J. ALLAM

344

1o-6

1o-8

73.0

74.0 75.0 76.0 reverse bias (V)

77.0

FIG. 19. (a) Current-voltage characteristic of Si p+-n diode as a function of pressure. (b) Normalized breakdown voltage as a function of pressure for three different devices. Solid symbols are for increasing pressure, hollow symbols for decreasing pressure. The solid line is a linear fit to the experimenThe tal data. The broken line is a theoretical fit ignoring variation in Eth. dashed line is a theoretical fit assuming dE,h/dp = dE,ldp. After [57], by permission of Gordon and Breach Science Publishers.

agreement with experiment. Note that ionization in the L valleys with (dEth/dp) = ( d E L / d p>) 0 is inconsistent with the previously described experimental results (at least as interpreted by this simple model). Hence, ionization processes with final states in the lowest (X) minima dominate in Si due to the large intervalley separation.

4.

“UNIVERSAL” DEPENDENCE OF AVALANCHE BREAKDOWN BANDSTRUCTURE

ON

The breakdown voltage is an important parameter for the design of electronic devices, and hence it would be useful to have a simple means of predicting Vbfor “new” semiconductors and their alloys. In 1966,Sze and Gibbons [82] presented approximate formulae relating V ,to the bandgap E, for p+-n and graded p-n junctions, based on experimental measurements of Ge, Si, GaAs, and Gap. These expressions, widely utilized in electronic

345

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

'...

- - - fit, dE,h/dp = 0

I

0.95- .________ fit, dEth/dp = dE,/dp'-., I

,

I

I

I

,

I

,

I

I

I

I

I

I

1 . I

-

device design, are based on the implicit assumptions that Ethscales with E, and that the variation of Ethdominates the dependence of vb on the material composition. However, the above hydrostatic pressure measurements have clearly shown that neither of these assumptions is universally correct. Based on the differing pressure dependence of InAs and GaAs, w e can classify tetrahedral semiconductors according to whether the conduction band intervalley separation Al.-X-L is small or large compared with E,. Many 111-V and 11-VI compounds, such as InP, AIAs, Gap, CdTe, and ZnSe, exhibit (Ar-x-LIEg) < 1, so that ionization should yield final electron states distributed throughout the Brillouin zone. Further experiments are required to see if these materials show the GaAs-like pressure dependence. In the absence of such experimental data, we can search for chemical trends by investigating the scaling of Vb with Eg and ( E ) . Published values for ionization coefficients in Si, Ge, Gap, InP, GaAs, Gao.471no.53As, A10.481n0.52A~, A1,Gal-,AsYSbl-,, Al,Gal-,As, and (Al,Gal-x)o.521no.48P were used to calculate v b for a 1-pm p-i-n diode. Accurate values for the energy gaps were obtained for 24 diamond and quasi-particle band zinc blende semiconductors, using published ab inirio structure calculations. Full details can be found in Ref. [62]. Figure 20a shows the dependence of v b on E,.Although a general trend of increasing

A. R. ADAMS, M. SILVER, AND J. ALLAM

346

n

>

W

t

80

70 60

50 40

30 20 10 0 -

0

0.5 1

1.5

2

2.5 3

3.5

(E) (ev) FIG.20. Chemical trends in v b for a 1-pn p-i-n diode, calculated from experimental ionization coefficient measurements. v b is plotted against (a) minimum energy gap E, and (b) average indirect energy gap ( E ) .

5

SEMICONDUCTOR OFTOELECTRONIC DEVICES

347

Vb is followed, deviations of up to 30% are observed compared with the formulae of Sze and Gibbons. In marked contrast, as shown in Fig. 20b, there is a simple linear relationship between v b and ( E )that is obeyed by all the wide-bandgap materials ( E , > 1 eV, ( E )> 1.5 eV). The relation is described by Vb = 45.8

X

( ( E )- 1.01)

(33)

As shown in Fig. 20b, narrow-gap materials, including GaSb and presumably InAs and InSb, do not scale with ( E ) but apparently with E,. Although we have discussed the scaling of V , in terms of an effective ionization threshold, the electron kinematics and electron-phonon scattering also influence Vb. The phonon energy is found to scale with an average direct bandgap [62], which is related to the Penn gap that determines the dielectric response and cohesive energies [83]. The use of the average indirect gap for impact ionization is a consequence of momentum exchange between the initial carrier and final electron-hole pair. The use of an average bandgap seems to be generally appropriate for high-field transport in wide-gap semiconductors, in which the large rates for isotropic intervalley scattering cause carriers to sample the entire Brillouin zone. The value of ( E ) can be used as a guide in the choice of materials for use in a wide variety of semiconductor electronic devices. Many attempts have been made to modify ionization coefficients by “band-structure engineering,” for example, in semiconductor multiple quantum wells [52]. The dependence of v b on ( E ) rather than E, for wide-gap materials, including GaAs, suggests that the band structure must be engineered over the whole Brillouin zone rather than only in a small region in the vicinity of the minimum energy gap.

5. CONCLUSIONS High pressure has proved an extremely useful technique for studying impact ionization in semiconductor photodiodes. The results for InAs, GaAs, Ge, and Si were markedly different. Despite the complex dependence of high-field transport on the band structure and scattering environment, the results could nevertheless be interpreted in terms of a simple picture involving the location in k-space of final electron states of the ionization process. For InAs, transport occurs predominantly in the r valley, and the breakwith pressure, scaling with the direct bandgap. For down voltage increases Si, final electron states of the ionization process are located primarily in

348

A.

R. ADAMS, M. SILVER,AND J. ALLAM

the X valleys, the effective threshold energy scales with the indirect energy gap, and the breakdown voltage decreases with pressure. For GaAs and Ge, final electron states are distributed between the r, X,and L valleys, and the pressure dependence of the breakdown voltage is small. Ge shows a nonmonotonic dependence of Vbon pressure, which is tentatively associated with the lowest-energy ionization process becoming forbidden as the bandgap increases. This picture has led to the proposal that V ,should depend on a Brillouinzone-averaged energy gap (E) for wide-bandgap semiconductors. A survey of measured ionization coefficients suggests that this “universal” formula is accurately followed by most wide-gap tetrahedrally coordinated semiconductors. In the work presented here, we have measured only the breakdown voltage. More detailed information on the field dependence of the ionization rates a and fl could be obtained by measuring the photocurrent multiplication M under appropriate illumination conditions to inject either electrons or holes into the high-field region. Measurements of other materials such as InP and GaSb should be performed to explore materials with different values of bandgaps and intervalley separations. In0.53Ga0.47As is of particular interest, since it seems to be on the boundary of wide-gap and narrow-gap materials. Measurement of the ionization rates in p-n junctions formed in narrow-bandgap materials is hampered by the large dark currents due to band-to-band tunneling. The ionization rates at very low fields can be determined by measuring only the secondary carriers (i.e., M - 1) via studying the base current or substrate current in transistors [84]. We have used simple intuitive models and analytical calculations to interpret our experimental data. Accurate numerical calculations employing full microscopic scattering rates and Monte Carlo simulation of the electron transport are desirable to test our conclusions regarding the influence of band structure on impact ionization [85,86]. Such studies can be expected to further improve our understanding of high-field electron transport and breakdown in semiconductors.

VI. summary Hydrostatic pressure measurements have provided an extremely powerful technique for examining the dependence of semiconductor optoelectronic devices on the electronic band structure. A clear trend is observed in the behavior of semiconductor lasers. At bandgaps of less than about 1 eV, the threshold current is dominated by the process of Auger recombi-

5 SEMICONDUCTOR OPTOELECTRONIC DEVICES

349

nation, but this effect decreases as the bandgap increases so that by about 1.3 eV it is small compared with spontaneous emission. As the direct bandgap is increased further, in materials grown on GaAs, it approaches the indirect bandgap of the AlGaAs carrier confining layers, and electrons are able to escape by thermal excitation into the X minima. All of the deleterious effects - Auger recombination, spontaneous emission, and carrier leakage -can be reduced by reorganizing the electronic band structure by the introduction of strained-layer quantum wells. This allows stimulated recombination to produce lasing action at lower current densities. Equally clear trends are observable in the process of impact ionization used in avalanche photodiodes. The value of the direct bandgap energy is the controlling parameter while it is less than about half the lowest indirect bandgap. However, in wider-bandgap materials, it is the weighted average bandgap of the r, X, and L points that dominates. This basic understanding of the physics of the impact ionization process has allowed us to propose a “universal model” that accurately describes the results in a wide range of materials and should allow accurate prediction of the onset of impact ionization in all semiconductor devices in which large electric fields are employed.

Acknowledgments

The experimental work carried out at the University of Surrey was supported by the Science and Engineering Research Council and by BT, Nortel, and Philips research laboratories. It is a pleasure to acknowledge valuable contributions from many collaborators, including I. K. Czajkowski, J. P. R. David, M. A. Gell, M. D. A. MacBean, A. T. Meney, E. P. O’Reilly, A. Phillips, M. A. Pate, J. S. Roberts, S. Ritchie, P. J. A. Thijs, S. Sweeney, and A. Valster. REFERENCES 1 . Adams, A. R., and Dunstan, D . J. (1990). Semiconductor Sci.Tech.5, 1194. 2. Kirchoefer, S. W., Holonyak, Jr. N., Hess, K.. Gulino, D. A.. Drickamer, H. G., Coleman, J. J., and Dapkus, P. D . (1982). SolidStateCommunications 42, 633. 3. Lambkin, J. D., Adams, A. R., Dunstan, J. D., Dawson, P., and Foxon, C. T. (1989). Phys.Rev.B 39, 5546. 4. Van de Walk, C. G., and Martin, M. M. (1987). Phys.Rev. B 35,8154. 5. Patel, D., Menoni, C. S., Temkin, H., Tome, C., Logan, R. A., and Coblentz, D. (1993). J.Appl.Phys.74, 737. 6 .Langer, D . W., and Montalvo, R. A. (1968). J . Chem.Phys.49,2836. 7. Heasman, K. C., Adams, A. R., and Plumb, R. (1987). Electronics Letters 23, 555.

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8. McMahon. C. H., Bae, J. W., Menoni, C. S., Patel, D., Temkin, H., Brusenbach, P., and Leibenguth, R. (1995). Appl. Phys.Lett. 66,2171. 9. Heasman, K. C., Adams, A. R., Greene, P. D., and Henshall, G. D. (1983). HPTL Conference on Physics at High Pressure (Harlow); Heasman, K. C. (1985). PhD thesis, University of Surrey. 10. Bernard, M. G. A., and Duraffourg, B. (1961). Phys.Stat. Sol.1,699. 11. Adams, A. R., Asada, M., Suematsu, Y.. and Arai, S. (1980). Japanese J. Appl.Phys. 19, L621. 12. Thompson. G. H. B. (1980). Physics of Semiconductor LaserDevices, John Wiley, New York. 13. Casey, H. C. (1978). J.Appl. Phys.49, 3684. Physics B 44, 151. 14. Haug, A. (1987). Applied 15. Kane, E. 0. (1982). In Handbookon Semiconductors (Moss, T. S . ,ed.), Vol. 1, p. 193, North-Holland, Amsterdam. 16. Coleman, J. J., Beernink, K. J., and Givens, M. E. (1992). ZEEE J.Quantum Electronics 28,1983. 17. Agrawal, G. P., and Dutta. N. K. (1986). Long Wavelength Semiconductor Lasers. Van Nostrand Reinhold Company, New York. 18. Ohkubo. M., Ijichi. T., Iketani, A., and Kikuta, T. (1994). ZEEE J. Quantum Electronics 30,408. 19. Asonen, H.. Ovtchinnikov, A., Zhaung, G., Nappi, J., Savolainen, P., andPessa, M. (1994). IEEE J. Quantum Electronics 30, 415. 20. Meney, A. T., Prins. A. D., Phillips. A. F., Sly, J. L., O’Reilly, E. P., Dunstan, D. J., Adams, A. R., and Valster. A. (1995). IEEE J.Selected Topics inOptoelectronics 1, 697. 21. Blood, P.. Fletcher. E. D.. Woodbridge, K., Heasman, K. C., and Adams, A. R. (1989). IEEE J. Quantum Electronics 25, 1459. 22. Page, H. Meney, A. T., Adams, A. R., and van der Poel, C. J. (1997). To be published. 23. Agrawal, G. P. (1992). Fibre-Optic Communications Systems, Wiley Series in Microwave and Optical Engineering (Chang, K., ed.), John Wiley and Sons, New York. 24. Thijs, P. J. A.. Binsma, J. J. M.. Tiemeijer. L. F., and van Dongen, T. (1994). ZEEE J. Quantum Electronics 30, 477. 25. Chuang, S. L., O’Gorman. J., and Levi, A. F. J. (1993). ZEEE J. Quantum Electronics 29,1631. 26. Ackerman, D. A., Shtengel, G. E.. Hybertsen, M. S., Morton, P. A., Kazarinov, R. F., Tanbun-Ek, T., and Logan, R. A. (1995). IEEE Journalof Selected Topics in Quantum Electronics 1, 250. 27. Bernussi, A. A,, Pikai, J.. Temkin, H.. Colentz, D. A,, and Logan, R. A. (1995). Appl. 66,3606. Phys.Lett. 28. Yamakoshi, S., Sanada. T.. Wada, O., Umebu, I., and Sakurai, T. (1982). Appl. Phys. Lett. 40, 144. 29. Patel. D., Menoni, C. S., Bernussi, A. A., andTemkin, H. (1996). Phys. Stat. Sol. (b) 198,375. 30. Dutta, N. K., and Nelson, R. J. (1982). J . Appf. Phys.53,74. 31. Zou. Y., Osinski, J. S.. Godzinski, P.. Dapkus. P. D., Rideout, W., Shartin, W. F., and Logan, R. A. (1993). IEEE J. Quantum Electronics 29, 1565. 32. Fuchs, G.. Schiedel. C.. Hangleiter, A.. Harle, V., and Scholz, F. (1993). Appl. Phys.Lett. 62, 396. 33. Adams, A. R., Silver, M., O’Reilly, E. P., Gonul, B., Phillips, A. F., Sweeney, S. J., and Thijs, P. J. A. (1996). Phys.Stat. Solidi ( b )198, 381. 34. Silver, M.. Phillips, A. F., O’Reilly, E. P., and Adams. A. R. (1996). SPZE Proc. 2693,592. 52,1945. 35. Valhala, K. J.. and Zah, C. E. (1988). Appl.Phys.Lett.

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Streifer, B., Scifres, D. R., and Burnham, R. D. (1979). Appl.Opt.18, 3547. Physics 16, 4159. Haug, A. (1983). J. Phys.C: SolidState 49, 599. Haug, A. (1988). J. Phys.Chem.Solids, Sci.Technol. 7, 1337. Haug, A. (1992). Semicond. Silver, M., O’Reilly, E. P., and Adams, A. R. (1997). IEEEJ.QuantumElectronics 33,1557. Brand, S., and Abram, R. A. (1984). J. Phys. C: SolidState Physics 17, L201. 63,3318. O’Reilly. E. P., and Silver, M. (1993). Appl.Phys.Lett. J.132,364. Taylor, R. I., Abram, R. A,, Burt. M. G., and Smith, C. (1985). IEEE Proc. 44. Adams, A. R. (1986). Electronics Letters 22, 249. 45. Yablonovitch, E., and Kane, E. 0. (1986). J. Lightwave Technology 4,504. 30,366. 46. O’Reilly, E. P., and Adams, A. R. (1994). IEEE J. Quantum Electronics 47. O’Reilly, E. P., Jones, G., Ghiti, A., and Adams. A. R. (1991). Electronics Leners27,1417. 48. Ghiti, A,, OReilly, E. P., and Adams, A. R. (1989). Electronics Letters 25,821. 49. Tiemeijer, L. F., Thijs, P. J. A., van Dongen, T., Slootweg, R. W. M., van der Heijden, J. M. M., Binsma, J. J. M., and Krijn. M. P. C. M. (1993). Appl.Phys. Left. 62, 826. 67, 2904. 50. Silver, M., Greene, P. D., and Adams. A. R. (1995). Appl.Phys. Lett. 51. Sze, S. M. (1981). Physics of Semtconducror Devices. Wiley, New York. 52. Capasso, F. (1986). In Semiconductors and Semimetals (Tsang, W. T., ed.), Vol. 22D, p. 1. Academic Press, New York. 53. Pearsall, T. P., Capasso, F., Nahory. R. E., Pollack, M. A., and Chelikowsky, J. R. (1978). Solid-State Electron. 21, 297. 52, 296. 54. Robbins, V. M., Smith, S. C., and Stillman, G. E. (1988). Appl.Phys. Lett. 55. Shichijo, H., and Hess, K. (1981). Phys. Rev.B 23,4197. 56. Allam, J., Adams, A. R., Pate, M. A., and Roberts, J. S. (1990). Inst. Phys.Con$ Ser. 112,375. 57. Allam, J. (1992). High Pressure Research 9, 231. 58. Czajkowski, I. K., Allam, J., Adams, A. R., and Gell, M. A. (1992).J. Appl. Phys.71,3821. Lett. 30,907. 59. David, J. P. R., Hopkinson, M., and Pate. M. A. (1994). Electron. 60. David, J. P. R., Allam, J., Adams, .\. R.. Roberts, J. S., Grey, R., Rees, G. J., and Robson, P. N. (1995). Appl.Phys.Len. 66,2876. 61. Allam, J., Adams, A. R., Pate, M. A,, and Roberts, J. S. (1995). Appl.Phys.Lett. 67, 3304. 62. Allam, J. (1997). Japan.J.Appl.Phys.36,1529. 63. Allam, J., and David, J. P. R. (1996). In Hot Carriers in Semiconductors (Hess, K., Leburton, J.-P., and Ravaioli, U., eds.), p. 343. Plenum, New York. 64. Anderson, C. L., and Crowell, C. R. (1972). Phys.Rev.B 5,2267. 65. Bude, J., and Hess, K. (1992). J. Appl.Phys.72, 3554. 66. Kane, E. 0. (1967). Phys. Rev.159, 624. 67. Stobbe, M., Redmer, R., and Schattke, W. (1994). Phys. Rev.B 49,4494. 68. Keldysh, L. V. (1959). Zh.Eksp.Teor.Fiz.37, 713. [Translated as Sov.Phys.JETP 10, 509 (1960).] 69. Robbins, D. J. (1980). Phys.Stat. Sol. (6)97, 9. 70. Martinez, G. (1980). In Handbook on Semiconductors (Balkanski, M., ed.), Vol. 2, p. 182, Elsevier. 71. Hess, K. (1991). Monte CarloDeviceSimulation: FullBand and Beyond(Hess. K., ed.), Kluwer, Boston. 72. Pickering, C., Adams, A. R., and Pitt, G. D. (1975). SolidState Communications 16, 1359. 73. El-Sabbahy, A., Adams, A. R., and Young, M. L. (1978). Solid-State Electronics 21,83. Electronics 27, 347. 74. Brennan, K., and Hess, K. (1986). Solid-State 75. Stratton, S. (1958). Proc. R. Soc. 2464 406. 76. Dumke, W. P. (1968). Phys.Rev 167, 783.

36. 37. 38. 39. 40. 41. 42. 43.

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Curby, R. C., and Ferry, D. K. (1973). Phys. Stat. Sol. (a) 15,319. Ridley, B. K. (1983). 1. Phys. C: Solid State Phys. 16,3373. Ridley, B. K. (1983). J.Phys. C: Solid State Phys. 16, 4733. Bulman, G. E., Robbins, V. M., Brennan, K. F.. Hess, K., and Stillman, G. E. (1983). IEEE Electron. Dev. Lett. 4, 181. 81. Wolford, D. J., and Bradley, J. A. (1985). Solid State Commun. 53, 1069. 82. Sze, S. M.,and Gibbons, G. 1966). Appl.Phys. Lett. 8, 111. 83. Phillips, J. C. (1973). Bonds and Bands in Semiconductors. Academic Press, New York. 84. Canali, C., Forzan. C., Neviani, A., Vendrame, L., Zanoni, E., Hamme, R. A,, Ma&, R. J., Capasso. F., and Chandrasekhar, S. (1995). Appl.Phys. Lett. 66,1095. 85. Jung, H. K., Taniguchi, K., and Hamaguchi, C. (19%). J. Appl.Phys. 79,2473. 86. Yoder. P. D.. and Hess, K. (1994). Semicond. Sci.Technol. 9, 852.

77. 78. 79. 80.

SEMICONDUCTORS AND SEMIMETALS. VOL. 55

CHAPTER6

The Application of High Nitrogen Pressure in the Physics and Technology of 111-N Compounds S. Porowski and I.Grzegory HIGHPRESSURE RESEARCH CENTER POLISH ACADEMY OF SCIENCES WARSAW. POLAND

INTRODUC~ION .............................................................................................................. THERMAL STABILITY OF AIN, GaN, AND InN ......................................................... SOLUBILITY OF N IN L I Q U I D Al, Ga, AND In ........................................................... KINETICLIMITATIONS OF DISSOLUTION OF NITROGENIN LIQUID Al, Ga, AND In ........................................................................................................................... V. HIGHN2 PRESSURE SOLUTION GROWTH OF GaN ....... 1. Experimental ............................................................................................................ 2. Crystals.....................................................................................................................

1. 11. 111. IV.

VII. WET ETCHING AND SURFACE PREPARATION ................ ..................... VIII. HOMOEPITAXY .............................................................................................................. IX. CONCLUSIONS .....

354 355 357 359 362 362 363 365 370 374 377 378 378

GaN is currently considered as the most important material for blue and ultraviolet optoelectronics. The device structures are usually grown on foreign substrates, which results in a high density of dislocations above 10' cm-2. The application of high N2 pressure gives the unique possibility of growing GaN single-crystalline substrates, which allows one to lower the dislocation density in epitaxial layers by three to four orders of magnitude. This chapter presents the results of high nitrogen pressure study of properties of A1-N, Ga-N, and In-N systems. The results include phase diagrams in large ranges of pressures and temperatures (up to 2 GPa and 2000 K) and also growth of GaN single crystals from atomic nitrogen solution in liquid gallium. The kinetic limitations of the dissolution of N2 in liquid Al, Ga, and In will also be discussed. The best conditions for crystal 353 Vol. 55 ISBN 0-12-752163-1

SEMICONDUCTORS AND SEMIMETALS Copyright 0 1998 by Academic Press A11 rights of reproduction in any form reserved. oOB0/8784/98 $25.00

354

S. POROWSKI AND I. GRZEGORY

growth at available pressures and temperatures that can be achieved for GaN follow from these results. The high nitrogen pressure experimental system equipped with multizone internal furnace was used for growth of high-quality GaN crystals. A t present both n-type and semi-insulating substrate quality GaN crystals with surface areas up to 1 cm2 and with dislocation densities below lo5 cm-' are routinely obtained and successfully used for homoepitaxy. Some results concerning homoepitaxial growth by the metalorganic chemical vapor deposition (MOCVD) and molecular beam epitaxy (MBE) methods are reviewed briefly. In particular, it is shown that perfectly matched (strain-free) GaN layers can be deposited on the highly resistive GaN : Mg substrates.

I. Introduction AIN, GaN, and InN are large, direct energy gap semiconductors that attract a lot of interest as optimum materials for short-wavelength optoelectronics and high-temperature electronic applications. Currently, thin layers of these compounds are grown on foreign substrates [l, 21 by molecular beam epitaxy and metalorganic chemical vapor deposition. Both n- and p-type layers can be crystallized, and the commercial blue diode based on InGaN/GaN is now available. Nakamura and his group [3] have demonstrated that GaN-based, room-temperature-operating laser diodes can be constructed on heteroepitaxially grown GaN layer structures. This proves that GaN is an excellent material for short-wavelength optoelectronics. However, it was shown that the layers on sapphire or S i c substrates grow as columns of perfect crystallites separated by dislocations forming low-angle boundaries. The lowest dislocation density for heteroepitaxial layers of GaN is as high as 10' cm2 [4].The homoepitaxial growth of layers on nitride single-crystalline substrate is the best way to reduce the dislocation density. Unfortunately, bulk crystals of nitrides cannot be obtained by known methods such as Czochralski or Bridgman growth from stoichiometric melts because of their extremely high melting temperatures and very high decomposition pressures at melting (Table I). Growth of semiconductor-quality crystals at these conditions would be practically impossible. Therefore the crystals have to be grown by methods requiring lower temperatures. In our experiments we use the high temperature solution method to grow A"% crystals from liquid Al, Ga, and In.

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS355 TABLE I MELTING CONDITIONS FOR 111-N COMPOUNDS

-3500 1.51 -2800 IS] -2200 (5, 61

AIN GaN InN

0.2 [7) 45 [8,9]

60 [61

11. Thermal Stability of AN, GaN, and InN By comparing temperature dependencies of the Gibbs free energy G ( T ) of nitrides and their constituents, we can illustrate the analogies and differences in the properties of AlN, GaN, and InN. Figure 1shows G(T) curves [lo, 111 for 111-V nitrides and their constituents for N2 pressure of 1 bar. temperature, K 1000

oO

2000

temperature, K 1000

3000

2000

3000

s

P

112N,

c)

I

.

-

.

*

I

b

a

temperature, K 1WO

2000

temperature.K 0

3000

HP loo^.

C

A

I000

2000

3000

-501

-100

d

FIG. 1. Gibbs free energy change with temperature for 111-V nitrides and their constituents. (a), (b), and (c) NZpressure of 1 bar. (d) G (T) for the constituents is the sum of the averaged energy of the metal and the free energy of nitrogen at pressures of 1 bar and 11 kbar.

