Solid State Physics: Advances in Research and Applications, Vol. 46

SOLID STATE PHYSICS VOLUME 46

Founding Editors FREDERICK SEITZ DAVID TURNBULL

SOLID STATE PHYSICS Advances in Research and Applications

Editors HENRY EHRENREICH DAVID TURNBULL

Division of Applied Sciences Harvard University, Cambridge, Massachusetts

VOLUME 46

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

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ISBN 0-12-607746-0 ISSN 0081-1947 PRINTED IN THE UNITED STATES OF AMERICA 92 93 94 95 96 91 EB 9

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Contents

CONTRIBUTORS TO VOLUME 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE .............................................

vii ix

Band Offsets In Semiconductor Heterojunctlons

I. I1. 111.

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

EDWARD T . Yu. JAMES0. MCCALDIN.AND THOMAS C. MCGILL Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theories and Empirical Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IIILV Material Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-VI Material Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain Effects in Lattice-Mismatched Heterojunctions . . . . . . . . . . . . . Heterovalent Material Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

2 8 26 33 59 74 103 130 145 146

Physical Properties of Macroscopically lnhornogeneous Medla

DAVID J. BERGMAN AND DAVIDSTROUD 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. DC Electrical Properties-General Theory and Calculational Techniques . 111. DC Electrical Properties-Applications to Specific Problems . . . . . . . . .

. .

IV . Electromagnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Nonlinear Properties and Flicker Noise . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148 150 192 220 246 269

Fundamental Magnetlzatlon Processes in Thln-Film Recording Medla

H . NEALBERTRAM AND JIAN-GANG ZHU I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Microstructures and Magnetic Properties of Thin-Film Materials . . . . . . . 111. Micromagnetic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Reversal Processes and Domain Structures . . . . . . . . . . . . . . . . . . . . V . Simulations of the Magnetic Recording Process . . . . . . . . . . . . . . . . . VI . Self-organized Behavior in Magnetic Systems. . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 281 299 320

345 362 371

CONTENTS INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AUTHOR SUBJECT INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVE

373 391 399

Contributors

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

DAVIDJ. BERGMAN (147), Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel H. NEALBERTRAM (271), Department of Electrical and Computer Engineering and Center for Magnetic Recording Research, University of California at Sun Diego, La Jolla, C A 92003 JAMES 0. MCCALDIN (l), California Institute of Technology, Pasadena, California 91125

THOMAS C. MCGILL (I), California Institute of Technology, Pasadena, California 91125 DAVIDSTROUD(147), Department of Physics, The Ohio State University, Columbus, Ohio 43210-1106 EDWARD T . YU (l), California Institute of Technology, Pasadena, California 91125 JIAN-GANG ZHU (271), Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455

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Preface

This volume contains three articles, each addressed to the physics underlying a different area of materials science and technology. The values and systematic trends of band offsets in semiconductor heterostructures, that is, the energy separation between the valence and conduction bands of the bulk semiconductors constituting a heterostructure, are discussed in the article by Yu, McCaldin and McGill. They are crucial to the materials choices for electronic and optoelectronic micro- and nanodevices. An understanding of the electrical and electromagnetic properties of composites, the subject of the article by Bergman and Stroud, is important in a variety of applications ranging from high temperature superconductivity to the exploration of geological porous sedimentary rocks for oil. And a detailed comprehension of static and dynamic magnetization phenomena in thin film granular magnetic materials, as described by Bertram and Zhu, is prerequisite to the ever increasing recording densities (a doubling every three years) in commercial computers. Band offset energies are used to characterize many of the physical aspects of an interface separating two semiconductors. The conceptualization of an interface represents an idealization of its real structure and its effects, which can be extremely complex. The notion of an abrupt discontinuity of the band edges is inconsistent with the uncertainty principle. Its effects, further complicated by structural and chemical changes, extend over a small number of unit cells. Nevertheless, the model of a well-defined band offset has proved to be extremely useful in predicting the performance of a given device structure. An excellent example is provided in the article concerning electronic states in semiconductor heterostructures by Bastard and collaborators in Volume 44.A reliable yet relatively simple way to obtain band offset value for heterostructures involving a variety of different semiconductors is of great importance. As Yu, McCaldin, and McGill point out, however, the goal is elusive. A number of current theories seem to yield band offset values in reasonable agreement with experiment, even though the physical ideas underlying these theories can be quite different. These ideas include electron affinities, Schottky barrier heights, bulk band structures on the same energy scale and the definition of effective midgap energies corresponding to charge neutrality for each bulk constituent. None of these ideas is incorrect. They are probably all important in some way as is the chemically and growth-induced interface morphology. ix

X

PREFACE

The authors therefore adopt an approach that they describe as “enlightened empiricism.” In this review they examine a variety of theoretical and experimental approaches and develop a balanced perspective that leads to new insights which the existing literature has not made evident. The information base provided here, together with the results for a wide range of semiconductors, will remain useful for the foreseeable future. Macroscopically inhomogeneous media, the subject of the article by Bergman and Stroud, are composites or granular or porous materials that display inhomogeneity on a macroscopic scale. In such materials there are small, yet much larger than atomic, regions exhibiting macroscopic homogeneity. Different regions may display quite different properties. Examples considered in this article include metal-insulator composites, normal-superconducting metal composites and brine-saturated porous rocks. Research interest in these materials has increased enormously in recent years. This increase is associated with the ever greater technological importance of synthetic composites, the search for more efficient techniques for oil exploration, and also with the discovery of new physical phenomena. These include the quantum Hall effect, anomalous diffusion near the percolation threshold of metal-insulator composites, and high temperature superconductivity which frequently occurs in polycrystalline anisotropic ceramic materials. This article’s focus is on the dc and ac electrical properties of composites. As the authors point out, the theoretical approaches discussed here are useful in the description of some other physical properties such as magnetic permeability and thermal conductivity. These connections are pointed out specifically. Mechanical properties such as elastic stiffness are not included here. In view of already existing reviews, most notably that by Landauer (Reference 1 of their article), which describe the development of the field to the latter 1970s, the authors stress more recent developments. Emphasis is given to aspects that are still under development, so that those entering the field will have a sense of the areas of opportunity. The overall perspective is theoretical. Experimental results are presented by way of illustration of basic ideas. On the whole, the article presents a balanced overview containing enough of the fundamentals to make it accessible to experimentalists and theorists alike. The article begins with a review of the basic theory for the dc electrical properties with emphasis on the calculation of the equivalent bulk effective dielectric constant and conductivity. The discussion includes both exact results and the more mathematically accessible and physically approachable mean field approach termed the effective medium approximation (EMA). The electrical behavior of composites near the percolation threshold has

PREFACE

xi

attracted considerable attention in recent years. The sketch of percolation theory, a subject which is discussed extensively in many reviews and books, suffices to convey an understanding of mean field critical exponents. The general results obtained from renormalization group calculations and measurements are clearly delineated for random resistor networks (RRNs) of various sorts and continuum systems. The dc conductivity of a continuum composite can be described by RRN models provided they are carefully constructed. Other physical properties of interest discussed in this context include magnetotransport, thermoelectricity and superconductivity. The discussion of the electromagnetic or ac electrical properties includes applications to metal-insulator composites at low concentrations, superconducting composites and anisotropic media, for example, intercalated graphite and quasi-linear organic conductors. These are considered both in the context of the quasi-static approximation (in which the inductive term in Faraday’s law can be neglected) and in the context of more refined approximation schemes. The generalization of the Maxwell-Garnett or effective field approaches is also discussed. The article concludes with a discussion of two types of nonlinear effects and llfnoise. The first nonlinear effect is associated with departures from the linear relationship between the electric and displacement fields occurring in lasers and ceramic varistors; the second with dielectric breakdown. Flicker or llf noise appears to be more sensitive than the ohmic conductivity to the details of the microstructure. As the authors note, this field has begun to receive attention only recently and is therefore ripe with research opportunities. The final article by Bertram and Zhu considers both the physics underlying magnetization processes in thin-film recording media and the results of numerical simulations that exhibit some of the complexities of the magnetic recording technology. Because of its strong links to high performance computing, this industry has become very large. The relative absence of emphasis, at least in this country, on more basic, but nevertheless goaloriented, research is therefore surprising. The contents of the Solid State Physics series reflects trends in the field even though it aims to cover scientific advances, whether trendy or not, both comprehensively and authoritatively. In spite of this, surveys on magnetism have not appeared in the series since the 1960s. Most notable from this period were articles by P. W. Anderson and C. Kittel on magnetic and indirect exchange interactions respectively (Volumes 14 and 22) and another by B. R. Cooper surveying magnetic properties of rare earth metals (Volume 21). Important articles concerning dilute magnetic alloys, the Kondo effect and heavy fermion metals, which have also appeared in the series, are peripheral to Magnetism, as defined by the four volume treatise by that name edited by

xii

PREFACE

G. T. Rado and H. Suhl and published by Academic Press during this same period. The final article of this volume is therefore particularly welcome after this long hiatus. It focuses on magnetic hysteresis, reversal processes and domain patterns in hard magnetic materials consisting of closely packed thin film crystallites utilized as magnetic recording media. The basic physics problem, which is explored in considerable detail, is understanding the effects of longrange magnetostatic and short-range exchange fields on assemblies of anisotropic grains. As the authors point out, these interactions lead to very complicated magnetization processes and pattern fluctuations whose understanding requires large-scale numerical simulation. The dramatic increase in recording densities (exceeding both optic and magneto-optic recording) has come about in large part as a result of the development of the requisite thin film technology and the understanding of the underlying physics. The authors provide concise overviews of the magnetic write and read process and of the basic magnetism required for understanding it. The microstructures and magnetic properties of both longitudinal and transverse thin film materials are discussed in detail, as are the single and multple particle reversal mechanisms and the domain structures. The simulations exhibit a fascinating interplay between physics and complexity. For example, in a typical in-plane isotropic film, nucleation of magnetization reversal occurs by vortex formation. The expansion of reversed regions during hysteresis is achieved through vortex motion. The vortices are better defined if the magnetostatic interaction strength is large relative to the grain anisotropy ; they are larger and more distantly separated if the intergranular exchange coupling is large. Strongly interacting assemblies of magnetic grains exhibit self-organized behavior. Because of its emphasis on both intrinsically interesting and technologically important physics this volume should be useful to many members of the emerging broad interdisciplinary scientific community. Henry Ehrenreich David Turnbull

SOLID STATE PHYSICS. VOLUME 46

Band Offsets in Semiconductor Heterojunctions EDWARD T . Yu JAMES 0. MCCALDIN THOMAS C. MCGILL California Institute of Technology Pasadena. California

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Theories and Empirical Rules . . . . . . . . . . . . . . . . . . . . . . 1. Empirical Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Band Offsets Calculated as Bulk Parameters . . . . . . . . . . . 3. Self-Consistent Calculations for Specific Interfaces . . . . . . . 4. Comparisons among Theories . . . . . . . . . . . . . . . . . . . 111. Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 6. XPS and Related Techniques . . . . . . . . . . . . . . . . . . . . 7. Electrical Measurements . . . . . . . . . . . . . . . . . . . . . . . IV . 111-V Material Systems. . . . . . . . . . . . . . . . . . . . . . . . . . 8. GaAs/AlAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. InGaAs/InAlAs/InP . . . . . . . . . . . . . . . . . . . . . . . . . 10. InAs/GaSb/AlSb . . . . . . . . . . . . . . . . . . . . . . . . . . . V. 11-VI Material Systems. . . . . . . . . . . . . . . . . . . . . . . . . . 11. HgTe/CdTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. CdSe/ZnTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Other 11-VI Heterojunctions . . . . . . . . . . . . . . . . . . . VI. Strain Effects in Lattice-Mismatched Heterojunctions . . . . . . . . 14. Influence of Strain on Electronic Structure . . . . . . . . . . . 15. Critical Thickness for Strain Relaxation . . . . . . . . . . . . . 16. Si/Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. InGaAs/GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. Other Lattice-Mismatched Heterojunctions . . . . . . . . . . . VII. Heterovalent Material Systems . . . . . . . . . . . . . . . . . . . . . 19. GaAs/Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. GaAs/ZnSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. AlSb/GaSb/ZnTe . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Other Heterovalent Interfaces . . . . . . . . . . . . . . . . . . . VIII . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Theory versus Experiment in Lattice-Matched Heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. Unresolved Issues . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 8 8 12 21 24 26 21 29 31 33 33 41 52 59 60 66 69 74 14 78 81 92 98 103 103 109 114 119 130 131 143 144 145 146

1 Copyright 01992 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-607746-0

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E.T. YU. J.O. McCALDIN AND T.C. McGILL

1. Introduction

Investigations concerning the physics and applications of semiconductor heterojunctions have resulted in tremendous progress since the initial theoretical proposals of G ~ b a n o v , ' - Shockley,4 ~ and Kroemer' over 30 years ago. The use of heterojunctions in the design and fabrication of semiconductor devices has yielded dramatic improvements in the performance attainable with existing semiconductor device concepts5-' and has led to the development of a wealth of new structures that could not have been realized using simple homojunction techn~logy.~-~' Among the most important physical parameters for a given heterojunction system are the conduction- and valence-band offsets; indeed, the quality and even the feasibility of heterojunction device concepts often depend crucially on the values of these band offsets. As shown in Fig. 1, the band offset is defined simply as the discontinuity in the band edge at the interface between two semiconductors. Current epitaxial crystal growth techniques such as molecular-beam epitaxy (MBE) are capable of producing abrupt, atomically sharp heterojunction interfaces; in addition, theoretical calculations indicate that the electronic structure in each layer of a heterojunction becomes very nearly bulklike even a single atomic layer away from the interface, lending credence to the idealized notion of an abrupt band-edge discontinuity. Figure 1 also shows the effects of electrostatic band bending that occurs because of charge redistribution near a heterojunction interface. Figure 2 shows the various types of band alignments that can arise in semiconductor interfaces: type I, type I1 staggered, type I1 broken-gap (or misaligned), and

'A. I. Gubanov, Zh. Tekh. Fiz. 21, 304 (1951). 2A. I. Gubanov, Zh. Eksp. Teor. Fiz. 21, 721 (1951). 3 A . I. Gubanov, Zh. Tekh. Fiz. 22, 729 (1952). 4W. Shockley, US. Patent 2,569,347 (1951). 5H. Kroemer, Proc. IRE 45, 1535 (1957). 6H. Kroemer, Proc. ZEEE 70, 13 (1982) W. P. Dumke, J. M. Woodall, and V. L. Rideout, Solid-State Electron. 15, 1339 (1972). H. C. Casey and M. B. Panish, "HeterostructureLasers," Academic Press, New York, 1978. R. Tsu and L. Esaki, Appl. Phys. Lett. 22, 562 (1973). loL. L. Chang, L. Esaki, and R. Tsu, Appl. Phys. Lett. 24, 593 (1974). "G. A. Sai-Halasz, R. Tsu, and L. Esaki, Appl. Phys. Lett. 30,651 (1977). "J. N. Schulman and T. C. McGill, Appl. Phys. Lett. 34, 663 (1979). 13D. L. Smith, T. C. McGill, and J. N. Schulman, Appl. Phys. Lett. 43, 180 (1983). I4G. C. Osbourn, J . Appl. Phys. 53, 1586 (1982). 15L.Esaki, ZEEE J. Quantum Electron. QE-22, 1611 (1986). 16F.Capasso, Annu. Rev. Muter. Sci.16, 263 (1986). "F. Capasso, K. Mohammed, and A. Y. Cho, IEEE J. Quantum Electron. QE-22,1853 (1986). 18D.L. Smith and C. Mailhiot, J. Appl. Phys. 62, 2545 (1987).

'

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

Ec--_Ec Ef----------------.

Ev->v

3

Ef

Ev

FIG. 1. Conduction- and valence-band offsets in a semiconductor heterojunction. E, and E , are the conduction- and valence-band edges, respectively, and E , is the Fermi level. The band offsets AEc and AEv are abrupt discontinuities in the band edges at the heterojunction interface. Electrostatic band bending also occurs because of charge redistribution near the heterojunction interface.

type 111. A type I alignment, in which the band gap of one semiconductor lies completely within the band gap of the other, occurs in a large number of heterojunction systems, e.g., GaAs/Al,Ga, -,As and GaSb/AlSb. A type I1 staggered alignment occurs when the band gaps of the two materials overlap but one does not completely enclose the other, and it is characteristic of interfaces such as ZnSe/ZnTe and CdSe/ZnTe. A type I1 broken-gap alignment occurs when the band gaps of the two materials do not overlap at all in energy, as occurs in the InAs/GaSb heterojunction. A type 111 alignment occurs in heterojunctions containing a semimetallic compound such as HgTe or a-Sn, with HgTe/CdTe being the most extensively studied type I11 heterojunction system. The device concepts that can be implemented successfully in a given heterojunction system will depend very strongly on the type of band alignment characteristic of that heterojunction, and heterojunction device performance will often depend critically on the exact values of the band offsets. One would like to have a reliable yet relatively simple way to obtain accurate band offset values for a wide variety of heterojunction systems to help determine their suitability for various device applications. A number of theoretical approaches and empirical rules for calculating band offsets have been developed, beginning with the electron affinity rule proposed by

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E.T. YU, J.O. McCALDIN AND T.C. McGILL

(4 Type

(b) Type II staggered

I Ec2

Ec2

Ec 1

Ec 1

Ev2 EVl Ev2

(c) Type 11 broken-gap

EVl

(d) Type Ill Ec2

Ec2

Ev2 EVl

Ecl

Ec 1

EV2 EVl

FIG.2. Possible types of band alignments at a semiconductor interface. Conduction- and valence-band-edge positions for each material have been labeled E , and E,, respectively, with the shaded regions indicating the energy band gap in each material.

Anderson” in which the conduction-band offset was assumed to be given simply by the difference in the electron affinities of the two heterojunction constituents. The central concept of the Anderson model was that properties of a semiconductor interface could be deduced from the properties of free semiconductor surfaces; other empirical models have been proposed in which I9R. L. Anderson, Solid-State Electron. 5, 341 (1962).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

5

band offset values are related to experimentally measured properties of semiconductor-metal” or other semiconductor-semiconductor interfaces.21 Several purely theoretical methods have also been developed. In these models, band offset values are obtained using approaches such as calculations of bulk band structure on an absolute energy calculations of effective “midgap,” “pinning,” or “charge neutrality” reference energy levels in each semiconductor that align at an or actual ab initio calculations of electronic structure for each semiconductor i n t e r f a ~ e . ~ ~ - ~ ~ Despite this wealth of theoretical work, a thorough and precise understanding of the physics of band offsets has yet to be attained. Tremendous progress has been made beyond the earliest notions, such as the electron affinity rule,lg but substantial work remains to be done-a testament to the extraordinary difficulty and complexity of the problem of accurately calculating the electronic structure of semiconductor interfaces. Current theories are often able to confirm experimentally determined band offset values to within approximately ?_: 0.05-0.10 eV, and in some cases, such as HgTe/CdTe, qualitatively correct predictions of unexpected experimental results have been made.28 However, the ability to predict band offset values reliably and consistently for heterojunctions in which the band offset has not already been measured experimentally has yet to be demonstrated. To be of use in the quantitative evaluation and design of a semiconductor heterostructure ’OJ. 0. McCaldin, T. C. McGill, and C. A. Mead, Phys. Rev. Lett. 36, 56 (1976). ”A. D. Katnani and G. Margaritondo, Phys. Rev. B 28, 1944 (1983). ”W. R. Frensley and H. Kroemer, J. Vac. Sci. Technol. 13, 810 (1976). 23W.R. Frensley and H. Kroemer, Phys. Rev. B 16, 2642 (1977). 24W. A. Harrison, J. Vac. Sci. Technol. 14, 1016 (1977). 25C.Tejedor and F. Flores, J. Phys. C 11, L19 (1978). 26F. Flores and C. Tejedor, J. Phys. C 12, 731 (1979). ”5. Tersoff, Phys. Rev. B 30,4874 (1984). ”5. Tersoff, Phys. Rev. Lett. 56, 2755 (1986). 29W.A. Harrison and J. Tersoff, J. Vuc. Sci. Technol. B 4, 1068 (1986). 30A.Zunger, Annu. Rev. Muter. Sci. 15, 411 (1985). 31A.Zunger, Solid State Phys. 39, 275 (1986). 32J. M. Langer and H. Heinrich, Phys. Rev. Lett. 55, 1414 (1985). 33G.A. Baraff, J. A. Appelbaum, and D. R. Hamann, Phys. Rev. Lett. 38, 237 (1977). 34G.A. Baraff, J. A. Appelbaum, and D. R. Hamann, J. Vuc. Sci. Technol. 14, 999 (1977). 35W.E. Pickett, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 39, 109 (1977). 36W.E. Pickett, S. G. Louie, and M. L. Cohen, Phys. Rev. B 17, 815 (1978). 37C.G. Van de Walle and R. M. Martin, J. Vac. Sci. Technol. B 3, 1256 (1985). 38C.G. Van de Walle and R. M. Martin, Phys. Rev. B 34, 5621 (1986). 39M.Cardona and N. E. Christensen, Phys. Rev. B 35, 6182 (1987). 40N. E. Christensen, Phys. Rev. B 37, 4528 (1988). 4’N. E. Christensen, Phys. Rev. B 38, 12687 (1988). 42W.R. L. Lambrecht and B. Segall, Phys. Rev. Lett. 61, 1764 (1988). 43W. R. L. Lambrecht, B. Segall, and 0. K. Andersen, Phys. Rev. B 41, 2813 (1990).

6

E.T. YU, J.O. McCALDIN AND T.C.McGILL

device, a band offset must typically be known to an accuracy of approximately +0.10eV or better; however, predicted band offset values for many heterojunctions of current interest extend over a range of 1 eV or more, rendering them of limited use in determining the viability of various device structures in these material systems. The large discrepancies among theoretically predicted band offset values for many heterojunction systems make apparent the need for reliable experimental determinations of band offsets. Such measurements provide valuable data crucial to the development of a satisfactory theoretical understanding of band offsets and are currently the only way to obtain trustworthy band offset values for novel, previously unstudied heterojunction systems. A large number of experimental techniques have been used to measure band offsets for various heterojunction systems.4656 However, the technical difficultyand often indirect nature of these measurements have been at least partially responsible for sizable discrepancies in measured band offset values. Despite several years of research, the value of the valence-band offset even for the extensively studied and supposedly well-understood GaAs/AI,Ga, -,As heterojunction system remained a subject of considerable controv e r s y 4 6 5 2 . 5 7-64 until the middle 1980s. In addition to the complexity 44R. Dingle, W. Wiegmann, and C.H. Henry, Phys. Rev. Lett. 33, 827 (1974). 4sR. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975). 46R. Dingle, in “Festkorperprobleme XV” (H. J. Queisser; ed.), p. 21, Pergamon Vieweg, Braunschweig, 1975. 47R. C.Miller, A. C.Gossard, D. A. Kleinman, and 0. Munteanu, Phys. Rev. B 29,3740 (1984). 48J. Batey and S. L. Wright, J. Appl. Phys. 59, 200 (1985). 49J. Batey and S. L. Wright, Surf: Sci. 174, 320 (1986). ’OW. I. Wang and F. Stern, J. Vuc. Sci. Technol. B 3, 1280 (1985). ”D. J. Wolford, T. F. Kuech, J. A. Bradley, M. A. Cell, D. Ninno, and M. Jaros, J. Vuc. Sci. Technol. B 4, 1043 (1986). 52G.Abstreiter, U. Prechtel, G. Weimann, and W. Schlapp, Physica B 134,433 (1985). s3E. A. Kraut, R.W. Grant, J. R. Waldrop, and S. P. Kowalczyk, Phys. Rev. Lett. 44, 1620 ( 1980). 54E. A. Kraut, R.W. Grant, J. R. Waldrop, and S. P. Kowalczyk, Phys. Rev. B 28, 1965 (1983). 55H. Kroemer, W. Y. Chien, J. S. Harris, Jr., and D. D. Edwall, Appl. Phys. Lett. 36, 295 (1980). 56J.Mentndez, A. Pinczuk, D. J. Werder, A. C. Gossard, and J. H . English, Phys. Rev. B33,8863 (1986). 57J. R. Waldrop, R.W. Grant, and E. A. Kraut, J. Vac. Sci. Technol. B 5, 1209 (1987). ”R. C. Miller, D. A. Kleinman, and A. C. Gossard, Phys. Rev. B 29,7085 (1984). 59T. W. Hickmott, P. M. Solomon, R. Fischer, and H. MorkoG, J. Appl. Phys. 57,2844 (1985). 6nD. Arnold, A. Ketterson, T.Henderson, J. Klem, and H. MorkoT, Appl. Phys. Lett. 45, 1237 (1984). 61H.Okumura, S. Misawa, S. Yoshida, and S . Gonda, Appl. Phys. Lett. 46, 377 (1985). 62M.0. Watanabe, J. Yoshida, M. Mashita, T.Nakanisi, and A. Hojo, J. Appl. Phys. 57,5340 (1985). 63G.Duggan, J. Vuc. Sci. Technol. B 3, 1224 (1985). 64E. T. Yu, D. H. Chow, and T.C.McGill, Phys. Rev. B 38, 12764 (1988).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

7

inherent in these measurements, evidence exists that the actual value of a heterojunction band offset can depend on the detailed conditions under which the interface was An understanding of these effects could provide insight into the factors that are most relevant in determining band offset values and might also allow band offset values to be adjusted, within limited ranges, to optimize various aspects of heterostructure device performance. However, such dependences can also severely complicate interpretations of band offsets in certain material systems. Despite these difficulties, experimental measurements have yielded fairly reliable band offset values for a large set of semiconductor heterojunction systems. In this article, we present an overview of the various theoretical approaches and experimental measurement techniques for determining band offset values and then discuss experimental and theoretical data reported for a number of specific heterojunction systems. An attempt is made to evaluate the credibility and accuracy of the experimental measurements and to provide a tabulation of reliable band offset values for as many heterojunctions as possible, as Kroemer7’ had done for a much more restricted set of heterojunctions and before the full complexity of the problem was realized, even for the GaAs/Al,Ga, -,As heterojunction; from these measurements we attempt to extract some general trends that seem to be especially relevant in governing band offset values. We also address some issues that are specific to particular interfaces, such as strain in lattice-mismatched heterojunctions and diffusion and chemical reactivity in heterovalent semiconductor interfaces. Our approach has been to undertake a critical examination of both experimental and theoretical approaches to the subject of band offsets and to attempt to blend these into a balanced perspective one might refer to as “enlightened empiricism.” From the work that has been reported to date we draw some general conclusions regarding the observed theoretical and experimental trends, discuss some effects that now seem to be well understood, and present some issues that remain unresolved and provide possibilities for further investigation.

65S. P. Kowalczyk, E. A. Kraut, J. R. Waldrop, and R. W. Grant, J. Vac. Sci. Technol. 21, 482

(1982). 66D.W. Niles, G. Margaritondo, P. Perfetti, C. Quaresima, and M. Capozi, Appl. Phys. Letf.47, 1092 (1985). 67D.W. Niles, E. Colavita, G. Margaritondo, P. Perfetti, C. Quaresima, and M. Capozi, J. Vac. Sci. Technol. A 4, 962 (1986). 68P. Perfetti, C. Quaresima, C. Coluzza, C. Fortunato, and G. Margaritondo, Phys. Rev. Left. 57, 2065 (1986). 69J. R. Waldrop, E. A. Kraut, S. P. Kowalczyk, and R. W. Grant, Surf: Sci. 132, 513 (1983). 70H. Kroemer, J. Vac. Sci. Technol. B 2,433 (1984).

8

E.T. YU, J.O. McCALDIN AND T.C. McGILL

11. Theories and Empirical Rules

We have divided the various theoretical treatments of semiconductor heterojunction band offsets into three categories. The first consists of empirical rules based on experimentally determined properties of semiconductor-vacuum, semiconductor-metal, or semiconductor-semiconductor interfaces. In the second category are theoretical calculations of semiconductor band structure that inherently treat the band offset as a bulk parameter independent of the detailed structure of a specific interface, and that therefore effectively assume the electronic states in every semiconductor can be placed on a single common energy scale. The third category comprises theories that include both heterojunction constituents in a single calculation, thereby yielding explicitly the electronic structure of each heterojunction considered and allowing the influence of variables such as strain and crystal orientation to be studied. Our theoretical overview is not intended to be completely exhaustive, but it does include most of the more widely quoted theories that have been proposed.

1. EMPIRICAL RULES a. Electron Afinity Rule The earliest attempts to describe band offsets theoretically assumed that band offset values were determined by the intrinsic properties of each individual semiconductor, and therefore attempted to place the electronic levels in every material on a single absolute energy scale. Band offsets were then determined by the relative position of each material on this absolute energy scale. The first such model was the so-called electron affinity postulated by Anderson, which states that the conduction-band offset AE is given simply by the difference in the electron affinities of the two heterojunction constituents. Experimentally determined electron affinities are used to obtain values for conduction-band offsets. Because the electron affinity is an experimental measure of the energy of the conduction-band edge in a semiconductor relative to the vacuum level, the essential assumption of the electron affinity rule is that the vacuum level serves as a valid common energy reference level for all materials. A major conceptual weakness of this rule is that electron affinities reflect potential shifts arising from surface electronic structure, rather than shifts that are due to charge redistribution at an actual interface. A more practical consideration ”A. G. Milnes and D. L. Feucht, “Heterojunctions and Metal-Semiconductor Junctions,”

Academic Press, New York, 1972.

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

3 AIAs

ZnTe

7

r

n

AlSb

1

>,

W

x

F P)

C W

0

-1

5.4

5.6

5.8

6

6.2

6.4

6.6

Lattice con sta nt (Angstroms) FIG. 3. Energy band gaps of selected semiconductors plotted as a function of lattice constant on an absolute energy scale as determined by the electron affinity rule. Electron affinity data were taken from the compilation of Ref. 71. The origin of the energy scale is taken to be the valenceband edge of GaAs. Direct band gaps are indicated by solid vertical lines, indirect gaps by dashed lines. Conduction- and valence-band offsets can be determined directly from the figure.

is that large uncertainties in experimental electron affinity values for many materials produce correspondingly large ambiguities in predicted band offset values. Figure 3 shows the energy band gaps of several semiconductors plotted as a function of lattice constant on the absolute energy scale given by the electron affinity rule. Such a diagram ref. 71a allows one to examine heterojunction pairs based on considerations of both lattice match and band alignment simultaneously. Electron affinity data used to produce Fig. 3 were taken from the compilation of Ref. 71. The origin of the energy scale is taken to be the valence-band edge of GaAs. For each material, the position of the energy band gap is indicated by a vertical line, with solid lines representing direct band gaps and dashed lines representing indirect gaps. The conduction- and valence-band edges are indicated by horizontal bars. The conduction- and valence-band offsets given by the electron affinity rule can be determined directly from this figure. We will use diagrams of this type to summarize the predictions of a number of band offset theories. An implicit 'laR. H. Miles, J. 0. McCaldin, and T. C. McGill, J . Cryst. Growth 85, 188 (1987).

10

E.T. YU, J.O. McCALDIN AND T.C. McGILL

assumption of these diagrams is that band offsets are transitive; i.e., for three semiconductors labeled A, B, and C, the following rule should be obeyed: AE,(A/B)

+ AE,(B/C) + AE,(C/A)

= 0.

(1.1)

This rule has been verified experimentally and theoretically for a number of material systems; however, for heterojunction systems in which interfacial reactions occur, and for unusual materials such as CuBr, the transitivity rule may not necessarily be valid. b. Common Anion Rules In addition to the electron affinity rule, a number of other semiempirical rules have been proposed as at least qualitative guides for predicting band offset values. Among the more widely quoted of these have been the so-called common anion rules, proposed originally by McCaldin, McGill, and Mead2' The physical with a modified form later postulated by Menendez et motivation for the original common anion rule arises from theoretical evidence that, in compound semiconductors, the valence-band states are derived predominantly from p-like atomic orbitals of the anion.70 One might then expect that the position of the semiconductor valence-band edge on an absolute energy scale would be determined principally by the energies of the outermost (valence) electrons of the anion. In their original paper, McCaldin et al. pointed out that the Schottky barrier height for a large number of III-V and II-VI semiconductors depended primarily on the electronegativity of the anion. It was later proposed" that this correlation might extend to valenceband offset values as well, leading to the postulate that, for a large number of compound semiconductors (materials containing A1 being a notable exception), the valence-band offset in a heterojunction should depend only on the difference in anion electronegativity for the two constituent materials. Early theories of band offsets, such as those of Harrison24 and of Frensley and K r ~ e m e r ,were ~ ~ .in~ general ~ agreement with the common anion rule, even for compounds such as AlAs for which the common anion rule was not claimed to be valid. Wei and Z ~ n g e have r ~ ~performed calculations suggesting that deviations from the common anion rule arise largely from cation dorbital contributions to the valence-band structure, which were generally omitted in early theoretical studies of band offsets. Figure 4 shows the energy band gaps of several semiconductors plotted as a function of lattice constant using valence-band-edge positions determined by the common anion rule. "5. Menendez, A. Pinczuk, D. J. Werder, J. P. Valladares, T. H. Chiu, and W. T. Tsang, Solid State Cornniun. 61, 703 (1987). "S.-H. Wei and A. Zunger, Phys. Rev. Lett. 59, 144 (1987).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

11

2

C W

0

-1

c

4 5.4

5.6

5.8

6

6.2

6.4

6.6

Lattice constant (Angstroms) FIG. 4. Energy band gaps of selected semiconductors plotted as a function of lattice constant on an absolute energy scale determined by anion electronegativity, as postulated by the common-anion rule of McCaldin et aLZ0Compounds containing A1 have been omitted from the plot. The origin of the energy scale is taken to be the valence-band edge of GaAs. Direct band gaps are indicated by solid vertical lines, indirect gaps by dashed lines. Conduction- and valenceband offsets can be determined directly from the figure.

The modified version of the common anion rule proposed by MenCndez et ~ l . ~states ’ that the valence-band offset in a heterojunction system in which the constituents share a common anion is determined primarily by the two cations. This rule was proposed on the basis of experimental evidence for the GaAs/Al,Ga, -,As and GaSb/Al,Ga, -,Sb heterojunctions and the theoretical predictions of T e r s ~ f f ’ ~ *for ’ ~these heterojunctions and for the GaP/InP, GaAs/InAs, and GaSb/InSb material systems. Unfortunately, considerations of lattice match severely limit the number of heterojunction systems for which this rule can be tested: substitutions of A1 for Ga in III-V compounds or Hg for Cd in II-VI semiconductors are the only cation changes that preserve the lattice constant. In lattice-mismatched heterojunctions, strain effects will most likely overwhelm any potential intrinsic deviation from the modified common anion rule of Menkndez. However, the approximate validity of this rule for the GaAs/Al,Ga, -,As and GaSb/Al,Ga, -,Sb heterojunctions might provide some useful insight into the basic physics of band offsets.

12

E.T. Y U , J.O. McCALDIN A N D T.C. McGILL

c. Empirical Compilation of Katnani and Margaritondo A more recent attempt to relate band offset values to empirical data was made by Katnani and M a r g a r i t ~ n d o , ’ ~who . ~ ~measured band offsets for a large number of heterojunctions formed by depositing Si or Ge on various semiconductor substrates. It was hoped that these measurements would yield reliable valence-band-edge energies, relative to the Si and Ge valence-band edges, for a large number of semiconductors. The results of these measurements were combined with other experimental band offset values to produce a set of valence-band-edge energies optimized to give the best agreement with the available experimental data. This scheme is somewhat analogous to the earlier semiempirical electron affinity and common anion rules, except that experimental data for semiconductor-semiconductor interfaces, rather than metal-semiconductor or vacuum-semiconductor interfaces, were used to provide a common energy reference for all materials. As acknowledged by the authors, however, this compilation was obtained simply by optimizing agreement with the experimental data available at the time and therefore provides limited insight in the physical basis responsible for determining band offset values. Figure 5 shows the energy gaps of several selected semiconductors plotted as a function of lattice constant using valence-bandedge energies proposed by Katnani and Margaritondo on the basis of the experimental data available to them.

2. BANDOFFSETS CALCULATED AS BULKPARAMETERS

It has long been recognized75 that, at least from a purely theoretical perspective, one would like to be able to calculate band offset values directly from the properties of each heterojunction constituent, rather than rely upon quantities determined experimentally for semiconductor-vacuum, semiconductor-metal, or semiconductor-semiconductor interfaces. A number of theories relating band offset values to the calculated electronic properties of bulk semiconductors have been proposed. In theories that extract band offset values simply from electronic properties of bulk semiconductors, the bulk band structure of each material is typically obtained relative to a reference level, determined in various theories by factors such as atomic potentials or cancellation of interfacial dipoles. A suitable alignment of the reference levels in each semiconductor then yields values for the band offsets. 74A.D. Katnani and G. Margaritondo, J. Appl. Phys. 54, 2522 (1983). 75H. Kroemer, Crit. Rev. Solid State Sci. 5, 555 (1975).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

13

n

2

W

x

F P,

C

w

5.4

5.6

5.8

6

6.2

6.4

6.6

Lattice constant (Angstroms) FIG.5. Energy band gaps of selected semiconductors plotted as a function of lattice constant using valence-band-edge energies proposed by Katnani and Margaritondo.*' The origin of the energy scale is taken to be the valence-band edge of GaAs. Direct band gaps are indicated by solid vertical lines, indirect gaps by dashed lines. Conduction- and valence-band offsets can be determined directly from the figure.

a. Pseudopotential Theory of Frensley and Kroemer The first attempt to calculate band offset values without the use of experimental data for vacuum-semiconductor or metal-semiconductor interfaces was the pseudopotential theory of Frensley and K r ~ e m e r . In ~ ~this .~~ theory, a pseudopotential calculation was used to obtain the bulk band structure of each semiconductor, with a self-consistency condition enforced for the electrostatic potential and the charge density calculated from valenceband wave functions. Once the bulk band structure was obtained for each individual semiconductor, the band offsets in a heterojunction could be obtained by matching the energies of the interstitial potentials for the two heterojunction constituents. This theory was subsequently refined to include an approximate correction to account for heterojunction dipoles induced by charge redistribution near the interfa~e.'~An approximate calculation of heterojunction dipole effects was made by estimating the effective charge on atoms near the

14

E.T. YU, J.O. McCALDIN AND T.C. McGILL

interface using the electronegativities of each atom and its nearest neighbors. This scheme leads to an effective “electronegativity potential” for each semiconductor. The correction to the band offsets for heterojunction dipole effects was then taken to be simply the difference in the electronegativity potential for the two heterojunction constituents. A consequence of this result is that, even when corrections for heterojunction dipole effects are included, band offset values are still predicted to obey the transitivity rule, Eq. (1.1). Although the validity of this rule is obvious for treatments such as the common anion and electron affinity rules, which do not include any effects specific to a particular interface, it is not as obvious that band offsets should be transitive when the detailed properties of each heterojunction interface are taken into account. The theory of Frensley and Kroemer, and in particular the ability to define an effective electronegativity potential for each material that accounts for interfacial dipole effects, suggested that the concept of band offsets as quantities determined primarily by properties of bulk semiconductors may be physically sound. Figure 6a and b show the band gaps of several semiconductors plotted using valence-band-edge energies determined by the Frensley-Kroemer pseudopotential theory. Corrections arising from heterojunction dipole effects have been neglected in Fig. 6a, and included in Fig. 6b. As discussed by Frensley and Kr~erner,’~ the dipole correction for most lattice-matched heterojunction pairs is typically a few tenths of an electron volt or less. For lattice-mismatched heterojunctions, the apparent dipole corrections can be much larger; however, the dipole corrections were not claimed to be valid for lattice-mismatched heterojunctions, and the applicability of Frensley and Kroemer’s electronegativity potential in lattice-mismatched heterojunctions is probably questionable.

b. LCAO Theory of Harrison Another early theory of heterojunction band offsets was the linear combination of atomic orbitals (LCAO) theory of Harri~on.’~ In Harrison’s approach, the electronic states in a semiconductor are constructed as a superposition of individual atomic orbitals. It is claimed that an adequate description of the relevant electronic structure in each material can be obtained using four orbitals-a single atomic s state and three atomic p states-for each of the two atoms in the zincblende primitive cell. The position of the valence-band edge is then given

2 76D.J. Chadi and M. L. Cohen, Phys. Sfatus Solidi ( B J 68,405 (1975).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS I....

(a)

2

.....

...

l . . ~ . . . - ~ I " . " . . . . I . . . . . ~ . . . ~ . . . . ~

T

AIAs

GQP ......................

___.

15

sly.:--

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

I I

T

I '

ZnTe - 1 ;

AlSb

I

n

% Fx

U

9)

c

W

1

::'

Ga~e~znse

- . . . . . . . . . . I I 'I...................................................... I '

GaSbCdSei

-1

I I

0 -............ -'......................................

5.4

5.6

5.8

.......

: ,

CdTe]

nAa.......-.. .............................

6

6.2

6.4

._

6.6

Lattice c o nst a nt (Angst rorns)

FIG.6. Energy band gaps of several selected semiconductors plotted as a function of lattice constant using valence-band-edge energies determined by the Frensley-Kroemer pseudopotential The two plots have been constructed (a) neglecting and (b) including the electronegativity potential, which provides an approximate correction for effects arising from the formation of heterojunction dipoles.

16

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

where E; is the on-site energy of the cation p state and E; is the on-site energy of the anion p state, both taken from calculated atomic values. V,, is a matrix element between atomic p states for adjacent atoms and is given approximately byz4 V,,

= 2.16h2/md2,

(2.2)

where m is the free electron mass and d the bond length; the numerical coefficient was chosen to agree with values obtained by Chadi and C ~ h e n ~ ~ to fit the true bands of Si and Ge. The valence-band-edge energies obtained using the LCAO approach are therefore automatically given on the common energy scale determined by the individual atomic state energies, and valenceband offsets are computed simply by taking the difference between the valence-band-edge energies on this common energy scale for the two heterojunction materials. Conduction-band offsets can be determined from the valence-band offset and experimental band gaps for each material. As was the case for the early semiempirical rules and the original theory of Frensley and Kroemer, Harrison’s LCAO theory does not include any correction for heterojunction dipole effects. Figure 7 shows the energy band gaps of several semiconductors plotted as a function of lattice constant on the common energy scale determined by Harrison’s LCAO theory. The values shown in the figure were calculated by Harrison using atomic p state energies taken from the calculations of Herman and Skillman.77Harrison later modified his theory by adding excited s states to his basis set and adjusting the interatomic matrix elements.78 These modifications and the use of Hartree-Fock atomic energies yielded a better description of the conduction-band structure and allowed both dielectric and elastic properties to be described using a single set of parameters, which had not been possible in the earlier version of his theory. Kraut7’ recalculated the valence-band-edge energies using Harrison’s LCAO theory of band offsets in conjunction with Hartree-Fock neutral atom ionization energies computed by Mann.80 However, valence-band offsets obtained in this way were found by Kraut to be in poorer agreement with available experimental results than 76D.J. Chadi and M. L. Cohen, Phys. Status Solidi ( B ) 68,405 (1975). 77F. Herman and S. Skillman, “Atomic Structure Calculations,” Prentice Hall, Englewood Cliffs, NJ, 1963. ”W. A. Harrison, Phys. Rev. B 24, 5835 (1981). 79E. A. Kraut, J. Vac. Sci. Technol. B 2,486 (1984). 8oJ. B. Mann, “Atomic Structure Calculations, I: Hartree-Fock Energy Results for Elements Hydrogen to Lawrencium,” Clearing House for Technical Information, Springfield, VA, 1967.

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

7

ZnTe

2

17

AlSb

T

n

%

W

1

InAs

x

P 0)

C

w

0

-1

~ . .

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

5.4

5.6

5.8

6

6.2

6.4

6.6

Lattice constant (Angstroms) FIG.7. Energy band gaps of selected semiconductors plotted as a function of lattice constant on the energy scale calculated using the LCAO theory of Harrison.24 The origin of the energy scale is taken to be the valence-band edge of GaAs. Direct band gaps are indicated by solid vertical lines, indirect gaps by dashed lines. Conduction- and valence-band offsets can be determined directly from the figure.

the original calculations of Harrison using the Herman-Skillman atomic energies. c. Interface Dipole Theories

A number of more recent theories, such as those of T e r s ~ f fand ~ ~ of *~~ Harrison and Ter~off,'~ have argued that interfacial dipoles, rather than bulk semiconductor energy levels, are the dominant factor in determining band offset values. The physical principle underlying these theories is that the alignment of the energy gaps at a semiconductor heterojunction will be such that the interface dipole is minimized. It is argued that by analyzing the band structure of each individual semiconductor, a midgap, pinning, or charge neutrality reference energy level at which evanescent states in the band gap are composed equally of conduction-band-like and valence-band-like states can be determined for each material. Band offsets are then determined by an

18

E.T. YU, J.O. McCALDIN AND T.C. McGILL

appropriate alignment of the midgap energies of each material at a heterojunction interface, a procedure that effectively minimizes the electrostatic dipole formed at the interface. The basic concept of a charge neutrality energy level was proposed by Tejedor et aL8 for calculating Schottky barrier heights in metal-semiconductor interfaces. This idea was later e~tended’~.’~ to the calculation of band offsets in semiconductor heterojunctions. The influence of interfacial dipoles, and the relevance of the charge neutrality level, is perhaps most easily understood by first considering the case of a metal-metal junction.28 At a metal-metal interface, a misalignment of the work functions, or equivalently of the electronegativities, of the two metals will induce a charge transfer, resulting in the formation of a dipole. However, the essentially infinite dielectric constant of metal will screen any discontinuity in the electronegativity and yield a simple alignment of the Fermi level across the interface. argued for the existence of an effective For semiconductors, midgap energy level corresponding to the energy in the band gap at which a surface, interface, or defect state induced in the band gap would contain equal conduction- and valence-band character; despite its name, this effective midgap energy very rarely coincides with the actual midpoint of the energy band gap. A discontinuity in the effective midgap energy would induce charge transfer and the formation of a dipole at the interface; discontinuities in the effective midgap energy would therefore be screened by the dielectric constant of the semiconductors. As a result, the band offsets at a semiconductor heterojunction should be within an energy V / Eof the offsets determined by a rigorous alignment of the effective midgap energies, where V is the discontinuity in the midgap energy corresponding to theories in which interfacial dipoles are neglected. Because typical values of V were on the order of 0.5 eV or less, it was argued that with dipole effects included the effective midgap energies should be aligned to within 0.05 eV for typical values of semiconductor dielectric constants. The ability to determine an effective midgap energy for each individual semiconductor, independent of the material with which it is to form an interface, is again equivalent to placing the energy gaps of each material on a single common energy scale, although the common energy scale in this case is not determined by any external reference energy as was the case in, for example, Harrison’s LCAO theory. The theories of Ter~off’~.’~ and of Harrison and Ter~off,’~which differ principally in the method used to calculate the effective midgap energy, therefore provide theoretical evidence

-

‘lC. Tejedor, F. Flores, and E. Louis, J. Phys. C 10,2163 (1977). ”J. Tersoff, Phys. Rev. Lett. 52, 465 (1984)

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

19

that even when heterojunction dipole effects are taken into account, and in fact are assumed to be the dominant factor in determining band offset values, the transitivity condition given by Eq. (1.1) should remain valid. Figure 8a and b show the energy gaps of several selected semiconductors plotted using the valence-band-edge alignments determined by the theories of TersoF8 and Harrison and Tersoff,” respectively. Cardona and Christensen3’ have proposed a model in which band offset values are determined by an approximate alignment of the dielectric midgap energy (DME)-the midpoint between the conduction and valence energies at the Penn gap-calculated for each material; the deviation from an exact alignment of the dielectric midgap energies is due to finite screening with an effective dielectric constant, obtained by averaging the long-wavelength dielectric constants for each material:

AE,(A/B)

= E: -

Et

-

where E t and E: are the valence-band-edge energies in materials A and B, respectively, E b and EE are the dielectric midgap energies, and E is the effective dielectric constant. A reasonable approximation to the band offset values predicted by this model can be obtained by assuming E = 3.5 for all material^;^^.^^ this approximation then allows a common energy scale to be defined for all materials. Figure 9 shows the energy gaps of several selected semiconductors plotted using valence-band offsets predicted by the model of Christensen and Cardona. An empirical correlation noted by Langer and Heinrich3’ and by Z ~ n g e r ~ ’ *has ~ l been showns3 to be related to the concept of the effective midgap energy alignment. It was pointed out by Langer and Heinrich and by Zunger that alignment of transition-metal impurity levels in compound semiconductors appeared to yield fairly accurate values for valence-band offsets in isovalent (i.e., 111-V on 111-V or 11-VI on 11-VI) heterojunctions. Model calculations of Tersoff and Harrisons3 indicate that, for cationsubstitutional impurities, requiring charge neutrality in the impurity d shell yields a correlation, for a given transition-metal impurity, of the impurity level with a characteristic energy level in the semiconductor very close to the effective midgap energy as defined by Harrison and Ter~off.’~ Alignment of the effective midgap energy at a heterojunction should therefore also produce an approximate alignment of the transition-metal impurity levels across the interface. 83J. Tersoff and W. A. Harrison, Phys. Rev. Lett. 58, 2367 (1987).

E.T. YU. J.O. McCALDIN AND T.C. McGILL

AlAs r GaAs f Znse

r

AlSb

1

I

....

CdTe

GaSb..O8' ...................................... , I

I I

..........-

I

1

-

t

-1

Lattice consta nt (Angst rorns)

I

T

. SIT: GaAs ' 1 ,

n

%

1

- .........

' I

~ ~ ZnSe 7 -

GaSb

I

i

' I

x

P-

ZnTe

!I .................................................... ' I

W

AlSb

I I I I ................................ I

I

' I I

Q

C W

0

-...

.............. I

-1

5.4

5.6

5.8

6

6.2

6.4

6.6

Lattice constant (Angstroms) FIG.8. Energy band gaps of several selected semiconductors plotted as a function of lattice constant using the valence-band-edge alignments determined by the interface dipole theories of (a) Tersoff2' and (b) Harrison and T e r ~ o f f The . ~ ~ origin of the energy scale is taken to be the valence-band edge of GaAs. Direct band gaps are indicated by solid vertical lines, indirect gaps by dashed lines. Conduction- and valence-band offsets can be determined directly from the figure.

21

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS v . .

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

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

2

%

ZnTe 7

AlSb

1

v

cdTelln

x

F Q,

C w

0

-1

5.4

5.6

5.8

6

6.2

6.4

6.6

Lattice constant (Angstroms) FIG.9. Energy band gaps of selected semiconductors plotted as a function of lattice constant using band offsets corresponding to the screened dielectric midgap energy alignment proposed by Cardona and C h r i ~ t e n s e n The . ~ ~ origin of the energy scale is taken to be the valence-band edge of GaAs. Direct band gaps are indicated by solid vertical lines, indirect gaps by dashed lines. Conduction- and valence-band offsets can be determined directly from the figure.

3. SELF-CONSISTENT CALCULATIONS FOR SPECIFIC INTERFACES A number of theories have also been developed that include effects arising from the detailed electronic structure of the specific semiconductor interface under consideration. The typical approach is to calculate the electronic band structure for a so-called supercell geometry, essentially a superlattice with a unit cell consisting of n monolayers of one semiconductor followed by nmonolayers of the other. Because the electronic structure in each layer becomes bulklike very rapidly as one moves away from the interface,3 3,34,37.38 it is possible to determine the position of the valenceband edge in each layer, and therefore the value of the band offset, in structures with only 5-10 monolayers of each material in the supercell. The first approaches of this type were reported by Baraff, Appelbaum, and Hamann33.34and by Pickett, Louie, and C ~ h e n .Baraff ~ ~ ,et~al.~and Pickett

22

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

et al. calculated the electronic structure of the GaAs/Ge interface and obtained valence-band offsets of 0.9 eV and 0.35 eV, respectively. Presumably at least in part because of the prohibitive amount of computation that would have been required, neither group applied its methods to a large number of other heterojunctions. A number of investigators subsequently developed more comprehensive band offset theories based on supercell calculations for specific semiconductor interfaces. The information obtained from full calculations for a large number of specific heterojunction systems is often used to develop or justify simpler, more intuitive model theories for band offset values, often based primarily on bulk semiconductor properties. Van de Walle and Martin37*38,84,85 have calculated band offset values by using a self-consistent local density functional theory and ab initio pseudopotentials to compute the electronic structure in each layer of a superlattice. The calculations of Van de Walle and Martin were also the first to incorporate the effects of strain in lattice-mismatched heterojunctions, with the Si/Ge heterojunction being considered as a prototypical lattice-mismatched material system. On the basis of their self-consistent local density functional calculations for Si/Ge and other heterojunction systems, Van de Walle and Martin proposed a model solid theory of band offset^.'^-^^ In their model solid theory, an absolute electrostatic potential is computed for each material by constructing the solid as a superposition of neutral atoms.' The atomic potentials can be placed on an absolute energy scale common to all materials, and an average electrostatic potential relative to the atomic potentials can be defined in the solid. This procedure was to yield good agreement with the results of the full self-consistent interface calculations. The bulk band structure relative to the average electrostatic potential within the solid is calculated using ab initio pseudopotentials. It is then possible to define the position of the valence-band edge in each material on the common energy scale given by the atomic potentials and consequently to derive band offset values for various heterojunctions. The relatively good agreement between the results of the model solid theory and the fully selfconsistent calculations suggests that band offsets can be considered, at least approximately, to be determined primarily by characteristics of the bulk constituent materials. Van de WalleS6has also calculated absolute deformation potentials using the model solid approach, allowing band offsets in

-

84C.G. Van de Walle and R. M. Martin, Phys. Rev. B 35, 8154 (1987). s5C.G. Van de Walle and R. M. Martin, J. Vac. Sci. Technol. B 4, 1055 (1986). 86C.G. Van de Walle, Phys. Rev. B 39, 1871 (1989).

23

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

strained heterojunctions to be predicted. Figure 10 shows the energy gaps of several selected semiconductors plotted using valence-band offsets given by the model solid theory of Van de Walle and Martina6 have also performed selfC h r i s t e n ~ e n ~ ~and y ~ lLambrecht et consistent calculations of band offsets for superlattice geometries, using linear muffin-tin orbital (LMTO) methods rather than ab initio pseudopotentials. Christensen used the results of his calculations to argue in favor of the dielectric midgap energy model proposed by Cardona and C h r i ~ t e n s e n . ~ ~ For a large number of heterojunctions, Christensen found that interfacial dipoles did indeed tend to drive band offsets toward the values predicted by alignment of dielectric midgap energies, although the presence of interface states for certain heterojunctions tended to produce band offsets in disagreement with the dielectric midgap energy model. ~

1

.

2

n

%

W

~

~

AlSb CdTe

lnSl

Si

F

lnAs

Q)

c

9

GaSb

x

W

~

-r

AlAs ZnSe 1

~

I

0

5.4

5.6

5.8

6

6.2

6.4

6.6

Lattice con sta nt (Angstroms) FIG. 10. Energy band gaps of selected semiconductors plotted as a function of lattice constant using valence-band-edge energies corresponding to the model solid theory of Van de Walle and MartinB6The origin of the energy scale is taken to be the valence-band edge of GaAs. Direct band gaps are indicated by solid vertical lines, indirect gaps by dashed lines. Conduction- and valence-band offsets can be determined directly from the figure.

24

E.T. YU, J.O. McCALDIN AND T.C. McGILL

Lambrecht and Segall used the results of their self-consistent dipole calculations to develop their interface bond polarity In this model, the bulk electronic structure in each material is first calculated with respect to an average reference potential using LMTO methods. Band offsets are obtained by calculating the difference between the average reference potentials for each material and applying a screened dipole correction to account for bond polarity and charge transfer at the heterojunction interface. An analysis of their self-consistent dipole calculations indicated that a relatively simple and accurate approximation for the screened dipole correction could be obtained by estimating the charge transfer in bonds formed between the two materials from bond polarities and calculating the resulting dipole potential with screening given by the long-wavelength dielectric constant. The resulting band offset values do not rigorously obey the transitivity relation, Eq. (l.l), and this model therefore does not allow all materials to be placed on a single energy scale. For most heterojunctions, the interface bond polarity model was found to agree reasonably well with the fully self-consistent LMTO calculations of Chri~tensen~'.~' and the selfconsistent dipole calculations of Lambrecht et ~

4.

COMPARISONS AMONG

1

.

~

~

9

~

~

THEORIES

Despite the extensive theoretical efforts discussed in the previous sections, band offsets in novel heterojunction systems cannot yet be reliably predicted to a high degree of accuracy by any existing theory. Figure 11 shows valenceband offsets calculated using several different theories for the AlAs/GaAs, GaPJSi, and ZnSeJGe heterojunctions. The shaded regions indicate the approximate range of experimental values thought to have been valid at the time each theory was developed. Until the middle 1980s, band offsets were thought to be fairly well understood, both theoretically and e~perimentally.~' Reliable band offset measurements were apparently available for several material systems, including GaAs/Al,Ga, -,AS, InAsJGaSb, and Ge/GaAs/ ZnSe, and these experimental values were in good agreement with the predictions of Harrison's LCAO theory.z4 Renewed interest in band offsets in the middle 1980s was inspired largely by the unexpected discovery that the AlAsJGaAs valence-band offset was much larger than had previously been thought. The correct value of the AlAs/ GaAs valence-band offset was in disagreement with the prediction of

87W.R. L. Lambrecht and B. Segall, Phys. Rev. B 41,2832 (1990).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

25

0.75 0.5

CI

% 0.25 i a 0

-0.25 1.5

1

u’

a

0.5

0 2.5

W’

a

1.5

H T HT VWM US DME LMTO SCD IBP MF FK 1972 1977 1977 1986 1986 1987 1987 1987 1988 1990 1990 FIG. 11. A comparison of theoretical and experimental band offset values for the AlAs/GaAs, GaP/Si, and ZnSe/Ge heterojunctions. Different theories are plotted in approximately chronological order, and the range of experimental values thought to have been valid at the time each theory was developed has been shaded. For extensively studied systems such as AlAs/GaAs, recent theories agree well with experiment. Agreement between theory and experiment, and even among theoretical values, is poorer for less studied systems such as GaP/Si and ZnSe/Ge. Theoretical predictions are from the following sources; MF 1972, Ref. 71; FK 1977, Ref. 23; H 1977, Ref. 24; T 1986, Ref. 28; HT 1986, Ref. 29; VWM 1987, Ref. 84; MS 1987, Ref. 84; DME 1987, Ref. 39; LMTO 1988, Ref. 40; SCD 1990, Ref. 43; IBP 1990, Ref. 87.

26

E.T.YU, J.O. McCALDIN AND T.C. McGILL

Harri~on,’~ as shown in Fig. 11. Several new theories have since appeared, most of which, as shown in the figure, are in close agreement with the experimental AlAs/GaAs valence-band offset. For less extensively studied heterojunctions, however, the agreement between experiment and theory, and even among different theoretical treatments, is considerably worse. For GaP/Si, the values predicted by the more recent theories (starting with Tersoff’* in 1986) differ by as much as 0.38 eV. These theories predict values much smaller than the few experimentally measured values that have been reported. For the ZnSe/Ge heterojunction, the recent theoretical values encompass a range of approximately 0.64 eV, and agreement with even the very wide range of reported experimental values, indicated by the shaded region in the bottom graph of Fig. 11, is tenuous. Figure 11 illustrates a general observation that for heterojunctions that are well understood experimentally, theories can reproduce experimental band offset values quite accurately. For more complicated or less extensively studied material systems, however, theoretical band offset values are probably best used as qualitative guides rather than accurate quantitative predictions. The widely disparate physical principles underlying current band offset theories, which nevertheless often yield rather similar band offset values, are probably also an indication that much remains to be understood about the electronic structure of semiconductor interfaces.

111. Experimental Techniques

The capabilities of current theoretical treatments of band offsets are such that consistently reliable theoretical predictions of band offset values in novel semiconductor heterojunctions cannot yet be obtained; band offsets must therefore be determined experimentally for each new material system of interest. Various methods have been devised to measure band offsets in semiconductor heterojunctions. These techniques can be divided into three categories: optical spectroscopy, in which optical absorption, photoluminescence, or photoluminescence excitation spectra from quantum-well or superlattice structures are analyzed with the band offsets as fitted parameters; various types of electron spectroscopy, such as x-ray photoelectron spectroscopy (XPS), ultraviolet photoelectron spectroscopy (UPS), or synchrotron photoemission spectroscopy, that provide a direct measure of the valenceband offset; and electrical (device-like) techniques, in which either the conduction - or the valence-band offset is extracted from measurements, such as C - Vor I - Vcharacteristics, on electrical device structures. An overview of several common band offset measurement techniques is presented in this section.

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

27

5. OPTICAL SPECTROSCOPY One of the earliest methods used to measure band offsets to a high degree of precision was optical spectroscopy performed on quantum wells and superlattices. These measurements have remained in widespread use despite the considerable care required to avoid well-known pitfalls of the technique. The extraction of band offsets from optical spectra was first performed for the GaAs/Al,Ga, -,As heterojunction, and the pioneering efforts of Dingle et a1.44-46 in this area, while providing the basis for much subsequent work, also illustrate the weaknesses and potential dangers of the technique. Analysis of optical absorption spectra from Al,Ga, -,As/GaAs/ Al,Ga, -,As quantum-well heterostructures by Dingle et al.4-6 yielded what was for several years the most widely accepted value for the GaAs/Al,Ga, -,As band offsets: BE, x 0.85 AEg and AEv x 0.15 AEg, where AEg is the difference between the GaAs and A1,Gal - ,As energy band gaps. In this technique, calculated quantum-well bound-state energies are fitted to the observed structure in the quantum-well absorption spectrum, with the band offset as an adjustable parameter. Unfortunately, the calculated bound-state energies for these structures depend quite strongly on parameters such as the well width and carrier effective mass, as well as on the band offset. Subsequent photoluminescence experiments on parabolic quantum ~ e l l s , ~in ~ .which ’~ the bound-state energies are more sensitive to the band offset values, demonstrated the importance of using accurate material parameters, such as effective masses, and yielded a conduction-band offset AEc % 0.57 A E g , in closer agreement with the currently accepted values. However, considerable care was required in the growth of the parabolic quantum-well samples. Because control of the A1 flux during growth was not sufficient to yield truly parabolic compositional profiles,63 an effective parabolic potential was synthesized by deposition of alternating layers of GaAs and Al,Ga, -,As; the widths of the Al,Gal -,As layers were increased quadratically with distance from the center of the quantum well, and the GaAs layer widths were correspondingly reduced. Optical techniques have also been applied, with somewhat mixed results, to other material systems, including InP/Ino~,3Gao,47As88~89 and Gao,471n,,s3As/A10,481no~52~s.90 A related technique, in which a light-scattering method is used to determine conduction-band offsets, has been developed by Menendez and coworkers and applied to a number of heterojunctions, including 88B.I. Miller, E. F. Schubert, U. Koren, A. Ourmazd, A. H. Dayem, and R. J. Capik, Appl. Phys. Lett. 49, 1384 (1986). 89R. Sauer, T. D. Harris, and W. T. Tsang, Phys. Rev. B 34,9023 (1986). ’OD. F. Welch, G. W. Wicks, and L. F. Eastman, J. Appl. Phys. 55, 3176 (1984).

28

E.T. YU, J.O. McCALDIN AND T.C. McGILL

GaAs/Al,Ga, -,Ass6 and GaSb/Al,Ga, -xSb.72 Backscattering spectra obtained from photoexcited carriers in multiple-quantum-well structures were found to contain peaks arising from inelastic light scattering, with energy shifts corresponding to transition energies between bound states in the quantum wells.9t Values for the conduction-band offset in these multiple quantum wells were obtained by fitting the observed transition energies to a theoretical model for quantum-well bound-state energies, with the conduction-band offset as an adjustable parameter. Band offsets obtained using this technique for the GaAs/AI,Ga, -,As and GaSb/Al,Ga, - ,Sb heterojunctions were found to be in fairly good agreement with other reported results. The GaAs/Al,Ga, -,As valence-band offset has also been measured by studying the pressure dependence of photoluminescence from GaAs/Al,Ga, -,As quantum wells and super lattice^.^^ This technique exploited the proximity in energy of the r-point and X-point conduction-band minima in AI,Ga, -,As; by applying hydrostatic pressure to GaAs/Al,Ga, -,As heterostructure samples, the GaAs quantum-well confined states were shifted above the X-point conduction-band minimum in the Al,Ga, -,As barriers, resulting in a sharp reduction in photoluminescence intensity from the r-confined quantum-well states. By analyzing the pressure dependence of the Al,Ga, -,As energy band gap and of the photoluminescence energies for pressures 5 60 kbar, a GaAs/Al,Ga, -,As valence-band offset AEv = (0.32 & 0.02) AE; was deduced, in reasonable agreement with currently accepted values. As stated by the authors in the original study,’l an assumption made in this experiment was that the valence-band edges of GaAs and Al,Ga, -,As move together with pressure, i.e., that the valenceband offset is approximately independent of pressure. Later measurements by Lambkin et aL9’ indicated that the GaAs/AlAs valence-band offset increased slightly with pressure, d(AE,)/dP w 1 meV/kbar. This dependence would yield only a relatively small shift of the GaAs/AlAs valence-band offset of approximately 0.06 eV over the range of pressures studied by Wolford et al. For other material systems, however, the valence-band offset has been found to depend more strongly on pressure. Magneto-optical studies of InAs/GaSb super lattice^^^ indicated that the InAs/GaSb valence-band offset varies considerably with pressure; the separation between the InAs conductionband edge and the GaSb valence-band edge was found to decrease at a rate of 5.8meVlkbar. Given an increase in the InAs energy band gap of 10 meV/kbar,93 this yields a valence-band offset that increases with pressure at a rate of 4.2 meV/kbar. 9’A. Pinczuk, J. Shah, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 46, 1341 (1981). 92J. D. Lambkin, A. R. Adams, D. J. Dunstan, P. Dawson, and C. T. Foxon, Phys. Rev. B 39, 5546 (1989).

93L.M. Claessen,J. C. Maan, M. Altarelli, P. Wyder, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 57,2556 (1986).

BAND OFFSETS I N SEMICONDUCTOR HETEROJUNCTIONS

6. XPS

AND

29

RELATEDTECHNIQUES

Various types of electron spectroscopies have been used to measure band offsets in a large number of material systems. In particular, x-ray photoelectron spectroscopy has been used to measure band offsets in several heterojunction systems. The basic XPS method, illustrated in Fig. 12, requires first

Material 1 Material 2

I Ec

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

hv

e-

FIG.12. A schematic energy band diagram illustrating the basic principle of the XPS band offset measurement. Reference core-level binding energies are measured in thick films of each material, and the separation between the two reference core levels is measured in heterojunction samples. These three quantities can then be combined to yield a value for the valence-band offset.

30

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

the measurement of reference core-level binding energies, e.g., the Ga 3d core level in GaAs or the A1 2 p level in AlAs, relative to the valence-band edge in two semiconductors. Heterojunctions consisting of a thin ( - 2 5 A) layer of one material deposited on the other are then grown in which the separation between the reference core levels in each material is measured; the separation between the reference core levels can be translated directly into a value for the valence-band offset using the previously measured core-level to valenceband-edge binding energies:

Related techniques, such as the direct measurement of valence-band separations using ultraviolet photoelectron spectroscopy (UPS) or synchrotron photoemission, which are characterized by higher energy resolution but greater surface sensitivity, have also been reported. Unlike many optical and electrical methods for determining band offsets, photoemission techniques provides a direct measurement of the band offset value, so interpretation of the experimental results is fairly straightforward. In addition, the XPS method is well suited to the study of novel material systems because the structures required are quite simple. XPS measurements can also provide valuable information regarding issues such as interface reactions and commutativity (independence of growth sequence). The primary source of uncertainty in XPS measurements of band offsets arises from the need to determine accurately (to within approximately f0.05 eV) the position of the valence-band edge in the XPS spectrum. Kraut, Grant, Waldrop, and K o w a l ~ z y kdeveloped ~ ~ , ~ ~ the high-precision analysis techniques that allow band offsets to be measured by XPS to a high degree of accuracy and pioneered the use of these techniques to study band offsets in heterojunctions such as G ~ / G ~ A s , ~ ’G, ’~~A S / A ~ A and S , ~1nA~/GaAs.’~ ~#~~ In addition, XPS has been used to measure valence-band offsets for the nearly lattice-matched ZnSe/GaAs/Ge material s y ~ t e m , 6 ~ . ’the ~ InAs/GaSb heterojun~tion,’~ and the GaSb/AlSb h e t e r o j ~ n c t i o nBand . ~ ~ offset measure94R.W. Grant, J. R. Waldrop, and E. A. Kraut, J. Vac. Sci. Technol. 15, 1451 (1978). 95J. R. Waldrop, S. P. Kowalczyk, R. W. Grant, E. A. Kraut, and D. L. Miller, J. Vac. Sci. Technol. 19, 573 (1981).

96S.P. Kowalczyk, W. J. Schaffer, E. A. Kraut, and R. W. Grant, J. Vac. Sci. Technol. 20, 705 (1 982).

97F.Xu, M. Vos, J. P. Sullivan, Lj. Atanasoska, S. G. Anderson, J. H. Weaver, and H. Cheng, Phys. Rev. B 38, 7832 (1988). 98G.J. Gualtieri, G. P. Schwartz, R. G. Nuzzo, R. J. Malik, and J. F. Walker, J. Appl. Phys. 61, 5337 (1987). 99G. J. Gualtieri, G. P. Schwartz, R. G. Nuzzo, and W. A. Sunder, Appl. Phys. Lett. 49, 1037 (1986).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

31

ments for the HgTe/CdTe heterojunction'" and the CdTe/HgTe/ZnTe material system"' have also been reported, although in the latter work strain effects were neglected despite the large lattice mismatch (- 6.5%) of CdTe and HgTe with ZnTe. The XPS technique has also been extended to the study of strain-'~~ dependent band offsets in the Si/Ge (001) material s y ~ t e m . ' ~ ~To measure strain-dependent band offset values, effective deformation potentials must be determined either t h e ~ r e t i c a l l y 'or ~ ~e ~ p e r i m e n t a l l y ' ~for ~ ~the '~~ strain-induced shifts between atomic core levels and the valence-band edge in bulk material and between core levels in different materials in strained heterojunctions. Alternatively, the positions of the conduction- and valenceband edges relative to the Fermi level in undoped Si and Si, -,Ge, can be monitored in strained, heavily doped samples and core-level energies subsequently measured in undoped heterostructures. lo' Interface reactions in certain heterojunction systems can also be studied by XPS. The sensitivity of XPS core-level binding energies to different chemical bonding states can produce shifted peak components in core-level XPS or synchotron photoemission spectra obtained from heterojunctions in which an intermediate, chemically reacted layer is present at the i n t e r f a ~ e . ' ~ ~ -Studies '~' of core-level peak intensity as a function of overlayer coverage can also yield information regarding the abruptness of heterojunction interfaces.'" Additional studies of band offsets in a variety of heterojunctions using XPS will be discussed in relation to specific material systems in Sections IV through VII.

7. ELECTRICAL MEASUREMENTS

Various electrical measurement techniques have been used to determine band offset values. Thermionic emission across a single barrier has been used '@'S. P. Kowalczyk, J. T. Cheung, E. A. Kraut, and R. W. Grant, Phys. Rev. Lett. 56,1605 (1986). lolT. M. Duc, C. Hsu, and J. P. Faurie, Phys. Rev. Lett. 58, 1127 (1987). '02W.-X.Ni, J. Knall, and G. V. Hansson, Phys. Rev. E 36, 7744 (1987). lo3G. P. Schwartz, M. S. Hybertsen, J. Bevk, R. G. Nuzzo, J. P. Mannaerts, and G. J. Gualtieri, Phys. Rev. B 39, 1235 (1989). lWE.T. Yu, E. T. Croke, T. C. McGill, and R. H. Miles, Appl. Phys. Lett. 56, 569 (1990). Io5E.T. Yu, E. T. Croke, D. H. Chow, D. A. Collins, M. C. Phillips, T. C. McGill, J. 0.McCaldin, and R. H. Miles, J. Vac. Sci. Technol. B 8, 908 (1990). lo6W.G. Wilke and K. Horn, J. Vuc. Sci. Technol. B 6, 1211 (1988). Io7W. G. Wilke, R. Seedorf, and K. Horn, J. Vuc. Sci. Technol. E 7, 807 (1989). loSK.J. Mackey, P. M. G. Allen, W. G. Herrenden-Harker, R. H. Williams, C. R. Whitehouse, and G. M. Williams, Appl. Phys. Lett. 49, 354 (1986).

32

E.T. YU, J.O. McCALDIN A N D T.C. McCILL

to measure band offsets in a number of h e t e r o j u n c t i ~ n sl o. The ~~~~~~~~ thermionic current density J at temperature T over a single barrier is given approximately by J z A*T2 exp( - 4/k,T),

(7.1)

where A* is the effective Richardson constant, kB is the Boltzmann constant, and 4 is the barrier height, given by CAE, + ( E , - E")] for holes and [AEc + ( E , - Ef)]for electrons. By analyzing the temperature dependence of current-voltage characteristics in single-barrier heterostructures, it is possible to determine the barrier height and therefore the band offset values. This technique has been applied with considerable success to, among others, the and HgTe/CdTe' l o material systems. GaAs/A1,Gal - xA~48349,59*60 Capacitance-voltage measurements have also been applied to the determination of band offset values. Kroemer eta1.55 showed that it is possible, by performing C- Vprofiling through a heterojunction, to extract a value for the band offset in that heterojunction. This technique has been used to determine band offsets in a number of heterojunction systems, including GaAs/Al,Ga, -xAs,55961,62 lattice-matched InGaAsP/InP,' 1 , 1 1 2 and In,Al, ~xAs/Ino,,,Gao,,7As."3~'14 A number of investigators have also applied a somewhat simplified version of the C - Vprofiling technique to the measurement of band offsets. For structures in which the doping level is constant in each heterojunction layer, a plot of (1/C2) as a function of V should yield a straight line. The intercept voltage Knt should then yield the total built-in voltage of the heterojunction and consequently the band offset value, assuming that the doping level in each layer is known. This technique has been used to determine band offsets in several heterojunction systems, including InAs/AlSb," InAso,9sSbo,o,/GaSb,"6 and CdS/InP.'17,"8 A charge transfer method has also been used to determine band offset values. In this technique, the sheet density is measured in the two-dimensional carrier gas formed at a heterojunction interface; a method has been '''A. C. Gossard, W. Brown, C. L. Allyn, and W. Wiegmann, J. Vac. Sci. Technol. 20,694 (1982). ''OD. H. Chow, J. 0. McCaldin, A. R. Bonnefoi, T. C. McGill, I. K. Sou, and J. P. Faurie, Appl. Phys. Lett. 51, 2230 (1987). lllS. R. Forrest and 0. K. Kim, J. Appl. Phys. 52, 5838 (1981). liZS. R. Forrest, P. H. Schmidt, R. B. Wilson, and M. L. Kaplan, Appl. Phys. Lett. 45, 1199 (1984).

Il3R. People, K. W. Wecht, K. Alavi, and A. Y. Cho, Appl. Phys. Lett. 43, 118 (1983). 'I4P. Z. Lee, C. L. Lin, J. C. Ho, L. G. Meiners, and H. H. Wieder, J. Appl. Phys. 67,4377 (1990). llSA. Nakagawa, H. Kroemer, and J. H. English, Appl. Phys. Lett. 54,1893 (1989). II6A.K. Srivastava, J. L. Zyskind, R. M. Lum, B. V. Dutt, and J. K. Klingert, Appl. Phys. Lett. 49, 41 (1986). 'I7J.

IISJ.

L. Shay, S. Wagner, K. J. Bachmann, and E. Buehler, J. Appl. Phys. 47, 614 (1976). L. Shay, S. Wagner, and J. C. Phillips, Appl. Phys. Lett. 28, 31 (1976).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

33

d e ~ e l o p e d ~ ~to* 'estimate '~ the band offset value from the measured sheet carrier density. This method has been used to measure the valence-band offset in GaAs/AlAs ( and GaAs/Al,Ga, -,As (100)1'9 heterojunctions; the orientation dependence of the GaAs/Al,Ga,-, valence-band offset has also been studiedlz0 by this technique, with no measurable difference found between the valence-band offsets in the (100) and (311) orientations. Finally, internal photoemission measurements have been used to determine band offsets for the GaAs/Al,Ga, -,As material system. In these experiments, threshold photon energies determined from photocurrent measurements in p+-GaAs/n--Al,Ga, -,As h e t e r o j u n ~ t i o n s ~ ~ *or' ~ 'in a GaAs/Al,Ga, -,As heterojunction combined with a Mo-GaAs Schottky barrier'" were used to obtain values for the conduction-band offset. GaAs/Al,Ga -,As conduction-band offset values have also been derived from measurements of photo current^^^ and photo voltage^'^^ arising from internal photoemission from the two-dimensional electron gas in GaAs/Al,Ga, -,As heterostructures.

,

IV. Ill-V Material Systems

In reviewing the available experimental data on band offsets, we have divided the various heterojunction systems for which data are available into a number of categories: lattice-matched 111-V heterojunctions, lattice-matched 11-VI heterojunctions, lattice-mismatched heterojunctions, and heterovalent material systems. By doing so we hoped to isolate to some degree chemical trends within each group of isovalent heterojunctions, effects arising from strain, and effects that are due to chemical reactivity at heterovalent interfaces. We begin with a review in this section of the available experimental band offset data for lattice-matched 111-V heterojunctions. 8. GAASIALAS

a. Experimental Data Any discussion of heterojunction band offset measurements begins most naturally with the GaAs/AlAs interface. The GaAs/Al,Ga,.-,As heterojunction system is currently the most important technologically, and has therefore 'I9W. I. Wang, E. E. Mendez, and F. Stern, Appl. Phys. Lett. 45, 639 (1984). lz0W. I. Wang, T. S. Kuan, E. E. Mendez, and L. Esaki, Phys. Rev. B 31,6890 (1985). Iz1M.A. Haase, M. A. Emanuel, S. C. Smith, J. J. Coleman, and G. E. Stillman, Appl. Phys. Lett. 50, 404 (1987). lZ2M.Heiblum, M. I. Nathan, and M. Eizenberg, Appl. Phys. Lett. 47, 503 (1985). lZ3K.W. Goossen, S. A. Lyon, and K. Alavi, Phys. Rev. B 36,9370 (1987).

34

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

been the subject of by far the most extensive experimental investigation. Despite these efforts, there existed for some time considerable controversy regarding the actual value of the GaAs/Al,Ga, -,As valence-band offset and its possible dependence on substrate orientation and growth sequence. For several years the value of the GaAs/Al,Ga, -,As valence-band offset was thought to have been well established. A 1974 experiment in which Dingle et u1.44-46 measured optical absorption spectra from Al,Ga, -,As/GaAs/Al,Ga, -,As quantum-well heterostructures with x = 0.2 f 0.01 yielded a conduction-band offset AEc = 0.85 A E g , corresponding to a valence-band offset AEv = 0.15 A E g , where AEg is the difference in band gap between GaAs and A1,GaI-+4s. Several other experiments seemed to confirm this measurement. Gossard et ul.'09 obtained a conduction-band offset AEc = 0.85 AEg from current-voltage ( J - V ) measurements performed on GaAs/Al,Ga, -,As single barrier heterostructures with square, triangular, and sawtooth-shaped barriers. People et ul.' l 3 measured a conduction-band offset AEc = (0.88 f 0.04) AEg by capacitance-voltage ( C - V ) profiling, and Welch et ~ 1 . ~ 'obtained GaAs/Al,Ga, -,As quantum-well photoluminescence spectra consistent with AEc = 0.85 A E g . Confidence in the so-called 85 : 15 rule was such that in experiments in which substantially smaller conduction-band offsets were the authors attributed the discrepancy to compositional grading at the GaAs/Al,Ga, -,As heterojunction during crystal growth, rather than considering the possibility that their band offset value might be more correct than that given by the 85 : 15 rule. Subsequent experiments, however, provided strong evidence that the 85 : 15 rule is indeed incorrect and that the GaAs/Al,Ga, -,As valence-band offset is substantially larger than had been thought at first. A number of these experiments also provided detailed data relating the GaAs/Al,Ga, -,As band offset to alloy composition x and energy gap difference AEg over the entire range of alloy compositions x E [0, 13. The early experiments supporting the 85 : 15 rule were conducted for only a limited range of alloy compositions, typically x 5 0.45. Several of the more recent band offset measurements reported for the GaAs/Al,Ga, -,As heterojunction are shown in Fig. 13, and experimental measurements for the GaAs/Al,Ga, -,As valence-band offset are summarized in Table I. The first indication that the 85 : 15 rule might be incorrect was provided by photoluminescence measurements on parabolic quantum wells reported by Miller, Gossard, Kleinman, and M ~ n t e a n u Earlier . ~ ~ measurements of band offsets using quantum-well photoluminescence or optical absorption techniques utilized data obtained from square quantum wells; in square quantum-well structures, the energies of the bound states depend quite 12%.

M. Wu and E. S. Yang, J. Appl. Phys. 51,2261 (1980).

35

BAND OFFSETS I N SEMICONDUCTOR HETEROJUNCTIONS

W

0

a E

GaAs

C

E GaAs V

n

%

W

W

>

d

FIG. 13. Summary of experimental band offset data for the GaAs/AI,Ga, -,As heterojunction. Data are from Miller et aL5' (a);Wang et a1.50s119s120 (0); Arnold et aL6' (m); Okumura et aL6' (0); Hickmott et aL5' (A);Watanabe et aL6' (A);Batey and Wright48*49 Wolford et aL5' (V); Menendez et ~ 1 ( +. ); and ~ Yu ~ et al.64( x ).

(v);

sensitively on parameters such as the quantum-well width and carrier effective mass as well as on the band offset. For parabolic wells, the boundstate energies depend more strongly on the band offset values than for square wells; however, controlling the sample growth to ensure that the wells are indeed parabolic is quite difficult. The data obtained by Miller et al. indicated that the conduction-band offset was smaller than had previously been thought, AEc z 0.50 A E g . A subsequent analysis of exciton transitions in

TABLE I. EXPERIMENTAL VALENCE-BAND OFFSETVALUESFOR GaAs/AI,Ga,,As

SOURCE Dingle et al. (1974)44-46 Kroemer et al. (1980)55 Wu and Yang (1980)’2s Gossard et al. (1982)’” People et al. (1983)’13 Welch et al. (1984)90 Miller et al. (1984)58 Arnold et al. (1984)60 Wang et al. (1985)50.119,’20

Okumura et al. (1985)61 Hickmott et al. (1985)” Watanabe et al. (1985)62

HETEROJUNCTION GaA~/A~o.20Gao.80As GaAs/Al,.,Ga,,,As GaAs/AIO.

35Ga0. 6

GaAs/AI,Ga,,As GaAs/A10,3Gao.7As GaAs/AIO,

33Ga0. 6

GaAs/AI,Ga,,As GaAs/AI,Ga,,As GaAs/&.i?6Ga0.74As GaAs/Alo,,Ga0.,As GaAs/AIAs GaAs/AI,Ga ,&s GaAs/A10,6Gao,,As GaAs/Al,Ga,,As GaAs/A10,30Ga0.70As

Batey and Wright (1985)48*49 Wolford et al. (1986)’’ Menindez et al. (1986)56 Dawson et al. (1986)lZ6 Katnani and Bauer (1986)’28 Waldrop et al. (1987)57 Yu et a1 (1988)64 Hirakawa et al (1990)’29

C(O.15 f 0.03)AEJ

GaAs/Al,Ga ,-As GaAs/Al,Ga,,As

C(0.12 k 0.04)AEg] [0.15AE,] 0.43 AE, 0.35 AEg 0.126 f 0.04 eV 0.21 0.03 eV 0.45 f 0.05 eV

-

0.38 AE, 0 . 5 5 ~eV (0.32 f 0.02)AEE

6

GaAs/Al As GaAs/AlAs GaAs/AIAs GaAs/AlAs

-

0.67AE, 0.63AE, 0.62AEg ~

-

-

0.69AEg

0.342 f 0.004 eV 0.38 eV 0.36-0.46 eV 0.46 & 0.07 eV 0.44 f 0.05 eV

-

GaAs/A10.06Ga0.94As AIO. 37 GaO.

+

C(0.85 0.03)AEJ 0.66AEg 0.64AE, [0.85AEJ C(0.88 f 0.04)AEg] [0.85AEg] 0.57AE,

“Values in square brackets are in disagreement with currently accepted values.

b 2

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

37

both square and parabolic quantum wells by Miller, Kleinman, and G o ~ s a r dyielded ~ ~ a conduction-band offset AE, % 0.57 AEg. The results of Miller et al. led to renewed interest in the determination of the GaAs/A1,Gal -,As band offset, and a large number of subsequent measurements48-5 1 , 5 6 , 5 9 - 6 2 , 1 1 9 , 1 2 0 , 1 2 6 , 1 2 7 yielded GaAs/Al,Ga, -,As band offset values consistent with the results of Miller et al. rather than those of ' , ~ using a charge transfer technique, reported Dingle et al. Wang et ~ l . , ~ 19,120 valence-band offset values of 0.126 f 0.04 eV, 0.21 f 0.03 eV, and 0.45 f 0.05 eV for the GaAs/A10,26Gao,74As,GaAs/Al,~,Ga,.,As, and GaAs/AlAs heterojunctions, respectively, corresponding to AEv % 0.28-0.39 AEg. Arnold et ~ 1 . ~ used ' current-voltage measurements as a function of temperature to deduce a valence-band offset AEv = 0.35 AEg over a range of alloy composition x~C0.3,1.01. Okumura et aL61 obtained a conduction-band offset AE, = 0.67 AEg for x < 0.42 from capacitance-voltage measurements. Hickmott et combined C-V and J-V measurements to obtain a conduction-band ' C-V measurements offset AE, = 0.63 AEgfor x = 0.4. Watanabe et ~ 1 . ~used to determine the conduction- and valence-band offsets independently and obtained values AE, = 0.62 AEg for x E [0.15,0.30] and AEv = 0.38 AEg for x = 0.30. Batey and Wright48,49 used J-V techniques to measure the GaAs/A1,Gal -,As valence-band offset as a function of alloy composition x and obtained AEv = 0 . 5 5 ~for x E [O, 11. Wolford et aL51 studied quantumwell photoluminescence in GaAs/A1,Gal -,As heterostructures as a function of pressure; from an analysis of the critical pressure for crossing in energy of the r-like bound states in GaAs and the X-like states in Al,Ga,-,As for various alloy compositions, a valence-band offset AEv = 0.32 & 0.02 AE: was . a~ light-scattering ~ deduced for x % 0.28 and 0.70. Menendez et ~ 1 used method to obtain AE, = 0.69 AEg for x = 0.06. Photoluminescence measurements by Dawson et ul.lz6 yielded evidence of a staggered band alignment corresponding to AEv = 342 4 meV in Al,Ga, -,As/AlAs heterostructures with x = 0.37, and data for other composition^'^^ were consistent with AEv = 0.55~.In all cases, measurements indicated that the energy gap difference was divided more equally between the conduction- and valenceband offsets than indicated by the 85 : 15 rule. Most of these experiments were conducted for GaAs/A1,Gal -,As heterojunctions in which the band gap of Al,Ga, -,As was still direct, i.e., x 5 0.4-0.45, and yielded conductionand valence-band offsets AE, % (0.55-0.65) AEg and AEv % (0.35-0.45) AEg. For heterojunctions involving Al,Ga, -,As layers with direct band gaps, the conduction- and valence-band offsets could be measured independently; the requirement that AE, + AEv = AEg then provided convincing confirmation that the band offset measurements were indeed valid. Iz6P. Dawson, B. A. Wilson, C. W. Tu, and R. C. Miller, Appl. Phys. Lett. 48, 541 (1986). "'B. A. Wilson, P. Dawson, C. W. Tu, and R. C. Miller, J. Vac. Sci. Technol. B 4, 1037 (1986).

38

E.T. YU, J.O. McCALDIN AND T.C. McGILL

The variation of the GaAs/Al,Ga, -,As valence-band offset with allo: composition x was also a subject of some debate. The initial approach was tc assume that the conduction-band offset was proportional to the energy gal difference at the heterojunction, even though the measurements of Dinglc et al. were carried out at only a single alloy composition. Arnold et aL6' measured current-voltage characteristics for p+-GaAs/ Al,Ga, -,As p--GaAs capacitor structures and deduced a valence-band offset AEv = 0.35AEi for x E c0.3, 1.01. Because the direct band gap difference AE; is no linear in x,this result was taken to imply a nonlinear dependence of AEvon x In contrast, measurements by Batey and Wright48,49of current-voltagi characteristics as a function of temperature yielded a GaAs/AI,Ga, -,A: valence-band offset linear in alloy composition, AEv z 0 . 5 5 ~ Uncertaintie . and inconsistencies in the actual band offset values obtained by varioui investigators, however, render the exact form of this dependence a secondar! issue. Additional complications can arise from characteristics such as substratc orientation and interface quality. Using x-ray photoelectron spectroscopy observed a dependence of the valence-band offset on botl Waldrop et a1.57*95 substrate orientation and growth sequence. For samples grown on (100 substrates, valence-band offsets of 0.46 eV and 0.36 eV were measured foi AlAs grown on top of GaAs (AlAs-GaAs) and GaAs grown on top of AlA! (GaAs-AlAs), respectively. For samples grown on (110) subtrates, valence, band offsets of 0.55 eV and 0.42 eV were reported for the AlAs-GaAs anc GaAs-A1As growth sequences, respectively. These data were contradicted however, by photoemission measurements of Katnani and Bauer"* or heterojunctions involving GaAs, AlAs, and Ge. Katnani and Bauer did no) observe any dependence of the GaAs/AlAs valence-band offset on growtk sequence, obtaining a value of 0.39 f 0.07 eV for both GaAs-A1As anc AlAs-GaAs heterojunctions. Measurements of Yu et and of Hirakawz et ~ l . ' ~ also ' provided evidence that, for interfaces of sufficiently high quality the GaAs/AlAs band offset is commutative. In addition, band offset measure. ments for GaAs/A1,,,6Ga,,74As (100) and (31 1) heterojunctions"' and foi ' ~ ~ to show anj GaAs/AlAs (loo), (1lo), and (1 11) B h e t e r o j u n c t i ~ n s failed dependence of band offset values on substrate orientation. For ideal, abrupt interfaces, one would expect that band offset value: should not exhibit any dependence on growth sequence; that band offsetr should be independent of growth sequence is referred to as the commutativitj property. The dependence of the valence-band offset on growth sequenct was thought to result from detailed micro. observed by Waldrop et scopic differences in epitaxial growth on various semiconductor surfaces. A dependence on growth sequence of band offset values in a given materia ~

1

.

~

~

3

'

~

'"A. D. Katnani and R. S. Bauer, Phys. Rev. B 33, 1106 (1986). '29K. Hirakawa, Y. Hashimoto, and T. Ikoma, Appl. Phys. Lett. 57, 2555 (1990).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

39

system large enough to affect device properties, as the effects observed by Waldrop et al. were, could have significant implications for the analysis and design of heterojunction devices in that material system. Such a dependence could also provide insight into the properties of an interface that are most relevant in determining band offset values. In light of the confirmation of ~ ~ ~ ’ ~ ~it would appear commutativity by other i n v e s t i g a t ~ r s , ~ ~ *h’owever, that noncommutativity of the band offset is not an inherent feature of the GaAs/AlAs material system but is instead a consequence of insufficient control over interface quality. Figure 13 summarizes several of the experimentally measured conductionand valence-band offsets that have been reported for GaAs/Al,Ga, -,As heterojunctions. Assuming that the valence-band offset is a linear function of composition x,a least-squares fit to the data yields AEv = 0 . 4 8 ~eV, as shown in the figure. The corresponding conduction-band offsets, for both the r and X valleys in Al,Ga, -,As, are also shown in the figure. The r-point (direct) and X-point (indirect) band gaps for Al,Ga,-,As as a function of alloy composition x have been taken from the review of Adachi.I3O b. Comparison with Theory Theoretical predictions for the GaAs/AlAs band offset extend over a wide range of values. The early band offset theories generally predicted small values for the valence-band offset, the physical justification being that, near the valence-band edge, the valence-band wave functions were derived primarily from p-like atomic orbitals of the anion. Because the anion in GaAs and AlAs is the same, it was thought that the valence-band edges in the two materials should be at approximately the same position in energy. For all cases in which a nonzero valence-band offset was predicted, the AlAs valenceband edge was correctly predicted to be lower in energy than the GaAs valence-band edge. Predictions of several of the more widely quoted band offset theories are summarized in Table 11. Figure 11 contains a graphical comparison of theoretically predicted valence-band offsets with the currently accepted experimental values for the GaAs/AlAs heterojunction. The electron affinity rule” in conjunction with the electron affinity data of Milnes and Fuecht” yielded a valence-band offset of 0.15 eV, while the LCAO theory of Harrisonz4 predicted a valence-band offset of 0.04 eV. Frensley and K r ~ e m e r , ~using ~ . ’ ~a pseudopotential band structure calculation, obtained a valence-band offset of 0 eV without interfacial dipole corrections and an offset of 0.69 eV with their dipole correction included. With the exception of the dipole-corrected value of Frensley and Kroemer, these predictions were all in general agreement with the value measured by is often erronDingle et al. The common anion rule of McCaldin et dzo eously cited as predicting a very small GaAs/AlAs valence-band offset; 130S. Adachi, J.

Appl. Phys. 58, R1 (1985).

40

E.T. YU, J.O. McCALDIN AND T.C. McGILL

TABLE 11. THEORETICAL PREDICTIONS FOR THE GaAs/AlAs VALENCE-BAND OFFSET SOURCE Electron affinity rule (1972)’l Frensley and Kroemer (1977)23 Frensley and Kroemer (1977)23(with dipole correction) Harrison (1977)24 Tersoff (1986)2s Harrison and Tersoff (1986)29 Van de Walle and Martin (1987)84 (self-consistent supercell) Van de Walle and Martin (1987)84 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy mcidel) Christensen (1988)40 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall ( 1990)87(interface bond polarity model)

AEJtheor.) (eV) 0.15 0.00 0.69

0.04 0.55 0.12 0.37 0.60

0.43 0.53 0.53

0.46

because Schottky barriers on AlAs and AlSb did not follow the behavior of the other materials studied, compounds containing A1 were not included in the data on which the rule was based and the rule was not claimed to be valid for A1 compounds. Following the experimental realization that Dingle’s result was substantially in error, a number of new theories predicting generally larger valenceband offsets were developed. Theories based on interface dipoles were developed by T e r s ~ f f and ~ ~ by * ~ Harrison ~ and TersoffZ9 these theories predicted valence-band offsets of 0.55 eV and 0.12 eV, respectively. The dielectric midgap energy model of Cardona and C h r i ~ t e n s e nyielded ~~ a valence-band offset of 0.43 eV, and the interface bond polarity model of Lambrecht and Sega1187predicted AE, = 0.46 eV. A number of calculations taking into account the electronic structure at specific interfaces were also developed. A self-consistent interface calculation based on ab initio pseudopotentials was developed by Van de Walle and Martin85 and predicted a valence-band offset of 0.37 eV; the model solid theory86 derived from these calculations yielded a valence-band offset of 0.60 eV. Self-consistent theories based on linear muffin-tin orbital (LMTO) calculations were developed and applied to the GaAs/AlAs heterojunction by Christensen4’ and by Lambrecht and Sega11;’ with both calculations yielding valence-band offsets of 0.53 eV. These more recent theories are all in relatively good agreement with the currently accepted experimental band offset values. As discussed in the previous section, Waldrop et a1.5’,95observed a dependence of the GaAs/ AlAs valence-band offset on substrate orientation. In contrast, measurements by Wang et al.lZ0and by Hirakawa et a l l z 9did not reveal any dependence of the GaAs/AlAs valence-band offset on substrate orientation. Such a dependence might at first seem fairly plausible, given the variations in atomic structure and chemical bonding for different interface orientations. However, theoretical calculations in which the detailed structure of sDecific, albeit ideal,

41

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

interfaces is taken into account generally predict a very small dependence of the band offset value on substrate ~ r i e n t a t i o n . ~ ~ ’ ~ ~ ’ ~ ~ 9. INGAAs/INALAs/INP Much effort has been devoted to the measurement of band offsets in the lattice-matched In,,,,Gao,4,As/In,,,2Alo,48As/InP material system. Ino,,,Ga,,4,As has a room-temperature band gap of 0.74 eV (1.67 pm), close to the 1.55-pm wavelength needed to minimize attenuation in optical fiber transmissions; the effective energy for optical absorption and emission can be increased considerably by quantum confinement effects. The In,,,3Ga,,4,As/In,~,zAlo~48As and In,,,,Ga,,,,As/InP heterojunctions, from which In,,, ,Ga,,,,As quantum wells can be fabricated, are therefore of considerable technological interest. A number of measurements have been made of band offsets for lattice-matched Ino,,,Gao.47As/In,,,2Alo,48As, In, _.Ga,As, -,P,/InP, and In0~,,A1,,,,As/InP heterojunctions. A summary of selected experimentally measured band offsets for the Ino,,3Ga,~4,As/Ino,,zAl,,4,As/ InP material system is given in Fig. 14, and

- 1

%

W

W

I

InGaAs

InAlAs

InP

InGaAs 0

EC

EC

A

AEV( InP/lnAIAs)

-1

J

FIG. 14. Summary of experimental band offset data for the In,,,,Ga,,,,As/ InO.,,A1,,,,As/InP material system. For the In,,,,Ga,,,,As/In,.,~A~o.4aAsheterojunction, data are from Morgan et a1.149( 0 ) ;People et a1.l” (0); Welch et aLgO(W); Weiner eta/.’,’ ( 0 ) ; Wagner et a1.152 Sugiyama et a1.153(V);Lee et a1.114 (A); and Waldrop et a1.141(A). For ( 0 ) ;Miller et a1.”’ the In,,,,Ga,,,,As/InP heterojunction, data are from Forrest et aL111*112 (0);Lang et a1.136(W); Westland et ~ ~ 1(0); . l Haase ~ ~ et a1.I4O (A); Waldrop et a1.141 (A);and The lnO,,,A1,.,,As/lnP valence-band offset is from the measurement of Cavicchi et a1.13’ Waldrop et a1.158Consistency of the experimental data with band offset transitivity can be seen in the figure.

(v);

(v).

42

E.T. YU, J.O. McCALDIN AND T.C. McGILL

theoretical band offset values are plotted in Fig. 15. The available experimental and theoretical band offset values are also summarized in Tables I11 and IV, respectively. a. InGaAslhP For the 1n0,,,Ga,~,,As/1nP material system, band offset values ranging from AEc = 190 & 30 meV % 0.32 AE, to AE, = 600 meV % AE, have been reported.88,89.111 , 1 1 2 , 1 3 1-143 The more widely accepted measurements for this heterojunction typically yield AE, % 0.4 A E g . Figure 14 shows the band alignments given by several of these measurements, and the available ~ 'the~ ~ ' ~ experimental data are summarized in Table 111. Forrest et ~ 1 . ~used capacitance-voltage technique to measure the conduction-band offset in lattice-matched In, -.Ga,As, -,P,/InP, and found AE, = 0.39 AE, over the entire compositional range of In, -,Ga,Asl -,P, alloys lattice. ~ ~photoluminescence energies in matched to InP. Miller et ~ 1 measured In,,,3Gao.4,As/InP quantum wells as a function of quantum-well width and obtained results generally consistent with AE, % 0.40 AE,, although for certain well widths the experimental results could be modeled theoretically using conduction-band offsets ranging from 20 to 50% of the total band gap l ~ ~ both the conduction- and the valencedifference. Lang et ~ 1 . measured band offset for the ln,,,,GaO,,,As/lnP heterojunction by using admittance spectroscopy to analyze p - n junctions containing In,~,3Ga,,4,As/InP superlattices and obtained AEv = 0.346 0.010 eV and AE, = 0.250 0.010 eV = 0.42 AE,. The sum of these measured conduction- and valence-

*

*

131R.Chin, N. Holonyak, Jr., s. W. Kirchoefer, R. M. Kolbas, and E. A. Rezek, Appl. Phys. Lett. 34, 862 (1979). 132Y.Guldner, J. P. Vieren, P. Voisin, M. Voos, M. Razeghi, and M. A. Poisson, Appl. Phys. Lett. 40, 877 (1982). '33P. E. Brunemeier, D. G. Deppe, and N. Holonyak, Jr., Appl. Phys. Lett. 46, 755 (1985). 134H.Temkin, M. B. Panish, P. M. Petroff, R. A. Hamm, J. M. Vandenberg, and S. Sumski, Appl. Phys. Lett. 47, 394 (1985). 135W.T. Tsang and E. F. Schubert, Appl. Phys. Lett. 49,220 (1986). 136D.V. Lang, M. B. Panish, F. Capasso, J. Allam, R. A. Hamm, A. M. Sergent, and W. T. Tsang, Appl. Phys. Lett. 50, 736 (1987). I3'D. J. Westland, A. M. Fox, A. C. Maciel, J. F. Ryan, M. D. Scott, J. I. Davies, and J. R. Riffat, Appl. Phys. Lett. 50, 839 (1987). 138M.S. Skolnick, L. L. Taylor, S. J. Bass, A. D. Pitt, D. J. Mowbray, A. G. Cullis, and N. G. Chew, Appl. Phys. Lett. 51, 24 (1987). 139R.E. Cavicchi, D. V. Lang, D. Gershoni, A. M. Sergent, J. M. Vandenberg, S. N. G. Chu, and M. B. Panish, Appl. Phys. Lett. 54,739 (1989). I4OM. A. Haase, N. Pan, and G. E. Stillman, Appl. Phys. Lett. 54, 1457 (1989). I4lJ. R. Waldrop, E. A. Kraut, C. W. Farley, and R. W. Grant, J. Appl. Phys. 69, 372 (1991). I4*J.R. Waldrop, R.W. Grant, and E. A. Kraut, Appl. Phys. Lett. 54, 1878 (1989). 143J.R. Waldrop, R.W. Grant, and E. A. Kraut, J. Vac. Sci. Technol. B 7, 815 (1989).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

InP

InGaAs

43

InAlAs

-

FIG. 15. Summary of theoretical band offset values for the In,,,,Ga,,47As/In,,,~Al~,4HAs/InP material system. For the In,,,,Ga,,47As/In,,,,AI,.4,As heterojunction, calculated values are from the model solid theory of Van de Walle and MartinH4(a);Cardona and C h r i ~ t e n s e n ~ ~ (0);and Hybert~en'~,(m). For the In,,,,Ga,,47As/InP heterojunction, calculated values are from the model solid theory of Van de Walle and Martins4 ( 0 ) ;Cardona and C h r i ~ t e n s e n ~ ~ (0);Hybertsen'61.162 (m); the common-anion rule of McCaldin et aLZ0(0);self-consistent dipole calculations of Lambrecht et aL4, (A);and the interface bond polarity model of Lambrecht and Sega1IH7(A). The theories shown here all obey the transitivity rule to within kO.01 eV. The shaded regions indicate the approximate range of commonly accepted experimental band offset values.

band offsets yields a total band-gap discontinuity AE,, = 0.596 f 0.015 eV, in excellent agreement with the independently measured band-gap differe n ~ e ' ,of ~ 0.613 eV at 4.2 K and 0.600 f 0.010 eV at 300 K. The consistency of these results is a reassuring indication of the validity of the measurements reported in Ref. 136. Cavicchi et aZ.139used the same technique to measure the conduction-band offset for In,Ga, -,As grown on InP and obtained AE, = 0.21 f 0.02 eV for x = 0.53. Measurements of optical absorption in Ino,,,Gao,,,As/InP multiple-quantum-well structures by Westland et ~ 1 . yielded ' ~ ~ a conduction-band offset AE, w 0.45 AEg. Optical absorp~ a lower bound for the tion measurements by Skolnick et ~ 2 1 . ' ~yielded Ino,,,Gao,,,As/InP conduction-band offset (based on the lack of absorption features at photon energies above 1.075 eV) of 0.235 & 0.020 eV, or AEc 2 (0.38 & 0.03) AEg. Internal photoemission measurements by Haase et ~ 1 . ' ~ ' yielded a conduction-band offset AEc = 0.203 k 0.015 eV = 0.34 AE,, at

-

44

0

I

112

d

E.T. YU. J.O. McCALDIN AND T.C. McGILL

2

8 s

c!

I l+II 0

TABLE IV. THEORETICAL BANDOFFSETVALUES FOR InP/In~,5,Ga~,47As/In~,5,AI~,48As

A 4 (theor.) (eV) . ,

SOURCE

McCaldin et al. (1976)20 Van de Walle and Martin (1987)84 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)87 (interface bond polarity model) Hybertsen (1990)16','62

0.28

Van de Walle and Martin (1987)84 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Hybertsen (1991)163

0.21

Van de Walle and Martin (1987)84 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Hybertsen (1991)163

0.34 0.26

W

i2U v

v)

5

vj

9 v)

0.22 0.17 0.41

InP/In0.

52 AIO .4SAS

0.13

1nP/ln0.52A10.48As

0.11

1nP/In0.52A10.48As

0.25

tn

Ei zU

c1 0

54 B C

z

2 0

z, P

v,

46

E.T. YU, J.O. McCALDIN AND T.C. McGILL

room temperature for the In,,,,Ga,,,,As/InP heterojunction. In addition, Haase et al. measured a temperature dependence of the conduction-band ’ XPS offset d(AE,)/aT = -0.2 f 0.1 meV/K. Finally, Waldrop et ~ 1 . ’ ~ used to measure the valence-band offset for the In,~,,Ga,,,,As/InP heterojunction, obtaining AEv = 0.34 eV = 0.57 A E g . In an earlier study, Waldrop et a1.1429143used XPS to measure band offsets for the lattice-mismatched GaAs/InP and InAs/InP heterojunctions and interpolated those results to obtain a valence-band offsetAEv = 0.42 AE,, corresponding to a conductionband offset AE, = 0.58 AE,, for the lattice-matched In,,,,Ga,,,,As/InP system. However, the ability to measure an effective “unstrained” valenceband offset value for a lattice-mismatched heterojunction by the XPS method is somewhat questionable, and the interpolated In,,,,Ga,,,,As/InP valenceband offset obtained by Waldrop et al., which does not agree with the value measured directly, should be viewed with some caution. A number of other measurements have been reported that yield band offsets differing considerably from the currently accepted values. In many cases, however, these discrepancies appear to arise from complications in experimental interpretation and are indicative of the difficulty of performing accurate band offset measurements rather than an actual variation in band ’ ~ ~ on the basis of offset values. Forrest and Kim’,, and Ogura et ~ 1 . claimed C- V measurements that the In,,,,Ga,,,,As/InP conduction-band offset dropped sharply and approached zero as the temperature was decreased, behavior they both attributed to the filling of interface trap levels with electrons at low temperature. Subsequently, Kazmierski et al. 146 pointed out the importance of accounting for trap levels in the actual interpretation of C - V measurements and claimed that in their measurements on In,,,,Ga,,,,As/InP heterojunctions the apparent conduction-band offset was in reality attributable to the presence of interface traps and that the actual conduction-band offset was nearly zero; however, their calculations incorrectly assumed147that donor trap levels were only partially filled at low temperature and completely filled at high temperature. A later analysis by Leu and Forrest’,, of the effects of temperature and measurement frequency on C - V profiles finally appeared to explain the apparent reduction in the conduction-band offset at low temperature. Assuming that el, the rate of emission for charge in a trap, is proportional to a Boltzmann occupation factor, el cc exp( - EJk,T), Leu and Forrest deduced that for frequencies o G el the measured value of AEc was nearly independent of the interface trap R. Forrest and 0.K. Kim, J. Appl. Phys. 53, 5738 (1982). 145M.Ogura, M. Mizuta, K. Onaka, and H. Kukimoto, Jpn. J. Appl. Phys. 22, 1502 (1983). 146K.Kazmierski, P. Philippe, P. Poulain, and B. de Cremoux, J. Appl. Phys. 61, 1941 (1987). I4’L. Y. Leu and S. R Forrest, J. Appl. Phys. 64, 5030 (1988).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

47

density and therefore very close to the value measured in the absence of interface traps, whereas for high frequencies w 9 e, the measured conductionband offset decreased with increasing trap density. It was therefore postulated that the apparent drop in the measured conduction-band offset at low temperature was actually a transition from the low-frequency to the highfrequency measurement regime induced by the temperature dependence of the Boltzmann factor in the trap response rate e,. Several other types of experiments have also been reported in which band offset values inconsistent with the currently accepted values were obtained. Chin et ~ 1 . ' ~ ' analyzed photoluminescence and laser spectra from In,,,7Gao,13AS,,,,P,,71/InP quantum wells grown by liquid phase epitaxy and obtained a conduction-band offset A E , x 0.67 AE,. Brunemeier et ~ 1 . l later used the same technique and obtained AE, x 0.65 A E , over the entire compositional range of lattice-matched In, -,GaxAsl -,,P,,/InP heterojunctions. These two results, however, should be viewed with some caution in light of the fact that this technique was also used to obtain a valence-band , ~ ~ ~ is offset AEv = 0.15 A E , for the GaAs/Al,Ga, -,As h e t e r o j u n ~ t i o nwhich now known to be a clear underestimation of the size of the valence-band offset or, equivalently, an overestimation of the conduction-band offset. Guldner et ~ 1 . ' ~ 'deduced a conduction-band offset of 0.53 eV from Shubnikov-de Haas and cyclotron resonance measurements on the two-dimensional ~ ~ electron gas at the 1n0,,,Ga,,,,As/1nP interface. Temkin et ~ 1 . lused optical absorption and photoluminescence spectra to deduce that the conduction-band offset was larger than the valence-band offset for the ' photoluminIn,,,,Ga,,47As/InP heterojunction, and Sauer et ~ 1 . ~analyzed escence excitation spectra and obtained AE, x 0.60 AE,. However, the difficulty of obtaining reliable band offset values from analyses of optical spectra has been clearly demonstrated for the case of GaAs/Al,Ga, AS,"^*^^ and the same considerations are a factor in determining band offsets in other heterojunction systems, including In,,,,Ga,~,,As/InP. b. InGaAslZnAlAs Band offsets in lattice-matched In,~,,Ga,,4,As/In,,,,Alo.48As and heterojunctions have been measured Ino, ,Ga,,,,As/Ino, ,(Gal - xA1x)0,48As by a number of techniques. The experimental data are summarized in Fig. 14 and Table 111. Morgan et ~ 2 1 . ' measured ~~ the temperature dependence of current-voltage characteristics in In,~,,Ga,,47As/In,,,~Alo~48Asheterostructures and obtained a conduction-band offset AE, = 0.72 AE, = 0.52 eV.

,

,

I4'R. D. Dupuis, P. D. Dapkus, R. M. Kolbas, N. Holonyak, Jr., and H. Shichijo, Appl. Phys. Lert. 33, 596 (1978). I4'D. V. Morgan, K. Board, C. E. C. Wood, and L. F. Eastman, Phys. Sturus Solidi ( A ) 72,251 (1982).

~ ~

48

E.T. YU. J.O. McCALDIN AND T.C. McGILL

People et a1.lS0 used the C-V profiling technique to measure the

Ino,,,Ga,,,,As/Ino~,zAlo,48Asconduction-band offset and at room temperature (297 K) obtained AEc = (0.50 f 0.05) eV = (0.71 f 0.07) AE,. Welch et aL9' analyzed photoluminescence spectra from Ino,,,Gao,,,As/ ~ n o , s z ~ ~ osingle , 4 8 quantum ~s wells and obtained a conduction-band discontinuity AEc z 0.7 AE, z 0.52 eV. Weiner et al."' measured optical absorption spectra at room temperature in In,,,,Ga,,,,As/ Ino,szAlo,48As multiple quantum wells; using both the conduction- and valence-band offsets as adjustable parameters to fit a theoretical model to the observed absorption spectra, Weiner et al. obtained a conduction-band offset AEc = 0.44 eV = 0.60 AEg and a valence-band offset AEv = 0.29 eV. Wagner et ~ 1 . 'mea~ ~ sured photoluminescence excitation spectra at low temperature (5-30 K) for In,~,,Gao,,,As/Ino~,zAlo,48Asquantum wells and obtained conduction- and valence-band discontinuities BE, z 0.5 eV z 0.7 AEg and AEv z 0.2 eV. Sugiyama et al.' s 3 analyzed the temperature dependence of current-voltage single-barrier hecharacteristics in Ino,,,Ga,,,,As/In,,,z(Ga, -xA1x)0.48A~ terostructures and obtained a conduction-band offset AEc = (0.53 f 0 . 0 5 ) eV ~ = (0.72 f 0.07) AEg(x);in other s t ~ d i e sthe ~ energy ~ ~ , band ~ ~ ~gap of In,,,,(Ga, -xA1x)0.48A~ quaternary alloys had been found to depend ' ~ conduction-band offsets linearly on the composition x. Lee et ~ 1 . l measured in In,Al, ~xAs/Ino~,3Ga,,,,Asheterojunctions for x E [0.47,0.52] using C-V profiling and obtained AEc = 0.50 eV = 0.70 AE, for the lattice-matched ' XPS to measure the composition, x = 0.52. Finally, Waldrop et ~ 1 . ' ~ used I n o , , , ~ a o , , , ~ s / I n o , , z ~ ~valence-band ~,48~s offset, with their measurements yielding AE, = 0.22 eV = 0.32 AE,, corresponding to AEc = 0.68 AE,. These measurements appear to converge on values of 0.70 AE, % 0.50 eV for the conduction-band offset and 0.30 AEg z 0.20 eV for the valence-band offset. The relative sizes of the conduction- and valence-band offsets appear therefore to be fairly close to those found in the GaAs/AlGaAs material system, with the fraction of the total band gap discontinuity found in the conduction-band offset being slightly greater in the Ino,,,Ga,,,,As/ ~ n o , s z ~ ~ heterojunction. o~48~s

People, K. W. Wecht, K. Alavi, and A. Y. Cho, Appl. Phys. Lett. 43, 118 (1983). '"J. S. Weiner, D. S. Chemla, D. A. B. Miller, T. H. Wood, D. Sivco, and A. Y. Cho, Appl. Phys. Lett. 46, 619 (1985). lS2J. Wagner, W. Stolz, and K . Ploog, Phys. Rev. B 32, 4214 (1985). '"Y. Sugiyama, T. Inata, T. Fujii, Y. Nakata, S. Muto, and S. Hiyamizu, Jpn. J. Appl. Phys. 25, L648 (1986). 154T.Fujii, Y. Nakata, Y. Sugiyama, and S. Hiyamizu, Jpn. J. Appl. Phys. 25, L254 (1986). 155T.Fuji, Y. Nakata, S. Muto, and S. Hiyamizu, Jpn. J. Appl. Phys. 25, L598 (1986).

ISOR.

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

49

c. InAlAslInP Relatively few measurements of band offset values have been performed for the In0,,,A1,,,,As/InP heterojunction. The available experimental data are summarized in Table 111. Caine et ~ 1 .measured l ~ ~ bias-tunable luminescence from lattice-matched p-In0~,,A1,,,,As/n-InP heterojunctions and obtained conduction- and valence-band offsets of 0.52 eV and 0.40 eV, respectively. Aina et al.' 5 7 measured photoluminescence from In0,,,A1,,,,As/InP multiple quantum wells and single heterojunctions, from which they deduced a staggered band alignment with an effective band gap for the single heterojunction of 1.06 eV, corresponding to conduction- and valence-band offsets of 0.39 eV and 0.29 eV, respectively. Waldrop et al.15' used XPS to measure the valence-band offset in lattice-matched In0,,,A1,,,,As/InP heterojunctions and obtained AEv = 0.16 eV. It is worth noting that a linear interpolation between the purportedly "strain-free" band offset values measured by Waldrop et al. for G ~ A s / I ~ P ' ~and ~,'~~ A I A s / I ~ P 'heterojunctions ~~ yields an In0~,,A1,,,,As/InP valence-band offset of approximately 0.02 eV, which does not agree well with the direct measurement of the In0,,,A1,,,,As/InP valence-band offset. The source of the disagreement among these measurements is not entirely clear. However, factors that apparently have not been taken into account in the photoluminescence studies of In0,,,A1,~,,As/InP heterojunctions include the effects of electrostatic band bending and of below-band-gap recombination on the energy of photon emission due to recombination at the interface. It is possible that electrostatic band bending or below-band-gap recombination could reduce the photon energy for emission due to recombination at the In0,,,A1,,,,As/InP interface, increasing the apparent value of the In0,,,A1,,,,As/InP valence-band offset. This effect could provide an explanation for the observed discrepancy between the band offset values obtained by XPS and those obtained by analysis of photoluminescence. For this reason, and also because of the known reliability of band offset values obtained from careful XPS measurements, we believe the valence-band offset obtained in Ref. 158, AE,(In0,,,A1,,,,As/InP) = 0.16 eV, to be the most trustworthy of the values cited here.

156E.J. Caine, S. Subbanna, H. Kroemer, J. L. Merz, and A. Y. Cho, Appl. Phys. Lett. 45, 1123 (1984). I5'L. Aina, M. Mattingly, and L. Stecker, Appl. Phys. Lett. 53, 1620 (1988). "'J. R. Waldrop, E. A. Kraut, C. W. Farley, and R. W. Grant, J. Vac. Sci. Technol. B 8, 768 ( 1990).

50

E.T. YU, J.O. McCALDIN AND T.C. McGILL

d. Transitivity

A test of the transitivity condition, Eq. (1. l), can provide information regarding the possible influence of interface effects on band offset values in the Ino,,,Gao,,,As/Ino, ,,Al,~,,As/InP material system. Figure 14 shows experimental band offset data for the Ino,,,Gao.47As/Ino~,zAlo,48A~, Ino,,,Gao,,,As/InP, and lnO,,,A1,~,,As/lnP heterojunctions; only the selected measurements described in the preceding sections have been included. As shown in the figure, these experimental band offset values clearly obey the transitivity relation; i.e., combining the Ino,,,Gao,,,As/Ino,,zAlo,48Asand Ino,,,Gao,,,As/InP band offset values yields the correct value, to within experimental error, for the lnO,,,A1,,,,As/lnP band offset. In addition, Waldrop et ~ 1 . ' ~ 'have used XPS to verify directly that band offset transitimaterial system. vity is satisfied for the Ino., 3Gao,,,As/Ino,,,Alo,48As/InP These results provide considerable evidence for the validity of the transitivity rule for heterojunctions with abrupt, high-quality interfaces; the experimental confirmation of transitivity and the close agreement among the selected experimental results also suggest that variation of band offset values with growth conditions is probably not a significant effect in this material system. Another effect of potential interest in the Ino,,,Gao~4,As/Ino,,zAlo,48As/ InP material system is the presence of interface strain at the Ino,,,Gao~,,As/ InP and lnO~,,A1,,,,As/lnP heterojunctions. As shown in Fig. 16, the interor 1n0,52A10,48, face in an MAs/InP heterojunction, where M = Ino~53Gao,47 can be either InAs-like (Fig. 16a) or MP-like (Fig. 16b). These two different types of interfaces have been distinguished experimentally in high-resolution x-ray diffraction studies of Ino~,,Gao,,,As/InP super lattice^.'^^ In addition, x-ray diffraction measurements performed on InAlAs/InP superlattices have revealed evidence of anion intermixing and the presence of single-monolayer strained layers at the InAlAs/InP interfaces.l6' H y b e r t ~ e n ' ~ ' - 'studied ~~ the effect of interface strain theoretically in the Ino,,,Gao,,,As/Ino~,zAlo,48As/ InP material system and found that, as long as the interfacial strain minimizes the total energy, the exact atomic configuration of the interface has little influence on the band offset values. If the energy was not minimized, however, the band offsets were found to depend quite significantly on the atomic configuration of the interface. Variations in growth conditions and consequently in interfacial quality for the Ino,,,Gao,,,As/InP and 159J. M. Vandenberg, M. B. Panish, H. Temkin, and R. A. Hamm, Appl. Phys. Lett. 53, 1920 (1988). I6OJ.

C. P. Chang, T. P. Chin, K. L. Kavanagh, and C. W. Tu, Appl. Phys. Lett. 58, 1530 (1991).

S. Hybertsen, Phys. Rev. Lett. 64, 555 (1990). 16'M. S. Hybertsen, J. Vuc. Sci. Technol. B 8, 773 (1990). 163M.S. Hybertsen, Appl. Phys. Lett. 58, 1759 (1991).

161M.

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

51

-I I I I I I I I M As

M As In

P

In

P

I I I I I I I I U

InAs- like interface

I

I

'I I I I"I

As

M As

M

P

I I I'

In

P

In

U

MP-like

interface

FIG. 16. Schematic diagrams of atomic planes in the MAs/InP heterojunction, where M = Ino,53Gao,4,or Ino,52AIo,4,.The InAs-like interface structure is shown in (a), and the MP-like interface is shown in (b).

In0,, ,Al,,,,As/InP heterojunctions may therefore be partially responsible for the discrepancies found among the experimental band offset measurements for these material systems. It is perhaps suggestive that very few inconsistencies were found among the experimental band offset measurements for the Ino,,,Gao,4,As/Ino~,2Alo,48As heterojunction, for which interface strain is not a consideration. Further study of this issue will be required to determine conclusively the effect of interface strain and chemical intermixing on band offset values. e. Comparison with Theory Relatively few theoretical calculations of band offsets are available for the

Ino,,,Gao,,,As/Ino,,~Alo,4,As/InP material system, because most calculations have been performed only for elemental and pure (binary) compound semiconductors. The available theoretical predictions are summarized in ~ ~s calculated band offsets for this Fig. 15 and Table IV. H y b e r t ~ e n ' ~ ' - 'ha

52

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

material system using pseudopotentials in a local density functional approach; the alloys are treated in the virtual-crystal approximation. As discussed in the previous section, Hybertsen's calculations indicate that interface strain does not exert a significant influence on band offsets in the In,~,,Ga,.,,As/InP and Ino.,~A1,,,,As/InP heterojunctions, and he obtains valence-band offsets of 0.41 eV, 0.17 eV, and 0.25 eV for the and 1n0.52A10.48As/1nP 1n0.53Ga0.47As/1nP,1n0.53Ga0.47As/1n0.52A10.48As~ 4~ a valence-band offset of interfaces, respectively. Lambrecht et ~ 1 . calculated 0.22 eV for the In,~,Gao~,As/InPheterojunction using their self-consistent dipole theory. The interface bond polarity model of Lambrecht and Sega1P7 yields a valence-band offset of 0.17 eV for the In,,,Ga,,,As/InP interface. Predictions of other theories for band offset values in alloy heterojunctions can be obtained by interpolation from results for pure binary compounds. An energy level Ei in a ternary alloy A,B, -,C can be estimated as164 Ei(A,B, -,C) = xEi(AC)

+ (1 - x)Ei(BC)

where ai is the deformation potential for Ei and a, is the lattice constant. An interpolation from calculations for binary compound semiconductors using the model solid theory of Van de Walle and yields valenceband offsets AEv(In,,,3Ga,,4,As/InP) = 0.34 eV, AEv(Ino,,,Ga,,,,As/ = 0.21 eV, and AEv(In,~,~Al,,,,As/InP) = 0.13 eV. A similar Ino~47Alo,48As) procedure for the dielectric midgap energy theory of Cardona and C h r i ~ t e n s e n ~ yields ~ . ' ~ ~ valence-band offsets AEv(Ino.53Ga,,,,As/InP) = 0.26 eV, AEV(In0.,3Ga0.47As/In0.5 2A10.4&) = 0.15 eV, and AE,(In0,,,A1,,,,As/InP) = 0.11 eV. From these results we see that the best agreement with the accepted experimental values is given by the model solid theory of Van de Walle and Martin, although most of the other theories appear to agree with the experimental results to within the combined experimental and theoretical error bars.

Another nearly lattice-matched material system that has been a subject of considerable study is InAs/GaSb/AlSb. Interest in this material system arises to a considerable degree from the unusual band alignments that are obtainable. In particular, it has been well established through direct measurements IMM. Cardona and N. E. Christensen, Phys. Rev. B 37, 1011 (1988).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

53

of the InAs/GaSb band o f f ~ e t , ~163165 ~ ~ ~electrical ~,' transport studies in and the prediction and observaInAs/GaSb/AlSb hetero~tructures,~~~-~~~ tion of a semiconductor-semimetal transition in InAs/GaSb super lattice^^^^^'^^-' 77 that the InAs/GaSb interface is characterized by a type I1 broken-gap band alignment, as shown in Fig. 2c. This band alignment has been exploited in the development of novel electrical devices'67-' 72 and narrow-band-gap strained-layer superlattices for detection of infrared radiation,' 78-1 8 0 and for numerous studies of fundamental physical interest.170.173-177,181-184 Band alignments given by several experimental measurements are shown in Fig. 17, and the experimental results have been compiled in Table V. Theoretical band offset values for the InAs/GaSb/AlSb material system are summarized in Fig. 18 and Table VI. a. Experiment Explicit values for the InAs/GaSb band offsets have been obtained by a number of investigators. Sakaki et measured lattice constants and 165G.A. Sai-Halasz, L. L. Chang, J.-M. Welter, C.-A. Chang, and L. Esaki, Solid Stare Commun. 27, 935 (1978). 166H.Sakaki, L. L. Chang, R. Ludeke, C.-A. Chang, G. A. Sai-Halasz, and L. Esaki, Appl. Phys. Lett. 31, 211 (1977). 167M.Sweeny and J. Xu, Appl. Phys. Lett. 54, 546 (1989). 16'J. R. Soderstrom, D. H. Chow, and T. C. McGill, Appl. Phys. Lett. 55, 1094 (1989). L69L.F. Luo, R. Beresford, and W. I. Wang, Appl. Phys. Lett. 55, 2023 (1989). 170G.A. Sai-Halasz, R. Tsu, and L. Esaki, Appl. Phys. Lett. 30,651 (1977). I7'H. Munekata, T. P. Smith 111, and L. L. Chang, J. Vac. Sci. Technol. B 7, 324 (1989). I7'D. A. Collins, D. H. Chow, E. T. Yu, D. Z.-Y. Ting, J.R. Soderstrom, Y. Rajakarunanayake, and T. C. McGill, in "Resonant Tunneling in Semiconductors," ed. by L. L. Chang and E. E. Mendez (Plenum Press, New York, 1991). p. 515. 173L.L. Chang, N. J. Kawai, G. A. Sai-Halasz, R. Ludeke, and L. Esaki, Appl. Phys. Lett. 35,939 (1979). 174Y.Guldner, J. P. Vieren, P. Voisin, M. Voos, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 45, 1719 (1980). I7'J. C. Mann, Y. Guldner, J. P. Vieren, P. Voisin, M. Voos, L. L. Chang, and L. Esaki, Solid State Commun. 39, 683 (1981). 176G.Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, J. Vac. Sci. Technol. 21, 531 (1982). 177H.Munekata, E. E. Mendez, Y. Iye, and L. Esaki, Surf Sci. 174,449 (1986). I7'D. L. Smith and C. Mailhiot, J. Appl. Phys. 62, 2545 (1987). I7'C. Mailhiot and D. L. Smith, J. Vac. Sci. Technol. A 7,445 (1989). 'OD. H. Chow, R. H. Miles, J. R. Soderstrom,andT. C. McGill, Appl. Phys. Lett.56,1418(1990). "'L. Esaki, IEEE J. Quantum Electron. QE-22, 1611 (1986). IS2E.E. Mendez, L. L. Chang, C.-A. Chang, L. F. Alexander, and L. Esaki, Surf Sci. 142, 215 (1984). lS3S.Washburn, R. A. Webb, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 31, 1198 (1985). ls4E. E. Mendez, L. Esaki, and L. L. Chang, Phys. Rev. Lett. 55, 2216 (1985).

54

E.T. YU, J.O. McCALDIN AND T.C. McGILL

21

GaSb

2

W

P

GaSb

T

n

x

AlSb

lnAs

AEc(lnAs/AISb)

1

0)

K

w

*) ........................... 0.0

........................... .,..l .lll.l...9 ........l..l..l....l....l.l........ . *

FIG. 17. Summary of experimental band offset data for the InAs/GaSb/AlSb material system. Conduction band edges are represented by solid lines and valence band edges by dotted lines. For the InAs/GaSb heterojunction, data are from Sakaki et a1.'66 Sai-Halasz et al.'65 (0); (A).For the GaSb/AlSb Claessen et aLg3(m); Gualtieri et aL9' (0);and Srivastava et heterojunction, data are from Gualtieri et aLg9 (a); Yu et al.lgO( 0 ) ;MenCndez et al." (m); and Cebulla et al.'91 (0).The InAs/AlSb conduction-band offset value is from the measurement of Nakagawa et al. ''

(a);

TABLEV. EXPERIMENTAL VALENCE-BANDOFFSETSFORInAs/GaSb/AlSb

SOURCE

Sakaki et al. (1977)166 Sai-Halasz et al. (1978)'65 Claessen et al. (1986)93 Srivastava et al. (1986)'16 Gualtieri et al. (1987)98 Voisin et al. (1984)'" Tejedor et al. (1985)lS9 Gualtieri et al. (1986)99 Menendez et al. (1987)72 Cebulla et al. (1988)"l Yu et al. (1991)"O Nakagawa et al. (1989)"'

0.50 In As/GaSb 0.56 In As/GaSb 0.56 InAs/GaSb ~ ~ A ~ o , 9 s ~ b o , o s / G a ~0.67 b f 0.04 0.51 0.1 InAs/GaSb

AlSb/GaSb AlSb/GaSb AlSb/GaSb AlXGal,Sb/GaSb AlSb/GaSb AlSb/GaSb InAs/AlSb

-

0.04 - 0.08 >0.267 0.40 f 0.15 (0.45 0.08)~ 0.35 0.39 f 0.07

-

-

1.35 f 0.05

-

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

55

0.75

n

0.5

h

W

W’

Q

0.25

0 0.75

I

0.5 0

0.25

t

Ir

As/

h

0

W’

a

0.25

0

;

0

0

I

MF FK 1972 1977

L

T HT VWM MS DME LMTO SCD IBP 1986 1986 1987 1987 1987 1988 1990 1990

FIG. 18. A comparison of theoretical band offset values for the InAs/GaSb/AlSb material system. Different theories are plotted in approximately chronological order; the approximate range of experimental values throught to have been valid when each theory was developed has been shaded. Theoretical predictions are from the following sources: M F 1972, Ref. 71; FK 1977, Ref. 23; H 1977, Ref. 24; T 1986, Ref. 28; HT 1986, Ref. 29; VWM 1987, Ref. 84; MS 1987, Ref. 84; DME 1987, Ref. 39; LMTO 1988, Ref. 40;SCD 1990, Ref. 43; IBP 1990, Ref. 87.

56

E.T. YU, J.O. McCALDIN AND T.C. McGILL TABLE VI. THEORETICAL BANDOFFSET VALUES FOR InAs/GaSb/AlSb

SOURCE

Milnes and Feucht (1972)71 McCaldin et al (1976)'O Frensley and Kroemer (1977)23 Frensley and Kroemer (1977)23 (with dipole correction) Harrison (1977)24 Freeouf and Woodall (1981)lg3 Katnani and Margaritondo (1983)" Tersoff (1986)" Harrison and Tersoff (1986)" Van de Walle and Martin (1987)84 (self-consistent supercell) Van de Walle and Martin (1987)84 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Christensen (1988)40 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)87 (interface bond polarity model)

AE" (InAs/GaSb) (eV)

AE" (AlSb/GaSb) (eV)

AE" (InAs/AlSb) (eV)

0.52 0.41 0.49 0.26

0.51

0.01

0.05 0.73

- 0.47

0.52 0.54 0.12 0.43 0.33 0.38

-0.02 0.44

0.54 0.10

0.38 0.09 0.38

0.05 0.24

0.56

0.46

0.10

0.54

0.30

0.23

0.46 0.44

0.45 0.40

0.07

0.59

0.3 1

0.27

0.44

energy band gaps in In, -,Ga,As and GaSb,-,As, alloys and current-voltage characteristics in n-In, -,Ga,As/p-GaSb, -,Asy heterojunctions; these measurements yielded A e (EPSb- E r A s ) = 0.14 eV, corresponding to a va~ ~ lence-band offset AE,(InAs/GaSb) = 0.50 eV. Sai-Halasz et ~ 1 . 'measured optical absorption at low temperature in In, -,Ga,As/GaSb, -,Asy superlattices, from which they deduced A = 0.15 eV, corresponding to a valence-band offset for the InAs/GaSb heterojunction of 0.56 eV. Claessen et ~ 1 mea. ~ sured the pressure dependence of the InAs/GaSb band offset at 4.2 K, obtaining A = 0.15 eV at atmospheric pressure and dA/dP = - 0.0058 eV/kbar. Gualtieri et al.,' measured the InAs/GaSb valence-band offset by XPS, obtaining AE,(InAs/GaSb) = 0.51 f 0.1 eV. These results are ~ capaciall in very close agreement. However, Srivastava et ~ 1 . l 'measured tance-voltage characteristics for n-InAs,,,,Sb,~,,/n-GaSb heterojunctions and deduced a valence-band offset AEv(InAso,,,Sb,,,,/GaSb) = 0.67 f 0.04 eV, which differs significantly from the other reported results; the small amount of Sb in the InAso,,,Sbo,o, layer should not be sufficient to produce the variation observed here. The source of this discrepancy is not clear, but

~

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

57

the wealth of evidence supporting a valence-band offset AE,(InAs/GaSb) z 0.51 eV leads us to believe that value to be the most trustworthy. For the GaSb/AlSb heterojunction, a number of band offset measurements are available. On the basis of common-anion arguments, in early measurements it was often thought, incorrectly, that the valence-band offset should be small. As discussed by McCaldin et UZ.,*~ the presence of A1 in the heterojunction precludes the use of the common-anion rule for GaSb/AlSb. Mendez et al.lSSmeasured photoluminescence spectra from GaSb/AlSb superlattices that appeared to be consistent with a small valence-band offset ( f0.05 eV). Griffiths et al.ls6 obtained photoluminescence data from GaSb/AlSb multiple quantum wells that appeared to be consistent with a small valence-band offset, provided that a nonparabolic mode11S7was used for the conductionband structure. Voisin et al. lS8performed optical absorption measurements on GaSb/AlSb superlatices and obtained spectra consistent with valenceband offsets of 0.04 eV and 0.08 eV for the heavy-hole and light-hole valence bands, respectively (the valence bands are actually split because of the slight lattice mismatch between AlSb and GaSb, -0.6%). A number of other experiments, however, have provided quite convincing evidence that there is a substantial valence-band offset at the GaSb/AlSb used resonant Raman scattering to study the interface. Tejedor et al. electronic structure of GaSb/AlSb superlattices and concluded that AE,(GaSb/AlSb) > 0.267 eV. Gualtieri et aLg9 obtained AE,(GaSb/AlSb) = 0.40 f 0.15 eV using x-ray photoelectron spectroscopy, and Yu et ~ 1 . used ' ~ ~ XPS to obtain a valence-band offset AE,(GaSb/AlSb) = 0.39 f 0.07 eV. Band offset commutativity was verified in both XPS experiments. MenCndez et ~ 1 . ' used ~ a light-scattering method to obtain AEv = (0.45 f 0.08)~eV for GaSb/Al,Ga, -,Sb heterojunctions. Measurements of optical absorption and excitation in GaSb/AlSb multiple quantum wells by Cebulla et al.lgl yielded a valence-band offset AE,(GaSb/AlSb) = 0.35 eV. Finally, Beresford et ~ 1 . ' ~ ~ IS5E.E. Mendez, C.-A. Chang, H. Takaoka, L. L. Chang, and L. Esaki, J. Vac. Sci. Technol. B 1, 152 (1983). lS6G.Griffiths, K. Mohammed, S. Subbanna, H. Kroemer, and J. L. Merz, Appl. Phys. Lert. 43, 1059 (1983). 18'G. Bastard, Phys. Rev. B 24, 5693 (1981). ISsP.Voisin, C. Delalande, M. Voos, L. L. Chang, A. Segmuller, C.-A. Chang, and L. Esaki, Phys. Rev. B 30, 2276 (1984). lS9C.Tejedor, J. M. Calleja, F. Meseguer, E. E. Mendez, C.-A. Chang, and L. Esaki, Phys. Rev. B 32, 5303 (1985). lS0E.T. Yu, M. C. Phillips, D. H. Chow, D. A. Collins, M. W. Wang, J. 0. McCaldin, and T. C. McGill. Submitted to Phys. Rev. B. lglU.Cebulla, G. Trankle, U. Ziem, A. Forchel, G. Griffiths, H. Kroemer, and S. Subbanna, Phys. Rev. B 37, 6278 (1988). lg2R Beresford, L. F. Luo, and W. I. Wang, Appl. Phys. Lett. 55,694 (1989).

58

E.T. YU, J.O. McCALDIN AND T.C. McGILL

observed resonant tunneling of holes in GaSb/AlSb/GaSb/AlSb/GaSb heterostructures, consistent with a substantial valence-band offset, AE,(GaSb/AlSb) 0.4 eV. On the basis of these results, it appears that the GaSb/AlSb heterojunction is indeed characterized by a fairly large valenceband offset, AE,(GaSb/AlSb) % 0.4 eV. For the last remaining combination, InAs/AlSb, only a single band offset measurement has been reported. Nakagawa et al." used capacitancevoltage measurements to obtain a conduction-band offset AE,(InAs/AlSb) = 1.35 f 0.05 eV, corresponding to a valence-band offset AE,(InAs/AlSb) = 0.09 f 0.05 eV. Strain effects were not included in this measurement; however, the lattice mismatch of 1.1% between InAs and AlSb could exert a significant influence on band offsets in coherently strained InAs/AlSb heterojunctions. The relative positions of the energy bands in the InAs/GaSb/AlSb material system, based on the band offset measurements described in this section, are shown in Fig. 17. Combining the band offset measurements for the InAs/ GaSb, GaSb/AlSb, and InAs/AlSb heterojunctions, one can see that the transitivity rule is satisfied to a high degree of accuracy:

-

-

AE,(InAs/GaSb)

+ AE,(GaSb/AlSb) + AE,(AlSb/InAs) = 0.51 - 0.40 - 0.09 = 0.02 eV.

(10.1)

From Eq. (10.1) one can argue that deviations from ideal interface structure, if present, exert a relatively small influence on band offsets for this material system. One might have expected such effects to be particularly prominent for the InAs/GaSb interface because of the possibility of forming two distinct types of interface structures (InSb-like or GaAs-like). XPS measurements by Gualtieri et aLg8yielded a valence-band offset of 0.53 eV for InAs deposited on GaSb and a value of 0.48 eV for GaSb deposited on InAs. One might expect that the detailed structure of the interface could be somewhat different for these two growth sequences, and this result therefore suggests that interfacial strain at the InAs/GaSb heterojunction exerts a very limited ( <0.05 eV) influence on band offset values. However, the data of Srivastava et al.' l 6 yielded a valence-band offset AEv = 0.67 eV for the InAs/GaSb heterojunction; interface strain or possibly even a nonabrupt interface in the InAs/GaSb heterojunction may be at least partially responsible for the discrepancy between the results of Srivastava et al. and the other measurements that have been b. Theory Most of the available predictions for band offsets in the InAs/GaSb/AlSb material system are in good agreement with each other and with experimen-

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

59

tal measurements, the major exceptions being the predictions of early theories for the GaSb/AlSb and InAs/AlSb valence-band offsets. Theoretical band offset values for the InAs/GaSb/AlSb material system are summarized in Fig. 18 and Table VI. The pseudopotential theory of Frensley and K r ~ e m e yields r ~ ~ valence-band offsets of 0.49 eV, 0.05 eV, and 0.44 eV for InAs/GaSb, AlSb/GaSb, and InAs/AlSb, respectively. Harrison’s LCAO theory24predicts AE,(InAs/GaSb) = 0.52 eV, AE,(AlSb/GaSb) = -0.02 eV (valence-band edge of GaSb below that of AlSb), and AE,(InAs/AlSb) = 0.54 eV. Tersop’ obtains valence-band offsets of 0.43 eV for InAs/GaSb, 0.38 eV for AlSb/GaSb, and 0.05 eV for InAs/AlSb, whereas Harrison and T e r ~ o f fpredict ~~ AE,(InAs/GaSb) = 0.33 eV, AE,(AlSb/GaSb) = 0.09 eV, and AEv = (InAs/AlSb) = 0.24 eV. The dielectric midgap energy model of Cardona and Christensen3’ yields AE,(InAs/GaSb) = 0.54 eV, AE,(AlSb/GaSb) = 0.30 eV, and AE,(InAs/AlSb) = 0.23 eV, and the model solid theoryE6 predicts valence-band offsets of 0.56 eV for InAs/GaSb, 0.46eV for AlSb/GaSb, and 0.10eV for InAs/AlSb. The interface bond polarity model of Lambrecht and Sega1187 yields valence-band offsets of 0.59 eV, 0.31 eV, and 0.27 eV for the InAs/GaSb, AlSb/GaSb, and InAs/AlSb heterojunctions, respectively. The electron affinity ruleIg AE,(AlSb/AlSb) = 0.51 eV, and yields AE,(InAs/GaSb) = 0.52 eV, AE,(InAs/AlSb) = 0.01 eV using the electron affinity data of Milnes and Feucht,” and AE,(InAs/GaSb) = 0.54 eV, AE,(AlSb/GaSb) = 0.44 eV, and AE,(InAs/AlSb) = 0.10 eV using the compilation of Freeouf and W0oda11.’~~ The empirical compilation of Katnani and Margaritondo” predicts AE,(InAs/GaSb) = 0.12 eV, and the common anion rule2’ yields a valenceband offset of 0.41 eV for the InAs/GaSb heterojunction. Self-consistent calculations of Van de Walle and MartinE4 yield valence-band offsets of 0.38 eV for both the InAs/GaSb and the AlSb/GaSb heterojunctions, and self-consistent LMTO calculations by Christensen4’ predict AE,(InAs/GaSb) = 0.46 eV and AE,(AlSb/GaSb) = 0.45 eV. Self-consistent dipole calculations by Lambrecht et al.43 yield valence-band offsets of 0.44 eV for InAs/GaSb, 0.40 eV for AlSb/GaSb, and 0.07 eV for InAs/AlSb. V. Il-VI Material Systems

There has been considerable interest in 11-VI heterojunction systems both for long-wavelength infrared applications and for wide-band-gap visible light-emitting devices. However, with a few exceptions, e.g., HgTe/CdTe, far fewer measurements are available for 11-VI heterojunction systems than for 111-V interfaces. In this section we review the available data for band offsets in lattice-matched 11-VI heterojunction systems. lg3J.

L. Freeouf and J. M. Woodall, Appl. P h p . Lett. 39, 727 (1981).

60

E.T. YU, J.O. McCALDIN AND T.C. McGILL

1 1. HGTE/CDTE

HgTe/CdTe superlattices have been proposed as a promising material for fabrication of long-wavelength infrared detector^.'^^^'^^ Pure HgTe is a semimetal with an inverted band structure: the conduction band has Ts (light-hole-like) symmetry and is degenerate with the heavy-hole valence band at the Brillouin zone center; the light-hole band has Ts (conductionband-like) symmetry and is approximately 0.26 eV below the heavy-hole valence-band edge in energy. Figure 19 shows schematic energy band diagrams for HgTe and CdTe. Hg,-,Cd,Te alloys can have very narrow band gaps and are therefore of interest for long-wavelength infrared applications. Narrow-band-gap HgTe/CdTe superlattices were proposed as an alternative to Hg, -,Cd,Te alloys. The superlattice band gap is determined primarily by the HgTe layer thickness in the superlattice, whereas the Hg, -,Cd,Te alloy

HgTe

CdTe

FIG. 19. Schematic energy band diagrams for HgTe and CdTe. CdTe is characterized by a normal band structure, with a Tsconduction band and Ts light-hole and heavy-hole valence bands. HgTe is characterized by an inverted band structure. The Tsd i k e band is below the Ta valence bands, has negative effective mass, and acts as a filled valence band. The Tslight-hole band has a positive effective mass and acts as the empty conduction band, producing a semiconductor with zero band gap. 194J. N. Schulman and T. C. McGill, Appl. Phys. Lett. 34,663 (1979). 195D.L. Smith, T. C. McGill, and J. N. Schulman, Appl. Phys. Lett. 43, 180 (1983).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

61

band gap is determined by the alloy composition x. The superlattice structure would therefore allow better control over the detector cutoff wavelength, because the superlattice layer thickness can be controlled more easily than the composition of a comparable Hg, -,Cd,Te alloy.’” Properties of these superlattices, such as the superlattice band gap, can depend quite sensitively on the value of the HgTe/CdTe band offset,lg6 particularly if the valenceband offset deviates significantly from the very small discontinuity originally postulated on the basis of the common-anion rule.” a. Experiment The available experimental data for the HgTe/CdTe valence-band offset characare summarized in Fig. 20 and Table VII. Kuech and M~Caldin’’~ terized the electrical behavior of HgTe/CdTe lattice-matched heterojunctions grown by a low-temperature metal organic chemical vapor deposition technique. Photoresponse and J- Vmeasurements indicated that the HgTe/ CdTe heterojunctions exhibited Schottky-barrier-like behavior, with the barrier height, i.e., the conduction-band offset, ranging from 0.65 to 0.92 eV, depending on the effective donor concentration in the CdTe layer. This -.........................................

0-

-0.25

-

-0.5

-

Harrison /Guldner et al. kEerroir et al. C , hristensen -Model Solid Kowalczyk et al. Shih and Splcer Sporken ef al. \Duc el al. \Chow et 01. \Interface Bond Polarity \Larnbrecht et 01. Harrison and Tersoff Tersoff -Cardona and Christensen

2 ........................................

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

i

-0.751

HgTe

CdTe

FIG. 20. Summary of experimental and theoretical valence-band offsets for the HgTe/CdTe heterojunction. For the CdTe valence-band edge, experimental measurements and theoretical calculations are indicated by solid and dotted lines, respectively. Ig6G.Y.Wu and T. C . McGill, J. Appl. Phys. 58, 3914 (1985). ”’T. F. Kuech and J. 0. McCaldin, J. Appl. Phys. 53, 3121 (1982).

62

E.T. YU, J.O. McCALDIN AND T.C. McGILL TABLEVII. EXPERIMENTAL VALENCE-BAND OFFSETVALUES FOR CdTe/HgTe

SOURCE

Guldner et al. (1983)'98 Berroir et al. (1986)'99 Reno et al. (1986)200 Yang et al. (1988)201 Kowalczyk et al. (1986)"' Duc et al (1987)'" Shih and Spicer (1987)202 Chow et al. (1987)"' Sporken et al. (1989)206

1.6 1.6 2 5

0.04 & 0.01 0-0.100 0.04 0.063 L 0.005

300 300 300 300 50-300

0.35 & 0.06 0.36 f 0.05 0.35 0.06 0.39 & 0.075" 0.35 f 0.05

"Evidence was observed that the valence-band offset decreases at low temperature.

dependence was taken as evidence for the formation of an inversion layer a the HgTe/CdTe interface, and it was postulated that the actual barrier heigh was large, corresponding to a small HgTe/CdTe valence-band offset, and tha the inversion layer reduced the effective barrier height deduced from thc electrical measurements. Guldner et performed the first actual measurement of the band offse for HgTe/CdTe, obtaining a very small valence-band offset, AE, = 0.04 3 0.01 eV, from an analysis of far-infrared magnetoabsorption measurement on a semimetallic HgTe/CdTe (1 11) superlattice. Subsequent magnetoab sorption measurements by Berrior et uL199 on an n-type semiconductin; CdTe (111) superlattice produced results consistent with a valence-banc offset AEv = 0.04 eV and yielded lower and upper limits for the valence-banc offset of 0 eV and 0.1 eV, respectively. Reno et ul.zOOstudied the room temperature band gap as a function of HgTe layer thickness in HgTe/CdTN superlattices and obtained results consistent with a small valence-band offsel AEv = 0.04 eV. Yang and Furdyna"' deduced a valence-band offset AE, = 0.063 & 0.005 eV by analyzing far-infrared magnetoabsorption measure ments and relating the energy difference between the first heavy-hole subbanc 198Y.Guldner, G. Bastard, J. P. Vieren, M. Voos, J. P. Faurie, and A. Million, Phys. Rev. Lett. 51 907 (1983). 199J. M. Berroir, Y. Guldner, J. P. Vieren, M. Voos, and J. P. Faurie, Phys. Rev. E M ,891 (1986) zOOJ. Reno, I. K. Sou, J. P. Faurie, J. M. Berrior, Y.Guldner, and J. P. Vieren, Appl. Phys. Left.4! 106 (1986). "'Z. Yang and J. K. Furdyna, Appl. Phys. Left. 52,498 (1988).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

63

and the first light-hole subband in two HgTe/CdTe (100) superlattices to the valence-band offset. A number of measurements, however, have yielded substantially larger valence-band offsets for the HgTe/CdTe heterojunction. Kowalczyk et al. l o o used XPS to measure the HgTe/CdTe (TTT) valence-band offset directly and obtained AEv = 0.35 f 0.06 eV. Duc et al.lol also performed XPS measurements for the HgTe/CdTe heterojunction and obtained AEv = 0.36 f 0.05 eV. Shih and Spicer”’ used synchrotron photoemission spectroscopy to measure a “natural” valence-band offset, a quantity they claimed to be the change in the energy of the valence-band edge that is intrinsic to the bulk band structure of the heterojunction constituents; their measurements yielded a natural valence-band offset of 0.35 f 0.06 eV. Current-voltage measurements in Hg,, &do, ,Te/CdTe/Hg,, 22Te single-barrier heterostructures by Chow et ul.ll0 yielded a valence-band offset of 0.39 f 0.075 eV for the HgTe/CdTe heterojunction at 300 K. There was some evidence in these experiments that the valence-band offset might decrease at low temperature, possibly accounting for the discrepancy between the roomtemperature photoemission data and the low-temperature magnetoabsorption experiments. The observation by Chow et aL203of negative differential resistance in a HgCdTe/CdTe/HgCdTe single-barrier heterostructure also suggests that at low temperature the HgTe/CdTe valence-band offset is small-less than 0.1 eV. Faurie et aL2O4 also suggested that a temperature dependence of the valence-band offset might be responsible for the disagreement between the two types of experiments, and Malloy and Van Vechten”’ performed theoretical model calculations that predicted a strong temperature dependence for the HgTe/CdTe valence-band offset. Sporken et aL206 however, studied the temperature dependence of the HgTe/CdTe valence-band offset using XPS and UPS and found that the valence-band offset changed by only a few milli-electron-volts between 300 K and 50 K; their measurements yielded a valence-band offset of 0.35 L- 0.05 eV. A possible resolution of the conflicting results obtained in the photoemission and current-voltage measurements at room temperature and the low-temperature magnetoabsorption measurements has been proposed by Johnson, Hui, and E h r e n r e i ~ h . ~Calculations ~ ~ ~ ~ ’ ~ of superlattice band 202C.K. Shih and W. E. Spicer, Phys. Rev. Lett. 58, 2594 (1987). ‘03D. H. Chow, T. C. McGill, I. K. Sou, J. P. Faurie, and C. W. Nieh, Appl. Phys. Lett. 52, 54 (1988). ’04J. P. Faurie, C. Hsu, and T. M. Duc, J. Vuc. Sci. Technol. A 5, 3074 (1987). ’”K. J. Malloy and J. A. Van Vechten, Appl. Phys. Lett. 54,937 (1989). ’06R. Sporken, S. Sivananthan, J. P. Faurie, D. H. Ehlers, J. Fraxedas, L. Ley, J. J. Pireaux, and R. Caudano, J. Vac. Sci. Technol. A 7,427 (1989). ’”N. F. Johnson, P. M. Hui, and H. Ehrenreich, Phys. Rev. Lett. 61, 1993 (1988). *08P.M. Hui, H. Ehrenreich, and N. F. Johnson, J. Vuc. Sci. Technol. A 7, 424 (1989).

64

E.T. YU, J.O. McCALDIN AND T.C. McGILL

structure using an envelope-function a p p r o a ~ h ’ ~ demonstrated ~.~’~ that the magnetoabsorption data of Berrior et ul. 199 are actually better described by a large valence-band offset (-0.35 eV) than by the small valence-band offset ( 0.04 eV) that was proposed originally. The calculations of Johnson et al. showed that for the superlattice structure used in the experiments of Ref. 199 the superlattice is semiconducting, as observed experimentally, for small valence-band offsets; the superlattice becomes semimetallic as the valenceband offset is increased above 0.23 eV but reverts to semiconducting behavior for valence-band offsets greater than 0.295 eV. The superlattice band gap and electron cyclotron mass calculated for a valence-band offset of 0.35 eV were in good agreement with the experimental values reported by Berroir et ~ 1 . ” ~ The results reported to date for the HgTe/CdTe heterojunction indicate quite clearly that that valence-band offset at room temperature is large, AE,(HgTe/CdTe) = 0.35 eV. At low temperature, however, the situation is suggest that the more ambiguous. The measurements of Sporken et UI.’’~ HgTe/CdTe valence-band offset remains nearly constant as the temperature is decreased from room temperature to 50 K. Near 4.2 K, the measurements of Chow et u1.110*203suggest that the valence-band offset is small, but the magnetoabsorption data’99 from HgTe/CdTe superlattices appear to be consistent with both large (-0.35 eV) and small valence-band offsets.

-

TABLEVIII. THEORETICAL PREDICTIONS FOR CdTe/HgTe VALENCE-BAND OFFSET

THE

SOURCE Harrison (1977)24 Tersoff (1986)” Harrison and Tersoff (1986)29 Van de Walle and Martin (1987)84 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Christensen (1988)40 Langer and Heinrich (1988)*15 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall(1990)*’ (interface bond polarity model)

-- 0.0 1

0.51 0.49 0.23 0.61 0.22 0.35 0.48 0.44

’09N. F. Johnson, H. Ehrenreich, K. C. Hass, and T. C. McGill, Phys. Rev. Lett. 59,2352 (1987). ’‘ON. F. Johnson, H. Ehrenreich, G. Y. Wu, and T. C. McGill, Phys. Rev. B 38, 13095 (1988).

65

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

b. Comparison with Theory Theoretical values for the HgTe/CdTe valence-band offset are shown in Fig. 20 and Table VIII. Early treatments of the HgTe/CdTe heterojunction system typically assumed, on the basis of the predictions of the commonanion rule,” that the HgTe/CdTe valence-band offset was small. This was done despite the fact that in the original work of McCaldin et aLZ0it was not claimed that the common-anion rule would be applicable to compounds containing Hg. The original HgTe/CdTe superlattice p r o p o ~ a l ’ ~ ~as*~ sumed, on the basis of the electron affinity rule” and available electron affinity data for HgTe’” and CdTe,212,2’3that the HgTe/CdTe valenceband offset was zero. However, the available electron affinity measurements for these materials yield values for the HgTe/CdTe valence-band offset ranging from -0.1 eV (valence band of HgTe below that of CdTe) to 0.3 eV, suggesting that a substantial value for the valence-band offset might be plausible. Only a few of the early theoretical and empirical treatments of band offsets addressed the HgTe/CdTe heterojunction, and these typically predicted small valence-band offsets for HgTe/CdTe. If the atomic energies given by Harrison214are used to calculate valence-band-edge energies for HgTe and CdTe in Harrison’s LCAO theory,24 a valence-band offset of only -0.01 eV is obtained. As we have discussed, the electron affinity rule predicts valenceband offsets ranging from -0.1 eV to 0.3 eV. Later treatments typically yield substantially larger valence-band offset values. The theory of Tersoff,” developed concurrently with but indepen’ a valence-band dently of the XPS measurement of Kowalczyk et ~ l . , ’ ~ yields offset AE,(HgTe/CdTe) = 0.51 eV. The theory of Harrison and Tersoff” yields a very similar result, AE,(HgTe/CdTe) = 0.49 eV. The transition-metal impurity alignment postulated by Langer and Heinrich3’ predicts a valenceband offset AE,(HgTe/CdTe) = 0.35 eV, based on an alignment of the Fe impurity level in HgTe and CdTe.”’ The model solid theory of Van de Walle and Martin86 predicts a valence-band offset for HgTe/CdTe of 0.23 eV. The dielectric midgap energy model of Cardona and C h r i ~ t e n s e npredicts ~~ AE,(HgTe/CdTe) = 0.64 eV, but C h r i ~ t e n s e nhas ~ ~performed self-consistent

’’

21’J. N. Schulman and T. C. McGill, J. Vac. Sci. Technol. 16, 1513 (1979). 212N.J. Shevchik, J. Tejeda, M. Cardona, and D. W. Langer, Phys. Status Solidi ( B ) 59, 87 (1973). 213R.K. Swank, Phys. Rev. 153, 844 (1967). 214W. A. Harrison, “Electronic Structure and the Properties of Solids,” pp. 552-553, Dover Publications, New York, 1989. 215H.Heinrich, in “New Developments in Semiconductor Physics,” Proceedings of the Third Summer School Held at Szeged, Hungary (G.Ferenczi and F. Beleznay, eds), p. 126, SpringerVerlag, New York, 1988.

66

E.T. YU, J.O. McCALDIN AND T.C. McGILL

LMTO calculations of the electronic structure at the HgTe/CdTe heterojunction and found that, because of anomalous charge transfer associated with the formation of interface states, the dielectric midgap energy model should not be expected to be accurate. Christensen’s LMTO calculations yield a valence-band offset AE,(HgTe/CdTe) = 0.22 eV. The self-consistent dipole calculations of Lambrecht et a1.43yield a HgTe/CdTe valence-band offset 0: 0.48 eV, and the interface bond polarity model” derived from these calcula. tions predicts a value of 0.44 eV. The scatter among these theoretical values i! relatively large, but the qualitative conclusion that a substantial valence, band offset exists in the HgTe/CdTe heterojunction is strongly supported bj the more recent theories. 12. CDSE/ZNTE a. Experiment Although CdSe occurs naturally in the wurtzite crystal structure, i number of investigators have demonstrated that high-quality CdSe in thl cubic zincblende structure can be deposited by molecular-beam epitaxy 0 1 ZnTe with a lattice mismatch of only -0.44%.’16-218In addition, a numbe of studies of the CdSe/ZnTe heterojunction system were performed severa years prior to these demonstrations, but the samples used for these investiga tions were probably of low quality compared to those that can be grown b, current techniques such as molecular-beam epitaxy. The available experi mental data on the CdSe/ZnTe valence-band offset are included in Table 13 Gashin and Simashkevich’ 19,220 fabricated CdSe/ZnTe heterojunctions b evaporating CdSe onto (1 10) ZnTe, obtaining cubic CdSe with the sam crystallographic orientation as the substrate for substrate temperature ranging from 450 to 700°C. On the basis of electrical measurements and th electron affinity rule for conduction-band offsets,” Gashin and Simashke vich proposed a valence-band offset of -0.13 eV and a conduction-ban1 offset of -0.4 eV, corresponding to a type I band lineup. However, thej 2’6N.Samarth, H. Luo, J. K. Furdyna, S. B. Qadri, Y.R. Lee, A. K. Ramdas, and N. Otsuk Appl. Phys. Lett. 54, 2680 (1990). 217M.C. Phillips, E. T. Yu, Y. Rajakarunanayake, J. 0. McCaldin, D. A. Collins, and T. McGill, J. Cryst. Growth 111, 820 (1991). 218E.T. Yu, M. C. Phillips, J. 0. McCaldin, and T. C. McGill, J. Vuc. Sci. Technol. B 9, 22: (1991).

219P.A. Gashin and A. V . Simashkevich, Phys. Status Solidi ( A ) 19, 379 (1973). 220P.A. Gashin and A. V . Simashkevich, Phys. Status Solidi ( A ) , 19, 615 (1973). 221F.Buch, A. L. Fahrenbruch, and R. H. Bube, Appl. Phys. Lett. 28, 593 (1976). 222F.Buch, A. L. Fahrenbruch, and R. H. Bube, J. Appl. Phys. 48, 1596 (1977).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

67

TABLEIX. EXPERIMENTAL AND THEORETICAL VALENCE-BAND OFFSETS FOR CdSe/ZnTe

SOURCE

Buch et al. (1976)221.222 Yu et ul. (1991)218 Milnes and Feucht (1972)'l McCaldin et al. (1976)" Frensley and Kroemer (1977)23 Frensley and Kroemer (1977)23 (with dipole correction) Harrison (1977)24 Freeouf and Woodall (1981)193 Katnani and Margaritondo (1983)" Harrison and Tersoff (1986)29 Cardona and Christensen (1987)39

A 4 (expt.) (eV)

AE, (theor.) (ev)

0.23 0.64 f 0.07 0.89 0.55 0.55 0.64 0.85 0.86 0.35 0.31 0.61

estimate of the band offset assumed an electron affinity for CdSe of 3.93eV, whereas Milnes and Feucht" give an electron affinity of 4.95eV for CdSe. Buch et a1.221,222 studied photovoltaic properties of n-CdSe/p-ZnTe heterojunctions and proposed a CdSe/ZnTe band diagram with a valence-band offset of 0.23 eV and a conduction-band offset of 0.75 eV, corresponding to a type I1 band alignment. Several of these early investigations also yielded evidence for interdiffusion of atoms and the resulting formation of an intermediate compound or solid solution at the CdSe/ZnTe interface. However, these studies were performed using material prepared by relatively primitive methods, and the resulting samples were probably of poor quality compared to those grown by modern observed diffusion of Cd, Zn, and techniques such as MBE. Fedotov et Te atoms in CdSe/ZnTe and CdS/ZnTe heterojunctions and the resulting formation of a region of solid solution, with apparent diffusion lengths of up postulated the formation of a to several micrometers. Buch et a1.221,222 CdSe,-,Te, layer at the CdSe/ZnTe interface on the basis of evidence of absorption near the heterojunction interface at energies as low as 1.45eV. Photocurrent and cathodoluminescence studies by Simashkevich and T s i u l y a n ~suggested ~ ~ ~ that a CdSe/ZnTe solid solution was formed at the CdSe/ZnTe heterojunction interface. Finally, Senokosov and UsatyizZ5 2Z3Ya.A. Fedotov, S. G. Konnikov, V. A. Supalov, N. M. Kondaurov, A. N. Kovalev, and A. V. Vanyukov, translated from Izv. Akad. Nauk SSSR Neorg. Muter. 11, 2148 (1975). 224A.V. Simashkevich and R. L. Tsiulyanu, J. Cryst. Growth 35, 269 (1976). 225E. A. Senokosov and A. N . Usatyr, Sou. Phys. Semicond. 12, 575 (1978).

68

E.T. YU, J.O. McCALDIN AND T.C. McGILL

postulated, on the basis of electrical and photoresponse measurements, that a high-resistivity layer of CdTe was formed by interdiffusion at an n-CdSe/pZnTe heterojunction. The advent of crystal growth techniques such as molecular-beam epitaxq has enabled investigators to prepare samples of much higher quality than those used in the aforementioned experiments. Samples can be grown and characterized under carefully controlled conditions, and the resulting measurements are of much greater reliability. Yu et uLZ1’ have used XPS to measure the CdSe/ZnTe valence-band offset for structures grown by molecular-beam epitaxy. Pure ZnTe and cubic zincblende CdSe samples, and heterojunctions consisting of either CdSe deposited on ZnTe or ZnTe deposited on CdSe, were grown by molecular-beam epitaxy; XPS measurements on these samples yielded a valence-band offset AE, = 0.64 0.07 eV, corresponding to a conduction-band offset AEc = 1.22 eV and a type I1 band alignment. Commutativity of the band offset was confirmed, although there was also some evidence that the band offset can depend on growth conditions. We consider this measurement to be the most reliable determination ol the CdSe/ZnTe valence-band offset.

*

b. Theory Theoretical predictions for the CdSe/ZnTe valence-band offset extend over a wide range of values. Calculated band offset values for CdSe/ZnTe are included in Table IX. The electron affinity rule” predicts AE, = 0.89 eV using the electron affinity data of Milnes and Feucht7’ and AE, = 0.86 eV il the compilation of Freeouf and W ~ o d a l l ” ~is used. Harrison’s LCAO theory24yields a value of 0.85 eV for the CdSe/ZnTe valence-band offset, and Kraut’s adaptation of Harrison’s theory79 using Hartree-Fock neutral atom energies predicts AE, = 1.11 eV. Frensley and Kroemer’s pseudopotential theory yields a valence-band offset of 0.55 eV without interfacial dipole corrections and 0.64 eV with dipole corrections incl~ded.’~ The theory of Harrison and TersoP9 predicts AE, = 0.31 eV. The dielectric midgap energy model of Cardona and Christensen3’ yields a valence-band offset AE, = 0.61 eV.226Among the more empirical theories, the common-anion rule2’ predicts AE, = 0.55 eV, and Katnani and Margaritondo” obtain AE, = 0.35 eV. The number of theoretical predictions for the CdSe/ZnTe band offset is somewhat limited compared to the number available for other heterojunctions because of the natural wurtzite crystal structure of CdSe; in particular, self-consistent calculations for specific interfaces, such as those of Van de Walle and of C h r i ~ t e n s e n , ~ ’and . ~ ~ of Lambrecht et u1.,42*43 226N.E. Christensen, I. Gorczyca, 0.B. Christensen, U. Schmid, and M. Cardona, J. Cryst Growth 101, 318 (1990).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

69

0.00

Buch et 01.

-0.25 ........................................ ........................................

n

%

Horrison and Tersoff Kotnoni and Margaritondo

W

a, (r

V

-0.50 ........................................

Q)

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

V

c

McColdin. McGill and Mead [Frensley ond Kroemer ,Cardono and Christensen Yu et 01.

0

a

a,

-0.75

0

c

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

a, 0

>

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

,Harrison 'Freeouf and Woodoll Milnes and Feucht

- 1 .oo ........................................

Kraut

- 1.25

Z nTe

CdSe

FIG. 21. Valence-band offset values for the CdSe/ZnTe heterojunction predicted by several theories,'9~22~24~z9~39.71.79~193 and the valence-band offset values obtained from the measureand of Yu et a1.218 ments of Buch et a1.221,222

have not been published for the CdSe/ZnTe interface. Figure 21 shows the experimental CdSe/ZnTe valence-band offsets of Yu et a1.218and of Buch et a ~ * 2 2 ' , 2 2 2 amidst several theoretical predictions for the CdSe/ZnTe valenceband offset; Fig. 22 shows the band alignment corresponding to the measurements of Yu et al. for the CdSe/ZnTe heterojunction. 13. OTHER11-VI HETEROJUNCTIONS a. Dilute Magnetic Semiconductors Materials Hg, -,Mn,Te,

such as Cd, -,Mn,Se, Zn, -,Mn,Te, Zn, -,Mn,Se, and Hg, -,Mn,Se are members of a class of materials known

70

E.T. YU, J.O. McCALDIN AND T.C. McGILL

CdSe

ZnTe

tC

1.67eV

EV

1

Fv ‘

E

FIG. 22. Conduction- and valence-band alignments determined from the measurements of Yu et a1.218for the CdSe/ZnTe heterojunction. Yu et dobtained AEv = 0.64 f 0.07 eV, corresponding to AEc = 1.22 ? 0.07 eV and yielding a type 11, staggered band alignment. Taken from E. T. Yu, M. C. Phillips, J. 0. McCaldin, and T. C. McGill, J . Vac. Sci. Technol. B. 9, 2233 (1991).

as semimagnetic (or dilute magnetic) semiconductors. The presence of Mn in these materials gives rise to a number of interesting magnetic and magnetooptic properties. Cd, -,Mn,Te e p i l a y e r ~ , ~CdTe/Cd, ~ ~ . ~ ~ -,Mn,Te ~ super lattice^,^^^^^^^ ZnSe/Zn, -,Mn,Se,231 ZnCdSe/ZnMnSe,232and ZnTe/ MnTe h e t e r o s t r u ~ t u r e sand ~ ~ ~Hg, -,Mn,Te e p i l a y e r ~ ’have ~ ~ been grown successfully using molecular-beam epitaxy, and Cd, -,Mn,Te/Cd, -,Mn,Te 227L.A. Kolodziejski, T. Sakamoto, R. L. Gunshor, and S. Datta, Appl. Phys. Lett. 44,799 (1984). 228M.Pessa and 0. Jylha, Appl. Phys. Lett. 45, 646 (1984). 229R. N. Bicknell, R. W. Yanka, N. C. Giles-Taylor, D. K. Blanks, E. L. Buckland, and J. F. Schetzina, Appl. Phys. Lett. 45, 92 (1984). z30L.A. Kolodziejski, T. C. Bonsett, R. L. Gunshor, S. Datta, R. B. Bylsma, W. M. Becker, and N. Otsuka, Appl. Phys. Lett. 45, 440 (1984). z31Y.Hefetz, J. Nakahara, A. V. Nurmikko, L. A. Kolodziejski, R. L. Gunshor, and S. Datta, Appl. Phys. Lett. 47, 989 (1985). 23zW.J. Walecki, A. V. Nurmikko, N. Samarth, H. Luo, J. K. Furdyna, and N. Otsuka, Appl. Phys. Lett. 57,466 (1990). 233N. Pelekanos, J. Ding, Q. Fu, A. V. Nurmikko, S. M. Durbin, M. Kobayashi, and R. L. Gunshor, Phys. Rev. B 43, 9354 (1991). 234J. P. Faurie, J. Reno, S. Sivananthan, I. K. Sou, X.Chu, M. Boukerche, and P. S. Wijewarnasuriya, J. Vac. Sci. Technol. A 4, 2067 (1986).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

71

and Hg, -,Mn,Te/Cd, -,Mn,Te superlattices have been proposed as possibilities for magnetically turnable laser and infrared materials, respectively. A number of studies of CdTe/Cd, -,Mn,Te multiple quantum wells and superlattices have been reported. The lattice constant a, for Cd, -,Mn,Te decreases linearly with x and is given by a, = (6.487 - 0.149x)A. CdTe/Cd, -,Mn,Te heterojunctions can therefore be grown with little strain for low Mn concentrations, but for large values of x strain effects could be significant. The crystal structure of Cd, -,Mn,Te is cubic zincblende for x 5 0.7-0.75 and hexagonal NiAs for larger values of x. The energy band gap of Cd, -,Mn,Te increases with composition x; electroreflecyielded an energy band gap of approximately tance 1.50 1 . 4 4 ~eV at room temperature, and studies of piezomodulated and photomodulated reflectivity spectra from Cd, - ,Mn,Te yielded a free exciton energy 1.528 1 . 3 1 6 ~eV at 300 K.236 Experimental studies of band offsets in the CdTe/Cd, - ,Mn,Te heterojunction have generally yielded fairly small values for the valence-band offset. Pessa and Jylha228 studied angle-resolved photoemission from Cd,,,Mn,,,Te/CdTe interfaces and found that the energy of the valenceband maximum did not shift (within kO.05 eV) upon alloying, suggesting ~ ’ that the Cd,,,Mn,,,Te/CdTe valence-band offset is small. Shih et ~ 1 . ~used photoemission measurements on Cd, -,Mn,Te alloys to determine that the “natural” valence-band offset for the CdTe/MnTe heterojunction was zero to ~ ~ ’ within the limits of their experimental uncertainty. Chang et ~ 1 . performed photoluminescence excitation spectroscopy on CdTe/Cd, -,Mn,Te multiple quantum wells and for x = 0.24 deduced a type I band alignment with a heavy-hole valence-band offset = 0.025 eV, corresponding to a conduction-band offset of approximately 0.36 eV. Gregory et ~ 1 . ~studied ~ ’ photoluminescence spectra from CdTe/Cd, - ,Mn,Te multiple quantum wells in the presence of a magnetic field and for x = 0.25 obtained a valence-band offset in the range 0.030-0.055 eV, also with a type I band alignment. Deleporte et ~ 1 . ~studied ~ ’ photoluminescence and photoluminescence excitation spectra from CdTe/Cd,,,,Mn,,,,Te superlattices in the presence of a magnetic field and deduced a value for the valence-band offset, once the uniaxial strain

+

+

235N.Bottka, J. Stankiewicz, and W. Giriat, J. Appl. Phys. 52, 4189 (1981). 236Y. R. Lee, A. K. Ramdas, and R. L. Agganval, Phys. Rev. B 38, 10600 (1988). 237C.K. Shih, W. E. Spicer, J. K. Furdyna, and A. Sher, J. Vac. Sci. Technol. A 5, 3031 (1987). ’%.-K. Chang, A. V. Nurmikko, J.-W. Wu, L. A. Kolodziejski, and R. L. Gunshor, Phys. Rev. B 37, 1191 (1988). 239T. J. Gregory, C. P. Hilton, J. E. Nicholls, W. E. Hagston, J. J. Davies, B. Lunn, and D. E. Ashenford, J. Cryst. Growth 101, 594 (1990). 240E. Deleporte, J. M. Berroir, G. Bastard, C. Delalande, J. M. Hong, and L. L. Chang, Superlattices Microstructures 8, 171 (1990).

72

E.T. YU, J.O. McCALDIN AND T.C. McGILL

splittings were eliminated, of approximately 15-20% of the total band-gap used optical measurements to obtain difference. Finally, Pelekanos et band offset values AEc = 1.28 eV, AEhh= 0.34 eV, and AE,, = 0.16 eV for CdTe/MnTe quantum wells coherently strained to CdTe. In these studies, MnTe was grown in the cubic zincblende form rather than its natural crystal structure, hexagonal NiAs. Band offsets have also been reported for the ZnSe/Zn, -xMnxSe,231 Zn, -,Cd,Se/Zn, -xMnxSe,232and ZnTe/MnTe233 heterojunctions. Hefetz et ~ 1 . studied ~ ~ ' optical spectra from ZnSe/Zn, -,Mn,Se multiple quantum wells for x = 0.23 to x = 0.51. Measurements of optical spectra in the presence of a magnetic field suggested that the average valence-band offset AEv,aywas small, less than 0.02 eV for x = 0.23, at which composition the lattice mismatch between ZnSe and Zn, -,Mn,Se was estimated to be 1.05%; the band-gap difference for unstrained material at this composition was estimated to be approximately 0.11 eV. Walecki et aLZ3' used magnetooptical studies of nearly lattice-matched Zn - ,Cd,Se/Zn, - .Mn,Se quantum wells to deduce a valence-band offset of 0.030 rf: 0.005 eV. Pelekanos et al.233 studied luminescence spectra from a ZnTe/MnTe quantum well and deduced a valence-band offset AEv = 0.10 eV. However, the way in which strain effects were incorporated was not stated; given the lattice mismatch between ZnTe and cubic zincblende MnTe of approximately 4%, the valence-band splittings in the MnTe barrier layers would be expected to be quite large. Nevertheless, the estimate of Pelakanos et al. suggests that the valence-band offset is small compared to the total band-gap difference of approximately 0.8 eV between ZnTe and cubic zincblende MnTe. The band offset theory of TersoffZ8predicts values for the CdTe/MnTe and ZnTe/MnTe band offsets, although without including effects arising from strain. In these calculations, the cubic zincblende structure was assumed for MnTe and the Mn d orbitals were not allowed to mix with the conductionand valence-band states. For CdTe/MnTe and ZnTe/MnTe, valence-band offsets of 0.75 eV and 0.76 eV, respectively, were obtained. These values appear to be somewhat larger than those obtained experimentally, although it was noted in Tersoff's paper that the approximations made to treat MnTe would probably reduce the accuracy of the MnTe band offset values. The absence of strain effects in these predictions would also affect their expected accuracy; it has been noted by Cardona and Christensen3' that strain effects, including uniaxial strain components, can strongly influence even the average

,

241N.Pelekanos, Q. Fu, J. Ding, W. Walecki, A. V. Nurrnikko, S . M. Durbin, J. Han, M. Kobayashi, and R.L. Gunshor, Phys. Rev. B 41,9966 (1990).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

73

valence-band offset predicted by midgap energy theories for lattice-mismatched heterojunctions.

b. HgTelZnTe Experimental studies of band offsets for HgTe/ZnTe heterojunctions have been performed, although these studies have not explicitly included the effects of strain arising from the lattice mismatch of approximately 6.5% between ’ XPS to measure the HgTe/ZnTe (TIT) ZnTe and HgTe. Duc et ~ 1 . ’ ~ used valence-band offset without accounting for strain effects and obtained a value of 0.25 f 0.05 eV from their measurements; measurements for samples grown along the (100) orientation yielded a value of 0.36 eV. As discussed in more detail in Section 17, the experimental procedure followed by Duc et al. probably yields a rough estimate of the average valence-band offset for the performed photoemission meaHgTe/ZnTe heterojunction. Shih et a1.z37~242 surements on Hg, -,Zn,Te alloys and obtained a “natural” valence-band ~ ~ offset between HgTe and ZnTe of 0.17 f 0.06 eV. Marbeuf et ~ 1 . ’determined a “natural” valence-band offset for HgTe/ZnTe from XPS measurements of core-level energies over the entire compositional range of Hg, -,Zn,Te alloys and obtained the same value. In each case, the valenceband edge of HgTe was found to be higher in energy than that of ZnTe. Very few theoretical predictions are available for the HgTe/ZnTe valenceband offset, and the predictions that have been made are probably best considered as band offset values for hypothetical “unstrained” heterojunctions. For most theories, in which energy levels are calculated on an absolute energy scale, this value should correspond approximately to the average valence-band offset; for midgap energy theories such as that of Tersoff,” however, strain effects can shift the average valence-band offset in a strained heterojunction away from the hypothetical unstrained value. The electron affinity rulelg yields a valence-band offset of approximately -0.14 eV (valence-band edge of HgTe below that of ZnTe) using reported electron affinity data for HgTezlZ and ZnTe.71 Harrison’s LCAO theory24 predicts a valence-band offset of 0.17 eV when the atomic energies given by Harrisonz14 are used to calculate the position of the HgTe valence-band edge. Tersoff‘s theoryz8 yields BE, = 0.50 eV. The model solid theory of Van de Walle and Martins6 predicts an unstrained valence-band offset of 0.34 eV.

242C.K. Shih, A. K. Wahi, I. Lindau, and W. E. Spicer, J. Vac. Sci. Technol. A 6,2640 (1988). 243A.Marbeuf, D. Ballutaud, R. Triboulet, and Y. Marfaing, J. Cryst. Growth 101, 608 (1990).

74

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

VI. Strain Effects in Lattice-Mismatched Heterojunctions

Advances in epitaxial growth techniques have stimulated great interest in strained-layer heterostructures realized in lattice-mismatched material s y s t e m s . ’ 4 ~ ’ 8 ~ ’ 7 9 ~ 1 8 0 ~The 2 4 ~ 2study 4 7 of heterostructures in lattice-mismatched material systems involves a number of issues that do not arise for lattice-matched heterojunctions, chief among these being the conditions under which coherently strained epitaxial structures can be grown and the effect of strain on electronic structure in a coherently strained semiconductor heterojunction. 14.

INFLUEhCE OF STRAIN ON

ELECTRONIC STRUCTURE

The effects of strain on band offset values can be understood by examining first the effect of strain on the electronic structure of a bulk semiconductor. We use the Si/Ge material system to illustrate several specific points in this section; however, the ideas presented are applicable to all lattice-mismatched heterojunctions. In k * p theory, the band structure for a crystal with cubic symmetry near a level that is threefold degenerate (neglecting spin degeneracy) at the r point (k = 0) is given by a Hamiltonian of the formz4’

where the basis elements have been taken to be { X , Z } , valence-band wave functions that transform as p-like atomic orbitals. Bir and P i k ~ s ’have ~~ shown that, because strain in a cubic crystal can be described by a strain tensor cij having the same symmetry as the quadratic tensor kikj in E q . (14.1)7 the perturbation Hamiltonian describing strain effects can be written

244G.C. Osbourn, R. M. Biefeld, and P. L. Gourley, Appl. Phys. Lett. 41, 172 (1982). 2451. J. Fritz, L. R. Dawson, and T. E. Zipperian, Appl. Phys. Lett. 43, 846 (1983). 246D.L. Smith, Solid State Commun. 57, 919 (1986). 247D.L. Smith and C. Mailhiot, J. Appl. Phys. 63, 2717 (1988). 248E.0. Kane, J . Phys. Chem. Solids 1, 82 (1956). 249G. L. Bir and G. E. Pikus, “Symmetry and Strain-Induced Effects in Semiconductors,” p. 310ff, John Wiley and Sons, New York, 1974.

75

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

where I, m, and n are phenomenological deformation potentials. The more familiar deformation potential^^^^*^^^ a, b, and d are given by a=-

1

+ 2m

b=-

l-m

d=-

n

(14.3) 3 ’ 3 ’ $‘ The deformation potential a describes the effect of hydrostatic strain on the average position of the light-hole, heavy-hole, and split-off valence bands, SE,,,, = a(&,, + cyy + &J; b describes valence-band splittings induced by uniaxial strain along the [OOl] direction, and d describes splittings induced by [11 11 strain. To obtain the valence-band splittings with spin-orbit effects included, one must transform from the {X, X Z } basis to an angular momentum basis lj, m) given, for example, byz5’ ( 14.4)

(14.6)

1%

-5)

13, -3)

= (l/JZ)(X - iY)

= (l/$)[ZJ.

1,

+(X - iY)f].

(14.7)

(14.9)

Spin-orbit effects split the six bands into a fourfold degenerate p3,2 multiplet and a twofold degenerate pljZmultiplet; at the Brillouin zone center, 13, k3) correspond to the heavy-hole band, I$, ki) to the light-hole band, and If, ki) to the split-off band. In the presence of strain in the [Ool] direction, the positions of these three valence bands at the r point, relative to the average position of the three bands, (14.10)

AElh =

+ $SEoo, + +,/A; + Ao6Eool + $6E;,,,

(14.

+ @Eoo, - J1 A ; + Ao6Eoo, + $6Eio,,

(14.

- &Ao

AEso = -&Ao

250F.H. Pollak and M. Cardona, Phys. Rev. 172, 816 (1968). 251C.Kittel, “Quantum Theory of Solids,” p. 282, John Wiley and Sons, New York, 1963.

76

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

where A . is the spin-orbit splitting and 6Eoo, = 2b(c2, - cXJ. For strain in the [ l l l ] direction, the band edge positions are given by Eqs. (14.10), (14.1l), and (14.12) with 6Eoo, replaced by 6 E , , , = 2J? dc,,,. Strain-induced splittings for lattice-mismatched heterojunctions can be quite large. For example, experimentally measured values of the deformation potential b in si252 and Ge253yield linear multiplet splittings GEool that can be as large as a few tenths of an electron volt for highly strained Si and Ge crystals. To illustrate the effect of strain on electronic band structure, we again consider the Si/Ge material system; effects in other lattice-mismatched heterojunctions are similar. Figure 23 shows the valence-band structure in unstrained and strained Si and Ge. In Si/Ge heterojunctions, Si is typically under (in-plane) tensile strain, and Ge is typically under (in-plane) compressive strain. As shown in the figure, in-plane tensile strain shifts the light-hole band above the heavy-hole band, whereas for in-plane compressive strain the heavy-hole band is above the light-hole band in energy. The band structures in Fig. 23 are calculated for uniaxial strain in the [OOl] direction and are shown for k in the (001) direction in the Brillouin zone. For the conduction-band valleys, strain effects are described by the deformation potentials Ef;and El.,. The shift of a conduction-band valley k is given by2549255

+

AEt = Ef;sii Eke:ejkeij,

( 14.13)

where ek is the unit vector parallel to the k vector for valley k, and a sum is taken over repeated indices. From Eq. (14.13) it can be seen that the average posit.ion of the conduction-band edges for valley k is shifted by an energy (Ef; + the shift in energy of the average valence-band-edge position is acii, yielding a shift in the energy gap between the average positions of the conduction-band edges associated with the conduction-band valley k and of the valence-band edges AE,,,,

+

= (Sf;

- u)cii.

(14.14)

In addition, degeneracies of indirect conduction-band minima, e.g., the A valleys in Si or the Lvalleys in Ge, can be lifted by strain. Strain-induced conduction-band splittings can be calculated from Eq. (14.13). These considerations indicate that the effect of strain on the electronic band structure in bulk material is quite large; one would therefore expect 252L.D. Laude, F. H. Pollak, and M. Cardona, Phys. Rev. E 3,2623 (1971). 253M.Chandrasekhar and F. H. Pollak, Phys. Rev. B 15, 2127 (1977). 254C.Herring and E. Vogt, Phys. Rev. 101,944 (1956). 2551. Balslev, Phys. Rev. 143, 636 (1966).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

77

(a) Si: in-plane tensile strain u nstrained strained n

OV5

m

) . o

W

r

F -0.5

C W

so

-1

.o

0

0.1

0.2

-

(b) Ge: in-plane compressive strain strained unstrained 0.5

n

).

o -0.5

%

p - 1 .o W

~

C

-1.5

so

-2.0 0

0.1

0.2

0

0.1

0.2

FIG. 23. Effects of strain on bulk band structure in Si and Ge. For Si/Ge heterojunctions, Si is typically under in-plane tensile strain and Ge under in-plane compressive strain. For in-plane tensile strain, the light-hole band is above the heavy-hole band in energy; for compressive strain, the heavy-hole band is higher in energy. Band structures are calculated for uniaxial strain in the [00l] direction and are shown for k in the (001) direction in the Brillouin zone.

effects of at least comparable magnitude on the electronic structure at a semiconductor heterojunction interface. Calculations by Van de Walle and mar ti^^,^^.^^ in which the effect of strain on band offsets is considered explicitly, predicted that strain should exert a strong influence on band offset values in the Si/Ge heterojunction, and experimental results have verified this p r e d i c t i ~ n . ' ~ ~ -Experiments '~~ on other material systems, such as

78

E.T. YU, J.O. McCALDIN AND T.C. McGILL

In,Ga, -.As/GaAs, have also demonstrated the influence of strain on electronic structure in semiconductor heterojunctions. A few theoretical treatments, notably the model solid theory of Van de Walle and provide explicit formalisms for estimating band offsets in coherently strained heterojunction systems.

15. CRITICAL THICKNESS FOR STRAIN RELAXATION An additional issue that arises in studies of lattice-mismatched heterojunctions is the ability to grow actual coherently strained, dislocation-free heterojunctions. Each layer in a coherently strained heterostructure must be kept below the critical thickness for strain relaxation, beyond which strain in the layer will be relieved via the formation of dislocations. Several theoretical models have been developed to predict the critical thickness for and critical thicknesses for a variety of material strain ~ ~this ~ . section ~ ~ ~ ~we ~~ systems have been measured e ~ p e r i m e n t a l l y . In present a brief historical overview of the current understanding of critical thickness, from the early thermodynamic equilibrium theories of Van der Merwe through the current nonequilibrium theories in which critical thicknesses depend on conditions such as growth temperature. Experimental data that strongly support the concept of metastable strain states and nonequilibrium critical thicknesses are also discussed. The early theories of critical thickness, developed primarily by Van der M e r ~ e , ~assumed ~ ~ , ~that ~ ’the crystal would reach thermodynamic equilibrium and settle into the state of lowest energy. Hence, for film thicknesses 256J.H. Van der Merwe, J. Appl. Phys. 34, 117 (1963). 257J.H. Van der Merwe, J. Appl. Phys. 34, 123 (1963). “‘J. H. Van der Merwe and C. A. B. Ball, in “Epitaxial Growth” (J. W. Matthews, ed.), Part b, Academic Press, New York, 1975. 259J.W. Matthews and A. E. Blakeslee, J . Cryst. Growth 27, 118 (1974). 260R.People and J. C. Bean, Appl. Phys. Lett. 47, 322 (1985). 261B.W. Dodson and J. Y. Tsao, Appl. Phys. Lett. 51, 1325 (1987). 262B.W. Dodson and J. Y. Tsao, Appl. Phys. Left.52,852 (1988). 263J.Y. Tsao, B. W. Dodson, S. T. Picraux, and D. M. Cornelison, Phys. Rev. Lett. 59, 2455 (1987). 2MJ.C. Bean, L. C. Feldman, A. T. Fiory, S. Nakahara, and I. K. Robinson, J . Vac. Sci. Technol. A 2,436 (1984). 26sA.T. Fiory, J. C. Bean, R. Hull, and S . Nakahara, Phys. Rev. B 31,4063 (1985). 266R.H. Miles, T. C. McGill, P. P. Chow, D. C. Johnson, R. J. Hauenstein, C. W. Nieh, and M. D. Strathman, Appl. Phys. Lett. 52, 916 (1988). 267J.-P.Reithmaier, H. Cerva, and R. Losch, Appl. Phys. Lett. 54, 48 (1989). 268H.-J.Gossmann, G. P. Schwartz, B. A. Davidson, and G. J. Gualtieri, J. Vac. Sci. Technol. B 7, 764 (1989).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

79

below the critical thickness the film should be coherently strained to match the in-plane lattice parameter of the substrate, and above the critical thickness the mismatch should be accommodated by a network of misfit dislocations. The validity of Van der Merwe’s model depends, however, on the crystal being able to reach a state of thermodynamic equilibrium. It may be possible for crystals grown epitaxially at low temperatures to exist in a metastable state. Consider, for example, an interface for which the critical thickness t , has some finite value. When the thickness of the overlayer is less than t,, the film will be coherently strained. As the overlayer thickness increases beyond t,, the film will remain coherently strained until dislocations nucleate to accommodate the misfit. Formation of these dislocations, however, may often require a nonzero activation energy; if the temperature of the crystal is much lower than this activation energy, the crystal may either remain in the metastable, coherently strained state or else form fewer dislocations than needed to minimize the total energy for film thicknesses greater than the critical thickness. An alternate approach to calculating critical thicknesses was proposed by Matthews and Blake~lee.’~~ Rather than minimizing the total energy of strain and dislocations in a crystal, Matthews and Blakeslee considered the forces on dislocation lines in deriving an expression for the critical thickness. The critical thickness in their model depended on the balance between the tension of threading dislocation lines and the lateral force exerted by the misfit strain. For small misfit strain forces, the layers will remain coherently strained; for sufficiently large forces, however, the threading dislocation line will expand by elongating in the plane of an interface, producing a misfit dislocation line along the interface connecting the threading dislocations in each layer. When this occurs, the mismatch is no longer accommodated completely by coherent strain but is accommodated by a combination of misfit dislocations and strain. Another model for calculating critical thicknesses was proposed by People and Bean.260This model is similar to the original theory of Van der Merwe, in that the critical thickness is determined by a minimization of energy in the entire crystal (thermodynamic equilibrium), rather than by requiring that dislocation lines be in mechanical equilibrium. People and Bean considered the energy densities of various types of dislocations and assumed that all misfit dislocations would be of the type with the lowest energy-the screw dislocation. By equating expressions for the energy density in a strained crystal and the energy density in a crystal with misfit dislocations, an expression for the critical thickness was obtained. A phenomenological parameter related to the effective lateral extent of the strain field created by a dislocation was chosen to yield the best agreement with the experimental ~ ~ the resulting agreement with experimental data of Bean et ~ 1 . ’ Despite

80

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

measurements, the physical basis for this theory and to some extent all thermodynamic equilibrium theories appears to be somewhat incomplete. Experimental data indicate that factors such as growth temperature and the existence of metastable strain states are of central importance in the relaxation of strained films. Attempts have been made to include the effects of growth temperature and metastability in models of critical thickness. Dodson and Tsao2613262 have developed a model for strain relaxation in latticemismatched heterojunctions via plastic flow; combining this model with the effects of finite instrumental r e s o l u t i ~ n , ~ good ~ ~ ,agreement ~~~ with the ~ of Kasper et ~ 2 1 . ~was ~ ’ obtained. experimental data of Bean et ~ 2 1 . ’ ~and ~ measured the temperature dependence of strain relaxation Tsao et ~ 2 1 . ’ ~also in Si/Si,Ge, --x heterojunctions, and proposed a model for strain relaxation based on temperature and the difference between stress in the sample arising from misfit strain and that arising from dislocation line tension. A number of experiments have been performed in which critical thicknesses were measured experimentally. To test their predictions of critical thickness and strain accommodation by dislocations, Matthews and Blakeslee examined GaAs/GaAs,,,P,., superlattice samples with layer thicknesses ranging from 75 8, to 700A using transmission and scanning electron microscopy. They found that the critical thicknesses for generation of misfit dislocations were between 160 8, and 350 A; this result is in agreement with their theory, which predicts a critical thickness of about 250A for their structures. For layer thicknesses above 350 A, however, the amount of lattice mismatch accommodated by dislocations, as determined by measurements of Burgers vector orientations and average distances between dislocation lines, was found to be much smaller than predicted. This discrepancy suggests that processes inhibiting the formation of stable dislocations, e.g., interactions between dislocations and barriers for nucleation of dislocations, can be of ~ transmission electron considerable importance. Bean et ~ 2 1 . ’ ~performed microscopy, x-ray diffraction, and Rutherford backscattering studies of Si, - ,Gex/Si superlattices; the observed critical thicknesses were significantly larger than predicted by the equilibrium model of Van der Merwe. This result, like that of Matthews and Blakeslee, suggests that nonequilibrium effects such as barriers to dislocation formation can play a major role in determining critical thickness values. There exists considerable other evidence that nonequilibrium effects are indeed of great importance in strain relaxation. Studies by Fiory et ~ 2 1 . ~ ~ ~ have shown that annealing of Si, -,Ge, films grown on Si (100) substrates at 2691. J. Fritz, P. L. Gourley, and L. R. Dawson, Appl. Phys. Lett. 51, 1004 (1987). 2701. J. Fritz, Appl. Phys. Lett. 51, 1080 (1987). 271E.Kasper, H. J. Herzog, and H. Kibbel, Appl. Phys. 8, 199 (1975).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

81

550°C increases the number of strain-relieving dislocations in the film. These experiments strongly suggest that the as-grown Si,-,Ge, film exists in a metastable state and that annealing at 800-1000°C allows the film to relax gradually toward the minimum-energy equilibrium state. Tsao et ~ 1 . ’ ~ ~ measured the temperature dependence of strain relaxation in Si,Ge, - ,layers grown on Ge substrates and found that the onset of strain relaxation depended quite strongly on temperature. Studies by Miles et al.266 of Si/Ge,,,Si,,, superlattices demonstrated a dependence of critical thickness and dislocation density on growth temperature. For superlattices with the same layer thicknesses and same total thickness grown at substrate temperatures ranging between 365°C and 530”C,the dislocation density was shown to increase with growth temperature, indicating that the superlattices, as grown, were in metastable states with fewer dislocations than would have been necessary to minimize the total energy. These studies indicate that a satisfactory theory of critical thicknesses will need to account for the effects of dislocation interactions, barriers to dislocation formation, and other nonequilibrium processes that will, in general, depend on crystal growth parameters and on the growth temperature in particular. 16. SI/GE The Si/Ge (001) heterojunction has been chosen by a number of investigators as a prototypical material system in which to study the effects of strain on band offset values.37,389103,104 The Si/Ge material system is especially appropriate for this type of study, because of the large lattice mismatch (4.18%) between Si and Ge and because conditions under which coherently strained epilayers can be grown are well In addition, theoretical calculations of the Si/Ge (001) valence-band offset have been performed that explicitly incorporate the effects of these calculations have been confirmed by several experimental results’02-’04 indicating that strain strongly influences the value of the valence-band offset. The Si/Ge interface is also of great technological interest because of the possibility of integrating devices utilizing Si/Si, -,Ge, heterojunctions directly into existing Si-based structures. The behavior of Si/Si -,Ge, heterostructure devices depends critically on the values of the conduction- and valence-band offsets.

,

a. Experimental Measurements Most of the early Si/Ge band offset measurements were performed on heterojunctions in which the strain configuration was not known. A compilation of these early experimental results is included in Table X. Kuech et ul.”’ 272T.F. Kuech, M. Maenpaa, and S. S. Lau, Appl. Phys. Lett. 39, 245 (1981).

82

E.T. YU, J.O. McCALDIN AND T.C. McGILL TABLE x. EARLY EXPERIMENTAL AND THEORETICAL VALENCE-BAND OFFSETSFORSi/Ge SOURCE

Kuech et al. (1981)272 Margaritondo et al. (1982)273 Katnani and Margaritondo (1983)2’ Mahowald et al. (1985)274 Milnes and Feucht (1972)’l Harrison (1977)24 Tersoff (1986)” Harrison and Tersoff (1986)29

0.39 f 0.04 0.2 0.17 0.4 f 0.1 0.33 0.38 0.18 0.29

obtained AEv = 0.39 f 0.04 eV for Ge deposited on Si (001) from reversebias capacitance measurements. Using photoemission spectroscopy, Margar~ ~ AEv = 0.2 eV for Ge deposited on Si ( l l l ) , and itondo et ~ 1 . ’measured ~ ~ AEv = 0.4 0.1 eV for Si deposited on Ge Mahowald et ~ 1 . ’ obtained (1 11). Photoemission measurements of Katnani and Margaritondo” yielded AEv = 0.17 eV for both Si deposited on Ge and Ge deposited on Si. In all cases, the valence-band edge of Ge was found to be above that of Si. However, strain effects were not accounted for in these measurements, and subsequent experiments have demonstrated that strain exerts a profound influence on band offset values in Si/Ge heterojunctions. The first Si/Ge band offset measurements in which strain effects were considered were performed by Ni et al.loZ In these experiments, XPS was used to measure the Si 2 p and Ge 3d core-level binding energies relative to the conduction- and valence-band edges in heavily doped Si and Si, - %Gex samples with varying degrees of strain. p-type and n-type doping were used to move the Fermi level to the valence- and conduction-band edges, respectively, so that the measured absolute core-level binding energies were automatically referenced to the band edges; one to two monolayers of Sb or In were evaporated on each sample to prevent Fermi-level pinning and the concomitant band bending at the sample surfaces. The Si 2 p and Ge 3d core-level binding energies were then measured in Si/Si, -xGex heterojunctions and combined with the previous measurements to yield values for the conductionand valence-band offsets. This technique also allowed band gaps in strained Si and Si, -,Ge, samples to be determined. The strain-dependent band gaps 273G.Margaritondo, A. D. Katnani, N. G. Stoffel, R. R. Daniels, and T.-X. Zhao, Solid State Commun. 43, 163 (1982). 274P.H. Mahowald, R. S. List, W. E. Spicer, J. Woicik, and P. Pianetta, J. Vac. Sci.Technol. B 3, 1252 (1985).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

83

measured by Ni et al. were in good agreement with photocurrent measurements performed by Lang et al.,275and the conduction-and valence-band offsets measured for Si/Si0.74Ge0.26on Si, Si/Sio,52Geo,48on Si, and Si/Sio,52Geo,48on Sio,75Geo,25 were in good agreement with band offset values obtained theoretically by i n t e r p o l a t i ~ n from ~ ~ . ~the ~ ~ pure Si/Ge heterojunction calculations of Van de Walle and Martin.37s38For the case of Si/Sio,74Geo,26on Si, band offset values AEc = 0.00 f 0.06 and AEv = 0.18 f 0.06 eV were measured; for Si/Sio,52Geo,48on Si, AEc = 0.03 f 0.06 eV and AEv = 0.36 f 0.06 eV were obtained; and for Si/Si0.52Ge0.48on Sio,75Geo,25, AEc = 0.13 0.06 and AEv = 0.24 f 0.06 eV were measured. The strain dependence of the valence-band offset in pure Si/Ge heterojunctions has also been studied using XPS. In these experiments it was necessary to account explicitly for the strain dependence of the core-level and valenceband-edge binding energies.277Schwartz et u1.'O3 used XPS to measure Si 2 p to Ge 3d core-level binding energy separations in heterojunctions consisting of either six monolayers of Ge coherently strained to Si (001) or six monolayers of Si coherently strained to Ge (001). Because the samples needed to be transferred through atmosphere to the XPS chamber, each sample was capped with 12 monolayers of the unstrained material to prevent possible alterations in strain in the coherently strained layer arising from oxidation. Strain in these samples was determined from Raman scattering measurements; the results were not inconsistent with the heterostructures being coherently strained. Core-level to valence-band-edge binding energies for unstrained Si and Ge were obtained from previously published results, and the strain dependence of the core-level to valence-band-edge binding energies in bulk material was then calculated using a linear muffin-tin orbital method. The resulting strain-dependent core-level to valence-band-edge binding energies were then combined with the heterojunction measurements to obtain strain-dependent valence-band offsets for the Si/Ge (001) heterojunction. For Ge coherently strained to Si (001) and Si coherently strained to Ge (OOl), Schwartz et al. obtained valence-band offsets of 0.74 f 0.13 eV and 0.17 f 0.13 eV, respectively. Yu et al.104*105 also applied the XPS technique to measure the strain dependence of the Si/Ge (001) valence-band offset. In these experiments, Yu et ul. determined experimentally the strain dependence of the Si 2 p and Ge 3d core-level to valence-band-edge binding energies in Si and Ge, respectively, by measuring core-level to valence-band-edge binding energies in strained films of pure Si and pure Ge. Strain in these films was varied by growing layers of Si and Ge coherently strained to Si, -xGex alloy layers of varying 275D.V. Lang, R. People, J. C. Bean, and M. Sergent, Appl. Phys. Lett. 47, 1333 (1985). 276R.People and J. C Bean, Appl. Phys. Lett. 48, 538 (1986). 277J. Tersoff and C. G. Van de Walle, Phys. Rev. Lett. 59, 946 (1987).

84

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

composition, allowing the Si 2p and Ge 3d core-level to valence-band-edge binding energies to be measured directly as functions of strain. The Si 2p to Ge 3d core-level binding energy separation was then measured as a function of strain in Si/Ge superlattices; by varying the Si and Ge layer thicknesses in the superlattices, heterojunctions with different levels of strain were obtained, allowing the strain dependence of the heterojunction core-level energy separation to be measured. Strain configurations for all samples were measured using x-ray diffraction. The XPS measurements were then combined to yield strain-dependent values for the Si/Ge (001) valence-band offset. For Ge coherently strained to Si (001) and Si coherently strained to Ge (OOl), the measured valence-band offsets were 0.83 0.11 eV and 0.22 0.13 eV, respectively. The Si/Ge heterojunction measurements of Schwartz et al.’03 and of Yu et d 1 0 4 can be compared to the Si/Si,_,Ge, measurements of Ni et ~ 1 . ’ ~ ’ and to modulation doping results reported by People et ~ 1 . ’ and ~ ~ by ~ ~ an interpolation scheme adapted from the method Abstreiter et ~ 1 . ’using proposed by Van de Walle and Martin.38 TableXI shows band offsets measured for various Si/Si -,Ge, heterojunctions; for the alloy heterojunctions, band offsets interpolated from the Si/Ge measurements of Refs. 103 and 104 are listed. Figures 24 and 25 show conduction- and valence-band offsets for Si, -,Ge,/Si, -yGe, heterojunctions coherently strained to either Si (001) or Ge (OOl), obtained by interpolation from the measurements of Yu et d 1 0 4 Semiquantitative information about band offsets at Si/Si, -,Ge, heterojunctions has been deduced from modulation doping measurements in ~ ~ modulation doping Si/Si, -,Ge, heterostructures. People et ~ 1 . ’observed effects for holes in Si/Sio,,Ge0,, heterojunctions coherently strained to Si (001) substrates; this result, combined with their failure to observe modulation doping effects for n-type samples, indicated that BE, B AE, for Si/Si,,,Ge,,, heterojunctions coherently strained to Si (001). Interpolation from the Si/Ge (001) valence-band offsets measured by Yu et aL104 yields AE, = 0.16 eV and AEc = -0.02 eV for this heterojunction; these values are consistent with People’s results. The band alignment corresponding to the measured values of Yu et al. and the observed modulation doping behavior are shown schematically in Fig. 26a. In another experiment, Abstreiter et ~ 1 . ’ ’ ~ observed enhanced electron mobilities in Si/Si,,,Ge,,, superlattices coherently strained to a Sio,75Geo.z5 278R. People, J. C. Bean, D. V. Lang, A. M. Sergent, H. L. Stormer, K. W. Wecht, R. T. Lynch, and K. Baldwin, Appl. Phys. Lett. 45, 1231 (1984). 279G.Abstreiter, H. Brugger, T. Wolf, H. Jorke, and H. J. Herzog, Phys. Rev. Lett. 54, 2441 (1985).