356

S. POROWSKI AND I. GRZEGORY

The reference state for G ( T )is the sum of the energy of one mole of gas of metal atoms and 1 g.at. of atomic nitrogen, that is, the reference state for bonding energies. AIN, GaN, and InN are crystals of high bonding energies compared with other III-V compounds. Bonding energy in III-V nitrides is 11.52, 8.92, and 7.72 eV per atom pair for AlN, GaN, and InN, respectively, whereas for GaAs this energy is only 6.52 eV per atom pair [El. The consequences of this higher bonding energy are high melting temperatures and good thermal stability for the considered compounds, especially AIN and GaN. On the other hand, the strong triple bond in the N2 molecule (9.8 eVlmolecule) lowers the Gibbs free energy G of the system of nitride constituents to AI(Ga, In) + 1/2 N2, approaching its energetic position to that of the nitride. Since the free energy of the constituents, which contains the free energy of gaseous nitrogen, decreases with temperature faster than the corresponding free energy of the crystal, nitride lose their stability at high temperatures. The diagrams of Fig. 1 indicate large differences in the thermal stability of III-V nitrides: Figures la, b to l c show different decomposition temperatures at 1 bar N2 pressure, which is caused mainly by the differences in bonding energy of the crystals. Figure Id illustrates the influence of N2 pressure on the stability of these nitrides. The figure shows G ( T )curves for all three nitrides and the average dependence of G for their constituents at two N2 pressures of 1 bar and 11 kbar. With increasing pressure, the free energy of nitrogen increases significantly faster than the free energy of condensed phases in the system. Therefore, the application of high nitrogen pressure allows the stability range of nitrides to be extended. As can be seen on the diagram, the increase of the thermal stability range is smallest for InN and greatest for AIN. Figure 2 shows the equilibrium pN,-T curves for AIN, GaN, and InN. The curve for AlN has been calculated by Slack and McNelly [13]. The one for GaN has been determined by Karpinski and Porowski 191 on the basis of the experiments performed by Karpinski et af. [8] with the use of gas pressure and Bridgman anvil techniques. The curve for InN has been proposed by Grzegory et af. [ l l , 141 and follows from differential thermal analysis and annealing of InN at high N2 pressure. The curves for GaN and InN at higher pressures deviate from linear dependence. This is caused mainly by the nonideal behavior of N2 gas. The diagrams indicate that at N2 pressure of 1 bar, AIN is thermodynamically stable up to temperatures approaching 3000 K and GaN up to -1200 K, whereas InN loses its stability at temperatures as low as -600 K. The technical capabilities of our experimental system limit the accessible parameters for crystallization experiments in the following ways:

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 357

P",)! FIG. 2.

bar

Equilibrium Nz pressure over 111-V nitrides.

1. For AlN, the only technical limit is related to the maximum available temperature of about 2000 K in our gas pressure system. The available pressures are sufficient even for melting. 2. For InN, the equipment pressure limit of 20 kbar determines the maximum temperature of InN stability. This is lower than 900 K, which is very far from the expected melting point. For GaN, the maximum equilibrium temperature determined by nitrogen pressure of 20 kbar is 1960 K, which comes closest to melting and suggests the best conditions for crystallization.

111. Solubility of N in Liquid Al, Ga, and In The melting temperatures of nitrides (Table I) are significantly higher than the maximum temperature available for each of the compounds in the gas pressure chamber. This causes a very low solubility of ALN, GaN, and InN in the corresponding liquid metals.

S. POROWSKI AND I. GRZEGORY

358

15001/ 7 Ann r ww

u .

1

1

0,001 0,002

I

I

I

0,003

0,004

0,005

xN' at.fr. FIG. 3. Liquidus line for a Ga-GaN system: The solid line was calculated in the ideal solution approximation.

Figure 3 shows the temperature dependence of N concentration in liquid Ga, for a Ga-GaN system. The experimental values have been obtained [ll] for the conditions corresponding to the equilibrium pN,-Tcurve (Fig. 2). The data indicate that in the high-pressure experimental system, the nitrogen content in Ga can be increased up to -1 atomic % (at.%) which is sufficient for effective crystallization from the solution (Fig. 4). The solid curve in Fig. 3 has been calculated assuming ideal behavior of the solution and the melting parameters of GaN proposed by Van Vechten [5]:TM = 2790 K and the entropy change at melting ASM = 16 kcallmole. Similar calculations performed for AlN and InN suggest that the highest solubilities, at temperatures and pressures available in the gas pressure chamber, can be expected for the Ga-GaN system (Fig. 4).The maximum nitrogen content in the solution in the Al-AlN system is determined by the technical limit for temperature (2000K), which is relatively far from the expected melting temperature of AlN. For an In-InN system, the maximum concentration is determined by the available N2 pressure (20 kbar), which limits the stability range of the nitride to T < 900 K. At such low temperatures, the expected concentration of N in liquid In is lower than 0.001 at.%.

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 359 3000

AIN

GaN ...-..-7.. --. . ...-.....-..

InN

decomposition of InN at 20 kbar

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

500 0.005

0.010

0.015

0.020

x(N), mole fraction FIG. 4. Liquidus line for AI-AIN. Ga-GaN, and In-InN systems calculated in the ideal solution approximation.

IV. Kinetic Limitations of Dissolution of N2in Liquid Al, Ga, and In In our crystallization process, GaN is synthesized from its constituents: liquid G a and N2 gas. The first stage of this process is the dissolution of nitrogen in liquid metal. This process was analyzed by ab initio quantum mechanical calculations using density functional theory (DFT) approximation [15, 161. It was shown that when the N2 molecule approaches the metal surface, the interaction between nitrogen and metal atoms results in a small increase of the distance between the nitrogen atoms (Figs. 5a, b, and d). When the distance between the N2 molecule and the surface decreases to d = 1.6 A, the dissociation of the N2 molecule and formation of the bonds between nitrogen and metal atoms takes place (Fig. 5c and d). It was also shown that an N2 molecule approaching the metal surface is repelled by the surface, which results in a potential barrier. These barriers have been calculated by Krukowski ef al.[17] for Al, Ga, and In surfaces (Fig. 6). If the energy of the N2 molecule is sufficient to overcome the potential barrier, the dissociation takes place. The potential barriers are

360

--_

S. POROWSKI AND I. GRZEGORY

-.. ... .. .-

.._.-. .....

9

4 I1

4

a

I:

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 361

In

-x-

---&--

-,. , - , . , 0,8

I

-

.

I

1,2 1,6 2,O 2:4 2:8 3'2

-

3:6 4:O

d [A1 FIG. 6. Energy of interaction of an N2 molecule with Al,Ga, and In surfaces.

considerably lower than the bonding energy of the N2 molecule. However, their values are quite high, which suggests that the dissociation process can be kinetically controlled even for relatively high temperatures. The barrier is lowest for A1 and highest for In. From this we can conclude that the dissociation process and subsequent nitrogen dissolution will require higher temperatures for In than Al. The efficiency of the dissolution process at certain T and p can be estimated from simple statistical evaluation of the stream of N2 molecules that have a higher energy than the corresponding barrier. It will increase with the temperature, following the Arrhenius formula. Since in a first approximation the stream will be proportional to the density of N2 gas, it will also increase with pressure. Figure 7 shows the temperature dependence of the rate of dissociation for two pressures (1 kbar, 20 kbar) calculated using this assumption. The dotted line of the diagram corresponds to the dissociation rate I(T ) ,which is solow that below this point the synthesis of A"'N nitride is practically impossible. As a practical limit we used the efficiency of 10 mg of nitride for 100 h process per 1 cm2.The crossing of the dotted line with the calculated reaction rates therefore determines the lowest temperatures for direct synthesis of A'"N nitrides from N2 gas and liquid metal. For 20 kbar these temperatures are 1000, 1500, and 1800K for

(a) d = 2.6 8, (b) d = 2.6 molecule and the surface.

A,

(c) d = 2.6 A, (d) N-N spacing: d is distance between NZ

362

S. POROWSKI A N D I. GRZEGORY

T IKI FIG. 7. N2 dissociation rate I on Al, Ga, and In surfaces. Solid line: p = 20 kbar. Dashed line: p = 1 kbar.

AlN, GaN, and InN, respectively. These estimations are consistent with experimental observation of combustion direct synthesis of A1N at 1300 K [18], crystallization of GaN from the solution of N in liquid gallium [19], and the absence of InN synthesis from its constituents. The kinetic barrier for InN can probably be avoided by the use of InN powder as a source of nitrogen. However, as was shown in the previous section, the estimated solubility of nitrogen in liquid indium is extremely small for temperatures below 900 K, which makes the growth of the InN crystals in our experimental system practically impossible.

V. High N2Pressure Solution Growth of GaN 1. EXPERIMENTAL The cross-section of a high-pressure reactor for crystallization of GaN is shown in Fig. 8. The three-zone cylindrical graphite furnace is placed inside the high-pressure chamber and is immersed in pressurized nitrogen gas. The maximum pressure reaches 20 kbar, and the maximum temperature is 2000 K. Gas pressure chambers with volume up to 1500 cm3 are used. The working volume of the furnace allows the use of crucibles containing up to 100 cm3 of liquid Ga. The crucibles are made either from graphite or hexagonal BN. In most cases they are of cylindrical shape and can be used in both vertical and horizontal configurations. The high-pressure reactor is equipped with additional systems necessary annealing in vacuum, cooling of the pressure chamber, electronic for in situ

6 APPLICATION OF HIGHNITROGEN PRESSURE TO III-N

d

7 '8

'9

Y

COMPOUNDS

363

8 / 7 / 6

FIG. 8. Cross-section of the high-pressure chamber for crystal growth experiments: (1) sample, (2) crucible, ( 3 )thermocouples, (4) thermal insulation, (5) heater, (6) metal gasket, (7) O-ring, (8) electrical lead, (9) steel tube, (10) steel plug.

stabilization, and programming of pressure and temperature. Pressure in the chamber is stabilized with a precision better than 10 bar. The temperature is measured by a set of thermocouples arranged along the furnaces and coupled with the standard input power control electronic systems based on Eurotherm units. This allows stabilization of temperature to within ?0.2 degrees and programmable changes of temperature gradients in the crucible. The supersaturation in the growth solution is created by the application of a temperature gradient of 2 to 20"Clcm along the axis of the crucible. The nitrogen dissolves in the hotter end of the crucible, and GaN crystallizes in the cooler end. The slow cooling method was not applied due to small concentrations of nitrogen in the liquid gallium (Fig. 3). The GaN crystals presented in this chapter were grown from the solutions in pure liquid gallium and in Ga alloyed with 0.2 to 0.5 at.% of Mg, Ca, or Zn. Mg, Ca, and Zn - the most common acceptors in GaN -were added to the growth solutions to reduce the concentration of free electrons in the crystals by compensation of residual donors. The crystallization experiments were performed without an intentional seeding, and the crystals nucleated spontaneously on the crucible walls at the cooler zone of the solution. The typical duration of the process was 120 to 150 hours. 2. CRYSTALS

The GaN crystals grown by this method are of wurtzite structure, mainly in the form of hexagonal platelets or hexagonal needles. High supersatura-

364

S . POROWSKI A N D I. GRZEGORY

tions favor growth in the c direction, which leads to the needlelike forms. The supersaturation in the growth solution is determined mainly by the growth temperature, temperature gradients, and mass transport mechanisms in gallium, and also by the local surrounding for a particular crystal (neighboring crystals). The crystals in the form of hexagonal platelets that are grown slowly, with a rate of < 0.1 mm/h in (1010) directions (perpendicular to the c-axis), are usually single crystals of peffect morphology, suggesting stable layerby-layer growth. They are transparent, with flat, mirrorlike faces. In some cases three-dimensional, hexagonal-growth figures are observed growing on the crystal surfaces during cooling of the system after crystallization in the temperature gradient. The GaN platelets grown without an intentional doping are shown in Fig. 9. As one can deduce from the form of the crystals, the growth is strongly anisotropic, being much faster in directions perpendicular to the c-axis. This relation is valid at supersaturations corresponding to the average growth rate in (1010) directions of 0.05 to 0.1 mm/h. For crystallization of large GaN crystals, it is crucial to control the supersaturation in order to avoid acceleration of the growth near edges and corners of the growing crystal. If supersaturation is too high, edge nucleation on the hexagonal faces of GaN platelets is often observed, which is the first step to unstable growth on those faces. The result of such a growth is shown in Fig. 10a. In extreme cases of very high supersaturation, the growth in the c direction nucleated at the edges of the plate becomes very fast, which leads to the formation of well-developed (1010) faces. Since the lateral growth on the c face is still slow, the resultinccrystals are hollow needles elongated in the c direction (Fig. lob). The tendency for unstable growth is stronger for one of the polar (0001) faces of the platelets. O n this side morphological features such as macrosteps, periodic inclusions of solute, or cellular growth are observed. The opposite surface is always mirrorlike and often atomically flat. Only this side of the crystals grown from solutions in pure Ga can be etched by alkaline water solutions [20]. It was identified by X-ray photoelectron diffraction (XPD) measurements 1211 as an (OOO1) N-terminated surface. The crystals grown from the solutions containing Mg, Ca, or Zn have the form of hexagonal platelets, similar to the undoped samples. This result indicates that the addition of these impurities into the growth solution does not significantly influence the relative growth rates. It is observed, however, that crystals containing Mg grow slightly slower in directions perpendicular to the c-axis and faster in the c directions, resulting in relatively smaller but thicker platelets. For crystals doped with Mg, the etching behavior is different, which means that the flat face is always inert to etching and only the opposite

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 365

FIG. 9. GaN crystals grown from the solution in liquid Ga at high N2 pressure. The distance between two grid lines corresponds to 1 mm.

side can be etched. However, the polarity of the Mg-doped crystals has not yet been determined and it is still unclear if Mg doping changes the growth mechanism or etching mechanism.

VI.

Physical Properties of Pressure-Grown GaN Crystals

GaN crystals grown from solutions in pure gallium are highly conductive, showing metallic behavior (Table 11) in the whole temperature range of 4.2 to 300 K. High free-electron concentrations of 3-6 X cm-3 with

S. POROWSKI AND I. GRZEGORY

366

b

a

FIG. 10. (a) Edge nucleation on (OOOI) face of pressure-grown GaN crystal. (b) Hollow GaN crystal with (1010) face formed at increased supersaturation.

mobilities of 30 to 90 cm2/Vs [22] have been found. The main residual impurity detected in the crystals by secondary ion mass spectroscopy (SIMS) [23] is oxygen. Its concentration was estimated to be in the range of 1OI8 to 10" ~ m - It~ is. well established that oxygen is a single donor in GaN. However, since the concentration of free electrons in the crystals is higher than the estimated concentration of the impurity, the presence of an additional donor cannot be excluded. The N vacancy is therefore often proposed as the additional source of free electrons [24,25]. Following the theory [24, 261, the most probable native defects in GaN crystals that are highly n-type are Ga vacancies (V&).This is due to their low formation energy even at strongly Ga-rich conditions of crystallization. The presence of the negatively charged Ga vacancies at concentrations of lo1' in the n-type pressure-grown GaN crystals has been detected by positron annihilation experiments [27]. The introduction of 0.2 to 0.5 at.% of Ca or Zn into the growth solution does not change the character of the electrical properties of the crystals. Crystals still show the metallic type of conductivity, although free-electron concentration decreases by a small factor. In contrast, the addition of Mg into the growth solution drastically changes the electrical properties of GaN crystals. Their resistivity increases by orders of magnitude (Table 11). TABLE I1 ELECTRICAL PROPERTIES OF GaN CRYSTALS ___

Crystal GaN GaN :Mg

Conductivity type Metallic Hopping

(acm, 300 K)

P

Carrier concentration (cm-')

10 2-10-z 104 -106

3-6 x loi9, n-type -

6 APPLICATION OF HIGHNITROGEN PRESSURE TO III-N COMPOUNDS 367

la,

Q "

10

h 1

0.a

1.0

1.1

1.2

1.3

1.4

(Torn"4

FIG. 11. Temperature dependence of resistivity for GaN crystals grown from the solutions in both pure liquid Ga and in Ga alloyed with Mg.

The increase of electrical resistivity in GaN:Mg crystals is related to a drastic decrease of the free-electron concentration. The temperature dependence of resistivity for these samples is typical for hopping conductivity [28] (Fig. l l ) , which suggests that the Fermi level lies within the gap. Also, the optical absorption data [28] indicate that the free-carrier concentration in the Mg-doped GaN is very low. Figure 12 compares the optical absorption of both undoped and Mg-doped pressure-grown GaN crystals. 2

E

2 I-

sY

u,

5

8 2 0

E

a

Q:

$ m T

500

1000

1500

2000

2500

WAVELENGTH (nm)

FIG. 12. Optical absorption for GaN crystals grown from the solutions in both pure liquid Ga and in Ga alloyed with Mg.

368

S. POROWSKI A N D I. GRZEGORY

The free-carrier absorption that dominates the low-energy part of the absorption spectra for the undoped GaN disappears completely for crystals grown from Mg-containing solutions. The absence of free electrons in GaN :Mg crystals is also reflected in their Raman spectra [29]. In particular, a strong peak corresponding to the longitudinal optical (LO) phonon is observed, in contrast to the undoped crystals, in which this peak is not visible at all due to the interaction with free electrons. In comparison with the undoped crystals, a number of additional peaks are visible in the Raman spectra for GaN :Mg samples, which are analyzed in detail in Ref. [29]. In contrast to the highly conductive n-type crystals, no Ga vacancies have been detected by positron annihilation [30] in the highly resistive GaN :Mg crystals. This shows that doping of GaN substantially modifies its stoichiometry, which is in qualitative agreement with the results of ab inifio calculations [24,26] that predict the decrease of V,, concentration with decreasing Fermi energy. Crystals grown from N solutions in pure Ga show strong yellow photoluminescence (PL) and relatively weak PL signal (Fig. 13a) close to the energy gap. Both signals disappear in crystals obtained from solutions doped with Mg, Zn, and Ca. Instead, a strong broad PL signal occurs at energies of 2.8 to 3.2 eV (at 80 K). Figure 13b shows examples of the PL spectra for crystals doped with Mg, Zn, and Ca. For different samples doped with the same impurity, the maximum of the peak can shift as much as 0.1 eV. The mechanism of this luminescence is not well understood. The structure of the crystals has been investigated by X-ray diffraction methods [31]. Table I11 summarizes the data characterizing the crystal structure of both highly conductive (grown from pure Ga, Ga + Ca, and Ga + Zn solutions) and nonconductive (grown from Ga + Mg alloys) crystals. For the conductive crystals, the shape of the X-ray rocking curves [(0002) CuKa reflection] depends on the size of the crystal. The full widths at half maximum (FWHM) are 20 to 30 arcsec for I-mm crystals and 30 to 40 arcsec for 1- to 3-mm ones. For larger platelets, the rocking curves often split into a few approximately 30- to 40-arcsec peaks showing a presence of low-angle (1-3 arcmin) boundaries separating grains of 0.5 to 2 mm in size. Misorientation between grains increases monotonically from one end of the crystal to the other [31]. Liliental-Weber et af. [32] showed by transmission electron microscope examination that one of the polar surfaces of GaN crystals (especially for the smaller ones) is often atomically flat (two to three monolayer steps present) and that the crystal under this surface is practically free of extended defects. Under the opposite surface a number of extended defects, such as stacking faults, dislocation loops, and Ga microprecipitates, are observed.

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 369

1.8

1.0

2.0

2.2

2.4

2.1

2.0

3.0

3.2

3.4

B

3.8

Energy [eVl

a

OaNPn

1.0

0.8

. g u)

0.6

-

0.4

c

0.2

0.0 1.0

2.0

2.2

2.4

26

2.8

3.0

Energy [eW

3.2

3.4

3.6

b

FIG. 13. (a) Low-temperature(80 K) photoluminescence spectrum of GaN crystals grown from the solutions in pure gallium. (b) Low-temperature (80 K ) photoluminescence spectra of GaN crystals grown from the solutions containing 0.2-0.5 at.% of Mg, Zn. and Ca.

The relative thickness of this part usually consists of 10% of the entire thickness of the platelet. Such peculiarities result from different growth modes in the two polar directions of the crystal in Ga-rich conditions of the crystallization process. It was observed [31] that the lattice constants for undoped GaN and the highly conductive Ga :Zn and GaN : Ca bulk crystals vary slightly for different crystals as well as for different areas of the individual crystals. This variation is caused by the differences in free-electron concentration and can lead to certain strains in the crystals.

S. POROWSKI AND I. GRZEGORY

370

TABLE I11 LAITICEPARAMETERS FOR GaN CRYSTALS AND HOMOEPITAXJAL LAYERS Sample GaN conductive. bulk Undoped homoepitaxial GaN on undoped GaN bulk GaN : Mg, bulk undoped homoepitaxial GaN Undoped homoepitaxial GaN on GaN :Mg bulk a

c (4

FWHM X-ray rocking curve (0002) refl.

3.188-3.1890

5.1856-5.1864

30-40 arcseconds

3.1881"

5.1844

30-40 arcseconds

3.1876

5.1846

20-30 arcseconds

3.1876

5.1846

20-30 arcseconds

a

(A)

Lattice parameter a for the substrate and layer is always the same.

There are no free electrons for the Mg-doped crystals and therefore the lattice parameters are uniform for individual crystals and the same for different samples [33]. Elimination of the strains in the GaN :Mg crystals improves their structure (Fig. 14). The rocking curves are narrower and no low-angle boundaries are observed even for 8-mm single-crystalline GaN :Mg platelets. The strains in the undoped GaN crystals are extremely small (< 0.02%), but their presence can adequately explain the wider rocking curves than those seen for the Mg-doped samples. Comparative studies of diffusion of Zn into the heteroepitaxial GaN layers on sapphire and bulk pressure-grown crsytals have been reported [34]. It was shown that dislocations play very important roles in the diffusion process in heteroepitaxial GaN layers. Figure 15 shows the Zn profiles in both the GaN crystal and the layer after 1 h annealing at N2 + Zn atmosphere at a temperature of 1350°C and a pressure of 10 kbar. As can be seen, the penetration of Zn into the bulk crystal is a few orders of magnitude slower than in the heteroepitaxial layer. A similar effect of reduced diffusion due to the reduced density of dislocations can be expected for homoepitaxial GaN layers. This can improve the quality of GaN p-n junctions [35].

VII. Wet Etching and Surface Preparation

The surfaces of the pressure-grown GaN crystals are often covered by the growth figures or surface layer resulting from the cooling of the system after crystallization at high temperature. Therefore, to obtain epi-ready surfaces of GaN substrates it is necessary to subject them to mechanical and mechanochemical polishing.

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 371

X-ray rocking curves, 00.4 reflection I

I

i !

-

1 I

undoped bulk GaN

.a

-

v)

C

a -

U

.-C

.Mg-doped bulk GaN 5 mm

LWHM=PO arc set

-

I

I

I

I

angle (degrees) FIG. 14. X-ray rocking curves for GaN crystals.

Mechanical polishing with diamond micropowders leads to the formation of highly damaged surfaces with scratches of 200 A in depth. The thickness of the damaged layer under such mechanically polished surfaces is usually 2000 to 2500 A, as was shown by the Rutherford backscattering (RBS) measurements [36] (Fig. 16). It was shown that bulk pressure-grown GaN crystals (highly conductive, undoped) can be etched in aqueous solutions (10N-1N) of KOH and NaOH [20]. However, only one of the polar (0001) surfaces of these bulk crystals is attacked by the applied etchants. The free etching of this surface ((OOOl),, as identified by XPD [21]) is strongly anisotropic and results in the formation of numerous stable pyramids 100 to 200 nm high. As was mentioned before, the Mg-doped crystals only one (0001) surface etches as well: however, it is not yet known what polarity it is. The same aqueous solutions (10N-1N) of KOH and NaOH have been used for mechanochemical polishing of the chemically active GaN surface [20]. It was shown that the use of a soft polishing pad and pressure above

S. POROWSKI AND I. GRZEGORY

3 72

S i M S Profiles 107

10s

105

-m

lo(

C

.-cn

16

v)

z

v,

102

10’

100

1W

0

500

1 m

1500

2000

Depth [s]

-

1000 seconds 1 prn of sputter depth FIG. 15. Zn profiles in GaN pressure-grown crystal and GaN heteroepitaxial layer after 1 h annealing at N2 + Zn atmosphere at a temperature of 1350°C and a pressure of 10 kbar.

2 kglcm’ removes all surface irregularities from the polished surface. When the mechanochemical polishing procedure is applied, atomically flat surfaces of bulk GaN (RMS = 0.1 nm as estimated by atomic force microscope) are reproducibly obtained. Figure 17 shows the X-ray reflectivity spectrum for an 80 mmz mechanochemically polished GaN substrate measured in the European Synchrotron Radiation Facility Grenoble [37], confirming that the surface is atomically flat. The RBS measurements [36] (Fig. 16) also indicate that the applied procedure allows removal of the subsurface damage resulting from the mechanical polishing. The procedures of wet etching and mechanochemical polishing with alkaline water solutions can also be applied for GaN heteroepitaxial layers, but only to those that show a tendency to develop hexagonal hillocks if grown by MOCVD. The orientation of these chemically active epitaxial GaN layers is the same as the orientation of the chemically active surface of the conductive GaN crystals.

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 373 35

I

I

I

I

I

30

surface-

25

-

-

-

-

- mechano-chemicalpohhing 0

-

I\

shucfure n m damaged

A

I

I

I

I

-

1

I

channel FIG. 16. RBS signals for polished GaN surfaces. (From Ref. [36].)

Reflectivity curves 1000000

ps .s

.

I

’7 '

\

lo00

GaN 10 A roughness 100 in

I

0.0

.

'

0.5

.

'

1.0

.

1.5

20

2.5

fhefa (degrees)

FIG. 17. X-ray reflectivity for mechanochemically polished surface of an 80-mm2GaN substrate. (From Ref. [37].)