E

U

TABLEXI.

EXPERIMENTAL BANDOFFSET MEASUREMENTS FOR VARIOUS Si/Si,-,Ge,

HETEROJUNCTIONS v)

M

AE" (ev)

4

AEc (eV)

v)

2

HETEROJUNCTION Ref. 102 Si/Ge on Si Si/Ge on Ge si/siO.

74Ge0. 2 6

Ref. 103" 0.74

-

0.13

Ref. 104"

Ref. 102

0.83 & 0.1 1

Ref. 103"

~

~

-

-

Ref. 104" ~

R 0

-

0.17 f 0.13

0.22 & 0.13

0.18 & 0.06

0.18

0.21

0.00

0.36 f 0.06

0.35

0.39

0.24

0.23

0.26

0.06

z ~

-0.05

-0.03

0.03 & 0.06

- 0.05

-0.01

0.13 & 0.06

0.10

U

s

on Si si/si0.52Ge0.48

on Si si/si0.52Ge0.48

0.06

On si0.75Ge0.25

"For alloy heterojunctions, the band offsets shown were obtained by interpolation from measured results for pure Si/Ge heterojunctions.

E4

0.14

zz 2 0

z

v)

86

E.T. YU, J.O. McCALDIN AND T.C. McGILL

AEc(Sil-xGex/Si,-yGey)

on Si (001)

Alloy composition x

on Si (001)

AEv(Si,-xGex/Sil-yGeY)

"0

0.2

0.4

0.6

0.8

1

Alloy composition x

FIG.24. Contour plots of conduction-band (upper) and valence-band (lower) offsets for Si, -.GeX/Si, -yGe, (001) alloy heterojunctions coherently strained to a Si (001) substrate, ' ~ an ~ interpolation calculated using Si/Ge (001) valence-band offsets measured by Yu et ~ 1 . and scheme adapted from that of Van de Walle and Martin.38 The signs of the band offsets are such that the valence- and conduction-band offsets are positive if the band edge in the Si, -yGeylayer is higher than in the Si,-,Ge, layer. In the contour plots, positive band offset values are indicated by the solid contour lines and negative values by the dashed lines.

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

AEc(Si,-xGex/Sil-yGey)

87

on Ge (001)

AEV(Sil-xGex/Si, -yGey) on Ge (001)

Alloy composition x

FIG.25. Contour F ts of conduction-band (upper) and valence-bani (lower) offsets for Si, -,GeJSi, -yGey (001) alloy heterojunctions coherently strained to a Ge (001) substrate, calculated using Si/Ge (001) valence-band offsets measured by Yu et al.'04 and an interpolation scheme adapted from that of Van de Welle and Martin3*The signs of the band offsets are such that the valence- and conduction-band offsets are positive if the band edge in the Si, -?GeYlayer is higher than in the Si,-,Ge, layer. In the contour plots, positive band offset values are indicated by the solid contour lines and negative values by the dashed lines.

88

(4

E.T. YU, J.O. McCALDIN AND T.C. McGILL

Si

(b)

Sio.8Geo.2

-0.02 eV

-___0.16 e V h

0000

E

E

Si

Sio.5Geos

q o Y+ ,+. v+.+ + 0.28 eV

EC

E

FIG.26. Schematic diagrams of modulation doping effects reported in the literature, and the band alignments obtained by interpolation from Si/Ge valence-band offsets measured by Yu et a1.1°4 For the heterojunction shown in (a), People et a1.278observed modulation doping effects when the Si layer was doped p-type but not when the doping was n-type, indicating that AE, + A E c . The band alignment shown in the figure is consistent with People’s results. For the (001) buffer layei, heterojunction shown in (b), which is coherently strained to a Si0.75Geo.25 Abstreiter et ~ 6 . ~observed ~ ’ enhanced electron mobilities when the Sio,,Ge,., layer was doped n-type. As seen in the figure, the band offsets obtained by interpolation are consistent with this observation.

(001) buffer layer when the Sio,,Geo,, superlattice layers were doped n-type. For this heterojunction system, interpolation from the measurements of Yu et a1.1°4 yields a valence-band offset of 0.28 eV and a conduction-band offset of 0.14 eV, consistent with Abstreiter’s results. The band alignment corresponding to the measured values of Yu et al. and the modulation doping behavior observed by Abstreiter et al. are shown schematically in Fig. 26b. b. Theoretical Calculations Early theoretical predictions for the Si/Ge band offset, shown in Table X, typically did not account for the effects of strain. The electron affinity rule,” applied using the electron affinity data of Milnes and F e ~ c h t , ~yielded ’ AEc = 0.12 eV and AE, = 0.33 eV for an unstrained Si/Ge heterojunction. Harrison’s LCAO theoryz4 yielded AE, = 0.38 eV, and Harrison and Teryielded AE, = soffs theoryz9 yielded AEv = 0.29 eV. Tersoffs 0.18 eV. The first theoretical treatment to consider explicitly the effects of strain on band offsets was that of Van de Walle and M a r t i r ~ , ~who ~ * ~performed * self-consistent local-density-functional calculations, using ab initio pseudopotentials, of the electronic structure of Si/Ge interfaces. Their results indicated that, in coherently strained heterojunctions, strain should exert a strong influence on band offset values; however, the discontinuity in the average position of the light-hole, heavy-hole, and split-off valence bands at the interface should be approximately independent of strain. For a Si/Ge

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

89

heterojunction coherently strained to Si (OOl), a valence-band offset of 0.84 eV was calculated; for a Si/Ge heterojunction coherently strained to Ge (OOl), a value of 0.31 eV was obtained. The discontinuity at the Si/Ge interface in the average position of the light-hole, heavy-hole, and split-off valence bands was calculated to be approximately 0.53 eV, and this quantity was found to be nearly independent of strain and substrate orientation. These predictions were subsequently confirmed by the experimental measurements described in the previous section. Figure 27 shows the conduction- and valence-band alignments predicted by Van de Walle and Martin for Si/Si, -,Ge, heterojunctions coherently strained to Si (001) (Fig. 27a) and to Ge (001) (Fig. 27b). Band alignments for pure Si/Ge heterojunctions were calculated explicitly, and results for Si, -,Ge, alloys were obtained by interpolation. The interpolation scheme was based on the observation that, in their calculations, the discontinuity across the Si/Ge interface in the average position of the light-hole, heavyhole, and split-off valence bands, AE,,,,, was nearly independent of strain and substrate orientation. AE",,, was taken to be a linear function of alloy composition, as shown in Fig. 27; this assumption was based on the validity of the model solid which yields valence-band offsets that are rigorously linear in alloy composition. The apparent linear dependence of the GaAs/Al,Ga, -,As valence-band offset on alloy composition, discussed in Section 8, also lends support to this assumption. The valence-band splittings were calculated using Eqs. (14.10)-(14.12), with values for 6Eool and A. for Si, -,Gex alloys obtained by linear interpolation from the values for pure Si and Ge. To obtain conduction-band offsets, the average position of the A conduction bands was determined in each layer from experimentally measured2*' band gaps, adjusted to account for hydrostatic strain via the deformation potential given by Eq. (14.14). The conduction-band splittings were calculated via Eq. (14.13), yielding the actual positions of the strain-split conduction-band edges. Conduction- and valence-band offsets can be read directly from Fig. 27. For heterojunctions coherently strained to Si (OOl), one can see that the conduction-band offsets are always quite small because of the splitting of the A conduction-band valleys, but that the valence-band offsets are enhanced by strain-induced valence-band splitting. For heterojunctions coherently strained to Ge (OOl), the conduction-band offsets are often somewhat larger than the valence-band offsets; in this case, the strain-induced splittings tend to reduce the valence-band offsets while increasing the conduction-band offsets.

"'R. Braunstein, A. R. Moore, and F. Herman, Phys. Rev. 109, 695 (1958).

90

E.T. YU. J.O. McCALDIN AND T.C. McGILL

Conduction Bands

- - / -

-I

I.0t 0.84

-

0.0

I , , , , , , , , , I

0

si

0

si

0.5 Ge fraction, x

0.5 Ge fraction, x

1

Ge

1

Ge

FIG.27. Conduction- and valence-band alignments for Si, -xGe, alloy layers coherently strained to (a) Si (001) and (b) Ge (001). Solid lines represent actual conduction- and valenceband edges, and dashed lines represent weighted averages of the light-hole, heavy-hole, and splitoff valence-band edges or of the A conduction-band edges. Taken from C. G. Van de Walle and R. M. Martin, Phys. Rev. B 34,5621 (1986).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

91

Another interpolation scheme, based on early calculations of Van de Walle and Martin37 that did not include spin-orbit splitting, was proposed by People and Bean.276In this approach, a bilinear interpolation, i.e., linear in heterojunction alloy composition and in substrate alloy composition, was used to obtain the valence-band offset at a Si/Si, -,Ge, heterojunction coherently strained to a Si,-,Ge, (001) substrate. Indirect band gaps for Si, -,Ge, layers with various levels of strain were determined using a phenomenological deformation potential theory,281 and these were used to obtain conduction-band offsets. This interpolation scheme and the method of Van de Walle and Martin both yield very similar results for alloy heterojunction band offsets. The interpolation scheme of People and Bean281predicts the following band offset values: A E , = 0.15 eV and A E , = -0.02 eV for Si/Sio.8Geo.2 coherently strained to Si (001); AE, = 0.37 eV and AE, = -0.02 eV for Si/Sio,,Geo,, coherently to Si (001); and A E , = 0.30 eV and AE, = 0.15 eV for Si/Sio,,Ge,,, coherently strained to Sio,75Geo,25. Van de Walle and Martin38 predicted the following band offsets for various Si/Si, -,Ge, heterojunctions: AE, = 0.17 eV and AE, = 0.00 eV for Si/Sio,,Ge0., coherently strained to Si (001); A E , = 0.38 eV and BE, = -0.02 eV for Si/Sio,,Geo,, coherently strained to Si (001); and AE, = 0.28 eV and AE, = 0.13 eV for Si/Sio,,Geo,, coherently strained to Sio,75Geo.25. A comparison with the results in TableXI shows that these theoretical values are very clearly confirmed by experimental measurements. The model solid theory of Van de Walle and can also be used to calculate band offsets for the Si/Ge heterojunction system. In the model solid theory, band-edge positions are calculated on an absolute energy scale, as described in Section 3; this model also allows absolute deformation potentials for hydrostatic shifts and uniaxial splittings to be calculated. For a strained crystal, the average valence-band-edge energy Eva,, is given by

where E:,,, is the average valence-band-edge energy in the absence of strain, Tr E = (E,, E,, E,,), and a, is the hydrostatic deformation potential for the average valence-band-edge energy. The valence-band splittings given by Eqs. (14.10)-(14.12) then yield the energies of the actual valence-band edges. Conduction-band offsets can be obtained using experimental values for energy band gaps and calculated hydrostatic and uniaxial deformation potentials for the conduction bands. The model solid theory yields AE, = 0.88 eV and AE, = 0.31 eV for Ge coherently strained to Si (001) and Si

+ +

'"R. People and J. C. Bean, Appl. Phys. Lett. 48, 538 (1986).

92

E.T.YU, 3 . 0 . McCALDlN AND T.C.McGlLL

coherently strained to Ge (OOl), respectively; these calculations are in good agreement with the full self-consistent interface calculations from which the model solid theory was derived.

The lattice-mismatched In,Ga, -,As/GaAs heterojunction has also been subject of considerable study; much of the interest in the In,Ga, -,As/GaAs material system is due to potential high-speed electronic and optoelectronic device applications. A number of studies of In,Ga, -,As/GaAs heterojunctions have also been directed toward achieving a basic physical understanding of strained-layer quantum-well and superlattice structures. a. Experiment Experimental measurements of the In,Ga, -$s/GaAs band offset are summarized in Table XII. The earliest experimental measurement of band offsets in a GaAs/In,Ga, -,As heterojunction was that of Kowalczyk et ~ l . , ’ ~ who measured the valence-band offset in the InAs/GaAs heterojunction, for which the lattice mismatch is approximately 7%, using XPS. In these experiments, the In 4d and Ga 3d core-level to valence-band-edge binding energies were measured in unstrained InAs and GaAs, respectively, and the In 4d to Ga 3d heterojunction core-level energy separation was measured in heterojunctions assumed to consist of InAs coherently strained to GaAs. Their measurements yielded an InAs/GaAs valence-band offset AE, = 0.17 f 0.07 eV, with the valence-band edge of InAs above that of GaAs. Although these experiments did not account for the effects of strain, one might hope that the valence-band offset measured in this way would still provide some information about band offsets in strained heterojunctions. This could be true because experiments on strained heterojunctions have indicated that hydrostatic shifts in core-level binding energies are relatively small compared to strain-induced splittings of the valence-band-edge Combining core-level energy separations measured in strained heterojunctions with core-level to valence-band-edge binding energies measured in bulk material should therefore yield an approximate determination of the average valence-band offset AE,,,,, once a correction for spin-orbit splitting has been included. Thus, it is possible that the valenceband offset determined by Kowalczyk et al. could be used to provide a 282W.-X.Ni and G. V. Hansson, Phys. Rev. B 42, 3030 (1990). z83R. W. Grant, J. R. Waldrop, E. A. Kraut, and W. A. Harrison, J. Vac. Sci. Technol. B 8, 136 ( 1990).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

93

TABLEXII. EXPERIMENTAL BANDOFFSETVALUES FOR In,Ga,,As/GaAs SOURCE

HETEROJUNCT~ON

RESULT

~

Kowalczyk et al. (1982)96

Marzin et al. (1985)284

Ji et al (1987)285

InAs/GaAs 15Ga0.85As/GaAs

AEv = 0.17 f 0.07 eV AEu = 0.052eV AE,, = - 0.018 eV AEc = 0.68AE8 AEc = 0.110eV AE,, = 0.060eV BE,, = - 0.019 eV AEc = 0.68AE, AEo = 0.126eV AE, = 0.70AEB

1

1

Ramberg et al. (1987)293

AEv = 0 . 9 7 ~eV x 0.8-0.9AEg

Mentndez et al. (1987)294

AEv = 0.035-0.041 eV = (0.60 & 0.04)AE8 AE,,, = (0.49 rl. 0.1)~eV

Joyce et al. (1988)295 Andersson et al. (1988)286

Pan et al. (1988)287

Yu et al. (1989)296 Niki et ol. (1989)”’

AEc = (0.36 & 0.12)AEg AEc = (0.62 f 0.08)AEB AEc = (0.83 f 0.06)AEB AEv = (0.30 k 0.05)AE, AEv = 0.6AE8 AEv = 0.3AE8 E8(In,Ga,-xAs) = 1.424 - 1 . 1 5 ~ eV AEc = 0.65AE,

Huang et al. (1989)289

AEc = 0.70AE8

Reithrnaier et al. (1990)290

AEo = 0.6AE,

Zou et al. (1991)291

AEo = 0.60AEB

Letartre et al. (1991)292

AEc = 0.141

0.06 eV x OSAE,

meaningful estimate of a quantity such as AE,,,, for the InAslGaAs heterojunction. Most measurements of band offsets in strained In,Ga, -,As/GaAs heterojunctions have been performed for heterostructures coherently strained to a GaAs (001) substrate; in this situation, the GaAs layer in the heterojunction is relaxed, and the In,Ga, - a s layer experiences biaxial compression. Such a scheme facilitates the growth of In,Ga, -,As/GaAs superlattice samples with varying quantum-well thicknesses but equal levels of strain, an important consideration in certain optical band offset measurement techniques. For samples in which only the In,Ga,-,As layer is strained, the heavy-hole and light-hole valence bands are degenerate at the zone center in the GaAs layers

94

E.T. YU, J.O. McCALDIN AND T.C. McGILL

(a)

GaAs InGaAs

(b)

GaAs InGaAs

FIG.28. Two types of band alignments that have been postulated for the In,Ga, -,As/Gak heterojunction strained to GaAs. In each case, uniaxial strain splits the In,Ga, -,As heavy-ho ( E v , J and light-hole (Ev,,,,) valence bands; the valence-band offset AEv is defined to be ti separation between the valence-band edge in GaAs and the heavy-hole valence-band edge 1 In,Ca, -,As.

and are split in the In,Ga, -,As layers. The conduction and valence band can be aligned at the heterojunction in a number of ways. Figure 28 shows th two types of alignments that have been postulated on the basis of experimer tal measurements. Marzin et U I . ~ ' ~ measured optical absorption spectra in In,Ga, -,As GaAs superlattices with x x 0.15 coherently strained to GaAs. The position of excitonic absorption peaks were fitted using the ratio of the conductior band offset to the band-gap difference, AEJAE,, as the only adjustabl parameter; the Bastard envelope function model,' 87 with strain effects addec was used to calculate the superlattice electronic structure, and an excito binding energy of 8 meV was assumed. For x = 0.17, Marzin et al. obtainel AE, = 0.126 eV, AEhh = 0.06 eV, and AE,h = -0.019 eV; for x = 0.15, th measured band offsets were b E , = 0.110 eV, AEhh = 0.052 eV, and AElh = -0.018 eV. These measurements correspond to AE,/AE, = 0.68. Ji et a1." measured low-temperature optical transmission spectra for In,Ga, -,As GaAs multiple quantum wells strained to GaAs substrates. Transmissioi spectra were measured for samples with varying In,Ga, -,As compositioi and quantum-well width, and the energies of the observed features wer compared to transition energies calculated theoretically with the ratio of th valence-band offset to the total band-gap difference, AE,/AE,, as an adjust 284J.-Y.Marzin, M. N. Charasse, and B. Sermage, Phys. Reo. B 31, 8298 (1985). zs5G. Ji, D. Huang, U. K. Reddy, T. S. Henderson, R.HoudrB, and H. MorkoC, J. Appl. Phys. 6: 3366 (1987).

BAND OFFSETS I N SEMICONDUCTOR HETEROJUNCTIONS

95

able parameter. For x = 0.13-0.193, Ji et al. obtained AEJAE, = 0.30, ~ ~ photoluminesindependent of composition. Andersson et ~ 1 . ' measured cence and photoconductivity spectra from In,Ga, - ,As/GaAs quantum wells coherently strained to GaAs (001) with compositions x = 0.073-0.36 and obtained AEJAE, = 0.83 f 0.06, independent of x. Pan et a1.287measured photoreflectance spectra for In,Ga, -,As/GaAs multiple quantum wells coherently strained to GaAs (001); these measurements yielded AE,/AE, = 0.30 f 0.05 for x = 0.1 1 and 0.12. Niki et a1.288 measured band gaps of strained In,Ga, -,As layers in In,Ga, -,As/GaAs multiple quantum wells coherently strained to GaAs and, for x = 0.12 to 0.25, obtained a linear relation between the composition and band gap, E,(In,Ga, -,As) = 1.424 1.15~ eV. In these experiments, the energy of the excitonic absorption feature was taken to be the band-gap energy; corrections for quantum confinement effects were estimated to be less than 3% of the total band gap. Combining these results with previously published measurements of the In,Ga, -,As/GaAs conduction-band offset, Niki et al. deduced AEc = 0 . 7 5 ~eV, or AEc/AE, = 0.65, independent of x. Photoluminescence measurements by Huang et al.289in In,Ga, -,As/GaAs quantum wells analyzed as a function of In,Ga, -,As composition and quantum-well width yielded a conduction-band offset AEJAE, = 0.70 for x = 0.09-0.20. Reithmaier et ~ l . ' ~ ' used optical absorption, photoluminescence, and electronic Raman scattering to determine band offsets in In,Ga, -,As/GaAs quantum wells with x ranging from 0.18 to 0.25, obtaining AEc/AE, = 0.6 independent of x. Photoluminescence measurements by Zou et ~ 1 . yielded ~ ~ ' AEJAE, = 0.60 for In,Ga, -,As/GaAs quantum wells coherently strained to GaAs with x = 0.09-0.28. Letartre et aLZ9' used deep-level transient spectroscopy (DLTS) and C-Vprofiling to determine the conduction-band offset in a GaAs/In,,,,Ga,~,,As/GaAs quantum well, obtaining AEc = 0.141 f 0.006 eV x 0.5 AE,. These measurements generally seem to indicate that, with the effects of strain included, the conduction-band offset is comparable to or somewhat larger than the valence-band offset for In,Ga, -,As/GaAs heterojunctions. 286T. G . Andersson, Z. G. Chen, V. D. Kulakovskii, A. Uddin, and J. T. Vallin, Phys. Rev. E 37, 4032 (1988). '"S. H. Pan, H. Shen, Z . Hang, F. H. Pollak, W. Zhuang, Q. Xu, A. P. Roth, R. A. Masut, C. Lacelle, and D. Morris, Phys. Rev. B 38, 3375 (1988) 288S. Niki, C. L. Lin, W. S. C. Chang, and H. H. Wieder, Appl. Phys. Lett. 55, 1339 (1989). 289K.F. Huang, K. Tai, S. N. G. Chu, and A. Y. Cho, Appl. Phys. Lett. 54,2026 (1989). 290J.-P.Reithmaier, R. Hoger, H. Riechert, A. Heberle, G. Abstreiter, and G . Weimann, Appl. Phys. Lett. 56, 536 (1990). 291Y. Zou, P. Grodzinski, E. P. Menu, W. G. Jeong, P. D. Dapkus, J. J. Alwan, and J. J. Coleman, Appl. Phys. Lett. 58, 601 (1991). 292X. Letartre, D. Stievenard, and E. Barbier, Appl. Phys. Lett. 58, 1047 (1991).

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E.T. YU. J.O. McCALDIN AND T.C. McGILL

A number of measurements have been reported, however, that appear to contradict this conclusion. Ramberg et ~ 1 . ” ~ measured current gain in npn GaAs/In,Ga, -,As/GaAs heterojunction bipolar transistor structures and eV for x = 0-0.08; this measurededuced a valence-band offset AE, = 0 . 9 7 ~ ment corresponds to AE, = = 0.8-0.9 AE,, using the dependence of the In,Ga, -,As energy gap on x measured by Niki et a1.288However, evidence was observed in these structures of some strain relaxation in the In,Ga, -,As layers, which would affect the values of the band offsets in these heterojunctions. Menkndez et ~ 1 . ” ~ used a light-scattering method to measure band offsets in In,Ga, -,As/GaAs quantum wells strained to GaAs for x = 0.05. By fitting the energies of the intersubband transitions by electrons in the In,Ga, -,As conduction band to a theoretical model with the ratio of the conduction-band offset to the total band-gap difference, AEc/AEg, as an adjustable parameter, Menendez et al. obtained AEJAE, = 0.40 k 0.04; their results corresponded to a valence-band offset AEv = 0.035-0.041 eV. These measurements, both for very low In concentrations in the alloy, suggest that the valence-band offset in In,Ga, - ,As/GaAs heterojunctions is larger than the conduction-band offset. Mentndez et al. also used their measurements to extrapolate a value for the average valence-band offset, AEv,av= (0.49 & 0.1)~ eV. A possible reconciliation of these results has been proposedzg5based on measurements that demonstrate a concentration dependence of the In,Ga, -,As/GaAs band offset. Joyce et al.295analyzed photoluminescence measurements for In,Ga, -,As/GaAs quantum wells and obtained conduction-band offsets AEJAE, = 0.62 k 0.08 for x = 0.12 k 0.002 and AEJAE, = 0.36 ? 0.12 for x = 0.04 f 0.01. Photocurrent measurements by Yu et aLZg6yielded AE,/AE, = 0.6 for x = 0.093 and AEv/AE, = 0.3 for x = 0.138. These two measurements suggest that the In,Ga,-,As/GaAs band offset might be concentration-dependent, and the concentration dependence measured by Joyce et al.295and Yu et aLZg6is consistent with other experimental measurements performed for more restricted compositional ranges.28~290~292-294 The compositional dependence of AE,/AE, observed by Joyce et aLZ9’and by Yu et aLZg6also casts doubt on the validity of extrapolating the average valence-band offset obtained by MenCndez et dZg4 for In,.o,Ga,,,,As/GaAs to the case of InAs/GaAs. If AEv/AE, is indeed 293L.P. Ramberg, P. M. Enquist, Y.-K. Chen, F. E. Najjar, L. F. Eastman, E. A. Fitzgerald, and K. L. Kavanagh, J. Appl. Phys. 61, 1234 (1987). 294J. Menendez, A. Pinczuk, D. J. Werder, S. K. Sputz, R. C. Miller, D. L. Sivco, and A. Y.Cho, Phys. Rev. B 36, 8165 (1987). 295M.J. Joyce, M. J. Johnson, M. Gal, and B. F. Usher,’Phys. Reo. B 38, 10978 (1988). 296P.W. Yu, G. D. Sanders, K. R. Evans, D. C. Reynolds, K. K. Bajaj, C. E. Stutz, and R. L. Jones, Appl. Phys. Lett. 54, 2230 (1989).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

97

dependent on composition, the extrapolated average valence-band offset AE,,,(InAs/GaAs) = 0.49 _+ 0.1 eV obtained by Menendez et al. would be substantially larger than the actual value. b. Theory Priester et al.297 calculated band offsets for GaAs/In,Ga, -,As heterojunctions using a self-consistent tight-binding method. These calculations yielded AE, = 0.020 eV for x = 0.05, which differs somewhat from the value ~ ~x = 0.15, Priester AEv = 0.035-0.041 eV measured by Mentndez et ~ 1 . ; ’for et al. obtained AEhh= 0.056 eV and AElh= -0.015 eV, in good agreement ~ the ~ ~ ~ ~ ~ ~ with a number of experimental m e a ~ ~ r e m e n t ~ .For pure GaAs/InAs heterojunction, Priester et al. obtained an average valenceband offset AE,,,, that depends slightly on strain. For InAs coherently strained to GaAs AEv,av= 0.090 eV, whereas AEv,av= 0.185 eV for GaAs coherently strained to InAs. These values are in reasonable agreement with the measurements of Kowalczyk et for InAs/GaAs heterojunctions, which would have yielded an average valence-band offset of approximately 0.16 eV, but disagree with the average valence-band offset obtained by extrapolation from the measurements of Mentndez et al.,294 AE,,,,(InAs/ GaAs) = 0.49 f 0.1 eV. In light of the possible compositional dependence of AEJAE, proposed by Joyce et aLZ9’and Yu et ~ l . , however, ’~~ the extrapolation performed by Menendez et al. of AEv,avfor InAs/GaAs from measurements on In,,,,Ga,,,,As/GaAs may be somewhat dangerous. On the basis of their calculations, Priester et al. also predicted that the band alignment for a GaAs/ln,Ga, -,As heterojunction strained to GaAs would be type I for both light and heavy holes when x = 1, as shown in Fig. 28a, but would be type I for heavy holes and type 11 for light holes when x = 0.15, as shown in Fig. 28b. The model solid theory of Van de Walle and Martina6 also yields predictions for the GaAs/In,Ga, -,As band offset. The model solid theory yields AEhh= 0.023 eV and AElh = -0.001 eV for x = 0.05; for x = 0.15, the predicted band offsets are AEhh= 0.068 eV and AElh= 0.002 eV. These predictions indicate that approximately 40% of the band-gap discontinuity appears as the valence-band offset and that the ratio AEJAE, depends only very weakly on x.These calculations are in reasonable agreement with ~ ~GaAs/ ’~~-’~~ the measurements of a number of i n v e s t i g a t ~ r s ~ ~ ~ ~ ’ ~for In,Ga,-,As heterojunctions with x ranging approximately from 0.09 to 0.28. The measurements of Anderson et aLZa6yielded a conduction-band offset somewhat larger than that predicted by the model solid theory, whereas 297C.Priester, G. Allan, and M. Lannoo, Phys. Rev. B 38, 9870 (1988).

98

E.T. YU, J.O. McCALDIN AND T.C. McGILL

Ramberg et al.293and Menendez et ~ 1 . obtained ’ ~ ~ conduction-band offsets considerably smaller than the values given by the model solid theory. The average valence-band offset AE,,,, predicted by the model solid theory for the pure GaAs/InAs heterojunction depends only slightly on strain. For InAs coherently strained to GaAs, AEv,av= 0.18 eV; for GaAs coherently strained to InAs, AEv,av= 0.16 eV. These compare quite favorably with the average valence-band offset AEv,av= 0.16 eV one would obtain from the measurements of Kowalczyk et a1.96 on pure InAs/GaAs heterojunctions but are in ~~ = 0.49 f 0.1 eV. disagreement with the value of Menendez et ~ l . , ’ AEv,av Again, however, one should recall the caveat regarding the extrapolation of the measurements of Menendez et al. from 1n0~,,Ga,,,,As/GaAs to InAs/ GaAs. Cardona and C h r i ~ t e n s e nhave ~ ~ also calculated values for the InAs/GaAs valence-band offset using their model based on dielectric midgap energies. For the InAs/GaAs heterojunction, their calculations yield an average valence-band offset AEv,av= 0.18 eV. This value is in good agreement with the value estimated from the measurements of Kowalczyk et uZ.,’~ AE,,,, = 0.16 eV. In making this comparison, it is necessary to note that the quantity measured by Kowalczyk et al. is probably representative more of the average valence-band offset AE,,,, than the actual valence-band offset AE, as had been assumed by Cardona and Christensen in comparing their calculations to experimental measurements. 18. OTHERLATTICE-MISMATCHED HETEROJUNCTIONS a. CdTejZnTe CdTe and ZnTe have cubic lattice constants of 6.48 A and 6.10A, respectively, yielding a lattice mismatch of approximately 6%.A number of studies have been performed of band offsets in CdTe/ZnTe heterojunctions, the general conclusion being that the average valence-band offset between CdTe and ZnTe is quite small. Menendez et ~ 1 . ’studied ~ ~ resonance Raman spectra from CdTe/ZnTe strained-layer superlattices grown on Cd,. ,Zn,,,Te buffer layers. Their experiments indicated that the heavy-hole wave functions were not well localized in the CdTe layers, suggesting that the CdTe/ZnTe valence-band offset was small. Duc et al.lol used XPS to measure the CdTe/ZnTe valence-band offset, not accounting for effects of strain, for the (TIT) orientation and obtained a value of 0.10 f 0.06 eV. A difference in the Cd 4d,,, to Zn 3d,/2 core-level energy separation of 0.047 eV was observed 298J. Menendez, A. Pinczuk, J. P. Valladares, R. D. Feldman, and R. F. Austin, Appl. Phys. Lett. 50, 1101 (1987).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

99

between the cases of CdTe grown on ZnTe and ZnTe grown on CdTe, suggesting that the core-level binding energies were not strongly affected by strain. For the (100) orientation, Hsu and FaurieZg9used the same technique to obtain a CdTe/ZnTe valence-band offset of 0.07 eV, with a difference in core-level energy separations for the two heterojunction growth sequences of 0.17 eV. As discussed in Section 17, the quantities measured by Duc et al. and by Hsu and Faurie might provide a rough estimate of the average valenceband offset AEv,avfor the CdTe/ZnTe heterojunction. For both orientations, this average valence-band offset appears to be considerably smaller than the total band-gap difference between unstrained CdTe and ZnTe of 0.75 eV. Mathieu et u1.300*301performed theoretical calculations of the subband structure for CdTe/ZnTe strained-layer superlattices, accounting for the effects of strain, and used their calculations to analyze photoluminescence spectra from a number of CdTe/ZnTe superlattices. Their analysis yielded zero-strain valence-band offsets (i.e., values obtained after band-edge shifts arising from both uniaxial and hydrostatic strain have been removed) of 0.060 f 0.020 eV300 and 0.055 f 0.040 eV,301depending on the parameters used to calculate the subband structure in the superlattices. Studies of optical spectra from CdTe/Cd, -,Zn,Te quantum wells and superlattices with x = 0.08-0.15 grown on Cd,,g,Zn,,,4Te buffer layers indicated that the average valence-band offset was small and that the superlattice band structure was actually type I for the heavy-hole valence band and type I1 for the light-hole valence band.3023303 These measurements all suggest that the valence-band offset for CdTe/ZnTe is quite small compared to the total band-gap difference between CdTe and ZnTe. Most band offset theories are in qualitative agreement with this conclusion. It is probably most appropriate to compare the early theories of band offsets, in which strain effects were not considered, to the average valenceband offsets deduced from experimental measurements. The electron affinity rule'' predicts valence-band offsets of 0.04 eV using the electron affinity data of Milnes and Feucht71 and 0.07 eV using the data compiled by Freeouf and Woodall.' 9 3 The common-anion rulez0 predicts that the CdTe/ZnTe valence-band offset should be zero. The theory of Frensley and KroemerZ3 yields a valence-band offset of -0.16 eV (type I1 band alignment), and Harrison's LCAO theory24 predicts 0.18 eV. Tersoff s interface dipole 299C.Hsu and J. P. Faurie, J. Vuc. Sci. Technol. B 6, 773 (1988). 300H.Mathieu, J. Allegre, A. Chatt, P. Lefebvre, and J. P. Faurie, Phys. Rev. B 38, 7740 (1988). 'O'H. Mathieu, A. Chatt, J. Allegre, and J. P. Faurie, Phys. Rev. B 41, 6082 (1990). 'O*H. Tuffigo, A. Wasiela, N. Magnea, H. Mariette, and Y. Merle d'Aubign6, Superlattices Microstructures 8, 283 (1990). 'O'Y. Merle d'Aubigne, H. Mariette, N . Magnea, H. Tuffigo, R. T. Cox, G. Lentz, L. S. Dang, J.-L. Pautrat, and A. Wasiela, J. Cryst. Growth 101, 650 (1990).

100

E.T. YU, J.O. McCALDIN AND T.C. McGILL

theory28predicts a valence-band offset of -0.01 eV, while the theory of Harrison and TersoffZ9yields -0.03 eV. The empirical results of Katnani and Margaritondo’l yield a CdTe/ZnTe valence-band offset of 0.10 eV. Among the more recent band offset theories, the model solid theory of Van de Walle and Martin yieldsE6average valence-band offsets of 0.05-0.08 eV and with strain splittings included predicts a type I band alignment for the heavyhole valence bands and a type I1 alignment for the light-hole bands, consistent with the experimental observations of Tuffigo et ~ 1 . ~ ~ ’ The available experimental and theoretical data on the CdTe/ZnTe band offset indicate that the average valence-band offset is quite small, in agreement with the general trend predicted for heterojunction systems with a common anion by early theories such as the common-anion rule and the LCAO theory of Harrison. Strain-induced splittings in the valence bands, however, could still produce substantial offsets for the individual valence bands in coherently strained CdTe/ZnTe heterojunctions. Because of the small average valence-band offset, the conduction-band offset should be quite large for all levels of strain; the model solid theory, for example, predicts CdTe/ZnTe conduction-band offsets ranging approximately from 0.36 eV to 0.54 eV, depending on the strain configuration of the heterojunction.

b. ZnSelZnTe

A few studies have also been performed of band offsets in the ZnSe/ZnTe material system; the lattice mismatch between ZnSe and ZnTe is approximately 7%. Experimental values for the average or in some cases “unstrained” valence-band offset for ZnSe/ZnTe are shown in Table XI11 and Fig. 29.Fujiyasu et d 3 0 4 studied photoluminescence from ZnTe/ZnSe superlattices grown on GaAs (100) substrates and deduced from the variation in photoluminescenceenergy with the ZnSe and ZnTe layer thicknesses that the used theoretical band alignment was type 11. Rajakarunanayake et uZ.305*306 calculations of band structure in ZnSe/ZnTe strained-layer superlattices to analyze photoluminescence data obtained by Kobayashi et d 3 0 7 Depending on whether the superlattice photoluminescence was attributed to Te, isoelectronic traps in ZnSe or to band-to-band transitions, valence-band offsets of 304H. Fujiyasu, K. Mochizuki, Y. Yamazaki, M. Aoki, A. Sasaki, H. Kuwabara, Y. Nakanishi, and G. Shimaoka, SurJ Sci. 174, 543 (1986). 305Y. Rajakarunanayake, R. H. Miles, G. Y. Wu, and T. C. McGill, Phys. Rev. B37,10212 (1988). 306Y.Rajakarunanayake, R. H. Miles, G. Y. Wu, and T. C. McGill, J. Vac. Sci.Technol. B 6,1354 (1988). 307M.Kobayashi, N . Mino, H. Katagiri, R. Kimura, M. Konagai, and K. Takahashi,Appl. Phys. Lett. 48, 296 (1986).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

101

0.00-

-0.25 Harrison ond Tersoff Frensley ond Kroemer

n

%

W

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

Kotnoni ond Morgoritondo

W

-0.50. ..............................

a

a,

McColdin. McGill and Meod

-0.75

0

C a, 0

>

Rojokorunonoyoke et 01. ..-Milnes ond Feucht

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

-1.00

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

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

/Harrison Fu et 01. ‘Von de Wolle ond Mortin \wu et 01.

- 1.25

ZnTe

ZnSe

FIG. 29. Summary of experimental and theoretical values for either the average or the “unstrained” valence-band offset in the ZnSe/ZnTe heterojunction.

AND THEORETICAL VALUESFOR EITHERAVERAGEOR TABLEXIII. EXPERIMENTAL “ UNSTRAINED” VALENCE-BAND OFFSETS FOR ZnSe/ZnTe

SOURCE

Rajakarunanayake et al. (1988)305.306 Wu et al. (1990)308 Fu et at. (1990)310 Milnes and Feucht (1972)’l McCaldin et al. (1976)’O Frensley and Kroemer (1977)23 Harrison (1977)24 Katnani and Margaritondo (1983)21 Harrison and Tersoff (198QZ9 Van de Walle and Martin (1987)86 (model solid theory)

A& (expt.1 (eV)

A 4 (theor.) (eV)

0.97 ?c 0.1 1.14 f 0.1

-

1.1

1.o 0.55

0.33 1.08 0.43 0.29 1.36 (“unstrained”) 1.10- 1.15 (average)

102

E.T. YU, J.O. McCALDIN AND T.C. McGILL

0.97 f 0.10 eV or 1.20 0.13 eV, respectively, were deduced. The valenceband offsets quoted by Rajakarunanayake et al. are for the average position of the heavy-hole and light-hole band edges in each material, a quantity that was assumed to be independent of strain; the actual discontinuities in the valence bands will depend on the strain-induced splittings in each layer. A similar analysis by Wu et aL308yielded an unstrained valence-band offset of ; ~ ~ ~ 1.136 f 0.1 eV, using photoluminescence data of Konagai et ~ 1 . their analysis assumed that the observed photoluminescence was attributable to excitonic recombination in the superlattice. Photoluminescence studies by Fu et aL3" also suggested that the band alignment for the ZnSe/ZnTe heterojunction was type 11; their measurements were consistent with the average valence-band offset predicted by Van de Walle and MartinE6 of approximately 1.1 eV. These measurements suggest that, before strain-induced valence-band splittings are included, the ZnSe valence-band edge is approximately 1 eV lower in energy than the ZnTe valence-band edge. However, uncertainties regarding the nature of luminescence from ZnSe/ZnTe superlattices, and in particular the role of Te isoelectronic traps in ZnSe,311,31zcould affect the band offset values extracted from these experiments. Theoretical predictions for the unstrained ZnSe/ZnTe valence-band offset extend over a wide range of values. Frensley and KroemerZ3 predict a valence-band offset of 0.33 eV without interfacial dipole corrections. Harrison's LCAO theoryz4 yields AE, = 1.08 eV. Tersoff's theoryz8 yields an unstrained valence-band offset of 0.86 eV, while Harrison and Tersoff obtain a value of 0.29 eV. The "unstrained" value from Tersoff's theory should be viewed with caution, however, because strain can shift the average valence-band offset to a value several tenths of an electron volt away from the simple unstrained band offset. The model solid theory of Van de Walle and Martin86 yields an unstrained valence-band offset of 1.36 eV. For actual strained ZnSe/ZnTe heterojunctions, the model solid theory yields average valence-band offsets of approximately 1.10- 1.15 eV. The electron affinity rule" predicts a valence-band offset of approximately 1 eV, and the common-anion ruleZoyields AEv = 0.55 eV. Finally, the empirical compilation of Katnani and Margaritondo" predicts a valence-band offset of 0.43 eV. It is apparent that many of the theoretical predictions for the ZnSe/ZnTe 'OSY. Wu, S. Fujita, and S. Fujita, J. Appl. Phys. 67, 908 (1990). 309M.Konagai, M. Kobayashi, R. Kimura, and K. Takahashi, J. Cryst. Growth 86, 290 (1988). 310Q.Fu, N. Pelekanos, W. Walecki, A. V. Nurmikko, S. Durbin, J. Han, M. Kobayashi, and R. L. Gunshor, Proceedings of the International Conference on Semiconductor Physics, Greece, p. 1353, 1990. "ID. Lee, A. Mysyrowicz, A. V. Nurmikko, and B. J. Fitzpatrick, Phys. Rev. Lett. 58, 1475 (1987). 312T.Yao, M. Kato, H. Tanio, and J. J. Davis, J. Cryst. Growth 86, 552 (1988).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

103

valence-band offset are in relatively poor agreement with each other and with the available experimental data. The theoretical models of Tersoff,28and Van de Walle and appear to yield values that are in good agreement with experiment, but the others deviate significantly from the experimental results. Again, however, it should be noted that most of the predicted band offsets do not incorporate the effects of strain. As discussed by Cardona and C h r i ~ t e n s e n strain , ~ ~ can strongly influence the value of even the average valence-band offset in certain models for predicting band alignments, such as their own dielectric midgap energy and the model of Tersoff.28 VII. Heterovalent Materlal Systems

Studies of heterovalent material systems, e.g., 111-V/II-VI or compound/ elemental semiconductor heterojunctions, are often complicated by chemical or structural imperfections at the heterojunction interface. Early theoretical studies by Harrison et d 3 I 3 demonstrated that intermixing is required in polar heterovalent interfaces, such as GaAs/Ge (Ool), to prevent large charge accumulations at the heterojunction interface. Experimental studies, particularly of 111-V/II-VI interfaces, suggest that interfacial reactivity may be endemic in heterovalent material systems. In this section we review the experimental data available on band offsets at heterovalent interfaces and discuss effects such as interfacial reactivity and antiphase disorder that can affect structural quality and band offset values in these material systems.

19. GAAs/GE a. Experiment The most widely studied of the heterovalent material systems has been GaAs/Ge. In one of the initial applications of the XPS technique for measuring band offsets, Grant et al.94 estimated a valence-band offset of 0.3 0.3 eV for Ge deposited on (110) GaAs. These experiments also revealed that the valence-band offset varied by as much as f O . l eV for Ge deposited on GaAs layers with different crystal orientations or surface reconstructions. A later, more refined measurement of bulk core-level binding energies for GaAs and Ge53 yielded a valence-band offset of 0.55 f 0.03 eV for GaAs/Ge (1 10) and valence-band offsets ranging from 0.48 eV to 0.66 eV for other GaAs substrate orientations and surface reconstruction^.^^ Bauer and McMenamin3l 4 used synchrotron radiation photoemission spectro'"W. A. Harrison, E. A. Kraut, J. R. Waldrop, and R. W. Grant, Phys. Rev. B 18, 4402 (1978). 314R.S. Bauer and .I. C. McMenamin, J. Vuc. Sci. Technol. 15, 1444 (1978).

104

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

scopy to measure valence-band offsets for Ge deposited on GaAs (1 10). For heterojunctions formed at substrate temperatures of 430°C and above, in which significant interdiffusion occurs, a valence-band offset of 0.2 k 0.1 eV was obtained; for abrupt interfaces obtained at a growth temperature of 35OoC, however, a valence-band offset of approximately 0.7 eV was measured. At an intermediate growth temperature of 420°C, Perfetti et ~ 1 . ~used ~ ' angle-resolved ultraviolet photoemission spectroscopy to measure BE, = 0.25 eV for Ge deposited on GaAs (110). These measurements demonstrate that the structural quality of the Ge overlayer and the detailed chemical nature of the interface can exert a substantial influence on band offset values. Other studies, however, suggest that the valence-band offset may be less sensitive to the conditions under which the interface is prepared than would be indicated by the aforementioned results. Katnani et u1.316-319used photoemission to obtain a valence-band offset of 0.47 f 0.05 eV for the GaAs/Ge heterojunction and performed a number of experiments indicating that the surface reconstruction on which the interface is grown, changes in interface orientation, and even the introduction of A1 interlayers at the GaAs/ Ge interface all shifted the value of the valence-band offset by not more than & 0.05 eV. Results of selected experimental band offset measurements for GaAs/Ge are shown in Fig. 30. Measured band offsets for purportedly abrupt GaAs/Ge heterojunctions are also compiled in Table XIV. Growth sequence can also influence the structural quality of a GaAs/Ge heterojunction and consequently the band offset value. Photoemission measurements by Zurcher and Bauer3" yielded a valence-band offset of 0.23-0.26 eV for GaAs deposited on Ge (1 lo), significantly lower than the values measured for abrupt interfaces formed by depositing Ge on GaAs (110). Structural studies of GaAs/Ge h e t e r o s t r u ~ t u r e sdemonstrated ~~~ that for growth of GaAs on (100) or (111) Ge, antiphase boundaries form in the GaAs layers and the GaAs surface becomes rough on an atomic scale; for growth of GaAs on (110) Ge, however, the GaAs surface was seen to remain fairly smooth, suggesting that if antiphase boundaries are formed, their effect on surface morphology is less significant than for the (100) and (111) N

315P.Perfetti, D. Denley, K. A. Mills, and D. A. Shirley, Appl. Phys. Lett. 33, 667 (1978). 316A.D. Katnani, P. Chiaradia, H. W. Sang, Jr., and R. S. Bauer, J. Vac. Sci. Technol. B 2, 471 (1984). 3L7A.D. Katnani, H. W. Sang, Jr., P. Chiaradia, and R. S. Bauer, J. Vac.Sci. Technol. B 3, 608 (1985). 318A.D., Katnani, P. Chiaradia, H. W. Sang, Jr., P. Zurcher, and R. S. Bauer, Phys. Rev. B 31, 2146 (1985). 319A.D. Katnani and R. S. Bauer, J. Vac. Sci. Technol. B 3, 1239 (1985). 320P.Zurcher and R. S. Bauer, J. Vuc. Sci. Technol. A 1, 695 (1983). 321C.-A.Chang and T.-S. Kuan, J. Vac.Sci. Technol. B 1, 315 (1983).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

GoAs 1.5

105

Ge

-I .......................... .......................

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

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

1

-2 1 W

w 0.5

1

-Chambers and Irwin et al. ‘Grant ‘Katnani and Bauer ‘Monch et al. ‘Katnoni and Margaritondo

FIG. 30. Band alignments obtained from experimental measurements on purportedly abrupt GaAs/Ge heterojunctions. Ge valence- and conduction-band edge energies are indicated by solid and dotted lines, respectively. TABLE XIV. EXPERIMENTAL BAND OFFSETS FOR PURPORTEDLY ABRUPT GaAs/Ge HETEROJUNCTIONS

SOURCE

Grant et al. (1978, 1980)94,53 Bauer et al. (197Q3I4 Monch et al. (1982)323 Katnani and Margaritondo (1983)” Katnani et al. (1984)316-319 Chambers and Irwin (1988)324

GaAs/Ge ( 1 10) GaAs/Ge (110) GaAs/Ge (1 10) GaAs/Ge (110) GaAs/Ge (IOO), (110), ( l l l ) , (211) GaAs/Ge (001)

0.55 & 0.03 0.7 0.42 f 0.1 0.35 0.47 f 0.05 0.60 0.05

orientations. Because the (100) and (111) surfaces of GaAs consist of alternating planes of pure Ga and As, the initial layer of GaAs on (100) or (111) Ge will have regions of pure Ga and of pure As; further growth of GaAs then leads to the formation of antiphase boundaries between these regions. For the (110) orientation, K r ~ e m e r ~ has ~ ’discussed a similar mechanism for ”’H. Kroemer, Surf: Sci. 132, 543 (1983).

106

E.T. YU, J.O. McCALDIN AND T.C. McCILL

the formation of antiphase domains. When Ge is deposited on GaAs, formation of antiphase domains is not a consideration and epilayers of high structural quality can be grown. A number of other measurements of the GaAs/Ge band offset have been ’ ~ photoemission to measure valence-band reported. Monch et ~ 1 . ~used offsets in GaAs/Ge (1 10) heterojunctions. For epitaxial Ge deposited on GaAs at 300°C, a valence-band offset of 0.42 k 0.1 eV was obtained; for amorphous Ge deposited at 20°C, the measured valence-band offset was 0.65 & 0.1 eV. Katnani and Margaritondo” used photoemission spectroscopy to measure the valence-band offset for Ge deposited on a cleaved GaAs used XPS to substrate and obtained AE, = 0.35 eV. Chambers and study GaAs/Ge (001) heterojunctions and obtained a valence-band offset of 0.60 k 0.05 eV. The GaAs/Ge band offset has also been determined from electrical measurements. Ballingall et al. studied J- Vcharacteristics for n-Gel n-GaAs heterojunctions and deduced conduction-band offsets of 0.08 k 0.01 eV3” and 0.05 k 0.03 eV,326corresponding to a valence-band offset of approximately 0.70 eV; however, some grading at the Ge/GaAs interface was thought to be present in those samples. Combining the measurements just described, it would appear that the experiments performed using heterojunctions thought to be abrupt and of high quality yield valence-band offsets ranging approximately from 0.35 eV to 0.7 eV, with the band alignment being type I. Figure 30 summarizes the GaAs/Ge energy band alignments measured by various investigators. It is quite clear from the wide range of band offset values reported for heterojunctions prepared under different conditions that the detailed atomic structure of the GaAs/Ge interface can have a substantial effect on band offset values. The dependence of the GaAs/Ge valence-band offset on growth conditions may be partially responsible for the rather wide range of experimental valenceband offsets reported even for supposedly abrupt GaAs/Ge heterojunctions. b. Theory Theoretical predictions for the GaAs/Ge valence-band offset will not, of course, account for the nonideal aspects of interfacial structure discussed earlier and are therefore best compared to the experimental measurements performed using abrupt, high-quality heterojunctions. For all of the predictions in the following discussion, the valence-band edge of Ge is higher in

323W.Monch, R. S. Bauer, H. Cant, and R. Murschall, J. Vac. Sci. Technol. 21, 498 (1982). 324S.A. Chambers and T. J. Irwin, Phys. Rev. B 38, 7484 (1988). 325J. M. Ballingall, R. A. Stall, C. E. C. Wood, and L. F. Eastman, J. Appl. Phys. 52,4098 (1981). 326J. M. Ballingall, C. E. C. Wood, and L. F. Eastman, J. Vac. Sci. Technol. B 1, 675 (1983).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

107

0.00

-0.25

lzEfAi ond -Pickett et

A

2

Morgoritondo

01.

W

W

Cordono and Christensen Lornbrecht et 01.

W

lnterfoce Bond Polarity

,” -0.50 -0

-Van de Wolle and Mortin \Harrison and Tersoff Milnes ond Feucht Y F r e n s l e y and Kroerner

c

0

13 a,

-0.75

0

C

W -

Boroff et

2

01.

-1.00

- 1.25

Ge

GaAs

FIG. 3 1. Theoretical values for the GaAs/Ge valence-band offset. The shaded region indicates the approximate range of experimental values reported for the valence-band offset in abrupt GaAs/Ge heterojunctions.

energy than that of GaAs. Theoretical values for the GaAs/Ge valence-band offset are shown in Fig. 31 and Table XV. Frensley and Kroemer’s pseudopotential theoryz3 predicts AEv = 0.71 eV, and Harrison’s LCAO theoryz4 yields AEv = 0.41 eV. Tersoffs interface dipole theoryz8 predicts a valenceband offset of 0.32 eV, and Harrison and Tersoffi9 obtain a value of 0.66 eV. The dielectric midgap energy model of Cardona and C h r i ~ t e n s e npredicts ~~ a valence-band offset of 0.45 eV, and the model solid theory of Van de Walle and Martins6 yields a value of 0.56 eV. The interface-bond-polarity model of Lambrecht and Sega1lS7predicts a valence-band offset of 0.48 eV. Among the empirically derived models, the electron affinity rule” yields a valence-band offset of 0.71 eV using the electron affinity data of Milnes and Feucht,’l and the empirical compilation of Katnani and Margaritondoz’ predicts a valence-band offset of 0.33 eV. In addition, a number of self-consistent interface and supercell calculations have been performed for the GaAs/Ge interface. Among the earliest of these calculations were those reported by Baraff

108

E.T. YU. J.O. McCALDIN AND T.C. McGILL

PREDICTIONS FOR THE GaAs/Ge TABLE XV. THEORETICAL VALENCE-BAND OFFSET SOURCE

Milnes and Feucht (1972)71 Baraff et al. (1977)33.34 Pickett e l al. (1977)35 Frensley and Kroemer (1977)23 Harrison (1977)24 Katnani and Margaritondo (1983)21 Tersoff (1986)28 Harrison and Tersoff (1986)29 Van de Walle and Martin (1987)84 (self-consistent supercell) Van de Walle and Martin (1987)86 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Christensen (1988)40 Lambrecht et a/. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)87 (interface bond polarity model)

'

h

0.71 0.9 0.35 0.71 0.41 0.33 0.32 0.66 0.63 0.56 0.45 0.46 0.45 0.48

and by Pickett et Baraff et performed calculations for the GaAs/Ge (100) interface and obtained a valence-band offset of -0.9 eV, and Pickett et ~ 1 obtained . ~ AE, ~ = 0.35 eV for the GaAs/Ge (110) interface. Self-consistent interface calculations by Van de Walle and Martin 84 yielded BE, = 0.63 eV. Self-consistent calculations performed by Christensen4' using LMTO methods applied in a supercell geometry yielded AE, = 0.46 eV, and ~ ' ~ ~a valence~ the self-consistent dipole theory of Lambrecht et ~ 1 . predicted band offset of 0.45 eV. The valence-band offsets predicted by most theories appear to be within the range of reported experimental values, although the experimental range is uncomfortably wide. Most of the available theories, despite their widely disparate physical justifications, predict values ranging approximately from 0.3eV to 0.7eV. This is in contrast to the situation for many other heterojunction systems, in which the predicted band offset values extend over a range of several tenths of an electon volt. However, the wide range of experimental values reported makes difficult any judgement regarding the relative accuracy of different band offset theories for the GaAs/Ge heterojunction. et

~

1

.

~

~

3

~

~

~

1

.

~

~

3

~

~

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

109

An added complication is that, as pointed out by Harrison et al.,j13 atomic intermixing must occur to prevent large charge accumulations at the GaAs/Ge interface. None of the theoretical treatments currently available account for this intermixing. In some sense, therefore, the heterojunctions treated theoretically may be quite different from those realized experimentally. Studies of other material systems such as A1Sb/GaSb/ZnTe'05~'90have demonstrated that interfacial reactivity can have a substantial effect on band offsets,but a quantitative understanding of these effects has yet to be attained. 20. GAAs/ZNSE

a. Experiment The GaAsfZnSe material system has been the most extensively studied of the lattice-matched 111-VfII-VI heterojunctions. Interest in 111-V/II-VI material systems arises from both the desire to use 111-V substrate materials for epitaxial growth of 11-VI semiconductors and the possibility of developing novel electronic and optoelectronic devices utilizing 111-VfII-VI heterojunctions. Several investigators have reported studies of the structural, chemical, and electronic properties of 111-VfII-VI interfaces, with the most effort having been devoted to the GaAsfZnSe material system. Tu and Kahnlz4 studied interfaces between GaAs and ZnSe or Se in the (100) and (1 10) orientations and observed evidence that a compound containing Ga and Se was formed at the GaAs/ZnSe interface. For ZnSe deposited on GaAs, Auger electon spectroscopy (AES) data indicated that, for both the (1 10) and (100) orientations, a Se-rich region of approximately 5 A thickness was formed at the interface. For the (110) orientation, annealing at 460°C transformed a 3 0 A ZnSe epilayer into Ga,Se, in the hexagonal wurtzite structure, as determined from low-energy electron diffraction (LEED) patterns. AES and LEED data for Se deposited on GaAs helped confirm that the intermediate compound was formed via Se-As interchange. K ~ b a y a s h ifound ~ ~ ~ that a Se treatment of Ga-terminated GaAs (001) surfaces at a substrate temperature of 500°C strongly suppressed Ga diffusion into ZnSe epilayers deposited on GaAs; AES analysis indicated that this suppression was due to the formation of several monolayers of GaAsSe on ~ ' GaAsfZnSe (100) the GaAs surface upon exposure to Se. Li et ~ 1 . ~studied interfaces using transmission electron microscopy and observed evidence of the presence of a Ga,Se, layer at the heterojunction interface, and Krost 327N.Kobayashi, Jpn. J. Appl. Phys. 21, L1597 (1988). 328D.Li, J. M. Gonsalves, N. Otsuka, J. Qiu, M. Kobayashi, and R. L. Gunshor, Appl. Phys. Lett. 57,449 (1990).

110

E.T. YU, J.O. McCALDIN AND T.C. McGILL

et aL3” observed evidence of the formation of Ga2Se3 at the GaAsJZnSe interface using Raman spectroscopy. In studies of GaAsJZnSe electrical were able to produce GaAsJZnSe interdevice structures, Qian et faces with interface state densities comparable to those found in GaAs/Al,Ga, -,As heterojunctions, and Qiu et a1.332demonstrated that the interface state density was strongly dependent on the stoichiometry of the GaAs surface on which ZnSe was deposited. Early measurements of the GaAsJZnSe valence-band offset suggested that the detailed structure of the heterojunction interface could have a substantial effect on the band offset value. Experimental measurements of the GaAsJ ZnSe band offsets are summarized in Fig. 32 and Table XVI. Kowalczyk ~

1

.

~

~

~

9

~

~

’

Eowolek and Wessels

01.

Kowalczyk et

01.

- 1.25 I

GaAs

ZnSe

FIG. 32. Reported experimental values for the GaAs/ZnSe valence-band offset. In all cases the band alignment at the GaAs/ZnSe heterojunction is type I. 329A.Krost, W. Richter, D. R.T. Zahn, K. Hingerl, and H. Sitter, Appl. Phys. Lett. 57, 1981 ( 1990). 330Q.-D.Qian, J. Qiu, M. R. Melloch, J. A. Cooper, Jr., L. A. Kolodziejski, M. Kobayashi, and R. L. Gunshor, Appl. Phys. Lett. 54, 1359 (1989). 33’Q.D. Qian, J. Qiu, M. Kobayashi, R. L. Gunshor, M. R.Melloch, and J. A. Cooper, Jr., J. Vat. Sci. Technol. B 7 , 793 (1989). 332J. Qiu, Q.-D. Qian, R. L. Gunshor, M. Kobayashi, D. R. Menke, D. Li, and N. Otsuka, Appl. Phys. Leu. 56, 1272 (1990).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

111

TABLEXVI. EXPERIMENTAL BANDOFFSET VALUES FOR GaAs/ZnSe

SOURCE ~~

Kowalczyk et al. (1982)65 Bawolek and Wessels (1985)334 Colak et al. (1989)333 Olego (1989)335.336 Kassel et al. (1990)j3'

0.96-1.10 ~

-

~

~

0.69 0.2-0.3 0.31 0.05 0.30 f 0.05

et ~ 1 . ~measured ' the GaAs/ZnSe ( 1 10) valence-band offset using XPS and found that the value depended on the method used to prepare the heterojunction. For ZnSe grown on GaAs at 300°C a valence-band offset of 0.96 eV was obtained, whereas for ZnSe deposited on GaAs at room temperature and crystallized at 300°C the measured valence-band offset was 1.10 eV. Electrical 1 for . metalln-ZnSe/n+-GaAs ~ ~ ~ measurements performed by Colak et ~ heterostructures yielded a conduction-band offset of approximately 0.2-0.3 eV, corresponding to a valence-band offset of approximately 1 eV. J - I/ measurements on n-n ZnSe/GaAs heterojunctions by Bawolek and W e ~ s e l yielded s ~ ~ ~ a conduction-band offset of 0.69 eV, corresponding to a valence-band offset of 0.55 eV. Olego33s,336measured a conduction-band offset of approximately 0.31 f 0.05 eV, corresponding to AE, = 0.97 eV, for the GaAs/ZnSe (100) heterojunction using a Raman scattering technique. Finally, optical studies of ZnSe/GaAs heterojunctions performed by Kassel et ~ 1 yielded . a~ conduction-band ~ ~ offset AEc = 0.30 k 0.05 eV, corresponding to AE, = 0.98 eV. In all of these measurements the band alignment was type I, and most of the experimental band offset measurements yielded results very close to AE, = 0.97 eV. One explanation proposed by Bawolek and W e s s e l ~for ~ ~the ~ discrepancy between their measurements and those of other investigators was the possible presence of a high interface state density in their heterojunctions.

-

b. Theory A large number of theoretical predictions are available for the GaAs/ZnSe valence-band offset. In all cases, the GaAs valence-band edge is predicted to 333S.Colak, T. Marshall, and D. Cammack, Solid-state Electron. 32, 647 (1989). 334E.J. Bawolek and B. W. Wessels, Thin Solid Films 131, 173 (1985). 335D.J. Olego, Phys. Rev. B 39, 12743 (1989). 336D.J. Olego and D. Carnmack, J. Cryst. Growth 101, 546 (1990). 337L.Kassel, H. Abad, J. W. Garland, P. M. Raccah, J. E. Potts, M. A. Haase, and H. Cheng, Appl. Phys. Lett. 56, 42 (1990).

112

E.T. YU, J.O. McCALDIN AND T.C. McGILL

be higher in energy than that of ZnSe, with most of the theories yielding a type I band alignment. Theoretical values for the GaAs/ZnSe valence-band offset are summarized in Fig. 33 and Table XVII. The pseudopotential theory of Frensley and KroemerZ3predicts a valence-band offset of 1.11 eV without interfacial dipole corrections and 1.21 eV if dipole corrections are included, both corresponding to a type I band alignment. Harrison’s LCAO theoryz4 predicts AEy = 1.05 eV, yielding a type I band alignment. Tersoffs interface dipole theory2* predicts AEv = 1.20 eV and a type I alignment, and Harrison and Tersoff s theory” yields a valence-band offset of 1.35 eV, corresponding to a type I1 staggered band alignment. Cardona and Christensen’s dielectric 0.00

-0.25

-2

-0.50

v

e,

,”

-0.75

Q)

McColdin, McGill, and Mead -0 C 0

13 ~

-1.00

V

C e,

-

F

>”

F

r

e

n

Horrison Christensen ‘Kotnoni and Morgoritondo s l e y and Kroerner Lornbrecht et 01.

- 1.25

-1.5 Von de Wolle and Martin

-1.75

GaAs

ZnSe

FIG.33. Theoretical values for the GaAs/ZnSe valence-band offset. The shaded region indicates the approximate range of experimental values reported for the GaAs/ZnSe valenceband offset.

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

113

TABLEXVII. THEORETICAL PREDICTIONS FOR THE GaAs/ZnSe BANDOFFSET AE: (theor.) (eV)

SOURCE

Electron affinity rule (1972)" McCaldin et al. (1976)20 Frensley and Kroemer (1977)23 Frensley and Kroemer (1977)23 (with dipole correction) Harrison (1977)24 Katnani and Margaritondo (1983)21 Tersoff (1986)" Harrison and Tersoff (1986)29 Van de Walle and Martin (1987)84 (self-consistent supercell) Van de Walle and Martin (1987)86 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Christensen (1988)40 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)87 (interface bond polarity model)

1.26 0.83 1.11 1.21

-0.02 0.41 0.13 0.03

1.05 1.08 1.20 1.35 1.59

0.19 0.16 0.04 -0.11 -0.35

1.42

-0.18

1.13

0.11

1.07 1.12

0.17 0.12

1.28

- 0.04

"Conduction-band offsets are obtained assuming room-temperature band gaps of 1.43 eV and 2.67 eV for GaAs and ZnSe, respectively. A positive value of AEc corresponds to a type I alignment, a negative value to a type I1 staggered alignment.

midgap energy theory39 predicts a valence-band offset of 1.13 eV, a type I band alignment, and the model solid theory of Van de Walle and MartinE6 yields AEv = 1.42 eV, corresponding to a type I1 alignment. The interface bond polarity model of Lambrecht and SegallE7yields AE, = 1.28 eV and a type I1 band alignment. Among the more empirical treatments, the electron affinity r ~ l e ' ~yields , ~ ~a valence-band offset of 1.26eV and a type I1 alignment. The common-anion ruleZopredicts AE, = 0.83 eV and the empirical treatment of Katnani and Margaritondo'l yields a valence-band offset of 1.08 eV, both producing type I alignments. The more recent self-consistent interface and supercell calculations have also been applied to the GaAs/ZnSe heterojunction, typically for the (1 10) orientation. Self-consistent interface calculations by Van de Walle and MartinE4 yielded AE, = 1.59 eV, corresponding to a type I1 alignment. Self-consistent calculations of Christensen4' using LMTO methods yielded a valence-band offset of 1.07 eV, and Lam-

114

E.T. YU, J.O. McCALDIN AND T.C. McGILL

brecht et obtained AEv = 1.12 eV from their self-consistent dipole calculations; these predictions both yield a type I band alignment. As seen in Fig. 33, the theoretically predicted valence-band offsets span a wide range of values, and relatively few values are in good agreement with the experimental band offset values. Most theories appear to predict valenceband offsets that are substantially larger than the experimental values; the calculations of Van de Walle and Martin84386are especially prominent in this respect. Given that interfacial reactions are known to occur in the GaAs/ ZnSe heterojunction, the tendency of theories to predict valence-band offsets consistently larger than the experimental values may be suggestive. As discussed in Section 19, theoretical treatments of band offsets typically consider abrupt interfaces, which in the case of the GaAs/ZnSe heterojunction fail to describe the actual interface that is produced experimentally. Experiments described in Sections 21 and 22 suggest that interfacial reactivity is fairly common in 111-V/II-VI heterojunctions. An understanding of this tendency and its effect on band offset values may help to provide insight into the fundamental physics of band offsets.

a. Experiment Much of the interest in the AlSb/GaSb/ZnTe material system has been due to the desire to obtain visible light emission from wide-gap 11-VI semiconductors such as ZnTe. The inability to dope ZnTe n-type led to the proposal of visible light-emitting diode (LED) structures based on heterojunction injection of minority carriers into wide-gap 11-VI materials, rather than recombination in simple p-n homojunction structures. The AlSb/ZnTe heterojunction was proposed as a particularly promising material system for this In addition, the nearly lattice-matched AlSb/GaSb/ZnTe material system has proved to be a convenient system for studying the physics and chemistry of heterovalent interfaces. Wilke and Hornlo6 used photoemission to study GaSb/ZnTe (110) heterojunctions and observed evidence of the formation of a reacted layer containing Ga and Te at the GaSb/ZnTe interface; in addition, Sb from the GaSb substrate was observed on top of the ZnTe overlayer. The interface reaction appeared to occur as the first layers of ZnTe were deposited on the GaSb surface. These measurements also yielded a value for the GaSblZnTe (1 10) valence-band offset, AEv = 0.34 k 0.05 eV, with the band alignment 338J. 0. McCaldin and T. C. McGill, J. Vuc. Sci. Technol. B 6, 1360 (1988).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

115

being type I. Schwartz et ~ 1 . measured j ~ ~ the AlSb/ZnTe (100) valence-band used offset using XPS and obtained a value of 0.35 k 0.11 eV. Yu et al.i0s*190 XPS to investigate band offsets and interfacial reactions in the AlSb/GaSb/ ZnTe (100) material system. For the AlSb/ZnTe and GaSb/ZnTe interfaces, Yu et al. observed evidence for the formation of an interfacial layer containing Te reacted with the Group III(A1 or Ga) atoms from the 111-V substrate. The observation of analogous behavior at the GaAs/ZnSe interface suggests that the compounds most likely to have been formed at the AlSb/ZnTe and GaSb/ZnTe interfaces are Al,Te, and Ga2Te3, respectively. Yu et ul. obtained the following values for valence-band offsets in the AlSb/GaSb/ZnTe material system: AE,(AlSb/ZnTe) = 0.42 f 0.07 eV, AE,(GaSb/ZnTe) = 0.60 k 0.07 eV, and AE,(GaSb/AlSb) = 0.39 k 0.07 eV. In each case the band alignment was found to be type I. These measured band offsets can be combined to determine whether the transitivity condition, Eq. (l.l), is obeyed for the AlSb/GaSb/ZnTe material system. and e ~ p e r i m e n t a l ' evidence ~ ~ , ~ ~ ~indicates that for abrupt, high-quality heterojunctions in most lattice-matched material systems, the transitivity condition should be obeyed; a deviation from transitivity would therefore imply that structural or chemical anomalies at specific interfaces were affecting band offset values. Combining the band offset values measured by Yu et u1.,105*190 one can see that the transitivity condition is violated for the AlSb/GaSb/ ZnTe material system by 0.21 f 0.05 eV. The violation of transitivity is shown explicitly in Fig. 34. This discrepancy is far larger than the experimen-

ZnTe

AlSb

GoSb

10.59 eV 0.42 eV

ZnTe

I

0.60 eV

f0.21

eV

FIG.34. Nontransitivity in the AlSb/GaSb/ZnTe material system. The valence-band offsets labeled in the figure were measured by Yu et uI.10s*190Combining these measurements demonstrates that transitivity is violated by 0.21 0.05 eV. 339G.P. Schwartz, G . J. Gualtieri, R. D. Feldman, R. F. Austin, and R. G. Nuzzo, J. Vac. Sci. Technol. B 8,741(1990).

116

E.T, YU, J.O. McCALDIN AND T.C. McGILL

tal uncertainty of the measurements. From Eq. (6.1) one can see that the transitivity rule can be verified simply from the measured core-level energy separations in the AlSb/ZnTe, GaSb/ZnTe, and AlSb/GaSb heterojunctions; if transitivity were valid for the AlSb/GaSb/ZnTe material system, one would expect Eq. (1.1) to be obeyed to within k0.05 eV or better. The measurements of Yu et al. therefore demonstrate a clear violation of the transitivity rule, suggesting that reacted layers at the AlSb/ZnTe and GaSb/ZnTe interfaces can exert a significant influence on band offset values. Comparing the results of band offset measurements for the AlSb/GaSb/ ZnTe material system, we see that the values measured for the AlSb/ZnTe ~ ’ by Yu et ul.,105,1900.35 valence-band offset by Schwartz et ~ 1 . ~and 0.11 eV and 0.42 ? 0.07 eV, respectively, are in good agreement. However, the GaSb/ZnTe valence-band offset measured by Wilke and Horn,lo60.34 f 0.05 eV, and the value of 0.60 i 0.07 eV obtained by Yu et ul.190 are in relatively poor agreement. The origin of this discrepancy is not known, but a number of possibilities exist. Wilke et al. and Yu et al. both observed evidence of a GaSb/ZnTe interfacial reaction, which could have affected the values of the band offsets differently for the two measurements. In addition, the difference in crystal orientation-(100) for Yu et al. and (I 10) for Wilke et a/.-would have affected the detailed structure of the interfaces and could therefore have influenced the measured band offset values.

+

b. Theory Theoretical predictions for the AlSb/GaSb heterojunction were discussed in Section 10. For the AlSb/ZnTe and GaSb/ZnTe heterojunctions, a large number of predictions are available. These predictions are summarized in Table XVIII. For AlSb/ZnTe, Frensley and Kroemer’s pseudopotential theory23 yields a valence-band offset of 0.80 eV without interfacial dipole corrections and 0.13 eV if the dipole correction is included. Harrison’s LCAO theory24 predicts AEv = 0.83 eV. Tersoffs theory** yields a valence-band offset of 0.39 eV, and the treatment of Harrison and Tersoff” predicts a value of 1.17 eV. Cardona and Christensen’s dielectric midgap energy model3’ predicts a valence-band offset of approximately 0.41 eV, and the model solid theorys6 yields AEv = 0.43 eV. The interface-bond-polarity model of Lambrecht et aL8’ predicts a valence-band offset of 0.66 eV. Among the empirical rules for determining band offset values, the electron affinity rule’’ yields a valence-band offset of 0.51 eV using the electron affinity values of Milnes and Feucht’l and a value of 0.59eV using the compilation of Freeouf and Woodall.’ 9 3 Among those who have made self-consistent calculations of band offsets for specific interfaces, only Lambrecht et have treated the AlSb/ZnTe interface, obtaining AEv = 0.60 eV. If one assumes transitivity

117

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

TABLE XVIII. THEORETICAL VALENCE-BAND OFFSETSFOR ZnTe/AISb AND ZnTe/GaSb

SOURCE

Milnes and Feucht (1972)71 McCaldin et al. (1976)20 Frensley and Kroemer (1977)23 Frensley and Kroemer (1977)23 (with dipole correction) Harrison (1977)24 Freeouf and Woodall (1981)lg3 Katnani and Margaritondo (1983)21 Tersoff (1986)28 Harrison and Tersoff (1986)29 Van de Walle and Martin (1987)86 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Christensen (1988)40 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)87 (interface bond polarity model)

AE,(ZnTe/AIS b) (theor.) (eV)

AE,(ZnTe/GaSb) (theor.) (eV)

0.51

1.02 0.69 0.85 0.86

0.80 0.13

0.83 0.59 0.39 1.17 0.43

0.8 1 1.03 0.77 0.77 1.26 0.89

0.41

0.71

0.38 0.60

0.83 0.96

0.66

1.01

and combines calculated band offsets for the GaSb/AlSb and GaSb/ZnTe interfaces, one obtains a valence-band offset of 0.38 eV from the calculations of Chri~tensen.~' In all cases the valence-band edge of AlSb is predicted to be higher in energy than that of ZnTe. Figure 35 shows the band alignments measured by Yu et ul.105,'90and by among values predicted by several theories. In general, the Schwartz et theoretically predicted valence-band offsets are larger than the experimentally measured values. For some of the early theories, notably those of Harrisonz4and of Frensley and Kroemer,' this discrepancy appears to arise largely from the early belief that the common anion rulez0 should be valid for compounds containing Al. If the valence-band offsets predicted by those theories for the GaSb/ZnTe interface are combined with the experimentally measured AlSb/GaSb band ~ f f s e t ,considerably ~ ~ , ~ better ~ ~agreement ~ ~ ~ ~ ~ ~ ~ with experiment is obtained. Most of the other theoretical predictions for the AlSb/ZnTe valence-band offset are within approximately 0.3 eV of the experimental values, although the theoretical values still tend to be larger than the measured band offsets. The source of this trend is not known at the present time, but it could be related to the interfacial reactions known to occur at AlSb/ZnTe heterojunctions.

118

E.T. YU, J.O. McCALDIN AND T.C. McGlLL

AlSb

ZnTe

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

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

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

Schwartt et al. Christensen Tersoff Cardona and Christensen Yu et 01. Van de Walle and Martin Milnes and Feucht

et al. Frensley and Kroerner Harrison

FIG.35. Band alignments measured experimentally by Schwartz et and by Yu et ul.,10s.190 and alignments predicted theoretically.23~24~28.29~39*40~43~71~84-87~193 For the ZnTe conduction- and valence-band edges, experimentally determined band alignments are indicated by solid lines and theoretical values by dotted lines. The indirect conduction-band edge of AlSb is indicated by the dashed line.

For the GaSb/ZnTe interface, Frensley and K r ~ e m e r *obtain ~ valenceband offsets of 0.85 eV and 0.86 eV with interface dipole corrections neglected and included, respectively, and Harrison's LCAO theoryz4 predicts AEv = 0.81 eV. Tersoffs theory" yields a valence-band offset of 0.77 eV, and Harrison and Tersoff obtained 1.26 eV. The dielectric midgap energy yields a valence-band offset of 0.71 eV, and the model solid theorya6 predicts AE, = 0.89 eV. The interface-bond-polarity model" yields AE, = 1.01 eV. The electron affinity rule19 yields a valence-band offset of approximately 1.02 eV, and the common-anion rulez0predicts a value of 0.69 eV. The empirical compilation of Katnani and Margaritondo *l yields a valence-band offset of 0.77 eV. The self-consistent supercell calculations of C h r i s t e n ~ e n ~ ~ yield a valence-band offset of 0.83 eV, and the self-consistent dipole theory of Lambrecht et predicts AE, = 0.96 eV. In all cases the valence-band edge of GaSb is predicted to be higher in energy than that of ZnTe. Figure 36 shows the band alignments measured by Wilke and Hornlo6 and by Yu et al.l9O among values predicted by various theories. As with the AlSb/ZnTe interface, the theoretical predictions for the GaSb/ZnTe valence-

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

GaSb

119

ZnTe

’1 .......................... ........................... ........................... ................ ......................

n

1-

2

-

v

>

P

r

Ec

-

C

w

0-

E”

-1

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

-

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

Wilke et al. Yu et 01. Cardona and Christensen Tersoff Harrison Christensen Frensley and Kroerner Van de Walk and Martin Larnbrecht et 01. (SCD) Larnbrecht et 01. (IBP) ‘Milnes and Feucht ‘Harrison and Tersoff

FIG.36. Band alignmentsmeasured experimentally by Wilke et a1.1°6 and by Yu et ~ l . , and ’ ~ ~ For the ZnTe conduction- and alignments predicted theoretically.23~24~28~2Q~3Q~40*43*71*84-87 valence-band edges, experimentally determined band alignmentsare indicated by solid lines, and theoretical values by dotted lines.

band offset are consistently larger than the experimental values. The origin of this trend is unclear, but its presence for both AlSbJZnTe and GaSb/ZnTe suggests that interfacial reactivity in these heterojunctions may be a contributing factor.

INTERFACES 22. OTHERHETEROVALENT a. ZnSelGe Figure 37 shows theoretical and experimental values for the ZnSeJGe valence-band offset, and theoretical predictions are compiled in Table XIX. Kowalczyk et aL6’ used XPS to measure the valence-band offset in ZnSeJGe (1 10) heterojunctions formed by deposition at room temperature and subsequent crystallization at 300°C. For ZnSe deposited on Ge, a valence-band offset of 1.29 eV was obtained, whereas for Ge deposited on ZnSe the measured valence-band offset was 1.52 eV. Katnani and Margaritondo” measured a valence-band offset of 1.40 eV for Ge deposited on ZnSe using

-

120

E.T. YU, J.O. McCALDIN AND T.C. McGILL

J

a,

g

-1-

et 01. I Kowalczyk Kotnoni and Morgaritonda

a, D C 0

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

n

lHorrison Kowalczyk et al. Tersoff

-2.5

Ge

ZnSe

FIG.37. Experimental and theoretical values for the ZnSe/Ge valence-band offset. Experimental results are indicated by solid lines, theoretical predictions by dotted lines.

synchrotron photoemission spectroscopy, and Xu et used photoemission spectroscopy to deduce a valence-band offset of 1.65 f 0.1 eV for Ge deposited on ZnSe (100). In all cases the valence-band edge of Ge was found to be above that of ZnSe. As seen in Fig. 37, the various band offset theories generally predict very large values for the ZnSe/Ge valence-band offset. The electron affinity rule7' yields a value of 1.97 eV. Frensley and K r ~ e m e rpredict ~ ~ a valence-band offset of 1.82 eV, and Harrison's LCAO theory24 yields AEv = 1.46 eV. Tersoff2' obtains a valence-band offset of 1.52 eV, and Harrison and Tersoff2' predict a value of 2.01 eV. The dielectric midgap energy model of Cardona and C h r i ~ t e n s e nyields ~~ AEv = 1.57 eV, and the model solid theorya6 predicts a valence-band offset of 1.98 eV. The interface bond polarity model of Lambrecht and Sega11a7 yields AEv = 1.73 eV. Early self-consistent calculations by Pickett and C ~ h e n ~predicted ~' a valenceband offset of 2.0 0.3 eV. Self-consistent interface calculations of Van de Walle and Martina4yield a valence-band offset of 2.17 eV, and self-consistent

*

340W.E. Pickett and M. L. Cohen, Phys. Rev. B 18, 939 (1978).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

PREDICTIONS FOR TABLE XIX. THEORETICAL ZnSe/Ge VALENCE-BAND OFFSET

SOURCE

Electron affinity rule (1972)” Frensley and Kroemer (1977)23 Harrison (1977)24 Pickett and Cohen (1978)340 Tersoff (1986)” Harrison and Tersoff (1986)*’ Van de Walle and Martin (1987)84 (self-consistent supercell) Van de Walle and Martin (1987)86 (model solid theory) Cardona and Christensen ( 1987)39 (dielectric midgap energy model) Christensen (1988)40 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall(1990)87 (interface bond polarity model)

121

THE

BE” (theor.) (eV)

1.97 1.82 1.46 2.0 1.52 2.01 2.17 1.98 1.57 1.58 1.53 1.73

supercell calculations by Christensen4’ predict a value of 1.58 eV. Finally, the self-consistent dipole calculations of Lambrecht et al.43yield AEv = 1.53 eV. From Fig. 37 it is apparent that the theoretical valence-band offsets are quite consistently larger than the experimental values. This trend is similar to that observed in the 111-V/II-VI heterojunctions discussed in Sections 20 and 21. b. GaPISi The GaP/Si band offset was measured by Katnani and Margaritondo” using synchrotron photoemission spectroscopy; a valence-band offset of 0.95 eV was obtained for Si deposited on Gap. Perfetti et a1.341*342 used synchrotron photoemission to measure both AE, and AEc for Si deposited on GaP (1 10) and obtained AE, = 0.8 f 0.1 eV. In both cases the valence-band edge of GaP was found to be lower in energy than that of Si. Theoretical predictions for the GaP/Si valence-band offset encompass a wide range of 341P.Perfetti, F. Patella, F. Sette, C. Quaresima, C. Capasso, A. Savoia, and G. Margaritondo, Phys. Rev. B 29, 5941 (1984). 342P. Perfetti, F. Patella, F. Sette, C. Quaresima, C. Capasso, A. Savoia, and G. Margaritondo, Phys. Rev. B 30,4533 (1984).

122

E.T. YU, J.O. McCALDIN A N D T.C. McGILL

values. Milnes and Feucht7' predict AE, = 1.43 eV. The pseudopotential theory of Frensley and K r ~ e m e ryields ~ ~ a valence-band offset of 0.96 eV, and Harrison's LCAO theory24predicts a value of 0.50 eV. Tersoff's theory2' predicts a valence-band offset of 0.45 eV, and Harrison and Tersoff2' obtain a value of 0.69eV. The dielectric midgap energy model of Cardona and Christensen3' yields a valence-band offset of approximately 0.57 eV, and the model solid theory of Van de Walle and Martins6 predicts AE, = 0.36 eV. The interface bond polarity modelE7yields a valence-band offset of 0.33 eV. Self-consistent interface calculations by Van de Walle and Martin 84 yielded a valence-band offset of 0.6 1 eV. Christensen4' performed self-consistent supercell calculations using LMTO methods and obtained AE, = 0.27 eV, and self-consistent dipole calculations by Lambrecht et al.43 yielded a valence-band offset of 0.31 eV. Experimental and theoretical values for the GaP/Si valence-band offset are shown in Fig. 38, and theoretical values are also compiled in Table XX. c. CdSjlnP The CdS/InP heterojunction system has been studied as a possible candidate for the fabrication of photovoltaic detectors and solar cells. Experimental and theoretical values for the CdS/InP valence-band offset are shown in Fig. 39 and Table XXI. Shay et al."7*118used C-Vmeasurements on n-CdS/p-InP heterojunctions to deduce a conduction-band offset of 0.56 eV, corresponding to a valence-band offset of 1.63 eV and a type 11, staggered band alignment. C - V measurements by Yoshikawa and Sakai343 yielded a conduction-band offset of 0.1 12 eV, corresponding to a valenceband offset of 1.19eV. Wilke et ul.'07 used photoemission to measure a valence-band offset of 0.77 eV for the CdS/InP (1 10) heterojunction, corresponding to AEc = 0.30 eV and a type I band alignment. In the measurethe CdS layers appeared to have ments of Shay et al. and of Yoshikawa et d., been grown in their equilibrium wurtzite crystal structure; the measurements of Wilke et al. were performed using samples in which CdS in the metastable zincblende crystal structure was deposited on InP. In all of these measurements, the valence-band edge of InP was found to be higher in energy than that of CdS. However, the measured values for the CdS/InP valence-band offset are in relatively poor agreement. Theoretically, the situation is similar. Frensley and K r ~ e m e predict r ~ ~ valence-band offsets of 0.84 eV without their dipole correction and 1.13 eV with the correction, while Harrison's LCAO theory24 predicts AE, = 1.48 eV. Harrison and Tersop' obtain a valenceband offset of 1.37 eV. The electron affinity rule" yields a valence-band offset 343A.Yoshikawa and Y. Sakai, Solid-state Electron. 20, 133 (1977).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

123

0.00

-0.2:

,Christensen ,Larnbrecht et 01 -Interface Bond Polarity \Model Solid

cI\

2

v

-0.50

a, 0

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

Tersoff Horrison

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

Cordon0 and Christensen Von de Wolle and Mortin

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

Horrison ond Tersoff Freeouf ond Woodoll

-0

a,

-0.7:

Perfetti et 01.

0

a

a, 0

MKotnoni and Morgoritondo ‘Frensley and Kroerner

c

-

-1.00

0

>

-1.2e

Milnes ond Feucht -1.:

Si

Go P

FIG. 38. Summary of experimental and theoretical valence-band offsets for the GaP/Si heterojunction. Experimental and theoretical values are indicated by solid and dotted lines, respectively.

of 1.46eV using the data compiled by Freeouf and Wooda1l,lg3 and the common-anion ruleZopredicts AEy = 0.91 eV. Finally, the empirical compilation of Katnani and Margaritondo’l yields a valence-band offset of 1.04 eV. Most other theoretical treatments have not considered the CdS/InP heterojunction system because the equilibrium crystal structure of CdS is wurtzite rather than cubic zincblende. d. InSblCdTela-Sn

A number of studies have been devoted to band alignments in the nearly lattice-matched InSb/CdTe/a-Sn material system. InSb and CdTe are conventional semiconductors with energy band gaps of approximately 0.18 eV and 1.50 eV, respectively, at room temperature; a-Sn is a semimetal with a

124

E.T. YU, J.O. McCALDIN AND T.C. McGILL

TABLE XX. THEORETICAL PREDICTIONSFOR GaP/Si VALENCE-BAND OFFSET

THE

SOURCE

Milnes and Feucht (1972)71 Frensley and Kroemer (1977)23 Harrison (1977)24 Freeouf and Woodall (1981)lg3 Tersoff (1986)" Harrison and Tersoff (1986)29 Van de Walle and Martin (1987)84 (self-consistent supercell) Van de Walle and Martin (1987)86 (model solid theory) Cardona and Christensen (1987)39 (dielectric midgap energy model) Christensen (1988)40 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall(1990)*' (interface bond polarity model)

TABLE XXI.

1.43 0.96 0.50 0.74 0.45 0.69 0.61 0.36 0.57 0.27 0.31 0.33

EXPERIMENTAL AND THEORETICAL VALENCE-BAND OFFSETSFOR CdS/InP

SOURCE

Shay et al. (1976)117*118 Yoshikawa and Sakai (1977)343 Wilke et al. (1989)'07 Freeouf and Woodall (1981)lg3 McCaldin et al. (1976)'O Frensley and Kroemer (1977)23 Frensley and Kroemer (1977)23 (with dipole correction) Harrison (1977)24 Katnani and Margaritondo (1983)" Harrison and Tersoff (1986)29

1.63 1.19 0.77 1.46 0.9 1 0.84 1.13 1.48 1.04 1.37

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

125

0.00

-0.25

-0.50 h

%

v

W

-0.75

Wilke et a1

W

Frensley and Kroemer

V C 0 13

McColdin. McCiII. and Mead

-1.00

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

0

Katnoni and Morgoritondo

C

W -

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

4

Frensley ond Kroemer (dipole) Yoshikawa and Sokai

- 1.25 Harrison and Tersoff ,Freeouf

-1.5

and Woodall

'Harrison

Shoy et 01

- 1.75

InP

CdS

FIG.39. Summary of experimental and theoretical valence-band offsets for the CdS/InP heterojunction. Experimental and theoretical values are indicated by solid and dotted lines, respectively.

negative energy band gap, E,(T; - ri) = -0.41 eV. Experimental and theoretical band offset values for the InSb/CdTe/a-Sn material system are shown in Fig. 40 and Table XXII. Mackey et al."' used photoemission to measure a valence-band offset of 0.87 0.10 eV for the CdTe/InSb heterojunction, with the valence-band edge of InSb above that of CdTe; their measurements also revealed evidence that an interfacial compound containused ing In and Te was formed at the CdTe/InSb interface. Forster et ultraviolet photoemission spectroscopy (UPS) to study the Sn/InSb (110) 344A.Forster, A. Tulke, and H. Liith, J. Vac. Sci. Technol. B 5, 1054 (1987).

126

E.T. YU, J.O. McCALDIN AND T.C. McGILL

...

HT

...

IBP

n

>

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

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

HT IBP MF MS

/H \LSA

T

DME

\HNH HNH

C LSA

H M T KM DME MMM

0-

E"

lnSb

CdTe

a-Sn

FIG.40. Experimental and theoretical values for valence-band offsets in the InSb/CdTe/a-Sn material system. Experimental values for the CdTe/InSb and a-Sn/CdTe valence-band offsets are indicated by solid lines, theoretical values by dotted lines. The arrows show measured values for the InSb/a-Sn valence-band offset. The theories shown obey transitivity to better than +0.05 eV. For the three interfaces, the values shown are from the following sources: HT, Ref. 29; IBP, Ref. 87; MF, Ref. 71; MS, Ref. 86; C, Ref. 40; LSA, Ref. 43; H, Ref. 24; M, Ref. 108; T, Ref. 28; KM, Ref. 21; DME, Ref. 39; MMM, Ref. 20; HNH, Ref. 346; FTL, Ref. 344; JMC, Ref. 345.

interface and obtained a valence-band offset of approximately 0.28 eV for amorphous or polycrystalline u-Sn deposited on (1 10) InSb. John et used synchrotron photoemission to measure the valence-band offset for a-Sn/ InSb (100) and obtained AEv = 0.40 & 0.06 eV for Sn deposited on both the Sb-stabilized c(4 x 4) and the In-stabilized c(8 x 2) surfaces of InSb (100). Hochst et al.346used photoemission to study the a-SnlCdTe (110) heterojunction and measured a valence-band offset of 1.1 eV assuming zero band gap for the u-Sn film, or approximately 0.95 eV using indirect evidence of the development of a nonzero band gap in thin a-Sn films. Most of the available theoretical band offset values are in reasonably good agreement with these experimental results. For the CdTe/InSb heterojunc345P. John, T. Miller, and T.-C. Chiang, Phys. Reo. B 39, 3223 (1989). 346H.Hochst, D. W. Niles, and I. HernAndez-Calder6n, J. Vac. Sci. Technol. B 6, 1219 (1988).

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

TABLE XXII. EXPERIMENTAL AND FOR

127

THEORETICAL VALENCE-BAND OFFSET VALUES InSb/CdTe/u-Sn

SOURCE

Mackey et al. (1986)'08 Forster et al. (1987)344 John et al. (1989)345 Hochst et al. (1988)346

CdTe/InSb InSb/u-Sn (1 10) InSb/u-Sn (100) CdTe/u-Sn (1 10)

0.87 f 0.1 0.28 0.40 0.06 0.95-1.10

Milnes and Feucht (1972)71 McCaldin et al. (1976)" Harrison(1977)24 Katnani and Margaritondo (1983)2' Tersoff (1986)28 Harrison and Tersoff (1986)2y Cardona and Christensen (1987)39 Christensen (1988)40 Van de Walle and Martin (1986)86 (model solid theory) Lambrecht et al. (1990)43 Lambrecht and Segall (1990)87

CdTe/InSb CdTe/InSb CdTePnSb CdTe/InSb CdTe/InSb CdTeJnSb CdTePnSb CdTe/InSb CdTe/InSb

0.96 0.69 0.91 0.77 0.84

CdTe/InSb CdTe/InSb

0.92 1.07

Harrison (1977)24 Harrison and Tersoff (1986)29 Cardona and Christensen (1987)39 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)87 (interface bond polarity model)

InSb/u-Sn InSb/u-Sn InSb/u-Sn InSbIu-Sn

0.37 0.61 0.39 0.39

InSblu-Sn

0.48

Harrison (1977)24 Harrison and Tersoff (1986)29 Cardona and Christensen (1987)3y Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)87 (interface bond polarity model)

CdTe/u-Sn CdTe/u-Sn CdTe/u-Sn CdTe/u-Sn

1.28 1.76 1.12 1.28

CdTeIu-Sn

1.51

1.15

0.73 0.93 0.94

tion, Harrisonz4 predicts a valence-band offset of 0.91 eV. Tersoffs theoryz8 yields AEv = 0.84 eV, and Harrison and Tersoff" obtain a valence-band offset of 1.15 eV. Cardona and Christensen predict a value of 0.73 eV for the CdTe/InSb valence-band offset, and the model solid theory of Van de Walle and Martin86 yields AEv = 0.94 eV. The interface bond polarity model of Lambrecht and Sega1187predicts a valence-band offset of 1.07 eV. Among the empirical models for band offset values, the electron affinity rule71 yields a valence-band offset of 0.96 eV, and the empirical compilation of Katnani and

128

E.T.-YU, J.O. McCALDIN A N D T.C. McGILL

Margaritondo" predicts AE, = 0.77 eV. The common-anion rule" yields a valence-band offset of 0.69 eV. Finally, Christensen's self-consistent supercell calculations using LMTO methods4' yielded a valence-band offset of 0.93 eV, and the self-consistent dipole calculations of Lambrecht et predicted AEv = 0.92eV, As shown in Fig. 40, these values are all in reasonably good agreement with the reported experimental values. Considering the complex nature of the CdTe/InSb interface,'" however, it is interesting that theoretical treatments that do not account for interfacial reactivity still yield values in good agreement with experiment. Fewer theoretical predictions are available for the a-Sn/InSb and a-Sn/ CdTe interfaces. For the a-Sn/InSb interface, Harrison24 predicts a valenceband offset of 0.37 eV, while the theory of Harrison and TersoffZ9yields a value of 0.61 eV. The interface bond polarity model of Lambrecht et aLs7 predicts a valence-band offset of 0.48 eV. The dielectric midgap energy model AEv = 0.39 eV, and self-consistent of Cardona and C h r i ~ t e n s e npredicts ~~ yield a valence-band offset of dipole calculations of Lambrecht et 0.39 eV. The predicted values of Harrison, of Cardona and Christensen, and of Lambrecht et al. are in reasonable agreement with the experimental results, while the value proposed by Harrison and Tersoff is somewhat larger than the measured values. For the a-Sn/CdTe interface, Harrisonz4 obtains a valence-band offset of 1.28 eV, and Harrison and Tersoff2' predict AE, = 1.76 eV. The interface bond polarity model of Lambrecht et al?' yields a valence-band offset of 1.51 eV. Cardona and C h r i ~ t e n s e nobtain ~ ~ 1.12 eV for the a-Sn/CdTe valence-band offset, and the self-consistent dipole calculations yield AE, = 1.28 eV. The results of Harrison, of of Lambrecht et Cardona and Christensen, and of the self-consistent dipole calculations by Lambrecht et al. are in reasonable agreement with the measured band offset values; the valence-band offset predictions of Harrison and Tersoff and of the interface bond polarity model are considerably larger than the measured value. One should recall, however, that the structural and chemical complexity of heterovalent interfaces is ignored in current theoretical treatments. It may not be prudent to attach great meaning to detailed quantitative correlations between experimental measurements and theoretical predictions for band offsets in heterovalent material systems. e. CuBrlGaAslGe Waldrop et have studied band offsets in the nearly latticematched CuBr/GaAs/Ge heterojunction system using XPS. Initial ~

1

.

~

~

~

9

~

~

~

347J. R. Waldrop and R. W. Grant, Phys. Rev. Lett. 43, 1686 (1979). 348J. R. Waldrop, R. W. Grant, S. P. Kowalczyk, and E. A. Kraut, J. Vac. Sci.Technol. A 3, 835 (1985).

129

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS

demonstrated that valence-band offsets for the Ge/CuBr, CuBr/GaAs, and GaAs/Ge heterojunctions were highly nontransitive. Although it is possible that this nontransitivity arises from nonideal structure at specific interfaces, Christensen4' has performed theoretical calculations indicating that CuBr can produce interface-specific electronic states that influence band offset values and yield an intrinsic deviation from transitivity even for abrupt, ideal ~ ' valence-band interfaces. The later measurements of Waldrop et ~ 1 . ~yielded offsets of 0.85 eV and 0.7 eV for the CuBr/GaAs and CuBr/Ge interfaces, respectively. The CuBr valence-band edge was found to be lower in energy than both the GaAs and the Ge valence-band edges. Combining these results with their value of 0.56 f 0.04 eV for the GaAs/Ge valence-band offset,348 one sees that transitivity is violated by approximately 0.7 eV. Only a few theoretical predictions are available for heterojunctions containing CuBr. Harrison 24 predicts valence-band offsets of 2.78 eV and 2.37 eV for CuBr/Ge and CuBr/GaAs, respectively. The interface bond polarity model of Lambrecht and Sega11a7yields AE,,(CuBr/Ge) = 1.48 eV and AE,(CuBr/GaAs) = 1.04 eV. Self-consistent supercell calculations by Christensen41 yield valenceband offsets of 1.10 eV for CuBr/Ge and 0.82 eV for CuBr/GaAs, and self. ~AE,(CuBr/Ge) ~ = consistent dipole calculations by Lambrecht et ~ 1predict 1.16 eV and AE,(CuBr/GaAs) = 0.74 eV. Except for the CuBr/GaAs valenceband offsets predicted by Christensen4' and by Lambrecht et u Z . , ~ ~ , ' ~ these theoretical values are in relatively poor agreement with the reported experimental results. Experimental and theoretical values for band offsets in the CuBr/GaAs/Ge material system are summarized in Table XXIII.

TABLE XXIII. EXPERIMENTAL AND THEORETICAL BAND OFFSET VALUES FOR CuBr/GaAs/Ge

SOURCE

Waldrop et al. (1985)"348 Harrison (1977)24 Christensen (1988)41 Lambrecht et al. (1990)43 (self-consistent dipole) Lambrecht and Segall (1990)'' (interface bond polarity model)

0.85 2.37 0.82 0.74

0.56 0.41 0.46 0.45

0.7 2.78 1.10 1.16

1.04

0.48

1.48

'The values of Waldrop et aL348are experimental; all others are theoretical calculations.

130

E.T. YU, J.O. McCALDIN AND T.C. McGILL

VIII. Discussion

If there is any trend that becomes obvious upon examining the wealth of experimental and theoretical work devoted to band offsets, it is that the subject remains one of considerable uncertainty. The inherent complexity of the subject has led us to adopt an approach to the band offset problem that may best be described as “enlightened empiricism.” By examining a wide variety of theoretical and experimental approaches to the band offset problem, as we have done in this review, we have sought to develop a balanced perspective that might allow us to gain new insights that have not previously been made evident in the literature. As our review of the available literature has revealed, both triumphs and failures have been experienced in the experimental study of band offsets. The successes have been achieved in spite of, and the failures in many cases have occurred because of, the notoriously difficult nature of many band offset measurement techniques. For material systems such as GaAs/Al,Ga, - xAs, GaSb/AlSb, and HgTe/CdTe, there have been relatively sudden shifts in the experimentally accepted band offset values. These shifts can certainly be attributed in part to difficulties in interpreting and analyzing experimental data, but it is also possible that some early experimental results were colored by early theoretical predictions of band offset values that later were found to be incorrect. Despite some early lapses, however, experimental techniques such as photoelectron spectroscopy, J- V and C - V measurements, and even optical spectroscopy have been developed to a level such that, with care, reliable measurements of band offsets can be obtained quite routinely. Although enormous progress has been achieved in the theoretical study of band offsets, a consistently reliable predictive theory has yet to be developed. In a few cases such as HgTe/CdTe, a theoretical treatment2’ has succeeded in predicting a reasonably accurate band offset value independently of an accurate experimental determination and in fact in conflict with earlier experimental and theoretical values. More often, however, theories have been successful primarily in reproducing band offset values after reliable experimental measurements have been performed. This tendency can be seen quite clearly in Figs. 11 and 18. Although the ability to reproduce a set of reliable experimental band offset values is a laudable and important achievement, predictive-rather than “postdictive”-ability is a much more stringent test of the power and validity of a given band offset theory. Another curious issue is that a number of current theories, such as those of Van de Walle and MartinsLE6 and of Cardona and Chri~tensen,~’ appear to yield band offset values in reasonable agreement with experiment, even though the physical ideas upon which these theories are based vary considerably. Theories and empirical rules have been proposed based on surface

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131

properties such as electron affinity , 1 9 g 7 ’ interface properties such as Schottky barrier heights,” calculations of bulk band structure on an absolute energy scale,22--24.84 effective “midgap” or charge neutrality energy points,25,26,28,39 or other interfacial dipole effectsa7As will be shown in this section, however, there is no single theory that yields band offset values that are substantially more accurate than those predicted by any other theory. Throughout this discussion one must keep in mind that the concept of a band offset is in reality an idealization of an extremely complex situation at a semiconductor interface. The concept of a perfectly abrupt discontinuity in the band edges is inconsistent with the uncertainty principle; the abrupt discontinuity actually represents an energy shift that occurs over a small number of unit cells near the interface. Furthermore, structural and chemical changes occur in the region very near the interface that can substantially affect the detailed electronic structure at the interface. Despite these caveats, the concept of a well-defined band offset has proved to be an enormously useful tool in predicting the viability and expected performance of a given device structure. It has been the inherent complexity of the subject that has frustrated attempts to attain a detailed quantitative understanding of band offsets. In the following sections we have attempted to arrange band offset values in several heterojunction systems into an organized structure and to identify trends that might act as guides in evaluating the literature and perhaps improve our ability to predict band offset values for unstudied heterojunction systems. A further motivation was the thought that the success of a particular theory for a large number of heterojunctions might serve to validate the approach and suggest that the physical concepts underlying that theory might be the dominant ones. We have tried to look objectively at the accuracy of several theories in predicting band offsets in a set of latticematched heterojunctions for which reasonably well-defined experimental values are available, and thereby to extract some insights into the factors that exert an especially strong influence over band offset values.

23. THEORY VERSUS EXPERIMENT IN LATTICE-MATCHED HETEROJUNCTIONS In Figs. 41 through 55 we have plotted, for a number of the more widely quoted band offset theories, the difference between theoretical and experimental valence-band offset values, A = AE:, - AEFP‘, for several latticematched heterojunction systems. The figures have been placed in approximately chronological order. Because of the complications that are known to arise in coherently strained heterojunctions, we have chosen to make this

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FIG.41. Comparison of experimental and theoretical valence-band offsets for predictions of Milnes and Feucht."

FIG.42. Comparison of experimental and theoretical valence-band offsets for predictions of McCaldin et al." Filled and open circles represent predictions for compounds that are, respectively,within and outside the scope of the original predictions as made by McCaldin et al.

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133

FIG.43. Comparison of experimental and theoretical valence-band offsets for predictions of Frensley and Kr0erner.2~

FIG.44. Comparison of experimental and theoretical valence-band offsets for dipole-corrected predictions of Frensley and K r ~ e r n e r . ~ ~

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FIG. 45. Comparison of experimental and theoretical valence-band offsets for predictions of Harrison.24

FIG. 46. Comparison of experimental and theoretical valenceband offsets for predictions of Katnani and Margaritondo.''

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FIG. 47. Comparison of experimental and theoretical valence-band offsets for predictions of Tersoff.’’

FIG.48. Comparison of experimental and theoretical valence-band offsets for predictions of Harrison and Tcrsoff.2’

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E.T. YU, J.O. McCALDIN AND T.C. McGILL

FIG.49. Comparison of experimental and theoretical valence-band offsets for predictions corraqponding to self-consistent calculations of Van dc Walle and Martin.'&

tC FIG.50. Comparison of experimental and theoretical valence-band offsets for predictions of the model solid theory of Van de Walle and Martin.84

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137

FIG.5 1. Comparison of experimental and theoretical valence-band offsets for predictions of the dielectric midgap energy model of Cardona and Chri~tensen.~~

FIG.52. Comparison of experimental and theoretical valence-band offsets for predictions of Chri~tensen.~'

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FIG.53. Comparison of experimental and theoretical valence-band offsets for predictions of the self-consistentdipole calculations of Lambrecht et a1.43

FIG.54. Comparison of experimental and theoretical valence-band offsets for predictions of the interface bond polarity model of Lambrecht and Sega11.”

BAND OFFSETS IN SEMICONDUCTOR HETEROJUNCTIONS Hvbertsen

d

%

139

39'

0.8

v f

0.4

X

a'

W

0

U

. b

I

f

-0.4

W'

a

-0.8

FIG.55. Comparison of experimental and theoretical valence-band offsets for predictions of Hybertsen.161,'6 2

comparison only for lattice-matched heterojunctions; in addition, we have included only systems for which the experimental band offset value is reasonably well known. The experimental values we have chosen are compiled in Table XXIV. For values obtained from a single reference, the error bars in the table are quoted directly from that reference; for values obtained by averaging the results of several experiments, the error bars represent the standard deviation of the collection of experimental values. The heterojunctions included in Figs. 41 through 55 have been divided into three categories: 111-V/III-V systems (AlAs/GaAs, InP/ InGaAs, InAlAs/InGaAs, InP/InAlAs, InAs/GaSb, AlSb/GaSb, InAs/AISb), 11-VI/II-VI systems (CdTe/HgTe, CdSe/ZnTe), and heterovalent systems (GaAs/Ge, ZnSe/GaAs, ZnTe/AlSb, ZnTe/GaSb, GaP/Si, CdTe/InSb). Multiple points for a single heterojunction represent different experimental band offset values, as listed in Table XXIV. If a theory did not predict a band offset value for a given heterojunction system, no value is shown. For the 111-V/III-V heterojunction systems, the clearest trend is chronological. The early theories (Figs. 41 through 46) show considerable disagreement with the experimental band offset values that are now accepted; the later theories, beginning with that of Tersoffz8 (Figs. 47 through 55), generally agree much better with the currently available experimental values. The early theories were particularly bad for heterojunctions containing Al; however, most of the early treatments (with the exception of the dipolecorrected theory of Frensley and Kroemerz3and the empirical compilation of

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TABLEXXIV. SELECTED EXPERIMENTAL BANDOFFSETVALUES FOR LATTICE-MATCHED HETEROJUNCTIONS HETEROJUNCTION

A& (eV)

Al As/GaAs InP/InGaAs InAlAs/InGaAs InP/InAlAs InAs/GaSb AISb/GaSb InAs/AlSb

0.55" 0.37 f 0.03b 0.20 f 0.03' 0. 16d, 0.29' 0.53 f 0.03' 0.40 0.04g 0.09 f 0.05h

CdTe/HgTe CdSe/ZnTe

0.36 f 0.02' 0.64 0.07'

GaAs/Ge ZnSe/GaAs ZnTe/ AlSb ZnTe/GaSb GaP/Si CdTe/InSb

0.52 f 0.13' 0.98 f 0.07' 0.35 f 0.11", 0.42 f 0.07" 0.34 f 0.05", 0.60 f 0.07" 0.8 & 0.lq, 0.95' 0.87 f 0.lV

+

"Refs. 48 and 49. bStatistical average of values from Refs. 88, 111, 112, 136, 137, 138, 139, 140, and 141. 'Statistical average of values from Refs. 90, 114, 141, 149, 150, 151, 152, and 153. dRef. 158. 'Ref. 157. 'Statistical average of values from Refs. 93, 98, 165, and 166. %tatistical average of values from Refs. 72, 99, 188, 190, and 191. hRef. 115. 'Statistical average of values from Refs. 100, 101, 110, 202, and 206. 'Ref. 218. 'Statistical average of values from Refs. 21, 53, 94, 314, 316, 317, 318,319,323, and 324. 'Statistical average of values from Refs. 65, 333, 335, 336 and 337. "Ref. 339. "Ref. 105. "Ref. 106. PRef. 190. qRefs. 341 and 342. 'Ref. 21. "Ref. 108.

Katnani and Margaritondo") were quite accurate in predicting the InAs/ GaSb valence-band offset. For 11-VI/II-VI heterojunctions, the number of lattice-matched systems for which experimental data are available is very small. For the HgTe/CdTe heterojunction, no theory is in extremely close agreement with the experi-

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mental value, but the more recent theories, beginning with that of Tersoff,z8 are in general agreement with the qualitative conclusion that the valenceband offset is not small; the early LCAO theory of Harrisonz4 had predicted a very small valence-band offset for HgTe/CdTe. It is also useful to examine the CdSe/ZnTe heterojunction, because a reliable experimental band offset value was not available until quite recently.'l Surprisingly, the early theories of McCaldin et dZ0 (Fig. 42) and of Frensley and KroemerZ3(Fig. 43) are in good agreement with the experimental value. Because the equilibrium crystal structure of CdSe is wurtzite rather than cubic zincblende, few of the more recent theories treat the CdSe/ZnTe heterojunction; however, the dielectric midgap energy model of Cardona and C h r i ~ t e n s e nis~in ~ good agreement with experiment. These theoretical values were all obtained prior to the experimental measurement of Yu et ~ 1 . ~ ' ~ Predictions for the heterovalent material systems are generally in poorer agreement with experimental values than predictions for pure 111-V/III-V or 11-VI/II-VI heterojunctions. In fairness to the theorists, this tendency may be due in part to greater uncertainties in the experimental band offset values. For heterojunctions for which a large number of experiments have been performed, such as GaAs/Ge, the range of experimental values is quite large-approximately 0.35 eV, as shown in Table XIV-and even the statistically reduced error bar is f0.13 eV. In addition, the experimental band offsets for heterovalent material systems shown in Table XXIV and Figs. 41 through 54 are often based on only one or two experimental measurements. The greater ambiguity of the experimental situation for heterovalent material systems compared to that for isovalent heterojunctions may be due only in part to a lack of extensive experimental data for each heterojunction. As discussed in Section VII, many, if not all, heterovalent material systems are characterized by some degree of chemical reactivity at the interface. Experiments by Kowalczyk et aL6' demonstrated that the GaAs/ZnSe valence-band offset can depend on the conditions under which the interface was ' that interfacial reactions affected prepared, and work of Yu et ~ 1 . l ~indicated band offset values in the AlSb/GaSb/ZnTe material system. An examination of theoretical and experimental valence-band offset values for 111-V/II-VI heterojunctions reveals that, in every case except CdTe/InSb, the theoretical values tend to be quite consistently larger than the experimental values. For the calculations of Van de Walle and Martin,84 shown in Figs. 49 and 50, the theoretical values for ZnSe/GaAs are much larger than the experimental value. This general trend among theoretical predictions can be seen quite clearly in Figs. 33 (ZnSe/GaAs), 35 (ZnTe/AlSb), and 36 (ZnTe/ GaSb). The same tendency is also observed for the ZnSe/Ge heterojunction, as can be seen in Fig. 37. For the one exception to this trend, CdTe/InSb, the experimental value is based on a single measurement.'08 Given the expected

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complexity of the CdTe/InSb interface, it is quite possible that the measured band offset could depend significantly on the detailed structure of the interface. This systematic trend between the predicted and observed values of valence-band offsets in 111-V/II-VI heterojunctions suggests that the chemical reactions and atomic intermixing that are known to occur may tend to ~ ’ ~ very reduce the value of the valence-band offset. Harrison et ~ 1 . realized early that, for polar orientations such as (loo), some atomic intermixing was required in heterovalent interfaces to prevent large charge accumulations at the heterojunction. Thus, one of the driving forces for interfacial chemical reactions may be the minimization of interfacial charge. This minimization may be partially responsible for the somewhat curious fact that for many heterovalent systems, theoretical treatments that do not include any effects of interfacial reactivity predict valence-band offsets in fairly close agreement with experimental values. Because the effect of heterovalent interface reactions may in fact be to minimize interfacial charge, idealized “model” theories, which is what current theories are, may be able to provide a qualitatively and perhaps even semiquantitatively accurate description of band offsets in heterovalent material systems. The final atomic arrangement that results may yield a valence-band offset that is close to, but consistently smaller than, what would be expected on the basis of a simplified, idealized description of bulk energy levels or minimization of interfacial dipoles. Regarding the successes and failures of specific theories for the entire range of heterojunction systems considered here, there is little that can be said. Most of the recent theories yield band offset values in good agreement with experiment for 111-V/III-V heterojunctions; the dielectric midgap energy ~ ~ in Fig. 51, was also able to model of Cardona and C h r i ~ t e n s e n ,shown predict the value of the CdSe/ZnTe valence-band offset quite accurately. This suggests that, for isovalent heterojunctions, the more modern theories may have a limited degree of actual predictive power. Among the early theories and empirical rules, the predictions of the common-anion rule of McCaldin et uL2’ shown in Fig. 42, may yet be of some interest. The actual predictions of the common-anion rule, based on measured Schottky barrier heights, are shown as solid circles in Fig. 42 and were quite good at the time the predictions were made. Predictions for heterojunctions not claimed by McCaldin et al. to be within the scope of the rule, i.e., for compounds containing A1 or Hg, are shown as open circles in the figure. With these additions the accuracy of the common-anion rule predictions is comparable to that of its contemporaries-the electron affinity rule7’ (Fig. 41), Frensley and KroemerZ3(Figs. 43 and 44), and Harrison24 (Fig. 45). Thus, within its known limitations the common-anion rule, as seen in Fig. 42, may still serve as a useful qualitative guide to band offset values in lattice-matched heterojunctions.

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The final point to note in comparing the currently available theories is that no single theory yields band offset values in good agreement with experimental measurements for all or even most of our selected set of heterojunctions. In addition, even the more recent theories are based on widely disparate underlying physical concepts, e.g., calculated band structures referred to atomic potentials, various charge neutrality or effective midgap energy levels, or screened interfacial dipoles. However, these theories all appear to predict band offset values with approximately the same degree of accuracy. On the basis of these observations it would seem that none of the current theories includes all of the relevant physics governing band offsets to the accuracy required to yield reliable values for all heterojunctions. Nevertheless, the success of both model theories and ab initio calculations for 111-V/III-V heterojunctions, which are well understood experimentally and in which interfacial reactions are not an important factor, suggests that current theories can provide predictive guidance for a limited range of heterojunction systems.

24. STRAIN As described in Section VI, strain effects have an extremely strong influence on band offset values in lattice-mismatched heterojunction systems. Theoretical and experimental studies of band offsets in the Si/Ge material system indicate that the dominant effect of strain is the splitting of the heavyhole, light-hole, and split-off valence-band edges induced by uniaxial strain, rather than shifts in the average valence-band-edge positions arising from hydrostatic strain. Thus, the discontinuity in the average position of the lighthole, heavy-hole, and split-off valence-band edges in a coherently strained Si/ Ge heterojunction depends only weakly on strain, and the actual band-edge positions are determined primarily by the uniaxial splitting of the valenceband edges. Calculations of absolute deformation potentials using the model solid theory of Van de Walle and Martin86 suggest that the weak dependence of the average valence-band offset on strain should be characteristic of other lattice-mismatched heterojunctions as well. The success of the calculations of Van de Walle and Martin37,38for the Si/Ge heterojunction suggests that the effects of strain on band offsets are well understood theoretically and also adds to the evidence that current theoretical treatments can, in limited circumstances, exhibit considerable predictive power. Detailed investigations of strain effects may also provide some insight into the respective roles of absolute energy scales and effective midgap or charge neutrality energy levels in determining band offset values. As pointed out by Cardona and C h r i ~ t e n s e nstrain , ~ ~ in lattice-mismatched heterojunctions can strongly influence the calculated average valence-band offset in theories based on effective midgap energy levels. Detailed investigations of quantities

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E.T. YU. J.O. McCALDIN AND T.C. McGILL

such as the pressure dependence of band offsets or, if possible, "unstrained" band offsets for lattice-mismatched heterojunctions might provide one way to distinguish between theories based on absolute energy scales and those based on effective midgap or charge neutrality levels. 25. UNRESOLVED ISSUES

At this point it must surely be evident to the reader that many issues remain unresolved in the study of band offsets. One issue of interest is the extent to which band offsets can be controlled by altering the atomic composition and structure of the heterojunction interface. Investigations of interfacial reactivity in heterovalent material systems have demonstrated that interfacial reactions can exert a substantial influence on band offset values. Studies described in Section VII of material systems such as GaAs/Ge, ZnSe/ GaAs, and AlSb/GaSb/ZnTe suggest that band offsets can shift by up to a few tenths of an electron volt by this mechanism. Another possibility is to alter the effective barrier height for carriers in a heterostructure device by &doping near a heterojunction i n t e r f a ~ e .Ho ~~ wever, ~ , ~ it~ will ~ also be necessary to determine the degree to which such adjustments can be made without inducing severe deterioration in other aspects of device performance. Another issue that deserves investigation is the degree to which fluctuations in interfacial structure can affect band offsets. Ballistic electron emission microscopy (BEEM) has been used to study Au-GaAs (100) interfaces with nanometer spatial resolution, and the electronic structure of the interface was found to be highly h e t e r o g e n e ~ u s . Similar ~ ~ ' ~ ~effects ~ ~ might be observed in semiconductor heterojunctions, particularly in heterovalent material systems in which chemical reactions are known to occur at the heterojunction interface. Such inhomogeneities could have significant implications for nanometer-scale device structures such as quantum wires and quantum dots. Perhaps the most important issue that remains unresolved is the question of what actually determines band offset values. A number of current theories appear to yield band offset values that are, for a limited set of heterojunctions, in reasonable agreement with experiment. However, the physical ideas on which these theories are based vary considerably, ranging, for example, from careful calculations of band structure on energy scales referenced to atomic energies to alignment of effective midgap or charge neutrality energy levels.

349F.Capasso, A. Y. Cho, K. Mohammed, and P. W. Foy, Appl. Phys. Lett. 46,664 (1985). "@T.-H.Shen, M. Elliott, R. H. Williams, and D. Westwood, Appl. Phys. Lett. 58, 842 (1991). 35'W. J. Kaiser and L. D. Bell, Phys. Reo. Lett. 60,1406 (1988). 352M.H. Hecht, L. D. Bell, W. J. Kaiser, and F. J. Grunthaner, Appl. Phys. Lett. 55,780 (1989).

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Studies such as those described in Section 24 might provide some information. There also remain several relatively unstudied material systems that may eventually become technologically relevant and at the very least could add to the available data used to develop and test new theories of band offsets. Relatively unexplored 111-V materials include most nitride compounds. A wide range of 11-VI compounds remain uncharted territory; these include compounds containing Be, Mg, and Ca, as well as oxygen compounds and, to a lesser extent, sulfides. It is possible, however, that the prediction of band offset values to a high degree of accuracy (e.g., kO.1 eV or better) may remain an intractable problem for some time to come. Given the known variability of band offsets with the detailed structure of a heterojunction interface, it will probably be difficultto account for all the factors that can influence band offsets at the 0.1 eV energy scale and below. However, it would be desirable and certainly of considerable importance to be able to develop a quantitative understanding of correlations between certain deviations from ideal interface structure and corresponding shifts in band offset values. Such an understanding would probably provide a great deal of insight into the fundamental physics governing band offsets.

IX. Conclusions

In this article we have reviewed the available experimental and theoretical studies of band offsets in semiconductor heterojunctions. Our approach has been to attempt to blend experimental observations and theoretical insights and develop a perspective that might best be referred to as enlightened empiricism. Experimentally, reliable band offset values are now known for a fairly large set of semiconductor heterojunctions, and experimental techniques have been sufficiently well developed that, with care, reliable band offset measurements can be performed quite consistently. The current level of theoretical understanding is such that a number of model theories, as well as some ab initio calculations, are able to provide band offset values in fairly good agreement with experimental values, and even with a limited amount of predictive power, for a restricted set of heterojunctions-primarily isovalent (111-V/III-V, II-VI/IITVI, or IV/IV) heterojunctions rather than heterovalent material systems. The inherent chemical complexity of heterovalent interfaces and the limited amount of experimental data available have hindered attempts to develop an accurate quantitative description of band offsets in these material systems. At the current stage of the subject’s development, it will probably be difficult to develop a detailed, comprehensive, and quantitatively accurate (to

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within f O . l eV or better) theory of band offsets, simply because the underlying physics governing band offsets is not understood in detail. The reactive nature of heterovalent interfaces will make even an accurate structural description of these heterojunctions quite difficult. However, it will be important to develop or enhance our understanding of a number of issues: the role of chemical reactions and detailed interfacial atomic structure in determining band offset values; how, in detail, strain and lattice mismatch affect band offsets; band offset values in previously unstudied heterojunctions; and possible inhomogeneities in electronic structure on nanometer spatial scales arising from details of atomic structure at interfaces. These and other issues are likely to continue to provide substantial challenges to both experimentalists and theorists in the years to come.

ACKNOWLEDGMENTS It is a pleasure to acknowledge our many valuable discussions with the students and faculty members associated with the McGill research group at Caltech. We would also like to thank Professor Henry Ehrenreich for his support and encouragement during the writing of this article, and for his critical reading of the manuscript. Part of this work was supported by the Defense Advance Research Projects Agency, monitored by the Office of Naval Research, under grant N00014-90-J- 1742.

SOLID STATE PHYSICS. VOLUME 46

Physical Properties of Macroscopically lnhomogeneous Media DAVID J . BERGMAN Raymond and Beverly Sackler Faculty of Exact Sciences School of Physics and Astronomy Tel Aviv University. Ramat Aviv Tel Aviv. Israel

DAVIDSTROUD Department of Physics The Ohio State University Columbus. Ohio

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. DC Electrical Properties-General Theory and Calculational Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Definitions and Basic Ideas . . . . . . . . . . . . . . . . . . . . 2. Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Simple Approximations . . . . . . . . . . . . . . . . . . . . . . 4. Better Approximations . . . . . . . . . . . . . . . . . . . . . . . 5 . Exact Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Analytical Properties and Series Expansions . . . . . . . . . . 7. Effective-Medium Approximation . . . . . . . . . . . . . . . . 8. Numerical Techniques for Continuum Composites . . . . . . 9. Numerical Techniques for Discrete Networks . . . . . . . . . I11. DC Electrical Properties-Applications to Specific Problems . . . 10. Percolation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 11 . Dielectric Behavior Near a Percolation Threshold . . . . . . . 12. Nonuniversal Conductivity Near a Percolation Threshold . . 13. Hall Effect and Magnetoresistance . . . . . . . . . . . . . . . . 14. Duality in Two Dimensions . . . . . . . . . . . . . . . . . . . . 15. Thermoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . 16. Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Electromagnetic Properties . . . . . . . . . . . . . . . . . . . . . . . 17. Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. Quasi-Static Approximation . . . . . . . . . . . . . . . . . . . . 19. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . Beyond the Quasi-Static Approximation . . . . . . . . . . . . 22. Other Approximations . . . . . . . . . . . . . . . . . . . . . . .

148 150 150 152 152 155 158 161 165 174 186 192 192 198 199 201 206 211 213 220 220 222 223 232 234 240

147 Copyright 0 1992 by Academic Press. Inc. All rights of reproduction in any form reserved.

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V. Nonlinear Properties and Flicker Noise . . . . . . . . . . . . . . . . 23. Flicker Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Weak Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . 25. Strong Nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . 26. Nonlinear Optical Effects . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

246 246 253 258 267 269

1. Introduction

When a material is homogeneous over the length scales that are important in a particular experiment, it is usually possible to characterize its physical properties by a small number of parameters. For example, in order to describe the macroscopic electrical properties of a crystal or a glass it is enough to specify its conductivity 0 and dielectric coefficient E, both of which are second-rank tensors in general. Because the basic, atomic-scale properties are governed by quantum mechanics, while the macroscopic properties are governed by classical mechanics or electromagnetism, the connection between the microscopic and the macroscopic properties is difficult to work out-most of condensed-matter physics is devoted to such problems. The present review is devoted to another class of materials-composites or granular or porous materials in which there exists a macroscopic scale of inhomogeneity. In such a material, there are small, yet much larger than atomic, regions where macroscopic homogeneity prevails and where the foregoing macroscopic parameters suffice to characterize the physics, but different regions may have quite different values for those parameters. If we are interested in the physical properties only at scales that are much larger than those regions and at which the material appears to be homogeneous, then the macroscopic behavior can again be characterized by bulk effective values oerE , of the conductivity and dielectric coefficient. Because both the small-scale behavior (inside the different homogeneous regions) and the overall large-scale behaviour are now governed by the same laws of classical physics, the connection between the properties on the two scales is now much stronger. Although the detailed structure of a crystal or even a glass is largely determined by the properties of the constituent atoms, the microstructure or microgeometry of a composite can be much more varied. In some cases the microstructure can be tailored to produce a material with desirable properties. In other cases it is so complicated (as when it is random) that we do not know it or cannot describe it precisely. This has led to the development of a class of approximations for g e , E , and other bulk effective moduli that do not depend on precise details of the microstructure. These approximations

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

149

include self-consistent embedding procedures as well as exact upper and lower bounds. Composite materials appear both in nature and in technology, but although they have existed for a long time, the interest in them among physicists has increased dramatically in recent years. This is due to a number of developments. One is the increased reliance on synthetic composites in technology. Another is the discovery of new physical phenomena in composite systems, such as the quantum Hall effect and ballistic transport in quantum well heterostructures and anomalous diffusion near the percolation threshold of a metal-insulator composite. Even the recently discovered high-T, superconductors often come in the form of polycrystalline ceramics, so that many of their properties are strongly affected by the composite microstructure. In writing this review we omit almost all discussion of the historical development of the field and try instead to describe its more recent aspects. We therefore refer the reader to the very thorough review by Landauer,’ which includes all the important developments and references, starting with Maxwell, up to about 1975. Even among recent developments, we have had to be selective in our coverage. We describe relatively briefly those topics that appear to have reached maturity or are already well documented and easily accessible in the archival literature. We discuss in greater detail those topics which, in our opinion, are still in the process of being developed and to which the reader who wishes to enter this field may well make original contributions. We also emphasize theoretical developments, describing experimental results primarily in the interest of having illustrative examples, without any intention of completeness. This article is mostly about dc and ac electrical properties of composite media. Some other physical properties, such as magnetic permeability and thermal conductivity, can be discussed using exactly the same methods, and this is pointed out where applicable. Yet other properties, such as elastic stiffness, must be discussed separately, even though there exist many methodological similarities. The article is organized into five sections. Section I1 describes the basic theory and results for dc electrical properties, including two sections about numerical techniques. It also includes a discussion of the effective-medium approximation, electrostatic resonances, exact bounds, and analytical properties. Section 111 describes a number of static physical properties of ‘A detailed summary of the status of this field of physics up to 1978, including an exhaustive list of historical references, can be found in the review article Electrical Conductivity in Inhomogeneous Media by R. Landauer, in “Electrical Transport and Optical Properties of Inhomogeneous Media,” Eds. J. C. Garland and D. B. Tanner, AIP Conf. Proc. No. 40, pp. 2-45, 1978.

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D.J. BERGMAN AND D. STROUD

composites that have attracted attention in recent years. Those include electrical conductivity and dielectric behavior near a percolation threshold, magnetotransport, thermoelectricity, superconductivity, and duality in twodimensional composites. We have not tried to cover in detail the topic of percolation, which has been reviewed quite thoroughly elsewhere. Only the aspects of percolation that are relatively new, are still being studied, and have not been reviewed elsewhere are described. Section IV discusses the extension of these methods to the ac electrical properties of composites and includes a number of illustrative examples. Section V is concerned with nonlinear properties and flicker noise in composites. This field has only recently begun to be intensively studied, and we expect many new developments in the years to come. II. DC Electrical Properties-General Theory and Calculational Techniques

1. DEFINITIONS AND BASICIDEAS An inhomogenous composite dielectric is characterized by a positiondependent dielectric constant &@), which takes different values in the different components. However, on length scales large compared to the inhomogeneities, we can characterize the dielectric behavior by a single number, the bulk effective dielectric constant E,. If the composite dielectric is used to fill the entire volume of a large parallel-plate capacitor, E , is defined so that the capacitance (as well as the stored charge and energy) will be the same as that of a homogeneous dielectric with E(r) =- E,. To calculate E , we then need to know the local field E(r) for a given applied voltage A 4 or average field E o = A+/L in the capacitor2 ( L = distance between the plates, A = area of a plate, V = total volume of the capacitor):

2A detailed discussion of these definitions and the various approximations that have been developed to calculate E~ can be found in a series of published lecture notes prepared by one of the authors: Bulk Physical Properties of Composite Media by D. J. Bergman, in “Les Methodes de 1’Homogenkisation:Theorie et Applications en Physique,” &ole d’Ett d’Analyse Numerique, pp. 1-128, Edition Eyrolles, Paris, 1985.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

15 1

The first of these expressions is based on a calculation of the stored energy. The last one involves calculating the charge stored on one of the plates and can be obtained from the first by using the divergence theorem and E = -V+(r), V (EE)= 0 to transform to a surface integral over the capacitor plates, where may be replaced by q!J0(r) = - E, r, the potential field in a homogeneous dielectric. The middle result is then obtained by transforming back to a volume integral, but keeping +o instead of the actual +. Had we been discussing a composite conductor, we would have defined a bulk effective conductivity 0,in an analogous way by focusing our attention either on the total resistance of a large sample placed between two flat, infinitely conducting plates, or on the total electric current, or on the rate of production of Joule heat. We would then have obtained expressions identical to (1.1) but with E(r), E , replaced by a(r), 0,. The same definitions and formalism also describe the magnetic permeability, heat conductivity, and diffusivity of a composite medium. All of these physical phenomena have in common the occurrence of two vector fields, analogous to E(r) and D(r), that satisfy

-

-

+

V x E = 0,

-

V D

= 0,

D

= E(r)E,

E

=

-V+.

(1.2)

A variety of boundary conditions can be imposed on these fields. The most common, however, are to specify the values of or of the normal component of D at the surface. When the composite medium is macroscopically uniform, then the bulk effective dielectric constant is an intensive material parameter -its value is independent of the volume and of the precise nature of the macroscopic boundary conditions, provided those too are macroscopically uniform. Consequently, the particular choice of boundary conditions used in solving for does not affect the value of E, obtained from (1.1). When some of the components are anisotropic, the definitions must be modified appropriately. For other physical properties, such as thermoelectric power, magnetotransport, and elastic stiffness, a similar discussion can be developed. While strictly valid only in a dc situation, in practice the concepts used in this section are useful somewhat beyond that. It is well known that the dielectric constant E, electrical conductivity 0,and magnetic permeability p can be extended to nonzero frequencies, where they usually acquire a nonzero imaginary part and where the real part may violate some dc restrictions like E 2 1, (r 2 0, p 2 0. If the frequency is not too large, the field E will still be approximately curl free, so that E = -V+ and V (KV+) = 0, where K now includes, besides the real and imaginary parts of E, also the (complex) conductivity 0

+

+

4x0 K=E--.

i o

152

D.J. BERGMAN AND D. STROUD

The criterion for this to be valid is that all the inhomogeneity length scales be small compared to the electromagnetic lengths, namely the wavelength and the skin depth (in the case of metal). When this is not the case, the full Maxwell equations must be invoked for a correct description of the physics. Calcluations of E , are hampered by two obstacles: (1) Often we do not know the detailed form of E(r), but only that it takes on a number of known values cl, E ' , . . . , E, corresponding to a number of different homogeneous components that appear with given volume fractions p i , p z , . . . . ,p,. (2) Even when we do know the detailed microgeometry of the mixture, and hence the spatial dependence of E(r), it is usually extremely difficult to calculate E(r), except for a few special cases. 2. EXACTRESULTS

Those special cases are as follows: (a) When the components are arranged as parallel cylinders (not necessarily circular) aligned along E,. (b) When the components are arranged as flat, parallel slabs perpendicular to E,. (c) When the composite is made up entirely of multicoated ellipsoids, in which all the interfaces are confocal ellipsoidal surfaces. The ellipsoids must come in all sizes in order to fill up the entire volume, but they must all have the same ratios of axes a : b : c, as well as the same arrangement and volume fractions of the various components, and they must all be similarly oriented. Cases (a) and (b) and the two-component-coated-spheres or Hashin-Shtrikman example of case (c) are shown in Fig. 1, along with the results for E,. 3. SIMPLEAPPROXIMATIONS

For other cases, we must resort to approximations. The simplest of these are expansions in terms of a small parameter, either E~ - E, or pi. For calculating a generic term in these expansions, one again needs to know the detailed microgeometry. However, the low-order terms can usually be obtained rather easily and with only limited information about the microgeometry. Thus, the expansion of E, - E, in powers of E, - E~ is r Er

- Ee =

1 Pi(&, - Ei)

i= 1

PROPERTIES OF MACROSCOPICALLY INHOMGENEOUS MEDIA

E(s)

- -5

E(s)

-

F(s)

153

€2 Pa

P d- I

s- T P ,

FIG.1. The simple types of microgeometry for which E, can be found exactly, along with the expressions for s,, F(s), and E(s). (a) Parallel cylinders; (b) parallel slabs; (c) Hashin-Shtrikman coated-spheres microgeometry; (d) same as (c) but the roles ofs, and s2 are interchanged. Taken from D. J. Bergman, in “Les Methodes de I’Homogtntisation: Thtorie at Applications en Physique,” Ecole dfite d’Analyse Numtrique, pp. 1- 128, Edition Eyrolles, Paris, 1985.

154

D.J. BERGMAN A N D D. STROUD

where the first term is always valid, and the second term is valid when the microgeometry has either isotropic or cubic point symmetry. The higherorder terms depend on finer details of the microstructure and can be expressed as multidimensional integrals involving microgeometric correlation functions and dipole-dipole interaction terms.3 Alternatively, these terms can be expressed as integrals over corrections to the zero-order field E,, which can be calculated by solving a hierarchy of boundary value problems in the composite m e d i ~ mA. ~low density expansion to linear order in pi, i < r, which is a good approximation when p, 1, is also easy to obtain if the sparse components (i = 1 , . . . ,r - 1) appear in the form of simply shaped (i.e., spherical or ellipsoidal) inclusions in the host component (i = r). In that case we can calculate the field E(r) inside each inclusion as though there were no other inclusions at all. This is then used in the middle expression of (1.1), after transforming it into

which requires values of E(r) only inside the inclusions. The result depends on the shapes and orientations of the inclusions through the depolarization factor n (n = for a sphere and satisfies 0 I n 5 1 for other shapes; see Ref. 1)

4

where pin is the volume fraction of i-component inclusions with depolarization factor n in the direction E,. To calculate higher-order terms in pi, i < r, we need to include interactions between different inclusions. Those are due to the fact that any inclusion is now subject not to E,, but to a field already distorted by the other inclusions. Those distortions are caused by charge dipoles and higher-order multipoles induced in the other inclusions. The induced dipole moments cause the longest-range distortions ( l/r3), and those are included in the famous Maxwell Garnett (MG) or Clausius-Mossotti approximation for a collection of spherical inclusions c1 in a host c2 (see Ref. 1)

-

‘W. F. Brown, Jr., J. Chem. Phys. 23, 1514 (1955). 4D.J. Bergman, Phys. Rep. 43, 378 (1978); also published in “Willis E. Lamb, a Festschrift on the Occasion of his 75th Birthday” (D. ter-Haar and M. 0.Scully, Eds.), pp. 377-407, NorthHolland, Amsterdam, 1978.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

155

When written in this implicit form, we can easily convince ourselves that the MG approximation agrees with (3.3) to order p1 and with (3.1) to order ( E ~- EJ’. It is also the same as the exact results for the coated-spheres geometry of Fig. lc. Another popular approximation is the symmetric effective-medium approximation (EMA),’?’ which can be written in the form

for a collection of spheres. This is an implicit algebraic equation of order r for E,, but only one of the solutions is physical, namely that which approaches E, when all zi = E,. Again, the physical solution agrees with the expansions (3.1) and (3.3) to the orders exhibited. This approximation is elaborated upon in Section 7.

4. BETTERAPPROXIMATIONS A number of methods have been developed to go beyond these approximations in the case of two-component composites. One of these goes back to Lord Rayleigh6 but was put in practice only recently. The electric potential in a periodic array of dielectric spheres is expanded in spherical harmonics around the center of one of the spheres. Continuity conditions at the spherical interface are then used to set up an infinite system of linear algebraic equations for the expansion coefficients.’ Another approach focuses on the phenomenon of electrostatic resonances. These are eigenstates of the electric potential problem, namely nontrivial solutions 4(r) of V ( E V ~=)0 that vanish on the surface of the composite. Such solutions correspond to an electric field that would exist inside the capacitor even though the voltage across it is zero. Such a state, which is analogous to a series resonance in an LC (inductor-capacitor) circuit, can occur only for special values of which have to be real and negative. This can never occur in a strictly dc experiment, because then E~ 2 1. However, in a metal-dielectric mixture there is usually a regime of frequencies between l/z (the conductivity mean free time) and wp (the plasma frequency of the metal) where is very nearly real and negative. Thus these resonances are observable. It turns out that these resonances possess convenient orthogonality and completeness properties that allow us to use them to expand E(r) of

-

’D. A. G. Bruggeman, Ann. Physik (Leipzig) 24,636 (1935). S. Rayleigh, Phil. Mug. 34,481 (1892) (Scientific Papers of Lord Rayleigh, Cambridge University Press, Glasgow, 1903, Vol. 4, p. 19). ’R. C. McPhedran and D. R. McKenzie, Proc. R. SOC.Lond. Ser. A 359,45 (1978). 6J. W.

156

D.J. BERGMAN AND D. STROUD

the real composite, and consequently to express E,, in the following f ~ r m : ~ . ~ , ~ (4.1a) where E2

s=-E2

,

OIS,
(4.lb)

-El

The poles of F(s) occur at the resonance values of E1/EZ, and the residues F, are obtained from integrals over the eigenstates q5,(r): (4.1~) where the integration is restricted to the subvolume V, occupied by the cl component. Another type of resonance or eigenstate for this problem is a nontrivial solution of V . ( E V ~=) 0 for which the total charge on the capacitor plates vanishes, that is, 0 = J d A E a$/an. Such an eigenstate is analogous to a parallel resonance in an LC circuit and can occur only for special values of cl, E~ such that < 0. At the appropriate values of s, which we donate as S,, E, vanishes and F(S,) = 1. Exactly one $, is found between every pair of neighbouring s,'s, and they appear as the simple poles in an expansion of a function similar to F(s) and related to

4

E, 1 - sF(s) E ( s ) r 1--=c7El E, ,s - s, s(1 - F(s))'

(4.2a)

where

O<E,
O<s",
(4.2b)

4

Both $, and E , may be calculated from the appropriate resonance solution (see Ref. 2). Note that the relation between E(s) and F(s) is symmetric and can be written in the implicit form (1 - E(s)(l

1

-

F(s)) = 1 - -.

*D. J. Bergman, J. Phys. C 12,4947 (1979). 9D. J. Bergman, Phys. Rev. B 19, 2359 (1979). loD. J. Bergman, Ann. Phys. 138,78 (1982).

S

(4.2~)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

157

FIG. 2. Schematic graph of the functions F(s), E(s) for real values of s. Note that the poles are all simple and lie on the semiclosed segment [O, 1) and that the functions are monotonic decreasing everywhere. The same graph also describes the generic form of the functions C(')(s), D(''(s) defined in Section 8.

Note also that both functions satisfy F(s*)

= F*(s),

E(s*)

= E*(s),

(4.3)

where the asterisk (*) denotes complex conjugation. In Fig. 2 we show schematic graphs of F(s) and E(s) for real values of s. Note that these functions are monotonically decreasing for real values of s at which they are nonsingular. A systematic procedure for evaluating E , as well as the eigenstates and s, has been developed that starts from the eigenstates of an eigenvalues 4n, individual spherical inclusion and the two-body interactions or overlap integrals of eigenstates from different These quantities are calculated analytically and then used in a numerical calculation of the eigenstates and E, of a periodic array of spheres. It is possible by this approach not only to calculate the continuation of the power series of (3.1) but also, by a direct to get good results for E , even close to a calculation of the eigenstates resonance, where it diverges. The power series itself is well behaved: All the

158

D.J. BERGMAN AND D. STROUD

coefficients are positive and the singularities are known a priori to be restricted to a well-defined portion of the real axis.2 Consequently a sequence of Pade approximants can be constructed that converges extremely rapidly to These matters are discussed further in Section 8. E , . ' ' ? ' ~

5. EXACTBOUNDS

When detailed information about the microgeometry is unavailable, an exact result for E , is even in principle unattainable. Nevertheless, it is found that exact upper and lower bounds can be calculated based on a variety of types of partial information. The simplest of these is the trivial bound Min(E,) IE , I Max(&,)

for E~ 2 0,

(5.1)

for which no information about the microstructure is necessary. Using the volume fractions, these improve to the Wiener bounds'

(i:)-' i=r

< E, <

i:PiEi. i=r

For isotropic composites, these improve further to become the HashinShtrikman (HS) bounds'4

It is interesting to note that all of these rigorous bounds are actually attained in solvable geometries: The bounds of (5.1) correspond to a homogeneous one-component composite, those of (5.2) to the parallel slabs or cylinders microgeometries, and those of (5.3) to the Hashin-Shtrikman microgeometry (see Fig. 1). One way to improve on these bounds is by knowing more than just the first two terms of (3.1). As mentioned earlier, these can be used to produce tighter upper and lower bounds by a method related to Pade approximants (see Section 8 ) . " ~ ' ~ "G. W. Milton Appl. Phys. A 26, 207 (1981). I'D. J. Bergman, in "Homogenization and Effective Media" (J. L. Ericksen, D. Einderlehrer, R. Kohn, and J.-L. Lions, eds.) p. 37, Springer-Verlag, New York 1986. "0.Wiener, Abh. Sachs. Akad. Wiss. Leipzig Math.-Naturwiss. KI. 32, 509 (1912). I4Z. Hahsin and S. Shtrikman, J . Appl. Phys. 33, 3125 (1962).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

159

Another way is by exploiting a knowledge of the bulk properties of other composites with the same microgeometry. By that we mean either a different material made from different components that are mixed together in the same way or the same material but measured differently. That might mean measuring E, at a different frequency or temperature in order to change the component values of E ' , c Z . It could also mean measuring other properties of the same material (such as thermal conductivity or magnetic permeability) that are described by the same mathematical framework.' These bounds are best described in terms of the functions F(s), E(s) defined earlier. The upper and lower HS bounds of (5.3) become, in this language,

for s 3

1

(5.4)

0'

The bounds for E,(E', E ~ ) when , its values are known at other pairs of values of e l , c Z , are best described as bounds for F(s), E(s) when their values are known for s = sl,sz, . . . . The recipe is then to construct a version F,(s) of F(s) with the minimum number of poles such that it takes the right values at all the measured points si. This will have the form M

c

(5.5a) for an even number of measured points 2M, and the form (5.5b)

+

for an odd number of measured points 2M 1. A similar procedure is used to construct a version Eb(s) of E(s), which may be translated into a version F,,(s) of F(s) by using (4.2~).The two functions Fa(s),F,,(S) then provide exact bounds for F(s)-one from above, the other from below."

"S. Prager, J. Chem. Phys. 50,4305 (1969).

160

D.J. BERGMAN A N D D. STROUD

These different types of bounds can all be generalized to the case when and E , / E 2 , and hence s and F(s), are complex.'0~'2"6 The meaning of a bound in that case is a restricted portion of the complex plane inside which the appropriate quantity must lie. The complex analogs of the real bounds (5.1)-(5.3) are shown in Fig. 3, along with explicit parametric representations

E1/E1

0.11 .

I

1

o.'.... .....b I

-0.1-

.

'.

\

.,'-t_

'\.,

\

\

f

'

\ \

-0.5-

-

'

*I f A

-p

e,

5., , /*'/

\

&--

\

\

1

.--..._.. \ D--+.-"....._.

\

-03-

1

I

B

__,,,,

-

I

-___--I

a.

/

-

€ -0.7-

-

3d (a)

S=l*Oli p =p=I& I

-09-

-11

-01

'

I

0.1

-

t

I . I 0.3 05

.

1 . 0.7 0.I9

1.1

FIG.3. Quantitative graphic description of the allowed regions in the complex F plane for a 3D composite for the particular choices (a) s = 1 + 0.li and p1 = p 2 = $, (b)s = 81 - i), and p1 = p z = f. The arc a and the straight segment b define the region to which F is restricted if only 16G.W. Milton, J . Appl. Phys. 52, 5294 (1981).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

161

of their boundaries. All of these boundaries, as well as those of many other of the complex bounds, are made of circular arcs.16,10,12They can thus be represented also by specifying three points that lie on each circle.

6. ANALYTICALPROPERTIES AND SERIESEXPANSIONS The functions F ( s ) and E(s) are both analytic in most of the complex s plane, including the point s = 00. Their singularities are confined to the real segment [0, 11. For a composite of finite size with interfaces that are entirely smooth (i.e., no corners and no points of contact between different parts of the interface), the singularities are a sequence of discrete, simple poles on the real, semiclosed segment [O, 1) with one accumulation point at s = $2 In special cases, e.g., for the solvable microgeometries of Fig. 1, only a finite number of these poles have nonzero residues. information about s (i.e., about and E ~ is) used. The arcs c and d define the region to which F is restricted when the information about the volume fractions p1 and p2 = 1 - p1 is also used. The arcs e andfdefine the corresponding region if the isotropic sum rule for the first moment of the pole spectrum pl [see Eq. (6.6c)I is also used. The lines a, c-fare all circular arcs, while b is a straight line segment. They can all be constructed (with ruler and compass only, if desired) from a knowledge of the points 0, 1, and A-E (note that all the formulas are given for a general dimensionality d, while the bounds in the drawing refer to the case d = 3): A = l/s, B = pl/s, C = pl/(s - p2), D = CPJS- ( l / d ) ~ ~El , = {plCs - (d - l)/dl/sCs - 1 + ( l / d ) ~ ~ l A } . parametric representation for the lines a-J where so is the parameter, is

1 -so s - so

o<so
%,

o<so
F,

= -,

F,

=

S

F,

PI

= -,

s - so

0 < so < p2;

P2 < s o < 1;

0 < so < (d - l)/d;

(d - l)/d < SO < 1. Taken from D. J. Bergman, Ann. Phys. 138, 78 (1982).

162

D.J. BERGMAN AND D. STROUD

Even when the interface is singular, as in the case of a checkerboard composite in two or three dimensions, the singular points of F(s) and E(s) still remain confined to the segment [O,l], though they may now include a continuum. The same is true if we average those functions over an ensemble of composites, or if we allow the volume to become infinite. The functions F(s), E(s), which are real at all nonsingular real points s [see Eq. (4.3)], usually have a nonvanishing imaginary part at this continuum. Moreover, this imaginary part is discontinuous, jumping to minus its value when the real axis is crossed: Im F(s

=x-

i0) = -1m F(s = x

+ i0) 2 0

for x E [OJ],

(6.1)

and the same for E(s). The continuum of singular points may thus be viewed as a cut in the complex s-plane. Elsewhere these functions are analytic. It is instructive to consider the manner in which this cut is produced when F(s), E(s) are averaged over an ensemble of composites. In order to have an intuitive physical picture, we think of the composite as a mixture of two conductors (i.e., cl, c 2 , E , are replaced by al,a2,a,). A pole of F(s) corresponds to a value of al/azfor which ae/aZis infinite. This will occur at s = 0 if az = 0 < a1 and a, > 0, that is, for a mixture of conductor with perfect insulator where the conductor percolates (i.e., when there is an uninterrupted path from end to end of the sample through the conducting component). Other poles, at small nonzero values of s, can occur if alis nearly percolating. In that case, if I a1I % I a21, then the total conductance of the sample will be dominated by clusters of a1 material that are prevented from percolating by narrow barriers of a2 material (see Fig. 4a). The contribution of one such cluster to the total resistance has the form

PI

pc

PI

’ pc

FIG.4. Schematic pictures of a two-component composite conductor with ul B u2 (a)just below and (b) just above the percolation threshold of the good conductor u,.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

163

where the positive coefficients A and B clearly depend on the precise microgeometry but satisfy B 4 A. A pole of F(s) will occur when R = 0, that is, when s=

(

l - - ;;)-1=(l+;)-141.

(6.3)

The exact position of such a pole not only depends on details of the cluster but is also affected by the proximity of other clusters. Also, the farther the cluster is from actual percolation, the smaller the ratio A/B, and consequently the farther the pole is away from s = 0. Because the precise positions of these nonzero poles will be different in another sample, averaging over an ensemble should produce a cut. Nonzero poles of F ( s ) can also occur if o lis just barely percolating. In that case, the total conductance of the sample is dominated by the percolating clusters (see Fig. 4b). The contribution of one of these and its neighborhood to the total conductance has the form

G = Aol

+ Bo2,

(6.4)

where the positive coefficients A , B again depend on the precise microgeometry but satisfy A $ B. A pole of E(s) will now occur when G = 0, that is, when s = ( l l-);+

4 1.

By (4.2c), these poles correspond to values of s where F(s) = 1, and between any two such adjacent values of s there lies a single pole of F(s). As before, these poles will tend to move away from s = 0 as the percolating cluster becomes more massive due to the sample being farther and farther above the threshold. Also as before, they will get smeared into a cut when F(s) is averaged over an ensemble of samples. The only pole that survives and is never smeared out by such ensemble averaging is the pole at s = 0, which appears in F(s) whenever oI percolates and in E(s) whenever o1 is nonpercolating. In Section 11 we will see how these analytical properties can be exploited to gain useful information about critical behavior near a percolation threshold and other conducting-nonconducting transition points. If we use F(s) to represent E,, then the expansion (3.1) becomes a power series in u = l/s. The relation between this series and the resonance expansion of F(s) is easily found by expanding each term in (4.la) as an infinite

164

D.J. BERGMAN A N D D. STROUD

geometric series

c prur+ m

F(s) =

(6.6a)

0

where (6.6b) is the rth moment of the pole spectrum. As in (3.1), the first coefficient in (6.6a) depends only on the volume fractions, and the second coefficient also has a universal form for composites with either isotropic or cubic point symmetry (also triangular symmetry in two-dimensional composites), namely Po = P1,

Pl = PlP2Jd

(6.6~)

where d is the dimensionality. The higher-order moments r > 1 depend on details of the microgeometry and are in general very difficult to calculate. For periodic arrays of spheres they have been calculated numerically up to a rather large value of r. This has been used to investigate the detailed analytical properties of F(s) for a cubic array of identical spheres.8 Another system for which higher-order pr's have been calculated is the two-component, independent bond, random-resistor network (RRN). A diagrammatic formalism for calculating pr at any value of r was developed and used for determining p r up to r = 6.l' The calculation of each pr involves numerical summation of products of lattice Green functions, the discrete analog of the multidimensional integrals of Ref. 3. These sums eventually take up prohibitive amounts of computing time, and this determines the largest value of r that can be achieved in practice. Another approach for RRNs focuses on the dependence of E, or F(s) on the volume fraction p l . When c2 = 0, that is, s = 0, a diagrammatic formalism was developed for expanding F ( 0 ) in powers of pl. Many terms have been calculated in this expansion, and this has enabled the singularities of F(0) as function of p1 to be investigated numerically.'8719 In principle, one could also set up a power series in p1 for F(s) at any value of s. This has not yet been done-in any case, it will not be a simple extension of the diagrammatic formalism developed for F(0). This is so because when s = 0 the conductance between two points of a network depends only on a ol I7D. J. Bergman and Y. Kantor, J. Phys. C 14, 3365 (1981). "R. Fisch and A. B. Harris, Phys. Rev. B 18,416 (1978). 19J. Adler, Y. Meir, A. Aharony, A. B. Harris, and L. Klein. J. Stat. Phys. 58, 511 (1990).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

165

cluster that connects them, whereas when s # 0 the oz bonds are also important and the entire network contributes to this conductance. It will be interesting to have such a series for F(s), because a way has now been found to connect between the two possible expansions-a power series in l/s and a power series in p1 for F(s).”

7. EFFECTIVE-MEDIUM APPROXIMATION Another approach to calculating oewas introduced by Bruggeman in 1935 under the name of the effective-medium approximation (EMA). 5 ~ 2 1The EMA is a mean-field theory closely related to similar approaches in the theory of magnetism (the Weiss molecular-field approach)” and the theory of electronic states in disordered alloys (the coherent potential appro~imation).’~ Like these, the EMA has the virtue of relative mathematical simplicity and of conceptual simplicity as well. It also has the disadvantages characteristic of mean-field theories in other contexts. For example, it gives mean-field, dimension-independent critical exponents near the critical points of the theory (the percolation threshold, in the context of binary composites, and the ferromagnetic phase transition, in Weiss molecular field theory). It is also rather difficult to improve systematically. Despite these drawbacks, its predictions are usually sensible physically and offer a means of quick insight into some problems that are difficult to attack by other means. The simplest means of introducing the EMA is to imagine two conductors, with conductivities oA and og, present in volume fractions p A and p B = 1 - pa. The components are imagined to exist as compact grains distributed in some random fashion, as shown schematically in Fig. 5. The effective conductivity oe is defined by the equation

” 0 . Bruno, Comm. Pure Appl. Math. 43, 769 (1990). ”R. Landauer, J. Appl. Phys. 23, 779 (1952). See also Ref. 1. ”The resemblance between the EMA and the Weiss molecular field theory of magnetism is really only superficial, because the latter is applied to a true phase transition while the EMA describes a conductivity transition in a medium without a Hamiltonian, in which temperature is not a relevant parameter. M. Stephen [Phys. Rev. B 17,4444 (1978)l has shown that the true analog of the Weiss molecular field theory in heterogeneous media is not the EMA but rather a much more complicated theory that becomes exact only in six or more dimensions, rather than the upper critical dimension of 4 that is typical of conventional phase transitions. 23For a review of the coherent potential approximation, see, for example, H. Ehrehreich and L. M. Schwartz, “Solid State Physics” (H. Ehrenreich, F. Seitz, and D. Turnbull, eds), Vol. 31, p. 1, Academic Press, New York, 1976.

166

D.J. BERGMAN AND D. STROUD

FIG.5 . (a) Schematic diagram of a two-component composite made up of grains of components A and B, with conductivities cA and cg, distributed at random. (b)Illustration of self-consistent embedding procedure adopted in the effective-medium approximation. The grain i (= A or B) is assumed to have a convenient shape (e.g., spherical) and to be embedded, not in its actual environment, but in a self-consistently determined effective medium of conductivity ue.

where the brackets denote volume averages. In the presence of an applied field, both currents and fields in the composite will be highly inhomogeneous and statistically random, so that the fields are very difficult to calculate exactly. In the EMA, one typically makes two approximations: (1) the grains are embedded in some “effective medium” that is to be determined selfconsistently (see Fig. 5) and (2) the grains are taken to have some convenient shapes (e.g., ellipsoidal). Of these two assumptions, the second, though convenient, can be dispensed with in principle. If the grains are assumed to be spherical, the field and current in a grain of the ith species are uniform and given by

Ein,i= (1 - r6ai)-’ Era,,

(7.2)

where 6, is the conductivity of the effective medium, Efaris the field far from the inclusion, and i = A or B. The averages (J) and (E) take the forms (7.6)

(7.7)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

167

and the requirement (7.1) leads to the self-consistency condition (7.8) For a two-component composite, Eq. (7.8) is a quadratic equation,

(7.8a) which can readily be solved for the effective conductivity oe as a function of composition. The physical root of the quadratic equation can invariably be determined from requirements of positivity of 0,. Equation (7.8) can be rearranged to give the EMA a seemingly different physical origin. We write it as

(P) = 0

(7.9)

P = J - 0,E

(7.10)

where

is the position-dependent “polarization” in the composite. According to Eq. (7.9), therefore, 0, is exactly determined by the condition that the space average of P shall vanish. In the EMA, P is approximated in the ith cell by

-

-

P,= J ~- O~ e ~,i , , i~= aOi (1 - r aai)-l E ~ ~ ~ (7.1 . 1) Setting the average of this quantity equal to zero gives Eq. (7.8). Figure 6 shows the conductivity of a composite as a function of p for various values of the ratio C T ~ / Oas ~ , calculated in the EMA. The extreme limits r ~ ~=/ 0oand ~ az/al = 00 are described by the simple equations (see also Fig. 15 in Section 10) (7.12) 0, =

3 4 %-

(7.13)

respectively, where pc = 3 is the EMA percolation threshold corresponding to the onset of connectivity in medium 1. Equations (7.12) and (7.13) are valid respectively above and below pc. In the language of Section 10, the EMA thus predicts that s and t , the percolation critical exponents [see Eq. (tO.t)], are

168

D.J. BERGMAN AND D. STROUD

0

0.5

1.o

P FIG.6. Effective conductivity ue for a composite containing volume fraction p of conductivity u1 and 1 - p of uz,as calculated in the effective-mediumapproximation. In all curves, uz = 1. Note that vertical scale is logarithmic.

both equal to unity. Generalization of the EMA to d = 2 shows that these values are independent of dimension, as is characteristic of mean-field theories in other contexts. The self-consistency argument just described can be cast in a number of other forms. BergmanZ4 has described an “unambiguous effective-medium approximation” in which one considers a single inclusion (A or B) of volume u embedded in an effective medium of volume V and calculates to order (u/V) the change 60, in the effective conductivity due to the inclusion. The self-consistency requirement is that the average of 60, shall vanish. The approximation is “unambiguous” because it does not matter whether one sets the average of SO,, or 6(1/oe), or the fluctuation of any other moment of o,, equal to zero. The EMA can be widely generalized to treat ellipsoids rather than spheres, and the individual constituents and the self-consistent effective medium can all be described by tensor rather than scalar conductivities. Even in the most general case, when the principal axes of the ellipsoid and the interior and effective conductivity tensors form three different sets of Cartesian coordinates, it is known that the electric field and polarization inside the ellipsoid 24D.J. Bergman, Phys. Rev. B 39,4598 (1989).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

169

remain uniform in the presence of a uniform externally applied field.z5This results permits an EMA to be written down in simple closed form asz6

where all the quantities appearing are tensors (that is, 3 x 3 matrices), the angular brackets denote a volume average, doi = oi- ae,and the matrix Ti is defined by (7.15)

where Siis the surface of the ith particle, assumed to be centered at the origin, and ns is the fl component of the unit outward normal from this surface. The Green's function G is in turn defined as the solution of

V * [(T,VG(X - x')]

=

-6(x

- x')

(7.16)

in free space. In this form, the EMA can be applied not only to tensorial conductivities and ellipsoidal shapes but also to multicomponent composites, or even a medium where the different values of (T form a continuum. One useful application of this formalism is to materials that are seemingly not composite media at all, namely polycrystalline samples of anisotropic media. Many important materials are electrically anisotropic. In all such media, the conductivity is not simply a multiple of the unit second-rank tensor. A polycrystal of such a conductor will, in effect, be a composite medium, each crystallite having a different conductivity tensor of the form (T

= R~,R-'

(7.17)

where od is the diagonal matrix of principal conductivities and R is a 3 x 3 rotation matrix describing the orientation of a given crystallite relative to the laboratory axes. If the grains of the composite are randomly oriented, then the effective conductivity (T, will be a scalar (that is, a multiple of the unit tensor) and the effective-medium approximation we have described can be shown to reduce to the simple equationz6

(7.18) 25N.Kinoshita and T . Mura, Phys. Status Solidi 5, 759 (1971). 26D.Stroud, Phys. Rev. B 12, 3368 (1975).

170

D.J. BERGMAN AND D. STROUD

where crii denotes one of the three principal components of the tensor 6. Equation (7.1 8) is identical to the effective-medium equations describing a composite made up of equal parts of three isotropic components, having conductivities gii( i = 1, 2, 3). Another potentially useful application of the tensor effective-medium approximation is to treat the behavior of composite media in a magnetic field, when even isotropic media acquire nondiagonal (and nonsymmetric) conductivity tensors. A symmetric EMA.for this case was carried out first by S t a ~ h o w i a kand ~ ~ somewhat later, though independently, by Cohen and Jortner for the Hall coefficient.28The latter applied their theory to interpret measurements of the Hall effect in metal vapors near a metal-insulator t r a n ~ i t i o n . ~Stroud " ~ ~ and his co-workers generalized these results to treat magnetoresistance measurements in polycrystalline m e t a l ~ and ~ ~ in. ~nor~ mal-superconducting composite^.^'.^^ The basic physics can already be seen by considering a Drude metal in a magnetic field B = Bh. For such a metal, the nonzero elements of the conductivity tensor take the form gzz

= go,

gxx

= g y y = g,/C1

bxy= - byx=

+ (w,z>21,

w,z/[ 1

+ (O,T)2],

(7.19)

where o,= eB/mc is the cyclotron frequency for free electrons and z is the conductivity relaxation time. In a homogeneous metal, the diagonal elements of the resistivity tensor p = 6'are independent of field-that is, there is no magnetoresistance. On the other hand, when a composite is prepared of a free-electron metal and some other constituent (or even of two free-electron metals with different values of no, o,,and T), then the EMA predicts that there is a nonzero magnetoresistance. In Fig. 19 (see Section 16), we show a schematic diagram of the magnetoresistance of a composite of perfect conductor and normal metal. The corresponding result has been calculated within the effective-medium approximation3' in good agreement with the schematic picture shown in Fig. 19.33 "H. Stachowiak, Physica 45,481 (1970). '*M. H. Cohen and J. Jortner, Phys. Rev. Lett. 30, 696 (1973). 29M.H. Cohen and J. Jortner, Phys. Rev. A 10,978 (1974). 30D.Stroud and F. P. Pan, Phys. Rev. B 20,455 (1979). 31D.Stroud, Phys. Rev. Lett. 44, 1708 (1980). "T. K. Xia and D. Stroud, Phys. Rev. B 37, 118 (1988). 33Theenhanced resistivity in the presence of a magnetic field is due to the distortion of current in the vicinity of the spherical defects. In a strong magnetic field, such current distortions ). extend a large distance away from the sphere that produces them (a distance of order 0 , ~ See J. Sampsell and J. C. Garland, Phys. Rev. B 13, 583 (1976).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

17 1

One objection to the EMA, as formulated here, is that it does not allow for correlations between particles-that is, it assumes that each component is surrounded by the same effective medium. In reality, many composite media are strongly correlated, in the sense that constituent 1, say, is far more likely to be surrounded by 2 than by 1. This type of correlation is typical of ordered as well as disordered arrangements of spheres. One way to treat such correlations within the EMA is to consider the “particles” to be coated spheres rather than simply bare spheres of one constituent only. These coated spheres can then be embedded in an effective medium, whose conductivity ue is chosen so that the average polarization [as defined in Eq. (7.11)] shall vanish. Because the fields and currents within a coated sphere exposed to a uniform external field can be calculated analytically, this approximation can be executed by hand. If all the spheres have the same ratio of inner to outer radius, and if all space in the composite is entirely filled with such spheres, then it can be shown that the resulting EMA is actually identical to the Maxwell Garnett (MG) approximation [Eq. (3.4)]. Furthermore, the MG in this geometry is exact and not an approximation. The coating material plays the role of the host. This amusing result suggests that the M G is particularly appropriate for treating such correlated systems, made up of two components nonsymmetrically distributed. Further elaborations of such methods, involving distributions of coating widths or a mixture of coated and uncoated particles, can also be developed. In such extensions, there are potentially many adjustable parameters that ideally should be fitted to measured quantities, such as the percolation threshold or the observed geometry. A number of calculations of this kind have been developed to treat, quite successfully, the optical response of composite^.^^'^^ A particularly ingenious version of the EMA has been developed by Sen, Scala, and C ~ h e to n ~treat ~ materials with a zero percolation threshold for conductivity. Such materials might seem bizarre and unlikely, but there is ample experimental evidence that porous sedimentary rocks behave in this manner when their pore space is entirely filled (i.e., “saturated”) with conducting brine. Such rocks have long been described by an empirical rule known as Archie’s law.37 In present notation, Archie’s law can be written 6,=

Ao,p”

(7.20)

34P.Sheng, Phys. Rev. Lett. 45, 60 (1980). ”P. Sheng and A. J. Callegan, in “The Physics and Chemistry of Porous Media” (D. L. Johnson and P. N. Sen, eds.), AIP Conf. Proc. No. 107, p. 144, American Institute of Physics, New York, 1983. 36P.N. Sen, C. Scala, and M. H. Cohen, Geophysics 46,781 (1981). See also the review by P. N. Sen, W. C. Chew, and D. Wilkinson, in “The Physics and Chemistry of Porous Media” (D. L. Johnson and P. N . Sen, eds.), AIP Conf. Proc. No. 107, p. 52, American Institute of Physics, New York, 1983. 37G.E. Archie, Trans. AZME 146, 54 (1942).

172

D.J. BERGMAN AND D. STROUD

where o1 is the brine conductivity, p is the volume fraction of pore space (entirely occupied by brine), A is a constant of order unity, and n is Archie’s exponent (corresponding to the exponent t mentioned earlier). Equation (7.20) implies that the pore space in the rock is infinitely connected down to zero porosity-a remarkable result that Sen et al. took to imply a fractal or self-similar structure. In order to model this self-similarity, they extended the coated-sphere model mentioned earlier in a hierarchical way. In their picture (see Fig. 7) one begins with a sphere coated by a conducting fluid. This fluid is itself composed of spheres coated with the next-smaller level of conducting fluid, and so on. It is evident that such a model system will be conducting no matter how small the volume fraction of conductor. To estimate the conductivity of this model, Sen et al. developed a differential effective-medium approximation, according to which the system is built up step by infinitesimal step. For example, suppose that at some stage in the buildup there is volume V of composite containing volume pV of brine and (1 - p)V of rock matrix. The conductivity at this stage is ae(p).Now add an infinitesimal volume 6V of rock matrix. The volume fraction of brine is now p + 6 p = p V / ( V + 6 V ) z p(1 - h V / V ) . On the other hand, the conductivity becomdes o,(p 6 p ) = oe(p)[l 3(6V/V)(ar - ae(P))/(ar + 2ae(P))I. Because or, the conductivity of the rock matrix, is zero, these equations simplify to

+

+

6P P

a,(p + 6 p ) - o,(p) = 60 = -$o,(p) -,

(7.21)

which can be integrated to give ae = const

p3/2

(7.22)

FIG.7. Schematic illustrating the self-similar effective-medium scheme of Ref. 36. The model considers coated particles (Fig. 7.5a) such that the coating itself is made up of coated particles (Fig. 7.5b). The iterative coating procedure is continued indefinitely.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

173

corresponding to Archie’s law with n = 3/2. Other geometrical parameters can be introduced to give many other values of the Archie exponent. The method does not fully account for certain other experimental anomalies, however, notably the strongly increasing behavior of the real dielectric constant at low frequencies, seen in many of these porous rocks. Another possible difficulty is the fact that the rock matrix does not appear to percolate in this model, suggesting that a rock so constructed would not exhibit the observed rigidity. One can, however, postulate various small, but rigid, attachments between grains, sufficient to produce the observed elastic stiffness but not large enough to interfere with the self-similar structure needed to produce Archie’s law in this approximation. Milton3’ has shown that the EMA becomes exact in an unusual hierarchical geometry quite unlike the cellular picture for which the EMA is usually assumed to be appropriate. For a two-component composite (containing volume fraction p of component A, conductivity oA and 1-p of component B, conductivity oB),the zeroth level of the hierarchy is a composite of cells with typical length scale a and effective conductivity go. Next, one adds volume fraction x 1 of grains of size tlna;a fraction p of these is type A and 1 - p of type B, and x 1 4 1. In general, the kth level has particles a scale tl larger than the (k - 1)th and a volume fraction x k of new particles is added at this level. In the limit n + 00, under the conditions Lim nx, = 0, n+

(7.23)

00

a+m,

(7.24)

Milton has shown that the effective conductivity CT,of the nth level of the hierarchy approaches the effective-medium prediction, i.e., the solution of Eq. (7.8a). Thus there is a physically achievable, albeit strange, geometry that is exactly described by the EMA. Presumably this argument extends to all the other versions of the EMA described above for linear media. The emphasis in this description has been on the EMA for continuum systems. Analogous approximations can be developed for random conductance or admittance networks that are the discrete analogs of continuum composites, as first described by K i r k p a t r i ~ k .The ~ ~ idea here (for a bond percolation model) is to replace all the conductors surrounding a given conductor g = gA or gB by self-consistently chosen effective conductors ”G. W. Milton, in .“The Physics and Chemistry of Porous Media,” (D. L. Johnson and P. N. Sen, eds.), AIP Conf. Proc. No. 107, p. 66, American Institute of Physics, New York, 1983. ”S. Kirkpatrick, Rev. Mod. Phys. 45, 573 (1973).

174

D.J. BERGMAN AND D. STROUD

g = g,. An extra current I is injected into the network at one end of the conductor and removed from the other end, so as to restore the potentials in all nodes to the values they would have if the network were uniform. This current can be calculated by elementary arguments, for g = gA or gB. Setting its average value to zero gives a self-consistent equation for ge:

where z is the number of nearest neighbors. For a simple cubic lattice, (7.25) is identical to (7.8a). A generalization of this lattice EMA to site percolation has been given by Watson and Leath.40 Finally, we mention that several authors have extended the EMA so that it can be applied to nonellipsoidal inclusion^.^^ In this case, the individual particles, instead of having a depolarization tensor that has three principal components as in the case of an ellipsoid, are characterized by a large number of depolarization factors, corresponding to the many resonances that have nonzero dipole moment. The corresponding self-consistency condition is, of course, more complicated than that for spherical particles.

8. NUMERICAL TECHNIQUES FOR CONTINUUM COMPOSITES From (1.1) it is clear that a detailed knowledge of the local fields E(r) or $(r) is needed for evaluating E,. The oldest numerical method for calculating #(r) is to choose a finite mesh of points in the medium and solve a discretized version of the equation

v

*

(E

V$)

= 0.

(8.1)

In this way one gets a system of equations for the potentials of a conducting network made by inserting a conductor between every pair of neighboring mesh points. This approach has limited value for a medium where e(r) changes discontinuously when one moves between adjacent components: A very fine mesh is required in order to properly handle the discontinuities. Consequently, this approch is used mainly in investigations of the critical behavior of random percolating media, where the exact details of the microgeometry are sometimes unimportant. This approach is tantamount to replacing the continuum medium by a network of discrete circuit elements. Numerical methods for such networks are discussed in Section 9. 40B.P. Watson and P. L. Leath, Phys. Rev. B 9,4893 (1974). 41R.S. Koss and D. Stroud, Phys. Rev. B 32, 3456 (1985).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

175

The first systematic numerical method for calculating E , was developed by Lord Rayleigh6 for a periodic array of spheres. Rayleigh’s method was based on expanding the potential field +(r) in terms of the polarization multipoles of the individual spheres. An infinite system of linear equations is thus obtained for the expansion coefficients, and truncated versions of these may then be solved numerically. Only with the advent of digital computers was this program first carried beyond the first two orders, by McPhedran and McKenzie and their c o - ~ o r k e r s . ’ . ~ ~ a. Electrostatic Resonances-the

Formalism

Another approach, which sometimes leads to the same equations, is based on the expansion of both 4(r) and E , in the geometric resonances of the composite, as described in Section 4. In this approach, the boundary value problem for 4 is first transformed into an integral equation

where the linear integral operator

r4 =

s

r is given by

-

d V O(r’) V‘ G(r, r’) Vq5(r’).

(8.2b)

Here E , is the external field, assumed to lie in the z direction and henceforth taken to be 1, O(r) is the indicator function for component 1

O(r)

=

1 0

if r is inside otherwise,

(8.2~)

and G(r, r’) is a Green function for the Laplace operator

V2G(r,r’) G The operator defined by

=

-ad(r - r’)

=0

on the boundary.

(8.2d)

r is self-adjoint if the scalar product of any two functions is

42D.R.McKenzie, R. C. McPhedran, and G. H. Derrick, Proc. R. SOC.Lond. A 362,211(1978).

176

D.J. BERGMAN AND D. STROUD

Note that the geometric resonances the eigenstates of r

4n(r) described in Section 4 are simply

r4n= S n 4 n .

(8.4)

The expansion of (4.1) can be obtained by first noting that, because

we can rewrite the second line of (1.1) as

We then solve (8.2a) symbolically by

and substitute this in (8.6) to get

The results of (4.1) are finally obtained by inserting in this the expansion of the unity operator in terms of the eigenstates of r

b. Collection of Grains Computing the eigenstates 4n is not easier than computing the actual potential field 4 in the composite. A practical computational technique can, however, be evolved by using a different basis in Hilbert space-that of the eigenfunctions of the individual grains, considered as isolated inclusions in an otherwise homogeneous host. For such an isolated inclusion a we define an indicator function 8, and an appropriate raoperator (8.10)

PROPERTIES OF MACROSCOPICALLY INHOMGENEOUS MEDIA

177

with a complete set of orthogonal eigenstates

r 4 a u = saa4aa.

(8.1 1)

Clearly, we have

e = c en, r=cr,,

(8.12)

and when we expand 4 or 4,, in terms of 4,a we must be careful to use each 4aaonly within the volume of its grain a (8.13) The superscript (+) appearing with the indicator functions 8, 8, means that they are taken to equal 1 over a volume that is infinitesimally greater than that of 8, 8, themselves. This is needed so as to prevent spurious delta functions from arising at interfaces when derivatives are applied to these functions. This point, though technical, is important because it dictates that the grains must truly be unconnected. Using the new basis 4,,, which we denote by laa), the integral equation (8.2a) is transformed into an infinite set of linear algebraic equations for the expansion coefficients A,, of (8.13):

Here

(8.14b) and the matrix elements of

are given by

178

D.J. BERGMAN AND D. STROUD

When a = b this matrix element is trivial, and when a # b it can be calculated in terms of an overlap integral of the states 4,,, 4 b f l over the volume of one of the grains a, b

This is a useful scheme whenever the individual grains are sufficiently simple that the eigenstates 4,, can be easily calculated; that is the situation in the case of spherical grains. Once the matrix (aa I r I b j ) has been calculated, it can be used in a variety of ways to yield useful information. (a) By solving a finite subset of the equations (8.14a) for the coefficients

A,, we can determine not only the field 4 but also, by using (8.6), the function F(s):

F(s) =

--I A,, (zlaa). v U

(8.15)

a,

(b) If we truncate the matrix (aalrlb~)so as to make it finite, we can find its eigenvalues s, and eigenvectors A= and use them to calculate the resonance expansion of F(s):

(8.16)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

(c) Starting from (8.8), the matrix of powers of u = l/s, as in (6.6),

179

can also be used to expand F(s) in

(8.17)

” C.

aa

Periodic Arrays

In practice, these techniques have been used only for periodic arrays of spherical inclusions.238For spherical grains, all the eigenstates are obtained in terms of simple spherical harmonics. When the system is a periodic lattice, Bloch’s theorem allows a partial diagonalization of the r matrix to be achieved by combining the individual grain eigenstates of a certain type from all the (identical) grains. We can now choose the eigenfunctions to satisfy

where a is the position vector of the grain a. Consequently, the expansion coefficientsof z depend only on the state index a

and the matrix element (aalrlbp) depends on the grain indices a, b only through the relative position vector a - b; that is, it has the translational symmetry of the lattice. Defining A,(k)

1

= - 1A,, N a

e-ik

’

a,

(8.20)

where N is the total number of unit cells and k lies in the first Brillouin zone of the reciprocal lattice, we can now rewrite (8.14a) as a set of equations for

180

D.J. BERGMAN AND D. STROUD

This is of course just the standard Bloch treatment that decouples the equations for the Fourier coefficients A,(k) with different k vectors. Substituting from (8.20) into (8.15)-(8.17), we now find that we only need the k = 0 Fourier coefficients in order to calculate F(s) by any of the methods (8.22) and (8.23)

(8.24)

In the case of spherical grains, a further simplification arises because one of the eigenfunctions is equal to z inside the inclusion. Consequently, there is only one nonzero z , coefficient and the sums in these equations reduce to a single term. Further simplifications can arise from the point symmetry of the lattice. A calculation of F(s) for a periodic array of spheres thus involves the following steps: (a) The matrix elements (aal rlbp) must be calculated; these are obtained analytically in closed form by calculating the appropriate overlap integrals using the properties of the spherical harmonics. (b) The zeroth Fourier components of these matrix elements Tap(0)must be calculated numerically. Because the intergrain elements (aa I r I b p ) decrease with some of these sums distance as an inverse power of a - b that depends on a, j, are not absolutely convergent and must be calculated carefully, using methods similar to that used in calculating the Lorentz local field due to an array of dipoles. (c) The Fourier-transformed matrix Tap(0)can be used either to calculate the moments pr from (8.24), or to find its eigenvectors and eigenvalues for use in (8.23) and (8.16), or to solve (8.21) for the field coefficients A,(O) to be used in (8.22). A detailed treatment of a simple cubic lattice of identical spheres can be found in Ref. 8. d. Use of Bounds Once a certain number of the moments pr are known, they can also be used to produce exact bounds for F(s), even outside the convergence radius of the

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

181

power series expansion. For a real value of s outside the segment (0, l), these bounds start from the relation for real s $ (0, 1). A similar inequality can be written for E(s), whose zeroth moment related to po by KO

+ Po = 1.

(8.25) I C ~is

(8.26)

This can be translated back to another inequality for F(s), namely

F(s) 2

PO

for real s 4 (0, 1).

s-1++0

(8.27)

Starting from C'O) = I;, D(O) = E , we define two hierarchies of functions C(I)(s),D@)(s),each of which has the same analytical properties as F(s), E(s) (i.e., they are all sums of simple poles on the segment [0, 1) with positive residues and are all less than 1 at s = 1):l' (8.28)

where &), are the zeroth moments of C"), D(r).It is easy to show that these moments depend on the moments p o ... p r of F(s) and that

(1 - C(r)(s))(1

I

-

D(r)(s))= 1 - -, S

in analogy with (8.26) and (4.2~).By analogy with (8.25) and (8.27) we can now produce the bounds P(r) 0

s-

P? < C"'(s) I-

1+ p p -

for real s $ (0, 1).

(8.30)

S

When these are translated back into bounds for I;(s), we get bounds that depend on the moments p o . . . p r of F(s). In Fig. 8 we show some bounds for

182

D.J. BERGMAN AND D. STROUD

FIG.8. Upper and lower bounds on the bulk effective conductivity u, of a simple cubic array of spherical inclusions with conductivity u1 and volume fractionf, = 0.495 embedded in a host medium with conductivity uz = 1, as a function of 0,.The method used to obtain these bounds is equivalent to the one described in the text that culminates in Eq. (8.30). The parameter L that labels each pair of bounds is the same as r + 1, where r is the order of the highest moment of the pole spectrum pr [see (8.24)] that was used to obtain those bounds. Taken from G. W. Milton, Appl. Phys. A 26, 1207 (1981). E , of a cubic array of dielectric spheres that were calculated by utilizing an approach that is equivalent to the one described here." A similar procedure leads to a hierarchy of exact bounds for F ( s ) when s is complex, except that the basic inequality is now

Re F(s) +

Re s Im F(s) I 0 Im s

~

(8.31)

instead of (8.25). This restricts F ( s ) to lie on one side of a straight line in the complex plane and is independent of any moments. The analogous inequality for E(s) restricts F ( s ) to the inside of a circle. Thus the intersection of that

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

183

circle and the half-plane represents the bound. We can also produce halfplane bounds for P ( s ) and D(*)(s)that are similar to (8.31). The resulting bounds for F(s) are the intersection of two circles that depend on the moments p o . . . pr- ,of F(s).",'~ e. Fourier Methods Such a method has been proposed for periodic ~ o m p o s i t e s . ~In ~ ,this ~~" approach the quasi-static electric field E(r) is expanded in its reciprocal lattice Fourier components E,. The wave equation for E(r) is then recast into an infinite system of linear algebraic equations for E,. A truncated version of those is solved and used to calculate E,. The intputs required for setting up the equations are the reciprocal lattice Fourier components of E(r), namely E,. In practice, the method requires solving a rather large system of linear algebraic equations and yields accurate results only if the spread of values E(r) is not too great. Its main advantage is that one is not limited to spherical grains. A much improved version of this approach, based on the integral equation (8.2a), has recently been developed.43b f. Methods for Disordered Systems One approach is based on the power series of (6.6a) (8.32a) where the coefficients are the nth moments of the pole spectrum of F(s) and are given by [see Eq. (8.17)] (8.32b) The evaluation of p, involves a multiple integral where the integrand is a product of n + 1 8-functions [see (8.2c)l and n dipole-dipole interaction terms of the form VV'G(r,r'). Averaging over an ensemble of random samples turns the product of 6 functions into an (n + 1)-point correlation function that characterizes the average microstructure of the composite. In order to calculate the moments beyond p,, and pl,we thus need to know correlation 43R. Tao, 2. Chen, and P. Sheng, Phys. Rev. B 41,2417 (1990). 43aL.C. Shen, C. Liu, J. Korringa, and K. J. Dunn, J. Appl. Phys, 67, 7071 (1990). 43bD.J. Bergman and K. J. Dunn, Phys. Rev. B 45, 13262 (1992).

184

D.J. BERGMAN A N D D. STROUD

functions of three or more points. Such information has been gathered for the three-point functions for models of porous m a t e r i a l ~and ~ ~ also . ~ ~for some real porous materials by the processing of digital images of planar sections of such That information was then used to obtain p2, from which improved bounds were derived for E , of those material^.^^. There are some technical problems in evaluating the integrals because the dipoledipole interactions decay slowly with distance ( l/lr - r’I3) and are singular when r = r’. For a discussion of those difficulties and methods for surmounting them, see Ref. 50. Another approach is to create an ensemble of periodic systems with a large unit cell that varies from sample to sample. The methods of subsection (a) are used to evaluate the pole spectrum for every sample, and those spectra are then averaged over the ensemble. The average spectrum is used to calculate E,. This approach has been used in Ref. 51. A different approach to the calculation of E,, or the bulk effective electrical conductivity o,, is based on the connection between electrical conduction and the diffusion process of a classical random walker.52.The diffusion. current density of a collection of such noninteracting walkers is given by

-

where y(r) is the local diffusion constant, p(r) is the walker density, and V(r) is the electrochemical potential, equal to the sum of the externally applied electric potential 4 and the chemical potential ,u

The chemical potential is, in turn, related to the walker density and the temperature ?: assumed to be a constant, by (8.35)

“S. Torquato and G. Stell, J. Chem. Phys. 79, 1505 (1983). 45S. Torquato and G. Stell, J. Chem. Phys. 82, 980 (1985). 46P.B. Corson, J. Appl. Phys. 45, 3159 (1974). 47J. G. Berryman, J. Appl. Phys. 57, 2374 (1985). 48J. G. Berryman and S. C. Blair, J. Appl. Phys. 60,1930 (1986). 49J. G. Berryman and G. W. Milton, J. Phys. D 21, 87 (1988). 50J. G. Berryman, J. Comput. Phys. 75, 86 (1988). ”Y.Kantor and D. J. Bergman, J. Phys. C 15, 2033 (1982). ’*L.M. Schwartz, J. R. Banavar, and B. I. Halperin, Phys. Rev. B 40, 9155 (1989).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

185

where po(r) is the equilibrium density, i.e., the value of p(r) when q5 = J = 0. If Vq5(r) is nonzero but small, we can expand p(r) around po(r) and linearize (8.33) to get

-

This equation identifies the local conductivity a(r) with y(r) po(r). We note that the role of the local electric field in this problem is played by -VV(r) and not by - Vq5(r), which is just the externally applied field. For simplicity we can take y(r) = y o = const, and then the equilibrium function p&) represents the local conductivity. When this problem is implemented on a discrete mesh, the requirement of detailed balance entails that the transition probability from a site t to a neighboring site s is given by

Here E = -Vq5 is the externally applied electric field, rs and rt are the site position vectors, and $(r) is a bias potential that has a constant value in each component, such that po(r)

- e-8$(r)

for r inside the component i.

(8.38)

The local flux J(r) can now be calculated by allowing a set of random walkers to diffuse through the system with periodic boundary conditions. This must be done for a weak imposed field E so that (8.38) is satisfied. In Ref. 52 this approach, called biased diffusion, was applied for calculating the bulk effective conductivity (or formation factor) of a porous insulator that is saturated with a homogeneous conductor. The method appears to converge much faster than a calculation based on unbiased diffusion of a set of random walkers. It is especially suited for calculating the conductivity in continuum percolating systems like the one described earlier. All of the foregoing approaches require detailed information about the microgeometry of specific samples and involve major computing efforts. A different approach is to exploit measurements of E, for composites made from a different pair of components but having the same microstructure as the one under consideration. Those measurements provide values of F(s) at a number of other points, and those can be used to get improved bounds for F(s) by the methods of Section 5. [See 'Eq. (5.5) and the accompanying dis~ussion."~'~] This approach has been tested on some real samples of brine-saturated

186

D.J. BERGMAN AND D. STROUD

sandstone and was found to work quite well.53 In Fig. 9 we reproduce a sequence of measurements of the complex E , of this sandstone at different frequencies, along with the improved bounds that were calculated by using one of these measurements.

9. NUMERICAL TECHNIQUES FOR DISCRETE NETWORKS Here the main interest lies in random-resistor networks (RRNs) as simplified models for continuum systems near a percolation threshold. (See Section 10 for a brief summary of percolation theory.) a. Series Expansion for Randomly Diluted Network One approach, pioneered by Fisch and Harris," is to consider a hypercubic network in which a fraction p of the bonds are occupied randomly and independently. One then defines correlation functions of the form

where N is the total number of unit cells, R i j is the resistance between sites i and j of the lattice, and C i j is the cluster function, defined by

c..=

1 0

if i a n d j are on the same cluster, otherwise.

9.2)

The average is a configurational average over the different possible realizations of the network, and the sums are over all pairs of lattice sites. When C i j = 0 we always take CijRij = 0, even if R i j = cc and r > 0. For p close to p,, these correlation functions decay to 0 over a characteristic length l Ip - p c I -, the percolation correlation length, which diverges as p + p,. For that reason, the susceptibilities

-

(9.3)

-

will also diverge at p c as xr Ip - p,I y r , and their values represent the properties of the system on the scale of 5. Thus, the ratio xl/xo represents the configuration-averaged resistance between two connected sites at a distance 5. Likewise, the ratio x -Jx0 represents the average conductance between two 53J.

Korringa and G. A. LaTorraca, J. Appl. Phys. 60,2966 (1986).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

187

t

5'00p,

21000

0.08 MHz

9000

0

17000

74000

36000

,

3600

6

r

>

>

1800

30 MHz

2.0MHz 0

0

I800

3600

5400

0

120

240

I80

360

lOOMHz

]I 0

,

, 3

6

8 ' OOMHz 9

1

2

FIG.9. Measured values (small filled circles) and bounds for the complex E , / E ~ of brinesaturated sandstones at various frequencies as indicated (cO is the permittivity of free space). Two bounds are shown at every frequency except for 250 MHz. The wide bounds use information about (T and E of the brine and the rock, as well as about the porosity (volume fractions) and formation factor (0, at zero frequency). The narrow bounds are obtained by using the measured complex value of E, at 250 MHz as an additional constraint. Taken from J. Korringa and G. A. LaTorraca, J . Appl. Phys. 60, 2966 (1986).

188

D.J. BERGMAN AND D. STROUD

such points. This approach has been used extensively to study the critical behavior of such networks near pc.18*19 b. Relaxation

Another widely used approach is numerical simulation of RRNs, namely solving one or more sample networks numerically. Early studies used relaxation methods or over-relaxation techniques to find the potentials at all sites of the network (see e.g., Ref. 54). These methods have the serious drawback of failing when the network contains isolated clusters. Although this problem can be alleviated by assigning a small nonzero conductance to the absent bonds, it nevertheless continues to plague these techniques, both slowing them down and reducing their accuracy. The conjugate gradient method has been used to accelerate the relaxation to the exact solution, and Fourier transformations that utilize fast Fourier transform algorithms have been used to accelerate these relaxation methods even more. These matters are discussed in Refs. 55 and 56, including many references, and we will not describe them here in any detail as they are technically rather intricate. c. Transfer Matrix A totally different approach to the simulation of networks, known as the transfer-matrix method, was pioneered by Derrida and c o - w o r k e r ~ . ~The ~-~~ idea here is to build up the d-dimensional network gradually by adding successive (d - 1)-dimensional layers in a certain direction, which we call z (see Fig. 10). Focusing attention on the most recently added layer, we allow arbitrary voltages V, to be applied at each of its surface sites. The currents I i that then flow into these sites are linearly related to those voltages by means of a nonnegative, symmetric admittance matrix A i j (9.4)

As we add new bonds to the network in order to complete the next layer, Aij changes: New surface sites are created (see Fig. 10a), while previous surface sites become internal sites and are therefore discarded (see Fig. lob). If the new bonds are added one by one, it is very easy to calculate the resulting 541. Webman, J. Jortner, and M. H. Cohen, Phys. Rev. B 11,2885(1975). "G. G.Batrouni, A. Hansen, and M. Nelkin, Phys. Rev. Lett. 57, 1336 (1986). 56G.G.Batrouni and A. Hansen, J. Stat. Phys. 52, 747 (1988). 57B.Derrida and J. Vannimenus, J. Phys. A 15,L557 (1982). "B. Derrida, D. Stauffer, H. J. Herrmann, and J. Vannimenus, J. Phys. Lett. 44,L701 (1983). 59H. J. Hernnann, B. Derrida, and J. Vannimenus, Phys. Rev. B 30,4080 (1984).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

(a1

189

(b)

FIG. 10. Transfer-matrix method applied to a 2D square network of width 3. The sites that appear as indices of the admittance matrix are emphasized by circles. (a) Bond g l zis added to the network, along with a new site denoted by 1. The new admittance matrix Aij therefore has one additional row and column with the index 1. The transformation A + A' is given by

AIj = Aij

for all other i, j .

(b) Bond g I 3is added to the network and the site 3, which becomes an internal site, is discarded. The new admittance matrix A: therefore has the same number of rows and columns as A,. The transformation A' + A" is given by A!'. = A!, "

"

A13Akj ~

A;, + g 1 3

for i, j # 1,

changes in Aij. When the network is complete, we usually want to place it between two (d - 1)-dimensional equipotential hyperplanes (perpendicular to the x axis in Fig. lo), with no currents flowing at the other boundaries. Because of that, the sites on one side of the network are usually tied together and treated as a single site, while those on the opposite side are put at V = 0 and do not appear in the A matrix. The sites on the initial and final layers belong to the zero-current boundaries-this is achieved by beginning the calculation with A i j = 0 and adding an extra layer of zero-conductance bonds after the last layer is in place. This method has two advantages: (1) because there is no relaxation process and no large matrix inversions and errors tend to decay rather than build up as the calculation progresses, the accuracy of the results is limited only by roundoff errors, and (2) because the size of the admittance matrix A is determined by the number of sites in a single layer, which remains fixed throughout the calculation, it is possible to build up networks that are extremely long in the z direction. The only limitation is on computer time, not memory.

190

D.J. BERGMAN AND D. STROUD

This technique gives only the total ohmic conductance, not the potentials on the internal nodes. For some purposes, however, the detailed voltage distribution is required. This can be obtained by storing the A matrices for each completed layer while solving the system from left to right. Each such matrix gives the response of the network to the left of the appropriate section. A similar procedure gives the response of the network on the right. From the pair of matrices AL, AR thus generated, one obtains a set of linear algebraic equations for the site potentials of each section (see Fig. 11). It even proves possible to dispense with the AR matrices.60 Both methods lead to stable solutions. They have been used to obtain extremely accurate bond voltage distributions in both two- and three-dimensional (2D and 3D) R R N s . ~ ' , ~ ' d. Network Reduction Another useful method for calculating the total conductivity of a 2D RRN of the square lattice type employs a sequence of transformations that reduces it exactly to a single bond.63 The basic step is the so-called Y-V transforma-

0

I

2 n-i n

ntl n t 2

L

vi I .I

I

"i

( 0 )

(b)

FIG. 11. (a) Modified transfer-matrix method for calculating the internal site voltages aJ J' = feX,di,,. (b) Another modified transfer-matrix method for calculating the internal site voltages for the nth layer Vyl, assuming those of the previous layer I/?+') are known, as well as the left-hand admittance matrix AL for the nth layer:

v: fk = c AL.V: fy = c ,455;fk +

f; =

cj

vy1 = g,( VI"

+

1'

-

vy1,.

MD. J. Bergman, E. Duering, and M. Murat, J . Sfat. Phys. 58, 1 (1990). 61E.Duering and D. J. Bergman, J. Stat. Phys. 60, 363 (1990). 62E. Duering, R. Blumenfeld, D. J. Bergman, A. Aharony, and M. Murat, J. Stat. Phys. 67,113 ( 1992). 63D.J. Frank and C . J. Lobb, Phys. Rev. B 37, 302 (1988).

PROPERTIES O F MACROSCOPICALLY INHOMOGENEOUS MEDIA

191

FIG. 12. The two three-terminal ohmic networks are entirely equivalent if the conductances are appropriately chosen. The relation between G,G2G, and GAGBGccan be expressed either as

GA =

G2G3

,

~

G,=--,

G,G,

G

G,=-

GlG, G '

G = G,

+ G2 + G,,

1

1

1 +-+-.

1

GA

GB

GC

or as GBGC

G, =- G'

2

G,=-

GCGA

G',

GAGB G 3 = 7 .

-G,--

tion, shown in Fig. 12 along with the transformation equations. Starting from the upper left-hand corner of a rectangular sample, the two corner resistors are first transformed into a diagonal resistor, which is then propagated diagonally downwards by a sequence of Y-V transformations until it reaches the bottom edge of the sample (see Figs. 13 and 14). This procedure is repeated until all the unit cells of the top row have been eliminated. The same process is applied to the other rows in sequence. In the end one is left with a single resistor whose resistance is that of the original network. This method is very accurate, stable, fast, and economical of memory. It is applicable, with

( 0 )

(b)

(C

1

(d)

FIG. 13. Sequence of Y-V transformations that result in the propagation of a diagonal resistor downward and to the right by one lattice unit. The triangle in (a) is replaced by a Y in (b). This is redrawn in (c) to emphasize the new Y-structure in the lower right-hand corner, which is then transformed into a new triangle in (d). Taken from D. J. Frank and C. J. Lobb, Phys. Reo. B 37, 302 (1988).

192

D.J. BERGMAN AND D. STROUD

FIG. 14. Sequence of transformations starting from an arbitrary 2 x 3 square network and ending with a single linear chain. Taken from D. J. Frank and C. J. Lobb, Phys. Rev. B 37, 302 (1988).

some modifications, to other 2D networks63 (triangular, honeycomb, Kagome), as well as to some types of nonlinear networks.64 Its only drawback is that it is restricted to 2D networks.

111. DC Electrical Properties-Applications Problems

to Specific

10. PERCOLATION THEORY The percolation phenomenon became a respectable subject of scientific research as a result of the pioneering work of Broadbent and Hammersley6’ (see Ref. 66 for a personal account of this early work). Since then, much effort has gone into studying both the geometrical and the physical aspects of this phenomenon. Its appearance in many real physical systems is partly responsible for this, as is the fact that percolation is MM.Octavio, A. Octavio, J. Aponte, R. Medina, and C. J. Lobb., Phys. Rev. B 37,9292 (1988). 65S. R. Broadbent and J. M. Hammersley, Proc. Cambridge Philos. SOC.53, 629 (1957). 66J. M. Hammersley, in “Percolation Structures and Processes” (G. Deutscher, R. Zallen, and J. Adler, eds.), Ann. Israel Phys. Soc. 5, 47-57 (1983).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

193

a very simple type of mechanism for a critical point where physical properties have singularities and exhibit critical behavior. The geometrical aspects of percolation are reviewed in great depth in the book and in the excellent article by S t a ~ f f e r , ~and ~ ’ ~many * physical properties connected with percolation are discussed in a special volume edited by Deutscher et aL6’ Here we give a brief summary of the theory of percolation, and in the two following sections we discuss some physical aspects of percolation that have not been reviewed elsewhere. Even before the concept of percolation was introduced into physics explicitly, a conductivity threshold caused by a percolation process was known to occur in the symmetric effective-medium approximation (EMA) of Br~ggeman.~ In that simple approximation, a random mixture of conductor and insulator has a bulk effective conductivity 0, that changes its behavior abruptly as the conductor volume fraction pM increases through the value 1/3 (for a general dimensionality d , the appropriate value is l/d). At that critical value, called the percolation threshold and denoted in general by p c , the conductivity begins to rise above 0, as shown in Fig. 15. By measuring the conductance as a function of pMin a sheet of randomly perforated carbon paper, Last and T h o u l e ~ showed s ~ ~ convincingly for the first time that in a real percolating system the conductivity does not increase linearly with pM, as predicted by EMA.

0

I

3

I

PM

FIG.15. Symmetric effective-mediumapproximation for the conductivity B , of an isotropic metalkinsulator mixture. For metal fraction pM< 4,B , = 0.For pM> $,the dependenceis linear: us = &e - 3).

67D.Stauffer and A. Aharony, “Introductionto Percolation Theory, Second Edition,” Taylor and Francis, London, 1992. 68D.Stauffer, Phys. Rep. 54, 1 (1979). 69G. Deutscher, R. Zallen, and J. Adler, eds., “Percolation Structures and Processes,” Israel Phys. SOC.,Vol. 5, Adam Hilger, Bristol, 1983. ’OB. J. Last and D. J. Thouless, Phys. Rev. Lett. 27, 1719 (1971).

194

D.J. BERGMAN AND D. STROUD

In fact, if we make a two-component random composite mixture of a good conductor and a bad conductor, with conductivities oM9 ol,then the bulk effective conductivity oeis found to have a power law dependence on pM- p, close to p,:

The first of these expressions would describe the critical behavior of a metal-nonmetal mixture above the percolation threshold of the metal, and the second would describe the critical behavior of a superconductor-normal conductor mixture below the percolation threshold of the superconductor. The values oft and s are found, from calculations on percolating networks, to be

i

1.30, t = 2.0 3 (exact),

d =2 d =3 d26

(Refs. 59, 63, 71, 72), (Ref. 58), (Refs. 73, 74),

1.30, 0.76 0 (exact),

d =2 d=3 d26

(Refs. 59,63,71,72), (Ref. 59), (Refs. 73,74),

s=

{

(10.2)

and these values may be contrasted with the values t = s = 1 that are predicted by EMA. If o,/oMis not strictly zero, the behavior described by (10.1) breaks down when APM = (pM- pc) tends to zero, because oecan never truly vanish or become infinite. This crossover in the critical behavior is believed to be described by the scaling expression'

(10.3a) A P = lPM-Pcl, where the scaling function Az) has the following asymptotic behavior: 71J. G. Zabolitzky, Phys. Rev. B 30,4077 (1984). 72C. J. Lobb and D. J. Frank, Phys. Rev. B 30,4090 (1984).

73J. P. Straley, J. Phys. C 10, 3009 (1977). 74M.J. Stephen, Phys. Rev. B 17,4444 (1978). "5. P. Straley, J. Phys. C 9, 783 (1976).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

+ Bz + . . . A‘z + B’zZ + . . . A

A”z‘/(‘+S)+

...

195

for z 4 1 and pM > pc (Regime I), for z

< 1 and pM< pc (Regime 11),

for z

$

1, pM

(10.3b)

3 p c (Regime 111).

The first two forms are needed if (10.3a) is to agree with (10.1); the last form is needed to make oe independent of pM and finite when pM -+ p c . Of course, (10.3a) only holds asymptotically when both pM- p c and oJaM are small. The scaling behavior described by (10.3a) is obtained explicitly in the effectivemedium approximation (see Section 7) and in the mean-field theory of p e r ~ o l a t i o nSimilar . ~ ~ behavior is exhibited by thermodynamic systems near critical points of phase transitions. Thus arose the idea that such behavior might also be expected near a percolation t h r e ~ h o l d .This ~ ~ . has been verified in some numerical sir nu la ti on^.^^^^^ Most of the theoretical work on percolation has actually dealt with random networks rather than with continuum composites, in the hope that the important critical properties are universal, i.e., independent of the precise details of the model. Some cases in which this is not true are discussed in Section 12. In network models, much of the research has dealt with simple (i.e., periodic) networks of identical resistors a certain fraction of which have been discarded at random. In such a “diluted” network, the conductivity is entirely determined by the so-called backbone of the percolating (or networkspanning) cluster-the subset of resistors that carry a nonzero current. The links-nodes-blobs (LNB) picture of Stanley7* and Coniglio7’ provides a useful intuitive picture of the backbone and also enables good quantitative estimates to be made of the critical behavior. In this picture, the backbone is viewed as an irregular supernetwork of nodes and links (see Fig. 16): A node is any site of the backbone that is connected to the boundary by at least three independent paths, while a link is the set of backbone bonds between two adjacent nodes. The average distance between adjacent nodes is the percolation correlation length

5

-

AP-”, 4 (exact),

3 (exact),

d = 2 (Ref. 80), d = 3 (Ref. 81), d26

(10.4)

(Refs. 67,68),

76J. P. Straley, Phys. Reo. B 15, 5733 (1977). 771. Webman, J. Jortner, and M. H. Cohen, Phys. Rev. B 16,2593 (1977). 78H.E. Stanley, J . Phys. A 10, L211 (1977). 79A. Coniglio, in “Disordered Systems and Localization” (C. Costellani, C. DiCastro, and L. Peliti, eds.), Lecture Notes in Physics, Vol. 149, Springer-Verlag, Berlin, 1981. ‘OB. Nienhuis, J. Phys. A 15, 199 (1982). ‘lD. W. Heerman and D. Stauffer, Z . Phys. B 44, 339 (1981).