374

S. POROWSKI AND I. GRZEGORY

VIII. Homoepitaxy There are no systematic data on homoepitaxy because the effective method of mechanochemical polishing has only recently been developed for one of the polar surfaces of GaN. Nevertheless, many important results on homoepitaxy reported up to now indicate new possibilities for GaN physics and technology. The homoepitaxial growth of GaN has been studied by both MOCVD [35, 38, 391 and MBE [40, 411. The experiments have been performed on both polar surfaces of GaN crystals. The “as-grown’’ [38, 391 as well as the mechanically [42] and mechanochemically [41] polished surfaces were used. The X-ray diffraction from MOCVD homoepitaxial layers and from GaN substrates (as-grown, highly conductive substrates were used) have been measured [38]. The curves of the layers were narrower than for the substrates, indicating improvement of crystal quality during the epitaxial growth. The lattice constant parallel to the interface was the same for both substrate and layer, whereas the c constant (perpendicular to the interface) of the layer was smaller by about 0.01% (Fig. 18). This effect is related to the fact that the lattice constants for undoped GaN crystals are slightly swallowed due to the presence of a high density of free electrons. It was observed [33] (Table 111, Fig. 18) that for Mg-doped substrates the lattice constants of the substrates and the undoped layers are perfectly identical. It seems that such unstrained homoepitaxial layers are the best available standard for GaN. The high quality of homoepitaxial layers grown by MOCVD was confirmed by photoluminescence measurements [38].Very narrow (-0.5 meV) lines corresponding to bound exciton and close donor-acceptor pair transitions have been reported. Very narrow excitonic lines have been also observed for homoepitaxial GaN grown by MBE on (OOOl), as-grown surfaces of GaN substrates [40]. A lower-energy line related 70 acceptor-bound excitons (ABE), lying at 3.4663 eV, had the half width of 0.5 meV. The two next lines, at 3.4709 and 3.718 eV, had similar widths. These lines, attributed to the excitons bound to two different donors (DBE), were distinctly separated despite a small (1 meV) distance between them. In the measurements [43] of PL at 2 K for MOCVD layers, the relative chemical shift for ABE has also been observed. Two acceptor-bound exciton lines with relative chemical shift of 0.8 meV were resolved. The lines were the narrowest ever reported for GaN (FWHM of 0.3 meV). Reflectance and photoluminescence spectra of exciton-polaritons in GaN homoepitaxial layers grown by MOCVD have been reported [44]. High-quality samples allowed well-resolved reflectance spectra in the free

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 375

Triple axis X-ray diffractionscans

72.00

72.90

72.92

72.94

2 theta (degrees)

I 5.1868

I

5.1856

I 5.1844

I 5.1832

c(A) FIG.18. Lattice constants of GaN substrates and homoepitaxial layers.

exciton region to be measured and the polariton structure of GaN to be studied. The energies of the transverse excitons were found to be ETA = 3.4767 2 0.0003 eV, Em = 3.4815 2 0.0003 eV, and ETc = 3.4986 2 0.0008 eV. Using the obtained exciton energies, the parameters describing the valence band splitting Aso = 17.9 2 1.2 meV and A,, = 8.8 2 0.8 meV have been estimated. The data obtained for GaN homoepitaxial layers in the cited paper [44] might be used as a reference to determine the shift of excitonic resonances in heteroepitaxial layers. Near-ideal atomic step-flow growth mode (Fig. 19) has been achieved by Cohen etal.[41] using MBE with NH3 as the nitrogen source. The growth experiments were performed on mechanochemically polished GaN pressure-grown crystals with an intentional misorientation of 0.5 to 2". The

S. POROWSKI AND I. GRZEGORY

376

STM of MBE arown GaN on misoriented (2-3")Polish GaN wafer 06/26/97-slowlycooled and vacuum transfered to STM

140

120

100

80

60

40

20

0 0

20

40

60

80

100

120

140 nm

ammonia rich (backaround pressure=4xlO(-7) Torr) atoms/sec/cm-(-Z), substrate temp about 760°C File name: 06301007.001 Ga flux=l.45~10"(14)

FIG. 19. Scanning tunneling microscope scan of a GaN homoepitaxial layer grown by MBE [41] on the (OOOl) mechanochemicallypolished surface of G a N pressure-grownsubstrate.

distance between monolayer steps (height of 2.7 A) corresponded to the misorientation angle. For the growth on the substrates without an intentional misorientation, the ideal reflection high-energy electron diffraction (RHEED) behavior was observed by the same group [41]. The period of the RHEED oscillations corresponded to the growth rate, in contrast to the growth of GaN on sapphire, in which defects reduce the effective Ga flux. No pinholes were observed, in contrast to GaN on sapphire.

6 APPLICATION OF HIGHNITROGEN PRESSURE TO 111-N COMPOUNDS 377

The growth of A1,Gal_,N ternaries has been also studied by MBE with an RF plasma source of nitrogen [37].The experiments were performed in the European Synchrotron Radiation Facility Grenoble, equipped with a vacuum transfer of the samples from the MBE chamber to the X-ray diffractometer. The A1,Gal-,N layers grown on the bulk GaN crystals RMS), as measured using X-ray exhibited a very low roughness (3-5 reflectivity. Using X-ray diffraction, it was found that the layers of x = 0.2 were not relaxed even to a thickness of 3000 A. For x = 1, the layer of about 1500 A was partially relaxed, whereas the layer of about 100 was fully strained. A comprehensive study of homoepitaxial homojunction GaN LEDs grown on bulk pressure-grown crystals has been performed [35]. Singlepeak blue emission at 420 nm with a linewidth of 60 nm was obtained. The high quality of the homoepitaxial diodes was revealed by a comparison with diodes heteroepitaxially grown on sapphire. The homoepitaxial devices were twice as bright as LEDs grown on sapphire.

A

A

IX. Conclusions High nitrogen pressure allows the range of stability of AlN, GaN, and InN to be extended for higher temperatures. The use of higher temperatures decreases both the thermodynamic barrier due to low solubility and the kinetic barrier for N2 dissociation. This allows synthesis and crystal growth of GaN and AlN. However, these barriers prevent growth of InN single crystals using nitrogen pressures up to 20 kbar. High-quality GaN single-crystalline platelets with a surface area of 60 to 100 mm2have been grown by the high-pressure method in a reproducible way. The residual donors in the crystals were compensated by doping 0.2 to 0.5 at.% of Mg, Ca, and Zn added to the growth solutions. Full compensation was achieved for crystallization from solutions containing Mg. Therefore, GaN single crystals of both low and high conductivity can be grown under pressure. Atomically flat surfaces of GaN substrates have been obtained by mechanochemical polishing with alkaline water solutions. It was demonstrated that near-ideal growth of homoepitaxial layers by the propagation of monoatomic steps is possible by both the MOCVD and MBE methods. First results on homoepitaxial LEDs are encouraging for further development of homoepitaxy-based device technology.

378

S. POROWSKI A N D I. GRZEGORY

Acknowledgments

The work on GaN crystallization is supported by Polish Committee for Scientific Research grants no. 7 7834 95C/2399 and 7T08A 007 13.

REFERENCES [l] H. Amano, M. Kito, K. Hiramatsu, and 1. Akasaki, Jpn. J. Appl.Phys.28,2112 (1989). [2] S. Nakamura, T. Mukai, M. Senoh, and N. Iwasa, Jpn. J.Appl.fhys. 31, 139 (1992). [3] S. Nakamura. M. Senoh. S. Nagahama. N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, and Y. Sugimoto. Jpn.J. Appl.fhys. 35, L74 (1996). [4] S. Nakamura, to appear in f r o c2nd . Int. Conf:on Nitride Semicond. ICNS '97, Tokushima, 1997. [ S ]J. A. Van Vechten. fhys. Rrv.5 7, (1973). [ S ]1. Grzegory, S. Krukowski. J. Jun. M. BoCkowski, M. Wroblewski, and S. Porowski, AIP Conference Proceedings 309,565 (1994). [7] Class W, Contract Rep., NASA-Cr-1171 (1968). (81 J. Karpinski. J. Jun, and S. Porowski. J. Crysr. Growth, 66,1 (1984). 191 J. Karpinski and S. Porowski, J. Cryst. Growth66,11 (1984). [lo] W. P. Glushko et al., in Termodinamitcheskije Svojstwa lndividualnych Veshtchestv, Nauka, Moscow. 1979 [in Russian]. [ l l ] I. Grzegory. J. Jun, M. Bockowski, S. Krukowski, M. Wroblewski, B. Lucznik, and S. Porowski, J. fhys. Chem.Solids 56, 639 (1995). [12] W. A. Harrison. Electronic Structure and Properties of Solids, Freeman, San Francisco. 1980. [13] G. A. Slack and T. F. McNelly, 1. Cryst. Growth34,276 (1976). [14] I. Grzegory. in froc. o f h i nXtV A I R A f T a n dX X X V E H f R G Conference, (W. Trzeciakowski, ed.). p. 14. World Scientific, Singapore, 1996. [15] P. Hohenberg and W. Kohn. fhys. Rev.136, 864 (1964). [16] W. Kohn and L. J. Sham. Phys.Rev.140, 1133 (1965). [17] S. Krukowski, Z. Romanowski, I. Grzegory and S. Porowski. to be published. [I81 M. BoCkowski, 1. Grzegory. M. Wroblewski, A. Witek, J. Jun, S. Krukowski, S. Porowski, R. M. Ayral-Marin, and J . C. Tedenac, AIP Conference Proceedings 309,1255 (1994). [19] S. Porowski and 1. Grzegory, J. Crystal Growrh178, 174 (1997). Growth182,17-22 (1977). [20] J. L. Weyher. S. Miiller, 1. Grzegory, and S. Porowski, J .Cryst. [21] M. Seelmann-Eggebert. J. L. Weyher. H. Obloh, H. Zimmermann, A. Rar, and S. Porowski, Appl. fhys. Lett. 71, 18 (107). [22] P. Perlin. J. Camasel, W. Knap. T. Talercio, J. C. Chervin. T. Suski, I. Grzegory. and S. Porowski, Appl.Phys.Lett. 67, 2524 (1995). [23] A. Barcz and T. Suski, unpublished. [24] P. Boguslawski, E. Briggs. and J. Bemholz, fhys. Rev.B 51,17255 (1995). [25] W. Kim, A. E. Botchkarev, A. Salvador. G. Popovici. H. Tang. and H. Morkoc, J. Appl. fhys. 82,I (1997). [26] J. Neugebauer and C .G. Van de Walk, fhys. Rev.B 50, 8067 (1994). [27] K. Saarinen. T. Laine, S. Kuisrna. P. Hautojarvi, L. Dobrzyriski, J. M. Baranowski, K. PakuIa. R. Stqpniewski, M. Wojdak. A. WysmoIek, T. Suski, M. Leszczynski, 1. Grzegory. and S. Porowski. Phys.Rev.Lett. 79,3030 (1997). [28] S. Porowski. M. Bockowski. B. Lucznik. I. Grzegory, M. Wroblewski, H. Teisseyre,

6 APPLICATION OF HIGHNITROGEN PRESSURE TO III-N

COMPOUNDS

379

M. Leszczynski, E. Litwin-Staszewska, T. Suski, P. Trautman, K. Pakula, and J. M. Baranowski, ActaPhysica Polonica, A Vol. 92, No. 5, 958 (1997). [29] W. Gembicki and H. Teisseyre, unpublished. presented at MRS Fall Meeting, Boston, 1997. [30] K. Saarinen etal., [31] M. Leszczynski, I. Grzegory, H. Teisseyre, T. Suski, M. Bockowski, J. Jun. J. M. Baranowski, S. Porowski, and J. Domagala, J. Cryst. Growth169,235 (1996). [32] 2. Liliental-Weber, S. Ruvimov, C. Kisielowski, Y. Chen, W. Swider, J. Washborn, N. Newman, A. Gassmann, X. Liu, L. Schloss, E. R. Weber, I. Grzegory, M. Bockowski, J. Jun, T. Suski, K. Pakula, J. Baranowski, S. Porowski, H. Arnano, and I. Akasaki, Mat. Res.SOC. Symp.Proc.395,351 (1996). [33] J. Domagala, P. Prystawko, M. Leszczynski, T. Suski, and S. Porowski, unpublished. 1341 T. Suski etal., presented at MRS Fall Meeting, Boston, 1997. (351 A. Pelzmann, C. Kirchner, M. Mayer, M. Schauler, M. Kamp, K. J. Ebeling, I. Grzegory, M. Leszczynski, G. Nowak, and S. Porowski, presented at ICNS’97, Tokushima, Japan, 1997. [36] M. Conway, J. S. Williams, and C. Jagadish, private communication. [37] R. Langer, A. Barski, M. Leszczynski, I. Grzegory and S. Porowski, unpublished. 1381 J. M. Baranowski, Mar. Res.Soc. Symp.Proc.449,393 (1997). [39] K. Pakula, A. Wysmolek, K. P. Korona, J . M. Baranowski, R. Stepniewski. I. Grzegory, M. Bockowski, J. Jun, S. Krukowski. M. Wroblewski, and S. Porowski, Solid State Comm. 97,919 (1996). [40] H. Teisseyre, G. Nowak, M. Leszczynski, I. Grzegory, M. BoCkowski, S. Krukowski, S. Porowski, M. Mayer, A. Pelzmann, M. Kamp, K. J. Ebeling, and G. Karczewski, MRS J . Nitride Semicond. Res.1, 13 (1996). Internet [41] R. Held, S. M. Seutter, B. E. lshaug, A. Parhamovsky, A. M. Dabiran, P. I. Cohen, C. J. Palmstroem, G. Nowak, I. Grzegory, and S. Porowski, submitted to MRS Fall Meeting, Boston, 1997. [42] F. A. Ponce, D. P. Bour, W. T. Young, M. Saunders, and J. W. Steeds, Appl.Phys.Lett. 69,2 (1996). [43] I. Ivanov, A. Henry, B. Monemar, and J. M. Baranowski, unpublished result. [44]R. Stqpniewski, K. P. Korona, A. Wysmolek, J. M. Baranowski, K. Pakula, M. Potemski, G. Martinez, I. Grzegory, and S. Porowski, Phys. Rev.B 356, 15151 (1997).

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SEMICONDUCTORS AND SEMIMETALS. VOL. 55

CHAPTER7

Diamond Anvil Cells in High-pressure Studies of Semiconductors Mohammad Yousuf MATERIAISSCIENCE DIVISION RESEAR~H INDIRA GANDHICENTREFOR ATOMIC KALPAKKAM, TAMIL NADU,INDIA

I. DAC: ANAPPARATUS PAREXCELLENCE TO ACHIEVE HIGHEST STATIC PRESSURE ............................................................................................... 1. Diamond as an Anvil Material ................................................................................ 2. Precision Fabrication of a 3. Principle of the Alignment 4. Gasketing: A Turning Point in DAC Use ............................................................... 5. Preparation of the Sample 6. Pressure Scale and Pressure Calibration .............................................. .................. 7. Pressure-Transmitting Medium and the Limit of Hydrostaticity ........................ 8. Pressure Combined with Other Thermodynamic Fields ....................................... 11. CONDENSED MATIER PHYSICS TECHNIQUES COUPLED TO A DAC ........................... 2. X-Ray Diffraction ...................................................................................................... 3. Transport Properties 111. HIGH-PRESSURE STUDIES OF SEMICONDUCTORS ...........................................................

IV. CONCLUDING REMARKS ................................................................................................... ...................... Acknowledgments ......

382 382 396 400 404 405 406 409 41 1 414 415 420 424 424 425 427 429 430

The present status of high-pressure research with the diamond anvil cell (DAC) is reviewed in this chapter, mainly from the point of view of a technique. After a brief description of the diamond anvil, the different types of DACs that are currently in use are discussed. Design considerations for the DAC developed at our laboratory are described next. A summary of aspects such as the anvil alignment, gasketing, sample loading in the DAC, pressure calibration, and pressure-transmitting media then follows. Integration of other field variables such as temperature and magnetic field is an important landmark in the history of the development of the DAC. 381 Vol. 55 ISBN 0-12-752163-1

SEMICONDUCTORS AND SEMIMETALS Copyright 0 195'8 by Academic Press AU rights of reproduction in any form reserved. 08018784198 $25.00

382

M. Y o u s u ~

An attempt has been made to bring out this aspect. Almost all the condensed-matter physics techniques have been coupled to the DAC. Techniques such as photoluminescence, Raman and Brillouin scatterings, X-ray diffraction, and transport property measurements are discussed in this chapter from the point of view of semiconductor research. Some typical results on conventional and strongly correlated semiconductors are described next. The concluding section briefly deals with speculations regarding ultrahighpressure research with the DAC in the future.

I. The DAC: An Apparatus Par Excellence for Achieving Highest Static Pressure

The evolution of the diamond anvil cell - from its conception by Lawson and Tang (1950), its development by Jamieson etal. (1959), Weir etal. (1959. 1965), Piermarini and Weir (1962), and Bassett etal.(1967), to the turning points by first Van Valkenburg (1965), who introduced the metal gasket, and then by Barnett etal. (1973), who developed ruby pressure calibration - makes a fascinating story (see Jayaraman, 1983,1986). Various designs of the DAC followed these milestones; the one conceived by Piermarini etal. (1975), which came to be known in the high-pressure literature as the NBS (National Bureau of Standards, now NIST) DAC, has a maximum pressure capability of -50 GPa. A very significant step in subsequent pressure-cell design was the development of the DAC by Mao and Bell (1978), who introduced the long-body, detachable piston-cylinder assembly and employed cylindrical rockers instead of the hemispherical ones used in the NBS DAC. Using this cell, a pressure of -100 GPa can be generated. The introduction of bevels on the diamond anvils was yet another turning point (Moss etal., 1986; Moss and Goettel, 1987a,b; Goettel etal., 1985), which raised the pressure limit to -500 GPa (Xu eral., 1986; Ruoff etal., 1992). The phenomenal advances achieved by innovative modifications in the mechanical design have turned the DAC into a unique and reliable static-pressure apparatus for carrying out detailed study of condensed-matter behavior under extreme pressure conditions. 1. DIAMOND AS AN ANVIL MATERIAL

Brilliant-cut, gem-quality diamond stones are generally used as anvils, although synthetic diamond and isotopically pure synthetic diamond anvils have been used to produce multimegabar pressure (Vohra, 1997). The

7 DIAMOND ANVIL CELLS IN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 383

surface shapes of the stones are customized for the pressures at which the experiments have to be performed. For experimental pressures below 50 GPa, the pointed culet of the gemstone is truncated into a flat face that ranges from 500 to 600 pm, and for experiments at still higher pressures, diamonds having a 5 to 8.5" bevel on a 300-pm culet with a 30- to 75-pm flat face are employed [Bevel of 7" is found to give best performance (Takemura, 1998)l. As a point of reference, a standard sheet of paper is -50 pm thick and a human hair is -100 pm in diameter (Akella, 1996). If the diamond anvil is required to be as transparent as possible throughout the infrared region, then the purest diamond (type 11) is needed. The purest diamond, having the greatest spectroscopic range of transparency, tends to cleave rather easily, however, and hence does not sustain ultrahigh pressure. The stone containing a reasonable concentration of evenly dispersed nitrogen platelet impurity (type I) is stronger, but has considerable absorption in the mid-infrared region (Spain, 1987). The features of the two types of diamonds are summarized in Table I. At multimegabar pressures, fluorescence from the diamond becomes significant in the range of wavelengths at which spectroscopic measurements are performed. A combination of chemical vapor deposition and highpressure/high-temperature techniques has yielded gem-quality synthetic diamonds that are isotopically pure 12Cand stay low fluorescent to pressures as high as 300 GPa with blue-green laser excitation and above 370 GPa with red excitation (Vohra, 1997). a. Physical and Mechanical Properties: Appticaiion-Based Criteria forSelection The selection of a suitable diamond stone for use as an anvil is the first step in the task of fabricating a DAC. All four characteristics, namely, the TABLE 1

TYPES OF DIAMOND (Seal, 1984) Type

Impurities

IA

N platelets and aggregates up to -lo00 ppm Single substitutional N up to 400 PPm Almost free from N Single substitutional boron, <0.25 ppm

IB IIA IIB

Relative abundance

UV abs. edge

Color

98%

-300 nm

-

1%

-300 nm

Yellow

Very rare (
2230 nm

Blue

-

384

M.Y o u s u ~

cut, color, clarity, and carat, are of critical importance. Diamond crystallizes in the cubic lattice, with S.G. Fd3m. Each atom is tetrahedrally coordinated by four like atoms, and the structure can be pictured as a stacking of sheets composed of continuously linked puckered hexagonal rings of carbon atoms parallel to the (111) planes of the cubic crystal. The bonding between the atoms is predominantly covalent in character and is due to sp3 hybrid formation. The electron pair bonds are strongest in diamond, making it the hardest material known to mankind. Unfortunately diamond is quite brittle too. A three-point-cut diamond anvil, that is, with the culet and the table parallel to the (111) plane, is expected to sustain maximum pressure. Because of the difficulty in grinding the (111) plane, which is the hardest plane, and because of cleavage energy considerations, most of the diamonds used in DACs are four-point cut, that is, having the culet and the table parallel to the (100) plane. It is of interest to note that the cleavage energy is minimum for the (111) plane, -10.6 Jm-2, and maximum for the (100) plane, -18.4 Jm-2, making the four-point-cut stone most suitable for use as an anvil (see Table 11). The color attribute of a diamond is an interesting study in itself. Color is due to the absorption band in the visible region, arising as a consequence of the characteristic impurities (nitrogen, boron, aluminum, etc.). Nearcolorless, light-brown, or light-yellow diamonds are acceptable from the point of view of strength. However, for spectroscopic experiments in the visible region, optically transparent diamond is essential. The clarity of the diamond is important for its use in optical spectroscopy. Since the firstTABLE I1 THEORETICAL CLEAVAGE ENERGIES FOR DIAMOND" Plane

Angle with respect to the (111) plane

Cleavage energy (Jm-*)

111 332 22 1 331 110 322 321 21 1 320 210 31 1 100

0 and 70" 32' 10" 15" 48' 22" 35" 16' and 90" 11" 24' 22" 12' 19"28' 36" 48' 39" 14' 29" 30' 54" 44'

10.6 11.7 12.2 12.6 13.0 13.4 14.3 15.0 15.3 16.4 16.6 18.6

Note. The cleavage energies are computed by calculating the number of bonds that cross the unit area of a chosen plane and multiplying this by the C-C bond strength. The C-C bond strength is taken as 5.8 X J (Pauling, 1966). The fracture surface energy is half the cleavage energy.

a

7 DIAMOND ANVIL CELLSI N HIGH-PRESSURE STUDIES OF SEMICONDU(TIORS 385

order Raman peak at 1333 cm-‘ is quite intense, a comparison of the fluorescence background and second-order Raman peak (appearing around 2500 cm-l) is done. All stones for which the ratio of the intensities of the second-order Raman peak to the fluorescence exceeds three are acceptable for optics experiments. An important feature that is crucial for achieving ultrahigh pressure is the shape of the diamond anvil. A perfectly circular culet (for a routine use to -60 GPa) or a beveled culet (for ultrahigh-pressure work) is the most desirable requirement, the achievement of which has been a daunting and most disappointing task. Gem-quality diamonds have facets and their intersections are inimical to stress concentrations. It is, therefore, desirable that diamond with a highly polished tapered surface, acquired by precision bruiting, be obtained for ultrahigh-pressure work. This bruited diamond will mimic the Bridgman anvil and will avoid any stress concentration. Trench fabrication in the conical surface, close to the culet, is expected to arrest the gasket material flow and hence to enhance the pressure-sustaining capability of the diamond anvils (DAs). It is observed that under loading, the pressure increases sharply from the culet edge to the culet center, where it maximizes, with a parabolic dependence P(r/u) = P(O)(l - (du)’). Above a certain pressure (stress) limit, a part of the applied load is used up in deforming the DAs. The compressive yield strength, measured for a perfect diamond crystal, is -400 GPa (Nelson and Ruoff, 1979). Plastic deformation followed by anvil failure is expected to begin when pressures close to half the yield strength of the diamond are reached (Adams and Shaw, 1982). It is interesting to note that ab initio molecular dynamics simulations show that upon hydrostatic compression, mechanical failure of diamond occurs only at -3000 GPa. The simulation work indicates that above 3000 GPa, the diamond collapses into a denser and metallic form of carbon termed as SC4. This simulation work also indicates that under uniaxial stress, a mechanical instability sets in at -550 GPa (Scandolo etal., 1995, 1996). Microscopic plastic deformation has been observed in DAs, but this failure mechanism is not a serious limitation of the anvil design. The resistance of diamond to plastic deformation at very high pressure is due to the presence of nitrogen platelets in type IA diamonds, and it is plausible that maximum pressure can be generated with DAs having larger nitrogen concentration. Seal (1984) has observed that type I1 diamond is probably harder than type I and therefore it should be possible to sustain larger pressure with type I1 DAs, provided cleavage failure due to poor anvil alignment is avoided. Under loading, once the deformation is initiated, the anvil in the central part starts cupping. Further increase in the load is then shared in two aspects, namely, in increasing the cupped area and in increasing the stress

386

M. Y o u s u ~

around the remaining part of the culet, which is an annulus. When this annular area becomes small and is around the edge only, the D A fails due to chipping. The introduction of a bevel in the D A is an important landmark in enhancing the pressure-sustaining capability of diamond anvils, since bevels postpone the cupping of the DA. Two designs that are commonly used in DAC work are shown in Fig. 1. The working faces may range in size from 300 to 400 p m across, bounded by 16 facets that approximate a 1987; Yousuf etal., 1996). Figure 2 reprocircle (A or C). (Jephcoat etal., duces the pressure profiles of DAs with design parameters 6,= & = 5", A = 300 pm, and B = 50 p m at various loads. Clearly, the pressure gradient is much larger during unloading, which is a major cause of failure (Ma0 eta/., 1985, Hemeley etal., 1997). Adams and Shaw (1982), Moss and Goettel (1987a,b), and Adams etal. (1993) calculated the stress pattern inside the anvil. The most vulnerable points are at the top of the anvil, their positions are dependent on the anvil design. High tensile stresses comparable with the tensile strength of the diamond (-3 GPa) are generated at unsupported regions of the table. The design considerations converge on two conclusions: It is most desirable that the culet and the table be nearly totally parallel, -1 minute of an arc, and that the surface finish be -2.5 nm.

6.

The GenericTypesofDiamond AnvilCells

The basic principle of the DAC is depicted in Fig. 3. In contrast to the use of a nonmetallic gasket in a Bridgman anvil cell, here a metal gasket is compressed between the culets of the DAs. The gasket has a central hole that is filled with a pressure-transmitting medium, the sample to be investigated, and the pressure calibrant, such as a ruby chip of -5 p m size. Since the area of the culet is very small, (-0.01 mm2), a modest load of -lo4 N is sufficient to generate a pressure in excess of 100 GPa. Success of DAC experiments to be carried out at ultrahigh pressures lies in several factors: accurately centering the gasket hole with respect to the culet centers, the parallelism between the culets of the DAs, use of very hard gasketing material (to avoid compressive yielding, although hard gasketing material does not support the anvil better), hardened backup support, and performing the experiment on not very soft material. Cells have been designed in different ways depending on the need and ingenuity of the designers. Six major types of DACs are shown in Fig. 4. NBS cell.Fig. 4a illustrates the Piermarini-Block DAC (1975), also known as the NBS cell. The main body of this cell consists of a rectangular

I

PAT

CENTRAL

FIG. 1. Schematic diagram of diamond anvils. (a) Beveled diamond anvil. The culet face comprises a central flat (diameter B) surrounded by a bevel that extends to the other rim of the culet at an angle Oto the plane of the flat (diameter A). (b) Flat culet with no bevel (diameter C). The table face shown is parallel to the culet flat and rests against a zirconium shim on the tungsten carbide support. The girdle is defined by the largest circumference of the diamond between the culet and table faces (after Jephcoat etal., 1987).

387

M. YOUSUF

388 \

/

Diamond anvil

\

2s

i

5

a

lS

I

150

100

50

0

50

100

150

Distancm from Csnter. prn

(b) FIG. 2. (a) Cross-section and plan view of the diamond anvils, gasket, and distribution of powdered ruby crystals. 0, and & are the bevel angles; A and B are the diameters of the bevel and flat areas of the culet tips of the diamonds,respectively.

7 DIAMOND ANVILCELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 389

160

100

60

0

50

100

150

D i r t a m from center, pm (C)

The matrix of points superimposed on the plan view gives the locations of pressure measurements. (b) Examples of pressure profiles measured in the high-pressure cell (with 0, and 6 = 5 ” ) during an increasing pressure cycle. (c) Examples of pressure profiles measured in the high-pressure cell with the same anvil configuration during a decreasing 1985). pressure cycle (after Mao er al.,

plate, -15 cm X 6 cm X 2 cm, made of AISI 4340 or Vascomax 300 steel, with provision for holding the stationary and moving pistons. The stationary diamond mount can be translated for centering and locking, once the DA is centered. The hemispherical mount of the moving piston has a tilt provision that can be used to obtain parallelism between the centers of the culets of the two DAs. The parallelism between the bevels is checked by the silver iodide technique or by the light fringe technique, discussed briefly in Sections I.3.a and I.3.b. Load is applied through the lever arm and controlled by the belleville spring washers. The cell is capable of generating a pressure of -50 GPa. Mao-Bellcell.This cell has its origin in the NBS cell. Modifications include an increased travel length of the piston within the cylinder and the

FIG. 3. Opposed diamond anvil configurations, with a metal gasket for sample confinement in a pressure medium.

use of a 5 : 1 lever mechanism. The angular displacement at the piston is reduced such that the load is applied vertically along the axis of the piston with negligible horizontal motion. These features improve the linear stability of the piston within the cylinder and permit the use of higher loads. Consequently, for achieving a certain given pressure, it is possible to use a large surface area and hence more sample volume (Ma0 and Bell, 1978). Like the NBS cell. the load is generated by turning the drive screw and is guided through the compression of the belleville spring washers (type B1000-05-5, Associated Spring Company, Bristol, Connecticut, USA). For alignment of the DAs, two tungsten carbide half-cylinders act as DA mounts. Their axes are set orthogonal to each other to provide the necessary

7 DIAMOND ANVILCELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 391

1.

DIAMOND A N V I L S

2 . T R A N S L A T I N G DIAMOND MOUNT P L A T E

3. TILTING DIAMOND MOUNT HEMISPHERE

4. EXTENDED PISTON

5 . PRESSURE P L A T E I

6 . PRESSURE P L A T E BEARING

7. B E L L E V I L L E SPRING WASHERS 8. G A S K E T

9. 10.

FULCRUM DRIVE SCREW

(a) NBS Cell. 1. CYLINDER 2. PISTON 3. THRUST BLOCK 4 . FULCRUM

5. LEVER 6 . MAIN BODY 7. DRIVE SCREW 8 . BELLEVILLE WASHERS

(b) Mao-Bell Cell Cross Sectional View. FIG. 4. Constructions of the best-known cells. (After (a) Piermarini etal., 1975; (b) Mao 1967; (e) LeToullec etaf., 1988; and Bell, 1977; (c) Huber etal., 1977; (d) Bassett etal., (f) Merrill and Bassett (1974).) (Eremets, 1996 and reproduced with permission of Oxford University Press, UK).

M. YOUSUF

392

ALF CYLINDER

HALF CYLINDER

-

HARDENED STEEL PISTON, LAPPED, FITTED INTO HARDENED STEEL CY LlNDER

0

5 cm

(b) Mao-Bell Cell Piston-CylinderAssembly. 24"

(e) Huber-Syassen-HolzapfelCell. FIG. 4. continued

7 DIAMOND ANVIL CELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 393

DIAMOND SAMPLE

ROCKER PISTON

DRIVING

NUT

(a) Bassett-Takahashi-StookCell.

Duwlw ANVILS

(e) LeToullec-Pinceaux-LoubeyreCell. FIG.4. continued

tilts and minut translations in a controlled fashion. Boro carbide halfcylinders are preferable for X-ray work because a diffracted beam can easily come out through them, thus enhancing the range of the diffraction angle. This cell can sustain a pressure of -100 GPa with ordinary DAs and has been found to generate pressures in excess of 500 GPa when beveled DAs are used (Mao, 1989). The Mao-Bell cell is depicted in Fig. 4b.

394

M. Y o u s u ~ 1.60 c m

l---i

3.INCONEL

GASKET

4.OIRMONO ANVIL5

(9 Merril-Bassett Cell. FIG. 4. continued

Huber-Syassen-Holzapfel cell.The cell designed by Huber, Syassen, and Holzapfel(1977), shown in Fig. 4c, employs a lever arm system actuated by a thread-and-nut mechanism for applying the load. The special geometry of the brackets, levers, and threads results in a large multiplication factor, permitting generation of forces of -5 kN on the DAs. Since the thrust acts strictly parallel to the axis of the cell, the device has a very good alignment stability. A special mount on the upper piston is used for parallel alignment of the diamond. This cell can sustain a pressure of -50 GPa. BassettTakahashi-Stook cell.Figure 4d shows a compact cell based on the design by Bassett, Takahashi, and Stook (1967). The movable piston

7 DIAMOND ANVIL CELLS IN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 395

is pushed by driving the nut, and the rotation of the nut is arrested by a key in the slot. Hardened steel half-cylinders are used to achieve centering and alignment of the DAs. In an ungasketed configuration, a pressure of -40 GPa can be reached with a 0.3-mm culet. LeToullec-Pinceaux-Loubeyre cell.This D A C uses a membrane system actuated by gas and allows for changing the load very smoothly (LeToullec etal., 1988). Using this cell, accurate measurement can be done in the vicinity of phenomena such as phase transitions, equilibrium phase lines, crystal growth, and so forth (Pruzan, 1994). The cell is depicted in Fig. 4e. A gas pressure of -15 MPa over a membrane area of m2 suffices to generate a load of -lo4 N. The membrane, driven by helium, works efficiently down to 4.2 K. The cell has been adapted for X-ray studies (LeToullec etal., 1992), Raman spectroscopy (Chervin etal., 1993), and infrared (Chervin etal., 1995). With beveled anvils, the cell can be operated at multimegabar pressure.

Merril-Bassett cell.In this cell, the main body consists of two triangular platens that are pulled together by three loaded belleville spring washers. Alignment of DAs is achieved by sliding the upper platen on three guiding pins anchored to the lower platen. This cell is small enough to be mounted on a goniometer head for single-crystal X-ray diffraction (XRD) experiments. The DAs sit on the support block (pressed into the platens), allowing wider-angle diffraction spot coverage. By choosing Be-Cu as the material for the cell fabrication, the cell can be coupled to magnetic fields. Because of its miniature size, it can be placed inside a helium cryostat for lowtemperature studies. Figure 4f shows a Merril-Bassett cell (1974). Othercells.Apart from these generic types, many laboratories have made their own cells, the guiding factor being the application of the principle of massive support. Variations in the design of the DAC arise mostly from differences in the mechanisms for generating force and for anvil alignment. In Japan, the cell developed by Takemura etal.(1979) is very popular. The cell developed by Deivasigamani etal.(1995) is being made available to many laboratories in India. For those growing number of scientists who are becoming aware of the possible value of high-pressure methods but who do not have the background necessary to start producing their own cells, several companies now exist that produce DACs for sale in a standard commercial sense. For instance, David Adams (“Dia cell products”) in the United Kingdom and Van Valkenburg (High Pressure Diamond Optics) in the United States are major commercial suppliers. Some Firms make the cell available commer-

cially but in collaboration with laboratories such as van der Waals’ laboratory, Amsterdam, the University of Ramot, Tel Aviv, and the Indira Gandhi Centre for Atomic Research (IGCAR), Kalpakkam. Because these commercially available cells have to satisfy sufficient safety demands and standards, these DACs are of lower pressure range. For instance, the IGCAR cell with a culet size of 500 p m can be used only to -60 GPa routinely and can be taken to 100 GPa with 200 to 300 pm culets. Commercialization of cells capable of generating multimegabar pressure has not flourished, since there are only a few laboratories, mostly in the United States, and Japan employing multimegabar pressures. The daunting factor is the failure of the DAs. Most of the time, anvils can be taken to multimegabar pressure only once; during unloading, the anvils with bevels undergo either a massive crack or chipping or become powdered.

2. PRECISION FABRICATION OF A TYPICAL DAC In the following section, the precision fabrication of the DAC developed at IGCAR is discussed in some detail (Deivasigamani etal., 1995). The DAC has the following main components: Piston-cylinder assembly Pressure-cell holder Piston-cylinder rockers Diamond anvils Belleville spring washers X-ray collimator Figure 5 is a photograph of the DAC. Since the DAC has to sustain a load in the range of lo4 N, high-strength steel, such as maraging steel, is used as the construction material for the piston-cylinder assembly and the pressure-cell holder. To mount the DAs and achieve a high degree of alignment so that the cell axis passes through the center of the diamond anvil culets and is perpendicular to the culet centers, cylindrical rockers have been used. A combination of a spherical rocker and a circular plate has also been tried and is found to make the anvil alignment easy and fast (Takemura, 1979;Sahu, 1998). Further, because diamond has a very high compressive strength (a few hundred GPa) and that of the hardened maraging steel is quite low (-2 GPa), an intermediate material is required between them.

7 DIAMOND ANVIL CELLS IN H I G H - P R E S S U R E STUDIES OF SEMICONDUCTORS 397

FIG. 5 . An assembled megabar diamond anvil cell (MDC) of the Mao-Bell design (Ma0 and Bell, 1978). The distinctive features are the cylinder, the lever arm, and the drive screw; the length of the lever arm is 16 cm. The silhouette of the piston diamond can usually be seen through the hole in the cylinder wall.

Tungsten carbide, due to its desired mechanical property and compressive yield strength of -6 GPa, is ideally suited and has been used as a material for the rockers. Diamond is selected according to the criteria discussed in Section 1.l.a. To carry out D AC XRD experiments in the parallel beam geometry, there is a provision for attaching an X-ray collimator. Lead-glass capillary is used as collimator material to withstand the heavy radiation damage. The glass capillary is contained in a stainless-steel housing having a provision for translation and tilts. Collimator system consisting of two pairs of semicylinders (with a flat groove of a few tens of p m ) mounted perpendicular to each other has been adopted for the DA C XRD experiments using synchrotron radiation source (Somayazulu, 1998; Hu, 1998; Yousuf ef af., 1998). Steps for the precision fabrication of various components are given in the following sections.

398

M.Y o u s u ~

a. Piston-Cylinder Assembly

Rough turning of piston and cylinder is carried out on the precision lathe, leaving 1/2 mm of material all over for finish grinding. Two M3 holes are drilled using a CNC machine. To provide the mount for the gasket, tappings are done using an extended attachment. After age hardening, the components are finish ground on a cylindrical grinding machine. Precision fabrication of the rocker cavities is very crucial because the degree of the alignment of the DAs onto these cavities decides the pressure-attaining capability of the DAC and the life of the DAs. Cylindrical cavities are made using an electric discharge machine (EDM). The precision of the cavities depends on the precision electrodes used and the rate of spark erosion. Cylindrical tungsten-copper electrodes having a dimensional tolerance within 2 p m are used, and a rate of erosion of -2pm/s is maintained. These rocker cavities are provided with two perfectly orthogonal tapped holes for adjusting the cylindrical rockers axially (Fig. 6).

I-,33.34-I

\

Alldimensions

are in mm. 2 FIG. 6. Subassembly of the piston-cylinder. (1) Cylinder (maraging steel). (2) cylinder rocker (tungsten carbide), ( 3 )diamond anvils, (4) piston rocker (tungsten carbide). and (5) piston (maraging steel).

7 DIAMOND ANVIL CELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 399

SIDE PLATE

BELLEVILLE S P R I N ~ S DRIVE SCREW

FIG. 7. Exploded view of the pressure-cell holder.

b. Pressure-Cell Holder There are seven parts to the pressure-cell holder in all, as shown in Fig. 7. Components such as the main body, side plate, press plate, and drive plate have parallelism of opposite faces, with a perfect 90" angle between the adjacent faces to ensure proper transfer of load coaxially to the pistoncylinder assembly. Extra care is taken while machining the press plate and the drive plate to have the side pins in one line without any shift in the axis that may cause asymmetric loading of the piston and hence lead to a premature failure of the DAs.

c. Piston and Cylinder Rockers The starting material for the matched pair of rockers is a precision tungsten carbide cylinder with a diameter variation within 1.5 p m over its length. Cylindrical segments are cut using a precision crystal cutting machine fitted with a diamond disc and having provision for precision translation and tilt adjustments. These half-cylinders are mounted on an EDM compound stage. With the corresponding electrodes, a conical hole and a tapered slot are made in the piston and the cylinder rockers, respectively. The flatness of the surface achieved is -10pm and its parallelism to the

M. Y o u s u ~

400 17"

CYLINDER ROCKER

All dimensrons an In m m . @Y

FIG. 8. Rockers for piston and cylinder cavities.

axis is -7 minutes of an arc (Fig. 8). By subsequent lapping, these numbers are improved by a factor of -4. 3. PRINCIPLE OF THE ALIGNMENT OF DIAMOND ANVILS

Once an appropriate pair of DAs is selected, they are mounted on the tungsten carbide rockers with a zirconium shim in between. The rockers provide adequate degrees of freedom for achieving precision alignment of the piston and cylinder DAs. The rocker with the mounted anvil is clamped into a miniature glass vise with a polished face to provide optical access through the rocker's pinhole (piston rocker) or slot (cylinder rocker)

7 DIAMOND ANVILCELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 401

(Fig. 9). Very fine adjustment to the position of the anvil is done under a 1OOX stereomicroscope using miniature tweezers, until it is seated perfectly flat on the rocker and symmetrically mounted about either the pinhole or slot, as the case may be. Epoxy resin (stycast) with suitable thermal properties and high hardneses is applied and is left for curing. This gluing maintains diamond anvil-zirconium shim-WC rocker combination in position and arrests the lateral movement of diamond anvil. After curing, the anvil-rocker combinations are checked for the presence of any extraneous material, such as a protruding burr of zirconium shim or stycast, and are removed with the help of a fine needle and ultrasonically cleaned. Each rocker is then transferred to an appropriate seat in the piston and cylinder assembly. Rockers are locked in place with set screws in the piston and cylinder located along the axes of each rocker. The piston and cylinder are matched together with the rocker axes orthogonal to each other to facilitate desired translational and rotational adjustments of the DAs about the plane of the culet faces. To avoid an accidental 180"relative rotation that may cause serious misalignment and DA breakage on contact, the orientation of the piston with respect to the cylinder is marked. When not in use, a soft spacer is inserted between the culets so that anvils do not come in direct contact and cause damage to the DAs. The following steps of alignment are strictly adhered to: 1. DAs are matched by translation of the rockers along the X and Y axes. 2. The piston is inserted into the cylinder, and with the piston-cylinder assembly rested on the V-block, the diamonds are viewed under a 40X stereomicroscope along the X and Y directions through the ports in the wall of the cylinder. 3. The gap between the anvils is reduced (but not closed) to estimate the offset of the culets along the X and Y axes. Adjustment is done by removing the piston from the cylinder and adjusting the set screws. The process is repeated until perfect matching of the culets is achieved. After the culets have been matched, angular orientation is adjusted by tapping on the off-axis surfaces of the culet. Final adjustment is obtained by one of the two following methods. a.

The Silver Iodide Technique

The piston-diamond culet is pressed on a glass slide containing finely powdered AgI. The piston is inserted into the cylinder. The piston-cylinder assembly is mounted into the pressure-cell holder. The AgI is viewed under

402

Fic;. 9. Mounting a diamond anvil onto a rocker. (a) A 25-pm-thick zirconium foil shim resting on the piston rocker before the diamond is mounted. The central hole is 380 p m in diameter, punched through with a needle. The rocker is shown resting in the piston for illustration only. (b) Diamond anvil clamped with a miniature glass vise onto a rocker with a zirconium shim in place prior to bonding to the rocker surface with an epoxy adhesive. The adhesive is spread around and over the diamond girdle. The rocker is resting in a hole in the body of the vise -1 cm in diameter. (c) Close-up view of the diamond shown in (b). (d) Appearance of the diamond after bonding to the rocker with epoxy resin. The rocker/ anvil has been transferred to the spark-eroded seat of the piston and is held in place with set screws ready for alignment. The screw holes located in either side of the rocker are used t o tix the gasket in place (after Jephcoat er a/., 1987).

7 DIAMOND ANVIL CELLSIN HIGH-PRESSIJRE STUDIES OF SEMICONDUCTORS 403

Fiti. 9. continued

a lOOX stereomicroscope. The load on the cell is increased by gently rotating the drive screw until the 0.3 GPa transition in AgI is observed. This transition is marked by the appearance of a black area that spreads out with increase in load. The symmetrical shape of the transition area as it spreads out with pressure is indicative of the alignment of the anvils. Further increase in pressure causes a second transition at -lOGPa, which for an aligned anvil arrangement should appear as a light halo circle. If either transition appears off-centered, it implies that the diamonds are not parallel but are wedge shaped and require corrective measures. For example, if a

404

M. Y o u s u ~

transition zone is deflected toward the I quadrant it indicates that the two culets are closer together in the I quadrant and farther apart in the I11 quadrant. Alignment is incorporated without loosening the screws of the rockers. The piston is removed from the cylinder and, by remembering the orientation of the deflected transition, either of the rockers is rotated by gently tapping it with a teflon-jacketed hammer, avoiding any damage to the tungsten carbide rockers. b. The LightFringeTechnique

The piston is inserted in the cylinder very gently and no pressure is applied. Using a microscope, interference light fringe patterns are observed. If the fringe patterns are concentrated in any area, the culets are not aligned. By tapping the rockers appropriately, one light fringe of parallelism is achieved.

4. GASKETING: A TURNING POINTIN DAC USE The introduction of the gasket into the DAC is a very important development in the history of the DAC, for it is this innovation that paved the way to making the DAC a quantitative tool for high-pressure research. Functionally, the gasket reduces the magnitude of pressure gradients and confines the sample, pressure calibrant, and pressure-transmitting medium. Over and above these functions, the gasket provides an additional support to the DAs, thereby enhancing their pressure-sustaining capability. Generally, hardened stainless-steel is used. However, for ultrahigh pressure work, high speed steel or rhenium or tungsten gaskets are used. Typically, the initial thickness of the gasket is 200 to 300 pm. A thick gasket is preferred because it gives a better lateral support, although this causes a part of the load to be consumed in the plastic deformation of the gasket. The optimum thickness of the gasket depends on the culet size, gasket hole size, the maximum intended pressure and the type of the sample to be investigated. a. Preindenting theGasket

For better performance and improved success of a DAC experiment, the gasket is preindented before hole drilling. Preindenting reduces the deformation of the hole during compression. The maximum preindenting pressure for a gasket depends on the maximum pressure intended in an experiment, the starting thickness of the gasket, and the pressure limit of the extrusion of the gasket. Generally, the gasket is preindented to half of

7 DIAMOND ANVILCELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 405

the maximum pressure intended for the experiment or less than the extrusion pressure, whichever is less. For instance, the initial thickness of Re gasket is 250 p m which should be prepressed to a thickness of 30-40 pm for ultrahigh pressure experiments. theSampleHoleintheGasket b. Drilling

After preindentation, a hole is drilled in the center of the gasket. This is done with a miniature drill with the aid of a microdrill station or by hand or by spark erosion. Laser melting is also employed by some experimenters. (1993) have reported a simple laboratory solution to the problem Sahu etal. of precision gasket-hole drilling.

5. PREPARATION OF THE SAMPLE IN A TYPICAL DAC EXPERIMENT The preindented gasket with a central hole is mounted onto the piston diamond. Single-crystal or powdered samples are loaded in the gasket hole. To avoid preferred orientation effect, fine powder particles of -1 pm are loaded. A ruby chip of -5pm dimension is mounted on the cylinder diamond. To provide a hydrostatic medium to the sample, either a liquid or a gas is loaded in the gasket hole. Chilled liquid such as methanol : ethanol :water in the ratio 16 :3 :1 is found to give a hydrostatic environment up to -20 GPa. For a higher-pressure hydrostatic environment, gas loading is desired. A specially designed apparatus is used to load compressed gas at -0.2 GPa into the DAC. This avoids the complexities of cryogenic (1980) loading, and higher initial fluid densities can be obtained. Mills efal. developed this design, which has been improved by Jephcoat etal. (1987). The basic elements of the apparatus are reproduced in Fig. 10. Inert gases such as neon and argon have been found to maintain a quasi-hydrostatic pressure condition to -80 GPa (Jephcoat etal., 1986). Although hydrogen and helium are the best hydrostatic media, for the reasons discussed in Section 1.5, use of hydrogen is avoided. Once the sample and the ruby chip have been loaded in the gasket hole, the piston-cylinder assembly is enclosed in an aluminum casing attached to a gear mechanism and both fit tightly into the pressure vessel. A two-stroke, air-driven, diaphragmtype compressor is used to raise the pressure in the vessel up to -0.2 GPa. To seal the gas at -0.2 GPa in the sample chamber, the anvils are closed down with a subsidiary pair of drive screws threaded into the cylinder through the piston baseplate. The load is maintained by -10 mm diameter belleville washers. The piston and cylinder are compressed to a pressure

3 cm FIG. 10. Cutaway view of the apparatus used for high-pressure gas loading. (a) Provision for packing gland and drive shaft, (b) pressure vessel, (c) O-ring, (d) wedge ring, (e) fill line from pump, (A) drive gear. (B) Satellite gear coupled to 10/24 drive screws threaded through the piston baseplate into the cylinder wall. The load is generated by compression of belleville washers (not shown) as the screws are turned. (C) Piston baseplate. (D) cylinder. (E) piston, (F) cylinder diamond (gasket is not shown for clarity). (G) rocker. The piston-cylinder assembly is enclosed in an aluminum casing (dashed line) that fits closely in the pressure vessel. Free space (e.g., the X-ray collimator housing within the piston) is filled with metal blanks (after Jephcoat, 1987).

of -2 GPa to seal the gas between the culets, the excess gas is vented, and the piston-cylinder assembly is mounted in the pressure-cell holder for carrying out the experiment.

6. PRESSURE SCALEAND PRESSURE CALIBRATION Direct calculation of pressure from measurement of the applied load is not possible because the distribution of the load over the anvils is unknown and losses due to internal friction cannot be quantified exactly. In the lower range, pressure can be measured by a free piston gauge and then with secondary gauges such as the resistivity gauges (Newhall etaf.,1963;Molinar etaf., 1986; Govinda Rajan, 1982). Absolute pressure can also be measured by simultaneously measuring the ultrasonic transit time and the length of a cubic crystal at small intervals and integrating (Ruoff etaf., 1973). Since the primary scales exist only to -3 GPa, all work above this pressure uses

7 DIAMOND ANVIL CELLS I N HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 407

extrapolation, the only guide being the self-consistency of each scale with others. This is obviously a somewhat hazardous procedure, analogous to extrapolating temperature scales into the region 1IT + m (Spain, 1987). Up to 7.7 GPa, pressure can be measured using a quasi-free piston method 1971). Those materials that undergo phase transitions (Haygrath er al., in this range can be used as a pressure calibrant. All measurements of static pressure above 7.7 GPa depend on the equation of state (EOS) of materials such as gold obtained in shock wave experiments (Jamieson et al., 1980). Once the EOS is known, using X-ray measurement, volume can be found and hence pressure. Using Brillouin scattering, the sound velocity Us can be measured, and using the equation PB - p(pA)

=

1

( Y s ~ U S ) ~ ~ ~

PA

an absolute measurement of pressure can be attempted (Jayaraman, 1983). A rapid and convenient method of pressure measurement uses ruby (1975) tied the ruby scale to the variation fluorescence shift. Piermarini etal. of the V/Voof NaCl via the Dekker EOS (Dekker, 1971) and showed that it is linear up to 19 GPa. Mao eral. (1986) calibrated ruby against copper as a standard under quasi-hydrostatic conditions in argon medium and tungsten in neon medium up to 110 GPa (Mao, 1989; Hemley and Mao, 1989). This can be written as P(GPa) = 1904/8 [{l + (AA1694.24))’ - 11, where AA is the ruby R, line wavelength shift in A, and B = 7.665 and 5 for quasi-hydrostatic and nonhydrostatic environments, respectively. Reduction of the intensity of ruby fluorescence at high pressure is the major problem with this technique. At -100GPa, the intensities of R lines decrease so much that they become comparable to that of an R; line. Figure 11 shows the fluorescence spectra of ruby R lines from 68 GPa to 177 GPa at 298 K. It can be noted that above 100 GPa, the R; line and R I and R2 combined have similar intensities. Another difficulty in the measurement of fluorescence of ruby above 100 GPa is caused by the enhanced luminescence of diamond, predominantly due to large stress in it. However, this problem can be circumvented to a great extent by designing a special detector system that can distinguish the contribution due to the large lifetime of ruby transition (-ms) and that due to the small lifetime of diamond (1-50 ns). The accuracy of the pressure is -0.03 GPa, when R1 and R2 lines are well resolved. The highest accuracy, -0.005 GPa, can be achieved at low temperature. The position of the ruby

600-300 0 300600 Relative Wave numbera (an-’) FIG. 11. Fluorescence spectra of ruby R lines in hydrogen from 68 to 177 GPa at 298 K. The frequency scale is relative to the RI line at each pressure. Spectra have been excited with the 4880-Aline of a continuous-wave Ar' laser. A background signal from diamond fluorescence comparable to the ruby R-line intensity has been subtracted from the two highestpressure spectra. (From Mao etal., 1992 and reproduced with permission of Oxford University Press, UK.)