196

D.J. BERGMAN AND D. STROUD

FIG.16. A sample diluted network between two equipotential plates. Shown are two compact or finite clusters (A, B) and a percolating cluster. In the percolating cluster, the backbone (i.e., the current-carrying) bonds are shown as heavy lines and the dangling clusters (C, D) are shown as regular lines. On the backbone we have identified the nodes by black circles (E, F, G), and the multiply connected pieces or blobs by hatching (H, I, J). All the other bonds of the backbone are singly connected bonds. A link comprises all the backbone bonds connecting between two adjacent nodes and usually includes both singly and multiply connected bonds.

-

and the average conductance of a link is gr Ap'. A link includes singly connected bonds (SCBs), i.e., resistors that carry the full current flowing in the link, as well as multiply connected "blobs." The average number N,,, of SCBs in a link satisfies N,,, 1/Ap (see Ref. 82). The number of SCBs provides a lower limit on the resistance of a link and hence 4' 2 1. The LNB picture can now be used to get the conductivity of the network (d is the dimensionality):

-

oe E g5/<"'

-

Ap< + ( d -

2)v

(10.5)

We thus get the following expression and bound for the critical exponent t of (10.1): t =(

+ (d

-

*'A. Coniglio, J. Phys. A 15, 3829 (1982).

2)v 2 1 + (d

-

2)v.

(10.6)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

197

Comparing (10.2) and (10.4), we see that the inequality (10.6) is indeed satisfied and becomes a strict equality for d 2 6. This shows that ford 2 6 the SCBs govern the critical behavior and the blobs are unimportant, whereas for d < 6 it is the blobs that determine the conductance of the network. Similar considerations can be used to analyze the critical behavior of other properties near p c . Other methods that have been used to study percolating networks are series expansion^,'^,'^, a field theory with an expansion around d = 6 in powers of E = 6 - d,83,84real-space renormalization groups of various kind^,^^-*^ and numerical simulations of large n e t ~ o r k s . ~Of~ ~ ~ ~ , ~ these, series expansions and simulation techniques were described in Section 9. Both the backbone and the percolation cluster are fractal objects. This means that, at p c , their masses increase as a noninteger power of the system size L:LDf,where D,is the fractal dimension. The finite or compact clusters are also fractal objects when their linear size is less than t. Since this was realized, many regular fractal models have been proposed to mimic various aspects of the critical behavior of percolating system^.^^.^^ Another related topic is the time-dependent phenomenon of anomalous diffusion: The distance covered by a random walker on a fractal cluster increases with time not like t”’, as it would on a uniform structure, but like t“, where the exponent k depends on the nature of the fractal and is related to the exponents [ and t defined earlier.91.92 When aJa, is not strictly zero, the conductance is determined not merely by the percolating backbone of aM bonds. The a1 bonds also make a contribution, as we saw in (10.3), and also the finite 0, clusters and the dangling ends of the percolation cluster. But even when aJoM = 0, the response of the system at finite frequencies is not determined entirely by the backbone, because the insulating component has a finite imaginary impedance due to its dielectric properties. We elaborate this point in the next section, where the dielectric behavior of a metal-dielectric composite near p c ”A. B. Harris and T. C. Lubensky, Phys. Rev. B 35, 6964 (1987). 84T.C. Lubensky and A. B. Harris, Phys. Rev. B 35, 6987 (1987). ”A. P. Young and R. B. Stinchcombe, J. Phys. C 8, L535 (1975). 86R. B. Stinchcombe and B. P. Watson, J. Phys. C 9, 3221 (1976). ”S. Kirkpatrick, Phys. Rev. B 15, 1533 (1977). 88R. Rosman and B. Shapiro, Phys. Rev. B 16, 5117 (1977). ”Y. Gefen, B. B. Mandelbrot, and A. Aharony, Phys. Rev. Lett. 45, 855 (1980). 90J. Feder, “Fractals,” Plenum, New York, 1988. 91Y.Gefen, A. Aharony, and S. Alexander, Phys. Rev. Lett. 50, 77 (1983). ’’(2. D. Mitescu and J. Roussenq, in “Percolation Structures and Processes” (G. Deutscher, R. Zallen, and J. Adler, eds.), Ann. Israel Phys. Soc. 5, 81-100 (1983).

198

D.J. BERGMAN AND D. STROUD

is discussed. Other physical properties that may exhibit interesting critical behavior near p c are magnetotransport, thermoelectricity, and superconductivity. These topics have not received nearly as much attention as the ohmic conductivity near p c . We have tried to describe what is known about them in Sections 13,15, and 16. We expect that more will be learned about them in the years to come.

11. DIELECTRIC BEHAVIOR NEARA PERCOLATION THRESHOLD In order to discuss the electrical properties of a metal-dielectric mixture at a finite frequency, we must use complex conductivities or dielectric constants. We then use the fact that the functions F(s), E(s) can be continued analytically to complex values of s to obtain results for the complex bulk coefficients of the composite. One way to obtain results for Re(&,)near p c is by constructing a simple form for the scaling function f ( z ) of (10.3). Such ansatzes were used in Ref. 93 to predict the critical behavior of the dielectric constant in a percolating system. Such a prediction can also be made simply by using (10.3), as we now proceed to show. Replacing aI= 0 by a, = i0&J4n,and s = 0 by s z' -iw&J4naM, we get from (10.3) [note that the exponent s is unrelated to the complex argument of F(s), E(s), even though the same symbol is used for both quantities]

111.

The real and imaginary parts of aegive us directly the critical behavior of the (real) bulk effective conductivity and the (real) bulk effective dielectric constant E , in the three regimes. In particular, the dielectric constant is preciicked to diverge as ApPs when p,is approached from either side, and at pc both Re E, and Re ce acquire a peculiar frequency dependence. The divergence of E , as pM + pc- was observed experimentally not long after this prediction was made.94 Later it was also verified that E, diverges as pM+ p,' , that is, from the other side of pc.95,96 93D.J. Bergrnan and Y. Irnry, Phys. Rev. Lett. 39, 1222 (1977). 94D. M. Grannan, J. C. Garland, and D. B. Tanner, Phys. Rev. Lett. 46, 375 (1981). 95S. Bhattacharya, J. P. Stokes, M. W. Kim, and J. S . Huang, Phys. Rev. Lett. 55, 1884 (1985). 96L.Benguigui, J . Phys. (Paris) Lett. 46, L1015 (1985).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

199

CONDUCTIVITY NEAR A PERCOLATION 12. NONUNIVERSAL THRESHOLD

a. Continuum Composites Although the analytical and computational treatments of the behavior near percolation are invariably made using RRN models, the systems on which real measurements are made are usually continuum composites. It has become clear that such composites sometimes deviate in their critical behavior from the binary R R N paradigm. When one tries to formulate a discrete network approximation for a continuum composite, one usually finds that a distribution of conductances g is called for, rather than just a. simple binary R R N with g = 0 or g = 1. Moreover, in some models of continuum percolation, the resulting probability density function f(g) has a power law s i n g ~ l a r i t y ~ ~ - ~ ~ f(g)

-

g-",

0 < a < 1,

as g -+ 0.

(12.1)

This causes the conductivity critical component to depend on a in the following way :' O0-l O 2 t(a) = Min(t, tl(a)),

(12.2a)

where t is the usual universal exponent of the binary R R N , while t,

= (d - 2)v

+ 1 -1a ~

(12.2b)

This occurs, for example, in the "Swiss cheese model," in which spherical voids are introduced with random and independent positions in an otherwise homogeneous conductor. As the volume fraction of the conductor decreases towards p , , the current is eventually carried mainly by a set of narrow conducting regions each of which is bounded by three voids (see Fig. 17). These connected regions can be replaced by a network of discrete conductors with a probability distribution of the form (12.1) with a = $. The physical reason for the change from t to t , is that, due to the abundance of low-g conductors, the conductance of a long one-dimensional 97B.I. Halperin, S. Feng, and P. N. Sen, Phys. Rev. Lett. 54,2391 (1985). 98P.N. Sen, J. N. Roberts, and B. I. Halperin, Phys. Rev. B 32, 3306 (1985). 99S. Feng, B. I. Halperin and P. N. Sen, Phys. Rev. B 35, 197 (1987). 'O0J. P. Straley, J. Phys. C 15, 2333 (1982). 'O'J. Machta, Phys. Rev. B 37, 7892 (1988). lo2T.C. Lubensky and A.-M. S. Tremblay, Phys. Rev. B 37, 7894 (1988).

200

D.J. BERGMAN AND D. STROUD

FIG. 17. (a) “Swiss cheese model” in two dimensions. Straight lines show the bonds of the superimposed discrete network; dotted lines are the missing bonds. (b) Narrow portion of a bond, passing between three overlapping spherical holes, in the three-dimensional Swiss cheese model. Taken from B. I. Halperin, S. Feng, and P. N. Sen, Phys. Rev. Lett. 54,2391 (1985).

chain is sometimes determined by just a single bond-the one with the lowest conductance. l o o Numerical simulations of RRNs, with probability distributions like (12.1) for the undiscarded conductors, have largely verified these values of t(cr), which were predicted by various theoretical argument^.^*^'^^ An important conclusion is that, if appropriate care is taken in constructing the RRN model, a real continuum composite can still be successfully described by such a model, at least as far as the ohmic conductivity is concerned. b. Wide Distributions Another situation arises when gf (9) is approximately constant and very small for a wide range of g, stretching over many orders of magnitude [but not including g = 0 of course, or else f(g) would be unnormalizable]. This occurs when the conductor network really models a process involving quantum mechanical tunneling and thermal activation over a barrier, as in the case of hopping conductivity in the regime of localized electronic states in In that case it has been argued that the a disordered conductivity of the network is determined essentially by the threshold conductance g c such that the bonds with g 2 gc just form a percolating network. It has also been argued that the conductance g, of the macroscopically equivalent homogeneous network is given by’O7

(12.3) lo3M. Murat, S. Marianer, and D. J. Bergman, J. Phys. A 19, L275 (1986). lWV. Ambegaoker, B. I. Halperin, and J. S. Langer, Phys. Rev. B 4, 2612 (1971). lo5B.I. Shklovskii and A. L. Efros, Zh. Eksp. Teo. Fiz. 60,867 (1971) [Sou. Phys. JETP 33,468 (1971)l. Io6M. Pollak J. Non-Crysr. Solids 8-10, 486 (1972). ‘O’P. le Doussal, Phys. Rev. B 39, 881 (1989).

201

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

Interestingly, the critical exponent that governs ge now depends entirely on the geometric critical exponent v. 13. HALLEFFECTAND MAGNETORESISTANCE The introduction of a magnetic field H , in addition to the applied electric field, makes the conductivity tensor anisotropic and nonsymmetric even if it was a scalar tensor for H =-0. If we choose k 11 z, this tensor has the form

e=

0

Bo

0

0

:)+ (-; 00

0 0 0

+

('f

0 60, 0

:)

(13.

6011

where go is the conductivity at H = 0, A is the Hall conductivity, and 60 60 are the transverse and longitudinal magnetoconductivities, respective] Although this form applies strictly only to an isotropic material, so that even crystals with cubic symmetry are excluded in general, the form of the antisymmetric part is correct at low magnetic fields also for cubic symmetry. The relationship between the volume averages of E and J can again serve to define the bulk effective conductivity tensor c,

where cenow depends on four parameters, as in (13.1). Note that in this case, if we tried to define ce by means of the production rate of Joule heat, we would lose the bulk effective Hall conductivity A, : That quantity appears only in the antisymmetric part of ceand consequently will not affect the rate of dissipation (J) (E). Early theoretical discussions of the Hall effect in composite system^'^^*'^^ produced important insights as well as some remarkable results, some of which were forgotten in the interim and later rediscovered (e.g., the weak-field Hall effect in a two-dimensional micr~geometry'~~). An important development was the extension of Bruggeman's ideas to produce a symmetric effective-medium approximation for ce.This was first done by Stachowiakz7 and, somewhat later, though independently, by Cohen and Jortner for the Hall coefficient.28.They applied their theory to interpret measurements of the Hall effect by metal vapors near a metal-insulator Stroud and his co-workers generalized those results and applied them to a discussion of magnetoresistance measurements in polycrystalline metal^'^,^' and in normal-superconducting composite^^'^^^ (see also Section 7).

-

'''5. Volger, Phys. Rev. 79, 1023 (1950). loYH. J. Juretschke, R. Landauer, and J. A. Swanson, J. Appl. Phys. 27,838 (1956).

202

D.J. BERGMAN AND D. STROUD

Further developments involve a restriction to low values of H , such that the three terms of the sum in (13.1) are proportional to successively higher powers of H , namely HO, H1, H2.This allows one to solve for the local electric field by a perturbation method and leads to the following expressions for ,Ie and 80,: (13.3)

+

&

- (A@’)

-

jdVSdl/‘(A(r) x E@)(r)) VV’G(r, r’la)

(1 3.4)

x E(b)(r’)).

These equations are generalizations of the expressions for two-component composites from Refs. 110 and 111 (see also Ref. 60). Here E(X)(r), etc. is the local electric field for the ohmic problem (i.e., when H = 0) in the composite when the average or applied field E, lies along the x-axis, etc., and G is Green’s function for the same problem and solves the following boundary value problem: V * a(r) VG(r, r’la) = -dd(r - r’), on the boundaries. G =0

(13.5)

Neither this Green function nor the local fields E(“)(r)are easy to find in any but the most trivial of systems. Nevertheless, these expressions are useful for a variety of reasons. The fact that E(“)(r) need to be known only in the absence of a magnetic field greatly facilitiates the numerical calculation of A,. This has been carried out on RRN models of the Rall effect in order to study the critical behavior near a percolation threshold.112g60The preferred method is thus to determine the detailed voltages on all the bonds of the percolating backbone (see Section 9 for a discussion of numerical methods for doing this) and then use those in an appropriate discretized version of (13.3). ‘I’D. J. Bergman, in “Percolation Structures and Processes” (G. Deutscher, R. Zallen, and J. Adler, eds.) Ann. Israel Phys. SOC.5, 297-321 (1983). ‘I’D. J. Bergman, Phil. Mag. B 56,983 (1987). ‘12D.J. Bergman, Y. Kantor, D. Stroud, and 1. Webman, Phys. Rev. Lett. 50, 1512 (1983).

203

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

The foregoing theory also leads to useful scaling ansatzes for I , and ha,. Specializing to a two-component composite, (13.3) and (13.4) can be transformed into'10,111~60 (13.6)

1 (do,),, =

- -

S d V (E@) da E',))

VEO

+

dV'(e, x E@)(r)) VV'G(r, r'lo)

-

vE;

dvJ"M

- (eHx E(b)(r')),

(13.7)

where eH is a unit vector along H, and all but one of the integrals are restricted to the subvolume V, of component M. Because the right-hand side of (13.6) depends only on ohmic properties of the composite, including the ratio of ohmic conductivities g.JoM, it is natural to assume that, near p , , it will depend on the microgeometry through the same scaling variable that appeared in (10.3a)l13 (13.8a) when

The asymptotic forms of the scaling function h(z) in the three principal critical regimes are

+ +

A+Bz .*. A'z' B'z3 ... A'tz(2t-g)l(t+s)+ . . .

+

I I1 I11

(13.8~)

where the form in regimes I and I1 needs some explanation: Whereas inside the poor conductor o, the local field E(r) is always of order E , , inside the good conductor o,, E(r) E , only on the percolating backbone (this is the

-

'I3D.J. Bergman and D. Stroud, Phys. Rev. B 32,6097 (1985).

204

D.J. BERGMAN AND D. STROUD

current-carrying part of the percolation cluster-dangling ends are excluded). Elsewhere in aMthe field will satisfy E(r) E , . aJaM< E , . Because the integration in (13.6) is confined to the aMsubvolume, the right-hand side will be O(1) when pM> p , , but O(o:/oi) when pM< p , . Using (13.8) and the relation

-

pH

= I/aZ

(13.9)

for the weak-field Hall resistivity pH,we get the following predictions for the critical behavior of pHe:’

where constants of order 1 have been omitted from all terms. In getting these results we assumed I, < AM, but we did not assume I , < I,, and indeed the second term in each line of (13.10) arises from the I , in the numerator of (13.8a). Which term dominates in any situation depends on the values of the relevant physical parameters. For instance, in regime I, although the poor conductor usually has a much larger Hall resistivity pHI% PHM, this is normally counteracted by the smallness of aJoM, so the coefficient of the second term in the first line of (13.10) is small, PHI(cJI/oM)’ < PHM. However, its diverging factor A P - ~ ‘grows much more rapidly than Ap-8 us Ap -+ 0, because g 0.4 while 2t s 4.0 in three dimension^.^^.^^ Thus, even though the good conductor always dominates the ohmic resistivity when pM> p , , either conductor may dominate the Hall resistivity, and one might even observe a crossover as Ap -,0. Similar effects can occur in the other regimes. ~ ~ This was dramatically verified in an experiment by Dai et ~ l . , ’who measured the Hall resistivity of an Al-Ge composite mixture. When this system was above the percolation threshold of A1 (i.e., the good conductor) they nevertheless found that the Hall effect was dominated by the Ge: The measured critical exponent agreed well with 2t, and the sign of the hall resistivity was positive, indicating that it was due to Al-doped Ge and not to A1 (see Fig. Ha). In order to observe the Ap- behavior and measure 9, they had to remove the A1 (by dissolving it in a solution of KOH), leaving behind a mixture of Ge (now the good conductor) and vacuum or glass substrate (the poor conductor). This made the coefficient of the second term in (13.10) small enough so that the first term could dominate (see Fig. 18b). lI4U. Dai, A. Palevski, and G. Deutscher, Phys. Rev. B 36, 790 (1987).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

0.03

0.1

X,,

205

0.3

-0. I5

FIG.18. (a) Log-log plot of the effective Hall coefficient R , versus xA,- 0.15 (here xA,is the volume percent A1 and x, = 0.15). The different symbols refer to different substrates. (b) Log-log plot of the Ge Hall coefficient versus xGc- 0.15 (filled circles) (here xGeis the volume percent Ge). The theoretical results from Ref. 112 are shown by solid triangles. Taken from U. Dai, A. Palevski, and G. Deutscher, Phys. Rev. B 36, 790 (1987).

206

D.J. BERGMAN AND D. STROUD

A scaling ansatz for 60, in weak magnetic fields has also been proposed that is based on (13.7) and on considerations similar to those made above for A,."' Its predictions have even more variety than those of (13.10), because a greater number of relevant physical parameters is involved. It remains to be seen whether these too can be checked experimentally. Expressions (13.6) and (13.7) can serve as a basis for attempting to calculate Ae and 60, for small H in real composites with a known and sufficiently simple microgeometry, such as an ordered array of spherical inclusions. In the regime of strong magnetic fields, when A and 60 are not small compared to 6,the best approach for nondilute composites remains the selfconsistent-effective-medium approximation. This has been used by Stroud and co-workers to predict some dramatic variations of the magnetoresistance with the volume fraction in metal-n~nrnetal~'as well as in normalsuperc~nducting~' mixtures. In the latter case, the predictions seem to be in agreement with 14. DUALITY IN TWO DIMENSIONS

Composites with a 2 D microgeometry possess a special symmetry called duality. One example of such a system is a 3 D composite with cylindrical symmetry; i.e., all interfaces are parallel to a fixed direction. Another example is a thin conducting film whose thickness is negligible compared to other microgeometric scales such as channel widths and grain sizes. In this case, even though the film is part of a 3 D system, the only important electric and current fields lie in the plane of the film. This symmetry was first discovered by Keller"6.1'7 for the case in which all components have scalar conductivity tensors. Keller's result was later generalized by Mendelson"8 to the case of general anisotropic conductivity tensors, including nonsymmetric conductivity sensors. A further generalization has been made by Milton,' who provides additional historical references. We will restrict ourselves here to the case in which all conductivity tensors (r are isotropic but may have an antisymmetric off-diagonal part due to the presence of a magnetic field perpendicular to the plane of the system, which we take to be the x,y plane,

.=( - A

A).

0

'"D. J. Resnick, J. C. Garland, and R. Newrock, Phys. Reu. Lett. 43, 1192 (1979). 'I6J. B. Keller, J. Appl. Phys. 34, 991 (1963). '"5. B., Keller, J. Math. Phys. 5, 548 (1964). '"K. S. Mendelson, J. Appl. Phys. 46, 917 (1975); J. Appl. Phys. 46,4740 (1975). 'I9G. W. Milton, Phys. Rev. B 38, 11296 (1988).

(14.1)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

207

The duality transformation consists of rotating both E(r) and J(r) by 90" in that plane. They then become, respectively, J'(r) and E'(r), which are the correct (current and electric) fields for a new problem in which the conductivity tensor a' is the reciprocal of a

l/a(r).

or@)=

(14.2)

This is shown by the following equations:

J = ( -1

0' ) E = e , x E ,

E = ( -1O 0 ')J=e,xE, 0 1 J=( -1 0)o-l(l

-

V J

=V

- (e, x E)

=

0

-e,

V x E = V x (e, x J) = e,(V

-1

(14.3)

(V x E) = 0,

- J) = 0.

The only tricky point is that the new fields E', J' may satisfy a different type of boundary condition than E, J. This causes no problem if the system is macroscopically homogeneous,' * or if the boundary conditions are chosen appropriately.' l o The bulk effective conductivity tensors ae and a: are defined by the relations between the volume averages of J, E and J , E , respectively (we cannot define them by means of the dissipation if we wish to include an antisymmetric part in ae,a:)

'

(14.4) For a microgeometry with either square, triangular, hexagonal, or isotropic point symmetry, ae and a: have the same form as (14.1). Denoting the (functional) dependence of aeon the local conductivity tensor a (r) for a fixed microgeometry by a,(a), we get from (14.3) a:

= a,(l/a) =

l/ae(a).

(14.5)

208

D.J. BERGMAN AND D. STROUD

This result holds for arbitrary values of H and leads to a number of useful exact relationships regarding the conductivity near a percolation threshold. First of all, when H = 0 and the composite is made of two components, (14.5) can be rewritten as

where all the conductivities are now scalars. From the fact that oe(aM, aI)is a homogeneous function of order 1 we now get

which goes under the name of phase exchange equality. By considering the case aI= 0 < aM,we immediately deduce from this that the percolation properties of the two components are mutually exclusive-when one component percolates, the other cannot and vice versa. Therefore the two components attain their percolation thresholds simultaneously, pIc = 1 - pMc,at which point we can write (see (10.3b))

(14.7) Using this in (14.6b), we immediately get t = s

-

ae(aM, aI) (aMa1)1/2 in Regime 111.

(14.8)

For a composite with a symmetric microgeometry, i.e.,

(14.9) even stronger results are obtained, namely PMc = PIC=

1

z?

a,(aM,a,) = ( O ~ C T , ) ' ~ ~ in Regime 111.

(14.10)

This happens in a continuum composite in which both components obey

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

209

Gaussian statistics with a delta function correlation and also in an independent random bond square lattice. The prediction that t = s in 2D systems has been verified in numerical simulations of random-resistor networks.” When a magnetic field is present, we first consider the special case when , = 1, = A. In the Hall conductivities of the two components are the same: 1 that case, because the Hall current 1(e, x V4) is automatically divergence free for any 4, the equation for 4 is independent of A:

O = V . J = -V.(OV~).

(14.11)

Consequently, the local electric field E(r) is independent of 1.Using (13.2), we then find

1, = A,

When fs, = co,that is, a superconductor, and pI < plc, it does not matter what we assume for 1,because V 4 = 0 inside that component. Therefore we can assume A, = AM in that case. Returning to the general case, we use (14.5) to write the components of the resistivity tensor ( 14.13a)

as

(14.13b)

We now let both o,, A, -+ 0, but take A,

-

fs:

so as to get

(14.14)

210

D.J. BERGMAN AND D. STROUD

It then follows from (14.12) and (14.6a) that (for pM> p , )

( 14.15b)

where pMxx,pMxyare the components of pM= l / a M and pe(pM,pr) is the bulk effective resistivity at H = 0 as a function of the resistivities of the two components. Equation (14.15b) implies that the Hall resistivity of the composite is the same as that of the conducting component. Similarly, (14.15a) implies that the relative magnetoresistance of the composite is just that of the metal: (14.16) These results were first obtained in this generality in Ref. 120, though the result of (14.15b) for the low (magnetic) field case (when A 6 cr) had already been obtained many years ago.'Og. That result has also been verified experimentally,'2' but (14.15a) and (14.16) still await an experimental check. For the weak magnetic field case, that is, I 4 cr, we can get explicit results even when both components have nonzero conductivities, that is, when A, and crl are nonzero. In that case, we can write (14.17)

A careful consideration of (14.5) together with the fact that the ratio (2, - A,)/(&, - A,) depends only on adoM[see (13.6)] leads to the result A, - I , AM-AI

: -a: -a

.;-A:'

(14.18)

'*OD. Stroud and D. J. Bergman, Phys. Reo. B 30,447 (1984). 12'A. Palevski, M. L. Rappaport, A. Kapitulnik, A. Fried, and G. Deutscher, J. Phys. Lett. 45, L367 (1984).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

21 1

Up to this point the discussion has focused on the duality transformation in a continuum 2D composite. In discrete network models a duality transformation is also possible. However, because the discretized versions of the local fields E(r) and J(r) are the voltage V, and current I , in a bond a, one must also rotate each bond by 90" when performing the duality transformation. In most cases this alters the structure of the network, whereas in the continuum case the structure was left unchanged. For example, even in the case of a simple square network, where the dual network is also a simple square network, if the original network was random, the dual network will usually be a different random square network. Thus one of the implicit assumptions used to derive (14.5) is not satisfied. Although this turns out to be unimportant for RRN models at H = 0, it happens to be an important consideration in the choice of an appropriate RRN model for H # 0. In particular, some RRN models for the Hall effect that are not self-dual have turned out to belong to a universality class that differs from that of continuum composites as far as concerns the critical behavior near p C . l l 2

15. THERMOELECTRICITY

A discussion of thermoelectric properties of a composite might seem at first to belong in a different article, because its mathematical framework is somewhat different from that of simple conductivity. In particular, there are now two curl-free fields-the electric field E = -Vc$ and the temperature gradient VT-and there are two conserved currents-the electric current (density) J, and the heat current (density) JQ.(Actually, the latter is only approximately conserved, but this nonconservation can be ignored when J, is small, because it is of second order in Vc$, VT). Each current depends on both Vc$ and VT, so that the coupling between the two fields can be expressed by a symmetric, positive definite, 2 x 2 matrix C :

(15.1)

Here e is the electron charge, ct is the thermoelectric coefficient, k, is Boltzmann's constant, and y is the thermal conductivity coefficient at zero electric field. Throughout this subsection we assume that the components as well as the composite are isotropic. Though these equations, as well as the symbols that appear in them, are quite standard, they have been written in a

212

D.J. BERGMAN A N D D. STROUD

special form so that the two current densities on the left-hand side have the same units (number of particles per unit time per unit area) and likewise for the two scalar fields on the right-hand side (energy). The symmetry of C is a consequence of Onsager’s relations, and its positive definiteness is necessary to ensure that entropy can only increase. In a two-component composite there are only two different C matrices, and they can both be diagonalized simultaneously by defining appropriate linear combinations of the physical currents and field^'^^-'^^:

(1 5.2)

where a,, a, depend on the material parameters of the two components. This separates the coupled-field problem into two uncoupled problems, each of which is like an ordinary composite conductivity problem. Consequently, all three bulk coefficients can be expressed linearly in terms of just two values of the characteristic conductivity function F(s). Because those are the only two unknowns in the problem, one deduces an exact linear relation between these three coefficients (15.3)

Thus a knowledge of oe and y e is sufficient to determine a, for a given composite. Similar arguments have been used to obtain some exact relations among the different effective moduli of coupled-field transport in multicomponent composite^.'^^ As another application of the separability property, we consider the socalled thermoelectric figure of merit for a two-component composite. The thermoelectric figure of merit 2, is defined by 2, = - > 1,

(15.4)

KP

P. Straley, J. Phys. D 14, 2101 (1981). lZ3Ourpresentation follows an unpublished derivation due to one of the present authors (D. J. Bergman, 1982, unpublished). IZ4M.Milgrom and S. Shtrikman, Phys. Rev. A 40,1568 (1989). Iz5M.Milgrom, Phys. Rev. B 41, 12484 (1990).

12’J.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

2 13

where K = y - Toa2 is the coefficient of heat conductivity at zero electric current (this is the heat conductivity that is usually measured). Because of its importance in the technology of thermoelectric heat pumps,lZ6it is of interest to calculate Z , also for composites. Using the separation trick and some of the exact bounds from Section 5, one can show that the largest value for Z,, in a two-component composite is always attained when it is made entirely of one component-obviously the one with the larger value of 2,. Because a large value of Z , is desirable for technological applications, this result shows that making a two-component composite will never result in a better thermoelectric material. 127 Use has been made of the separation trick to construct a scaling theory of the critical behavior of the thermoelectric coefficient near a percolation threshold. Similar to the case of electrical conductivity in the presence of a magnetic field (see Section 13), this case also yields a rich variety of possible behaviors, depending on the values of the physical parameters of the sy~tem.”~~ Note that only two-component thermoelectric composites can be reduced to uncoupled quasi-conductivity problems in the manner described and exploited here. Multicomponent thermoelectrics need to be treated differently. 7 ,

16. SUPERCONDUCTIVITY In this section, we briefly review what may be called the classical properties of superconducting composites. We shall largely omit discussion of the quantum properties, which depend on the continuity of the superconducting wave function and its behavior in a magnetic field. In particular, we do not review the properties that may be ascribed to Josephson or proximityeffect coupling; such behavior is complex enough to merit a separate review. a. Conductivity ;Magnetoconductivity We shall consider a random or disordered binary composite in d dimensions (d = 2 or 3) described by a scalar conductivity a(r) that can take on two values, a1 = 03 with probability p and a2 with probability 1 - p . The effective conductivity a&) must therefore diverge when p exceeds p c , the lZ6T.C. Harman and J. M. Honig, “Thermoelectric and Thermomagnetic Effects and Applications,” p. 40,McGraw-Hill, New York, 1967. I2’D. J. Bergman and 0. Levy, J. Appl. Phys. 70, 6821 (1991). 127a0. Levy and D. J. Bergman, J. Phys. A 25, 1875 (1992). ‘27bD.J. Bergman and 0. Levy, in “Modern Perspectives on Thermoelectrics and Related Materials,” MRS Symposium Proc. 234, 39-45 (1991).

214

D.J. BERGMAN AND D. STROUD

percolation threshold for the superconducting component. For p < p c , oe is finite, with an expected asymptotic behavior Lim 0, z c2(pC- p ) - " ,

(16.1)

P-Pc

where s is the exponent introduced in Section 10. As noted there and in Section 12, s may depend on both dimensionality and the microstructure of the composite. For example, in a lattice, s = 1.30 in d = 2 and s = 0.76 in d = 3. In the Swiss cheese model of Section 12 (random overlapping d-dimensional spherical holes of normal metal in a superconducting matrix) or in the inverse Swiss cheese model (in which the roles of matrix and holes are reversed), the value of s may possibly differ from the lattice values.99 Apart from an experimental study by Deutscher and Rappaport,12* there appears to be little experimental evidence confirming this picture. In either d = 2 or d = 3, an experimental check would be complicated by the proximity effect, which causes the superconducting region to grow a distance 5, into the surrounding normal region, where 5, is the temperature-dependent normal-metal coherence length. A more clear-cut example of such classical effects can be seen upon the application of a magnetic field Happ> Hcl, the lower critical field of the superconducting component. Such a field will fully penetrate the superconducting composite, and to a first approximation the magnetic induction B can be assumed to be uniform and equal to Happin the normal component. In a magnetic field, the conductivity oNof this component becomes a tensor, as does oe. Among the fields satisfying this condition, there are two ranges to consider. (1) At sufficiently low field, the off-diagonal elements of both o, and oNare linear in H,while the diagonal elements are unchanged to first order in H . Under such conditions, it can be shown that oe,ij= I T ~ ,i Zj. ~ ~This ~ o leads to the interesting prediction that the Hall coefficient of the composite below the percolation threshold for superconductivity obeys the relation113 [see (13.10)]

(16.2) for p near p c . This prediction also apparently remains to be tested in both d = 2 and d = 3. (2) At higher fields, one expects both a transverse and a longitudinal magnetoresistance in the composite when superconductor is added, even if the pure normal metal has no magnetoresistance, because the superconducting inclusions distort the nearby current lines, leading to lZ8G.Deutscher and M. L. Rappaport, J. Phys. (Paris) Lett. 40,L219 (1979).

~

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

21 5

P

Pc

FIG. 19. Schematic diagram of the resistivity pxxin a composite containing a volume fraction p of superconductor (S) and 1 - p of normal metal (N), in the presence of a magnetic field B = Bi.p E is the percolation threshold for the S component.

increased dissipation [see Eq. (13.7)]. This conclusion is borne out by measurements of Resnick et al. on a composite of Pb in an A1 matrix1I5 and has been confirmed by effective-medium calculation^^^ (see Fig. 19). b. Magnetic Properties: Penetration Depth, Critical Field, Diamagnetic Susceptibility We begin by considering the London penetration depth of a composite containing a volume fraction p of superconductor (S) and 1 - p of normal metal (N). In the London theory of superconductivity, the basic electrodynamic properties are described by two macroscopic equation^:'^' E = at d ( n,q2 Z J )' '

(16.3)

(1 6.4)

Equation (16.3) simply describes the undamped response of n, particles per unit volume, each of mass m* and charge q, to the force generated by an electric field E. Combining the second London equation with the Maxwell equation 4715 VXB=(16.5) C

lZ9F.London and H. London, Proc. R. SOC.(Lond.) Ser. A 149,71 (1935).

216

D.J. BERGMAN A N D D. STROUD

gives

(v - $)B

=0

(16.6)

where (16.7)

is the London penetration depth of the superconductor. Equation (16.6) implies that the magnetic field decays exponentially within the superconductor, with decay length A. To obtain the analogous equality in an N/S composite, we write Eq. (16.3) in the frequency domain as

Js = aSE,

iA

0s =-

(16.8) (16.9)

0'

where A = c2/(47cA2) and os is clearly an imaginary, frequency-dependent (inductive) conductivity of the S component. For p > p c , the S percolation threshold, the conductivity of the composite at sufficiently low frequencies is dominated by as.Using Eq. (lO.l), we therefore deduce that

(16.10) which implies that A, w A ( p - pC)' or

A, x A(p - pc)-'/*.

(16.11)

To test Eq. (16.11) experimentally would require making a series of samples of different p , then measuring the penetration depth of each. Like a measurement of s below p c , such a direct test would be complicated by the proximity effect. Some workers have interpreted temperature-dependent penetration depth measurements in a single sample of composite superconductor in terms of a temperature-dependent volume fraction of superconductor p(T).'303'31If one assumes a smooth P(T),then the N/S transition occurs '30Ch.Leemann, Ph. Fluckiger, V. Marsico, J. L. Gavilano, P. K. Srivastava, P. H. Lerch, and P. Martinoli, Phys. Rev. Lett. 64,3082 (1990). I3'G. Deutscher, 0.Entin-Wohlman, S. Fishman, and Y. Shapiro, Phys. Reo. B 21,5041 (1980).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

217

at a temperature T , such that p(T,) = p , . Near T,, a Taylor expansion gives p ( T ) - p(T,) = a(T- T,), where the Taylor coefficient c1 is presumably negative. This argument then gives A ( T ) a ( T , - T)’” near T,, in contrast to the usual Bardeen-Cooper-Schrieffer (BCS) temperature dependenceI3’ A ( T ) K ( T , - T)-’”. It also gives for the resistivity above T , the result p ( T ) a ( T - T,)”, which differs from the predictions of superconducting fluctuation theory. Both predictions rely, of course, on a somewhat arbitrary (linear) translation from volume fraction to temperature. Another basic quantity in the magnetic properties of composite superconductors is the critical field. Homogeneous superconductors are generally divided into two classes: type I and type 11. Type I superconductors have a single critical field, H,. Below H,,flux is excluded from the superconductor (except for a thin layer of thickness A). Above H,, flux penetrates fully and the superconductor becomes normal. Type I1 superconductors have two critical fields, H,, and H,, > H,,. Below H,,(the lower critical field), flux is excluded as from type I superconductors. Between H,, and H,,,flux penetrates partially and inhomogeneously, typically forming a lattice of quantized flux lines. Above H,,,flux penetrates homogeneously and the superconductor becomes normal. The variation of B with H in typical type I and type I1 superconductors is well known. This behavior can be interpreted in terms of a field-dependent magnetic permeability p defined as B / H . Below H,(or H,,)p = 0. Above H, in type I superconductors, p rises discontinuously to unity. In type I1 superconductors, p rises continuously above H,,, reaching unity at H,, . In this picture, H, or H,,may be viewed as a kind of breakdown field above which the S component starts to become normal. The problem is seemingly analogous to the problem of dielectric breakdown in a network of insulating and conducting bonds below the conductivity threshold, as discussed in Section 25 (and in greater detail in Ref. 145). The analog of E is the curl-free field H. (H is curl free because the screening currents are viewed as magnetization currents and therefore do not constitute a source for H.) In like manner, the analog of J is the divergenceless field B. With this analogy, H,(p) or H , , ( p ) can be deduced from known results of the dielectric breakdown problem. Thus, at fixed composite volume [see Section 25 and Eqs. (25.28) and (25.33)], we expect that

with y z v ; here p,* is the volume fraction of superconductor at which the 13’For a discussion, see, e.g., M. Tinkham “Introduction to Superconductivity,” McGraw-Hill, New York, 1975.

218

D.J. BERGMAN AND D. STROUD

normal metal first forms an infinite connected cluster, and v is the percolation correlation length exponent. In d = 3, and in some two-dimensional systems as well, p,* > p c ; so there exists a finite concentration regime, in this picture, where the composite is electrically a perfect conductor but has zero critical field. This is because at such concentrations the composite is bicontinuous and hence can transport both supercurrent (through the S component) and flux (through the N component). In a real N/S composite, the validity of this picture presumably depends on the manner in which the magnetic field is introduced. If a bicontinuous composite is cooled in a field, the flux expelled from the S component will probably still continue to thread the composite even below T,, just as in a single superconducting loop. But if the field is turned on below T,, it will be screened out by induced supercurrents. Also, if the flux through a given link of normal metal is sufficiently small, and if the superconducting grains are all large compared to the penetration depth A, of the superconducting materials (as assumed in the previous discussion), this picture must be modified by the requirement of flux quantization through each link. A discussion of such effects has been given, for example, by Alexander. 33 We conclude this section with a brief discussion of the differential diamagnetic susceptibility x of a superconducting composite near p , . It is well known that in bulk superconductors, x increases sharply in magnitude very near T , because of diagmagnetic fluctuations, i.e., momentary fluctuations of the normal metal into the superconducting state. A loosely analogous phenomenon occurs in composites, as was first discussed by de G e n n e ~ . ’ ~ ~ De Gennes noted that for p < p , , the S component is present in the form of only finite clusters, some of which contain closed loops. In the presence of an applied dc magnetic field, diagmagnetic screening currents can flow in these loops, giving rise to a finite diamagnetic susceptibility x. As p approaches p , , the loops become larger and larger, as does x. One therefore expects x to diverge near p , according to a power law of the form

’

(16.13)

where the exponent b depends on the dimensionality of the network and is a measure of the number and size of loops (“loopiness”) of the network. Equation (16.13) was first proposed by de Gennes, who also suggested that b = 2v - t in all dimensions. More recently, Rammal and ~ o - w o r k e r s ~ ~ ~ - ~ ~ modified this formula to b = 2v - t - /?,where /? is another standard 13’S. Alexander, Phys. Reu. B 27, 1541 (1983). ‘34P.G. De Gennes, C.R. Acud. Sci. 292, 701 (1981). I3’R. Rammal and J. C. Angles d’Auriac, J. Phys. C 16, 3933 (1983). ‘36R. Rammal, T. C. Lubensky, and G. Toulouse, J. Phys. Lett. 44, L65 (1983). 13’R. Rammal, T. C. Lubensky, and G. Toulouse, Phys. Rev. B 27, 2820 (1983).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

219

percolation exponent. The de Gennes formula agrees slightly better with numerical sir nu la ti on^'^^ in d = 2, while the Rammal et al. expression may agree better in d = 3. Lagar'kov et al. have proposed b = v in d = 2, b = 0 (a logarithmic divergence) in d = 3.'39 The latter authors have also argued that the divergence is removed if one considers the inductive interaction between screening currents induced on different clusters (an effect omitted in all the other estimates, which considered such clusters as independent). c. Critical Current Density Homogeneous superconductors have the property that they break down and become normal when the current density exceeds a critical value, known as the critical current density, J,. J , can be viewed as a temperaturedependent intrinsic property characteristic of a particular bulk superconducting material. In general, it is desirable to have J, as large as possible, so that a superconductor can carry a large current before it breaks down and becomes normal. The critical current of a composite superconductor can be simply estimated from the nodes-links-blobs picture of the percolating cluster (see Section lo), as illustrated in Fig. 16. We consider a lattice model of a percolating superconductor, in which each bond is either superconducting (S) or normal (N), with probability p or 1 - p . An S bond is assumed to have a critical current I , . The p , is that concentration of a superconductor below which the S component is present only in the form of isolated clusters. In the nodes-links-blobs picture, the infinite cluster consists of nodes connected by links of length tp,where 5, is the percolation correlation length. Every link includes segments that are only a single bond wide-these are the so-called singly connected bonds, each of which must carry the entire current flowing through that link. If a current density J is introduced into this percolating superconductor, the total current I passing through a given link is of order JI&--'. When I > I , , the singly connected bonds go normal. Evidently, this will occur when J = (;(d-l)Ic = J,, or, on using Eq. (10.4),

J,

%

Z , U - ( ~ - ' ) (P - P,)",

(16.14)

where u = (d

-

l)v

(1 6.15)

and a is the bond length. Using accepted values of v, (16.14) predicts that J, cc ( p - pc)1.33in d = 2 and J, cc ( p - P , ) ~ . ' * in d = 3. Equation (16.15) 138D.R. Bowman and D. Stroud, Phys. Rev. Lett. 52, 299 (1984). '"A. N. Lagar'kov, L. V. Panina, and A. K. Sarychev, Sou. Phys. JETP66,123 (1987) [Zh. Eksp. Teor. Fiz. 93, 215 (1987)l.

220

D.J. BERGMAN AND D. STROUD

was first proposed by Deutscher and RappaportlzMand has been well verified n~merically.’~~*’~~ Various attempts have been made to generalize this result to more realistic (nonlattice) models. A continuum percolation model has been described by 1) for the so-called Swiss cheese Lobb et aZ.,14’ who obtain u = (d - l)(v and u = (d - l)(v + $) for the inverse Swiss cheese model of N/S composites. The predictions for Swiss cheese have been numerically verified by Octavio et a1.64 The nodes-links-blobs model for critical currents, even when generalized to a continuum description, is obviously still quite oversimplified. As described earlier, the critical currents are calculated using average spacings between links. A real composite superconductor will obviously have a distribution of link spacings and hence fluctuations in the number of links per unit area. In regions of the superconductor where such links are relatively scarce, the critical current density will be relatively low. The larger the volume of superconductor, the more likely such a region of low critical current. Arguments of this kind suggest (at least qualitatively) that the critical current may actually vanish, even well above p c , in the limit of large volume. Such arguments have been made much more precise, for the related problem of dielectric breakdown, by Duxbury and collaborators. As noted in Section 25, they show that, for fixed concentration of dielectric, the critical field for dielectric breakdown vanishes like l/(log L)II [see (25.28)], where ct is some power less than unity, for a sample of linear dimensions L, in d dimension^.'^^-'^^. In two dimensions, where the dielectric breakdown problem can be exactly mapped onto the critical current problem,14’ we expect that J , will indeed vanish in the limit of large volume.

+

IV. Electromagnetic Properties

17. BASICEQUATIONS We now generalize the dc model of Section I1 to treat electromagnetic properties at finite frequencies. To be explicit, we will consider a binary Kirkpatrick, in “Inhomogeneous Superconductors- 1979 (Berkeley Springs, W.Va.)” (D. U. Gubser, T. L. Francavilla, I. R. Leibowitz, and S. A. Wolf, eds.) AIP Conf. Proc. No. 58, p. 79, American Institute of Physics, 1980. I4’C.J. Lobb and D. J. Frank, in “Inhomogeneous Superconductors- 1979 (Berkeley Springs, W. Va.)” (D. U. Gubser, T. L. Francavilla, J. R. Leibowitz, and S. A. Wolf, eds.), AIP Conf. Proc. No. 58, p. 308, American Institute of Physics, 1980. 142C.J. Lobb, P. M. Hui, and D. Stroud, Phys. Rev. B 36, 1956 (1987). 143P.M. Duxbury, P. D. Beale, and P. L. Leath, Phys. Rev. Lett. 57, 1052 (1986). IMP.M. Duxbury, P. L. Leath, and P. D. Beale, Phys. Rev. B 36, 367 (1987). 14’Y. S. Li and P. M. Duxbury, Phys. Rev. B. 36, 5411 (1987). I4OS.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

221

composite made up of components A and B, present in volume fractions p and 1 - p , respectively. Each component is assumed to exist as small pieces with homogeneous properties (see Fig. 5). The real dielectric function of the ) E ~ , ~ ( O and ), the real conductivity composite is then E1(r, o)= E ~ , ~ ( O or gl(r, o)= C T ~ , ~ (orO r~,,~(o). ) For simplicity, the magnetic permeabilities pA and pB are assumed equal to unity. The transport of electromagnetic waves through a binary composite is described by the macroscopic Maxwell equations. Assuming monochromatic fields of the form E(r, t ) = E(r) exp( -iot), and adopting the usual convention that the physical fields are the real parts of complex quantities, these take the form (in Gaussian units) V * ( E ~ E=) 4np,

(17.1)

V*B=O,

(17.2)

io

V x E = - B,

(17.3)

C

4.n

io

C

C

V xB=-ga,E--&IE,

(17.4)

which are to be supplemented by the continuity equation

-

V (a,E) - imp = 0.

(17.5)

Substituting (17.5) into (17.1) gives

V*D=O,

(17.6)

where the free current density and polarization current have been combined into a single effective displacement field (17.7)

With the introduction of a complex dielectric function (1 7.8)

222

D.J. BERGMAN AND D. STROUD

Maxwell’s equations take the form V * (EE)= 0

(17.9)

V.B=O

(17.10)

UB VxE=i-

(17.11)

C

V x B = -i-.

CO&E

(17.12)

C

Note that the entire linear response of the composite is now contained within the complex dielectric function E(r, o),which is defined to include the response of both free and bound charge within the composite medium. 18. QUASI-STATIC APPROXIMATION If the frequency is sufficiently low (in the sense to be described), the inductive term iwB/c in Faraday’s law can be neglected. The electric field and displacement then satisfy the equations

V.D=O,

(18.1)

VxE=O,

(18.2)

which are formally identical to the dc equations discussed in the previous section. The neglect of the induced Faraday emf is known as the quasi-static approximation (QSA). In general, this approximation is reasonable if a typical linear dimension of the particle, say a, is small compared to the wavelength or the penetration depth of the radiation in either constituent of the composite. For particles of order a few hundred angstroms in linear dimension, the QSA may be a reasonable approximation even at visible or near-ultraviolet frequencies. Given the validity of the QSA, we can use all our dc results to study electromagnetic wave propagation in composites, in the long-wavelength limit. The propagation of waves is properly described in terms of an effective

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

223

dielectric constant E , , which is a complex function of the two complex variables and cB, as well as of p and the microgeometry. Similarly, the Maxwell Garnett (MG) approximation and effective-medium approximation (EMA) can be extended immediately to treat isotropic three-dimensional composites, provided that conductivities are replaced by dielectric functions wherever they appear. 19. APPLICATIONS a. Metal-Insulator Composite at Low Concentrations As a first example, consider a composite with a volume fraction p of metal with a Drude dielectric function

and 1 - p of insulator with dielectric constant eB(w) = 1. Here wp is the plasma frequency and z is a characteristic relaxation time. In a typical bulk free-electron metal, such as Al, wp z lo" sec-l and wpz z 100. In a small particle of such bulk metal, wpz may be reduced from this value by surface scattering. When p < 1, E , is accurately given by the MG approximation, provided the metal particles can be assumed to be spherical. In this limit, (19.2)

This form has two interesting frequency regimes, which we discuss in turn. Frequencies such that + 2 z 0 correspond to the so-called surface plasmon r e ~ o n a n c e . 'Here ~ ~ the denominator in Eq. (19.2) approaches zero, and in consequence I Re(&,)I becomes very large. For a metal-insulator composite, this occurs near w = w p / f i .For particles with shapes other than spherical, this resonance splits into several peaks that occur at other frequencies. The surface plasmon resonance characteristically shows up as a strong absorption line. This absorption is responsible for the beautiful ruby colors seen in dilute suspensions of small gold particles in a transparent host such as glass. The absorption coefficient c1 is the fraction of energy absorbed per unit 146R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, P h p . Rev. B 8, 3689 (1973).

224

D.J. BERGMAN AND D. STROUD

length of material and is given by 0

a=2-1m&

(19.3)

C

or, when p -4 1, (19.4) on using Eq. (19.2). Note that in the quasi-static approximation, the sphere radius drops out (though the shape still plays a role). Figure 20 shows the absorption coefficient of a dilute suspension of metal spheres in a host of dielectric constant unity, as calculated from Eq. (19.4). The surface plasmon resonance indeed does show up as a strong absorption near w = wpJ,,h, as expected. A dilute suspension of spheres also shows interesting behavior at w~ -4 1. For most metals, this corresponds to the far infrared. By substituting Eq. (19.1) into Eq. (19.4), one finds (19.5)

0

1.o

0.5

1.5

o/op FIG.20. Absorption coefficient a, for a composite of volume fraction 0.01 of spheres of Drude metal ( O ~ T= 100) embedded in a host medium of dielectric constant unity, as calculated in the quasi-static approximation and the dilute limit.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

225

where o = o3/(4.n) is the static conductivity. E ~ p e r i m e n t 'does ~ ~ show an o2absorption coefficient that varies roughly linearly with filling fraction p , as predicted by Eq. (19.5), but the magnitude

(19.6) is much larger than predicted by the quasi-static approximation. The discrepancy is typically a factor of l o 5 or even larger. The possible causes of this discrepancy have been reviewed;'47a some of these are discussed further in the following. At higher concentrations of metal, the qualitative optical behavior of the composite shows a number of striking features as a function of concentration. We will illustrate these by simple calculations within the EMA,14' which serves as a useful interpolation scheme. The phenomena shown are, however, not restricted to the EMA but are quite general properties connected with the percolation process. Figure 21 shows Re[o,(o)] E w/4z Im[~,(w)], plotted versus frequency for several values of p , as calculated within the EMA. For p < pc (=+ in the EMA), Re(o,) shows a single broad peak, confined within the frequency This is a band of surface plasmon resonances, broadened range 0 < o < op. from the single sharp peak shown in Fig. 20 by electromagnetic interactions among individual grains. It is the analog of the impurity band seen in the electronic density of states of disordered alloy^.'^ For p > p c , a Drude peak, centered at o = 0, develops in addition to the surface plasmon band. The appearance of this peak is a signal that the composite is conducting at zero frequency. The peak appears at the percolation threshold of a metal-insulator composite, above which the composite has nonzero dc conductivity (see Section 10). As p increases further, the integrated strength of the Drude peak grows, consistent with an increasing dc conductivity. The surface plasmon peak eventually shrinks and narrows to a band centered at o = up,/$. This corresponds to a void resonance, representing oscillating charge in the vicinity of a spherical void in an otherwise homogeneous metal. Similar behavior is shown by the energy lossfunction, -Irn[l/~,(w)]. Like Re oe, -Im[l/~,] is found to show characteristic structure related to the connectedness of various components of the compo~ite.'~'In a homogeneous metal, -Im[l/~] shows peaks at the plasmon resonances, where Re E x 0. For a metal described by the Drude dielectric function, Eq. (19.1), this peak occurs at o = op.This peak persists in the composite for p > p:, the 14'D. B. Tanner, A. J. Severs, and R. A. Buhrman, Phys. Rev. B 11, 1330 (1975). 1 4 7 a F ~arcritical review of the various explanations for this effect, see R. P. Devaty and A. J. Severs, Phys. Rev. B 41, 7421 (1990). 148D.Stroud, Phys. Rev. B 19, 1783 (1979).

226

D.J. BERGMAN AND D. STROUD

I

0.5

f

=

0.999

n

I I I I I

0.9

I

0.1 /-

I I

I I

"

0

0.2

0.4

0.6

0.8

1.0

1.2

o/op FIG.21. Schematic of the real part of the ac conductivity, Re ueff(w), of a metal-insulator composite made up of a volume fractionfof Drude metal and 1 -fof insulator, as calculated in the effective medium approximation in the limit T + co.The curves are displaced vertically. The heavy vertical line at w = 0 denotes a delta function, which represents the Drude peak; the integrated strength of the delta function is proportional to the height of the delta function. The peak a t f = 0.999 is arbitrarily increased in height for clarity. (The notation differs from the text.) Taken from D. Stroud, Phys. Rev. B 19, 1783 (1979).

volume fraction of metal above which the insulator no longer forms an infinite connected cluster. Besides the sharp plasmon peak, there is also a broad band in -Im( l/&Jfrom localized surface plasmons. The disappearance of the bulk peak for p < p r can be understood from percolation theory. Near the plasma frequency, Re eA is approximately zero, while cB = 1. Thus the metallic component behaves like an insulator, with zero conductivity for displacement current, while the insulator behaves like a metal. The roles of the two components are thus reversed. The zero in E , at o = oppersists as long as the insulator component B is present only in the form of finite clusters, that is, p > p t . Structure in Re o,(o) similar to that shown in Fig. 21 has been seen in a number of experiments. Figure 22 shows measurements of Cummings et

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

227

Photon energy (eV)

0

10,000

20,000 30,000 40,000

Frequency (cm-I) FIG.22. Real conductivity Re a,(w) of a composite of volume fractionfof Ag and 1 -fof KCI as a function of frequency w , from measurements reported in Ref. 149.

al. ' 4 9 taken on three-dimensional Ag/KCl composites, which have a bicontinuous regime and a percolation threshold. The appearance of the Drude peak above p c is clearly visible. Because Ag is not a Drude metal, there is, in addition to the surface plasmon peak, a region of large oscillator strength at high frequencies corresponding to interband transitions in Ag, which persist in the composite. Most of the structure is reproduced in the effective-medium approximation. An EMA calculation149 that includes both interband and intraband contributions to the dielectric function of Ag is shown in Fig. 23. b. Superconducting Composite Optical absorption by a composite of normal metal and superconductor is of interest because of the possibility that the presence of normal metal will produce absorption below the superconducting energy gap, thereby causing that gap to appear smaller than it really is. In the BCS theory of superconductivity, Re a,(o) is characterized by two parts: (1) a delta function at w = 0, corresponding to the pure inductive response typical of infinite conductivity, and (2) a gap of width 2A/h, where A is the BCS energy gap, below which Re a,(o) = 0 and there is no absorption. As calculated by Mattis and Bardeen,' 50 al(w) = Re ~ ( wtakes ) the following form in the superconductI4'K. D. Cummings, J. C. Garland, and D. B. Tanner, Phys. Rev. B 30,4170 (1984). lS0D.C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958).

228

D.J. BERGMAN AND D. STROUD

-

10,000

E

3

-

v

-

Y

)r

+ .-

> .= 0

I

i

-

5,000-

3

I

I

ii

7

h

Ag in KC1 300 K f -- 0.057 0.192 -0.547 0.656 ........... 0.987

!\ !\

I\

____

ii

i. .\ Ii

i! \ \

-

FIG.23. Re a,(w) for the composite of Fig. 22, as calculated in the effective-medium approximation.Taken from K. D. Cummings, J. C. Garland, and D. B. Tanner, Phys. Rev. B 30, 4170 (1984).

ing state at temperature T = 0:

(%)==.

= (1

+g)E(k)

-

4 6 K(k),

ho 2 2A,

(19.7)

(19.8) where E and K are the standard elliptic integrals, and (T, is the conductivity of the superconductor in its normal state. For hw < 2A, nlSvanishes. Including the inductive delta function, the total conductivity in the superconductive state is

+

~ ~ oZs~is related to olS by the Kramers-Kronig where us= o l S i ( and relation. 5 0 In a composite of superconductor and insulator, a novel kind of surface plasmon resonance is possible that has so far been reported only in some of

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

229

the high-temperature superconductors.”’ To see the origin of this effect, we write Go = .nAa,/h, the correct BCS form for the inductive part at low temperature. Then the dielectric function of the superconductor is

where cph is that part of the dielectric function due to nonsuperconducting processes, such as phonon excitations. If we use Eq. (19.9), neglect the absorptive part, and approximate Eph by unity, we obtain E,(O)

w 1-

4n20,A w2h ’

(19.11)

~

which is identical in form to a Drude dielectric function with an infinite relaxation time. Thus, if a spherical grain of such a superconductor is placed in an insulator with dielectric constant E ~ ,we expect a surface plasmon resonance in the form of a sharp absorption line at E, 2ci w 0 or

+

4.n20,A

OSP=

‘I2

(x)

(19.12)

This line will be detectable only if it occurs below the superconducting gap, that is, wsp < 2A/h or

A/h > . n ’ ~ ~ / ( 3 ~ ~ ) ,

(19.13)

which implies a large A or a small 0,. In conventional low-T, superconductors, in which this inequality is not satisfied, the resonance is lost in the singleparticle excitations above the gap. But such an absorption spike has indeed been reported by Noh et a1.”’ in small spherical particles of YBa,Cu,O,_, and explained on the basis of the foregoing model, suitably generalized to include phonon contributions, absorption above the gap, and the optical anisotropy of the S spheres. c. Anisotropic Media

A number of optical media are composed of optically anisotropic constituents. Examples include intercalated graphites, many high-temperature superconductors, and quasi-linear organic conductors. In such materials, the dielectric function is a second-rank tensor with three different nonzero I5’T. W. Noh, S. G . Kaplan, and A. J. Sievers, Phys. Rev. Lett. 62, 599 (1989).

230

D.J. BERGMAN A N D D. STROUD

principal values. A polycrystal of such an anisotropic dielectric is, in effect, a composite medium, as discussed in Section 7, and can be treated by the effective-medium approximation, Eq. (7.18) (with all conductivities replaced by dielectric functions). Calculations based on this approach and various generalizations to nonspherical grains have been used by several workers' 52-1 5 5 to interpret the optical properties of the quasi-planar high temperature superconductor YBa,Cu,O,,, which is a planar material with a highly anisotropic conductivity in both its normal and its superconducting state. These calculations postulate ellipsoidal grains oriented so that the principal axes of the conductivity tensor are parallel to the principal axes of the ellipsoid within each grain. A conspicuous feature is the finite absorption found below 2A/h, manifest in a finite al(o)below the Walker and Scharnberg have done a similar treatment,' 54 but involving a slightly different self-consistent embedding condition than that used by Noh and co-workers. A treatment involving oriented anisotropic grains of high-temperature superconductor has been developed by Diaz-Guilera and T r e m b l a ~ . ' ~ ~ Optical anisotropy can also be produced, or changed, by the application of a magnetic field. One such change is the Faraday effect, which is the rotation of light (either on transmission or reflection) on passing through a medium in the presence of a magnetic field. Such a field is treated formally by adding to the zero-field dielectric tensor an additional antisymmetric contribution. If the medium is isotropic in the absence of the field, the problem is essentially an ac generalization of the Hall effect discussed in Section 13. In a nonabsorbing medium, the components of the antisymmetric tensor are usually purely imaginary, so that left and right circularly polarized waves travel with different velocities. Faraday rotation in composite media has been studied theoretically in the dilute limit using the Maxwell Garnett appro xi ma ti or^'^^ and for both magnetic and nonmagnetic particles at higher concentrations using the EMA.15, Experimental studies may be of practical interest because of the possibility of obtaining optical materials with large Faraday rotation per unit thickness and low absorption. A somewhat different type of anisotropy has been studied experimentally by Sherriff and Devaty'58 in small particles of Bi. Such particles are particularly complicated because, even in the absence of an applied magnetic ls2T. W. Noh, P. E. Sulewski, and A. J. Sievers, Phys. Rev. B 36, 8866 (1987). IS3P.E. Sulewski, T. W. Noh, J. T. McWhirter, and A. J. Sievers, Phys. Rev. B 36, 5735 (1987). ls4D.Walker and K. Scharnberg, Phys. Rev. B 42,2211 (1990). I5'A. Diaz-Guilera and A.-M. S. Tremblay. J. Appl. Phys. 69, 379 (1991). 156P.M. Hui and D. Stroud, Appl. Phys. Lett. 50, 950 (1987). '"T. K. Xia, P. M. Hui, and D. Stroud, J. Appl. Phys. 67, 2736 (1990). lS8R. E. Sherriff and P. P. Devaty, Phys. Rev. B 41, 1340 (1990).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

23 1

field, the nonspherical Fermi surface of Bi leads to optical anisotropy. In a magnetic field, it was found that the resulting Maxwell Garnett resonances are frequency shifted. Furthermore, magnetic dipole resonances were observed in addition to the usual electric dipole (Maxwell Garnett or surface plasmon) resonances. d. Dielectric Enhancement in Brine-Saturated Porous Rocks A completely different kind of ac problem, of considerable interest in oil exploration, is presented by a sedimentary rock whose pore space is entirely saturated with brine or 0il.159*'60The greatest practical interest naturally resides in the situation in which some of the pores are fully or partially occupied by oil. But a completely brine-saturated rock-that is, a rock whose pore space is entirely filled by brine-is a simpler first example, which falls into the category of two-component composites under discussion here. Because the brine is a conducting fluid. a brine-saturated rock is an example of a metal-insulator composite, but of a most unusual kind. Numerous experiments lead to the conclusion that this composite has a percolation threshold of zero p ~ r o s i t y ~ ~ , ~ ~ is, - t hthe a t pore space forms an infinite connected cluster even at porosities approaching zero. (The porosity is the volume fraction of pore space and is generally denoted 4.) Not only is the pore space fully connected, but there are no indications of any disconnected pores that are isolated from the infinite pore cluster. Because the rock matrix also forms an infinite cluster, this composite is always biconnected. At gigahertz frequencies, the (complex) rock and brine dielectric functions E, and cW can be approximated as53,161 (19.14)

E , = EL

EW

= E&

+ 4niaw 0 ~

(19.15)

where EL and E& are the real parts of the permittivity of the rock matrix and the brine and ow is the brine conductivity, all assumed to be frequencyindependent in this range. Over a broad range of frequencies, the effective dielectric function of the composite E, = E',

+ 4ni0, 0 ~

159R.N. Rau and R. P. Wharton, J. Pet. Techno1 34,2689 (1982). 160W. E. Kenyon, J. Appl. Phys. 55, 3153 (1984). l6ID. Stroud, G. W. Milton, and B. R. De, Phys. Rev. B 34, 5145 (1986).

(19.16)

232

D.J. BERGMAN AND D. STROUD

+

behaves singularly: E , varies approximately as w P 8 ,while a,(o) x a,(O) const. . 0 1 - 8 . 1 5 9 - 1 6 1 Various attempts have been made to understand this remarkable diverging behavior of E', at low frequency in terms of the percolating structure of the composite, or by using the pole sum rules described in Section I1 (see Sections 6 and 8). To understand E , in this limit. Stroud et ~ 1 . have ' ~ ~proposed a simple analytic formula for E , that takes into account two sum rules satisfied by the pole spectrum of the composite [see Eq. (6.6c)l. More elaborate analytic forms for the pole spectrum have been described by Holwech and Nost'62 and by Ghosh and F ~ c h s . 'Korringa ~~ and LaTorracaS3 have chosen, instead, to obtain limiting bounds to the complex dielectric function, as described in Sections 5 and 8. A connection between this divergent behavior and identifiable geometric features of the porous rock (such as fractal structural characteristics) remains elusive. Presumably this divergence is simply the analog of that seen in E,(w) at p c in a metal-insulator composite, suitably translated to p c = 0. But this insight has not yet been made more precise.

20.

SUM

RULES

Just as in homogeneous solids, sum rules can be written down for E , in a two-component composite. They are related to the pole sum rules discussed in Section 6 [see (6.6c)l and connect E,(o) to simple characteristics of the microgeometry, such as the volume fractions and macroscopic isotropy. Because these sum rules are otherwise independent of particular structural models for the composite, they are potentially useful for interpreting experimental data. The sum rules are easily derived.14' We assume that ~ ~ ( and 0~ )~ ( exhibit the high-frequency behavior

Thus at high frequencies I - cBI cc l/02and is small compared to either or eB. An expansion of E , - E,, in powers of - zBl is therefore rapidly Holwech and B. Nost, Phys. Rev. B 39, 12845 (1989). 163K.Ghosh and R. Fuchs, Phys. Rev. B 38, 5222 (1988). 16*I.

0 )

233

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

+

(1 - P)&B]. For an convergent at sufficiently high frequencies [ E =~ ~ isotropic composite, such an expansion yields [see (3.1)]

where p is the volume fraction of A. On expressing the right-hand side as an expansion in powers of l/w’, this becomes (20.4)

The sum rules for

E,

now follow from the requirement that BE,(o)

=

E,(o) - cav(o) be analytic in the upper-half complex-frequency plane, which

follows in turn from causality. This requirement leads to the KramersKronig relation

‘I

Re BE,(o) = P 7-c

Im BE,(o’)

-m

do‘

(20.5)

O’2-02

Expanding left- and right-hand sides in powers of 1/02 at large w, using Eq. l/04give

(20.4), and equating coefficients of 1/03 and m

JJ-

(20.6)

o’Im &,(o’) do’= 0, m

m

oT3 Im b~,(o’)do‘ = $cp(l - p ) ( M A- MB)2.

(20.7)

m

The Kramers-Kronig relations for

and

give

m

I-

o’Im E ~ ( o ’ ) do’= nMi,

i = A, B.

(20.8)

m

From this equation, and from the fact that the integrands in Eqs. (20.7) and (20.8) are even functions of o’, we finally obtain the desired sum rules:

Iom O’ Im

E,(w’)do‘ = &[pMA

+ (1 - p)MB],

(20.9)

234

D.J. BERGMAN A N D D. STROUD

Equation (20.9) states that the integrated oscillator strength of the composite is the average of the integrated oscillator strengths of the constituents. Equation (20.10)implies that the center of gravity of Im E , is pushed up to a higher frequency than that of Im caV. It is the analog of the second of the pole sum rules [Eq. (6.6c)l. Note that the sum rules (20.9) and (20.10) are derived on the assumption that QSA is still valid even at asymptotically high frequencies. Results analogous to Eqs. (20.9)and (20.10)can be obtained for the energy loss function -Im[l/~,(o)]. The results are rw

1

1

Again, the effect of inhomogeneity is to push up in frequency some of the oscillator strength in - Im ge- relative to that of - Im E & ~ . The sum rules may also be generalized to more elaborate dielectric functions. For example, Noh and Sievers have extended the sum rules to the important case in which and gB approach different high-frequency limits.164 21. BEYONDTHE QUASI-STATIC APPROXIMATION a. Higher-Order Multipoles One way to examine the validity of the QSA is to examine a dilute suspension of inclusions in an otherwise homogeneous medium, subjected to an incoming linearly polarized monochromatic plane electromagnetic wave with electric field E = E, exp(ikz - iot). For spherical inclusions with a ) a host of dielectric function E, = 1 , the complex dielectric function E ~ ( wand extinction coeficient rxtot, defined as the ratio of power lost out of the incident beam per unit volume to the incident power per unit area, is given by the standard expression atot=

4nn0 ~

kZ 1

Re S(O),

(21.1)

m

(21.2) 164T.Noh and A. J. Sievers, Phys. Rev. Lett. 63,1800 (1989).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

235

where S(0) is the forward scattering amplitude and a, and b, are the electric and magnetic multipole coefficients. These are given by

(21.3)

(21.4)

where y = wR&/c, x = wR/c, ,u = y/x, $,(x) = xj,(x), d,(x) = xh!,’)(x),j,(x) and hh’)(x) are spherical Bessel and Hankel functions, no is the number density of spheres, and R is the sphere radius. Equations (21.1)-(21.4) are obtained by a standard partial wave expansion of the incoming plane wave in electric and magnetic multipoles and use of the optical theorem to relate the total extinction coefficient to the forward scattering a m p 1 i t ~ d e . lThe ~~ coefficients a, and b, are proportional to the n-pole electric and magnetic portions of the wave scattered from the sphere. Although cltot represents the sum of losses due to scattering and absorption, scattering losses are negligible at long wavelengths (kR 4 1). The two dominant contributions to the absorption come from the electric and magnetic dipole terms, a , and b,. When x 4 1, y 4 1, an expansion of the spherical Bessel and Hankel functions gives

(21.5)

b,=

-

1).

(21.6)

All other coefficients (e.g., the electric quadrupole coefficient a,) vary as x5 or a higher power of x. Thus the dominant coefficient at low frequency is a,, the electric dipole term (except in one special case to be discussed). Substituting Eq. (21.5) into Eq. (21.1) yields

(21.7)

16%ee, for example, M. Born and E. Wolf, “Principles of Optics,” Fourth edition, Sect. 13.5, Pergamon, New York, 1970.

236

D.J. BERGMAN AND D. STROUD

This is identical to the absorption coefficient (19.4) calculated from the quasi-static Maxwell Garnett dielectric constant. Thus the quasi-static approximation is generally valid in the limit kR 6 1. In this limit the extinction coefficient is independent of R. This scale independence is expected because in the quasi-static limit a composite can be described by an effective dielectric function that is unaffected if all the dimensions of the composite are uniformly multiplied by a scale factor. In a metal-insulator composite at low frequencies, the contribution c,t to the extinction coefficient from b , can be comparable to the electric dipole contribution, which is the same as the quasi-static approximation or even much larger. Substituting (21.5) and (21.6) into (21.1) gives ct, = C,ozp, a, = C,ozp, where C, is given by (19.5) and

2mRZ c, = _ _ 5c3

(21.8)

*

The magnetic dipole contribution also varies as o ' p , but unlike a, it increases with particle radius because the induced eddy currents dissipate more energy in larger particles. For particles of conductivity comparable to that of A1 at room temperature, C, may exceed C, at a particle radius as small as 30 A. For 100-A particles, inclusion of the magnetic dipole absorption can give an enhancement of far-infrared absorption of order 10' over the quasi-static approximation. This extra magnetic dipole (or eddy current) absorption undoubtedly contributes to the well-known discrepancy between the observed and the quasi-static far-infrared absorption in small metal particle^.'^^.'^^" However, it is still inadequate to explain the factor of 104-106 seen in many experiments. A plot of the extinction calculated using the full Mie theory in conjunction with the Drude approximation for the dielectric function of a metal particle is shown in Fig. 24, along with the predictions of the quasi-static approximation.'66 Although it is not clear from the figure, the quasi-static approximation for the absorption coefficient actually works best at higher frequencies in this instance. Several authors have attempted to extend the effective-medium approximation to treat higher-order terms in the Mie expansion. Stroud and Pan'66 have proposed choosing the effective medium to have a dielectric function E, such that the forward scattering from the spheres, i.e., S(O), vanishes on the average. Mahan' 6 7 has avoided including the magnetic-dipole scattering in an effective dielectric function and has instead proposed introducing separate 166D.Stroud and F. P. Pan, Phys. Rev. B 17, 1602 (1978). 16'G. D. Mahan, Phys. Rev. B 38, 9500 (1988).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

237

I 0-3

I 0-4

I 0-5

10-6

I 0-7

10-8

I 0-9 I 0-4

I 0-3

10-2

FIG.24. Extinction coefficient per unit number density of metal spheres in units of rta', where a is the sphere radius, plotted for a dilute collection of metal spheres in vacuum. Except where shown, all calculations were carried out by summing the full Mie series to convergence. The corresponding coefficient, calculated with only the dominant parts of a, and b,, is shown as a dashed curve for up7= 100 and wpa/c = 1. Taken from D. Stroud and F. P. Pan, Phys. Rev. B 17, 1602 (1978).

effective dielectric functions and magnetic permeabilities, each of which is determined by independent, Bruggeman-like equations. Numerically, the two approximations seem to give rather similar results. The conditions under which one can treat the effective medium by separate effective dielectric functions and permabilities is discussed by Lamb et ~ 1 . ' ~ ~ I6*W.Lamb, D. M. Wood, and N. W. Ashcroft, Phys. Rev. E 21,2248 (1980).

238

D.J. BERGMAN AND D. STROUD

b. Large-Scale Structures Even when individual particles in a composite are small, there are sometimes large-scale structures that are of the same order as the wavelength /z of electromagnetic radiation in the composite, so that the quasi-static approximation again breaks down. Such structures could arise in a variety of circumstances. These include the formation of large (e.g., fractal) clusters of particles with cluster linear dimension z 1. Even in a randomly disordered composite, the percolation correlation length 5, diverges as the percolation threshold p c is approached from either side, so that sufficiently near the threshold 5 , > A. The optical properties of a composite in this regime have been discussed In their picture, the incident radiation near p , samples the by Yagil et dielectric constant not of the composite as a whole, but rather of only a small chunk of volume z Li, where L, is the smallest of the three lengths 1 (the electromagnetic wavelength in the composite), 5, and L(w). In turn, L(o)is the anomalous difision length, defined as the distance a charge carrier will travel within the metallic portion of the composite in one ac cycle. Such a small chunk deviates from the bulk in two ways: its dielectric function E, differs from that of an infinite sample, and it has a distribution of possible values for E, described by a probability density. Both effects arise from the large fluctuations exhibited by different finite-size samples of composite near p , . The finite-size dielectric constant E , for a chunk of size L, is assumed to be characterized by a finite-size scaling function near p c of the form

(21.9)

where F , and F - are two scaling functions expected to apply above and below p , . Equation (21.9) has been tested numerically, at least in d = 2,170*’71 and found to be valid. In like manner, fluctuations in E , in a block of size Li are assumed to be described by a scaling distribution function. Yagil et al. used these two forms to calculate the optical properties of a thin composite film, by averaging these properties over the expected distribution of dielectric functions for a finite system of size L,, and obtain good agreement with experiment for semicontinuous Au films. 169Y.Yagil, M. Yosefin, D. J. Bergrnan, G. Deutscher, and P. Gadenne, Phys. Rev. B. 43, 11342 (1 99 1). I7’A. Bug, G. S. Grest, I. Webrnan, and M. H. Cohen, J. Phys. A 19, L323 (1986). 17’K. S . Koss and D. Stroud, Phys. Rev. B 35,9004 (1987).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

239

C. Photon Diflision Regime The emphasis through most of this review has been on modeling a composite by an equivalent homogeneous medium with an effective dielectric function. But under some conditions, the propagation of photons is best described, not by an equivalent homogeneous medium, but rather by a diffusion equation. This diffusion description takes us beyond the scope of the present review, but we sketch some of the essential physics in order to make contact with other discussions in the optical l i t e r a t ~ r e . ~ ~ ~ , ' ~ ~ For simplicity we restrict our discussion to a nonabsorbing binary composite whose components have real, frequency-independent dielectric constants c1 and c2. In the limit where diffusion theory is applicable, the photon density n(x, t ) obeys the diffusion equation

[Vz

- D i]n(r, t ) = Q(r, t )

(2 1.1 0)

where Q is a source term and D is a diffusion constant, related to the elastic mean free path 1 by D = lv/3, v being the average photon velocity in the composite. Although 1 (or D) is generally difficult to compute, it can be accurately estimated in a few cases. For example, in a dilute suspension of spheres of medium 1 in medium 2, 1 = l/(no), where n is the number of spheres per unit volume and r~ is the Mie scattering cross section. The diffusion approximation is expected to be applicable in the regime

1616L

(21.11)

where 1 is the wavelength of light in the medium and L is a typical linear dimension of the sample (e.g., the thickness of a slab through which a beam of photons is propagating). The predictions of the diffusion approximation appear to have been well verified in this limit."3 When 1 becomes comparable to I , coherent interference phenomena among the multiply-scattered waves passing through the composite give rise to deviations from the diffusion approximation, analogous to the phenomena of weak and strong localization observed in the propagation of electrons in random media. These phenomena of photon localization have been extensively discussed in a number of a r t i ~ 1 e s . lIn ~ ~all cases, they are phenomena that cannot be 172P.W. Anderson, Phil. Mag. 52, 505 (1985). I7%ee,for example, K. M. Yoo, Feng Liu, and R. R. Alfano, Phys. Rev. Lett. 64,2647 (1990) and references cited therein.

240

D.J. BERGMAN AND D. STROUD

easily discussed in terms of a homogeneous effective medium for photon propagation. 22. OTHER APPROXIMATIONS

Numerous authors have proposed approximation^^'"^^^ for the electromagnetic response of composites. Although some are straightforward extensions of the Maxwell Garnett or effective-medium approaches,’82-’ 84 the majority are considerably more elaborate. In this section, we briefly review a few of these approaches. Because of the extent of the literature, we can do no more than sample the range of possible approaches. a. Distributions of Particles Embedded in a Host In many cases, the composite is a collection of particles of dielectric function and definite shape (e.g., spheres) embedded in a host of a different dielectric function, say E ~ The . positions of the A particles are generally random in some fashion. The composite thus resembles a macroscopic version of a structurally disordered atomic system, such as a liquid metal or an amorphous solid. The A particles play the role of the “atoms,” and the difference eA - cB corresponds to the scattering potential. Several authors have attempted to exploit this analogy more concretely. They generally proceed by writing out an integral equation for the electric field. In any random dielectric, the Maxwell equations (17.11) and (17.12) can be combined to give V x V x E - k2E = k2&E,

(22.1)

where k2 = kiE0, kZ, = co2/c2,6~ = (.$r) - c O ) / E O , E(r) is the position-dependent dielectric constant, and c0 is a reference dielectric constant to be chosen 174V.A. Davis and L. Schwartz, Phys. Rev. B. 31, 5155 (1985). 175V.A. Davis and L. Schwartz, Phys. Rev. B 33,6627 (1986). 176B.N . J. Persson and A. Liebsch, Solid State Commun. 44, 1637 (1982). 177A.Liebsch and B. M. J. Persson, J. Phys. C. 16, 5375 (1983). 178A.Liebsch and P. Villaseiior Gonzalez, Phys. Rev. B 29, 6907 (1984). L79B.U. Felderhof and R. B. Jones, Z . Phys. B 62, 43 (1986); Z . Phys. B 62,215 (1986). lS0B.U. Felderhof, Phys. Rev. B 39, 5669 (1989). IS’B.U. Felderhof, G. W. Ford, and E. G. D. Cohen, J. Stat. Phys. 28,135 (1982); J. Stat. Phys. 28, 649 (1982). ISzM. Hori and Y. Yonezawa, J. Phys. C 10,229 (1977). IS3C.G. Granqvist and 0. Hunderi, Phys. Rev. B 18, 1554 (1978). Ig4P.O’Neill and A. Ignatiev, Phys. Rev. B 18, 6540 (1978). IS5P.Clippe, R. Evrard, and A. A. Lucas, Phys. Rev. B 14, 1715 (1976). IE6F.Claro, Phys. Rev. B 25, 7875 (1982).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

241

in some convenient way. With the introduction of a 3 x 3 tensor Green’s function G defined by V x V x G - k2G = -1?’6~(r - r’)I,

(22.2)

where I is the 3 x 3 unit tensor, Eq. (22.1) can be converted into an integral equation for E. The result is E(r) = E,(r)

+

s

d3r‘ G(k;r - r’) &(r’)E(r’),

(22.3)

where E, is a solution to the related homogeneous equation. Equation (22.3) is a vector analog of the integral Schrodinger equation, with E playing the role of the quantum-mechanical wave function $. The Green’s function G can be calculated for many choices of boundary and initial conditions. For example, the outgoing-wave Green’s function is readily shown to be (R = r - r’) &kR

Gij(k; R) = -(k26ij

+ ViVj) 4nR , ~

(22.4)

where i and j denote Cartesian components. Given the solution to Eq. (22.3), a natural definition of a frequency- and wave number-dependent dielectric tensor E,(q, o)is the following: Let the homogeneous solution be a monochromatic field of the form E,(r, t ) = E, exp(iq r - iot).Then define Ee(q, o)by the relation

-

where (. . .), denotes an ensemble average. So defined, E , depends on q as well as w, and the longitudinal and transverse responses may differ. In the limit of small q, however, one expects the q dependence to disappear and the longitudinal and transverse dielectric functions to become equal. If we introduce a 3 x 3 scattering matrix T defined by dE(r)E(r) =

s

T(r, r‘)Eo(r’) dr‘,

(22.6)

then E(q, o)can be expressed in terms of T in the form

where G(q, o)is the Fourier-transformed solution of Eq. (22.2). Equation (22.7) suggests that the effective dielectric function in a disordered dielectric is

242

D.J. BERGMAN AND D. STROUD

analogous to the self-energy function in the theory of disordered electronic systems (a precise analogy is given in Ref. 174). Rather than seeking an effective dielectric function explicitly, one can, instead, look for an efective wave vector k,. The natural way to define k , is to look for a choice of k , such that the average scattering matrix (T) in (22.7) vanishes. The corresponding G then describes the average propagation characteristics of a wave in the disordered medium and the corresponding k , = k,. Likewise, the effective dielectric function E , is just E,, the reference dielectric constant that causes the average scattering matrix to vanish. At sufficiently long wavelengths, presumably the resulting dielectric constant E, will be the same for both longitudinal and transverse waves. In suspensions of identical spheres, the scattering matrix T can be expanded in a multiple-scattering series involving the scattering matrices ti of the individual spheres. Because this series has the same form as in liquid metals, one can adapt liquid-metal approximations to treat dielectric suspensions. Davis and S ~ h w a r t z ' ~have ~ , ' ~used ~ this approach to calculate E , in the long-wavelength regime kR 4 1 (where R is the sphere radius), essentially from Eq. (22.7). They borrowed several approximation schemes for (T) from the theory of liquid metals, including those known as the quasi-crystalline approximation and the effective-medium approximation of Roth' 8 7 (an approximation entirely distinct from the EMA discussed in this review). Figure 25 shows the effective conductivity a,(o) as calculated in Ref. 175, using three different "liquid-metal'' approximations of this kind, compared to experiment in Ag spheres embedded in gelatin. The Roth EMA gives better results than the Maxwell Garnett approximation, but none of the approximations yields fully satisfactory agreement with the measurements ofKreibig et az.188

Another multiple-scattering approach has been developed by Lamb, Wood, and Ashcroft.'68 These authors studied both ordered and disordered suspensions of spheres in the long-wavelength limit, including both two-body and three-body correlation functions, using a multiple-scattering approach expressed in the language of the Korringa-Kohn-Rostoker method of electronic band theory. Once again, this method yields results that differ substantially from those of the simpler Maxwell Garnett approaches and agree better (but not perfectly) with experiment. The integral equation (22.3) takes a particularly simple form in the quasistatic limit. If the argument kR < 1, G can be approximated by

1 3RiRj - R26ij. Gij(k;R)% - R5 471 9

"'L. Roth, Phys. Rev. B 9, 2476 (1974). "'U. Kreibig, A. Althoff,and H. Pressman, SwJ Sci. 106, 308 (1981).

(22.8)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

50

-

I

,

I

243

I

40 -

7

-

1

3

2

4

5

0 (eV> FIG.25. Absorption coefficient for a composite containing volume fraction 0.21 of Ag spheres in gelatin, as measured by Kreibig er a1.l’’ and as calculated in several approximations described in the text. Note that the EMA here is an approach derived from the theory of liquid metals and is unrelated to the effective-medium approximation described in Section 7. Taken from V. A. Davis and L. Schwartz, Phys. Rev. B 33, 6627 (1986).

471Gij is thus the ith component of the electric field at the origin due to an electric dipole of unit magnitude located at r and oriented in the j direction. Multiplying Eq. (22.3) by 8e(r) gives P(r) = 6e(r)EO(r)

+ 8e(r)

s

d3r‘G(k; r - r’) P(r’),

(22.9)

where P(r) = Ge(r)E(r). We can now make the further approximation that a given particle feels only electric dipole fields scattered from neighboring particles (“point dipole approximation”). Mathematically, we assume that, if r is in grain a and r’ is in a different grain 8, then G(k; r - r’) is approximated by G(k; Ra - Rp), where R, is the center of grain a. Defining the induced dipole moment pa of grain a by pa =

s

P(r)d3r

(22.10)

u.

we can manipulate the integral equation into the form of a set of coupled algebraic equations for the pa’s:

244

D.J. BERGMAN AND D. STROUD

where $) is the polarizability of the crth sphere, given by

(22.12) and cB being the dielectric functions of the host and inclusions. One can solve Eq. (22.11) either exactly (for special geometries) or approximately (by numerical and analytical methods) to obtain the induced dipole moment of a collection of spheres subjected to an external electric field. For example, Liebsch and c o - w o r k e r ~ 7~8 ~have ~ - ~done calculations for a “lattice gas” of spheres-that is, an ordered lattice from which some spheres are removed at random. The “site randomness” is treated using the coherent potential approximation of ailoy theory. The resulting absorption coefficient is shown in Fig. 26 where it is also compared to experiment. Once again, agreement is not perfect, presumably in part because of an oversimplification of the geometry. Numerical calculations for finite clusters of spheres, also in the point dipole approximation, have been carried out by Clippe et al.lsSClaro186has carried out related calculations for distributions of spheres, also in the quasi-

FIG.26. Absorption coefficient (arbitrary vertical scale) for a volume fraction of 0.21 of Ag spheres in gelatin, as calculated by Liebsch and P e r ~ s o n using ’ ~ ~ a lattice-gas coherent potential approximation and a face-centered cubic lattice (full line) and the Maxwell Garnett approximation (dashed line). The experimental results of Kreibig et ~ 1 . are ’ ~ arbitrarily ~ normalized.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

245

static approximation, but going beyond the point dipole approximation to include electric quadrupole and higher-order interactions, as well as simple analytical estimates of the pole spectrum. The higher-order electric multipole moments make an important difference for close spheres. Felderhof and co-workers’ 79-’81 have carried out studies of such suspensions, using cluster expansions adapted from the theory of classical fluids. They have also used the same type of cluster expansion to compute the density of surface plasmon resonances in such a suspension, using the pole representation discussed in Section 11. b. Complex Geometries; Impedance Networks Another approach is to model the binary composite as a random ac impedance network with, for instance, “metallic” and “insulating” bonds. The effective impedance of the network is calculated using the methods of Section 9. This approach has been extensively reviewed by Clerc et ~ l . ” The ~ basic idea is to assign to each bond an impedance characteristic of the material it is supposed to model. For example, a Drude metal is modeled by a resistance R and inductance L in series, the two in parallel with a capacitance C that admits displacement current, whereas an insulator can be modeled by a capacitance alone. The RLC resonances of this circuit are then the analogs of the surface plasmon resonances in a real metal-insulator composite. The resonance spectrum can be calculated numerically in two dimensions using the Y-A method discussed in Section 9, or other numerical methods. Threedimensional calculations are more difficult at present, because such elegant schemes as the Y-A method have not yet been extended beyond d = 2. The resulting resonance density can closely resemble that seen in the surface plasmon spectrum of real composites. The calculated spectrum for a model ~ the Y-A metal-insulator composite, as obtained by Zeng et ~ 1 . ” from algorithm, is shown in Fig. 27. Although calculations can be carried out, one can object that the networks seem to resemble the geometry of a real composite only superficially. In defense, there is another connection, first noted by Kirkpatri~k:~’The electrostatic potential in a random composite satisfies

v - (EVdD) = 0.

(22.13)

Equation (22.13) can be evaluated numerically as a set of coupled difference equations on a cubic lattice of lattice constant a, using a discrete approximation for the gradient operator. The resulting differenceequations are identical to Kirchhoffs laws for a random impedance network. Thus, the Kirchhoff 189J. P. Clerk, G. Giraud, J. M. Laugm, and J. M. Luck, Adv. Phys. 39, 191 (1990). C. Zeng, P. M. Hui, and D. Stroud, Phys. Rev. B 39, 1063 (1989).

19%.

246

D.J. BERGMAN AND D. STROUD

2 .c

I

I

p =0.6

I .E

h

3

u

b” 1.c

i

100 x 100 0

Simulation (One Realization ) EMA

-

aJ

m

0.E

C

I .o

0.5

1.5

w /w, FIG.27. Effective conductivity Re u,(w) of a model composite consisting of a volume fraction p = 0.6 of Drude metal and 1 - p of insulator, as calculated by numerical simulation on a

random impedance lattice (squares) and in the effective-medium approximation (full curve). Taken from X.C. Zeng, P. M. Hui, and D. Stroud, Phys. Rev. B 39, 1063 (1989).

equations for a random impedance network are simply the discrete version of Eq. (22.13). In this sense, such random networks really do resemble the composites they are intended to model. V. Nonlinear Properties and Flicker Noise

23. FLICKER NOISE The term fricker noise in the context of electrical conductivity refers to fluctuations of the total conductance. When a constant current I is driven

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

247

through the conductor, those fluctuations are manifested as fluctuations in the voltage drop, along with other fluctuations known as Johnson noise and shot noise. The three types of noise differ in how they scale with I : The rootmean-square (RMS) voltage fluctuation due to Johnson noise is an equilibrium phenomenon and hence independent of I. The RMS voltage fluctuation due to shot noise is proportional to I l ” , and that due to flicker noise is proportional to I . It was observed experimentally that the flicker noise in metal-insulator composites increases dramatically and often becomes the dominant type of noise as the conductivity or percolation threshold is approached (see Figs. 28 and 29).191-196It was also observed in these studies that the power spectrum of this noise has a l/o”frequency dependence with a of order 1; hence this noise is also sometimes called llfnoise (see Fig. 30). Because of the interest generated in the properties of flicker noise near a

FIG.28. The excess voltage noise as a function of sample voltage. The data were taken at 10 Hz and the zero-current noise has been subtracted from each measurement. The solid line of slope 2 is drawn through the points as a guide. The dashed line is the calculated Johnson noise level for this resistance. Taken from D. A. Rudman, J. J. Calabrese, and J. C. Garland, Phys. Rev. Lett. 55, 296 (1985). I9’R. V. Voss and J. Clarke, Phys. Reo. B 13, 556 (1976). 192J. V. Mantese, W. I. Goldburg, D. H. Darling, H. G. Craighead, U. J. Gibson, R. A. Buhrman, and W. W. Webb, Solid State Commun. 37, 353 (1981). 193G.A. Garfunkel and M. B. Weismann, Phys. Reo. Lett. 55, 296 (1985). 194D.A. Rudman, J. J. Calabrese, and J. C. Garland, Phys. Rev. B. 33, 1456 (1986). 195J.H. Scofield, J. V. Mantese, and W. W. Webb, Phys. Rev. B 32, 736 (1985). 196R.H. Koch, R. B. Laibowitz, E. I. Alessandrini, and J. M. Viggiano, Phys. Rev. B 32, 6932 (1985).

248

D.J. BERGMAN AND D. STROUD

lo0 10' RESISTANCE R (ARBITRARY UNITS) FIG.29. The scaling of S$RZ versus R for several samples. Taken from G. A. Garfunkel and M. B. Weisman, Phys. Rev. Lett. 55, 296 (1985).

percolation threshold, most of the theoretical treatments of this phenomenon are based on simple random network models of the actual continuum composite, e.g., the independent random bond resistor network. The basic quantity needed to describe the flicker noise is the power spectrum of the resistance or conductance fluctuations S,(o) =

S,(o) =

s s

dt e-'"'(dR(t)6R(O)), (23.1) dt e-i0'(6G(t)6G(0)),

where 6R(t) (dG(t)) is the time dependent fluctuation of the total resistance (conductance) of the sample. Using one of the standard results for the conductance of a composite [see (1.1)] of cross section A and length L, G

=

L

=

$j d V a ( r ) ( g )

2

,

(23.2)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

1 o2

10"

249

1014 -.13

10

1 oo 'N

r

Y

10-17

0-2

o - ~

1 f (Hz)

FIG.30. Reproducibility and inverse size scaling of the l / f noise of four Cr specimens fabricated from the same film. The symbols, lengths L (in micrometers) and widths w (in micrometers), are tabulated in the figure. (a) Log-log plot of L f S , ( f ) / p 2 I J = HZ versus specimen size N,. (b) Log-log plot of N,S,cf)/p2 versusf for all four specimens. Taken from J. H. Scofield, J. V. Mantese and W. W. Webb, Phys. Rev. B 32,736 (1985).

we can write for the instantaneous fluctuation of G (23.3) Note that even though 60 produces a fluctuation of the local field E(r), this does not affect G to first order in Sa because of the variational property of (23.2). Using (23.3) we get

We now assume that the correlation function of Sa has the following form: (So(r, t)ba(r', 0 ) ) = b(r)d3(r - r')g(t),

(23.5)

and this leads to (23.6a)

250

D.J. BERGMAN AND D. STROUD

where

g(o) =

s

dt e-i"'g(t).

(23.6b)

All is valid if the fluctuations 60 are small, i.e., if the local field fluctuation 6E is always small enough so that (23.3) is a good approximation. The relative power spectrum of the total conductance fluctuations is given by (23.7) For a macroscopically homogeneous composite this quantity is clearly inversely proportional to the volume. If the volume is large enough, SG/G2 will be small and can be equated with S J R 2 ; both quantities can then be identified as the relative power spectrum of the flicker noise. Moreover, having assumed that the time dependence g(t), and hence also g(w), is common to all components, the only part of the relative power spectrum that depends on microgeometry is the following quantity: (23.8a) Here (23.8b) is the usual bulk effective conductivity,

s

be = V d V b ( r ) ( g y ,

(23.8~)

and a(r) is the nonfluctuating or time-averaged local conductivity. We can also think of be as the bulk effective b-coefficient in an expression like (23.5) for the correlation function of time-dependent local fluctuations of the bulk effective conductivity 8ae(r,t). This may seem paradoxical, because a, and be are, by definition, independent of r. Nevertheless, it is possible to take this point of view if we only use the form (23.5) for the correlation function in convolutions with other functions that change appreciably only on length scales much greater than the scale of the inhomogeneities in the system.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

251

In a discrete network, where the conductance of different bonds ga fluctuates independently, gn <SgaSgb>

=(ga)

+

(23.9)

= Sab(Sd>,

we find analogously (23.1Oa)

(23.lob) where V , is the voltage across the bond a, Vis the voltage across the entire network, and the average can be understood either as a time average or as an ensemble average over different realizations of the network. Similar expressions can be obtained in terms of the bond resistances r,, bond currents I,, and total current I : (23.11)

(23.12a)

(23.12b) Finally, the network analog of (23.8) is

(SG’) G2

-

(SR’) - Sgz R2

Ngz’

(23.13a)

(23.13b)

(23.13~)

252

D.J. BERGMAN A N D D. STROUD

where N is the total number of unit cells and V, = V L , the voltage drop on a single conductor in a homogeneous network subject to the same boundary conditions as the actual, inhomogeneous one. Clearly, g e and 6ge are the (ensemble) average and RMS fluctuation of the bond conductances in a homogeneous network of independently fluctuating conductors whose macroscopic properties are the same as those of the actual, inhomogeneous network. As in the case of other properties near the percolation threshold of a randomly diluted resistor network, the characteristic noise parameter Sgf/gf exhibits a power law behavior as function of Ap = p - p , (23.14) Starting with Rammal and his c o - ~ o r k e r s , ' ~ ~this - ' ~behavior ~ has been studied using different methods, such as regular fractal network model^,'^' real space renormalization group t r a n s f o r m a t i ~ n s , ' ~ ~numerical -~~~ simulations,' 99,201,202,55and rigorous bounds.203 It is also natural to try to apply the ideas of Bruggeman to produce a selfconsistent effective-medium approximation (EMA) for 69, or 60,. One might have expected that this would be a good approximation for most values of p and would fail only in the vicinity of p , , as in the case of 9,. Those expectations are, however, not realized. When the EMA procedure is correctly applied to the flicker noise problem, it leads to results that make sense only when p is close to 1. As p decreases to about midway between 1 and p , , 60, diverges and remains infinite for all lower values of p down to pC.'O4 This arises because in the EMA procedure almost no restrictions are placed on the microgeometry of the system, so the result should, in principle, apply also to bizarre microstructures. Indeed, it can be shown that the EMA result is exactly correct for a special type of structure that includes an exponential hierarchy of inhomogeneity length For that structure, 60, may indeed be infinite. This is an indication that the flicker noise is more sensitive than the ohmic conductivity to details of the microstructure. This point will I9'R. Rammal, C. Tannous, and A.-M. S. Tremblay, Phys. Rev. A 31, 2662 (1985). "'R. Rammal, J. Phys. Lett. 46, L129 (1985). ly9R.Rammal, C. Tannous, P. Breton, and A.-M. S. Tremblay, Phys. Rev. Lett. 54, 1718 (1985). 2ooP.M. Hui and D. Stroud, Phys. Rev. B 34, 8101 (1986). '''A. Csordas, J. Phys. A 19, L613 (1986). 202L.de Arcangelis, S. Redner, and A. Coniglio, Phys. Rev. B 31,4725 (1985). *03D.C. Wright, D. J. Bergman, and Y. Kantor, Phys. Rev. B 33, 396 (1986). 204D.J. Bergman, Phys. Rev. B 39,4598 (1989).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

253

be discussed in greater detail in Section 24. A discussion of the behavior of ha, in continuum composites is also deferred to that section. 24. WEAKNONLINEARITY "Weak nonlinearity" refers to materials that have a small nonlinear correction to predominantly linear behavior. For example, consider a composite dielectric medium in which the local relation between E and D is

D

= EE

+ blEI"E,

(24.la)

and

bJEJ"'< E.

(24.lb)

We allow E and b to have different values in each component, but the exponent m has the same value in all of them. All isotropic dielectrics with inversion symmetry exhibit such behavior with rn = 2 for a certain range of field strengths, as is well known from experimental studies using lasers to produce strong fields. It is not hard to show that a composite of such weakly nonlinear dielectrics will exhibit a bulk behavior characterized by the same exponent m and by bulk effective coefficients E,, be:

where the angular brackets ( ) denote a volume average. To show this, we note first that, for an isotropic microgeometry and isotropic components, (D) will be parallel to E, = (E), the externally applied uniform field, by symmetry. We then note that the first term of the volume average of (24.la) can be transformed, as in (l.l), as follows:

V

j

'S

dVE(r) E(r) * Eo E - E; O-V

dVE(r) E2(r>E,, E; ~

(24.3)

and that the final form has the usual variational properties. Consequently, to first order in b we may replace the exact field E(r) on the right-hand side by the linear approximation El(r),which is calculated by putting b = 0. Having done that, we may transform back to the left-hand side of (24.3). Obviously, we can use E, instead of E also in the second term of the average of (24.la), so that we finally get

- E, E, + 1 dVb(r) IEII"(E1 Eo))IE,("E,. V IEOlm+2 '

E;

(24.4)

254

D.J. BERGMAN A N D D. STROUD

Because El(r)/lE , I is obviously independent of I E , I, this establishes the form of (24.2), as well as giving explicit expressions for E, [which are the same as in (1. l)] and for be

's

be = V

dVb(r)

-

I4I "(El Eo) IEO

(24.5) The second of these expressions is obtained from the first by the procedure used in connection with (1.1): The first integral is transformed to a surface in this case), and then integral, where 4o may be replaced by 4 (or transformed back to a volume integral. Comparing (24.5) with (23.8~)it is evident that the problem of a composite with a weak, cubic nonlinearity (m = 2) is mathematically identical to the problem of flicker noise. The fact that be can be calculated without having to know the solution E(r) of the nonlinear problem in the composite is very useful and makes the problem of weakly nonlinear composites a tractable From (24.5) it is evident that what is needed are moments of the field E,(r) of the linear problem. We henceforth omit the subscript 1 and refer to this field simply as E. Starting from expressions like (23.8~)and (24.5), some special results and approximations have been found for b e . These include a result for a low density of weakly nonlinear inclusions in a linear host,20s as well as a Maxwell Garnett-type approximation, a non-self-consistent effective medium approximation, and exact results for the solvable microgeometries of Fig. 1 (see Ref. 206). Considerable effort has gone into calculating the moments of E in the case of binary random-resistor networks (RRNs) near a percolation threshold, These have using numerical simulations and other techniques.'99~202~207*61 led to predictions concerning the critical behavior of be, including values for the critical exponents. In particular, the critical exponents characterizing be for different values of rn were found to be independent of each other. This has led to attempts to describe the local field or current distribution in an RRN at the percolation threshold by a multifractal f o r m a l i ~ m . ~ ~ * . ~ ~ ~

'O'D. Stroud and P. M. Hui, Phys. Rev. B 37, 8719 (1988). 206X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, Phys. Rev. B 38, 10970 (1988). '07R. Blumenfeld, Y. Meir, A. Aharony, and A. B. Harris, Phys. Rev. B 35, 3524 (1987). 208L. de Arcangelis, S. Redner, and A. Coniglio, Phys. Rev. B 34,4656 (1986).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

255

An experiment to detect directly the cubic nonlinearity in a synthetic 2D composite has been carried It utilizes the temperature dependence of the resistivity of a thin film of aluminized Mylar, which leads to a small cubic term in the I-Vcharacteristic of the film. The film itself is turned into a composite medium by cutting out parts of it in a random fashion. The analogy between a weak, m = 2 nonlinearity and the mean square fluctuation of the bulk effective bond conductance 6gz can be extended in the following way. If we consider higher-order cumulant averages* of 6g,, we can easily show that (24.6) where ( ), denotes a cumulant average. (For a discussion of cumulant averages and their basic properties see, e.g., Ref. 210.) The continuum analog of this is

where the nth cumulant of 6o(r) is assumed to satisfy

= 6o'"'(r,)

- d3(r1 - r2)d3(r2 - r3) . . . d3(r,-,

- r").