7 DIAMOND ANVIL CELLSI N HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 409

lines of a particular ruby particle, when compared with the bulk, wellannealed piece, show an error of -0.02 GPa, which is connected to the internal stress picked up by ruby particles during milling. 7. PRESSURE-TRANSMITTING MEDIUM AND THE LIMITOF HYDROSTATICITY Pressure (and not the stress) is the thermodynamic field (Griffiths and Wheeler, 1970), and hence the material investigation must be done under truly hydrostatic pressure conditions. Table I11 summarizes the pressure media that have been used. Mao (1989) and Loubeyre etal. (1994) have shown that hydrogen remains a good hydrostatic medium even in the megabar region. The ruby R line does not become significantly broad at 170 GPa and 77 K. Specific drawbacks of hydrogen are its high chemical reactivity and diffusion in anvil and gasket, but these are reduced at low temperatures. However, hydrogen embrittlement and the possibility of explosion always remain present. Burnett et al. (1990) showed that solidified helium is a suitable pressure medium, except for those cases where fragile samples are used. Helium is a preferable pressure-transmitting medium due to its lowest freezing point. Keeping the two criteria as the measure of degree of hydrostaticity namely (i) broadening of the ruby R1 fluorescence peak and (ii) more sensitive, broadening of the photoluminescence peak of the GaAs-AlGaAs quantum well, it is observed that Xe causes severe non-hydrostaticity, Ar is better than methanol-ethanol mixture whereas with the use of He, there is no perceptible peak broadening. Following the prescription given by Piermarini etal. (1973), and Asaumi and Ruoff (1986), the hydrostaticity of a medium in a DAC can be estimated in one of the following ways: 1. The pressure gradient in the medium across the anvil 2. The full width at half maximum (FWHM) of the ruby fluorescence lines 3. Separation between the ruby R1 and R2 lines 4. Variation of the interplanar spacings corresponding to different Miller indices hkl Detection of the hydrostatic limit has been performed in a viscosity measurement. At ultrahigh pressure (3100 GPa) and very low temperature, all the pressure media convert into the solid state. The freezing pressure or the glass transition pressure and the limit of hydrostaticity are found to

410

M.

YOUSUF

TABLE 111 SOMEPRESSURE TRANSMITTING MEDIAAND THEIR USEFULPRESSURE RANGES

Freezing pressure at RT (GP4

Pressure range of nearly hydrostatic behavior (GPa)

10.4"

-20.0

Easy filling

14.5"

-20.0

Easy filling

11.8

>60.0

Ne

4.7

16.0

Ar

1.2

16.0

Cryogenic or high-pressure filling needed Cryogenic or high-pressure filling needed Cryogenic or high-pressure filling needed Cooling below 165 Kneeded Cryogenic or high-pressure filling needed Cryogenic or high-pressure filling needed Cryogenic filling Cryogenic filling

Medium Methanol: ethanol 4: 1 Methanol :ethanol Water 16:3:1 He

30.0

Xe

>60.0

5.7

H2

5.3

D2

?

0.2

Nz

2.4

0 2

5.9

~

13.0

Remarks

Ref. Piermarini, Block. and Barnett (1973) Fujishiro ef al. (1982) Bell and Mao (1981)

Bell and Mao (1981)

Bell and Mao (1981)

Liebenberg (1979) Mao and Bell (1979)

LeSar etal. (1979) Nicol etal. (1979)

~

" Glass transition pressure

be quite different. For those experiments done in multimegabar range, no pressure medium is used. This is because above the hydrostatic limit, the medium will cause severe deviatoric stress on the sample. It has been noted that the use of hydrostatic medium makes the cell unstable at ultrahigh pressure, as it is difficult to achieve higher pressure with softer material inside it. A further stability requirement at ultrahigh pressure demands that even the gasket material should be very hard.

7 DIAMOND ANVIL CELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 411

8. PRESSURE COMBINED WITH OTHER THERMODYNAMIC FIELDS The whole gamut of condensed-matter physics phenomena, especially those that are cooperative in nature, involve low-energy excitations and often are overwhelmed by thermal energy. Hence, experiments necessarily have to be performed at low temperature. Melting phenomena, materials synthesis, and phase diagram studies can be done at high-pressure and high-temperature conditions. To understand a large number of cooperative phenomena, experiments have to be done under the simultaneous application of pressure, temperature, and magnetic field. a.

Low Temperature

At low temperature, the fine structure of the energy levels becomes clear and pressure produces shifts in energy peaks. In semiconductors, for instance, features like energy band crossing and energy gap closure can be observed. Pressure tunes the band offset of the valence band in quantum wells. Investigations of deep and shallow impurity states under pressure lead to better understanding of defects. Since the D A C is a small, compact apparatus, it can be directly immersed in a cryogenic fluid such as liquid nitrogen or oxygen. Operation at liquid helium temperatures demands special design of the cryostat and materials consideration of the DAC. Webb etal.(1976) have described the use of a Bassett cell in a 3He-4He dilution refrigerator for pressures to 10 GPa and temperatures down to 0.03 K. In this arrangement, pressure can be changed from outside by applying 4He gas pressure to a chamber with bellows, which in turn pushes the anvil piston. In the apparatus devised by Shaw and Nicol (1981), optical spectroscopy studies can be performed up to 10GPa and OK. The cryostat has provisions for optical windows, and pressure can be changed from outside by a helium-driven piston acting on a pivoted clamp. Mao-Bell cells have been used for cryogenic temperatures after modification (Fig. 12) (Ma0 and Bell, 1979). Force to the thrust block can be applied by advancing the four belleville-spring-loaded precision screws symmetrically into the threaded body of the cylinder block through a remote-control tightening mechanism. The cell can be used up to 15 GPa. A miniature low-temperature DAC made of nonmagnetic CrNiAl alloy has been designed by Eremets and Timofeev (1992). Since the coefficients of thermal expansion of the D A and the body are quite different, a large change in pressure occurs during cooling. The

M.

412

YOUSUF

Ioob

0

1

2

3cm

FIG. 12. Diamond-window, high-pressure cell for cryogenic experiments. The inset to the right is an expanded view of the tungsten carbide support pieces, showing directions of translation and rotation for alignment of the diamond anvils. The cell as shown operates to pressures of approximately 15 GPa by advancing the precision screws (not shown in the figure). Opposing screw directions are used to avoid torque on the remote-control system that could rupture the high-vacuum cryostat.

variation of pressure upon cooling depends on the cell design. Kawamura etal. (1985) observed that the pressure increased from 6.9 GPa at room temperature to 13.6 GPa at 4.5 K. Pressure calibration at low temperature has been attempted by several experimenters, such as Adams etal.(1976), Noack and Holzapfel (1978), Diatschenko and Chu (1975), Silvera and Wijngaarden (1985), and Vos and Schouten (1991). Figure 13 shows the temperature dependence of the ruby R1line for the zero-pressure reference point (Silvera and Wijngaarden, 1985). b. High Temperature

Two approaches for producing high temperature have been adopted: External heating by winding a heater immediately around the DAs has been adopted by Bassett and Takahashi (1965), Moore etal. (1970), Schiferl etal. (1990,1993), Arashi (1987), Bassett etal. (1993) and Ming etal. (1987). However, the temperature capability is limited due to graphitization and oxidation of diamond, rapid creep deformation of DAC materials, welding of the gasket to the anvils and so forth. Diamond begins to oxidize at 700°C in air. In vacuum, diamond can survive temperature of -1500°C; and above

7 DIAMOND ANVIL CELLSIN HIGH-PRESSURE STUDIESOF SEMICONDUCTORS 413

Pcom

AV

(kbrr)

(CIS’)

-b

-

- 10 -10

-

-15

- 20 0 A h m a e l EL

0

(lor@)

1

1

p

I00

200

300

30

Temperature. T(K) FIG. 13. Temperature dependence of the ruby RIline. The right-hand scale gives the thermal correction to the pressure, relative to 5 K. (After Silvera and Wijngaarden 1985 and reproduced with permission of Oxford University Press, UK.)

this it forms carbon eutectics and compounds with the metal in contact. Alternatively, internal heating can be achieved by using a very fine wire of the sample and passing current through it (Liu and Bassett, 1975). First demonstrated by Ming and Bassett (1974), internal heating by coupling the electromagnetic radiation to the sample is elegant and offers the possibility of reaching temperatures of an order of magnitude higher than those achieved by external heating. Laser heating in a DAC allows pressure and temperature to be controlled independently and has advantages over shock wave techniques. A wide P-T field is accessible, with P up to 200 GPa and T up to 5000 K (Boehler, 1992). A laser-heating system is reproduced in Fig. 14 (Takemura and Yusa, 1997). Novel semiconductor systems have been synthesized using this system (Chandra Shekar etal., 1997; Sahu et al., 1998). Pressure measurement is done by placing a ruby chip at the edge of the gasket, which is heated no more than 400 K because it is thermally well insulated by the pressure-transmitting medium, say argon (Boehler, 1992). Therefore, pressure is measured with an accuracy of 0.2 to 0.3 GPa even during heating. Sample pressure is calculated using the correction term due to thermal pressure, akTAT, where a,kT, and AT are the coefficient of thermal expansion, isothermal bulk modulus, and the temperature change, respectively. A temperature increase of about 4000 K causes a thermal

M. Y o u s u ~

414

Power meler

Power

Laser

Pyrarnldal rnlrror

1

Fro. 14. Schematic drawing of the optical system for laser-heating experiments. A pyramidal mirror is used for collecting multiple laser beams. Notice that the beam of the C 0 2 laser comes over the DAC and is reflected on the top surface of the pyramidal mirror (after Takemura and Yusa. 1996).

pressure of -20 GPa. In real experimental conditions, the pressure on the sample is somewhat less than the calculated thermal pressure, since the sample is in between the constant-volume and constant-pressure thermodynamic state (Williams and Knittle, 1991; Heinz, 1990). The temperature on the sample can be measured either by pyrometry or by measurement of the ratio of intensities of Stokes and anti-Stokes Raman lines. In the pyrometry method, the thermal spectrum is fitted to the Planck function y(A, T ) = yo(& T).E(A,T ) , where €(A, T ) is the emissivity of the sample. For the second method, the ratio of Stokes and anti-Stokes intensities is given by the ratio lantl-stokes/lStokes = [(ol+ os)/ ( w / - w , ) ] ’ . exp( -ho,/27rkB T ) , where o,is the phonon frequency and wl is the exciting laser wavelength.

11.

Condensed-Matter Physics Techniques Coupled to a DAC

The DAC has been coupled to almost all the techniques of condensedmatter physics. In the following section, some examples are given.

7 DIAMOND ANVIL CELLSIN HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 415

1. OPTICAL SPECTROSCOPY The human eye is the best detector and visible light the best probe, for this combination provides a direct perception of the underlying phenomena, and hence gives psychological satisfaction. For instance, the observation of a semiconductor-to-metal transition, say in SmS, loaded in a DAC and viewed through a microscope gives an inexpressible feeling. A D A C is the superior apparatus in experiments in which material behavior under pressure is investigated by probing through an incident photon and analyzing the outgoing photon. This is sobecause diamond is transparent to electromagnetic radiation in the infrared, visible, and near-ultraviolet spectra (-5 eV), and also to X-rays and y-rays (>lo keV). Limitations of the pressure up to which experiments can be performed arise due to the increase in absorption by the diamond. Type I and I1 diamonds show stress-induced optical absorption in the entire visible spectrum range of 1.5 to 3.5 eV above 200 GPa (Vohra, 1992). a. Absorption and Photoluminescence Spectroscopy When a photon of energy EP enters the material, a photon of energy EpLis detected (Fig. 15a). The incoming photon interacts with an electron, promoting it from the valence band (VB) to the conduction band (CB) (Fig. 15b). The minimum energy to achieve this can be deduced from absorption spectroscopy. The energy of the electron in the CB is quickly reduced by a nonradiative process to a value at the minimum of the CB. This electron can now recombine with a hole in the VB, emitting a photon, which is a measure of the bandgap E, (Fig. 15b). In a defect solid state, the excited electron may belong to an impurity level with an energy below the CB edge (Fig. 15c), such as observed in donor impurity, or it may recombine with a hole trapped on an impurity level above the VB maximum (Fig. 15d). The photons created characterize the impurity level. Pressure is capable of introducing changes in the CB and VB and hence of tuning the impurity levels. At sufficiently high pressure, when VB and CB overlap, all the features of the impurity levels can be destroyed. b. Raman Scattering Lattice properties of solids can be probed by Raman scattering and can be used to study the variations of the optical lattice modes under pressure. Figure 16a schematically illustrates the mechanism: An incident photon ( E p .k,) annihilates or creates a phonon (EL, 4);the detected photon

M.YOUSUF

416

Incident

photon

hk Elcclron

pholon

V Acceptor

FIG. 15. (a) Sketches illustrating the principle of photoluminescence spectroscopy. The symbol e refers to an excitation process that creates an electron-hole pair, and r to a recombination process that annihilates the pair. The mechanism of photoluminescence: (b) excitation by a photon of energy greater than the bandgap Eg.and the emission of a photon with energy equal to Eg;(c) electron-hole recombination from a donor impurity state to the valence band (d) recombination via a conduction band electron to a hole trapped on an acceptor state (after Spain, 1987and reproduced with permission of Taylor & Francis, UK).

(E,, k,)satisfies the energy conservation Eo = Ep 2 EL and momentum conservation k, = k, 2 q.Notations have their usual meaning. The phonon dispersion relation curve of a material with N atoms per unit cell has 3 N modes, of which three are acoustic (one longitudinal and two transverse) and the remaining 3 ( N - 1) are optic [(N - 1) longitudinal and 2(N - 1) transverse] modes. Depending on the symmetry, transverse modes can be degenerate. One such phonon dispersion curve is illustrated in Fig. 16b.The slopes of the curves give the lattice mode velocities, typically -lo3 ms-'. It is of interest to note that the speed of the electromagnetic

7 DIAMOND ANVIL CELLSI N HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 417 Phonon (EL) Incident

E

1

photon (E,-

EL)

(a) Photon

Brillouin zone bouniary

~

Longitudinal optic ( L O ) Transverse optic ( T O )

(b) Laser beam

Gasket

--

I

Sample

/ Diamonds

Slit

slop

(4 FIG. 16. (a) Sketch illustrating the principle of Raman spectroscopy. (b) A typical phonon dispersive curve. (c) A typical experimental system for Raman, fluorescence, and photoluminescence measurements in the backscatteringgeometry (after Jayaraman, 1986 and reproduced with permission of AIP, USA and after Spain, 1987 and reproduced with permission of Taylor & Francis, UK).

radiation in vacuum is 3 X lo8 ms-I, and in the scale of the figure the dispersion curve is very nearly a vertical line. Accordingly, in Raman spectroscopy the optic modes are probed near q = 0 for vertical transition or first-order Raman spectroscopy. Higher-order interactions occur with multiple-phonon annihilation or creation. For instance, second-order Ra-

man spectroscopy involves two phonons. The higher-order Raman spectroscopy permits probing of other phonon states. Selection rules on the basis of symmetry considerations make possible the probing of certain phonon modes only. Nowadays, a laser is used as the probing incident photon. Raman lines occur as satellite peaks on either side of the incident laser line. Because the Raman peaks are very weak less than the incident beam intensity), a double or triple monochromator is used to get a reasonable signal-to-noise ratio. A backscattering geometry is most frequently used in DAC Raman scattering experiments (Fig. 16c). The laser beam is focused to a fine spot of 20 to 5 0 p m (-5pm in the multimegabar range).

c. Brillouin Scattering

Raman scattering is the outcome of the interaction between photons and the optic modes, whereas Brillouin scattering corresponds to the interaction of photons with the acoustic modes. When an acoustic mode propagates in condensed matter, the wave sets up a series of density fluctuations of wave vector q, which can scatter light (Fig. 17a). This is close to the diffraction by fluctuations governed by the equation A = hc/E= 2d sin 8, where d corresponds to the wavelength of the acoustic mode. Thus, Aphoton = he/ Ep ho to n = 2Aphonon sin 8. However, there is a marked difference in that this process is inelastic, unlike diffraction. Since the diffracting disturbances travel with acoustic velocity, the outgoing photon wavelength is Doppler shifted. Unlike Raman spectra, the Brillouin peaks lie much closer to the laser line because the acoustic mode energies are of -10 meV. Moreover, the intensity of the Brillouin peak is very weak, for instance, ZBrillouinl I&,ylcigh Spectrometers with very high resolution, based on principles such as multiple-pass interferometry, are required. In Fig. 17b, the path of the laser beam leading to Brillouin scattering is traced. Generally, the experiment is carried out in forward-scattering geometry, so that the Brillouin shifts are independent of the refractive index of the intervening materials, such as diamond and the pressure-transmitting medium. Using this apparatus, Whitfield etal.(1976) obtained the speed of sound Us = A u A i / with ~ 8 = 45".Here, A u is the change in the frequency of photon and hi is the wavelength of the incident photon. Using the backscattering geometry, Polian and Grimsditch (1984) have achieved 67 GPa in a Brillouin scattering experiment, using a relatively small aperture and well-supported anvils. Probing of acoustic phonons in 1986; different directions can be done with large-aperture DACs (Lee etal., 1993). Shimizu and Sasaki, 1992; Zha etal.,

-

7 DIAMOND ANVIL CELLS IN HIGH-PRESSURE STUDIESOF SEMICONDUCTORS 419 E.

Pressure

maximum Acoustic wave speed

i A

FIG. 17. (a) A sketch showing the diffraction of a light wave by an acoustic wave. (b)The optical path through a pair of diamonds (not drawn to scale), compressing medium. and sample for Brillouin measure1976; after Jayaraman, 1986 and reproduced ment (from Whitfield eta/., with permission of AIP, USA and, after Spain, 1987 and reproduced with permission of Taylor & Francis. UK).

420

M. YOUSUF

2. X-RAYDIFFRACTION The DAC is the most widely used apparatus for high-pressure X-ray powder or single-crystaldiffraction studies. Crystal structure, compressibility, and pressure-induced phase transitions can be studied using the DAC X-ray diffraction. X-ray diffraction experiments on powdered samples can be done either in energy-dispersive or in angle-dispersive mode (Yousuf etal., 1996; Sahu etal., 1995, Chandra Shekar etal., 1993). Ultrahigh pressures and temperatures in a DAC are achieved at the expense of reducing the sample volume. The capacity for XRD with high spatial resolution is of paramount importance in probing microscopic samples at the maximum ( P ,T) conditions and for minimizing the effect of gradients. Polychromatic synchrotron radiation with energy-dispersive mode is ideal for the development of new classes of structural microprobes. Primary X-ray beams down to 3 pm can be produced with a microbeam slit system and optical devices. The microprobe can be routinely used for a variety of high-pressure experiments, including single-crystal XRD up to -50 GPa, powder diffraction up to -300 GPa, deviatoric strain measurements, and XRD at simultaneous high pressure and temperature (Ma0 and Hemley, 1996). a. Microbeam

Collimator design (Jephcoat etal., 1987; Yousuf etal., 1996) has briefly been mentioned in Section 1.2. Insitu microprobing capabilities are essential for studying microscopic samples at multimegabar and kilokelvin conditions. For a diamond culet of 20 pm or a laser-heating spot of 30 pm, or a single crystal of 5 pm, an X-ray beam of 5 to 10 pm is now collimated with a tapered pinhole (Ruoff, 1993) or a parallel slit system (Somyazulu, 1998;Hu, 1998;Yousuf et al., 1998) (Fig. 18a and b). Amajor advance in this area has been made by the development of a glancing-angle microfocusing system for polychromatic X-radiation. The system, designed by Yang etal. (1993, consists of two bent Kirkpatrick-Baez mirrors made of single-crystal silicon or a glass plate coated with a film of rhenium. At a 1-milliradian glancing angle, the 70-mm central portion of the mirror intercepts a 70ym-wide beam that is focused down to 3 pm (Fig. 18c). At the focal distance of 150 pm, the full convergence angle is -0.5 milliradian and the reflectivity is -90% up to 60 keV. b. DAC for Energy-Dispersive X-RayDiffraction

For ultrahigh pressure experiments, DACs and synchrotron radiation sources (SRS) go hand in hand. A schematic of a typical energy-dispersive

7 DIAMOND ANVIL CELLS IN HIGH-PRESSURE STUDIESOF SEMICONDUCTORS 421 Tapered Pinhole

Parallel Slits

FIG. 18. Micro X-ray beams produced by a tapered pinhole, parallel slits, and a glancing-angle focusing mirror.

X-ray diffraction (EDXD) system is depicted in Fig. 19. A narrow polychromatic X-ray beam passes through the DAC and impinges on the sample placed between the anvils. The diffracted beam is collimated at an angle 28 and received by an intrinsic Ge solid-state detector that disperses and counts diffracted X-ray photons according to their energies from 12 to 80 keV. The angle between the incident and diffracted beam is optimized keeping the region of interest of interplanar spacings in mind (Buras and Gerward, 1989; Yousuf etal., 1996). The expression for resolu-

FIG. 19. Schematic of DAC sample configuration for EDXD with synchrotron radiation source.

422

M. Y o u s u ~

tion in EDXD is given by SE/E = C - [ ( A E -sin d 6) + 5.546F~dCsin 6 + (C cot 6A6)2]”2, where SE = full width at half maximum, E = energy at which the diffraction peak appears, A E = dark current through the solidstate detector and the noise in the field-effect transistor (FET) amplifier, d = interplanar spacing of the diffraction peak, A6 = beam divergence, F = fano factor, C = 1/(6.999 keV), and t = the energy required for creating an electron-hole pair. Thus, for a small diffraction angle, the third term is the most important one, whereas at a large diffraction angle, the first two terms are important. For a given value of d , 6E/Eundergoes a monotonic decrease with the increase in diffraction angle, passes through a minimum, and then shows a parabolic increase. In an EDXD experiment, it is therefore necessary to set the diffraction angle at an optimum value. Textural information can be obtained by rotation around the X axis and by changing the w angle (Fig. 19). The “coarse-grain” effect not only can be eliminated by averaging around the X axis, but also can be separated from the effects of preferred orientation and deviatoric strain (Ma0 et al., 1997). X-RayDiffraction c. D A C forAngle-Dispersive Angle-dispersive X-ray diffraction (ADXD) experiments have been extensively done in a DAC, since the appartaus is simple and easy to align, and hence versatile. Buras etal. (1977) were the first to combine a DAC with synchrotron radiation, followed by Ruoff and Baublitz (1981), Baublitz etal.(1981), Spain etal.(1981), and Manghnani etal.(1981). An SRS plus DAC plus imaging plate system combination has become very popular 1989; Shimomura etal., 1993;Nelmes etal., 1992;Yousuf etal., (Yousuf etal., 1998). In IGCAR, a truly focusing X-ray diffractometer system, namely, a DAC in Guinier geometry, has been developed and is described next (Sahu etal., 1995; Yousuf elal., 1996). The schematic of the Guinier diffractometer setup is depicted in Fig. 20. The diffractometer system is in the vertical configuration and in symmetric transmission mode. The incident X-ray beam is derived from an 18-kW rotating anode X-ray generator, with a molybdenum target (AKa1= 0.70930 A). The curved quartz-crystal monochromator is of JohanssonGuinier type with the two parameters A and B equal to 118 mm and 355 mm, respectively. The reflecting surface of the monochromator is parallel to the crystallographic (lOil) plane. The monochromatic X-ray beam, Kal with the spectral purity of -95%, enters the DAC after passing through a beam reducer attached to the diffractometer. The D A C is mounted on a multiple stage system having X,Y ,Z ,xl and x2 movements for alignment with respect to the incident K,, beam. The DAC is positioned to have the

7

DIAMOND ANVIL CELLS IN HIGH-PRESSURE

STUDIES OF SEMICONDUCTORS

423

FIG.20. Schematic of DAC sample configuration for ADXD in Guinier geometry. (1) Rotating anode (Mo) X-ray source; (2) aperture diaphragm: ( 3 )quartz curved crystal monochromator; (4) scattering diaphragm: (5) direct beam stop; (6) DAC; (7) beam stop: (8) detector: ( 9 ) detector slit; and (10) Seeman-Bohlin focusing circle (Sahu etal., 1995 and reproduced with permission of AIP, USA).

sample inside it lying on the Seeman Bohlin circle with 114.6-mm diameter. The range of diffraction angle covered is from -30" to +2S0. To reap the full advantage offered by the Guinier diffractometer, it is essential to obtain perfect alignment of the X-ray beam optics, a nontrivial task. The first step is to align the monochromator to optimize the K,, component of the incident X-ray beam. Next, the diffractometer is aligned to the monochromatic beam. These two aspects are achieved by monitoring the intensity of the two fluorescent screens mounted rigidly at appropriate locations of the diffractometer. With a scintillation counter detector (SCD) monitoring the beam, the monochromator is fine tuned. Distances such as the X-ray source to the monochromator, the monochromator to the sample, and the sample to the SCD are adjusted, and finally the detector slit width is reduced to obtain an optimum Kal intensity. The width of the projected spot of the electron beam on the rotating anode target is -100pm, and hence the alignment procedure is carried out until the width of the focused X-ray beam at the focal circle is 100 pm. With the molybdenum target and an SCD slit width of 100 pm, the best FWHM is -0.04'. Although complete component is quite possible, this elimination of the Ka2 (A = 0.71359 also leads to a drastic reduction in the Kal intensity. A compromise is made between two opposing factors, namely, the spectral purity and the intensity of the monochromatic beam, and the ZKu2 is -S%. D A C X-ray diffraction with SCD is quite time consuming, and hence linear detectors such as

A)

424

M. Y o u s u ~

position-sensitive detectors (PSD) and area detectors such as X-ray film and imaging plates have been coupled to this diffractometer (Yousuf etal., 1996; Sahu etal., 1995; Purniah etal., 1997). The average FWHM of the peaks obtained with the PSD and imaging plates is -0.15'. 3. TRANSPORT PROPERTIES It is difficult to carry out transport properties measurements in a DAC because the sample cavity is very small and it is difficult to bring electrical leads from the high-pressure environment to the measuring apparatus. In spite of the underlying difficulties, phenomenal progress has taken place in carrying out electrical resistance measurements. For instance, a rather novel type of miniature diamond cell, fabricated entirely out of BeCu alloy has been developed for electrical transport measurements in high magnetic fields (Tozar 1993). Shimizu etaf. (1996) have carried out electrical resistance to 130 GPa at temperatures down to 50 mK using a DAC assembled on the 3He-4He dilution refrigerator. Babuskin and Ignatchenko (1996) have performed electrical resistance and thermoemf studies to 50 GPa in the temperature range of 77 to 400 K. The challenge of carrying out transport studies arises from two factors: the presence of a metallic gasket and the small size of the anvil flat. There is no insulating material that can yield like a metal and thus serve as a satisfactory gasket. Block etaf. (1977) tried to circumvent the problem by coating the metal gasket with a ceramic paint to insulate the leads. Mao and Bell (1981) devised a four-probe arrangement in which fine wires were supported in a prepressed MgO disk with the sample on it. Grzybowski and Ruoff (1984) have used sputtered electrical leads on the upper diamond anvil, insulated from the metal girdle by a mixture of A1203and NaCl. Figure 21 illustrates the sample, lead, insulated gasket, and shaped DAs developed by Pate1 etal. (1986) for carrying out electrical resistivity and Hall measurements to a pressure of -10 GPa.