(24.7b)

Clearly, the left-hand side of (24.7a) is independent of the volume for a macroscopically homogeneous sample, and its numerator is similar to be of (24.5). The cumulant averages of 6G are much easier to study, computationally as well as analytically, in binary network models than in continuum composites. For this reason most efforts have gone into calculations of (dg;), in simple, randomly diluted networks. For such networks, as in the calculation of ( g e ) itself, there are only two possible values of the individual bond 209M.A. Dubson, Y. C. Hui, M. B. Weismann, and J. C. Garland, Phys. Rev. B 39, 6807 (1989). 2'oS.-K. Ma, "Statistical Mechanics," Sects. 12.3 and 12.4, World Scientific, Philadelphia, 1985.

~

* The cumulant average of X",where X is a random variable, is denoted by (X"),and is defined in terms of regular averages by expanding both sides of the equation ( e A x- I), = h(e") in powers of I , and then equating them term by term.

256

D.J. BERGMAN AND D. STROUD

cumulant average, namely if the bond is present, if the bond is absent.

(24.8)

Expression (24.6) then becomes simply the (2n)th moment of the voltage distribution. Methods for calculating such voltage distributions were discussed in Section 9. Those calculations have shown that these moments have a power law dependence on p - p c , but that there is no apparent relationship between the exponents for the different moments.'99~z02*z08 One should use caution in applying the randomly diluted network results to real continuum composites, because the primitive microstructure of the networks may prevent them from reproducing important properties of the real composite. It might seem possible to discuss the behavior of flicker noise in continuum percolation by using the same R R N models, with singular distributions of the bond conductance, that were used in Section 12 to discuss the ohmic conductivity. However, one must then be careful to assign correctly the values of the individual bond cumulants or nth-order nonlinear coefficients. In particular, these coefficients often scale differently from the bond conductance ga when the microgeometric features are changed.21 For example, in the inverted Swiss Cheese model, where the solid material is insulating and the voids are filled with a conducting material oM,the conductivity near threshold is dominated by the narrow necks between slightly overlapping conducting spheres. If those necks are replaced by discrete conductors, then the bond conductance is related to the neck radius a,, the sphere radius a, and the specific conductivity aMof the spheres byzo4 9

-

anaM

for a, 4 a,.

(24.9)

At the same time, the mean-square fluctuation of g is related to the fluctuation parameter of the conductor b, byzo4 for a, 4 a,.

(24.10)

Obviously, g and (Sg') behave quite differently as the overlap between two spheres decreases to zero (i.e., as a, + 0)-the first also decreases to zero, while the second diverges. The reason for this different behavior is connected to the divergence of the field E(r) at the cusp of the intersection between neighboring spheres. The 'l'A.-M. S. Tremblay, S. Feng, and P. Breton, Phys. Rev. B 33,2077 (1986).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

257

singularity of E becomes more severe as a, -,0, and the appearance of E4 in the expression for be or (Sg;) is then just enough to make the integral diverge. At the same time, the integral in the expression for oe, which only includes E2,remains finite. With higher-order cumulants or nonlinearities, the integral in the expression for ( S g ~ ) , , n > 2, diverges even when a, is finite. What this means, of course, is that one must allow a small but finite radius of curvature c at the tip of the cusp, and the result for (Sg:), will depend crucially on c.204 These points are best illustrated by considering a homogeneous conductor, with a large volume r/; in which a single, thin, circular-disk-shaped void is introduced, with its plane perpendicular to the direction of current flow. The two important parameters are the disk radius a and the radius of curvature c at the tip of the cross section of the disk (see Fig. 31). The changes in the total resistivity p, i.e., in the second moment of E, as well as the changes in the fourth and sixth moments of E, can easily be evaluated for the case c < a. The results are, in order of magnitude, a3 a -In-, V c

az(a - c )

V

'

a4/c

__

V '

(24.1 1)

for the second, fourth and sixth moments (m = 0, 2, 4), r e s p e c t i ~ e l y . ~ ~ ~ ~ ~ ' ~ Clearly, whereas the resistivity (rn = 0) is quite insensitive to c and depends JO

JO

FIG.31. Cross section of a flat circular void or crack of radius a in a conducting medium. The edge of the crack has a small but finite radius of curvature c < a. The external uniform current density J , flows perpendicular to the plane of the crack. The actual local current field J is distorted by the presence of the crack. Taken from D. J. Bergman, Physica A 157,72 (1989). 'lZD. J. Bergman, Physica A 157, 72 (1989).

258

D.J. BERGMAN AND D. STROUD

mainly on a, the flicker noise or cubic nonlinearity coefficient (m = 2) is logarithmically divergent with the ratio a/c, while the fifth-power nonlinearity coefficient (rn = 4)diverges linearly with a/c. Thus, the higher the moment of E that is involved, the more sensitive is the physical property to details of the microgeometry that result in anomalously large local values of E.

25. STRONG NONLINEARITY

We shall be discussing two types of strong nonlinearity-a simple power law relation between, say, E and D, and an onset threshold where there is an abrupt crossover from linear behavior to something else, e.g., the phenomenon of dielectric breakdown or mechanical fracture. The second type of phenomenon is well known and has clear technological significance, but the first type is not as common. One example is the relation between E and D in a dielectric at high field intensities, such as those produced by lasers, when the linear approximation breaks down completely due to the preponderance of multiphoton processes. Another example is the I- I/ characteristic in certain classes of conductors, e.g, ZnO ceramics, which are used in the fabrication of varistor^."^ Few things can be taken for granted in the field of nonlinear phenomena. Therefore we begin by examining the question of uniqueness of the solution for the electric potential 4, given certain boundary conditions, when D = f ( I E I)E and E I) is a positive but otherwise arbitrary function. Suppose there are two solutions 4,, leading to El, D, and E,, D,. Then the integral

+,,

(25.1)

can be shown to vanish, by transforming it to a surface integral, if the boundary values either of 4,, 4, or of the normal components of D,, D, are the same. On the other hand, the integrand is easily seen to satisfy the inequality

where the equality holds only if El 11 E,. Clearly, if I E If( I E I) is a monotonically increasing function of I E I , the left-hand side will be positive definite *I3R. Einziger,

Annu. Rev. Muter. Sci. 17, 299 (1987).

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

259

wherever El # E,, and therefore the integral of (25.1) will be nonzero (and positive) unless 41= c$z + const. This proof, originally due to M i l g r ~ m , ” ~ does not use the superposition principle, which is often relied on to prove uniqueness in the linear case. A corresponding uniqueness proof of the solution of Kirchhoffs equations for a nonlinear resistor network can also be constructed along these lines. a. Power Law Relation between E and D We now restrict our discussion to the case in which in each component the relation between E and D is given by D

=E

(E(~E,

(25.3)

with the same value of fi but different values of E. This type of relation is found when a dielectric solid is subject to the intense electromagnetic fields found in focused laser radiation. Such a relation also exists between the electric field and the current density in ZnO ceramic^."^. The local fields are uniquely determined by the boundary conditions whenever /I > - 1. We now show that for an isotropic composite of this type, a similar relation also exists between the volume-averaged fields, namely” l 6 ’ 3 ’

(25.4)

We also find expressions for the bulk effective nonlinear dielectric coefficient E,. The analogous relations for nonlinear networks were first obtained by Kenkel and Straley.z17-218 To this end, we first note that because the infinitesimal electrostatic work is given by

dW=

J dPfE

*

6D) = (/I + 1) ~ V E ( E I B (* EdE),

J

214M. Milgrom, Asirophys. J. 302, 617 (1986). 2’5R. Blurnenfeld and D. J. Bergrnan, Physica A 157,428 (1989). 216R. Blurnenfeld, Ph.D. Thesis, Tel Aviv University, 1990, unpublished. 217S. W. Kenkel and J. P . Straley, Phys. Rev. Lett. 49, 767 (1982). 218J. P. Straley and S. W . Kenkel, Phys. Rev. B 29, 6299 (1984).

(25.5a)

260

D.J. BERGMAN AND D. STROUD

it follows that the total electrostatic energy is given by

Infinitesimal changes in E(r) and in the boundary values of the potential 40(r) lead to the following change in U :

If we take both 640 and be as arising from changes in a constant multiplicative factor, i.e.,

6P 640 = 40, P

61

(25.7)

6 E = -E ,

A

we get (25.8) which integrates out to U ( p , A) = U(1, l)Ap’+Z.

(25.9)

-

To be specific, let us assume the boundary condition 4o = - p(Eo r), which results in (E) = pEO.This immediately leads to (25.10) where

(25.1 1) is independent of (E) and is an order one homogeneous functional of E(r). Another form for U , obtained from (25.5b) by transforming to a surface integral, changing 4 to the boundary value function 40,and transforming

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

261

back to a volume integral, is @+2U = (E) @ + lv ~-

- (D).

(25.12)

Comparing this to (25.10), and noting that (E) 11 (D) by symmetry for an isotropic composite, we get the results (25.4). Using those results, we can rewrite the total energy U as

which can also be recast as

Substituting for E or for D in (25.12) from (25.3), we also get two more expressions for E , : (25.15)

(25.16.) Although none of the expressions (25.1l), (25.14)-(25.16) are easy to implement, they can at least serve as starting points for further studies. Similar expressions can also be derived for discrete nonlinear networks with a power law I- V characteristic. Because of the inapplicability of the superposition principle, many important and useful results from the theory of linear dielectrics do not extend to this case. Nevertheless, a few microgeometries are solvable even here, namely the parallel cylinders and parallel slabs microstructures216(see Fig. la and b).

262

D.J. BERGMAN AND D. STROUD

Another approach that shows promise is to expand the fields, as well as E,, around the results for a homogeneous medium, where c(r) = c0 = const.z'9~220 The idea is to write c(r)

= c0

+ BE

(25.17)

and then proceed to expand all the fields, as well as E,, in powers of SE, recalling from Section 6 that in the linear case this approach had great power. Allowing, for the moment, the more general local constitutive relation

D =AE2)E,

(25.18)

we can write, for the nth-order variation of D,

S"D =f(Ei)S"E

+ 2f'(E:)(E0

*

S"E)E, + g(Eo, SE, S2E, . . .,S"-lE), (25.19)

where E, = const is the solution for c(r) 3 E,, which we will take to lie along the z axis, and where the last term represents a complicated expression that includes only lower-order variations of E, namely S"E with rn < n. From the pair of equations

S"E = -VS"4,

V S"D = 0,

(25.20)

we now get the following equation for 8.4:

This is an infinite hierarchy of linear equations in which the right-hand side always depends on the solutions of previous equations of the hierarchy. By rescaling the z coordinate as

(25.22) (25.21) is transformed into a Poisson equation

V2S"= ~ F(Eo,SE, . . . ,S"-lE). *19R.Blumenfeld and D. J. Bergman, Phys. Rev. B 40, 1987 (1989). 220R.Blumenfeld and D. J. Bergman, Phys. Rev. B 44, 7378 (1991).

(25.23)

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

263

In the case of a power law constitutive relation, as in (25.3), we.find that the rescaling transformation (25.22) is independent of E , and becomes simply (25.24) Once this expansion of 4 has been carried out, the fields 40, 6"4 can be used to provide a similar type of expansion for E , . The nth-order term in this expansion has the form (25.25) which clearly depends on 40, 64 . . . dn-'4 but not on 8'4. As in the linear case, the terms up to n = 2 can be calculated explicitly for isotropic composites without requiring any information about the microgeometry other than the volume fractions. The result is219

where ( ) again denotes a simple volume average, so that

This result is valid for any j? > - 1 and reduces to (3.1) when P = 0. For negative /?,the square roots in (25.26a) become imaginary, but the final result is still real and positive. b. Electrical Breakdown The problems of dielectric breakdown and mechanical fracture have of course been studied for many years because of their importance in technology. Their connection to problems of composite media comes from the fact that in many cases the breakdown is due to macroscopic imperfections in an otherwise homogeneous material. For example, in a homogeneous solid with a single disk-shaped void like the one in Fig. 31, mechanical failure will commence at the sharp tip of the disc, because that is where the local stress, like the electric current density, is greatest. Moreover, because the failure

264

D.J. BERGMAN AND D. STROUD

tends to increase the total area of the disc-shaped crack, the stress at the tip will become even greater, and thus the process of increasing that area will continue catastrophically until total rupture of the sample takes place and releases the stress. For an isolated void, like the one in Fig. 31, the critical value of the macroscopic stress when failure begins depends on the ratio a/c, which determines the stress amplification factor at the tip of the In order to have a simple model for studying breakdown phenomena in composite media, it is convenient to consider a discrete random network of fuses-each bond represents an ohmic conductor as long as the current i flowing through it is less than some critical value i,. Above that value the conductor "burns out" irreversibly and becomes an insulator. One then studies the behavior of such a network by increasing the total current through it and causing a succession of its bonds to burn out. Duxbury and his co-workers have analyzed such models, as well as continuum models, for a dilute random distribution of defects. They noted that the breakdown of ohmic behavior (i.e., burning out of bonds) is determined by the worst defect cluster in the sample rather than by the common or most probable defect clusters. Therefore the statistical analysis of such a breakdown phenomenon involves the tail of the defect cluster distribution function and not its peak. An important parameter is the largest bond current in the still ohmic network, .,i They found that ,i scales with a power of In L, where L is the linear size of the ample'^^-'^^ i,

-

(In L)",

(25.27)

1


This has the important effect of making the average electric field at the breakdown threshold, E,, depend on the sample size. They found that instead of being an intensive material parameter, E, has the form'44

(25.28) where K is an intensive material parameter. They also found that the actual breakdown threshold, varies greatly from sample to sample and that its cumulative distribution has the Weibull form'44

W , )= 1 - exp( - c L ~ E ; ) ,

(25.29)

where c and m are material constants. This is an s-shaped function that

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

265

increases monotonically from 0 to 1 and gives the probability for E, to be less than a given value. In technology, where one is often interested in producing samples with a low probability of failure, the important range of E , is obviously the asymptotic one, where F(E,) -P 0. In systems that have a large concentration of defects, the failure behavior may be affected by the fractal microstructure that appears near a percolation threshold. This problem has been studied extensively, both analytically and computationally, on random-network models and also on some model experimental systems. We refer the reader to Ref. 221 for a detailed and authoritative review of these studies, including many references. In order to illustrate the use of network models for studying the failure of percolating systems, we consider the electrical breakdown of a network of insulating and conducting bonds in which the latter burn out irreversibly and become insulating, like actual fuses, when a certain threshold current i, is reached. In the links-nodes-blobs (LNB) picture, described in Section 10, the current-carrying backbone of the percolating cluster of conducting bonds is a supernetwork of links that carry the current between adjacent to the macroscopic n ~ d e ~ . The ~ ~typical ~ , link ~ ~current ~ , I, ~ is~connected ~ . density J by

I, = J l d - ',

(25.30)

-

where the percolation correlation length 5 ( p - p,)-' is equal to the average distance between adjacent nodes. Because every link has a certain number of singly connected bonds (SCBs) that carry the full link current (see Fig. 16), it is one of these that will burn out and disconnect the link when I, reaches the value i,. This immediately increases the currents through many other links, so the entire system undergoes a rapid. and total failure. The failure thresholds J,, E , for the macroscopic current density and electric field are easily seen to be

(25.3 1) (25.32)

221B.K. Chakrabarti, Reo. Solid State Sci 2, 559 (1988). '"The discussion presented here is based on a similar discussion of mechanical breakdown in an elastic network that was published in Ref. 223 and on a discussion of electrical breakdown published in Ref. 142. 223D.J. Bergman, in "Fragmentation,Form and flow in Fractured Media," (R. Englman and Z. Jaeger, eds.), Ann. Israel Phys. SOC.8, 266-272 (1986).

266

D.J. BERGMAN AND D. STROUD

<

where A p = p - p, and is the conductivity exponent governing the behavior of the link conductance gr ApC [see (10.4)-( 10.6)]. Interestingly, whereas J , always tends to zero as p + p c , E , actually diverges when d = 3, because then v - N -0.2, but tends to zero very, very slowly when d = 2, because then v - 5 N 0.03. The behavior of J , , as set forth in (25.31), may also serve to describe the macroscopic critical current density for a percolating network of superconducting bonds, if one ignores questions of the quantum phase coherence around superconducting loops (see Section 16). Another type of failure phenomenon that can be modeled in a similar fashion is the dielectric breakdown of a random metal-insulator mixture below the percolation threshold of the metal component when a sufficiently strong field is applied. In that case we invoke the dual of the LNB picture, namely a collection of finite clusters of conducting bonds of typical linear size 5, separated by thin insulating barriers. When p is near p , , these barriers always include regions whose thickness is just one bond-those have been called singly disconnecting bonds (SDBs).’03. When a time-independent external field is applied to the network, the largest bond voltages appear across the SDBs. If the insulating bonds break down and become conductors above some threshold voltage uc, we easily find that the macroscopic breakdown field E , is given by

-

<

(25.33) where now A p = p , - p . Because the foregoing discussion predicts intrinsic, size-independent values for the breakdown threshold, in contrast with the behavior found for dilute systems in Refs. 143-145, it is important to identify the conditions under which each of these descriptions is valid. The following analysis, previously presented as an oral comment,224is published here for the first time. The behavior found in Refs. 143-145 is appropriate when the largest defect cluster is of size -In L and is a rare occurrence; it is then much larger than the typical cluster size or LNB unit cell size 5. In the opposite case, when In L 4 5, there are many clusters of size 5 and the clusters of size In L are irrelevant. We therefore expect the intrinsic behavior described here and in Refs. 223 and 142 to hold when 5 > In L, whereas the behavior described earlier and in Refs. 143-145 should hold when t < In L. We have not done justice in this section to the many theoretical and experimental studies of breakdown and other nonlinear phenomena that have been made in recent years. This is currently a very active area of 224D.J. Bergman, unpublished.

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

267

research, summarized in part by Ref. 221. No doubt within a few years it will mature and deserve its own special article in this series of reviews. 26. NONLINEAR OPTICAL EFFECTS The preceding discussion is immediately relevant to finite frequencies, within the quasi-static approximation. Finite frequencies are important because of the possible use of random composites as nonlinear optical materials. Some such composites can have exceptionally large nonlinear susceptibilities. One example is “filter glass,” which consists of a dilute suspension of CdS,Se, - particles in a transparent, linear host. The semiconducting material has a large cubic nonlinear susceptibility.When the material is embedded in a glass host, this susceptibility can be greatly enhanced at certain frequencies-that is, the efective nonlinear susceptibility of the composite (per unit volume of the nonlinear inclusions) can be much larger than the susceptibility of a hypothetical bulk material made entirely of the nonlinear material. This effective nonlinear susceptibility is the analog of the nonlinear transport coefficients discussed in Section 24. The enhancement we refer to is caused by the same kinds of local field effects that produce the Mie resonances in the linear response. (Beyond these local field effects, there is additional enhancement from quantum size effects in the semiconductor microcrystallites.) The nonlinear response of composites at finite frequencies can be treated, in the quasi-static limit, by the methods of Section 24. Most nonlinear studies in composite^^^^^^^^ have involved the so-called optical Kerr effect, in which both D and E are monochromatic and connected by an intensity-dependent dielectric function. If the constituents have inversion symmetry, the leading intesntiy dependence will be of the form

where E(r) and X(r) are, in general, complex and frequency dependent. In the presence of inversion symmetry, Eq. (26.1) will have no terms quadratic in E. The volume-averaged electric fields and electric displacements (D) and (E) are related by a similar equation (26.2) Equation (26.2) defines the effective coefficients E, and

ze.

225See,for example, G. R. Olbright, N. Peyghambarian, S.W. Koch, and L. Banyai, Opt. Lett. 12,

413 (1987). 226R. Rossignol, D. Ricard, K. C. Rustagi, and C. Flytzanis, Opt. Commun. 55, 1431 (1985).

268

D.J. BERGMAN AND D. STROUD

Approximations for x, can be adapted from the static case, just as can formulas for E,. For example, when a small concentration p 4 1 of nonlinear material is embedded in a linear host, X, is given by225*226 (26.3) Analogous formulas can be obtained for a small concentration of linear inclusion in a nonlinear host. When both components have nonzero xi's and the concentrations are not small, no fully satisfactory approximations are as yet available. As discussed in Section 23, the scheme most closely analogous to the linear EMA is less successful than the EMA when extended to the nonlinear case. The high-concentration regime thus remains an important area for future study. Figure 32 shows Re 2, for a small concentration of model Drude particles in a linear dielectric, as obtained from Eq. (26.3). The very large enhancement seen near frequency w P / d is a result of a near-vanishing of the factor ( E ~+ 2cz) in Eq. (26.3). Because this denominator enters to the fourth power,

o/op FIG.32. Full curve and left-hand scale: real part of the cubic nonlinear susceptibility, Re x,(w), calculated for a dilute composite of volume fraction p of model Drude metal particles of

nonlinear susceptibilityy, and wpz = 12.5, embedded in a dielectric of dielectric constant unity. Dashed curve and right-hand scale: Real part of the effective conductivity, Re u,,for the same composite. Note evidence of a surface plasmon resonance in both quantities. Both graphs are renormalized (in different ways) by the volume fraction p .

PROPERTIES OF MACROSCOPICALLY INHOMOGENEOUS MEDIA

269

the enhancement can be several orders of magnitude for a material with a sharp Mie resonance. Thus there are potentially much larger enhancements for the nonlinear susceptibilities in composites than for the linear response. It is interesting that the same kind of local field enhancement plays a role in other optical properties of granular materials. A well-known example of this kind is surface-enhanced Raman scattering from rough surfaces. The rough surface can be treated like a granular medium, and the local field near sharp projections of the rough surface can produce enormous enhancement in the Raman signal from the ~urface.’’~ Clearly, there can be an enormous variety in the nonlinear response of composites at finite frequencies. The many possible coefficients are limited only by the symmetry of the underlying crystal lattices. Furthermore, like other nonlinear properties, the optical response may exhibit hysteresis, bistability, and chaotic time-dependences. The study of such properties in composite media is at only the very beginning stages.

ACKNOWLEDGMENTS The authors would like to acknowledge valuable conversations with many of their co-workers on aspects of this work. The authors are very grateful for the generous support of the U.S. National Science Foundation, Division of Materials Research, the U.S.Israel Binational Science Foundation, the Israel Academy of Sciences, and the Ohio Supercomputer Center, where several calculations described here were carried out. D.G.S. would like to acknowledge the hospitality of the School of Physics and Astronomy of Tel-Aviv University during several visits in the course of this work, and D.J.B. would like to acknowledge the hospitality of the Department of Physics of The Ohio State University in Columbus, Ohio during a number of visits both before and in the course of the period when this review was being written.

’”For a discussion of this enhancement, see, for example, S. McCall, P. Platzman, and Wolff, Phys. Lett. A 77, 381 (1980).

P. A.

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SOLID STATE PHYSICS, V O L U M E 46

Fundamental Magnetization Processes in Thin-Film Recording Media H. NEALBERTRAM Department of Electrical and Computer Engineering and Center for Magnetic Recording Research University of California at San Diego Ln Jolla, California

JIAN-GANG ZHU Department of Electrical Engineering University of Minnesota Minneapolis, Minnesota I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Physics Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Outline. . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . 11. Microstructures and Magnetic Properties of Thin-Film Materials. 4. Longitudinal Films. . . . . . . . . . . . . . . . . . . . . . . . . . 5. Perpendicular Films . . . . . . . . . . . . . . . . . . . . . . . . . 111. Micromagnetic Modeling . . . . . . , . . . . . . . . . . . . . . . . . 6. Review of Single- and Multiple-Particle Reversal Mechanisms 7. Numerical Modeling of Thin Films . . . . . . . . . . . . . . . . IV. Reversal Processes and Domain Structures . . . . . . . . . . . . . . 8. Longitudinal Films. . . . . . . . . . . . . . . . . . . . . . . . . . 9. Perpendicular Films . . . . . . . . . . . . . . . . . . . . . . . . . V. Simulations of the Magnetic Recording Process . . . . . . . . . . . 10. Single Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . 11. Transition Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Interacting Transitions . . . . . . . . . . . . . . . . . . . . . . . VI. Self-organized Behavior in Magnetic Systems . . . . . . . . . . . . 13. Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Reversal Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Correlations and Noise. . . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 1 27 1 276 28 1 28 1 282 297 299 299 313 320 320 334 345 345 353 357 362 363 365 369 371

1. Introduction

1. BACKGROUND Magnetism plays a central role in the technology of information storage and retrieval. Magnetic materials of high coercivity (hard magnets) form the basic structure (media) of magnetic disks for computer systems and magnetic 27 1 Copyright 01992 by Academic Press, Inc. All riihts of renrodiirtinn in I ~ Vh n n r e r e r v d

272

H. NEAL BERTRAM AND JIAN-GANG ZHU

tape for tape recorders. In the magnetic recording process the recording and retrieval of information are performed by magnetic materials (transducers or heads) of extremely small coercivity (soft magnets). In addition, hard magnetic materials with temperature-sensitive coercivities are utilized as media in magneto-optic recording, where laser heating provides the reversal energy for recording and Kerr magneto-optic rotation provides the playback signal. Furthermore, magnetic materials of extremely high coercivity are central, for example, to high-fidelity loudspeakers. Beyond information storage and transmission, high-energy product magnets are utilized in a variety of technical applications and are being considered for high-speed elevated transportation systems and magnetic imaging. Soft materials form the essential elements of transformers as well as active microwave devices. With the rapid growth of these technologies over the past several decades, there has been accompanying research into magnetic materials’ and associated magnetization processes.’ In addition, in the field of magnetic recording considerable research, theoretical as well as experimental, into the physics of magnetic recording processes has o ~ c u r r e d . ~ . ~ This review focuses on magnetic hysteresis, reversal processes, and domain patterns in hard magnetic materials utilized as thin-film recording media. The essential physics problem is understanding the effects of long-range mag netostatic and short-range exchange fields on assemblies of anisotropic grains. These interactions lead to extremely complicated magnetization processes and pattern fluctuations. Large-scale numerical simulation has proved necessary for in-depth understanding. The results should give insight into magnetic processes in hard materials in general. In the area of theoretical studies, considerable analytic results have been obtained, although simplifying assumptions must always be made. In numerical simulations of magnetization patterns and reversal processes in hard as well as soft magnetic materials, the ability to model accurately has been limited by current computational power. The inherent difficulty in accurate modeling of magnetic systems is the incorporation of the long-range magnetostatic fields. Any reasonable numerical spatial or temporal discretization requires significant computer storage and speed. In the past several years, with the availability of supercomputers, there have been considerable

’E. P. Wolhfarth, “Ferromagnetic, Materials,” Northern Holland, Amsterdam, 1980; A. E. Berkowitz and E. Kneller, “Magnetism and Metallurgy,” Academic Press, New York, 1969. 2G. T. Rado and H. Suhl, eds. “Magnetism,” Academic Press, New York, 1963; J. Smit and H. P. J. Wijn, “Ferrites,” Wiley, New York, 1959. C. D. Mee and E. Daniel, “Magnetic Recording Handbook,” (McGraw-Hill,New York, 1989. 4R. M. White, “Introduction to Magnetic Recording,” IEEE Press, New York, 1985.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

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advances in the micromagnetic characterization of magnetization processes in magnetic rnaterial~.~-'' Rapid increases in recording densities and information transfer rates have been key elements in the steady progress in advanced computers on all scales, from PCs to supercomputers. Recording densities in commercial computer products have increased continuously over the last two decades at a rate at which the density doubles approximately every 3 years.13 For example, recording densities in announced products varied from 30 Mbit/in.' in 1985 to 130 Mbit/in.' in 1991. Recently, 1-2 Gbit/in.' recording has been achieved These densities exceed optic as well as in laboratory demonstrations. 14,' magneto-optic recording,' which is limited by the wavelength of light. Following this trend, products with this density (1-2 Gbit total storage per single disk) should be introduced about the turn of the century. Products involving magnetic storage at 10 Gbit/in.' are currently under research and products are expected by the end of the first decade of the twenty-first century. A key technological development that has contributed to these advances has been the development of thin-film recording media, accompanied by an understanding of the underlying physics of thin-film magnetization processes. Thin-film recording media are universally cobalt-based films, possessing flux densities as high as 1 tesla (47cM, 10,000 G). High-performance recording films are polycrystalline with single crystal grains that are continuous through the film thickness and have in-plane grain diameters approximately equal to the film thickness (Fig. 1). These grains are strongly coupled by magnetostatic interactions and possible intergranular exchange coupling. Systems of individual (ideally with no exchange coupling) small grains yield high coercivities essential for optima1 signal levels and small spatial fluctuations necessary for minimal system noise. Thus, in essence, the magnetic structure of high-performance films resembles that of magnetic tape: a collection of separate magnetic particles. With sputtered or plated films recording properties can be optimized by depositing thin films with densely N

' G . F. Hughes, J. Appl. Phys. 54, 5306 (1983). J-G. Zhu and H. N. Bertram, J. Appl. Phys. 69,6084 (1991). 'E. Della Torre, ZEEE Trans. Magn. MAG-22, 484 (1986). 'M. E. Schabes and H. N. Bertram, J. Appl. Phys. 64, 1349 (1988). 'D. R Fredkin and T. R. Kohler, J. Appl. Phys. 67, 5544 (1990). 'OR. H. Victora, J. Appl. Phys. 62,4220 (1987); 63, 3423 (1988). "B. Middleton, ZEEE Trans. Magn. MAG-26, 2137 (1990). '*M. Mansuripur, R. C. Giles, and G. Patterson, J. Magn. SOC.Jpn. 15, 17 (1991). I3R. Wood, Spectrum 27, 32 (1990). I4T. D. Howell et al., ZEEE Trans. Magn. MAG-26, 2298 (1990). I5M. Futamota er al., ZEEE Trans. Magn. MAG-27, 3553 (1991).

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H. NEAL BERTRAM AND JIAN-GANG ZHU

FIG.1. Transmission electron microscope (TEM) image of a typical polycrystallinethin-film recording medium. This top view shows a system of well-defined single crystallite grains that are continuous through the film thickness. Taken from K. E. Johnson et al., J . Appl. Phys. 67,4686 (1990).

packed grains of extremely high magnetization. Demonstrations of 1 Gbitlin.' recording utilized media with grains -200 %, in size,16 so that approximately 1500 grains constituted a bit cell. In 10 Gbit/in.' recording, if stable homogenous media can be produced with grains of 100 A, only 600 grains will constitute a bit cell. If the grains in thin-film recording media were completely noninteracting, the signal-to-noise ratio (SNR) of the system would vary as the number of grains in a bit cell. (10 Gbit/in.' recording utilizing 600 grains in a bit cell yields an SNR 28 dB). The goal of media design is thus to make the grains as small as possible and the transitions as sharp as possible. The major complication for interacting particle assemblies is that the signal-to-noise ratio depends on grain cluster size.17 These cluster sizes are larger than the grain size and depend in a complicated way on the play-off of the crystalline microstructure and the types of magnetic interaction. Simply reducing the individual grain size may not proportionally reduce the medium noise.

-

N

16T.Yogi et al., IEEE Trans. Magn. MAG-26,2271 (1990). "H. N. Bertram and R. Arias, J. Appl. Phys., 71(7), 3439-3454 (1992).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

275

Recorded Transition Disc Medium

FIG.2. Diagram of recording geometry. B is the size of the bit cell or, alternatively,distance between recorded transitions; W is the recorded track width; S is the magnetic thickness of the recorded layer; d is the head-medium spacing; g is the gap length of the recording head. The medium moves with velocity u with respect to the head. The coordinate system utilized in the simulations is x along the recording direction, y normal to the film plane, and z along the crosstrack direction.

In the magnetic recording process a medium (tape or disk) moves at a fixed speed past a transducer that writes information over a track width W by applying temporally varying fields to the magnetic medium (Fig. 2). This relative motion of the head and medium translates temporal variations to spatial variations so that a changing pattern of magnetization is written along the track.18 In the digital magnetic recording process the applied field magnitude is fixed at a value large enough to saturate the medium and changes sign temporally according to the binary signal to be recorded. These temporal changes are restricted to lie in the center of “bit” cells of fixed time size. Digital input to the channel (analog information to the magnetic recording channel is first digitized) is generally encoded by a variety of schemes.19 The essence of digital magnetic recording is that, in a cell, a change in head field from one polarity to the other results in a spatial change of magnetization from one polarity to the opposite. This is termed the writing of a transition of magnetization and occurs only when the encoded information cell has a “1” of binary information. The physical “bit” cell area as recorded in the medium is W x B, where B denotes the minimum transition spacing or cell length in the medium. The time scale of magnetization reversal is on the order of low9sec and temporal cell intervals are, currently, on the order of sec (30 MBit/sec for computer and 100 Mbit/sec for tape storage). In this review all temporal field changes are considered instantaneous with respect to the motion of the medium; however, the effect of 18H. N. 19J.

Bertram, Proc. IEEE 74, 1494 (1986).

K.Wolf, Proc. 1990 IEEE Int. Conf: Comp. Design ICCD’90, 210 (1990).

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H. NEAL BERTRAM AND JIAN-GANG ZHU

dynamic fields on the recording process should be considered in the near future. Understanding signals and noise corresponding to written magetization transitions in thin-film media is critical to predicting error mechanisms in signal processing. Thin films currently in use are “longitudinal” films characterized by a net in-plane anisotropy. Thus, individual grain magnetizations lie predominantly in the film plane with the average film magnetization along the head-medium motion direction. Perpendicular recording, where the net grain anisotropy and magnetization lie perpendicular to the film plane, has been proposed for extremely high density recording.” To date, perpendicular recording media (with associated head and signal processing schemes) have not been introduced in a product. Nevertheless, in this review magnetic reversal and fluctuation mechanisms for both types of media are discussed.

2. PHYSICS OVERVIEW Magnetization processes and hysteresis are generally characterized by a magnetization curve or M - H loop. In Fig. 3 a hysteresis curve for typical longitudinal thin-film recording media is shown. The magnetization component along the applied field direction is plotted versus applied field, yielding a

C

0.8 -

Q,

I -3000

I

-2000

-1000

0

1000

I

I

2000

3000

Magnetic Field (Oe) FIG.3. Typical longitudinal planar isotropic thin-film hysteresis loop. Initial curve is labeled I . The field is applied in the film plane. Courtesy of Hewlett-Packard. ’OS. Iwasaki and Y. Nakamura, IEEE Trans. Mugn. MAG13, 1272 (1977).

MAGNETIZATION PROCESSES I N THIN-FILM RECORDING MEDIA

277

hysteretic, history-dependent family of curves. The curve labeled I is the initial magnetization curve starting from an “ac” demagnetized state. The fundamental parameters are the saturation magnetization M,, the zero field saturation remanence magnetization M,, and the coercivity H,,which is the reverse field to bring the magnetization to zero from initial saturation. In recording applications the loop slope at the coercive state (coercive squareness) is characterized by S* [dM/dH = M,/((l - S*)H,)] with the remanence squareness defined by S = M,/M,. The loop curvature where reversal begins and ends is also important.16,18 Medium hysteresis properties are important in determining the shape of a recorded transition. In the writing process the change of head current in a bit cell produces a magnetization that is not spatially sharp but varies over a distance (characterized by the transition length a) from one polarity to the opposite (Fig. 4a). Approximately, this finite length is determined by the spatial variation of the head field (head field gradient) and the magnetostatic fields generated by the transition itself (demagnetization fields). Magnetostatic fields may be derived from a variety of expressions; however, a useful form is one that gives the “poles” as the field source:

-

d2r’A M(r’)(r - r‘)/lr

,.

-

J

-

r’I3

d3r’v * M(r’)(r - r’)/lr - rr13.

V

;yxA Magnetization Transition

Output Voltage

11

; I ; :

42 a a

t

~

+ I

b

FIG.4. (a) Track width-averaged magnetization transition. a is the transition width. (b) Corresponding replay voltage. Narrow transitions yield large peak voltages and narrow pulse shapes.

278

H. NEAL BERTRAM AND JIAN-GANG ZHU

+

This vector expression (in cgs units where B = H 47cM) represents an integration over surfaces S and volumes Vfor any distribution of vector magnetization M(r’).’ For thin films, assuming the magnetization lies in the film plane in the direction of recording, the transition width is given approximately by 4Mr6(d

+ 6/2) ,

QH,

-

where 6 is the film thickness, d is the head-to-medium spacing (Fig. 2), and Q 0.75 is related to the head field gradient.22 A demagnetization-limited form of (1.2) independent of recording geometry can also be derived.” Equation (1.2) shows that to write a transition with the shortest possible length, the coercivity H , should be increased relative to the flux content M,6, and the head-to-medium spacing d should be decreased, ideally to contact. In Fig. 4b a sketch of the replay voltage from a magnetization transition is given. The shape of the transition for nominal longtudinal recording follows the spatial, x, derivative of the magnetization, but broadened in shape replay head field:

This expression derives from the principle of reciprocity’* and shows that the playback voltage can be expressed as an integral over the recording medium of a correlation of the normalized replay head field vector product with the vector-recorded magnetization. In (1.3) the voltage has a signal component that is represented by the average magnetization and a medium noise voltage that arises from fluctuations in the magnetization. The magnetization pattern shown in Fig. 4a is an average for each position x over the recorded track width W of a complicated vector magnetization pattern. The recorded bit cell is composed of many interacting grains whose magnetizations are set by a reversal process involving simultaneous rotation and reversal of all the particles. The magnetization transition varies across the track in a random manner, yielding fluctuations in the read-back voltage (Fig. 5).23,24 Even if (1.2) is a good representation of the length of the average W. F. Brown, Jr., “Magnetostatic Principles in Ferromagnetism,” North-Holland, Amsterdam, 1962. **M. L. Williams and R. L. Comstock, AIP Conf. Proc. No. 5, 137 (1973). 23T.C. Arnoldussen and H. C. Tong. IEEE Trans. Magn. MAG-22,889 (1986). 24M. R. Kahn et al., IEEE Trans. Magn. MAG-26,2715 (1990). 21

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279

FIG.5. SEMPA photo of recorded transitions in thin-film recording media. The gray scale represents the magnetization along the recording track direction. Transition boundaries across the track vary in a noisy “zigzag” manner. Taken from M. R. Kahn et al., IEEE Trans. Magn. MAG-26, 2715 (1990), 01990 IEEE.

magnetization transition, the detailed dependence on microstructure determines the noise. In addition, the writing of multiple transitions involves complicated processes for the average magnetization as well as the fluctuations. In addition to the magnetization processes that occur due to the application of spatially nonuniform fields, fundamental physics questions can be studied by considering hysteresis and fluctuations due to spatially uniform fields. A useful approach to the solution of these spin configurations is via the formalism of micro magnetic^.^^ In micromagnetic theory a macroscopic view is adopted where the atomic spin configuration has been averaged to a continuous magnetization M(r). The magnitude of M(r) is taken to be the spontaneous magnetization at the temperature of interest; only its direction coordinates at each r need to be determined. The total system energy is written as a sum of local macrostopic anisotropy energies (perhaps including magnetostriction), continuum (usually ferromagnetic) exchange energy, Zeeman energy, and magnetostatic energy. The first three energies are linear in z5W. F. Brown, Jr., “Micromagnetics,” Interscience, New York, 1963.

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the direction coordinates, whereas the magnetostatic energy [derivable from (1.1)] is quadratic in the magnetizations. The energy density is local for all terms except the magnetostatic energy, where an integration over the sample volume of this long-range interaction is involved. Usually ferromagnetic exchange in continuum formz6 is assumed, whose energy is reduced if spin angles vary slowly between neighbouring grains. Given the system energy, there are two common approaches for determining the spin configuration. One method utilizes a variational principle to cast the energy into differential equations (Brown’s equations”) with numerical solution by finite-element techniques. This technique has been utilized in studies of magnetic particles of somewhat large size that fall into a middle range between hard and soft.’ In this differential approach the magnetic volume is discretized as well as the surrounding nonmagnetic volume. Combined integral-differential techniques have been developed that allow for the use of finite-element techniques utilizing discretization only within the magnetic volume.27 An alternative approach is to solve the Landau-Lifshitz dynamic equations of motion with suitable damping.” This dynamic equation is a macroscopic extension of the spin Hamiltonian and may be written in a form useful for numerical integration as

I

dM(r) = - IyIM(r) x Hef,(r)- - M(r) x (M(r) x Heff(r)). dt M The first term represents gyromagnetic electron spin motion, whereas the second term is a phenomenological damping term. The form of this damping term ensures that the magnitude M = IM(r)I is conserved. An alternative damping expression due to Gilbert can be written,’’ but the two forms are mathematically equivalent and (1.4) as an explicit differential equation is convenient for numerical analysis. The effective vector magnetic field Heffis written as a generalized derivative of the system energy with respect to the magnetizations:

The solution of (1.4) involves discretization only interior to the sample volume so that 2N simulations equations must be solved ( N is the number of discretization elements). 26C.Kittel, Rev. Mod. Phys. 21, 541 (1949). *’D. R. Fredkin and T. R. Kohler, ZEEE Trans. Mugn. 26,415 (1990). *‘C. W. Haas and H. B. Callen, in “Magnetism” (G. T. Rado and H. Suhl, eds.), Vol. 1, Chap. 10, Academic Press, New York, 1963.

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28 1

In the study of polycrystalline films reviewed here the dynamic phenomenological equation formulation (1.4, 1.5) is utilized. To date, each grain has been assumed to be single domain where the magnetization vector does not vary over the grain volume. This assumption appears reasonable, because for most films the grain diameter is small compared to the exchange or coherence length. In addition, grain sizes are currently large compared to superparamagnetic volumes, so that thermal flucations may be n e g l e ~ t e d . ~ ~Thus, .~’ the energy of the polycrystalline system involves, in addition to applied fields, the local anisotropy of each grain, possible nearest-neighbor exchange interactions at the grain boundaries, and the complete magnetostatic energy of all pair couplings of these finite-size dipoles. 3. OUTLINE

This review is concerned entirely with magnetization processes in planar thin polycrystalline films. In Section I1 a review of materials and microstructure focusing on the use of alloying and underlayers to obtain desired magnetic properties is given. Emphasis is placed on the effect of composition and deposition conditions in producing true polycrystalline films with isolated grains. In Section I11 mathematical modeling is discussed. In the first part of Section I11 a review of simple analytic micromagnetic results is given in order to introduce the fundamentals of collective behavior as well as the concept of strong and weak interactions. In SectionIV the results of simulations of reversal and domain formation of the two common types of films, longitudinal and perpendicular, are presented. In this section the effect of interactions on the M-H loop ( H c ,M,, M,, S*) is discussed, corresponding to experimental results presented in Section 11. In Section V the results of film magnetization processes due to the application of spatially nonuniform fields of the type occurring in the recording process are presented. The writing of single and multiple transitions is discussed, as well as magnetization fluctuations or medium noise. In Section VI a discussion of self-organized behavior in magnetic thin films and related spatial fluctuations is given. II. Microstructures and Magnetic Properties of Thin-Film Materials

Virtually all magnetic films utilized as recording media are polycrystalline (except amorphous CoSm films) with coercivities varying from a few hundred 29E.Koster and T. C. Amoldussen, in Ref. 3, pp. 101-257. 301. S. Jacob and C. P. Bean, in “Magnetism” (G. T. Rado and H. Suhl, eds.), Vol. 3, pp. 271-350, Academic Press, New York, 1963.

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to several thousand o e r ~ t e d1,29 . ~ With proper deposition conditions and underlayer structure, coercivities above 3000 Oe have been achieved for CoCrPtB and CoSm film^.^^,^^ All films considered for recording applications to date are Co-based alloys with saturation magnetizations in the range 300 to 800 emu/cm3. Because film crystallites are closely packed with almost unity volume packing fraction, the saturation magnetization of the film is close to that of the individual grains (although high deposition pressure can yield grain packing densities of about 60-70%). The grains are individual crystallites extending through the film thickness with in-plane diameters and intergranular microstructure and magnetic structure depending on the deposition process as well as film thickness. Magnetic films are not directly deposited on disk substrates. Currently, NiP is plated on an aluminum substrate to provide a smooth deposition-surface with enhanced hardness. The NiP surface is textured to avoid head striction on the final disk surface. An enhancement underlayer is usually deposited prior to the magnetic film. Underlayers are designed either to improve initial growth of the magnetic film layer, to create interfacial polycrystalline epitaxy controlling crystalline orientation of magnetic grains, or to provide a proper morphology for desired microstructure of the magnetic film layer. In a practical recording disk, a thin layer of carbon is often deposited on top of the magnetic film layer for tribiological reasons. In Fig. 6 the structure of a typical thin-film recording disk is shown. In the next two sections, experimental studies of microstructural and material features, magnetic hysteresis, and related micromagnetic phenomena are discussed.

4. LONGITUDINAL FILMS In longitudinal thin-film recording media, magnetic film thicknesses are usually below 600 A and the average film magnetization lies predominantly in the film plane. a. Planar Isotropic Films In the film plane the macroscopic magnetic hysteresis properties of these films are isotropic. Material compositions usually are Co-based binary alloys, such as COP, Copt, and CoRe, developed in the last several decades, or ternary alloys, such as CoNiPt, CoCrPt, and CoCrTa, developed more recently. The crystalline structure is mainly hexagonal close packed (hcp) with the uniaxial crystalline anisotropy easy axis along the c axis. "T. C. Amoldussen, Proc. IEEE 74, 1526 (1986). 32N.Tani et al., IEEE Trans. Mugn. MAG-27, 4136 (1991). 33E.M. T. Velu and D. N. Lambeth, J. Appl. Phys. 69, 5175 (1991).

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10-30 nm

20-60 nm

50-150 nm

15 pm

2 SUBSTRATE ( A l , SiO,)

.75 mm

FIG.6. Schematic cross section of a thin-film recording disk.

(1) c-Axis Orientation Distribution For most thin films the orientation of crystalline c-axes is randomly distributed in all possible directions. A threedimensional (3D) random distribution yields isotropic magnetic properties only in the film plane. Crystallite c-axes can also be oriented at a specific angle with respect to the film plane due to epitaxial growth on an enhancement ~nderlayer.~"~'For example, the c-axes of the grains in CoCrPt films deposited on a thin Cr underlayer tend to lie in the film plane because of lattice matching epitaxy of the CoCrPt (11.0) plane with the Cr (100) plane.34338-40In CoCrTa films deposited on a thick Cr underlayer, for example, the matching of the Cr (100) plane with CoCrTa (10.1) leads to caxis orientation at an angle of 28" to the film plane. Epitaxial growth strongly depends on deposition conditions as well as underlayer thickness and underlayer surface morphology. For instance, it has been suggested that under certain deposition conditions, c-axis in-plane epitaxy in CoCrPt films on a Cr underlayer becomes maximized with Cr thickness around 150 A. Continual increase of the Cr underlayer thickness results in randomization of 34J. Daval and D. Randet, IEEE Trans. Magn. MAG-6, 768 (1970). 35T.Ohno et al., IEEE Trans. Mugn. MAG23, 2809 (1987). 36J. A. Christner et al., J. Appl. Phys. 63, 3260 (1988). 37T.Yogi et al., IEEE Trans. Magn. MAG24,2727 (1988). "K. E. Johnson et al., J. Appl. Phys. 67,4686 (1990). "T. Ohno et al., IEEE Trans. Magn. MAG-23, 2809 (1987). 40T. Yogi et al., IEEE Trans. Magn. MAG-24,2727 (1988).

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the c-axis orientation into three dimension^.^' Other researchers report that for CoCrTa on a thick Cr underlayer, the (10.1) texture is p r ~ m o t e d . ~ ~ ’ ~ ~ In most sputtered films, local correlation of c-axis orientation has not been observed. However, in electroless-plated COPfilms, local coherence of c-axis orientation has been seen?1 The c-axes of grains within a submicrometer region ( - 10-20 grains) are oriented along a common direction even though distinct nonmagnetic grain boundaries occur. Films with c-axis orientation clustering exhibit large spatial magnetization fluctuations and high recording noise.

(2) Morphological Separation of Magnetic Grain Boundaries An important role of the underlayer is to create morphological separations, or voids, between adjacent grain boundaries in the magnetic layer. Figure 7 illustrates the surface morphology resulting from columnar growth of a sputtered Cr

FIG.7. SEM image showing the topography and columnar structure of a sputtered Cr film. Taken from S. Agarwal, ZEEE Trans. Magn. MAG-21, 1527 (1986), 01986 IEEE.

41G.C. Rauch, Proceedings of Micromagnetic and Microstructural Characterization of Media and Heads for Magnetic Recording, San Diego, 1991.

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FIG.8. Cross-section TEM micrograph of a CoNiCr/Cr film with 1000-A-thick Cr underlayer. Morphological separation of the CoNiCr grains is evident. Taken from T. Yogi et al., IEEE Trans. Magn. MAG24, 2727 (1988), 01988 IEEE.

underlayer, exhibiting surface roughness of the order of tens of nanometer^.^' Co alloy grains of the magnetic layer nucleate on these “domes,” as shown in Fig. 8 for a CoCrPt/Cr film. For certain deposition conditions and Co alloys, magnetic films deposited on a Cr underlayer can have voided grain boundaries as shown in Fig. 9. Cr domes sometimes are elongated in the film plane and result in elongated CoCrPt grain chains, although the grains in each chain are also separated by voided boundaries. The surface diameter of the Cr underlayer columns determines the grain diameter of the magnetic layer. Increasing Cr underlayer thickness results in more pronounced dome tops as well as larger surface sizes of the Cr columns. Magnetic properties are changed by voided grain boundaries. In Fig. 10 two transmission electron micrographs of CoCrPt films grown on Cr underlayers of two different thicknesses are shown. For the film with a thin Cr underlayer (Fig. lOa), the grains in the magnetic film layer are closely packed and no voided grain boundaries can be observed. For films with a thicker Cr underlayer (Fig. lob), the magnetic grains are separated by pronounced voided grain boundaries and magnetic grain size is comparatively larger. The corresponding measured hysteresis loops are plotted in 42S. Agarwal, IEEE Trans. Magn. MAG-21, 1527 (1981).

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FIG.9. Plane-view TEM micrograph of a CoCrPt/Cr film with a thick Cr underlayer and higl deposition pressure. The film exhibits 1560-0e coercivity and low recording noise. 1 Gbit/in. recording was achieved with utilizing this medium. Taken from T. Yogi et al., IEEE Tranr Magn.26,2275 (1990), 01990 IEEE.

Fig. 11. In general, films with voided grain boundaries result in highei coercivity H,,lower saturation squareness S , and lower coercive squarenes! S*. These films also exhibit low recording noise.43 One direct result o creating voided grain boundaries is the disruption or weakening of ferromag netic exchange coupling between adjacent grains. Voided grain boundarie! can also result from changing deposition conditions, such as Ar pressure Similar changes in hysteresis properties have been found in comparison wit1 the effect of Cr ~ n d e r l a y e r .The ~ ~ .effect ~ ~ of intergranular exchange couplinl on magnetic phenomena is the main focus of this review. ( 3 ) compositional Segregation Alloying of magnetic films in addition tc morphological induced grain growth leads to compositional inhomogenei ties. The saturation magnetization of CoCr alloy films can have highe: 431. L. Sanders et al., J. Appl. Phys. 65, 1234 (1989). 44T. Chen, D. A. Rogowski and R. M. White, J. Appl. Phys. 49, 1816 (1978). 45T. Chen and T. Yamashita, IEEE Trans. M a p . MAG-24, 2700 (1988).

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FIG.10. TEM micrographs showing the effect of Cr underlayer thickness on a grain size and grain segregation on magnetic films grown on (a) 100-A-thick and (b) 1500-&thick Cr underlayers. With a thick Cr underlayer, voided grain boundaries become evident. Taken from K. Johnson et al., J. Appl. Phys. 67,4688 (1990).

-yxx)

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1000

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FIG.11. Hysteresis curves for thin and thick Cr underlayers for CoCrPt/Cr films (with micrographs shown in Fig. 10). A 1500-&thick Cr underlayer yields increased coercivity and decreased squarenesses S and S*. Taken from K. C. Johnson et al., J. Appl. Phys., 67,4688(1990).

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saturation magnetization than that in the bulk alloy for the same Cr content. For example, proper substrate heating during deposition can result in higher saturation magnetization as well as a higher anisotropy constant than that found without heating46(Fig. 12). The increase in M ,is believed to be due to segregation of Cr to the grain boundaries. Segregation of Cr has been directly observed by electron microscope4' and spin techniques in CoCr thin films with relatively high Cr content. Cr-rich grain boundaries are likely to be nonmagnetic, or weakly magnetic, which may disrupt or weaken the exchange coupling between neighboring grains. Films with Cr-rich grain boundaries exhibit identical trends in magnetic properties ( I f c , S, S*), as discussed earlier for grain isolation by the formation of morphological voids4' Similar phenomena occur in other alloy films. Grain boundaries in plated COP alloy films have been found to be phosphorus Increasing the phosphorous content from 0 to 6% in CoNiP films results in a large increase in film c~ercivity.~' CoCrTa is another interesting example. It has been found that increasing Ta content from 0 to 2% in CoCrTa films yields an increase in the saturation magnetization, suggesting that Ta may enhance Cr segregation to the grain b o u n d a r i e ~ .Lattice ~~ expansion has also been observed as Ta content increases.53The change of Ta content significantly increases the film coercivity and reduces coercive squareness and recorded transition noise.54 However, it is not currently clear whether changes in hysteresis properties are dominated by grain decoupling or anisotropy enhancement. Copt, CoNiPt, CoCrPt, and most recently CoCrPtB films all exhibit very high coercivity for Pt content in the range of 10 to 25%. It has been reported that in Copt films the high coercivity is correlated with the formation of a face-centred cubic (fcc)-hcp phase mixture, although fcc-hcp phase mixture has not been investigated in detail in more recent CoCrPt films. Correlation of the coercivity increase with an increase of lattice constant has been observed in CoCrPt films.ss These correlations suggest that the high coercivity achieved in these films is likely due to enhancement of the crystalline anisotropy in the grains. High coercivities, nearly 3000 Oe, have also been reported in Co,Sm films. It is suggested that such high coercivities may be 46H. Hoffmann, IEEE Trans. Magn. MAG-22, 472 (1986). 47J. N. Chapman, I. R. McFadyen,and J. P. C. Bernards,J. Mugn. Magn. Muter. 62,395 (1986). 48Y. Maeda and K. Takei, IEEE Trans. Mugn. MAG-27,3553 (1991). 49S. L. Duan et al., J. Appl. Phys. 67,4704 (1990). 50K. Hono and D. E. Laughlin, J. Mugn. Magn. Muter. SO, L137 (1989). "F. E. Luborsky, IEEE Trans. Mugn. MAG-6, 502 (1970). 52T. Reith, private communication and unpublished results. 53K.Johnson, private communication and unpublished results. 54J. A. Christner ef ul., J. Appl. Phys. 63, 3260 (1988). "K. Johnson, private communication and unpublished results.

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n

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I

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I

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16

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FIG.12. Dependence of (a) the crystalline anisotropy K and (b) the saturation magnetization M , on the Cr composition of CoCr perpendicular films for different substratetemperatures: 70°C and 200°C. Taken from H. Hoffmann, I E E E Trans. Magn. MAG22,472 (1986), 0 1 9 8 6 IEEE.

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due to a large crystalline anisotropy energy constant of the very fine CoSm crystallites in the film.56 (4) Effect of Film Thickness and Grain Diameter on Magnetic Properties

Magnetic film thickness and associated grain size have been found to have a strong effect on loop shape, coercivity, and squareness as well as on medium noise. Coercivity versus film thickness always exhibits a maximum, near 100-200 A for sputtered films, but dependent on material composition and film m i c r o s t r ~ c t u r e Plated . ~ ~ films have exhibited maxima at larger thicknesses, near 700 A.58At small thicknesses, with corresponding small grain volumes, superparamagnetic behavior is likely d ~ m i n a n t . 'For ~ large thicknesses and corresponding large volumes it is possible that the reduction in coercivity is due to multidomain behavior of the individual grains. However, for currently used planar isotropic films, grain sizes even at relatively large film thicknesses ( 500 A x 500 A x 600 A) are still much smaller than the critical dimension for multidomain grains ( N 1000-1400 A for Co-based alloys).59 Thus, the decrease in coercivity for relatively large thickness is complicated and is influenced primarily by grain interactions and possible morphological changes. For example, increasing deposition time to produce thicker magnetic layers often results in an increase in grain diameter. In addition, grain anisotropy axes gradually orient toward the film normal direction for films beyond 600 A. Films with the same alloy composition and same thickness but larger grain size usually exhibit higher ~ o e r c i v i t i e sThe . ~ ~increase ~ ~ ~ of grain size is usually achieved by either increasing substrate temperature during the deposition, increasing underlayer thickness, or reducing film deposition rate.31 The latter two procedures promote either voided or segregated grain boundaries, whereas increasing substrate temperature yields more continuous media. Thus, because grain segregation is affected in complicated ways by techniques that alter the grain size, identifying the main mechanism for the coercivity increase with increasing grain size becomes difficult through experimental studies. Micromagnetic modeling yields considerable insight into grain size phenomena, as discussed in Section 8a. N

( 5 ) Magnetization Configurations and Transition Patterns In Fig. 13, a Lorentz electron microscope image of a recorded transition in a CoRe film and an illustration of the magnetization configuration are shown. Away from 56E. M. T. Velu and D. N. Lambeth, J. Appf. Phys. 69, 5175 (1991). "Y. Hsu, J. M. Sivertsen, and J. H. Judy, ZEEE Trans. Mugn. 26, 1599 (1990). "J. S. Judge et al., J. Efecirochem. SOC.112, 681 (1965). "E. Kneller and F. E. Luborsky, J. Appl. Phys. 34,656(1963). 60N. Mahvan et al., ZEEE Trans. Mugn. MAG26,2277 (1990).

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29 1

FIG. 13. (a) Lorentz electron microscopy image of recorded transitions on CoRe films ( H , = 220 Oe) and (b) correspondingschematic illustration of vortex structure (at the transition center) and ripple patterns (away from the transition). Taken from T. Chen, IEEE Trans. Magn. MAG-17, 1181 (1981), 0 1 9 8 1 IEEE.

the transition center, the feather-like structure indicates rippling of the magnetization as would occur for uniformly magnetized media. Magnetization vortices, a form of localized magnetization flux closure, form at transition centres, resulting in low magnetization energy. Ripples and vortices are characteristic features of magnetization transition patterns in planar isotropic films. Figure 14 shows a Lorentz electron microscope image of a magnetization configuration near the remanent coercive state for a typical CoCrPt/Cr film. Magnetic domains are elongated along the applied field direction. Within each domain, the magnetization is either parallel or antiparallel to the applied field and a ripple structure is evident. The boundaries separating oppositely magnetized domains are irregular and each boundary consists of a series of vortices with the same sense of rotation. (6) Multilayer Films Transition noise, arising from spatial magnetization flucturations, is proportional to the effective magnetic cluster size in the film

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FIG. 14. Lorentz electron microscopy image of magnetization patterns near the coercive state on a CoCrPt/Cr film. Reverse domains are elongated in the applied field direction. Domain boundaries contains vortices. Courtesy of K. C. Johnson and M. Mirzamaani.

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plane.I7 With magnetostatic and possible exchange interactions this cluster size is larger than the individual grain. One way to reduce these interactions is to make a multilayer thin film by sequentially depositing magnetic films separated by nonmagnetic interlayers.61 Experimental films of CoCrTa and CoCrPt have magnetic layer thicknesses in the range 80-300 A with typically 25- 100 i% thick Cr separation layer^.^^-^^ These multilayer films exhibit noise reduction significantly below that predicted by the number of grains in each elemental surface area. Thus, interlayer magnetic interactions occur that are strongly dependent on the separation layer thickness. Reducing the Cr interlayer thickness results in a significant reduction of transition noise provided the ferromagnetic exchange coupling between adjacent magnetic layers is disrupted. It has been reported that a minimal Cr thickness near 10 to 20 i% is req~ired.~’ 6M curve measurements can be utilized to assess the strength and character of the The 6M curve is the difference between twice the initial magnetization curve (I in Fig. 3) and the major M-H loop; it is a very useful technique for assessing magnetic interactions (discussed in detail in Section 14). Measurements of a double-layer CoCrTa film are shown in Fig, 15.68 The 6M curve for the film with well-separated magnetic layers (acr= 300 A) is characterized by a large positive peak. As the interlayer thickness decreases, the positive peak is significantly reduced. Accompanied by the changes in the 6 M curves, reducing the Cr interlayer thickness from 300 to 25 A results in a substantial reduction of the film coercive squareness S* as well as a reduction in recorded transition noise. b. Oriented Films In oriented films, macroscopic hysteresis properties become anisotropic in the film plane: both saturation remanence and coercivity become higher in the orientation direction and lower in the transverse direction (in the film plane). Mechanisms for this anisotropy vary with film composition and deposition conditions. Three main mechanisms will be discussed here, beginning with the most recent experimental studies. In CoCrPt/Cr and CoCrTa/Cr films, it has been found that by texturing the film substrate, the 61H.Hata et al., J. Appl. Phys. 67,4692 (1990). 62E. S. Murdoc, B. R. Natarajan, and R. G. Walmsley, ZEEE Trans. Magn. MAG-26, 2700 ( 1990). 63S. E. Lambert, J. K. Howard, and I. L. Sanders, ZEEE Trans. Magn. MAG-26,2706 (1990). 64H. Hata et al., ZEEE Trans. Magn. MAG-26, 2706 (1990). 65A. Murayama and M. Miyamura, IEEE Trans. Magn. MAG-27, 5064 (1991). 66P. I. Mayo et al., J. Appl. Phys. 69, 4733 (1991). 67J-G.Zhu and H. N . Bertram, J. Appl. Phys. 69,4709 (1991). 68T.Min, J-G. Zhu, and J. H. Judy, ZEEE Trans. Magn. MAG27, 5058 (1991).

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1

single3ook layel_,

0.25

+1L-300Av/

+Tcr=300A

30.15 I

B 0.05

0

0.5

1

1.5

2

Normalized applied field HlHcm FIG.15. Measured 6 M curves of double magnetic layer thin CoCrTalCr films with various intermagnetic layer separations. Reducing the interlayer separation results in a reduction of the peak values. Taken from T. Min et al., IEEE Trans. Magn. MAG-27,5058 (1991), 01991 IEEE.

orientation ratio, defined as the ratio of the saturation remanence in the orientation direction to that in the transverse direction, becomes greater than unity and anisotropy r e s ~ l t s . ~Experimental ~-~~ studies suggest that this anisotropy is induced by anisotropic strains in the films even though the crystalline c axes show no preferred orientation in the film plane.73 In films with large orientation ratios, the crystalline lattice constant is smaller along the texturing direction than the transverse direction. These films always exhibit a negative magnetostriction constant so that the texturing direction becomes an easy axis. The induced anisotropy is approximately linearly related to the magnitude of the strain. Grain boundary alignment in the texture direction is also believed to be a mechanism for magnetic a n i s o t r ~ p y Films . ~ ~ with higher orientation ratio exhibit higher signal-tonoise ratios at low recording densities, whereas at very high recording densities, high-orientation-ratio films have significantly higher transition noise.75 69E.Tang and N. Ballard, IEEE Trans. Magn. MAG-22, 579 (1986). 70E. M. Simpson et al., IEEE Trans. Magn.,MAG-23, 3405 (1987). 71T.Lin, R. Alani, and D. N. Lambeth, J. Magn. Magn. Muter. 78, 213 (1989). 72M.Mirzamaani et al., J. Appl. Phys. 67,4695 (1990). 73A.Kawamoto, J. Appl. Phys. 69, 5151 (1991). 74E. M. Simpson et al., IEEE Trans. Magn. MAG-23, 3405 (1987). 75M.F. Doerner et al., Muter. Res. SOC.Proc. 232,27 (1991).

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FIG.16. Typical hysteresis loops of amorphous CoSm films, measured with in-plane external fields parallel and transverse to the anisotropy axis. Minor loops with zero reversible susceptibility are included in easy axis loop. Taken from U. Kullmann et al., l E E E Trans. Magn. M A E 2 9 420, (19841, 0 1 9 8 4 IEEE.

CoSm amorphous films deposited in the presence of a magnetic field exhibit large magnetic anisotropy along the field direction.76 Figure 16 shows typical measured hysteresis loops for amorphous CoSm films along the easy and transverse directions. Both saturation squareness and coercive squareness are virtually unity. Films with 25% samarium content exhibit coercivities of about 1100 Oe. Strong local ordering of Co-Co and Co-Sm pairs has been suggested to be the mechanism for the induced ani~otropy.~’ Evaporated FeCoCr films also exhibit a high degree of orientation due to elongated grains formed by the oblique evaporation p r o ~ e s s . ~Figure * , ~ ~ 17 shows a Lorentz microscopy image of transition patterns in an FeCoCr film.23Away from transition boundaries, the magnetization is oriented perfectly in the orientation direction and no ripple structure appears, in contrast to in-plane isotropic films. Transition boundaries are characterized by irregular zig zag patterns, which result from the reduction of magnetostatic energy with minimal anisotropy energy. (The zigzag boundaries can be thought of as stretched vortices.) 76U.Kullmann, E. Koester, and C. Dorsch, IEEE Trans. Magn. MAG-20, 420 (1984). 77C.D. Graham and T. Egami, Annu. Rev. Muter. Sci. 8,423 (1978). 78E. M. Rossi et al., J. Appl. Phys. 15, 2254 (1984). 79T. C. Arnoldussen et al., IEEE Trans. Magn. MAG-20, 821 (1984).

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FIG.17. Lorentz electron microscopy image of recorded transitions on highly oriented FeCoCr films. Zigzag patterns at transitions are very irregular, partly because of irregularity in film microstructure. Recording density at (a) 300 flux reversals/mm and (b) 600 flux reversals/ mm where the transition boundaries are percolated and island domains form. Courtesy of H. C. Tong and T. C. Arnoldussen.

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5. PERPENDICULAR FILMS Perpendicular films are designed so that grain magnetizations are normal to the film plane in order to minimize demagnetization at the center of a written transition.80-82 This result is achieved by producing grains with large aspect ratios in the film normal direction with corresponding perpendicular shape anisotropy and with large perpendicular crystalline anisotropy. Perpendicular films for magnetic recording applications are universally composed of CoCr. Figure 18 shows a scanning electron microscope (SEM)

FIG.18. SEM image of fractional cross section of a sputtered CoCr perpendicular thin film. Taken from T. Wielinga et al., IEEE Trans. Magn. MAG-18, 1107 (1982), 0 1 9 8 2 IEEE. '"S. Iwasaki and K. Takemura, IEEE Trans. Magn. MAG-11, 1173 (1975). Iwasaki, K. Ouchi, and N. Honda, IEEE Trans. Magn. MAG-16, 1111 (1980). 82S.Iwasaki, IEEE Trans. Magn. MAG-20,657 (1984).

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micrograph of a fractional cross section of a sputtered CoCr perpendicular thin film.83 The thickness of these films is in the range of 0.2 to 1 pm with columnar grain diameter around 5008,. The columns are CoCr magnetic crystallites and have hcp structure. Usually, without an enhancement underlayer, the crystalline orientation of CoCr crystallites in the initial layer (about 500 8,thick) at the substrate interface is random, just as for a planar isotropic longitudinal film.84,85As the film is grown thicker, the c axis of the columns becomes oriented in the film normal direction with very small dispersions, yielding a perpendicular crystalline anisotropy.86 A thin film magnetized uniformly perpendicular will produce a demagnetizing field that forces the magnetization to lie in the film plane if the demagnetization energy is greater than the perpendicular crystalline anisotropy energy (K < 2nMf). Thus, higher Cr content, from 17 to 22%, is usually used, as compared with 12 to 16% in longitudinal films. Calculation indicates that for Cr content at about 20%, the perpendicular crystalline anisotropy becomes about equal to the film shape anisotropy. However, in films with lower Cr content where K < 2nM,2, the magnetization will still remain perpendicular to the film plane.87 This is due to the magnetization reversal modes of these films that will be discussed in detail in Section IV. Measured hysteresis loops of a typical CoCr perpendicular film in the perpendicular direction are characteristically sheared by the thin-film demagnetization field, as shown in Fig. 19. Measurements have been made of

M, emukc

-500

+

FIG.19. Typical measured hysteresis loops of CoCr perpendicular thin films. Taken from E. Wuori and J. Judy, IEEE Trans. Magn. MAG20, 774 (1984), 01984 IEEE.

83T.Wielinga, J. C. Lodder, and J. Worst, IEEE Trans. Magn. MAG-18, 1107 (1982). 84M.Ohkushi and T. Kusuda, Jpn. J. Appl. Phys. 2, 2130 (1983). "P. J. Grundy, M. Ali, and C. A. Gaunce, IEEE Trans. Magn. MAG-20,794 (1984). 86Y. Hsu, J. M. Sivertsen, and J. H. Judy, IEEE Trans. Magn. MAG-26, 1599 (1990). "H. Hoffmann, IEEE Trans. Magn. MAG22,472 (1986).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

299

magnetic hysteresis on submicrometer-size CoCr perpendicular film samples, utilizing the anomalous Hall effect technique.88 The hysteresis loops consist of discrete fine jumps where each reversal step corresponds to the simultaneous magnetization reversal of one or a few columnar grains. Lorentz electron microscopy observations of the magnetization patterns at the saturation remanent state show that the reversed domains in the films with Cr-rich columnar boundaries are isolated single columns or single-columnwide narrow chain^.^^-^^ When the Cr segregation is absent, magnetic domains in the films are characterized by wide stripes, and magnetization reversal processes in these films exhibit dynamic domain expansion following n~cleation.~'Avalanche domain expansion results in a shoulder in the hysteresis loopg2 Thermally activated magnetization time decay in CoCr perpendicular films is character is ti^.^^-^^ Although the physical mechanism of this slow relaxation process is not yet entirely clear, perpendicular magnetized columns are only marginally stable due to the perpendicular demagnetizing field. The characteristics of the demagnetizing field as well as the micromagnetic reversal modes in the films could be responsible for the complexity of the magnetization decay, as it has been found that the decay measured over seven decades deviates significantly from a pure log t dependen~e.'~

111. Micromagnetic Modeling

6. REVIEWOF SINGLE- AND MULTIPLE-PARTICLE REVERSAL MECHANISMS a. Stoner- Wohlfarth Model: Uniform Rotation The classic calculation of magnetic reversal is that of uniform rotation against a uniaxial a n i ~ o t r o p y The . ~ ~ material is assumed to be uniformly magnetized (single domain with magnitude M) and to possess uniaxial anisotropy due either to intrinsic crystalline atomic ordering or to sample '*B. C. Webb and S. Schultz, ZEEE Trans. Magn. MAG-24, 3007 (1988). 89M.Ohkushi et al., J. Magn. Magn. Muter. 35, 266 (1983). 90K. Ouchi and S.-I. Iwasaki, ZEEE Trans. Magn. MAG-18, 1110 (1982). 9'F. Schmidt and A. Hubert, J. Magn. Magn. Muter. 61, 307 (1986). 92T. Wielenga, Ph.D. Thesis, Twente University of Technology, Enschede, The Netherlands, 1983. 93R. M. Kloepper, B. Finkelstein, and F. 0. Braunstein, IEEE Trans. Magn. MAG-20, 757 (1984). 94S. B. Oseroff et al., ZEEE Trans. Magn. MAG-21, 1495 (1985). 95D.K. Lottis et al., J. Appl. Phys. 63, 2920 (1988). 96B.C. Webb, and S. Schultz, J. Appl. Phys. 63, 2923 (1988). 97E.Stoner and E. P. Wohlfarth, Philos. Trans. R. SOC.Lond. Ser. A 240, 74 (1948).

300

H. NEAL BERTRAM AND JIAN-GANG ZHU

M

FIG. 20. Diagram of single-domainparticle with uniaxial anisotrophy K and applied field H . 0 is magnetization angle with the applied field and 0, is the applied field angle with the anisotropy easy axis.

shape anisotropy. For uniformly magnetized particles with a general ellipsoidal (prolate or oblate) shape, (1.1) can be utilized to yield a uniaxial anisotropy whose magnitude is proportional to M 2 and that varies as the square of the direction cosine to the long axis direction.'* If a magnetic field H is applied at angle 8, to the anisotropy axis, the system energy density is given (Fig. 20) by E

=

- H M cos 6'

+ K sin'(8,

-

e),

(6.1)

where K represents the magnitude of the anisotropy and 8 is the angle of the magnetization with respect to the applied field. In this simple one-variable problem the magnetization lies in the plane formed by the applied field and the anisotropy axis. In Fig. 21 the variation of energy is plotted versus magnetization angle 8 for various magnitudes of the applied field H (8, = 20"). 6' = 0" and 180" represents magnetization along the field direction, whereas 6' = 20" and 200" represents magnetization along the anisotropy axis. For large positive fields ( h = l.OH/M) the energy variation exhibits only one minimum or equilibrium state at a magnetization angle between the applied field and anisotropy direction. With the application of a large field the magnetization will follow a dynamic evolution given by (1.4) to an equilibrium state given by the single minimum according to (6.1). The equilibrium condition for any system, single or multivariable, is that the magnetization be parallel to the effective field Heff(1.5) or, equivalently, that the net torque M(r) x Heff(r)vanishes. If the field magnitude is reduced, the magnetization follows (1.4) to a new equilibrium, given, for example, in Fig. 21 by the energy curve for h = 0. At "B. D. Cullity, "Introduction to Magnetic Materials,"p. 240, Addison-Wesley, Reading, MA, 1972.

MAGNETIZATIONPROCESSES IN THIN-FILM RECORDING MEDIA

301

h=1.0

h = l .O h=- 1 .O h=- 1.5

-1.5

-100

\/

I

I

I

I

1

I

I

I

0 100 200 300 Magnetization angle 0 (degree)

FIG.21. Energy versus magnetization angle 8 for various normalized applied fields, h = HIH,

’

vanishing field two equilibria exist; however, the magnetization will stay in the equilibrium state that is closest to the original state. It will not cross the energy barrier to the alternative minimum. For this single-particle system neither the rate of decrease of the field nor the damping ratio ;l/y affects final equilibrium. In general systems, as in the M-H loop example in Fig. 3, the field is reduced slowly compared to dynamic processes so the magnetization is always in “quasi-equilibrium” with the field. In the reduction of the field from positive saturation as discussed in this example, each incremental decrease in field yields a new incremental equilibrium, which can be determined by the vanishing of the torque without dynamics. This is a reversible process in that incremental changes of the field in the positive direction will cause the system to trace out the same states. Even for the single particle, as the field is decreased, the component of magnetization along the applied field direction decreases corresponding to the M - H loop example in Fig. 3. Further reduction of the field continues the sequence of quasi-equilibrium states or reversible magnetization rotation until a field, in this example approximately equal to H = - 1.0 K / M , occurs. At this field the energy barrier that maintains the magnetization in the original state nearest the applied field has almost completely vanished. Subsequent field increase in the negative direction will yield a system with only one equilibrium, and the dynamic equations will take the magnetization irreversibly to this new equilibrium. After irreversible switching a slight change in the field toward

H. NEAL BERTRAM AND JIAN-GANG ZHU

302

the original direction will not return the magnetization to the state that existed prior to this dynamic process; the magnetization will only move reversibly (along a “minor loop”) to a state very close to the new equilibrium. The field at which a saddle point in the energy occurs (a2E/a02= 0) for this simple example is called the nucleation jield or switching j e l d . Further increases in the applied field in the negative direction yield reversible changes in the magnetization toward the field direction. In Fig. 22, M - H loops are plotted for several angles of the applied field with respect to the anisotropy axis.99For an applied field coincident with the anisotropy axis (6, = 0, easy direction), (6.1) yields a magnetization along the field direction until the field has been reduced to H = - 2 K / M . At this

1.o

a = 0”

I

+m

7

7

0.5

m

+h

O

-0.5

-1.0

h FIG.22. Hysteresis loops of magnetization component along the applied field for various applied field angles with the easy axis. a in the figure corresponds to 0 in the text. Taken from B. D. Cullity, “Introduction to Magnetic Materials.” Addison-Wesley, Reading, MA, 1972, p. 336.

99B.D. Cullity, “Introduction to Magnetic Materials,” p. 240, Addison-Wesley, Reading, MA, 1972.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

303

nucleation field the magnetization undergoes irreversible rotation to the opposite direction and a “square loop” is obtained. For small field angles the loop follows the behavior discussed for 8, = 20”. As the field is reduced from an initial saturating state, the magnetization undergoes reversible rotation away from the field direction. It is characteristic, although not entirely general, that reversible rotation corresponds to M-H loop derivatives (incremental susceptibilities) that are not large. During this process the magnetization rotates away from the field direction, becomes coincident with the anisotropy direction at H = 0, and then rotates farther away from the field direction toward the reverse direction. At a reverse field equal to the nucleation field, the M-H loop becomes steep corresponding to large irreversible angular changes in the magnetization direction. After nucleation the magnetization is in the single state between the applied field and anisotropy direction. Even though this example is for the simplest case of uniform rotation, the character of the loop resembles that shown in Fig. 3. In Fig. 22, for small field angles, the nucleation field corresponds to the coercivity. For large field angles (0, = 70” in Fig. 22), as the field magnitude is reduced from initial saturation and reversed, the magnetization rotates reversibly away from the field direction to a direction orthogonal to the applied field and beyond before nucleation occurs. In this case the coercivity is determined by reversible rotation. For ensembles of noninteracting randomly oriented particles undergoing uniform rotation, and also for complex systems such as polycrystalline thin films, the coercivity results from a combination of reversible and irreversible effects. Note that for applied fields in the “hard” direction orthogonal to the anisotropy axis, 8, = 90”, nucleation occurs at a positive field, H = 2 K / M . However, hysteresis does not occur. There is no coercivity and open M-H loop; the magnetization simply rotates reversibly away from the hard direction toward the easy direction as the field is reduced below the nucleation field. It is noteworthy for the discussion in Section 8 that for distributions of noninteracting grains totally randomly oriented the loop coercivity is H , = 0.48 H , ( H , = 2 K / M is the uniaxial anisotropy field). The squareness is S = M,/M = 0.5 for a truly random distribution. S* is difficult to define because the loop derivative changes at the coercive state.’” For particles with shape anisotropy, H , depends on the aspect ratio and approaches 2nM for long particles. For particles with coincident shape and crystalline anisotropy axes the net anisotropy is simply the sum of the individual anisotropies.

looB. D. Cullity, “Introduction to Magnetic Materials,” p. 240, Addison-Wesley, Reading, MA, 1972.

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H. NEAL BERTRAM AND JIAN-GANG ZHU

b. Nonun$orm Magnetization Reversal in Single Grains

Fine grains, in general, do not possess spatially uniform magnetizations. If the particle shape is not ellipsoidal, large deviations in the vector magnetostatic fields occur, particularly at sharp corners, and if the particle is not too small compared to the exchange length, the magnetization will deviate from a nominal bulk direction in regions of high magnetostatic fields.”’ Even if the magnetization is nominally uniform the reversal process at nucleation need not be by uniform rotation. For the discussion of perpendicular recording media in Section9 it is sufficient here to review one generic mode of nonuniform rotation: the curling process.’02 The general problem of nucleation and magnetization reversal involves a multidimensional version of (6.1). If a grain has N distinct spin units, corresponding often to atomic sites, then a general form of the energy (and the dynamic equations as well) will have 2N variables (or 3N with extra conditions conserving the magnitude of each spin). Thus, there will be a 2Ndimensional energy surface and 2N equilibrium states (many of which are degenerate). If the system is placed in a given state (perhaps the saturation state corresponding to a large field), perturbations of the spins from equilibrium will cause the energy to increase no matter what the vector direction of the 2N magnetization spin space. Reduction of the field will cause reversible rotation of all the magnetizations. Eventually a nucleation field will be reached at which irreversible change of the 2N coupled spin system occurs. The corresponding “energy saddle point” for nucleation corresponds to a vanishing of the energy barrier for one particular spin state vector deviation. This direction in the spin vector state may be called the lowest eigenmode of the system, and the corresponding nucleation field is termed the lowest eigenvalue. Because there are, in general, many possible final system equilibrium states, (1.4) must be evaluated to determine the final equilibrium state. This discussion also applies to coupled systems of grains such as constitute polycrystalline thin films. To avoid lengthy computations it is helpful if the nucleation mode can be guessed. For long particles where the crystalline anisotropy is also the long shape anisotropy direction, the “curling” mode was suggested and resulting nucleation field calculated analytically.l o 2 Modern computation has verified these model modes.lo3A long particle is sketched in Fig. 23. For large fields, or equilibrium in zero field, the magnetization is uniform along the long direction. The M - H loop is square but nucleation is complicated. If nucleation is tested by rotating the magnetization uniformly against the net lolM. E. Schabes and H. N. Bertram, J. Appl. Phys. 64, 1349 (1988). loZE.H. Frei, S. Shtrickman, and D. Treves, Phys. Rev. 106,446 (1957). lo3W. Chen and D. R. Fredkin, J. Appl. Phys. 67, 4508 (1990).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

305

H

FIG. 23. Schematic of curling reversal mode where magnetization deviates from equilibrium in azimuthal direction only.

anisotropy direction, the energy will increase until a reverse field magnitude IHI = 2(Kc K,)/M occurs where K , , K , represent the crystalline and shape anisotropy, respectively. However, in the curling mode sketched in the Fig. 23, nucleation is tested by allowing the magnetization to rotate slightly in the circumferential direction, forming a partial vortex with rotational symmetry and no variation along the long direction. Until an appropriate nucleation field is reached, the energy will increase due to rotations of magnetization away from the crystalline anisotropy K , and due to increased exchange energy in the formation of the vortex. However, no increase in shape anisotropy energy will occur. If the curling mode is described by a general radial functional of the angular variation, the nucleation field may be solved by an Euler-Legrange minimization of the energy density."' The functional at nucleation is a Bessel function with nucleation field given by

+

2K 2.16nA IH"I = L + M Mr2

IH"I =

2(K, + K , )

(curling) (uniform rotation),

where r is the radius of the particle and A is the exchange constant. The system will always choose the lowest eigenvalue or nucleation field. Comparing these two modes, for large particles, expenditure of exchange energy by curling raises the system energy less than uniform rotation, which involves shape anisotropy. For small particles uniform rotation will occur at nucleation. Note that the nucleation field due to crystalline anisotropy, 2 K c / M ,is additive in both expressions. Additive crystalline anisotropy occurs in general

306

H. NEAL BERTRAM AND JIAN-GANG ZHU

if the magnetization is uniform prior to nucleation and parallel to the crystalline anisotropy axis. c. Reversal in Simple Multiple-Particle Systems: Dipole Pair Here reversal mechanisms for several simple configurations of interacting grains are discussed. Simple configurations, in particular the dipole pair, give insight into the complicated reversal patterns seen in multigrain thin films and, for orientations of high symmetry, can be analyzed analytically. Both nucleation fields and subsequent dynamic reversal for i d e n t i ~ a l ’ ~and ~~’~~ nonidentical dipole pairs have been analyzed.’06 The dipole pair refers to a extension of the case studied in Section 6a for uniform rotation of a single grain with uniaxial anisotropy. A configuration of sufficient interest is shown in Fig. 24 with grains with parallel anisotropy axes and possibly different magnitudes K , and K,. The applied field will be taken to be along the anisotropy axis direction. The particle centers are separated by distance r and oriented with respect to the common anisotropy axes by bond angle 8. Each grain is assumed to have magnetizations with

2 Grain I

I

I I I I

I

I I

FIG.24. Diagram of interacting dipole pair. Crystalline anisotropy easy axes are parallel and along the applied field direction. B is the “bond” angle of the particles with separation r. ‘@‘H.N. Bertram and J. C . Mallinson, J. Appl. Phys. 40, 1301 (1969). Bertram and J. C . Mallinson, J. Appl. Phys. 41, 1102 (1970). lo6J-G. Zhu and H. N. Bertram, J Appl. Pys. 66, 1291 (1989).

loSH. N.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

307

magnitude M and, without loss of generality, orientations in the pair plane of 8, and 8,. The system energy is simply the sum of single-particle terms of the form (6.1) with the addition of the magnetostatic interaction term. Simple point dipolar interaction will be assumed of general form (cgs units where B = H + 471M)

Edipis the total energy and p = M V is the dipole moment of Volume I/. Assuming identical dipole moments except for orientation gives a total system energy density of

where Hi,, = M z V / r 3 is an effective interaction field. The energy surface now spans the two-dimensional spin space defined by - n < 8, 71, - 71 < 8, < 71. Equilibrium states are given by the vanishing of the first derivatives:

-=

Energy minima, maxima, and saddle points are given by examining the determinant of the second derivatives evaluated at each possible equilibrium point:

I 6EZ =

a2E

aZE

d2E

a2E

I

Analysis of the second variation of the energy can be considered as an eigenvalue problem so that nucleation occurs when the variation vanishes through the zero of the smallest eigenvalue with corresponding eigenmode. Physically, nucleation from a given equilibrium state occurs at the lowest (reverse) field where a saddle point develops in the energy surface in a specific direction in 8,, 8, space away from equilibrium. First consider identical ( K ; = K , = K ) aligned grains with vanishing bond angle, 8 = 0. For a large positive field only one equilibrium occurs with 8, = 8, = 0. It can be shown analytically that this positive aligned state

308

H. NEAL BERTRAM AND JIAN-GANG ZHU

remains stable as the field is reduced until a nucleation field opposite in direction to the original saturation field occurs of magnitude

The initial reversal mode is in the spin direction of “fanning” where

60, = -60,.

(6.8)

Equation (6.8) represents initial rotation of grain magnetizations away from equilibrium at nucleation. In this case the variations are equal in magnitude but opposite in direction. That (6.8) represents a low-energy mode can be seen by examining (6.3) because only the magnetostatic energy determines the mode direction. There are two final equilibrium states 0, = n, O2 = n and 0, = n, 0, = - n (or 0, = -n, 0, = n). Analysis of the dynamic^"^ shows that for all interactions strengths Hint,only the parallel reversed spin state occurs (at least for relative damping coefficient A/y = l), even though the spins begin their irreversible rotation by motions toward an antiparallel state. Chains of interacting spheres with no crystalline anisotropy have been analyzed, yielding an asymmetric fanning mode and nucleation fields for long chains IH,, M . l o 7 Another analytic result of interest is adjacent placement of spins where 0 = n/2. For H , > 3Hint the ferromagnetic state 0, = 0, = 0 is stable with nucleation occurring at a field magnitude opposite in direction to initial saturation of

-

The nucleation mode is “coherent” rotation:

60,

=

so,.

(6.10)

The magnetostatic energy is minimized by the spins remaining parallel as they rotate to the bond direction. The final state is the parallel reversed state for H K > 3Hint.For H , < 3Hintas the field is reversed from initial saturation, the moments fan reversibly, reaching an antiparallel state along the bond direction at H = 0. Note that for bond angles along the applied field direction the magnetostatic field is positive and the effect of interactions is to raise the nucleation field. For bond directions orthogonal to the field the effect of interactions is to reduce the nucleation field. lo71.S. Jacobs and C. P.Bean, Phys. Rev. 100, 1060 (1955).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

309

Grains in polycrystalline fields are not identical in that the anisotropy field magnitude and direction as well as the magnetostatic field vary from grain to grain. The effect of this dispersion is examined here in a simple way by considering nonidentical grains. First consider a pair with differing anisotropy constants oriented at zero bond angle (0 = 0). The general energy expression (6.4) can be utilized to yield a nucleation field for reversal from an initial (equilibrium) saturation ( 6 , = O2 = 0) state. The magnitude of the nucleation field is

I H, I

=

H K + 2Hinl- Jm(e = 01,

+

5

where H K = (HK, HK2)/2 and AHK mode is given by

(6.1 1)

(HK, - HK,)/2. The nucleation

(6.€2) For identical particles, AH, = 0, (6.11), (6.12) reduce to (6.7), (6.8). These results permit a reasonable definition of weak and strong interactions with respect to collective processes. Interaction strength is defined relative to the dispersion in intrinsic properties: Hint< AHK Hint$ AHK

(weak interactions), (strong interactions).

(6.13)

For strong interactions (6.1 l), (6.12) reduce to (6.7), (6.8). The particles nucleate by a fanning mode at a nucleation field given by the mean anisotropy. The final state will be the completely reversed parallel state. For weak interactions (assuming HK, 4 H K 1 )the nucleation field and switching mode are, respectively,

I H , IE' HK,

+ 2~~~~

(e = 01,

(6.14) (6.15)

The nucleation field is close to that of the anisotropy field of grain 2, which has the smaller anisotropy field. The nucleation mode is asymmetric fanning % - 1. At nucleation grain 2 begins to rotate because 602/68, E' - HK,/Hint much more rapidly away from equilibrium than does grain 1. Subsequent dynamic motion will lead to reversal of only grain 2: The magnetization of

310

H. NEAL BERTRAM A N D JIAN-GANG ZHU

grain 1 with the higher anisotropy will rotate only slightly away from initial equilibrium, returning to the initial state after dynamic equilibrium is achieved. The final state will be antiparallel magnetizations with only grain 2 reversed. Reversal of grain 1 will occur with subsequent increase of the reverse field magnitude. The composite M - H loop will consist of two nucleation jumps. The previous example of a weakly interacting pair permits clarification of the concepts of “local” and “collective” field models. The foregoing discussions have considered the pair as a truly collective system. In a local field approach to an interacting system, the nucleation of one grain is determined by fixing the magnetizations of all the other grains and minimizing the energy of the subsequent one-variable problem.” The interaction effects of the other grains enter now as an additive field. Using (6.3) or (6.4), even for the strongly interacting case, the effective fixed interaction field is 2Hint. The weakly interacting case nucleation field (6.14), (6.15) is, as expected, equivalent to a local field model; during nucleation of the first grain, the magnetization of the second grain with the larger anisotropy remains essentially fixed. The nucleation field is simply the sum of the nucleation field of the isolated weaker particle plus the fixed field from the grain with the higher anisotropy field. When the particles are strongly interacting so that both particles rotate equally in the fanning mode, the interaction field is reduced by the relative motion and the nucleation field is correspondingly reduced (6.7). The essential point is that in terms of assessing collective phenomena in interacting systems, the strength of the interacting fields should be measured in terms of the dispersion of intrinsic properties, not in relation to, for example, the mean anisotropy field. In magnetic tape with well-dispersed particles, interactions may be weak enough so that local field models, which require significantly less computation, might be utilized. In polycrystalline thin films the interactions are strong, so complete collective analysis must be undertaken. Nucleation from initial saturation for the nonidentical pair with bond direction orthogonal to the field direction (0 = n / 2 ) may be summarized as

IH , I

= H , - H~~~-

Jm(e

= 421,

(6.16)

(6.17) For the strongly interacting case (6.16), (6.17) reduce to (6.9), (6.10) with the anisotropy term replaced by the mean anisotropy in the nucleation field. The mode is coherent rotation (68,/68, rz 1). For weak interactions (6.16), (6.17)

31 1

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

reduce to

(6.19) Note that the nucleation mode is asymmetric coherent rotation. Initially, at nucleation before large angular deviations take place, a large deviation from equilibrium occurs for grain 2 compared to grain 1, but both deviations are in the same direction. During dynamic evolution grain 2 rotates beyond 90" with little rotation of grain 1, thereby yielding an interaction field that is positive on grain 1 and negative (in the direction of the reversed applied field) on grain 2. The dynamic process is therefore driven to a final antiparallel state where grain 2 has reversed and grain 1 has returned to its original equilibrium direction. Analysis of dynamics subsequent to nucleation for the orthogonal bond direction shows that even for strong interactions the antiparallel state can occur.1o5 An illustration is given in Fig. 25, where dispersion has been introduced through slightly different applied fields and only the case of infinite damping R/y S- 1 has been considered.lo6Because the interactions are 1.o

I

k 0.8

Ha-H,

-

I

7

I

v

AH/H,=0.025

0

2M

0.6

02

Hi,,/H,=0.2

5 d 0.4 0

.,-I

4

i d

4

01

0.2

I 0.0 0

50

100

150

relaxation time

200

250

To(

FIG.25. Time evolution of magnetizations at nucleation for a pair of interacting dipoles in slightly different applied fields. The bond angle is 0 = 4 2 .

312

H. NEAL BERTRAM AND JIAN-GANG ZHU

strong, the magnetizations initially rotate almost coherently, but eventually, when the angles approach 742, a turning point occurs with one spin returning to its original direction and the other reversing. d. Reversal in Simple Multiple-Particle Systems: Column Chains In Section IV the reversal processes in perpendicular films will be discussed. The structure is a two-dimensional planar array of grains with virtually identical crystalline anisotropy. The simplified corresponding dipole pair configuration is the identical pair with bond direction orthogonal to the common applied field and anisotropy direction. However, in real films a dispersion occurs due to fluctuations in the interaction field as well as slight deviations in perpendicular alignment of the anisotropy axes. The interactions are strong compared to the dispersion, but as discussed earlier, for this configuration, state splitting can occur. Dispersion relative to the interaction field required for the antiparallel state to occur has been studied.lo6 A somewhat more applicable study has been made of a one-dimensional chain of strongly interacting, identical columnar grains where the chain direction is orthogonal to the common anisotropy and applied field direction.’06 Dispersion occurs in the interaction field due to the finite chain length. The dynamic evolution at nucleation is shown in Fig. 26. The

+ ’

P 2 P 3 P

4

4

5

FIG.26. Time evolution ( 1 , . .. , 5) at nucleation of the magnetizations of an interacting chain of columnar grains.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

3 13

numbers 1, . . . , 5 denote time steps. Even though the interactions are strong, dispersion in spin rotation occurs during reversal, with the center two spins eventually reversing and the outer four returning to their original equilibrium' direction. The nucleation field is slightly below the crystalline anisotropy field H , . Even though each column magnetization is assumed to rotate uniformly, the nucleation field does not include the columnar shape anisotropy (6.2); simultaneous rotation of grain magnetizations in the chain direction almost completely cancels shape effects.

7. NUMERICAL MODELING OF THINFILMS In this section details of the numerical model utilized for all the dynamic micromagnetic calculations are discussed. The geometric structure is presented first, followed by an elaboration of all magnetic energies that enter into the effective field (1.5). Parameterization and scaling laws are discussed, followed by a review of essential numerical details. a. Geometric Structure Polycrystalline thin films will be represented by a two-dimensional array of hexagonally shaped grains on a hexagonal lattice (Fig. 27). D is the grain diameter. The intergranular spacing is d and 6 is the film thickness. The volume of each grain is therefore V = f i D 2 6 / 2 . The coordinate system for

7-

4 FIG.27. Diagram of 2D hexagonal array grain configuration. D is the grain diameter in the film plane, d is the intergranular boundary separation, and 6 is the grain height coincident with the film thickness.

3 14

H. NEAL BERTRAM AND JIAN-GANG ZHU

the calculations is x along the recording or field direction, y normal to the film plane, and z in the direction across the recorded track. This regular pattern most closely resembles thin-film recording media, at least highquality films such as those shown in Fig. 1, which exhibit well-defined polycrystallinity. A hexagonal structure with six nearest neighbors exhibits the highest symmetry for a regular pattern. Thin-film recording media are composed of tightly packed grains. However, with this array the volume packing fraction of magnetic material can be set by only one parameter, the common intergranular spacing d. The intergranular spacing strongly affects the magnetostatic coupling, so that sample control of this phenomenon is important. A periodic structure is useful for large-scale numerical simulations because, as discussed later, fast Fourier transform (FFT) techniques may be utilized for computation of the magnetostatic fields, permitting substantial time saving. b. Energy Density There are two primary assumptions in the model. (1) Each grain is assumed to be a single crystal with uniform properties. (2) The magnetization of each grain is assumed to be uniform over the grain, not only in equilibrium but during dynamic reversal as well. Thus, single-grain uniform rotation, as discussed in Section 6, applies, except that now the magnetostatic interactions are more complex and intergranular exchange interactions will be included. The four main energy contributions are discussed here. The energy density will be given in terms of the normalized magnetization:

mi = M(ri)/M, where the subscript i denotes the grain index ( i = 1, . . . ,N) and N is the total number of grains. All expressions are given in cgs units ( B = H 471M). In these densely packed films the saturation magnetization of the film M , is approximately equal to the saturation magnetization of the each grain and related by M , = M D 2 / ( D+ d)’.

+

(1) Crystalline Anisotropy In the results presented in the review only uniaxial anisotropy has been utilized, which closely approximates the hexagonal symmetry of the Co-based magnetic grains. Other anisotropy models, such as cubic, can easily be substituted. To first order, the crystalline anisotropy energy density of each grain may be written as

where K i is the magnitude of the anisotropy of each grain and

lidenotes the

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

3 15

orientation of the anisotropy easy axis. In this review only grains with constant anisotropy magnitude will be considered: K i= K for all i. The f i will generally be set randomly from grain to grain representing a 2D or 3D distribution of axes. Equation (7.2) is often written in terms of the angle 8 between the magnetization and the anisotropy axis (6.1): Eanis= K sin’ 8,.

(7.3)

(2) Zeeman Energy The external applied field may be either uniform or, for recording simulations, spatially varying. The general case of the energy density is (7.4) where the integration is taken over the volume of the ith grain. (3)Magnetostatic Energy The magnetostatic field may be written in a variety of forms,” all of which lead to a quadratic tensor. A convenient form is to write the energy in terms of the interaction field and then to express the field in terms of magnetic poles:

EL,, = -

1

-

d3r‘ mi Hmag(r’),

(7.5)

d2r”A * rnj(r“)(r‘ - r“)/lr‘ - r‘’1 3.

(7.6)

2v vi

where, utilizing ( l . l ) ,

The factor of 1/2 in (7.5) is to prevent double counting in this pair interaction energy. In (7.6) the field from each of thejth grains involves only the surface poles because the magnetization is presumed uniform in each volume. Incorporating (7.6) into (7.5) yields

EL^^ =

- M2mi

- [ c oij- mi + j+ 1

where

Dij =

jv3 j d3r‘

Sj

d’r” al(r”)(r’ - r”)/lr’ - r”I3,

(7.7)

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H. NEAL BERTRAM AND JIAN-GANG ZHU

is the magnetostatic field tensor. Note that (7.7) is the energy for one grain; the total system energy is a sum of (7.7) over all grains avoiding double counting. The self-energy term includes a factor 112 because the variation of this quadratic term to obtain the self-field produces a factor of 2. Self-energy must be included for grains of general rectangular shape. For grains with cubic symmetry the term is only an additive constant and should not be utilized because numerical destabilization can occur. The demagnetization field tensor may be expressed in analytic form but is complicated.53108For application of this micromagnetic formalism to other microstructures that require 3D discretization, the coefficients (7.8) have been calculated analytically for a cubic discretizati~n.’~~

(4)Exchange Energy To describe the possible exchange coupling between adjacent crystallites in the films, a phenomenological intergranular ferromagnetic exchange energy density is introduced. In the case of a continuous magnetization distribution the exchange energy density may be written asz6

where A is the exchange constant related to the interatomic exchange and a,

P, y are the direction cosines of the continuous magnetization components: a = MJM, fi = M , / M , = M,/M. (7.10) In terms of the reduced magnetization m:

(7.11) In the case of polycrystalline films each grain is assumed to be uniformly magnetized so that exchange torques within a grain are not considered. However, at the grain boundaries, intergranular exchange coupling might occur if a nonmagnetic intergranular layer were not present. To apply (7.11) to the case of neighboring grains, the derivatives are discretized, to first order, by assuming a linear variation of the magnetization between neighboring grains. For example, the contribution to E,, for grain i and grainj displaced in the x direction a distance D (center-to-center distance between adjacent grains with d = 0) is EYx E A[m(ri

+ Dfi) - m(ri)I2/D2.

(7.12)

losJ.-G.Zhu, “Interactive Phenomena in Magnetic Thin Films,” Ph.D. Thesis, University of California, San Diego 1989. lo9M. E. Schabes and A. Aharoni, IEEE Trans. Magn. MAG-23, 3882 (1987).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

317

Keeping only the term that varies with magnetization,

E:! z

-

2A -m(ri D2

+ Dfi) - m(ri).

(7.13)

Generalizing to all nearest neighbors j of grain i, the exchange energy density is given to lowest order by (7.14)

-

erg/cm. For For perfectly couples grains A would be on the order of films with well-defined grain boundaries (d > ~3 A), exchange coupling is unlikely. However, for d < z 3 A it is possible that inhomogeneous grain boundaries might develop during grain growth. Therefore, a phenomenological constant A* IA is assumed to characterize the exchange coupling: (7.15)

(5) Total EfSectiueField The total effective field (1.5) acting on each grain is written in normalized form: (7.16) where the field has been normalized by the anisotropy field: h = H/H,. Summing all the energy terms and taking the appropriate variation yields, in normalized form,

-

hLff = hLx, + ki(ki mi) + h,

N

C1 9),

j=

nn

mj

+ he 1mj,

(7.17)

j

where the normalized magnetostatic and exchange coefficients are h, = M/H,, he = A* / K D 2 ,respectively. Note that in many figures in this review as well as in early works of the authors C* = he is utilized. The Zeeman contribution assumes that any spatially varying field does not vary appreciable over the grain; if sufficiently high-gradient head fields are utilized, then the volume-averaged field should be used. The normalized form of the dynamic equations (1.4) is written as

dmi ~

dz

=

-

mi x hLff - ami x (mi x hLff)

(i = 1,. . . ,N ) ,

(7.18)

318

H. NEAL BERTRAM A N D JIAN-GANG ZHU

where z = Iy(tH, is a normalized time and c1 = A//lyl is the normalized damping coefficient. Equations (7.17), (7.18) represent 3 N simultaneous firstorder differential equations.