III. High-pressure Studies of Semiconductors A large volume of data has been obtained on various classes of semiconductors by performing almost all the techniques of condensed-matter physics in a DAC. Scholarly reviews by Jayaraman (1983) and by Spain (1987) are very educative. Two specialized journals -High Temperature High and HighPressure Research, the regular series of AIRAPT conferPressure ences on high-pressure science and technology, and the special series of international conferences on semiconductors under pressure document the rapid growth of the field.

7 DIAMOND A N V I L CELLS IN HIGH-PRESSURE STUDIES OF SEMICONDUmORS 425

,

/GASKET LOCATING

UPPER

ELECTRICAL

SAMPLE

JE'JELLED EDGE

FIG. 21. A sketch of the arrangement developed for electrical resistivity and Hall measurements. (From Patel etal., 1986 and reproduced with the permission of Taylor & Francis, UK.)

1. CONVENTIONAL SEMICONDUCTORS The structural systematics of group IV elements, and 111-V and 11-VI compounds have been discussed in detail by McMahon and Nelmes (1996). The almost classical semiconductor to metal transition of elements Si (12GPa), Ge (11 GPa), Se (20GPa), Te (4GPa), Br (lOOGPa), I (17 GPa), and Xe (-160 GPa) - naturally leads to the question as to whether all the other elements from the upper right-hand comer of the periodic table will transform into the metallic state within an accessible range of pressure (Holzapfel, 1996; Bandyopadhyay etal., 1996). Bandstructure estimates for semiconducting to metallic transitions are available for B (120 GPa), C (diamond, 1.1 TPa), N (100 GPa), Br (33 GPa), and C1 (67 GPa). For Br and C1, where metallization has been confirmed, the values are underestimates. Nitrogen has been studied up to 180 GPa and does not show metallic transition, probably pointing to the conclusion that a large range of metastability may exist for very strong intramolecular bonding elements. Reflectivity measurements on Br up to 170 GPa give evidence for metallic behavior above 70 GPa (Shimizu etal., 1996), and by the scaling rules for halogens, C1 is expected to metallize above 150 GPa (Siringo etal., 1990). Most of the semiconductor to metal transitions, for instance, in Si, Ge, Se, and Te, as well as in 111-IV compounds, are linked to strongly first-order structural transitions. Akahama etal. (1996) found that black phosphorus metallizes through bandgap closure at -1.7 GPa, without a structural trans-

426

M. Y o u s u ~

tion (Akai etaf., 1989). One question, therefore, that is paramount to ask is whether a continuous bandgap closure in the solid is compatible with the prediction of a first-order semiconductor-to-metal transition in a fluid state ending in a critical point, as has been predicted for Xe at conditions of around 50 GPa and 5000 K (Ebeling etaf., 1988). It is of interest to note that the second-order character of the semiconductor-to-metal transition in I around 17.5 GPa is doubtful (Vladimirov, 1984;Pasternak etaf., 1987a,b; Olijnyk etaf., 1992, 1994). Experiments on amorphous semiconductors prove new aspects of electronic processes that differ from those in their crystalline counterparts. The electronic properties depend on the gap states arising from the defects. Generally, metallisation in the amorphous semiconductor occurs at much lower pressure than in the bulk (Minomura, 1992,1985; Yousuf etal., 1992; Chandra Shekar etal., 1993). It is observed that elemental a-Se and most of its compounds in amorphous states undergo irreversible amorphous-to-crystalline transitions, but in certain cases, the transitions are reversible (Minomura, 1985). There is an interesting class of materials termed dilute magnetic semiconductors in which the role of pressure has been exploited to understand the magnetism of the material (Jayaraman, 1983; Qadri etaf., 1983a,b, 1985, l987,1989,1992a,b). High pressure results on semiconducting compounds have been growing at a fast pace and the proceedings of the International Conference on Semiconductor under Pressure bear the testimony. Here, a few recent results are highlighted. Beryllium chalcogenides have been found to exhibit a structural transformation from zinc-blende to the NiAs structure type in that the transition pressures are 54.9, 61.3, and 39.0 GPa for BeS, BeSe, and BeTe respectively (Victor Jaya, 1997, Narayana e faL, 1997). Ab initio electronic band structure calculations done recently exhibit a good agreement with the experimental values (Kalpana etaf., 1997). Ab initio electronic band structure calculations on strontium chalcogenides are also in good agreement with the experimental values (Shameem Banu etaf., 1997, 1998; Luo etaf., 1994). Metallization of H is estimated to occur around 200 ( 2 50) GPa (Ramaker etaf., 1975; Chacham and Louie, 1991; Barbee and Cohen, 1991). Quantum Monte Carlo calculations predict for metallic atomic hydrogen a sequence of structural transitions from CF8 (diamond) + t14 (p-tin) + CP1 + C12 in the range 200-900GPa (Natoli etaf., 1993). Metallic quantum liquid phase is expected at 0 K and >5 TPa (McDonald and Burgess, 1992). Figure 22 depicts the P-T phase diagram of hydrogen (Holzapfel, 1996). EOS data at 0 K indicates that only minor volume discontinuities may occur at phase transitions between phases I, 11, 111, and IV. However, Edwards and Hansel (1997) argue that the existence of spontaneous asymmetry of molecules in the solid hydrogen may be the reason for the resistance to its

7 DIAMOND ANVIL CELLSI N HIGH-PRESSURE STUDIES OF SEMICONDUCTORS 427

,

,

,

,

,

I

I

,

,

? 0

: ‘.

molecular fluid

.o :?

4 - 3 - 2 - 1

metallic fluid

0

1

2

.. ? __:.

3

4 log (PIGPa)

I

Frc. 22. P-Tphase diagram of hydrogen (after Holzapfel, 1996 and reproduced with permission of IOP. UK).

metallization. Edwards and Ashcroft (1997) found that at around nine fold compression a spontaneous polarisation develops due to the movement of protron pairs away from their ideal lattice site. Weir etal.(1996) reported the decrease of the resistivity of hydrogen and deuterium shock compressed to 180 GPa and 4400 K. This was interpreted as metallization of molecular hydrogen above 140 GPa and 3000 K. At the highest pressures reported in their experiments the current density was about 500 A/cm2 and the resistivity was evaluated from the voltage drop across the electrodes, assuming ohmic behavior. In conducting fluids at high temperature such as electric arcs or plasmas for fusion research, nonohmic behavior is consistently observed even for lower field strengths and current densities than those which are reported by these authors. Thus the resistance which was measured at high intensity could be quite different from the intrinsic zerocurrent resistivity of the sample, so that direct comparison of its absolute value with that of metals might be misleading (Besson, 1997). Nevertheless, no experimental data exist, apparently, on the electric behavior of high density media such as those reported in by Weir etal.(1996) so the possibility of nonohmic characteristics can only be inferred from the behavior of low-density plasmas, and this point is speculative. While no clear picture is in sight, Nellis and Weir (1997) mention that it remains to be determined how this primarily molecular system transforms to an “atomic” metal at higher densities and temperatures.

M. Y o u s u ~

428

(EL 1 ) 8.6 GPa 10.7 G Po

2 n d cycle 6.1 GPO

3.6 GPa

w

Zl 8.9 GPa

250

350

350

450

PHONON FREQUENCY ( ~ 6 ’ ) FIG. 23. Raman spectra of bulklike AlAs ( a ) and GaAs (b) epitaxial films at various pressures. Visual specimen appearance and laser location are indicated by the shading and solid circles in the rectangular inset sketches. The top two tracesshow reversal to the zinc blende phase on decompression and subsequent recompression. Spectra during the first pressure cycle were excited with 6471 A, and the reversal spectra with 5682 A (after Weinstein et al., 1981 and reproduced with permission of Plenum, USA).

2. STRONGLY CORRELATED S E M I C O N D U SYSTEMS ~OR Study of low-dimensional systems is the current trend in semiconductor research. These studies pertain to lattice-matched 2D layered systems, namely, AIGaAs/GaAs and GaAs/InP, and 1D quantum wires or OD quantum dots. Pressure studies of such systems may lead to new directions in research. The combination of lattice-mismatch strain and lower-dimensional confinement is expected to answer questions pertaining to growth rate and band-structure engineering. Since pressure directly tunes the strain in low-dimensional systems without needing a myriad of different samples, these experiments give feedback on the conduction band, the valence band,

7 DIAMOND ANVILCELLSI N HIGH-PRESSURE STUDIESOF SEMICONDUCTORS 429

the hole effective mass that controls the light polarization in optoelectronic devices, and relaxation effects in mismatched systems (Adams, 1991). A few typical examples of use of DACs in the strongly confined semiconductor systems are highlighted here. In AlAs/GaAs multilayers and superlattices (SLs), it is found that zinc blende AlAs could be overpressed above its bulk boundary for transition to the rock salt structure. In the regime 12.3 < P < 17.2 GPa, between bulk AlAs and GaAs transitions, AlAs/ GaAs SLs enter a metastable state. Figure 23 reproduces the results (Weinstein el af., 1991). Pressure can tune the electronic energy bands and has been used to study the band offset at heterointerfaces, giving a possibility of tunable quantum well lasers and indirect gap-related DX defects (Venkateswaran etal., 1986; Kirchoefer et a/., 1982; Cui et at., 1991). Pressure-tuned lattice-mismatch strain has opened a wide field of applications in device technology. Photoluminescence studies of a Cd,,34Zno.66Te/ZnTe strained-layer superlattice indicate that the Cdo.34Zno.66Te layer (type I) has confined electrons and heavy holes, whereas the light holes are confined in the ZnTe layer (type 11). The low-temperature photoluminescence is typically dominated by a strong feature due to the decay of the high-hole exciton; a weaker feature is due to the decay of the heavy-hole exciton and to the luminescence from an excited quantum well state (Williamson et af., 1991).

IV. Concluding Remarks Advances in DAC techniques have enabled a variety of condensedmatter physics studies to be performed at very high pressure, up to -560 GPa. Valuable information on phase stability, EOS, band structure and metal to insulator transitions has been obtained on a large number of semiconductors. The introduction of the laser-heating techniques has opened up a new era in high-pressure, high-temperature research, for experiments that were previously possible only through shock wave techniques. This technique is presently being exploited to synthesize novel materials hitherto only theoretically predicted. The search for metallic hydrogen is still on, and DAC experiments under low temperature and high pressure have been performed to draw the phase diagram of hydrogen. A question that is paramount in every mind concerns the ultimate structure of condensed matter (Young, 1991a,b). Many elemental systems appear to adopt the body-centered cubic phase. Further many elemental insulators and semiconductors turn metallic at high pressures. Very recently sulfur has been shown to become metallic and superconducting near 93 GPa (Struzkhin etaf., 1998). This suggests that elements such as Br,C1 may turn metallic at pressures reachable today with the diamond anvil cell. In certain compounds, partial reduction with hydrogen is favored at

megabar pressures, although it is most unlikely at 0.1 MPa. An example is the reduction of A1203by hydrogen above 140 GPa. Many reactions are greatly accelerated under extreme compression, and examples have been observed in diamond plus fluorine, and iron plus oxygen (Ruoff, 1991,1993). Impressive advances in the theoretical calculations of band theory both by the ab initio pseudo-potential and the ab inirio molecular dynamics methods allow accurate determination of thermodynamic functions that can be experimentally verified. The question of how accurately the pressure is known, especially above 30 GPa rankles in the mind. Important developments have addressed this question and the answer is not far to seek. Three measurements in the same experiment - namely, determination of sound velocity from Brillouin scattering, relative volume change from X-ray diffraction, and the ruby line shift -answer this question. High-pressure studies on semiconductors have led to a basic understanding of their electronic band structure, crystal structure and the role of strains in heterostructures. Now through high pressure studies it may become possible to design semiconductor heterostructures for applications. For instance the use of strongly confined semiconductors in optoelectronic devices can be improved by high-pressure studies, because pressure directly tunes the energy levels and alters the band offset and the strain in the heterostructure. It is satisfying to know that study of heterostructures with respect to their transport, electronics, and structure is becoming the current trend. The Biennial International Conferences on high-pressure semiconductor physics have had a major share of studies on heterostructure (especially those in 1996, 1994, and 1992). Improvements in DAC capability (up to -560 GPa), the fabrication of micro X-ray beams (-3 pm), coupling the DAC with the SRS (brilliance photons/s/mrad/100 mA/O.l% AA/A), the possibility of laser heating (-7000 K) simultaneously with pressure, and the advances in area detectors such as the imaging plate system and charged-coupled devices open up new areas for semiconductor physics research that need to be judiciously exploited. For DAC electrical resistivity work, a tough problem has been in attaching good ohmic contacts to the sample. The mechanical contacts often used for performing measurements on bulk would severely degrade the device properties of the heterostructure. Until advancement occurs in this area, the information obtained through electrical resistivity, Hall mobility, and thermoelectric power cannot be added to the probeless measurements.

Acknowledgments It is my privilege to work with my teacher, Dr. K. Govinda Rajan, Head, Advanced Materials Laboratory, who introduced me to the field of high

7 DIAMOND ANVIL CELLSI N HIGH-PRESSURE STUDIESOF SEMICONDUCTORS 431

pressure and taught me the nuances of the technique. I have benefited immensely from the stimulating discussions I have had with him during the preparation of this chapter. Working on this chapter has been possible due to his encouragement. I am ever grateful to him for his guidance. It is my pleasure to work with a dedicated team of scientsists, Dr. P. C. Sahu, Mr. N. V. Chandra Shekar, Mr. N. Subramanian, Mr. M. Sekar, and Mr. L. M. Sundaram, who have worked very hard to establish a world-class instrumentation on the DAC X-ray diffraction system at this high-pressure laboratory of the Materials Science Division of Indira Gandhi Centre for Atomic Research. This development has been possible due to the whole hearted support and encouragement we have been receiving from Dr. Placid Rodriguez, our director and Dr. Baldev Raj, our group director. I convey my heartfelt gratitude to them. Further, I wish to convey my sincere thanks to Professor A. Jayaraman and Dr. J. Akella for giving us guidance during the development of the DAC at our Centre. Also, I wish to express my gratitude to Professor A. Jayaraman for a critical reading of the manuscript. Last but not least, I acknowledge my indebtedness to the authors and publishers for allowing me to reproduce figures from their articles. REFERENCES Adams, A. R. (1991). In Frontiers ofHighPressure Research (Hochheimer, H. D., and Etters, R. D., eds.), p. 281, Plenum Press, New York. Adams, D. M., and Shaw, A. C. (1982). J. Phys.D: Appl.Phys.15, 1609. Instrum. 9,1140. Adams, D. M., Appleby, R., and Sharma. S. K. (1976). J.Phys.E: Scientific & Adams, D. M., Christy, A. G., and Norman, A. J. (1993). J .Phys.E: MeasurementScience Technology 4, 422. Akahama, Y., Kawamura, H., Hausermann, P., Hanfland, M., and Shimomura, 0. (1996). In High Pressure Science and Technology, Proc.JointX V A I R A P T and X X X l l l EHPRG Inter. Conf. (Trzeciakowski, W. A,, ed.), p. 360, World Scientific, Singapore. Akai, T., Endo, S., Akahama, Y., Koto. K.. and Murayama, Y. (1989). HighPress. Res.1,115. Akella, J. (1996). Scienceand Technology Review,Lawrence Livermore National Laboratory, USA. Arashi, H. (1987). In High Pressure Researchin MineralPhysics(Manghnani, M. H., and Sgyono, Y., eds.), p. 335, American Geophysical Union, Washington. D.C. Asaumi, K., and Ruoff, A. L. (1986). Phys.Rev.B 33,5633. Science and Technology, Babuskin, A. N., and Ignatchenko, 0. A. (1996). In High Pressure Conf. (Trzeciakowski, W. A.. ed.), Proc.JointX V A I R A P T and XXXIII EHPRG Inter. p. 606, World Scientific, Singapore. Bandyopadhyay, A. K., and Ming, L. C. (1996). Phys.Rev.B,54, 12049. Barbee, T. W., and Cohen, M. L. (1991). Phys.Rev.B 43, 5269. Barnett, J. D., Block, S., and Piermarini, G. J. (1973). Rev.Sci.Instrum. 44,1. Bassett, W. A., and Takahashi, T. (1965). Am. Mineral50,1576. Bassett, W. A., Shen, A. H., Bucknum, M., and Chou, I. M. (1993). Rev.Sci. Instrum. 64,2340. 38,37. Bassett, W. A., Takahashi, T., and Stook, P. (1967). Rev.Sci.Instrum. Baublitz, M. A,, Arnold, V., and Ruoff, A. L. (1981). Rev.Sci.Insfrum. 52,1616. Institution of Washington YeurBook 80, 404. Bell, P. M., and Mao, H. K. (1981). Curnrgie

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Index A Ab initio calculations, 191, 192-193 tetrahedral semiconductors and, 158-161 A1 kinetic limitations of dissolution of N2 in liquid, 359-362 solubility of nitrogen in liquid, 357-358 AIGaAstGaAs, intrinsic bistability, 54, 74-76 Alloy scattering, 12 AIN, thermal stability of, 355-357 AlSb barriers, 101 Angle-dispersive X-ray diffraction (ADXD), 422-424 Angular momentum operators, 123, 124 Anharmonic parameters, 191-194 Auger recombination, 317-324 Avalanche photodiodes (APDs), hydrostatic pressure and GaAs, 335-339 Ge, 339-341 impact ionization and band-structure dependence, 327-332 InAs and InAsP, 332-335 Si, 341-344 universal dependence of breakdown on band structure, 344-347

pure, 134, 140-141 superlattices (SLs), 136-137 Blackman-Brout sum rule, 152 Bridgman, P. W., 118-119 Bridgman anvil apparatus, 65, 66, 67, 356, 385 Bridgman growth, 354 Brillouin scattering, 407, 418 Brillouin zone, 57, 58, 124-127

C Carrier concentration pressure dependence of band structure and changes in 2DEG concentration in quantum wells, 10-18 pressure dependence of donor levels of, in quantum wells, 5-9 CdTe/InSb, 16, 18 Centrosymmetric cubic crystals region of opacity, 176-177 region of transparency, 172-175 Composite fermions (CFs), 26-29 Conduction electron high conduction band (CHCC), 318-324 Conduction hole spin-orbit valence band (CHSH), 318-324 Crystal symmetry phonons and, 127-132 strains and stresses and, 132-138 Czochralski growth, 354

B Backscattering geometry, 183-184 Band mixing, 57-63 Bardeen transfer Hamiltonian, 51, 53 Bassett-Takahashi-Stook DAC, 394-395 Ben-Daniel Hamiltonian, 48, 49, 57, 59 Berry phase techniques, 191 Bisotropic, 134 heterojunctions (HJs), 135- 136

D Damping factor, amplitude of oscillations and, 37-39 Debye frequency, 152, 155 Density functional theory (DFT), 359 Diamond anvil cell (DAC), 120, 141-142 alignment, 400-404 Bassett-Takahashi-Stook cell, 394-395

431

438

INDEX

clarity, 384-385 color, 384 conventional semiconductors and, 425-427 cut, 384 evolution of, 382 gasketing, 404-405 high temperature, 412-414 Huber-Syassen-Holzapfel cell, 394 LeToullec-Pinceaux-Loubeyre cell. 395 light fringe technique, 404 low temperature, 41 1-412 Mao-Bell cell, 389-390,393 Menill-Bassett cell, 395 NBS (National Bureau of Standards) cell, 382, 386, 389 optical spectroscopy and, 415-419 other makers of, 395-3% piston and cylinder rockers, 399-400 piston-cylinder assembly, 398 precision fabrication of, 396-400 preparation of sample, 405-406 pressure-cell holder, 399 pressure media and limit of hydrostaticity, 409-410 pressure scale and calibration, 406-409 selecting diamonds, 383-386 shapes and types of diamonds, 382-383, 385 silver iodide technique, 401-404 strongly confined semiconductors and, 427-429 transport properties, 424 X-ray diffraction and, 420-424 Double-barrier structures (DBSs), 46 calculating I-V, 51-54 charge distribution calculations, 54-57 intrinsic bistability, 54, 74-76 negative differential resistance, 70-72 resonant tunneling in, 68-101 Ricco-Azbel formulae, 47-48,68 Duke Hamiltonian, 48, 49, 57. 59

E Elastic compliance, 121, 133 Elastic stiffness, 121, 133 Electrons, effects of stress and strain on, in crystals, 121-127 Electro-reflectance, 120 Electrostriction coefficients, 1%

Energy-dispersive X-ray diffraction (EDXD) ,420-422 Envelope functions, 48 band mixing, 57-63 Epitaxial growth, 141 European Synchrotron Radiation Facility Grenoble, 372, 377

F Fabry-Perot analysis, 54 Fabry-Perot lasers, 308 Fang-Howard potential, 55 Far-infrared spectroscopy of polar phonons in cubic materials, 188-190 Fermi-Dirac function, 51 Fermi energy, 14, 16 Forward-scattering geometry, 178-179 Fourier transforming of magnetooscillations, 16 Fractional quantum Hall effect (FQHE), 2 composite fermions, 26-29 Lande g-factor, 19 overview of, 25-26 Franz-Keldysh oscillations, 277 Frohlich interaction, 204-205

G Ga kinetic limitations of dissolution of N2in liquid, 359-362 solubility of nitrogen in liquid, 357-358 GaAs avalanche breakdown, 335-339 bulk, single quantum wells, 283-288 piezo-optical response of, in opaque region, 266-269 GaAslAlAs band alignments, 68-73 band mixing, 57-63 intrinsic bistability, 54, 74-76 magnetic field studies, 87-92 pressure dependence of current unrelated to band, 73 superlattices conduction states, 288-290 zero magnetic field, 76-86 GaAs/AIGa, pressure dependence of current unrelated to band, 73 GaAs/AIGaAs band alignments, 68-73

INDEX Land6 g-factor, 19 magnetophonon resonance effect, 33, 34 parallel conduction, 7-8 pressure dependence of band structure and changes in 2DEG concentration in quantum wells, 10-18 pressure dependence of donor levels of camer concentration in quantum wells, 5-9 2DEG in, 5-18 GaAs/AIInAs pressure dependence of donor levels of camer concentration in quantum wells, 5-9 2DEG in, 5, 7 GaAslGaAlAs double asymmetric quantum wells, 278-283 single quantum wells, 277-278, 283-288 GaAs/GaAs structures, Raman spectroscopy, 217-219 GaInAs/AIIn, magnetophonon resonance effect, 33-34 GaInAslInP, 7 magnetophonon resonance effect, 33, 35-37 pressure dependence of band structure and changes in 2DEG concentration in quantum wells, 10-18 GaN high N2pressure solution growth of, 362-365 homoepitaxy, 374-377 physical properties of pressure-grown crystals, 365-370 thermal stability of, 355-357 wet etching and surface preparation, 370-373 GaSb/InAs, 18 zero Hall resistance in semimetallic, 25 Ge avalanche breakdown, 339-341 piezo-optical response of, in opaque region, 266-269 Griineisen formula, 37, 119, 120 Griineisen parameters, mode, 148-152 thermal expansion (quantum effects), 152-155 Gunn diode, 46, 66

439

H Harmonic parameters, 191 Heterojunctions (HJs) growth of cubic crystal, 135-136 hydrostatic pressure and, 184-188 piezoelectric fields and elastic aspects, 199-205 pressure and temperature dependence of strains, 205-213 strained, 180-184 High nitrogen pressure background, 354 homoepitaxy, 374-377 kinetic limitations of dissolution of N1 in liquid Al, Ga, and In, 359-362 physical properties of pressure-grown GaN crystals, 365-370 solubility of nitrogen in liquid Al, Ga, and In, 357-358 solution growth of GaN, 362-365 thermal stability of AIN, GaN, and InN, 355-35 7 wet etching and surface preparation, 370-373 Homoepitaxy, 374-377 Hooke's law, 132-133, 199,206 Huber-Syassen-Holzapfel DAC, 394 Hydrostatic pressure advantages of, 4 camer concentration reduction, 4 DACs and limit of, 409-410 dependence of band structure and changes in 2DEG concentration in quantum wells, 10-18 dependence of donor levels of camer concentration in quantum wells, 5-9 effects on elastic state of heterojunctions and superlattices, 206-209 effects on low-dimensional semiconductor structures, 2-4 heterojunctions under, 184-188 laser dependence, 308-326 strain and stress, 137-138 tuning of Land6 g-factor by, 18-21 Hydrostatic pressure, avalanche photodiodes (APDs) and GaAs, 335-339 Ge, 339-341 impact ionization and band-structure dependence, 327-332

440

INDEX

InAs and InAsP, 332-335 Si, 341-344 universal dependence of breakdown on band structure, 344-347 Hydrostatic pressure, magnetophonon resonance (MRP) and, 12,30 amplitude of oscillations and damping factor, 37-39 pressure dependence of effective mass, 31-37 Hydrostatic pressure, phonons (optical) and ab initio calculations, 158-161 effects of, 144-163 linewidths and lifetimes of phonons, 155-156 mode Griineisen parameters, 148-152 phase transitions, 156-163 Rarnan spectra, 162-163 thermal expansion (quantum effects), 152-155 Hysteresis, 159

I Impact ionization and band-structure dependence, 327-332

In kinetic limitations of dissolution of Nz in liquid, 359-362 solubility of nitrogen in liquid, 357-358 InAs and InAsP, avalanche breakdown, 332-335 Indira Gandhi Centre for Atomic Research (IGCAR), 3% Indirect gap semiconductors,external stress on, 264-266 InGaAslGaAs single quantum wells, 272-277 superlattices, miniband dispersion, 278 InN, thermal stability of, 355-357 Integer quantum Hall effect (IQHE), 2 LandC g-factor, 19 overview of, 21-22 spin texture excitations, 22-25 zero Hall resistance in semimetallic GaSbIlnAs, 25 Interband tunneling AlSb barriers, 101 forward bias, 104-110 reverse bias, 102-104 Intervalley electron-phonon (EP) and

hole-phonon (HP) interactions, 264-266 Intervalley splitting, 125-127 Intravalley splitting, 125-127 Intrinsic bistability, 54, 74-76 Irreducible representations (IRRS), phonons and, 128-132

K Kane Hamiltonian, 62 Keating model, 191, 192, 193 Keldysh equation, 336 Kohn's theorem, 22-23 Kronecker delta, 164-166

L Landau levels, 101,108-110 composite fermion, 26-29 magnetic field studies, 87 Landau quantization, 22 LandC g-factor tuning, by hydrostatic pressure, 18-21, 101 Laser Auger recombination, 317-324 concepts, 305-308 lasing wavelength pressure dependence, 308-309 long-wavelength devices, 315-325 pressure and temperature dependence, 308-326 threshold current dependence, 309-325 visible, 313-315 L-band tunneling, 98-101 LeToullec-Pinceaux-Loubeyre DAC, 395 Local density approximation (LDA), 192 Longitudinal optical (LO) phonons, 131 Longitudinal optical-transverse optical (LO-TO) splitting, 119, 121-122, 124-125, 131-132

M Magnetic field studies, 87-92 Magnetophonon resonance (MRP), hydrostatic pressure and, 12, 30 amplitude of oscillations and damping factor, 37-39 pressure dependence of effective mass, 31-37 Magnetotransport effects, 4 Magnetotunneling, 89

INDEX Many-body effects, 25, 192 Mao-Bell DAC, 389-390,393 Memll-Bassett DAC, 395 Metalorganic chemical vapor deposition (MOCVD), 354, 372, 374,377 Metalorganic vapor phase epitaxy (MOVPE), 25 Metal-oxide semiconductor (MOS) structures, 2 Mode Gruneisen parameters, 148-152 thermal expansion (quantum effects), 152-155 Modulation spectroscopy, 120 Molecular beam epitaxy (MBE), 2, 82, 354, 374,375,377 Multiple-barrier structures, resonant tunneling in, 68-101 Multiple quantum wells (MQWs), stressing, 142- 143 Murnaghan equation, 145-146.338

N NBS (National Bureau of Standards) DAC, 382, 386, 389 Negative differential resistance (NDR) electrical measurements, 64-65 first semiconductor, 46 Negative differential resistance, high pressure studies of band alignments, 68-73 early work, 66-68 intrinsic bistability, 54, 74-76 L-band tunneling, 98-101 magnetic field studies, 87-92 peak to valley ratios (PVRs), 68, 70 pressure dependence of current unrelated to band, 73 quantum beats, 50, 92-94 resonant interband tunneling, 101-110 in single-, double-, and multiple-barrier structures, 68-101 2D+2D tunneling, 92-98 zero magnetic field, 76-86 Noncentrosymmetric cubic crystals region of opacity, 179-180 region of transparency, 177-179 Non-Hermitian Hamiltonian, 61

0 Optical spectroscopy, DAC and, 415-419

441

Optoelectronic devices avalanche photodiodes, 327-348 experimental considerations, 302-305 laser concepts, 305-308 laser pressure and temperature dependence, 308-326 role of, 301-302 uniaxial strain effects, 326-327

P Phonons crystal symmetry, 127-132 effects of stress and strain on, in crystals, 121-127 irreducible representations (IRRS), 128-132 longitudinal optical (LO), 131 nonpolar, 169 polar, 128 silent, 128 soft mode, 127 transverse optical (TO), 131 Phonons (optical), hydrostatic pressure and ab initio calculations, 158-161 effects of, 144-163 linewidths and lifetimes of phonons, 155-156 mode Griineisen parameters, 148-152 phase transitions, 156-163 Raman spectra, 162-163 thermal expansion (quantum effects), 152-155 Phonons (optical), strain and centrosymmetric cubic crystals, region of opacity, 176-177 centrosymmetric cubic crystals, region of transparency, 172- 175 control experiments, 171-191 deformation potentials (PDPs), 164-166 deformation potentials, models and trends, 191-194 deformation potentials, uses of, 194-198 effects of, 163-198 far-infrared spectroscopy of polar phonons in cubic materials, 188-190 fibers under tensile stresses, 190-191 heterojunctions under hydrostatic pressure, 184-188 noncentrosymmetric cubic crystals, region of opacity, 179-180

442

INDEX

noncentrosymmetric cubic crystals, region of transparency, 177-179 secular equation, 167-171 singlet/doublet phonon components. 167, 169 strained superlattices and heterojunctions. 180- 184 Phonon deformation potentials (PDPs), 164-166 availability of, 214 models and trends, 191-194 uses of, 194-198 Phonon secular equation in cubic crystals, 167-170 in noncubic crystals, 170-171 Photoelastic coefficients, 194-196 Photoluminescence, 204,206,415 Photoreflectance spectroscopy, 204, 272 Piezobirefringence in transparent region, 270-271 Piezoelectric (PZ) fields and elastic aspects in heterojunctions and superlattices, 199-205 Piezo-optical response of Ge and GaAs in opaque region, 266-269 Pikus-Bir Hamiltonian, 124 Poisson's equation, 54, 55, 80 Polar axes, 204 Polycrystalline materials, phonon deformation potentials for, 196-198 Porto's notation, 173 Pseudomorphic growth, 135 Pure bisotropic, 134 stress. 140-141

Q Quantum beats, 50, 92-94 Quantum-confined Stark effect (QCSE), 272-277 Quantum wells band alignment in SiGe/Si, 290-295 Land6 g-factor, 19-20 pressure dependence of band structure and changes in 2DEG concentration in, 10-18 pressure dependence of donor levels of camer concentration in, 5-9 stressing multiple, 142-143 Quantum wells. effects of stress, 271 band alignment in SiGe/Si, 290-295

bulk GaAs and GaAs/GaAlAs single, 283-288 GaAs/AIAs superlattices conduction states, 288-290 GaAs/GaAIAs double asymmetric, 278-283 GaAs/GaAlAs single, 277-278 InGaAs/GaAs single, 272-277 InGaAslGaAs superlattices, miniband dispersion. 278

R Raman spectroscopy angular dispersion, 214-215 centrosymmetric cubic crystals, region of opacity, 176-177 centrosymmetric cubic crystals, region of transparency, 172-175 DAC and, 415-418 GaAs and other compounds, 217-219 heterojunctions under hydrostatic pressure, 184-188 noncentrosymmetric cubic crystals, region of opacity, 179-180 noncentrosymmetric cubic crystals, region of transparency, 177-179 phonons and, 128-132,162-163 piezoelectric fields and, 204-205 WGe structures, 217 strain characterization of heterojunctions and superlattices through, 213-220 strained superlattices and he teroj unctions, 180-1 84 strain mapping, 215-217 use of, 118, 119-120 Reflection high-energy electron diffraction (RHEED), 376 Resonant t u n n e h g See also Negative differential resistance, high pressure studies of band mixing, 57-63 calculating I-V, 51-54 charge distribution calculations, 54-57 double-barrier, 46 electrical measurements, 64-65 interband tunneling, 101-110 Ricco-Azbel formulae, 47-48, 68 in single-, double-, and multiple-barrier structures, 68-101 space charge, 54-57

INDEX strain due to pseudomorphic growth, 63-64 transfer matrix method, 48-51, 61 uses for, 45-46 Ricco-Azbel formulae, 47-48, 68 Ridley-Watkins mechanism of electron transfer, 66 Ruby fluorescence shift, 407 Rutherford backscattering, 141

S Schottky barrier formation, 67 Schrodinger-Poisson calculation, 80, 87, 90. 92 Schroedinger equation, 56 Semimetal-to-semiconductor transition (SMSCT), 104 Shear modulus, 200 Shock compression experiments, 143- 144 Shubnikov-de Haas oscillations, 4, 5, 10, 16, 18 Si, avalanche breakdown, 341-344 SiGelSi quantum wells, band alignment in, 290-295 SUGe structures, Raman spectroscopy, 217 Single-barrier structures negative differential resistance, 72 resonant tunneling in, 68-101 Skyrmions, 23-25 Spin-orbit Hamiltonian, 239 Spin texture excitations, 22-25 Star of k, 125 Strain(s) crystal symmetry and, 132-138 due to pseudomorphic growth, 63-64 effects of, on electrons and phonons in crystals, 121-127 heterojunctions and bisotropic, 135- 136 historical review of effects, 118-121 hydrostatic pressure, 137- 138 intervalley splitting, 125-127 intravalley splittings, 125-127 mapping, 215-217 superlattices and bisotropic, 136-137 uniaxial, and optoelectronic devices, 326-327 Strain, optical phonons and centrosymmetric cubic crystals, region of opacity, 176-177

443

centrosymmetric cubic crystals, region of transparency, 172-175 control experiments, 171-191 deformation potentials (PDPs), 164-166 deformation potentials, models and trends, 191-194 deformation potentials, uses of, 194-198 effects of, 163-198 far-infrared spectroscopy of polar phonons in cubic materials, 188190 fibers under tensile stresses, 190-191 heterojunctions under hydrostatic pressure, 184-188 noncentrosymmetric cubic crystals, region of opacity, 179-180 noncentrosymmetric cubic crystals, region of transparency, 177-179 secular equation, 167-171 singletidoublet phonon components, 167, 169 strained superlattices and heterojunctions, 180-184 Strain characterization of heterojunctions and superlattices critical pressure, 207 piezoelectric fields and elastic aspects, 199-205 pressure and temperature dependence of strains, 205-213 Raman spectroscopy and, 213-220 Stress crystal symmetry and, 132-138 diamond anvil cell (DAC), 120,141-142 effects of, on electrons and pbonons in crystals. 121-127 experimental techniques for applying, 138-144 heterojunctions (HJs) and pure bisotropic, 135- 136 hydrostatic pressure, 137-138 pure bisotropic, 140-141 superlattices (SLs) and biaxial, 136-137 uniaxial compressive, 138-140 uniaxial stress techniques, 120-121 Stress, effects of external uniaxial bands at k # 0, 254-264 conduction band minima at A,, 255-257 conduction band minima at LI, 257-258

444

INDEX

critical pointsiband structure (diamond zinc blende semiconductors), 238-244 critical pointsiband structure (wurtzitetype materials), 244-249 exciton effects at k # 0, 262-264 indirect gap semiconductors,264-266 indirect transitions, 258-260 interband transitions, 260-262 piezobirefringence in transparent region, 270-271 piezo-optical response of Ge and GaAs in opaque region, 266-269 spin-exchange effects, 249-253 spin-exchange effects (diamondzinc blende semiconductors),250-252 spin-exchange effects (wurtzite materials), 252-253 Stress and quantum wells, effects of, 271 band alignment in SiGelSi quantum wells, 290-295 bulk GaAs and GaAdGaAlAs single quantum wells, 283-288 GaAs/AlAs superlattices conduction states, 288-290 GaAdGaAIAs double asymmetric quantum wells, 278-283 GaAdGaAlAs single quantum wells, 277-278 InGaAdGaAs single quantum wells, 272-277 InGaAdGaAs superlattices, miniband dispersion, 278 Superlattices (SLs) GaAdAIAs superlattices conduction states, 288-290 growth of cubic crystal, 136-137 piezoelectric fields and elastic aspects, 199-205

pressure and temperature dependence of strains, 205-213 strained, 180-184

T Temperature dependence, laser, 308-326 Temperature effects on elastic state of heterojunctions and superlattices, 210-213 Tetragonal distortion, 200, 207 Thermal expansion (quantum effects), 152-155 Transfer matrix method, 48-51,61, 80 Transverse optical (TO) phonons, 131 Tunnel diode, 46,66 Tunneling. See Resonant tunneling Z h 2 D tunneling, 92-98

U Ultrasonic waves, 143 Uniaxial strain effects, optoelectronic devices and, 326-327 Uniaxial stress techniques, 120-121 See also Stress, effects of external uniaxial compressive stress, 138-140

V Valence force. field (VFF), 191 Voigt-Reuss-Hill (VRH) average, 197

X X-ray diMaction angle-dispersive, 422-424 DAC and, 420-424 energy-dispersive,420-422 microbeam, 420

Z Zeeman energy, 19,22,26,28 Zero-point quantum correction, 154

Contents of Volumes in This Series

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

Volume 2 Physics of 111-V Compounds M. G. Holland, Thermal Conductivity

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

445 Vol. 55 ISBN 0-12-752163-1

SEMICONDUCTORS AND SEMIMETALS Copyright 0 1998 by Academic Press All rights of reproduction in any form reserved. 0080/8784/98$25.00

446

CONTENTS OF VOLUMES IN THISSERIES

Volume 3 Optical of Properties 111-V Compounds Marvin Hass, Lattice Reflection William G. Spitzer, Multiphonon Lattice Absorption D. L. Stienvalt and R. F. Poner, Emittance Studies H. R. Philipp and H. Ehrenveich, Ultraviolet Optical Properties Manuel Cardona, Optical Absorption above the Fundamental Edge Earnest J.Johnson, Absorption near the Fundamental Edge John 0. Dimmock, Introduction to the Theory of Exciton States in Semiconductors B. Lax and J. G. Mavroides, Interband Magnetooptical Effects H. Y. Fan, Effects of Free Cames on Optical Properties Edward D. Pafik and George B. Wright, Free-Camer Magnetooptical Effects Richard H. Bube, Photoelectronic Analysis B. 0. Seraphin and H. E. Bennett, Optical Constants

Volume 4 Physics of III-V Compounds N. A. Goryunova, A. S. Borschevskii, and D. N. Trefiakov, Hardness N. N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds A"'BV Don L. Kendall, Diffusion A. G. Chynowerh, Charge Multiplication Phenomena Robert W. Keyes, The Effects of Hydrostatic Pressure on the Properties of 111-V Semiconductors L. W. Aukerman, Radiation Effects N. A. Goryunova, F. P. Kesamanly, and D. N. Nasledov, Phenomena in Solid Solutions R. T. Bare, Electrical Properties of Nonuniform Crystals

Volume 5 Infrared Detectors Henry Levinstein, Characterization o f Infrared Detectors Paul W.Kruse. Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. B. Prince, Narrowband Self-Filtering Detectors Ivars Melngalis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides Donald Long and Joseph L. Schmidt, Mercury-CadmiumTelluride and Closely Related Alloys E. H. Purley, The Pyroelectric Detector Norman B. Stevens, Radiation Thermopiles R. J.Keyes and T.M. Quist, Low Level Coherent and Incoherent Detection in the Infrared M. C. Teich, Coherent Detection in the Infrared F. R. Arums, E. W. Sard, B. J. Peyton, and F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers, Jr., Macrowave-Based Photoconductive Detector Robert Sehr and Rainer Zuleeg, Imaging and Display

Volume 6 Injection Phenomena Murray A. Lamperr and Ronald B. Schilling, Current Injection in Solids:The Regional Approximation Method Richard Williams, Injection by Internal Photoemission Allen M. Barnefr, Current Filament Formation

CONTENTS OF VOLUMES IN THISSERIES

447

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, M O S Transistors G. R. Antell, Gallium Arsenide Transistors T. L. Tansley, Heterojunction Properties

Part B T. Misawa, I MP ATI Diodes H. C. Okean, Tunnel Diodes Robert B. Campbell and Hung-Chi Chang, Silicon Junction Carbide Devices R. E. Ensfrom, H. Kressel, and L. Krussner, High-Temperature Power Rectifiers of GaAs,.,P,

Volume 8 Transport and Optical Phenomena Richard J. Slim, 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 11: Gallium Arsenide

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

Volume 10 Transport Phenomena R. L. Rhode, Low-Field Electron Transport J.D. Wiley, Mobility of Holes in 111-V Compounds C. M. Wolfe and G. E. Stillman, Apparent Mobility Enhancement in Inhomogeneous Crystals Robert L. Petersen, The Magnetophonon Effect

448

CONTENTS OF VOLUMES IN THISSERIES

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 Infrared Detectors (II) W. L. Eiseman, J.D. Merriam, and R. F. Poner, Operational Characteristicsof Infrared Photodetectors Peter R. Bratt. Impurity Germanium and Silicon Infrared Detectors E. H. Putley, InSb Submillimeter Photoconductive Detectors G. E. Stillman, C. M. Wolfe, 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. Putley, The Pyroelectric Detector-An Update

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

Volume 14 Lasers, Junctions, Transport N. Holonyak, Jr.and M. H. Lee, Photopumped 111-V Semiconductor Lasers Henry Kressel and Jerome K. Butler, Heterojunction Laser Diodes A Van der Ziel, Space-Charge-Limited Solid-state Diodes Peter 1. Price, Monte Carlo Calculation of Electron Transport in Solids

Volume 15 Contacts, Junctions, Emitters B. L. Sharma, Ohmic Contacts to 111-V Compounds Semiconductors Allen Nussbaum, The Theory of Semiconducting Junctions John S. Escher, NEA Semiconductor Photoemitters

Volume 16 Defects, (HgCd)Se, (HgCd)Te Henry Kressel. The Effect of Crystal Defects on Optoelectronic Devices C. R. Whitsett, J. G. Broerman, and C. J . Summers, Crystal Growth and Properties of Hgl.,CdxSe alloys M. H. Weiler, Magnetooptical Properties of Hg,.,Cd,Te Alloys Paul W. Kruse and John G. Ready. Nonlinear Optical Effects in Hg,,Cd,Te

Volume 17 CW Processing of Silicon and Other Semiconductors James F. Gibbons, Beam Processing of Silicon Arto Lietoila, Richard B. Gold, James F. Gibbons, and Lee A . Christel,Temperature Distributions and Solid Phase Reaction Rates Produced by Scanning CW Beams

CONTENTS OF VOLUMES IN THISSERIES

449

Arto Leitoila and James F. Gibbons, Applications of CW Beam Processing to Ion Implanted Crystalline Silicon N. M. Johnson, Electronic Defects in CW Transient Thermal Processed Silicon K. F. Lee, T. J. Stultz, and James F. Gibbons, Beam Recrystallized Polycrystalline Silicon: Properties, Applications, and Techniques T. Shibata, A. Wakita, T. W . Sigmon, and James F. Gibbons, Metal-Silicon Reactions and Silicide Yves I. Nissim and James F. Gibbons, CW Beam Processing of Gallium Arsenide

Volume 18 Mercury Cadmium Telluride Paul W. Kruse, The Emergence of (Hg,.,Cd,)Te as a Modern Infrared Sensitive Material H. E. Hirsch, S. C.Liang, and A. G. White, Preparation of High-Purity Cadmium, Mercury, and Tellurium W. F. H. Micklethwaite, The Crystal Growth of Cadmium Mercury Telluride Paul E. Petersen, Auger Recombination in Mercury Cadmium Telluride R. M. Broudy and V. J.Mazurczyck, (HgCd)Te Photoconductive Detectors M. B. Reine, A. K. Soad, and T. J. Tredwell, Photovoltaic Infrared Detectors M. A. Kinch, Metal-Insulator-SemiconductorInfrared Detectors

Volume 19 Deep Levels, GaAs, Alloys, Photochemistry G. F. Neumark and K. Kosai, Deep Levels in Wide Band-Gap 111-V Semiconductors David C.Look, The Electrical and Photoelectronic Properties of Semi-Insulating GaAs R. F. Brebrick, Ching-Hua Su, and Pok-Kai Liao, Associated Solution Model for Ga-In-Sb and Hg-Cd-Te Yu.Ya. Gurevich and Yu. V. Pleskon, Photoelectrochemistry of Semiconductors

Volume 20 Semi-Insulating GaAs R. N. Thomas, H. M. Hobgood, G. W. Eldridge, D. L. Barrett, T.T. Braggins, L. B. Ta, and S. K. Wang, High-Purity LEC Growth and Direct Implantation of GaAs for Monolithic Microwave Circuits

C.A. Stolte, Ion Implantation and Materials for GaAs Integrated Circuits C.G. Kirkpatrick, R. T. Chen, D. E. Holmes, P. M.Asbeck, K. R. Elliott, R. D. Fairman, and J. R. Oliver, LEC GaAs for Integrated Circuit Applications J.S. Blakemore and S. Rahimi, Models for Mid-Gap Centers in Gallium Arsenide

Volume 21 Hydrogenated Amorphous Silicon Part A Jacques I. Pankove, Introduction Masataka Hirose, Glow Discharge; Chemical Vapor Deposition Yoshiyuki Uchida, di Glow Discharge T. D. Moustakas, Sputtering lsao Yumada, Ionized-Cluster Beam Deposition Bruce A. Scott, Homogeneous Chemical Vapor Deposition

450

CONTENTS OF VOLUMES IN THISSERIES

Frank J. Kampas,Chemical Reactions in Plasma Deposition Paul A. Longeway, Plasma Kinetics Herberr A. Weakliem, Diagnostics of Silane Glow Discharges Using Probes and Mass Spectroscopy Lester Glunman.Relation between the Atomic and the Electronic Structures A. Chenevas-Paule. Experiment Determination of Structure S. Minomura, Pressure Effects on the Local Atomic Structure David Adler, Defects and Density of Localized States

Part B Jacques I. Pankove, Introduction G. D.Cody, The Optical Absorption Edge of a-Si: H Nabil M. Amer and Warren B.Jackson, Optical Properties of Defect States in a-Si: H P. J. Zanzucchi, The Vibrational Spectra of a-Si: H Yoshihiro Hamakawa, Electroreflectance and Electroabsorption Jeffrey S. Lannin, Raman Scattering of Amorphous Si, Ge, and Their Alloys R. A. Street, Luminescence in a-Si: H Richard S. Crandall, Photoconductivity J. Tauc, Time-Resolved Spectroscopy of Electronic Relaxation Processes P. E. Vanier, IR-Induced Quenching and Enhancement of Photoconductivity and Photoluminescence H. Schade, Irradiation-Induced Metastable Effects L. Ley. Photoelectron Emission Studies

Part C Jacques I. Pankove. Introduction J. David Cohen, Density of States from Junction Measurements in Hydrogenated Amorphous Silicon P. C. Taylor, Magnetic Resonance Measurements in a-Si: H K. Morigaki, Optically Detected Magnetic Resonance J. Dresner. Camer Mobility in a-Si: H T. Tiedje, Information about band-Tail States from Time-of-Flight Experiments Arnold R. Moore, Diffusion Length in Undoped a-Si: H W. Beyer and J. Overhof, Doping Effects in a-Si: H H. Fritzche, Electronic Properties of Surfaces in a-Si: H C. R. Wronski. The Staebler-Wronski Effect R. J. Nemanich. Schottky Barriers on a-Si: H B. Abeles and T. Tiedje, Amorphous Semiconductor Superlattices

Part D Jacques 1. Pankove, Introduction D. E. Carlson, Solar Cells G. A. Swartz, Closed-Form Solution of I-V Characteristic for a a-Si: H Solar Cells IsamuShimizu, Electrophotography Sachio Ishioka, Image Pickup Tubes

CONTENTS OF VOLUMES IN THISSERIES

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P. G. LeComber and W. E. Spear, The Development of the a-Si: H Field-Effect Transistor and Its Possible Applications D.G. Ast, a-Si: H FET-Addressed LCD Panel S. Kaneko, Solid-state Image Sensor Masakiyo Matsumura, Charge-Coupled Devices M. A. Bosch, Optical Recording A. D Amicoand G. Fortunato, Ambient Sensors Hiroshi Kukimoto, Amorphous Light-Emitting Devices Robert J.Phelan, Jr., Fast Detectors and Modulators Jacques I. Pankove, Hybrid Structures P. G. LeComber, A. E. Owen, W. E. Spear, J. Hajto, and W. K. Choi, Electronic Switching in Amorphous Silicon Junction Devices

Volume 22 Lightwave Communications Technology Part A Kazuo Nakajima, The Liquid-Phase Epitaxial Growth of IngaAsp W. T.Tsang, Molecular Beam Epitaxy for 111-V Compound Semiconductors G. B.Stringfellow, Organometallic Vapor-Phase Epitaxial Growth of 111-V Semiconductors G. Beuchet, Halide and Chloride Transport Vapor-Phase Deposition of InGaAsP and GaAs Manijeh Razeghi, Low-Pressure Metallo-Organic Chemical Vapor Deposition of Ga,in,,AsP,, Alloys P. M. Petroff, Defects in 111-V Compound Semiconductors

Part B J.P. van der Ziel, Mode Locking of Semiconductor Lasers Kam Y.Lau and Ammon Yariv, High-Frequency Current Modulation of Semiconductor Injection Lasers Charles H. Henry, Special Properties o f Semiconductor Lasers Yasuharu Suematsu, Katsumi Kishino, Shigehisa Arai, and Fumio Koyama. Dynamic SingleMode Semiconductor Lasers with a Distributed Reflector W. T. Tsang, The Cleaved-Coupled-Cavity (C3) Laser

Part C R. J.Nelson and N. K. Dutta, Review o f InGaAsP InP Laser Structures and Comparison of Their Performance N. Chinoneand M . Nakamura, Mode-Stabilized Semiconductor Lasers for 0.7-0.8- and 1.1-1.6pm Regions Yoshiji Horikoshi, Semiconductor Lasers with Wavelengths Exceeding 2 pm B. A. Dean and M. Dixon, The Functional Reliability of Semiconductor Lasers as Optical Transmitters R. H. Saul, T. P. Lee, and C. A. Burus. Light-Emitting Device Design C. L. Zipfel, Light-Emitting Diode-Reliability TienPei Lee and Tingye Li, LED-Based Multimode Lightwave Systems Kinichiro Ogawa, Semiconductor Noise-Mode Partition Noise

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CONTENTS OF VOLUMES IN THISSERIES

Part D Federico Capasso, The Physics of Avalanche Photodiodes T. P. Pearsall and M. A . Pollack, Compound Semiconductor Photodiodes Taka0 Kaneda, Silicon and Germanium Avalanche Photodiodes S. R. Forrest, Sensitivity of Avalanche Photodetector Receivers for High-Bit-Rate LongWavelength Optical Communication Systems J. C. Campbell, Phototransistors for Lightwave Communications

Part E Shyh Wang, Principles and Characteristics of Integrable Active and Passive Optical Devices Shlomo Margalit and Amnon Yariv, Integrated Electronic and Photonic Devices Takaoki Mukai, Yoshihisa Yamamoto, and TatsuyaKimura, Optical Amplificationby Semiconductor Lasers