( 6 ) Scaling Laws The normalized magnetization during dynamic evolution, and in final equilibrium, scales with the normalized effective field (7.18). The normalized field (7.17) depends on the distribution of anisotropy axes, the interaction field strengths he and h,, and geometric factors that determine Dij. By normalizing the integration limits and variables in (7.8) it can be shown that Dijscales functionally as G(6/D, d/D). In fact for 6 I D, G varies approximately as 6/D.I7 All results of the solution of (7.18) may be written in general functional form using scaling parameters. For example, functional forms for the hysteresis loop, the loop coercivity, and the magnetization fluctuations may be expressed, respectively, as (7.19)

(7.20)

(7.21)

where F , , F , , F , represent the appropriate functional forms. For example, for longitudinal thin films where 6 I D and G varies as 6/D, a natural scaled plotting of the coercivity versus material magnetization will be M 6 / H K . In general, increasing the interactions reduces the coercivity, although it is clear from (7.20) that the grain aspect ratio 6 / D is important. Noise in the magnetic recording process results from magnetization fluctuations that are averaged across the track width by the replay head. The total noise depends on the number of effective grains or correlation domains in the sampled region. Generally, an increase of exchange he leads to isotropic expansion of correlation or domain size. The general noise reduction technique is to reduce the diameter D. However, for fixed A * / K , reducing D increases the and correlation domain size so that noise reduction might not O C C U ~Thus, . ’ ~ from scaling laws alone, it is clear that medium noise reduction can occur substantially by grain diameter reduction only if no exchange coupling occurs: A* = 0.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

3 19

(7) Range of Parameters Values of magnetic parameters are given in Section 11. For Co-based films the magnetization varies between 300 and 800 emu/cm3. The crystalline anisotropy of Co is on the order of K 5-7 x 106erg/cm3, but for typical CoCr with Cr concentrations in the range 1-2 x lo6 erg/cm3. This yields a range of 12-20% K is in the range K magnetostatic interaction parameters: 0.05 < h, < 0.7. For grains diameters erg/cm, the range of exchange paramin the range 15-60 nm and A eters becomes 0 < he < 1. Because slight increases in the exchange interaction result in profound changes in magnetic behavior, the range utilized in these studies is 0 < he < 0.25.

-

-

( 8 ) Numerical Method The set of coupled first-order differential equations is integrated using the Adams method with strict error control.’ l o The Adams method is a varying order and varying step size method, and the computation codes were optimized for vectorization on the Cray X-MP supercomputer. A general M-H loop simulation starts with a saturated array magnetization along the applied field direction. A spatially uniform applied field is reduced stepwise from a large saturating initial value to opposite saturation utilizing a very small step size: AH,,, < 0.002HK.At each field step, the applied field is kept constant until an equilibrium condition is satisfied in the magnetization orientations: (7.22)

-

where E is the error tolerance. In order to avoid energy surface saddle points, the equilibrium status of an obtained magnetization configuration is tested by introducing a small random perturbation in the magnetization orientation of each particle. If the perturbed magnetization orientations evolve back to the initially obtained configuration, then the equilibrium configuration is accepted and the calculation proceeds. A similar algorithm is utilized for recording transitions with spatially nonuniform fields. It has been found that the calculation results are insensitive to the variation of the reduced damping constant a at least for o! > 1. Due to the strong magnetostatic interactions between the grains and the extremely slow temporal variation in applied field, the angle between the direction of the magnetization and the effective field for each grain is always very small. Away from equilibrium, magnetization precession is always constrained to a very small cone angle about the effective field. Small values of a < 1 change the W. Gear, “Numerical Initial Value Problems on Ordinary Differential Equations,” Prentice-Hall, Englewood Cliffs, NJ, 1971.

IloC.

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H. NEAL BERTRAM AND JIAN-GANG ZHU

results somewhat'" but yield a significant increase in computation time. Therefore, the precession term in the differential equation (7.18) is neglected in the computations presented in this review. Direct computation of the magnetostatic field scales the computation time as N Z .Fast Fourier transform techniques reduce the computation time with scaling as N In N."' FFT techniques require periodicity and careful consideration of the boundary conditions. A technique that can be used for nonregular structures (such as random placement of grains) involves multipole expansions.' 13s114 For large arrays multipole techniques yield computation times that vary linearly with the number of discretization elements N; however, FFT methods are considerably faster, at least for the scale of problems under current investigation.

IV. Reversal Processes and Domain Structures

8. LONGITUDINAL FILMS In this section .the effect of grain interactions and medium microstructure on material properties such as coercivity H , , squareness S, and loop shape S* will be discussed. In addition, detailed grain spin orientations and dynamic processes will be reviewed. General hysteresis phenomena were shown first by Hughes, utilizing slightly different m~deling."~The size of the simulation array for all results presented in this section is 64 x 64 grains. a. Hysteresis Properties Both magnetostatic and exchange interactions strongly affect hysteresis. In Fig. 28, calculated hysteresis loops"6 for the case of solely magnetostatic interactions along with a noninteracting array are plotted. The grain anisotropy orientations are completely (3D) random. In the noninteracting case, reversal is governed by the external applied field relative to local crystalline anisotropies. The resulting hysteresis loop has a coercivity HJHK = 0.48, a squareness S = 0.5, and a sloped loop so that S* is small and difficult to define (Section 6a). With magnetostatic interactions (h, = 0.3) the coercivity decreases to about half the noninteracting value; H,/HK 0.24;

-

'"J.-G. Zhu and H. N . Bertram, J. Appl. Phys. 63, 3248 (1988). l12M. Mansuripur and R. Giles, IEEE Trans. Magn. MAG-24,2326 (1988). '13J. L. Blue and M. R. Scheinfein, IEEE Trans. Mugn. MAG-27, 4778 (1991). '14J. Carrier, L. Greengard and V. Rokhlin, SIAM J. Sci.Star. Comput. 9,669 (1988). lI5G. F. Hughes, J. Appl. Phys. 54, 5306 (1983). '16J.-G.Zhu and H. N. Bertram, unpublished.

321

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA 1 .o

f

\

0.5

I

.-

N

c,

Q

c 01

2

-0.5

-1.0

-0.4

0.0

0.4

Applied field H,/Hk FIG.28. Hysteresis loops for non-exchange-coupled in-plane isotropic films calculated with h, = 0.3 (solid curve) and h, = 0 (dashed curve). Magnetostatic interactions yield an increase of the remanence and coercive squareness and a reduction of the coercivity. Simulation array size: 64 x 64 grains.

-

-

the squareness increases, S 0.7; and the loop becomes quite square, S* 0.8. These hysteresis properties represent general behavior in that the coercivity always decreases with interactions for both 2D planar and 3D random anisotropy orientations. In Fig. 29 coercivity is plotted versus magnetostatic interaction strength for both 2D and 3D random anisotropy orientations and no intergranular exchange interaction. The decrease is almost linear for 0 < h, < 0.3 with very similar coercivities for the two orientation distributions. For k , = 0 the noninteracting limit of H J H K 0.5 for both distributions occurs. In Fig. 30 coercivity versus normalized medium thickness is plotted for fixed magnetostatic interaction h, = 0.4. For a 2D distribution of anisotropy axes the coercivity decreases monotonically from the noninteracting limit. Following the discussion in Section 7b(6), for 2D distributions, the curves in Figs. 29 and 30 should be approximately universal if plotted as km6/D. For 3D random anisotropy a monotonic decrease of coercivity with thickness 6 / D occurs, but 6/D = 0 does not correspond to vanishing interactions. With out-of-plane anisotropy, as 6 / D + 0, the shape demagnetization (47cM) for each thin grain keeps the magnetization in the plane, enhancing reversible rotation and reducing the coercivity. For k , < 0.6 an increase of saturation squareness S occurs with increasing In magnetostatic interactions along with the decrease in coercivity.'

-

'

ll'J.-G. Zhu and H. N. Bertram, J. Appl. Phys. 63, 3248 (1988).

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H. NEAL BERTRAM AND JIAN-GANG ZHU

:0.2

---__

he=O 6/D=0.75

0

d/D=0.02

0.1

0

z

0.0 0.2

0.0

0.4

0.6

Magnetostatic field constant h, FIG.29. Coercivity versus magnetostatic interaction field constant h,. The dashed curve represents crystaline easy axes randomly oriented in the film plane (2D-random) and the solid curve easy axes randomly oriented in three dimensions (3D-random). Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 63, 3248 (1988).

0.6

-

----. 2D-random 0.5 "\

0.0

3D- random

0.2

0.4 0.6 0.8 Normalized thickness 6 / D

1.0

FIG.30. Coercivity versus normalized film thickness S/D. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 63, 3248 (1988).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

323

general, for fixed magnetostatic interaction h,, films with 2D random crystalline easy axis orientation exhibit higher coercivities, larger saturation squarenesses S , and slightly higher coercive squarenesses S* than for 3D axis orientations. For increasing magnetostatic interaction beyond h, = 0.6 (and negligible exchange interaction), a monotonic decrease in saturation squareness occurs: Large magnetostatic interactions coupled with randomly oriented crystalline easy axes result in local magnetization flux closure in the absence of an external field. The width of the nonmagnetic intergranular boundary also affects the coercivity. Increasing grain separation d / D , while keeping 6 / D constant, results in a coercivity increase because the magnetostatic interaction strength is reduced. For small changes 0 < d / D < 0.25 the increase is linear with slope A ( H c / H K ) / A ( d / D ) 0.48.117 The effect of intergranular exchange coupling is shown in Fig. 31. The calculated hysteresis loops represent films with fixed magnetostatic coupling and finite intergranular exchange coupling (he = 0.12) compared to nonexchanged coupled films (he = O ) . l I 7 Exchange coupling strongly affects hysteresis properties with increased saturation squareness S, increased coercive squareness S*, and lowered coercivity. Figure 32 shows the dependence of hysteresis loop squarenesses S and S* on the intergranular exchange coupling constant he. High coercive squareness (S* > 0.9) is a characteristic feature of exchange-coupled planar isotropic films. In Fig. 33 the dependence of coercivity on exchange interaction is shown. These numerical results provide

-

3D-random h,=0.3

II

c 0 .t;

.-N

-_----

-/---7I

0.0

4

W

K

m o

x

-0.5

-1

.o -0.4

0.0

0.4

Applied field H,/H, FIG.31. Calculated hysteresis loops for a non-exchange-coupled film (dashed curve) and an exchange-coupled film (solid curve). h, = 0.3 is common to both.

324

H. NEAL BERTRAM AND JIAN-GANG ZHU

0.95 m

0.90 0.85

v, v) v)

?

2 u-

0.80 0.75

0.70

m

0.65 I

I

0.60 0.00

I

I

I

0.08

0.04

I

I

I

0.1 6

0.1 2

I nte rg ran ula r e x c h a n g e coupling he FIG.32. Calculated remanent squareness S and coercive squareness S* versus intergranular exchange coupling constant h, (h, = 0.2). The crystalline anisotropy easy axes are completely (3D) randomly oriented. J. G . Zhu and H. N. Bertram, unpublished results.

N = *

0.10 - d/D=0.02

E6 0.05 -

3D-random

z

I

I

I

I

I

I

I

1

FIG.33. Coercivity versus intergranular exchange coupling he. The easy axes are completely randomly oriented (3D).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

325

verification that changes of experimental hysteresis properties as films are produced with voided or segregated grains (Section 4) are, indeed, mainly due to the disruption or weakening of the intergranular exchange coupling. Experimental film coercivities generally increase with increasing film thickness, reaching a maximum at about 150 8, and decreasing thereafter (Section 4). The initial increase is likely due to the decrease of superparamagnetic effects with increasing grain volume.''* The subsequent decrease most likely follows the behavior shown in Fig. 30, because the grain diameter D,at least initially, is set by the underlayer Cr columnar diameter. At large thicknesses possible grain diameter growth can lead to coalescing grains with subsequent exchange interaction and enhanced coercivity decrease. Scaling laws, as developed in Section 7b(6), can be applied to these simulations. For example, increasing film grain diameter with no change in effective intergranular exchange interaction A* leads to decreased exchange coupling he and decreased magnetostatic coupling. Thus, increasing D will lead to increased coercivity and decreased S and S* as seen experimentally [Section 4a(4)]. b. Magnetization Patterns and Reversal Processes Magnetization reversal processes have been analyzed in detail through computer simulation for in-plane isotropic films."9,'20 These studies showed that in a typical in-plane isotropic film, nucleation of magnetization reversal occurs by vortex formation. Relative motion of interacting, neighboring vortices yields reverse domains elongated in the applied field direction. Domains in films with intergranular exchange coupling expand in size through large-scale vortex motion. The magnetization reversal process is completed by vortex annihilation. All the results shown have been calculated assuming a completely random (3D) orientation distribution of crystalline easy axes. Here magnetization patterns and dynamic processes are illustrated for each step along the major hysteresis loop. (1) Magnetization Cluster-Ripple Structure The saturation remanence state is formed by initially aligning all the grain magnetizations in the field direction (either by applying a large field or simply by numerical specification) and then allowing the system to relax to equilibrium (or removing the field). An example of the magetization configuration at the saturation remanence state for typical magnetic parameters is shown in Fig. 34. Even lI8B. D. Cullity, "Introduction to Magnetic Materials," pp. 385-389, Addison-Wesley, Reading, MA, 1972. lI9J.-G.Zhu and H. N. Bertram, J. Appl. Phys. 69,6089 (1991). lZ0J.-G.Zhu and H. N. Bertram, IEEE Trans. Mugn. MAG-27, 3553 (1991).

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f

Mr

FIG.34. Typical magnetization configuration at the remanence state, simulated with h, = 0.4 and he = 0.1. Normalized film thickness 6 / D = 1 and intergranular boundary separation d / D = 0.02 were used. Each arrow represents the in-plane projection of the magnetization of a single grain (or hexagon). Taken from J. G. Zhu and H. N. Bertram, I E E E Trans. Magn. M A G 27, 3553 (1991), 01991 IEEE.

though the orientation of the crystalline anisotropy easy axes is completely random, local coherence of grain magnetization orientations is apparent; clusters of grains have common magnetization orientations and large changes in cluster magnetization orientation occur between adjacent clusters. Along the initial saturation direction, magnetization orientation of the clusters alternates in sign in the transverse direction with quasi-periodicity. This cluster-ripple structure characterizes the saturation remanent state in planar isotropic films and produces a feather-like picture in Fresnel mode Lorentz electron microscopy imaging.' 08,121 This ripple pattern results from a trade-off of the randomly oriented crystalline easy axes and the magnetostatic interactions. Intergranular exchange coupling increases cluster size, thereby increasing the wavelength of the quasi-periodicity along the initial saturation direction.49 The effect of intergranular exchange on ripple size in thin-film recording media has been observed experimentally.l o 8

(2) Vortex Formation The magnetization vortex is the most elementary structure resulting from nucleation and magnetization reversal. In Fig. 35 lZ1I. A.

Beardsley and J.-G. Zhu, J. Appl. Phys. 67, 5352 (1990).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

1

327

2

4

FIG.35. Typical vortex formation process, calculated with h,

= 0.4 and he = 0.1.6/D= 1 and d / D = 0.02 were used. Pictures 1,2, and 3 are static states with increasing field values. Pictures 3-6 represent a transient process during vortex formation. Taken from J. G . Zhu and H. N. Bertram, IEEE Trans. Magn. MAG-27, 3553 (1991), 01991 IEEE.

magnetization reversal by vortex formation is shown for small increments of reverse applied field from the initial remanence state. The six pictures show the spin configurations of a small subset of the simulated film area. Picture 1 represents the magetization configuration at the saturation remanent state with the typical ripple pattern shown in Fig. 34. As the magnitude of the external field is increased in the reverse direction, the ripple pattern develops reversibility to a " c " structure (pictures 1-2-3) by rotation of the grain magnetizations toward the reverse direction. Note that this rotation is substantial only at boundaries between adjacent clusters. Thus, ripple periodicity is unchanged. Further increase of the applied field yields an unstable c structure, and the magnetization at the open end of the c undergoes transient irreversible rotation (pictures 4-56), completing vortex formation. Magnetization reversal at the open end of the c is by fanning, as discussed in Section 6. A vortex structure develops naturally from the cluster-ripple pattern. Vortex formation results in nucleation or domain formation of a reversed region. Before vortex nucleation, the grain magnetization directions on either the top or bottom of a c pattern do not vary in the transverse direction. After vortex formation and closure of the c, the magnetization orientations circulate, reducing the magnetostatic energy. Increasing magnetostatic interaction strength relative to grain anisotropy h,

328

H. NEAL BERTRAM AND JIAN-GANG ZHU

results in better-defined vortex structures; increasing the intergranular exchange coupling yields larger vortices and increased intervortex separation.”’ (3) Vortex Motion Expansion of reversed regions during hysteresis is achieved through vortex motion. For zero or weak intergranular exchange coupling, vortex motion is characterized by a two-step process: elongation of the vortex center followed by contraction of the elongated vortex center to a new position, as demonstrated in Fig. 36. The sequence in Fig. 36 can occur dynamically in one field step or as equilibrium states over a series of field steps, depending on the local anisotropy configuration. Vortex motion is always in the direction transverse to the reversal field, resulting in expansion of the reversed region. Vortex motion distance in a single process is determined by the elongation size. This distance varies throughout the film because of the spatial randomness of the anisotropy axes. With zero intergranular exchange coupling, the distance is always small, although it is increased somewhat by large magnetostatic interactions. Intergranular exchange coupling significantly enhances this distance. For strong intergranu-

4

5

6

FIG.36. Typical vortex motion process in a weakly exchange-coupled film (he = 0.1, h, = 0.4). Pictures 1-4 illustrate elongation process of the vortex center and pictures 4-6 illustrate vortex center contraction. Taken from J. G. Zhu and H. N. Bertram, IEEE Trans. Magn. MAG27, 3553 (1991), 01991 IEEE. 122J.-G.Zhu and H. N. Bertram, ZEEE Trans. Magn. MAG27, 3553 (1991).

330

H. NEAL BERTRAM AND JIAN-GANG ZHU

1

2

4 FIG.38. Formation process of an elongated reverse domain for a non-exchange-coupled film. Pictures 1-4 are static states with increasing field values. The boundaries on opposite sides of the final elongated reverse domain (picture 4) consist of vortices with opposite sense of rotation. h, = 0.6, S / D = I, and d / D = 0.02 were utilized with 3D random easy axes. Taken from J. G. Zhu and H. N. Bertram, J. Appl. Phys. 69, 6084,(1991).

involve densely formed, elongated narrow domains. Figure 39 shows domain patterns during magnetization reversal over a large film area calculated with a typical value of magnetostatic interaction (h, = 0.3) and zero intergranular exchange coupling (he = 0). The gray scale represents the magnetization component along the direction of the external field. Within each elongated domain, the magnetization exhibits a ripple structure. Figure 40 shows the formation process of an elongated reverse domain for a film with strong intergranular exchange coupling: he = 0.2 and h, = 0.4. The sequence 1 , 2 , 3 , 4 represents time progression at a fixed field. For films with intergranular exchange coupling, elongated reverse domains form in a manner similar to that for non-exchange-coupled films. However, with

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

331

FIG. 39. Static magnetization patterns along a major hysteresis loop for h, = 0 and h, = 0.3. Gray scale represents the magnetization component along the field direction will full bright in the initial saturation direction. Elongated reverse domains near the coercive state are evident. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 69,6084,(1991).

intergranular exchange, not only is the vortex size increased but vortex separation along the domain boundaries increases as well. This is a natural consequence of the increased wavelength in the ripple structure due to exchange coupling. In between adjacent vortices along each domain boundary, magnetization cross-tie patterns occur, similar to the cross-tie structures observed in soft films but on a smaller . ~ c a l e . ’ ~ In~films ~ ’ ~with ~ relatively strong intergranular exchange coupling (he > 0.15), the vortices in a domain boundary translate collectively so that the boundary moves approximately as lZ3R.M. Moon, J. Appl. Phys. 30, 82s (1959). lZ4D. J. Craik and R.S. Tebble, “Ferromagnetism and Ferromagnetic Domains,” p. 337, North-Holland, Amsterdam, 1965.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

333

FIG. 41. Expansion of a reverse domain during magnetization reversal in an exchangecoupled film, h, = 0.2, h, = 0.3. Gray scale represents the magnetization component along the field direction with full bright in the initial saturation direction. In this particular case, the transient expansion process occurs without field increment, resulting in a unity coercive squareness, S* = 1. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 69, 6084,(1991).

( 5 ) Vortex Annihilation Magnetization reversal is completed by vortex annihilation. Contractions of unreversed domains lead to merging domain boundaries, resulting in the formation of vortex-vortex or vortex-cross-tie pairs. Unreversed domains vanish through annihilation of these pair structures. Figure 42 presents a time sequence ast fixed field of the annihilation of a vortex-cross-tie pair, calculated with h, = 0.1 and h, = 0.4. Picture 1 shows the vortex-cross-tie pair prior to annihilation. Annihilation begins as the magnetization in between the vortex and the cross-tie rotates into the transverse direction (pictures 1-2), yielding a c structure. The transversely oriented magnetizations in the c rotate into the reverse field direction,

334

H. NEAL BERTRAM AND JIAN-GANG ZHU

Ha

1

2

FIG.42. Typical vortex-cross-tie annihilation process, calculated with he = 0.1, h, = 0.4, S/D = 1, d / D = 0.02, and 3D random easy axes orientation. Taken from J. G . Zhu and H. N. Bertram, IEEE Trans. Magn. MAG27, 3553 (1991), 0 1 9 9 1 IEEE.

completing the annihilation process (pictures 2-3-4). Two vortices with opposite sense of rotation (from opposite sides of a domain) often form a vortex-vortex pair followed by a vortex-vortex pair annihilation, especially in non-exchange-coupled films. A representative annihilation process is shown in Fig. 43. The annihilation starts as the vortices align themselves in the applied field direction and then open, forming a pair of c and 2 structures aligned along the field direction (pictures 1-4). The process completes as the transversely oriented magnetizations rotate toward the applied field direction (pictures 5 and 6).

9. PERPENDICULAR FILMS

In this section, simulation studies of the magnetization reversal process in perpendicular thin films will be discussed. Although the predominant candidate for perpendicular media is CoCr, the analysis presented here refers to films with long columnar grains, 6 >>D, and crystalline anisotropy along the normal direction (with a 5” angular dispersion) to the film plane. Only the effect of fields applied along the perpendicular direction will be discussed. A logical division for a discussion of reversal in perpendicular media is, as for longitudinal films, into solely magnetostatic coupled films and films with both magnetostatic and exchange coupling.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

335

9 FIG.43. Typical vortex-vortex annihilation process, calculated with he = 0.1, h, = 0.4, 6 / D = 1, d / D = 0.02, and 3D random easy axes orientation. Taken from J. G. Zhu and H. N. Bertram, ZEEE Trans. M a p . MAG27, 3553 (1991), 0 1 9 9 1 IEEE.

a. Chain Nucleation Mode in Non-Exchange-Coupled Films

A typical simulated hysteresis loop with zero intergranular exchange coupling (he = C* = 0) is shown in Fig. 44 (h, = 0.2). The coercivity is virtually equal to the crystalline anisotropy field H , independent of the film magnetization M . The loop consists of discrete fine magnetization jumps, with each jump corresponding to a single nucleation process. The loop is smoothly sheared because of the perpendicular demagnetizing field - N,M,, where N , is the demagnetizing factor, with N,, = 4.n for an infinitely wide thin-film array. The dashed curve in the figure represents the result of deshearing the loop with a demagnetizing factor corresponding to the array geometry.12’ The desheared loop indicates that the nucleation fields for individual irreversible processes (Barkhausen jumps) are all approximately equal to the crystalline anisotropy field H , except near the beginning and end of reversal. All nucleation reversal processes follow the basic chain reversal mode discussed in Section 6d. In Fig. 45 a nucleation process is shown near the beginning of film reversal where the magnetization of the majority of the grains is still in the initial saturation direction. This figure presents a dynamic lZ5J.-G.Zhu and H. N. Bertram, J. Appl. Phys. 66, 1291 (1988).

336

H. NEAL BERTRAM AND JIAN-GANG ZHU

-3

-2

-1 0 1 Applied field H/H,

2

3

FIG.44. Simulated hysteresis loop (solid curve) for CoCr perpendicular thin film with h, = 0.2 and h, = 0. The dashed curve is a desheared loop using an array size average demagnetizing factor. The fine jumps in the hysteresis curve correspond to discrete nucleation processes. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 66, 1291 (1989).

process at a fixed field, and only a small area of the calculated array is shown. Before the nucleation process, the magnetizations of the columnar grains in the film are virtually all in the film normal direction. Nucleation begins by collective rotation of the magnetization of a chain of grains into the film plane. Magnetizations of neighboring grains not in the nucleated chain remain fixed. The magnetizations of neighboring grains in the chain form a head-to-tail configuration (picture 2). The nucleation chain is basically one grain wide. As time progresses (pictures 2-5), large dispersions develop in the magnetization orientation angles so that magnetization rotation is most pronounced in the center of the chain segments. As the magnetizations of the center grains rotate beyond the film plane toward the reverse film normal direction, the magnetizations of the neighboring grains rotate back into their original film normal direction. The new static state (picture 6 ) shows that again the magnetizations of the grains all become virtually perpendicular to the film plane. In this example only a total of five grains have reversed their magnetization, although many more grains participated in the nucleation process. Initial coherent rotation of chain magnetizations removes the shape anisotropy of each grain; thus, the nucleation field for closely spaced grains simply equals the crystalline anisotropy field H,. Film dispersion, primarily in magnetostatic fields from neighboring grains not involved in the chain process, leads to subsequent reversal of only a few grains. The chain

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

337

FIG.45. Transient magnetization configurations during a chain nucleation process at M x 0.8MSas viewed in the film normal direction. The pictures are time ordered and each arrow represents the magnetization of a columnar grain. Only a portion of the simulation array is shown. Pictures 1 and 6 represent static configurations before and after the nucleation process. Taken from J. G. Zhu and H. N. Bertram, J. Appl. Phys. 66, 1291 (1989).

nucleation mode occurs at each nulcleation process in the entire magnetization reversal. Figure 46 shows a nucleation process near the coercive state where half of the columnar grain magnetizations are in the reverse direction. The nucleation process shown in this figure involves a small circular chain of grains. The magnetizations of the six grains in the chain rotate into the film plane, forming an in-plane magnetization vortex (picture 2). Again, the dispersion in magnetization rotations results in magnetization reversal of only four grains in the chain. The magnetizations of the other two grains rotate back to the original perpendicular direction (pictures 6 , 7 , 8 , and 9). A nucleation process can also contain a reversed grain. This usually occurs where there is a dense distribution of previously reversed grains, often near the end of the hysteresis loop. During the nucleation process, the magnetization of the reverse grain (or grains) in the chain rotates back to the film plane

338

H. NEAL BERTRAM AND JIAN-GANG ZHU

FIG.46. Transient magnetization configurations during a nucleation process near the coercive state. The pictures are time ordered and show a circular nucleation chain. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 66, 1291 (1989).

along the chain rotation direction, still resulting in a head-to-tail magnetization configuration minimizing the magnetostatic energy. The underlying physics of the chain nucleation mode has been discussed in Section 6d. It is of interest to compare this nucleation mode with the curling mode (Section 6b), which is considered to be the reversal mechanism of individual columnar grains and has been suggested for coupled grains in CoCr perpendicular films.lZ6For columnar grains with large aspect ratio, the curling mechanism is more energetically favorable than uniform rotation (6.2). However, intergranular magnetostatic interactions via the chain reversal mode significantly reduce the nucleation field. In Fig. 47, film coercivity is plotted as a function of intergranular boundary separation for h, = 0.246 and he = 0. The nucleation field for the curling mode is plotted for comparison. Nucleation of curling does not depend on the intergranular boundary separation because magnetostatic interactions are not incorporated. The coercivity for single-particle uniform rotation is H,/HK 2.1 and 1.7, due to shape anisotropy (6.2). The exceeds that of curling, HJHK chain nucleation mechanism yields a coercivity approximately equal to the

-

IZ6K.Ouchi and S. lwasaki, ZEEE Trans. Magn. MAG-23,2443 (1987).

-

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339

CURLING

0.8

I

I

I

I

I

0.4

0.0

Inter-columnar

0.8

I

I

1.2

I

I

1.6

boundary separation d/D

FIG.47. Coercivity versus intergranular boundary separation d / D with fixed aspect ratio 6/ D = 10. The coercive field correspondingto the curling mode (6.2) is plotted as a dashed line and is independent of d/D. For each separation the system follows the nucleation mode with the lowest nucleation field. J. G. Zhu and H. N. Bertram, unpublished result.

-

crystalline anisotropy, H J H , 1. Simulated film coercivity significantly increases as intergrain separation increases , becoming larger than that for curling at d / D 0.85. A system always reverses by the lowest possible mode. Therefore, chain reversal of uniformly magnetized grains occurs for dense arrays of grains that would individually reverse by curling. The chain reversal process relies on coherent rotation of adjacent grains removing the shape anisotropy of each individual grain. With intergranular separation, rotation of adjacent grain magnetizations causes magnetostatic fields due to side poles of opposite polarity on adjacent grain surfaces. With large separations, individual shape anisotropy must be considered and the system is best modeled by an array of interacting curling grains.’” In this case the film coercivity is simply that of curling, because interactions resemble a mean field. Because of the chain nucleation mechanism, the number of grains having their magnetization reversed in a single nucleation process is very limited. In the hysteresis loop shown in Fig. 44, in over 88% of the total nucleation processes the number of grains reversed in each nucleation process is less than three. The average number of grains reversed per nucleation process is approximately two. Because a reversal virtually always occurs where the

-

I2’G. T. A. Huysmans, J. C. Lodder, and J. Wakui, J. Appl. Phys. 64, 2016 (1988).

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H. NEAL BERTRAM AND JIAN-GANG ZHU

FIG.48. Static magnetization patterns at various states along a major hysteresis loop for a non-exchange-coupledfilm. The gray scale represents the perpendicular component of magnetization. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 66, 1291 (1989).

demagnetizing field is largest, the resulting static magnetization pattern is that of a homogeneous distribution in the film of reversed columns. In Fig. 48 magnetization patterns are shown at various static states during reversal with the gray scale representing the perpendicular magnetization component of the grains. The domain patterns can be characterized as dots and singlecolumn-wide chains and are similar to those observed by Lorentz microscopy.''' Note that noninteracting grains would yield a random distribution of spacings between reversed grains. '"P. J. Grundy, M. Ali, and C. A. Faunce, ZEEE Trans. Magn. MAG20, 794 (1984).

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b. Reversal in Exchange-Coupled Films Introducing intergranular exchange coupling changes the hysteresis properties, the magnetitation reversal process, and the resulting static magnetization patterns. Figure 49 shows calculated hysteresis loops for various intergranular exchange couplings (C* = he). The film coercivity significantly decreases with increasing exchange coupling constant. In addition, the loop for he = 0.25 shows a well-defined shoulder at the beginning of the hysteresis loop. The effect of exchange coupling will be characterized by regions of weak and strong coupling. ( 1 ) Weak Exchange Coupling (0 < he < 0.2) In weakly exchange-coupled films, the chain nucleation mode still characterizes the nucleation process. In static states, magnetizations of the columnar grains are still virtually in the film normal directions. However, the width of the nucleation chain and the number of grains reversed in a single nucleation process increase with an increase of the intergranular exchange coupling. In addition, a reversed domain can expand through chain nucleation of unreversed adjacent grains as illustrated in Fig. 50. Pictures 1 and 4 in the figure represent static states at the beginning and end of the nucleation process. During nucleation, a wide chain nucleates adjacent to the previously reversed region (picture 2). Because of the exchange interaction, grains in the chain adjacent to the reversed region rotate more than the other grains (rotation dispersion). When their magnetization orientations pass the film plane, the other grains in the chain are driven back to the original unreversed state (pictures 3 and 4). Another example, shown in Fig. 51, demonstrates a chain nucleation process resulting in the connection of two previously reversed regions. The nucleation chain in this case is 3-4 columnar grains wide, and the nucleation process yields magnetization reversal of 11 grains in the nucleation chain. These two nucleation processes characteristically represent the chain nucleation processes in weakly exchange-coupled films. As a consequence, wider static stripe

_ _ _ _ C'=

0

-C'=O.

-C'=O.2

-C'=0.25

1

FIG.49. Calculated hysteresis loops for various values of intergranular exchange coupling constant (C* = he). Taken from J. G . Zhu and H. N. Bertram, J . Appl. Phys. 66, 1291 (1989).

342

H. NEAL BERTRAM AND JIAN-GANG ZHU

FIG.50. Transient magnetization configurations during a chain nucleation process for an exchange-coupled film (he = 0.1).The chain nucleation process yields magnetization reversal of two grains adjacent to previously reversed grains. Taken from J. G. Zhu and H. N. Bertram, J. Appl. Phys. 66, 1291 (1989).

8 0 0 0 0 0 $) 0 D 0 0 0 0 0 0 0

D

s 8

0

0

0 0

0

0

0

0 0 0 B 0

0 0 0 0 0 0 @ 8 0 0 0 0 0 0

Q e 0 0 0 0 0 0

u 0 0 0 El@ 0 B 0 0 0 0 0 P e s

!I 0 0 13 Q Q 8 8 0 0 0 0 0 B 8 0

*€.El

0

L? CI

€ % 8 l 3 0 0 0 0 0

@

Q

e3

0 0 0 0 0 0

w

B

u

Q

0 0 0 0 0 0

0 0

0 0 0 0 0 0 0 0

5

0 0 0 0 0

0 0

Li

e 0 0

0 0 0 0 0

6

FIG.51. Simulated transient magnetization configurations of a nucleation process for an exchange-coupled film with he = 0.15. This nucleation results in the connection of the two previously reversed regions. S / D = 5 and d / D = 0.02 were utilized. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 66, 1291 (1989).

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FIG.52. Simulated static magnetization patterns at various states along the major hysteresis loop for a weakly exchange-coupled film, he = 0.1.The gray scale represents the perpendicular component of magetization. Compared to Fig. 48, the reversed domains are wider. Taken from J. G. Zhu and H. N. Bertram, J. Appl. Phys. 66, 1291 (1989).

magnetization patterns occur as shown in Fig. 52. Increasing the intergranular exchange coupling increases the stripe domain width.

(2) Strong Exchange Coupling (he > 0.2) In the case of strong intergranular exchange coupling, a reversed domain still nucleates through the chain nucleation mode, but a nucleation chain becomes much wider than that in weakly or non-exchange-coupled films. Static magnetization configurations

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H.NEAL BERTRAM AND JIAN-GANG ZHU

of grains in between oppositely magnetized grains exhibit large in-plane components. A Bloch-like domain wall is evident with wall width approximately two to three columnar grains. A wall motion-like expansion of reversed domains, shown in Fig. 53, reduces the nucleation field. Intergranular exchange coupling does not significantly change the nucleation field for initial nucleation in the hysteresis process; initial reversal occurs at almost the same external field value for films with and without intergranular exchange coupling (Fig. 49). Dynamic expansion of reversed domains after nucleation in strongly exchange-coupled films results in the shoulder seen in Fig. 49 for he = 0.25. In the saturation remanence state the perpendicular magnetization has been sheared by the film perpendicular demagnetizing factor to a value approximately equal to the coercivity (Fig. 44). Thus, in perpendicular media, measured films exhibit squarenesses S Hc/47cM,. The formation of a saturation remanent magnetization of magnitude HJ4n is due to chain nucleation processes. Thus, because all the states, for at least weak exchange coupling, consist of reverse columns or clusters, the fraction of reverse clusters in the remnant state is given by n- OS(1 - Hc/47cM,). The cluster size relative to the grain diameter is set by the interaction strengths h,, he.

-

-

-

FIG.53. Magnetization configurations of a transient expansion process for a reversed domain for a strongly exchange-coupled film, h, = 0.25. Pictures 1 and 6 represent static states before and after nucleation. Taken from J. G. Zhu and H. N. Bertram, J. Appi. Phys. 66, 1291 (1989).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

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Magnetization fluctuations and medium noise are simply related to the ratio H J M , (Section 15). In this section the general effect of increasing intergranular exchange coupling for both longitudinal and perpendicular films has been shown to yield a continuous spectrum of behavior from “particulate” to “continuous”. For small or vanishing exchange coupling the reversal process is dominated by magnetostatic interactions yielding reversal fine structure on the order of a few grains: small vortices in longitudinal media and single or double column reversal in perpendicular media. With exchange coupling, reversal domain sizes become large, with either large vortices aligning into clear domain wall boundaries in longitudinal media or circular column reversal domains containing many grains in perpendicular media. In general, static processes occurring with small field increments in non-exchange-coupled media become dynamic wall motion processes in highly exchange-coupled media.

V. Simulations of the Magnetic Recording Process

10. SINGLETRANSITIONS The writing of a single transition in digital magnetic recording results from the application of a spatially varying head field. In the ideal case the medium has been dc erased by application of a saturating field, leaving the magnetization in a saturation remanent state. The head current is such that the field direction is opposite to initial saturation. Instantaneous application of the head field results in a transition (or two, one on either side of the gap). The transition relaxes as the medium moves, because for fixed current the field experienced decreases away from the gap center. Eventually at a fixed time interval (or motion distance) a second transition will be written. As the medium leaves the head, demagnetization field imaging disappears. The subsequent increase in demagnetization fields may broaden the transition if the original transition is sharp enough. For longitudinal media the relaxation of magnetization as the recorded transition moves away from the immediate vicinity of the gap region is small. For perpendicular recording this motion results in significant changes in the recorded pattern.”’ In the results presented here for perpendicular recording, a recording field that simulates the effect of medium motion is utilized in the writing of a tran~iti0n.l~’ Iz9J.-G.Zhu and H. N. Bertram, ZEEE Trans. Mugn. MAG-22, 379 (1986). IS0O.Lopez and D. A. Clark, J. Appl. Phys. 57, 3943 (1985).

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a. Transitions in Longitudinal Films The fundamental magnetization process in the formation of a transition is studied by examining the relaxation of an initially perfect transition. Except for extraordinarily small interactions parameters H,, he, a perfect transition corresponds to a energy maximum due to the large demagnetization fields (1.4)." Demagnetization-limited transitions result from recording with very large head field gradients, which occur, for example, when the head-medium spacing is extremely small (Fig. 2). A demagnetization-limited transition is obtained by letting a perfect head-on magnetization configuration relax following the micromagnetic equations (7.17), (7.18) in zero applied field. In Fig. 54 normalized time steps in the dynamic relaxation process for a perfectly oriented film are shown (h, = 0.65, he = 0.1). In this calculation, the crystalline easy axes of the grains are oriented exactly in the field direction. The initial state consists of all grain magnetizations oriented along the recording direction, forming a perfect head-on transition, uniform across the track or simulation array width. In order to initiate reversal, perturbations are applied to the initial magnetization orientation of each grain with slight random deviations ( <0.5") from the transition direction. Relaxation begins with the formation of vortices and flux saddle points at the transition boundary ( t = 3.6). As time progresses, the vortices expand and elongate along the transition direction ( t = 11.6,24.4). The final static transition boundary is an approximately regular zigzag structure ( t = 72.8) in which the magnetostatic interaction energy is minimized. The final equilibrium zigzag wall in Fig. 54 has slightly irregular spacings across the track as well as irregular transition widths in the recording direction. The wall angles are set by the hexagonal array symmetry, but the irregularities in wall structure result from the randomness in initial conditions. These complex systems, in general, possess many close-lying metastable states. A simpler, but approximate, alternative method of analysis is the cellular automata model (Section VI), which exploits the self-organizing behavior of highly interactive thin films. In Fig. 55 the three energy terms are plotted versus normalized time corresponding to Fig. 54. It is clear that the relaxation process is driven mainly by the magnetostatic fields. During the relaxation process, the crystalline anisotropy energy increases initially, due to magnetization rotation away from perfect alignment along the anisotropy orientation direction. When vortex elongation occurs, the magnetizations rotate back toward the recording direction and the anisotropy energy decreases. The intergranular exchange coupling decreases initially as vortices form at the wall boundary, reducing the head-on grain exchange energy. Subsequent vortex formation, elongation, and final zigzag wall structure correspond to the large decreases in magnetostatic energy accompanied by an increase in the exchange

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347

t’tz = 1 9’11 = 1

FIG.54. Transient magnetization configurations during relaxation from a head-on domain boundary in an exchange-coupled array: h, = 0.1. The crystalline easy axes of the grains are perfectly aligned in the transition direction (x-axis direction). t is normalized time. Other parameters utilized were h, = 0.65,6/D = 0.75, and d / D = 0.02. Taken from J. G. Zhu, “Interactive Phenomena in Magnetic Thin Films,” Ph.D. Thesis, University of California at San Diego, 1989.

energy.lo8 Different initial perturbations yield different detailed final patterns, but with the same energy progression. With no exchange coupling (dashed curves in Fig. 5 9 , similar temporal development of system energy occurs, although equilibrium results (generally) much more rapidly. In planar isotropic thin films, the final static transition configuration consists of magnetization vortices due to the randomly oriented crystalline

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H. NEAL BERTRAM AND JIAN-GANG ZHU

0.1

0.0

_ _ _ _ _ c*= 0 - C’=O.l

-0.1

M,/H,=O.

i

65

6/D=0.75

e,

a -0.2

d/D=O

h

M &

e,

d w -0.3

0

10

20

30

40

50

60

70

80

R e d u c e d time t FIG.55. Normalized system energies versus normalized time during the relaxation process shown in Fig. 54. Dashed curves represent the results for a non-exchange-coupledarray. Taken from J. G. Zhu, “Interactive Phenomena in Magnetic Thin Films,” PhD. Thesis, University of California at San Diego, t989.

easy axes. Two typical transition patterns are shown in Fig. 56 for (a) an exchange-coupled film with he = 0.1 and (b) a non-exchange-coupled film with he = 0. The transitions were formed, as in Fig. 54, by relaxation from a perfectly aligned transition in zero head field with h, = 0.52. The average magnetization integrated across the track or array width follows the pattern in Fig. 4a. Far from the region of the transition center the film is in the saturation remanence state and ripple patterns, as in Fig. 34, occur. The region in the transition centers where the average magnetization vanishes contains vortices and domain boundaries as discussed in Section 8 for uniformly magnetized films. With no intergranular exchange coupling, magnetization vortices are small in size and densely distributed. The vortex size in the exchange-coupled case is increased and their centers are well separated. Although the crystalline easy axes are completely randomly (3D) distributed, grain magnetizations lie virtually in the film plane because of the large magnetostatic fields. Magnetization flux closure in the film plane through vortex formation significantly reduces the magnetostatic energy, and the randomly oriented crystalline easy axes provide natural sites for the vortex centers. The head-on relaxation method generates a transition whereby a local energy minimum relative to the initial center is reached. Transitions generated by application of a spatial1 varying positive head field to an initially

MAGNETIZATION PROCESSES I N THIN-FILM RECORDING MEDIA

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FIG.56. Simulated transition patterns for (a) an exchange-coupled film, he = 0.1, and (b) a non-exchange-coupled film, he = 0. The crystalline each axes of the grains are randomly oriented in three dimensions. The magnetization configuration at the transition centers is characterized by vortices. Intergranular exchange coupling significantly increases the size of the vortices and separations between adjacent vortices. Taken from J. G. Zhu and H. N. Bertram, l E E E Trans. Magn. MAG-24, 2706 (1988), 0 1 9 8 8 IEEE.

reverse-saturated medium are generally broader than a demagnetizationlimited transition. However, the characteristics of the magnetization patterns in the transition region remain the same because they are determined mainly by intrinsic micromagnetic processes. A significant difference, discussed in Section 11, is that actual recorded transitions have enhanced fluctuations in the track width-averaged center positions.

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H. NEAL BERTRAM AND JIAN-GANG ZHU

The variation of array width-averaged magnetization with recording direction, (10.1)

is remarkably independent of interaction details as long as the ratio M J H , is fixed. Thus, the magnetization profiles of Fig. 56a and b are similar except that the exchange-coupled film has a slightly lower coercivity. A better comparison is shown in Fig. 57, where array-averaged magnetizations are plotted for exchange- and non-exchange-coupled films. If the magnetization, via h,, is changed along with the exchange he to keep a constant M J H , ratio, then the transition slopes at the center are identical with slight differences in the approach to saturation. The exchange-coupled film has a faster approach to saturation, somewhat like an error function. These results suggest that simple models for the transition width as given by (1.2) are accurate, whereas micromagnetic modeling is required to understand the fluctuations. b. Transitions in Perpendicular Films Figure 58 shows a typical simulated transition for a perpendicular film with h, = 0.1 but no exchange coupling. The transition is simulated by

-10

-5

0

5

10

position x/D FIG.57. Transition profiles for an exchange-coupled film (he = 0.1) and a non-exchangecoupled film. An arctangent function and an error function with common slope at the transition center are plotted in the figure to characterize the shape of the transition profile. Taken from J. G. Zhu, “Interactive Phenomena in Magnetic Thin Films,’’ Ph.D. Thesis, University of California at San Diego, 1989.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

.

-1.0' 0

351

. "

10

'

'

20

'

'

30

.

'

40

.

'

50

"

60

position x FIG. 58. Magnetization pattern of a recorded transition and corresponding transition profile in a non-exchange-coupled perpendicular film, he = 0. Other parameters used in the calculation are h, = 0.1,6/D = 5, and d/D = 0.02. The crystalline easy axes are randomly oriented in a solid cone angle centered in the perpendicular direction with half-cone angle A0 = 5". Taken from J. G. Zhu and H. N. Bertram, J. Appl. Phys. 66,6084 (1991).

application of a magnetic field produced by a pole tip recording head to a medium initially in the reverse remanent state. The gray scale in the figure represents the perpendicular component of the magnetization, with full bright representing the initial saturation direction. The magnetizations of the grains are all virtually perpendicular to the film plane, as indicated by the high contrast in the figure. Away from the transitions, oppositely magnetized isolated columns and single-column-wide narrow chains are characteristic of the saturation remanent state due to the demagnetization, as discussed in Section 9. Near the center of the transition on the side, where the magnetization is reversed by the head field, x 37, the magnetization is nearly saturated because of the small demagnetizing field.lo6 The overshoot in the transition profile is characteristic of perpendicular recording.

-

352

H. NEAL BERTRAM AND JIAN-GANG ZHU

The track width-averaged magnetization, -

M,(x) = -

w

1""

(10.2)

M,(x, z ) dz,

-w/2

-

is plotted in Fig. 58. The remanence magnetization at either side of the transition corresponds to M , HJ4n. At saturation remanence, on either side of the transition the remanence level is M J M , 0.8. Since for nonexchange-coupled perpendicular films H , H,, and for this example h, = M J H , = 0.1, the remanence ratio is M J M s = H,/4nM, = 1/4nh, 0.8. In Section 9 it was shown that exchange coupling in perpendicular media results in reversal patterns consisting of grain clusters. A simulated transition along with the average magnetization profile is shown in Fig. 59 with

1.0 \ 2

-

-

-

L

0.0 -

v

-

1

.

0

0

10

-

" ~ 20

'

30

"

" " 40

~ ~ 50

80

position x FIG.59. Magnetization pattern of a recorded transition and corresponding transition profile in an exchange-coupled perpendicular film, he = 0.1. Other parameters used in the calculation were h, = 0.1, a/D = 5, and d / D = 0.02. The crystalline easy axes are randomly oriented in a solid angle centered at the perpendicular direction with the half-cone angle A0 = 5". Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 66, 6084 (1991).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

353

he = h, = 0.1. The general character follows the pattern for non-exchangecoupled films. The average magnetization profile essentially follows the head field shape,'30 scales as H,,and, as in longitudinal films, is approximately independent of interactions. However, the large cluster size leads to large magnetization fluctuations. 11. TRANSITION NOISE Spatial magnetization fluctuations in thin-film recording media result in noise in reproduced voltage pulses, referred to as medium noise. Noise in the transition region is of particular interest because the recorded signal is maximum at the transition center (Fig. 3). In this section, computer simulation studies of transition noise in both in-plane isotropic longitudinal films and perpendicular films will be reviewed. Magnetization fluctuations in uniformly magnetized media will be discussed in Section 15 in terms of the self-organizing properties of these films. The only intrinsic source of magnetization fluctuations investigated to date in these simulations has been the dispersion of grain easy axis 3 3 To mimic transitions recorded at different positions and orientations.' to study their statistical properties, a large number of transitions are simulated, each with a different random distribution of the easy axis orientations. The variances of the transition profiles are calculated from these ensembles utilizing

'-'

(11.1) where (11.2) is the ensemble mean. M,(x) is the track width-averaged magnetization (lO.l), (10.2) and i is the ensemble index. a. Planar Isotropic Longitudinal Films Figure 60 shows the ensemble mean magnetization transition (1 1.2) (top) and the ensemble variance (11.1) (bottom). The four cases are for nonexchange-coupled grains, he = 0, and exchange-coupled grains, he = 0.1. TWO 13'J.-G. Zhu and H. N. Bertram, ZEEE Trans. Mugn. MAG24, 2706 (1988). "'J.-G. Zhu and H. N. Bertram, ZEEE Trans. Mugn. MAG-26, 2140 (1990). '33J.-G.Zhu, ZEEE Trans. Mugn. MAG-27, 5040 (1991).

3 54

H. NEAL BERTRAM AND JIAN-GANG ZHU

1.0 -

s-

0.5

-

6/D=1.5

\ A

‘j; 0.0 Y

5

-0.5

-

-40 -30 -20 -10 1.4r

I

I

0

10

I

I

0 10 position x

-40 -30 -20 -10

20

30

I

1

20

30

FIG.60. Ensemble mean and variance of the transition profiles for planer isotropic longitudinal films for different intergranular exchange coupling and both 2D and 3D random easy axis orientations. A large variance occurs at the transition center for he = 0.1 with either type of easy axis orientation. For he = 0, the variance is small for both 2D and 3D cases. Taken from H. N. Bertram and J. G. Zhu, IEEE Trans. Magn. MAG27, 5043, (1991), 01991 IEEE.

different random orientation ensembles, 2D and 3D, are also shown.134The Karlqvist head field function was utilized to record the transitions numerically.13sAs discussed in Section 10a, the transition ensemble average is virtually the same for all four cases because the transition shape depends primarily on M J H , and not on details of the microstructure and magnetic interactions. The common feature of the ensemble variance in Fig. 60 is that it is maximum at the transition center as expected from configurations of maximum d i ~ 0 r d e r . Increasing l~~ the intergranular exchange coupling significantly increases the variance for both 2D and 3D cases. With the same exchange interaction he, the variances for the 2D and 3D cases are virtually identical. 134H.N. Bertram and J.-G. Zhu, IEEE Trans. Magn. MAG-27, 5043 (1991). M. White, “Introduction to Magnetic Recording,” p. 23, IEEE Press, New York, 1985. 136T. J. Silva and H. N. Bertram, IEEE Trans. Magn. MAG-26, 3129 (1990). 13’R.

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In exchange-coupled films, large magnetization vortices and large separations between adjacent vortices in the transition region (Fig. 56) yield large irregular "zigzag "-like transition boundaries. This is illustrated by the gray scale plots in Fig. 61a and b. (The gray scale in the figure represents the magnetization component along the transition direction.) For solely magnetostatic coupled grains (Fig. 61b) the transition boundaries exhibit a closely spaced wall oscillation period consisting of a narrow fingering pattern. The scale of the irregular zigzag structure in the transition boundaries increases with an increase of the intergranular exchange coupling, as shown in Fig. 61a. The noise arises from fluctuations in these patterns from transition to transition. However, as emphasized in Section 15, the noise depends on the number of coherent subunits across the track. Non-exchange-coupled films, as in Fig. 61a, have many more of these coherent regions across the track width than exchange-coupled films (Fig. 61b) and consequently less noise. As a result, increasing intergranular exchange coupling significantly increases transition noise. Therefore, disrupting or weakening the intergranular exchange coupling by creating either voided or chemically segregated grain boundaries reduces transition noise, as verified in extensive experimental studies (Section 4). b. Perpendicular Films Figure 62 presents statistical behavior of ensembles of perpendicular transitions for non-exchange-coupled and exchange-coupled (he = 0.1) films. The randomness is initiated by the slight (< 0.5') dispersion in perpendicular anisotropy axes. In contrast to lingitudinal films, perpendicular films have the highest variance away from the transition center. Fluctuations in perpendicular media arise from the occurrence of reversed magnetization clusters (Figs. 58 and 59). Near the transition center, the variance decreases and reaches its lowest value at the transition overshoot, where the magnetization reaches saturation. The variance away from the transition center is higher in the exchange-coupled case than in non-exchange-coupled films because of the large domains that occur in exchange-coupled films. The density of oppositely magnetized domains at the saturation remanent state is determined by the ratio of film coercivity and perpendicular demagnetizing field ( M J M , = H,/47cMS).For zero exchange coupling, simulations of magnetization patterns at remanence have been performed for various 47cM,/H, ratios. It is shown that magnetization variance ( A M 2 ( x ) ) varies approximately in proportion to the quantity 47cM: - Hf., which is the result for a random distribution of oppositely magnetized columns (Section 15). Thus, increasing the film coercivity results in a decrease of the fluctuations until H , = 47cM,. A similar relation is expected for exchange-coupled films

356

H. NEAL BERTRAM AND JIAN-GANG ZHU

FIG.61. Magnetization patterns of recorded single transitions in planar isotropic longitudinal films for (a) he = 0.1 and (b) he = 0. Different h, values were chosen so that the two transition regions have approximately the same width (similar transition parameter a): h, = 0.2 for the exchange-coupled case and h, = 0.3 for the non-exchange-coupled case. The crystalline easy axes of the grains are randomly oriented in three dimensions. J. G. Zhu and H. N. Bertram, unpublished results.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

I -20

I

I

I

-10

I

I

0

I 10

I

357

I

20

position x/D FIG.62. Mean and variance of transition profile ensembles for CoCr perpendicular thin films with different exchange coupling constants: he = 0 (dashed curves) and he = 0.1 (solid curves). In contrast to longitudinal films, the variances are higher away from the transition center. The variance is lowest where the overshoot occurs. The variance level is higher for exchange coupling than for nonexchange coupled films. J. G. Zhu and H. N. Bertram, J . Appl. Phys. 69,6084 (1991).

except that the magnitude of the noise is increased in proportion to the cluster size. In perpendicular recording the average magnetization scales as the medium c o e r ~ i v i t y ' ~ ~and, ~ ' ~ 'as shown here, the medium noise decreases with increasing coercivity. This is in agreement with experimental studies that show a strong increase of signal-to-noise ratio verus increasing film coercivity. " 12. INTERACTING TRANSITIONS

In planar isotropic longitudinal thin films, transition noise has been shown to increase with recording d e n ~ i t i e s . ' ~ ~ .In ' ~ 'this section, simulations of 137J.-G.Zhu and H. N. Bertram, ZEEE Trans. Magn. MAG-22, 379 (1986). I3'S. B. Luitjens et al., ZEEE Trans. Magn. MAG-24, 2338 (1988). "'R. A. Baugh et al., ZEEE Trans. Magn. MAG-19, 1122 (1983). I4'N. R. Belk et al., J. Appl. Phys. 59, 557 (1986).

358

H.NEAL BERTRAM AND JIAN-GANG ZHU

pairs of sequentially recorded transitions (dibits) will be re~iewed.'~'A pair of transitions is calculated by simulating the corresponding recording process: Initially, the magnetization of the film is at a reverse remanent saturation state. A positive head field is applied to create the first transition. The motion of the medium is stepwise, with step size equal to the grain diameter D. After the medium is moved a distance B (recorded bit or cell length), the sign of the applied field is reversed and the second transition is written. The final magnetization pattern is obtained by setting the head field to zero. This process is repeated for a statistical ensemble of arrays with different random distributions of anisotropy axes. The essential conclusion is that intergranular exchange coupling results in a significant enhancement of the transition noise at small transition intervals. In Fig. 63, the calculated ensemble mean (top) and ensemble variance 1.0L

0.5 \ A Q W

I

h= ,O.

----

1

1

- 6/D=1.5 2nd tran.

0.0 - 1st tran. L

v -0.5

h,=O.

40

(Y

60

80

h II

x 2.0 1.5 1 .o 0.5

0.0 20

40 60 position x (D)

80

FIG.63. Mean and variance of simulated transition pair ensembles with he = 0 (solid curve) and he = 0.1 (dashed curve) for planar isotropic longitudinal films. A common h, = 0.1 is used in both cases. Crystalline easy axes are randomly oriented in the film plane (2D-random). For the exchange-coupled case, the variance peak at the second transition is significantly higher than that at the first transition. The variance for the non-exchange-coupled case is significantly lower at both transition centers with similar values. Taken from J. G. Zhu, IEEE Trans. Magn. M A G 27, 5040, (1991), 01991 IEEE. I4'J.-G. Zhu, IEEE Trans. Magn. MAG27, 3553 (1991).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

359

(bottom) of transition pairs are plotted for an exchange-coupled case (he = 0.1) and a non-exchange-coupled case (he = 0) for transition-writing separation B = 120. The ensemble average is virtually the same for the two cases except in the dibit tails. As expected from the discussion in Section 8a, the saturation remanence ( S ) is higher for exchange-coupled films. The variance peak at the center of both transitions is in accordance with the results for the single transition (Fig. 60). However, for the exchangecoupled film, the peak value of the variance at the second transition is much larger than that at the first transition. For the non-exchange-coupled case, the variances at both transition centers are similar and small. The increase of the variance at the second transition in the exchange-coupled case is a function of the intertransition interval B. In fig. 64 the peak of the transition variance is shown for both the first and second peaks of the exchange-coupled film versus the transition interval. For the non-exchange-coupled case only one curve representing the common variance peak height is shown. For wellseparated transitions the variance peaks correspond to that of the isolated transitions shown in Fig. 60. As the distance between transitions is reduced sufficiently, the variance peak at the second transition significantly increases and reaches a maximum followed by a gradual decrease. Figure 65 shows a typical simulated transition pair for both exchange-coupled (a) and non-exchange-coupled(b) films at intertransition interval B = 120 [where the variance peak at the second transition is near maximum in the exchange-coupledcase (Fig. 64)]. The gray

trans.

FIG.64. Peak value of the variance at the first and second transitions shown in Fig. 64 versus intertransition interval B.

360

H. NEAL BERTRAM AND JIAN-GANG ZHU

FIG.65. Magnetization patterns of sequentially recorded transition pairs (dibits) for (a) an exchange-coupled film (he = 0.1) and (b) a non-exchange-coupled film at B / D = 12 where B is the recording distance between the two transitions. The gray scale represents the magnetization component along the recording direction (x direction). For the exchange-coupled case, transition boundaries show large-scale irregular zigzag-like patterns, whereas for the non-exchangecoupled case, the transition boundaries are characterized by narrow finger patterns. Taken from J. G. Zhu, IEEE Trans. Magn. MAG-27, 5040, (1991), 01991 IEEE.

scale in the figure represents the longitudinal component of magnetization. In the exchange-coupled case, the first transition boundary consists of large irregular zigzag patterns. It is difficult to see the transition variance asymmetry in this plot, but the enhanced fluctuations in the exchange-coupled case are clearly evident. Noise enhancement of the second transition is due to fluctuations in the magnetostatic field acting on the second transition. These fluctuations across

MAGNETIZATIONPROCESSES IN THIN-FILM RECORDING MEDIA

361

the width of the second recorded transition modulate the recording field of the second transition, resulting in increased variance. The magnetostatic interaction field variance is increased both by reducing the transition separation and by increasing the exchange coupling. Increasing the intergranular exchange increases the correlation or cluster size in the transition zigzag pattern and hence the magnetostatic field fluctuations. When the intertransition interval becomes very small, percolation between adjacent transition boundaries occurs, as illustrated in Fig. 66 for exchangecoupled and non-exchange-coupled films. The percolated transition boun-

FIG. 66. Magnetization patterns of sequentiallyrecorded transition pairs for (a) an exchangecoupled film (he = 0.1) and (b) a non-exchange-coupled film at B / D = 8. At this intertransition interval, the transition boundaries percolation occurs. Taken from J. G. Zhu, IEEE Trans. Magn. MAG-27, 5040, (1991), 01991 IEEE.

362

H. NEAL BERTRAM AND JIAN-GANG ZHU

daries yield large island “domains” in the exchange-coupled case, while the percolation channels in the non-exchange-coupled case are much finer.

VI. Self-organized Behavior in Magnetic Systems

Many physical systems exhibit self-organized behavior. These systems are characterized by a large number of degrees of freedom that yield numerous, virtually identical, metastable states. Thus, marginal stability of the coupled dynamical equations gives rise to system fluctuations that exhibit no time or length scale over extremely large variable ranges. The driving force for state formation can be randomness in intrinsic properties (e.g., classic “sand piles”)’42 or fluctuations in initial conditions (e.g., earthquake^").'^^ Selforganized behavior can also occur in completely deterministic systems.’44 Thin-film recording media, which consist of strongly interacting assemblies of magnetic grains, can exhibit self-organized behavior. The system equilibrium states are virtually identical because of strong magnetostatic and exchange interactions that dominate the dispersion in intrinsic anisotropy. A magnetic system is brought to marginal stability or criticality by the application of an applied field of magnitude approximately equal to the coercive force. Magnetic states near the coercive state in highly interacting systems, which exhibit avalanche reversal processes, show self-organized behavior. In the recording process the magnetic states of an initially sharp transition are unstable because of self-demagnetization fields. Self-organized criticality was first demonstrated numerically in magnetic systems by studying the relaxation of a sharp t r a n ~ i t i 0 n . Instability l~~ and subsequent motion of the system to a demagnetization-limited stable state were shown; neither time scaling in the relaxation process nor length scales in the distribution of reversal domains along the magnetization direction resulted. The initial conditions of the computer model were a perfectly sharp transition; reversal was initiated by the random reversal of a spin of greatest instability somewhere across the track adjacent to the transition boundary. Experimentally self-organized behavior in magnetic systems has been studied for uniaxial perpendicular garnet films.146 For the polycrystalline thin films studied in this review, under uniform applied fields, a cellular automata model has been utilized and self-organized I4’P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987). 143J. M. Carlson and J. S. Langer, Phys. Rev. Lett. 62, 2632 (1989). 144K.Wiesenfeld, J. Theiler, and B. McNamara, Phys. Rev. Lett. 65, 949 (1990). 14’X. Che and H. Suhl, Phys. Rev. Lett. 64, 1670 (1990). 146K.L. Babcock and R. M. Westervelt, Phys. Rev. Lett. 64, 2168 (1990); R. M. Westevelt et al., J. Appl. Phys. 69, 5436 (1991).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

363

behavior in the reversal process has been observedl4’ and correlations studied.17 In the following three sections, the essential characteristics of self-organized behavior in hard polycrystalline films will be reviewed. This type of approach generally involves the use of local field models with assumptions that retain the collective nature of the processes. This simplification allows considerable computation time reduction, so that truly large-scale problems, N lo5, can be studied (with moderate computation level) and reasonable statistical characterization obtained.

-

13. MODEL

The thin-film structure is identical to that described in Section 7 (Fig. 27). In this simplified modeling only two directions of the magnetization of each cell are allowed: + or - along the direction of the applied field with magnitude equal to the saturation remanence M , . A log-normal distribution f(HK) of the magnitudes of nucleation fields is assumed (with dispersion width parameter a), randomly assigned to the cells. This distribution is given by (13.1)

Physically, H K represents a distribution in crystalline anisotropy fields either in magnitude or direction. The normalization and characterization of fields, including magnetostatic and exchange, are identical to those formulated in Section 7 except that the normalization parameter is the anisotropy distribution median Ho. Magnetostatic and exchange fields are included in a simple way. In a reversal process the magnetostatic and exchange fields on an unreversed cell arise from neighbor cells that have previously undergone reversal. These fields are added to the external applied field. Figure 67a illustrates the magnetostatic fields at the center cell from three cells assumed to be reversed (in the same direction as the applied field). The two upper cells produce the fields h: that are in the same direction as their reversed magnetization, while the right cell produces h i in the opposite direction. It is this characteristic property of the magnetostatic interaction fields where h i opposes the reversal of side cells that results in the formation of elongated, narrow reverse domains. The exchange fields he, illustrated in Fig. 67b at the center cell, arise from the same assumed reversed cell, as in Fig. 67a and are all in the reversed direction. This symmetry yields isotropic expansion of reversed domains. 147J.-G.Zhu and H. N. Bertram, J. Appl. Phys. 69,4709 (1991).

364

H. NEAL BERTRAM AND JIAN-GANG ZHU

Magnetostatic Interaction

Exchange Interaction

b

a

FIG.67. Diagram of local field approximation for (a) magnetostatic coupling and (b) exchange coupling. Taken from J. G. Zhu and H. N. Bertram, J. Appl. Phys. 69,4709 (1991).

Configurations analogous to those illustrated are easily inferred. Including the interactions in this manner is believed to incorporate the essential collective nature of the reversal processes yet to simplify substantially the computation. For simulating the M - H loops, the grains are first set parallel (saturated state), and a reversing field is then applied. The external applied field magnitude is increased until the net applied field on a cell is greater than its nucleation field, and then the moment is reversed. The interaction fields on the nearest neighbors are then recalculated, often resulting in a sequence of reversals. The condition that the net applied field be greater than the nucleation field is written as

h,,, = ha +

1h: nn

-

1h, + 1he 2 h,, nn

(13.2)

nn

where the sums are over only the reversed nearest neighbors (nn). Simultaneous multiple switching is not allowed; the grain satisfying (13.2) with the highest net reversing field will be reversed first. Subsequently, the interaction fields that this reversal produces will be taken into account, in general yielding subsequent neighbor reversals (domain growth). After the reversals have settled, the process is repeated, increasing the external applied field,

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

365

allowing the formation of magnetization patterns and associated hysteresis loops. In practice, the applied field is given stepwise changes that are sufficiently large to make the next grain reverse after previous domain reversals have settled.

14. REVERSALPATTERNS In this section domain growth during magnetization reversal along the major M-H loop is discussed. In addition, characterization of interactions by iim curves is presented. For given values of the exchange and magnetostatic interaction fields (he, h:, and h i ) and the dispersion parameter CJ of the log-normal distribution of nucleation fields, magnetization patterns corresponding to applied fields h, may be computed. The medium is initially saturated in one direction, which corresponds to a normalized magnetization m = + 1 or - 1; then the field is applied in the opposite direction. The field magnitude is increased, initially to a value at which the first reversal occurs. After each configuration settles, a pair (fi, h 3 is stored as well as the that magnetization configuration. The calculation ends when f i = - 1 or is, all grains are reversed.

+,

a. Magnetization Patterns Figure 68a-c shows magnetization patterns at f i = -0.75, 0.0, and 0.75, respectively, for an appropriate applied field after initial saturation at f i = - 1.0. The parameters were h: = h; = 0.5, with he = 0 . 1 , ~ =~0.25, and array size 100 x 400. At f i = -0.75 the magnetization has changed only slightly from initial (negative) saturation, with few reversed domains having formed. The reversed domains nucleate at random locations in the array, as expected for the randomly assigned nucleation fields. Increasing the field to the next nucleation either increases the size of previously nucleated domains or initiates new ones. Typically the domains grow by acquiring approximately a constant width and then expand along the field direction, acquiring elongated forms. At the coercive state, iE = 0.0, the domains have coalesced to a disordered antiferromagnetic state, somewhat similar to the ac-erased state.67 Subsequent increase of the applied field yields growth of reversed domains so that, for example, at f i = 0.75, a pattern containing only a few remaining unreversed domains occurs. Increasing the magnitude of the intergranular exchange increases the domain width and length distribution. l 7 Distributions of domain sizes show no evidence of length scales, especially near the coercive state.67With increasing intergranular exchange, patterns as shown in Fig. 68 occur but the domain widths are increased (conversely with decreasing exchange). The patterns do not simply scale to a larger size, however. The

366

H. NEAL BERTRAM A N D JIAN-GANG ZHU

FIG.68. Magnetization patterns along the hysteresis loop for m = -0.75 (a), 0.0 (b), and 0.75 (c). Initial state was % = -i1. Parameters were h: = h i = 0.5, with he = 0.1, u = 0.25, and array size 100 x 400. Taken from H.N. Bertram and R. Arias, J . Appl. Phys. 71(7), 3439-3454, (1992).

domain length distribution does not increase and, in fact, decreases somewhat; magnetostatic interactions favor long narrow domains, whereas exchange interactions favor more isotropic domain sizes.

b. 6m curves In the study of interacting particle systems, a simple measurement to assess interactions by examining the difference between a scaled initial remanent magnetization curve and the major remanent M - H curve has been ~ u g g e s t e d . l ~ *This . ' ~ ~difference defines the 6m curve: (14.1) P. Wohlfarth, J. Appl. Phys. 29,595 (1958). 1490. Hankel, Phys. Status Solidi 7,919 (1964). I4'E.

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

367

where ir(hext)and mr(hext)are the normalized (to M , ) remanence curves for the initial and major loop magnetization curves, respectively. These curves are illustrated in Fig. 3, where remanence curves result from the removal of the applied field at each step. For a noninteracting system 6m = 0; interactions either by a mean field or by the collective processes discussed in this review yield 6m # 0. Depending on the microstructure and magnetic interactions, either positive or negative 6m curves occur, and it has been suggested that measurement of this simple parameter allows assessment of eventual medium noise in recording configurations.' Simplified models are required for the calculation of 6m curves because extensive numerical computation when utilizing the full micromagnetic model is required to obtain accurately the acerased state. In addition, an extremely large array size is necessary in order to obtain sufficient accuracy, because small differences between two magnetization curves are being assessed. In Fig. 69 experimental measurements of 6m curves are shown for a series of thin polycrystal films with different noise levels.'50 These media were prepared with varying Cr underlayer thicknesses in order to control grain segregation while not appreciably changing other magnetic layer properties (Section 4). As discussed in Section 6 and also in the next section, increased

I Delta M 1.26

t

-

Underlayer

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

1 c ...............................

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

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

,

100 A

.....

1000 A

2000 A

0.76

0.6

0.26

0

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

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

-0.26 0

600

1000

IWO

2000

FIG.69. Measured 6m curves versus field for a series of planer CoCrPt/Cr longitudinal films with varying Cr underlayer thickness. Taken from P. I. Mayo et al., J . Appl. Phys. 69, 4733 (1 99 1).

'''P. I. Mayo et al., J. Appl. Phys. 69, 4733 (1991).

368

H. NEAL BERTRAM AND JIAN-GANG ZHU

grain segregation reduces intergranular exchange coupling and consequently magetization fluctuations or medium noise. In Fig. 69 increasing Cr underlayer thickness changes the measured Sm curve from strongly positive to mainly negative. Figure 70 shows Srn curves calculated utilizing the cellular automata The initial magnetization curve was calculated from an ac-magnetized state simulated by analogy to a conventional ac demagnetizing process. In Fig. 70, calculated Srn(h) is plotted for four different exchange field strengths. The asterisks in the figure mark the corresponding remanent coercive states. Fr the case he = 0.25, Sm is mainly positive. As the exchange field strength decreases, the peak decreases and broadens. For zero exchange field, he = 0, Sm is mainly negative. Note that the extremum in each curve occurs at a field slightly less than the remanent coercivity. The predominant sign of a Srn curve for polycrystalline films can be understood qualitatively. With only magnetostatic coupling, the equilibrium energy minima for the ac-erased state are deeper than the equilibria at saturation (rn = - 1 on the major loop): In the ac erased state the domains comprise dense antiparallei chains of spins with minimized magnetostatic energy. Therefore, application of a field results in less magnetization change for the initial curve than from the saturated state. This yields a negative Srn curve; the negative extent of the Sm curve for he = 0 depends on the ratio of h; and h i . With h: constant, increasing the value of h; results in a more

75

0.8

0.0

0.2

I

n

0.4

- - - h,=0.2 "

__--_

0.6

h,=0.1

0.8

1.0

1.2

normalized applied field ha FIG.70. Calculated 6m curves versus field for various exchange couplings he. Taken from J. G. Zhu and H. N. Bertram, J . Appl. Phys. 69, 4709 (1991).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

369

pronounced negative portion because the antiferromagnetic state of ac erasure becomes even more stable. With exchange coupling, the formation of reversed domains involves many grains. In the ac-erased state an antiferromagnetic pattern again occurs but with wide domains. Application of a field yields large magnetization changes because the magnetization process with exchange is predominantly by wall motion. From the saturated state the initial nucleation of reverse domains is not appreciable changed. Thus, with exchange, the ac-erased state is less stable compared to nucleation from a saturated state and positive Srn curves occur. The 6rn curves are very sensitive to microstructure as well as detailed magnetic interactions; however, a direct relation to magetization fluctuations and noise may occur only for particular structures and parameter variations. 15.

CORRELATIONS AND

NOISE

Magnetization fluctuations are readily assessed utilizing less computationally intensive models, because large array size for sufficient statistical averaging is possible. Uniformly magnetized media provided a fundamental basis for assessing fluctuations because the random processes are stationary in all directions in the recording plane. For complete stationarity, magnetic films may be characterized by a two-dimensional correlation f~nction:’~’ P(X, z ) =

(rn(x’, z’)rn(x’

1

+ x , z’ + z ) ) - if?

(15.1)

-m2

+

+

where m is the array average magnetization and (rn(x’, z’)rn(x’ x , z’ z ) ) is the two-dimensional autocorrelation function. The variance d of this two-state random process with oriented magnetic grains (rn = f 1) is

A useful parameter for the discussion of the variation of correlation coefficient with magnetic interactions is an integration in the direction orthogonal to the moment and applied field direction (cross-track):” (15.3)

15’W. B. Davenport, Jr., and W. L. Root, “An Introduction to the Theory of Random Signals and Noise,” p. 55, McGraw-Hill, New York, 1958.

3 70

H. NEAL BERTRAM AND JIAN-GANG ZHU

where W is the array width or replay head width in the recording process. S(0) represents the correlation or domain width, whereas S(x) yields an array width-averaged decay of correlation in the magnetization or field direction. The general behavior of correlations can be inferred from Fig. 68a-c. The spatial decay of transverse correlation (or domain width in Fig. 68) yields an S(0) that is a slight function of the state of magnetization. However, because increased exchange yields large correlated magetization clusters or domains, S(0) increases strongly with increasing exchange. The correlation length in the recording direction [perhaps defined by S(I,) = 0.51 increases with increasing exchange but also depends on the state of magnetization: Maximum correlation length occurs at the coercive state, E = 0. Experimental studies of these correlations have been performed.' 36*15 2 The noise power of the magnetization fluctuations averaged over the total area of the array is17 NP,

=

lim L

-03

w Ldx (m(x) - E)'

-

L

where m(x)

3 -

w

s,

S""

dz m(x, z).

(15.4)

(15.5)

-w/2

It can be shown that this definition of noise is directly related to the magnetization variance and the cross-track integration of correlation function S(0): NP,

= S(O)(l -

m2).

(1 5.6)

An example of the variation of NP and S(0) with state of magnetization is given in Fig. 71. The curves are plotted versus iii and the field variation takes the system along a major loop from one state of saturation to the other. Because S(0) is a slight function of the state of magnetization, the total noise power defined by (1 5.1) vanes, approximately, as the simple statistical disorder variance 1 - iii2 (15.2). In Section 11 noise in recorded transitions for both longitudinal and perpendicular media has been reviewed. Even for recorded transitions where the average magnetization varies spatially in the recording direction, crosstrack fluctuations in both types of materials, to first order, follow the simple variance formula 1 - E2,where iii + m(x) as defined by (15.5). Thus, medium noise varies with position along the track approximately as 1 - r n ( ~ ) ~In. longitudinal media, therefore, noise increases with position into the transition lSzG.J. Tarnopolsky, H. N. Bertram, and L. T. Tran, J. Appl. Phys. 69, 4730 (1991).

MAGNETIZATION PROCESSES IN THIN-FILM RECORDING MEDIA

37 1

1.4r

1.2 1 .o

0.8 0.6 0.4

0.2 0.0 -0.8

-0.4

0.0

-

0.4

0.8

m FIG.71. Noise power S(O)(l major hysteresis loop.

- f i 2 ) and

transverse correlation width S(0) versus ii along the

center as the magnetization decreases (Fig. 60). In perpendicular recording the noise variation is more complicated. In the “saturated” regions away from the transition center the magnetization is not saturated but demagnetized to a value limited by the coercivity (Section 11): Iiii(+oo)l = H,/47cM, < 1. Thus, in perpendicular recording, the predominant contribution to medium noise occurs in the saturated regions on either side of the transition center (Fig. 62). Thus, in perpendicular recording, the noise varies approximately as 1 - f i ( c ~ M) 1~ - (Hc/47cM,)* and therefore decreases with increasing coercivity. In both longitudinal and perpendicular recording the magnitude of the noise level varies as the domain or cluster size, and cluster sizes increase strongly with increasing intergranular exchange coupling.

ACKNOWLEDGMENTS The work was supported by the Center for Magnetic Recording Research, a fellowship from the IBM Corporation, and NSF grant DMR-87-07421. The authors would like to thank Tom Arnoldussen and Tadashi Yogi for helpful technical suggestions and Samuel Yuan and Xiandong Che for careful reading of many sections. Ann Bertram carefully edited the entire manuscript. The San Diego Supercomputer Center at UCSD provided utilization of their Cray X-MP and Y-MP for these computations.

This Page Intentionally Left Blank

Author Index

Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his or her name is not cited in the text.

A

Abad, H., 111(337), 111, 140(337) Abeles, B., 223(146), 223 Abstreiter, G.,.6(52), 6, 33(52), 36(52), 84(279), 84, 88(279), 93(290), 95-97(290), 95 Adachi, S., 39(130), 39 Adams, A. R., 28(92), 28 Adler, J., 164(19), 164, 188(19), 193(69), 193, 197(19) Afros, A. L., 200(105), 200 Agarwal, S., 284, 285(42), 285 Aggarwal, R. L., 71(236), 71 Aharony, A,, 164(19), 164, 188(19), 190(62), 190, 197(19, 89, 91), 197, 254(207), 254, 316(109), 316 Aina, L., 44(157), 49(157), 49, 140(157) Alani, R., 294(71), 294 Alavi, K., 32(113), 32, 33(123), 33, 34(113), 36(113), 41(150), 44(150), 48(150), 48, 140(150) Alessandrini, E. I., 247(196), 247 Alexander, L. F., 53(182), 53 Alexander, S., 197(91), 197, 218(133), 218 Alfano, R. R., 239(173), 239,242(173) Ali, M., 298(85), 298, 340(128), 340 Allam, J., 41-44(136), 42, 140(136) Allan, G., 97(297), 97 Allegre, J., 99(300-301), 99 Allen, P. M. G., 31(108), 31, 125-128(108), 140(108), 142(108) Allyn, C. L., 32(109), 32, 34(109), 36(109) Altarelli, M., 28(93), 28, 53-54(93), 56(93), 58(93), 140(93) Althoff, A., 242(188), 242 Alwan, J. J., 93(291), 95(291), 95,97(291) Ambegaoker, V., 200(104), 200 Anderson, P. W., 239(172), 239 Anderson, R. L., 4-5(19), 4, 8(19), 39(19), 59(19), 65-66(19), 68-69(19), 73(19),

88(19), 99(19), 107(19), 113(19), 116(19), 118(19), 122(19), 131(19) Anderson, S. G., 30(97), 30, 120(97) Andersson, T. G., 93(286), 95-97(286), 95 Angles d’Auriac, J. C., 218(135), 218 Aoki, M., 100(304), 100 Aponte, J., 192(64), 192,220(64) Appelbaum, J. A., 5(33-34), 5, 22(33-34), 108(33-34) Archie, G. E., 171(37), 171, 231(37) Arias, R., 274(1’J), 274, 293(17), 318(17), 363(17), 365(17), 366, 369-370(17) Arnold, D., 6(60), 6, 32(60), 35-38(60) Amoldussen, T. C., 278(23), 278, 281(29), 281, 282(29, 31), 282, 290(29, 31), 295(23, 79), 295 Ashcroft, N. W., 237(168), 237,242(168) Ashenford, D. E., 71(239), 71 Atanasoska, Lj., 30(97), 30, 120(97) Austin, R. F., 98(298), 98, 115-118(339), 115, 140(339)

B Babcock, K. L., 362(146), 362 Bachmann, K. J., 32(117), 32, 122(117), 124(1 17) Bajaj, K. K., 93(296), 96-97(296), 96 Bak, P., 362(142), 362 Baldwin, K., 84(278), 84, 88(278) Ball, C. A. B., 78(258), 78 Ballard, N., 294(69), 294 Ballingall, J. M., 106(325-326), 106 Ballutaud, D., 73(243), 73 Balslev, I., 76(255), 76 Banavar, J. R., 184-185(52), I84 Banyai, L., 267(225), 267 Baraff, G. A., 5(33-34), 5, 22(33-34), 108(33-34) Barbier, E., 93(292), 95-97(292), 95

373

374

AUTHOR INDEX

Bardeen, J., 227(150), 227 Bass, S. J., 42-44(138), 42, 140(138) Bastard, G., 53(176), 53, 57(187), 57, 62(198), 62, 71(240), 71, 94(187) Batey, J., 6(48-49), 6, 32(48-49), 35- 38(48-49), 140(48-49) Batrouni, G. G., 188(55-56), 188, 252(55) Bauer, R. S., 36(128), 38-39(128), 39, 103(314), 103, 104(316-320), 104, 105(314, 316-319, 323), 106(323), 106, 115(128), 140(314, 316-319, 323) Baugh, R. A., 357(139), 357 Bawolek, E. J., 111(334), 111 Beale, P. D., 22q143-144), 220, 264(143-144), 266(143-144) Bean, C. P., 281(30), 281, 308(107), 308 Bean, J. C., 78(260,264-265), 78,79(260, 264), 8q264-265), 83(275-276), 83, 84(278), 84, 88(278), 91(276) Becker, W. M., 70(230), 70 Belk, N. R., 357(140), 357 Bell, L. D., 144(351-352), 144 Benguigui, L., 198(96), 198 Beresford, R., 53(169), 53, 57(192), 57 Bergman, D. J., 150(2), 150, 154(4), 153-154, 156(2, 8-10), 156, 157(8-9), 158-159(2, 12), 158, 160-161(10, 12), 161, 164(8, 17), 164, 168(24), 168, 171(10), 179(2,8), 180(8), 181(12), 183(12), 184(5l), 184, 185(12), 190(60-62), 190, 198(93), 198, 200(103), 200, 202(60, 110-112), 202, 203(60, 110-111, 113), 203, 204(113), 206(111), 207(110), 210(120), 210, 211(110, 112), 212(123), 212, 213(127), 213,214(113), 238(169), 238,247-248, 252(203-204), 252,254(61,206), 254, 256(204), 257(204, 212), 257,259(215), 259,262(219-220), 262, 263(219), 265(223), 265, 266(203, 223-224), 266, 268(215) Berkowitz, A. E., 272(1), 272 Bernards, J. P. C., 288(47), 288 Berroir, J. M., 62(199-200), 62, 64(199), 71(240), 71 Berryman, J. G., 184(47-50), 184 Bertram, H. N., 273(6, 8), 273, 274(17), 274, 275(18), 275,278(18), 293(17, 67), 293, 304(101), 304, 306(104-106), 306, 311-312(106), 318(17), 320(111, 116),

320, 321(117), 321-322, 323(117), 324, 325(119-120), 325-327, 328(122), 328-334, 335(125), 335-344, 345(129), 345, 346( 18), 349, 351-352, 353(131-132), 353, 354(134, 136), 354, 356, 357(137), 357, 363(17, 147), 363-364, 365(17, 67), 366, 368(67), 368, 369(17), 370(17, 136, 152), 370 Bevk, J., 31(103), 31, 77(103), 81(103), 83-85(103) Bhattacharya, S., 198(95), 198 Bicknell, R. N., 70(229), 70 Biefield, R. M., 74(244), 74 Bir, G. L., 74-75(249), 74 Blair, S. C., 184(48), 184 Blakeslee, A. E., 78-79(259), 78 Blanks, D. K., 70(229), 70 Blue, J. L., 320(113), 320 Blumenfeld, R., 190(62), f90,254(207), 254, 259(215-216), 259, 261(216), 262(219-220), 262,263(219), 268(215) Board, K., 44(149), 47(149), 47, 140(149) Bonnefoi, A. R., 32(110), 32,62-64(110), 140(110) Bonsett, T. C., 70(230), 70 Born, M., 235(165), 235 Bottka, N., 71(235), 71 Boukerche, M., 70(234), 70 Bowman, D. R., 219(139), 219 Bradley, J. A., 6(51), 6, 28(51), 37(51) Braunstein, F. O., 299(93), 299 Braunstein, R., 89(280), 89 Breton, P., 252(199), 252, 254(199), 256(199), 257(211), 257 Broadbent, S. R., 192(65), 192 Brown, W., 32(109), 32, 34(109), 36(109) Brown, W. F., Jr., 154(3), 154, 164(3), 278(21), 278, 279(25), 279, 310(25), 315(21) Bruggeman, D. A. G., 155(5), 155, 165(5), 193(5) Brugger, H., 84(279), 84, 88(279) Brunemeier, P. E., 42(133), 42,44(133), 47(133) Bruno, O., 165(20), 165 Bube, R. H., 65,67(221-222), 69(221-222) Buch, F., 66, 67(221-222), 69(221-222) Buckland, E. L., 70(229), 70 Buehler, E., 32(117), 32, 122(117), 124(117)

AUTHOR INDEX

Bug, A,, 238(170), 238 Buhrman, R. A., 225(147), 225, 236(147), 247(912), 247 Bylsma, R. B., 70(230), 70

C Caine, E. J., 44(156), 49(156), 49 Calabrese, J. J., 247(194), 247 Callegari, A. J., 171(35), 171 Calleja, J. M., 54(189), 57(189), 57, 117(189) Callen, H. B., 280(28), 280 Cammack, D., lll(333, 336), 111, 14q333, 336) Capasso, C., 121(341-342), 121 Capasso, F., 2(16-17), 2,41-44(136), 42, 140(136, 341-342), 144(349), 144 Capik, R. J., 27(88), 27, 41-42(88), 44(88), 140(88) Capozi, M., 7(66-67), 7, 54(66) Cardona, M., 5(39), 5, 19(39), 21(39), 23(39), 25(39), 40(39), 43(39), 45(39), 52(39, 164), 52, 55-56(39), 59(39), 64(39), 65(39, 212), 65, 67(39), 68(39, 226), 68, 69(39), 72(39), 75(250), 75, 76(252), 76, 98(39), 103(39), 107-108(39), 113(39), 116-122(39), 124(39), 126-128(39), 130-131(39), 137(39), 141-142(39), 144(39) Carlson, J. M., 362(143), 362 Carrier, J., 320(114), 320 Casey, H. C., 2(8), 2 Caudano, R., 62-64(206), 63, 14q206) Cavicchi, R. E., 41-44(139), 42, 140(139) Cebulla, U., 54(191), 57(191), 57, 140(191) Cerva, H., 78(267), 78 Chadi, D. J., 16(76), 16 Chakrabarti, B. K., 265(221), 262, 267(221) Chambers, S. A., 105-106(324), 106, 140(324) Chandrasekhar, M., 76(253), 76 Chang, C.-A., 53(165-166, 182), 53, 54(165-166, 188-189), 56(165), 57( 185, 188-189), 57, 58(165-166), 104(321), 104, 117(189), 14q165-166, 188) Chang, J. C. P., 50(160), 50 Chang, L. L., 2(10), 2, 28(93), 28, 53(93, 165-166, 171, 173-176. 182-184). 53. 54(93, 165-166, 188), 56(93, 165),

315

57(185, 188), 57, 58(93, 165-166), 71(240), 71, 140(93, 165-166, 188) Chang, S.-K., 71(238), 71 Chang, W. S. C., 93(288), 95-97(288), 95 Chapman, J. N., 288(47), 288 Charasse, M. N., 93-94(284), 94,96-97(284) Chatt, P., 99(300-301), 99 Che, X., 362(145), 362 Chemla, D. S., 41(151), 44(151), 48(150), 48, 140(151) Chen, T., 286(44-45), 286,291 Chen, W., 304(103), 304 93(293), 96(293), 96, 98(293) Chen, Y.-K., Chen, Z., 183(43), 183 Chen, Z. G., 93(286), 95-97(286), 95 Cheng, H., 30(97), 30, 111(337), 111, 120(97), 140(337) Cheung, J. T., 31(100), 31, 62-63(100), 65(100), 140(100) Chew, N. G., 42-44(138), 42, 140(138) Chiaiadia, P., 104-lOS(316-318), 104, 140(316-3 18) Chiang, T.-C., 126-127(345), 126 Chien, W. Y., 6(55), 6, 32(55), 34(55), 36(55) Chin, R., 42(131), 42,44(131), 47(131) Chin, T. P., 50(160), 50 Chiu, T. H., 10-11(72), 10, 28(72), 54(72), 57(72), 117(72), 140(72) Cho, A. Y., 2(17), 2, 32(113), 32, 34(113), 36( 113), 41( 150-15 l), 44(150-151, 156), 48(150-151), 48, 49(156), 49, 93(289, 294), 95(289), 95, 96-97(289, 294), 95-96,98(294), 14q150- 151), 144(349), 144

Chow, D. H., 6(64), 6, 31(105), 31, 32(110), 32, 35-36(64), 38-39(64), 53(168, 172, 180), 53, 54(190), 57(190), 57, 62(110, 203), 63-64(110, 203), 63, 74(180), 83(105), 109(105, 190), 115-118(105, 190), 119(190), 140(105, 110, 190), 141(190) Chow, P. P., 78(266), 78, 81(266) Christensen, N. E., S(39-41), 5, 21(39), 23(39-41), 24(40-41), 25(39-40), 40(39-40), 41(40), 43(39), 45(39), 52(39, 164), 52, 55-56(39-40), 59(39-40), 64-65(39-40). 67(39), 68(39-41, 226), 68, 69(39), 72(39), 98(39), 103(39), 107(39), 108(39-40), 113(39-40), 115(41), 116(39),

376

AUTHOR INDEX

117-1 19(39-40), 120(39), 121-122(39-40),124(39-40), 126-128(39-40), 129(41), 130-131(39), 137(39-40), 141-142(39), 144(39) Christensen, 0. B., 68(226), 68 Christner, J. A., 284(36), 288(54), 288 Chu, S. N. G., 41-44(139), 42,93(289), 95-97(289), 95, 140(139) Chu, X., 70(234), 70 Claessen, L. M., 28(93), 28, 53-54(93), 56(93), 58(93), 140(93) Clark, D. A,, 345(130), 345, 353(130) Clarke, J., 247(191), 247 Claro, F., 240(186), 240, 244(186) Clerk, J. P., 245(189), 245 Clippe, P., 240(185), 240, 244(185) Cody, G. D., 223(146), 223 Cohen, E. G. D., 240(181), 240, 245(181) Cohen, M. H., 170(28-29), 170, 171(36), 171, 188(54), 188, 195(77), 195, 201(28-29), 231(36), 238(170), 238 Cohen, M. L., 5(35-36), 5, 16(76), 16, 22(35-36), 108(35), 120-121(340), 120 Cohen, R. W., 223(146), 223 Colak, S., 111(333), 111, 140(333) Colavita, E., 7(67), 7 Coleman, J. J., 33(121), 33, 93(291), 95(291), 95,97(291) Collins, D. A., 31(105), 31, 53(172), 53, 54(190), 57(190), 57,66(217), 66, 83(105), 109(105, 190), 115-118(105, 190), 119(190), 140(105, 190), 141(190) Coluzza, C., 7(68), 7 Comstock, R. L., 278(22), 278 Coniglio, A., 195(79), 195, 196(82), 196, 252(202), 252, 254(202, 208), 254, 256(202, 208) Cooper, Jr., J. A., llO(330-331), 110 Cornelison, D. M., 78(263), 78, 80-81(263) Corson, P. B., 184(46), 184 Coutts, M. D., 223(146), 223 Cox, R. T., 99(303), 99 Craighead, H. G., 247(192), 247 Craik, D. J., 331(124), 331 Croke, E. T., 31(104-105), 31,77(104), 81(104), 83(104- 105), 84-88(104), 92(104), 109(105), 115-118(105), 140(105) Csordas, A., 252(201), 252,256(201) Cullis, A. G., 42-44(138), 42, 140(138)

Cullity, B. D., 300(98), 300, 302(99), 302, 303(100), 303, 325(118), 325 Cummings, K. D., 227(149), 227-228

D Dai, U., 204(114), 204,205 Dang, L. S., 99(303), 99 Daniel, E., 272(3), 272 Daniels, R. R., 82(273), 82 Dapkus, P. D., 47(148), 47, 93(291), 95(291), 95, 97(291) Darling, D. H., 247(192), 247 Datta, S., 70(227, 230-231), 70, 72(231) Daval, J., 283(34), 283 Davidson, B. A., 78(268), 78 Davies, J. I., 41-44(137), 42, 140(137) Davies, J. J., 71(239), 71 Davis, J. J., 102(312), 102 Davis, V. A., 240(174-175), 240, 242(174-175), 243 Dawson, L. R., 74(245), 74, 80(269), 80 Dawson, P., 28(92), 28, 36(126-127), 37(127), 37 Dayem, A. H., 27(88), 27,41-42(88), 44(88), 140(88) De, B. R., 231-232(161), 231 de Arcangelis, L., 252(202), 252, 254(202, 208), 254, 256(202, 208) de Cremoux, B., 46( 146), 46 De Gennes, P. G., 218(134), 218 Delalande, C., 54(188), 57(188), 57, 71(240), 71, 140(188) Deleporte, E., 71(240), 71 Della Torre, E., 273(7), 273 Denley, D., 104(315), 104 Deppe, D. G., 42(133), 42,44(133), 47(133) Derrick, G. H., 175(42), 175 Derrida, B., 188(57-59), 188, 194(58-59), 197(58-59), 209(59) Deutscher, G., 193(69), 193,204(114), 204-205, 210(121), 210,214(128), 214, 220(128), 238(169), 238 Deutscher, O., 216(131), 216 Devaty, R. P., 225(147), 225(147a), 225, 236(147a), 230(158), 230 Diaz-Guilera, A., 230(155), 230 Ding, J., 70(233), 70, 72(233, 241), 72

377

AUTHOR INDEX

F

Dingle, R., 6(44-46), 6, 27(44-46), 34(44-46), 36(46)

Dodson, B. W., 78(261-263), 78, 8q261-263), 81(263) Doerner, M. F., 294(75), 294 Dorsch, C., 295(76), 295 Duan, S. L., 288(49), 288 Dubson, M. A,, 255(209), 255 Duc, T. M., 31(101), 31, 62(101), 63(101, 204), 63, 73(101), 98(101), 104(101) Duering, E., 190(60-62), 190, 202-203(60), 254(61) Duggan, G., 6(63), 6, 27(63), 37(63) Dumke, W. P., 2(7), 2 Dunstan, D. J., 28(92), 28 Dupuis, R. D., 47(148), 47 Durbin, S., 101-102(310), 102 Durbin, S. M., 70(233), 70, 72(233, 241), 72 Dutt, B. V., 32(116), 32, 53-54(116), 56(116) Duxbury, P. M., 217(145), 220(143-145), 220, 264( 143- 145),266( 143- 145)

E Eastman, L. F., 27(90), 27, 34(90), 36(90), 41(90), 44(90, 149), 47(149), 47,48(90), 93(293), 96(293), 96,98(293), 106(325-326), 106, 140(90, 149) Edwall, D. D., 6(55), 6, 32(55), 34(55), 36(55) Egami, T., 295(77), 295 Ehlers, D. H., 62-64(206), 63, 140(206) Ehrenreich, H., 63(207-208), 63, 64(209-210), 64, 165(23), 165, 225(23) Einziger, R., 258-259(213), 258 Eizenberg, M., 33(122), 33 Elliott, M., 144(350), 144 Emanuel, M. A., 33(121), 33 English, J. H., 6(56), 6, 28(56), 32(115), 32, 35-37(56), 54(115), 58(115), 14q115) Enquist, P. M., 93(293), 96(293), 96, 98(293) Entin-Wohlman, S., 216(131), 216 Esaki, L., 2(9-11, 15), 2, 28(93), 28, 33(120), 33, 35-38(120), 40(120), 53(93, 165-166, 170, 173-177, 181-184), 53, 54(93, 165-166, 188-189), 56(93, 165), 57(185, 188-189), 57, 58(93, 165-166), 117(189), 140(93, 165-166, 188) Evans, K. R., 93(296) Evrard, R., 240(185), 240, 244(185)

Fahrenbruch,A. L., 65, 67(221-222), 69(221-222)

Farley, C. W., 41-42(141), 42, 44(141, 158), 46(141), 49(158), 49, 50(141), 115(141), 140(141, 158) Faunce, C. A., 340(128), 340 Faurie, J. P., 31(101), 31, 32(110), 32, 62(101, 110, 198-200, 206), 62, 63(101, 110, 203-204,206), 63, 64( 110, 199,203,206), 70(234), 70, 73(101), 98(101), 99(299-301), 99, 104(101), 140(206) Feder, J., 197(90), 197 Fedotov, Ya. A., 67(223), 67 Felderhof, B. U., 240(179-181), 240, 245( 179-18 1) Feldman, L. C., 78-80(264), 78 Feldman, R. D., 98(298), 98, 115-118(339), 115, 14q339) Feng, S., 199(97,99), 199-200, 214(99), 257(211), 257 Finkelstein, B., 299(93), 299 Fiory, A. T., 78(264-265), 78, 79(264), 8q264) Fisch, R., 164(18), 164, 186(18), 188(18), 197(18) Fischer, R., 6(59), 6, 32(59), 35-37(59) Fitzgerald, E. A,, 93(293), 96(293), 96, 98(293) Fitzpatrick, B. J., 102(311), 102 Flores, F., 5(25-26), 5, 18(25-26), 18(81), 18, 19(26), 130(25-26) Fluckiger, Ph., 216(130), 216 Flytzanis, C., 267(226), 267 Forchel, A,, 54(191), 57(191), 57, 140(191) Ford, G. W., 240(181), 240, 245(181) Forrest, S. R., 32(111-112), 32, 41-42( 111-1 12), 44( 111- 112), 46(144, 147), 46, 140(111-112) Forster, A., 125(344), 125, 127(344) Fortunato, C., 7(68), 7 FOX,A. M., 41-44(137), 42, 140(137) Foxon, C. T., 28(92), 28 Foy, P. W., 144(349), 144 Frank, D. J., 190(63), 190-192, 194(63, 72), 194, 197(63, 71), 220(141), 220 Fraxedas, J., 62-64(206), 63, 140(206) Fredkin, D. R., 273(9), 273, 280(9, 27), 280, 304(103), 304

378

AUTHOR INDEX

Freeouf, J. L., 56(193), 59(193), 59, 67-68(193), 99(193), 116-1 18(193), 123-124(193) Frei, E. H., 304-305(102), 304 Frensley, W. R., 5(22-23), 5, lO(22-23), 13- 14(22-23), 25(23), 39(22-23), 40(22), 55-56(23), 59(23), 64-65(23), 67-68(23), 69(22), 99(23), 101-102(23), 107-108(23), 112-113(23), 116-122(23), 124(23), 131(22-23), 133(23), 141(23), 143(23) Fried, A,, 210(121), 210 Fritz, I. J., 74(245), 74, 80(269-270), 80 Fu, Q., 70(233), 70, 72(233, 241), 72, 101-102(310), 102 Fuchs, R., 232(163), 232 Fujii, T., 41(153), 44(153), 48(153-155), 48, 140(153) Fujita, S., 101-102(308), 102 Fujiyasu, H., 100(304), 100 Furdyna, J. K., 62(201), 62, 66(216), 66, 70(232), 70, 71(237), 71, 72(232), 73(237) Futamota, M., 273(15), 273

C Gal, M., 93(295), 96-97(295), 96 Cant, H., 105-106(323), 106, 140(323) Garfunkel, G. A., 247(193), 247 Garland, J. C., 170(33), 170, 198(94), 198, 206(115), 206, 215(115), 227(149), 227-228,247(194), 247, 255(209), 255 Garland, J. W., 111(337), 111, 140(337) Gashin, P. A., 66(219-220), 66 Gaunce, C. A., 298(85), 298 Gavilano, J. L., 216(130), 216 Gear, C. W., 318(110), 318 Gefen, Y.,197(89, 91), 197 Gell, M. A., 6(51), 6, 28(51), 37(51) Gershoni, D., 41-44(139), 42, 140(139) Ghosh, K., 232(163), 232 Gibson, U. J., 247(192), 247 Giles, R. C., 273(12), 273, 320(112), 320 Giles-Taylor, N. C., 70(229), 70 Giraud, G., 245(189), 245 Giriat, W., 71(235), 71 Goldburg, W. I., 247(192), 247 Gonda, S., 6(61), 6, 32(61), 35-37(61) Gonsalves, J. M., 109(328), 109 Goossen, K. W., 33(123), 33