Volume 23 Pulsed Laser Processing of Semiconductors R.F. Wood, C. W. White, and R. T.Young, Laser Processing of Semiconductors: An Overview C. W. White, Segregation, Solute Trapping, and Supersaturated Alloys G. E.Jellison, Jr., Optical and Electrical Properties of Pulsed Laser-Annealed Silicon R. F. Wood and G. E. Jellison, Jr.,Melting Model of Pulsed Laser Processing R. F.Wood and F. W. Young,Jr., NonequilibriumSolidificationFollowingPulsed Laser Melting D. H. Lowndes and G. E. Jellison, Jr., Time-Resolved Measurement During Pulsed Laser Irradiation of Silicon D. M. Zebner, Surface Studies of Pulsed Laser Irradiated Semiconductors D. H. Lowndes, Pulsed Beam Processing of Gallium Arsenide R. B.James, Pulsed C 0 2 Laser Annealing of Semiconductors R. T. Young and R. F. Wood, Applications of Pulsed Laser Processing

Volume 24 Applications of Multiquanhun Wells, Selective Doping, and Superlattices C. Weisbuch, Fundamental Properties of 111-V Semiconductor Two-DimensionalQuantized Structures: The Basis for Optical and Electronic Device Applications H. Morkoc and H. Unlu,Factors Affecting the Performance of (Al, Ga)As/GaAs and (Al, Ga)As/InGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications N. T. Linh, Two-Dimensional Electron Gas FETs: Microwave Applications M. Abe et al., Ultra-High-speed HEMT Integrated Circuits D. S. Chemla, D. A. B. Miller, and P. W. Smith, Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing F. Capasso, Graded-Gap and Superlattice Devices by Band-Gap Engineering W. T. Tsang, Quantum Confinement Heterostructure Semiconductor Lasers G. C. Usbourn et al., Principles and Applications of Semiconductor Strained-Layer Superlattices

CONTENTS OF VOLUMES IN THIS SERIES

453

Volume 25 Diluted Magnetic Semiconductors W. Giriat and J. K. Furdyna, Crystal Structure, Composition, and Materials Preparation of Diluted Magnetic Semiconductors W. M. Becker, Band Structure and Optical Properties of Wide-Gap AI.,Mn,B'V Alloys at Zero Magnetic Field SaulOseroff and Pieter H. Keesum,Magnetic Properties: Macroscopic Studies Giebultowicz and T.M. Holden, Neutron Scattering Studies of the Magnetic Structure and Dynamics of Diluted Magnetic Semiconductors J. Kossut, Band Structure and Quantum Transport Phenomena in Narrow-Gap Diluted Magnetic Semiconductors C. Riquaux, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors 1.A. Caj,Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. Myciefski, Shallow Acceptors in Diluted Magnetic Semiconductors: Splitting, Boil-off, Giant Negative Magnetoresistance A. K . Ramadas and R. Rodriquez, Raman Scattering in Diluted Magnetic Semiconductors P. A. Wolff, Theory of Bound Magnetic Polarons in Semimagnetic Semiconductors

Volume 26 111-V Compound Semiconductors and Semiconductor Properties of Superionic Materials Zou Yuanxi, 111-V Compounds H. V. Winston, A. T.Hunter, H . Kimura,and R. E. Lee,InAs-Alloyed GaAs Substrates for Direct Implantation P. K. Bhattachary and S. Dhar,Deep Levels in 111-V Compound Semiconductors Grown by MBE Yu.Yu.Gurevich and A.K .Ivanov-Shifs. Semiconductor Properties of Supersonic Materials

Volume 27 High Conducting Quasi-One-Dimensional Organic Crystals E. M. Conweff, Introduction to Highly Conducting Quasi-One-Dimensional Organic Crystals 1.A. Howard,A Reference Guide to the Conducting Quasi-One-Dimensional Organic Molecular Crystals J. P. Pouquet, Structural Instabilities E. M. Conwefl, Transport Properties C. S.Jacobsen, Optical Properties J.C. Scott, Magnetic Properties L.Zuppirofi, Irradiation Effects: Perfect Crystals and Real Crystals

Volume 28 Measurement of High-speed Signals in Solid State Devices J.Freyand D. loannou, Materials and Devices for High-speed and Optoelectronic Applications H. Schumacherand E. Strid, Electronic Wafer Probing Techniques D.H. Auston, Picosecond Photoconductivity: High-speed Measurements of Devices and Materials J .A. Valdmanis, Electro-Optic Measurement Techniques for Picosecond Materials, Devices, and Integrated Circuits.

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CONTENTS OF VOLUMES I N THISSERIES

J. M. Wiesenfeld and R. K. Ja%,Direct Optical Probing of Integrated Circuits and HighSpeed Devices G. Plows, Electron-Beam Probing A. M. Weiner and R. E. Marcus, Photoemissive Probing

Volume 29 Very High Speed Integrated Circuits: Gallium Arsenide LSI M. Kuzuhara and T. Nazaki, Active Layer Formation by Ion Implantation H. Hasimotu, Focused Ion Beam Implantation Technology T. Nozaki and A. Higashisaka. Device Fabrication Process Technology M. In0 and T.Takada, GaAs LSI Circuit Design M. Hirayama, M. Ohmori, and K. Yamasaki. GaAs LSI Fabrication and Performance

Volume 30 Very High Speed Integrated Circuits: Heterostructure H. Watanabe, T. Mizutani, and A. Usui,Fundamentals of Epitaxial Growth and Atomic Layer Epitaxy S. Hiyamizu, Characteristicsof Two-DimensionalElectron Gas in 111-V Compound Heterostructures Grown by MBE T. Nukanisi, Metalorganic Vapor Phase Epitaxy for High-Quality Active Layers T. Nimura, High Electron Mobility Transistor and LSI Applications T. Sugera and T. Ishibashi, Hetero-Bipolar Transistor and LSI Application H. Marsueda, T. Tanaku, and M. Nakamura. Optoelectronic Integrated Circuits

Volume 31 Indium Phosphide: Crystal Growth and Characterization J. P. Farges, Growth of Discoloration-free InP M. J. McColIurn and G. E.Stillman, High Purity InP Grown by Hydride Vapor Phase Epitaxy T. Inada and T. Fukuda, Direct Synthesis and Growth of Indium Phosphide by the Liquid Phosphorous Encapsulated Czochralski Method 0. Oda, K. Katagiri, K. Shinuhara, S. Katsuru, Y. Takahashi, K. Kainosho, K. Kohiro, and R. Hirano. InP Crystal Growth, Substrate Preparation and Evaluation K. Tuda, M. Tatsumi, M. Morioka, T. Araki, and T. Kawase, InP Substrates: Production and Quality Control M. Razeghi, LP-MOCVD Growth, Characterization, and Application of InP Material T. A. Kennedy and P. J. Lin-Chung, Stoichiometric Defects in InP

Volume 32 Strained-Layer Superlattices: Physics T.P. Pearsall, Strained-Layer Superlattices Fred H. Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors J .Y. Marzin, J. M . Gerrird, P. Voisin. and 1.A. Brum, Optical Studies of Strained Ill-V Heterolayers R. People and S. A. Jackson, Structurally Induced States from Strain and Confinement M. Juros, Microscopic Phenomena in Ordered Superlattices

CONTENTS OF VOLUMESI N THISSERIES

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Volume 33 Strained-Layer Superlattices: Materials Science and Technology R. Hull and J. C. Bean, Principles and Concepts of Strained-Layer Epitaxy William J.Schaff; Paul J. Tasker, Marc C. Foisy, and Lester F. Eastman, Device Applications of Strained-Layer Epitaxy S. T. Picraux, B. L. Doyle, and J . Y. Tsao, Structure and Characterization of StrainedLayer Superlattices E. Kasper and F. Schaffer, Group IV Compounds Dale L . Martin, Molecular Beam Epitaxy of IV-VI Compounds Heterojunction Robert L. Gunshor, Leslie A. Kolodziejski, Arto V. Nurmikko, and Nobuo Otsuka, Molecular Beam Epitaxy of 11-VI Semiconductor Microstructures

Volume 34 Hydrogen in Semiconductors J.1. Pankove and N. M. Johnson, Introduction to Hydrogen in Semiconductors C. H. Seager, Hydrogenation Methods J.I. Pankove, Hydrogenation of Defects in Crystalline Silicon J . W . Corbetf, P. Deak, U. V. Desnica, and S. J. Pearton, Hydrogen Passivation of Damage Centers in Semiconductors S. J.Pearton, Neutralization of Deep Levels in Silicon J.I. Pankove, Neutralization of Shallow Acceptors in Silicon N. M. Johnson, Neutralization of Donor Dopants and Formation of Hydrogen-Induced Defects in n-Type Silicon M. Stavola and S. J .Pearton, Vibrational Spectroscopy of Hydrogen-Related Defects in Silicon A. D. Marwick, Hydrogen in Semiconductors: Ion Beam Techniques C. Herring and N. M. Johnson, Hydrogen Migration and Solubility in Silicon E. E. Haller, Hydrogen-Related Phenomena in Crystalline Germanium J. Kakalios, Hydrogen Diffusion in Amorphous Silicon J.Chevalier, B. Clerjaud, and B. Pajot, Neutralization of Defects and Dopants in Ill-V Semiconductors G. G. DeLeo and W . B. Fowler, Computational Studies of Hydrogen-Containing Complexes in Semiconductors R. F. Kiefl and T. L. Estle, Muonium in Semiconductors C. G. Van de Walle,Theory of Isolated Interstitial Hydrogen and Muonium in Crystalline Semiconductors

Volume 35 Nanostructured Systems Mark Reed, Introduction H. van Houten, C. W. J.Beenakker, and B. J. van Wees, Quantum Point Contacts G. Timp, When Does a Wire Become an Electron Waveguide? M. Butfiker, The Quantum Hall Effects in Open Conductors W . Hansen, J.P. Kotthaus, and U.Merkr, Electrons in Laterally Periodic Nanostructures

Volume 36 The Spectroscopy of Semiconductors D. Heiman, Spectroscopy of Semiconductors at Low Temperatures and High Magnetic Fields Arto V. Nurmikko, Transient Spectroscopy by Ultrashort Laser Pulse Techniques

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CONTENTS OF VOLUMES IN THISSERIES

A. K. Ramdas and S. Rodriguez, Piezospectroscopy of Semiconductors Orest J. Glembocki and Benjamin V. Shanabrook, Photoreflectance Spectroscopy of Micro-

structures David G.Seiler, Christopher L. Littler, and Margaret H. Wiler. One- and Two-Photon MagnetoOptical Spectroscopy of InSb and Hg,.,Cd,Te

Volume 37 The Mechanical Properties of Semiconductors A.-B. Chen, Arden Sher and W. T.Yost, Elastic Constants and Related Properties of Semiconductor Compounds and Their Alloys David R. Clarke, Fracture of Silicon and Other Semiconductors Hans Siethoff, The Plasticity of Elemental and Compound Semiconductors Sivaraman Guruswamy, Katherine T. Faber and John P.Hirih, Mechanical Behavior of Compound Semiconductors Subhanh Mahajan, Deformation Behavior of Compound Semiconductors John P. Hirih, Injection of Dislocations into Strained Multilayer Structures Don Kendall, Charles B. Fleddermann, and Kevin J. Malloy, Critical Technologies for the Micromachining of Silicon Ikuo Matsuba and Kinji Mokuya, Processing and Semiconductor Thermoelastic Behavior

Volume 38 Imperfections in IIW Materials Udo Scherz and Matihias Schefflr, Density-FunctionalTheory of sp-Bonded Defects in IIIl V Semiconductors Maria Kaminska and Eicke R. Weber, El2 Defect in GaAs David C. Look, Defects Relevant for Compensation in Semi-Insulating GaAs R. C. Newman, Local Vibrational Mode Spectroscopy of Defects in IIIN Compounds Andrzej M. Hennel, Transition Metals in IIIN Compounds Kevin J. Malloy and Ken Khachaturyan, DX and Related Defects in Semiconductors V. Swaminathan and Andrew S. Jordan, Dislocations in IIIN Compounds Krzysztof W . Nauka, Deep Level Defects in the Epitaxial IIIN Materials

Volume 39 Minority Carriers in IU-V Semiconductors: Physics and Applications Niloy K. Duna, Radiative Transitions in GaAs and Other 111-V Compounds Richard K. Ahrenkiel, Minority-Camer Lifetime in 111-V Semiconductors Tomofumi Furuta, High Field Minority Electron Transport in p-GaAs Mark S. Lundstrom, Minority-Camer Transport in 111-V Semiconductors Richard A. Abram, Effects of Heavy Doping and High Excitation on the Band Structure of GaAs David Yevick and Wiiold Bardyszewski, An Introduction to Non-Equilibrium Many-Body Analyses of Optical Processes in 111-V Semiconductors

Volume 40 Epitaxial Microstructures E.F.Schuber?, Delta-Dopingof Semiconductors:Electronic, Optical, and Structural Properties of Materials and Devices

CONTENTS OF VOLUMES IN THISSERIES

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A. Gossard, M. Sundaram, and P. Hopkins, Wide Graded Potential Wells P. Perroff, Direct Growth of Nanometer-Sue Quantum Wire Superlattices E. Kapon, Lateral Patterning of Quantum Well Heterostructures by Growth of Nonplanar Substrates H. Temkin, D. Gershoni, and M. Punish, Optical Properties of Gal-,In,As/InP Quantum Wells

Volume 41 High Speed Heterostructure Devices F. Capasso, F. Beltram, S. Sen, A. Pahlevi, and A. Y. Cho, Quantum Electron Devices: Physics and Applications P. Solomon, D. J.Frank, S.L. Wright, and F. Canora, GaAs-Gate Semiconductor-InsulatorSemiconductor FET M. H. Hashemi and U.K. Mishra, Unipolar InP-Based Transistors R. Kiehl, Complementary Heterostructure FET Integrated Circuits T. Ishibashi, GaAs-Based and InP-Based Heterostructure Bipolar Transistors H. C. Liu and T. C. L. C.Sollner, High-Frequency-TunnelingDevices H. Ohnishi, T. More, M. Takarsu, K . Imamura, and N. Yokoyama, Resonant-Tunneling HotElectron Transistors and Circuits

Volume 42 Oxygen in Silicon F. Shimura, Introduction to Oxygen in Silicon W. Lin,The Incorporation of Oxygen into Silicon Crystals T. J.Schaffner and D. K. Schroder, Characterization Techniques for Oxygen in Silicon W . M. Bullis, Oxygen Concentration Measurement S. M. H u , Intrinsic Point Defects in Silicon B. Pajor, Some Atomic Configurations of Oxygen J.Michel and L. C. Kimerling, Electical Properties of Oxygen in Silicon R. C.Newman and R. Jones, Diffusion of Oxygen in Silicon T. Y. Tan and W . J.Taylor, Mechanisms of Oxygen Precipitation: Some Quantitative Aspects M. Schrems, Simulation of Oxygen Precipitation K. Simino and 1. Yonenaga, Oxygen Effect on Mechanical Properties W. Bergholz, Grown-in and Process-Induced Effects F. Shimura, Intrinsichternal Gettering H. Tsuya, Oxygen Effect on Electronic Device Performance

Volume 43 Semiconductors for Room Temperature Nuclear Detector Applications R. B. James and T. E. Schlesinger, Introduction and Overview L. S. Darken and C.E. Cox, High-Purity Germanium Detectors A. Burger, D. Nason, L. Van den Berg, and M. Schieber, Growth of Mercuric Iodide X. J.Bao, T. E. Schlesinger, and R. B. James, Electrical Properties of Mercuric Iodide X.J.Bao, R. B. James, and T. E. Schlesinger, Optical Properties of Red Mercuric Iodide M. Huge-Ali and P. Siffert, Growth Methods of CdTe Nuclear Detector Materials M. Huge-Ali and P Sifferf, Characterization of CdTe Nuclear Detector Materials M. Huge-Ali and P. Siffert, CdTe Nuclear Detectors and Applications R. B. James, T. E. Schlesinger, 1.Lund, and M. Schieber, Cdl.,ZnxTe Spectrometers for Gamma and X-Ray Applications

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CONTENTS OF VOLUMES IN THISSERIES

D. S.McGregor, J.E.Kammeraad, Gallium Arsenide Radiation Detectors and Spectrometers J . C. Lund, F. Olschner, and A. Burger, Lead Iodide M. R. Squillante, and K. S. Shah, Other Materials: Status and Prospects V.M. Gerrish, Characterization and Quantification of Detector Performance J.S. Iwancryk and B. E. Pan, Electronics for X-ray and Gamma Ray Spectrometers M. Schieber, R. B. James, and T. E. Schlesinger, Summary and Remaining Issues for Room Temperature Radiation Spectrometers

Volume 44 11-IV Blue/Green Light Emitters: Device Physics and Epitaxial Growth J.Han and R. L. Gunshor, MBE Growth and Electrical Properties of Wide Bandgap ZnSebased 11-VI Semiconductors Shizuo Fujifa and Shzgeo Fujita, Growth and Characterization of ZnSe-based 11-VI Semiconductors by MOVPE Easen Ho and Leslie A. Kolodziejski, Gaseous Source UHV Epitaxy Technologies for Wide Bandgap 11-VI Semiconductors Chris G. Van de Walle, Doping of Wide-Band-Gap 11-VI Compounds-Theory Roberro Cingolani, Optical Properties of Excitons in ZnSe-Based Quantum Well Heterostructures A. Ishibashiand A. V. Nurmikko. 11-VI Diode Lasers: A Current View of Device Performance and Issues Supratik Guha and John Petruzello, Defects and Degradation in Wide-Gap 11-VI-based Structures and Light Emitting Devices

Volume 45 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Electrical and Physiochemical Characterization Heiner Ryssel. Ion Implantation into Semiconductors: Historical Perspectives You-Nian Wang and Teng-Cai Ma. Electronic Stopping Power for Energetic Ions in Solids Sachiko T. Nakagawa, Solid Effect on the Electronic Stopping of Crystalline Target and Application to Range Estimation G. Miiiler. S.Kalbitzer and G. N. Greaves. Ion Beams in Amorphous Semiconductor Research Jumana Boussey-Said, Sheet and Spreading Resistance Analysis of Ion Implanted and Annealed Semiconductors M.L. Polignano and G. Queirolo, Studies of the Stripping Hall Effect in Ion-Implanted Silicon J.Sroemenos, Transmission Electron Microscopy Analyses Roberta Nipori and Marco Servidori, Rutherford Backscattering Studies of Ion Implanted Semi-conductors P. Zaumseil. X-ray Diffraction Techniques

Volume 46 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Optical and Photothermal Characterization M. Fried, T. Lohner and J. Gyulai, Ellipsometric Analysis Anionios Seas and Constantinos Christofides. Transmission and Reflection Spectroscopy on Ion Implanted Semiconductors

CONTENTS OF VOLUMES IN THISSERIES

459

Andreas Othonos and Constantinos Christofides, Photoluminescence and Raman Scattering of Ion Implanted Semiconductors. Influence of Annealing Constantinos Chrkrofides, Photomodulated Thermoreflectance Investigation of Implanted Wafers. Annealing Kinetics of Defects U. Zammit, Photothermal Deflection Spectroscopy Characterization of Ion-Implanted and Annealed Silicon Films Andreas Mandelis, Arief Budiman and Miguel Vargas, Photothermal Deep-Level Transient Spectroscopy of Impurities and Defects in Semiconductors R. Kafish and S. Charbonneau, Ion Implantation into Quantum-Well Structures Alexandre M. Myasnikov and Nikolay N. Gerasirnenko, Ion Implantation and Thermal Annealing of 111-V Compound Semiconducting Systems: Some Problems of 111-V Narrow Gap Semiconductors

Volume 47 Uncooled Infrared Imaging Arrays and Systems R. G. Buser and M. P. Tompsett, Historical Overview P. W. Kruse, Principles o f Uncooled Infrared Focal Plane Arrays R. A. Wood, Monolithic Silicon Microbolometer Arrays C. M. Hanson, Hybrid Pyroelectric-Ferroelectric Bolometer Arrays D. L. Polla and J. R. Choi, Monolithic Pyroelectric Bolometer Arrays N. Teranishi, Thermoelectric Uncooled Infrared Focal Plane Arrays M. F. Tompsett, Pyroelectric Vidicon T. W. Kenny, Tunneling Infrared Sensors J.R. Vig, R. L.Filler and Y. Kim, Application of Quartz Microresonators to Uncooled Infrared Imaging Arrays P. W. Kruse, Application of Uncooled Monolithic Thermoelectric Linear Arrays to Imaging Radiometers

Volume 48 High Brightness Light Emitting Diodes G. B. Stringfellow, Materials Issues in High-Brightness Light-Emitting Diodes M. G. Craford, Overview of Device issues in High-Brightness Light-Emitting Diodes F. M. Sreranka, AlGaAs Red Light Emitting Diodes C. H. Chen, S. A. Stockman, M. J.Peanasky, and C. P. Kuo, OMVPE Growth of AlGaInP for High Efficiency Visible Light-Emitting Diodes F. A. Kish and R. M. Fletcher, AlGaInP Light-Emitting Diodes M. W. Hodapp, Applications for High Brightness Light-Emitting Diodes I. Akasaki and H. Amano, Organometallic Vapor Epitaxy of GaN for High Brightness Blue Light Emitting Diodes S. Nakamura, Group 111-V Nitride Based Ultraviolet-Blue-Green-Yellow Light-Emitting Diodes and Laser Diodes

Volume 49 Light Emission in Silicon: from Physics to Devices David J. Lockwood, Light Emission in Silicon Gerhard Absrreiter, Band Gaps and Light Emission in SifSiGe Atomic Layer Structures Thomas G. Brown and Dennis G. Hall, Radiative Isoelectronic Impurities in Silicon and Silicon-Germanium Alloys and Superlattices J.Michel, L. V. C. Assali, M. T. Morse, and L. C. Kimerling, Erbium in Silicon

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CONTENTS OF VOLUMES IN THISSERIES

Yoshihiko Kanemitsu, Silicon and Germanium Nanoparticles Philippe M. Faucher, Porous Silicon: Photoluminescence and Electroluminescent Devices C. Delerue, G. Allan, and M. Lannoo, Theory of Radiative and Nonradiative Processes in Silicon Nanocrystallites Louis Brus,Silicon Polymers and Nanocrystals

Volume 50 Gallium Nitride (GaN) J. I. Pankove and T. D. Moustakas, Introduction S. P.DenBaars and S. Keller, Metalorganic Chemical Vapor Deposition (MOCVD) of Group I11 Nitrides W. A. Bryden and T. J.Kistenmacher, Growth of Group 111-A Nitrides by Reactive Sputtering N. Newman, Thennochemistry of 111-N Semiconductors S. J. Pearton and R. J. Shul,Etching of 111 Nitrides S. M. Bedair, Indium-based Nitride Compounds A. Tramperr, 0. Brandr, and K. H. Ploog, Crystal Structure of Group I11 Nitrides H. Morkoc, F. Hamdani, and A. Salvador, Electronic and Optical Properties of 111-V Nitride based Quantum Wells and Superlattices K. Doverspike and J. I. Pankove, Doping in the 111-Nitrides T. Suski and P. Perlin, High Pressure Studies of Defects and Impurities in Gallium Nitride B. Monemar, Optical Properties of GaN W. R. L.Lambrecht, Band Structure of the Group 111 Nitrides N. E. Christensen and P. Perlin, Phonons and Phase Transitions in GaN S. Nakamura, Applications of LEDs and LDs I. Akasaki and H. Amano, Lasers J.A. Cooper, Jr., Nonvolatile Random Access Memories in Wide Bandgap Semiconductors

Volume 51A Identification of Defects in Semiconductors George D. Warkins, EPR and ENDOR Studies of Defects in Semiconductors J.-M. Spaerh, Magneto-Optical and Electrical Detection of Paramagnetic Resonance in Semiconductors T.A. Kennedy and E. R. Glaser, Magnetic Resonance of Epitaxial Layers Detected by Photoluminescence K. H. Chow, B.Hirri, and R. F. Kiep, 'kSR on Muonium in Semiconductorsand Its Relation to Hydrogen Kimmo Saarinen, Pekka Haurojarvi, and Catherine Corbel, Positron Annihilation Spectroscopy of Defects in Semiconductors R. Jones and P. R. Briddon. The A b Inirio Cluster Method and the Dynamics of Defects in Semiconductors

Volume 51B Identification of Defects in Semiconductors Gordon Davies, Optical Measurements of Point Defects P. M. Mooney, Defect Identification Using Capacitance Spectroscopy MichaelStavola, Vibrational Spectroscopy of Light Element Impurities in Semiconductors P. Schwander, W . D. Rau, C. Kisielowski, M. Gribelyuk, and A. Ourmazd, Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy Nikos D. Jager and Eicke R. Weber, Scanning Tunneling Microscopy of Defects in Semiconductors

CONTENTS OF VOLUMES IN Tnrs SERIES

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Volume 52 Sic Materials and Devices K. Jarrendahl and R. E Davis, Materials Properties and Characterization of Sic V. A. Dmitriev and M. G. Spencer, SIC Fabrication Technology: Growth and Doping V. Saxena and A. J. Steckl, Building Blocks for Sic Devices: Ohmic Contacts, Schottky Contacts, and p-n Junctions M. S. Shur, Sic Transistors C. D. Brand!, R. C. Clarke, R. R. Siergiej, J. B. Casady, A. W . Morse, S. Sriram, and A. K. Agarwal, S i c for Applications in High-Power Electronics R. J. Trew, SIC Microwave Devices J.Edmond, H. Kong, G. Negley, M. Leonard, K. Doverspike, W . Weeks, A. Suvorov, D. Waltz, and C. Carter, Jr., Sic-Based UV Photodiodes and Light-Emitting Diodes H. Morkoc, Beyond Silicon Carbide! 111-V Nitride-Based Heterostructures and Devices

Volume 53 Cumulative Index Volume 54 High Pressure in Semiconductor Physics I William Paul, High Pressure in Semiconductor Physics: A Historical Overview N. E. Christensen, Electronic Structure Calculations for Semiconductors under Pressure R. J.Nelmes and M. I.McMahon, Structural Transitions in the 111-V and 11-VI and GroupIV Semiconductors under Pressure A. R. Goni and K. Syassen, Optical Properties of Semiconductors under Pressure P. Traurman, M. Baj, and J. M. Baranowski, Hydrostatic Pressure and Uniaxial Stress in Investigations of the EL2 Defect in GaAs M. L i and P. Y. Y u , High Pressure Study of DX Centers Using Capacitance Techniques T. Suski, Spatial Correlations of Impurity Charges in Doped Semiconductors N. Kuroda, Pressure Effects on the Electronic Properties of Diluted Magnetic Semiconductors

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