Gorczyca, I., 68(226), 68 Gossard, A. C., 6(45, 47, 56, 58), 6, 27(45, 47, 58), 28(56, 91), 28, 32(109), 32, 34(45, 47, 109), 35(56, 58), 36(56, 58, 109), 37(56, 58), 47(47, 58) Gossmann, H.-J., 78(268), 78 Gourley, P. L., 74(244), 74,80(269), 80 Graham, C. D., 295(77), 295 Grannan, D. M., 198(94), 198 Granqvist, C. G., 240(183), 240 Grant, R. W., 6(53-54, 57), 6, 7(65, 69), 7, 30(53-54, 57, 65,69,94-96), 30, 31(100), 31, 35(53), 36(57), 38(57, 9 3 , 40(57, 9 9 , 42(142-143), 42,44(158), 46(142-143), 49(142-143, 158), 49,62-63(100), 65(100), 92(96, 283), 92, 93(96), 97-98(96), 103(53, 69, 94, 313), 103, 105(53), 109(313), 111(65), 119(65), 128(347-348), 128, 129(348), 140(53, 65, 94, 100, 141, 158), 141(65), 142(313) Greengard, L., 320(114), 320 Gregory, T. J., 71(239), 71 Crest, G. S., 238(170), 238 Griffiths, G., 54(191), 57(186, 191), 57, 140(191) Grodzinski, P., 93(291), 95(291), 95,97(291) Grundy, P. J., 298(85), 298, 340(128), 340 Grunthaner, F. J., 144(352), 144 Gualtieri, G. J., 30(98-99), 30, 31(103), 31, 53(98), 54(98-99), 56(98), 57(99), 58(98), 77(103), 78(268), 78, 81(103), 83-85(103), 115-1 16(339), 115, 117(99, 339), 118(339), 140(98-99, 339) Gubanov, A. I., 2(1-3), 2 Guldner, Y.,42(132), 42, 44(132), 47(132), 53(174-175), 53, 62(198-200), 62, 64(199) Gunshor, R. L., 70(227, 230,233), 70, 71(238), 71, 72(231-233, 241), 72, 101-102(310), 102, 109(328), 109, 1 lO(330-332), I I0

H Haas, C. W., 280(28), 280 Haase, M. A., 33(121), 33,41-42(140), 42, 44(140), 111(337), 111, 140(337), 140(140) Hagston, W. E., 71(239), 71

AUTHOR INDEX Hahsin, Z., 158(14), 158 Halperin, B. I., 184-185(52), 184, 199(97-99), 199,200(98, 104), 200,214(99) Hamann, D. R., 5(33-34), 5, 22(33-34), 108(33-34) Hamm, R. A., 41(136), 42(134, 136), 42, 43-44(136), 47(134), 50(160), 50, 140(136) Hammersley, J. M., 192(65-66), 192 Han, J., 72(241), 72, 101-102(310), 102 Hang, Z., 93(287), 95-97(287), 95 Hankel, O., 366(149), 366 Hansen, A,, 188(55-56), 188,252(55) Hansson, G. V., 31(102), 31, 77(102), 81-82(102), 84-85(102), 92(282), 92 Harman, T. C., 213(126), 213 Harris, A. B., 164(18), 164, 186(18), 188(18), 197(18, 83-84), 197,254(207), 254 Harris, Jr., J. S., 6(55), 6, 32(55), 34(55), 36(55) Harris, T. D., 27(89), 27, 42(89), 44(89), 47(89) Harrison, W. A., 5(24,29), 5, 10(24), 14(24), 16(24, 78), 16, 17(24, 29), 19(29, 83), 19, 21(29), 24(24), 25(24,29), 26(24), 39(24), 40(24,29), 55-56(24, 29), 59(24, 29), 64(24, 29), 65(24, 29, 214), 65, 67-69(24, 29), 73(24, 214), 82(24, 29), 88(24, 29), 92(283), 92, 99(24), 100(29), 101-102(24, 29), 103(24, 313) 103, 107-108(24, 29), 109(313), 112-113(24,29), 116-122(24, 29), 124(24,29), 126-128(24,29), 129(24), 131(24), 134(24), 135(29), 141 (24), 142(313), 143(24) Hashimoto, Y.,36(129), 38-40(129), 39 Hass, K. C., 64(209), 64 Hata, H., 293(61), 293,293(64), 293 Hauenstein, R. J., 78(266), 78, 81(266) Heberle, A., 93(290), 95-97(290), 95 Hecht, M. H., 144(351-352), 144 Heerman, D. W., 195(81), I95 Hefetz, Y.,70(231), 70, 72(231) Heiblum, M., 33(122), 33 Heinrich, H., 5(32), 5, 19(32), 64(215), 65(32, 215), 65 Henderson, T. S., 6(60), 6, 32(60), 36-38(60), 93-94(285), 94,96-97(285) Henry, C. H., 6(44), 6, 27(44), 34(44) Herman, F., 16(77), 16, 89(280), 89 Hernindez-Calderon, I., 126-127(346), 126

379

Herrenden-Harker, W. G., 31(108), 31, 125- 128(108), 140(108), 142(108) Herring, C., 76(254), 76 Herrmann, H. J., 188(58-59), 188, 194(58-59), 197(58- 59), 209(59) Herzog, H. J., 80(271), 80,84(279), 84, 88(279) Hickmott, T. W., 6(59), 6, 32(59), 35-37(59) Hilton, C. P., 71(239), 71 Hinged, K., 110(329), 110 Hirakawa, K., 36(129), 38-40(129), 39 Hiyamizu, S., 41(153), 44(153), 48(153-155), 48, 14q153) Ho, J. C., 32(114),32,41(114), 44(114), 48(114), 140(114) Hochst, H., 126-127(346), 126 Hoffmann, H., 288(46), 288-289,298(87), 298 Hoger, R., 93(290), 95-97(290), 95 Hojo, A., 6(62), 6, 32(62), 35-37(62) Holonyak, Jr., N., 42(131, 133), 42, 44(131, 133), 47(131, 133, 148), 47 Holwech, I., 232(162), 232 Honda, N., 297(81), 297 Hong, J. M., 71(240), 71 Honig, J. M., 213(126), 213 Hono, K., 288(50), 288 Hori, M., 240(182), 240 Horn, K., 31(106-107), 31, 114(106), 116(106), 118- 119(106), 122(107), 124(107), 140(106) Houdre, R., 93-94(285), 94,96-97(285) Howard, J. K., 293(63), 293 Howell, T. D., 273(14), 273 Hsu, C., 31(101), 31,62(101), 63(101,204), 63, 73(101), 98(101), 99(299), 99, 104(101) Hsu, Y.,290(57), 290,298(86), 298 Huang, D., 93-94(285), 94, 96-97(285) Huang, J. S., 198(95), 198 Huang, K. F., 93(289), 95-97(289), 95 Hubert, A., 299(91), 299 Hughes, G. F., 273(5), 273, 316(5), 320(115), 320 Hui, P. M., 63(207-208), 63, 220(142), 220, 230( 156- 157), 230, 245( 190), 245-246, 249,252(200), 252,254(205-206), 254, 265-266( 142) Hui, Y.C., 255(209), 255 Hull, R., 78(265), 78, 80(265) Hunderi, O., 240(183), 240

380

AUTHOR INDEX

Huysmans, G. T. A., 339(127), 339 Hybertsen, M. S., 31(103), 31, 43(161-163), 45( 161- 163), 50- 5 1(161-163), 50, 77(103), 81(103), 83-85(103), 139(161-162)

1

Ignatiev, A., 240(184), 240 Ikoma, T., 36(129), 38-40(129), 39 Imry, Y., 198(93), 198 Inata, T., 41(153), 44(153), 48(153), 48, 140(153) Irwin, T. J., 105-106(324), 106, 140(324) Iwasaki, S.-I., 276(20), 276, 297(80-82), 297, 299(90), 299, 338(126), 338 Iye, Y., 53(177), 53

J Jacobs, I. S., 281(30), 281, 308(107), 308 Jaros, M., 6(51), 6, 28(51), 37(51) Jeong, W. G., 93(291), 95(291), 95,97(291) Ji, G., 93-94(285), 94,96-97(285) John, P., 126-127(345), 126 Johnson, D. C., 78(266), 78,81(266) Johnson, K., 287,288(53, 55), 288 Johnson, K. E., 283-284(38), 283, 290(38) Johnson, M. J., 93(295), 96-97(295), 96 Johnson, N. F., 63(207-208), 63, 64(209-210), 64 Jones, R. B., 240(179), 240,245(179) Jones, R. L., 93(296), 96-97(296), 96 Jorke, H., 84(279), 84, 88(279) Jortner, J., 170(28-29), 170, 188(54), 188, 195(77), 195,201(28-29) Joyce, M. J., 93(295), 96-97(295), 96 Judge, J. S., 290(58), 290 Judy, J. H., 290(57), 290, 293(68), 293, 298(86), 298 Juretschke, H. J., 201(109), 201,210(109) JylhL, O., 70-71(228), 70

K Kahn, A., 33(124), 33, 109(124) Kahn, M. R., 278(24), 278-279

Kaiser, W. J., 144(351), 144 Kane, E. O., 74(248), 74 Kantor, Y., 164(17), 164, 184(51), 184, 202(112), 202, 211(112), 252(203), 252, 266(203) Kapitulnik, A,, 210(121), 210 Kaplan, M. L., 32(112), 32, 41-42(112), 44(112), 140(112) Kaplan, S. G., 229(151), 229 Kasper, E., 80(271), 80 Kassel, L., 111(337), lZ1, 140(337) Katagiri, H., 100(307), 100 Katnani, A. D., 5(21), 5, 12(21, 74), 12, 13(21), 36(128), 38-39(128), 39,56(21), 59(21), 67-69(21), 82(21,273), 82, 100- 102(2I), 104(316-3 19), 104, 105(21, 316-319), 106(21), 108(21), 113(21), 115(128), 117-119(21), 121(21), 123- 124(21), 126- 128(21), 134(21), 140(21, 316-319), 141(21) Kato, M., 102(312), 102 Kavanagh, K. L., 50(160), 50, 93(293), 96(293), 96,98(293) Kawai, N. J., 53(173), 53 Kawamoto, A,, 294(73), 294 Kazmierski, K., 46(146), 46 Keller, J. B., 206(116-117), 206 Kenkel, S. W., 259(217-218), 259 Kenyon, W. E., 231-232(160), 231 Ketterson, A., 6(60), 6,32(60), 35-38(60) Kibbel, H., 80(271), 80 Kim, M. W., 198(95), 198 Kim, 0. K., 32(111), 32,41-42(111), 44(111), 46(144), 46, 140(111) Kimura, R., 100(307), 100, 102(309), 102 Kinoshita, N., 169(25), 169 Kirchoefer, S. W., 42(131), 42,44(131), 47(131) Kirkpatrick, S., 173(39), 173, 197(87), 197, 220(140), 220,245(39) Kittel, C., 75(251), 75, 280(26), 280, 316(26) Kleinman, D. A., 6(47, 58), 6, 27(47, 58), 34(47), 35-37(58), 47(47, 58) Klem, J., 6(60), 6, 32(60), 36-38(60) Klingert, J. K., 32(116), 32, 53-54(116), 56( 116) Kloepper, R. M., 299(93), 299 Knall, J., 31(102), 31, 77(102), 81-82(102), 84-85( 102) Kneller, E., 272(1), 272,290(59), 290

381

AUTHOR INDEX

Kobayashi, M., 70(233), 70,72(233,241), 72, 100(307), 100, 10l(3 lo), 102(309-3lo),

102,109(327-328), 109,llO(330-332), 110

Koch, R. H., 247(196), 247 Koch, S. W., 267(225), 267 Koester, E., 295(76), 295 Kohler, T. R.,273(9), 273,280(9, 27), 280 Kolbas, R. M., 42(131), 42,44(131), 47(131,

148),47

Krost, A,, 110(329), 110 Kuan, T. S., 33(120), 33,35-38(120), 40(120), 104(321), 104 Kuech, T. F., 6(51), 6,28(51), 37(51), 61(197),

61,81-82(272),81 Kukimoto, H., 46(145), 46 Kulakovskii, V.D., 93(286), 95-97(286), 95 Kullmann, U., 295(76), 295 Kusuda, T., 298(84), 298 Kuwabara, H., 100(304), 100

Kolodziejski, L. A., 70(227, 230), 70,71(238),

71,72(231), 110(330), 110 Konagai, M., 100(307), 100, 102(309), 102 Kondaurov, N.M., 67(223), 67 Konnikov, S. G., 67(223), 67 Koren, U., 27(88), 27,41-42(88),44(88),

140(88) Korringer, J., 186(53), 186-187,231-232(53) Koss, R. S., 174(41), 174,238(171), 238 Koster, E., 281-282(29), 281,290(29) Kovalev, A. N., 67(223), 67 Kowalczyk, S. P., 6(53-54), 6,7(65, 69), 7,

30(53-54,65,69,95-96),30,31(100), 31, 38(95), 40(95), 62-63(100), 65(100), 92-93(96),97-98(96), 103(53,69), 105(53), 111(65), 119(65), 128-129(348), 128,140(53, 65,loo), 141(65) Kraut, E. A., 6(53-54,57), 6,7(65, 69),7, 16(79), 16,30(53-54,57,65,69,94-96), 30, 31(100), 31,35(53), 36(57), 38(57, 95),40(57, 95),41(141), 42(141-143), 42, 44(141, 158), 46(141-143), 49(142-143, 158), 49,50(141), 62-63(100), 65(100), 68(79), 92(96,283), 92,93(96), 97-98(96), 103(53,69,94,313), 103, 105(53), 109(313), 111(65), 115(141), 119(65), 128-129(348), 128,140(53,65, 94,100,141,158), 141(65), 142(313) Kreibig, U., 242(188), 242 Kroemer, H., 2(5-6),2,5(22-23), 5, 6(55), 6, 7(70), 7,lO(22-23,70), 12(75), 12, 13-14(22-23),24(70), 25(23), 32(55, 115), 32,34(55), 36(55), 39(22-23), 40(22), 44(156), 49(156), 49, 54(115, 191), 55-56(23), 57(186, 191),57, 58( 11 5), 59(23), 67-68(23),69(22), 99(23), 101-102(23), 105(322), 105, 107-108(23), 112-113(23), 116-122(23), 124(23), 131(22-23), 133(23), 140(115, 191), 141(23), 143(23)

L Lacell, C., 93(287), 95-97(287),95 Lagar’kov, A. N., 219(138), 219 Laibowitz, R.B., 247(196), 247 Lamb, W., 237(168), 237,242(168) Lambert, S. E., 293(63), 293 Lambeth, D. N., 282(33), 282,290(56),290,

294(71), 294 Lambkin, J. D., 28(92), 28 Lambrecht, W.R. L., 5(42-43), 5, 23(42-43),

24(42-43,87), 25(43, 87), 24,40(42-43, 87),43(43, 87), 45(43, 87), 52(43, 87), 55-56(43,87), 59(43, 87),64(43,87), 66(43,87),68(42-43), 107(87), 108(42-43,87), 113(43,87), 114-119(43), 116-120(87), 121(43, 87), 122(43), 124(43, 87), 126-127(43,87), 128(42-43, 87), 129(43, 87), 131(87), 138(43, 87) Landauer, R., 149(1), 149,154-155(1), 165(21), 165,201(109),201,210(109) Lang, D. V., 41-44(136,139), 42,83(275), 83, 84(278), 84, 88(278), 140(136, 139) Langer, J. M., 5(32), 5, 19(32), 65(32, 212), 65 Langer, J. S., 200(104), 200,362(143), 362 Lannoo, M., 97(297), 97 Last, B. J., 193(70), I93 LaTorraca, G. A,, 186(53), 186-187, 231-232(53) Lau, S. S., 81-82(272), 81 Laude, L. D., 76(252), 76 Ladghlin, D. E., 288(50), 288 Laugier, J. M., 245(189), 245 le Doussal, P., 200(107), 200 Leath, P.L., 174(40), 174,220(143-144),220, 264(143-144),266(143- 144) Lee, D., 102(311), 102

382

AUTHOR INDEX

Lee, P. Z., 32(114), 32,41(114), 44(114), 48( 1 14), 140(114) Lee, Y. R., 66(216), 66, 71(236), 71 Leemann, Ch., 216(130), 216 Lefebvre, P., 99(300), 99 Lentz, G., 99(303), 99 Lerch, Ph., 216(130), 216 Letartre, X., 93(292), 95-97(292), 95 Leu, L. Y., 46(147), 46 Levy, O., 213(127), 213 Ley, L., 62-64(206), 63, 140(206) Li, D., 109(328), 109, 110(332), 110 Li, Y. S., 217(145), 220(145), 220, 264(145), 266(145) Liebsch, A,, 240(176-178), 240, 244(176-178) Lin, C. L., 32(114), 32, 41(114), 44(114), 48(114), 93(288), 95-97(288), 95, 140(114) Lin, T., 294(71), 294 Lindau, I., 73(242), 73 List, R. S., 82(274), 82 Liu, Feng, 239( 173), 239, 242( 173) Lobb, C. J., 190(63), 190-191, 192(64), 192, 194(63, 72), 194, 197(63, 72), 220(64, 141-142), 220,265-266(142) Lodder, J. C., 298(83), 298, 339(127), 339 London, F., 215(129), 215 London, H., 215(129), 215 Lopez, O., 345(130), 345, 353(130) Losch, R., 78(267), 78 Lottis, D. K., 299(95), 299 Louie, S. G., 5(35-36), 5 , 22(35-36), 108(35) Louis, E., 18(81), 18 Lubensky, T. C., 197(83-84), 197, 199(102), 199,218(136-137), 218 Luborsky, F. E., 288(51), 288, 290(59), 290 Lucas, A. A,, 240(185), 240, 244(185) Luck, J. M., 245(189), 245 Ludeke, R., 53(166, 173), 53, 54(166), 58(166), 140(166) Luitjens, S. B., 357(138), 357 Lum, R. M., 32(116), 32, 53-54(116), 56(116) Lunn, B., 71(239), 71 Luo, H., 66(216), 66, 70(232), 70, 72(232) Luo, L. F., 53(169), 53, 57(192), 57 Liith, H., 125(344), 125, 127(344) Lynch, R. T., 84(278), 84, 88(278) Lyon, S. A., 33(123), 33

M Ma, S.-K., 255(210), 255 Maan, J. C., 28(93), 28, 53-54(93), 56(93), 58(93), 140(93) Machta, J., 199(101), 199 Maciel, A. C., 41-44(137), 42, 140(137) Mackey, K. J., 31(108), 31, 125-128(108), 140(108), 142(108) Maeda, Y., 288(48), 288 Maenpaa, M., 81-82(272), 81 Magea, N., 99(302-303), 99, lOO(302) Mahan, G. D., 236(167), 236 Mahowald, P. H., 82(274), 82 Mahvan, N., 290(60), 290 Mailhiot, C., 2(18), 2, 53(178-179), 53, 74(179,247), 74, 84( 178) Malik, R. J., 30(98), 30, 53-54(98), 56(98), 58(98), 140(98) Mallinson, J. C., 306(104-105), 306 Malloy, K. J., 63(205), 63 Mandelbrot, B. B., 197(89), I97 Mann, J. B., 17(80), 17 Mann, J. C., 53(175), 53 Mannaerts, J. P., 31(103), 31, 77(103), 81(103), 83-85(103) Mansuripur, M., 273(12), 273, 320(112), 320 Mantese, J. V., 247(192, 195), 247 Marbeuf, A., 73(243), 73 Marfaing, Y., 73(243), 73 Margaritondo, G., 5(21), 5, 7(66-68), 7, 12(21, 74), 12, 13(21), 54(66), 56(21), 59(21), 67-69(21), 82(21,273), 82, 100-102(21), 105-106(21), 108(21), 113(21), 117-119(21), 121(21, 341-342), 121, 123-124(21), 126-128(21), 134(21), 140(21, 341-342), 141(21) Marianer, S., 200(103), 200 Marshall, T., 111(333), 111, 140(333) Marsico, V., 216(130), 216 Martin, B., 282(33), 282 Martin, R. M., 5(37-38), 5, 22(37-38, 84-85), 22,25(84), 40(84-85), 41(37, 84), 43(84), 45(84), 52(84), 55-56(84), 59(84), 64(84), 68(37-38), 75(38), 77(37-38), 78(84-85), 81(37-38), 83(37-38), 84(38), 86-87(38), 88(37-38), 89(84-85), 91(37-38, 84-85), 103(84-85), 108(84), 113-1 14(84), 115(84), 118-1 19(84-85), 120-122(84),

383

AUTHOR INDEX

124(84), 130(84-85), 131(84), 136(84), 142(84), 144(37-38) Martinoli, P., 216(130), 216 Marzin, J.-Y., 93-94(284), 94, 96-97(284) Mashita, M., 6(62), 6, 32(62), 35-37(62) Masriette, H., 99(302-303), 99, lOO(302) Masut, R. A., 93(287), 95-97(287), 95 Mathieu, H., 99(3OO-301), 99 Matthews, J. W., 78-79(259), 78 Mattingly, M., 44(157), 49(157), 49, 140(157) Mattis, D. C., 227(150), 227 Mayo, P. I., 293(66), 293, 367(150), 367 McCaldin, J. O., 5(20), 5, 10-11(20), 31(105), 31, 32(110), 32, 39(20), 43(20), 45(20), 54(190), 56(20), 57(20, 190), 57, 59(20), 61(197), 61, 62-64(1 lo), 65(20), 66(217-218), 66, 67-68(20, 218), 69-70(218), 83(105), 99(20), 101-102(20), 109(105, 190), 113(20), 114(338), 114, 115-1 16(105, 190), 117-1 18(20, 105, 190), 119(190), 123-124(20), 126- 128(20), 131- 132(20), 140(105, 110, 190,218), 141(20, 190, 218), 142(20) McCall, S., 269(227), 269 McFadyen, I. R., 288(47), 288 McGill, T. C., 2(12-13), 2, 5(20), 5, 6(64), 6, 10-11(20), 31(104-105), 31, 32(110), 32, 35-36(64), 38(64), 39(20, 64), 43(20), 45(20), 53(168, 172, 180), 53, 54(190), 56(20), 57(20, 190), 57, 59(20), 60(194-195), 60, 61(195-196), 61, 62( 1lo), 63( 110, 203), 64( 110, 203, 209-210), 64, 65(194), 65(20, 211), 65, 66(217-218), 66, 67-68(20, 218), 69-70(218), 77(104), 78(266), 78, 81(104, 266), 83(104-105), 84-88(104), 92(104), 99(20), 1OO( 305 - 306), 10l(20, 305-306). 102(20), 109(105, 190), 113(20), 114(338), 114, 115-116(105, 190), 117-118(20, 105, 190), 119(190), 123-124(20), 126- 128(20), 132-132(20), 140(105, 110, 190, 218), 141(20, 190, 218), 142(20) McKenzie, D. R., 155(7), 155, 175(7, 42), 175 McMenamin, J. C., 103(313), 103, 105(314), 140(314) McNamara, B., 362(144), 362 McPhedran, R. C., 155(7), 155, 175(42), 175 McWhirter, J. T., 230( 153), 230

Mead, C . A., 5(20), 5 , 10-11(20), 39(20), 43(20), 45(20), 56-57(20), 59(20), 65(20), 67-68(20), 99(20), 101-102(20), 113(20), 1 17- 1 18(20), 123- 124(20), 126- 128(20), 131 132(20), 141(20), 142(20) Medina, R., 192(64), 192,220(64) Mee, C . D., 272(3), 272 Meiners, L. G., 32(114), 32,41(114), 44(114), 48( 1 14), 140(1 14) Meir, Y., 164(19), 164, 188(19), 197(19), 254(207), 254 Melloch, M. R., llO(330-331), 110 Mendelson, K. S., 206-207(118), 206 Mendez, E. E., 33(119-120), 33, 35 -38( 119- 120), 40( 120), 53( 176- 177, 182-184), 53, 54(189), 57(185, 189), 57, 140(189) Menendez, J., 6(56), 6, 10-11(72), 10, 28(56, 72), 35-37(56), 54(72), 57(72), 93(294), 96-97(294), 96, 98(294, 298), 98, 117(72), 140(72) Menke, D. R., 1 lO(332). 110 Menu, E. P., 93(291), 95(291), 95, 97(291) Merle d’Aubigne, Y., 99(302-303), 99, 100(302) Merz, J. L., 44(156), 49(156), 49, 57(186), 57 Meseguer, F., 54(189), 57(189), 57, 140(189) Middleton, B., 273( 1 l), 273 Miles, R. H., 31(104-105), 31, 53(180), 53, 74(180), 77(104), 78(266), 78, 81(104, 266), 83(104-105), 84-88(104), 92(104), 100- lOl(305-306), 100, 109(105), 1 15- 118(109, 140( 105) Milgrom, M., 212(124-125), 212,259(214), 259 Miller, B. I., 27(88), 27, 41-42(88), 44(88), 140(88) Miller, D. A. B., 41(151), 44(151), 48(151), 48, 140(15 1 ) Miller, D. L., 30(95), 30, 38(95), 40(95) Miller, R. C., 6(47, 58), 6, 27(47, 58), 34(47), 35(58), 36(58, 126-127), 37(58, 127), 37, 47(47, 58), 93(294), 96-98(294), 96 Miller, T., 126-127(345), 126 Million, A,, 62(198), 62 Mills, K. A,, 104(315), 104 Milnes, A. G., 8-9(71), 8, 25(71), 39-40(71), 55-56(71), 59(71), 67-69(71), 73(71), 82(71), 88(71), 99(71), 101-102(71), -

384

AUTHOR INDEX

107-108(71), 113(71), 116-119(71), 121-122(71), 124(71), 126- 127(71), 131- 1 32(71), 143(71) Milton, G . W., 158(11), 158, 160-161(16), 160, 173(38), 173, 182(11), 182, 183(16), 184(49), 184, 185(11), 204(38), 206(119), 206,231-232(161), 231, 252(38) Min, T., 293(68), 293-294 Mino, N., 100(307), 100 Mirzamaani, M., 294(72), 294 Misawa, S., 6(61), 6, 32(61), 35-37(61) Mitescu, C. D., 197 Miyamura, M., 293(65), 293 Mizuta, M., 46(145), 46 Mochizuki, K., 100(304), 100 Mohammed, K., 2(17), 2, 57(186), 57, 144(349), 144 Monch, W., 105-106(323), 106, 14q323) Moon, R. M., 331(123), 331 Moore, A. R., 89(280), 89 Morgan, D. V., 44(149), 47(149), 47, 140(149) Morkoc, H., 6(59-60), 6, 32(59-60), 35-37(59-60), 38(60), 93-94(285), 94, 96-97(285) Morris, D., 93(287), 95-97(287), 95 Mowbray, D. J., 42-44(138), 42, 140(138) Munekata, H., 53(171, 177), 53 Munteanu, O., 6(47), 6, 27(47), 34(47), 47(47) Mura, T., 169(25), 169 Murat, M., 190(62), 190, 200(103), 200 Murayama, A., 293(65), 293 Murdoc, E. S., 293(62), 293 Murschall, R., 105-106(323), 106, 14q323) Muto, S., 41(153), 44(153), 48(153), 48, 140(153) Mysyrowicz, A,, 102(31I), 102

N Najjar, F. E., 93(293), 96(293), 96, 98(293) Nakagawa, A., 32(115), 32, 54(115), 58(115), 140(115) Nakahara, J., 70(231), 70, 72(231) Nakahara, S., 78(264265), 78, 79-8q264) Nakamura, Y., 276(20), 276 Nakanishi, Y., 100(304), 100 Nakanisi, T., 6(62), 6, 32(62), 35-37(62)

Nakata, Y., 41(153), 44(153), 48(153-155), 48, 140(153) Natarajan, B. R., 293(62), 293 Nathan, M. I., 33(122), 33 Nelkin, M., 188(55), 188, 252(55) Newrock, R., 206(115), 206,215(115) Ni, W.-X., 31(102), 31, 77(102), 81-82(102), 84-85(102), 92(282), 92 Nicholls, J. E., 71(239), 71 Nieh, C. W., 63-64(203), 63, 78(266), 78, 81(266) Nienhuis, B., 195(80), 195 Niki, S., 93(288), 95-97(288), 95 Niles, D. W., 7(66-67), 7, 54(66), 126-127(346), 126 Ninno, D., 6(51), 6, 28(51), 37(51) Noh, T. W., 229(151), 229,230(152-153), 230,234(164), 234 Nost, B., 232(162), 232 Nurmikko, A. V., 7q231-233), 70,71(238), 71, 72(231-233, 241), 72, 101(310), 102(310-311), I02 NUZZO, R. G., 30(98-99), 30,53(98), 54(98-99), 56(98), 57(99), 58(98), 115-116(339), 115, 117(99, 339), 118(339), 115, 140(98-99, 339)

0 Octavio, A., 192(64), 192,220(64) Ogura, M., 46(145), 46 Ohkushi, M., 298(94), 298, 299(89), 299 Ohno, T., 283(39), 283,283(35), 283 Okumura, H., 6(61), 6, 32(61), 35-37(61) Olbright, G. R., 267(225), 267 Olego, D. J., lll(335-336), 111, 140(335-336) Onaka, K., 46(145), 46 O’Neill, P., 240(184), 240 Osbourn, G. C., 2(14), 2, 74(244), 74, 74(14) Oseroff, S. B., 299(94), 299 Otsuka, N., 66(216), 66, 7q230, 232), 70, 72(232), 109(328), 109, 110(332), 110 Ouchi, K., 297(81), 297,299(90), 299, 338(126), 338 Ourmazd, A., 27(88), 27, 41-42(88), 44(88), 140(88)

385

AUTHOR INDEX

P Palevski, A., 204(114), 204-205, 210(121), 210 Pan, F. P., 170(30), 170, 201(30), 206(30), 236(166), 236-237 Pan, N., 41-42(140), 42,44(140), 140(140) Pan, S. H., 93(287), 95-97(287), 95 Panina, L. V., 219(138), 219 Panish, M. B., 2(8), 2, 41(136, 139), 42(134, 136, 139), 42, 43-44(136, 139), 47(134), 50(160), 50, 140(136, 139) Patella, F., 121(341-342), 121, 140(341-342) Patterson, G., 273(12), 273 Pautrat, J.-L., 99(303), 99 Pelekanos, N., 70(233), 70, 72(233, 241), 72, 101-102(310), 102 People, R., 32(113), 34(113), 36(113), 41(150), 44(150), 48(150), 48, 78-79(260), 78, 83(275-276), 83, 84(278), 84, 88(278), 91(276, 281), 91, 140(150) Perfetti, P., 7(66-68), 7, 54(66), 104(315), 104, 12l(341-342), 121, 140(341-342) Persson, B. N. J., 240(176-177), 240, 244( 176- 177) Pessa, M., 70-71(228), 70 Petroff, P. M., 42(134), 42, 47(134) Peyghambarian, N., 267(225), 267 Philippe, P., 46(146), 46 Phillips, J. C., 32(118), 32, 122(118), 124(118) Phillips, M. C., 31(105), 31, 54(190), 57(190), 57, 66(217-218), 66, 67-70(218), 83(105), 109(105, 190), 115-118(105, 190), 119(190), 140(105, 190, 218), 141(190, 218) Pianetta, P., 82(274), 82 Pickett, W. E., 5(35-36), 5, 22(35-36), 108(35), 120- 121(340), 120 Picraux, S. T., 78(263), 78, 80-81(263) Pikus, G. E., 74-75(249), 74 Pinczuk, A., 6(56), 6, 10-11(72), 10,28(56, 72, 91), 28, 35-37(56), 54(72), 57(72), 93(294), 96-97(294), 96,98(294, 298), 98, 117(72), 140(72) Pireaux, J. J., 62-64(206), 63, 140(206) Pitt, A. D., 42-44(138), 42, 140(138) Platzman, P., 269(227), 269 Ploog, K., 41(152), 44(152), 48(152), 48, 140(152)

Poisson, M. A,, 42(132), 42,44(132), 47(132) Pollak, F. H., 75(250), 75, 76(252-253), 76, 93(287), 95-97(287), 95 Pollak, M., 200( 106), 200 Potts, J. E., 111(337), 111, 140(337) Poulain, P., 46( 146), 46 Prager, S., 159(15), 159 Prechtel, U., 6(52), 6, 33(52), 36(52) Pressman, H., 242(188), 242 Priester, C., 97(297), 97

Q Qian, Q.-C., 110(330-332), 110 Qiu, J., llO(330-332), I10 Quadri, S. B., 66(216), 66 Quaresima, C., 7(66-68), 7, 54(66), 121(341-342), 121, 140(341-342)

R

Raccah, P. M., 111(337), I l l , 140(337) kddo, G. T., 272-273(2), 272 Rajakarunanayake, Y., 53(172), 53,66(217), 66, 100-lOl(305-306), I00 Ramberg, L. P., 93(293), 96(293), 96,98(293) Ramdas, A. K., 66(216), 66, 71(236), 71 Rammal, R., 218(135-137), 218, 244(198), 252(197-199), 252, 254(199), 256(199) Randet, D., 283(34), 283 Rappaport, M. L., 210(121), 210, 214(128), 214, 220(128) Rau, R. N., 231-232(159), 231 Rauch, G. C., 284(41), 284 Rayleigh, J. W. S., 155(6), 155, 175(6-7) Razeghi, M., 42(132), 42,44(132), 47(132) Reddy, U. K., 93-94(285), 94,96-97(285) Redner, S., 252(202), 252,254(202, 208), 254, 256(202, 208) Reith, T., 288(52), 288 Reithmaier, J.-P., 78(267), 78, 93(290), 95-97(290), 95 Reno, J., 62(200), 62, 70(234), 70 Resnick, D. J., 206(115), 206, 215(115) Reynolds, D. C., 93(296) Rezek, E. A., 42(131), 42, 44(131), 47(131) Ricard, D., 267(226), 267

386

AUTHOR INDEX

Richter, W., 110(329), 110 Ridout, V. L., 2(7), 2 Riechert, H., 93(290), 95-97(290), 95 Riffat, J. R., 41-44(137), 42, 140(137) Roberts, J. N., 199-200(98), 199 Robinson, I. K., 78-80(264), 78 Rogowski, D. A,, 286(44), 286 Rokhlin, V., 320(114), 320 Rosman, R., 197(88), 197 Rossi, E. M., 295(78), 295 Rossignol, R., 267(226), 267 Roth, A. P., 93(287), 95-97(287), 95 Roth, L., 242(187), 242 Roussenq, J., 197(92), 197 Rudman, D. A,, 247(194), 247 Rustagi, K. C., 267(226), 267 Ryan, J. F., 41-44(137), 42, 14q137)

S

Sai-Halasz, G. A,, 2(11), 2, 53(165-166, 170, 173), 53, 54(165-166), 56(165), 58( 165- 166), 140(165-166) Sakai, Y.,122(343), 122, 124(343) Sakaki, H., 53-54(166), 53, 58(166), 140(166) Sakamoto, T., 70(227), 70 Samarth, N., 66(216), 66, 70(232), 70, 72(232) Sampsell, J., 170(33), 170 Sanders, G. D., 93(296), 96-97(296), 96 Sanders, I. L., 286(43), 286, 293(63), 293 Sang, Jr., H. W., 104-105(316-318), 104 Sarychev, A. K., 219(138), 219 Sasaki, A., 100(304), I00 Sauer, R., 27(89), 27, 42(89), 44(89), 47(89) Savoia, A., 121(341-342), 121, 140(341-342) Scala, C., 171(36), 171, 231(36) Schabes, M. E., 273(8), 273, 304(101), 304, 316(109), 316 Schaffer, W. J., 30(96), 30, 92-93(96), 97-98(96) Scharnberg, K., 230( 154), 230 Scheinfein, M. R., 320(113), 320 Schetzina, J. F., 70(229), 70 Schlapp, W., 6(52), 6, 33(52), 36(52) Schmid, U., 68(226), 68 Schmidt, F., 299(91), 299 Schmidt, P. H., 32(112), 32,41-42(112), 44(112), 140(112)

Schubert, E. F., 27(88), 27, 41(88), 42(88, 135), 42,44(88), 140(88) Schulman, J. N., 2(12-13), 2,60(194-195), 60, 61(195), 65(194, 211), 65 Schultz, S., 299(88, 96), 299 Schwartz, G. P., 30(98-99), 30, 31(103), 31, 53(98), 54(98-99), 56(98), 57(99), 58(98), 77(103), 78(268), 78, 81(103), 83-85(103), 115-116(339), 115, 117(99, 339), 118(339), 140(98-99, 339) Schwartz, L. M., 184-185(52), 184, 240( 174- 175), 240,242( 174- 1 75), 243 Scofield, J. H., 247(195), 247 Scolnick, M. S., 42-44(138), 42, 140(138) Scott, M. D., 41-44(137), 42, 140(137) Seedorf, R., 31(107), 31, 122(107), 124(107) Segall, B., 24-25(87), 24, 40(87), 43(87), 45(87), 52(87), 55-56(87), 59(87), 64(87), 66(87), 107-108(87), 113(87), 116-121(87), 124(87), 126-129(87), 131(87), 138(87) Segmuller, A., 54(188), 57(188), 57, 140(188) Sen, P. N., 171(36), 171, 199(97-99), 199, 200(98), 200, 214(99), 231(36) Senokosov, E.A., 67(225), 67 Sergent, A. M., 41-44(136, 139), 42, 83(275), 83, 84(278), 84, 88(278), 140(136, 139) Sermage, B., 93-94(284), 94,96-97(284) Sette, F., 121(341-342), 121, 140(341-342) Shah, J., 28(91), 28 Shapiro, B., 197(88), I97 Shapiro, Y.,216(131), 216 Shay, J. L., 32(117-118), 32, 122(117-118), 124(1 17- 118) Shen, H., 93(287), 95-97(287), 95 Shen, T.-H., 144(350), 144 Sheng, P., 171(34-35), 171, 183(43), 183 Sher, A., 71(237), 71, 73(237) Sherriff, R. E., 230(158), 230 Shevchik, N. J., 65(212), 65, 73(212) Shichijo,, H., 47(148), 47 Shih, C. K., 62-63(202), 63, 71(237), 71, 73(237, 242), 73, 14q202) Shimaoka, G., 100(304), 100 Shirley, D. A., 104(315), 104 Shklovskii, B. I., 200(105), 200 Shockley, W., 2(4), 2 Shtrikman, S., 158(14), 158, 212(124), 212, 304-305(102), 304

387

AUTHOR INDEX

Sievers, A. J., 225(147, 147a), 225, 229(151), 229,230(152-153, 156-157), 230, 234(164), 234,236(147, 147a) Silva, T. J., 354(136), 354, 370(136) Simashkevich, A. V., 66(219-220), 66, 67(224), 67 Simpson, E. M., 294(70, 74), 294 Sitter, H., 1 l0(329), 110 Sivananthan, S., 62-64(206), 63, 70(234), 70, 140(206) Sivco, D. L., 41(151), 44(151), 48(151), 48, 140(151), 93(294), 96-98(294), 96 Sivertsen, J. M., 290(57), 290, 298(86), 298 Skillman, S., 16(77), 16 Smit, J., 272-273(2), 272 Smith, D. L., 2(13, 18), 2, 53(178-179), 53, 74(179, 246-247), 74, 84(178) Smith, S. C., 33(121), 33 Smith, 111, T. P., 53(171), 53 Soderstrom, J. R., 53(168, 172, 180), 53, 74( 180) Solomon, P. M., 6(59), 6, 32(59), 35-37(59) Sou, I. K., 32(110), 32, 62(110, 200), 62, 63-64(110, 203), 63, 70(234), 70, 140(110) Spicer, W. E., 62-63(202), 63, 71(237), 71, 73(237, 242), 73, 82(274), 82, 140(202) Sporken, R., 62-64(206), 63, 140(206) Sputz, S. K., 93(294), 96-98(294), 96 Srivastava, A. K., 32(116), 32, 53-54(116), 56(116) Srivsatava, P. K., 216(130), 216 Stachowiak, H., 170(27), 170, 201(27) Stall, R. A., 106(325), 106 Stankiewicz, J., 71(235), 71 Stanley, H. E., 195(78), 195 Stauffer, D., 188(58), 188, 193(67-68), 193, 194(58), 195(67-68, 81), 195, 197(58) Stecker, L., 44(157), 49(157), 49, 140(157) Stell, G., 184(44-45), 184 Stephen, M. J., 165(22), 165, 194-195(74), I94 Stern, F., 6(50), 6, 32(50), 33(119), 33, 35-37(50, 119), 38(50), 140(50) Stievenard, D., 93(292), 95-97(292), 95 Stillman, G. E., 33(121), 33, 42(140), 42, 44(140), 140(140) Stinchcombe, R. B., 197(85-86), 197 Stoffel, N. G., 82(273), 82

Stokes, J. P., 198(95), 198 Stolz, W., 41(152), 44(152), 48(152), 48, 140(152) Stoner, E., 299(97), 299 Stormer, H. L., 84(278), 84, 88(278) Straley, J. P., 194(73, 75), 194, 195(75-76), 195, 199-200(100), 199, 212(122), 212, 259(217-218), 259 Strathman, M. D., 78(266), 78, 81(266) Stroud, D., 169(26), 169, 170(30-32), 170, 174(41), 174, 201(26, 30-32), 202(112), 202, 203-204(113), 203, 206(30-31), 210(120), 210, 211(112), 214(113), 215(31), 219(139), 219,220(142), 220, 225(148), 225-226,231(161), 231, 232(148, 161), 236(166), 236-237, 238(171), 238,245(190), 245246,249, 252(200), 252,254(205-206), 254, 265-266(142) Stutz, C. E., 93(296), 96-97(296), 96 Subbanna, S., 44(156), 49(156), 49, 54(191), 57(186, 191), 57, 140(191) Sugiyama, Y.,41(153), 44(153), 48(153-154), 48, 140(153) Suhl, H., 272-273(2), 272, 362(145), 362 Sulewski, P. E., 230(152-153), 230 Sullivan, J. P., 30(97), 30, 120(97) Sumski, S., 42(134), 42,47(134) Sunder, W. A., 30(99), 30, 54(99), 57(99), 140(99), 177(99) Supalov, V. A., 67(223), 67 Swank, R. K., 65(213), 65 Swanson, J. A., 201(109), 201,210(109) Sweeny, M., 53(167), 53

T

Tai, K., 93(289), 95-97(289), 95 Takahashi, K., 100(307), 100, 102(309), 102 Takaoka, H., 57(185), 57 Takei, K., 288(48), 288 Takemura, K., 297(80), 297 Tang, C., 362(142), 362 Tang, E., 294(69), 294 Tani, N., 282(32), 282 Tanio, H., 102(312), 102 Tanner, D. B., 198(94), 198, 225(147), 225, 227(149), 227-228, 236(147)

388

AUTHOR INDEX

Tannous, C., 252(197), 252,252(199), 254(199), 256(199) Tao, R., 183(43), 183 Tarnopolsky, G. J., 370(152), 370 Taylor, L. L., 42-44(138), 42, 140(138) Tebble, R. S., 331(124), 331 Tejeda, J., 65(212), 65, 73(212) Tejedor, C., S(25-26) 5, 18(25-26, 81), 18, 19(26), 54(189), 57(189), 57, 117(189), 130(25-26) Temkin, H., 42(134), 42, 47(134), 50(160), 50 Tersoff, J., S(27-29), 5, 1l(27-28), 17(27-29), 18(27-28, 82), 18, 19(27-29, 83), 19, 21(28-29), 25(28-29), 26(28), 40(27-29), 55-56(28-29), 59(28-29), 64-65(28-29), 72-73(28), 82(28-29), 83, 88(27-29), lOO(28-29), 101(29), 102(28-29), 103(28), 107-108(28-29), 112-1 13(28-29), 116-122(28-29), 124(28-29), 126- 128(28-29), 130-1 31(28), 135(28-29), 139(28), 141(28) Theiler, J., 362(144), 362 Thouless, D. J., 193(70), 193 Ting, D. Z.-Y., 53(172), 53 Tinkham, M., 217(132), 217 Tong, H. C., 278(23), 278, 295(23) Torquato, S., 184(44-45) 184 Toulouse, G., 218(136-137), 218 Tran, L. T., 370(152), 370 Trankle, G., 54(191), 57(191), 57, 140(191) Tremblay, A.-M. S., 199(102), 199, 230(155), 230,252(197, 199), 252,254(199), 256(199), 257(211), 257 Treves, D., 304-305(102), 304 Triboulet, R., 73(243), 73 Tsang, W. T., 10-11(72), 10,27(89), 27, 28(72), 41(136), 42(89, 135-136), 42, 43(136), 44(89, 136), 47(89), 54(72), 57(72), 117(72), 140(72, 136) Tsao, J. Y., 78(261-263), 78, 80(261-263), 8 l(263) Tsiulyanu, R. L., 67(224), 67 TSU,R., 2(9-ll), 2, 53(170), 53 Tu, C. W., 36(126-127), 37(127), 37, 50(160), 50

Tu, D.-W., 33(124), 33, 109(124) Tuffigo, H., 99(302-303), 99, lOO(302) Tullke, A,, 125(344), 125, 127(344)

U Uddin, A,, 93(286), 95-97(286), 95 Usatyi, A. N., 67(225), 67 Usher, B. F., 93(295), 96-97(295), 96

V Valladares, J. P., 10-11(72), 10, 28(72), 54(72), 57(72), 98(298), 98, 117(72), 140(72) Vallin, J. T., 93(286), 95-97(286), 95 Van de Walle, C. G., S(37-38), 5, 22(37-38, 84-86), 22,23(86), 25(84), 40(84-86), 41(37, 84), 43(84), 45(84), 52(84, 86), 55-56(84), 59(84, 86), 64(84), 68(37-38), 73(86), 75(38), 77(37-38), 78(84-86), 81(37-38), 83(37-38), 83, 84(38), 86-87(38), 88(37-38), 89(84-86), 91(37-38, 84-86), 97-98(86), 100- 102(86), 103(84-86), 107(86), 108(84, 86), 113-114(84, 86), 115(84), 116- 117(86), 118-1 19(84-86), 120-122(84, 86), 124(84, 86), 126- 127(86), 13q84-86), 131(84), 136(84), 142(84), 143(86), 144(37-38) Van der Merwe, J. H., 78(256-258), 78, 81(256-257) Van Vechten, J. A., 63(205), 63 Vandenberg, J. M., 41(139), 42(134, 139), 42, 43-44(139), 47(134), 50(160), 50, 140(139) Vannimenus, J., 188(57-59), 188, 194(58-59), , 197(58-59), 209(59) Vanyukov, A. V., 67(223), 67 Velu, E. M. T., 282(33), 282, 290(56), 290 Victora, R. H., 273(10), 273 Vieren, J. P., 42(132), 42, 44(132), 47(132), 53(174-175), 53, 62(198-200), 62, 64(199) Viggiano, J. M., 247(196), 247 Villasenor Gonzalez, P., 240(178), 240, 244( 178) Vogt, E., 76(254), 76 Voisin, P., 42(132), 42, 44(132), 47(132), 53(174-175), 53, 54(188), 57(188), 57, 140(188) Volger, J., 201(108), 201

AUTHOR INDEX

Voos, M., 42(132), 42,44(132), 47(132), 53(174-175), 53, 54(188), 57(188), 57, 62(198-199), 62, 64(199), 140(188) Vos, M., 30(97), 30, 120(97) Voss, R. V., 247(191), 247

W

Wagner, J., 41(152), 44(152), 48(152), 48, 140(152) Wagner, S., 32(117-118), 32, 122(117-118), 124(117-118) Wahi, A. K., 73(242), 73 Wakui, J., 339(127), 339 Waldrop, J. R.,6(53-54, 57), 6, 7(65, 69), 7, 30(53-54, 57, 65, 69, 94-95), 30, 35(53), 36(57), 38(57, 9 9 , 40(57, 95), 41(141), 42(141-143), 42,44(141, 158), 46(141-143), 49(142-143, 158), 49, 50(141), 92(283), 92, 103(53, 69, 94, 313), 103, 105(53), 109(313), 111(65), 115(141), 119(65), 128(347-348), 128, 129(348), 140(53, 65, 94, 141, 158), 141(65), 142(313) Walecki, W. J., 70(232), 70, 72(232), 72(241), 72, 101-102(310), I02 Walker, D., 230(154), 230 Walker, J. F., 30(98), 30, 53-54(98), 56(98), 58(98), 140(98) Walmsley, R. G., 293(62), 293 Wang, M. W., 54(190), 57(190), 57, 109(190), 115- 119(190), 140- 141(190) Wang, W. I., 6(50), 6, 32(50), 33(119-120), 33, 35-37(50, 119-120), 38(50, 120), 40(120), 53(169), 53, 57(192), 57, 140(50) Washburn, S., 53(183), 53 Wasiela, A., 99(302-303), 99, lOO(302) Watanabe, M. O., 6(62), 6, 32(62), 35-37(62) Watson, B. P., 174(40), 174, 197(86), 197 Weaver, J. H., 30(97), 30, 120(97) Webb, B. C., 299(88, 96), 299 Webb, R. A., 53(183), 53 Webb, W. W., 247(192, 195), 247 Webman, I., 188(54), 188, 195(77), 195, 202(112), 202, 211(112), 238(170), 238 Wecht, K. W., 32(113), 34(113), 36(113), 41(150), 44(150), 48(150), 48, 84(278), 84, 88(278), 140(150)

389

Wei, S.-H., 10(73), 10 Weimann, G., 6(52), 6, 33(52), 36(52), 93(290), 95-97(290), 95 Weiner, J. S., 41(151), 44(151), 48(151), 48, 140(151) Weismann, M. B., 247(193), 247, 255(209), 255 Welch, D. F., 27(90), 27, 34(90), 36(90), 41(90), 44(90), 48(90), 140(90) Welter, J.-M., 53-54(165), 53, 56(165), 58(165), 140(165) Werder, D. J., 6(56), 6, 10-11(72), 10, 28(56, 72), 35-37(56), 54(72), 57(72), 93(294), 96-98(294), 96, 117(72), 140(72) Wessels, B. W., 111(334), 111 Westervelt, R. M., 362(146), 362 Westland, D. J., 41-44(137), 42, 140(137) Westwood, D., 144(350), 144 Wharton, R. P., 231-232(159), 231 White, R. M., 272(4), 272, 286(44), 286, 354(135), 354 Whitehouse, C. R., 31(108), 31, 125-128(108), 140(108), 142(108) Wicks, G. W., 27(90), 27, 34(90), 36(90), 41(90), 44(90), 48(90), 140(90) Wieder, H. H., 32(114), 32,41(114), 44(114), 48(114), 93(288), 95-97(288), 95, 140(114) , Wiegmann, W., 6(44-45), 6, 27(44-45), 28(91), 28, 32(109), 32, 34(44-45, 109) Wielinga, T., 297, 298(83), 298, 299(92), 299 Wiener, O., 158(13), 158 Wiesenfeld, K., 362(142, 144), 362 Wijewarnasuriya, P. S., 70(234), 70 Wijn, H. P. J., 272-273(2), 272 Wilke, W. G., 31(106-107) 31, 114(106), 116(106), 118- 119(106), 122(107), 124(107), 140(106) Williams, G. M., 31(108), 31, 125-128(108), 140(108), 142(108) Williams, M. L., 278(22), 278 Williams, R. H., 31(108), 31, 125-128(108), 140(108), 142(108), 144(350), 144 Wilson, B. A,, 36(126-127), 37(127), 37 Wilson, R. B., 32(112), 32, 41-42(112), 44( 112), 140(112) Wohlfarth, E. P., 299(97), 299, 366(148), 366 Woicik, J., 82(274), 82 Wolf, E., 235(165), 235

390

AUTHOR INDEX

Wolf, J. K., 275(19), 275, 277(19) Wolf, T., 84(279), 84, 88(279) Wolff, P. A,, 269(227), 269 Wolford, D. J., 6(51), 6, 28(51), 37(51) Wolhfarth, E. P., 272(1), 272 Wood, C. E. C., 44(149, 151), 47(149), 47, 106(325-326), I06 Wood, D. M., 237(168), 237, 242(168) Wood, R., 273(13), 273 Wood, T. H., 41(151), 48(151), 48, 140(149, 151) Woodall, J. M., 2(7), 2, 56(193), 59(193), 59, 67-68( 193), 99( 193), 116- 1 18(193), 123- 124(193) Worst, J., 298(83), 298 Wright, D. C., 248, 252(203), 252, 266(203) Wright, S. L., 6(48-49), 6, 32(48-49), 35-38(48-49), 14q48-49) Wu, C. M., 34(125), 34, 36(125) Wu, G. Y., 61(196), 61, 64(210), 64, 100-lOl(305-306), 100 WU, J.-W., 71(238), 71 WU, Y., 101-102(308), 102 Wuori, E., 298 Wyder, P., 28(93), 28, 53-54(93), 56(93), 58(93), 14q93)

X Xia, T. K., 170(32), 170, 201(32), 230(157), 230 Xu, F., 30(97), 30, 120(97) Xu, J., 53(167), 53 XU,Q.,93(287), 95-97(287), 95

Y Yagil, Y., 238(169), 238 Yamashita, T., 286(45), 286 Yamazaki, Y., 100(304), 100 Yang, E. S., 34(125), 34, 36(125) Yang, Z., 62(201), 62 Yanka, R. W., 70(229), 70 Yantor, Y., 248

Yao, T., 102(312), 102 Yogi, T., 274(16), 274, 277(16), 283(36-37,40), 283, 284(37), 285-286 Yonezawa, Y., 240(182), 240 Yoo, K. M., 239(173), 239, 242(173) Yorhida, J., 6(62), 6, 32(62), 35-37(62) Yosefin, M., 238(169), 238 Yoshida, S., 6(61), 6, 32(61), 35-37(61) Yoshikawa, A., 122(343), 122, 124(343) Young, A. P., 197(85), 197 Yu, E. T., 6(64), 6, 31(104-105), 31, 35-36(64), 38-39(64), 53(172), 53, 54(190), 57(190), 57,66(217-218), 66, 67-70(218), 77(104), 81(104), 83((104-105), 84-88(104), 92(104), 109(105, 190), 115-118(105, 190), 119(190), 140(105, 190,218), 141(190, 218) Yu, P. W., 93(296), 96-97(296), 96

Z

Zabolitzky, J. G., 194(71), 194 Zahn, D. R. T., 110(329), I10 Zallen, R., 193(69), 193 Zeng, X. C., 245(190), 245-246,254(206), 254 Zhao, T.-X., 82(273), 82 Zhu, J-G., 273(6), 273, 293(67-68), 293, 306( 106), 306, 311-3 12(106), 316( log), 316, 320(111, 116), 320, 321(117), 321-322, 323(117), 324, 325(119-120), 325-326, 326(108), 327, 328(122), 328-334, 335(125), 335-344, 345(129), 345, 347(108), 347-352, 353(131-133), 353, 354(134), 354, 356, 357(108, 137), 357, 358(141), 358, 360-361, 363(147), 363-364, 365(67), 368(67), 368 Zhuang, W., 93(287), 95-97(287), 95 Ziem, U., 54(191), 57(191), 57, 140(191) Zipperian, T. E., 74(245), 74 Zou, Y., 93(291), 95(291), 95, 97(291) Zunger, A,, S(30-31), 5, 10(73), 10, 19(30-31) Zurcher, P., 104(318, 320), 104, 105(318), 140(3 18) Zyskind, J. L., 32(116), 32, 53-54(116), 56(116)

Subject Index

A Absorption coefficient, 224-225,242-244 AlAs/GaAs heterojunctions, band offset values, 24-26 AlSb/GaSb/ZnTe, band offsets, 114-1 19 experimental values, 114-116 theoretical values, 116-1 19 Anisotropic media, electromagnetic properties, 229-23 1 Anomalous diffusion length, 238 Ansatz, 203, 206 Archie’s law, 171-172

unresolved issues, 144-145 XPS and related techniques, 29-31 BCS theory, 227 Bessel function nucleation, 305 spherical, 235 Biased diffusion, macroscopically inhomogeneous media, 185 Brown’s equations, 280 Bulk effective conductivity, 151

C

B Band alignments, semiconductor interface, 2-4 Band offsets, 1-146 calculated as bulk parameters, 12-21 interface dipole theories, 17-21 LCAO theory of Harrison, 14, 16-17 pseudopotential theory of Frensley and Kroemer, 13- 15 calculating, 3-5 capacitance-voltage measurements, 32 charge transfer method, 32-33 comparisons among theories, 24-26 electrical measurements, 31-33 empirical rules, 8-12 common anion rules, 10-1 1 electron affinity rule, 8-10 empirical compilation of Katnani and Margaritondo, 12 interface bond polarity model, 24 internal photoemission measurements, 33 lattice-matched heterojunctions, theory versus experiment, 131-143 linear muffin-tin orbital methods, 23-24 model solid theory, 22-23 optical spectroscopy, 27-28 self-consistent calculations for specific interfaces, 21-24 strain-dependent, 3 1 strain effects, 74-78, 143-144 theoretically predicted values, 6-7

CdSe/ZnTe, band offsets, 66-70 CdS/InP, band offsets, 122-125 CdTe/Cd, -,Mn,Te heterojunctions, band offsets, 71-72 CdTe/MnTe, band offsets, 72 CdTe/ZnTe, band offsets, 98-100 Chain nucleation non-exchange-coupled films, 335-340 transient magnetization configurations, 335-337, 341-342 Charge neutrality energy level, 18 CoCr perpendicular thin films, transition profile ensembles, 355, 357 CoCrPt films, morphological separation of grain boundaries, 284-287 Common anion rules, 10-11 Composite conductor, two-component, 162 Composite media, see Macroscopically inhomogeneous media Conduction-band offsets, 2-3 Conduction-band valley, shift, 76 Continuum composites dc electrical properties, macroscopically inhomogeneous media, 174-186 nonuniversal conductivity near percolation threshold, 199-201 Critical current density, macroscopically inhomogeneous media, 219-220 Critical field, 217 Critical thickness, strain relaxation, latticemismatched heterojunctions, 78-81

39 1

392

SUBJECT INDEX

Crystalline anisotropy, thin film recording media, numerical modeling, 3 14-315 CuBr/GaAs/Ge, band offsets, 128-129 Current density, thermionic, 32

D dc electrical properties Clausius-Mossotti approximation, 154 Maxwell-Garnett approximation, 154-155 symmetric effective-medium approximation, 155 Defects, dilute random distribution, 264 Deformation potentials, 75 6m curves, 366-369 Diamagnetic susceptibility, 218-219 Dielectric coefficient, bulk effective nonlinear, 259 Dielectric function, superconductor, 229 Dielectric midgap energy, 19 Diffusion approximation, 239 Diffusion current density, collection of noninteracting walkers, 184 Diffusion equation, 239 Dilute magnetic semiconductors, band offsets, 69-73 Dipole chains, thin film recording media, 312-313 Dipole pair system energy density, 307 thin film recording media, 306-312 Discrete network, dc electrical properties, macroscopically inhomogeneous media, 186-192 Disordered systems, macroscopically inhomogeneous media, 183-186 Duality, 206-21 1

E Effective conductivity, 165, 168 macroscopically inhomogeneous media, 181-182 Effective conductivity tensor, 201 Effective-medium approximation, 164-174, 167, 169, 242 conductivity tensor, 170 Green’s function, 169

objection to, 171 percolation threshold, 167 self-consistency condition, 167- 168 self-consistent equation, 174 self-similar, 172 tensor, 169-170 Effective midgap energy level, 18-19,21 Effective wave vector, 42 Eigenstates, orthogonal, macroscopically inhomogeneous media, 177 Electrical breakdown, macroscopically inhomogeneous media, 263-268 Electrical measurements, band offsets, 3 1-33 Electric multipole coefficient, 235 Electron affinity rule, 8-10, 39 Electronic structure, strain effect, 74-78 Electrostatic potential, 245 Electrostatic resonances, macroscopically inhomogeneous media, 155-156, 175-176 Empirical compilation, Katnani and Margaritondo, 12 Energy band gaps, semiconductors, as function of lattice constant, 9, 11, 13, 15, 17, 19-21, 23 Energy loss function, 225, 234 Exchange-coupled films, reversal in, 341-345 strong exchange coupling, 343-345 weak exchange coupling, 341-343 Exchange coupling, local field approximation, 363-364 Extinction coefficient, 234, 236-237 F FeCoCr film, Lorentz microscopy image of transition patterns, 215-216 Flicker noise, macroscopically inhomogeneous media, 246-253 power law behavior, 252 Fourier methods, macroscopically inhomogeneous media, 183 Frensley and Kroemer, pseudopotential theory, 13-15

G GaAs/AIAs, band offsets, 33-41 comparison with theory, 39-41

393

SUBJECT INDEX

dependence on growth sequence, 38-39 85:15 rule, 34-35, 37 electron affinity rule, 39 experimental data, 33-39 variation with alloy composition, 38 GaAs/Al,Ga, -,As, valence-band offset, 28 GaAs/Ge, band offsets, 103-109 experimental values, 103-106 theoretical values, 106- 109 GaAs/ZnSe, band offsets, 108-1 14 experimental values, 108-1 11 theoretical values, 111-1 14 c-axis orientation distribution, thin film recording media, 283-284 GaP/Si, band offsets, 24-26, 121-124 Geometric structure, thin film recording media, numerical modeling, 3 13-3 14 Grains, collection, macroscopically inhomogeneous media, 176- 179 Green’s function, 202, 241

H Hall effect, magnetoresistance, 201-206 Hankel function, spherical, 235 Hashin-Shtrikman bounds, 158-159 Head-on relaxation method, thin film recording media, 346-348 Heterojunctions, lattice-matched, 131-143 see also Band offsets; Latticemismatched heterojunctions Heterovalent material systems band offsets, 103-129 CdS/InP, 122-123 CuBr/GaAs/Ge, 128-129 GaAs/Ge, 103-109 GaAs/ZnSe, 108-114 GaP/Si, 121-122 InSb/CdTe/a-Sn, 123, 125-128 ZnSe/Ge, 119-121 HgTe/CdTe superlattices, band offsets, 60-66 experimental values, 61-64 theoretical values, 61, 64-66 HgTe/ZnTe, band offsets, 73 Hysteresis loop amorphous CoSm films, 295 CoCr perpendicular thin films, 298-299, 335-336

intergranular exchange coupling constant, 341 longitudinal films, 320-325 magnetization patterns along, 365-366 static magnetization patterns, 330-331, 340-341. 343

I

11-VI material systems band offsets, 59-73 dilute magnetic semiconductors, 69-73 HgTe/CdTe superlattices, 60-66 HgTe/ZnTe, 73 SdSe/ZnTe, 66-70 111-Vmaterial systems band offsets, 33-59 GaAs/AIAs, 33-41 InAs/GaSb/AlSb, 52-49 InGaAs/InAlAs/InP, 41-52 Impedance networks, electromagnetic properties, 245-246 InAs/GaSb/AlSb, band offsets, 52-59 experimental values, 53-58 theoretical values, 55-56, 58-59 InGaAs/GaAs, band offsets, 92-98 experimental values, 92-97 theoretical values, 97-98 InGaAs/InAlAs, band offsets, 47-48 InGaAs/InAlAs/InP, band offsets, 41 -52 comparison with theory, 51-52 experimental values, 44 InAlAs/InP, 49 InGaAs/InAIAs, 47-48 InGaAs/InP, 42-43, 46-47 theoretical values, 45 transitivity, 50-51 yield band offsets, 46 InGaAs/InP, band offsets, 42-43,46-47 InSb/CdTe/a-Sn, band offsets, 123, 125-128 Interaction strength, 309 Interface bond polarity model, 24 Interface dipole theories, 17-21 Interfaces, self-consistent calculations, 21-24

J Johnson noise, 247

394

SUBJECT INDEX

K Katnani and Margaritondo, empirical compilation, 12 Kramers-Kronig relation, 233

L Landau-Lifshitz dynamic equations of motion, 280 Large-scale structures, electromagnetic properties, 238 Lattice-matched heterojunctions, theory versus experiment, 131-143 Lattice-mismatched heterojunctions, 22, 74- 103 CdTelZnTe, 98-100 critical thickness for strain relaxation, 78-81 InGaAs/GaAs, 92-98 Si/Ge, 81-92 strain effects, 74-78, 143-144 ZnSe/ZnTe, 100-103 LCAO theory, 14, 16-17 Linear combination of atomic orbitals theory of Harrison, 14, 16-17 Linear muffin-tin orbital methods, 23-24 Links-nodes-blobs, 265-266 Liquid-metal approximations, 242 LMTO methods, 23-24 London penetration depth, 216 London theory of superconductivity, 215

M Macroscopically inhomogeneous media, 147-269, see also Numerical techniques Clausius-Mossotti approximation, 154 dc electrical properties analytical properties and series expansions, 161-165 better approximations, 155- 158 definitions and basic ideas, 150- 152 dielectric behavior near percolation threshold, 198 duality in two dimensions, 206-21 1

effective-medium approximation, 164-174 electrostatic resonances, 155-156 exact bounds, 158-161 exact results, 152-153 Hall effect and magnetoresistance, 201 nonuniversal conductivity near percolation threshold, 199-201 percolation theory, 192-198 random-resistor network, 164 simple approximations, 152, 154-155 superconductivity, 213-220 thermoelectricity, 21 1-213 electromagnetic properties anisotropic media, 229-231 basic equations, 220-222dielectric enhancement in brinesaturated porous rocks, 231-232 distributions of particles embedded in host, 240-245 electrostatic energy, 260 higher-order multipoles, 234-237 impedance network, 245-246 large-scale structures, 238 metal-insulator composite at low concentrations, 223-227 photon diffusion regime, 239-240 quasi-static approximation, 222-223 sum rules, 232-234 superconducting composite, 227-229 flicker noise, 246-253 power law behavior, 252 Maxwell-Garnett approximation, 154- 155 nonlinear optical effects, 267-269 strong nonlinearity, 258-267 electrical breakdown, 263-267 power law relation between E and D, 259-263 symmetric effective-medium approximation, 155 weak nonlinearity, 253-258 Magnetic field, effective vector, 280 Magnetic grain boundaries, morphological separation, 284-287 Magnetic multipole coefficient, 235 Magnetic parameters, thin film recording media, numerical modeling, 319 Magnetic properties, macroscopically inhomogeneous media, 2 15-219 Magnetism, 271-276

395

SUBJECT INDEX

Magnetization configurations and transition patterns, 290-291 continuous, 279-280 fluctuations intrinsic source, 353 noise power, 370-371 nonuniform, reversal in single grains, 304-306 physics, 276-281 time evolution, at nucleation, 311-312 transition, thin film recording media, 277-279 Magnetization cluster-ripple structure, thin film recording media, 325-326 Magnetization patterns planar isotropic longitudinal films, 356 sequentially recorded transition pairs, 359-361 thin film recording media, 365-366 Magnetization vortex annihilation, 333-335 formation, thin film recording media, 326-328 motion, thin film recording media, 328-329 Magnetoconductivity, macroscopically inhomogeneous media, 213-215 Magnetoresistance, Hall effect, 201 -206 Magnetostatic coupling, local field approximation, 363-364 Magnetostatic energy, thin film recording media, numerical modeling, 315-316 Maxwell equations, 221-222, 240 Maxwell-Garnett approximation, 171, 223, 242 weak nonlinearity, 254 Metal-insulator composite electromagnetic properties, 223-227 percolation threshold, 225 Micromagnetic theory, 279 Model solid theory, Van De Walle and Martin, 92-98 Multilayer films, microstructures and magnetic properties, 29 1-293 Multiple-particle systems, reversal in, 306-3 13 Multipoles, higher-order, electromagnetic properties, 234-237

N Networks randomly diluted, series expansion, 186, 188 reduction, discrete network, 190-192 Nodes-links-blobs model, 195, 220 Noise, see also Flicker noise Johnson, 247 thin film recording media, 369-371 transition, perpendicular films, 355, 357 planar isotropic longitudinal films, 353-356 Non-exchange-coupled films, chain nucleation mode, 335-340 Nucleation, see also Chain nucleation Bessel function, 305 energy saddle point, 304 field, 302-303 transient magnetization configurations during, 337-338 Numerical techniques continuum composites, macroscopically inhomogeneous media, 174-186 bounds use, 180-183 collection of grains, 176-179 disordered systems, 183-186 electrochemical potential, 184 electrostatic resonances, 175-1 76 Fourier methods, 183 periodic arrays, 179-180 discrete networks, macroscopically inhomogeneous media, 186- 192 network reduction, 190-192 relaxation, 188 series expansion, 186, 188 transfer matrix, 188-190

0 Optical anisotropy, macroscopically inhomogeneous media, 230 Optical effects, nonlinear, macroscopically inhomogeneous media, 267-269 Optical Kerr effect, 267 Optical spectroscopy, band offsets, 27-28 Oriented films, microstructures and magnetic properties, 293-296

396

SUBJECT INDEX

P Percolation theory, macroscopically inhomogeneous media, 192-198 correlation length, 195 fractal clusters, 197 singly connected bonds, 196-197 threshold, 194-195 Percolation threshold dielectric behavior near, 198 flicker noise, 247-248 metal-insulator composite, 225 nonuniversal conductivity near, 199-201 zero porosity, 231 Periodic arrays, macroscopically inhomogeneous media, 179- 180 Perpendicular films microstructures and magnetic properties, 297-299 transition noise, 355, 357 Perturbation Hamiltonian, strain effects, 74 Photon diffusion regime, electromagnetic properties, 239-240 Planar isotropic films microstructures and magnetic properties, 282-293 CoCrPt films, 284-287 compositional separation, 286, 288-290 film thickness and grain diameter on magnetic properties, 290 c-axis orientation distribution, 283-284 magnetization configurations and transition patterns, 290-29 1 morphological separation of magnetic grain boundaries, 284-287 multilayer films, 291 -293 transition noise, 353-356 Poisson equation, 262 Power law, relation between E and D, 259-263 Pseudopotential theory, Frensley and Kroemer, 13-15

Q Quasi-static approximation, 222-223 higher-order multipoles, 236

R Random ac impedance network, 245-246 Random-resistor network, 164 models, 199-200 weak nonlinearity, 254-256 Relaxation discrete networks, 188 normalized system energies, thin film recording media, 347-348 transient magnetization configurations during, thin film recording media, 346-347 Reverse domains, elongated, thin film recording media, 329-333

S Scaling function, 203 finite-size, 238 Scaling laws, thin film recording media, numerical modeling, 318 Sedimentary rock, brine-saturated, dielectric enhancement, 23 1-232 Semiconductor interface, band alignments, 2-4 Self-consistent calculations, interfaces, 21-24 Self-consistent effective-medium approximation, 252 Self-consistent equation, 174 Semiconductors, energy band gaps, as function of lattice constant, 9, 11, 13, 15, 17, 19-21,23 Si/Ge, band offsets, 81-92 contour plots, 86-87 experimental values, 81-88 modulation doping effects, 88 strain dependence, 31, 83 strain effect on bulk band structure, 76-77 theoretical values, 88-92 Singly disconnecting bonds, 266 Stoner-Wohlfarth model, 299-303 Strain relaxation, critical thickness, latticemismatched heterojunctions, 78-81 Strong exchange coupling, exchange-coupled film, 343-345 Strong nonlinearity, macroscopically inhomogeneous media, 258-267

SUBJECT INDEX Sum rules, macroscopically inhomogeneous media, 232-234 Supercell geometry, 22 Superconducting composite, electromagnetic properties, 227-229 Superconductivity, macroscopically inhomogeneous media, 213-220 conductivity, 213-215 critical current density, 219-220 critical field, 217 diamagnetic susceptibility, 218-219 magnetic properties, 215-219 penetration depth, 216 Superconductor, dielectric function, 229 Surface plasmon resonance, 223-224 Swiss cheese model, 199-201 Switching field, 302-303

T

Thermionic current density, 32 Thermoelectricity, macroscopically inhomogeneous media, 21 1-213 Thin film recording media, 271-371 change of head current in bit cell, 277 coercivity versus intergranular boundary separation, 338-339 intergranular exchange coupling, 324-325 magnetostatic interaction field constant, 321-322 effective vector magnetic field, 280 exchange energy density, 316 grains, 273-274 longitudinal films, 320-335 elongated domain formation, 329-333 hysteresis properties, 320-325 magnetization patterns and reversal processes, 325-33 5 magnetization ripple-cluster structure, 325-326 transitions, 346-350 vortex annihilation, 333-335 vortex formation, 326-328 vortex motion, 328-329 magnetic recording process simulations interacting transitions, 357-362

397

intergranular exchange coupling, 358-359 single transitions, 345-353 transition noise, 353-357 magnetism, 271-276 M - H IOOPS, 364 micromagnetic modeling coherent rotation, 308 fanning mode, 308-309 hysteresis loops, 302-303 M - H loop, 301,303 nonuniform magnetization reversal in single grains, 304-306 nucleation field, 302-303 nucleation mode, 309-311 reversal in simple multiple-particle systems, 306-313 Stoner-Wohlfarth model, 299-303 system energy density, 300-301 microstructures and magnetic properties, 281-282 oriented films, 293-296 perpendicular films, 297-299 planar isotropic films, 282-293 numerical modeling crystalline anisotropy, 314-315 energy density, 314-320 exchange energy, 316-317 geometric structure, 313-314 magnetostatic energy, 315-316 numerical method, 319-320 range of parameters, 319 scaling laws, 3 18 total effective field, 317-318 Zeeman energy, 315 perpendicular films, 334-345 chain nucleation mode non-exchangecoupled films, 335-340 reversal in exchange-coupled films, 341-345 transitions, 350-353 recording densities, 273 recording geometry, 275 self-organized behavior in magnetic systems, 362-371 correlations and noise, 369-371 6m curves, 366-369 magnetization patterns, 365-366 model, 363-365

398

SUBJECT INDEX

Thin film recording media (cont’d) reversal patterns, 365-369 signal-to-noise ratio, 274 transition width, 278 Total effective field, thin film recording media, numerical modeling, 317-318 Transfer matrix, discrete networks, 188-190 Transition noise, thin film recording media, 353-357 Transitivity, InGaAs/InAlAs/InP, 50-51

Weak nonlinearity, macroscopically inhomogeneous media, 253-258 Weiss molecular field theory, 165 Wiener bounds, 158

X X-ray photoelectron spectroscopy, band offsets, 29-31

V Valence-band edge, position, 16 Valence-band offsets, 2-3 Valence-band splittings, 75-76 Van der Merwe’s model, 78-79 Void resonance, 225

W Weak exchange coupling, exchange-coupled films, 341-343

Z Zeeman energy, thin film recording media, numerical modeling, 315-316 ZnSe/Ge, band offsets, 24-26, 119-121 ZnSe/ZnTe, band offsets, 100-103 ZnSe/Zn, -.Mn,Se multiple quantum wells, band offsets, 72 ZnTe/MnTe, band offsets, 72

Cumulative Author Index, Volumes 1-46 A

Abrikosov, A. A.: Supplement 12Introduction to the Theory of Normal Metals Adler, David: Insulating and Metallic States in Transition Metal Oxides, 21, 1 Adrian, Frank J.: see Gouray, B. S. Akamatu, Hideo: see Inokuchi, H. Alexander, H., and Haasen, P.: Dislocations and Plastic Flow in the Diamond Structure, 22,28 Allen, Philip B., and MitroviC, Boiidar.: Theory of Superconducting T,, 37, 1 Amelinckx, S., and Dekeyser, W.: The Structure and Properties of Grain Boundaries, 8, 327 Amelinckx, S.: Supplement 6-The Direct Observation of Dislocations Anderson, Philip W.: Theory of Magnetic Exchange Interactions: Exchange in Insulators and Semiconductors, 14, 99 Appel, J.: Polarons, 21, 193 Ashcroft, N. W., and Stroud, D.: Theory of the Thermodynamics of Simple Liquid Metals, 33, 1 B

Bastard, G., Brum, J. A., and Ferreira, R.: Electronic States in Semiconductor Heterostructures, 44, 229 Becker, J. A.: Study of Surfaces by Using New Tools, 7, 379 Beenakker, C. W. J., and van Houten, H.: Quantum Transport in Semiconductor Nanostructures, 44,1 Beer, Albert C.: Supplement 4Galvanomagnetic Effects in Semiconductors Bendow, Bernard: Multiphonon Infrared Absorption in the Highly Transparent Frequency Regime of Solids, 33,249 Bergman, David J.: Physical Properties of Macroscopically Inhomogeneous Media, 46, 147 Bertram, H. Neal: Fundamental Magnetiza-

tion Processes in Thin-Film Recording Media, 46, 271 Beyers, R.,and Shaw, T. M.: The Structure of Y,Ba,Cu,O,, and Its Derivatives, 42, 135 Blatt, Frank J.: Theory of Mobility of Electrons in Solids, 4, 200 Blount, E. I.: Formalisms of Band Theory, 13, 306 Borelius, G.: Changes of State of Simple Solid and Liquid Metals, 6, 65 Borelius, G.: The Changes in Energy Content, Volume, and Resistivity with Temperature in Simple Solids and Liquids, 15, 1 Bouligand, Y.: Liquid Crystals and Their Analogs in Biological Systems, in Supplement 14-Liquid Crystals, 259 Boyce, J. B.: see Hayes, T. M. Brill, R.: Determination of Electron Distribution in Crystals by Means of X-Rays, 20, 1 Brown, E.: Aspects of Group Theory in Electron Dynamics, 22, 313 Brown, Frederick C.: Ultraviolet Spectroscopy of Solids with the Use of Synchrotron Radiation, 29, 1 Brum, J. A.: see Bastard, G. Bube, Richard H.: Imperfection Ionization Energies in CdS-Type Materials by Photoelectronic Techniques, 11, 223 Bullet, D. W.: The Renaissance and Quantitative Development of the TightBinding Method, 35, 129 Bundy, F. P.,and Strong, H. M.: Behavior of Metals at High Temperatures and Pressures, 13, 81 Busch, G., and Giintherodt, H.-J.: Electronic Properties of Liquid Metals and Alloys, 29, 235 Busch, G. A,, and Kern, R.: Semiconducting Properties of Gray Tin, 11, 1 C

Callaway, Joseph: Electron Energy Bands in Solids, 7, 100 Callaway, J., and March, N. H.: Density Functional Methods: Theory and Applications, 38, 136.

400

CUMULATIVE AUTHOR INDEX. VOLUME 1-46

Cardona, Manuel: Supplement 11-Optical Modulation Spectroscopy of Solids Cargill, G. S., 111: Structure of Metallic Alloy Glasses, 30,227 Carlsson, A. E.: Beyond Pair Potentials in Elemental Transition and Semiconductors, 43, 1

Charvolin, Jean, and Tardieu, Annette: Lyotropic Liquid Crystals: Structures and Molecular Motions, in Supplement 14Liquid Crystals, 209 Chou, M. Y.: see de Heer, Walt A. Clendenen, R. L.: see Drickamer, H. G. Cohen, Jerome B.: The Internal Structure of Guinier-Preston Zones in Alloys, 39, 131 Cohen, M. H., and Reif, F.: Quadrupole Effects in Nuclear Magnetic Resonance Studies of Solids, 5, 322 Cohen, Marvin L., and Heine, Volker: The Fitting of Pseudopotentials to Experimental Data and Their Subsequent Application, 24, 38 Cohen, Marvin L.: see de Heer, Walt A. Cohen, Marvin L.: see Joannopoulos, J. D. Compton, W. Dale, and Rabin, Herbert: FAggregate Centers in Alkali Halide Crystals, 16, 121 Conwell, Esther M.: Supplement 9-High Field Transport in Semiconductors Cooper, Bernard R.: Magnetic Properties of Rare Earth Metals, 21, 393 Corbett, J. W.: Supplement 7-Electron Radiation Damage in Semiconductors and Metals Corciovei, A., Costache, G., and Vamanu, D.: Ferromagnetic Thin Films, 27, 237 Costache, G.: see Corciovei, A. Currat, R., and Janssen, T.: Excitations in Incommensurate Crystal Phases, 41, 202 D

Dalven, Richard: Electronic Structure of PbS, PbSe, and PbTe, 28, 179 Das, T. P., and Hahn, E. L.: Supplement 1Nuclear Quadrupole Resonance Spectroscopy Davison, S. G., and Levine, J. D.: Surface States, 25, 1 Dederichs, P. H.: Dynamical Diffraction Theory By Optical Potential Methods, 27, 136

de Fontaine, D.: Configurational Thermodynamics of Solid Solutions, 34, 74 de Gennes, P. G.: Macromolecules and Liquid Crystals: Reflections on Certain Lines of Research, in Supplement 14-Liquid Crystals, 1 de Heer, Walt A,, Knight, W. D., Chou, M. Y., and Cohen, Marvin L.: Electronic Shell Structure and Metal Clusters, 40, 94 de Jeu, W. H.: The Dielectric Permittivity of Liquid Crystals, in Supplement 14-Liquid Crystals, 109 Dekeyser, W.: see Amelinckx, S. Dekker, A. J.: Secondary Electron Emission, 6, 251 de Launay, Jules: The Theory of Specific Heats and Lattice Vibrations, 2, 220 Deuling, H. J.: Elasticity of Nematic Liquid Crystals, in Supplement 14-Liquid Crystals, 77 Devreese, J. T.: see Peeters, F. M. de Wit, Roland: The Continuum Theory of Stationary Dislocations, 10, 249 Dexter, D. L.: Theory of the Optical Properties of Imperfections in Nonmetals, 6, 355 Dimmock, J. 0.: The Calculation of Electronic Energy Bands by the Augmented Plane Wave Method, 26, 104 Doran, Donald G., and Linde, Ronald K.: Shock Effects in Solids, 19, 230 Drickamer, H. G.: The Effects of High Pressure on the Electronic Structure of Solids, 17, 1 Drickamer, H. G., Lynch, R. W., Clendenen, R. L., and Perez-Albuerne, E. A.: X-Ray Diffraction Studies of the Lattice Parameters of Solids under Very High Pressure, 19, 135 Dubois-Violette, E., Durand, G., Guyon, E., Manneville, P., and Pieranski, P.: Instabilities in Nematic Liquid Crystals, in Supplement 14-Liquid Crystals, 147 Duke, C. B.: Supplement 10-Tunneling in Solids Durand, G.: see Dubois-Violette, E.

E Echenique, P. M., Flores, F., and Ritchie, R. H.: Dynamic Screening of Ions in Condensed Matter, 43, 230

CUMULATIVE AUTHOR INDEX, VOLUME 1-46

Ehrenreich, H., and Schwartz, L. M.: The Electronic Structure of Alloys, 31, 150 Einspruch, Norman, G.: Ultrasonic Effects in Semiconductors, 17,217 Eshelby, J. D.: The Continuum Theory of Lattice Defects, 3, 79 F

Fan, H. Y.: Valence Semiconductors Germanium and Silicon, 1, 283 Ferreira, R.: see Bastard, G. Flores, F.: see Echenique, P. M. Frederikse, H. P. R.: see Kahn, A. H. Fulder, Peter, Keller, Joachim, and Zwicknagl, Gertrud: Theory of Heavy Fermion Systems, 41, 1 G

Galt, J. K.: see Kittel, C. Geballe, Theodore, H.: see White, Robert M. Gilman, J. J., and Johnston, W. G.: Dislocations in Lithium Fluoride Crystals, 13, 148 Givens, M. Parker: Optical Properties of Metals, 6, 313 Glicksman, Maurice: Plasmas in Solids, 26, 275 Goldberg, I. B.: see Weger, M. Gomer, Robert: Chemisorption on Metals, 30, 94 Gourary, Barry S., and Adrian Frank, J.: Wave Functions for Electron-Excess Color Centers in Alkali Halide Crystals, 10, 128 Gschneidner, Karl A., Jr.: Physical Properties and Interrelationships of Metallic and Semimetallic Elements, 16, 275 Guinier, Andre: Heterogeneities in Solid Solutions, 9, 294 Guntherodt, H.-J.: see Busch, G. Guttman, Lester: Order-disorder Phenomena in Metals, 3, 146 Guyer, R. A.: The Physics of Quantum Crystals, 23, 413 Guyon, E.: see Dubois, Violette, E. H

Haasen, P.: see Alexander, H. Hahn, E. L.: see Das, T. P.

40 1

Halperin, B. I., and Rice, T. M.: The Excitonic State at the Semiconductor-Semimetal Transition, 21, 116 Ham, Frank S.:The Quantum Defect Method, 1, 127 Hashitsume, Ndtsuki: see Kubo, R. Hass, K. C.: Electronics Structure of CopperOxide Superconductors, 42, 213 Haydock, Roger: The Recursive Solution of the Schrodinger Equation, 35, 216 Hayes, T. M., and Boyce,J. B.: Extended X-Ray Absorption Fine Structure Spectroscopy, 37, 173 Hebel, L. C., Jr.: Spin Temperature and Nuclear Relaxation in Solids, 15,409 Hedin, Lars, and Lundqvist, Stig: Effects of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids, 23, 2 Heeger, A. J.: Localized Moments and Nonmoments in Metals: The Kondo Effects, 23, 284 Heer, Ernst, and Novey, Theodore B.: The Interdependence of Solid State Physics and Angular Distribution of Nuclear Radiations, 9, 200 Heiland, G., Mollwo, E., and Stockmann, F.: Electronic Processes in Zinc Oxide, 8, 193 Heine, Volker: see Cohen, M. L. Heine, Volker: The Pseudopotential Concept, 24, 1

Heine, Volker, and Weaire, D.: Pseudopotential Theory of Cohesion and Structure, 24, 250 Heine, Volker: Electronic Structure from the Point of View of the Local Atomic Environment, 35, 1 Hensel, J. C., Phillips, T. G., and Thomas, G. A,: The Electron-Hole Liquid in Semiconductors: Experimental Aspects, 32, 88 Herzfeld, Charles M., and Meijer, Paul H. E.: Group Theory and Crystal Field Theory, 12, 2 Hideshima, T.: see Saito, N. Huebener, R. P.: Thermoelectricity in Metals and Alloys, 27,64 Huntington, H. B.: The Elastic Constants of Crystals, 7, 214 Hutchings, M. T.: Point-Charge Calculations of Energy Levels of Magnetic Ions in Crystalline Electric Fields, 16, 227

402

CUMULATIVE AUTHOR INDEX, VOLUME 1-46

I Inokuchi, Hiroo, and Akamutu, Hideo: Electrical Conductivity of Organic Semiconductors, 12, 93 Ipatova, I. P.: see Maradudin, A. A. Isihara, A.: Electron Correlations in Two Dimensions, 42, 271 Iwayanagi, S.: see Saito, N. J James, R. W.: The Dynamical Theory of X-Ray Diffraction, 15, 55 Jan, J.-P.: Galvanomagnetic and Thermomagnetic Effects in Metals, 5, 3 Janssen, T.: see Currat, R. Jarrett, H. S.: Electron Spin Resonance Spectroscopy in Molecular Solids, 14, 215 Joannapoulos, J. D., and Cohen, Marvin L.: Theory of Short-Range Order and Disorder in Tetrahedrally Bonded Semiconductors, 31,71 Johnston, W. G.: see Gilman, J. J. Jsrgensen, Kluxbiill, Chr.: Chemical Bonding Inferred from Visible and Ultraviolet Absorption Spectra, 13, 376 Joshi, S. K., and Rajagopal, A. K.: Lattice Dynamics of Metals, 22, 160

K Kanzig, Werner: Ferroelectrics and Antiferoelectrics, 4, 5 Kahn, A. H., and Frederikse, H. P. R.: Oscillatory Behavior of Magnetic Susceptibility and Electronic Conductivity, 9, 257 Keller, Joachim, see Fulde, P. Keller, P., and Liebert, L.: Liquid-Crystal Synthesis for Physicists, in Supplement 14Liquid Crystals, 19 Kelly, M. J.: Applications of the Recursion Method to the Electronic Structure from an Atomic Point of View, 25, 296 Kelton, K. F.: Crystal Nucleation in Liquids and Glasses, 45, 75 Kern, R.: see Busch, G. A. Keyes, Robert W.: The Effects of Elastic Deformation on the Electrical Conductivity of Semiconductors, 11, 149

Keyes, Robert W.: Electronic Effects in the Elastic Properties of Semiconductors, 20,37 Kittel, C., and Galt, J. K.: Ferromagnetic Domain Theory, 3,439 Kittel, C.: Indirect Exchange Interactions in Metals, 22, 1 Klemens, P. G.: Thermal Conductivity and Lattice Vibrational Modes, 7, 1 Klick, Clifford C., and Schulman, James H.: Luminescence in Solids, 5,97 Knight, W. D.: Electron Paramagnetism and Nuclear Magnetic Resonance in Metals, 2, 93 Knight, W. D.: see de Heer, Walt A. Knox, Robert S.: Bibliography of Atomic Wave Functions, 4,413 Knox,R. S.: Supplement 5-Theory ofExcitons Koehler, J. S.: see Seitz, F. Kohn, W.: Shallow Impurity States in Silicon and Germanium, 5,258 Kondo, J.: Theory of Dilute Magnetic Alloys, 23, 184 Koster, G. F.: Space Groups and Their Representations, 5, 174 Kothari, L. S., and Singwi, K. S.: Interaction of Thermal Neutrons with Solids, 8, 110 Kroger, F. A., and Vink, H. J.: Relations between the Concentrations of Imperfections in Crystalline Solids, 3, 310 Kubo, Ryogo, Miyake, Satoru J., and Hashitsume, Natsuki: Quantum Theory of Galvanomagnetic Effect at Extremely Strong Magnetic Fields, 17, 270 Kwok, Philip C. K.: Green’s Function Method in Lattice Dynamics, 20, 214

L Lagally, M. G.: see Webb, M. B. Lang, Norton D.: The Density-Functional Formalism and the Electronic Structure of Metal Surfaces, 28, 225 Laudise, R. A., and Nielsen, J. W.: Hydrothermal Crystal Growth, 12, 149 Lax, Benjamin, and Mavroides, John G.: Cyclotron Resonance, 11, 261 Lazarus, David: Diffusion in Metals, 10, 71 Liebfried, G., and Ludwig, W.: Theory of Anharmonic Effects in Crystals, 12, 276 Levine, J. D.: see Davison, S. G.

CUMULATIVE AUTHOR INDEX. VOLUME 1-46

Lewis, H. W.: Wave Packets and Transport of Electrons in Metals, 7, 353 Liebert, L.: see Keller, P. Linde, Ronald K.: see Doran, D. G. Liu, S. H.: Fractals and Their Applications in Condensed Matter Physics, 39, 207 Lobb, C. J.: see Tinkham, M. Low, William: Supplement 2-Paramagnetic Resonance in Solids Low, W., and Offenbacher, E. L.: Electron Spin Resonance of Magnetic Ions in Complex Oxides. Review of ESR Results in Rutile, Perovskites, Spinel, and Garnet Structures, 17, 136 Ludwig, G. W., and Woodbury, H. H.: Electron Spin Resonance in Semiconductors, 13, 223 Ludwig, W.: see Leibfried, G. Lundqvist, Stig.: see Hedin, L. Lynch, R. W.: see Drickamer, H. G.

M McCaldin, James 0.:see Yu, Edward T. McClure, Donald S.: Electronic Spectra of Molecules and Ions in Crystals. Part I. Molecular Crystals, 8, 1 McClure, Donald S.: Electronic Spectra of Molecules and Ions in Crystals. Part 11. Spectra of Ions in Crystals, 9,400 McGill, Thomas C.: see Yu, Edward T. McGreevy, Robert L.: Experimental Studies of the Structure and Dynamics of Molten Alkali and Alkaline Earth Halides, 40, 247 MacKinnon, A,: see Miller, A. MacLaughlin, Douglas E.: Magnetic Resonance in the Superconducting State, 31, 1 McQueen, R. G.: see Rice, M. H. Mahan, G. D.: Many-Body Effects on X-Ray Spectra of Metals, 29, 75 Manneville, P.: see Dubois-Violette, E. Maradudin, A. A., Montroll, E. W., Weiss, G. H., and Ipatova, I. P.: Supplement 3Theory of Lattice Dynamics in the Harmonic Approximation Maradudin, A. A.: Theoretical and Experimental Aspects of the Effects of Point Defects and Disorder on the Vibrations of Crystals-I, 18, 274 Maradudin, A. A.: Theoretical and Experimental Aspects of the Effects of Point

403

Defects and Disorder on the Vibrations of Crystals-2, 19, 1 March, N. H.: see Callaway, J. Markham, Jordan J.: Supplement 8-FCenters in Alkali Halides Mavroides, John G.: see Lax, B. Meijer, Paul H. E.: see Herzfeld, C. M. Mendelssohn, K., and Rosenberg, H. M.: The Thermal Conductivity of Metals at Low Temperatures, 12,223 Miller, A,, MacKinnon, A., and Weaire, D.: Beyond the Binaries-The Chalcopyrite and Related Semiconducting Compounds, 36, 119 Mitra, Shashanka S.: Vibration Spectra of Solids, 13, 1 Mitrovii;, Boiidar: see Allen, P. B. Miyake, Satoru J.: see Kubo, R. Mollwo, E.: see Heiland, G. Montgomery, D. J.: Static Electrification of Solids, 9, 139 Montroll, E. W.: see Maradudin, A. A. Muto, Toshinosuke, and Takagi, Yutaka: The Theory of Order-Disorder Transitions in Alloys, 1, 194 N Nagamiya, Takeo: Helical Spin Ordering-] Theory of Helical Spin Configurations, 20, 306 Nelson, D. R., and Spaepen, Frans: Polytetrahedral Order in Condensed Matter, 42, 1 Newman, R., and Tyler, W. W.: Photoconductivity in Germanium, 8, 50 Nichols, D. K., and van Lint, V. A. J.: Energy Loss and Range of Energetic Neutral Atoms in Solids, 18, 1 Nielsen, J. W.: see Laudise, R. A. Nilsson, P. 0.: Optical Properties of Metals and Alloys, 29, 139 Novey, Theodore B.: see Heer, E. Nussbaum, Allen: Crystal Symmetry, Group Theory, and Band Structure Calculations, 18, 165 0

Offenbacher, E. L.: see Low, W. Okano, K.: see Saito, N.

404

CUMULATIVE AUTHOR INDEX, VOLUME 1-46

P Pake, G. E.: Nuclear Magnetic Resonance, 2, 1 Parker, R. L.: Crystal Grown Mechanisms: Energetics, Kinetics, and Transport, 25, 152 Peercy, P. S.: see Samara, G. A. Peeters, F. M., and Devreese, J. T.: Theory of Polaron Mobility, 38, 82 Perez-Albuerne, E. A.: see Drickamer, H. G. Peterson, N. L.: Diffusion in Metals, 22, 409 Pettifor, D. G.: A Quantum-Mechanical Critique of the Miedema Rules for Alloy Formation, 40, 43 Pfann, W. G.: Techniques of Zone Melting and Crystal Growing, 4,424 Phillips, J. C.: The Fundamental Optical Spectra of Solids, 18, 55 Phillips, J. C.: Spectroscopic and Morphological Structure of Tetrahedral Oxide Glasses, 37,93 Phillips, T. G.: see Hensel, J. C. Pieranski, P.: see Dubois-Violette, E. Pines, David: Electron Interaction in Metals, 1, 368 Piper, W. W., and Williams, F. E.: Electroluminescence, 6, 96 Platzman, P. M., and Wolff, P. A.: Supplement 13-Wares and Interactions in Solid State Plasmas

R Rabin, Herbert: see Compton, W. D. Rajagopal, A. K.: see Joshi, S. K. Rasolt, M.: Continuous Symmetries and Broken Symmetries in Multivalley Semiconductors and Semimetals, 43,94 Reif, F.: see Cohen, M. H. Reitz, John R.: Methods of the One-Electron Theory of Solids, 1, 1 Rice, M. H., McQueen, R. G., and Walsh, J. M.: Compression of Solids by Strong- Shock Waves, 6, 1 Rice, T. M.: see Halperin, B. I. Rice, T. M.: The Electron-Hole Liquid in Semiconductors: Theoretical Aspects, 32, 1 Ritchie, R. H.: see Echenique, P. M. Roitburd, A. L.: Martensitic Transformation

as a Typical Phase Transformation in Solids, 33, 317 Rosenberg, H. M.: see Mendelssohn, K.

S Safran, S. A,: Stage Ordering in Intercalation Compounds, 40, 183 Saito, N., Okano, K., Iwayanagi, S., and Hideshima, T.: Molecular Motion in Solid State Polymers, 14, 344 Samara, G. A.: High-pressure Studies of Ionic Conductivity in Solids, 38, 1 Samara, G. A., and Peercy, P. S.: The Study of Soft-Mode Transitions at High Pressure, 36, 1 Scanlon, W. W.: Polar Semiconductors, 9, 83 Schafroth, M. R.: Theoretical Aspects of Superconductivity, 10, 295 Schnatterley, S. E.: Inelastic Electron Scattering Spectroscopy, 34, 275 Schulman, James H.: see Klick, C. C. Schwartz, L. M.: see Ehrenreich, H. Seitz, Frederick: see Wigner, E. P. Seitz, Frederick, and Koehler, J. S.: Displacement of Atoms during Irradiation, 2, 307 Sellmyer,D. J.: Electronic Structure of Metallic Compounds and Alloys: Experimental Aspects, 33, 83 Sham, L. J., and Ziman, J. M.: The ElectronPhonon Interface, 15, 223 Shaw, T. M.: see Beyers, R. Shull, G. C., and Wollan, E. 0.:Applications of Neutron Diffraction to Solid State Problems, 2, 138 Singh, Jai: The Dynamics of Excitons, 38,295 Singwi, K. S.: see Kothari, L. S. Singwi, K. S., and Tosi, M. P.: Correlations in Electron Liquids, 36, 177 Slack, Glen A.: The Thermal Conductivity of Nonmetallic Crystals, 34, 1 Smith, Charles S.: Macroscopic Symmetry and Properties of Crystals, 6, 175 Sood, Ajay K.: Structural Ordering in Colloidal Suspensions, 45, 1 Spaepen, Frans: see Nelson, D. R. Spector, Harold N.: Interaction of Acoustic Waves and Conduction Electrons, 19, 291 Stern, Frank: Elementary Theory of the Optical Properties of Solids, 15, 300

CUMULATIVE AUTHOR INDEX. VOLUME 1-46

Stockmann, F.: see Heiland, G. Strong, H. M.: see Bundy, F. P. Stroud, D.: see Ashcroft, N. W. Stroud, David: see Bergman, David J. Sturge, M. D.: The Jahn-Teller Effect in Solids, 20, 92 Swenson, C. A.: Physics at High Pressure, 11, 41

T Takagi, Yutaka: see Muto, T. Tardieu, Annette: see Charvolin, Jean Thomas, G. A,: see Hensel, J. C. Thomson, Robb: PHysics of Fracture, 39, 1 Tinkham, M., and Lobb, C . J.: Physical Properties of the New Superconductors, 42, 91 Tosi, Mario P.: Cohesion of Ionic Solids in the Born Model, 16, 1 Tosi, M. P.: see Singwi, K. S. Turnbull, David: Phase Changes, 3, 226 Tyler, W. W.: see Newman, R. V

Vamanu, D.: see Corciovei, A. van Houten, H.: see Breenakker, C. W. J. van Lint, V. A. J.: see Nichols, D. K. Vink, H. J.: see Kroger, F. A.

W Wallace, Duane C.: Thermoelastic Theory of Stressed Crystals and Higher-Order Elastic Constants, 25, 302 Wallace, Philip R.: Positron Annihilation in Solids and Liquids, 10, 1 Walsh, J. M.: see Rice, M. H. Weaire, D.: see Heine, V. Weaire, D.: see Miller, A. Weaire, D.: see Wooten, F. Webb, M. B., and Lagally, M. G.: Elastic Scattering of Low-Energy ElCctrons from Surfaces, 28, 302 Weger, M., and Goldberg, I. B.: Some Lattice and Electronic Properties of the pTungstens, 28, 1 Weiss, G. H.: see Maradudin, A. A. Weiss, H.: see Welker, H.

405

Welker, H., and Weiss, H.: Group 111-Group V Compounds, 3, 1 Wells, A. F.: The Structures of Crystals, 7,426 White, Robert M., and Geballe, Theodore H.: Supplement 15-Long Range Order in Solids Wigner, Eugene P., and Seitz, Frederick: Qualitative Analysis of the Cohesion in Metals, 1,97 Williams, F. E.: see Piper, W. W. Wokaun, Alexander: Surface-Enhanced Electomagnetic Processes, 38, 224 Wolf, E. L.: Nonsuperconducting Electron Tunneling Spectroscopy,30,1 Wolf, H. C.: The Electronic Spectra of Aromatic Molecular Crystals, 9, 1 Wolff, P. A,: see Platzman, P. M. Wollan, E. 0.:see Shull, C. G. Woodbury, H. H.: see Ludwig, G. W. Woodruff, Truman 0.: The Orthogonalized Plane-Wave Method, 4, 367 Wooten, F., and Weaire, D.: Modeling Tetrahedrally Bonded Random Networks by Computer, 40, 1

Y Yafet, Y.: g Factors and Spin-Lattice Relaxation of Conduction Electrons, 14, 1 Yeomans, Julia: The Theory and Application of Axial king Models, 41, 151 Yonezawa, Fumiko: Glass Transition and Relaxation of Disordered Structures, 45, 179 Yu, Edward T.: Band Offsets in Semiconductor Heterojunctions, 46, 1 Z

Zak, J.: The kq-Representation in the Dynamics of Electrons in Solids, 27, 1 Zheludev, I. S.: Ferroelectricity and Symmetry, 26,429 Zheludev, I. S.: Piezoelectricity in Textured Media, 29, 315 Zhu, Jian-Gang: see Bertram H. Neal Ziman, J. M.: see Sham, L. J. Ziman, J. M.: The Calculation of Bloch Functions, 26, 1 Zunger, Alex: Electronic Structure of 3d Transition-Atom Impurities in Semiconductors, 39, 276 Zwicknagl, Gertrud: : see Fulde, P.

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